0$ means $\\alpha $ prefers $i$ to $j$ and $Y_{ij}^{\\alpha }\\le 0$ otherwise.","The magnitude of $Y_{ij}^\\alpha $ can represent the degree of preference and it varies in applications.","It can be dichotomous choice $\\lbrace \\pm 1\\rbrace $ , $k$ -point Likert scale (e.g.","$k=3,4,5$ ), or even real values.","In this paper, consider the following linear model: $ Y_{ij}^\\alpha = \\theta _i - \\theta _j + z_{ij}^\\alpha $ where $\\theta : V\\rightarrow \\mathbb {R}$ is some common score on $V$ and the residue $z_{ij}^\\alpha $ may have interesting structures in crowdsourcing settings.","The annotators might have different effects on the residues.","While for most annotators, the deviations from the common score are due to random noise; occasionally the annotators deviate from the common behavior regularly – some careless ones always choose the left or the right candidate in comparisons, but others only do this when they get too confused to decide.","Such behaviors can be modeled in the following way, $ z_{ij}^\\alpha =\\gamma ^\\alpha + \\varepsilon _{ij}^\\alpha ,$ where $\\gamma ^\\alpha $ measures an annotator's position bias in a fixed effect, and the remainder $\\varepsilon _{ij}^\\alpha $ measures the random effect in sampling which is assumed to be sub-gaussian noise.","For example, a positive value of $\\gamma ^\\alpha $ means the annotator $\\alpha $ is more likely to prefer the left choice.","Under the random design of pairwise comparison experiments, a candidate should be placed on the left or the right randomly, so the position should not affect the choice of a careful (good) annotator.","Therefore $\\gamma ^\\alpha $ is assumed to be sparse, i.e., zero for most of annotators, and a nonzero position bias $\\gamma ^\\alpha $ means the annotator $\\alpha $ is either always choosing one position over the other (bad) or occasionally incurring this when they get too confused or tired (ugly).","We note that (REF ) should not be confused with recent studies in [12], [30] on outlier detection problem, $z_{ij}^\\alpha =\\gamma _{ij}^\\alpha + \\varepsilon _{ij}^\\alpha $ , where $\\gamma _{ij}^\\alpha $ models sparse outliers for each sample $(\\alpha ,i,j)$ , which only measures the random effect of samples rather than the fixed effect of annotators.","By modeling the annotator's fixed effect on position bias, one can systematically classify the annotators into the good, the ugly, and the bad according to their behaviors."],["ISS/LBI","Define the gradient operator by $\\delta _0:\\mathbb {R}^{|V|}\\rightarrow \\mathbb {R}^{|E|}$ such that $(\\delta _0 \\theta )(i,j,\\alpha ) = \\theta _i - \\theta _j$ , and the annotator operator $A:\\mathbb {R}^{|\\mathcal {A}|} \\rightarrow \\mathbb {R}^{|E|}$ by $(A\\gamma )(i,j,\\alpha ) = \\gamma ^\\alpha $ , then the model above can be rewritten as: $Y = \\delta _0 \\theta + A \\gamma + \\varepsilon ,$ In this case, detecting the annotators affected by position bias can be reformulated as: learning a sparse vector $\\gamma $ from given data $(\\delta _0,A,Y)$ .","To solve such a problem, in this paper, we choose a new approach based on the following dynamics, $\\frac{dp}{dt} &= A^T(Y-\\delta _0 \\theta -A\\gamma ) \\\\0 &= \\delta _0^T (Y-\\delta _0 \\theta -A\\gamma ) \\\\p &\\in \\partial \\Vert \\gamma \\Vert _1.","$ Its solution path can be easily solved by a sequence of nonnegative least squares, see [24] and references therein.","In this paper we use the free R-package [29].","In [24], it has been shown that the dynamics above has several advantages over the traditional LASSO approach, which can be formulated as follows in our setting $\\min _{\\theta ,\\gamma } \\frac{1}{2}\\Vert Y - \\delta _0 \\theta - A \\gamma \\Vert _2^2 + \\lambda \\Vert \\gamma \\Vert _1.$ First of all, the dynamics above is statistically equivalent to LASSO in terms of model selection consistency but may render oracle estimator which is bias-free, while the LASSO estimator is well-known biased.","In this sense the ISS path can be better than the LASSO path.","Here the solution path $\\hat{\\gamma }(t)_{t:0\\rightarrow \\infty }$ plays the same role of the regularization path of LASSO $\\hat{\\gamma }(\\lambda )_{\\lambda :\\infty \\rightarrow 0}$ with roughly $t=1/\\lambda $ , where the important features (variables) are selected before the noisy ones.","Following the tradition in image processing, such a dynamics is called Inverse Scale Space (ISS).","Beyond the charming statistical properties, ISS also admits an extremely simple discrete approximation, i.e., the Linearized Bregman Iteration (LBI), which has been widely used in image reconstruction with TV-regularization.","Adapted to our setting, the discretized algorithm is illustrated in Algorithm REF , which is scalable, easy for parallelization, and particularly suitable for large scale crowdsourced ranking data analysis.","LBI in correspondence to (REF ) Initialization: Given parameter $\\kappa $ and $\\triangle t$ , define $k=0, w^0=0, \\theta ^0 = (\\delta _0^T\\delta _0)^{\\dag }\\delta _0^TY,\\gamma ^0=0$ .","Iteration: $w^{k+1} &= w^k + A^T(Y - \\delta _0\\theta ^k - A\\gamma ^k)\\triangle t. \\\\\\gamma ^{k+1}&=\\kappa \\,\\mathrm {shrink}(w^{k+1}).","\\\\\\theta ^{k+1} &= \\theta ^k + \\kappa \\delta _0^T (Y-\\delta _0 \\theta ^k-A\\gamma ^k)\\triangle t. $ Stopping: exit when $k\\triangle t > t $ .","where $\\mathrm {shrink}(x) := \\mathrm {sign}(x)\\max \\lbrace |x|-1,0\\rbrace $ ."],["FDR Control and New Knockoff Filter","A crucial question for LASSO and ISS is how to choose the regularization parameter $\\lambda $ and $t$ in real-world data.","After all, different parameters can give different bad or ugly annotator sets.","Traditional methods either require a prior knowledge on the amount of such annotators which is often unknown in practice, or some statistically optimal choice of such regularization parameters.","Extensive studies in statistics have shown that such parameter tuning typically lead to an over estimation of the sparse signal, therefore False Discovery Rate (FDR) control is necessary [3] which is adopted in this paper.","FDR is defined as the expected proportion of false discoveries among the discoveries.","Putting in a mathematical way, here we consider $FDR = \\mathbb {E}\\left[\\frac{\\#\\lbrace \\alpha :\\gamma ^\\alpha =0,\\hat{\\gamma }^\\alpha \\ne 0\\rbrace }{\\#\\lbrace \\alpha :\\hat{\\gamma }^\\alpha \\ne 0\\rbrace \\wedge 1}\\right].$ To control the FDR means to control the accuracy of the bad/ugly annotators we detected to see if they are reasonable ones.","Recently, a new method called knockoff filter [3] is proposed to automatically control FDR in standard linear regression without a prior knowledge on the sparsity.","In this paper, such an approach will be extended to our linear model (REF ) and the algorithms, where both non-sparse $\\theta $ and sparse $\\gamma $ co-exist in the model.","The extended method consists of the same three steps as in [3], where the key difference lies in the knockoff feature construction adapted to our setting.","Construct knockoff features: let $\\tilde{A}$ be knockoff features that satisfy $\\tilde{A}^T\\tilde{A} = A^TA,~ A^T\\tilde{A} = A^TA- \\mathrm {diag}(s),~\\delta _0^T\\tilde{A} = \\delta _0^TA$ where $s$ is positive and can be solved by SDP: $\\max _s & &\\sum _j s_j\\\\s.t.","& &0 \\le s_j \\le 1\\\\& &\\mathrm {diag}(s) \\preceq 2A^T(I - H)A,$ with $H:=\\delta _0(\\delta _0^T\\delta _0)^{\\dag }\\delta _0^T$ .","Let $Q\\in \\mathbb {R}^{|E|\\times |\\mathcal {A}|}$ be an orthonormal matrix such that $\\delta _0^TQ=0, A^TQ=0$ , which requires $|E|\\ge 2|\\mathcal {A}|+|V|$ easily met in crowdsourcing.","Then (REF ) can be satisfied by defining $\\tilde{A} := A - (I - H)A(A^T(I - H)A)^{-1}\\mathrm {diag}(s) +QC$ where $C\\in \\mathbb {R}^{|\\mathcal {A}|\\times |\\mathcal {A}|}$ satisfies $C^TC = 2\\mathrm {diag}(s) - \\mathrm {diag}(s)(A^T(I - H)A)^{-1}\\mathrm {diag}(s)$ .","Now define the extended design matrix $A_{ko} = [A,\\tilde{A}]$ and $\\gamma _{ko} = [\\gamma ,\\tilde{\\gamma }]^T$ , then replace $A$ with $A_{ko}$ and $\\gamma $ with $\\gamma _{ko}$ in (REF ) , (REF ) or Alg.","REF , we can get solution path $\\hat{\\gamma }_{ko}(\\lambda )$ (or $\\hat{\\gamma }_{ko}(t)$ ).","Generate knockoff statistics for every original feature: define $Z_{j}$ to be the first entering time for $A_j$ , i.e., $Z_{j} = \\sup \\lbrace \\lambda : \\hat{\\gamma }_j(\\lambda ) \\ne 0\\rbrace $ for LASSO (or $\\sup \\lbrace 1/t: \\hat{\\gamma }_j(t) \\ne 0\\rbrace $ for ISS/LBI) and $\\tilde{Z}_j$ can be defined similarly.","Then the knockoff statistics becomes $W_j = \\max (Z_j,\\tilde{Z}_j) \\mathrm {sign}(Z_j - \\tilde{Z}_j)$ Choose variables based on the knockoff statistics: define the selected variable set $\\hat{S} = \\lbrace j: W_j\\ge T_{0/1}\\rbrace $ , where $T_{0/1} = \\min \\lbrace t: \\frac{0/1+\\#\\lbrace j:W_j\\le -t\\rbrace }{\\#\\lbrace j:W_j\\ge t\\rbrace }\\le q\\rbrace .$ $T_0$ is knockoff cut and $T_1$ is knockoff+ cut.","It can be shown that the new knockoff filter above indeed controls FDR in the following sense, whose proof is similar to that of [3] (collected in Supplementary Materials for completeness).","Theorem 1 If $\\epsilon $ is i.i.d $N(0,\\sigma ^2)$ and $|E|\\ge 2|\\mathcal {A}|+|V|$ , then for any $q\\in [0,1]$ , the knockoff filter with ISS/LBI (or LASSO) satisfies $\\mathbb {E}\\left[\\frac{\\#\\lbrace j:\\gamma _j =0~and~j\\in \\hat{S}\\rbrace }{\\#\\lbrace j:j\\in \\hat{S}\\rbrace +q^{-1}}\\right] \\le q$ and the knockoff+ method satisfies $\\mathbb {E}\\left[\\frac{\\#\\lbrace j:\\gamma _j =0~and~j\\in \\hat{S}\\rbrace }{\\#\\lbrace j:j\\in \\hat{S}\\rbrace }\\right] \\le q$ Remark 1 There is an equivalent reformulation of (REF ) to eliminate the non-sparse structure variable $\\theta $ and convert it to a standard LASSO.","Let $\\delta _0$ have a full SVD decomposition $\\delta _0 = U \\Sigma V^T$ and $U=[U_1,U_2]$ , where $U_1$ is an orthonormal basis of the column space $\\rm {col}(\\delta _0)$ and $U_2$ becomes an orthonormal basis for $\\rm {ker} (\\delta _0^T)$ .","Then $U_2^TY = U_2^TA \\gamma + U_2^T\\varepsilon .$ Let $y = U_2^TY, X = U_2^TA, e = U_2^T\\varepsilon $ , then $e$ is i.i.d $N(0,\\sigma ^2)$ $y = X \\gamma + e.$ Based on this, we can use the original knockoff filter $\\tilde{X}$ in [3] to select the position-biased annotators.","A shortcoming of this approach lies in the full SVD decomposition which might be too expensive for large scale problem.","The former approach will not suffer from this.","However, one can see in the following theorem both approaches are in fact equivalent.","Therefore such a reformulation provides us a conceptual insight in understanding the construction of knockoff filters.","Theorem 2 The approach in Remark 1 is equivalent to what we proposed above in the following sense: The knockoff features of (REF ) satisfies $\\tilde{X} = U_2^T \\tilde{A}$ and $\\tilde{A} = U_2\\tilde{X} + U_1U_1^TA$ ; The knockoff statistics constructed by ISS (or LASSO) for both procedures are exactly the same.","Both knockoff filters above can choose variables with FDR control but the estimator $(\\hat{\\theta },\\hat{\\gamma }_{ko})$ consists of knockoff features, so we need to reestimate $\\hat{\\theta },\\hat{\\gamma }$ after bad annotator detection by passing to a least square while only keeping those nonzero parameters and features.","Suppose that $\\hat{S}$ is the set of bad or ugly annotators given by knockoff filters, then one can find the final estimators by $(\\hat{\\theta },\\hat{\\gamma }) = \\arg \\min _{\\theta ,\\gamma _{\\hat{S}}} \\Vert Y - \\delta _0 \\theta - A_{\\hat{S}}\\gamma _{\\hat{S}}\\Vert _2^2.$ 0pt Table: Knockoff with q=10%q = 10\\% via ISS."],["EXPERIMENTS","In this section, five examples are exhibited with both simulated and real-world data to illustrate the validity of the analysis above and applications of the methodology proposed.","The first example is with simulated data while the latter four exploit real-world data collected by crowdsourcing."],["Simulated Study","Settings We first validate the proposed algorithm on simulated binary data labeled by 150 annotators.","Of the 150 annotators we have 100 good annotators (annotators 1 to 100 without position bias) and 50 bad/ugly annotators (annotators 101 to 150 with position bias).","We note that for good annotators, it does not mean that each worker always present the correct labels.","Instead, it means that they also have the probability to make incorrect judgements due to certain reasons, rather than position effect.","Specifically, we first create a random total order on $n$ candidates $V$ as the ground-truth and add paired comparison edges $(i,j)\\in E$ to graph $G=(V,E)$ until a complete graph, with the preference direction following the ground-truth order.","Here we choose $n=|V|=16$ , which is consistent with the third real-world dataset with smallest node size.","Then, for good annotators, they make judgements with an incorrect probability $p_1$ (i.e., $p_1 \\%$ of $E$ is reversed in preference direction), while for bad/ugly annotators, they are with a probability of $p_2$ disturbed by position effect.","Evaluation metrics Two metrics are employed to evaluate the performance of the proposed algorithms.","The first one is Control of Actual FDR, the second is Number of True Discoveries.","Experimental results With different choices of $p_1$ and $p_2$ , the mean Control of Actual FDR and Number of True Discoveries with $q = 10\\%$ over 100 runs are shown in Table REF to measure the performance of knockoff filter via ISS in position biased annotator detection.","It can be seen that via knockoff filter, ISS can provide an accurate detection of position biased annotators (indicated by control of actual FDR around $10\\%$ and Number of True Discoveries around 50).","Comparable results of LASSO with $q = 10\\%$ can be found in Table REF .","It can be seen that via knockoff filter, both LASSO and ISS can provide an accurate detection of position biased annotators.","This result is consistent with the theoretical comparison between LASSO and ISS discussed in [24], where ISS/LBI has similar theoretical guarantees as LASSO, but with bias-free and simpler implementation (the 3 line algorithm in Sec.","REF ) properties.","0pt Table: Knockoff with q=10%q = 10\\% via LASSO."],["Real-world Datasets","As there is no ground-truth for position biased annotators in real-world data, one can not compute control of actual FDR and Number of True Discoveries as in simulated data to evaluate the detection performance here.","In this subsection, we inspect the annotators returned by knockoff filter via ISS/LASSO under $q = 10\\%$ to see if they are reasonably good position biased workers.","3pt Table: Position biased annotators detected in Human age dataset, together with the click counts of each side (i.e., Left and Right).Figure: ISS regularization path of four real-world datasets (Green: the good; Red: the bad; Blue: the ugly)."],["Human Age","In this dataset, 30 images from human age dataset FG-NET http://www.fgnet.rsunit.com/ are annotated by a group of volunteer users on ChinaCrowds platform.","The groundtruth age ranking is known to us.","The annotator is presented with two images and given a binary choice of which one is older.","Totally, we obtain 14,011 pairwise comparisons from 94 annotators.","By adopting the knockoff-based algorithm we proposed, LASSO and ISS identify exactly the same set of abnormal annotators (i.e., 17 users) at q=10%, as is shown in Table REF .","It is easy to see that these annotators can be divided into two types: (1) the bad: click one side all the time (with ID in red); (2) the ugly: click one side with high probability (with ID in blue).","Besides, the regularization paths of ISS can be found in Figure REF (a), where the position biased annotators detected mostly lie outside the majority of the paths.","Note that since we allow a small percentage of false positives, some ugly annotators might be good in reality as well.","To see the effect of position biased annotators on global ranking scores, Table REF shows the outcomes of two ranking algorithms, namely original and corrected.","The original is calculated by least squares problems on all of the pairwise comparisons, while the corrected is obtained by the correction step via knockoff illustrated in Section REF .","It is easy to see that the removal of position biased annotators often changes the orders of some competitive images, such as ID=11 and ID=21, ID=30 and ID=8, etc.","To see which ranking is more reasonable, Table REF shows the groundtruth ranking of these competitive images.","We can find from this table that, compared with the original ranking, the corrected one is in more agreement with the groundtruth ranking, which further shows that: i) position biased annotators may disturb the ranking to a departure from the real ranking.","ii) pairs with little differences are more likely to lead to position biased annotations.","From this viewpoint, we can see that the knockoff-based FDR-controlling method indeed effectively selects the position biased annotators.","0pt Table: Comparison of original vs. corrected rankings on Human age dataset.","The integer represents the ranking position and the number in parenthesis represents the global ranking score returned by the corresponding algorithm.0pt Table: Groundtruth ranking of the competitive images highlighted with red color in Table ."],["Reading Level","The second dataset is a subset of reading level dataset [9], which contains 490 documents.","8,000 pairwise comparisons are collected from 346 annotators using CrowdFlower crowdsourcing platform.","More specifically, each annotator is asked to provide his/her opinion on which text is more challenging to read and understand.","Table REF shows the position biased annotators detected from this dataset, together with the ISS regularization path shown in Figure REF (b).","It is easy to see that LASSO and ISS picked out the same 6 annotators as position biased ones.","In terms of the small number of bad annotators detected, we can say that the overall quality of annotators on this task is relatively high.","0pt Table: Position biased annotators detected in Reading level."],["Image Quality Assessment","The third dataset is a pairwise comparison dataset for subjective image quality assessment (IQA), which contains 15 reference images and 15 distorted versions of each reference, for a total of 240 images which come from two publicly available datasets LIVE, [2] and IVC [1].","Totally, 342 observers, each of whom performs a varied number of comparisons via Internet, provide $52,043$ paired comparisons for crowdsourced subjective image quality assessment.","Note that the number of responses each reference image received is different in this dataset.","To validate whether the annotators we detected are good position biased annotators or not, we randomly take reference image 1 as an illustrative example while other reference images exhibit similar results.","Table REF shows the annotators with position bias picked by knockoff filter and the ISS regularization path is shown in Figure REF (c).","In this dataset, the abnormal annotators picked out by LASSO and ISS are also exactly the same.","It is easy to see that annotators picked out are mainly those clicking on one side almost all the time.","Besides, it is interesting to see that all these bad annotators highlighted with red color in Table REF click the left side all the time.","We then go back to the crowdsourcing platform and find out that the reason behind this is a default choice on the left button thus induces some lazy annotators cheat for the annotation task.","0pt Table: Position biased annotators detected in reference image 1."],["WorldCollege Ranking","We now apply the knockoff filter to the WorldCollege dataset, which is composed of 261 colleges.","Using the Allourideas crowdsourcing platform, a total of 340 distinct annotators from various countries (e.g., USA, Canada, Spain, France, Japan) are shown randomly with pairs of these colleges, and asked to decide which of the two universities is more attractive to attend.","Finally, we obtain a total of 8,823 pairwise comparisons.","We then apply knockoff filter to the resulting dataset and find out that both LASSO and ISS selected 36 annotators as position biased ones, as is shown in Table REF and Figure REF (d).","It is easy to see that similar to the human age dataset, the annotators picked out are either clicking one side all the time, or clicking one side with high probability.","0pt Table: Position biased annotators detected in WorldCollege."],["Discussion","Someone may argue that setting a threshold on the ratio of left/right answers can be an easy way to detect position biased annotators.","To illustrate why simply setting a threshold does not work, Figure REF shows the click counts of each side (i.e., X-axis: number of left clicks; Y-axis: number of right clicks), where each color $\\circ /\\times $ represents one annotator.","It is easy to see that there are indeed some overlaps between abnormal and normal annotators.","For example, in reading level dataset, annotators with ID=69 and ID=57 both provide 6:0 on the ratio of left/right clicks.","However, ID=69 is detected as abnormal annotator, while ID=57 as normal one.","To figure out the reason behind this, we further compute the Match Ratio (MR) of these two annotators with the global ranking scores obtained by all pairwise comparisons and find that $MR_{ID=69}=3/6$ and $MR_{ID=57}=5/6$ .","This indicates that the position biased annotator (i.e., ID=69) we picked out is the one not only with one-side click but also with a large deviation with the majority.","Similar results can be easily found in other three datasets."],["Outlier Detection","Outliers are often referred to as abnormalities, discordants, deviants, or anomalies in data.","Generally speaking, there can be two types of outliers: (1) samples as outliers; (2) subjects as outliers.","Hawkins formally defined in [15] the concept of an outlier as follows: “An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism.\"","Outliers are rare events, but once they have occurred, they may lead to a large instability of models estimated from the noisy data.","For type (1), many methods have been developed for outlier detection, such as distribution-based [4], depth-based [19], distance-based [20], [21], density-based [6], and clustering-based [17] methods.","For subject-based outlier detection, some sophisticated methods have been proposed to model annotators' judgements.","Recently, [9] propose a Crowd-BT algorithm to detect spammers and malicious annotators: spammers assign random labels while malicious annotators assign the wrong label most of the time.","Besides, [25] defines a score to rank the annotators for crowdsourced labeling tasks.","Furthermore, [26] presents an empirical Bayesian algorithm called SpEM to eliminate the spammers and estimate the consensus labels based only on the good annotators.","However, a phenomenon that has annoyed researchers who have used paired comparison tests is position bias or testing order effects.","Until now, little work have been found for such kind of position biased annotator detection, which is our main focus in this paper."],["FDR Control and Knockoff Method","Most variable selection techniques in statistics such as LASSO suffer from over-selection as picking up too many false positives by leaving out few true positives.","In order to offer guarantees on the accuracy of the selection, it is desired to control the false discovery rate (FDR) among all the selected variables.","The Benjamini-Hochberg (BH) procedure [5] is a typical method known to control FDR under independence scenarios.","Recently, [3] developed a new knockoff filter filtermethod for FDR control for general dependent features as long as the sample size is larger than that of parameters.","In this paper, we extend this method to our setting with mixed parameters of both nonsparse and sparse ones to achieve the same FDR control."],["Inverse Scale Space and Linearized Bregman Iteration","Linearized Bregman Iteration (LBI) has been widely used in image processing and compressed sensing [23], [31] even before its limit form as Inverse Scale Space (ISS) dynamics [7].","ISS/LBI at least have two advantages over the popular LASSO in variable selection: (1) ISS may give unbiased estimator [24], under nearly the same condition for model selection consistency as LASSO whose estimators are however always biased [11].","(2) LBI, regarded as a discretization of ISS dynamics, is an extremely simple algorithm which combines an iterative gradient descent algorithm together with a soft thresholding.","It only runs in a single path and regularization is achieved by early stopping like boosting algorithms [24], which may save the computational cost greatly and thus suitable for large scale implementation [32]."],["Conclusion","Annotator's position bias is ubiquitous in crowdsourced ranking data, which, up to our knowledge, has not been systematically addressed in literature.","In this paper, we propose a statistical model for annotator's position bias with pairwise comparison data on graphs, together with new algorithms to reach statistically good estimates with a FDR control based on some new design of knockoff filters.","FDR control here does not need a prior knowledge on the sparsity of position bias, i.e., the amount of bad or ugly annotators.","Such a framework is valid for both traditional LASSO estimator and the new dynamic approach based on ISS/LBI with debiased estimator and scalable implementations which is desired for crowdsourcing experiments.","Experimental studies are conducted with both simulated examples and real-world datasets.","Our results suggest that the proposed methodology is an effective tool to investigate annotator's abnormal behavior in modern crowdsourcing data."],["Acknowledgements","The research of Qianqian Xu was supported in part by National Natural Science Foundation of China (No.","61422213, 61402019, 61390514, 61572042), China Postdoctoral Science Foundation (2015T80025), “Strategic Priority Research Program\" of the Chinese Academy of Sciences (XDA06010701), and National Program for Support of Top-notch Young Professionals.","The research of Jiechao Xiong and Yuan Yao was supported in part by National Basic Research Program of China under grant 2015CB85600, 2012CB825501, and NSFC grant 61370004, 11421110001 (A3 project), as well as grants from Baidu and Microsoft Research-Asia.","Xiaochun Cao and Yuan Yao are the corresponding authors.","We would like to thank Yongyi Guo for helpful discussions and anonymous reviewers who gave valuable suggestions to help improve the manuscript.","Sketchy Proof of Theorem REF .","Similar to the treatment in [3], we only need to prove that the knockoff statistics $W_j$ satisfy the following two properties: sufficiency property: $W = f([\\delta _0,A_{ko}]^T[\\delta _0,A_{ko}], [\\delta _0,A_{ko}]^TY)$ , which indicates $W$ depends only on $[\\delta _0,A_{ko}]^T[\\delta _0,A_{ko}]$ and $[\\delta _0,A_{ko}]^TY$ .","antisymmetry property: Swapping $A_j$ and $\\tilde{A}_j$ has the effect of switching the sign of $W_j$ .","The second property is obvious because $W_j$ is constructed using entering time difference.","Now we go to prove the first property.","For ISS and LBI, the whole path is only determined by $A_{ko}^T(Y-\\delta _0 \\theta -A_{ko}\\gamma _{ko}) &= A_{ko}^TY - A_{ko}^T[\\delta _0 ,A_{ko}][\\theta ^T,\\gamma _{ko}^T]^T), \\\\\\delta _0^T (Y-\\delta _0 \\theta -A_{ko}\\gamma _{ko}) &= \\delta _0^TY - \\delta _0^T[\\delta _0 ,A_{ko}][\\theta ^T,\\gamma _{ko}^T]^T),$ which is only based on $[\\delta _0,A_{ko}]^T[\\delta _0,A_{ko}]$ and $[\\delta _0,A_{ko}]^TY$ , so is the entering time $Z_{j}$ The same reasoning holds for LASSO since $\\min _{\\theta ,\\gamma } \\frac{1}{2}\\Vert Y - [\\delta _0,A_{ko}] [\\theta ^T,\\gamma _{ko}^T]^T\\Vert _2^2 + \\lambda \\Vert \\gamma _{ko}\\Vert _1$ is equivalent to $\\min _{\\theta ,\\gamma } &\\frac{1}{2}(\\Vert Y\\Vert _2^2 + [\\theta ^T,\\gamma _{ko}^T] [\\delta _0,A_{ko}] ^T[\\delta _0,A_{ko}] [\\theta ^T,\\gamma _{ko}^T]^T\\\\&-2 [\\theta ^T,\\gamma _{ko}^T] [\\delta _0,A_{ko}] ^TY )+ \\lambda \\Vert \\gamma _{ko}\\Vert _1$ So the entire path is determined by $[\\delta _0,A_{ko}]^T[\\delta _0,A_{ko}]$ and $[\\delta _0,A_{ko}]^TY$ .","Proof of Theorem REF .","Suppose $\\tilde{X}$ is the knockoff statistics for (REF ), then it satisfies $\\tilde{X}^T\\tilde{X} = X^TX, X^T\\tilde{X} = X^TX - \\mathrm {diag}(s).$ Let $B = A + U_2(\\tilde{X} - X)$ , then $\\tilde{X} = U_2^TB$ and it can be verified $B^TB= A^TA, A^TB = A^TA-\\mathrm {diag}(s), \\delta _0^TB=\\delta _0^TA$ which means $B$ is a valid knockoff feature matrix for (REF ).","On the reverse, let $\\tilde{A}$ be knockoff features for (REF ), it is also easy to verify $\\tilde{X} = U_2^T\\tilde{A}$ satisfies condition (REF ).","This establishes an injection between $\\tilde{X}$ and $\\tilde{A}$ .","The equivalence of knockoff statistics comes from the equivalence of solution paths in both approaches.","To see this, () actually means $\\hat{\\theta } = (\\delta _0^T\\delta _0)^{\\dag }\\delta _0^T(Y - A_{ko}\\gamma _{ko})$ , plugging $\\hat{\\theta }$ in (REF ), we get $\\frac{dp}{dt} &=&A_{ko}^T (Y-\\delta _0 \\hat{\\theta }-A_{ko}\\gamma _{ko}) \\\\&=& A_{ko}^T(U_2U_2^T(Y-A_{ko}\\gamma _{ko}))\\\\&= & (U_2^TA_{ko})^T(U_2^TY-U_2^TA_{ko}\\gamma _{ko})$ This is equivalent to the ISS for the second procedure model (REF ) in Remark 1.","So in both approaches, the two ISS solution paths are identical.","The same reasoning holds for LASSO, the derivative of (REF ) w.r.t.","$\\theta $ is zero at the optimal estimator which means $0 = \\delta _0^T (Y-\\delta _0 \\hat{\\theta }-A_{ko}\\gamma _{ko})$ this is actually ().","So plugging $\\hat{\\theta }$ in (REF ), we get $&&\\Vert Y-\\delta _0 \\theta -A_{ko}\\gamma _{ko}\\Vert _2^2 \\\\&=& \\Vert (I - \\delta _0(\\delta _0^T\\delta _0)^{\\dag }\\delta _0^T)^T(Y-A_{ko}\\gamma _{ko})\\Vert _2^2\\\\&= & \\Vert U_2U_2^T(Y-A_{ko}\\gamma _{ko})\\Vert _2^2\\\\&= & \\Vert U_2^TY-U_2^TA_{ko}\\gamma _{ko})\\Vert _2^2.$ This is in fact the $l_2$ loss for the second procedure in Remark 1.","Finally identical paths lead to the same knockoff statistics which ends the proof."]],"string":"[\n [\n \"False Discovery Rate Control and Statistical Quality Assessment of\\n Annotators in Crowdsourced Ranking\"\n ],\n [\n \"Abstract With the rapid growth of crowdsourcing platforms it has become easy and relatively inexpensive to collect a dataset labeled by multiple annotators in a short time.\",\n \"However due to the lack of control over the quality of the annotators, some abnormal annotators may be affected by position bias which can potentially degrade the quality of the final consensus labels.\",\n \"In this paper we introduce a statistical framework to model and detect annotator's position bias in order to control the false discovery rate (FDR) without a prior knowledge on the amount of biased annotators - the expected fraction of false discoveries among all discoveries being not too high, in order to assure that most of the discoveries are indeed true and replicable.\",\n \"The key technical development relies on some new knockoff filters adapted to our problem and new algorithms based on the Inverse Scale Space dynamics whose discretization is potentially suitable for large scale crowdsourcing data analysis.\",\n \"Our studies are supported by experiments with both simulated examples and real-world data.\",\n \"The proposed framework provides us a useful tool for quantitatively studying annotator's abnormal behavior in crowdsourcing data arising from machine learning, sociology, computer vision, multimedia, etc.\"\n ],\n [\n \"Introduction\",\n \"In applications, building good predictive models is challenging primarily due to the difficulties in obtaining annotated training data.\",\n \"A traditional way for data labeling is to hire a small group of experts to provide labels for the entire set of data.\",\n \"However, such an approach can be expensive and time consuming for large scale data.\",\n \"Thanks to the wide spread of crowdsourcing platforms (e.g., MTurk, Innocentive, CrowdFlower, CrowdRank, and Allourideas), a much more efficient way is to post unlabeled data to a crowdsourcing marketplace, where a big crowd of low-paid workers can be hired instantaneously to perform labeling tasks [27], [28], [16], [22], [8].\",\n \"Despite of its high efficiency and immediate availability, crowd labeling raises many new challenges.\",\n \"Since typical crowdsourced tasks are tedious and annotators usually come from a diverse pool including genuine experts, novices, biased workers, and malicious annotators, labels generated by the crowd suffer from low quality.\",\n \"Thus, all crowdsourcers need strategies to ensure the reliability of answers.\",\n \"In other words, outlier detection is a critical task in order to achieve a robust labeling results.\",\n \"Various methods have been developed in literature for outlier detection, of which majority voting strategy [14], [18] is the most typical one.\",\n \"In this setting, each pair is allocated to multiple annotators and their opinions are averaged over so as to identify and discard noisy data provided by unreliable raters.\",\n \"They thus require large amount of pairwise labels to be collected.\",\n \"More importantly as a local outlier detection method, majority voting is ineffective in identifying outliers that can cause global ranking inconsistencies [12], [13].\",\n \"The work in [30] attacks this problem and formulates the outlier detection as a LASSO problem based on sparse approximations of the cyclic ranking projection of paired comparison data in Hodge decomposition.\",\n \"Regularization paths of the LASSO problem could provide an order on samples tending to be outliers.\",\n \"However, these work all treat pairwise comparison judgements as independent random outliers, which are typically defined to be data samples that have unusual deviations from the remaining data.\",\n \"In this paper, instead of modeling the random effect of sample-wise outliers, we are primarily interested in the fixed effect where the annotators are influenced by positions when labeling in pairwise comparison setting.\",\n \"Such an annotator's position bias [10] is ubiquitous in uncontrolled crowdsourced ranking experiments.\",\n \"In our studies, annotator's position bias typically arises from: i) the ugly: one typically clicks one side more often than another.\",\n \"As some pairs are highly confusing or annotators get too tired, in these cases, some annotators tend to click one side hoping to simply raise their record to receive more payment; while for pairs with substantial differences, they click as usual.\",\n \"ii) the bad: some extremely careless annotators, or robots pretending to be human annotators, actually do not look at the instances and click one side all the time to quickly receive pay for work.\",\n \"Such kinds of annotators may significantly deteriorate the quality of crowdsourcing data and increase the cost of acquiring annotations (since each raw feedback comes with a cost: the task requestor has to pay workers a pre-specified monetary reward for each labeling they provide, usually, regardless of the feedback correctness).\",\n \"Although it might be relatively easy to identify the bad annotators above by inspecting their inputs, it is impossible for eye inspection to pick up those ugly annotators with mixed behaviors.\",\n \"Therefore it is desired to design a statistical framework to quantitatively detect and eliminate annotator's position bias for crowdsourcing platforms in market.\",\n \"Such a systematic study, up to the author's knowledge, however has not been seen in literature.\",\n \"In this paper, we propose a linear model with annotator's position bias and new algorithms to find good estimates with an automatic control on the false discovery rate (FDR) – the expected fraction of false discoveries among all discoveries.\",\n \"To understand FDR, imagine that we have a detection method that has just made 100 discoveries.\",\n \"Then, if our method is known to control the FDR at the 10% level, this means that with high probability, we can expect at most 10 of these discoveries to be false and, therefore, at least 90 to be true and replicable.\",\n \"Such a FDR control is desired when we don't have a prior knowledge about the amount of bad or ugly annotators and typical statistical estimates will lead to an over-identification of them.\",\n \"Specifically, our contributions in this work are highlighted as follows: (A) A linear model with annotator's position bias as fixed effects; (B) New algorithms to find good estimates of such position bias, etc., based on Inverse Scale Space dynamics and its discretization Linearized Bregman Iteration; (C) New knockoff filters for FDR control adapted to our setting, which aims to mimic the correlation structure found within the original features for position bias; (D) Extensive experimental validation based on one simulated and four real-world crowdsourced datasets.\"\n ],\n [\n \"Methodology\",\n \"In this section, we systematically introduce the methodology for annotator's position bias estimation.\",\n \"Specifically, we first start from a basic linear model with different types of noise models, which have been successfully used widely in literature.\",\n \"Then we introduce a new dynamic approach with unbiased estimator called Inverse Scale Space (ISS).\",\n \"Based on this, we present the modified knockoff filter for FDR control in details.\"\n ],\n [\n \"Basic Linear Model\",\n \"Let $V = \\\\lbrace 1,2,\\\\dots ,n\\\\rbrace $ be the set of nodes and $E = \\\\lbrace (\\\\alpha ,i,j): i,j\\\\in V, \\\\alpha \\\\in U\\\\rbrace $ be the set of edges, where $U$ is the set of all annotators.\",\n \"Suppose the pairwise ranking data is given as $Y: E\\\\rightarrow R$ .\",\n \"$Y_{ij}^\\\\alpha >0$ means $\\\\alpha $ prefers $i$ to $j$ and $Y_{ij}^{\\\\alpha }\\\\le 0$ otherwise.\",\n \"The magnitude of $Y_{ij}^\\\\alpha $ can represent the degree of preference and it varies in applications.\",\n \"It can be dichotomous choice $\\\\lbrace \\\\pm 1\\\\rbrace $ , $k$ -point Likert scale (e.g.\",\n \"$k=3,4,5$ ), or even real values.\",\n \"In this paper, consider the following linear model: $ Y_{ij}^\\\\alpha = \\\\theta _i - \\\\theta _j + z_{ij}^\\\\alpha $ where $\\\\theta : V\\\\rightarrow \\\\mathbb {R}$ is some common score on $V$ and the residue $z_{ij}^\\\\alpha $ may have interesting structures in crowdsourcing settings.\",\n \"The annotators might have different effects on the residues.\",\n \"While for most annotators, the deviations from the common score are due to random noise; occasionally the annotators deviate from the common behavior regularly – some careless ones always choose the left or the right candidate in comparisons, but others only do this when they get too confused to decide.\",\n \"Such behaviors can be modeled in the following way, $ z_{ij}^\\\\alpha =\\\\gamma ^\\\\alpha + \\\\varepsilon _{ij}^\\\\alpha ,$ where $\\\\gamma ^\\\\alpha $ measures an annotator's position bias in a fixed effect, and the remainder $\\\\varepsilon _{ij}^\\\\alpha $ measures the random effect in sampling which is assumed to be sub-gaussian noise.\",\n \"For example, a positive value of $\\\\gamma ^\\\\alpha $ means the annotator $\\\\alpha $ is more likely to prefer the left choice.\",\n \"Under the random design of pairwise comparison experiments, a candidate should be placed on the left or the right randomly, so the position should not affect the choice of a careful (good) annotator.\",\n \"Therefore $\\\\gamma ^\\\\alpha $ is assumed to be sparse, i.e., zero for most of annotators, and a nonzero position bias $\\\\gamma ^\\\\alpha $ means the annotator $\\\\alpha $ is either always choosing one position over the other (bad) or occasionally incurring this when they get too confused or tired (ugly).\",\n \"We note that (REF ) should not be confused with recent studies in [12], [30] on outlier detection problem, $z_{ij}^\\\\alpha =\\\\gamma _{ij}^\\\\alpha + \\\\varepsilon _{ij}^\\\\alpha $ , where $\\\\gamma _{ij}^\\\\alpha $ models sparse outliers for each sample $(\\\\alpha ,i,j)$ , which only measures the random effect of samples rather than the fixed effect of annotators.\",\n \"By modeling the annotator's fixed effect on position bias, one can systematically classify the annotators into the good, the ugly, and the bad according to their behaviors.\"\n ],\n [\n \"ISS/LBI\",\n \"Define the gradient operator by $\\\\delta _0:\\\\mathbb {R}^{|V|}\\\\rightarrow \\\\mathbb {R}^{|E|}$ such that $(\\\\delta _0 \\\\theta )(i,j,\\\\alpha ) = \\\\theta _i - \\\\theta _j$ , and the annotator operator $A:\\\\mathbb {R}^{|\\\\mathcal {A}|} \\\\rightarrow \\\\mathbb {R}^{|E|}$ by $(A\\\\gamma )(i,j,\\\\alpha ) = \\\\gamma ^\\\\alpha $ , then the model above can be rewritten as: $Y = \\\\delta _0 \\\\theta + A \\\\gamma + \\\\varepsilon ,$ In this case, detecting the annotators affected by position bias can be reformulated as: learning a sparse vector $\\\\gamma $ from given data $(\\\\delta _0,A,Y)$ .\",\n \"To solve such a problem, in this paper, we choose a new approach based on the following dynamics, $\\\\frac{dp}{dt} &= A^T(Y-\\\\delta _0 \\\\theta -A\\\\gamma ) \\\\\\\\0 &= \\\\delta _0^T (Y-\\\\delta _0 \\\\theta -A\\\\gamma ) \\\\\\\\p &\\\\in \\\\partial \\\\Vert \\\\gamma \\\\Vert _1.\",\n \"$ Its solution path can be easily solved by a sequence of nonnegative least squares, see [24] and references therein.\",\n \"In this paper we use the free R-package [29].\",\n \"In [24], it has been shown that the dynamics above has several advantages over the traditional LASSO approach, which can be formulated as follows in our setting $\\\\min _{\\\\theta ,\\\\gamma } \\\\frac{1}{2}\\\\Vert Y - \\\\delta _0 \\\\theta - A \\\\gamma \\\\Vert _2^2 + \\\\lambda \\\\Vert \\\\gamma \\\\Vert _1.$ First of all, the dynamics above is statistically equivalent to LASSO in terms of model selection consistency but may render oracle estimator which is bias-free, while the LASSO estimator is well-known biased.\",\n \"In this sense the ISS path can be better than the LASSO path.\",\n \"Here the solution path $\\\\hat{\\\\gamma }(t)_{t:0\\\\rightarrow \\\\infty }$ plays the same role of the regularization path of LASSO $\\\\hat{\\\\gamma }(\\\\lambda )_{\\\\lambda :\\\\infty \\\\rightarrow 0}$ with roughly $t=1/\\\\lambda $ , where the important features (variables) are selected before the noisy ones.\",\n \"Following the tradition in image processing, such a dynamics is called Inverse Scale Space (ISS).\",\n \"Beyond the charming statistical properties, ISS also admits an extremely simple discrete approximation, i.e., the Linearized Bregman Iteration (LBI), which has been widely used in image reconstruction with TV-regularization.\",\n \"Adapted to our setting, the discretized algorithm is illustrated in Algorithm REF , which is scalable, easy for parallelization, and particularly suitable for large scale crowdsourced ranking data analysis.\",\n \"LBI in correspondence to (REF ) Initialization: Given parameter $\\\\kappa $ and $\\\\triangle t$ , define $k=0, w^0=0, \\\\theta ^0 = (\\\\delta _0^T\\\\delta _0)^{\\\\dag }\\\\delta _0^TY,\\\\gamma ^0=0$ .\",\n \"Iteration: $w^{k+1} &= w^k + A^T(Y - \\\\delta _0\\\\theta ^k - A\\\\gamma ^k)\\\\triangle t. \\\\\\\\\\\\gamma ^{k+1}&=\\\\kappa \\\\,\\\\mathrm {shrink}(w^{k+1}).\",\n \"\\\\\\\\\\\\theta ^{k+1} &= \\\\theta ^k + \\\\kappa \\\\delta _0^T (Y-\\\\delta _0 \\\\theta ^k-A\\\\gamma ^k)\\\\triangle t. $ Stopping: exit when $k\\\\triangle t > t $ .\",\n \"where $\\\\mathrm {shrink}(x) := \\\\mathrm {sign}(x)\\\\max \\\\lbrace |x|-1,0\\\\rbrace $ .\"\n ],\n [\n \"FDR Control and New Knockoff Filter\",\n \"A crucial question for LASSO and ISS is how to choose the regularization parameter $\\\\lambda $ and $t$ in real-world data.\",\n \"After all, different parameters can give different bad or ugly annotator sets.\",\n \"Traditional methods either require a prior knowledge on the amount of such annotators which is often unknown in practice, or some statistically optimal choice of such regularization parameters.\",\n \"Extensive studies in statistics have shown that such parameter tuning typically lead to an over estimation of the sparse signal, therefore False Discovery Rate (FDR) control is necessary [3] which is adopted in this paper.\",\n \"FDR is defined as the expected proportion of false discoveries among the discoveries.\",\n \"Putting in a mathematical way, here we consider $FDR = \\\\mathbb {E}\\\\left[\\\\frac{\\\\#\\\\lbrace \\\\alpha :\\\\gamma ^\\\\alpha =0,\\\\hat{\\\\gamma }^\\\\alpha \\\\ne 0\\\\rbrace }{\\\\#\\\\lbrace \\\\alpha :\\\\hat{\\\\gamma }^\\\\alpha \\\\ne 0\\\\rbrace \\\\wedge 1}\\\\right].$ To control the FDR means to control the accuracy of the bad/ugly annotators we detected to see if they are reasonable ones.\",\n \"Recently, a new method called knockoff filter [3] is proposed to automatically control FDR in standard linear regression without a prior knowledge on the sparsity.\",\n \"In this paper, such an approach will be extended to our linear model (REF ) and the algorithms, where both non-sparse $\\\\theta $ and sparse $\\\\gamma $ co-exist in the model.\",\n \"The extended method consists of the same three steps as in [3], where the key difference lies in the knockoff feature construction adapted to our setting.\",\n \"Construct knockoff features: let $\\\\tilde{A}$ be knockoff features that satisfy $\\\\tilde{A}^T\\\\tilde{A} = A^TA,~ A^T\\\\tilde{A} = A^TA- \\\\mathrm {diag}(s),~\\\\delta _0^T\\\\tilde{A} = \\\\delta _0^TA$ where $s$ is positive and can be solved by SDP: $\\\\max _s & &\\\\sum _j s_j\\\\\\\\s.t.\",\n \"& &0 \\\\le s_j \\\\le 1\\\\\\\\& &\\\\mathrm {diag}(s) \\\\preceq 2A^T(I - H)A,$ with $H:=\\\\delta _0(\\\\delta _0^T\\\\delta _0)^{\\\\dag }\\\\delta _0^T$ .\",\n \"Let $Q\\\\in \\\\mathbb {R}^{|E|\\\\times |\\\\mathcal {A}|}$ be an orthonormal matrix such that $\\\\delta _0^TQ=0, A^TQ=0$ , which requires $|E|\\\\ge 2|\\\\mathcal {A}|+|V|$ easily met in crowdsourcing.\",\n \"Then (REF ) can be satisfied by defining $\\\\tilde{A} := A - (I - H)A(A^T(I - H)A)^{-1}\\\\mathrm {diag}(s) +QC$ where $C\\\\in \\\\mathbb {R}^{|\\\\mathcal {A}|\\\\times |\\\\mathcal {A}|}$ satisfies $C^TC = 2\\\\mathrm {diag}(s) - \\\\mathrm {diag}(s)(A^T(I - H)A)^{-1}\\\\mathrm {diag}(s)$ .\",\n \"Now define the extended design matrix $A_{ko} = [A,\\\\tilde{A}]$ and $\\\\gamma _{ko} = [\\\\gamma ,\\\\tilde{\\\\gamma }]^T$ , then replace $A$ with $A_{ko}$ and $\\\\gamma $ with $\\\\gamma _{ko}$ in (REF ) , (REF ) or Alg.\",\n \"REF , we can get solution path $\\\\hat{\\\\gamma }_{ko}(\\\\lambda )$ (or $\\\\hat{\\\\gamma }_{ko}(t)$ ).\",\n \"Generate knockoff statistics for every original feature: define $Z_{j}$ to be the first entering time for $A_j$ , i.e., $Z_{j} = \\\\sup \\\\lbrace \\\\lambda : \\\\hat{\\\\gamma }_j(\\\\lambda ) \\\\ne 0\\\\rbrace $ for LASSO (or $\\\\sup \\\\lbrace 1/t: \\\\hat{\\\\gamma }_j(t) \\\\ne 0\\\\rbrace $ for ISS/LBI) and $\\\\tilde{Z}_j$ can be defined similarly.\",\n \"Then the knockoff statistics becomes $W_j = \\\\max (Z_j,\\\\tilde{Z}_j) \\\\mathrm {sign}(Z_j - \\\\tilde{Z}_j)$ Choose variables based on the knockoff statistics: define the selected variable set $\\\\hat{S} = \\\\lbrace j: W_j\\\\ge T_{0/1}\\\\rbrace $ , where $T_{0/1} = \\\\min \\\\lbrace t: \\\\frac{0/1+\\\\#\\\\lbrace j:W_j\\\\le -t\\\\rbrace }{\\\\#\\\\lbrace j:W_j\\\\ge t\\\\rbrace }\\\\le q\\\\rbrace .$ $T_0$ is knockoff cut and $T_1$ is knockoff+ cut.\",\n \"It can be shown that the new knockoff filter above indeed controls FDR in the following sense, whose proof is similar to that of [3] (collected in Supplementary Materials for completeness).\",\n \"Theorem 1 If $\\\\epsilon $ is i.i.d $N(0,\\\\sigma ^2)$ and $|E|\\\\ge 2|\\\\mathcal {A}|+|V|$ , then for any $q\\\\in [0,1]$ , the knockoff filter with ISS/LBI (or LASSO) satisfies $\\\\mathbb {E}\\\\left[\\\\frac{\\\\#\\\\lbrace j:\\\\gamma _j =0~and~j\\\\in \\\\hat{S}\\\\rbrace }{\\\\#\\\\lbrace j:j\\\\in \\\\hat{S}\\\\rbrace +q^{-1}}\\\\right] \\\\le q$ and the knockoff+ method satisfies $\\\\mathbb {E}\\\\left[\\\\frac{\\\\#\\\\lbrace j:\\\\gamma _j =0~and~j\\\\in \\\\hat{S}\\\\rbrace }{\\\\#\\\\lbrace j:j\\\\in \\\\hat{S}\\\\rbrace }\\\\right] \\\\le q$ Remark 1 There is an equivalent reformulation of (REF ) to eliminate the non-sparse structure variable $\\\\theta $ and convert it to a standard LASSO.\",\n \"Let $\\\\delta _0$ have a full SVD decomposition $\\\\delta _0 = U \\\\Sigma V^T$ and $U=[U_1,U_2]$ , where $U_1$ is an orthonormal basis of the column space $\\\\rm {col}(\\\\delta _0)$ and $U_2$ becomes an orthonormal basis for $\\\\rm {ker} (\\\\delta _0^T)$ .\",\n \"Then $U_2^TY = U_2^TA \\\\gamma + U_2^T\\\\varepsilon .$ Let $y = U_2^TY, X = U_2^TA, e = U_2^T\\\\varepsilon $ , then $e$ is i.i.d $N(0,\\\\sigma ^2)$ $y = X \\\\gamma + e.$ Based on this, we can use the original knockoff filter $\\\\tilde{X}$ in [3] to select the position-biased annotators.\",\n \"A shortcoming of this approach lies in the full SVD decomposition which might be too expensive for large scale problem.\",\n \"The former approach will not suffer from this.\",\n \"However, one can see in the following theorem both approaches are in fact equivalent.\",\n \"Therefore such a reformulation provides us a conceptual insight in understanding the construction of knockoff filters.\",\n \"Theorem 2 The approach in Remark 1 is equivalent to what we proposed above in the following sense: The knockoff features of (REF ) satisfies $\\\\tilde{X} = U_2^T \\\\tilde{A}$ and $\\\\tilde{A} = U_2\\\\tilde{X} + U_1U_1^TA$ ; The knockoff statistics constructed by ISS (or LASSO) for both procedures are exactly the same.\",\n \"Both knockoff filters above can choose variables with FDR control but the estimator $(\\\\hat{\\\\theta },\\\\hat{\\\\gamma }_{ko})$ consists of knockoff features, so we need to reestimate $\\\\hat{\\\\theta },\\\\hat{\\\\gamma }$ after bad annotator detection by passing to a least square while only keeping those nonzero parameters and features.\",\n \"Suppose that $\\\\hat{S}$ is the set of bad or ugly annotators given by knockoff filters, then one can find the final estimators by $(\\\\hat{\\\\theta },\\\\hat{\\\\gamma }) = \\\\arg \\\\min _{\\\\theta ,\\\\gamma _{\\\\hat{S}}} \\\\Vert Y - \\\\delta _0 \\\\theta - A_{\\\\hat{S}}\\\\gamma _{\\\\hat{S}}\\\\Vert _2^2.$ 0pt Table: Knockoff with q=10%q = 10\\\\% via ISS.\"\n ],\n [\n \"EXPERIMENTS\",\n \"In this section, five examples are exhibited with both simulated and real-world data to illustrate the validity of the analysis above and applications of the methodology proposed.\",\n \"The first example is with simulated data while the latter four exploit real-world data collected by crowdsourcing.\"\n ],\n [\n \"Simulated Study\",\n \"Settings We first validate the proposed algorithm on simulated binary data labeled by 150 annotators.\",\n \"Of the 150 annotators we have 100 good annotators (annotators 1 to 100 without position bias) and 50 bad/ugly annotators (annotators 101 to 150 with position bias).\",\n \"We note that for good annotators, it does not mean that each worker always present the correct labels.\",\n \"Instead, it means that they also have the probability to make incorrect judgements due to certain reasons, rather than position effect.\",\n \"Specifically, we first create a random total order on $n$ candidates $V$ as the ground-truth and add paired comparison edges $(i,j)\\\\in E$ to graph $G=(V,E)$ until a complete graph, with the preference direction following the ground-truth order.\",\n \"Here we choose $n=|V|=16$ , which is consistent with the third real-world dataset with smallest node size.\",\n \"Then, for good annotators, they make judgements with an incorrect probability $p_1$ (i.e., $p_1 \\\\%$ of $E$ is reversed in preference direction), while for bad/ugly annotators, they are with a probability of $p_2$ disturbed by position effect.\",\n \"Evaluation metrics Two metrics are employed to evaluate the performance of the proposed algorithms.\",\n \"The first one is Control of Actual FDR, the second is Number of True Discoveries.\",\n \"Experimental results With different choices of $p_1$ and $p_2$ , the mean Control of Actual FDR and Number of True Discoveries with $q = 10\\\\%$ over 100 runs are shown in Table REF to measure the performance of knockoff filter via ISS in position biased annotator detection.\",\n \"It can be seen that via knockoff filter, ISS can provide an accurate detection of position biased annotators (indicated by control of actual FDR around $10\\\\%$ and Number of True Discoveries around 50).\",\n \"Comparable results of LASSO with $q = 10\\\\%$ can be found in Table REF .\",\n \"It can be seen that via knockoff filter, both LASSO and ISS can provide an accurate detection of position biased annotators.\",\n \"This result is consistent with the theoretical comparison between LASSO and ISS discussed in [24], where ISS/LBI has similar theoretical guarantees as LASSO, but with bias-free and simpler implementation (the 3 line algorithm in Sec.\",\n \"REF ) properties.\",\n \"0pt Table: Knockoff with q=10%q = 10\\\\% via LASSO.\"\n ],\n [\n \"Real-world Datasets\",\n \"As there is no ground-truth for position biased annotators in real-world data, one can not compute control of actual FDR and Number of True Discoveries as in simulated data to evaluate the detection performance here.\",\n \"In this subsection, we inspect the annotators returned by knockoff filter via ISS/LASSO under $q = 10\\\\%$ to see if they are reasonably good position biased workers.\",\n \"3pt Table: Position biased annotators detected in Human age dataset, together with the click counts of each side (i.e., Left and Right).Figure: ISS regularization path of four real-world datasets (Green: the good; Red: the bad; Blue: the ugly).\"\n ],\n [\n \"Human Age\",\n \"In this dataset, 30 images from human age dataset FG-NET http://www.fgnet.rsunit.com/ are annotated by a group of volunteer users on ChinaCrowds platform.\",\n \"The groundtruth age ranking is known to us.\",\n \"The annotator is presented with two images and given a binary choice of which one is older.\",\n \"Totally, we obtain 14,011 pairwise comparisons from 94 annotators.\",\n \"By adopting the knockoff-based algorithm we proposed, LASSO and ISS identify exactly the same set of abnormal annotators (i.e., 17 users) at q=10%, as is shown in Table REF .\",\n \"It is easy to see that these annotators can be divided into two types: (1) the bad: click one side all the time (with ID in red); (2) the ugly: click one side with high probability (with ID in blue).\",\n \"Besides, the regularization paths of ISS can be found in Figure REF (a), where the position biased annotators detected mostly lie outside the majority of the paths.\",\n \"Note that since we allow a small percentage of false positives, some ugly annotators might be good in reality as well.\",\n \"To see the effect of position biased annotators on global ranking scores, Table REF shows the outcomes of two ranking algorithms, namely original and corrected.\",\n \"The original is calculated by least squares problems on all of the pairwise comparisons, while the corrected is obtained by the correction step via knockoff illustrated in Section REF .\",\n \"It is easy to see that the removal of position biased annotators often changes the orders of some competitive images, such as ID=11 and ID=21, ID=30 and ID=8, etc.\",\n \"To see which ranking is more reasonable, Table REF shows the groundtruth ranking of these competitive images.\",\n \"We can find from this table that, compared with the original ranking, the corrected one is in more agreement with the groundtruth ranking, which further shows that: i) position biased annotators may disturb the ranking to a departure from the real ranking.\",\n \"ii) pairs with little differences are more likely to lead to position biased annotations.\",\n \"From this viewpoint, we can see that the knockoff-based FDR-controlling method indeed effectively selects the position biased annotators.\",\n \"0pt Table: Comparison of original vs. corrected rankings on Human age dataset.\",\n \"The integer represents the ranking position and the number in parenthesis represents the global ranking score returned by the corresponding algorithm.0pt Table: Groundtruth ranking of the competitive images highlighted with red color in Table .\"\n ],\n [\n \"Reading Level\",\n \"The second dataset is a subset of reading level dataset [9], which contains 490 documents.\",\n \"8,000 pairwise comparisons are collected from 346 annotators using CrowdFlower crowdsourcing platform.\",\n \"More specifically, each annotator is asked to provide his/her opinion on which text is more challenging to read and understand.\",\n \"Table REF shows the position biased annotators detected from this dataset, together with the ISS regularization path shown in Figure REF (b).\",\n \"It is easy to see that LASSO and ISS picked out the same 6 annotators as position biased ones.\",\n \"In terms of the small number of bad annotators detected, we can say that the overall quality of annotators on this task is relatively high.\",\n \"0pt Table: Position biased annotators detected in Reading level.\"\n ],\n [\n \"Image Quality Assessment\",\n \"The third dataset is a pairwise comparison dataset for subjective image quality assessment (IQA), which contains 15 reference images and 15 distorted versions of each reference, for a total of 240 images which come from two publicly available datasets LIVE, [2] and IVC [1].\",\n \"Totally, 342 observers, each of whom performs a varied number of comparisons via Internet, provide $52,043$ paired comparisons for crowdsourced subjective image quality assessment.\",\n \"Note that the number of responses each reference image received is different in this dataset.\",\n \"To validate whether the annotators we detected are good position biased annotators or not, we randomly take reference image 1 as an illustrative example while other reference images exhibit similar results.\",\n \"Table REF shows the annotators with position bias picked by knockoff filter and the ISS regularization path is shown in Figure REF (c).\",\n \"In this dataset, the abnormal annotators picked out by LASSO and ISS are also exactly the same.\",\n \"It is easy to see that annotators picked out are mainly those clicking on one side almost all the time.\",\n \"Besides, it is interesting to see that all these bad annotators highlighted with red color in Table REF click the left side all the time.\",\n \"We then go back to the crowdsourcing platform and find out that the reason behind this is a default choice on the left button thus induces some lazy annotators cheat for the annotation task.\",\n \"0pt Table: Position biased annotators detected in reference image 1.\"\n ],\n [\n \"WorldCollege Ranking\",\n \"We now apply the knockoff filter to the WorldCollege dataset, which is composed of 261 colleges.\",\n \"Using the Allourideas crowdsourcing platform, a total of 340 distinct annotators from various countries (e.g., USA, Canada, Spain, France, Japan) are shown randomly with pairs of these colleges, and asked to decide which of the two universities is more attractive to attend.\",\n \"Finally, we obtain a total of 8,823 pairwise comparisons.\",\n \"We then apply knockoff filter to the resulting dataset and find out that both LASSO and ISS selected 36 annotators as position biased ones, as is shown in Table REF and Figure REF (d).\",\n \"It is easy to see that similar to the human age dataset, the annotators picked out are either clicking one side all the time, or clicking one side with high probability.\",\n \"0pt Table: Position biased annotators detected in WorldCollege.\"\n ],\n [\n \"Discussion\",\n \"Someone may argue that setting a threshold on the ratio of left/right answers can be an easy way to detect position biased annotators.\",\n \"To illustrate why simply setting a threshold does not work, Figure REF shows the click counts of each side (i.e., X-axis: number of left clicks; Y-axis: number of right clicks), where each color $\\\\circ /\\\\times $ represents one annotator.\",\n \"It is easy to see that there are indeed some overlaps between abnormal and normal annotators.\",\n \"For example, in reading level dataset, annotators with ID=69 and ID=57 both provide 6:0 on the ratio of left/right clicks.\",\n \"However, ID=69 is detected as abnormal annotator, while ID=57 as normal one.\",\n \"To figure out the reason behind this, we further compute the Match Ratio (MR) of these two annotators with the global ranking scores obtained by all pairwise comparisons and find that $MR_{ID=69}=3/6$ and $MR_{ID=57}=5/6$ .\",\n \"This indicates that the position biased annotator (i.e., ID=69) we picked out is the one not only with one-side click but also with a large deviation with the majority.\",\n \"Similar results can be easily found in other three datasets.\"\n ],\n [\n \"Outlier Detection\",\n \"Outliers are often referred to as abnormalities, discordants, deviants, or anomalies in data.\",\n \"Generally speaking, there can be two types of outliers: (1) samples as outliers; (2) subjects as outliers.\",\n \"Hawkins formally defined in [15] the concept of an outlier as follows: “An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism.\\\"\",\n \"Outliers are rare events, but once they have occurred, they may lead to a large instability of models estimated from the noisy data.\",\n \"For type (1), many methods have been developed for outlier detection, such as distribution-based [4], depth-based [19], distance-based [20], [21], density-based [6], and clustering-based [17] methods.\",\n \"For subject-based outlier detection, some sophisticated methods have been proposed to model annotators' judgements.\",\n \"Recently, [9] propose a Crowd-BT algorithm to detect spammers and malicious annotators: spammers assign random labels while malicious annotators assign the wrong label most of the time.\",\n \"Besides, [25] defines a score to rank the annotators for crowdsourced labeling tasks.\",\n \"Furthermore, [26] presents an empirical Bayesian algorithm called SpEM to eliminate the spammers and estimate the consensus labels based only on the good annotators.\",\n \"However, a phenomenon that has annoyed researchers who have used paired comparison tests is position bias or testing order effects.\",\n \"Until now, little work have been found for such kind of position biased annotator detection, which is our main focus in this paper.\"\n ],\n [\n \"FDR Control and Knockoff Method\",\n \"Most variable selection techniques in statistics such as LASSO suffer from over-selection as picking up too many false positives by leaving out few true positives.\",\n \"In order to offer guarantees on the accuracy of the selection, it is desired to control the false discovery rate (FDR) among all the selected variables.\",\n \"The Benjamini-Hochberg (BH) procedure [5] is a typical method known to control FDR under independence scenarios.\",\n \"Recently, [3] developed a new knockoff filter filtermethod for FDR control for general dependent features as long as the sample size is larger than that of parameters.\",\n \"In this paper, we extend this method to our setting with mixed parameters of both nonsparse and sparse ones to achieve the same FDR control.\"\n ],\n [\n \"Inverse Scale Space and Linearized Bregman Iteration\",\n \"Linearized Bregman Iteration (LBI) has been widely used in image processing and compressed sensing [23], [31] even before its limit form as Inverse Scale Space (ISS) dynamics [7].\",\n \"ISS/LBI at least have two advantages over the popular LASSO in variable selection: (1) ISS may give unbiased estimator [24], under nearly the same condition for model selection consistency as LASSO whose estimators are however always biased [11].\",\n \"(2) LBI, regarded as a discretization of ISS dynamics, is an extremely simple algorithm which combines an iterative gradient descent algorithm together with a soft thresholding.\",\n \"It only runs in a single path and regularization is achieved by early stopping like boosting algorithms [24], which may save the computational cost greatly and thus suitable for large scale implementation [32].\"\n ],\n [\n \"Conclusion\",\n \"Annotator's position bias is ubiquitous in crowdsourced ranking data, which, up to our knowledge, has not been systematically addressed in literature.\",\n \"In this paper, we propose a statistical model for annotator's position bias with pairwise comparison data on graphs, together with new algorithms to reach statistically good estimates with a FDR control based on some new design of knockoff filters.\",\n \"FDR control here does not need a prior knowledge on the sparsity of position bias, i.e., the amount of bad or ugly annotators.\",\n \"Such a framework is valid for both traditional LASSO estimator and the new dynamic approach based on ISS/LBI with debiased estimator and scalable implementations which is desired for crowdsourcing experiments.\",\n \"Experimental studies are conducted with both simulated examples and real-world datasets.\",\n \"Our results suggest that the proposed methodology is an effective tool to investigate annotator's abnormal behavior in modern crowdsourcing data.\"\n ],\n [\n \"Acknowledgements\",\n \"The research of Qianqian Xu was supported in part by National Natural Science Foundation of China (No.\",\n \"61422213, 61402019, 61390514, 61572042), China Postdoctoral Science Foundation (2015T80025), “Strategic Priority Research Program\\\" of the Chinese Academy of Sciences (XDA06010701), and National Program for Support of Top-notch Young Professionals.\",\n \"The research of Jiechao Xiong and Yuan Yao was supported in part by National Basic Research Program of China under grant 2015CB85600, 2012CB825501, and NSFC grant 61370004, 11421110001 (A3 project), as well as grants from Baidu and Microsoft Research-Asia.\",\n \"Xiaochun Cao and Yuan Yao are the corresponding authors.\",\n \"We would like to thank Yongyi Guo for helpful discussions and anonymous reviewers who gave valuable suggestions to help improve the manuscript.\",\n \"Sketchy Proof of Theorem REF .\",\n \"Similar to the treatment in [3], we only need to prove that the knockoff statistics $W_j$ satisfy the following two properties: sufficiency property: $W = f([\\\\delta _0,A_{ko}]^T[\\\\delta _0,A_{ko}], [\\\\delta _0,A_{ko}]^TY)$ , which indicates $W$ depends only on $[\\\\delta _0,A_{ko}]^T[\\\\delta _0,A_{ko}]$ and $[\\\\delta _0,A_{ko}]^TY$ .\",\n \"antisymmetry property: Swapping $A_j$ and $\\\\tilde{A}_j$ has the effect of switching the sign of $W_j$ .\",\n \"The second property is obvious because $W_j$ is constructed using entering time difference.\",\n \"Now we go to prove the first property.\",\n \"For ISS and LBI, the whole path is only determined by $A_{ko}^T(Y-\\\\delta _0 \\\\theta -A_{ko}\\\\gamma _{ko}) &= A_{ko}^TY - A_{ko}^T[\\\\delta _0 ,A_{ko}][\\\\theta ^T,\\\\gamma _{ko}^T]^T), \\\\\\\\\\\\delta _0^T (Y-\\\\delta _0 \\\\theta -A_{ko}\\\\gamma _{ko}) &= \\\\delta _0^TY - \\\\delta _0^T[\\\\delta _0 ,A_{ko}][\\\\theta ^T,\\\\gamma _{ko}^T]^T),$ which is only based on $[\\\\delta _0,A_{ko}]^T[\\\\delta _0,A_{ko}]$ and $[\\\\delta _0,A_{ko}]^TY$ , so is the entering time $Z_{j}$ The same reasoning holds for LASSO since $\\\\min _{\\\\theta ,\\\\gamma } \\\\frac{1}{2}\\\\Vert Y - [\\\\delta _0,A_{ko}] [\\\\theta ^T,\\\\gamma _{ko}^T]^T\\\\Vert _2^2 + \\\\lambda \\\\Vert \\\\gamma _{ko}\\\\Vert _1$ is equivalent to $\\\\min _{\\\\theta ,\\\\gamma } &\\\\frac{1}{2}(\\\\Vert Y\\\\Vert _2^2 + [\\\\theta ^T,\\\\gamma _{ko}^T] [\\\\delta _0,A_{ko}] ^T[\\\\delta _0,A_{ko}] [\\\\theta ^T,\\\\gamma _{ko}^T]^T\\\\\\\\&-2 [\\\\theta ^T,\\\\gamma _{ko}^T] [\\\\delta _0,A_{ko}] ^TY )+ \\\\lambda \\\\Vert \\\\gamma _{ko}\\\\Vert _1$ So the entire path is determined by $[\\\\delta _0,A_{ko}]^T[\\\\delta _0,A_{ko}]$ and $[\\\\delta _0,A_{ko}]^TY$ .\",\n \"Proof of Theorem REF .\",\n \"Suppose $\\\\tilde{X}$ is the knockoff statistics for (REF ), then it satisfies $\\\\tilde{X}^T\\\\tilde{X} = X^TX, X^T\\\\tilde{X} = X^TX - \\\\mathrm {diag}(s).$ Let $B = A + U_2(\\\\tilde{X} - X)$ , then $\\\\tilde{X} = U_2^TB$ and it can be verified $B^TB= A^TA, A^TB = A^TA-\\\\mathrm {diag}(s), \\\\delta _0^TB=\\\\delta _0^TA$ which means $B$ is a valid knockoff feature matrix for (REF ).\",\n \"On the reverse, let $\\\\tilde{A}$ be knockoff features for (REF ), it is also easy to verify $\\\\tilde{X} = U_2^T\\\\tilde{A}$ satisfies condition (REF ).\",\n \"This establishes an injection between $\\\\tilde{X}$ and $\\\\tilde{A}$ .\",\n \"The equivalence of knockoff statistics comes from the equivalence of solution paths in both approaches.\",\n \"To see this, () actually means $\\\\hat{\\\\theta } = (\\\\delta _0^T\\\\delta _0)^{\\\\dag }\\\\delta _0^T(Y - A_{ko}\\\\gamma _{ko})$ , plugging $\\\\hat{\\\\theta }$ in (REF ), we get $\\\\frac{dp}{dt} &=&A_{ko}^T (Y-\\\\delta _0 \\\\hat{\\\\theta }-A_{ko}\\\\gamma _{ko}) \\\\\\\\&=& A_{ko}^T(U_2U_2^T(Y-A_{ko}\\\\gamma _{ko}))\\\\\\\\&= & (U_2^TA_{ko})^T(U_2^TY-U_2^TA_{ko}\\\\gamma _{ko})$ This is equivalent to the ISS for the second procedure model (REF ) in Remark 1.\",\n \"So in both approaches, the two ISS solution paths are identical.\",\n \"The same reasoning holds for LASSO, the derivative of (REF ) w.r.t.\",\n \"$\\\\theta $ is zero at the optimal estimator which means $0 = \\\\delta _0^T (Y-\\\\delta _0 \\\\hat{\\\\theta }-A_{ko}\\\\gamma _{ko})$ this is actually ().\",\n \"So plugging $\\\\hat{\\\\theta }$ in (REF ), we get $&&\\\\Vert Y-\\\\delta _0 \\\\theta -A_{ko}\\\\gamma _{ko}\\\\Vert _2^2 \\\\\\\\&=& \\\\Vert (I - \\\\delta _0(\\\\delta _0^T\\\\delta _0)^{\\\\dag }\\\\delta _0^T)^T(Y-A_{ko}\\\\gamma _{ko})\\\\Vert _2^2\\\\\\\\&= & \\\\Vert U_2U_2^T(Y-A_{ko}\\\\gamma _{ko})\\\\Vert _2^2\\\\\\\\&= & \\\\Vert U_2^TY-U_2^TA_{ko}\\\\gamma _{ko})\\\\Vert _2^2.$ This is in fact the $l_2$ loss for the second procedure in Remark 1.\",\n \"Finally identical paths lead to the same knockoff statistics which ends the proof.\"\n ]\n]"},"id":{"kind":"string","value":"1605.05860"}}},{"rowIdx":1866791,"cells":{"text":{"kind":"list like","value":[["Magnetoresistivity in a Tilted Magnetic Field in p-Si/SiGe/Si\n Heterostructures with an Anisotropic g-Factor: Part II"],["Abstract The magnetoresistance components $\\rho_{xx}$ and $\\rho_{xy}$ were measured in two p-Si/SiGe/Si quantum wells that have an anisotropic g-factor in a tilted magnetic field as a function of temperature, field and tilt angle.","Activation energy measurements demonstrate the existence of a ferromagnetic-paramagnetic (F-P) transition for a sample with a hole density of $p$=2$\\times10^{11}$\\,cm$^{-2}$.","This transition is due to crossing of the 0$\\uparrow$ and 1$\\downarrow$ Landau levels.","However, in another sample, with $p$=7.2$\\times10^{10}$\\,cm$^{-2}$, the 0$\\uparrow$ and 1$\\downarrow$ Landau levels coincide for angles $\\Theta$=0-70$^{\\text{o}}$.","Only for $\\Theta$ > 70$^{\\text{o}}$ do the levels start to diverge which, in turn, results in the energy gap opening."],["Introduction","Magnetotransport measurements on dilute p-Si/SiGe/Si structures, with two-dimensional hole gas (2DHG) densities of about 10$^{11}$ cm$^{-2}$ , have revealed an unusual phenomenon at filling factor $\\nu $ =3/2, the so-called \"re-entrant\" metal-insulator transition.","[1], [2], [3], [4], [5], [6] This phenomenon manifests itself as an additional peak of the magnetoresistance $\\rho _{xx}(T, \\Theta )$ at $\\nu $ =3/2.","The peak demonstrates an insulator type behavior, i.e.","its magnitude increases with decreasing sample temperature.","[3], [5] The authors of Ref.","2 explained this appearance by the presence of smooth long-range potential fluctuations having a magnitude comparable to the Fermi energy.","However, in Refs.","3,4,5 the magnetoresistance anomaly was attributed to a crossing of Landau levels (LLs) with different spin directions 0$\\uparrow $ and 1$\\downarrow $ as the magnetic field increased.","It appears that some p-Si/SiGe/Si systems show a magnetoresistance anomaly at $\\nu =3/2$ that depends on the tilt angle between the magnetic field and sample normal, [6] whereas in other p-Si/SiGe/Si systems this anomaly is not manifested at all.","[4] A third set of p-Si/SiGe/Si systems have such anomaly in $\\rho _{xx}$ at $\\nu =3/2$ , but it does not depend on the tilt angle.","[3] In our earlier article [7], we analyzed the conductivity at $\\nu $ =2 in tilted magnetic fields in a sample with $p$ =2$\\times $ 10$^{11}$ cm$^{-2}$ and demonstrated the presence of a ferromagnetic-paramagnetic (F-P) transition at a tilt angle of about 60$^{\\text{o}}$ .","It should be noted that at $\\nu $ =3/2 we did not observe any significant variation of the conductivity, instead a resistivity peak of the re-entrant-transition-type occurred in this region of filling factor.","We therefore focused our research on the $\\nu $ =2 region, i.e.","in the vicinity of the ferromagnetic-paramagnetic transition.","The magnetoresistance components $\\rho _{xx}$ and $\\rho _{xy}$ for the p-Si/SiGe/Si structure were measured in a tilted magnetic field, from which the conductivity $\\sigma _{xx}$ was calculated together with its dependence on temperature $T$ , magnetic field, and the tilt angle $\\Theta $ .","Such an approach allowed us to approximately calculate values of the Landau level energies, rather than just providing a qualitative description of the phenomenon, as was presented in Refs. 1,2,3,4,5,6.","The F-P phase transition seen at $\\nu \\cong $ 2, $T$ =0.3 K, and $\\Theta \\approx $ 60$^{\\text{o}}$ , is the result of crossing of the 0$\\uparrow $ and 1$\\downarrow $ LLs.","This transition is characterized by a jump in the filling factor and by a coexistence of both phases in the transition region.","A F-P transition has previously been reported in p-Si/SiGe/Si at $\\nu $ =4 and 6 in a tilted magnetic field by the authors of Ref. 8.","The present paper is an continuation of our previous article [7] and has three aims: (i) to study the dependence of the energy gap between LLs 0$\\uparrow $ and 1$\\downarrow $ on the magnetic field tilt angle $\\Theta $ to provide further confirmation of the crossing of these levels, in the p-Si/SiGe/Si sample with $p$ =2$\\times $ 10$^{11}$ cm$^{-2}$ ; (ii) to investigate the conductivity anisotropy in this sample, by measuring the conductivity at different orientations of the magnetic field component in the sample plane with respect to the current: $B_{\\parallel }\\parallel I$ and $B_{\\parallel }\\perp I$ , and comparing this with the theoretical model proposed in [9]; (iii) to measure the magnetoresistance in a tilted magnetic field for another p-Si/SiGe/Si sample with a lower density of $p$ =7.2$\\times $ 10$^{10}$ cm$^{-2}$ and compare it with the experimental data obtained by other groups on similar samples [3], [4], [6], with the hope of clearing up the inconsistency of the previous results mentioned above."],["Experiment and Discussion","In this research we studied two p-Si/SiGe/Si systems grown on a Si (100) substrate that consisted of a 300 nm Si buffer layer followed by a 30 nm Si$_{(1-x)}$ Ge$_{x}$ layer, 20 nm undoped Si spacer, and 50 nm layer of B-doped Si with a doping concentration of 2.5$\\times $ 10$^{18}$ cm$^{-3}$ .","One sample had x=0.08, yielding $p$ =7.2$\\times $ 10$^{10} $ cm$^{-2}$ , and the second had x=0.13, with $p$ =2$\\times $ 10$^{11}$ cm$^{-2}$ .","Both samples had a hole mobility of about 1$\\times $ 10$^4$ cm$^2$ /Vs at liquid-helium temperatures.","Figure: Dependence of the activation energy on tilt angleΘ\\Theta .","Inset: Dependence of the conductivity σ xx \\sigma _{xx} onΘ\\Theta at ν≈\\nu \\approx 2; TT=0.3 K.In the sample with $p$ =2$\\times $ 10$^{11}$ cm$^{-2}$ we measured the temperature dependence of the conductivity at different tilt angles $\\Theta $ over the temperature range 20 mK to 1 K, from which we were able to determine the activation energy $\\Delta E$ at various angles via the slope of the Arrhenius curves: $\\ln \\sigma _{xx} \\propto 1 /T$ .","The dependence of the activation energy on the tilt angle $\\Theta $ is shown in Figure REF , where it can clearly be seen that the activation energy achieves a minimum at $\\Theta \\approx $ 60$^{\\text{o}}$ .","The conductivity $\\sigma _{xx}$ ($\\Theta $ ) at the minima of oscillations at $\\nu \\cong $ 2, also shows a maximum as a function of tilt angle at $\\Theta \\approx $ 60$^{\\text{o}}$ , as shown in the inset to Figure REF .","It is worth noting that when the measurements are performed with the magnetic field normal to the sample plane the energy gap related to $\\nu \\cong $ 2 is about 3.2 K (0.28 meV).","Thus, we are justified in extracting the energy gap value from the temperature range of 200 mK - 1 K. When the tilt angle approaches 60$^{\\text{o}}$ the size of the energy gap is very small, due to the LLs crossing.","So, whilst the actual gap value obtained here is subject to considerable uncertainty, the observation of a minimum of the energy gap value at about 60$^{\\text{o}}$ qualitatively supports our model.","Figure: Energies of the LLs 0↑\\uparrow and 1↓\\downarrow vs.angle Θ\\Theta for the sample withpp=2×\\times 10 11 ^{11} cm -2 ^{-2}.These facts confirm that the observed F-P transition is indeed associated with the crossing of the LLs 0$\\uparrow $ and 1$\\downarrow $ at 60$^{\\textrm {o}}$ .","Now, knowing the activation energy dependence on $\\Theta $ and using the value $\\Delta E$ =0.28 meV found in Ref.","7 for $\\Theta $ =0, we can get a more accurate angle dependence of the energies of the levels 0$\\uparrow $ and 1$\\downarrow $ .","It is presented in Figure REF .","The F-P transition is expected to be accompanied by the formation of ferromagnetic domains.","According to Ref.","9, the domain formation should be manifested in an anisotropy of the magnetoresistance, i.e.","in a tilted field the value of the magnetoresistance should depend on the orientation of $B_{\\parallel }$ , the in-plane projection of the magnetic field, with respect to the current.","For example, an anisotropy in the region where LLs cross has been reported in several papers for GaAs/AlGaAs [10] and n-Si/SiGe [11], [12] heterostructures.","We tilted the sample in the two possible orientations, keeping the field projection ($B_{\\parallel } \\parallel I$ ) parallel and ($B_{\\parallel } \\perp I$ ) perpendicular to the current, but did not observe any anisotropy of the magneto-resistance in the vicinity of the transition.","Figure REF illustrates the dependence of the conductivity on the normal component of the applied magnetic field $B_{\\perp }$ at different angles and for both orientations of the in-plane projection of $B$ relative to the current.","Figure: Dependences of the σ xx \\sigma _{xx} on the normal componentof the magnetic field for different tilt angles shown for twoorientations of the magnetic field B ∥ ∥IB_{\\parallel } \\parallel I andB ∥ ⊥IB_{\\parallel } \\perp I at T=0.3T=0.3 K. The curves for each angle areshifted by 5×\\times 10 -6 ^{-6} Ω -1 \\Omega ^{-1} for clarity.As seen in Figure REF , the curves for the different directions of the in-plane projection of the magnetic field ($B_{\\parallel }\\parallel I$ and $B_{\\parallel } \\perp I$ ) virtually coincide, i. e. in our case the anisotropy of the conductivity is absent with a high degree of accuracy.","We also carried out similar studies at $T$ =(18 - 200) mK for the lower density p-Si/SiGe/Si sample with $p=7.2\\times 10^{10}$ cm$^{-2}$ .","The dependence of the resistivity $\\rho _{xx}$ on the magnetic field for different tilt angles are shown in Figure REF .","We particularly notice that, at tilt angles $\\Theta >$ from $0^{\\text{o}}$ to 70$^{\\text{o}}$ , the oscillations corresponding to $\\nu $ =2 are extremely weak.","They only start manifesting themselves for $\\Theta >$ 70$^{\\text{o}}$ .","At $\\nu $ =3/2, a maximum of resistance appears similar to the one we observed in the other sample, with a magnitude that depends strongly on the tilt angle.","Figure: Dependences of the ρ xx \\rho _{xx} on the normal component ofthe magnetic field for different tilt angles.","TT=0.2 K.Yet the oscillations at $\\nu $ =2 are clearly visible in another way of measuring the magnetoresistance: when the sample is rotated in a fixed total magnetic field, the perpendicular field component $B_{\\perp }$ causes oscillations at the angles determined by the concentration of charge carriers in the sample.","Figure REF shows such an angle dependence of the magnetoresistance measured at several fixed magnetic fields, where the oscillation related to $\\nu $ =2 can be seen to move from a tilt of about 9$^{\\text{o}}$ at 10 T to 5$^{\\text{o}}$ at 18 T. This corresponds to $B_{\\perp }$ = 1.7 T in each case, as shown in the Figure REF inset.","[11] The field value for $\\nu $ =2 $B$ =1.7 T is slightly different from data shown above.","This is probably a result of an ageing of the sample as the experiments of Ref.","11 were done much earlier.","Figure: Resistance ρ xx \\rho _{xx} as a function of the field tiltangle with respect to the plane of the 2D layer at different valuesof the total magnetic field, T≈T \\approx 0.4 K. Inset: ρ xx \\rho _{xx}as a function of the normal component of the magnetic field BB.The dependence of the conductivity $\\sigma _{xx}$ on the normal component of the magnetic field $B_{\\perp }$ is shown in Figure REF at different tilt angles, with $B_{\\parallel } \\parallel I$ .","Since the oscillations of $\\rho _{xx}$ at high tilt angles are observed against a background of high resistance with $\\rho _{xx} \\gg \\rho _{xy}$ , it turns out that $\\sigma _{xx} \\sim 1/\\rho _{xx}$ , so minima in $\\rho _{xx}$ correspond to maxima in $\\sigma _{xx}$ , as observed at $B_{\\perp }\\approx $ 1.5 T in Figure REF .","Figure: Dependences of the σ xx \\sigma _{xx} on the normal componentof the magnetic field for different tilt angles for the samplep=7.2×10 10 p=7.2 \\times 10^{10} cm -2 ^{-2}; T=0.2T=0.2 K.The absence of oscillations at magnetic fields corresponding to $\\nu $ =2 in the range of angles (0-70)$^{\\text{o}}$ indicates that the 0$\\uparrow $ and 1$\\downarrow $ LLs coincide.","The appearance of these oscillations for $\\Theta > $ 70$^{\\text{o}}$ is due, in our opinion, to the fact that the levels begin to diverge, resulting in the energy gap opening up.","Apparently, the gap opening in the sample with $p=7.2 \\times 10^{10}$ cm$^{-2}$ is associated with the angle dependence of the g-factor.","The g-factor in this material is anisotropic [1] and depends on the magnetic field tilt angle relative to the sample surface normal.","If the g-factor had an axial symmetry we could write $g^*=\\sqrt{g_{\\perp }^2\\cos ^2(\\Theta )+g_{\\parallel }^2 \\sin ^2(\\Theta )}$ , where $g_{\\perp }$ is the g-factor with the magnetic field perpendicular to the 2DHG, and $g_{\\parallel }$ is with the magnetic field parallel to the 2DHG.","For strong anisotropy, when $g_{\\parallel }$ =0 (as it should be in our structure) this reduces to $g^* = g_{\\perp } \\cos \\Theta $ .","However, if such a dependence of the g-factor were to occur, then the F-P transition should not be observed.","Unfortunately, we are unable to make reliable calculations and determine the width of the gap appearing in the sample with $p=7.2\\times 10^{10}$ cm$^{-2}$ due to the large magnetoresistance produced by the parallel magnetic field in this sample.","[13] It should be noted that the values of $\\rho _{xx}(B)$ and $\\sigma _{xx}(B)$ , on which background the oscillations develop, strongly depend on the magnetic field, and the greater the angle the stronger is this dependence.","So, it does not seem to be possible to reliably separate the small oscillations at $\\Theta > $ 70$^{\\text{o}}$ from the smooth background of $\\rho _{xx}(B)$ , which is about 10$^6$ ohms.","(Such problem for the sample with $p=2 \\times 10^{11}$ cm$^{-2}$ did not arise because the overall change $\\rho _{xx}(B)$ /$\\rho _{xx}$ (0) in a parallel magnetic field of 18 T did not exceed a factor of 4, and the in-plane resistance was only about 10$^4$ ohms).","Thus, the complete F-P transition in the sample with $p=7.2 \\times 10^{10}$ cm$^{-2}$ is not observed in tilted fields.","In a wide range of angles $\\Theta $ =(0-70)$^{\\text{o}}$ the 0$\\uparrow $ and 1$\\downarrow $ LLs are still coinciding, and only for $\\Theta >$ 70$^{\\text{o}}$ is there a gap in the hole energy spectrum arising as a result of a divergence of the LLs."],["Conclusion","The ferromagnetic-paramagnetic transition is observed in a p-Si/GeSi/Si sample with $p=2 \\times 10^{11}$ cm$^{-2}$ at a magnetic field corresponding to filling factor $\\nu \\approx $ 2.","It appears as a result of a change in the relative position of the 0$\\uparrow $ and 1$\\downarrow $ LLs as a function of the tilt angle $\\Theta $ .","This fact was first demonstrated in Ref.","7 and is confirmed in this paper by measurements of the energy gap dependence on the angle $\\Theta $ .","For this sample we also demonstrate an absence of anisotropy of xx with respect to the magnetic field projection on to the sample plane, despite such an anisotropy having been proposed in Ref. 9.","At the same time, in the sample with $p=7.2\\times 10^{10}$ cm$^{-2}$ the ferromagnetic-paramagnetic transition is not observed.","In a wide range of angles $\\Theta =$ 0-70$^{\\text{o}}$ the LLs 0$\\uparrow $ and 1$\\downarrow $ coincide, and only for $\\Theta > $ 70$^{\\text{o}}$ does a gap open in the hole spectrum as a result of the LLs diverging.","Ambiguity in the results observed by various authors [1], [2], [3], [4], [5], [6], as well as ourselves, on different p-Si/GeSi/Si samples is due, in our opinion, to dissimilar dependences of the g-factors on the magnetic field tilt angle.","This is caused by different levels of disorder in all these samples, since disorder can lead to breaking of the axial symmetry."],["Acknowledgments","The authors are grateful to E. Palm, T. Murphy, J.H.","Park, and G. Jones for their help with the experiments.","This work was supported by grant of RFBR 11-02-00223, grant of the Presidium of the Russian Academy of Science, the Program \"Spintronika\" of Branch of Physical Sciences of RAS.","The NHMFL is supported by the NSF through Cooperative Agreement No.","DMR-0654118, the State of Florida, and the US Department of Energy."]],"string":"[\n [\n \"Magnetoresistivity in a Tilted Magnetic Field in p-Si/SiGe/Si\\n Heterostructures with an Anisotropic g-Factor: Part II\"\n ],\n [\n \"Abstract The magnetoresistance components $\\\\rho_{xx}$ and $\\\\rho_{xy}$ were measured in two p-Si/SiGe/Si quantum wells that have an anisotropic g-factor in a tilted magnetic field as a function of temperature, field and tilt angle.\",\n \"Activation energy measurements demonstrate the existence of a ferromagnetic-paramagnetic (F-P) transition for a sample with a hole density of $p$=2$\\\\times10^{11}$\\\\,cm$^{-2}$.\",\n \"This transition is due to crossing of the 0$\\\\uparrow$ and 1$\\\\downarrow$ Landau levels.\",\n \"However, in another sample, with $p$=7.2$\\\\times10^{10}$\\\\,cm$^{-2}$, the 0$\\\\uparrow$ and 1$\\\\downarrow$ Landau levels coincide for angles $\\\\Theta$=0-70$^{\\\\text{o}}$.\",\n \"Only for $\\\\Theta$ > 70$^{\\\\text{o}}$ do the levels start to diverge which, in turn, results in the energy gap opening.\"\n ],\n [\n \"Introduction\",\n \"Magnetotransport measurements on dilute p-Si/SiGe/Si structures, with two-dimensional hole gas (2DHG) densities of about 10$^{11}$ cm$^{-2}$ , have revealed an unusual phenomenon at filling factor $\\\\nu $ =3/2, the so-called \\\"re-entrant\\\" metal-insulator transition.\",\n \"[1], [2], [3], [4], [5], [6] This phenomenon manifests itself as an additional peak of the magnetoresistance $\\\\rho _{xx}(T, \\\\Theta )$ at $\\\\nu $ =3/2.\",\n \"The peak demonstrates an insulator type behavior, i.e.\",\n \"its magnitude increases with decreasing sample temperature.\",\n \"[3], [5] The authors of Ref.\",\n \"2 explained this appearance by the presence of smooth long-range potential fluctuations having a magnitude comparable to the Fermi energy.\",\n \"However, in Refs.\",\n \"3,4,5 the magnetoresistance anomaly was attributed to a crossing of Landau levels (LLs) with different spin directions 0$\\\\uparrow $ and 1$\\\\downarrow $ as the magnetic field increased.\",\n \"It appears that some p-Si/SiGe/Si systems show a magnetoresistance anomaly at $\\\\nu =3/2$ that depends on the tilt angle between the magnetic field and sample normal, [6] whereas in other p-Si/SiGe/Si systems this anomaly is not manifested at all.\",\n \"[4] A third set of p-Si/SiGe/Si systems have such anomaly in $\\\\rho _{xx}$ at $\\\\nu =3/2$ , but it does not depend on the tilt angle.\",\n \"[3] In our earlier article [7], we analyzed the conductivity at $\\\\nu $ =2 in tilted magnetic fields in a sample with $p$ =2$\\\\times $ 10$^{11}$ cm$^{-2}$ and demonstrated the presence of a ferromagnetic-paramagnetic (F-P) transition at a tilt angle of about 60$^{\\\\text{o}}$ .\",\n \"It should be noted that at $\\\\nu $ =3/2 we did not observe any significant variation of the conductivity, instead a resistivity peak of the re-entrant-transition-type occurred in this region of filling factor.\",\n \"We therefore focused our research on the $\\\\nu $ =2 region, i.e.\",\n \"in the vicinity of the ferromagnetic-paramagnetic transition.\",\n \"The magnetoresistance components $\\\\rho _{xx}$ and $\\\\rho _{xy}$ for the p-Si/SiGe/Si structure were measured in a tilted magnetic field, from which the conductivity $\\\\sigma _{xx}$ was calculated together with its dependence on temperature $T$ , magnetic field, and the tilt angle $\\\\Theta $ .\",\n \"Such an approach allowed us to approximately calculate values of the Landau level energies, rather than just providing a qualitative description of the phenomenon, as was presented in Refs. 1,2,3,4,5,6.\",\n \"The F-P phase transition seen at $\\\\nu \\\\cong $ 2, $T$ =0.3 K, and $\\\\Theta \\\\approx $ 60$^{\\\\text{o}}$ , is the result of crossing of the 0$\\\\uparrow $ and 1$\\\\downarrow $ LLs.\",\n \"This transition is characterized by a jump in the filling factor and by a coexistence of both phases in the transition region.\",\n \"A F-P transition has previously been reported in p-Si/SiGe/Si at $\\\\nu $ =4 and 6 in a tilted magnetic field by the authors of Ref. 8.\",\n \"The present paper is an continuation of our previous article [7] and has three aims: (i) to study the dependence of the energy gap between LLs 0$\\\\uparrow $ and 1$\\\\downarrow $ on the magnetic field tilt angle $\\\\Theta $ to provide further confirmation of the crossing of these levels, in the p-Si/SiGe/Si sample with $p$ =2$\\\\times $ 10$^{11}$ cm$^{-2}$ ; (ii) to investigate the conductivity anisotropy in this sample, by measuring the conductivity at different orientations of the magnetic field component in the sample plane with respect to the current: $B_{\\\\parallel }\\\\parallel I$ and $B_{\\\\parallel }\\\\perp I$ , and comparing this with the theoretical model proposed in [9]; (iii) to measure the magnetoresistance in a tilted magnetic field for another p-Si/SiGe/Si sample with a lower density of $p$ =7.2$\\\\times $ 10$^{10}$ cm$^{-2}$ and compare it with the experimental data obtained by other groups on similar samples [3], [4], [6], with the hope of clearing up the inconsistency of the previous results mentioned above.\"\n ],\n [\n \"Experiment and Discussion\",\n \"In this research we studied two p-Si/SiGe/Si systems grown on a Si (100) substrate that consisted of a 300 nm Si buffer layer followed by a 30 nm Si$_{(1-x)}$ Ge$_{x}$ layer, 20 nm undoped Si spacer, and 50 nm layer of B-doped Si with a doping concentration of 2.5$\\\\times $ 10$^{18}$ cm$^{-3}$ .\",\n \"One sample had x=0.08, yielding $p$ =7.2$\\\\times $ 10$^{10} $ cm$^{-2}$ , and the second had x=0.13, with $p$ =2$\\\\times $ 10$^{11}$ cm$^{-2}$ .\",\n \"Both samples had a hole mobility of about 1$\\\\times $ 10$^4$ cm$^2$ /Vs at liquid-helium temperatures.\",\n \"Figure: Dependence of the activation energy on tilt angleΘ\\\\Theta .\",\n \"Inset: Dependence of the conductivity σ xx \\\\sigma _{xx} onΘ\\\\Theta at ν≈\\\\nu \\\\approx 2; TT=0.3 K.In the sample with $p$ =2$\\\\times $ 10$^{11}$ cm$^{-2}$ we measured the temperature dependence of the conductivity at different tilt angles $\\\\Theta $ over the temperature range 20 mK to 1 K, from which we were able to determine the activation energy $\\\\Delta E$ at various angles via the slope of the Arrhenius curves: $\\\\ln \\\\sigma _{xx} \\\\propto 1 /T$ .\",\n \"The dependence of the activation energy on the tilt angle $\\\\Theta $ is shown in Figure REF , where it can clearly be seen that the activation energy achieves a minimum at $\\\\Theta \\\\approx $ 60$^{\\\\text{o}}$ .\",\n \"The conductivity $\\\\sigma _{xx}$ ($\\\\Theta $ ) at the minima of oscillations at $\\\\nu \\\\cong $ 2, also shows a maximum as a function of tilt angle at $\\\\Theta \\\\approx $ 60$^{\\\\text{o}}$ , as shown in the inset to Figure REF .\",\n \"It is worth noting that when the measurements are performed with the magnetic field normal to the sample plane the energy gap related to $\\\\nu \\\\cong $ 2 is about 3.2 K (0.28 meV).\",\n \"Thus, we are justified in extracting the energy gap value from the temperature range of 200 mK - 1 K. When the tilt angle approaches 60$^{\\\\text{o}}$ the size of the energy gap is very small, due to the LLs crossing.\",\n \"So, whilst the actual gap value obtained here is subject to considerable uncertainty, the observation of a minimum of the energy gap value at about 60$^{\\\\text{o}}$ qualitatively supports our model.\",\n \"Figure: Energies of the LLs 0↑\\\\uparrow and 1↓\\\\downarrow vs.angle Θ\\\\Theta for the sample withpp=2×\\\\times 10 11 ^{11} cm -2 ^{-2}.These facts confirm that the observed F-P transition is indeed associated with the crossing of the LLs 0$\\\\uparrow $ and 1$\\\\downarrow $ at 60$^{\\\\textrm {o}}$ .\",\n \"Now, knowing the activation energy dependence on $\\\\Theta $ and using the value $\\\\Delta E$ =0.28 meV found in Ref.\",\n \"7 for $\\\\Theta $ =0, we can get a more accurate angle dependence of the energies of the levels 0$\\\\uparrow $ and 1$\\\\downarrow $ .\",\n \"It is presented in Figure REF .\",\n \"The F-P transition is expected to be accompanied by the formation of ferromagnetic domains.\",\n \"According to Ref.\",\n \"9, the domain formation should be manifested in an anisotropy of the magnetoresistance, i.e.\",\n \"in a tilted field the value of the magnetoresistance should depend on the orientation of $B_{\\\\parallel }$ , the in-plane projection of the magnetic field, with respect to the current.\",\n \"For example, an anisotropy in the region where LLs cross has been reported in several papers for GaAs/AlGaAs [10] and n-Si/SiGe [11], [12] heterostructures.\",\n \"We tilted the sample in the two possible orientations, keeping the field projection ($B_{\\\\parallel } \\\\parallel I$ ) parallel and ($B_{\\\\parallel } \\\\perp I$ ) perpendicular to the current, but did not observe any anisotropy of the magneto-resistance in the vicinity of the transition.\",\n \"Figure REF illustrates the dependence of the conductivity on the normal component of the applied magnetic field $B_{\\\\perp }$ at different angles and for both orientations of the in-plane projection of $B$ relative to the current.\",\n \"Figure: Dependences of the σ xx \\\\sigma _{xx} on the normal componentof the magnetic field for different tilt angles shown for twoorientations of the magnetic field B ∥ ∥IB_{\\\\parallel } \\\\parallel I andB ∥ ⊥IB_{\\\\parallel } \\\\perp I at T=0.3T=0.3 K. The curves for each angle areshifted by 5×\\\\times 10 -6 ^{-6} Ω -1 \\\\Omega ^{-1} for clarity.As seen in Figure REF , the curves for the different directions of the in-plane projection of the magnetic field ($B_{\\\\parallel }\\\\parallel I$ and $B_{\\\\parallel } \\\\perp I$ ) virtually coincide, i. e. in our case the anisotropy of the conductivity is absent with a high degree of accuracy.\",\n \"We also carried out similar studies at $T$ =(18 - 200) mK for the lower density p-Si/SiGe/Si sample with $p=7.2\\\\times 10^{10}$ cm$^{-2}$ .\",\n \"The dependence of the resistivity $\\\\rho _{xx}$ on the magnetic field for different tilt angles are shown in Figure REF .\",\n \"We particularly notice that, at tilt angles $\\\\Theta >$ from $0^{\\\\text{o}}$ to 70$^{\\\\text{o}}$ , the oscillations corresponding to $\\\\nu $ =2 are extremely weak.\",\n \"They only start manifesting themselves for $\\\\Theta >$ 70$^{\\\\text{o}}$ .\",\n \"At $\\\\nu $ =3/2, a maximum of resistance appears similar to the one we observed in the other sample, with a magnitude that depends strongly on the tilt angle.\",\n \"Figure: Dependences of the ρ xx \\\\rho _{xx} on the normal component ofthe magnetic field for different tilt angles.\",\n \"TT=0.2 K.Yet the oscillations at $\\\\nu $ =2 are clearly visible in another way of measuring the magnetoresistance: when the sample is rotated in a fixed total magnetic field, the perpendicular field component $B_{\\\\perp }$ causes oscillations at the angles determined by the concentration of charge carriers in the sample.\",\n \"Figure REF shows such an angle dependence of the magnetoresistance measured at several fixed magnetic fields, where the oscillation related to $\\\\nu $ =2 can be seen to move from a tilt of about 9$^{\\\\text{o}}$ at 10 T to 5$^{\\\\text{o}}$ at 18 T. This corresponds to $B_{\\\\perp }$ = 1.7 T in each case, as shown in the Figure REF inset.\",\n \"[11] The field value for $\\\\nu $ =2 $B$ =1.7 T is slightly different from data shown above.\",\n \"This is probably a result of an ageing of the sample as the experiments of Ref.\",\n \"11 were done much earlier.\",\n \"Figure: Resistance ρ xx \\\\rho _{xx} as a function of the field tiltangle with respect to the plane of the 2D layer at different valuesof the total magnetic field, T≈T \\\\approx 0.4 K. Inset: ρ xx \\\\rho _{xx}as a function of the normal component of the magnetic field BB.The dependence of the conductivity $\\\\sigma _{xx}$ on the normal component of the magnetic field $B_{\\\\perp }$ is shown in Figure REF at different tilt angles, with $B_{\\\\parallel } \\\\parallel I$ .\",\n \"Since the oscillations of $\\\\rho _{xx}$ at high tilt angles are observed against a background of high resistance with $\\\\rho _{xx} \\\\gg \\\\rho _{xy}$ , it turns out that $\\\\sigma _{xx} \\\\sim 1/\\\\rho _{xx}$ , so minima in $\\\\rho _{xx}$ correspond to maxima in $\\\\sigma _{xx}$ , as observed at $B_{\\\\perp }\\\\approx $ 1.5 T in Figure REF .\",\n \"Figure: Dependences of the σ xx \\\\sigma _{xx} on the normal componentof the magnetic field for different tilt angles for the samplep=7.2×10 10 p=7.2 \\\\times 10^{10} cm -2 ^{-2}; T=0.2T=0.2 K.The absence of oscillations at magnetic fields corresponding to $\\\\nu $ =2 in the range of angles (0-70)$^{\\\\text{o}}$ indicates that the 0$\\\\uparrow $ and 1$\\\\downarrow $ LLs coincide.\",\n \"The appearance of these oscillations for $\\\\Theta > $ 70$^{\\\\text{o}}$ is due, in our opinion, to the fact that the levels begin to diverge, resulting in the energy gap opening up.\",\n \"Apparently, the gap opening in the sample with $p=7.2 \\\\times 10^{10}$ cm$^{-2}$ is associated with the angle dependence of the g-factor.\",\n \"The g-factor in this material is anisotropic [1] and depends on the magnetic field tilt angle relative to the sample surface normal.\",\n \"If the g-factor had an axial symmetry we could write $g^*=\\\\sqrt{g_{\\\\perp }^2\\\\cos ^2(\\\\Theta )+g_{\\\\parallel }^2 \\\\sin ^2(\\\\Theta )}$ , where $g_{\\\\perp }$ is the g-factor with the magnetic field perpendicular to the 2DHG, and $g_{\\\\parallel }$ is with the magnetic field parallel to the 2DHG.\",\n \"For strong anisotropy, when $g_{\\\\parallel }$ =0 (as it should be in our structure) this reduces to $g^* = g_{\\\\perp } \\\\cos \\\\Theta $ .\",\n \"However, if such a dependence of the g-factor were to occur, then the F-P transition should not be observed.\",\n \"Unfortunately, we are unable to make reliable calculations and determine the width of the gap appearing in the sample with $p=7.2\\\\times 10^{10}$ cm$^{-2}$ due to the large magnetoresistance produced by the parallel magnetic field in this sample.\",\n \"[13] It should be noted that the values of $\\\\rho _{xx}(B)$ and $\\\\sigma _{xx}(B)$ , on which background the oscillations develop, strongly depend on the magnetic field, and the greater the angle the stronger is this dependence.\",\n \"So, it does not seem to be possible to reliably separate the small oscillations at $\\\\Theta > $ 70$^{\\\\text{o}}$ from the smooth background of $\\\\rho _{xx}(B)$ , which is about 10$^6$ ohms.\",\n \"(Such problem for the sample with $p=2 \\\\times 10^{11}$ cm$^{-2}$ did not arise because the overall change $\\\\rho _{xx}(B)$ /$\\\\rho _{xx}$ (0) in a parallel magnetic field of 18 T did not exceed a factor of 4, and the in-plane resistance was only about 10$^4$ ohms).\",\n \"Thus, the complete F-P transition in the sample with $p=7.2 \\\\times 10^{10}$ cm$^{-2}$ is not observed in tilted fields.\",\n \"In a wide range of angles $\\\\Theta $ =(0-70)$^{\\\\text{o}}$ the 0$\\\\uparrow $ and 1$\\\\downarrow $ LLs are still coinciding, and only for $\\\\Theta >$ 70$^{\\\\text{o}}$ is there a gap in the hole energy spectrum arising as a result of a divergence of the LLs.\"\n ],\n [\n \"Conclusion\",\n \"The ferromagnetic-paramagnetic transition is observed in a p-Si/GeSi/Si sample with $p=2 \\\\times 10^{11}$ cm$^{-2}$ at a magnetic field corresponding to filling factor $\\\\nu \\\\approx $ 2.\",\n \"It appears as a result of a change in the relative position of the 0$\\\\uparrow $ and 1$\\\\downarrow $ LLs as a function of the tilt angle $\\\\Theta $ .\",\n \"This fact was first demonstrated in Ref.\",\n \"7 and is confirmed in this paper by measurements of the energy gap dependence on the angle $\\\\Theta $ .\",\n \"For this sample we also demonstrate an absence of anisotropy of xx with respect to the magnetic field projection on to the sample plane, despite such an anisotropy having been proposed in Ref. 9.\",\n \"At the same time, in the sample with $p=7.2\\\\times 10^{10}$ cm$^{-2}$ the ferromagnetic-paramagnetic transition is not observed.\",\n \"In a wide range of angles $\\\\Theta =$ 0-70$^{\\\\text{o}}$ the LLs 0$\\\\uparrow $ and 1$\\\\downarrow $ coincide, and only for $\\\\Theta > $ 70$^{\\\\text{o}}$ does a gap open in the hole spectrum as a result of the LLs diverging.\",\n \"Ambiguity in the results observed by various authors [1], [2], [3], [4], [5], [6], as well as ourselves, on different p-Si/GeSi/Si samples is due, in our opinion, to dissimilar dependences of the g-factors on the magnetic field tilt angle.\",\n \"This is caused by different levels of disorder in all these samples, since disorder can lead to breaking of the axial symmetry.\"\n ],\n [\n \"Acknowledgments\",\n \"The authors are grateful to E. Palm, T. Murphy, J.H.\",\n \"Park, and G. Jones for their help with the experiments.\",\n \"This work was supported by grant of RFBR 11-02-00223, grant of the Presidium of the Russian Academy of Science, the Program \\\"Spintronika\\\" of Branch of Physical Sciences of RAS.\",\n \"The NHMFL is supported by the NSF through Cooperative Agreement No.\",\n \"DMR-0654118, the State of Florida, and the US Department of Energy.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1144"}}},{"rowIdx":1866792,"cells":{"text":{"kind":"list like","value":[["A stratification on the moduli spaces of symplectic and orthogonal\n bundles over a curve"],["Abstract A symplectic or orthogonal bundle $V$ of rank $2n$ over a curve has an invariant $t(V)$ which measures the maximal degree of its isotropic subbundles of rank $n$.","This invariant $t$ defines stratifications on moduli spaces of symplectic and orthogonal bundles.","We study this stratification by relating it to another one given by secant varieties in certain extension spaces.","We give a sharp upper bound on $t(V)$, which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces.","In particular, we compute the dimension of each stratum.","We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite.","We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case."],["Introduction","Let $X$ be a smooth algebraic curve over $\\mathbb {C}$ .","A vector bundle $V$ of rank two over $X$ determines a ruled surface $\\mathbb {P}V$ .","Such surfaces have been studied since the 19th century.","A line subbundle $L$ of $V$ gives a section $\\sigma _L$ of $\\mathbb {P}V$ , called a directrix curve.","The self-intersection number of $\\sigma _L$ is given by $\\sigma _{L} \\cdot \\sigma _{L} \\ = \\ \\deg (V/L) - \\deg L \\ = \\ \\deg V - 2 \\deg L.$ The Segre invariant of $\\mathbb {P}V$ is defined as the minimal value of $\\sigma _{L} \\cdot \\sigma _L$ over all $L \\subset V$ .","Via the above formula, this invariant also provides a measure of the difference of the slopes of $V$ and $L$ .","The Segre invariant yields a natural stratification on the moduli space of vector bundles of rank two over $X$ , which was studied by Lange and Narasimhan [13].","A generalization of this stratification to the moduli of vector bundles over $X$ of arbitrary rank was considered by Lange [11] and the details were settled in Brambila-Paz–Lange [2], and Russo–Teixidor i Bigas [19].","The aim of the present article is to establish parallel results for symplectic and orthogonal bundles over $X$ .","Let us define a few notions and fix notations.","A vector bundle $V$ over $X$ of rank $2n$ is called symplectic (resp., orthogonal) if there is a nondegenerate alternating (resp., symmetric) bilinear form $\\omega \\colon V \\otimes V \\rightarrow \\mathcal {O}_{X}$ .","A subbundle $E$ of $V$ is called isotropic if $\\omega |_{E \\otimes E} = 0$ .","By linear algebra, an isotropic subbundle of $V$ has rank $\\le n$ .","When $V$ is symplectic, a rank $n$ isotropic subbundle of $V$ is often called a Lagrangian subbundle.","We say that a symplectic or orthogonal bundle $V$ is stable (resp., semistable) if for every isotropic subbundle $E$ of $V$ , we have $\\mu (V) = \\frac{ \\deg V}{\\mathrm {rk}V} \\ > \\ \\frac{ \\deg E}{\\mathrm {rk}E} = \\mu (E) \\ \\ \\ (\\text{resp., } \\mu (V) \\ge \\mu (E)).$ Note that this is a priori weaker than the stability condition for $V$ as a vector bundle; compare with Ramanathan [18].","However, Ramanan [17] proved that semistability as an orthogonal bundle is equivalent to semistability of the underlying vector bundle, and moreover that a general stable orthogonal bundle is a stable vector bundle.","The same argument (worked through in [5]) shows that the analogous statement is true for symplectic bundles.","We denote by $SU_X(2n,\\mathcal {O}_{X})$ the moduli space of semistable vector bundles of rank $2n$ and trivial determinant, and write $\\mathcal {M}S_X(2n)$ (resp., $\\mathcal {M}O_X(2n)$ ) for the sublocus in $SU_X(2n,\\mathcal {O}_{X})$ of bundles admitting a symplectic (resp., orthogonal) structure.","In the symplectic case it has been proven by Serman [20] that the forgetful map $\\mathcal {M}_X(\\mathrm {Sp}_{2n}\\mathbb {C}) \\rightarrow SU_X(2n, \\mathcal {O}_{X})$ associated to the extension of the structure group $\\mathrm {Sp}_{2n}\\mathbb {C}\\subset \\mathrm {SL}_{2n} \\mathbb {C}$ , is an embedding, where $\\mathcal {M}_X(\\mathrm {Sp}_{2n}\\mathbb {C})$ is the moduli space of semistable principal $\\mathrm {Sp}_{2n}\\mathbb {C}$ -bundles over $X$ .","So $\\mathcal {M}S_X(2n)$ coincides with the embedded image of $\\mathcal {M}_X(\\mathrm {Sp}_{2n}\\mathbb {C})$ .","The orthogonal case is more delicate.","By [20], the forgetful map $\\mathcal {M}_X(\\mathrm {SO}_{2n}\\mathbb {C}) \\rightarrow SU_X(2n, \\mathcal {O}_{X})$ is generically two-to-one, amounting to forgetting the data of an orientation on a principal $\\mathrm {SO}_{2n}\\mathbb {C}$ -bundle.","On the other hand, the map $\\mathcal {M}_X(\\mathrm {O}_{2n}\\mathbb {C}) \\rightarrow \\mathcal {M}_X(\\mathrm {GL}_{2n}\\mathbb {C})$ is an embedding.","The moduli space $\\mathcal {M}_X(\\mathrm {O}_{2n}\\mathbb {C})$ of semistable principal $\\mathrm {O}_{2n}\\mathbb {C}$ -bundles over $X$ has several components, which are indexed by the first and second Stiefel–Whitney classes $(w_{1}, w_{2}) \\in H^{1}(X, \\mathbb {Z}_{2}) \\times H^{2}(X, \\mathbb {Z}_{2})$ .","The class $w_1$ corresponds to the determinant, and there are two components of $\\mathcal {M}_X(\\mathrm {O}_{2n}\\mathbb {C})$ with $w_1$ trivial.","We write $\\mathcal {M}O_{X}(2n)^\\pm $ for the embedded images of these components in $SU_X(2n, \\mathcal {O}_{X})$ .","Let $V$ be a symplectic or orthogonal bundle of rank $2n$ .","Generalizing the Segre invariant for bundles of rank two, we define $t(V) := -2\\max \\lbrace \\deg E : E \\hbox{ a rank $n$ isotropic subbundle of } V \\rbrace .$ In particular, if $V$ is stable (resp., semistable), then $t(V) > 0$ (resp., $t(V) \\ge 0$ ).","(Note that when $V$ is a symplectic bundle of rank $>2$ , this differs from the invariant $s_{\\mathrm {Lag}}$ defined in [3] by a constant: $t(V) = \\frac{2}{n+1} s_{\\mathrm {Lag}}(V) $ .)","For a symplectic or orthogonal bundle $V$ , we denote by $M(V)$ the space of rank $n$ isotropic subbundles $E \\subset V$ such that $t(V) = -2 \\deg E$ .","It can be viewed as a closed subscheme of a Quot scheme, so it has a natural structure of projective variety.","The invariant $t(V)$ induces stratifications on moduli spaces of semistable symplectic and orthogonal bundles over $X$ .","For each positive even integer $t$ , we define $\\mathcal {M}S_X(2n;t) := \\lbrace V \\in \\mathcal {M}S_X(2n) : t(V) = t \\rbrace $ and $\\mathcal {M}O_X(2n;t) := \\lbrace V \\in \\mathcal {M}O_X(2n) : t(V) = t \\rbrace .$ By semi-continuity of the invariant $t(V)$ , these subloci are constructible sets.","For the symplectic case, we show: Theorem 1.1 Let $X$ be a smooth algebraic curve of genus $g \\ge 2$ .","For any symplectic bundle $V$ of rank $2n$ , we have $t(V) \\le n(g-1) +1$ .","This is sharp in the sense that for a general $V \\in \\mathcal {M}S_X(2n)$ , $n(g-1) \\le t(V) \\le n(g-1 ) +1.$ For each even integer $t$ with $ 2 \\le t \\le n (g-1)$ , the stratum $\\mathcal {M}S_X(2n;t)$ is nonempty and irreducible, and $\\dim \\mathcal {M}S_X(2n;t) \\ = \\ \\frac{1}{2} \\left( n(3n+1)(g-1) + (n+1)t \\right).$ Furthermore, if $t < n(g-1)$ , then $\\mathcal {M}S_X(2n;t)$ is contained in the closure of $\\mathcal {M}S_X(2n;t+2)$ in $\\mathcal {M}S_X(2n)$ .","For each positive even integer $t < n(g-1)$ , the space $M(V)$ is a single point for a general $V \\in \\mathcal {M}S_X(2n;t)$ .","On the other hand, for a general $V \\in \\mathcal {M}S_X(2n)$ , $\\dim M(V) \\ = \\ {\\left\\lbrace \\begin{array}{ll}0& \\text{when} \\ n(g-1) \\text{ is even, } \\\\\\frac{n+1}{2}& \\text{when} \\ n(g-1) \\text{ is odd}.\\end{array}\\right.","}$ Let us give some historical remarks.","For vector bundles, the Segre stratifications are well understood; see Lange–Narasimhan [13], Hirschowitz [4], Brambila-Paz–Lange [2] and Russo–Teixidor i Bigas [19].","Holla and Narasimhan [8] defined a generalized Segre invariant for principal $G$ -bundles for an arbitrary reductive group $G$ , and obtained a bound on the invariant in general.","This bound was sharpened for symplectic bundles ($G = \\mathrm {Sp}_{2n}\\mathbb {C}$ ) by the present authors [3], where Theorem REF (1) was proven using the Terracini lemma in projective geometry.","Also Theorem REF was proven for symplectic bundles of rank four in [3].","The special case when the curve has genus two has earlier been studied in detail by the second named author [7].","In this paper, we provide another simpler proof of (1), and prove (2) and (3) for arbitrary rank $n$ and genus $g$ .","In the orthogonal case, the moduli space $\\mathcal {M}O_X(2n)$ has two connected components, as was discussed before.","We denote by $\\mathcal {M}O_{X}(2n)^+$ (resp.","$\\mathcal {M}O_X(2n)^-$ ) the component consisting of bundles of trivial (resp., nontrivial) second Stiefel–Whitney class.","We first prove the following.","Theorem 1.2 Let $E_1$ and $E_2$ be isotropic rank $n$ subbundles of an orthogonal bundle of rank $2n$ .","Then $\\deg E_1$ and $\\deg E_2$ have the same parity.","A semistable orthogonal bundle $V$ belongs to $\\mathcal {M}O_{X}(2n)^+$ (resp., $\\mathcal {M}O_{X}(2n)^-$ ) if and only if its isotropic rank $n$ subbundles have even degree (resp., odd degree).","We then show the following on the stratification on each component.","Theorem 1.3 Let $X$ be a smooth algebraic curve of genus $g \\ge 2$ .","For any orthogonal bundle $V$ of rank $2n$ , we have $t(V) \\le n(g-1) + 3$ .","This is sharp in the sense that two even numbers $t$ with $n(g-1) \\le t \\le n(g-1)+3$ correspond to the values of $t(V)$ for general $V$ in $\\mathcal {M}O_X(2n)^+$ and $\\mathcal {M}O_X(2n)^-$ .","For each even integer $t$ with $2 \\le t \\le n (g-1) $ , the stratum $\\mathcal {M}O_X(2n;t)$ is nonempty and irreducible, and $\\dim \\mathcal {M}O_X(2n;t) = \\frac{1}{2} \\left( n(3n-1)(g-1) + (n-1)t \\right).$ Furthermore, if $t < n(g-1)$ then $\\mathcal {M}O_X(2n;t)$ is contained in the closure of $\\mathcal {M}O_X(2n;t+4)$ in the relevant component $\\mathcal {M}O_X(2n)^{\\pm }$ .","For each positive even integer $t < n(g-1)$ , the space $M(V)$ is a single point for a general $V \\in \\mathcal {M}S_X(2n;t)$ .","We also compute the dimension of $M(V)$ for general bundles in each component $\\mathcal {M}O_{X}(2n)^{\\pm }$ .","It turns out to depend on the class of $n(g-1)$ modulo 4; the precise statement is set out in §5.3.","A more detailed description of the top strata of $\\mathcal {M}O_X(2n)$ is given in §5.4.","To prove these statements, we consider families of symplectic and orthogonal extensions.","The main strategy is to relate the invariant $t(V)$ to the geometry of certain higher secant varieties in the projectivized extension spaces.","This idea goes back to the work of Lange and Narasimhan [13], where Theorem REF is proven for rank two bundles using higher secant varieties of the curve $X$ embedded in the extension spaces.","In §2, we generalize the geometric framework in [13] to our situation in Criterion REF .","The embedded curve $X$ in the extension space is replaced by the quadric bundle $\\mathbb {P}E$ and the Grassmannian bundle $\\mathrm {Gr}(2, E)$ for symplectic and orthogonal cases respectively.","To work with bundles of rank $2n \\ge 4$ , one has also to understand the situation when two rank $n$ isotropic subbundles intersect non-transversely.","It turns out that this case can also be understood geometrically (Criterion REF ).","This is a key advance upon the methods in [3], which enables us to argue for symplectic and orthogonal bundles of arbitrary rank.","In §3, we construct “universal extension spaces” parameterizing all the extensions of fixed type, and show that the rational classifying maps to $\\mathcal {M}S_X(2n)$ or $\\mathcal {M}O_X(2n)$ are defined on dense subsets.","In §4 and §5, we prove the main results for symplectic and orthogonal bundles respectively, using dimension counts based on the geometric information from the preceding sections.","In §5, we also observe some interesting properties of certain families of orthogonal bundles, stemming from the richer topological structure of the moduli space.","Acknowledgements: The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.","2011-A423-0004).","Also he would like to thank Høgskolen i Vestfold, Bakkenteigen for the invitation in June 2011.","The second author gratefully acknowledges the hospitality and generous financial support of Konkuk University, Seoul.","Both authors thank Christian Pauly and Olivier Serman for helpful discussions."],["Symplectic and orthogonal extensions and lifting criteria","In this section we discuss symplectic and orthogonal extensions.","Most of the results in this section were obtained for symplectic extensions in [3].","Those results will be restated here for the reader's convenience, and the modified versions for orthogonal extensions will be proven in detail.","Let $V \\rightarrow X$ be a vector bundle of rank $2n$ equipped with a nondegenerate bilinear form $\\omega \\colon V \\otimes V \\rightarrow \\mathcal {O}_{X}$ , and let $E \\subset V$ be a subbundle.","Then there is an exact sequence $ 0 \\rightarrow E^{\\perp } \\rightarrow V \\rightarrow E^{*} \\rightarrow 0, $ where $E^\\perp $ is the orthogonal complement of $E$ .","If $E$ is isotropic of rank $n$ , then $E = E^\\perp $ and $V$ defines a class $\\delta (V) \\in H^1(X, \\mathrm {Hom}(E^{*}, E)) \\cong H^1(X, E \\otimes E)$ .","Criterion 2.1 Suppose $E$ is simple.","An extension $0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0$ is induced by a symplectic (resp., orthogonal) structure on $V$ with respect to which $E$ is isotropic if and only if the extension class $\\delta (V)$ belongs to the subspace $ H^{1}(X, \\mathrm {Sym}^{2}E) \\quad \\left( \\hbox{resp.,} \\quad H^{1}(X, \\wedge ^{2}E ) \\right) $ of $H^{1}(X, E \\otimes E)$ .","This is due to S. Ramanan.","A detailed proof for the symplectic case is given in [6], and the proof for the orthogonal case is practically identical."],["Cohomological criterion for lifting","Here we recall the notion of a bundle-valued principal part (see Kempf [10] for corresponding results on line bundles).","A locally free sheaf $W$ on $X$ has the flasque resolution $ 0 \\rightarrow W \\rightarrow \\underline{\\mathrm {Rat}}(W) \\rightarrow \\mathrm {\\underline{Prin}}(W) \\rightarrow 0, $ where $\\underline{\\mathrm {Rat}}(W)$ is the sheaf of rational sections of $W$ and $\\mathrm {\\underline{Prin}}(W)$ the sheaf of $W$ -valued principal parts.","We denote their groups of global sections by $\\mathrm {Rat}(W)$ and $\\mathrm {Prin}(W)$ respectively.","Taking global sections, we obtain $ 0 \\rightarrow H^0(X, W) \\rightarrow \\mathrm {Rat}(W) \\rightarrow \\mathrm {Prin}(W) \\rightarrow H^{1}(X, W) \\rightarrow 0.$ For a principal part $p \\in \\mathrm {Prin}(W)$ , we write its class in $H^{1}(X, W)$ as $[p]$ .","Now consider an extension $0 \\rightarrow E \\rightarrow V \\rightarrow E^* \\rightarrow 0$ , and an elementary transformation $F$ of $E^*$ defined by the sequence $ 0 \\rightarrow F \\stackrel{\\mu }{\\rightarrow } E^* \\rightarrow \\tau \\rightarrow 0 $ for some torsion sheaf $\\tau $ .","We say that $F$ lifts to $V$ if there is a sheaf injection $F \\rightarrow V$ such that the composition $F \\rightarrow V \\rightarrow E^*$ coincides with $\\mu $ .","The following statements are proven in [6]: Lemma 2.2 Suppose $h^{0}(X, \\mathrm {Hom}(E^{*}, E)) = 0$ , and consider an extension $0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0$ with class $\\delta (V) \\in H^{1}(X, E \\otimes E)$ .","There is a bijection between principal parts $p \\in \\mathrm {Prin}(E \\otimes E)$ such that $\\delta (V) = [p]$ , and elementary transformations of $E^*$ lifting to subbundles of $V$ .","The bijection is given by $ p \\ \\leftrightarrow \\ \\mathrm {Ker}\\left(p \\colon E^{*} \\rightarrow \\mathrm {\\underline{Prin}}(E) \\right).","$ Suppose that $\\delta (V) = [p] \\in H^{1}(X, \\mathrm {Sym}^{2}E)$ , corresponding to a symplectic extension.","The subbundle corresponding to $\\mathrm {Ker}(p)$ is isotropic in $V$ if and only if $p$ is a symmetric principal part; that is, $^tp = p$ .","Suppose that $\\delta (V) = [p] \\in H^{1}(X, \\wedge ^{2}E)$ , corresponding to an orthogonal extension.","The subbundle corresponding to $\\mathrm {Ker}(p)$ is isotropic in $V$ if and only if $p$ is an antisymmetric principal part; that is, $^tp = -p$ .","$\\Box $"],["Subvarieties of the extension spaces","Given any vector bundle $W$ , consider the projectivization $\\pi \\colon \\mathbb {P}W \\rightarrow X$ .","Then we have a natural rational map $ \\mathbb {P}W \\dashrightarrow \\mathbb {P}H^{1}(X, W) $ defined as follows (a slightly different description was given in [3]): Consider the evaluation map $ X \\times H^{0}(X, K_X \\otimes W^{*}) \\rightarrow K_{X}\\otimes W^{*} $ .","Via Serre duality, the dual of this map is identified with $ W \\otimes T_{X} \\rightarrow X \\times H^{1}(X, W).","$ Projectivizing this map and then composing with the projection $X \\times \\mathbb {P}H^{1}(X, W) \\rightarrow \\mathbb {P}H^{1}(X, W)$ , we get a map $ \\phi \\colon \\mathbb {P}W \\dashrightarrow \\mathbb {P}H^{1}(X, W).","$ On a fibre $W|_x$ , this map is identified with the projectivized coboundary map in the sequence $ 0 \\rightarrow H^0 (X, W) \\rightarrow H^{0}(X, W(x)) \\rightarrow \\frac{W(x)}{W}|_{x} \\rightarrow H^{1}(X, W) \\rightarrow H^1(X, W(x)) \\rightarrow 0 .","$ Thus for $w \\in \\mathbb {P}W$ , the image $\\phi (w)$ may be realized as the cohomology class of a $W$ -valued principal part supported at $x$ with a simple pole along $w$ .","As discussed in [3], the rational map $\\phi $ is induced by the complete linear system of the line bundle $\\pi ^{*}K_{X}\\otimes \\mathcal {O}_{\\mathbb {P}W}(1)$ over $\\mathbb {P}W$ .","For $W = \\mathrm {Sym}^{2} E$ and $ W= \\wedge ^2 E$ respectively, we have the rational maps $ \\mathbb {P}(\\mathrm {Sym}^{2} E) \\dashrightarrow \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E) \\quad \\hbox{and} \\quad \\mathbb {P}(\\wedge ^2 E )\\dashrightarrow \\mathbb {P}H^{1}(X, \\wedge ^{2} E).","$ Note that both of these are restrictions of the map $\\mathbb {P}(E \\otimes E) \\dashrightarrow \\mathbb {P}H^{1}(X, E \\otimes E)$ .","In the symplectic case, we consider the chain of maps $\\mathbb {P}E \\ \\hookrightarrow \\ \\mathbb {P}(\\mathrm {Sym}^{2} E) \\dashrightarrow \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E) ,$ where the first inclusion is given by the Segre embedding $[v] \\mapsto [v \\otimes v]$ .","In the orthogonal case, we consider the chain of maps $\\mathrm {Gr}(2,E) \\ \\hookrightarrow \\ \\mathbb {P}(\\wedge ^{2} E) \\dashrightarrow \\mathbb {P}H^{1}(X, \\wedge ^{2} E),$ where the Grassmannian bundle $\\mathrm {Gr}(2,E)$ is embedded in $\\mathbb {P}(\\wedge ^{2} E)$ via the Plücker embedding.","These rational maps are denoted as $\\phi _s \\colon \\mathbb {P}E \\dashrightarrow \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E) \\quad \\hbox{and} \\quad \\phi _a: \\mathrm {Gr}(2,E) \\dashrightarrow \\mathbb {P}H^{1}(X, \\wedge ^{2} E).","$ Lemma 2.3 Let $W = E \\otimes E$ for a stable bundle $E$ over a curve of genus $g \\ge 2$ .","The rational map $\\phi \\colon \\mathbb {P}(W) \\dashrightarrow \\mathbb {P}H^{1}(X, W)$ is base point free if $\\mu (E) < - \\frac{1}{2}$ , and an embedding if $\\mu (E) < -1$ .","In particular, if $\\mu (E) < -1$ , the rational maps $\\phi _s$ and $\\phi _a$ in (REF ) are embeddings.","If $E$ is general of negative degree, $\\phi $ is injective on a general fiber of $\\mathbb {P}(W)$ .","If we further assume that $g \\ge 3$ then $\\phi $ separates two general fibers of $\\mathbb {P}(W)$ .","(1) (A similar version was stated in [3], unfortunately with a flawed proof.)","One can check that $\\phi $ is base point free (resp., an embedding) if $h^{0}(X, W(D)) = 0$ for all effective divisors $D$ of degree one (resp., of degree two) on $X$ .","Since $W = E \\otimes E$ is semistable, these vanishing results follow from the assumption that $\\mu (E) < - \\frac{1}{2}$ (resp., $\\mu (E) < -1$ ).","(2) Let $L$ and $M$ be line bundles with $\\deg L = \\deg E$ and $\\deg M = 0$ , and put $E_0 = L \\oplus M^{\\oplus (n-1)}$ , where $n = \\mathrm {rk}(E)$ .","One can check that for a general choice of $L$ and $M$ , the bundle $E_0 \\otimes E_0 (x) $ has no sections.","Deforming $E_0$ to a general stable bundle $E$ , we see that $h^{0}(X, W(x)) = 0$ for a general $x \\in X$ .","A similar argument shows that if $g \\ge 3$ , then $h^{0}(X, W(x+y)) = 0$ for general $ x,y \\in X$ ."],["Irreducibility of the space of principal parts","In this subsection, we prove some technical facts on principal parts.","Let $p$ be a principal part with values in $E \\otimes E$ .","Definition 2.4 The degree $\\deg (p)$ of $p$ is defined as the length of the torsion sheaf $ \\mathrm {Im}\\left( p \\colon E^{*} \\rightarrow \\mathrm {\\underline{Prin}}(E) \\right).","$ The support of $p$ is defined as the support of $\\mathrm {Im}(p)$ .","A symmetric (resp., antisymmetric) principal part $p$ of degree $k$ (resp., $2k$ ) is called general if it is supported at $k$ distinct points.","Obviously, the general symmetric (resp., antisymmetric) principal parts of degree $k$ (resp., $2k$ ) are parameterized by a quasi-projective irreducible variety.","We want to confirm that an arbitrary symmetric or antisymmetric principal part can be obtained as a limit of a continuous family of general ones.","Lemma 2.5 ([3]) Let $p $ be a $\\mathrm {Sym}^2 E$ -valued principal part of degree $k$ , supported at a single point $x$ .","Then there exists a local frame $e_{1}, \\ldots , e_n$ for $E$ in a neighborhood of $x$ , in terms of which $p$ is expressed as $ p = \\sum _{i=1}^{n} \\frac{e_{i} \\otimes e_{i}}{z^{k_i}} $ where $z$ is a uniformizer at $x$ , and $ k_{1} \\ge k_{2} \\ge \\cdots \\ge k_n \\ge 0$ , and $k = \\sum _{i=1}^{n} k_{i} $ .","$\\Box $ For antisymmetric principal parts, we have the following analogue: Lemma 2.6 Let $p$ be a $\\wedge ^{2}E$ -valued principal part, supported at a single point $x$ .","Then there exists a local frame $e_{1}, \\ldots , e_n$ for $E$ in a neighborhood of $x$ , in terms of which $p$ is expressed as $ p = \\sum _{i=1}^{s} \\frac{e_{2i-1} \\wedge e_{2i}}{z^{k_i}} $ where $z$ is a uniformizer at $x$ , and $ k_{1} \\ge k_{2} \\ge \\cdots \\ge k_s \\ge 0$ , and $\\deg (p) = 2 \\left( \\sum _{i=1}^{s} k_{i} \\right)$ .","In particular, any antisymmetric principal part has even degree.","This argument is adapted from the proof of [3].","Locally, $p$ can be expressed as $ p = \\frac{1}{z^{k_1}} A $ for some $n \\times n$ antisymmetric matrix $A$ with entries in the ring $R = \\mathbb {C}[z]/ (z^{k_1})$ , since we are concerned with the principal parts at $x$ only.","In this context, it suffices to show that there exists a matrix $P \\in M_{n} (R)$ , such that $\\det P$ is a unit in $R$ and $^t P A P = \\text{diag}(z^{d_1}J, z^{d_2}J, \\ldots , z^{d_s}J),$ where $0 = d_1 \\le d_2 \\le \\cdots \\le d_s$ and $J = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}$ .","This can be shown by mimicking the standard process to get the normal form of an antisymmetric matrix over $\\mathbb {C}$ .","The only difference lies in that $R$ has non-units contained in the principal ideal $(z)$ , and this can be taken care of by allowing the terms $z^{d_1}, \\ldots , z^{d_s}$ on the diagonal.","Alternatively, one may work instead over the formal power series ring $\\mathbb {C}[[z]]$ , which is a PID, and truncate the terms in the ideal $(z^{k_1})$ at the final step.","For the process over a PID, see Adkins–Weintraub [1].","Hence we have a frame for $E$ on this neighborhood in terms of which $p$ appears as $ \\sum _{i=1}^{s} \\frac{e_{2i-1} \\wedge e_{2i}}{z^{k_i}}, $ where $k_{i} = k_1 - d_i$ .","Since the terms in the sum (REF ) impose independent conditions on sections of $E$ , we have $ \\deg ( p ) = \\sum _{i=1}^{s} \\deg \\left( \\frac{e_{2i-1} \\wedge e_{2i}}{z^{k_i}} \\right) = 2 \\sum _{i=1}^{s} k_{i} , $ as required.","Corollary 2.7 For a fixed vector bundle $E$ and for each $k >0$ , the spaces of symmetric principal parts of degree $k$ and antisymmetric principal parts of degree $2k$ are irreducible.","We consider the antisymmetric case; the symmetric case can be proven similarly.","It suffices to show the irreducibility of the space of antisymmetric principal parts supported at a single point $x$ .","By Lemma REF , we may choose a trivialization of $E$ near $x$ with respect to which $p$ is expressed as $\\sum _{i=1}^{s} p_i$ , where $ p_{i} = \\frac{e_{2i-1} \\wedge e_{2i}}{z^{k_i}} $ and $2 \\sum _{i=1}^{s} k_{i} = k$ .","For each $i$ , choose distinct $\\lambda ^{i}_{1}, \\ldots , \\lambda ^{i}_{k_i} \\in \\mathbb {C}$ , and define a family of principal parts $ p_{i}(t) = \\frac{e_{2i-1} \\wedge e_{2i}}{(z - \\lambda ^{i}_{1}t) \\cdots (z - \\lambda ^{i}_{k_i}t)} $ where $t$ is a complex parameter.","We then put $p(t) = \\sum _{i=1}^{s}p_{i}(t)$ .","By construction, $p(0) = p$ , while for small $t \\ne 0$ we can rewrite $p(t)$ as a sum of $k$ antisymmetric principal parts of degree 2, supported at $k$ distinct points.","Hence we have shown that any antisymmetric principal part of degree $2k$ is a limit of a continuous family of general antisymmetric principal parts of degree $2k$ .","But we know that the general antisymmetric principal parts are parameterized by an irreducible variety.","This proves the claim."],["Geometric criteria for lifting","In this subsection, we find a geometric interpretation of the cohomological criterion on isotropic liftings in Lemma REF .","Throughout this subsection, $E$ is a general stable bundle of negative degree.","In particular, $E$ is simple and $h^0(X, \\mathrm {Hom}(E^*, E)) = 0$ .","Consider the rational maps $\\phi _s: \\mathbb {P}E \\dashrightarrow \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E) \\quad \\hbox{and} \\quad \\phi _a: \\mathrm {Gr}(2, E) \\dashrightarrow \\mathbb {P}H^1(X, \\wedge ^2 E).$ As discussed in Lemma REF , these maps are defined on dense subsets, and furthermore are embeddings if $\\mu (E)<-1$ .","In general, abusing notation, we denote by $\\mathbb {P}E$ and $\\mathrm {Gr}(2, E)$ the closures of the images $\\phi _s (\\mathbb {P}E)$ in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ and $\\phi _a (\\mathrm {Gr}(2, E))$ in $\\mathbb {P}H^1(X, \\wedge ^2 E)$ respectively.","Next, for a quasi-projective variety $Z \\subset \\mathbb {P}^N$ , we write $\\mathrm {Sec}^k Z$ for the $k$ -th secant variety of $\\bar{Z}$ , which is the closure of the union of linear subspaces spanned by $k$ general points of $Z$ .","In particular, $\\mathrm {Sec}^1 Z = \\bar{Z}$ .","Criterion 2.8 Consider an extension given by $ \\delta (V) : \\quad 0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0.","$ When $\\delta (V) \\in \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ , there is an elementary transformation $F$ of $E^*$ with $\\deg (E^*/F) \\le k$ lifting to a Lagrangian subbundle of $V$ if and only if $\\delta (V) \\in \\mathrm {Sec}^{k} \\mathbb {P}E$ .","When $\\delta (V) \\in \\mathbb {P}H^1(X, \\wedge ^2 E)$ , there is an elementary transformation $F$ of $E^*$ with $\\deg (E^*/F) \\le 2k$ lifting to a rank $n$ isotropic subbundle of $V$ if and only if $ \\delta (V) \\in \\mathrm {Sec}^{k} \\mathrm {Gr}(2,E)$ .","In this case, $\\deg E $ and $\\deg F$ have the same parity.","In the symplectic case, this is the content of [3].","The same idea works for antisymmetric case as follows: By Lemma REF (3), an elementary transformation $F \\subseteq E^*$ with $\\deg (E^*/F) \\le 2k$ lifts to an isotropic subbundle of $V$ if and only if the extension class $\\delta (V)$ is of the form $[p]$ where $p$ is an antisymmetric principal part of degree $2l \\le k$ .","If $p$ is general, it is of the form $ \\sum _{i=1}^{l} \\frac{e_{i} \\wedge f_i}{z_i}, $ where $z_1, z_2, \\ldots , z_l$ are uniformizers at $l$ distinct points $x_1, x_2,\\ldots , x_l \\in X$ .","According to the description of the map $\\phi _a$ in §REF , the class $[p] \\in \\mathbb {P}H^1(X, \\wedge ^2 E)$ lies on the secant plane spanned by $l$ distinct points of $\\mathrm {Gr}(2,E)$ .","Hence $[p] \\in \\mathrm {Sec}^l \\mathrm {Gr}(2, E)$ .","Conversely, a general point of $\\mathrm {Sec}^l \\mathrm {Gr}(2, E)$ corresponds to a class $\\delta (V) =[p]$ where $p$ is of the form (REF ).","From this correspondence on the general points, we get the desired statement by Corollary REF .","In this case, $F = \\mathrm {Ker}\\left(p \\colon E^{*} \\rightarrow \\mathrm {\\underline{Prin}}(E) \\right)$ with $\\deg p = 2l$ , so $\\deg F$ has the same parity as $\\deg E$ .","Corollary 2.9 Consider an extension given by $ \\delta (V) : \\quad 0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0.","$ Assume $\\delta (V) \\in H^1(X, \\mathrm {Sym}^2 E)$ .","If $\\delta (V) \\le \\mathrm {Sec}^k \\mathbb {P}E$ , then $t(V) \\le 2(k + \\deg E)$ .","Assume $\\delta (V) \\in H^1(X, \\wedge ^2 E)$ .","If $\\delta (V) \\le \\mathrm {Sec}^k \\mathrm {Gr}(2, E)$ , then $t(V) \\le 2(2k + \\deg E)$ .","By definition of the invariant, $t(V) \\le - 2 \\deg F$ for any rank $n$ isotropic subbundle $F \\subset V$ .","The above bounds follow as a direct consequence of Criterion REF .","The converse of the above Corollary is not true in general.","For instance, assume $\\delta (V) \\in H^1(X, \\mathrm {Sym}^2 E)$ .","The bound $t(V) \\le 2(k + \\deg E)$ tells us that there is a Lagrangian subbundle $F$ of degree $\\ge -\\deg E -k$ , but this does not imply that $F$ lifts from an elementary transformation of $E^*$ .","So in general, we are led to consider a diagram of the form ${ 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\0 [r] & H [r] [u] & F [r] [u] & G [r] [u] & 0 } $ where $H \\subset E$ is a subbundle of degree $-h \\le 0$ and rank $r \\ge 0$ , and $G$ is a locally free subsheaf of $E^*$ of rank $n-r$ .","When $r = 0$ so that $E|_x$ and $F|_x$ meet transversely for general $x \\in X$ , this reduces to the situation of Criterion REF of the lifting of elementary transformations.","The remaining part of this subsection will be devoted to finding a criterion for the existence of such a diagram with $r>0$ .","For $H \\subseteq E$ , let $q \\colon E \\rightarrow E/H$ be the quotient map.","Since $H$ is isotropic, $H^{\\perp }$ fits into the diagram ${ 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\0 [r] & E [r] @{=}[u] & H^{\\perp } [r] [u] & \\left( E/H \\right)^{*} [r] [u]^{^tq} & 0.","}$ For $\\delta (V) \\in H^1(X, \\mathrm {Hom}(E^*, E))$ , we have $\\delta \\left( H^{\\perp } \\right) = \\, ^tq^{*} (\\delta (V)) \\ \\in \\ H^{1}(X, \\mathrm {Hom}((E/H)^{*}, E)).$ Furthermore, $H^{\\perp }$ inherits a (degenerate) bilinear form from $V$ .","Since $(H^{\\perp })^{\\perp } = H$ , the quotient $H^{\\perp }/H$ is equipped with a nondegenerate bilinear form coming from $V$ .","In fact, $H^{\\perp }/H$ is a symplectic (resp., orthogonal) extension in the upper exact sequence of ${ 0 [r] & E/H [r] & H^{\\perp } /H [r] & \\left( E/H \\right)^{*} [r] & 0 \\\\0 [r] & E [r] [u]^q & H^{\\perp } [r] [u] & \\left( E/H \\right)^{*} [r] @{=}[u] & 0 }$ corresponding to the class $\\delta (H^{\\perp }/H) = q_{*}(^tq)^{*} (\\delta (V))$ in $ H^{1} \\left( X, \\mathrm {Sym}^{2} (E/H) \\right)$ (resp., $H^{1} \\left( X, \\wedge ^{2} (E/H) \\right)$ ).","Lemma 2.10 The induced maps $ q_{*}(^tq)^{*} \\colon H^{1}(X, \\mathrm {Sym}^{2} E ) \\rightarrow H^{1} \\left( X, \\mathrm {Sym}^{2} (E/H) \\right) $ and $ q_{*}(^tq)^{*} \\colon H^{1}(X, \\wedge ^{2} E ) \\rightarrow H^{1} \\left( X, \\wedge ^{2} (E/H) \\right) $ are surjective.","These maps are induced from $ (^tq)^{*} : E \\otimes E \\rightarrow (E/H) \\otimes E \\quad \\hbox{and} \\quad q_{*} \\colon (E/H) \\otimes E \\rightarrow (E/H) \\otimes (E/H) $ respectively.","By local computation, it can be seen that the images of $q_{*}(^tq)^{*}$ in $(E/H) \\otimes (E/H)$ are precisely $\\mathrm {Sym}^{2} \\left( E/H \\right)$ and $\\wedge ^{2} \\left( E/H \\right)$ respectively.","Hence on the first cohomology level, the induced maps are surjective.","We obtain the following geometric criterion on liftings: Criterion 2.11 Let $V$ be an extension of $E^*$ by $E$ with class $\\delta (V)$ .","Fix a subbundle $0 \\ne H \\subset E$ and write $\\deg (E) = -e, \\deg (H) = -h$ .","Let $f > 2h - e$ .","Assume $\\delta (V) \\in H^1(X, \\mathrm {Sym}^2 E)$ .","Then $V$ admits a Lagrangian subbundle $F$ of degree $\\ge -f$ inducing a diagram of the form (REF ) if and only if $q_{*}(^tq)^{*} (\\delta (V)) \\ \\in \\ \\mathrm {Sec}^{(e+f -2h)} \\mathbb {P}(E/H) .$ Suppose $\\delta (V) \\in H^1(X, \\wedge ^2 E)$ and $e \\equiv f \\mod {2}$ .","Then $V$ admits a rank $n$ isotropic subbundle $F$ of degree $\\ge -f$ inducing a diagram of the form (REF ) if and only if $q_{*}(^tq)^{*} (\\delta (V)) \\ \\in \\ \\mathrm {Sec}^{\\frac{1}{2}(e+f - 2h)} \\mathrm {Gr}(2, E/H) .$ Since the arguments for (1) and (2) are parallel, we prove (2) only.","The orthogonal bundle $V$ admits an isotropic subbundle $F$ inducing the diagram (REF ) if and only if $H^{\\perp }$ admits a subbundle $F$ of degree $\\ge -f$ which is isotropic with respect to the antisymmetric form inherited from $V$ , yielding the diagram ${ 0 [r] & E [r] & H^{\\perp } [r] & \\left( E/H \\right)^{*} [r] & 0 \\\\0 [r] & H [r] [u] & F [r] [u] & G [r] [u] & 0 }$ Since $G = F/H$ has the same rank as $(E/H)^*$ , the map $G \\rightarrow (E/H)^*$ is an elementary transformation whose quotient is a torsion sheaf of degree $\\le e+f - 2h$ .","Factorizing by $H$ , we get ${ 0 [r] & E/H [r] & H^{\\perp }/H [r] & \\left( E/H \\right)^{*} [r] & 0 \\\\& & F/H [r]^= [u] & G [r] [u] & 0.","}$ We are in this situation precisely when the orthogonal extension $H^{\\perp }/H$ with class $ q_{*}(^tq)^{*} \\delta (V) \\in H^1 (X, \\wedge ^2 (E/H)) $ admits an isotropic lifting of some elementary transformation of $(E/H)^*$ of degree $\\le e+f - 2h$ .","By Criterion REF (2), this is equivalent to the class $\\delta \\left( H^{\\perp }/H \\right)$ belonging to $\\mathrm {Sec}^{\\frac{1}{2}(e+f -2h)} \\mathrm {Gr}(2,E/H)$ in $\\mathbb {P}H^{1}(X, \\wedge ^{2}(E/H))$ ."],["Parameter spaces of extensions","Here we construct “universal extension spaces”, following Lange [12] (see also [3]), and investigate stability of the corresponding symplectic and orthogonal bundles."],["Construction of the families","For a positive integer $e$ , let $U_X(n,-e)^s$ denote the moduli space of stable vector bundles of rank $n$ and degree $-e < 0$ over $X$ .","By Narasimhan–Ramanan [16], there exist a finite étale cover $ \\pi _{e} \\colon \\tilde{U}_e \\rightarrow U_X(n, -e)^s $ and a bundle $\\mathcal {E}_e \\rightarrow \\tilde{U}_e \\times X$ with the property that $\\mathcal {E}_e|_{\\lbrace E \\rbrace \\times X} \\cong \\pi _e(E)$ for all $E \\in \\tilde{U}_e$ .","When $\\gcd (n, e)=1$ , it is well known that $\\pi _e$ reduces to the identity map.","By Riemann-Roch and semistability, for each $E \\in U_X(n, -e)^s$ , we have $ h^1(X, \\mathrm {Sym}^{2}E) = (n+1)e + \\frac{n(n+1)}{2}(g - 1).","$ Therefore, the sheaf $R^{1} {p}_{*} (\\mathrm {Sym}^2 (\\mathcal {E}_e))$ is locally free of rank $(n+1)(e + \\frac{1}{2}n(g - 1))$ on $\\tilde{U}_e$ .","We denote its projectivization by $\\mu \\colon \\mathbb {S}_{e} \\rightarrow \\tilde{U}_e$ .","We have a diagram $ {\\mathbb {S}_{e} [dd]_{\\mu } & \\mathbb {S}_{e} \\times X [d]^{\\mu \\times \\mathrm {Id}_X} [l]_{r} & \\\\& \\tilde{U}_{e} \\times X [dl]_{p} [dr]^{q} & \\\\\\tilde{U}_{e} & & X}$ We write $r \\colon \\mathbb {S}_e \\times X \\rightarrow \\mathbb {S}_e$ for the projection.","By Lange [12], there is an exact sequence of vector bundles $ 0 \\rightarrow (\\mu \\times \\mathrm {Id}_{X})^{*}\\mathcal {E}_e \\otimes r^{*}\\mathcal {O}_{\\mathbb {S}_e}(1) \\rightarrow \\mathcal {W}_{e} \\rightarrow (\\mu \\times \\mathrm {Id}_{X})^{*}\\mathcal {E}_e^{*} \\rightarrow 0 $ over $\\mathbb {S}_{e} \\times X$ , with the property that for $\\delta \\in \\mathbb {S}_e$ with $\\mu (\\delta ) = E$ , the restriction of $\\mathcal {V}_{e}$ to $\\lbrace \\delta \\rbrace \\times X$ is isomorphic to the extension of $E^*$ by $E$ defined by $\\delta \\in \\mathbb {P}H^1(X, \\mathrm {Sym}^{2}E)$ .","By Lemma REF , the space $\\mathbb {S}_{e} $ classifies all symplectic extensions of $E^*$ by $E$ for all $E \\in U(n, -e)^s$ , up to homothety.","In the same way, we define a bundle $\\mathbb {A}_{e} \\rightarrow \\tilde{U}_e$ whose fibre at $E$ is $\\mathbb {P}H^{1}(X, \\wedge ^{2}E)$ , a projective space of dimension $(n-1)(e + \\frac{1}{2}n(g - 1)) - 1$ .","There is a sequence of vector bundles $ 0 \\rightarrow (\\mu \\times \\mathrm {Id}_{X})^{*}\\mathcal {E}_e \\otimes r^{*}\\mathcal {O}_{\\mathbb {A}_e}(1) \\rightarrow \\mathcal {V}_{e} \\rightarrow (\\mu \\times \\mathrm {Id}_{X})^{*}\\mathcal {E}_e^{*} \\rightarrow 0 $ over $\\mathbb {A}_{e} \\times X$ , with the property that for $\\delta \\in \\mathbb {A}_e$ with $\\mu (\\delta ) = E$ , the restriction of $\\mathcal {W}_{e}$ to $\\lbrace \\delta \\rbrace \\times X$ is isomorphic to the extension of $E^*$ by $E$ defined by $\\delta \\in \\mathbb {P}H^1(X, \\wedge ^{2}E)$ .","Again by Lemma REF , the space $\\mathbb {A}_{e} $ classifies all the orthogonal extensions of $E^*$ by $E$ for all $E \\in U(n, -e)^s$ , up to homothety.","For each $e>0$ , the universal bundles $\\mathcal {W}_e \\rightarrow \\mathbb {S}_e \\times X$ and $\\mathcal {V}_e \\rightarrow \\mathbb {A}_e \\times X$ induce classifying maps $\\sigma _e: \\mathbb {S}_e \\dashrightarrow \\mathcal {M}S_X(2n)\\ \\ \\text{and} \\ \\ \\alpha _e: \\mathbb {A}_e \\dashrightarrow \\mathcal {M}O_X(2n)$ respectively.","The indeterminacy loci of these maps consist of precisely the points whose associated symplectic/orthogonal bundles are not semistable."],["Stability of extensions","The universal extension spaces $\\mathbb {S}_e$ and $\\mathbb {A}_e$ provide a natural way to study the Segre stratification on the moduli space of symplectic/orthogonal bundles, via the classifying maps $\\sigma _e$ and $\\alpha _e$ .","In order to proceed in this direction, we must verify that a general bundle with extension class represented in $\\mathbb {S}_e$ (resp., $\\mathbb {A}_e$ ) is a stable symplectic (resp., stable orthogonal) bundle.","The same question for vector bundles was formulated by Lange [11], and solved in Brambila-Paz–Lange [2] and Russo–Teixidor i Bigas [19].","In both papers, elementary transformations were used to construct stable bundles with the prescribed Segre invariant.","In this subsection, we prove the existence of a stable orthogonal/symplectic bundle in $\\mathbb {S}_e$ and $\\mathbb {A}_e$ for each $e >0$ .","Elementary transformations are used, but in a somewhat different context.","We begin by establishing the statement for $e = 1$ and $e = 2$ .","We will need the following bound on the classical Segre invariants of vector bundles, due to Hirschowitz [4]: Lemma 3.1 Let $E$ be a general stable vector bundle of rank $n$ and degree $-e$ .","Let $H \\subset E$ be a subbundle of rank $r$ and degree $-h$ .","Then $r(n-r) (g-1) \\ \\le \\ nh -re \\ < \\ r(n-r) (g-1) +n .$ Also we need the following result called the Hirschowitz lemma and its variant.","Lemma 3.2 Let $H_1$ and $H_2$ be general stable bundles, $\\mathrm {rk}(H_i) = r_i$ and $\\deg (H_i) = d_i$ for $i = 1,2$ .","If $r_1 d_2 + r_2 d_1 \\ge r_1 r_2(g-1)$ so that $\\mu (H_1 \\otimes H_2) \\ge g-1$ , then $h^1(X, H_1 \\otimes H_2) = 0$ .","Let $F$ be a general stable bundle of rank $n$ .","If $\\deg F \\ge \\frac{1}{2}n(g-1)$ so that $\\mu (F \\otimes F) \\ge g-1$ , then $h^1(X, F \\otimes F) = 0$ .","Part (1) was proven by Hirschowitz [4]; see also Russo–Teixidor i Bigas [19].","The variant (2) is [3].","Proposition 3.3 Let $E \\in U(n, -e)^s$ be general, $e = 1,2$ .","For $e=1$ , every point in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ outside a sublocus $Y$ with $\\dim Y \\le n = \\dim \\mathbb {P}E$ corresponds to a stable symplectic bundle.","Hence a general point of $\\mathbb {S}_1$ represents a stable symplectic bundle.","For $e=1$ , every point in $\\mathbb {P}H^1(X, \\wedge ^2 E)$ corresponds to a stable orthogonal bundle.","For $e=2$ , every point of $\\mathbb {P}H^1(X, \\wedge ^2 E)$ outside a sublocus $Z$ with $\\dim Z \\le 2(n-2) +1 = \\dim Gr(2,E)$ corresponds to a stable orthogonal bundle.","Hence a general point of $\\mathbb {A}_1$ or $\\mathbb {A}_2$ represents a stable orthogonal bundle.","Consider a symplectic or orthogonal extension $0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0$ where $E$ is a general stable bundle of rank $n$ and degree $-e \\in \\lbrace -1,-2 \\rbrace $ .","Assume that $V$ is not stable, so there is an isotropic subbundle $F$ of $V$ of rank $r (\\le n)$ and degree $\\ge 0$ .","The intersection of $E$ and $F$ contains a subbundle $F_1$ of rank $r_1$ (possibly zero), and the image of $F$ in $E^*$ is a locally free subsheaf $F_2$ of rank $r_2$ , yielding a diagram $ { 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\0 [r] & F_{1} [r] [u] & F [r] [u] & F_{2} [r] [u] & 0. }","$ We will bound the dimension of the locus of the extensions $V$ admitting this kind of diagram.","First assume $r_{1} \\ne 0$ and so $r_2 < n$ .","Since $E$ is general, by Lemma REF we have $ \\deg F_{1} \\le - \\frac{r_1}{n} \\left( e + (n-r_{1})(g-1) \\right) \\ <0 .$ In the same way (if $r_2 \\ne 0$ ), $ \\deg F_{2} \\le \\frac{r_2}{n} \\left( e - (n-r_{2})(g-1) \\right),$ which implies $\\deg F_2 \\le 0$ since $r_{2} < n$ .","Therefore, $\\deg F < 0$ .","Next, assume $r_1 = 0$ and $ r = r_{2} = n$ .","In this case, $F$ is an elementary transformation of $E^*$ .","If $\\deg F \\ge 0$ , then the torsion sheaf $E/F$ has degree $\\le e$ .","In the symplectic case, we need only consider the case when $e = 1$ .","By Criterion REF (1), if $F$ lifts to $V$ as a Lagrangian subbundle, then $\\delta (V) \\in \\phi _s(\\mathbb {P}E ) $ in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ .","In the orthogonal case, $e \\le 2$ .","By Criterion REF (2), if $F$ lifts to $V$ as an isotropic subbundle, then in fact $e=2$ and $\\delta (V) \\in { \\mathrm {Gr}(2, E) }$ in $\\mathbb {P}H^1(X, \\wedge ^2 E)$ .","Finally assume $r_1 = 0$ and $r=r_2 < n$ .","From the inequality (REF ), $\\deg F = \\deg F_2 \\le 0$ .","The possibility $\\deg F = 0$ appears only for the following special cases: $e = 1$ ; $r = n - 1$ ; $g = 2$ $e = 2$ ; $r = n - 1$ ; $g = 3$ $e = 2$ ; $r = n - 1$ ; $g = 2$ $e = 2$ ; $r = n - 2$ ; $g = 2$ .","From now on, we show that in each of these cases, the dimension of the locus of extensions admitting a lifting of a subsheaf $F$ of $E^*$ as an isotropic subbundle of $V$ of degree zero and rank $r 0$ , a general point of $\\mathbb {S}_e$ (resp., $\\mathbb {A}_e$ ) represents a stable symplectic (resp., orthogonal) bundle.","We consider the orthogonal case first: For each value of $e>0$ , we will exhibit a stable orthogonal bundle represented in $\\mathbb {A}_e$ .","The statement will then follow from the openness of the stable objects in families.","Let $E \\in U_{X}(n,-1)$ be general.","For any $k \\ge 2$ , choose a general antisymmetric principal part $p$ of degree $2k$ , which defines an element of $\\mathrm {Sec}^k \\mathrm {Gr}(2, E)$ .","Since $\\mathrm {Gr}(2,E)$ is nondegenerate and properly contained in $\\mathrm {Sec}^{k} \\mathrm {Gr}(2,E)$ , we may assume that $[p]$ does not lie on the image of $\\mathrm {Gr}(2, E)$ .","By Lemma REF , the sheaf $F := \\mathrm {Ker}\\left( p \\colon E^{*} \\rightarrow \\mathrm {\\underline{Prin}}(E) \\right)$ lifts to a rank $n$ isotropic subbundle of the extension $V$ with $\\delta (V) = [p]$ .","Deforming $p$ if necessary, we can assume that $F$ is stable.","By Proposition REF (2), moreover, $V$ is a stable orthogonal bundle.","Since $V$ fits into an orthogonal extension $ 0 \\rightarrow F \\rightarrow V \\rightarrow F^{*} \\rightarrow 0, $ it is associated to a point in $\\mathbb {A}_{2k-1}$ .","In the same way, consider a general bundle $E \\in U_X(n, -2)$ and choose a general antisymmetric principal part of degree $2k+2$ for each $k \\ge 2$ , defining a point of $\\mathrm {Sec}^{k+1} \\mathrm {Gr}(2,E)$ .","By Proposition REF (2), we may assume that the extension $V$ associated to the point $[p]$ is a stable orthogonal bundle, if we avoid a subvariety $Z$ with $\\dim Z \\le \\dim \\mathrm {Gr}(2,E)$ ; and this is possible since $\\mathrm {Gr}(2,E)$ is nondegenerate and properly contained in $\\mathrm {Sec}^{k+1} \\mathrm {Gr}(2,E)$ .","As in the previous case, $V$ fits into an orthogonal extension (REF ) where this time $\\deg (F) = -2k$ , so $V$ is represented in $\\mathbb {A}_{2k}$ .","In the symplectic case, we argue similarly.","For each $k \\ge 2$ , we can choose a general symmetric principal part $q$ of degree $k+1$ which defines a point of $\\mathrm {Sec}^{k+1} \\mathbb {P}E $ .","By the above argument, the extension $V$ determined by $q$ is a stable symplectic bundle which is represented in $\\mathbb {S}_{k}$ ."],["Description of the Segre strata for symplectic bundles","In this section, we use the map $\\sigma _e: \\mathbb {S}_e \\dashrightarrow \\mathcal {M}S_X(2n)$ discussed in the previous section to prove Theorem REF .","Recall that for each positive even integer $t$ , we denote by $\\mathcal {M}S_X(2n;t)$ the sublocus of $\\mathcal {M}S_X(2n)$ consisting of symplectic bundles $V$ with $t(V) = t$ .","Also, for $V \\in \\mathcal {M}S_X(2n)$ , we write $M(V)$ for the space of Lagrangian subbundles of $V$ of (maximal) degree $-\\frac{1}{2}t(V)$ .","Theorem 4.1 Consider a positive even integer $ t=2e \\le n (g-1)+1$ .","A general point of $\\mathbb {S}_e$ corresponds to a bundle $V$ with $t(V) = t$ ; in particular, $\\mathcal {M}S_X(2n;t)$ is nonempty.","Furthermore, $\\mathcal {M}S_X(2n;t)$ is irreducible.","If $t < n(g-1)$ , then $M(V)$ is a single point for a general $V \\in \\mathcal {M}S_X(2n;t)$ .","If $t \\in \\lbrace n(g-1), \\: n(g-1) +1 \\rbrace $ and $V$ is general in $\\mathcal {M}S_X(2n;t)$ then any pair of Lagrangian subbundles in $M(V)$ intersect transversely in $V$ in a general fiber.","Remark 4.2 Statements (2) and (3) can be viewed as a symplectic analogue of Lange–Newstead [14].","Let $E$ be a general bundle in $U(n, -e)^s$ for $e \\ge 1$ .","By Theorem REF , the moduli map $\\mathbb {S}_{e}|_{E} \\cong \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E) \\dashrightarrow \\mathcal {M}S_X(2n)$ is defined on a dense subset.","We want to compute a bound on the dimension of the locus in $H^{1}(X, \\mathrm {Sym}^{2}E)$ of extensions $V$ which admit a Lagrangian subbundle $F$ of degree $\\ge -e$ other than $E$ .","There are two possibilities: either $F$ is an elementary transformation of $E^*$ lifting to $V$ , or $F$ fits into a diagram of the form (REF ).","Step 1.","We show first that the latter situation does not arise for a general $V$ in $H^1(X, \\mathrm {Sym}^2 E)$ for $t = 2e \\le n(g-1) +1$ .","As before, we write $\\deg (H) = -h$ .","Recall from the diagram (REF ) that $G$ is an elementary transformation of $(E/H)^*$ whose quotient is a torsion sheaf of degree $\\deg (E/H)^* - \\deg G = (e-h) - (\\deg F + h) \\le 2(e-h).$ In particular, $e-h \\ge 0.$ Now for each fixed $H \\subset E$ , Criterion REF says that the locus of extensions in $H^1(X, \\mathrm {Sym}^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\ge -e$ is bounded by $ \\dim \\left( \\mathrm {Sec}^{2(e-h)}\\mathbb {P}(E/H) \\right) + 1 + \\dim \\mathrm {Ker}\\left( q_{*} \\, ^tq^{*} \\right).","$ The secant variety has dimension bounded by $ 2(e-h)(n-r) + 2(e-h) - 1 = 2(e-h)(n-r+1) - 1.","$ Also by Proposition REF , we have $ \\dim \\mathrm {Ker}\\left(q_{*} \\, ^tq^{*} \\right) &= h^{1}(X, \\mathrm {Sym}^{2}E) - h^{1} \\left(X, \\mathrm {Sym}^{2}(E/H) \\right) \\\\&\\le h^{1}(X, \\mathrm {Sym}^{2}E) - (n-r+1)(e-h) - \\frac{1}{2} (n-r)(n-r+1)(g-1).","$ Finally we take account of the deformations of $H$ by computing the dimension of the appropriate Quot scheme of $E$ .","Since $E$ is general, by Lemma REF we have $nh - re \\ \\ge \\ r(n-r)(g-1).$ We may furthermore assume that $H$ and $E/H$ are general, and so $ h^1(X, \\mathrm {Hom}(H, E/H)) = 0 $ by Lemma REF (1).","Therefore $[H]$ is a smooth point of the Quot scheme (cf.","Le Potier [15]), and the dimension of the Quot scheme is given by $ h^{0}(X, \\mathrm {Hom}(H, E/H)) = nh - re - r(n-r)(g-1).","$ Adding up these three terms, we see that the dimension of the locus of extensions in $H^1(X, \\mathrm {Sym}^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\ge -e$ is bounded by $\\dim \\left( \\mathrm {Sec}^{2(e-h)} \\mathbb {P}(E/H) \\right) + 1 + \\dim \\mathrm {Ker}(q_* \\, ^t q^*) + h^0(X, \\mathrm {Hom}(H, E/H)) \\\\\\le \\ (e-h)(n-r+1) - \\frac{1}{2} (n-r)(n+r+1) (g-1) + nh-re + h^1(X, \\mathrm {Sym}^2 E) \\\\= \\ e(n-r) - (e-h)(r-1) - \\frac{1}{2} (n-r)(n+r+1) (g-1) + h^1(X, \\mathrm {Sym}^2 E).$ By the assumption $e \\le \\frac{1}{2} (n (g-1)+1)$ and the inequality (REF ), this is smaller than the dimension of the whole extension space $h^1(X, \\mathrm {Sym}^2 E)$ .","This shows that for $t = 2e \\le n (g-1) +1$ , a general $V \\in H^1(X, \\mathrm {Sym}^2 E)$ does not admit a Lagrangian subbundle $F$ of degree $\\ge -e$ which fits into a diagram of the form (REF ).","Step 2.","Next we consider the case $r = 0$ , so $F$ is an elementary transformation of $E^*$ lifting to $V$ isotropically.","Let $\\deg F = -f \\ge -e$ .","By Criterion REF , the dimension of the locus of extensions in $H^1(X, \\mathrm {Sym}^2 E)$ admitting such an $F$ is bounded by $\\dim \\left( \\mathrm {Sec}^{e+f} \\mathbb {P}E \\right) +1 \\ \\le \\ (e+f)(n+1).$ If $f < \\frac{1}{2} n (g-1)$ , this bound is smaller than $h^1(X, \\mathrm {Sym}^2 E) = (n+1)e + \\frac{1}{2}n(n+1)(g-1).$ Step 3.","Combining the dimension counts for two possibilities in the above, we conclude: (i) if $t=2e < n (g-1)$ , a general $V \\in H^1(X, \\mathrm {Sym}^2 E)$ does not have a Lagrangian subbundle of degree $\\ge -e$ other than $E$ itself.","This shows (2).","(ii) if $t=2e \\in \\lbrace n (g-1), \\: n(g-1) + 1 \\rbrace $ , a general $V \\in H^1(X, \\mathrm {Sym}^2 E)$ does not have a Lagrangian subbundle of degree $> -e$ .","Also, any Lagrangian subbundle of degree $-e$ different from $E$ intersects $E$ transversely at a general fiber.","This shows (3).","Hence for a general $V \\in H^1(X, \\mathrm {Sym}^2 E)$ with $t = 2e \\le n(g-1) +1$ , we have $t(V) = t$ .","It follows that the rational map $\\sigma _e : \\mathbb {S}_e \\dashrightarrow \\mathcal {M}S_X(2n)$ sends a general point of $\\mathbb {S}_e$ to $\\mathcal {M}S_X(2n;t)$ , so $\\mathcal {M}S_X(2n;t)$ is non-empty for each $t \\le n (g-1) +1$ .","Since $\\mathbb {S}_e$ is irreducible, so is its image.","To see that $\\mathcal {M}S_X(2n;t)$ is irreducible, it suffices to show that every point of $\\mathcal {M}S_X(2n;t)$ is in the closure of the image of $\\mathbb {S}_e$ .","Any $V$ in $\\mathcal {M}S_X(2n;t)$ admits a Lagrangian subbundle $E$ of degree $-e$ which might be unstable.","But every such $E$ is contained in an irreducible family of bundles whose general member is a stable bundle in $U_X(n, -e)$ .","This shows (1).","To finish the proof of Theorem REF , we need the following description of the tangent space of $M(V)$ .","Lemma 4.3 For a symplectic bundle $V \\in \\mathcal {M}S_X(2n)$ and a point $[\\eta : E \\subset V]$ of $M(V)$ , the Zariski tangent space of $M(V)$ at $\\eta $ is identified with $H^0(X, \\mathrm {Sym}^2 E^*)$ .","It is well known that the Zariski tangent space to the Quot scheme of a vector bundle $V$ at a point $[E\\subset V]$ is given by $H^0(X, \\mathrm {Hom}(E, V/E))$ .","Intuitively, this can be explained as follows: A rank $n$ subbundle of $V$ is equivalent to a global section of the Grassmannian bundle $\\mathrm {Gr}(n, V)$ over $X$ .","A tangent vector to the Quot scheme of $V$ at $[E \\subset V]$ corresponds, at each fiber $x \\in X$ , to a tangent vector to the Grassmannian $\\mathrm {Gr}(n, V_x)$ at $[E_x \\subset V_x]$ .","Therefore, the tangent vector is a global section of the bundle $\\mathrm {Hom}(E, V/E)$ .","When $E$ is a Lagrangian subbundle of a symplectic bundle $V$ , we have $V/E \\cong E^*$ and so $\\mathrm {Hom}(E, V/E) \\cong E^* \\otimes E^*$ .","In this case, a similar argument shows that a tangent vector to $M(V)$ at $[E \\subset V]$ corresponds to a global section of $\\mathrm {Sym}^2 E^*$ , since for each $x \\in X$ the tangent space of the Lagrangian Grassmannian at $[E_x \\subset V_x]$ is identified with $\\mathrm {Sym}^{2}E^{*}|_x$ .","Proof of Theorem REF We begin with part (3) of the statement.","The first part of (3) is Theorem REF (2).","For the latter part, we invoke Lemma REF : $\\dim M(V) = h^0(X, \\mathrm {Sym}^2 E^*)$ for a smooth point $[E \\subset V]$ of $M(V)$ .","By Lemma REF (2), for a general $E \\in U_X(n,-e)$ , we have $\\dim H^0(X, \\mathrm {Sym}^2 E^*) \\ = \\ {\\left\\lbrace \\begin{array}{ll}0& \\text{if} \\ 2e = n(g-1), \\\\\\frac{n+1}{2}& \\text{if} \\ 2e=n(g-1)+1.\\end{array}\\right.","}$ (1) A straightforward computation shows that when $ n(g-1) \\le 2e \\le n(g-1)+1 $ , $ \\dim M(V) = \\dim \\mathbb {S}_e - \\dim \\mathcal {M}S_X(2n).","$ Since $ \\dim M(V) \\ge \\dim \\sigma _{e}^{-1}(V)$ , this equality implies that $\\sigma _e$ is dominant.","This shows that for a general $V \\in \\mathcal {M}S_X(2n)$ , we have $n(g-1) \\le t(V) \\le n(g-1) +1.$ By semicontinuity, $t(V) \\le n(g-1) +1$ for any symplectic bundle $V$ .","(2) If $t = 2e \\ge n(g-1) $ , then $\\dim \\mathcal {M}S_X(2n;t) = \\dim \\mathcal {M}S_X(2n)$ by (1).","Assume $t = 2e < n(g-1)$ .","By Theorem REF (2), the map $\\sigma _e \\colon \\mathbb {S}_{e} \\dashrightarrow \\mathcal {M}S_X(2n;t)$ is generically finite (of degree $\\deg \\pi _e$ ), and so $\\dim \\mathcal {M}S_X(2n;t) \\ = \\ \\dim \\mathbb {S}_e \\ = \\ \\frac{1}{2}n(3n+1)(g-1) + (n+1)e.$ For the last part of (2), we can use Criterion REF to show that the Segre stratification matches the stratification given by the higher secant varieties: For a general $V \\in \\mathcal {M}S_X(2n;2e)$ , let $E \\in M(V)$ so that $E$ is general in $U_{X}(n, -e)^s$ .","Let $k$ be the smallest integer satisfying $ \\mathrm {Sec}^{k} \\mathbb {P}E = \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E)$ .","Then by Criterion REF , there is some elementary transformation $F$ of $E^*$ with $\\deg E/F = k$ lifting to $V$ as a Lagrangian subbundle.","By deforming $V$ and $E$ inside $\\mathbb {S}_e$ , we may assume that $F$ is general in $U_X(n,e-k)^s$ .","Now consider the symplectic extension $ 0 \\rightarrow F \\rightarrow V \\rightarrow F^* \\rightarrow 0.","$ In the same way, $\\delta (V) \\in \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}F)$ belongs to $\\mathrm {Sec}^{k}\\mathbb {P}F$ , since the elementary transformation $E \\rightarrow F^*$ lifts to $V$ .","Note that $\\dim \\mathrm {Sec}^k \\mathbb {P}F \\le k(n+1) -1 \\ < \\ (n+1)(k-e) + \\frac{1}{2}n(n+1)(g-1) -1 = \\dim \\mathbb {P}H^1 (X, \\mathrm {Sym}^2 F),$ since we assumed $2e < n(g-1)$ .","Therefore, $\\mathrm {Sec}^{k}\\mathbb {P}F$ is properly contained in $\\mathbb {P}H^1 (X, \\mathrm {Sym}^2 F)$ , and certainly it is inside the closure of $\\mathrm {Sec}^{k+1} \\mathbb {P}F \\setminus \\mathrm {Sec}^k \\mathbb {P}F$ .","Thus, again by Criterion REF , the class $\\delta (V)$ belongs to the closure of a family of bundles admitting liftings of elementary transformations of degree $(k-e)-(k+1) = -(e+1)$ .","In particular, $V$ belongs to the closure of $\\mathcal {M}S_{X}(2n;2e+2)$ in $\\mathcal {M}S_{X}(2n)$ .","By the irreducibility of $\\mathcal {M}S_{X}(2n;2e)$ , the same holds for arbitrary $V \\in \\mathcal {M}S_X(2n;2e)$ .","$\\Box $ Remark 4.4 Suppose $t=2e < n(g-1)$ .","If $\\gcd (n,e) = 1$ , then $\\tilde{U}_e = U_X(n,-e)^s$ and $\\mathcal {M}S_X(2n;t)$ is birational to the fibration $\\mathbb {S}_e$ over $U_X(n,-e)^s$ whose fiber at $ E$ is $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ .","Finally we give a geometric interpretation of the cardinality of $M(V)$ , when it is finite.","Assume $t =2e = n (g-1) $ .","Let $V$ be general in $\\mathcal {M}S_X(2n;t)$ and choose $E \\in M(V)$ with $\\deg E = -e = -\\frac{1}{2} n(g-1)$ .","Consider the subvariety $\\mathbb {P}E \\subset \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ .","When $g \\ge 4$ , we have $\\mu (E) < -1$ , and by Lemma REF , the map $\\phi _s : \\mathbb {P}E \\rightarrow \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ is an embedding.","Furthermore, it was proven in [3] that the secant variety $\\mathbb {P}E$ is not defective in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ in the sense that $\\dim \\mathrm {Sec}^{k} \\mathbb {P}E \\ = \\ \\min \\lbrace (n+1)k -1, \\ \\dim \\mathbb {P}H^1 (X, \\mathrm {Sym}^2 E) \\rbrace .$ In particular, for $k = n(g-1)$ we have $\\dim \\mathrm {Sec}^{k} \\mathbb {P}E \\ = \\ (n+1)k -1 \\ = \\ \\dim \\mathbb {P}H^1 (X, \\mathrm {Sym}^2 E).","$ By a $k$ -secant space of $\\mathbb {P}E$ , we mean a linear subspace in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ spanned by some $k$ points of $\\mathbb {P}E$ .","By (REF ), there are finite number of $n(g-1)$ -secant spaces of $\\mathbb {P}E$ which pass through a general point of $\\mathbb {P}H^1 (X, \\mathrm {Sym}^2 E)$ .","The following result generalizes Lange-Narasimhan [13] for rank 2 bundles.","Theorem 4.5 Assume $g \\ge 4$ and $n(g-1)$ is even.","Let $V \\in \\mathcal {M}S_X(2n)$ be a general symplectic bundle with $t(V) = n(g-1)$ , and let $E \\in M(V)$ .","There is a one-to-one correspondence between $M(V) \\setminus \\lbrace E\\rbrace $ and $n(g-1)$ -secant spaces of $\\mathbb {P}E$ passing through the point $[E \\subset V] \\in \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ .","By Criterion REF , a general class $\\delta (V)$ lies on a $n(g-1)$ -secant space of $\\mathbb {P}E$ if and only if there is an associated elementary transformation $F$ of $E^*$ with $\\deg (E^*/ F) = n(g-1)$ , lifting to a Lagrangian subbundle of $V$ .","Since $t(V) = n(g-1)$ , the Lagrangian subbundle $F$ is an element of $M(V)$ .","Conversely, by Theorem REF (3) every element of $M(V)$ other than $E$ appears in this way, since $V$ is general.","Remark 4.6 It is an interesting problem to compute the cardinality of $M(V)$ explicitly.","Let $N_{n,g}$ denote the cardinality of $M(V)$ for a general symplectic bundle $V$ of rank $2n$ over a curve of genus $g$ so that $n(g-1)$ is even.","It is well known that $N_{1,g} = 2^g$ .","The same problem for maximal subbundles of vector bundles was solved by Holla [9] (see also Lange–Newstead [13]).","As far as we are aware, the number $N_{n,g}$ is not known in general."],["Description of Segre strata for orthogonal bundles","In this section, we investigate the geometry of the Segre stratification on the moduli space of orthogonal bundles $\\mathcal {M}O_{X}(2n)$ .","In contrast to $\\mathcal {M}S_{X}(2n)$ , this has two connected components $\\mathcal {M}O_{X}(2n)^{\\pm }$ , corresponding to bundles of trivial and nontrivial second Stiefel–Whitney class (see Serman [20] for more details).","We begin by determining the component to which each stratum $\\mathcal {M}O_{X}(2n;t)$ belongs."],["Topological classification","Theorem 5.1 (1) Let $E_1$ and $E_2$ be isotropic rank $n$ subbundles of an orthogonal bundle $V$ of rank $2n$ .","Then $\\deg E_1$ and $\\deg E_2$ have the same parity.","(2) A semistable orthogonal bundle $V$ belongs to $\\mathcal {M}O_{X}(2n)^+$ (resp., $\\mathcal {M}O_{X}(2n)^-$ ) if and only if its isotropic rank $n$ subbundles have even degree (resp., odd degree).","By Theorem REF , the moduli maps $\\alpha _{e} \\colon \\mathbb {A}_{e} \\dashrightarrow \\mathcal {M}O_{X}(2n)$ are defined on dense subsets for each $e>0$ .","Since each family $\\mathbb {A}_e$ is connected, the image of each $\\mathbb {A}_e$ is entirely contained in either $\\mathcal {M}O_{X}(2n)^+$ or $\\mathcal {M}O_{X}(2n)^-$ .","As was already noted in the proof of Theorem REF , whenever $e_1$ and $e_2$ have the same parity, the images of $\\mathbb {A}_{e_1}$ and $\\mathbb {A}_{e_2}$ have nonempty intersection.","Hence they lie on the same component of $\\mathcal {M}O_X(2n)$ .","This is essentially due to the fact that any antisymmetric principal part has an even degree (see Lemma REF ).","For each $k>0$ , the image of $\\mathbb {A}_{2k}$ is contained in $\\mathcal {M}O_{X}(2n)^+$ since the trivial bundle has trivial second Stiefel–Whitney class.","Thus the image of $\\mathbb {A}_{2k-1}$ is contained in $\\mathcal {M}O_{X}(2n)^-$ .","This shows the above statements (1) and (2) for the case when $V$ is a stable orthogonal bundle and the isotropic subbundles are stable bundles.","By deforming an arbitrary bundle to a stable one, we see that (1) and (2) hold in general."],["An extra stratum","The local deformations of an orthogonal bundle $V$ are given by $H^1(X, \\wedge ^2 V)$ .","Since $V$ is self-dual, $\\wedge ^2 V \\subset \\mathrm {End}(V)$ .","Hence both components $\\mathcal {M}O_{X}(2n)^{\\pm }$ have dimension $ h^{1}(X, \\wedge ^{2}V) = n(2n-1)(g-1), $ since we may assume that $V$ is simple.","On the other hand, $\\dim \\mathbb {A}_e & = & \\dim U_X(n,-e) + \\dim \\mathbb {P}H^1 (X, \\wedge ^2 E) \\\\& = & (n-1)e + \\frac{1}{2} n (3n-1) (g-1)$ for a general $E \\in U_X(n,-e)$ .","Comparing dimensions, we see that in order for an $\\mathbb {A}_e$ to cover either component of $\\mathcal {M}O_{X}(2n)$ , we need $e \\ge \\left\\lceil \\frac{1}{2}n(g-1) \\right\\rceil $ .","However, a connected set $\\mathbb {A}_e$ can cover at most one component.","Hence we are led to consider at least two top strata, corresponding to distinct values of $e \\ge \\left\\lceil \\frac{1}{2}n(g-1) \\right\\rceil $ .","This is one important difference between the stratifications in the symplectic and orthogonal cases."],["Geometry of the strata","The arguments in this subsection are essentially the same as those appearing in §4, except for the complication coming from the topological property described in Theorem REF .","Theorem 5.2 Consider a positive even integer $t = 2e \\le n(g-1)+3$ .","A general point of $\\mathbb {A}_e$ corresponds to a bundle $V$ with $t(V) = t$ ; in particular, $\\mathcal {M}O_{X}(2n;t)$ is nonempty and irreducible.","If $t < n(g-1)$ then $M(V)$ is a single point for general $V \\in \\mathcal {M}O_{X}(2n;t)$ .","Let $E$ be a general bundle in $U(n, -e)^s$ for $e \\ge 1$ .","As before, we will bound the dimension of the locus in $H^{1}(X, \\wedge ^{2}E)$ of extensions $V$ which admit a rank $n$ isotropic subbundle $F$ of degree $\\ge -e$ other than $E$ .","Again, either $F$ is an elementary transformation of $E^*$ lifting to $V$ , or $F$ fits into a diagram of the form (REF ).","Step 1.","We first show that the latter situation does not arise for a general $V$ in $H^1(X, \\wedge ^2 E)$ for $t \\le n(g-1) - 1$ (not $t \\le n(g-1) + 1$ as in the symplectic case; see Remark REF ).","For each fixed $H \\subset E$ , by Criterion REF (2), the locus of extensions fitting into a diagram of the form (REF ) for some $F$ of degree $\\ge -e$ is bounded by $ \\dim \\left( \\mathrm {Sec}^{e-h}\\mathrm {Gr}(2, E/H) \\right) + 1 + \\dim \\mathrm {Ker}\\left( q_{*} \\, ^tq^{*} \\right).","$ As in the symplectic case, we obtain $e-h \\ge 0$ .","Since $ \\dim \\mathrm {Gr}(2,E/H) = 2(n-r-2) + 1 = 2(n-r) - 3, $ the secant variety has dimension bounded by $ (e-h)(2(n-r) - 3) + (e-h) - 1 = 2(e-h)(n-r-1) - 1.","$ By Proposition REF , we have $ \\dim \\mathrm {Ker}\\left(q_{*} \\, ^tq^{*} \\right) &= h^{1}(X, \\wedge ^{2}E) - h^{1} \\left(X, \\wedge ^{2}(E/H) \\right) \\\\&\\le h^{1}(X, \\wedge ^{2}E) - (n-r-1)(e-h) - \\frac{1}{2} (n-r)(n-r-1)(g-1).","$ As before, the subbundle $H$ of $E$ varies in a Quot scheme of dimension $ h^{0}(X, \\mathrm {Hom}(H, E/H)) = nh - re - r(n-r)(g-1).","$ Thus the dimension of the locus of extensions in $H^1(X, \\wedge ^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\ge -e$ is bounded by $\\dim \\left( \\mathrm {Sec}^{e-h} \\mathrm {Gr}(2, E/H) \\right) + 1 + \\dim \\mathrm {Ker}(q_* \\, ^t q^*) + h^0(X, \\mathrm {Hom}(H, E/H)) \\\\\\le \\ (e-h)(n-r-1) - \\frac{1}{2} (n-r)(n-r-1) (g-1) + nh-re + h^1(X, \\wedge ^2 E) \\\\= \\ e(n-r) - (e-h)(r+1) - \\frac{1}{2} (n-r)(n-r-1) (g-1) + h^1(X, \\wedge ^2 E).$ Since $e-h \\ge 0$ and we have assumed $e \\le \\frac{1}{2} (n (g-1)-1)$ , this is smaller than $h^1(X, \\wedge ^2 E)$ .","Therefore, for $t = 2e \\le n (g-1) -1$ , a general $V \\in H^1(X, \\wedge ^2 E)$ does not admit a rank $n$ isotropic subbundle $F$ of degree $\\ge -e$ which fits into a diagram of the form (REF ).","Step 2.","Next we consider the case $r = 0$ , so $F$ is an elementary transformation of $E^*$ lifting to $V$ isotropically.","Write $\\deg F = -f \\ge -e$ .","By Criterion REF (3) and Lemma REF , we have $e+f \\equiv 0 \\mod {2}$ .","By Criterion REF (2), the dimension of the locus of extensions in $H^1(X, \\wedge ^2 E)$ admitting such an $F$ is bounded by $\\dim \\left( \\mathrm {Sec}^{\\frac{1}{2}(e+f)} \\mathrm {Gr}(2, E) \\right) + 1 \\ \\le \\ \\frac{1}{2}(e+f) (2(n-2)+2) = (e+f)(n-1).$ If $f < \\frac{1}{2} n (g-1)$ , this bound is smaller than $h^1(X, \\wedge ^2 E) = (n-1)e + \\frac{1}{2}n(n-1)(g-1)$ so a general extension in $H^{1}(X, \\wedge ^{2}E)$ does not admit any such lifting.","Step 3 of the proof of Theorem REF carries over, mutatis mutandis, to the orthogonal case, to prove statement (2), and (1) for $t \\le n(g-1)-1$ .","To prove (1) for $n(g-1) \\le t \\le n(g-1)+3$ , we just repeat the arguments in Steps 1 and 2 under the assumption $-f \\ge -e + 2$ , to show that for general $E \\in U_{X}(n, -e)$ , a general $V \\in H^{1}(X, \\wedge ^{2}E)$ does not admit a rank $n$ isotropic subbundle $F$ of degree $\\ge -e + 2$ .","Remark 5.3 Note that we do not prove an analogue of Theorem REF (3) for the orthogonal case.","A dimension count analogous to that in the proof of Theorem REF (3) does not exclude the possibility that two maximal isotropic rank $n$ subbundles of a general bundle in $\\mathcal {M}O_{X}(2n)$ intersect in a line bundle, or in a rank 2 subbundle if $g = 2$ .","It is unclear to us at this stage whether or not this in fact happens.","Next, we compute the dimension of $M(V)$ for a general $V \\in \\mathcal {M}O_X(2n)^{\\pm }$ .","We will use the following analogue of Lemma REF , which is proven in the same way as in the symplectic case: Lemma 5.4 For an orthogonal bundle $V \\in \\mathcal {M}O_X(2n)$ and a point $[\\eta : E \\subset V]$ of $M(V)$ , the Zariski tangent space of $M(V)$ at $\\eta $ is identified with $H^0(X, \\wedge ^2 E^*)$ .","$\\Box $ Combining with Lemma REF (2), we obtain: Corollary 5.5 Let $t = 2e = n(g-1) + \\varepsilon $ with $\\varepsilon \\in \\lbrace 0,1,2,3 \\rbrace $ .","Then for a general $V \\in \\mathcal {M}O_X(2n;t)$ , we have $ \\dim M(V) = \\frac{\\varepsilon (n-1)}{2}.","$ $\\Box $ Proof of Theorem REF .","(1) Consider the rational map $\\alpha _e: \\mathbb {A}_e \\rightarrow \\mathcal {M}O_X(2n)$ .","For each $t = 2e = n(g-1) + \\varepsilon $ where $\\varepsilon \\in \\lbrace 0,1,2,3 \\rbrace $ , a direct computation shows that $\\dim M(V) = \\dim \\mathbb {A}_e - \\dim \\mathcal {M}O_X(2n).$ Since $\\dim M(V) \\ge \\dim \\alpha _e^{-1} (V)$ for any $V \\in \\mathcal {M}O_X(2n;2e)$ , this equality implies that $\\alpha _e$ is dominant to a component of $\\mathcal {M}O_X(2n)$ .","This shows that for general $V$ in some one of the components $ \\mathcal {M}O_X(2n)^{\\pm }$ , we have $n(g-1) \\le t(V) \\le n(g-1) +3.$ By semicontinuity, $t(V) \\le n(g-1) +3$ for any orthogonal bundle $V$ .","(2) The nonemptiness and irreducibility were proven in Theorem REF (1).","The remaining part can be proven in the same way as Theorem REF (2).","(3) This was proven in Theorem REF (2).","$\\Box $"],["Configuration of the dense strata","For each $2e = n(g-1 ) + \\varepsilon $ with $0 \\le \\varepsilon \\le 3$ , the locations of the images of the $\\alpha _{e}$ depend on the congruence class of $n(g-1)$ modulo 4: the trivial bundle of rank $2n$ is contained in $\\mathcal {M}O_X(2n)^+$ , hence for each $k$ , we have $\\mathcal {M}O_X(2n;4k) \\subset \\mathcal {M}O_X(2n)^+$ and $\\mathcal {M}O_X(2n;4k+2) \\subset \\mathcal {M}O_X(2n)^-$ .","We may summarize the situation for the dense strata as follows: $ n(g-1) \\equiv 0 \\mod {4}: \\quad \\quad \\begin{array}{c|c|c} t & \\hbox{Component} & \\dim M(V) \\\\ \\hline n(g-1) & \\mathcal {M}O_{X}(2n)^{+} & 0 \\\\n(g-1) + 2 & \\mathcal {M}O_{X}(2n)^{-} & \\hbox{$\\frac{1}{2}(n-1)$} \\end{array} $ $ n(g-1) \\equiv 1 \\mod {4}: \\quad \\quad \\begin{array}{c|c|c} t & \\hbox{Component} & \\dim M(V) \\\\ \\hline n(g-1) + 1 & \\mathcal {M}O_{X}(2n)^{-} & \\hbox{$\\frac{1}{2}(n-1)$} \\\\n(g-1) + 3 & \\mathcal {M}O_{X}(2n)^{+} & \\hbox{$n-1$} \\end{array} $ $ n(g-1) \\equiv 2 \\mod {4}: \\quad \\quad \\begin{array}{c|c|c} t & \\hbox{Component} & \\dim M(V) \\\\ \\hline n(g-1) & \\mathcal {M}O_{X}(2n)^{-} & \\hbox{0} \\\\n(g-1) + 2 & \\mathcal {M}O_{X}(2n)^{+} & \\hbox{$\\frac{1}{2}(n-1)$} \\end{array} $ $ n(g-1) \\equiv 3 \\mod {4}: \\quad \\quad \\begin{array}{c|c|c} t & \\hbox{Component} & \\dim M(V) \\\\ \\hline n(g-1) + 1 & \\mathcal {M}O_{X}(2n)^{+} & \\hbox{$\\frac{1}{2}(n-1)$} \\\\n(g-1) + 3 & \\mathcal {M}O_{X}(2n)^{-} & \\hbox{$n-1$} \\end{array} $ Remark 5.6 In [3], symplectic bundles $W$ of rank four were studied which satisfy $s_{2}(W) < t(W)$ ; that is, whose maximal vector subbundles of half rank are all nonisotropic.","Orthogonal bundles $V$ belonging to the top stratum $\\mathcal {M}O_{X}(2n;n(g-1)+2)$ provide a different example of this phenomenon.","Due to the Hirschowitz bound [4], all such $V$ have a vector subbundle of degree at least $- \\left\\lceil \\frac{1}{2}n(g-1) \\right\\rceil $ , but no isotropic rank $n$ subbundle of this degree or greater.","When $n(g-1)$ is even, such an orthogonal $V$ is nongeneral as a vector bundle (compare with Lange–Newstead [14]): Since any maximal rank $n$ subbundle $F \\subset V$ is nonisotropic, we must have $h^{0}(X, \\mathrm {Sym}^{2}F^{*}) > 0$ .","But in this case $\\mu (\\mathrm {Sym}^{2}F^{*}) \\le g-1$ .","By Lemma REF (2), this is a condition of positive codimension on $F$ , so none of the maximal vector subbundles of $V$ are general.","Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-Gu, Seoul 143-701, Korea.","E-mail: ischoe@konkuk.ac.kr Høgskolen i Oslo og Akershus, Postboks 4, St. Olavs plass, 0130 Oslo, Norway.","E-mail: george.hitching@hioa.no"]],"string":"[\n [\n \"A stratification on the moduli spaces of symplectic and orthogonal\\n bundles over a curve\"\n ],\n [\n \"Abstract A symplectic or orthogonal bundle $V$ of rank $2n$ over a curve has an invariant $t(V)$ which measures the maximal degree of its isotropic subbundles of rank $n$.\",\n \"This invariant $t$ defines stratifications on moduli spaces of symplectic and orthogonal bundles.\",\n \"We study this stratification by relating it to another one given by secant varieties in certain extension spaces.\",\n \"We give a sharp upper bound on $t(V)$, which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces.\",\n \"In particular, we compute the dimension of each stratum.\",\n \"We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite.\",\n \"We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.\"\n ],\n [\n \"Introduction\",\n \"Let $X$ be a smooth algebraic curve over $\\\\mathbb {C}$ .\",\n \"A vector bundle $V$ of rank two over $X$ determines a ruled surface $\\\\mathbb {P}V$ .\",\n \"Such surfaces have been studied since the 19th century.\",\n \"A line subbundle $L$ of $V$ gives a section $\\\\sigma _L$ of $\\\\mathbb {P}V$ , called a directrix curve.\",\n \"The self-intersection number of $\\\\sigma _L$ is given by $\\\\sigma _{L} \\\\cdot \\\\sigma _{L} \\\\ = \\\\ \\\\deg (V/L) - \\\\deg L \\\\ = \\\\ \\\\deg V - 2 \\\\deg L.$ The Segre invariant of $\\\\mathbb {P}V$ is defined as the minimal value of $\\\\sigma _{L} \\\\cdot \\\\sigma _L$ over all $L \\\\subset V$ .\",\n \"Via the above formula, this invariant also provides a measure of the difference of the slopes of $V$ and $L$ .\",\n \"The Segre invariant yields a natural stratification on the moduli space of vector bundles of rank two over $X$ , which was studied by Lange and Narasimhan [13].\",\n \"A generalization of this stratification to the moduli of vector bundles over $X$ of arbitrary rank was considered by Lange [11] and the details were settled in Brambila-Paz–Lange [2], and Russo–Teixidor i Bigas [19].\",\n \"The aim of the present article is to establish parallel results for symplectic and orthogonal bundles over $X$ .\",\n \"Let us define a few notions and fix notations.\",\n \"A vector bundle $V$ over $X$ of rank $2n$ is called symplectic (resp., orthogonal) if there is a nondegenerate alternating (resp., symmetric) bilinear form $\\\\omega \\\\colon V \\\\otimes V \\\\rightarrow \\\\mathcal {O}_{X}$ .\",\n \"A subbundle $E$ of $V$ is called isotropic if $\\\\omega |_{E \\\\otimes E} = 0$ .\",\n \"By linear algebra, an isotropic subbundle of $V$ has rank $\\\\le n$ .\",\n \"When $V$ is symplectic, a rank $n$ isotropic subbundle of $V$ is often called a Lagrangian subbundle.\",\n \"We say that a symplectic or orthogonal bundle $V$ is stable (resp., semistable) if for every isotropic subbundle $E$ of $V$ , we have $\\\\mu (V) = \\\\frac{ \\\\deg V}{\\\\mathrm {rk}V} \\\\ > \\\\ \\\\frac{ \\\\deg E}{\\\\mathrm {rk}E} = \\\\mu (E) \\\\ \\\\ \\\\ (\\\\text{resp., } \\\\mu (V) \\\\ge \\\\mu (E)).$ Note that this is a priori weaker than the stability condition for $V$ as a vector bundle; compare with Ramanathan [18].\",\n \"However, Ramanan [17] proved that semistability as an orthogonal bundle is equivalent to semistability of the underlying vector bundle, and moreover that a general stable orthogonal bundle is a stable vector bundle.\",\n \"The same argument (worked through in [5]) shows that the analogous statement is true for symplectic bundles.\",\n \"We denote by $SU_X(2n,\\\\mathcal {O}_{X})$ the moduli space of semistable vector bundles of rank $2n$ and trivial determinant, and write $\\\\mathcal {M}S_X(2n)$ (resp., $\\\\mathcal {M}O_X(2n)$ ) for the sublocus in $SU_X(2n,\\\\mathcal {O}_{X})$ of bundles admitting a symplectic (resp., orthogonal) structure.\",\n \"In the symplectic case it has been proven by Serman [20] that the forgetful map $\\\\mathcal {M}_X(\\\\mathrm {Sp}_{2n}\\\\mathbb {C}) \\\\rightarrow SU_X(2n, \\\\mathcal {O}_{X})$ associated to the extension of the structure group $\\\\mathrm {Sp}_{2n}\\\\mathbb {C}\\\\subset \\\\mathrm {SL}_{2n} \\\\mathbb {C}$ , is an embedding, where $\\\\mathcal {M}_X(\\\\mathrm {Sp}_{2n}\\\\mathbb {C})$ is the moduli space of semistable principal $\\\\mathrm {Sp}_{2n}\\\\mathbb {C}$ -bundles over $X$ .\",\n \"So $\\\\mathcal {M}S_X(2n)$ coincides with the embedded image of $\\\\mathcal {M}_X(\\\\mathrm {Sp}_{2n}\\\\mathbb {C})$ .\",\n \"The orthogonal case is more delicate.\",\n \"By [20], the forgetful map $\\\\mathcal {M}_X(\\\\mathrm {SO}_{2n}\\\\mathbb {C}) \\\\rightarrow SU_X(2n, \\\\mathcal {O}_{X})$ is generically two-to-one, amounting to forgetting the data of an orientation on a principal $\\\\mathrm {SO}_{2n}\\\\mathbb {C}$ -bundle.\",\n \"On the other hand, the map $\\\\mathcal {M}_X(\\\\mathrm {O}_{2n}\\\\mathbb {C}) \\\\rightarrow \\\\mathcal {M}_X(\\\\mathrm {GL}_{2n}\\\\mathbb {C})$ is an embedding.\",\n \"The moduli space $\\\\mathcal {M}_X(\\\\mathrm {O}_{2n}\\\\mathbb {C})$ of semistable principal $\\\\mathrm {O}_{2n}\\\\mathbb {C}$ -bundles over $X$ has several components, which are indexed by the first and second Stiefel–Whitney classes $(w_{1}, w_{2}) \\\\in H^{1}(X, \\\\mathbb {Z}_{2}) \\\\times H^{2}(X, \\\\mathbb {Z}_{2})$ .\",\n \"The class $w_1$ corresponds to the determinant, and there are two components of $\\\\mathcal {M}_X(\\\\mathrm {O}_{2n}\\\\mathbb {C})$ with $w_1$ trivial.\",\n \"We write $\\\\mathcal {M}O_{X}(2n)^\\\\pm $ for the embedded images of these components in $SU_X(2n, \\\\mathcal {O}_{X})$ .\",\n \"Let $V$ be a symplectic or orthogonal bundle of rank $2n$ .\",\n \"Generalizing the Segre invariant for bundles of rank two, we define $t(V) := -2\\\\max \\\\lbrace \\\\deg E : E \\\\hbox{ a rank $n$ isotropic subbundle of } V \\\\rbrace .$ In particular, if $V$ is stable (resp., semistable), then $t(V) > 0$ (resp., $t(V) \\\\ge 0$ ).\",\n \"(Note that when $V$ is a symplectic bundle of rank $>2$ , this differs from the invariant $s_{\\\\mathrm {Lag}}$ defined in [3] by a constant: $t(V) = \\\\frac{2}{n+1} s_{\\\\mathrm {Lag}}(V) $ .)\",\n \"For a symplectic or orthogonal bundle $V$ , we denote by $M(V)$ the space of rank $n$ isotropic subbundles $E \\\\subset V$ such that $t(V) = -2 \\\\deg E$ .\",\n \"It can be viewed as a closed subscheme of a Quot scheme, so it has a natural structure of projective variety.\",\n \"The invariant $t(V)$ induces stratifications on moduli spaces of semistable symplectic and orthogonal bundles over $X$ .\",\n \"For each positive even integer $t$ , we define $\\\\mathcal {M}S_X(2n;t) := \\\\lbrace V \\\\in \\\\mathcal {M}S_X(2n) : t(V) = t \\\\rbrace $ and $\\\\mathcal {M}O_X(2n;t) := \\\\lbrace V \\\\in \\\\mathcal {M}O_X(2n) : t(V) = t \\\\rbrace .$ By semi-continuity of the invariant $t(V)$ , these subloci are constructible sets.\",\n \"For the symplectic case, we show: Theorem 1.1 Let $X$ be a smooth algebraic curve of genus $g \\\\ge 2$ .\",\n \"For any symplectic bundle $V$ of rank $2n$ , we have $t(V) \\\\le n(g-1) +1$ .\",\n \"This is sharp in the sense that for a general $V \\\\in \\\\mathcal {M}S_X(2n)$ , $n(g-1) \\\\le t(V) \\\\le n(g-1 ) +1.$ For each even integer $t$ with $ 2 \\\\le t \\\\le n (g-1)$ , the stratum $\\\\mathcal {M}S_X(2n;t)$ is nonempty and irreducible, and $\\\\dim \\\\mathcal {M}S_X(2n;t) \\\\ = \\\\ \\\\frac{1}{2} \\\\left( n(3n+1)(g-1) + (n+1)t \\\\right).$ Furthermore, if $t < n(g-1)$ , then $\\\\mathcal {M}S_X(2n;t)$ is contained in the closure of $\\\\mathcal {M}S_X(2n;t+2)$ in $\\\\mathcal {M}S_X(2n)$ .\",\n \"For each positive even integer $t < n(g-1)$ , the space $M(V)$ is a single point for a general $V \\\\in \\\\mathcal {M}S_X(2n;t)$ .\",\n \"On the other hand, for a general $V \\\\in \\\\mathcal {M}S_X(2n)$ , $\\\\dim M(V) \\\\ = \\\\ {\\\\left\\\\lbrace \\\\begin{array}{ll}0& \\\\text{when} \\\\ n(g-1) \\\\text{ is even, } \\\\\\\\\\\\frac{n+1}{2}& \\\\text{when} \\\\ n(g-1) \\\\text{ is odd}.\\\\end{array}\\\\right.\",\n \"}$ Let us give some historical remarks.\",\n \"For vector bundles, the Segre stratifications are well understood; see Lange–Narasimhan [13], Hirschowitz [4], Brambila-Paz–Lange [2] and Russo–Teixidor i Bigas [19].\",\n \"Holla and Narasimhan [8] defined a generalized Segre invariant for principal $G$ -bundles for an arbitrary reductive group $G$ , and obtained a bound on the invariant in general.\",\n \"This bound was sharpened for symplectic bundles ($G = \\\\mathrm {Sp}_{2n}\\\\mathbb {C}$ ) by the present authors [3], where Theorem REF (1) was proven using the Terracini lemma in projective geometry.\",\n \"Also Theorem REF was proven for symplectic bundles of rank four in [3].\",\n \"The special case when the curve has genus two has earlier been studied in detail by the second named author [7].\",\n \"In this paper, we provide another simpler proof of (1), and prove (2) and (3) for arbitrary rank $n$ and genus $g$ .\",\n \"In the orthogonal case, the moduli space $\\\\mathcal {M}O_X(2n)$ has two connected components, as was discussed before.\",\n \"We denote by $\\\\mathcal {M}O_{X}(2n)^+$ (resp.\",\n \"$\\\\mathcal {M}O_X(2n)^-$ ) the component consisting of bundles of trivial (resp., nontrivial) second Stiefel–Whitney class.\",\n \"We first prove the following.\",\n \"Theorem 1.2 Let $E_1$ and $E_2$ be isotropic rank $n$ subbundles of an orthogonal bundle of rank $2n$ .\",\n \"Then $\\\\deg E_1$ and $\\\\deg E_2$ have the same parity.\",\n \"A semistable orthogonal bundle $V$ belongs to $\\\\mathcal {M}O_{X}(2n)^+$ (resp., $\\\\mathcal {M}O_{X}(2n)^-$ ) if and only if its isotropic rank $n$ subbundles have even degree (resp., odd degree).\",\n \"We then show the following on the stratification on each component.\",\n \"Theorem 1.3 Let $X$ be a smooth algebraic curve of genus $g \\\\ge 2$ .\",\n \"For any orthogonal bundle $V$ of rank $2n$ , we have $t(V) \\\\le n(g-1) + 3$ .\",\n \"This is sharp in the sense that two even numbers $t$ with $n(g-1) \\\\le t \\\\le n(g-1)+3$ correspond to the values of $t(V)$ for general $V$ in $\\\\mathcal {M}O_X(2n)^+$ and $\\\\mathcal {M}O_X(2n)^-$ .\",\n \"For each even integer $t$ with $2 \\\\le t \\\\le n (g-1) $ , the stratum $\\\\mathcal {M}O_X(2n;t)$ is nonempty and irreducible, and $\\\\dim \\\\mathcal {M}O_X(2n;t) = \\\\frac{1}{2} \\\\left( n(3n-1)(g-1) + (n-1)t \\\\right).$ Furthermore, if $t < n(g-1)$ then $\\\\mathcal {M}O_X(2n;t)$ is contained in the closure of $\\\\mathcal {M}O_X(2n;t+4)$ in the relevant component $\\\\mathcal {M}O_X(2n)^{\\\\pm }$ .\",\n \"For each positive even integer $t < n(g-1)$ , the space $M(V)$ is a single point for a general $V \\\\in \\\\mathcal {M}S_X(2n;t)$ .\",\n \"We also compute the dimension of $M(V)$ for general bundles in each component $\\\\mathcal {M}O_{X}(2n)^{\\\\pm }$ .\",\n \"It turns out to depend on the class of $n(g-1)$ modulo 4; the precise statement is set out in §5.3.\",\n \"A more detailed description of the top strata of $\\\\mathcal {M}O_X(2n)$ is given in §5.4.\",\n \"To prove these statements, we consider families of symplectic and orthogonal extensions.\",\n \"The main strategy is to relate the invariant $t(V)$ to the geometry of certain higher secant varieties in the projectivized extension spaces.\",\n \"This idea goes back to the work of Lange and Narasimhan [13], where Theorem REF is proven for rank two bundles using higher secant varieties of the curve $X$ embedded in the extension spaces.\",\n \"In §2, we generalize the geometric framework in [13] to our situation in Criterion REF .\",\n \"The embedded curve $X$ in the extension space is replaced by the quadric bundle $\\\\mathbb {P}E$ and the Grassmannian bundle $\\\\mathrm {Gr}(2, E)$ for symplectic and orthogonal cases respectively.\",\n \"To work with bundles of rank $2n \\\\ge 4$ , one has also to understand the situation when two rank $n$ isotropic subbundles intersect non-transversely.\",\n \"It turns out that this case can also be understood geometrically (Criterion REF ).\",\n \"This is a key advance upon the methods in [3], which enables us to argue for symplectic and orthogonal bundles of arbitrary rank.\",\n \"In §3, we construct “universal extension spaces” parameterizing all the extensions of fixed type, and show that the rational classifying maps to $\\\\mathcal {M}S_X(2n)$ or $\\\\mathcal {M}O_X(2n)$ are defined on dense subsets.\",\n \"In §4 and §5, we prove the main results for symplectic and orthogonal bundles respectively, using dimension counts based on the geometric information from the preceding sections.\",\n \"In §5, we also observe some interesting properties of certain families of orthogonal bundles, stemming from the richer topological structure of the moduli space.\",\n \"Acknowledgements: The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.\",\n \"2011-A423-0004).\",\n \"Also he would like to thank Høgskolen i Vestfold, Bakkenteigen for the invitation in June 2011.\",\n \"The second author gratefully acknowledges the hospitality and generous financial support of Konkuk University, Seoul.\",\n \"Both authors thank Christian Pauly and Olivier Serman for helpful discussions.\"\n ],\n [\n \"Symplectic and orthogonal extensions and lifting criteria\",\n \"In this section we discuss symplectic and orthogonal extensions.\",\n \"Most of the results in this section were obtained for symplectic extensions in [3].\",\n \"Those results will be restated here for the reader's convenience, and the modified versions for orthogonal extensions will be proven in detail.\",\n \"Let $V \\\\rightarrow X$ be a vector bundle of rank $2n$ equipped with a nondegenerate bilinear form $\\\\omega \\\\colon V \\\\otimes V \\\\rightarrow \\\\mathcal {O}_{X}$ , and let $E \\\\subset V$ be a subbundle.\",\n \"Then there is an exact sequence $ 0 \\\\rightarrow E^{\\\\perp } \\\\rightarrow V \\\\rightarrow E^{*} \\\\rightarrow 0, $ where $E^\\\\perp $ is the orthogonal complement of $E$ .\",\n \"If $E$ is isotropic of rank $n$ , then $E = E^\\\\perp $ and $V$ defines a class $\\\\delta (V) \\\\in H^1(X, \\\\mathrm {Hom}(E^{*}, E)) \\\\cong H^1(X, E \\\\otimes E)$ .\",\n \"Criterion 2.1 Suppose $E$ is simple.\",\n \"An extension $0 \\\\rightarrow E \\\\rightarrow V \\\\rightarrow E^{*} \\\\rightarrow 0$ is induced by a symplectic (resp., orthogonal) structure on $V$ with respect to which $E$ is isotropic if and only if the extension class $\\\\delta (V)$ belongs to the subspace $ H^{1}(X, \\\\mathrm {Sym}^{2}E) \\\\quad \\\\left( \\\\hbox{resp.,} \\\\quad H^{1}(X, \\\\wedge ^{2}E ) \\\\right) $ of $H^{1}(X, E \\\\otimes E)$ .\",\n \"This is due to S. Ramanan.\",\n \"A detailed proof for the symplectic case is given in [6], and the proof for the orthogonal case is practically identical.\"\n ],\n [\n \"Cohomological criterion for lifting\",\n \"Here we recall the notion of a bundle-valued principal part (see Kempf [10] for corresponding results on line bundles).\",\n \"A locally free sheaf $W$ on $X$ has the flasque resolution $ 0 \\\\rightarrow W \\\\rightarrow \\\\underline{\\\\mathrm {Rat}}(W) \\\\rightarrow \\\\mathrm {\\\\underline{Prin}}(W) \\\\rightarrow 0, $ where $\\\\underline{\\\\mathrm {Rat}}(W)$ is the sheaf of rational sections of $W$ and $\\\\mathrm {\\\\underline{Prin}}(W)$ the sheaf of $W$ -valued principal parts.\",\n \"We denote their groups of global sections by $\\\\mathrm {Rat}(W)$ and $\\\\mathrm {Prin}(W)$ respectively.\",\n \"Taking global sections, we obtain $ 0 \\\\rightarrow H^0(X, W) \\\\rightarrow \\\\mathrm {Rat}(W) \\\\rightarrow \\\\mathrm {Prin}(W) \\\\rightarrow H^{1}(X, W) \\\\rightarrow 0.$ For a principal part $p \\\\in \\\\mathrm {Prin}(W)$ , we write its class in $H^{1}(X, W)$ as $[p]$ .\",\n \"Now consider an extension $0 \\\\rightarrow E \\\\rightarrow V \\\\rightarrow E^* \\\\rightarrow 0$ , and an elementary transformation $F$ of $E^*$ defined by the sequence $ 0 \\\\rightarrow F \\\\stackrel{\\\\mu }{\\\\rightarrow } E^* \\\\rightarrow \\\\tau \\\\rightarrow 0 $ for some torsion sheaf $\\\\tau $ .\",\n \"We say that $F$ lifts to $V$ if there is a sheaf injection $F \\\\rightarrow V$ such that the composition $F \\\\rightarrow V \\\\rightarrow E^*$ coincides with $\\\\mu $ .\",\n \"The following statements are proven in [6]: Lemma 2.2 Suppose $h^{0}(X, \\\\mathrm {Hom}(E^{*}, E)) = 0$ , and consider an extension $0 \\\\rightarrow E \\\\rightarrow V \\\\rightarrow E^{*} \\\\rightarrow 0$ with class $\\\\delta (V) \\\\in H^{1}(X, E \\\\otimes E)$ .\",\n \"There is a bijection between principal parts $p \\\\in \\\\mathrm {Prin}(E \\\\otimes E)$ such that $\\\\delta (V) = [p]$ , and elementary transformations of $E^*$ lifting to subbundles of $V$ .\",\n \"The bijection is given by $ p \\\\ \\\\leftrightarrow \\\\ \\\\mathrm {Ker}\\\\left(p \\\\colon E^{*} \\\\rightarrow \\\\mathrm {\\\\underline{Prin}}(E) \\\\right).\",\n \"$ Suppose that $\\\\delta (V) = [p] \\\\in H^{1}(X, \\\\mathrm {Sym}^{2}E)$ , corresponding to a symplectic extension.\",\n \"The subbundle corresponding to $\\\\mathrm {Ker}(p)$ is isotropic in $V$ if and only if $p$ is a symmetric principal part; that is, $^tp = p$ .\",\n \"Suppose that $\\\\delta (V) = [p] \\\\in H^{1}(X, \\\\wedge ^{2}E)$ , corresponding to an orthogonal extension.\",\n \"The subbundle corresponding to $\\\\mathrm {Ker}(p)$ is isotropic in $V$ if and only if $p$ is an antisymmetric principal part; that is, $^tp = -p$ .\",\n \"$\\\\Box $\"\n ],\n [\n \"Subvarieties of the extension spaces\",\n \"Given any vector bundle $W$ , consider the projectivization $\\\\pi \\\\colon \\\\mathbb {P}W \\\\rightarrow X$ .\",\n \"Then we have a natural rational map $ \\\\mathbb {P}W \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, W) $ defined as follows (a slightly different description was given in [3]): Consider the evaluation map $ X \\\\times H^{0}(X, K_X \\\\otimes W^{*}) \\\\rightarrow K_{X}\\\\otimes W^{*} $ .\",\n \"Via Serre duality, the dual of this map is identified with $ W \\\\otimes T_{X} \\\\rightarrow X \\\\times H^{1}(X, W).\",\n \"$ Projectivizing this map and then composing with the projection $X \\\\times \\\\mathbb {P}H^{1}(X, W) \\\\rightarrow \\\\mathbb {P}H^{1}(X, W)$ , we get a map $ \\\\phi \\\\colon \\\\mathbb {P}W \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, W).\",\n \"$ On a fibre $W|_x$ , this map is identified with the projectivized coboundary map in the sequence $ 0 \\\\rightarrow H^0 (X, W) \\\\rightarrow H^{0}(X, W(x)) \\\\rightarrow \\\\frac{W(x)}{W}|_{x} \\\\rightarrow H^{1}(X, W) \\\\rightarrow H^1(X, W(x)) \\\\rightarrow 0 .\",\n \"$ Thus for $w \\\\in \\\\mathbb {P}W$ , the image $\\\\phi (w)$ may be realized as the cohomology class of a $W$ -valued principal part supported at $x$ with a simple pole along $w$ .\",\n \"As discussed in [3], the rational map $\\\\phi $ is induced by the complete linear system of the line bundle $\\\\pi ^{*}K_{X}\\\\otimes \\\\mathcal {O}_{\\\\mathbb {P}W}(1)$ over $\\\\mathbb {P}W$ .\",\n \"For $W = \\\\mathrm {Sym}^{2} E$ and $ W= \\\\wedge ^2 E$ respectively, we have the rational maps $ \\\\mathbb {P}(\\\\mathrm {Sym}^{2} E) \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, \\\\mathrm {Sym}^{2}E) \\\\quad \\\\hbox{and} \\\\quad \\\\mathbb {P}(\\\\wedge ^2 E )\\\\dashrightarrow \\\\mathbb {P}H^{1}(X, \\\\wedge ^{2} E).\",\n \"$ Note that both of these are restrictions of the map $\\\\mathbb {P}(E \\\\otimes E) \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, E \\\\otimes E)$ .\",\n \"In the symplectic case, we consider the chain of maps $\\\\mathbb {P}E \\\\ \\\\hookrightarrow \\\\ \\\\mathbb {P}(\\\\mathrm {Sym}^{2} E) \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, \\\\mathrm {Sym}^{2}E) ,$ where the first inclusion is given by the Segre embedding $[v] \\\\mapsto [v \\\\otimes v]$ .\",\n \"In the orthogonal case, we consider the chain of maps $\\\\mathrm {Gr}(2,E) \\\\ \\\\hookrightarrow \\\\ \\\\mathbb {P}(\\\\wedge ^{2} E) \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, \\\\wedge ^{2} E),$ where the Grassmannian bundle $\\\\mathrm {Gr}(2,E)$ is embedded in $\\\\mathbb {P}(\\\\wedge ^{2} E)$ via the Plücker embedding.\",\n \"These rational maps are denoted as $\\\\phi _s \\\\colon \\\\mathbb {P}E \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, \\\\mathrm {Sym}^{2}E) \\\\quad \\\\hbox{and} \\\\quad \\\\phi _a: \\\\mathrm {Gr}(2,E) \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, \\\\wedge ^{2} E).\",\n \"$ Lemma 2.3 Let $W = E \\\\otimes E$ for a stable bundle $E$ over a curve of genus $g \\\\ge 2$ .\",\n \"The rational map $\\\\phi \\\\colon \\\\mathbb {P}(W) \\\\dashrightarrow \\\\mathbb {P}H^{1}(X, W)$ is base point free if $\\\\mu (E) < - \\\\frac{1}{2}$ , and an embedding if $\\\\mu (E) < -1$ .\",\n \"In particular, if $\\\\mu (E) < -1$ , the rational maps $\\\\phi _s$ and $\\\\phi _a$ in (REF ) are embeddings.\",\n \"If $E$ is general of negative degree, $\\\\phi $ is injective on a general fiber of $\\\\mathbb {P}(W)$ .\",\n \"If we further assume that $g \\\\ge 3$ then $\\\\phi $ separates two general fibers of $\\\\mathbb {P}(W)$ .\",\n \"(1) (A similar version was stated in [3], unfortunately with a flawed proof.)\",\n \"One can check that $\\\\phi $ is base point free (resp., an embedding) if $h^{0}(X, W(D)) = 0$ for all effective divisors $D$ of degree one (resp., of degree two) on $X$ .\",\n \"Since $W = E \\\\otimes E$ is semistable, these vanishing results follow from the assumption that $\\\\mu (E) < - \\\\frac{1}{2}$ (resp., $\\\\mu (E) < -1$ ).\",\n \"(2) Let $L$ and $M$ be line bundles with $\\\\deg L = \\\\deg E$ and $\\\\deg M = 0$ , and put $E_0 = L \\\\oplus M^{\\\\oplus (n-1)}$ , where $n = \\\\mathrm {rk}(E)$ .\",\n \"One can check that for a general choice of $L$ and $M$ , the bundle $E_0 \\\\otimes E_0 (x) $ has no sections.\",\n \"Deforming $E_0$ to a general stable bundle $E$ , we see that $h^{0}(X, W(x)) = 0$ for a general $x \\\\in X$ .\",\n \"A similar argument shows that if $g \\\\ge 3$ , then $h^{0}(X, W(x+y)) = 0$ for general $ x,y \\\\in X$ .\"\n ],\n [\n \"Irreducibility of the space of principal parts\",\n \"In this subsection, we prove some technical facts on principal parts.\",\n \"Let $p$ be a principal part with values in $E \\\\otimes E$ .\",\n \"Definition 2.4 The degree $\\\\deg (p)$ of $p$ is defined as the length of the torsion sheaf $ \\\\mathrm {Im}\\\\left( p \\\\colon E^{*} \\\\rightarrow \\\\mathrm {\\\\underline{Prin}}(E) \\\\right).\",\n \"$ The support of $p$ is defined as the support of $\\\\mathrm {Im}(p)$ .\",\n \"A symmetric (resp., antisymmetric) principal part $p$ of degree $k$ (resp., $2k$ ) is called general if it is supported at $k$ distinct points.\",\n \"Obviously, the general symmetric (resp., antisymmetric) principal parts of degree $k$ (resp., $2k$ ) are parameterized by a quasi-projective irreducible variety.\",\n \"We want to confirm that an arbitrary symmetric or antisymmetric principal part can be obtained as a limit of a continuous family of general ones.\",\n \"Lemma 2.5 ([3]) Let $p $ be a $\\\\mathrm {Sym}^2 E$ -valued principal part of degree $k$ , supported at a single point $x$ .\",\n \"Then there exists a local frame $e_{1}, \\\\ldots , e_n$ for $E$ in a neighborhood of $x$ , in terms of which $p$ is expressed as $ p = \\\\sum _{i=1}^{n} \\\\frac{e_{i} \\\\otimes e_{i}}{z^{k_i}} $ where $z$ is a uniformizer at $x$ , and $ k_{1} \\\\ge k_{2} \\\\ge \\\\cdots \\\\ge k_n \\\\ge 0$ , and $k = \\\\sum _{i=1}^{n} k_{i} $ .\",\n \"$\\\\Box $ For antisymmetric principal parts, we have the following analogue: Lemma 2.6 Let $p$ be a $\\\\wedge ^{2}E$ -valued principal part, supported at a single point $x$ .\",\n \"Then there exists a local frame $e_{1}, \\\\ldots , e_n$ for $E$ in a neighborhood of $x$ , in terms of which $p$ is expressed as $ p = \\\\sum _{i=1}^{s} \\\\frac{e_{2i-1} \\\\wedge e_{2i}}{z^{k_i}} $ where $z$ is a uniformizer at $x$ , and $ k_{1} \\\\ge k_{2} \\\\ge \\\\cdots \\\\ge k_s \\\\ge 0$ , and $\\\\deg (p) = 2 \\\\left( \\\\sum _{i=1}^{s} k_{i} \\\\right)$ .\",\n \"In particular, any antisymmetric principal part has even degree.\",\n \"This argument is adapted from the proof of [3].\",\n \"Locally, $p$ can be expressed as $ p = \\\\frac{1}{z^{k_1}} A $ for some $n \\\\times n$ antisymmetric matrix $A$ with entries in the ring $R = \\\\mathbb {C}[z]/ (z^{k_1})$ , since we are concerned with the principal parts at $x$ only.\",\n \"In this context, it suffices to show that there exists a matrix $P \\\\in M_{n} (R)$ , such that $\\\\det P$ is a unit in $R$ and $^t P A P = \\\\text{diag}(z^{d_1}J, z^{d_2}J, \\\\ldots , z^{d_s}J),$ where $0 = d_1 \\\\le d_2 \\\\le \\\\cdots \\\\le d_s$ and $J = \\\\begin{pmatrix} 0 & 1 \\\\\\\\ -1 & 0 \\\\end{pmatrix}$ .\",\n \"This can be shown by mimicking the standard process to get the normal form of an antisymmetric matrix over $\\\\mathbb {C}$ .\",\n \"The only difference lies in that $R$ has non-units contained in the principal ideal $(z)$ , and this can be taken care of by allowing the terms $z^{d_1}, \\\\ldots , z^{d_s}$ on the diagonal.\",\n \"Alternatively, one may work instead over the formal power series ring $\\\\mathbb {C}[[z]]$ , which is a PID, and truncate the terms in the ideal $(z^{k_1})$ at the final step.\",\n \"For the process over a PID, see Adkins–Weintraub [1].\",\n \"Hence we have a frame for $E$ on this neighborhood in terms of which $p$ appears as $ \\\\sum _{i=1}^{s} \\\\frac{e_{2i-1} \\\\wedge e_{2i}}{z^{k_i}}, $ where $k_{i} = k_1 - d_i$ .\",\n \"Since the terms in the sum (REF ) impose independent conditions on sections of $E$ , we have $ \\\\deg ( p ) = \\\\sum _{i=1}^{s} \\\\deg \\\\left( \\\\frac{e_{2i-1} \\\\wedge e_{2i}}{z^{k_i}} \\\\right) = 2 \\\\sum _{i=1}^{s} k_{i} , $ as required.\",\n \"Corollary 2.7 For a fixed vector bundle $E$ and for each $k >0$ , the spaces of symmetric principal parts of degree $k$ and antisymmetric principal parts of degree $2k$ are irreducible.\",\n \"We consider the antisymmetric case; the symmetric case can be proven similarly.\",\n \"It suffices to show the irreducibility of the space of antisymmetric principal parts supported at a single point $x$ .\",\n \"By Lemma REF , we may choose a trivialization of $E$ near $x$ with respect to which $p$ is expressed as $\\\\sum _{i=1}^{s} p_i$ , where $ p_{i} = \\\\frac{e_{2i-1} \\\\wedge e_{2i}}{z^{k_i}} $ and $2 \\\\sum _{i=1}^{s} k_{i} = k$ .\",\n \"For each $i$ , choose distinct $\\\\lambda ^{i}_{1}, \\\\ldots , \\\\lambda ^{i}_{k_i} \\\\in \\\\mathbb {C}$ , and define a family of principal parts $ p_{i}(t) = \\\\frac{e_{2i-1} \\\\wedge e_{2i}}{(z - \\\\lambda ^{i}_{1}t) \\\\cdots (z - \\\\lambda ^{i}_{k_i}t)} $ where $t$ is a complex parameter.\",\n \"We then put $p(t) = \\\\sum _{i=1}^{s}p_{i}(t)$ .\",\n \"By construction, $p(0) = p$ , while for small $t \\\\ne 0$ we can rewrite $p(t)$ as a sum of $k$ antisymmetric principal parts of degree 2, supported at $k$ distinct points.\",\n \"Hence we have shown that any antisymmetric principal part of degree $2k$ is a limit of a continuous family of general antisymmetric principal parts of degree $2k$ .\",\n \"But we know that the general antisymmetric principal parts are parameterized by an irreducible variety.\",\n \"This proves the claim.\"\n ],\n [\n \"Geometric criteria for lifting\",\n \"In this subsection, we find a geometric interpretation of the cohomological criterion on isotropic liftings in Lemma REF .\",\n \"Throughout this subsection, $E$ is a general stable bundle of negative degree.\",\n \"In particular, $E$ is simple and $h^0(X, \\\\mathrm {Hom}(E^*, E)) = 0$ .\",\n \"Consider the rational maps $\\\\phi _s: \\\\mathbb {P}E \\\\dashrightarrow \\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E) \\\\quad \\\\hbox{and} \\\\quad \\\\phi _a: \\\\mathrm {Gr}(2, E) \\\\dashrightarrow \\\\mathbb {P}H^1(X, \\\\wedge ^2 E).$ As discussed in Lemma REF , these maps are defined on dense subsets, and furthermore are embeddings if $\\\\mu (E)<-1$ .\",\n \"In general, abusing notation, we denote by $\\\\mathbb {P}E$ and $\\\\mathrm {Gr}(2, E)$ the closures of the images $\\\\phi _s (\\\\mathbb {P}E)$ in $\\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ and $\\\\phi _a (\\\\mathrm {Gr}(2, E))$ in $\\\\mathbb {P}H^1(X, \\\\wedge ^2 E)$ respectively.\",\n \"Next, for a quasi-projective variety $Z \\\\subset \\\\mathbb {P}^N$ , we write $\\\\mathrm {Sec}^k Z$ for the $k$ -th secant variety of $\\\\bar{Z}$ , which is the closure of the union of linear subspaces spanned by $k$ general points of $Z$ .\",\n \"In particular, $\\\\mathrm {Sec}^1 Z = \\\\bar{Z}$ .\",\n \"Criterion 2.8 Consider an extension given by $ \\\\delta (V) : \\\\quad 0 \\\\rightarrow E \\\\rightarrow V \\\\rightarrow E^{*} \\\\rightarrow 0.\",\n \"$ When $\\\\delta (V) \\\\in \\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ , there is an elementary transformation $F$ of $E^*$ with $\\\\deg (E^*/F) \\\\le k$ lifting to a Lagrangian subbundle of $V$ if and only if $\\\\delta (V) \\\\in \\\\mathrm {Sec}^{k} \\\\mathbb {P}E$ .\",\n \"When $\\\\delta (V) \\\\in \\\\mathbb {P}H^1(X, \\\\wedge ^2 E)$ , there is an elementary transformation $F$ of $E^*$ with $\\\\deg (E^*/F) \\\\le 2k$ lifting to a rank $n$ isotropic subbundle of $V$ if and only if $ \\\\delta (V) \\\\in \\\\mathrm {Sec}^{k} \\\\mathrm {Gr}(2,E)$ .\",\n \"In this case, $\\\\deg E $ and $\\\\deg F$ have the same parity.\",\n \"In the symplectic case, this is the content of [3].\",\n \"The same idea works for antisymmetric case as follows: By Lemma REF (3), an elementary transformation $F \\\\subseteq E^*$ with $\\\\deg (E^*/F) \\\\le 2k$ lifts to an isotropic subbundle of $V$ if and only if the extension class $\\\\delta (V)$ is of the form $[p]$ where $p$ is an antisymmetric principal part of degree $2l \\\\le k$ .\",\n \"If $p$ is general, it is of the form $ \\\\sum _{i=1}^{l} \\\\frac{e_{i} \\\\wedge f_i}{z_i}, $ where $z_1, z_2, \\\\ldots , z_l$ are uniformizers at $l$ distinct points $x_1, x_2,\\\\ldots , x_l \\\\in X$ .\",\n \"According to the description of the map $\\\\phi _a$ in §REF , the class $[p] \\\\in \\\\mathbb {P}H^1(X, \\\\wedge ^2 E)$ lies on the secant plane spanned by $l$ distinct points of $\\\\mathrm {Gr}(2,E)$ .\",\n \"Hence $[p] \\\\in \\\\mathrm {Sec}^l \\\\mathrm {Gr}(2, E)$ .\",\n \"Conversely, a general point of $\\\\mathrm {Sec}^l \\\\mathrm {Gr}(2, E)$ corresponds to a class $\\\\delta (V) =[p]$ where $p$ is of the form (REF ).\",\n \"From this correspondence on the general points, we get the desired statement by Corollary REF .\",\n \"In this case, $F = \\\\mathrm {Ker}\\\\left(p \\\\colon E^{*} \\\\rightarrow \\\\mathrm {\\\\underline{Prin}}(E) \\\\right)$ with $\\\\deg p = 2l$ , so $\\\\deg F$ has the same parity as $\\\\deg E$ .\",\n \"Corollary 2.9 Consider an extension given by $ \\\\delta (V) : \\\\quad 0 \\\\rightarrow E \\\\rightarrow V \\\\rightarrow E^{*} \\\\rightarrow 0.\",\n \"$ Assume $\\\\delta (V) \\\\in H^1(X, \\\\mathrm {Sym}^2 E)$ .\",\n \"If $\\\\delta (V) \\\\le \\\\mathrm {Sec}^k \\\\mathbb {P}E$ , then $t(V) \\\\le 2(k + \\\\deg E)$ .\",\n \"Assume $\\\\delta (V) \\\\in H^1(X, \\\\wedge ^2 E)$ .\",\n \"If $\\\\delta (V) \\\\le \\\\mathrm {Sec}^k \\\\mathrm {Gr}(2, E)$ , then $t(V) \\\\le 2(2k + \\\\deg E)$ .\",\n \"By definition of the invariant, $t(V) \\\\le - 2 \\\\deg F$ for any rank $n$ isotropic subbundle $F \\\\subset V$ .\",\n \"The above bounds follow as a direct consequence of Criterion REF .\",\n \"The converse of the above Corollary is not true in general.\",\n \"For instance, assume $\\\\delta (V) \\\\in H^1(X, \\\\mathrm {Sym}^2 E)$ .\",\n \"The bound $t(V) \\\\le 2(k + \\\\deg E)$ tells us that there is a Lagrangian subbundle $F$ of degree $\\\\ge -\\\\deg E -k$ , but this does not imply that $F$ lifts from an elementary transformation of $E^*$ .\",\n \"So in general, we are led to consider a diagram of the form ${ 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\\\\\0 [r] & H [r] [u] & F [r] [u] & G [r] [u] & 0 } $ where $H \\\\subset E$ is a subbundle of degree $-h \\\\le 0$ and rank $r \\\\ge 0$ , and $G$ is a locally free subsheaf of $E^*$ of rank $n-r$ .\",\n \"When $r = 0$ so that $E|_x$ and $F|_x$ meet transversely for general $x \\\\in X$ , this reduces to the situation of Criterion REF of the lifting of elementary transformations.\",\n \"The remaining part of this subsection will be devoted to finding a criterion for the existence of such a diagram with $r>0$ .\",\n \"For $H \\\\subseteq E$ , let $q \\\\colon E \\\\rightarrow E/H$ be the quotient map.\",\n \"Since $H$ is isotropic, $H^{\\\\perp }$ fits into the diagram ${ 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\\\\\0 [r] & E [r] @{=}[u] & H^{\\\\perp } [r] [u] & \\\\left( E/H \\\\right)^{*} [r] [u]^{^tq} & 0.\",\n \"}$ For $\\\\delta (V) \\\\in H^1(X, \\\\mathrm {Hom}(E^*, E))$ , we have $\\\\delta \\\\left( H^{\\\\perp } \\\\right) = \\\\, ^tq^{*} (\\\\delta (V)) \\\\ \\\\in \\\\ H^{1}(X, \\\\mathrm {Hom}((E/H)^{*}, E)).$ Furthermore, $H^{\\\\perp }$ inherits a (degenerate) bilinear form from $V$ .\",\n \"Since $(H^{\\\\perp })^{\\\\perp } = H$ , the quotient $H^{\\\\perp }/H$ is equipped with a nondegenerate bilinear form coming from $V$ .\",\n \"In fact, $H^{\\\\perp }/H$ is a symplectic (resp., orthogonal) extension in the upper exact sequence of ${ 0 [r] & E/H [r] & H^{\\\\perp } /H [r] & \\\\left( E/H \\\\right)^{*} [r] & 0 \\\\\\\\0 [r] & E [r] [u]^q & H^{\\\\perp } [r] [u] & \\\\left( E/H \\\\right)^{*} [r] @{=}[u] & 0 }$ corresponding to the class $\\\\delta (H^{\\\\perp }/H) = q_{*}(^tq)^{*} (\\\\delta (V))$ in $ H^{1} \\\\left( X, \\\\mathrm {Sym}^{2} (E/H) \\\\right)$ (resp., $H^{1} \\\\left( X, \\\\wedge ^{2} (E/H) \\\\right)$ ).\",\n \"Lemma 2.10 The induced maps $ q_{*}(^tq)^{*} \\\\colon H^{1}(X, \\\\mathrm {Sym}^{2} E ) \\\\rightarrow H^{1} \\\\left( X, \\\\mathrm {Sym}^{2} (E/H) \\\\right) $ and $ q_{*}(^tq)^{*} \\\\colon H^{1}(X, \\\\wedge ^{2} E ) \\\\rightarrow H^{1} \\\\left( X, \\\\wedge ^{2} (E/H) \\\\right) $ are surjective.\",\n \"These maps are induced from $ (^tq)^{*} : E \\\\otimes E \\\\rightarrow (E/H) \\\\otimes E \\\\quad \\\\hbox{and} \\\\quad q_{*} \\\\colon (E/H) \\\\otimes E \\\\rightarrow (E/H) \\\\otimes (E/H) $ respectively.\",\n \"By local computation, it can be seen that the images of $q_{*}(^tq)^{*}$ in $(E/H) \\\\otimes (E/H)$ are precisely $\\\\mathrm {Sym}^{2} \\\\left( E/H \\\\right)$ and $\\\\wedge ^{2} \\\\left( E/H \\\\right)$ respectively.\",\n \"Hence on the first cohomology level, the induced maps are surjective.\",\n \"We obtain the following geometric criterion on liftings: Criterion 2.11 Let $V$ be an extension of $E^*$ by $E$ with class $\\\\delta (V)$ .\",\n \"Fix a subbundle $0 \\\\ne H \\\\subset E$ and write $\\\\deg (E) = -e, \\\\deg (H) = -h$ .\",\n \"Let $f > 2h - e$ .\",\n \"Assume $\\\\delta (V) \\\\in H^1(X, \\\\mathrm {Sym}^2 E)$ .\",\n \"Then $V$ admits a Lagrangian subbundle $F$ of degree $\\\\ge -f$ inducing a diagram of the form (REF ) if and only if $q_{*}(^tq)^{*} (\\\\delta (V)) \\\\ \\\\in \\\\ \\\\mathrm {Sec}^{(e+f -2h)} \\\\mathbb {P}(E/H) .$ Suppose $\\\\delta (V) \\\\in H^1(X, \\\\wedge ^2 E)$ and $e \\\\equiv f \\\\mod {2}$ .\",\n \"Then $V$ admits a rank $n$ isotropic subbundle $F$ of degree $\\\\ge -f$ inducing a diagram of the form (REF ) if and only if $q_{*}(^tq)^{*} (\\\\delta (V)) \\\\ \\\\in \\\\ \\\\mathrm {Sec}^{\\\\frac{1}{2}(e+f - 2h)} \\\\mathrm {Gr}(2, E/H) .$ Since the arguments for (1) and (2) are parallel, we prove (2) only.\",\n \"The orthogonal bundle $V$ admits an isotropic subbundle $F$ inducing the diagram (REF ) if and only if $H^{\\\\perp }$ admits a subbundle $F$ of degree $\\\\ge -f$ which is isotropic with respect to the antisymmetric form inherited from $V$ , yielding the diagram ${ 0 [r] & E [r] & H^{\\\\perp } [r] & \\\\left( E/H \\\\right)^{*} [r] & 0 \\\\\\\\0 [r] & H [r] [u] & F [r] [u] & G [r] [u] & 0 }$ Since $G = F/H$ has the same rank as $(E/H)^*$ , the map $G \\\\rightarrow (E/H)^*$ is an elementary transformation whose quotient is a torsion sheaf of degree $\\\\le e+f - 2h$ .\",\n \"Factorizing by $H$ , we get ${ 0 [r] & E/H [r] & H^{\\\\perp }/H [r] & \\\\left( E/H \\\\right)^{*} [r] & 0 \\\\\\\\& & F/H [r]^= [u] & G [r] [u] & 0.\",\n \"}$ We are in this situation precisely when the orthogonal extension $H^{\\\\perp }/H$ with class $ q_{*}(^tq)^{*} \\\\delta (V) \\\\in H^1 (X, \\\\wedge ^2 (E/H)) $ admits an isotropic lifting of some elementary transformation of $(E/H)^*$ of degree $\\\\le e+f - 2h$ .\",\n \"By Criterion REF (2), this is equivalent to the class $\\\\delta \\\\left( H^{\\\\perp }/H \\\\right)$ belonging to $\\\\mathrm {Sec}^{\\\\frac{1}{2}(e+f -2h)} \\\\mathrm {Gr}(2,E/H)$ in $\\\\mathbb {P}H^{1}(X, \\\\wedge ^{2}(E/H))$ .\"\n ],\n [\n \"Parameter spaces of extensions\",\n \"Here we construct “universal extension spaces”, following Lange [12] (see also [3]), and investigate stability of the corresponding symplectic and orthogonal bundles.\"\n ],\n [\n \"Construction of the families\",\n \"For a positive integer $e$ , let $U_X(n,-e)^s$ denote the moduli space of stable vector bundles of rank $n$ and degree $-e < 0$ over $X$ .\",\n \"By Narasimhan–Ramanan [16], there exist a finite étale cover $ \\\\pi _{e} \\\\colon \\\\tilde{U}_e \\\\rightarrow U_X(n, -e)^s $ and a bundle $\\\\mathcal {E}_e \\\\rightarrow \\\\tilde{U}_e \\\\times X$ with the property that $\\\\mathcal {E}_e|_{\\\\lbrace E \\\\rbrace \\\\times X} \\\\cong \\\\pi _e(E)$ for all $E \\\\in \\\\tilde{U}_e$ .\",\n \"When $\\\\gcd (n, e)=1$ , it is well known that $\\\\pi _e$ reduces to the identity map.\",\n \"By Riemann-Roch and semistability, for each $E \\\\in U_X(n, -e)^s$ , we have $ h^1(X, \\\\mathrm {Sym}^{2}E) = (n+1)e + \\\\frac{n(n+1)}{2}(g - 1).\",\n \"$ Therefore, the sheaf $R^{1} {p}_{*} (\\\\mathrm {Sym}^2 (\\\\mathcal {E}_e))$ is locally free of rank $(n+1)(e + \\\\frac{1}{2}n(g - 1))$ on $\\\\tilde{U}_e$ .\",\n \"We denote its projectivization by $\\\\mu \\\\colon \\\\mathbb {S}_{e} \\\\rightarrow \\\\tilde{U}_e$ .\",\n \"We have a diagram $ {\\\\mathbb {S}_{e} [dd]_{\\\\mu } & \\\\mathbb {S}_{e} \\\\times X [d]^{\\\\mu \\\\times \\\\mathrm {Id}_X} [l]_{r} & \\\\\\\\& \\\\tilde{U}_{e} \\\\times X [dl]_{p} [dr]^{q} & \\\\\\\\\\\\tilde{U}_{e} & & X}$ We write $r \\\\colon \\\\mathbb {S}_e \\\\times X \\\\rightarrow \\\\mathbb {S}_e$ for the projection.\",\n \"By Lange [12], there is an exact sequence of vector bundles $ 0 \\\\rightarrow (\\\\mu \\\\times \\\\mathrm {Id}_{X})^{*}\\\\mathcal {E}_e \\\\otimes r^{*}\\\\mathcal {O}_{\\\\mathbb {S}_e}(1) \\\\rightarrow \\\\mathcal {W}_{e} \\\\rightarrow (\\\\mu \\\\times \\\\mathrm {Id}_{X})^{*}\\\\mathcal {E}_e^{*} \\\\rightarrow 0 $ over $\\\\mathbb {S}_{e} \\\\times X$ , with the property that for $\\\\delta \\\\in \\\\mathbb {S}_e$ with $\\\\mu (\\\\delta ) = E$ , the restriction of $\\\\mathcal {V}_{e}$ to $\\\\lbrace \\\\delta \\\\rbrace \\\\times X$ is isomorphic to the extension of $E^*$ by $E$ defined by $\\\\delta \\\\in \\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^{2}E)$ .\",\n \"By Lemma REF , the space $\\\\mathbb {S}_{e} $ classifies all symplectic extensions of $E^*$ by $E$ for all $E \\\\in U(n, -e)^s$ , up to homothety.\",\n \"In the same way, we define a bundle $\\\\mathbb {A}_{e} \\\\rightarrow \\\\tilde{U}_e$ whose fibre at $E$ is $\\\\mathbb {P}H^{1}(X, \\\\wedge ^{2}E)$ , a projective space of dimension $(n-1)(e + \\\\frac{1}{2}n(g - 1)) - 1$ .\",\n \"There is a sequence of vector bundles $ 0 \\\\rightarrow (\\\\mu \\\\times \\\\mathrm {Id}_{X})^{*}\\\\mathcal {E}_e \\\\otimes r^{*}\\\\mathcal {O}_{\\\\mathbb {A}_e}(1) \\\\rightarrow \\\\mathcal {V}_{e} \\\\rightarrow (\\\\mu \\\\times \\\\mathrm {Id}_{X})^{*}\\\\mathcal {E}_e^{*} \\\\rightarrow 0 $ over $\\\\mathbb {A}_{e} \\\\times X$ , with the property that for $\\\\delta \\\\in \\\\mathbb {A}_e$ with $\\\\mu (\\\\delta ) = E$ , the restriction of $\\\\mathcal {W}_{e}$ to $\\\\lbrace \\\\delta \\\\rbrace \\\\times X$ is isomorphic to the extension of $E^*$ by $E$ defined by $\\\\delta \\\\in \\\\mathbb {P}H^1(X, \\\\wedge ^{2}E)$ .\",\n \"Again by Lemma REF , the space $\\\\mathbb {A}_{e} $ classifies all the orthogonal extensions of $E^*$ by $E$ for all $E \\\\in U(n, -e)^s$ , up to homothety.\",\n \"For each $e>0$ , the universal bundles $\\\\mathcal {W}_e \\\\rightarrow \\\\mathbb {S}_e \\\\times X$ and $\\\\mathcal {V}_e \\\\rightarrow \\\\mathbb {A}_e \\\\times X$ induce classifying maps $\\\\sigma _e: \\\\mathbb {S}_e \\\\dashrightarrow \\\\mathcal {M}S_X(2n)\\\\ \\\\ \\\\text{and} \\\\ \\\\ \\\\alpha _e: \\\\mathbb {A}_e \\\\dashrightarrow \\\\mathcal {M}O_X(2n)$ respectively.\",\n \"The indeterminacy loci of these maps consist of precisely the points whose associated symplectic/orthogonal bundles are not semistable.\"\n ],\n [\n \"Stability of extensions\",\n \"The universal extension spaces $\\\\mathbb {S}_e$ and $\\\\mathbb {A}_e$ provide a natural way to study the Segre stratification on the moduli space of symplectic/orthogonal bundles, via the classifying maps $\\\\sigma _e$ and $\\\\alpha _e$ .\",\n \"In order to proceed in this direction, we must verify that a general bundle with extension class represented in $\\\\mathbb {S}_e$ (resp., $\\\\mathbb {A}_e$ ) is a stable symplectic (resp., stable orthogonal) bundle.\",\n \"The same question for vector bundles was formulated by Lange [11], and solved in Brambila-Paz–Lange [2] and Russo–Teixidor i Bigas [19].\",\n \"In both papers, elementary transformations were used to construct stable bundles with the prescribed Segre invariant.\",\n \"In this subsection, we prove the existence of a stable orthogonal/symplectic bundle in $\\\\mathbb {S}_e$ and $\\\\mathbb {A}_e$ for each $e >0$ .\",\n \"Elementary transformations are used, but in a somewhat different context.\",\n \"We begin by establishing the statement for $e = 1$ and $e = 2$ .\",\n \"We will need the following bound on the classical Segre invariants of vector bundles, due to Hirschowitz [4]: Lemma 3.1 Let $E$ be a general stable vector bundle of rank $n$ and degree $-e$ .\",\n \"Let $H \\\\subset E$ be a subbundle of rank $r$ and degree $-h$ .\",\n \"Then $r(n-r) (g-1) \\\\ \\\\le \\\\ nh -re \\\\ < \\\\ r(n-r) (g-1) +n .$ Also we need the following result called the Hirschowitz lemma and its variant.\",\n \"Lemma 3.2 Let $H_1$ and $H_2$ be general stable bundles, $\\\\mathrm {rk}(H_i) = r_i$ and $\\\\deg (H_i) = d_i$ for $i = 1,2$ .\",\n \"If $r_1 d_2 + r_2 d_1 \\\\ge r_1 r_2(g-1)$ so that $\\\\mu (H_1 \\\\otimes H_2) \\\\ge g-1$ , then $h^1(X, H_1 \\\\otimes H_2) = 0$ .\",\n \"Let $F$ be a general stable bundle of rank $n$ .\",\n \"If $\\\\deg F \\\\ge \\\\frac{1}{2}n(g-1)$ so that $\\\\mu (F \\\\otimes F) \\\\ge g-1$ , then $h^1(X, F \\\\otimes F) = 0$ .\",\n \"Part (1) was proven by Hirschowitz [4]; see also Russo–Teixidor i Bigas [19].\",\n \"The variant (2) is [3].\",\n \"Proposition 3.3 Let $E \\\\in U(n, -e)^s$ be general, $e = 1,2$ .\",\n \"For $e=1$ , every point in $\\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ outside a sublocus $Y$ with $\\\\dim Y \\\\le n = \\\\dim \\\\mathbb {P}E$ corresponds to a stable symplectic bundle.\",\n \"Hence a general point of $\\\\mathbb {S}_1$ represents a stable symplectic bundle.\",\n \"For $e=1$ , every point in $\\\\mathbb {P}H^1(X, \\\\wedge ^2 E)$ corresponds to a stable orthogonal bundle.\",\n \"For $e=2$ , every point of $\\\\mathbb {P}H^1(X, \\\\wedge ^2 E)$ outside a sublocus $Z$ with $\\\\dim Z \\\\le 2(n-2) +1 = \\\\dim Gr(2,E)$ corresponds to a stable orthogonal bundle.\",\n \"Hence a general point of $\\\\mathbb {A}_1$ or $\\\\mathbb {A}_2$ represents a stable orthogonal bundle.\",\n \"Consider a symplectic or orthogonal extension $0 \\\\rightarrow E \\\\rightarrow V \\\\rightarrow E^{*} \\\\rightarrow 0$ where $E$ is a general stable bundle of rank $n$ and degree $-e \\\\in \\\\lbrace -1,-2 \\\\rbrace $ .\",\n \"Assume that $V$ is not stable, so there is an isotropic subbundle $F$ of $V$ of rank $r (\\\\le n)$ and degree $\\\\ge 0$ .\",\n \"The intersection of $E$ and $F$ contains a subbundle $F_1$ of rank $r_1$ (possibly zero), and the image of $F$ in $E^*$ is a locally free subsheaf $F_2$ of rank $r_2$ , yielding a diagram $ { 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\\\\\0 [r] & F_{1} [r] [u] & F [r] [u] & F_{2} [r] [u] & 0. }\",\n \"$ We will bound the dimension of the locus of the extensions $V$ admitting this kind of diagram.\",\n \"First assume $r_{1} \\\\ne 0$ and so $r_2 < n$ .\",\n \"Since $E$ is general, by Lemma REF we have $ \\\\deg F_{1} \\\\le - \\\\frac{r_1}{n} \\\\left( e + (n-r_{1})(g-1) \\\\right) \\\\ <0 .$ In the same way (if $r_2 \\\\ne 0$ ), $ \\\\deg F_{2} \\\\le \\\\frac{r_2}{n} \\\\left( e - (n-r_{2})(g-1) \\\\right),$ which implies $\\\\deg F_2 \\\\le 0$ since $r_{2} < n$ .\",\n \"Therefore, $\\\\deg F < 0$ .\",\n \"Next, assume $r_1 = 0$ and $ r = r_{2} = n$ .\",\n \"In this case, $F$ is an elementary transformation of $E^*$ .\",\n \"If $\\\\deg F \\\\ge 0$ , then the torsion sheaf $E/F$ has degree $\\\\le e$ .\",\n \"In the symplectic case, we need only consider the case when $e = 1$ .\",\n \"By Criterion REF (1), if $F$ lifts to $V$ as a Lagrangian subbundle, then $\\\\delta (V) \\\\in \\\\phi _s(\\\\mathbb {P}E ) $ in $\\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ .\",\n \"In the orthogonal case, $e \\\\le 2$ .\",\n \"By Criterion REF (2), if $F$ lifts to $V$ as an isotropic subbundle, then in fact $e=2$ and $\\\\delta (V) \\\\in { \\\\mathrm {Gr}(2, E) }$ in $\\\\mathbb {P}H^1(X, \\\\wedge ^2 E)$ .\",\n \"Finally assume $r_1 = 0$ and $r=r_2 < n$ .\",\n \"From the inequality (REF ), $\\\\deg F = \\\\deg F_2 \\\\le 0$ .\",\n \"The possibility $\\\\deg F = 0$ appears only for the following special cases: $e = 1$ ; $r = n - 1$ ; $g = 2$ $e = 2$ ; $r = n - 1$ ; $g = 3$ $e = 2$ ; $r = n - 1$ ; $g = 2$ $e = 2$ ; $r = n - 2$ ; $g = 2$ .\",\n \"From now on, we show that in each of these cases, the dimension of the locus of extensions admitting a lifting of a subsheaf $F$ of $E^*$ as an isotropic subbundle of $V$ of degree zero and rank $r 0$ , a general point of $\\\\mathbb {S}_e$ (resp., $\\\\mathbb {A}_e$ ) represents a stable symplectic (resp., orthogonal) bundle.\",\n \"We consider the orthogonal case first: For each value of $e>0$ , we will exhibit a stable orthogonal bundle represented in $\\\\mathbb {A}_e$ .\",\n \"The statement will then follow from the openness of the stable objects in families.\",\n \"Let $E \\\\in U_{X}(n,-1)$ be general.\",\n \"For any $k \\\\ge 2$ , choose a general antisymmetric principal part $p$ of degree $2k$ , which defines an element of $\\\\mathrm {Sec}^k \\\\mathrm {Gr}(2, E)$ .\",\n \"Since $\\\\mathrm {Gr}(2,E)$ is nondegenerate and properly contained in $\\\\mathrm {Sec}^{k} \\\\mathrm {Gr}(2,E)$ , we may assume that $[p]$ does not lie on the image of $\\\\mathrm {Gr}(2, E)$ .\",\n \"By Lemma REF , the sheaf $F := \\\\mathrm {Ker}\\\\left( p \\\\colon E^{*} \\\\rightarrow \\\\mathrm {\\\\underline{Prin}}(E) \\\\right)$ lifts to a rank $n$ isotropic subbundle of the extension $V$ with $\\\\delta (V) = [p]$ .\",\n \"Deforming $p$ if necessary, we can assume that $F$ is stable.\",\n \"By Proposition REF (2), moreover, $V$ is a stable orthogonal bundle.\",\n \"Since $V$ fits into an orthogonal extension $ 0 \\\\rightarrow F \\\\rightarrow V \\\\rightarrow F^{*} \\\\rightarrow 0, $ it is associated to a point in $\\\\mathbb {A}_{2k-1}$ .\",\n \"In the same way, consider a general bundle $E \\\\in U_X(n, -2)$ and choose a general antisymmetric principal part of degree $2k+2$ for each $k \\\\ge 2$ , defining a point of $\\\\mathrm {Sec}^{k+1} \\\\mathrm {Gr}(2,E)$ .\",\n \"By Proposition REF (2), we may assume that the extension $V$ associated to the point $[p]$ is a stable orthogonal bundle, if we avoid a subvariety $Z$ with $\\\\dim Z \\\\le \\\\dim \\\\mathrm {Gr}(2,E)$ ; and this is possible since $\\\\mathrm {Gr}(2,E)$ is nondegenerate and properly contained in $\\\\mathrm {Sec}^{k+1} \\\\mathrm {Gr}(2,E)$ .\",\n \"As in the previous case, $V$ fits into an orthogonal extension (REF ) where this time $\\\\deg (F) = -2k$ , so $V$ is represented in $\\\\mathbb {A}_{2k}$ .\",\n \"In the symplectic case, we argue similarly.\",\n \"For each $k \\\\ge 2$ , we can choose a general symmetric principal part $q$ of degree $k+1$ which defines a point of $\\\\mathrm {Sec}^{k+1} \\\\mathbb {P}E $ .\",\n \"By the above argument, the extension $V$ determined by $q$ is a stable symplectic bundle which is represented in $\\\\mathbb {S}_{k}$ .\"\n ],\n [\n \"Description of the Segre strata for symplectic bundles\",\n \"In this section, we use the map $\\\\sigma _e: \\\\mathbb {S}_e \\\\dashrightarrow \\\\mathcal {M}S_X(2n)$ discussed in the previous section to prove Theorem REF .\",\n \"Recall that for each positive even integer $t$ , we denote by $\\\\mathcal {M}S_X(2n;t)$ the sublocus of $\\\\mathcal {M}S_X(2n)$ consisting of symplectic bundles $V$ with $t(V) = t$ .\",\n \"Also, for $V \\\\in \\\\mathcal {M}S_X(2n)$ , we write $M(V)$ for the space of Lagrangian subbundles of $V$ of (maximal) degree $-\\\\frac{1}{2}t(V)$ .\",\n \"Theorem 4.1 Consider a positive even integer $ t=2e \\\\le n (g-1)+1$ .\",\n \"A general point of $\\\\mathbb {S}_e$ corresponds to a bundle $V$ with $t(V) = t$ ; in particular, $\\\\mathcal {M}S_X(2n;t)$ is nonempty.\",\n \"Furthermore, $\\\\mathcal {M}S_X(2n;t)$ is irreducible.\",\n \"If $t < n(g-1)$ , then $M(V)$ is a single point for a general $V \\\\in \\\\mathcal {M}S_X(2n;t)$ .\",\n \"If $t \\\\in \\\\lbrace n(g-1), \\\\: n(g-1) +1 \\\\rbrace $ and $V$ is general in $\\\\mathcal {M}S_X(2n;t)$ then any pair of Lagrangian subbundles in $M(V)$ intersect transversely in $V$ in a general fiber.\",\n \"Remark 4.2 Statements (2) and (3) can be viewed as a symplectic analogue of Lange–Newstead [14].\",\n \"Let $E$ be a general bundle in $U(n, -e)^s$ for $e \\\\ge 1$ .\",\n \"By Theorem REF , the moduli map $\\\\mathbb {S}_{e}|_{E} \\\\cong \\\\mathbb {P}H^{1}(X, \\\\mathrm {Sym}^{2}E) \\\\dashrightarrow \\\\mathcal {M}S_X(2n)$ is defined on a dense subset.\",\n \"We want to compute a bound on the dimension of the locus in $H^{1}(X, \\\\mathrm {Sym}^{2}E)$ of extensions $V$ which admit a Lagrangian subbundle $F$ of degree $\\\\ge -e$ other than $E$ .\",\n \"There are two possibilities: either $F$ is an elementary transformation of $E^*$ lifting to $V$ , or $F$ fits into a diagram of the form (REF ).\",\n \"Step 1.\",\n \"We show first that the latter situation does not arise for a general $V$ in $H^1(X, \\\\mathrm {Sym}^2 E)$ for $t = 2e \\\\le n(g-1) +1$ .\",\n \"As before, we write $\\\\deg (H) = -h$ .\",\n \"Recall from the diagram (REF ) that $G$ is an elementary transformation of $(E/H)^*$ whose quotient is a torsion sheaf of degree $\\\\deg (E/H)^* - \\\\deg G = (e-h) - (\\\\deg F + h) \\\\le 2(e-h).$ In particular, $e-h \\\\ge 0.$ Now for each fixed $H \\\\subset E$ , Criterion REF says that the locus of extensions in $H^1(X, \\\\mathrm {Sym}^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\\\ge -e$ is bounded by $ \\\\dim \\\\left( \\\\mathrm {Sec}^{2(e-h)}\\\\mathbb {P}(E/H) \\\\right) + 1 + \\\\dim \\\\mathrm {Ker}\\\\left( q_{*} \\\\, ^tq^{*} \\\\right).\",\n \"$ The secant variety has dimension bounded by $ 2(e-h)(n-r) + 2(e-h) - 1 = 2(e-h)(n-r+1) - 1.\",\n \"$ Also by Proposition REF , we have $ \\\\dim \\\\mathrm {Ker}\\\\left(q_{*} \\\\, ^tq^{*} \\\\right) &= h^{1}(X, \\\\mathrm {Sym}^{2}E) - h^{1} \\\\left(X, \\\\mathrm {Sym}^{2}(E/H) \\\\right) \\\\\\\\&\\\\le h^{1}(X, \\\\mathrm {Sym}^{2}E) - (n-r+1)(e-h) - \\\\frac{1}{2} (n-r)(n-r+1)(g-1).\",\n \"$ Finally we take account of the deformations of $H$ by computing the dimension of the appropriate Quot scheme of $E$ .\",\n \"Since $E$ is general, by Lemma REF we have $nh - re \\\\ \\\\ge \\\\ r(n-r)(g-1).$ We may furthermore assume that $H$ and $E/H$ are general, and so $ h^1(X, \\\\mathrm {Hom}(H, E/H)) = 0 $ by Lemma REF (1).\",\n \"Therefore $[H]$ is a smooth point of the Quot scheme (cf.\",\n \"Le Potier [15]), and the dimension of the Quot scheme is given by $ h^{0}(X, \\\\mathrm {Hom}(H, E/H)) = nh - re - r(n-r)(g-1).\",\n \"$ Adding up these three terms, we see that the dimension of the locus of extensions in $H^1(X, \\\\mathrm {Sym}^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\\\ge -e$ is bounded by $\\\\dim \\\\left( \\\\mathrm {Sec}^{2(e-h)} \\\\mathbb {P}(E/H) \\\\right) + 1 + \\\\dim \\\\mathrm {Ker}(q_* \\\\, ^t q^*) + h^0(X, \\\\mathrm {Hom}(H, E/H)) \\\\\\\\\\\\le \\\\ (e-h)(n-r+1) - \\\\frac{1}{2} (n-r)(n+r+1) (g-1) + nh-re + h^1(X, \\\\mathrm {Sym}^2 E) \\\\\\\\= \\\\ e(n-r) - (e-h)(r-1) - \\\\frac{1}{2} (n-r)(n+r+1) (g-1) + h^1(X, \\\\mathrm {Sym}^2 E).$ By the assumption $e \\\\le \\\\frac{1}{2} (n (g-1)+1)$ and the inequality (REF ), this is smaller than the dimension of the whole extension space $h^1(X, \\\\mathrm {Sym}^2 E)$ .\",\n \"This shows that for $t = 2e \\\\le n (g-1) +1$ , a general $V \\\\in H^1(X, \\\\mathrm {Sym}^2 E)$ does not admit a Lagrangian subbundle $F$ of degree $\\\\ge -e$ which fits into a diagram of the form (REF ).\",\n \"Step 2.\",\n \"Next we consider the case $r = 0$ , so $F$ is an elementary transformation of $E^*$ lifting to $V$ isotropically.\",\n \"Let $\\\\deg F = -f \\\\ge -e$ .\",\n \"By Criterion REF , the dimension of the locus of extensions in $H^1(X, \\\\mathrm {Sym}^2 E)$ admitting such an $F$ is bounded by $\\\\dim \\\\left( \\\\mathrm {Sec}^{e+f} \\\\mathbb {P}E \\\\right) +1 \\\\ \\\\le \\\\ (e+f)(n+1).$ If $f < \\\\frac{1}{2} n (g-1)$ , this bound is smaller than $h^1(X, \\\\mathrm {Sym}^2 E) = (n+1)e + \\\\frac{1}{2}n(n+1)(g-1).$ Step 3.\",\n \"Combining the dimension counts for two possibilities in the above, we conclude: (i) if $t=2e < n (g-1)$ , a general $V \\\\in H^1(X, \\\\mathrm {Sym}^2 E)$ does not have a Lagrangian subbundle of degree $\\\\ge -e$ other than $E$ itself.\",\n \"This shows (2).\",\n \"(ii) if $t=2e \\\\in \\\\lbrace n (g-1), \\\\: n(g-1) + 1 \\\\rbrace $ , a general $V \\\\in H^1(X, \\\\mathrm {Sym}^2 E)$ does not have a Lagrangian subbundle of degree $> -e$ .\",\n \"Also, any Lagrangian subbundle of degree $-e$ different from $E$ intersects $E$ transversely at a general fiber.\",\n \"This shows (3).\",\n \"Hence for a general $V \\\\in H^1(X, \\\\mathrm {Sym}^2 E)$ with $t = 2e \\\\le n(g-1) +1$ , we have $t(V) = t$ .\",\n \"It follows that the rational map $\\\\sigma _e : \\\\mathbb {S}_e \\\\dashrightarrow \\\\mathcal {M}S_X(2n)$ sends a general point of $\\\\mathbb {S}_e$ to $\\\\mathcal {M}S_X(2n;t)$ , so $\\\\mathcal {M}S_X(2n;t)$ is non-empty for each $t \\\\le n (g-1) +1$ .\",\n \"Since $\\\\mathbb {S}_e$ is irreducible, so is its image.\",\n \"To see that $\\\\mathcal {M}S_X(2n;t)$ is irreducible, it suffices to show that every point of $\\\\mathcal {M}S_X(2n;t)$ is in the closure of the image of $\\\\mathbb {S}_e$ .\",\n \"Any $V$ in $\\\\mathcal {M}S_X(2n;t)$ admits a Lagrangian subbundle $E$ of degree $-e$ which might be unstable.\",\n \"But every such $E$ is contained in an irreducible family of bundles whose general member is a stable bundle in $U_X(n, -e)$ .\",\n \"This shows (1).\",\n \"To finish the proof of Theorem REF , we need the following description of the tangent space of $M(V)$ .\",\n \"Lemma 4.3 For a symplectic bundle $V \\\\in \\\\mathcal {M}S_X(2n)$ and a point $[\\\\eta : E \\\\subset V]$ of $M(V)$ , the Zariski tangent space of $M(V)$ at $\\\\eta $ is identified with $H^0(X, \\\\mathrm {Sym}^2 E^*)$ .\",\n \"It is well known that the Zariski tangent space to the Quot scheme of a vector bundle $V$ at a point $[E\\\\subset V]$ is given by $H^0(X, \\\\mathrm {Hom}(E, V/E))$ .\",\n \"Intuitively, this can be explained as follows: A rank $n$ subbundle of $V$ is equivalent to a global section of the Grassmannian bundle $\\\\mathrm {Gr}(n, V)$ over $X$ .\",\n \"A tangent vector to the Quot scheme of $V$ at $[E \\\\subset V]$ corresponds, at each fiber $x \\\\in X$ , to a tangent vector to the Grassmannian $\\\\mathrm {Gr}(n, V_x)$ at $[E_x \\\\subset V_x]$ .\",\n \"Therefore, the tangent vector is a global section of the bundle $\\\\mathrm {Hom}(E, V/E)$ .\",\n \"When $E$ is a Lagrangian subbundle of a symplectic bundle $V$ , we have $V/E \\\\cong E^*$ and so $\\\\mathrm {Hom}(E, V/E) \\\\cong E^* \\\\otimes E^*$ .\",\n \"In this case, a similar argument shows that a tangent vector to $M(V)$ at $[E \\\\subset V]$ corresponds to a global section of $\\\\mathrm {Sym}^2 E^*$ , since for each $x \\\\in X$ the tangent space of the Lagrangian Grassmannian at $[E_x \\\\subset V_x]$ is identified with $\\\\mathrm {Sym}^{2}E^{*}|_x$ .\",\n \"Proof of Theorem REF We begin with part (3) of the statement.\",\n \"The first part of (3) is Theorem REF (2).\",\n \"For the latter part, we invoke Lemma REF : $\\\\dim M(V) = h^0(X, \\\\mathrm {Sym}^2 E^*)$ for a smooth point $[E \\\\subset V]$ of $M(V)$ .\",\n \"By Lemma REF (2), for a general $E \\\\in U_X(n,-e)$ , we have $\\\\dim H^0(X, \\\\mathrm {Sym}^2 E^*) \\\\ = \\\\ {\\\\left\\\\lbrace \\\\begin{array}{ll}0& \\\\text{if} \\\\ 2e = n(g-1), \\\\\\\\\\\\frac{n+1}{2}& \\\\text{if} \\\\ 2e=n(g-1)+1.\\\\end{array}\\\\right.\",\n \"}$ (1) A straightforward computation shows that when $ n(g-1) \\\\le 2e \\\\le n(g-1)+1 $ , $ \\\\dim M(V) = \\\\dim \\\\mathbb {S}_e - \\\\dim \\\\mathcal {M}S_X(2n).\",\n \"$ Since $ \\\\dim M(V) \\\\ge \\\\dim \\\\sigma _{e}^{-1}(V)$ , this equality implies that $\\\\sigma _e$ is dominant.\",\n \"This shows that for a general $V \\\\in \\\\mathcal {M}S_X(2n)$ , we have $n(g-1) \\\\le t(V) \\\\le n(g-1) +1.$ By semicontinuity, $t(V) \\\\le n(g-1) +1$ for any symplectic bundle $V$ .\",\n \"(2) If $t = 2e \\\\ge n(g-1) $ , then $\\\\dim \\\\mathcal {M}S_X(2n;t) = \\\\dim \\\\mathcal {M}S_X(2n)$ by (1).\",\n \"Assume $t = 2e < n(g-1)$ .\",\n \"By Theorem REF (2), the map $\\\\sigma _e \\\\colon \\\\mathbb {S}_{e} \\\\dashrightarrow \\\\mathcal {M}S_X(2n;t)$ is generically finite (of degree $\\\\deg \\\\pi _e$ ), and so $\\\\dim \\\\mathcal {M}S_X(2n;t) \\\\ = \\\\ \\\\dim \\\\mathbb {S}_e \\\\ = \\\\ \\\\frac{1}{2}n(3n+1)(g-1) + (n+1)e.$ For the last part of (2), we can use Criterion REF to show that the Segre stratification matches the stratification given by the higher secant varieties: For a general $V \\\\in \\\\mathcal {M}S_X(2n;2e)$ , let $E \\\\in M(V)$ so that $E$ is general in $U_{X}(n, -e)^s$ .\",\n \"Let $k$ be the smallest integer satisfying $ \\\\mathrm {Sec}^{k} \\\\mathbb {P}E = \\\\mathbb {P}H^{1}(X, \\\\mathrm {Sym}^{2}E)$ .\",\n \"Then by Criterion REF , there is some elementary transformation $F$ of $E^*$ with $\\\\deg E/F = k$ lifting to $V$ as a Lagrangian subbundle.\",\n \"By deforming $V$ and $E$ inside $\\\\mathbb {S}_e$ , we may assume that $F$ is general in $U_X(n,e-k)^s$ .\",\n \"Now consider the symplectic extension $ 0 \\\\rightarrow F \\\\rightarrow V \\\\rightarrow F^* \\\\rightarrow 0.\",\n \"$ In the same way, $\\\\delta (V) \\\\in \\\\mathbb {P}H^{1}(X, \\\\mathrm {Sym}^{2}F)$ belongs to $\\\\mathrm {Sec}^{k}\\\\mathbb {P}F$ , since the elementary transformation $E \\\\rightarrow F^*$ lifts to $V$ .\",\n \"Note that $\\\\dim \\\\mathrm {Sec}^k \\\\mathbb {P}F \\\\le k(n+1) -1 \\\\ < \\\\ (n+1)(k-e) + \\\\frac{1}{2}n(n+1)(g-1) -1 = \\\\dim \\\\mathbb {P}H^1 (X, \\\\mathrm {Sym}^2 F),$ since we assumed $2e < n(g-1)$ .\",\n \"Therefore, $\\\\mathrm {Sec}^{k}\\\\mathbb {P}F$ is properly contained in $\\\\mathbb {P}H^1 (X, \\\\mathrm {Sym}^2 F)$ , and certainly it is inside the closure of $\\\\mathrm {Sec}^{k+1} \\\\mathbb {P}F \\\\setminus \\\\mathrm {Sec}^k \\\\mathbb {P}F$ .\",\n \"Thus, again by Criterion REF , the class $\\\\delta (V)$ belongs to the closure of a family of bundles admitting liftings of elementary transformations of degree $(k-e)-(k+1) = -(e+1)$ .\",\n \"In particular, $V$ belongs to the closure of $\\\\mathcal {M}S_{X}(2n;2e+2)$ in $\\\\mathcal {M}S_{X}(2n)$ .\",\n \"By the irreducibility of $\\\\mathcal {M}S_{X}(2n;2e)$ , the same holds for arbitrary $V \\\\in \\\\mathcal {M}S_X(2n;2e)$ .\",\n \"$\\\\Box $ Remark 4.4 Suppose $t=2e < n(g-1)$ .\",\n \"If $\\\\gcd (n,e) = 1$ , then $\\\\tilde{U}_e = U_X(n,-e)^s$ and $\\\\mathcal {M}S_X(2n;t)$ is birational to the fibration $\\\\mathbb {S}_e$ over $U_X(n,-e)^s$ whose fiber at $ E$ is $\\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ .\",\n \"Finally we give a geometric interpretation of the cardinality of $M(V)$ , when it is finite.\",\n \"Assume $t =2e = n (g-1) $ .\",\n \"Let $V$ be general in $\\\\mathcal {M}S_X(2n;t)$ and choose $E \\\\in M(V)$ with $\\\\deg E = -e = -\\\\frac{1}{2} n(g-1)$ .\",\n \"Consider the subvariety $\\\\mathbb {P}E \\\\subset \\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ .\",\n \"When $g \\\\ge 4$ , we have $\\\\mu (E) < -1$ , and by Lemma REF , the map $\\\\phi _s : \\\\mathbb {P}E \\\\rightarrow \\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ is an embedding.\",\n \"Furthermore, it was proven in [3] that the secant variety $\\\\mathbb {P}E$ is not defective in $\\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ in the sense that $\\\\dim \\\\mathrm {Sec}^{k} \\\\mathbb {P}E \\\\ = \\\\ \\\\min \\\\lbrace (n+1)k -1, \\\\ \\\\dim \\\\mathbb {P}H^1 (X, \\\\mathrm {Sym}^2 E) \\\\rbrace .$ In particular, for $k = n(g-1)$ we have $\\\\dim \\\\mathrm {Sec}^{k} \\\\mathbb {P}E \\\\ = \\\\ (n+1)k -1 \\\\ = \\\\ \\\\dim \\\\mathbb {P}H^1 (X, \\\\mathrm {Sym}^2 E).\",\n \"$ By a $k$ -secant space of $\\\\mathbb {P}E$ , we mean a linear subspace in $\\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ spanned by some $k$ points of $\\\\mathbb {P}E$ .\",\n \"By (REF ), there are finite number of $n(g-1)$ -secant spaces of $\\\\mathbb {P}E$ which pass through a general point of $\\\\mathbb {P}H^1 (X, \\\\mathrm {Sym}^2 E)$ .\",\n \"The following result generalizes Lange-Narasimhan [13] for rank 2 bundles.\",\n \"Theorem 4.5 Assume $g \\\\ge 4$ and $n(g-1)$ is even.\",\n \"Let $V \\\\in \\\\mathcal {M}S_X(2n)$ be a general symplectic bundle with $t(V) = n(g-1)$ , and let $E \\\\in M(V)$ .\",\n \"There is a one-to-one correspondence between $M(V) \\\\setminus \\\\lbrace E\\\\rbrace $ and $n(g-1)$ -secant spaces of $\\\\mathbb {P}E$ passing through the point $[E \\\\subset V] \\\\in \\\\mathbb {P}H^1(X, \\\\mathrm {Sym}^2 E)$ .\",\n \"By Criterion REF , a general class $\\\\delta (V)$ lies on a $n(g-1)$ -secant space of $\\\\mathbb {P}E$ if and only if there is an associated elementary transformation $F$ of $E^*$ with $\\\\deg (E^*/ F) = n(g-1)$ , lifting to a Lagrangian subbundle of $V$ .\",\n \"Since $t(V) = n(g-1)$ , the Lagrangian subbundle $F$ is an element of $M(V)$ .\",\n \"Conversely, by Theorem REF (3) every element of $M(V)$ other than $E$ appears in this way, since $V$ is general.\",\n \"Remark 4.6 It is an interesting problem to compute the cardinality of $M(V)$ explicitly.\",\n \"Let $N_{n,g}$ denote the cardinality of $M(V)$ for a general symplectic bundle $V$ of rank $2n$ over a curve of genus $g$ so that $n(g-1)$ is even.\",\n \"It is well known that $N_{1,g} = 2^g$ .\",\n \"The same problem for maximal subbundles of vector bundles was solved by Holla [9] (see also Lange–Newstead [13]).\",\n \"As far as we are aware, the number $N_{n,g}$ is not known in general.\"\n ],\n [\n \"Description of Segre strata for orthogonal bundles\",\n \"In this section, we investigate the geometry of the Segre stratification on the moduli space of orthogonal bundles $\\\\mathcal {M}O_{X}(2n)$ .\",\n \"In contrast to $\\\\mathcal {M}S_{X}(2n)$ , this has two connected components $\\\\mathcal {M}O_{X}(2n)^{\\\\pm }$ , corresponding to bundles of trivial and nontrivial second Stiefel–Whitney class (see Serman [20] for more details).\",\n \"We begin by determining the component to which each stratum $\\\\mathcal {M}O_{X}(2n;t)$ belongs.\"\n ],\n [\n \"Topological classification\",\n \"Theorem 5.1 (1) Let $E_1$ and $E_2$ be isotropic rank $n$ subbundles of an orthogonal bundle $V$ of rank $2n$ .\",\n \"Then $\\\\deg E_1$ and $\\\\deg E_2$ have the same parity.\",\n \"(2) A semistable orthogonal bundle $V$ belongs to $\\\\mathcal {M}O_{X}(2n)^+$ (resp., $\\\\mathcal {M}O_{X}(2n)^-$ ) if and only if its isotropic rank $n$ subbundles have even degree (resp., odd degree).\",\n \"By Theorem REF , the moduli maps $\\\\alpha _{e} \\\\colon \\\\mathbb {A}_{e} \\\\dashrightarrow \\\\mathcal {M}O_{X}(2n)$ are defined on dense subsets for each $e>0$ .\",\n \"Since each family $\\\\mathbb {A}_e$ is connected, the image of each $\\\\mathbb {A}_e$ is entirely contained in either $\\\\mathcal {M}O_{X}(2n)^+$ or $\\\\mathcal {M}O_{X}(2n)^-$ .\",\n \"As was already noted in the proof of Theorem REF , whenever $e_1$ and $e_2$ have the same parity, the images of $\\\\mathbb {A}_{e_1}$ and $\\\\mathbb {A}_{e_2}$ have nonempty intersection.\",\n \"Hence they lie on the same component of $\\\\mathcal {M}O_X(2n)$ .\",\n \"This is essentially due to the fact that any antisymmetric principal part has an even degree (see Lemma REF ).\",\n \"For each $k>0$ , the image of $\\\\mathbb {A}_{2k}$ is contained in $\\\\mathcal {M}O_{X}(2n)^+$ since the trivial bundle has trivial second Stiefel–Whitney class.\",\n \"Thus the image of $\\\\mathbb {A}_{2k-1}$ is contained in $\\\\mathcal {M}O_{X}(2n)^-$ .\",\n \"This shows the above statements (1) and (2) for the case when $V$ is a stable orthogonal bundle and the isotropic subbundles are stable bundles.\",\n \"By deforming an arbitrary bundle to a stable one, we see that (1) and (2) hold in general.\"\n ],\n [\n \"An extra stratum\",\n \"The local deformations of an orthogonal bundle $V$ are given by $H^1(X, \\\\wedge ^2 V)$ .\",\n \"Since $V$ is self-dual, $\\\\wedge ^2 V \\\\subset \\\\mathrm {End}(V)$ .\",\n \"Hence both components $\\\\mathcal {M}O_{X}(2n)^{\\\\pm }$ have dimension $ h^{1}(X, \\\\wedge ^{2}V) = n(2n-1)(g-1), $ since we may assume that $V$ is simple.\",\n \"On the other hand, $\\\\dim \\\\mathbb {A}_e & = & \\\\dim U_X(n,-e) + \\\\dim \\\\mathbb {P}H^1 (X, \\\\wedge ^2 E) \\\\\\\\& = & (n-1)e + \\\\frac{1}{2} n (3n-1) (g-1)$ for a general $E \\\\in U_X(n,-e)$ .\",\n \"Comparing dimensions, we see that in order for an $\\\\mathbb {A}_e$ to cover either component of $\\\\mathcal {M}O_{X}(2n)$ , we need $e \\\\ge \\\\left\\\\lceil \\\\frac{1}{2}n(g-1) \\\\right\\\\rceil $ .\",\n \"However, a connected set $\\\\mathbb {A}_e$ can cover at most one component.\",\n \"Hence we are led to consider at least two top strata, corresponding to distinct values of $e \\\\ge \\\\left\\\\lceil \\\\frac{1}{2}n(g-1) \\\\right\\\\rceil $ .\",\n \"This is one important difference between the stratifications in the symplectic and orthogonal cases.\"\n ],\n [\n \"Geometry of the strata\",\n \"The arguments in this subsection are essentially the same as those appearing in §4, except for the complication coming from the topological property described in Theorem REF .\",\n \"Theorem 5.2 Consider a positive even integer $t = 2e \\\\le n(g-1)+3$ .\",\n \"A general point of $\\\\mathbb {A}_e$ corresponds to a bundle $V$ with $t(V) = t$ ; in particular, $\\\\mathcal {M}O_{X}(2n;t)$ is nonempty and irreducible.\",\n \"If $t < n(g-1)$ then $M(V)$ is a single point for general $V \\\\in \\\\mathcal {M}O_{X}(2n;t)$ .\",\n \"Let $E$ be a general bundle in $U(n, -e)^s$ for $e \\\\ge 1$ .\",\n \"As before, we will bound the dimension of the locus in $H^{1}(X, \\\\wedge ^{2}E)$ of extensions $V$ which admit a rank $n$ isotropic subbundle $F$ of degree $\\\\ge -e$ other than $E$ .\",\n \"Again, either $F$ is an elementary transformation of $E^*$ lifting to $V$ , or $F$ fits into a diagram of the form (REF ).\",\n \"Step 1.\",\n \"We first show that the latter situation does not arise for a general $V$ in $H^1(X, \\\\wedge ^2 E)$ for $t \\\\le n(g-1) - 1$ (not $t \\\\le n(g-1) + 1$ as in the symplectic case; see Remark REF ).\",\n \"For each fixed $H \\\\subset E$ , by Criterion REF (2), the locus of extensions fitting into a diagram of the form (REF ) for some $F$ of degree $\\\\ge -e$ is bounded by $ \\\\dim \\\\left( \\\\mathrm {Sec}^{e-h}\\\\mathrm {Gr}(2, E/H) \\\\right) + 1 + \\\\dim \\\\mathrm {Ker}\\\\left( q_{*} \\\\, ^tq^{*} \\\\right).\",\n \"$ As in the symplectic case, we obtain $e-h \\\\ge 0$ .\",\n \"Since $ \\\\dim \\\\mathrm {Gr}(2,E/H) = 2(n-r-2) + 1 = 2(n-r) - 3, $ the secant variety has dimension bounded by $ (e-h)(2(n-r) - 3) + (e-h) - 1 = 2(e-h)(n-r-1) - 1.\",\n \"$ By Proposition REF , we have $ \\\\dim \\\\mathrm {Ker}\\\\left(q_{*} \\\\, ^tq^{*} \\\\right) &= h^{1}(X, \\\\wedge ^{2}E) - h^{1} \\\\left(X, \\\\wedge ^{2}(E/H) \\\\right) \\\\\\\\&\\\\le h^{1}(X, \\\\wedge ^{2}E) - (n-r-1)(e-h) - \\\\frac{1}{2} (n-r)(n-r-1)(g-1).\",\n \"$ As before, the subbundle $H$ of $E$ varies in a Quot scheme of dimension $ h^{0}(X, \\\\mathrm {Hom}(H, E/H)) = nh - re - r(n-r)(g-1).\",\n \"$ Thus the dimension of the locus of extensions in $H^1(X, \\\\wedge ^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\\\ge -e$ is bounded by $\\\\dim \\\\left( \\\\mathrm {Sec}^{e-h} \\\\mathrm {Gr}(2, E/H) \\\\right) + 1 + \\\\dim \\\\mathrm {Ker}(q_* \\\\, ^t q^*) + h^0(X, \\\\mathrm {Hom}(H, E/H)) \\\\\\\\\\\\le \\\\ (e-h)(n-r-1) - \\\\frac{1}{2} (n-r)(n-r-1) (g-1) + nh-re + h^1(X, \\\\wedge ^2 E) \\\\\\\\= \\\\ e(n-r) - (e-h)(r+1) - \\\\frac{1}{2} (n-r)(n-r-1) (g-1) + h^1(X, \\\\wedge ^2 E).$ Since $e-h \\\\ge 0$ and we have assumed $e \\\\le \\\\frac{1}{2} (n (g-1)-1)$ , this is smaller than $h^1(X, \\\\wedge ^2 E)$ .\",\n \"Therefore, for $t = 2e \\\\le n (g-1) -1$ , a general $V \\\\in H^1(X, \\\\wedge ^2 E)$ does not admit a rank $n$ isotropic subbundle $F$ of degree $\\\\ge -e$ which fits into a diagram of the form (REF ).\",\n \"Step 2.\",\n \"Next we consider the case $r = 0$ , so $F$ is an elementary transformation of $E^*$ lifting to $V$ isotropically.\",\n \"Write $\\\\deg F = -f \\\\ge -e$ .\",\n \"By Criterion REF (3) and Lemma REF , we have $e+f \\\\equiv 0 \\\\mod {2}$ .\",\n \"By Criterion REF (2), the dimension of the locus of extensions in $H^1(X, \\\\wedge ^2 E)$ admitting such an $F$ is bounded by $\\\\dim \\\\left( \\\\mathrm {Sec}^{\\\\frac{1}{2}(e+f)} \\\\mathrm {Gr}(2, E) \\\\right) + 1 \\\\ \\\\le \\\\ \\\\frac{1}{2}(e+f) (2(n-2)+2) = (e+f)(n-1).$ If $f < \\\\frac{1}{2} n (g-1)$ , this bound is smaller than $h^1(X, \\\\wedge ^2 E) = (n-1)e + \\\\frac{1}{2}n(n-1)(g-1)$ so a general extension in $H^{1}(X, \\\\wedge ^{2}E)$ does not admit any such lifting.\",\n \"Step 3 of the proof of Theorem REF carries over, mutatis mutandis, to the orthogonal case, to prove statement (2), and (1) for $t \\\\le n(g-1)-1$ .\",\n \"To prove (1) for $n(g-1) \\\\le t \\\\le n(g-1)+3$ , we just repeat the arguments in Steps 1 and 2 under the assumption $-f \\\\ge -e + 2$ , to show that for general $E \\\\in U_{X}(n, -e)$ , a general $V \\\\in H^{1}(X, \\\\wedge ^{2}E)$ does not admit a rank $n$ isotropic subbundle $F$ of degree $\\\\ge -e + 2$ .\",\n \"Remark 5.3 Note that we do not prove an analogue of Theorem REF (3) for the orthogonal case.\",\n \"A dimension count analogous to that in the proof of Theorem REF (3) does not exclude the possibility that two maximal isotropic rank $n$ subbundles of a general bundle in $\\\\mathcal {M}O_{X}(2n)$ intersect in a line bundle, or in a rank 2 subbundle if $g = 2$ .\",\n \"It is unclear to us at this stage whether or not this in fact happens.\",\n \"Next, we compute the dimension of $M(V)$ for a general $V \\\\in \\\\mathcal {M}O_X(2n)^{\\\\pm }$ .\",\n \"We will use the following analogue of Lemma REF , which is proven in the same way as in the symplectic case: Lemma 5.4 For an orthogonal bundle $V \\\\in \\\\mathcal {M}O_X(2n)$ and a point $[\\\\eta : E \\\\subset V]$ of $M(V)$ , the Zariski tangent space of $M(V)$ at $\\\\eta $ is identified with $H^0(X, \\\\wedge ^2 E^*)$ .\",\n \"$\\\\Box $ Combining with Lemma REF (2), we obtain: Corollary 5.5 Let $t = 2e = n(g-1) + \\\\varepsilon $ with $\\\\varepsilon \\\\in \\\\lbrace 0,1,2,3 \\\\rbrace $ .\",\n \"Then for a general $V \\\\in \\\\mathcal {M}O_X(2n;t)$ , we have $ \\\\dim M(V) = \\\\frac{\\\\varepsilon (n-1)}{2}.\",\n \"$ $\\\\Box $ Proof of Theorem REF .\",\n \"(1) Consider the rational map $\\\\alpha _e: \\\\mathbb {A}_e \\\\rightarrow \\\\mathcal {M}O_X(2n)$ .\",\n \"For each $t = 2e = n(g-1) + \\\\varepsilon $ where $\\\\varepsilon \\\\in \\\\lbrace 0,1,2,3 \\\\rbrace $ , a direct computation shows that $\\\\dim M(V) = \\\\dim \\\\mathbb {A}_e - \\\\dim \\\\mathcal {M}O_X(2n).$ Since $\\\\dim M(V) \\\\ge \\\\dim \\\\alpha _e^{-1} (V)$ for any $V \\\\in \\\\mathcal {M}O_X(2n;2e)$ , this equality implies that $\\\\alpha _e$ is dominant to a component of $\\\\mathcal {M}O_X(2n)$ .\",\n \"This shows that for general $V$ in some one of the components $ \\\\mathcal {M}O_X(2n)^{\\\\pm }$ , we have $n(g-1) \\\\le t(V) \\\\le n(g-1) +3.$ By semicontinuity, $t(V) \\\\le n(g-1) +3$ for any orthogonal bundle $V$ .\",\n \"(2) The nonemptiness and irreducibility were proven in Theorem REF (1).\",\n \"The remaining part can be proven in the same way as Theorem REF (2).\",\n \"(3) This was proven in Theorem REF (2).\",\n \"$\\\\Box $\"\n ],\n [\n \"Configuration of the dense strata\",\n \"For each $2e = n(g-1 ) + \\\\varepsilon $ with $0 \\\\le \\\\varepsilon \\\\le 3$ , the locations of the images of the $\\\\alpha _{e}$ depend on the congruence class of $n(g-1)$ modulo 4: the trivial bundle of rank $2n$ is contained in $\\\\mathcal {M}O_X(2n)^+$ , hence for each $k$ , we have $\\\\mathcal {M}O_X(2n;4k) \\\\subset \\\\mathcal {M}O_X(2n)^+$ and $\\\\mathcal {M}O_X(2n;4k+2) \\\\subset \\\\mathcal {M}O_X(2n)^-$ .\",\n \"We may summarize the situation for the dense strata as follows: $ n(g-1) \\\\equiv 0 \\\\mod {4}: \\\\quad \\\\quad \\\\begin{array}{c|c|c} t & \\\\hbox{Component} & \\\\dim M(V) \\\\\\\\ \\\\hline n(g-1) & \\\\mathcal {M}O_{X}(2n)^{+} & 0 \\\\\\\\n(g-1) + 2 & \\\\mathcal {M}O_{X}(2n)^{-} & \\\\hbox{$\\\\frac{1}{2}(n-1)$} \\\\end{array} $ $ n(g-1) \\\\equiv 1 \\\\mod {4}: \\\\quad \\\\quad \\\\begin{array}{c|c|c} t & \\\\hbox{Component} & \\\\dim M(V) \\\\\\\\ \\\\hline n(g-1) + 1 & \\\\mathcal {M}O_{X}(2n)^{-} & \\\\hbox{$\\\\frac{1}{2}(n-1)$} \\\\\\\\n(g-1) + 3 & \\\\mathcal {M}O_{X}(2n)^{+} & \\\\hbox{$n-1$} \\\\end{array} $ $ n(g-1) \\\\equiv 2 \\\\mod {4}: \\\\quad \\\\quad \\\\begin{array}{c|c|c} t & \\\\hbox{Component} & \\\\dim M(V) \\\\\\\\ \\\\hline n(g-1) & \\\\mathcal {M}O_{X}(2n)^{-} & \\\\hbox{0} \\\\\\\\n(g-1) + 2 & \\\\mathcal {M}O_{X}(2n)^{+} & \\\\hbox{$\\\\frac{1}{2}(n-1)$} \\\\end{array} $ $ n(g-1) \\\\equiv 3 \\\\mod {4}: \\\\quad \\\\quad \\\\begin{array}{c|c|c} t & \\\\hbox{Component} & \\\\dim M(V) \\\\\\\\ \\\\hline n(g-1) + 1 & \\\\mathcal {M}O_{X}(2n)^{+} & \\\\hbox{$\\\\frac{1}{2}(n-1)$} \\\\\\\\n(g-1) + 3 & \\\\mathcal {M}O_{X}(2n)^{-} & \\\\hbox{$n-1$} \\\\end{array} $ Remark 5.6 In [3], symplectic bundles $W$ of rank four were studied which satisfy $s_{2}(W) < t(W)$ ; that is, whose maximal vector subbundles of half rank are all nonisotropic.\",\n \"Orthogonal bundles $V$ belonging to the top stratum $\\\\mathcal {M}O_{X}(2n;n(g-1)+2)$ provide a different example of this phenomenon.\",\n \"Due to the Hirschowitz bound [4], all such $V$ have a vector subbundle of degree at least $- \\\\left\\\\lceil \\\\frac{1}{2}n(g-1) \\\\right\\\\rceil $ , but no isotropic rank $n$ subbundle of this degree or greater.\",\n \"When $n(g-1)$ is even, such an orthogonal $V$ is nongeneral as a vector bundle (compare with Lange–Newstead [14]): Since any maximal rank $n$ subbundle $F \\\\subset V$ is nonisotropic, we must have $h^{0}(X, \\\\mathrm {Sym}^{2}F^{*}) > 0$ .\",\n \"But in this case $\\\\mu (\\\\mathrm {Sym}^{2}F^{*}) \\\\le g-1$ .\",\n \"By Lemma REF (2), this is a condition of positive codimension on $F$ , so none of the maximal vector subbundles of $V$ are general.\",\n \"Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-Gu, Seoul 143-701, Korea.\",\n \"E-mail: ischoe@konkuk.ac.kr Høgskolen i Oslo og Akershus, Postboks 4, St. Olavs plass, 0130 Oslo, Norway.\",\n \"E-mail: george.hitching@hioa.no\"\n ]\n]"},"id":{"kind":"string","value":"1204.0834"}}},{"rowIdx":1866793,"cells":{"text":{"kind":"list like","value":[["Entanglement Detection Using Majorization Uncertainty Bounds"],["Abstract Entanglement detection criteria are developed within the framework of the majorization formulation of uncertainty.","The primary results are two theorems asserting linear and nonlinear separability criteria based on majorization relations, the violation of which would imply entanglement.","Corollaries to these theorems yield infinite sets of scalar entanglement detection criteria based on quasi-entropic measures of disorder.","Examples are analyzed to probe the efficacy of the derived criteria in detecting the entanglement of bipartite Werner states.","Characteristics of the majorization relation as a comparator of disorder uniquely suited to information-theoretical applications are emphasized throughout."],["Introduction","Quantum measurements in general have indeterminate outcomes, with the results commonly expressed as a vector of probabilities corresponding to the set of outcomes.","In case of noncommuting observables, the joint indeterminacy of their measurement outcomes has an inviolable lower bound, in stark contrast to classical expectations.","This was discovered by Heisenberg in his desire to advance the physical understanding of the newly discovered matrix mechanics by relating the unavoidable disturbances caused by the act of measurement to the fundamental commutation relations of quantum dynamics [1].","Heisenberg's arguments relied on the statistical spread of the measured values of the observables to quantify uncertainty.","This gave rise to the variance formulation of the uncertainty principle which remains a powerful source of intuition on the structure and spectral properties of microscopic systems.","With the prospect of quantum computing and the development of quantum information theory in recent decades, on the other hand, the need for a measure of uncertainty that can better capture its information theoretical aspects, especially in dealing with noncanonical observables, has inspired new formulations.","Among these are the entropic measure developed in the eighties, and the majorization formulation proposed recently.","Uncertainty relations resulting from these formulations have found application to quantum cryptography, information locking, and entanglement detection, in addition to providing uncertainty limits [2], [3].","In this paper we develop applications of the majorization formulation of uncertainty introduced in Ref.","[3] to the problem of entanglement detection.","Deciding whether a given quantum state is entangled is a central problem of quantum information theory and known to be computationally intractable in general [4].","As a result, computationally tractable necessary conditions for separability, which provide a partial solution to this problem, have been the subject of active research in recent years.","Among these, the Peres-Horodecki positive partial transpose criterion actually provides necessary and sufficient separability conditions for $2\\otimes 2$ and $2\\otimes 3$ dimensional systems, and necessary conditions otherwise.","Other notable results are the reduction and global versus local disorder criteria, both necessary conditions in general [5].","An observable that has non-negative expectation values for all separable states and negative ones for a subset of entangled states provides an operational method of entanglement detection and is known as an entanglement witness [6].","It has also long been known that uncertainty relations can serve a similar purpose by providing inequalities that must be satisfied by separable states and if violated signal entanglement [7], [8].","In this paper we extend the majorization formulation of uncertainty developed in Ref.","[3] to the problem of entanglement detection.","As discussed in that paper (hereafter referred to as Paper I) and in the following, majorization as a comparator of uncertainty is qualitatively different from, and stronger than, scalar measures of uncertainty.","Consequently, the entanglement detection results developed here represent a qualitative strengthening of the existing variance and entropic results.","This will be evident, among other things, by the fact that they yield, as corollaries, an infinite class of entanglement detectors based on scalar measures.","As mentioned above, results of quantum measurements are in general probability vectors deduced from counter statistics, and an information theoretical formulation of uncertainty is normally based on an order of uncertainty defined on such vectors (see Paper I).","Thus a scalar measure of uncertainty is commonly a real, non-negative function defined on probability vectors whose value serves to define the uncertainty in question.","For example, the Shannon entropy function is the measure of uncertainty for the standard entropic formulation of uncertainty [9].","By contrast, majorization provides a partial order of uncertainty on probability vectors that is in general more stringent, and fundamentally stronger, than a scalar measure [10].","Note that, unlike scalar measures, majorization does not assign a quantitative measure of uncertainty to probability vectors, and as a partial order may find a pair of vectors to be incomparable.","A characterization of majorization that clarifies the foregoing statements can be attained by considering the quasi-entropic set of measures.","These were defined in Paper I as the set of concave, symmetric functions defined on probability vectors, and include the Shannon, Tsallis, and (a subfamily of) Rényi entropies as special cases [11].","It is important to realize that the uncertainty order determined by one member of the quasi-entropic set for a given pair of probability vectors may contradict that given by another.","While additional considerations may justify the use of, e.g., Shannon entropy in preference to the others, the foregoing observation clearly indicates the relative nature of the uncertainty order given by a specific measure, and immediately raises the following question: are there pairs of vectors for which all quasi-entropic measures determine the same uncertainty order?","The answer is yes, and the common determination of the quasi-entropic set in such cases defines the uncertainty order given by majorization.","What about the cases where there are conflicting determinations by the members of the quasi-entropic set?","The majorization relation defines such pairs as incomparable, whence the “partial” nature of the order defined by majorization.","We may therefore consider the majorization order to be equivalent to the collective order determination of the entire quasi-entropic set (see §IV).","This characterization of the majorization relation clearly shows its standing vis-à-vis the scalar measures of uncertainty.","More practical definitions of the majorization relation will be considered in §II.","The rest of this paper is organized as follows.","In §II we review elements of the majorization formulation of uncertainty needed for our work and establish the notation.","In §III we present the central results of this paper on entanglement detection, and in §IV we derive entire classes of entanglement detectors for quasi-entropic measures as corollaries to the theorems of §III.","Concluding remarks are presented in §V.","Details of certain mathematical proofs are given in the Appendix."],["Majorization formulation of uncertainty","This section presents a review of the basic elements of majorization theory and the formulation of the uncertainty principle based on it.","It follows the treatment given in Paper I."],["Majorization","The basic element of our formulation is the majorization protocol for comparing the degree of uncertainty, or disorder, among probability vectors, i.e., sequences of non-negative numbers summing to unity [10].","By definition, ${\\lambda }^{1}$ is no less uncertain than ${\\lambda }^{2}$ if ${\\lambda }^{1}$ equals a mixture of the permutations of ${\\lambda }^{2}$ .","Then ${\\lambda }^{1}$ is said to be majorized by ${\\lambda }^{2}$ and written ${\\lambda }^{1}\\prec {\\lambda }^{2}$ .","An equivalent definition that flushes out the details of the foregoing is based on the vector ${\\lambda }^{\\downarrow }$ which is obtained from $\\lambda $ by arranging the components of the latter in a nonincreasing order.","Then, ${\\lambda }^{1}\\prec {\\lambda }^{2}$ if ${\\sum }_{i}^{j} {\\lambda }^{1\\downarrow }_{i} \\le {\\sum }_{i}^{j} {\\lambda }^{2\\downarrow }_{i} $ for $j=1,2, \\ldots d-1$ , where $d$ is the larger of the two dimensions and trailing zeros are added where needed.","As stated earlier, the majorization relation is a partial order, i.e., not every two vectors are comparable under majorization.","As suggested in §I, two vectors are found to be incomparable when the difference in their degrees of uncertainty does not rise to the level required by majorization.","As evident from the second definition given above, majorization requires the satisfaction of $N-1$ inequalities if the number of non-zero components of the majorizing vector is $N$ .","This accounts for the strength of the majorization relation as compared to scalar measures of disorder.","Indeed as alluded to in §I, for any quasi-entropic function $F(\\lambda )$ , ${\\lambda }^{1} \\prec {\\lambda }^{2}$ implies $F({\\lambda }^{1}) \\ge F({\\lambda }^{2})$ , but not conversely.","On the other hand, if for every quasi-entropic function $F(\\lambda )$ we have $F({\\lambda }^{1}) \\ge F({\\lambda }^{2})$ , then ${\\lambda }^{1} \\prec {\\lambda }^{2}$ [10].","To establish uncertainty bounds, we need to characterize the greatest lower bound, or infimum, and the least upper bound, or supremum, of a set of probability vectors [12].","The infimum is defined as the vector that is majorized by every element of the set and in turn majorizes any vector with that property .","The supremum is similarly defined.","We will briefly outline the construction of the infimum here and refer the reader to Paper I for further details.","Given a set of probability vectors $\\lbrace {\\lambda }^{a} {\\rbrace }_{a=1}^{N}$ , consider the vector ${\\mu }^{inf}$ defined by ${\\mu }_{0}^{inf}=0&,\\,\\,\\,{\\mu }_{j}^{inf}=\\min \\big ( {\\sum }_{i=1}^{j} {\\lambda }^{1\\downarrow }_{i},{\\sum }_{i=1}^{j} {\\lambda }^{2\\downarrow }_{i},\\ldots , \\nonumber \\\\&{\\sum }_{i=1}^{j}{\\lambda }^{N\\downarrow }_{i} \\big ), \\,\\,\\, 1 \\le j \\le {d}_{max}, $ where ${d}_{max}$ is the largest dimension found in the set.","The infimum is then given by ${\\lambda }^{inf}_{i}={[\\inf ({\\lambda }^{1},{\\lambda }^{2}, \\ldots , {\\lambda }^{N})]}_{i}={\\mu }_{i}^{inf}-{\\mu }_{i-1}^{inf}, $ where $1 \\le i \\le {d}_{max}$ .","The construction of the supremum starts with ${\\mu }^{sup}$ in parallel with Eq.","(REF ), but with “max” replacing “min,” and may require further steps detailed in Paper I.","It is worth noting here that the infimum (supremum) of a pair of probability vectors will in general be more (less) disordered than either.","Important special cases are (i) one of the two majorizes the other, in which case the latter is the infimum and the former the supremum of the two, and (ii) the two are equal, in which case either is both the infimum and the supremum.","Furthermore, a useful qualitative rule is that the more “different” are two probability vectors, the further will the infimum or supremum of the two be from at least one of them.","Finally, we note that while the infimum or supremum of a set of probability vectors always exists, it need not be a member of the set."],["Measurement and uncertainty","A generalized measurement may be defined by a set of positive operators $\\lbrace {\\hat{\\mathrm {E}}}_{\\alpha } \\rbrace $ called measurement elements and subject to the completeness condition ${\\sum }_{\\alpha }{\\hat{\\mathrm {E}}}_{\\alpha } = \\hat{\\mathbb {1}}$ .","The probability that outcome $\\alpha $ turns up in a measurement of the state $\\hat{\\rho }$ is given by the Born rule ${P}_{\\alpha }(\\hat{\\rho })=\\textrm {tr} [\\hat{\\mathrm {E}}_{\\alpha } \\hat{\\rho } ]$ .","A generalized measurement can always be considered to be the restriction of a more basic type, namely a projective measurement, performed on an enlarged system to the system under generalized measurement [13].","A projective measurement is usually associated with an observable of the system represented by a self-adjoint operator $\\hat{M}$ , and entails a partitioning of the spectrum of $\\hat{M}$ into a collection of subsets $\\lbrace {b}_{\\alpha }^{M} \\rbrace $ called measurement bins.","We call a projective measurement maximal if each bin consists of a single point of the spectrum of the measured observable.","Having assembled the necessary concepts, we can now characterize uncertainty by means of majorization relations in a natural manner.","To start, we define the probability vector ${P}^{X}(\\hat{\\rho })$ resulting from a measurement $\\mathrm {X}$ on a state $\\hat{\\rho }$ to be uncertain if it is majorized by $\\mathcal {I}=(1,0,\\ldots ,0)$ but not equal to it.","As such, ${P}^{X}(\\hat{\\rho })$ is said to be strictly majorized by $\\mathcal {I}$ and written ${P}^{X}(\\hat{\\rho })\\prec \\prec \\mathcal {I}$ .","Similarly, given a pair of measurements $\\mathrm {X}$ and $\\mathrm {Y}$ on a state $\\hat{\\rho }$ , we say ${P}^{\\mathrm {X}}(\\hat{\\rho })$ is more uncertain, equivalently more disordered, than ${P}^{\\mathrm {Y}}(\\hat{\\rho })$ if ${P}^{\\mathrm {X}}(\\hat{\\rho }) \\prec {P}^{\\mathrm {Y}}(\\hat{\\rho })$ .","Further, we define the joint uncertainty of a pair of measurements $\\mathrm {X}$ and $\\mathrm {Y}$ to be the outer product ${P}^{\\mathrm {X}}\\otimes {P}^{\\mathrm {Y}}$ , i.e., ${P}_{\\alpha \\beta }^{\\mathrm {X} \\oplus \\mathrm {Y}}={P}^{\\mathrm {X}}_{\\alpha } {P}^{\\mathrm {Y}}_{\\beta }$ .","Since $H({P}^{\\mathrm {X}}\\otimes {P}^{\\mathrm {Y}})=H({P}^{\\mathrm {X}})+H({P}^{\\mathrm {Y}})$ , where $H(\\cdot )$ is the Shannon entropy function, this definition is seen to be consistent with its entropic counterpart.","As stated earlier, ${P}^{\\mathrm {X}} \\prec {P}^{\\mathrm {Y}}$ implies $H({P}^{\\mathrm {X}}) \\ge H({P}^{\\mathrm {Y}})$ but not conversely.","These definitions naturally extend to an arbitrary number of states and measurements.","We are now in a position to state the majorization statement of the uncertainty principle established in Paper I: “The joint results of a set of generalized measurements of a given state are no less uncertain than a probability vector that depends on the measurement set but not the state, and is itself uncertain unless the measurement elements have a common eigenstate.\"","In symbols, ${P}^{\\mathrm {X}}(\\hat{\\rho })\\otimes {P}^{\\mathrm {Y}}(\\hat{\\rho }) \\otimes \\ldots \\otimes {P}^{\\mathrm {Z}}(\\hat{\\rho }) \\prec {{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}} \\prec \\prec \\mathcal {I}, $ where ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}={\\sup }_{\\hat{\\rho }} [{P}^{\\mathrm {X}}(\\hat{\\rho })\\otimes {P}^{\\mathrm {Y}}(\\hat{\\rho }) \\otimes \\ldots \\otimes {P}^{\\mathrm {Z}}(\\hat{\\rho })] , $ unless the measurement elements $\\lbrace \\hat{\\mathrm {E}}^{\\mathrm {X}}, \\hat{\\mathrm {E}}^{\\mathrm {Y}}, \\ldots , \\hat{\\mathrm {E}}^{\\mathrm {Z}} \\rbrace $ have a common eigenstate in which case ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}} = \\mathcal {I}$ .","Note that ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ is the majorization uncertainty bound for the measurement set considered."],["majorization uncertainty bounds","As stated above, the uncertainty bound ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ given by Eq.","(REF ) depends on the measurement set but not the state of the system.","As the supremum of all possible measurement outcomes, it is the probability vector that sets the irreducible lower bound to uncertainty for the set.","As such, it is the counterpart of the variance product or entropic lower bound in the traditional formulations of the uncertainty principle.","Unlike the latter, however, ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ is in general not realizable or even approachable by any state of the system.","Therefore, there is in general no such thing as a “minimum uncertainty state” within the majorization framework.","This is a consequence of the fact, mentioned earlier, that the infimum or supremum of a set of vectors need not be a member of the set.","An obvious special case is the trivial example of zero uncertainty for which ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}= \\mathcal {I}$ , signaling the existence of a common eigenstate for the measurement elements.","Intuitively, we expect that mixing states can only increase their uncertainty, as is known to be the case for scalar measures of uncertainty.","In case of majorization, this expectation is manifested in the property that the uncertainty bound ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ is in general realized on the class of pure states.","Indeed for two and three mutually unbiased observables on a two-dimensional Hilbert space considered, we found in Paper I that the required maxima for the components of ${\\mu }^{sup}$ that serve to define the uncertainty bound are reached on pure states.","More specifically, as outlined in §IIA above and detailed in §IIB, IVA and IVB of Paper I, the calculation of ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ involves a component-wise determination of ${\\mu }^{sup}$ by a series of maximizations over all possible density matrices $\\hat{\\rho }$ .","The property in question then guarantees that the maximization process can be limited to density matrices representing pure states only.","We will prove this assertion in general, as well as for the class of separable states, in the Appendix.","As discussed earlier, an entanglement detector can be effective in deciding whether a given density matrix is separable by providing a condition that is satisfied by all separable states and if violated signals entanglement.","A large body of entanglement detection strategies have been developed in recent years which are primarily based on scalar conditions, including those based on variance and entropic type uncertainty relations, which can be found in Refs.","[5], [6], [7], [8].","Here we shall develop majorization conditions for entanglement detection based on the formulation of measurement uncertainty given in Paper I and outlined in §II above.","In particular, the entanglement condition given in Theorem 1 below is linear and susceptible to experimental implementation, so it can be formulated as an entanglement witness.","Nonlinear detectors developed below, on the other hand, rely on majorization-based uncertainty relations.","We will also introduce the notion of subsystem disorder and a sharpened version of the Nielsen-Kempe [14] separability condition as a nonlinear entanglement detector.","As majorization relations, our results in general entail multiple inequalities whose number will grow with the uncertainty levels involved.","As discussed in §IIA, this is an important feature of the majorization relation as comparator of disorder, one that sets it apart from scalar conditions and provides for the refinement needed in comparing highly disordered vectors.","As will be seen in §IV, this property has the consequence that each majorization condition yields a scalar condition for the entire quasi-entropic class of uncertainty measures."],["Linear detectors","Our detection strategy is based on the intuitive expectation that the measurement uncertainty bound for the class of separable states of a multipartite system must be majorized by the corresponding bound for all states, and that this hierarchy can be exploited for entanglement detection.","This is the majorization rendition of the strategy often used to derive separability conditions [7], [8].","We will first consider the case of one generalized measurement resulting in a linear detection condition.","Theorem 1.","Let the results of the generalized measurement ($\\mathrm {X}, \\lbrace \\hat{\\mathrm {E}}_{\\alpha }^{X} \\rbrace $ ) on a multipartite state ${\\hat{\\rho }}^{ABC \\dots F}$ of parties $(A, B, C, \\dots , F)$ defined on a finite-dimensional Hilbert space be bounded by ${{P}}_{sup}^{\\mathrm {X}}={\\sup }_{{\\hat{\\rho }}^{ABC \\dots F}}{{P}}^{\\mathrm {X}}({\\hat{\\rho }}^{ABC \\dots F}), $ and in the case of a separable state ${\\hat{\\rho }}^{ABC \\dots F}_{sep}$ by ${{P}}_{sep;sup}^{\\mathrm {X}}={\\sup }_{{\\hat{\\rho }}^{ABC \\dots F}_{sep}}{{P}}^{\\mathrm {X}}({\\hat{\\rho }}^{ABC \\dots F}_{sep}).","$ Then, (i) the suprema in the foregoing pair of equations may be taken over the class of pure and pure, product states, respectively, and (ii) given an arbitrary state ${\\hat{\\sigma }}^{ABC \\dots F}$ , the condition ${{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F}) \\nprec {{P}}_{sep;sup}^{\\mathrm {X}}$ implies that ${\\hat{\\sigma }}^{ABC \\dots F}$ is entangled [15].","Part (i) of Theorem 1 is the majorization version of the familiar result that uncertainty bounds are realized on pure states and is proved in the Appendix [16].","Part (ii) is the statement that the bound in Eq.","(REF ) is a necessary condition for separability and its violation signals the existence of entanglement.","As noted above, Theorem 1 provides an operational method of entanglement detection and can be reformulated as a set of entanglement witnesses.","Clearly, the efficacy of Theorem 1 in detecting entanglement depends on the choice of the measurement ${\\mathrm {X}}$ for a given quantum state.","This suggests looking for the optimum measurement, in parallel with the optimization problem for entanglement witnesses [17].","Note that here the choice of the generalized measurement ${\\mathrm {X}}$ affects both ${{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F})$ and ${{P}}_{sep;sup}^{\\mathrm {X}}$ , thus making the task of finding a fully optimized solution rather difficult.","It is therefore fortunate that we can achieve a partial optimization on the basis of Theorem 3 of Paper I, the relevant content of which we can state as follows: Lemma Under the conditions of Theorem 1 and with ${\\mathrm {X}}$ restricted to rank 1 measurements, we have ${{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F}) \\prec {{P}}^{{\\mathrm {X}}^{\\star }}({\\hat{\\sigma }}^{ABC \\dots F})={\\lambda }({\\hat{\\sigma }}^{ABC \\dots F}), $ where ${\\mathrm {X}}^{\\star }$ is a maximal projective measurement whose elements are the set of rank 1 orthogonal projection operators onto the eigenvectors of ${\\hat{\\sigma }}^{ABC \\dots F}$ and ${\\lambda }({\\hat{\\sigma }}^{ABC \\dots F})$ is the corresponding spectrum.","It should be noted here that while the choice of ${{\\mathrm {X}}^{\\star }}$ succeeds in optimizing ${{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F})$ which appears in part (ii) of Theorem 1, its effect on ${{P}}_{sep;sup}^{\\mathrm {X}}$ , the other object appearing therein, is unknown and may result in a poor overall choice.","Moreover, any degeneracy in the spectrum of ${\\hat{\\sigma }}^{ABC \\dots F}$ renders ${{\\mathrm {X}}^{\\star }}$ nonunique and subject to further optimization.","These features will be at play in the example considered below where, the foregoing caveats notwithstanding, the resulting detector turns out to be fully optimal.","The example in question is the Werner state ${\\hat{\\rho }}^{wer}_{d}(q)$ for a bipartite system of two $d$ -level parties $A$ and $B$ defined on a $d^2$ -dimensional Hilbert space [18], [19] ${\\hat{\\rho }}^{wer}_{d}(q)=\\frac{1}{d^2}({1-q})\\hat{{\\mathbb {1}}} + q \\mid {\\mathfrak {B}}_{1} \\rangle \\langle {\\mathfrak {B}}_{1}\\mid , $ where $\\mid {\\mathfrak {B}}_{1} \\rangle \\ =\\frac{1}{\\sqrt{d}} {\\sum }_{j=0}^{d-1} \\mid A,j \\rangle \\otimes \\mid B,j \\rangle .","$ Here $\\lbrace \\mid A,j \\rangle \\rbrace $ and $\\lbrace \\mid B,j \\rangle \\rbrace $ ) are orthonormal bases for subsystem $A$ and $B$ respectively.","Also, here and elsewhere we use the symbol $\\hat{{\\mathbb {1}}}$ to denote the identity operator of dimension appropriate to the context.","Note that $\\mid {\\mathfrak {B}}_{1} \\rangle $ is the generalization of the two-qubit Bell state $(\\mid 00 \\rangle + \\mid 11 \\rangle )/\\sqrt{2}$ .","As such, it is totally symmetric, as well as maximally entangled in the sense that its marginal states are maximally disordered and equal to $\\hat{{\\mathbb {1}}}/d$ .","An important fact to be used in the following is that ${\\hat{\\rho }}^{wer}_{d}(q)$ is separable if and only if $q \\le (1+d)^{-1}$ [19].","In order to probe the separability of ${\\hat{\\rho }}^{wer}_{d}(q)$ by means of Theorem 1, we must first choose a measurement.","We will use the above Lemma to guide our choice of the putative optimum measurement ${\\mathrm {X}}^{\\star }$ .","The Lemma requires that the measurement elements be equal to the orthogonal projections onto the eigenvectors of ${\\hat{\\rho }}^{wer}_{d}(q)$ .","However, a simple calculation shows that the spectrum of ${\\hat{\\rho }}^{wer}_{d}(q)$ is degenerate, consisting of a simple eigenvalue equal to $q+{d}^{-2}(1-q)$ and a $({d}^{2}-1)$ -fold degenerate eigenvalue equal to ${d}^{-2}(1-q)$ .","A natural choice of basis for the degenerate subspace is a set of $d^2-1$ generalized Bell states which, together with $\\mid {\\mathfrak {B}}_{1} \\rangle $ of Eq.","(REF ), constitute the orthonormal eigenbasis ${\\lbrace \\mid {\\mathfrak {B}}_{\\alpha } \\rangle \\rbrace }_{\\alpha =1}^{d^2}$ for ${\\hat{\\rho }}^{wer}_{d}(q)$ .","This generalized Bell basis is thus characterized by the fact that each $\\mid {\\mathfrak {B}}_{\\alpha } \\rangle $ is maximally entangled and possesses equal Schmidt coefficients.","The measurement elements of the optimal measurement ${\\mathrm {X}}^{\\star }$ are thus given by ${\\hat{\\mathrm {E}}}_{\\alpha }^{{X}^{\\star }}= \\mid {\\mathfrak {B}}_{\\alpha } \\rangle \\langle {\\mathfrak {B}}_{\\alpha } \\mid $ , $\\alpha =1, 2, \\dots , d^2$ .","With ${\\mathrm {X}}^{\\star }$ so defined, we readily find ${{P}}^{{\\mathrm {X}}^{\\star }}[{\\hat{\\rho }}^{wer}_{d}(q)]=&[q+{d}^{-2}(1-q), {d}^{-2}(1-q), \\nonumber \\\\&{d}^{-2}(1-q), \\ldots , {d}^{-2}(1-q)], $ which corresponds to the spectrum of ${\\hat{\\rho }}^{wer}_{d}(q)$ as expected.","Note that the Lemma guarantees that any other rank 1 measurement in place of ${\\mathrm {X}}^{\\star }$ would produce a more disordered probability vector on the right-hand side of Eq.","(REF ).","The next step is the calculation of ${{P}}_{sep;sup}^{{\\mathrm {X}}^{\\star }}$ of Theorem 1, which is the least disordered probability vector that can result from the measurement of ${\\mathrm {X}}^{\\star }$ on a pure, product state of two $d$ -level subsystems.","As can be seen in Eq.","(REF ) of the Appendix, this calculation requires finding the maximum overlap of every measurement element $\\mid {\\mathfrak {B}}_{\\alpha } \\rangle \\langle {\\mathfrak {B}}_{\\alpha } \\mid $ with the class of pure, product states.","This overlap is given by the square of the largest Schmidt coefficient of the respective Bell state $\\mid {\\mathfrak {B}}_{\\alpha }\\rangle $ [20].","Since all Schmidt coefficients are equal to $1/\\sqrt{d}$ for every Bell state $\\mid {\\mathfrak {B}}_{\\alpha }\\rangle $ , we can conclude that ${{P}}_{sep;sup}^{{\\mathrm {X}}^{\\star }}=(1/d, 1/d, \\ldots , 1/d,0,0, \\ldots ,0).","$ We are now in a position to apply the entanglement criterion of Theorem 1, which reads ${{P}}^{\\mathrm {{X}^{\\star }}}[{\\hat{\\rho }}^{wer}_{d}(q)] \\nprec {{P}}_{sep;sup}^{\\mathrm {{X}^{\\star }}}$ in this instance [21].","Using the information in Eqs.","(REF ) and (REF ), we readily see that this criterion translates to the single inequality $q+{d}^{-2}(1-q) > 1/d$ , or $q > 1/(1+d)$ , which is the necessary and sufficient condition for the inseparability of the Werner state ${\\hat{\\rho }}^{wer}_{d}(q)$ .","Thus every entangled Werner state of two $d$ -level systems is detected by the measurement ${\\mathrm {X}}^{\\star }$ defined above.","While the efficacy of the generalized Bell states for detecting entanglement in Werner states is well established [5], [6], [7], [8], we note its natural emergence from the general majorization results of this subsection.","It should also be noted that Theorem 1 and the foregoing analysis can be extended to multipartite states."],["Nonlinear detectors","As an example of nonlinear entanglement detectors, we will derive majorization conditions for separability based on uncertainty relations.","The method we will follow relies on the degeneracy properties of projective measurements on a single system versus the products of such measurements on a multipartite system [7], [8].","For example, with measurement ${\\mathrm {X}}^{A}$ on ${\\hat{\\rho }}^{A}$ and ${\\mathrm {X}}^{B}$ on ${\\hat{\\rho }}^{B}$ and each having a simple spectrum, the product projective measurement ${\\mathrm {X}}^{A} \\otimes {\\mathrm {X}}^{B}$ on the bipartite system ${\\hat{\\rho }}^{AB}$ may be degenerate.","A specific case is ${{\\hat{\\sigma }}_{x}}^{A}\\otimes {{\\hat{\\sigma }}_{x}}^{B}$ , whose spectrum $(+1/4,+1/4,-1/4,-1/4)$ is doubly degenerate, versus ${{\\hat{\\sigma }}_{x}}^{A}$ or ${{\\hat{\\sigma }}_{x}}^{B}$ each of which has the simple spectrum $(+1/2,-1/2)$ .","Note the fact that here we are following common practice by defining ${{\\hat{\\sigma }}_{x}}^{A}\\otimes {{\\hat{\\sigma }}_{x}}^{B}$ to be the projective measurement whose two elements project into the subspaces corresponding to the eigenvalues $+1/4$ and $-1/4$ (and not the rank 1 projective measurement that resolves the degeneracies by virtue of using all four product elements).","Consequently, it may happen that ${\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}$ and ${\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}$ have a common eigenstate while ${\\mathrm {X}}^{A}$ and ${\\mathrm {Y}}^{A}$ do not, and that the said common eigenstate is an entangled pure state.","In such cases, the corresponding probability vectors will reflect the stated differences and may be capable of detecting entanglement as in the case of linear detectors.","Consider two product projective measurements ${\\mathrm {X}}^{A} \\otimes {\\mathrm {X}}^{B}$ and ${\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}$ performed on the bipartite state ${\\hat{\\rho }}^{AB}$ .","The measurement results are then given by ${{P}}^{{\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}}({\\hat{\\rho }}^{AB})= \\textrm {tr}&[{\\hat{\\Pi }}^{{\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}} {\\hat{\\rho }}^{AB} ], \\nonumber \\\\ {{P}}^{{\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}}({\\hat{\\rho }}^{AB})= \\textrm {tr}&[{\\hat{\\Pi }}^{{\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}} {\\hat{\\rho }}^{AB} ], \\nonumber \\\\{P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\rho }}^{AB}) &={P}^{ {\\mathrm {X}}^{A} \\otimes {\\mathrm {X}}^{B} }({\\hat{\\rho }}^{AB})\\nonumber \\\\\\otimes {P}^{{\\mathrm {Y}}^{A} \\otimes {\\mathrm {Y}}^{B} }({\\hat{\\rho }}^{AB}), $ where ${\\hat{\\Pi }}^{{\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}}$ and ${\\hat{\\Pi }}^{{\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}}$ , both projection operators but not necessarily rank 1, represent the measurement elements of the two product measurements.","Note the majorization definition of joint uncertainty, given earlier, at work on the last line of Eq.","(REF ).","Under these conditions, we have the following general result.","Theorem 2.","The results of measurements ${\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}$ and ${\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}$ on a separable state ${\\hat{\\rho }}^{AB}_{sep}$ satisfy ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\rho }}^{AB}_{sep}) \\prec {P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}, $ where ${P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}$ is defined in Eq.","(REF ), hence the condition ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\sigma }}^{AB}) \\nprec {P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}$ implies that the state ${\\hat{\\sigma }}^{AB}$ is entangled.","To establish Theorem 2, consider product states of the form ${\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}$ .","Then Lemma 1 of Ref.","[8] asserts that ${P}^{ {\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B} }({\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}) &\\prec {P}^{ {\\mathrm {X}}^{A}}({\\hat{\\rho }}^{A}), \\nonumber \\\\{P}^{ {\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B} }({\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}) &\\prec {P}^{ {\\mathrm {Y}}^{A}}({\\hat{\\rho }}^{A}).","$ Using these relations and Eq.","(REF ), we find [22] ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}) \\prec {P}^{ {\\mathrm {X}}^{A}}({\\hat{\\rho }}^{A})\\otimes {P}^{ {\\mathrm {Y}}^{A}}({\\hat{\\rho }}^{A}).","$ Since the right-hand side of Eq.","(REF ) is by definition majorized by ${P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}$ , we conclude that ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}) \\prec {P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}.","$ At this point we note that ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}_{sep;sup}$ , which by definition majorizes all probability vectors that can appear on the left-hand side of Eq.","(REF ), can be found among pure, product states [16].","This assertion is established in Eq.","(REF ) et seq.","of the Appendix.","Consequently, the product state ${\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}$ in Eq.","(REF ) may be replaced by any separable state ${\\hat{\\rho }}^{AB}_{sep}$ , thereby establishing Eq.","(REF ) and Theorem 2.","We note in passing that, because the measurements considered in Theorem 2 are in general not rank 1, the Lemma which we used earlier for detector optimization cannot be applied here.","To illustrate Theorem 2, we will consider the case of three mutually unbiased observables measured on bipartite states of two-level systems, as in §IVB of Paper I.","In effect, this amounts to measuring products of the three spin components of a pair of spin-1/2 systems (or qubits).","For this case, Eq.","(REF ) reads ${P}^{ ({{\\sigma }_{x}}^{A}\\otimes {{\\sigma }_{x}}^{B}) \\oplus ({{\\sigma }_{y}}^{A}\\otimes {{\\sigma }_{y}}^{B})\\oplus ({{\\sigma }_{z}}^{A}\\otimes {{\\sigma }_{z}}^{B})}({\\hat{\\rho }}^{AB}_{sep})\\prec {P}_{sup}^{{\\sigma }_{x}\\oplus {\\sigma }_{y}\\oplus {\\sigma }_{z}}, $ where ${\\hat{\\rho }}^{AB}_{sep}$ is any separable state of two qubits.","Thus a violation of this relation by a two-qubit state implies that it is entangled.","For the set of states ${\\hat{\\rho }}^{AB}$ to be probed by Theorem 2, we will consider the Werner family of two-qubit states ${\\hat{\\rho }}^{wer}(q)=\\frac{1}{4}({1-q}){\\mathbb {1}}+ q \\mid {\\mathfrak {B}}_{1} \\rangle \\langle {\\mathfrak {B}}_{1} \\mid , $ which is known to be entangled if and only if $q >1/3$ .","Here $0 \\le q \\le 1$ and $\\mid {\\mathfrak {B}}_{1} \\rangle =(\\mid 00 \\rangle + \\mid 11 \\rangle )/\\sqrt{2}$ is the totally symmetric Bell state of Eq.","(REF ) for the present case.","A calculation of the probability vector for this measurement gives $(1 \\pm q)(1 \\pm q)(1 \\pm q)/8$ for the 8 components of ${P}^{({{\\sigma }_{x}}^{A}\\otimes {{\\sigma }_{x}}^{B}) \\oplus ({{\\sigma }_{y}}^{A}\\otimes {{\\sigma }_{y}}^{B})\\oplus ({{\\sigma }_{z}}^{A}\\otimes {{\\sigma }_{z}}^{B})}[{\\hat{\\rho }}^{wer}(q)]$ .","Arranged in a non-ascending order, these are ${(1+q)}^{3}/8$ , ${(1+q)}^{2}{(1-q)}/8$ and ${(1+q)}{(1-q)}^{2}/8$ both three-fold degenerate, and ${(1-q)}^{3}/8$ .","The supremum ${P}_{sup}^{{\\sigma }_{x}\\oplus {\\sigma }_{y}\\oplus {\\sigma }_{z}}$ , on the other hand, was calculated in §IVB of Paper I, where it was found as ${P}_{sup}^{{\\sigma }_{x}\\oplus {\\sigma }_{y}\\oplus {\\sigma }_{z}}=\\frac{1}{8}\\big [{(1+1/\\sqrt{3})}^{3},2{(1+1/\\sqrt{2})}^{2} \\nonumber \\\\-{(1+1/\\sqrt{3})}^{3}, 4-{(1+1/\\sqrt{2})}^{2}, 4-{(1+1/\\sqrt{2})}^{2}\\nonumber \\\\,0,0,0,0 \\big ].","$ According to Theorem 2, ${\\hat{\\rho }}^{wer}(q)$ violates the separability condition of Eq.","(REF ) if ${P}_{sup}^{{\\sigma }_{x}\\oplus {\\sigma }_{y}\\oplus {\\sigma }_{z}}$ fails to majorize ${P}^{({{\\sigma }_{x}}^{A}\\otimes {{\\sigma }_{x}}^{B}) \\oplus ({{\\sigma }_{y}}^{A}\\otimes {{\\sigma }_{y}}^{B})\\oplus ({{\\sigma }_{z}}^{A}\\otimes {{\\sigma }_{z}}^{B})}[{\\hat{\\rho }}^{wer}(q)]$ , which occurs for $q > 1/\\sqrt{3}=0.577$ in this example.","The entanglement in ${\\hat{\\rho }}^{wer}(q)$ is thus detected by Theorem 2 for $q > 0.577$ , and missed for the range $0.333 < q \\le 0.577$ .","By comparison, a similar method based on the Shannon entropy in Ref.","[7] detects entanglement in ${\\hat{\\rho }}^{wer}(q)$ for $q >0.65$ .","Reference [8], using the same method but relying on the family of Tsallis entropies, matches our condition $q > 0.577$ by means of a numerical calculation that searches over the Tsallis family for optimum performance.","We end this subsection by a brief description of a sharpened version of the Nielsen-Kempe theorem as a nonlinear entanglement detector [12].","This celebrated theorem is an elegant separability condition based on majorization relations.","It asserts that the spectrum of a separable bipartite state is majorized by each of its marginal spectra.","By extension, this theorem guarantees that the spectra of all possible subsystems of a separable multipartite system must majorize its global spectrum [14].","Our version of this theorem employs the notion of the infimum of a set of probability vectors discussed in §IIA, and is based on the observation that if a probability vector is majorized by each member of a set of probability vectors, then it must also be majorized by the infimum of that set.","Consider the multipartite state ${\\hat{\\rho }}^{ABC \\dots F}$ together with all its subsystem states ${\\lbrace {\\hat{\\rho }}^{{X}_{a}} \\rbrace }_{a=1}^{f}$ and subsystem spectra ${\\lbrace {\\lambda }^{{X}_{a}} \\rbrace }_{a=1}^{f}$ , where ${\\lbrace {{X}_{a}} \\rbrace }_{a=1}^{f}$ represent all proper subsets of the set of parties $(A, B, C, \\dots , F)$ .","Then the infimum of the subsystem spectra, ${\\Lambda }^{ABC \\dots F}={\\inf }[{\\lambda }^{{X}_{1}},{\\lambda }^{{X}_{2}}, \\ldots , {\\lambda }^{{X}_{f}}]$ , embodies the subsystem disorder of the state ${\\hat{\\rho }}^{ABC \\dots F}$ , as expressed in the following theorem.","Theorem 3 (Nielsen-Kempe).","A multipartite state is entangled if its system disorder fails to exceed its subsystem disorder, i.e., ${\\lambda }^{ABC \\dots F}$ is entangled if ${\\lambda }^{ABC \\dots F}\\nprec {\\Lambda }^{ABC \\dots F}$ ."],["Quasi-Entropic Detectors","The majorization based theorems of the previous section have direct corollaries that yield scalar entanglement detectors for the entire class of Schur-concave measures.","A Schur-concave function $G$ is defined by the property that ${\\lambda }^{1} \\prec {\\lambda }^{2}$ implies $G({\\lambda }^{1}) \\ge G({\\lambda }^{2})$ [23].","Equivalently, Schur-concave functions are characterized by being monotonic with respect to the majorization relation.","They include all functions $G(\\cdot )$ that are concave and symmetric with respect to the arguments, a subset which we have defined as quasi-entropic.","An important subclass of quasi-entropic functions is obtained if we restrict $G$ to have the following trace structure: $G(\\lambda )=\\mathrm {tr}[g(\\lambda )]= {\\sum }_{\\alpha } g({\\lambda }_{\\alpha }), $ where $g(\\cdot )$ is a concave function of a single variable and $\\lambda $ is treated as a diagonal matrix for the purpose of calculating the trace.","Note that $G(\\cdot )$ as constructed in Eq.","(REF ) is manifestly symmetric and, as a sum of concave functions, it is also concave.","We shall refer to this class of functions, which include the Shannon and Tsallis entropies, as trace type quasi-entropic.","An important fact regarding this class of measures is that if, for a given pair of probability vectors $\\lambda $ and $\\mu $ , we have $G({\\lambda })>G({\\mu })$ for every trace type quasi-entropic measure $G$ , then $\\lambda \\prec \\mu $ .","The standard entropic measure of uncertainty [9], which is based on the Shannon entropy function, is trace type quasi-entropic and corresponds to the choice $g(x)=H(x)=-x \\ln (x)$ in Eq.","(REF ).","An example of an information theoretically relevant measure that is quasi-entropic but not trace type is the Rényi subfamily of entropies of order less than one [24].","Accordingly, although the scalar entanglement detectors that follow from the theorems of §III actually hold for the entire class of Schur-concave measures, we will continue to focus on the quasi-entropic class as the most suitable for information theoretical applications.","Scalar entanglement detectors based on Shannon, Tsallis, and Rényi entropies are already well known, albeit with detection bounds that may differ from those reported here [7], [8], [5].","Our primary purpose here is to emphasize the natural and categorical manner in which majorization based entanglement detectors yield entire classes of scalar detectors.","With $G(\\cdot )$ a quasi-entropic function, the aforementioned monotonicity property implies the following corollaries to Theorems 1-3.","Corollary 1.","Under the conditions of Theorem 1, the multipartite state ${\\hat{\\sigma }}^{ABC \\dots F}$ is entangled if $G[{{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F})] < G[{{P}}_{sep;sup}^{\\mathrm {X}}].","$ Corollary 2.","Under the conditions of Theorem 2, the bipartite state ${\\hat{\\sigma }}^{AB}$ is entangled if $G[{P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B})}({\\hat{\\sigma }}^{AB})]+ G[{P}^{ ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})} ({\\hat{\\sigma }}^{AB})] < G[{P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}].","$ Corollary 3.","Under the conditions of Theorem 3, the multipartite state ${\\hat{\\sigma }}^{ABC \\dots F}$ is entangled if $G[{\\lambda }^{ABC \\dots F}] < G[{\\Lambda }^{ABC \\dots F}].","$ As an application of the above corollaries, we will consider the entanglement detection threshold for the bipartite Werner state considered in Eq.","(REF ) et seq.","Using Corollary 1 together with the Tsallis entropy function, we find that ${\\hat{\\rho }}^{wer}_{d}(q)$ is entangled if ${S}^{tsa}_{r}\\big [ {{P}}^{{\\mathrm {X}}^{\\star }}[{\\hat{\\rho }}^{wer}_{d}(q)] \\big ] < {S}^{tsa}_{r}\\big [ {{P}}_{sep;sup}^{{\\mathrm {X}}^{\\star }} \\big ], $ where ${S}^{tsa}_{r}(\\cdot )$ is the Tsallis entropy of order $r$ , and the two probability vectors appearing in Eq.","(REF ) are those in Eqs.","(REF ) and (REF ).","The Tsallis entropy function ${S}^{tsa}_{r}(\\cdot )$ corresponds to the choice $g(\\lambda )= (\\lambda -{\\lambda }^{r})/(r-1)$ , $1 1/(1+d)$ , the known entanglement threshold for a the two-qudit Werner state.","This limiting case corresponds to the bottom graph in Fig.","REF .","The foregoing discussion of how well Corollary 1 does in detecting entanglement notwithstanding, the aim of this section is not to promote quasi-entropic detectors given in Corollaries 1-3 per se, but rather to emphasize the reach of the majorization results of §III whence they are inherited.","In addition, the above example highlights the fact that, while the entanglement detection bounds given in Corollaries 1-3 are optimal for Schur-concave functions as a class, their performance need not be optimal for individual members of the class, a fact that was emphasized in paper I as well."],["Concluding Remarks","In this paper we have developed entanglement detection criteria within the majorization formulation of uncertainty presented in paper I.","These are inherently stronger than similar scalar conditions in the sense that they are equivalent to and imply infinite classes of such scalar criteria.","Majorization relations in effect envelope the set of scalar measures of disorder based on Schur-concave functions, a huge set that includes the information-theoretically relevant class of quasi-entropic measures as a subset.","This enveloping property elucidates the exceptional effectiveness of majorization relations in dealing with problems of quantum information theory.","Entanglement detection criteria that can be experimentally implemented are especially useful in studies involving entangled microsystems, hence the importance of linear detectors and entanglement witnesses.","As already mentioned, Theorem 1 can be formulated as an entanglement witness inasmuch as it is a linear majorization condition on measurement results.","What may not be so obvious is that in principle the spectrum of a state is also experimentally accessible on the basis of the Lemma of §IIIA.","That Lemma states that the spectrum of a quantum state is the supremum of probability vectors that may be deduced from the counter statistics of all possible rank 1 generalized measurements performed on the state.","In practice, this provides an experimental method for estimating the spectrum, since a high precision determination may require a search over a large number of possible measurements.","There is already an extensive literature on entanglement detection with many important results.","With the notable exception of the Nielsen-Kempe theorem, these results are not formulated within the majorization framework.","The present contribution in part serves to demonstrate the effectiveness of the majorization formulation of some of the existing methods.","One can reasonably expect a sharpening of the results of some of the other entanglement detection strategies when reformulated within the majorization framework.","It is also not unreasonable to expect useful, albeit computationally intractable, necessary and sufficient separability criteria to emerge from such formulations.","This work was in part supported by a grant from California State University, Sacramento.","*"],["majorization bounds are found on pure states","Our task here is to establish that majorization uncertainty bounds can be reached on pure states.","We will prove this for two important cases, first for the uncertainty bound on all states of the system, and second on the class of separable states.","To avoid unnecessary clutter and technical distractions, we will limit the number of generalized measurements to three and the underlying Hilbert spaces to finite dimensions in the following treatment.","Suppose generalized measurements ($\\mathrm {X}, \\lbrace \\hat{\\mathrm {E}}_{\\alpha }^{X} \\rbrace $ ), ($\\mathrm {Y}, \\lbrace \\hat{\\mathrm {E}}_{\\beta }^{Y} \\rbrace $ ), and ($\\mathrm {Z}, \\lbrace \\hat{\\mathrm {E}}_{\\beta }^{Z} \\rbrace $ ) are performed on a state $\\hat{\\rho }$ that is supported on a finite-dimensional Hilbert space.","What we will show below is that the search for the components of ${\\mu }^{sup}$ which serve to define ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}$ can be limited to pure states.","By definition, ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}={\\sup }_{\\hat{\\rho }}[{{P}}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })], $ where ${{P}}_{\\alpha \\beta \\gamma }^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })=\\textrm {tr}({\\hat{\\mathrm {E}}}_{\\alpha }^{X} \\hat{\\rho }) \\textrm {tr}({\\hat{\\mathrm {E}}}_{\\beta }^{Y} \\hat{\\rho }) \\textrm {tr}({\\hat{\\mathrm {E}}}_{\\gamma }^{Z} \\hat{\\rho }).","$ We recall from §IIA above and §IIB, IVA and IVB of Paper I that the supremum defined by Eqs.","(REF ) and (REF ) is found by calculating ${\\mu }^{sup}_{i}$ for $i=1,2,\\ldots $ (${\\mu }^{sup}_{0}=0$ by definition).","Furthermore, ${\\mu }^{sup}_{1}$ is the maximum value of a single component of ${{P}}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })$ as $\\hat{\\rho }$ is varied, ${\\mu }^{sup}_{2}$ is the maximum value of the sum of two (different) components of ${{P}}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })$ as $\\hat{\\rho }$ is varied, and ${\\mu }^{sup}_{i}$ is the maximum value of the sum of $i$ (different) components of ${{P}}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })$ as $\\hat{\\rho }$ is varied.","It is thus sufficient to prove that the said maximum for ${\\mu }^{sup}_{i}$ is in fact realized on a pure state.","Let $\\hat{\\rho }={\\sum }_{a}\\, {\\lambda }_{a}(\\hat{\\rho }) \\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{a} \\mid $ be the principal ensemble representation for $\\hat{\\rho }$ .","Then the above maximization process defines ${\\mu }^{sup}_{i}$ as the maximum of ${\\sum }_{a,b,c}{\\lambda }_{a}(\\hat{\\rho }){\\lambda }_{b}(\\hat{\\rho }){\\lambda }_{c}(\\hat{\\rho }) \\big [ \\langle {\\psi }_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{1}}^{X}\\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{1}}^{Y}\\mid {\\psi }_{b} \\rangle \\nonumber \\\\ \\times \\langle {\\psi }_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{1}}^{Z}\\mid {\\psi }_{c} \\rangle + \\langle {\\psi }_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{2}}^{X}\\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{2}}^{Y}\\mid {\\psi }_{b} \\rangle \\nonumber \\\\ \\times \\langle {\\psi }_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{2}}^{Z}\\mid {\\psi }_{c} \\rangle + \\ldots \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\nonumber \\\\+ \\langle {\\psi }_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{i}}^{X}\\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{i}}^{Y}\\mid {\\psi }_{b} \\rangle \\langle {\\psi }_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{i}}^{Z}\\mid {\\psi }_{c} \\rangle \\big ] \\,\\,\\,\\,\\,\\, \\nonumber \\\\+\\,\\, {\\xi }_{i} \\big (1-{\\sum }_{a} {\\lambda }^{\\rho }_{a}\\big )+{\\sum }_{a,b} {\\eta }_{i,ab} \\big ({\\delta }_{ab}-\\langle {\\psi }_{b}\\mid {\\psi }_{a} \\rangle \\big ), $ as the state vectors $\\lbrace \\mid {\\psi }_{a}\\rangle \\rbrace $ which are the eigenstates of $\\hat{\\rho }$ , the probabilities $\\lbrace {\\lambda }_{a}(\\hat{\\rho }) \\rbrace $ which constitute the spectrum of $\\hat{\\rho }$ , and the Lagrange multipliers ${\\xi }_{i}$ and ${\\eta }_{i,ba}$ which serve to enforce the structure of ${\\sum }_{a}\\, {\\lambda }^{\\rho }_{a} \\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{a} \\mid $ as an orthogonal ensemble are varied.","The symbol ${\\delta }_{ab}$ in the above expression is the Kronecker delta, and the three sets of indices $\\lbrace {({\\alpha }, {\\beta }, {\\gamma })}_{1}, {({\\alpha }, {\\beta }, {\\gamma })}_{2}, \\dots , {({\\alpha }, {\\beta }, {\\gamma })}_{i} \\rbrace $ are understood to be distinct, i.e., ${\\alpha }_{1} \\ne {\\alpha }_{2} \\ne \\ldots \\ne {\\alpha }_{i}$ , and similarly for the other two sets.","Note also that since $\\langle {\\psi }_{a}\\mid {\\psi }_{b} \\rangle $ is in general a Hermitian matrix (in the indices $a$ and $b$ ), the Lagrange multipliers ${\\eta }_{i,ba}$ may also be taken to constitute a Hermitian matrix.","A variation with respect to $\\langle {\\psi }_{a} \\mid $ gives ${\\lambda }_{a}(\\hat{\\rho }) \\big [ {\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z} \\big ]\\mid {\\psi }_{a} \\rangle ={\\sum }_{b} {\\eta }_{i,ba} \\mid {\\psi }_{b} \\rangle , $ where ${\\hat{\\mathcal {E}}}_{i}^{X}={\\sum }_{k=1}^{i} {P}^{\\mathrm {Y}}_{{\\beta }_{k}}(\\hat{\\rho }){P}^{\\mathrm {Z}}_{{\\gamma }_{k}}(\\hat{\\rho }){\\hat{\\mathrm {E}}}_{{\\alpha }_{k}}^{X}, $ and ${\\hat{\\mathcal {E}}}_{i}^{Z}$ and ${\\hat{\\mathcal {E}}}_{i}^{Y}$ are defined similarly.","Note that the three operators introduced in Eqs.","(REF ) and (REF ) are Hermitian and positive.","Let ${\\hat{\\mathcal {E}}}_{i}={\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z}$ and ${\\mathcal {E}}_{i,aa}=\\langle {\\psi }_{a} \\mid {\\hat{\\mathcal {E}}}_{i} \\mid {\\psi }_{a} \\rangle $ .","Then the maximum sought in (REF ), namely ${\\mu }^{sup}_{i}$ , is given by ${\\sum }_{a}{\\lambda }_{a}(\\hat{\\rho }){\\mathcal {E}}_{i,aa}/3=\\textrm {tr}[{\\hat{\\mathcal {E}}}_{i}\\hat{\\rho }]/3$ .","A key fact at this juncture is that the diagonal elements ${\\mathcal {E}}_{i,aa}$ do not depend on $a$ and are all equal.","In other words, each pure state $\\mid {\\psi }_{a} \\rangle $ in the ensemble contributes equally to ${\\mu }^{sup}_{i}$ .","This property follows from a variation of (REF ) with respect to ${\\lambda }_{a}(\\hat{\\rho })$ , which leads to ${\\xi }_{i}=\\langle {\\psi }_{a} \\mid {\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z} \\mid {\\psi }_{a} \\rangle ={\\mathcal {E}}_{i,aa}.","$ Clearly then, each pure state $\\mid {\\psi }_{a} \\rangle $ in the ensemble must realize the same maximum ${\\mu }^{sup}_{i}$ as the entire ensemble $\\hat{\\rho }$ .","We conclude therefore that the sought maximum can be found among the pure states of the system.","What if we are looking for ${\\mu }^{sup}_{i}$ but with $\\hat{\\rho }$ limited to separable states?","It turns out that here too the search can be limited to pure states, which would be pure product states in this instance.","To establish this result, we will modify the foregoing analysis by stipulating that $\\hat{\\rho }={\\sum }_{a}\\, {q}_{a} \\mid {\\phi }^{A}_{a} \\rangle \\langle {\\phi }^{A}_{a} \\mid \\otimes \\mid {\\phi }^{B}_{a} \\rangle \\langle {\\phi }^{B}_{a} \\mid $ representing a separable, bipartite state of parties $A$ and $B$ .","Then the corresponding ${\\mu }^{sup}_{sep,i}$ is the maximum of ${\\sum }_{a,b,c}{q}_{a}{q}_{b}{q}_{c} \\big [\\langle {\\phi }^{B}_{a}\\mid \\otimes \\langle {\\phi }^{A}_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{1}}^{X}\\mid {\\phi }^{A}_{a} \\rangle \\otimes \\mid {\\phi }^{B}_{a} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{b}\\mid \\otimes \\langle {\\phi }^{A}_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{1}}^{Y}\\mid {\\phi }^{A}_{b} \\rangle \\otimes \\mid {\\phi }^{B}_{b} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{c}\\mid \\otimes \\langle {\\phi }^{A}_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{1}}^{Z}\\mid {\\phi }^{A}_{c} \\rangle \\otimes \\mid {\\phi }^{B}_{c} \\rangle \\nonumber \\\\+ \\langle {\\phi }^{B}_{a}\\mid \\otimes \\langle {\\phi }^{A}_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{2}}^{X}\\mid {\\phi }^{A}_{a} \\rangle \\otimes \\mid {\\phi }^{B}_{a} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{b}\\mid \\otimes \\langle {\\phi }^{A}_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{2}}^{Y}\\mid {\\phi }^{A}_{b} \\rangle \\otimes \\mid {\\phi }^{B}_{b} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{c}\\mid \\otimes \\langle {\\phi }^{A}_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{2}}^{Z}\\mid {\\phi }^{A}_{c} \\rangle \\otimes \\mid {\\phi }^{B}_{c} \\rangle \\nonumber \\\\+ \\ldots \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\, \\nonumber \\\\+ \\langle {\\phi }^{B}_{a}\\mid \\otimes \\langle {\\phi }^{A}_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{i}}^{X}\\mid {\\phi }^{A}_{a} \\rangle \\otimes \\mid {\\phi }^{B}_{a} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{b}\\mid \\otimes \\langle {\\phi }^{A}_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{i}}^{Y}\\mid {\\phi }^{A}_{b} \\rangle \\otimes \\mid {\\phi }^{B}_{b} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{c}\\mid \\otimes \\langle {\\phi }^{A}_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{i}}^{Z}\\mid {\\phi }^{A}_{c} \\rangle \\otimes \\mid {\\phi }^{B}_{c} \\rangle \\nonumber \\\\+ {\\sum }_{a} {\\eta }_{i,a}^{A} \\big [ 1-\\langle {\\phi }^{A}_{a}\\mid {\\phi }^{A}_{a} \\rangle ]+{\\sum }_{a} {\\eta }_{i,a}^{B} \\big [ 1-\\langle {\\phi }^{B}_{a}\\mid {\\phi }^{B}_{a} \\rangle ] \\nonumber \\\\+ {\\xi }_{i} \\big (1-{\\sum }_{a} {q}_{a}\\big ), \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, $ as the state vectors $ \\lbrace \\mid {\\phi }^{A}_{a} \\rangle \\rbrace $ , $\\lbrace \\mid {\\phi }^{B}_{a} \\rangle \\rbrace $ , the probabilities $\\lbrace {q}_{a} \\rbrace $ , and the Lagrange multipliers ${\\eta }_{i,a}^{A}$ , ${\\eta }_{i,a}^{B}$ , and ${\\xi }_{i}$ which serve to enforce normalization conditions and probability conservation for the ensemble are varied.","Variations with respect to $ \\langle {\\phi }^{A}_{a} \\mid $ and $ \\langle {\\phi }^{B}_{a} \\mid $ now give ${q}_{a} \\big [ \\langle {\\phi }^{B}_{a}\\mid ({\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z}) \\mid {\\phi }^{B}_{a} \\rangle \\big ] \\mid {\\phi }^{A}_{a} \\rangle &={\\eta }_{i,a}^{A} \\mid {\\phi }^{A}_{a} \\rangle \\nonumber \\\\{q}_{a} \\big [ \\langle {\\phi }^{A}_{a}\\mid ({\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z}) \\mid {\\phi }^{A}_{a} \\rangle \\big ] \\mid {\\phi }^{B}_{a} \\rangle &={\\eta }_{i,a}^{B} \\mid {\\phi }^{B}_{a} \\rangle , $ where $({\\hat{\\mathcal {E}}}_{i}^{X},{\\hat{\\mathcal {E}}}_{i}^{Y},{\\hat{\\mathcal {E}}}_{i}^{Z})$ are defined as in Eq.","(REF ) et seq.","Equations (REF ) directly imply that ${\\eta }_{i,a}^{A}={\\eta }_{i,a}^{B}$ , and as in the general case above, the maximum sought in (REF ), namely ${\\mu }^{sup}_{sep,i}$ , is found to equal ${\\sum }_{a}{q}_{a} {\\mathcal {E}}_{i,aa}/3=\\textrm {tr}[{\\hat{\\mathcal {E}}}_{i}\\hat{\\rho }]/3$ , where ${\\hat{\\mathcal {E}}}_{i}={\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z}$ .","Furthermore, a variation with respect to the probabilities $\\lbrace {q}_{a} \\rbrace $ shows the equality of the matrix elements ${\\mathcal {E}}_{i,aa}$ just as in the general case above, whereupon we learn that each pure product state in the ensemble makes the same contribution to ${\\mu }^{sup}_{sep,i}$ .","In other words, the separable density matrix which maximizes (REF ) may be taken to be a pure product state [16].","It should be clear from the above arguments that the results hold for any number of generalized measurements as well as any number of parties in the multipartite state.","In summary, then, we have found that the joint majorization uncertainty bound, ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ , for a set of generalized measurements performed on an arbitrary quantum state supported on a finite-dimensional Hilbert space can be found on the class of pure states of the system, and in the case of a separable states, ${{P}}_{sep;sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ can be found on the class of pure product states of the system."]],"string":"[\n [\n \"Entanglement Detection Using Majorization Uncertainty Bounds\"\n ],\n [\n \"Abstract Entanglement detection criteria are developed within the framework of the majorization formulation of uncertainty.\",\n \"The primary results are two theorems asserting linear and nonlinear separability criteria based on majorization relations, the violation of which would imply entanglement.\",\n \"Corollaries to these theorems yield infinite sets of scalar entanglement detection criteria based on quasi-entropic measures of disorder.\",\n \"Examples are analyzed to probe the efficacy of the derived criteria in detecting the entanglement of bipartite Werner states.\",\n \"Characteristics of the majorization relation as a comparator of disorder uniquely suited to information-theoretical applications are emphasized throughout.\"\n ],\n [\n \"Introduction\",\n \"Quantum measurements in general have indeterminate outcomes, with the results commonly expressed as a vector of probabilities corresponding to the set of outcomes.\",\n \"In case of noncommuting observables, the joint indeterminacy of their measurement outcomes has an inviolable lower bound, in stark contrast to classical expectations.\",\n \"This was discovered by Heisenberg in his desire to advance the physical understanding of the newly discovered matrix mechanics by relating the unavoidable disturbances caused by the act of measurement to the fundamental commutation relations of quantum dynamics [1].\",\n \"Heisenberg's arguments relied on the statistical spread of the measured values of the observables to quantify uncertainty.\",\n \"This gave rise to the variance formulation of the uncertainty principle which remains a powerful source of intuition on the structure and spectral properties of microscopic systems.\",\n \"With the prospect of quantum computing and the development of quantum information theory in recent decades, on the other hand, the need for a measure of uncertainty that can better capture its information theoretical aspects, especially in dealing with noncanonical observables, has inspired new formulations.\",\n \"Among these are the entropic measure developed in the eighties, and the majorization formulation proposed recently.\",\n \"Uncertainty relations resulting from these formulations have found application to quantum cryptography, information locking, and entanglement detection, in addition to providing uncertainty limits [2], [3].\",\n \"In this paper we develop applications of the majorization formulation of uncertainty introduced in Ref.\",\n \"[3] to the problem of entanglement detection.\",\n \"Deciding whether a given quantum state is entangled is a central problem of quantum information theory and known to be computationally intractable in general [4].\",\n \"As a result, computationally tractable necessary conditions for separability, which provide a partial solution to this problem, have been the subject of active research in recent years.\",\n \"Among these, the Peres-Horodecki positive partial transpose criterion actually provides necessary and sufficient separability conditions for $2\\\\otimes 2$ and $2\\\\otimes 3$ dimensional systems, and necessary conditions otherwise.\",\n \"Other notable results are the reduction and global versus local disorder criteria, both necessary conditions in general [5].\",\n \"An observable that has non-negative expectation values for all separable states and negative ones for a subset of entangled states provides an operational method of entanglement detection and is known as an entanglement witness [6].\",\n \"It has also long been known that uncertainty relations can serve a similar purpose by providing inequalities that must be satisfied by separable states and if violated signal entanglement [7], [8].\",\n \"In this paper we extend the majorization formulation of uncertainty developed in Ref.\",\n \"[3] to the problem of entanglement detection.\",\n \"As discussed in that paper (hereafter referred to as Paper I) and in the following, majorization as a comparator of uncertainty is qualitatively different from, and stronger than, scalar measures of uncertainty.\",\n \"Consequently, the entanglement detection results developed here represent a qualitative strengthening of the existing variance and entropic results.\",\n \"This will be evident, among other things, by the fact that they yield, as corollaries, an infinite class of entanglement detectors based on scalar measures.\",\n \"As mentioned above, results of quantum measurements are in general probability vectors deduced from counter statistics, and an information theoretical formulation of uncertainty is normally based on an order of uncertainty defined on such vectors (see Paper I).\",\n \"Thus a scalar measure of uncertainty is commonly a real, non-negative function defined on probability vectors whose value serves to define the uncertainty in question.\",\n \"For example, the Shannon entropy function is the measure of uncertainty for the standard entropic formulation of uncertainty [9].\",\n \"By contrast, majorization provides a partial order of uncertainty on probability vectors that is in general more stringent, and fundamentally stronger, than a scalar measure [10].\",\n \"Note that, unlike scalar measures, majorization does not assign a quantitative measure of uncertainty to probability vectors, and as a partial order may find a pair of vectors to be incomparable.\",\n \"A characterization of majorization that clarifies the foregoing statements can be attained by considering the quasi-entropic set of measures.\",\n \"These were defined in Paper I as the set of concave, symmetric functions defined on probability vectors, and include the Shannon, Tsallis, and (a subfamily of) Rényi entropies as special cases [11].\",\n \"It is important to realize that the uncertainty order determined by one member of the quasi-entropic set for a given pair of probability vectors may contradict that given by another.\",\n \"While additional considerations may justify the use of, e.g., Shannon entropy in preference to the others, the foregoing observation clearly indicates the relative nature of the uncertainty order given by a specific measure, and immediately raises the following question: are there pairs of vectors for which all quasi-entropic measures determine the same uncertainty order?\",\n \"The answer is yes, and the common determination of the quasi-entropic set in such cases defines the uncertainty order given by majorization.\",\n \"What about the cases where there are conflicting determinations by the members of the quasi-entropic set?\",\n \"The majorization relation defines such pairs as incomparable, whence the “partial” nature of the order defined by majorization.\",\n \"We may therefore consider the majorization order to be equivalent to the collective order determination of the entire quasi-entropic set (see §IV).\",\n \"This characterization of the majorization relation clearly shows its standing vis-à-vis the scalar measures of uncertainty.\",\n \"More practical definitions of the majorization relation will be considered in §II.\",\n \"The rest of this paper is organized as follows.\",\n \"In §II we review elements of the majorization formulation of uncertainty needed for our work and establish the notation.\",\n \"In §III we present the central results of this paper on entanglement detection, and in §IV we derive entire classes of entanglement detectors for quasi-entropic measures as corollaries to the theorems of §III.\",\n \"Concluding remarks are presented in §V.\",\n \"Details of certain mathematical proofs are given in the Appendix.\"\n ],\n [\n \"Majorization formulation of uncertainty\",\n \"This section presents a review of the basic elements of majorization theory and the formulation of the uncertainty principle based on it.\",\n \"It follows the treatment given in Paper I.\"\n ],\n [\n \"Majorization\",\n \"The basic element of our formulation is the majorization protocol for comparing the degree of uncertainty, or disorder, among probability vectors, i.e., sequences of non-negative numbers summing to unity [10].\",\n \"By definition, ${\\\\lambda }^{1}$ is no less uncertain than ${\\\\lambda }^{2}$ if ${\\\\lambda }^{1}$ equals a mixture of the permutations of ${\\\\lambda }^{2}$ .\",\n \"Then ${\\\\lambda }^{1}$ is said to be majorized by ${\\\\lambda }^{2}$ and written ${\\\\lambda }^{1}\\\\prec {\\\\lambda }^{2}$ .\",\n \"An equivalent definition that flushes out the details of the foregoing is based on the vector ${\\\\lambda }^{\\\\downarrow }$ which is obtained from $\\\\lambda $ by arranging the components of the latter in a nonincreasing order.\",\n \"Then, ${\\\\lambda }^{1}\\\\prec {\\\\lambda }^{2}$ if ${\\\\sum }_{i}^{j} {\\\\lambda }^{1\\\\downarrow }_{i} \\\\le {\\\\sum }_{i}^{j} {\\\\lambda }^{2\\\\downarrow }_{i} $ for $j=1,2, \\\\ldots d-1$ , where $d$ is the larger of the two dimensions and trailing zeros are added where needed.\",\n \"As stated earlier, the majorization relation is a partial order, i.e., not every two vectors are comparable under majorization.\",\n \"As suggested in §I, two vectors are found to be incomparable when the difference in their degrees of uncertainty does not rise to the level required by majorization.\",\n \"As evident from the second definition given above, majorization requires the satisfaction of $N-1$ inequalities if the number of non-zero components of the majorizing vector is $N$ .\",\n \"This accounts for the strength of the majorization relation as compared to scalar measures of disorder.\",\n \"Indeed as alluded to in §I, for any quasi-entropic function $F(\\\\lambda )$ , ${\\\\lambda }^{1} \\\\prec {\\\\lambda }^{2}$ implies $F({\\\\lambda }^{1}) \\\\ge F({\\\\lambda }^{2})$ , but not conversely.\",\n \"On the other hand, if for every quasi-entropic function $F(\\\\lambda )$ we have $F({\\\\lambda }^{1}) \\\\ge F({\\\\lambda }^{2})$ , then ${\\\\lambda }^{1} \\\\prec {\\\\lambda }^{2}$ [10].\",\n \"To establish uncertainty bounds, we need to characterize the greatest lower bound, or infimum, and the least upper bound, or supremum, of a set of probability vectors [12].\",\n \"The infimum is defined as the vector that is majorized by every element of the set and in turn majorizes any vector with that property .\",\n \"The supremum is similarly defined.\",\n \"We will briefly outline the construction of the infimum here and refer the reader to Paper I for further details.\",\n \"Given a set of probability vectors $\\\\lbrace {\\\\lambda }^{a} {\\\\rbrace }_{a=1}^{N}$ , consider the vector ${\\\\mu }^{inf}$ defined by ${\\\\mu }_{0}^{inf}=0&,\\\\,\\\\,\\\\,{\\\\mu }_{j}^{inf}=\\\\min \\\\big ( {\\\\sum }_{i=1}^{j} {\\\\lambda }^{1\\\\downarrow }_{i},{\\\\sum }_{i=1}^{j} {\\\\lambda }^{2\\\\downarrow }_{i},\\\\ldots , \\\\nonumber \\\\\\\\&{\\\\sum }_{i=1}^{j}{\\\\lambda }^{N\\\\downarrow }_{i} \\\\big ), \\\\,\\\\,\\\\, 1 \\\\le j \\\\le {d}_{max}, $ where ${d}_{max}$ is the largest dimension found in the set.\",\n \"The infimum is then given by ${\\\\lambda }^{inf}_{i}={[\\\\inf ({\\\\lambda }^{1},{\\\\lambda }^{2}, \\\\ldots , {\\\\lambda }^{N})]}_{i}={\\\\mu }_{i}^{inf}-{\\\\mu }_{i-1}^{inf}, $ where $1 \\\\le i \\\\le {d}_{max}$ .\",\n \"The construction of the supremum starts with ${\\\\mu }^{sup}$ in parallel with Eq.\",\n \"(REF ), but with “max” replacing “min,” and may require further steps detailed in Paper I.\",\n \"It is worth noting here that the infimum (supremum) of a pair of probability vectors will in general be more (less) disordered than either.\",\n \"Important special cases are (i) one of the two majorizes the other, in which case the latter is the infimum and the former the supremum of the two, and (ii) the two are equal, in which case either is both the infimum and the supremum.\",\n \"Furthermore, a useful qualitative rule is that the more “different” are two probability vectors, the further will the infimum or supremum of the two be from at least one of them.\",\n \"Finally, we note that while the infimum or supremum of a set of probability vectors always exists, it need not be a member of the set.\"\n ],\n [\n \"Measurement and uncertainty\",\n \"A generalized measurement may be defined by a set of positive operators $\\\\lbrace {\\\\hat{\\\\mathrm {E}}}_{\\\\alpha } \\\\rbrace $ called measurement elements and subject to the completeness condition ${\\\\sum }_{\\\\alpha }{\\\\hat{\\\\mathrm {E}}}_{\\\\alpha } = \\\\hat{\\\\mathbb {1}}$ .\",\n \"The probability that outcome $\\\\alpha $ turns up in a measurement of the state $\\\\hat{\\\\rho }$ is given by the Born rule ${P}_{\\\\alpha }(\\\\hat{\\\\rho })=\\\\textrm {tr} [\\\\hat{\\\\mathrm {E}}_{\\\\alpha } \\\\hat{\\\\rho } ]$ .\",\n \"A generalized measurement can always be considered to be the restriction of a more basic type, namely a projective measurement, performed on an enlarged system to the system under generalized measurement [13].\",\n \"A projective measurement is usually associated with an observable of the system represented by a self-adjoint operator $\\\\hat{M}$ , and entails a partitioning of the spectrum of $\\\\hat{M}$ into a collection of subsets $\\\\lbrace {b}_{\\\\alpha }^{M} \\\\rbrace $ called measurement bins.\",\n \"We call a projective measurement maximal if each bin consists of a single point of the spectrum of the measured observable.\",\n \"Having assembled the necessary concepts, we can now characterize uncertainty by means of majorization relations in a natural manner.\",\n \"To start, we define the probability vector ${P}^{X}(\\\\hat{\\\\rho })$ resulting from a measurement $\\\\mathrm {X}$ on a state $\\\\hat{\\\\rho }$ to be uncertain if it is majorized by $\\\\mathcal {I}=(1,0,\\\\ldots ,0)$ but not equal to it.\",\n \"As such, ${P}^{X}(\\\\hat{\\\\rho })$ is said to be strictly majorized by $\\\\mathcal {I}$ and written ${P}^{X}(\\\\hat{\\\\rho })\\\\prec \\\\prec \\\\mathcal {I}$ .\",\n \"Similarly, given a pair of measurements $\\\\mathrm {X}$ and $\\\\mathrm {Y}$ on a state $\\\\hat{\\\\rho }$ , we say ${P}^{\\\\mathrm {X}}(\\\\hat{\\\\rho })$ is more uncertain, equivalently more disordered, than ${P}^{\\\\mathrm {Y}}(\\\\hat{\\\\rho })$ if ${P}^{\\\\mathrm {X}}(\\\\hat{\\\\rho }) \\\\prec {P}^{\\\\mathrm {Y}}(\\\\hat{\\\\rho })$ .\",\n \"Further, we define the joint uncertainty of a pair of measurements $\\\\mathrm {X}$ and $\\\\mathrm {Y}$ to be the outer product ${P}^{\\\\mathrm {X}}\\\\otimes {P}^{\\\\mathrm {Y}}$ , i.e., ${P}_{\\\\alpha \\\\beta }^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y}}={P}^{\\\\mathrm {X}}_{\\\\alpha } {P}^{\\\\mathrm {Y}}_{\\\\beta }$ .\",\n \"Since $H({P}^{\\\\mathrm {X}}\\\\otimes {P}^{\\\\mathrm {Y}})=H({P}^{\\\\mathrm {X}})+H({P}^{\\\\mathrm {Y}})$ , where $H(\\\\cdot )$ is the Shannon entropy function, this definition is seen to be consistent with its entropic counterpart.\",\n \"As stated earlier, ${P}^{\\\\mathrm {X}} \\\\prec {P}^{\\\\mathrm {Y}}$ implies $H({P}^{\\\\mathrm {X}}) \\\\ge H({P}^{\\\\mathrm {Y}})$ but not conversely.\",\n \"These definitions naturally extend to an arbitrary number of states and measurements.\",\n \"We are now in a position to state the majorization statement of the uncertainty principle established in Paper I: “The joint results of a set of generalized measurements of a given state are no less uncertain than a probability vector that depends on the measurement set but not the state, and is itself uncertain unless the measurement elements have a common eigenstate.\\\"\",\n \"In symbols, ${P}^{\\\\mathrm {X}}(\\\\hat{\\\\rho })\\\\otimes {P}^{\\\\mathrm {Y}}(\\\\hat{\\\\rho }) \\\\otimes \\\\ldots \\\\otimes {P}^{\\\\mathrm {Z}}(\\\\hat{\\\\rho }) \\\\prec {{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}} \\\\prec \\\\prec \\\\mathcal {I}, $ where ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}={\\\\sup }_{\\\\hat{\\\\rho }} [{P}^{\\\\mathrm {X}}(\\\\hat{\\\\rho })\\\\otimes {P}^{\\\\mathrm {Y}}(\\\\hat{\\\\rho }) \\\\otimes \\\\ldots \\\\otimes {P}^{\\\\mathrm {Z}}(\\\\hat{\\\\rho })] , $ unless the measurement elements $\\\\lbrace \\\\hat{\\\\mathrm {E}}^{\\\\mathrm {X}}, \\\\hat{\\\\mathrm {E}}^{\\\\mathrm {Y}}, \\\\ldots , \\\\hat{\\\\mathrm {E}}^{\\\\mathrm {Z}} \\\\rbrace $ have a common eigenstate in which case ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}} = \\\\mathcal {I}$ .\",\n \"Note that ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}$ is the majorization uncertainty bound for the measurement set considered.\"\n ],\n [\n \"majorization uncertainty bounds\",\n \"As stated above, the uncertainty bound ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}$ given by Eq.\",\n \"(REF ) depends on the measurement set but not the state of the system.\",\n \"As the supremum of all possible measurement outcomes, it is the probability vector that sets the irreducible lower bound to uncertainty for the set.\",\n \"As such, it is the counterpart of the variance product or entropic lower bound in the traditional formulations of the uncertainty principle.\",\n \"Unlike the latter, however, ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}$ is in general not realizable or even approachable by any state of the system.\",\n \"Therefore, there is in general no such thing as a “minimum uncertainty state” within the majorization framework.\",\n \"This is a consequence of the fact, mentioned earlier, that the infimum or supremum of a set of vectors need not be a member of the set.\",\n \"An obvious special case is the trivial example of zero uncertainty for which ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}= \\\\mathcal {I}$ , signaling the existence of a common eigenstate for the measurement elements.\",\n \"Intuitively, we expect that mixing states can only increase their uncertainty, as is known to be the case for scalar measures of uncertainty.\",\n \"In case of majorization, this expectation is manifested in the property that the uncertainty bound ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}$ is in general realized on the class of pure states.\",\n \"Indeed for two and three mutually unbiased observables on a two-dimensional Hilbert space considered, we found in Paper I that the required maxima for the components of ${\\\\mu }^{sup}$ that serve to define the uncertainty bound are reached on pure states.\",\n \"More specifically, as outlined in §IIA above and detailed in §IIB, IVA and IVB of Paper I, the calculation of ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}$ involves a component-wise determination of ${\\\\mu }^{sup}$ by a series of maximizations over all possible density matrices $\\\\hat{\\\\rho }$ .\",\n \"The property in question then guarantees that the maximization process can be limited to density matrices representing pure states only.\",\n \"We will prove this assertion in general, as well as for the class of separable states, in the Appendix.\",\n \"As discussed earlier, an entanglement detector can be effective in deciding whether a given density matrix is separable by providing a condition that is satisfied by all separable states and if violated signals entanglement.\",\n \"A large body of entanglement detection strategies have been developed in recent years which are primarily based on scalar conditions, including those based on variance and entropic type uncertainty relations, which can be found in Refs.\",\n \"[5], [6], [7], [8].\",\n \"Here we shall develop majorization conditions for entanglement detection based on the formulation of measurement uncertainty given in Paper I and outlined in §II above.\",\n \"In particular, the entanglement condition given in Theorem 1 below is linear and susceptible to experimental implementation, so it can be formulated as an entanglement witness.\",\n \"Nonlinear detectors developed below, on the other hand, rely on majorization-based uncertainty relations.\",\n \"We will also introduce the notion of subsystem disorder and a sharpened version of the Nielsen-Kempe [14] separability condition as a nonlinear entanglement detector.\",\n \"As majorization relations, our results in general entail multiple inequalities whose number will grow with the uncertainty levels involved.\",\n \"As discussed in §IIA, this is an important feature of the majorization relation as comparator of disorder, one that sets it apart from scalar conditions and provides for the refinement needed in comparing highly disordered vectors.\",\n \"As will be seen in §IV, this property has the consequence that each majorization condition yields a scalar condition for the entire quasi-entropic class of uncertainty measures.\"\n ],\n [\n \"Linear detectors\",\n \"Our detection strategy is based on the intuitive expectation that the measurement uncertainty bound for the class of separable states of a multipartite system must be majorized by the corresponding bound for all states, and that this hierarchy can be exploited for entanglement detection.\",\n \"This is the majorization rendition of the strategy often used to derive separability conditions [7], [8].\",\n \"We will first consider the case of one generalized measurement resulting in a linear detection condition.\",\n \"Theorem 1.\",\n \"Let the results of the generalized measurement ($\\\\mathrm {X}, \\\\lbrace \\\\hat{\\\\mathrm {E}}_{\\\\alpha }^{X} \\\\rbrace $ ) on a multipartite state ${\\\\hat{\\\\rho }}^{ABC \\\\dots F}$ of parties $(A, B, C, \\\\dots , F)$ defined on a finite-dimensional Hilbert space be bounded by ${{P}}_{sup}^{\\\\mathrm {X}}={\\\\sup }_{{\\\\hat{\\\\rho }}^{ABC \\\\dots F}}{{P}}^{\\\\mathrm {X}}({\\\\hat{\\\\rho }}^{ABC \\\\dots F}), $ and in the case of a separable state ${\\\\hat{\\\\rho }}^{ABC \\\\dots F}_{sep}$ by ${{P}}_{sep;sup}^{\\\\mathrm {X}}={\\\\sup }_{{\\\\hat{\\\\rho }}^{ABC \\\\dots F}_{sep}}{{P}}^{\\\\mathrm {X}}({\\\\hat{\\\\rho }}^{ABC \\\\dots F}_{sep}).\",\n \"$ Then, (i) the suprema in the foregoing pair of equations may be taken over the class of pure and pure, product states, respectively, and (ii) given an arbitrary state ${\\\\hat{\\\\sigma }}^{ABC \\\\dots F}$ , the condition ${{P}}^{\\\\mathrm {X}}({\\\\hat{\\\\sigma }}^{ABC \\\\dots F}) \\\\nprec {{P}}_{sep;sup}^{\\\\mathrm {X}}$ implies that ${\\\\hat{\\\\sigma }}^{ABC \\\\dots F}$ is entangled [15].\",\n \"Part (i) of Theorem 1 is the majorization version of the familiar result that uncertainty bounds are realized on pure states and is proved in the Appendix [16].\",\n \"Part (ii) is the statement that the bound in Eq.\",\n \"(REF ) is a necessary condition for separability and its violation signals the existence of entanglement.\",\n \"As noted above, Theorem 1 provides an operational method of entanglement detection and can be reformulated as a set of entanglement witnesses.\",\n \"Clearly, the efficacy of Theorem 1 in detecting entanglement depends on the choice of the measurement ${\\\\mathrm {X}}$ for a given quantum state.\",\n \"This suggests looking for the optimum measurement, in parallel with the optimization problem for entanglement witnesses [17].\",\n \"Note that here the choice of the generalized measurement ${\\\\mathrm {X}}$ affects both ${{P}}^{\\\\mathrm {X}}({\\\\hat{\\\\sigma }}^{ABC \\\\dots F})$ and ${{P}}_{sep;sup}^{\\\\mathrm {X}}$ , thus making the task of finding a fully optimized solution rather difficult.\",\n \"It is therefore fortunate that we can achieve a partial optimization on the basis of Theorem 3 of Paper I, the relevant content of which we can state as follows: Lemma Under the conditions of Theorem 1 and with ${\\\\mathrm {X}}$ restricted to rank 1 measurements, we have ${{P}}^{\\\\mathrm {X}}({\\\\hat{\\\\sigma }}^{ABC \\\\dots F}) \\\\prec {{P}}^{{\\\\mathrm {X}}^{\\\\star }}({\\\\hat{\\\\sigma }}^{ABC \\\\dots F})={\\\\lambda }({\\\\hat{\\\\sigma }}^{ABC \\\\dots F}), $ where ${\\\\mathrm {X}}^{\\\\star }$ is a maximal projective measurement whose elements are the set of rank 1 orthogonal projection operators onto the eigenvectors of ${\\\\hat{\\\\sigma }}^{ABC \\\\dots F}$ and ${\\\\lambda }({\\\\hat{\\\\sigma }}^{ABC \\\\dots F})$ is the corresponding spectrum.\",\n \"It should be noted here that while the choice of ${{\\\\mathrm {X}}^{\\\\star }}$ succeeds in optimizing ${{P}}^{\\\\mathrm {X}}({\\\\hat{\\\\sigma }}^{ABC \\\\dots F})$ which appears in part (ii) of Theorem 1, its effect on ${{P}}_{sep;sup}^{\\\\mathrm {X}}$ , the other object appearing therein, is unknown and may result in a poor overall choice.\",\n \"Moreover, any degeneracy in the spectrum of ${\\\\hat{\\\\sigma }}^{ABC \\\\dots F}$ renders ${{\\\\mathrm {X}}^{\\\\star }}$ nonunique and subject to further optimization.\",\n \"These features will be at play in the example considered below where, the foregoing caveats notwithstanding, the resulting detector turns out to be fully optimal.\",\n \"The example in question is the Werner state ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ for a bipartite system of two $d$ -level parties $A$ and $B$ defined on a $d^2$ -dimensional Hilbert space [18], [19] ${\\\\hat{\\\\rho }}^{wer}_{d}(q)=\\\\frac{1}{d^2}({1-q})\\\\hat{{\\\\mathbb {1}}} + q \\\\mid {\\\\mathfrak {B}}_{1} \\\\rangle \\\\langle {\\\\mathfrak {B}}_{1}\\\\mid , $ where $\\\\mid {\\\\mathfrak {B}}_{1} \\\\rangle \\\\ =\\\\frac{1}{\\\\sqrt{d}} {\\\\sum }_{j=0}^{d-1} \\\\mid A,j \\\\rangle \\\\otimes \\\\mid B,j \\\\rangle .\",\n \"$ Here $\\\\lbrace \\\\mid A,j \\\\rangle \\\\rbrace $ and $\\\\lbrace \\\\mid B,j \\\\rangle \\\\rbrace $ ) are orthonormal bases for subsystem $A$ and $B$ respectively.\",\n \"Also, here and elsewhere we use the symbol $\\\\hat{{\\\\mathbb {1}}}$ to denote the identity operator of dimension appropriate to the context.\",\n \"Note that $\\\\mid {\\\\mathfrak {B}}_{1} \\\\rangle $ is the generalization of the two-qubit Bell state $(\\\\mid 00 \\\\rangle + \\\\mid 11 \\\\rangle )/\\\\sqrt{2}$ .\",\n \"As such, it is totally symmetric, as well as maximally entangled in the sense that its marginal states are maximally disordered and equal to $\\\\hat{{\\\\mathbb {1}}}/d$ .\",\n \"An important fact to be used in the following is that ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ is separable if and only if $q \\\\le (1+d)^{-1}$ [19].\",\n \"In order to probe the separability of ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ by means of Theorem 1, we must first choose a measurement.\",\n \"We will use the above Lemma to guide our choice of the putative optimum measurement ${\\\\mathrm {X}}^{\\\\star }$ .\",\n \"The Lemma requires that the measurement elements be equal to the orthogonal projections onto the eigenvectors of ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ .\",\n \"However, a simple calculation shows that the spectrum of ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ is degenerate, consisting of a simple eigenvalue equal to $q+{d}^{-2}(1-q)$ and a $({d}^{2}-1)$ -fold degenerate eigenvalue equal to ${d}^{-2}(1-q)$ .\",\n \"A natural choice of basis for the degenerate subspace is a set of $d^2-1$ generalized Bell states which, together with $\\\\mid {\\\\mathfrak {B}}_{1} \\\\rangle $ of Eq.\",\n \"(REF ), constitute the orthonormal eigenbasis ${\\\\lbrace \\\\mid {\\\\mathfrak {B}}_{\\\\alpha } \\\\rangle \\\\rbrace }_{\\\\alpha =1}^{d^2}$ for ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ .\",\n \"This generalized Bell basis is thus characterized by the fact that each $\\\\mid {\\\\mathfrak {B}}_{\\\\alpha } \\\\rangle $ is maximally entangled and possesses equal Schmidt coefficients.\",\n \"The measurement elements of the optimal measurement ${\\\\mathrm {X}}^{\\\\star }$ are thus given by ${\\\\hat{\\\\mathrm {E}}}_{\\\\alpha }^{{X}^{\\\\star }}= \\\\mid {\\\\mathfrak {B}}_{\\\\alpha } \\\\rangle \\\\langle {\\\\mathfrak {B}}_{\\\\alpha } \\\\mid $ , $\\\\alpha =1, 2, \\\\dots , d^2$ .\",\n \"With ${\\\\mathrm {X}}^{\\\\star }$ so defined, we readily find ${{P}}^{{\\\\mathrm {X}}^{\\\\star }}[{\\\\hat{\\\\rho }}^{wer}_{d}(q)]=&[q+{d}^{-2}(1-q), {d}^{-2}(1-q), \\\\nonumber \\\\\\\\&{d}^{-2}(1-q), \\\\ldots , {d}^{-2}(1-q)], $ which corresponds to the spectrum of ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ as expected.\",\n \"Note that the Lemma guarantees that any other rank 1 measurement in place of ${\\\\mathrm {X}}^{\\\\star }$ would produce a more disordered probability vector on the right-hand side of Eq.\",\n \"(REF ).\",\n \"The next step is the calculation of ${{P}}_{sep;sup}^{{\\\\mathrm {X}}^{\\\\star }}$ of Theorem 1, which is the least disordered probability vector that can result from the measurement of ${\\\\mathrm {X}}^{\\\\star }$ on a pure, product state of two $d$ -level subsystems.\",\n \"As can be seen in Eq.\",\n \"(REF ) of the Appendix, this calculation requires finding the maximum overlap of every measurement element $\\\\mid {\\\\mathfrak {B}}_{\\\\alpha } \\\\rangle \\\\langle {\\\\mathfrak {B}}_{\\\\alpha } \\\\mid $ with the class of pure, product states.\",\n \"This overlap is given by the square of the largest Schmidt coefficient of the respective Bell state $\\\\mid {\\\\mathfrak {B}}_{\\\\alpha }\\\\rangle $ [20].\",\n \"Since all Schmidt coefficients are equal to $1/\\\\sqrt{d}$ for every Bell state $\\\\mid {\\\\mathfrak {B}}_{\\\\alpha }\\\\rangle $ , we can conclude that ${{P}}_{sep;sup}^{{\\\\mathrm {X}}^{\\\\star }}=(1/d, 1/d, \\\\ldots , 1/d,0,0, \\\\ldots ,0).\",\n \"$ We are now in a position to apply the entanglement criterion of Theorem 1, which reads ${{P}}^{\\\\mathrm {{X}^{\\\\star }}}[{\\\\hat{\\\\rho }}^{wer}_{d}(q)] \\\\nprec {{P}}_{sep;sup}^{\\\\mathrm {{X}^{\\\\star }}}$ in this instance [21].\",\n \"Using the information in Eqs.\",\n \"(REF ) and (REF ), we readily see that this criterion translates to the single inequality $q+{d}^{-2}(1-q) > 1/d$ , or $q > 1/(1+d)$ , which is the necessary and sufficient condition for the inseparability of the Werner state ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ .\",\n \"Thus every entangled Werner state of two $d$ -level systems is detected by the measurement ${\\\\mathrm {X}}^{\\\\star }$ defined above.\",\n \"While the efficacy of the generalized Bell states for detecting entanglement in Werner states is well established [5], [6], [7], [8], we note its natural emergence from the general majorization results of this subsection.\",\n \"It should also be noted that Theorem 1 and the foregoing analysis can be extended to multipartite states.\"\n ],\n [\n \"Nonlinear detectors\",\n \"As an example of nonlinear entanglement detectors, we will derive majorization conditions for separability based on uncertainty relations.\",\n \"The method we will follow relies on the degeneracy properties of projective measurements on a single system versus the products of such measurements on a multipartite system [7], [8].\",\n \"For example, with measurement ${\\\\mathrm {X}}^{A}$ on ${\\\\hat{\\\\rho }}^{A}$ and ${\\\\mathrm {X}}^{B}$ on ${\\\\hat{\\\\rho }}^{B}$ and each having a simple spectrum, the product projective measurement ${\\\\mathrm {X}}^{A} \\\\otimes {\\\\mathrm {X}}^{B}$ on the bipartite system ${\\\\hat{\\\\rho }}^{AB}$ may be degenerate.\",\n \"A specific case is ${{\\\\hat{\\\\sigma }}_{x}}^{A}\\\\otimes {{\\\\hat{\\\\sigma }}_{x}}^{B}$ , whose spectrum $(+1/4,+1/4,-1/4,-1/4)$ is doubly degenerate, versus ${{\\\\hat{\\\\sigma }}_{x}}^{A}$ or ${{\\\\hat{\\\\sigma }}_{x}}^{B}$ each of which has the simple spectrum $(+1/2,-1/2)$ .\",\n \"Note the fact that here we are following common practice by defining ${{\\\\hat{\\\\sigma }}_{x}}^{A}\\\\otimes {{\\\\hat{\\\\sigma }}_{x}}^{B}$ to be the projective measurement whose two elements project into the subspaces corresponding to the eigenvalues $+1/4$ and $-1/4$ (and not the rank 1 projective measurement that resolves the degeneracies by virtue of using all four product elements).\",\n \"Consequently, it may happen that ${\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}$ and ${\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B}$ have a common eigenstate while ${\\\\mathrm {X}}^{A}$ and ${\\\\mathrm {Y}}^{A}$ do not, and that the said common eigenstate is an entangled pure state.\",\n \"In such cases, the corresponding probability vectors will reflect the stated differences and may be capable of detecting entanglement as in the case of linear detectors.\",\n \"Consider two product projective measurements ${\\\\mathrm {X}}^{A} \\\\otimes {\\\\mathrm {X}}^{B}$ and ${\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B}$ performed on the bipartite state ${\\\\hat{\\\\rho }}^{AB}$ .\",\n \"The measurement results are then given by ${{P}}^{{\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}}({\\\\hat{\\\\rho }}^{AB})= \\\\textrm {tr}&[{\\\\hat{\\\\Pi }}^{{\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}} {\\\\hat{\\\\rho }}^{AB} ], \\\\nonumber \\\\\\\\ {{P}}^{{\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B}}({\\\\hat{\\\\rho }}^{AB})= \\\\textrm {tr}&[{\\\\hat{\\\\Pi }}^{{\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B}} {\\\\hat{\\\\rho }}^{AB} ], \\\\nonumber \\\\\\\\{P}^{ ({\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}) \\\\oplus ({\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B})}({\\\\hat{\\\\rho }}^{AB}) &={P}^{ {\\\\mathrm {X}}^{A} \\\\otimes {\\\\mathrm {X}}^{B} }({\\\\hat{\\\\rho }}^{AB})\\\\nonumber \\\\\\\\\\\\otimes {P}^{{\\\\mathrm {Y}}^{A} \\\\otimes {\\\\mathrm {Y}}^{B} }({\\\\hat{\\\\rho }}^{AB}), $ where ${\\\\hat{\\\\Pi }}^{{\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}}$ and ${\\\\hat{\\\\Pi }}^{{\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B}}$ , both projection operators but not necessarily rank 1, represent the measurement elements of the two product measurements.\",\n \"Note the majorization definition of joint uncertainty, given earlier, at work on the last line of Eq.\",\n \"(REF ).\",\n \"Under these conditions, we have the following general result.\",\n \"Theorem 2.\",\n \"The results of measurements ${\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}$ and ${\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B}$ on a separable state ${\\\\hat{\\\\rho }}^{AB}_{sep}$ satisfy ${P}^{ ({\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}) \\\\oplus ({\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B})}({\\\\hat{\\\\rho }}^{AB}_{sep}) \\\\prec {P}_{sup}^{ {\\\\mathrm {X}}\\\\oplus {\\\\mathrm {Y}}}, $ where ${P}_{sup}^{ {\\\\mathrm {X}}\\\\oplus {\\\\mathrm {Y}}}$ is defined in Eq.\",\n \"(REF ), hence the condition ${P}^{ ({\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}) \\\\oplus ({\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B})}({\\\\hat{\\\\sigma }}^{AB}) \\\\nprec {P}_{sup}^{ {\\\\mathrm {X}}\\\\oplus {\\\\mathrm {Y}}}$ implies that the state ${\\\\hat{\\\\sigma }}^{AB}$ is entangled.\",\n \"To establish Theorem 2, consider product states of the form ${\\\\hat{\\\\rho }}^{A} \\\\otimes {\\\\hat{\\\\rho }}^{B}$ .\",\n \"Then Lemma 1 of Ref.\",\n \"[8] asserts that ${P}^{ {\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B} }({\\\\hat{\\\\rho }}^{A} \\\\otimes {\\\\hat{\\\\rho }}^{B}) &\\\\prec {P}^{ {\\\\mathrm {X}}^{A}}({\\\\hat{\\\\rho }}^{A}), \\\\nonumber \\\\\\\\{P}^{ {\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B} }({\\\\hat{\\\\rho }}^{A} \\\\otimes {\\\\hat{\\\\rho }}^{B}) &\\\\prec {P}^{ {\\\\mathrm {Y}}^{A}}({\\\\hat{\\\\rho }}^{A}).\",\n \"$ Using these relations and Eq.\",\n \"(REF ), we find [22] ${P}^{ ({\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}) \\\\oplus ({\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B})}({\\\\hat{\\\\rho }}^{A} \\\\otimes {\\\\hat{\\\\rho }}^{B}) \\\\prec {P}^{ {\\\\mathrm {X}}^{A}}({\\\\hat{\\\\rho }}^{A})\\\\otimes {P}^{ {\\\\mathrm {Y}}^{A}}({\\\\hat{\\\\rho }}^{A}).\",\n \"$ Since the right-hand side of Eq.\",\n \"(REF ) is by definition majorized by ${P}_{sup}^{ {\\\\mathrm {X}}\\\\oplus {\\\\mathrm {Y}}}$ , we conclude that ${P}^{ ({\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}) \\\\oplus ({\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B})}({\\\\hat{\\\\rho }}^{A} \\\\otimes {\\\\hat{\\\\rho }}^{B}) \\\\prec {P}_{sup}^{ {\\\\mathrm {X}}\\\\oplus {\\\\mathrm {Y}}}.\",\n \"$ At this point we note that ${P}^{ ({\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B}) \\\\oplus ({\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B})}_{sep;sup}$ , which by definition majorizes all probability vectors that can appear on the left-hand side of Eq.\",\n \"(REF ), can be found among pure, product states [16].\",\n \"This assertion is established in Eq.\",\n \"(REF ) et seq.\",\n \"of the Appendix.\",\n \"Consequently, the product state ${\\\\hat{\\\\rho }}^{A} \\\\otimes {\\\\hat{\\\\rho }}^{B}$ in Eq.\",\n \"(REF ) may be replaced by any separable state ${\\\\hat{\\\\rho }}^{AB}_{sep}$ , thereby establishing Eq.\",\n \"(REF ) and Theorem 2.\",\n \"We note in passing that, because the measurements considered in Theorem 2 are in general not rank 1, the Lemma which we used earlier for detector optimization cannot be applied here.\",\n \"To illustrate Theorem 2, we will consider the case of three mutually unbiased observables measured on bipartite states of two-level systems, as in §IVB of Paper I.\",\n \"In effect, this amounts to measuring products of the three spin components of a pair of spin-1/2 systems (or qubits).\",\n \"For this case, Eq.\",\n \"(REF ) reads ${P}^{ ({{\\\\sigma }_{x}}^{A}\\\\otimes {{\\\\sigma }_{x}}^{B}) \\\\oplus ({{\\\\sigma }_{y}}^{A}\\\\otimes {{\\\\sigma }_{y}}^{B})\\\\oplus ({{\\\\sigma }_{z}}^{A}\\\\otimes {{\\\\sigma }_{z}}^{B})}({\\\\hat{\\\\rho }}^{AB}_{sep})\\\\prec {P}_{sup}^{{\\\\sigma }_{x}\\\\oplus {\\\\sigma }_{y}\\\\oplus {\\\\sigma }_{z}}, $ where ${\\\\hat{\\\\rho }}^{AB}_{sep}$ is any separable state of two qubits.\",\n \"Thus a violation of this relation by a two-qubit state implies that it is entangled.\",\n \"For the set of states ${\\\\hat{\\\\rho }}^{AB}$ to be probed by Theorem 2, we will consider the Werner family of two-qubit states ${\\\\hat{\\\\rho }}^{wer}(q)=\\\\frac{1}{4}({1-q}){\\\\mathbb {1}}+ q \\\\mid {\\\\mathfrak {B}}_{1} \\\\rangle \\\\langle {\\\\mathfrak {B}}_{1} \\\\mid , $ which is known to be entangled if and only if $q >1/3$ .\",\n \"Here $0 \\\\le q \\\\le 1$ and $\\\\mid {\\\\mathfrak {B}}_{1} \\\\rangle =(\\\\mid 00 \\\\rangle + \\\\mid 11 \\\\rangle )/\\\\sqrt{2}$ is the totally symmetric Bell state of Eq.\",\n \"(REF ) for the present case.\",\n \"A calculation of the probability vector for this measurement gives $(1 \\\\pm q)(1 \\\\pm q)(1 \\\\pm q)/8$ for the 8 components of ${P}^{({{\\\\sigma }_{x}}^{A}\\\\otimes {{\\\\sigma }_{x}}^{B}) \\\\oplus ({{\\\\sigma }_{y}}^{A}\\\\otimes {{\\\\sigma }_{y}}^{B})\\\\oplus ({{\\\\sigma }_{z}}^{A}\\\\otimes {{\\\\sigma }_{z}}^{B})}[{\\\\hat{\\\\rho }}^{wer}(q)]$ .\",\n \"Arranged in a non-ascending order, these are ${(1+q)}^{3}/8$ , ${(1+q)}^{2}{(1-q)}/8$ and ${(1+q)}{(1-q)}^{2}/8$ both three-fold degenerate, and ${(1-q)}^{3}/8$ .\",\n \"The supremum ${P}_{sup}^{{\\\\sigma }_{x}\\\\oplus {\\\\sigma }_{y}\\\\oplus {\\\\sigma }_{z}}$ , on the other hand, was calculated in §IVB of Paper I, where it was found as ${P}_{sup}^{{\\\\sigma }_{x}\\\\oplus {\\\\sigma }_{y}\\\\oplus {\\\\sigma }_{z}}=\\\\frac{1}{8}\\\\big [{(1+1/\\\\sqrt{3})}^{3},2{(1+1/\\\\sqrt{2})}^{2} \\\\nonumber \\\\\\\\-{(1+1/\\\\sqrt{3})}^{3}, 4-{(1+1/\\\\sqrt{2})}^{2}, 4-{(1+1/\\\\sqrt{2})}^{2}\\\\nonumber \\\\\\\\,0,0,0,0 \\\\big ].\",\n \"$ According to Theorem 2, ${\\\\hat{\\\\rho }}^{wer}(q)$ violates the separability condition of Eq.\",\n \"(REF ) if ${P}_{sup}^{{\\\\sigma }_{x}\\\\oplus {\\\\sigma }_{y}\\\\oplus {\\\\sigma }_{z}}$ fails to majorize ${P}^{({{\\\\sigma }_{x}}^{A}\\\\otimes {{\\\\sigma }_{x}}^{B}) \\\\oplus ({{\\\\sigma }_{y}}^{A}\\\\otimes {{\\\\sigma }_{y}}^{B})\\\\oplus ({{\\\\sigma }_{z}}^{A}\\\\otimes {{\\\\sigma }_{z}}^{B})}[{\\\\hat{\\\\rho }}^{wer}(q)]$ , which occurs for $q > 1/\\\\sqrt{3}=0.577$ in this example.\",\n \"The entanglement in ${\\\\hat{\\\\rho }}^{wer}(q)$ is thus detected by Theorem 2 for $q > 0.577$ , and missed for the range $0.333 < q \\\\le 0.577$ .\",\n \"By comparison, a similar method based on the Shannon entropy in Ref.\",\n \"[7] detects entanglement in ${\\\\hat{\\\\rho }}^{wer}(q)$ for $q >0.65$ .\",\n \"Reference [8], using the same method but relying on the family of Tsallis entropies, matches our condition $q > 0.577$ by means of a numerical calculation that searches over the Tsallis family for optimum performance.\",\n \"We end this subsection by a brief description of a sharpened version of the Nielsen-Kempe theorem as a nonlinear entanglement detector [12].\",\n \"This celebrated theorem is an elegant separability condition based on majorization relations.\",\n \"It asserts that the spectrum of a separable bipartite state is majorized by each of its marginal spectra.\",\n \"By extension, this theorem guarantees that the spectra of all possible subsystems of a separable multipartite system must majorize its global spectrum [14].\",\n \"Our version of this theorem employs the notion of the infimum of a set of probability vectors discussed in §IIA, and is based on the observation that if a probability vector is majorized by each member of a set of probability vectors, then it must also be majorized by the infimum of that set.\",\n \"Consider the multipartite state ${\\\\hat{\\\\rho }}^{ABC \\\\dots F}$ together with all its subsystem states ${\\\\lbrace {\\\\hat{\\\\rho }}^{{X}_{a}} \\\\rbrace }_{a=1}^{f}$ and subsystem spectra ${\\\\lbrace {\\\\lambda }^{{X}_{a}} \\\\rbrace }_{a=1}^{f}$ , where ${\\\\lbrace {{X}_{a}} \\\\rbrace }_{a=1}^{f}$ represent all proper subsets of the set of parties $(A, B, C, \\\\dots , F)$ .\",\n \"Then the infimum of the subsystem spectra, ${\\\\Lambda }^{ABC \\\\dots F}={\\\\inf }[{\\\\lambda }^{{X}_{1}},{\\\\lambda }^{{X}_{2}}, \\\\ldots , {\\\\lambda }^{{X}_{f}}]$ , embodies the subsystem disorder of the state ${\\\\hat{\\\\rho }}^{ABC \\\\dots F}$ , as expressed in the following theorem.\",\n \"Theorem 3 (Nielsen-Kempe).\",\n \"A multipartite state is entangled if its system disorder fails to exceed its subsystem disorder, i.e., ${\\\\lambda }^{ABC \\\\dots F}$ is entangled if ${\\\\lambda }^{ABC \\\\dots F}\\\\nprec {\\\\Lambda }^{ABC \\\\dots F}$ .\"\n ],\n [\n \"Quasi-Entropic Detectors\",\n \"The majorization based theorems of the previous section have direct corollaries that yield scalar entanglement detectors for the entire class of Schur-concave measures.\",\n \"A Schur-concave function $G$ is defined by the property that ${\\\\lambda }^{1} \\\\prec {\\\\lambda }^{2}$ implies $G({\\\\lambda }^{1}) \\\\ge G({\\\\lambda }^{2})$ [23].\",\n \"Equivalently, Schur-concave functions are characterized by being monotonic with respect to the majorization relation.\",\n \"They include all functions $G(\\\\cdot )$ that are concave and symmetric with respect to the arguments, a subset which we have defined as quasi-entropic.\",\n \"An important subclass of quasi-entropic functions is obtained if we restrict $G$ to have the following trace structure: $G(\\\\lambda )=\\\\mathrm {tr}[g(\\\\lambda )]= {\\\\sum }_{\\\\alpha } g({\\\\lambda }_{\\\\alpha }), $ where $g(\\\\cdot )$ is a concave function of a single variable and $\\\\lambda $ is treated as a diagonal matrix for the purpose of calculating the trace.\",\n \"Note that $G(\\\\cdot )$ as constructed in Eq.\",\n \"(REF ) is manifestly symmetric and, as a sum of concave functions, it is also concave.\",\n \"We shall refer to this class of functions, which include the Shannon and Tsallis entropies, as trace type quasi-entropic.\",\n \"An important fact regarding this class of measures is that if, for a given pair of probability vectors $\\\\lambda $ and $\\\\mu $ , we have $G({\\\\lambda })>G({\\\\mu })$ for every trace type quasi-entropic measure $G$ , then $\\\\lambda \\\\prec \\\\mu $ .\",\n \"The standard entropic measure of uncertainty [9], which is based on the Shannon entropy function, is trace type quasi-entropic and corresponds to the choice $g(x)=H(x)=-x \\\\ln (x)$ in Eq.\",\n \"(REF ).\",\n \"An example of an information theoretically relevant measure that is quasi-entropic but not trace type is the Rényi subfamily of entropies of order less than one [24].\",\n \"Accordingly, although the scalar entanglement detectors that follow from the theorems of §III actually hold for the entire class of Schur-concave measures, we will continue to focus on the quasi-entropic class as the most suitable for information theoretical applications.\",\n \"Scalar entanglement detectors based on Shannon, Tsallis, and Rényi entropies are already well known, albeit with detection bounds that may differ from those reported here [7], [8], [5].\",\n \"Our primary purpose here is to emphasize the natural and categorical manner in which majorization based entanglement detectors yield entire classes of scalar detectors.\",\n \"With $G(\\\\cdot )$ a quasi-entropic function, the aforementioned monotonicity property implies the following corollaries to Theorems 1-3.\",\n \"Corollary 1.\",\n \"Under the conditions of Theorem 1, the multipartite state ${\\\\hat{\\\\sigma }}^{ABC \\\\dots F}$ is entangled if $G[{{P}}^{\\\\mathrm {X}}({\\\\hat{\\\\sigma }}^{ABC \\\\dots F})] < G[{{P}}_{sep;sup}^{\\\\mathrm {X}}].\",\n \"$ Corollary 2.\",\n \"Under the conditions of Theorem 2, the bipartite state ${\\\\hat{\\\\sigma }}^{AB}$ is entangled if $G[{P}^{ ({\\\\mathrm {X}}^{A}\\\\otimes {\\\\mathrm {X}}^{B})}({\\\\hat{\\\\sigma }}^{AB})]+ G[{P}^{ ({\\\\mathrm {Y}}^{A}\\\\otimes {\\\\mathrm {Y}}^{B})} ({\\\\hat{\\\\sigma }}^{AB})] < G[{P}_{sup}^{ {\\\\mathrm {X}}\\\\oplus {\\\\mathrm {Y}}}].\",\n \"$ Corollary 3.\",\n \"Under the conditions of Theorem 3, the multipartite state ${\\\\hat{\\\\sigma }}^{ABC \\\\dots F}$ is entangled if $G[{\\\\lambda }^{ABC \\\\dots F}] < G[{\\\\Lambda }^{ABC \\\\dots F}].\",\n \"$ As an application of the above corollaries, we will consider the entanglement detection threshold for the bipartite Werner state considered in Eq.\",\n \"(REF ) et seq.\",\n \"Using Corollary 1 together with the Tsallis entropy function, we find that ${\\\\hat{\\\\rho }}^{wer}_{d}(q)$ is entangled if ${S}^{tsa}_{r}\\\\big [ {{P}}^{{\\\\mathrm {X}}^{\\\\star }}[{\\\\hat{\\\\rho }}^{wer}_{d}(q)] \\\\big ] < {S}^{tsa}_{r}\\\\big [ {{P}}_{sep;sup}^{{\\\\mathrm {X}}^{\\\\star }} \\\\big ], $ where ${S}^{tsa}_{r}(\\\\cdot )$ is the Tsallis entropy of order $r$ , and the two probability vectors appearing in Eq.\",\n \"(REF ) are those in Eqs.\",\n \"(REF ) and (REF ).\",\n \"The Tsallis entropy function ${S}^{tsa}_{r}(\\\\cdot )$ corresponds to the choice $g(\\\\lambda )= (\\\\lambda -{\\\\lambda }^{r})/(r-1)$ , $1 1/(1+d)$ , the known entanglement threshold for a the two-qudit Werner state.\",\n \"This limiting case corresponds to the bottom graph in Fig.\",\n \"REF .\",\n \"The foregoing discussion of how well Corollary 1 does in detecting entanglement notwithstanding, the aim of this section is not to promote quasi-entropic detectors given in Corollaries 1-3 per se, but rather to emphasize the reach of the majorization results of §III whence they are inherited.\",\n \"In addition, the above example highlights the fact that, while the entanglement detection bounds given in Corollaries 1-3 are optimal for Schur-concave functions as a class, their performance need not be optimal for individual members of the class, a fact that was emphasized in paper I as well.\"\n ],\n [\n \"Concluding Remarks\",\n \"In this paper we have developed entanglement detection criteria within the majorization formulation of uncertainty presented in paper I.\",\n \"These are inherently stronger than similar scalar conditions in the sense that they are equivalent to and imply infinite classes of such scalar criteria.\",\n \"Majorization relations in effect envelope the set of scalar measures of disorder based on Schur-concave functions, a huge set that includes the information-theoretically relevant class of quasi-entropic measures as a subset.\",\n \"This enveloping property elucidates the exceptional effectiveness of majorization relations in dealing with problems of quantum information theory.\",\n \"Entanglement detection criteria that can be experimentally implemented are especially useful in studies involving entangled microsystems, hence the importance of linear detectors and entanglement witnesses.\",\n \"As already mentioned, Theorem 1 can be formulated as an entanglement witness inasmuch as it is a linear majorization condition on measurement results.\",\n \"What may not be so obvious is that in principle the spectrum of a state is also experimentally accessible on the basis of the Lemma of §IIIA.\",\n \"That Lemma states that the spectrum of a quantum state is the supremum of probability vectors that may be deduced from the counter statistics of all possible rank 1 generalized measurements performed on the state.\",\n \"In practice, this provides an experimental method for estimating the spectrum, since a high precision determination may require a search over a large number of possible measurements.\",\n \"There is already an extensive literature on entanglement detection with many important results.\",\n \"With the notable exception of the Nielsen-Kempe theorem, these results are not formulated within the majorization framework.\",\n \"The present contribution in part serves to demonstrate the effectiveness of the majorization formulation of some of the existing methods.\",\n \"One can reasonably expect a sharpening of the results of some of the other entanglement detection strategies when reformulated within the majorization framework.\",\n \"It is also not unreasonable to expect useful, albeit computationally intractable, necessary and sufficient separability criteria to emerge from such formulations.\",\n \"This work was in part supported by a grant from California State University, Sacramento.\",\n \"*\"\n ],\n [\n \"majorization bounds are found on pure states\",\n \"Our task here is to establish that majorization uncertainty bounds can be reached on pure states.\",\n \"We will prove this for two important cases, first for the uncertainty bound on all states of the system, and second on the class of separable states.\",\n \"To avoid unnecessary clutter and technical distractions, we will limit the number of generalized measurements to three and the underlying Hilbert spaces to finite dimensions in the following treatment.\",\n \"Suppose generalized measurements ($\\\\mathrm {X}, \\\\lbrace \\\\hat{\\\\mathrm {E}}_{\\\\alpha }^{X} \\\\rbrace $ ), ($\\\\mathrm {Y}, \\\\lbrace \\\\hat{\\\\mathrm {E}}_{\\\\beta }^{Y} \\\\rbrace $ ), and ($\\\\mathrm {Z}, \\\\lbrace \\\\hat{\\\\mathrm {E}}_{\\\\beta }^{Z} \\\\rbrace $ ) are performed on a state $\\\\hat{\\\\rho }$ that is supported on a finite-dimensional Hilbert space.\",\n \"What we will show below is that the search for the components of ${\\\\mu }^{sup}$ which serve to define ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\mathrm {Z}}$ can be limited to pure states.\",\n \"By definition, ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\mathrm {Z}}={\\\\sup }_{\\\\hat{\\\\rho }}[{{P}}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\mathrm {Z}}(\\\\hat{\\\\rho })], $ where ${{P}}_{\\\\alpha \\\\beta \\\\gamma }^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\mathrm {Z}}(\\\\hat{\\\\rho })=\\\\textrm {tr}({\\\\hat{\\\\mathrm {E}}}_{\\\\alpha }^{X} \\\\hat{\\\\rho }) \\\\textrm {tr}({\\\\hat{\\\\mathrm {E}}}_{\\\\beta }^{Y} \\\\hat{\\\\rho }) \\\\textrm {tr}({\\\\hat{\\\\mathrm {E}}}_{\\\\gamma }^{Z} \\\\hat{\\\\rho }).\",\n \"$ We recall from §IIA above and §IIB, IVA and IVB of Paper I that the supremum defined by Eqs.\",\n \"(REF ) and (REF ) is found by calculating ${\\\\mu }^{sup}_{i}$ for $i=1,2,\\\\ldots $ (${\\\\mu }^{sup}_{0}=0$ by definition).\",\n \"Furthermore, ${\\\\mu }^{sup}_{1}$ is the maximum value of a single component of ${{P}}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\mathrm {Z}}(\\\\hat{\\\\rho })$ as $\\\\hat{\\\\rho }$ is varied, ${\\\\mu }^{sup}_{2}$ is the maximum value of the sum of two (different) components of ${{P}}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\mathrm {Z}}(\\\\hat{\\\\rho })$ as $\\\\hat{\\\\rho }$ is varied, and ${\\\\mu }^{sup}_{i}$ is the maximum value of the sum of $i$ (different) components of ${{P}}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\mathrm {Z}}(\\\\hat{\\\\rho })$ as $\\\\hat{\\\\rho }$ is varied.\",\n \"It is thus sufficient to prove that the said maximum for ${\\\\mu }^{sup}_{i}$ is in fact realized on a pure state.\",\n \"Let $\\\\hat{\\\\rho }={\\\\sum }_{a}\\\\, {\\\\lambda }_{a}(\\\\hat{\\\\rho }) \\\\mid {\\\\psi }_{a} \\\\rangle \\\\langle {\\\\psi }_{a} \\\\mid $ be the principal ensemble representation for $\\\\hat{\\\\rho }$ .\",\n \"Then the above maximization process defines ${\\\\mu }^{sup}_{i}$ as the maximum of ${\\\\sum }_{a,b,c}{\\\\lambda }_{a}(\\\\hat{\\\\rho }){\\\\lambda }_{b}(\\\\hat{\\\\rho }){\\\\lambda }_{c}(\\\\hat{\\\\rho }) \\\\big [ \\\\langle {\\\\psi }_{a} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\alpha }_{1}}^{X}\\\\mid {\\\\psi }_{a} \\\\rangle \\\\langle {\\\\psi }_{b} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\beta }_{1}}^{Y}\\\\mid {\\\\psi }_{b} \\\\rangle \\\\nonumber \\\\\\\\ \\\\times \\\\langle {\\\\psi }_{c} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\gamma }_{1}}^{Z}\\\\mid {\\\\psi }_{c} \\\\rangle + \\\\langle {\\\\psi }_{a} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\alpha }_{2}}^{X}\\\\mid {\\\\psi }_{a} \\\\rangle \\\\langle {\\\\psi }_{b} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\beta }_{2}}^{Y}\\\\mid {\\\\psi }_{b} \\\\rangle \\\\nonumber \\\\\\\\ \\\\times \\\\langle {\\\\psi }_{c} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\gamma }_{2}}^{Z}\\\\mid {\\\\psi }_{c} \\\\rangle + \\\\ldots \\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\, \\\\nonumber \\\\\\\\+ \\\\langle {\\\\psi }_{a} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\alpha }_{i}}^{X}\\\\mid {\\\\psi }_{a} \\\\rangle \\\\langle {\\\\psi }_{b} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\beta }_{i}}^{Y}\\\\mid {\\\\psi }_{b} \\\\rangle \\\\langle {\\\\psi }_{c} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\gamma }_{i}}^{Z}\\\\mid {\\\\psi }_{c} \\\\rangle \\\\big ] \\\\,\\\\,\\\\,\\\\,\\\\,\\\\, \\\\nonumber \\\\\\\\+\\\\,\\\\, {\\\\xi }_{i} \\\\big (1-{\\\\sum }_{a} {\\\\lambda }^{\\\\rho }_{a}\\\\big )+{\\\\sum }_{a,b} {\\\\eta }_{i,ab} \\\\big ({\\\\delta }_{ab}-\\\\langle {\\\\psi }_{b}\\\\mid {\\\\psi }_{a} \\\\rangle \\\\big ), $ as the state vectors $\\\\lbrace \\\\mid {\\\\psi }_{a}\\\\rangle \\\\rbrace $ which are the eigenstates of $\\\\hat{\\\\rho }$ , the probabilities $\\\\lbrace {\\\\lambda }_{a}(\\\\hat{\\\\rho }) \\\\rbrace $ which constitute the spectrum of $\\\\hat{\\\\rho }$ , and the Lagrange multipliers ${\\\\xi }_{i}$ and ${\\\\eta }_{i,ba}$ which serve to enforce the structure of ${\\\\sum }_{a}\\\\, {\\\\lambda }^{\\\\rho }_{a} \\\\mid {\\\\psi }_{a} \\\\rangle \\\\langle {\\\\psi }_{a} \\\\mid $ as an orthogonal ensemble are varied.\",\n \"The symbol ${\\\\delta }_{ab}$ in the above expression is the Kronecker delta, and the three sets of indices $\\\\lbrace {({\\\\alpha }, {\\\\beta }, {\\\\gamma })}_{1}, {({\\\\alpha }, {\\\\beta }, {\\\\gamma })}_{2}, \\\\dots , {({\\\\alpha }, {\\\\beta }, {\\\\gamma })}_{i} \\\\rbrace $ are understood to be distinct, i.e., ${\\\\alpha }_{1} \\\\ne {\\\\alpha }_{2} \\\\ne \\\\ldots \\\\ne {\\\\alpha }_{i}$ , and similarly for the other two sets.\",\n \"Note also that since $\\\\langle {\\\\psi }_{a}\\\\mid {\\\\psi }_{b} \\\\rangle $ is in general a Hermitian matrix (in the indices $a$ and $b$ ), the Lagrange multipliers ${\\\\eta }_{i,ba}$ may also be taken to constitute a Hermitian matrix.\",\n \"A variation with respect to $\\\\langle {\\\\psi }_{a} \\\\mid $ gives ${\\\\lambda }_{a}(\\\\hat{\\\\rho }) \\\\big [ {\\\\hat{\\\\mathcal {E}}}_{i}^{X}+ {\\\\hat{\\\\mathcal {E}}}_{i}^{Y}+{\\\\hat{\\\\mathcal {E}}}_{i}^{Z} \\\\big ]\\\\mid {\\\\psi }_{a} \\\\rangle ={\\\\sum }_{b} {\\\\eta }_{i,ba} \\\\mid {\\\\psi }_{b} \\\\rangle , $ where ${\\\\hat{\\\\mathcal {E}}}_{i}^{X}={\\\\sum }_{k=1}^{i} {P}^{\\\\mathrm {Y}}_{{\\\\beta }_{k}}(\\\\hat{\\\\rho }){P}^{\\\\mathrm {Z}}_{{\\\\gamma }_{k}}(\\\\hat{\\\\rho }){\\\\hat{\\\\mathrm {E}}}_{{\\\\alpha }_{k}}^{X}, $ and ${\\\\hat{\\\\mathcal {E}}}_{i}^{Z}$ and ${\\\\hat{\\\\mathcal {E}}}_{i}^{Y}$ are defined similarly.\",\n \"Note that the three operators introduced in Eqs.\",\n \"(REF ) and (REF ) are Hermitian and positive.\",\n \"Let ${\\\\hat{\\\\mathcal {E}}}_{i}={\\\\hat{\\\\mathcal {E}}}_{i}^{X}+ {\\\\hat{\\\\mathcal {E}}}_{i}^{Y}+{\\\\hat{\\\\mathcal {E}}}_{i}^{Z}$ and ${\\\\mathcal {E}}_{i,aa}=\\\\langle {\\\\psi }_{a} \\\\mid {\\\\hat{\\\\mathcal {E}}}_{i} \\\\mid {\\\\psi }_{a} \\\\rangle $ .\",\n \"Then the maximum sought in (REF ), namely ${\\\\mu }^{sup}_{i}$ , is given by ${\\\\sum }_{a}{\\\\lambda }_{a}(\\\\hat{\\\\rho }){\\\\mathcal {E}}_{i,aa}/3=\\\\textrm {tr}[{\\\\hat{\\\\mathcal {E}}}_{i}\\\\hat{\\\\rho }]/3$ .\",\n \"A key fact at this juncture is that the diagonal elements ${\\\\mathcal {E}}_{i,aa}$ do not depend on $a$ and are all equal.\",\n \"In other words, each pure state $\\\\mid {\\\\psi }_{a} \\\\rangle $ in the ensemble contributes equally to ${\\\\mu }^{sup}_{i}$ .\",\n \"This property follows from a variation of (REF ) with respect to ${\\\\lambda }_{a}(\\\\hat{\\\\rho })$ , which leads to ${\\\\xi }_{i}=\\\\langle {\\\\psi }_{a} \\\\mid {\\\\hat{\\\\mathcal {E}}}_{i}^{X}+ {\\\\hat{\\\\mathcal {E}}}_{i}^{Y}+{\\\\hat{\\\\mathcal {E}}}_{i}^{Z} \\\\mid {\\\\psi }_{a} \\\\rangle ={\\\\mathcal {E}}_{i,aa}.\",\n \"$ Clearly then, each pure state $\\\\mid {\\\\psi }_{a} \\\\rangle $ in the ensemble must realize the same maximum ${\\\\mu }^{sup}_{i}$ as the entire ensemble $\\\\hat{\\\\rho }$ .\",\n \"We conclude therefore that the sought maximum can be found among the pure states of the system.\",\n \"What if we are looking for ${\\\\mu }^{sup}_{i}$ but with $\\\\hat{\\\\rho }$ limited to separable states?\",\n \"It turns out that here too the search can be limited to pure states, which would be pure product states in this instance.\",\n \"To establish this result, we will modify the foregoing analysis by stipulating that $\\\\hat{\\\\rho }={\\\\sum }_{a}\\\\, {q}_{a} \\\\mid {\\\\phi }^{A}_{a} \\\\rangle \\\\langle {\\\\phi }^{A}_{a} \\\\mid \\\\otimes \\\\mid {\\\\phi }^{B}_{a} \\\\rangle \\\\langle {\\\\phi }^{B}_{a} \\\\mid $ representing a separable, bipartite state of parties $A$ and $B$ .\",\n \"Then the corresponding ${\\\\mu }^{sup}_{sep,i}$ is the maximum of ${\\\\sum }_{a,b,c}{q}_{a}{q}_{b}{q}_{c} \\\\big [\\\\langle {\\\\phi }^{B}_{a}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{a} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\alpha }_{1}}^{X}\\\\mid {\\\\phi }^{A}_{a} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{a} \\\\rangle \\\\nonumber \\\\\\\\\\\\times \\\\langle {\\\\phi }^{B}_{b}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{b} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\beta }_{1}}^{Y}\\\\mid {\\\\phi }^{A}_{b} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{b} \\\\rangle \\\\nonumber \\\\\\\\\\\\times \\\\langle {\\\\phi }^{B}_{c}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{c} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\gamma }_{1}}^{Z}\\\\mid {\\\\phi }^{A}_{c} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{c} \\\\rangle \\\\nonumber \\\\\\\\+ \\\\langle {\\\\phi }^{B}_{a}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{a} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\alpha }_{2}}^{X}\\\\mid {\\\\phi }^{A}_{a} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{a} \\\\rangle \\\\nonumber \\\\\\\\\\\\times \\\\langle {\\\\phi }^{B}_{b}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{b} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\beta }_{2}}^{Y}\\\\mid {\\\\phi }^{A}_{b} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{b} \\\\rangle \\\\nonumber \\\\\\\\\\\\times \\\\langle {\\\\phi }^{B}_{c}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{c} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\gamma }_{2}}^{Z}\\\\mid {\\\\phi }^{A}_{c} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{c} \\\\rangle \\\\nonumber \\\\\\\\+ \\\\ldots \\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\, \\\\,\\\\,\\\\,\\\\, \\\\nonumber \\\\\\\\+ \\\\langle {\\\\phi }^{B}_{a}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{a} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\alpha }_{i}}^{X}\\\\mid {\\\\phi }^{A}_{a} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{a} \\\\rangle \\\\nonumber \\\\\\\\\\\\times \\\\langle {\\\\phi }^{B}_{b}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{b} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\beta }_{i}}^{Y}\\\\mid {\\\\phi }^{A}_{b} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{b} \\\\rangle \\\\nonumber \\\\\\\\\\\\times \\\\langle {\\\\phi }^{B}_{c}\\\\mid \\\\otimes \\\\langle {\\\\phi }^{A}_{c} \\\\mid {\\\\hat{\\\\mathrm {E}}}_{{\\\\gamma }_{i}}^{Z}\\\\mid {\\\\phi }^{A}_{c} \\\\rangle \\\\otimes \\\\mid {\\\\phi }^{B}_{c} \\\\rangle \\\\nonumber \\\\\\\\+ {\\\\sum }_{a} {\\\\eta }_{i,a}^{A} \\\\big [ 1-\\\\langle {\\\\phi }^{A}_{a}\\\\mid {\\\\phi }^{A}_{a} \\\\rangle ]+{\\\\sum }_{a} {\\\\eta }_{i,a}^{B} \\\\big [ 1-\\\\langle {\\\\phi }^{B}_{a}\\\\mid {\\\\phi }^{B}_{a} \\\\rangle ] \\\\nonumber \\\\\\\\+ {\\\\xi }_{i} \\\\big (1-{\\\\sum }_{a} {q}_{a}\\\\big ), \\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\, $ as the state vectors $ \\\\lbrace \\\\mid {\\\\phi }^{A}_{a} \\\\rangle \\\\rbrace $ , $\\\\lbrace \\\\mid {\\\\phi }^{B}_{a} \\\\rangle \\\\rbrace $ , the probabilities $\\\\lbrace {q}_{a} \\\\rbrace $ , and the Lagrange multipliers ${\\\\eta }_{i,a}^{A}$ , ${\\\\eta }_{i,a}^{B}$ , and ${\\\\xi }_{i}$ which serve to enforce normalization conditions and probability conservation for the ensemble are varied.\",\n \"Variations with respect to $ \\\\langle {\\\\phi }^{A}_{a} \\\\mid $ and $ \\\\langle {\\\\phi }^{B}_{a} \\\\mid $ now give ${q}_{a} \\\\big [ \\\\langle {\\\\phi }^{B}_{a}\\\\mid ({\\\\hat{\\\\mathcal {E}}}_{i}^{X}+ {\\\\hat{\\\\mathcal {E}}}_{i}^{Y}+{\\\\hat{\\\\mathcal {E}}}_{i}^{Z}) \\\\mid {\\\\phi }^{B}_{a} \\\\rangle \\\\big ] \\\\mid {\\\\phi }^{A}_{a} \\\\rangle &={\\\\eta }_{i,a}^{A} \\\\mid {\\\\phi }^{A}_{a} \\\\rangle \\\\nonumber \\\\\\\\{q}_{a} \\\\big [ \\\\langle {\\\\phi }^{A}_{a}\\\\mid ({\\\\hat{\\\\mathcal {E}}}_{i}^{X}+ {\\\\hat{\\\\mathcal {E}}}_{i}^{Y}+{\\\\hat{\\\\mathcal {E}}}_{i}^{Z}) \\\\mid {\\\\phi }^{A}_{a} \\\\rangle \\\\big ] \\\\mid {\\\\phi }^{B}_{a} \\\\rangle &={\\\\eta }_{i,a}^{B} \\\\mid {\\\\phi }^{B}_{a} \\\\rangle , $ where $({\\\\hat{\\\\mathcal {E}}}_{i}^{X},{\\\\hat{\\\\mathcal {E}}}_{i}^{Y},{\\\\hat{\\\\mathcal {E}}}_{i}^{Z})$ are defined as in Eq.\",\n \"(REF ) et seq.\",\n \"Equations (REF ) directly imply that ${\\\\eta }_{i,a}^{A}={\\\\eta }_{i,a}^{B}$ , and as in the general case above, the maximum sought in (REF ), namely ${\\\\mu }^{sup}_{sep,i}$ , is found to equal ${\\\\sum }_{a}{q}_{a} {\\\\mathcal {E}}_{i,aa}/3=\\\\textrm {tr}[{\\\\hat{\\\\mathcal {E}}}_{i}\\\\hat{\\\\rho }]/3$ , where ${\\\\hat{\\\\mathcal {E}}}_{i}={\\\\hat{\\\\mathcal {E}}}_{i}^{X}+ {\\\\hat{\\\\mathcal {E}}}_{i}^{Y}+{\\\\hat{\\\\mathcal {E}}}_{i}^{Z}$ .\",\n \"Furthermore, a variation with respect to the probabilities $\\\\lbrace {q}_{a} \\\\rbrace $ shows the equality of the matrix elements ${\\\\mathcal {E}}_{i,aa}$ just as in the general case above, whereupon we learn that each pure product state in the ensemble makes the same contribution to ${\\\\mu }^{sup}_{sep,i}$ .\",\n \"In other words, the separable density matrix which maximizes (REF ) may be taken to be a pure product state [16].\",\n \"It should be clear from the above arguments that the results hold for any number of generalized measurements as well as any number of parties in the multipartite state.\",\n \"In summary, then, we have found that the joint majorization uncertainty bound, ${{P}}_{sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}$ , for a set of generalized measurements performed on an arbitrary quantum state supported on a finite-dimensional Hilbert space can be found on the class of pure states of the system, and in the case of a separable states, ${{P}}_{sep;sup}^{\\\\mathrm {X} \\\\oplus \\\\mathrm {Y} \\\\oplus \\\\ldots \\\\oplus \\\\mathrm {Z}}$ can be found on the class of pure product states of the system.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1289"}}},{"rowIdx":1866794,"cells":{"text":{"kind":"list like","value":[["New Lower Bounds for Matching Vector Codes"],["Abstract A Matching Vector (MV) family modulo $m$ is a pair of ordered lists $U=(u_1,...,u_t)$ and $V=(v_1,...,v_t)$ where $u_i,v_j \\in \\mathbb{Z}_m^n$ with the following inner product pattern: for any $i$, $< u_i,v_i>=0$, and for any $i \\ne j$, $< u_i,v_j> \\ne 0$.","A MV family is called $q$-restricted if inner products $< u_i,v_j>$ take at most $q$ different values.","Our interest in MV families stems from their recent application in the construction of sub-exponential locally decodable codes (LDCs).","There, $q$-restricted MV families are used to construct LDCs with $q$ queries, and there is special interest in the regime where $q$ is constant.","When $m$ is a prime it is known that such constructions yield codes with exponential block length.","However, for composite $m$ the behaviour is dramatically different.","A recent work by Efremenko [STOC 2009] (based on an approach initiated by Yekhanin [JACM 2008]) gives the first sub-exponential LDC with constant queries.","It is based on a construction of a MV family of super-polynomial size by Grolmusz [Combinatorica 2000] modulo composite $m$.","In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families.","When $q$ is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length.","When the modulus $m$ is constant (as it is in the construction of Efremenko) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over $\\mathbb{Z}_m$."],["Introduction","A Matching Vector Family (MV Family) is a combinatorial object that arises in several contexts including Ramsey graphs, weak representation of OR polynomials and recently in constant query locally decodable codes (LDCs).","It is defined by two ordered lists $U=\\left(u_1, \\cdots u_t\\right)$ and $V=\\left(v_1, \\cdots v_t\\right)$ where $u_i, v_j \\in \\mathbb {Z}_m^n$ and $m$ and $n$ are integers greater than 1.","The property that the two lists have to satisfy is the following: for all $i\\in [t]$ , $\\langle u_i, v_i \\rangle = 0\\ (mod\\ m)$ whereas for all $i \\ne j\\in [t]$ , $\\langle u_i, v_j \\rangle \\ne 0\\ (mod\\ m)$ .","By $\\langle \\cdot , \\cdot \\rangle $ we denote the standard inner product.","Let us call this the standard definition of a MV family.","If in addition, all the inner products $\\langle u_i, v_j \\rangle \\ (mod\\ m)$ lie in a set of size $q$ , then it is called a $q-restricted$ MV family.","Note that $q=m$ corresponds to the standard MV family.","The size of the MV family is $t$ , the number of vectors in the list.","In this paper, we shall prove upper bounds on $q-restricted$ MV families in the first part and on standard MV families in the later part.","Let $\\mathbf {MV}(m,n)$ denote the largest $t$ such that there exists a MV family of size $t$ in $\\mathbb {Z}_m^n$ .","Analogously, let $\\mathbf {MV}(m,n,q)$ denote the largest $t$ such that there exists a $q-restricted$ MV family of size $t$ in $\\mathbb {Z}_m^n$ .","The question of bounding $\\mathbf {MV}(m,n)$ (or $\\mathbf {MV}(m,n,q)$ ) is closely related to the well-known combinatorial problem of set systems with restricted modular intersections [2], [22], [12], [13] (in this setting the vectors $u_i,v_i$ are required to have entries that are either 0 or 1).","The systematic study of this more general problem, in the context of MV codes, was initiated in [8].","The setting of prime $m$ is well understood.","For large prime $m=p$ , it is known that $\\mathbf {MV}(p,n)=O\\left(p^{n/2}\\right)$ [8].","Infact, this is almost tight.","When $m$ is a small prime, again we have a tight upper bound of $O\\left(n^{p-1}\\right)$ [2].","Surprisingly, the setting of small composite $m$ leads to very useful constructions of Ramsey graphs and constant query LDCs.","This is due to a construction of MV family over $Z_6$ by Grolmusz [12] of superpolynomial size in contrast to a polynomial upper bound when $m$ is a small prime.","Thus, it is interesting to study the behavior of MV families for small composite $m$ , and more generally arbitrary general composites.","We will see later the connection between upper bounds on $\\mathbf {MV}(m,n,q)$ and lower bounds on the encoding lengths of MV Codes (a family of LDCs).","For general $m$ , the best upper bound known [8] is $\\mathbf {MV}(m,n) \\le m^{n-1 + o_m(1)},$ with $o_m(1)$ denoting a function that goes to zero when $m$ grows.","It was conjectured in [8] that an upper bound of $ \\sim m^{n/2}$ should hold for any $m$ (not just prime).","This would be tight for large $m$ as there are constructions of MV families almost meeting this bound [28].","However, the proof method used in [8] to prove the $O\\left(p^{n/2}\\right)$ bound does not extend to non primes.","In this work, we prove the conjecture for $q-restricted$ MV families in $\\mathbb {Z}_m^n$ , for any $m$ as long as $q= \\frac{o(n)\\log m}{\\log \\left(o(n)\\log m\\right)}$ (See Theorem REF ).","When $m=p$ is a fixed small prime, it follows from [2] that $\\mathbf {MV}(p,n) = O\\left(n^{p-1}\\right)$ .","On the other hand, when $m$ is a fixed composite, say $m=6$ , there exists a MV family of superpolynomial size $\\Omega \\left(\\exp \\left(\\log ^2n /\\log \\log n\\right)\\right)$ ) [12].","We prove a stronger upper bound on $\\mathbf {MV}(m,n)$ , compared to Theorem REF in such a case assuming a well known conjecture in additive combinatorics (see Theorem REF ).","Table REF lists the known and new upper bounds on MV families.","Table: List of upper bounds on 𝐌𝐕(m,n)\\mathbf {MV}(m,n), 𝐌𝐕(m,n,q)\\mathbf {MV}(m,n,q)Theorem 1 For all $m\\ge 2, n \\ge 1$ we have $ \\mathbf {MV}(m,n,q) \\le q^{O(q\\log q)}m^{n/2} $ Hence, Theorem REF resolves the conjecture of [8] for any $m$ and for $q = \\frac{o(n)\\log m}{\\log \\left(o(n)\\log m\\right)}$ .","When $m >> n$ , our bound is quite close to the best known construction of MV families which gives $\\mathbf {MV}(m,n) \\ge \\left( \\frac{m+1}{n-2} \\right)^{n/2 - 1}$ [28].","Our second result assumes the polynomial Freiman-Ruzsa conjecture (PFR) conjecture (discussed below) and gives a stronger upper bound on the size of MV families when $m$ is a constant and $n$ grows.","Before we state the conjecture, we need to define what a difference set is.","For an abelian group $G$ let $A \\subseteq G$ .","Then the difference set $A-A=\\lbrace a_1-a_2:a_1,a_2 \\in A\\rbrace $ Conjecture 1 (PFR Conjecture in $\\mathbb {Z}_m^n$ ) Suppose $A \\subseteq \\mathbb {Z}_m^n$ and $|A-A|\\le \\lambda \\cdot |A|$ .","Then one can find a subgroup $H$ of size at most $|A|$ such that $A$ can be covered by $\\lambda ^{\\prime } = \\lambda ^{c_m}$ many translates of $H$ , where $c_m$ depends only on $m$ .","We note that the PFR conjecture has already found several applications in computer science.","Ben-Sasson and Zewi [6] used it to construct two-source extractors from affine extractors; and Ben-Sasson, Lovett and Zewi [5] used it to bound the deterministic communication complexity of functions whose corresponding matrix has low rank.","Our work provides another application for the PFR and demonstrates its wide-reaching applicability.","We further note that a quasi-polynomial version of the PFR conjecture was recently proved by Sanders [21] (see also the exposition in [18]).","Unfortunately, all the applications discussed above require the truly polynomial version of the conjecture, and so cannot apply to Sanders' result.","We now state the second theorem.","Theorem 2 Assuming the PFR conjecture over $\\mathbb {Z}_m^n$ (Conjecture REF ) we have $\\mathbf {MV}(m,n) \\le \\exp \\left( c(m)\\frac{n}{\\log n} \\right), $ with $c(m)$ an explicit function of $m$ .","From a technical point of view, one of the ingredients in this work builds on the recent work of Ben-Sasson, Lovett and Zewi [5] who used the PFR conjecture to show that matrices over $\\mathbb {Z}_2$ with large bias (say, with many more ones than zeros) and small rank must contain a large monochromatic sub-matrix.","An important ingredient in our proof is a generalization of their results from $\\mathbb {Z}_2$ to $\\mathbb {Z}_m$ for all $m$ , not necessarily prime.","We note however that this is just one ingredient in our overall proof."],["Lower Bounds on LDCs: Motivation for MV Family","Locally Decodable Codes (LDCs) are a special kind of Error Correcting Codes (ECCs) that allow the receiver to decode a single symbol of the message by querying a small number of positions in a corrupted encoding.","More formally, an $(q,\\delta ,\\epsilon )$ -LDC encodes $K$ -symbol messages $x$ to $N$ -symbol codewords $C(x),$ such that for every $i\\in [K],$ the symbol $x_i$ can be recovered with probability $1-\\epsilon ,$ by a randomized decoding procedure that makes only $q$ queries, even if the codeword $C(x)$ is corrupted in up to $\\delta N$ locations.","Since the early 90's, LDC's have found exciting applications in various areas ranging from data transmission to complexity theory to cryptography/privacy.","We refer the reader to [23], [27] for more background.","A central research question, which is far from being solved, has to do with understanding the best possible `stretch' of an LDC with a constant number of queries.","That is, how large $N$ has to be as a function of $K$ for constant $q$ and with constant $\\delta ,{\\varepsilon }$ (these two last parameters are not our focus here and we will generally assume them to be small fixed constants).","For $q=1,2$ this question is completely answered.","There are no LDC's for $r=1$ [16] and the best LDC's with $q=2$ have exponential encoding length [10], [15].","For $q > 2$ there are huge gaps in our understanding.","Katz and Trevisan were the first to study this problem [16] and, today, the best general lower bounds on $N$ are slightly super-linear bounds of the form $\\tilde{\\Omega }\\left(K^{1+1/\\left(\\lceil r/2\\rceil -1\\right)}\\right)$ [24].","Notice that, when the number of queries is 3 or 4, these bounds are quadratic (see also [15], [25] for the $q=3,4$ case).","The upper bounds were, until recently, those coming from polynomial codes and were of the order of $N \\le \\exp \\left(K^{\\frac{1}{q-1}}\\right)$ .","Improved upper bounds, breaking this barrier slightly, were given in [3].","This state of affairs changed dramatically when, in a breakthrough paper, Yekhanin [26] developed a new approach for constructing LDCs, called MV codes, that have much shorter codeword length than polynomial codes.","Efremenko [9] was the first to show that this approach could yield codes with subexponential encoding length (Yekhanin's paper showed this under a number theoretic assumption).","More refinements and improvements to this new framework were obtained [20], [17], [14], [19], [8], [1] to give LDC's with $q$ queries and with encoding length that grows, when $q$ is a constant, roughly like $N \\sim \\exp \\exp \\left((\\log K)^{O(1/\\log q)}(\\log \\log K)^{1-1/\\log q}\\right).$ While significantly smaller than the length of polynomial codes, the codeword length of these new codes is still super polynomial in $K$ .","The most general setting of parameters was addressed in [8] where the authors had given a black box construction of $q$ query MV codes using $q-restricted$ MV families in $\\mathbb {Z}_m^n$ .","Using the standard definition of MV families, this implied $m$ query MV codes using MV families in $Z_m^n$ .","In this basic, yet general reduction, it was shown that upper bounds on MV families would lead to lower bounds on the encoding length of MV codes.","With this motivation in mind, the authors in [8] made a conjecture on the upper bound on the size of MV families which would lead to lower bounds on the encoding length of MV codes under the basic framework.","We note that Yekhanin in [26] used restricted MV families in $\\mathbb {Z}_p^n$ where $p$ is a very large Mersenne prime and used a specialized technique to reduce the number of queries from $p$ to 3.","Another instance of reduction in the number of queries from what the standard construction gives, was given by Efremenko [9] where he again used restricted MV families.","A certain gadget was discovered using computer search whereby the author worked in $\\mathbb {Z}_{511}$ but got down the number of queries to 3 from the basic bound of 511.","The following is a corollary of Theorem REF .","Corollary 3 For an arbitrary positive integer $m$ , consider an infinite family of $q$ -query Matching Vector code $C_n:{\\mathbb {F}}^k \\rightarrow {\\mathbb {F}}^N$ for $n \\in \\mathbb {N}$ , where $k(n)$ and $N(n)$ are growing functions of $n$ , constructed using the black box reduction from a $q$ -restricted Matching Vector Family in $\\mathbb {Z}_m^n$ ([8]).","For large enough $n$ , if $q=\\frac{o(n)\\log m}{\\log \\left(o(n)\\log m\\right)}$ , then $ N \\ge k^{2-o(1)}$ Specifically, if $q=O(1)$ , then $N = \\Omega \\left(k^{2}\\right)$ .","Next we have the following corollary from Theorem REF .","Corollary 4 For some arbitrary positive integer $m$ , assume the PFR conjecture over $\\mathbb {Z}_m^n$ (Conjecture REF ).","Consider an infinite family of $m$ -query Matching Vector code $C_n:{\\mathbb {F}}_q^k \\rightarrow {\\mathbb {F}}_q^N$ for $n \\in \\mathbb {N}$ , where $k(n)$ and $N(n)$ are growing functions of $n$ , constructed using the black box reduction from a standard Matching Vector Family in $\\mathbb {Z}_m^n$ ([8]).","For large enough $n$ , if $m=O(1)$ , then $ N = \\exp \\left(\\Omega _m\\left(\\log k \\ \\log \\log k\\right)\\right)$ Thus Corollary REF states that, assuming Conjecture REF , MV codes with constant number of queries must have super polynomial encoding length in the basic framework.","Note that we get the same bound in Efremenko's framework for 3 queries.","This is because the form of the superpolynomial bound is assuming a constant $m$ and applying our bound to Efremenko's work again leads to a superpolynomial bound as $m=511$ in his setting (another constant).","(He uses $\\mathbb {Z}_{511}$ to construct the MV family and further reduces the number of queries to 3.)","This essentially means that in order to construct polynomial length codes, one needs to construct MV families in $\\mathbb {Z}_m^n$ for non-constant $m$ and use some specialized gadget to reduce the number of queries.","One way is to ensure it is a $q-restricted$ (constant $q$ ) MV family.","This automatically ensures $q$ query decoding.","However, the quadratic lower bound continues to hold even in this scenario for constant $q$ .","To beat the quadratic lower bound for constant query MV codes, one needs to construct $q-restricted$ MV families for growing $m$ and $q=\\frac{o(n)\\log m}{\\log \\left(o(n)\\log m\\right)}$ and then develop some special gadget to get the number of queries down further from $q$ to some constant."],["Proof Overview","The proof of Theorem REF relies on intuitions coming from the theory of two-source extractors [7], which are functions of two variables $F(X,Y)$ such that the output of $F$ is distributed in a close-to-uniform fashion whenever the two inputs are drawn, independently, from two distributions of sufficiently high entropy.","Since our proof does not use two-source extractors explicitly we do not define them formally and just use them to explain the high level idea behind the proof.","It is a well known fact [7] that the inner product function $F(X,Y) = \\langle X,Y \\rangle $ , say over $\\mathbb {Z}_2^n \\times \\mathbb {Z}_2^n$ is a good two source extractor when the two inputs $X$ and $Y$ are both drawn uniformly from sets $S_X,S_Y \\subseteq \\mathbb {Z}_2^n$ of size larger $2^{n/2}$ .","This immediately suggests a connection to MV families, since, if we take $S_X = U$ and $S_Y = V$ for a MV family $U,V$ in $\\mathbb {Z}_2^n$ , we would get a completely non-uniform output (it will be zero with exponentially small probability).","This means that the size of $U,V$ is bounded from above by approximately $2^{n/2}$ .","If we try to use a similar argument over $\\mathbb {Z}_m$ we run into trouble since the inner product function modulo $m$ is not a good two source extractors for sources of size $m^{n/2}$ .","Take, for example, $S_X = S_Y=\\lbrace 0,2,4\\rbrace ^n \\subseteq \\mathbb {Z}_6^n$ and observe that $\\langle X,Y \\rangle $ is always divisible by 2 and so is far from being uniformly distributed over $\\mathbb {Z}_6$ .","It is, however, possible to show that this example is, in some sense, the only example and that, in general, we can always find a certain number of elements of either $S_X$ or $S_Y$ that `agree' modulo some factor of $m$ .","This observation suggests proving Theorem REF by induction on the number of factors of $m$ , which is the way we proceed.","The proof of Theorem REF uses a slightly different view of MV families as matrices with certain zero/non-zero pattern and small rank.","Specifically, for a MV family $U,V$ of size $t$ in $\\mathbb {Z}_m^n$ consider the $t \\times t$ matrix $P$ whose $(i,j)$ 'th entry is $\\langle u_i,v_j \\rangle \\mod {m}$ .","The definition of a MV family implies that $P$ has zeros on the diagonal and non-zeros everywhere else.","If $m$ was a prime, we could think of $\\mathbb {Z}_m$ as a field ${\\mathbb {F}}$ and say that, since $P$ is the inner product matrix of vectors of length $n$ over a field, it must have rank at most $n$ .","Conversely, every $t \\times t$ matrix over a field ${\\mathbb {F}}$ with these properties (zero on the diagonal and non-zero off the diagonal) and with rank $n$ gives a MV family of size $t$ in ${\\mathbb {F}}^n$ .","We can call a matrix with this pattern of zeros/non-zeros an MV matrix.","Thus, when $m$ is prime, the question of bounding the size of a MV family is the same as lower bounding the rank of a MV-matrixFor technical reasons, the actual proof will not be entirely using matrices and will keep the MV family in the background.","This is because we need to keep certain invariants throughout the proof and these are easier to define for families of vectors than for matrices..","When $m$ is composite, this whole approach should be re-examined since $\\mathbb {Z}_m$ is no longer a field and our familiar understanding of matrices and linear algebra over a field are no longer valid.","We do, however, manage to carry over this correspondence between the two problems by defining the notion of rank in a careful way (more on this issue below).","Assume for the purpose of this overview that the usual notion of rank and other intuitions from linear algebra are valid over $\\mathbb {Z}_m$ and let us proceed with sketching the proof of Theorem REF using the equivalent formulation as bounding (from below) the rank of a MV matrix $P$ .","The starting point is a generalization of a result of [5], mentioned above, from $\\mathbb {Z}_2$ to $\\mathbb {Z}_m$ .","We show that every matrix $P$ over $\\mathbb {Z}_m$ that is biased (i.e., its values are not distributed close to uniformly) and has low rank, contains a large monochromatic sub-matrix modulo some factor $m^{\\prime }$ of $m$.","The size of the sub-matrix is bounded from below by $\\sim |P|\\exp (-r^{\\prime }/\\log (r^{\\prime }))$ , where $r^{\\prime }$ is the rank of $P$ modulo $m^{\\prime }$ (this factor depends on the specific way the matrix is biased).","This generalizes the result of [5] which assumes $m=2$ and finds a large monochromatic sub-matrix (modulo 2).","We note that the sub-matrix lemma is the only component in the proof that relies on the PFR conjecture.","Let us refer to this result from now on as the sub-matrix lemma.","We can apply the sub-matrix lemma to a MV matrix $P$ since its values are far from uniform (the probability of zero is much less than $1/m$ ) and since its rank is assumed (towards a contradiction) to be low.","Suppose for the sake of simplicity that $m=p \\cdot q$ , with $p,q$ distinct primes (the proof for general $m$ is significantly more technical but relies on the same basic intuitions).","Applying the sub-matrix lemma we obtain a sub-matrix $P_1$ of $P$ that is constant modulo some factor $m_1$ of $m$ (so $m_1$ is either $p$ , $q$ or $m$ ) of size at least $|P|\\exp (-r_1/\\log (r_1))$ , where $r_1\\le n$ is the rank of $P \\mod {m}_1$ .","Using some matrix manipulations, and subtracting a rank one matrix, we can get a large sub-matrix $P_1^{\\prime }$ that does not intersect the diagonal of $P$ and s.t all of the entries of $P_1^{\\prime }$ are zero modulo $m_1$ .","Suppose $|P_1^{\\prime }| = t_1$ and consider the $2t_1 \\times 2t_1$ sub-matrix $P_1^{\\prime \\prime }$ of $P$ that has $P_1^{\\prime }$ as its top-right (or bottom-left) block and s.t the top-left and bottom-right blocks are taken to have zero diagonal elements.","Formally, if $P_1^{\\prime }$ is indexed by rows in $R$ and columns in $T$ with $R \\cap T = \\emptyset $ then the rows/columns of $P_1^{\\prime \\prime }$ will be indexed by $R \\cup T$ .","If we consider the matrix $P_1^{\\prime \\prime }$ modulo $m_1$ then it has top-right block which is all zero and so its rank (modulo $m_1$ ) will be the sum of the ranks of the top-left and bottom right blocks.","Thus, one of these blocks, w.l.o.g the top-left one, must have rank at most $n/2$ (over $\\mathbb {Z}_{m_1}$ ).","Notice also that both of these blocks are themselves MV matrices modulo $m$ since they are sub-matrices of $P$ with the same row and column sets.","Let $\\tilde{P}_1$ be the top-left block of $P_1^{\\prime \\prime }$ .","We can now apply, again, the monochromatic sub-matrix lemma to find a large sub-matrix $P_2$ of $\\tilde{P}_1$ which is constant modulo some other factor $m_2$ of $m$ .","The size of $P_2$ will be $t_1 \\cdot \\exp (-r_2/\\log (r_2)) = |P|\\cdot \\exp (-r_1/\\log (r_1) - r_2/\\log (r_2) ).$ The factor $m_2$ is also either $p$ or $q$ .","If it happens to be that $m_1=m_2$ then $r_2 \\le n/2$ and so we gain in the size of $P_2$ in this second step (the expression $r_2/\\log (r_2)$ is smaller than $n/2\\log (n/2)$ which is smaller by roughly a factor of two than our bound on $r_1/\\log (r_1)$ .","Suppose we continue with this iterative process of finding constant sub-matrices for $\\ell $ steps and that, by luck, all the factors $m_1,m_2, \\ldots $ are equal to the same factor of $m$ (say $p$ ).","Then, after roughly $\\log (n)$ iteration, we will reduce the rank modulo $p$ to one and still have at least $|P|\\cdot \\exp \\left(-\\sum _{i=1}^\\ell \\frac{n}{2^i\\log (n/2^i)}\\right)$ rows, which is close to the original size of $P$ if we assume (in contradiction) that $|P| >> \\exp ( n/\\log n)$ (this calculation is given in Claim REF ).","In this case we obtain a new large MV family $U^{\\prime },V^{\\prime }$ modulo $m$ such that all inner products $\\langle u_i^{\\prime },v_j^{\\prime } \\rangle $ of elements $u_i^{\\prime }\\in U^{\\prime }, v_j^{\\prime } \\in V^{\\prime }$ are fixed modulo $p$ .","From this we can easily construct a MV family of roughly the same size in $\\mathbb {Z}_q^n$ and then use the bounds on $\\mathbf {MV}(q,n)$ for primes to get a contradiction.","In the `unlucky' case we will have different factors $m_1,m_2,\\ldots $ in each stage, but we can adapt the analysis to consider the decrease in rank simultaneously for all factors of $m$ .","The full proof is by induction on the number of factors of $m$ and uses the iterative sub-matrix argument above to go from a MV family modulo $m$ to a MV family of roughly the same size modulo some proper factor of $m$ (and then uses the inductive hypothesis on this new MV family)."],["Matrix rank over $\\mathbb {Z}_m$","An important technical issue, which was already hinted at above, is in the definition of the rank of a matrix with entries in a ring $\\mathbb {Z}_m$ .","There are two main properties of matrix rank over a field that we relied on in the proof sketch above.","The first is that a rank $r$ matrix is always the inner product matrix of vectors in $r$ dimensions.","Equivalently, a $t \\times t$ matrix of rank $r$ can be written as a product of a $t \\times r$ matrix and an $r \\times t$ matrix.","This is important if we are to go back and forth between matrices and MV families.","Another property we used is that, if we have a $2t \\times 2t$ matrix composed of 4 blocks of size $t \\times t$ and the top-right block is zero, then the rank of the matrix is the sum of the ranks of the top-left block and the bottom right block.","Ideally, we would like to define rank over $\\mathbb {Z}_m$ so that both properties are satisfied.","This is, however, impossible as the following example shows: Consider the $2 \\times 2$ matrix with the two rows $(4,0)$ and $(0,3)$ over $\\mathbb {Z}_6$ .","This matrix can be written as the product of the two vectors $(2,3)^T$ and $(2,3)$ and so should have rank one, if we are to satisfy the first property.","However, if we are to satisfy the second property, its rank should be the sum of the ranks of the two $1 \\times 1$ matrices $(4)$ and $(3)$ , which clearly cannot have rank zero!","Our solution to this problem is to give two different definitions of rank, each satisfying one of the two properties.","We then show that the two definitions of rank can differ from each other by a multiplicative factor of $\\log m$ , which our proof can handle.","The first definition of rank is as the smallest $r$ such that our $t \\times t$ matrix can be written as a product of a $t \\times r$ matrix and an $r \\times t$ matrix.","Clearly this would satisfy the first property (but not the second).","The second definition of rank is termed column-rank and is defined as the logarithm to the base $m$ of the size of the additive subgroup of $\\mathbb {Z}_m^t$ generated by the columns of the matrix.","Notice that this definition of rank can result in the rank being non-integer.","For example, the rank of the matrix with a single column $(2,0)$ over $\\mathbb {Z}_6$ would be equal to $\\log _6(3)$ since the subgroup generated by this column is composed of the three vectors $(2,0),(4,0),(0,0)$ .","It is not hard to show (see Claim REF ) that this definition satisfies the second property described above regarding block matrices.","Clearly, the two definitions agree for matrices over a field.","We show (see Claim REF ) that the two notions of rank can differ by a multiplicative factor of at most $\\log m$ .","This allows us to use both definitions in different parts of the proof without losing too much in the transition.","We finish this discussion by noting that in no part of the proof do we use the characterization of rank using determinants, which is often very useful when working over a field."],["Organization","We begin with some preliminaries in Section .","We prove Theorem REF in Section .","Section contains some claims about matrices over $\\mathbb {Z}_m$ .","Section introduces collision free MV families.","Both Section and Section will be used in the proof of Theorem REF in Section .","The proof of Theorem REF also requires the sub-matrix lemma, whose proof appears in Section ."],["Notations:","Throughout the paper we will be handling ordered lists of elements.","A list $A$ of size $t$ over a finite set $\\Omega $ is an ordered $t$ -tuple $A=\\left(a_1,a_2,\\cdots ,a_t\\right)$ where each $a_i \\in \\Omega $ .","A list can have repetitions.","If it doesn't we say it is twin free.","When discussing sublists $A \\subseteq B$ with $B = (b_1,\\ldots ,b_t)$ we will use the convention that, unless specified otherwise, $A$ maintains the ordering induced by $B$ .","For a positive integer $t$ , we let $[t]$ denote the list $\\left(1,\\cdots t\\right)$ .","So, for example, when we say that $T \\subseteq [t]$ we mean that $T$ is a list of integers in increasing order belonging to $[t]$ .","We say that a list $A = (a_1,\\ldots ,a_t)$ over $\\Omega $ is constant if $a_i=a_j$ for all $i,j \\in [t]$ .","We assume all logarithms are in base 2 unless otherwise specified."],["MV Families: Basic Facts and Definitions","We now start with some basic definition and claims regarding MV families.","Definition 2.1 (Matching Vector Family) Let $U=\\left(u_1, u_2, \\cdots u_t\\right)$ and $V=\\left(v_1, v_2, \\cdots v_t\\right)$ be lists over $\\mathbb {Z}_m^n$ .","Then $\\left(U,V\\right)$ is called a matching vector family of size $t$ in $\\mathbb {Z}_m^n$ if $\\langle u_i,v_i \\rangle =0\\ \\left(mod\\ m\\right), \\quad \\forall i$ .","$\\langle u_i,v_j \\rangle \\ne 0\\ \\left(mod\\ m\\right), \\quad \\forall i \\ne j$ .","If in addition, we $|\\lbrace \\langle u,v \\rangle : u \\in U, v \\in V\\rbrace |=q$ , we call such a MV family an $q-restricted$ MV family.","We denote the size of $\\left(U,V\\right)$ by $\\left|\\left(U,V\\right)\\right|$ .","For instance, $\\left|\\left(U,V\\right)\\right|=t$ above.","Definition 2.2 (Subset of Matching Vector Family) Let $U=\\left(u_1, u_2, \\cdots u_t\\right), V=\\left(v_1, v_2, \\cdots v_t\\right)$ form a matching vector family in $\\mathbb {Z}_m^n$ of size $t$ .","By $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ , we mean there exists a sublist $T \\subseteq [t]$ such that $U^{\\prime }=\\left(u_i:i \\in T\\right), V^{\\prime }=\\left(v_i:i \\in T\\right)$ .","Observe that $\\left(U^{\\prime },V^{\\prime }\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ .","Definition 2.3 ($\\mathbf {MV}\\left(m,n\\right)$ ) We denote by $\\mathbf {MV}\\left(m,n\\right)$ the maximum size of a matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ .","Similarly, we denote by $\\mathbf {MV}\\left(m,n,q\\right)$ the maximum size of an $q-restricted$ matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ .","We shall use the following simple facts implicitly throughout the paper.","Fact 2.4 $\\mathbf {MV}\\left(m,n\\right)$ is an increasing function of $n$ .","For $n_1q$ .","Let $v_1,\\cdots ,v_q \\in Z_m$ be the $q$ possible nonzero values that the inner products $\\langle u,v \\rangle $ attain.","For each $v_j$ there is some prime $p_{i_j}$ where $v_j \\ne 0 \\left( \\mod {p}_{i_j}^{e_{i_j}}\\right)$ .","So, we can replace $m$ with just $\\prod _{j=1}^q p_{i_j}^{e_{i_j}}$ and discard all primes other than $p_{i_1}, \\cdots ,p_{i_q}$ .","Claim 3.4 If $N$ has $r$ prime factors, then $|\\lbrace x \\in Z_N: order(x) q^{10q}$.","By Lemma REF , we know that for $s\\ge m/q^{10q}$ , $\\mathbf {MV}_{1,s}\\left(m,n,q\\right)\\le 12q \\cdot q^{24\\log \\frac{m}{s}}\\left(\\frac{m}{s}\\right)^{n/2}\\le 12q \\cdot q^{24\\log q^{10q}}\\left(\\frac{m}{s}\\right)^{n/2}$ .","Similarly, we have for $s\\ge m/4q$ , $\\mathbf {MV}_{s,1}\\left(m,n,q\\right)\\le 12q \\cdot q^{24\\log q^{10q}}\\left(\\frac{m}{s}\\right)^{n/2}$ .","Now, suppose there is a $q-restricted$ MV family $(U,V)$ in $\\mathbb {Z}_m^n$ of size $t>12q \\cdot q^{24\\left(1+\\log q^{10q} \\right)}m^{n/2}$ .","Applying Lemma REF with $r_1=r_2=1$ , we get a $q-restricted$ MV family $(U^{\\prime },V^{\\prime })\\subseteq (U,V)$ of size $t^{\\prime } \\ge s^{-n/2}q^{-24}t >q^{24\\log q^{10q}}\\left(\\frac{m}{s}\\right)^{n/2}$ where $s\\ge m/q^{10q}$ such that $\\langle u^{\\prime },v^{\\prime } \\rangle =0 \\ \\left(mod\\ s\\right)$ for all $u^{\\prime } \\in U^{\\prime }, v^{\\prime } \\in V^{\\prime }$ .","Either $U^{\\prime \\left(s\\right)}$ is constant or $V^{\\prime \\left(s\\right)}$ is constant.","But, by the previous paragraph, we have for $s\\ge m/q^{10q}$ , $\\mathbf {MV}_{s,1}\\left(m,n,q\\right)$ and $\\mathbf {MV}_{1,s}\\left(m,n,q\\right)$ are at most $12q \\cdot q^{24\\log q^{10q}}\\left(\\frac{m}{s}\\right)^{n/2}$ .","This leads to a contadiction."],["Proof of Lemma ","By assumption we have that $\\langle u,v \\rangle =0 \\ \\left(mod\\ r_1r_2\\right)$ for all $u \\in U, v \\in V$ .","So, we can consider $\\frac{\\langle u,v \\rangle }{r_1r_2} \\in \\mathbb {Z}_{r_3}$ .","Also, by hypothesis, the inner products $\\frac{\\langle u,v \\rangle }{r_1r_2}$ occupy $q^{\\prime }\\le q$ residues in $\\mathbb {Z}_{r_3}$ .","We have that For $1 \\le i \\le t$ , $\\frac{\\langle u_i,v_i \\rangle }{r_1r_2}=0 \\ \\left(mod\\ r_3\\right)$ since $\\langle u_i,v_i \\rangle =0 \\pmod {m}$ .","For $1 \\le i,j \\le t$ , $i \\ne j$ , $\\frac{\\langle u_i,v_j \\rangle }{r_1r_2} \\ne 0 \\ \\left(mod\\ r_3\\right)$ since $\\langle u_i,v_j \\rangle \\ne 0 \\pmod {m}$ .","Let $\\mu $ denote the distribution over $\\mathbb {Z}_{r_3}$ defined by $\\frac{\\langle u_i,v_j \\rangle }{r_1r_2} \\mod {r}_3$ where $u_i, v_j$ are drawn independently and uniformly from $U,V$ respectively.","Case 1: $4q^{\\prime } \\ge r_3$.","Observe that $\\mu $ outputs 0 only when $i=j$ .","Therefore, $\\mathbf {Pr}[\\mu =0]=1/t \\le 1/12q^{\\prime } \\le 1/3r_3$ .","On the other hand, $\\mathbf {Pr}[\\mathcal {U}_{r_3}=0]=1/r_3$ .","This implies that $\\Delta \\left(\\mu ,\\mathcal {U}_{r_3}\\right) \\ge 1/3r_3$ .","Thus, applying Lemma REF with $\\omega =exp\\left(2\\pi i/r_3\\right)$ , we get that for some $1 \\le j \\le r_3-1$ , $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^j\\right)^{x} \\right]\\right|\\ge \\frac{2}{3r_3\\sqrt{r_3}} \\ge \\frac{1}{12q^{\\prime 3/2}}.$ Let $\\omega ^{\\prime }=\\omega ^j$ and $ord(\\omega ^{\\prime })$ (the order of $\\omega ^{\\prime }$ ) be $s=r_3/gcd\\left(r_3,j\\right)$ .","Also, note that as $j\\ge 1$ , we have $s \\ge 2$ .","Also, trivially, $s \\ge r_3/q^{\\prime 10q^{\\prime }} \\ge r_3/q^{10q}$ .","Case 2: $4q^{\\prime } 1$ and thus $\\omega ^{\\prime }$ is not trivial.","Now, we shall show the existence of $\\omega ^{\\prime }$ of much higher order provided $r_3>q^{\\prime 10q^{\\prime }}$ .","By Claim REF , for $S=q^{\\prime 10q^{\\prime }}$ and $N=r_3$ , and noting that $r_3$ has atmost $q$ prime factors by Claim REF , we have $Pr_j[ord\\left(\\omega ^j\\right)\\le r_3/S]\\le 1/4q^{\\prime }$ Thus, with probabilty at least $1/2q^{\\prime }-1/4q^{\\prime }=1/4q^{\\prime }$ , a random $j$ satisfies $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^j\\right)^{x} \\right]\\right| \\ge 1/\\sqrt{2q^{\\prime }} \\ge \\frac{1}{12q^{3/2}}$ $s=ord\\left(\\omega ^j\\right) \\ge r_3/S$ Also, as $r_3/4q^{\\prime }>1$ the above two conditions are true for some $j \\ne 0$ .","Now, we combine the above two cases as follows.","Let $\\omega ^{\\prime }=\\omega ^j$ and ${\\varepsilon }= \\frac{1}{12q^{3/2}}$ .","We have shown by the above case-by-case analysis that $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^{\\prime }\\right)^{x} \\right]\\right|\\ge \\epsilon $ $s=ord(\\omega ^{\\prime })$ is such that $s \\ge \\max \\lbrace 2,r_3/q^{10q}\\rbrace $ Using the Cauchy-Schwartz inequality twice we get $&&\\left|\\mathbb {E}_{u \\sim U, v \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle u,v \\rangle /r_1r_2}\\right]\\right|\\ge \\epsilon \\\\&\\Rightarrow & \\left|\\mathbb {E}_{u,\\tilde{u} \\sim U, v \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle u-\\tilde{u},v \\rangle /r_1r_2}\\right]\\right|\\ge \\epsilon ^2\\\\&\\Rightarrow & \\left|\\mathbb {E}_{u,\\tilde{u} \\sim U, v,\\tilde{v} \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle u-\\tilde{u},v-\\tilde{v} \\rangle /r_1r_2}\\right]\\right|\\ge \\epsilon ^4\\\\&\\Rightarrow & \\left|\\mathbb {E}_{u,\\tilde{u} \\sim U, v,\\tilde{v} \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle \\left(u-\\tilde{u}\\right)/r_1,\\left(v-\\tilde{v}\\right)/r_2 \\rangle }\\right]\\right|\\ge \\epsilon ^4.$ We need to explain the last expression.","Since by assumption $U^{\\left(r_1\\right)}$ and $V^{\\left(r_2\\right)}$ are constants, $\\left(u-\\tilde{u}\\right)/r_1 \\in \\mathbb {Z}_{m}^n$ and $\\left(v-\\tilde{v}\\right)/r_2 \\in \\mathbb {Z}_{m}^n$ are well defined.","Thus, we can fix $\\tilde{u}$ and $\\tilde{v}$ by an averaging argument such that $\\left|\\mathbb {E}_{u \\sim U, v \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle \\left(u-\\tilde{u}\\right)/r_1,\\left(v-\\tilde{v}\\right)/r_2 \\rangle }\\right]\\right|\\ge \\epsilon ^4.$ Let $U^{\\prime }=\\left(u_1^{\\prime },u_2^{\\prime }, \\cdots u_t^{\\prime }\\right), V^{\\prime }=\\left(v_1^{\\prime },v_2^{\\prime }, \\cdots v_t^{\\prime }\\right)$ where $u_i^{\\prime }= \\left(u_i -\\tilde{u}\\right)/r_1$ and $v_i^{\\prime }= \\left(v_i -\\tilde{v}\\right)/r_2$ .","Notice that $U^{\\prime }$ and $V^{\\prime }$ are not assumed to be a MV family (later we will derive from them a MV family).","We now define two probability distributions $\\mu ^{U^{\\prime }}$ and $\\mu ^{V^{\\prime }}$ over $\\mathbb {Z}_s^n$ .","For each $w \\in \\mathbb {Z}_s^n$ , let $\\mu ^{U^{\\prime }}\\left(w\\right)=\\left|B_s\\left(w,U^{\\prime }\\right)\\right|/\\left|U^{\\prime }\\right|$ and $\\mu ^{V^{\\prime }}\\left(w\\right)=\\left|B_s\\left(w,V^{\\prime }\\right)\\right|/\\left|V^{\\prime }\\right|$ .","That is, $\\mu ^{U^{\\prime }}\\left(w\\right)$ is the probability that $u^{\\prime (s)}=w$ where $u^{\\prime }$ is chosen uniformly in $U^{\\prime }$ , and similarly for $\\mu ^{V^{\\prime }}\\left(w\\right)$ .","Therefore, since the order of $w^{\\prime }$ is $s$ , we have that $ \\left|\\mathbb {E}_{w_1 \\sim \\mu ^{U^{\\prime }},w_2 \\sim \\mu ^{V^{\\prime }}}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle w_1,w_2 \\rangle }\\right]\\right|\\ge \\epsilon ^4.$ Recalling that $s$ is the order of $\\omega ^{\\prime }$ and applying Lemma REF , we get $\\textnormal {cp}\\left(\\mu ^{U^{\\prime }}\\right)\\textnormal {cp}\\left(\\mu ^{V^{\\prime }}\\right) \\ge \\epsilon ^8/s^n$ .","Therefore, one of $\\textnormal {cp}\\left(\\mu ^{U^{\\prime }}\\right)$ , $\\textnormal {cp}\\left(\\mu ^{V^{\\prime }}\\right)$ , say $\\textnormal {cp}\\left(\\mu ^{U^{\\prime }}\\right)$ , is at least $\\epsilon ^4/s^{n/2}$ .","Let $w^*$ be the point of maximum probability mass given by $\\mu ^{U^{\\prime }}$ .","Then, $\\mu ^{U^{\\prime }}\\left(w^*\\right)=\\mu ^{U^{\\prime }}\\left(w^*\\right)\\sum _{w \\in \\mathbb {Z}_s^n}\\mu ^{U^{\\prime }}\\left(w\\right)\\ge \\sum _{w \\in \\mathbb {Z}_s^n}\\mu ^{U^{\\prime }}\\left(w\\right)^2=\\textnormal {cp}\\left(\\mu ^{U^{\\prime }}\\right)\\ge \\epsilon ^4/s^{n/2}.$ Now, $\\mu ^{U^{\\prime }}\\left(w^*\\right) \\ge \\epsilon ^4/s^{n/2}$ means that $\\left|\\lbrace u \\in U: \\frac{u-\\tilde{u}}{r_1}=w^*\\ \\left(mod\\ s\\right)\\rbrace \\right|\\ge t \\epsilon ^4/s^{n/2} $ .","Equivalently, $\\bigg |\\big \\lbrace u \\in U: u-\\tilde{u}=r_1w^*\\ \\left(mod\\ r_1s\\right)\\big \\rbrace \\bigg |\\ge t \\epsilon ^4/s^{n/2}.$ Let $T^{\\prime }=\\left(i:u_i=\\tilde{u}+r_1w^*\\ \\left(mod\\ r_1s\\right)\\right)$ .","Now, define $U^{\\prime \\prime }=\\left(u_i:i \\in T^{\\prime }\\right)$ and $V^{\\prime \\prime }=\\left(v_i:i \\in T^{\\prime }\\right)$ .","Observe that $\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ such that $U^{\\prime \\prime \\left(r_1s\\right)}$ and $V^{\\prime \\prime \\left(r_2\\right)}$ are constants.","$\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right| \\ge t\\left(\\epsilon ^4/s^{n/2}\\right)$ .","The only thing left is to show that $\\langle u,v\\rangle =0\\ \\left(mod\\ r_1r_2s\\right)$ for all $u \\in U^{\\prime \\prime },v \\in V^{\\prime \\prime }$ .","This may not be true in general.","However, we can take a large subset of the matching vector family so that the resulting matching vector family satisfies this condition.","To see this, let $u \\in U^{\\prime \\prime }, v \\in V^{\\prime \\prime }$ be arbitrary.","Now, $u=r_1s \\cdot u^{\\prime }+u_0$ and $v=r_2 \\cdot v^{\\prime }+v_0$ where $u^{\\prime }, v^{\\prime }$ depend on $u,v$ respectively and $u_0,v_0$ are independent of $u,v$ .","Then, $\\langle u,v \\rangle =r_1r_2s \\langle u^{\\prime },v^{\\prime }\\rangle +r_1s \\langle u^{\\prime },v_0\\rangle +r_2 \\langle u_0,v^{\\prime }\\rangle + \\langle u_0,v_0\\rangle .$ As $u$ varies over $U^{\\prime \\prime }$ , $\\langle u^{\\prime },v_0\\rangle $ takes at most $q$ values modulo $r_2$ .","Hence, $r_1s\\langle u^{\\prime },v_0\\rangle $ takes at most $q$ values modulo $r_1r_2s$ .","Therefore, there exist at least $\\left(1/q\\right)\\left|U^{\\prime \\prime }\\right|$ elements of $U^{\\prime \\prime }$ such that $r_1s\\langle u^{\\prime },v_0\\rangle $ is a constant modulo $r_1r_2s$ .","We take the corresponding elements from $V^{\\prime \\prime }$ to form a matching vector family $\\left(U^{\\prime \\prime \\prime },V^{\\prime \\prime \\prime }\\right)\\subseteq \\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)$ .","We apply another round using the same idea on $U^{\\prime \\prime \\prime },V^{\\prime \\prime \\prime }$ , this time ensuring that $r_2\\langle u_0,v^{\\prime }\\rangle $ is constant modulo $r_1r_2s$ as $v$ varies over a large fraction of $V^{\\prime \\prime \\prime }$ .","Thus, we end up with $\\tilde{V}$ of size at least $\\left(1/q\\right)\\left|V^{\\prime \\prime \\prime }\\right|$ such that $r_2\\langle u_0,v_i\\rangle $ is a constant modulo $r_1r_2s$ .","We take the corresponding subset $\\tilde{U}$ from $U^{\\prime \\prime \\prime }$ so that $(\\tilde{U}, \\tilde{V})\\subseteq \\left(U^{\\prime \\prime \\prime },V^{\\prime \\prime \\prime }\\right)$ is a matching vector family.","Denote the size of $(\\tilde{U}, \\tilde{V})$ by $\\tilde{t}$ .","Note that $\\tilde{U}=\\left(\\tilde{u}_1,\\cdots ,\\tilde{u}_{\\tilde{t}}\\right), \\tilde{V}=\\left(\\tilde{v}_1,\\cdots ,\\tilde{v}_{\\tilde{t}}\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ of size at least $\\left(1/q^2\\right)t\\left(\\epsilon ^4/s^{n/2}\\right)=s^{-n/2}q^{-\\left(8+4\\log _q \\left(12\\right)\\right)}t\\ge s^{-n/2}q^{-\\left(8+4\\log _2 \\left(12\\right)\\right)}t \\ge s^{-n/2}q^{-24}t$ .","Also, as $\\langle u,v\\rangle $ is a constant modulo $r_1r_2s$ , for $u \\in \\tilde{U}, v \\in \\tilde{V}$ , and $\\langle \\tilde{u}_i,\\tilde{v}_i \\rangle =0\\ \\left(mod \\ r_1r_2s\\right)$ , we get that $\\langle u,v\\rangle =0 \\ \\left(mod\\ r_1r_2s\\right)$ , for $u \\in \\tilde{U}, v \\in \\tilde{V}$ .","This concludes the proof.","$\\Box $"],["Proof of Lemma ","We prove the lemma by backward induction on $r_1r_2|m$ .","That is, to prove the claim about $\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right)$ , we assume the inductive hypothesis for $\\mathbf {MV}_{r_1^{\\prime },r_2^{\\prime }}\\left(m,n,q\\right)$ where $r_1^{\\prime }r_2^{\\prime }>r_1r_2$ and $r_1^{\\prime }r_2^{\\prime }|m$ .","Base Case.","The base case of $r_1r_2=m$ is trivial.","To see this, observe that if $\\langle u,v \\rangle =0 \\ \\left(mod\\ m\\right)$ for all $u \\in U, v \\in V$ , then by the definition of a matching vector family in $\\mathbb {Z}_m^n$ , the size of such a family cannot exceed 1.","Hence, for $r_1r_2=m$ , $\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right)=1 \\le 12q\\cdot q^{24\\log \\frac{m}{r_1r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}$ .","Inductive Step.","Let $m=r_1r_2r_3$ with $r_1r_2r_1r_2$ and $r_1^{\\prime }r_2^{\\prime }|m$ .","We need to show that $\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right)\\le 12q \\cdot q^{24\\log \\frac{m}{r_1r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}$ .","Suppose this is false, so that there exists a $q-restricted$ matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ with $U=\\left(u_1, \\cdots u_t\\right),V=\\left(v_1,\\cdots v_t\\right)$ where $t >12q \\cdot q^{24\\log \\frac{m}{r_1r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}$ such that $U^{\\left(r_1\\right)}$ and $V^{\\left(r_2\\right)}$ are constants.","$\\langle u,v \\rangle =0 \\ \\left(mod\\ r_1r_2\\right)$ for all $u \\in U, v \\in V$ .","Note that $t \\ge 12q$ .","Therefore, applying Lemma REF , there exists $s|r_3$ with $s \\ge 2$ and matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ such that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right| \\ge s^{-n/2}q^{-24}t$ where $\\langle u^{\\prime },v^{\\prime } \\rangle =0 \\ \\left(mod\\ r_1r_2s\\right)$ for all $u^{\\prime } \\in U^{\\prime }, v^{\\prime } \\in V^{\\prime }$ .","either $U^{\\prime \\left(r_1s\\right)}$ is constant or $V^{\\prime \\left(r_2s\\right)}$ is constant.","Without loss of generality, we assume that $U^{\\prime \\left(r_1s\\right)}$ is a constant.","Therefore, $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|& > & s^{-n/2}q^{-24}\\cdot 12q\\cdot q^{24\\log \\frac{m}{r_1r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}\\\\&&=12q \\cdot q^{24\\left(\\log \\frac{m}{r_1r_2}-1\\right)}\\left(\\frac{m}{r_1r_2s}\\right)^{n/2}\\\\&&\\ge 12q \\cdot q^{24\\log \\frac{m}{r_1r_2s}}\\left(\\frac{m}{r_1r_2s}\\right)^{n/2},$ where the last inequality used the fact that $s \\ge 2$ .","This however contradicts the inductive hypothesis.","$\\Box $"],["Notations:","For a $t \\times s$ matrix $M$ over $\\mathbb {Z}_m$ and for lists $T \\subseteq [t], S \\subseteq [s]$ the $T \\times S$ submatrix of $M$ is the matrix with rows in $T$ and columns in $S$ .","For $i\\in [s]$ and $j \\in [t]$ we denote the $i$ 'th row of $M$ by $M(i:)$ and the $j$ 'th column by $M(:j)$ .","Definition 4.1 (Span of a set) For $A \\subseteq \\mathbb {Z}_m^n$ let $\\textnormal {span}\\left(A\\right)$ denote the additive subgroup generated by $A$ .","We say that a set $A$ spans $u \\in \\mathbb {Z}_m^n$ if $u \\in \\textnormal {span}(A)$ .","Definition 4.2 (Rank of a matrix over $\\mathbb {Z}_m$ ) Let $M$ be a $t \\times t$ matrix over $\\mathbb {Z}_m$ .","Then $\\textnormal {rank}\\left(M\\right)$ is the smallest $r$ such that $M=AB$ where $A$ is an $t \\times r$ martrix over $\\mathbb {Z}_m$ and $B$ is an $r \\times t$ matrix over $\\mathbb {Z}_m$ .","Definition 4.3 (Column rank of a matrix over $\\mathbb {Z}_m$ ) Let $M$ be a $t \\times t$ matrix over $\\mathbb {Z}_m$ .","Let $\\textnormal {colspan}\\left(M\\right)$ denote the subgroup of $\\mathbb {Z}_m^t$ generated by the columns of $M$ .","The column rank of $M$ over $\\mathbb {Z}_m$ is defined as $\\textnormal {colrank}\\left(M\\right)=\\log _m \\left|\\textnormal {colspan}\\left(M\\right)\\right|.$ The column rank is, in general, a real number in the range $[0,t]$ .","Since the rank can behave in unexpected ways over $\\mathbb {Z}_m$ , we make sure to prove some of the basic facts that we will be using later on.","Fact 4.4 Let $M$ be a $t \\times t$ matrix over $\\mathbb {Z}_m$ and let $M^{\\prime }$ be any submatrix of $M$ .","Then $\\textnormal {colrank}\\left(M^{\\prime }\\right)\\le \\textnormal {colrank}\\left(M\\right)$ .","Suppose $M^{\\prime }$ is given by the first $t^{\\prime }$ rows and the first $t^{\\prime \\prime }$ columns of $M$ .","We will define an injective map $f:\\textnormal {colspan}\\left(M^{\\prime }\\right) \\rightarrow \\textnormal {colspan}\\left(M\\right)$ .","Given any $x \\in \\textnormal {colspan}\\left(M^{\\prime }\\right)$ we can write $x = \\sum _{j=1}^{t^{\\prime \\prime }} \\alpha _j \\cdot M^{\\prime }(:j)$ in some fixed way (there might be several choices of $\\alpha _j$ ).","Define $f(x) = \\sum _{j=1}^{t^{\\prime \\prime }} \\alpha _j \\cdot M(:j)$ .","Then, $x$ is clearly the restriction of $f(x)$ to the first $t^{\\prime }$ indices and so the map is injective.","Fact 4.5 Let $M$ be a $t \\times t$ matrix over $\\mathbb {Z}_m$ and let $s|m$ .","Then $\\textnormal {rank}\\left(M^{\\left(s\\right)}\\right)\\le \\textnormal {rank}\\left(M\\right)$ .","Suppose there exist an $t \\times r$ matrix $A$ and an $r \\times t$ matrix $B$ over $\\mathbb {Z}_m$ such that $M=AB$ .","Then $M^{\\left(s\\right)}=A^{\\left(s\\right)}B^{\\left(s\\right)}$ and so the rank of $M^{(s)}$ is at most $r$ .","We will need the following claims relating the rank and the column rank of matrices over $\\mathbb {Z}_m$ .","Claim 4.6 Let $M$ be an $t \\times t$ matrix over $\\mathbb {Z}_m$ .","Then, $\\frac{\\textnormal {rank}\\left(M\\right)}{\\log m} \\le \\textnormal {colrank}\\left(M\\right) \\le \\textnormal {rank}\\left(M\\right).$ Let $r=\\textnormal {rank}\\left(M\\right)$ and $r^{\\prime }=\\textnormal {colrank}\\left(M\\right)$ .","We first prove that $r^{\\prime } \\le r$ .","This is equivalent to proving that $\\left|\\textnormal {colspan}\\left(M\\right)\\right| \\le m^r$ .","Let $M=AB$ where $A$ is an $t \\times r$ martrix over $\\mathbb {Z}_m$ and $B$ is an $r \\times t$ matrix over $\\mathbb {Z}_m$ .","Since the columns of $M$ are all in the span of the columns of $A$ we have that the column span of $M$ can contain at most $m^r$ elements.","We now prove that $r^{\\prime } \\ge r/\\left(\\log m\\right)$ or, equivalently, $\\left|\\textnormal {colspan}\\left(M\\right)\\right| \\ge 2^r$ .","Suppose in contradiction that $\\left|\\textnormal {colspan}\\left(M\\right)\\right| < 2^r$ .","Take a minimal spanning set $S$ of $\\textnormal {colspan}\\left(M\\right)$ (that is, a set that spans $\\textnormal {colspan}\\left(M\\right)$ and such that no proper subset of it does).","Suppose $|S| \\ge r$ and consider all linear combinations (over $\\mathbb {Z}_m$ ) of elements of $S$ with coefficients in $\\lbrace 0,1\\rbrace \\subseteq \\mathbb {Z}_m$ .","Since $\\left|\\textnormal {colspan}\\left(M\\right)\\right| < 2^r$ there are two distinct $0-1$ linear combinations that map to the same element.","This means that there is a linear combination with coefficients in $\\lbrace 1,-1\\rbrace $ of the elements of $S$ that is equal to zero.","Since both 1 and $-1$ are invertible modulo $m$ we can write one of the elements of $S$ as a linear combination of the other elements.","This contradicts the minimality of $S$ and so, we must have $|S| < r$ .","This implies that $\\textnormal {rank}(M) < r$ , a contradiction, since we can write $M$ as the product of the matrix with columns in $S$ with the matrix of coefficients giving the columns of $M$ .","Claim 4.7 Let $M$ be an $t \\times t$ matrix over $\\mathbb {Z}_m$ , let $r=\\textnormal {rank}\\left(M\\right)$ .","There exists $r^{\\prime }$ columns of $M$ that span the rest of $M^{\\prime }$ s columns such that $r^{\\prime } \\le r\\log m$ .","Take a minimal spanning set $S$ of the columns of $M$ (that is, a set that spans all other columns and such that no proper subset of it spans all columns).","If $2^{\\left|S\\right|} > m^r$ , then $2^{\\left|S\\right|} > \\left|\\textnormal {colspan}(M)\\right|$ (by Claim REF ) and we proceed as in the proof from Claim REF above.","If we look at all the $0-1$ combinations of the columns of $S$ , then there are two distinct $0-1$ linear combinations of the columns that map to the same element of $\\textnormal {colspan}\\left(M\\right)$ .","Thus, let $\\sum _i\\alpha _iS\\left(:i\\right)=\\sum _i\\beta _iS\\left(:i\\right)$ where $\\alpha _i \\ne \\beta _i$ for at least one $i$ , say $i_0$ .","Therefore, we have $\\sum _i\\left(\\alpha _i-\\beta _i\\right)S\\left(:i\\right)=0$ .","Note that $\\left(\\alpha _{i_0}-\\beta _{i_0}\\right)=\\pm 1$ and hence is invertible.","This lets us write $S\\left(:i_0\\right)$ as a linear combinations of the remaining columns contradicting the minimality of $S$ .","Thus, $r^{\\prime }=\\left|S\\right| \\le r\\log m$ .","The following claim shows that the column rank behaves similar to rank in terms of subadditivity.","Claim 4.8 Let $A, B$ be $t \\times t$ matrices over $\\mathbb {Z}_m$ .","Then, $\\textnormal {colrank}\\left(A+B\\right)\\le \\textnormal {colrank}\\left(A\\right)+\\textnormal {colrank}\\left(B\\right)$ .","We show that $\\left|\\textnormal {colspan}\\left(A+B\\right)\\right|\\le \\left|\\textnormal {colspan}\\left(A\\right)\\right|\\left|\\textnormal {colspan}\\left(B\\right)\\right|$ .","Note that $\\textnormal {colspan}\\left(A+B\\right) \\subseteq \\textnormal {colspan}\\left(A\\right)+ \\textnormal {colspan}\\left(B\\right) \\stackrel{def}{=} \\lbrace a+b|a\\in \\textnormal {colspan}\\left(A\\right), b \\in \\textnormal {colspan}\\left(B\\right)\\rbrace $ .","Therefore, $\\left|\\textnormal {colspan}\\left(A+B\\right)\\right| \\le \\left|\\textnormal {colspan}\\left(A\\right)+ \\textnormal {colspan}\\left(B\\right)\\right| \\le \\left|\\textnormal {colspan}\\left(A\\right)\\right|\\left|\\textnormal {colspan}\\left(B\\right)\\right|$ .","Claim 4.9 Let $M$ be a $2t \\times 2t$ matrix over $\\mathbb {Z}_m$ , such that $M=\\left(\\begin{array}{cc}A&0\\\\\\star &B\\end{array}\\right)$ where $A, B $ and $\\star $ are $t \\times t$ matrices.","Then, $\\textnormal {colrank}\\left(A\\right)+\\textnormal {colrank}\\left(B\\right) \\le \\textnormal {colrank}\\left(M\\right)$ .","We show that $\\left|\\textnormal {colspan}\\left(A\\right)\\right|\\left|\\textnormal {colspan}\\left(B\\right)\\right|\\le \\left|\\textnormal {colspan}\\left(M\\right)\\right|$ .","Let $\\textnormal {colspan}\\left(A\\right)=R_1$ , $\\textnormal {colspan}\\left(B\\right)=R_2$ , $\\textnormal {colspan}\\left(M\\right)=R$ .","We define $f:R_1 \\times R_2 \\rightarrow R$ and show that $f$ is injective.","Given $r_1 \\in R_1$ and $r_2 \\in R_2$ , let $\\alpha _1, \\cdots \\alpha _t$ and $\\beta _1,\\cdots \\beta _t$ denote coefficients for linear combinations of the columns of $A$ and $B$ respectively that give $r_1$ and $r_2$ .","There might be many such linear combinations but we fix one for each $r_i$ .","Then, $f\\left(r_1,r_2\\right)=\\sum _{i=1}^{t}\\alpha _iM\\left(:i\\right)+\\sum _{i=t+1}^{2t}\\beta _{i-t}M\\left(:i\\right)$ .","Now, given a column vector $f\\left(r_1,r_2\\right) \\in R$ , we uniquely identify $r_1$ and $r_2$ as follows.","We look at the first $t$ rows and call it $s_1$ .","Now $s_1=r_1$ and let $\\alpha _1, \\cdots \\alpha _{t}$ be the linear combination fixed for $r_1$ while defining $f$ .","Now, consider $f\\left(r_1,r_2\\right)-\\sum _{i=1}^{t}\\alpha _iM\\left(:i\\right)$ and call the last $t$ rows $s_2$ .","Note that $s_2=r_2$ .","Claim 4.10 Let $M$ be a $t \\times t$ square matrix over $\\mathbb {Z}_m$ with zero diagonal entries.","If for some $s |m$ , $\\textnormal {colrank}\\left(M^{\\left(s\\right)}\\right)\\le 2$ , then there exists at least $t^{\\prime }=t/m^2$ indices such that $M$ restricted to those indices as rows and columns is the all zero matrix modulo $s$ .","As $\\textnormal {colrank}\\left(M^{\\left(s\\right)}\\right)\\le 2$ , it follows that $\\left|\\textnormal {colspan}\\left(M^{\\left(s\\right)}\\right)\\right|\\le s^2 \\le m^2$ .","Hence, $M^{\\left(s\\right)}$ has at most $m^2$ distinct columns.","Therefore, there exists a set of indices $S$ of size $t^{\\prime }\\ge t/m^2$ with $S=\\lbrace r_1, r_2, \\cdots r_{t^{\\prime }}\\rbrace $ such that all the columns $M^{\\left(s\\right)}\\left(:r_i\\right)$ are identical.","Also, as the diagonal elements are zero modulo $m$ , they are zero modulo $s$ .","Thus, the $S \\times S$ submatrix is the all zero matrix modulo $s$ ."],["Collision-Free MV families","In the proof of Theorem REF it will be useful to assume that the elements of the MV family do not `collide' when reduced modulo an integer $s$ dividing $m$ .","In this section we develop the necessary machinery to allow for this assumption.","We start by defining a collision free matching vector family.","Definition 5.1 (Collision free MV family) A collision free matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ is a matching vector family such that for all $s|m, s \\ge 2$ , all elements of $U$ are distinct modulo $s$ , and all elements of $V$ are distinct modulo $s$ .","Note that if $\\left(U,V\\right)$ is a collision free matching vector family, then so is any $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ .","Lemma 5.2 Let $m\\ge 2$ be an arbitrary integer.","Let $s$ be a divisor of $m$ , such that $1d\\left(m/2\\right)\\log m$ $2^{\\left(1/2\\right)d\\left(m\\right)n/\\log n} \\ge 3m2^{d_2\\left(m\\right)n/\\log n}$ $d_2\\left(m\\right)n/\\log n \\ge 2\\log m+d_3\\left(m\\right)n/\\log n$ $d_3\\left(m\\right) \\ge d_1\\left(m\\right) \\cdot d_4 \\cdot m \\log m$ $d_4 \\ge 300$ $d_1\\left(m\\right)\\ge 2c\\left(m\\right)$ $d_2 \\ge d_3+1$ It can be verified that the following choice for the functions meets the above conditions.","$d\\left(m\\right)=1200 \\cdot c\\left(m\\right) \\cdot m^{6\\log m}$ $d_1\\left(m\\right)=2 \\cdot c\\left(m\\right)$ $d_2\\left(m\\right)=602 \\cdot c\\left(m\\right) \\cdot m \\log m$ $d_3\\left(m\\right)=600 \\cdot c\\left(m\\right)\\cdot m\\log m$ $d_4=300$ We shall explicitly mention which conditions of the above functions are being used in different parts of the proof."],["Base Case.","The base case is where $m=p$ is prime.","Lemma REF implies that $\\mathbf {MV}\\left(p,n\\right) \\le 1+{n+p-2 \\atopwithdelims ()p-1}<\\left(2\\max \\lbrace n,p\\rbrace \\right)^p$ .","If we show $\\left(2n\\right)^p \\le 2^{d\\left(p\\right)n/\\log n}$ and $\\left(2p\\right)^p \\le 2^{d\\left(p\\right)n/\\log n}$ we will be done.","Indeed, by the choice of $d\\left(m\\right)$ (Condition 2 and 3) both of the above will hold."],["Inductive Case.","Let $n \\ge 2, m \\ge 2$ be arbitrary positive integers.","Suppose, by induction, that $\\mathbf {MV}\\left(s,n\\right) < 2^{d\\left(s\\right)n/\\log n}$ for all $s|m, s 2$ for all $s^{\\prime }|m, s^{\\prime }\\ge 2$ .","Then, for some $s|m, s\\ge 2$ , there exists a collision free matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ satisfying $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge 2^{-d_1\\left(m\\right)r_s/\\log r_s}\\left|\\left(U,V\\right)\\right|$ where $r_s=\\textnormal {rank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ .","Either $\\textnormal {colrank}\\left(P_{U^{\\prime },V^{\\prime }}^{\\left(s\\right)}\\right) \\le \\left(3/4\\right)\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ or $\\textnormal {colrank}\\left(P_{U^{\\prime },V^{\\prime }}^{\\left(s\\right)}\\right) \\le 2$ .","Claim 6.6 Let $\\left(U,V\\right)$ be a collision free matching vector family in $\\mathbb {Z}_m^n$ such that $\\left|\\left(U,V\\right)\\right|\\ge 3m \\cdot 2^{d_2\\left(m\\right)n/\\log n}$ .","Then, there exists a collision free matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ satisfying $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge 2^{-d_2\\left(m\\right)n/\\log n}\\left|\\left(U,V\\right)\\right|$ .","$P_{U,V}^{\\left(s\\right)}$ is the all zero matrix for some $s|m, s \\ge 2$ .","Let us proceed with the proof of the inductive claim assuming these two claims.","We have a collision free matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)$ with $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge 2^{\\left(1/2\\right)d\\left(m\\right)n/\\log n} \\ge 3m \\cdot 2^{d_2\\left(m\\right)n/\\log n}$ .","(Condition 6 satisfied by the choice of $d\\left(m\\right), d_2\\left(m\\right)$ ) Applying Claim REF , there exists a collision free matching vector family $\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\subseteq \\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ satisfying $\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right|\\ge 2^{-d_2\\left(m\\right)n/\\log n}2^{\\left(1/2\\right)d\\left(m\\right)n/\\log n}$ .","$P_{U^{\\prime \\prime },V^{\\prime \\prime }}^{\\left(s\\right)}$ is the all zero matrix for some $s|m, s \\ge 2$ .","By the choice of $d\\left(m\\right)$ , it can be verified that $-d_2\\left(m\\right)+\\left(1/2\\right)d\\left(m\\right)>d\\left(m/2\\right)\\log m$ (Condition 5).","Thus, $\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right| > 2^{d\\left(m/2\\right)n\\log m/\\log n}$ .","We now show that this is enough to get a contradiction.","If $s=m$ , we have $\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right|\\le 1$ as $\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ .","If $s|R|/2$ or $|T| \\le |R|/2$ .","In both cases, we shall exhibit a matching vector family as required in the statement of the claim.","Case 1: $|T|>|R|/2$ .","For $U=\\left(u_1, u_2, \\cdots u_t\\right)$ , $V=\\left(v_1, v_2, \\cdots v_t\\right)$ , let $U^{\\prime }=\\left(u_j|j\\in T\\right)$ and $V^{\\prime }=\\left(v_j|j\\in T\\right)$ , and $P^{\\prime }=P_{U^{\\prime },V^{\\prime }}$ .","Now, as $P^{\\prime \\left(s\\right)}$ is monochromatic, and $\\langle u_j,v_j\\rangle =0\\ \\left(mod\\ s\\right)$ for $j \\in T$ , we have $\\langle u^{\\prime },v^{\\prime } \\rangle =0 \\left(mod\\ s\\right)$ for all $u^{\\prime } \\in U^{\\prime }, v^{\\prime } \\in V^{\\prime }$ .","Observe that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right| \\ge 2^{-1-c\\left(m\\right)r_s/\\log r_s}t \\ge 2^{-2c\\left(m\\right)r_s/\\log r_s}t\\ge 2^{-d_1\\left(m\\right)r_s/\\log r_s}t$ (by the choice of $d_1\\left(m\\right)$ , Condition 10) $\\textnormal {colrank}\\left(P_{U^{\\prime },V^{\\prime }}^{\\left(s\\right)}\\right) =0 \\le 2$ This finishes Case 1.","Case 2: $|T| \\le |R|/2$ .","Let $R^{\\prime }=R \\setminus T$ and $S^{\\prime }=S \\setminus T$ .","Note that $R^{\\prime } \\cap S^{\\prime }=\\emptyset $ and $|R^{\\prime }|=|S^{\\prime }|$ .","Consider the $R^{\\prime }\\cup S^{\\prime } \\times R^{\\prime }\\cup S^{\\prime }$ submatrix of $P_{U,V}$ .","Call it $P^{\\prime }$ .","Note that $P^{\\prime \\left(s\\right)}=\\left(\\begin{array}{cc}P_1^{\\prime }&C\\\\\\star &P_2^{\\prime }\\end{array}\\right)$ where $P_1^{\\prime }$ and $P_2^{\\prime }$ are the $R^{\\prime } \\times R^{\\prime }$ and the $S^{\\prime } \\times S^{\\prime }$ submatrices of $P_{U,V}^{\\left(s\\right)}$ respectively and $C$ is monochromatic.","We add a matrix of column rank at most 1 to $P^{\\prime \\left(s\\right)}$ to yield $P^{\\prime \\prime \\left(s\\right)}$ which is the same as $P^{\\prime \\left(s\\right)}$ except that $C$ is replaced by the all zero block matrix.","Thus, $P^{\\prime \\prime \\left(s\\right)}=\\left(\\begin{array}{cc}P_1^{\\prime }&0\\\\\\star &P_2^{\\prime }\\end{array}\\right)$ Note that by Claim REF , $\\textnormal {colrank}\\left(P^{\\prime \\prime \\left(s\\right)}\\right) \\le \\textnormal {colrank}\\left(P^{\\prime \\left(s\\right)}\\right)+1$ .","Now, using Claim REF , $\\textnormal {colrank}\\left(P_1^{\\prime }\\right)+\\textnormal {colrank}\\left(P_2^{\\prime }\\right)\\le \\textnormal {colrank}\\left(P^{\\prime \\left(s\\right)}\\right)+1 \\le \\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)+1 \\le \\left(3/2\\right)\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ as $\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right) >2$ .","Therefore, one of $P_1^{\\prime }, P_2^{\\prime }$ , say $P_1^{\\prime }$ satisfies $\\textnormal {colrank}\\left(P_1^{\\prime }\\right)\\le \\left(3/4\\right)\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ .","Construct the matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)$ as follows.","Let $U^{\\prime }=\\left(u_j|j\\in R^{\\prime }\\right)$ and $V^{\\prime }=\\left(v_j|j\\in R^{\\prime }\\right)$ .","Again, observe that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right| \\ge 2^{-1-c\\left(m\\right)r_s/\\log r_s}t \\ge 2^{-2c\\left(m\\right)r_s/\\log r_s}t\\ge 2^{-d_1\\left(m\\right)r_s/\\log r_s}t$ (by the choice of $d_1\\left(m\\right)$ , Condition 10).","$\\textnormal {colrank}\\left(P_{U^{\\prime },V^{\\prime }}^{\\left(s\\right)}\\right) \\le (3/4)\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ .","This completes the proof of Case 2.","$\\Box $"],["Proof of Claim ","We will use Claim REF iteratively.","For this, we first set up some notations."],["The setup.","Define a sequence of collision free matching vector families for $i=0,\\ldots ,z$ .","$\\left(U,V\\right)=\\left(U_0,V_0\\right), \\left(U_1,V_1\\right) \\cdots $ Let $t_i=\\left|\\left(U_i,V_i\\right)\\right|$ .","Each step $i$ has label $s_i|m$ (this label will be given by Claim REF ).","Let $cr_i:\\mathbb {Z}^+ \\rightarrow \\mathbb {R}$ be defined by $cr_i\\left(s\\right)=\\textnormal {colrank}\\left(P_{U_i,V_i}^{\\left(s\\right)}\\right).$ Let $r_i:\\mathbb {Z}^+ \\rightarrow \\mathbb {Z}$ be defined by $r_i\\left(s\\right)=\\textnormal {rank}\\left(P_{U_i,V_i}^{\\left(s\\right)}\\right).$"],["Invariants.","We will show how to go from step $i$ to step $i+1$ .","We stop after stage $z$ when $cr_z\\left(s\\right) \\le 2$ for some $s|m, s\\ge 2$ .","We shall maintain the following invariants for $0 \\le i \\le z-1$ .","$\\left(U_{i+1},V_{i+1}\\right)\\subseteq \\left(U_{i},V_{i}\\right)$ and hence is a collision free matching vector family in $\\mathbb {Z}_m^n$ .","$t_{i+1} \\ge 2^{-d_1\\left(m\\right)r_i\\left(s_i\\right)/\\log r_i\\left(s_i\\right)}t_i$ .","$cr_{i+1}\\left(s_i\\right)\\le \\left(3/4\\right)cr_i\\left(s_i\\right)$ or $cr_{i+1}\\left(s_i\\right)\\le 2$ .","$cr_{i+1}\\left(s^{\\prime }\\right)\\le cr_i\\left(s^{\\prime }\\right)$ for all $s^{\\prime }|m$ ."],["Step $i$ {{formula:ef557492-5072-4f74-b6af-082231e35300}} Step {{formula:d641f54b-beb7-403b-a4b6-d7eea3906b1e}} .","We state a claim that we will prove below.","Claim 6.7 $\\sum _{i=0}^{z-1}d_1\\left(m\\right)r_i\\left(s_i\\right)/\\log r_i\\left(s_i\\right)\\le d_3\\left(m\\right)n/log n$ .","In order to apply Claim REF , we need to satisfy $t_i \\ge 3m$ .","Observe that by Claim REF , $t_i\\ge t_z &\\ge & t_0\\prod _{j=0}^{z-1}2^{-d_1\\left(m\\right)r_j\\left(s_j\\right)/\\log r_j\\left(s_j\\right)}\\\\ &\\ge & 2^{-d_3\\left(m\\right)n/\\log n}t_0 \\\\&\\ge & 3m \\cdot 2^{-d_3\\left(m\\right)n/\\log n+d_2\\left(m\\right)n/\\log n}\\ge 3m,$ (by the choice of $d_2\\left(m\\right), d_3\\left(m\\right)$ in Condition 11).","Apply Claim REF to $\\left(U_{i},V_{i}\\right)$ to get label $s_i$ for step $i$ and $\\left(U_{i+1},V_{i+1}\\right)\\subseteq \\left(U_{i},V_{i}\\right)$ .","The first three invariants are maintained by the statement of Claim REF .","The last invariant follows from Fact REF .","Note that by the inequality we just established, $t_z\\ge 2^{-d_3\\left(m\\right)n/\\log n}t_0$ .","Also, by the stopping condition, $cr_z\\left(s^{\\prime }\\right) \\le 2$ for some $s^{\\prime }|m, s^{\\prime }\\ge 2$ .","Thus, applying Claim REF , we get another matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U_z,V_z\\right)\\subseteq \\left(U,V\\right)$ such that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge t_z/m^2 \\ge 2^{-2\\log m-d_3\\left(m\\right)n/\\log n}\\left|\\left(U,V\\right)\\right|\\ge 2^{-d_2\\left(m\\right)n/\\log n}\\left|\\left(U,V\\right)\\right|$ (Condition 7 satisfied by the choice of $d_2\\left(m\\right)$ and $d_3\\left(m\\right)$ ).","$P_{U^{\\prime },V^{\\prime }}^{\\left(s^{\\prime }\\right)}$ is the all zero matrix.","This finishes the Proof of Claim REF .","Proof of Claim REF : Let $t_s$ be the number of steps with label $s$ .","Note that as the column rank modulo $s$ goes down by a factor of at least $3/4$ each time we are in a step labeled $s$ , it is easy to see that $t_s \\le \\log _{4/3}cr_0\\left(s\\right) \\le \\log _{4/3}n$ .","We shall rely on the monotonic increasing nature of $x/\\log x$ when $x \\ge e$ .","As $cr_i\\left(s\\right)>2$ , by Claim REF , $r_i\\left(s\\right)\\ge cr_i\\left(s\\right)> 2$ which means $r_i(s) \\ge 3 > e$ as the rank is always an integer.","We thus have $&&\\sum _{i=0}^{z-1}d_1\\left(m\\right)\\frac{r_i\\left(s_i\\right)}{\\log r_i\\left(s_i\\right)}\\\\&\\le & d_1\\left(m\\right)\\log m\\sum _{i=0}^{z-1}\\frac{cr_i\\left(s_i\\right)}{\\log cr_i\\left(s_i\\right)}\\ \\ \\text{(by Claim \\ref {clm-rankcolrank}) and monotonicity of $x/\\log x$ as discussed above}\\\\&\\le & d_1\\left(m\\right)\\log m\\sum _{s|m, s\\ge 2}\\sum _{j=1}^{\\lfloor \\log _{4/3}n\\left(s\\right)\\rfloor }\\left(\\frac{cr_0\\left(s\\right)}{\\left(4/3\\right)^{j-1}\\log \\left(cr_0\\left(s\\right)/\\left(4/3\\right)^{j-1}\\right)}\\right)\\\\&\\le & d_1\\left(m\\right)\\log m\\sum _{s|m, s\\ge 2}d_4cr_0\\left(s\\right)/\\log cr_0\\left(s\\right)\\ \\ \\text{(by Claim \\ref {clm-sum} and Condition $9$ satisfied by $d_4$)}\\\\&\\le & d_1\\left(m\\right)\\log m\\sum _{s|m, s\\ge 2}d_4n/\\log n\\ \\ \\text{(as $cr_0\\left(s\\right)\\le r_0\\left(s\\right)\\le r_0\\left(m\\right)\\le n$, by Claim \\ref {clm-rankcolrank} and Fact \\ref {rankmon})}\\\\&\\le & d_4d_1\\left(m\\right)m\\left(\\log m\\right)n/\\log n\\\\&\\le & d_3\\left(m\\right)n/\\log n\\ \\ \\text{(by the choice of $d_3\\left(m\\right)$, Condition $8$)}$ This completes the proof.","$\\Box $"],["Monochromatic rectangles from low rank matrices","In this section we prove Lemma REF (the Sub-Matrix Lemma).","We begin with some preliminary definitions.","The following is a standard result in algebra and can be find in any introductory text.","Theorem 7.1 (Fundamental Theorem of finitely generated abelian groups) Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of prime power order and an infinite cyclic group.","More precisely, $G \\cong \\mathbb {Z}^n \\times \\mathbb {Z}_{q_1} \\times \\mathbb {Z}_{q_2} \\cdots \\times \\mathbb {Z}_{q_r}$ where $q_i$ 's are prime powers with $q_1 \\le q_2 \\cdots \\le q_r$ .","The decomposition is unique after applying this ordering on $q_i$ 's.","If the group $G$ is finite, then $n=0$ .","We will use the following two definitions regarding sumsets.","Definition 7.2 (Difference Set) For $A \\subseteq \\mathbb {Z}_m^n$ define its difference set as $A-A{=}\\lbrace a-a^{\\prime }|a,a^{\\prime }\\in A\\rbrace $ .","Definition 7.3 ($rep_S\\left(x\\right)$ ) For any $S \\subseteq \\mathbb {Z}_m^n$ and $x \\in \\mathbb {Z}_m^n$ , $rep_S\\left(x\\right)$ is the number of different representations of $x$ as an expression of the form $s-s^{\\prime }$ where $s,s^{\\prime } \\in S$ .","Next, we define the $\\epsilon $ -spectrum of $B$ with respect to a primitive root of unity of order $m$ .","Definition 7.4 (Spectrum) For $B \\subseteq \\mathbb {Z}_m^n$ , and $\\epsilon \\in [0,1]$ , the $\\epsilon $ -spectrum of $B$ with respect to $\\omega $ , a primitive root of unity of order $m$ , is the set $\\textnormal {Spec}_{\\epsilon }\\left(B\\right)=\\left\\lbrace x \\in \\mathbb {Z}_m^n:\\left|\\mathbb {E}_{b \\sim B}\\left[\\omega ^{\\langle x,b \\rangle }\\right]\\right| \\ge \\epsilon \\right\\rbrace .$ When $\\omega $ is implicit in the context, we will drop the phrase \"with respect to $\\omega $ \".","We start by proving the following lemma which is a generalization of a lemma from [5].","Lemma 7.5 Let $A,B \\subseteq \\mathbb {Z}_m^n$ be sets.","Let $\\omega $ be a primitive root of unity of order $m$ .","If $A \\subseteq \\textnormal {Spec}_{\\epsilon }\\left(B\\right)$ , then there exist sets $A^{\\prime }\\subseteq A, B^{\\prime }\\subseteq B$ , such that $|A^{\\prime }| \\ge |A|/m$ and $|B^{\\prime }| \\ge \\epsilon ^2\\frac{|A|}{|span\\left(A\\right)|}|B|$ such that $D_{\\omega }\\left(A^{\\prime },B^{\\prime }\\right)=1$ .","We start by setting up some notations.","Let $W=\\textnormal {span}\\left(A\\right)$ be the subgroup of $\\mathbb {Z}_m^n$ spanned by $A$ .","By Theorem REF , there exists an isomorphism $\\tau :\\prod _{i=1}^r\\mathbb {Z}_{q_i}\\rightarrow W$ .","Let $\\prod _{i=1}^r\\mathbb {Z}_{q_i}$ and note that we can think of elements of $ as vectors with integer coordinates where the $ i$^{\\prime }th coordinate is in $ Zqi$.","Let $ e1,e2, er where $e_i$ is the vector that has 1 in the $i$ 'th coordinate and 0 everywhere else.","Given $x \\in , $ 1, r$, with $ i Zqi$ such that $$x=\\sum _{i=1}^r\\alpha _ie_i.$$Then $ (x)=i=1ri(ei)$.","Let $ vi=(ei)$ for $ 1 i r$.","We can think of the $ vi$^{\\prime }s as a basis of $ W$.","Therefore, for $ =(1, 2, r) we have $\\tau \\left(\\alpha \\right)=\\sum _{i=1}^r\\alpha _iv_i$ .","Let $\\Theta =\\lbrace \\left(\\beta _1, \\cdots \\beta _r\\right) \\in \\mathbb {Z}_m^r|\\exists u \\in \\mathbb {Z}_m^n \\text{ such that } \\forall i, \\beta _i=\\langle v_i,u\\rangle \\rbrace .$ Claim 7.6 For $1 \\le i \\le r$ , $q_iv_i=0^n\\ \\left(mod \\ m\\right)$ .","Let $x =0^r \\in .","Now $ (x)=0n (mod m)$.","Note that $ x$ can also be written as $ x=qiei$.","Applying $$ on both sides, we get $ (x)=qivi$.","Thus, $ qivi=0n (mod m)$.$ Claim 7.7 For $\\beta \\in \\Theta $ , $1 \\le i \\le r$ , $q_i\\beta _i=0\\ \\left(mod \\ m\\right)$ .","As $\\beta \\in \\Theta $ , there is a $u \\in \\mathbb {Z}_m^n$ such that $\\forall i$ , $\\beta _i=\\langle v_i,u\\rangle $ .","Then, $q_i\\beta _i=q_i\\langle v_i,u\\rangle =0\\ \\left(mod\\ m\\right)$ by Claim REF .","For $\\alpha \\in \\beta \\in \\Theta $ we define their inner product $\\langle \\alpha , \\beta \\rangle \\in \\mathbb {Z}_m$ by considering $\\alpha _i \\in \\lbrace 0,\\ldots ,q_i-1\\rbrace , \\beta _i \\in \\lbrace 0,\\ldots ,m-1\\rbrace $ , taking the inner product over the integers and then reducing the result modulo $m$ .","This is indeed an inner product by Claim REF .","Claim 7.8 Given $\\beta \\in \\Theta \\setminus \\lbrace 0\\rbrace $ , $\\sum _{a \\in W}\\omega ^{\\langle \\tau ^{-1}\\left(a\\right),\\beta \\rangle }=\\sum _{\\alpha \\in \\omega ^{\\langle \\alpha ,\\beta \\rangle }=0.","}\\begin{proof}Let \\beta _i \\ne 0.","Then \\sum _{\\alpha \\in C}\\omega ^{\\langle \\alpha ,\\beta \\rangle }=0 whenever \\sum _{j=0}^{q_i-1}\\omega ^{j\\beta _i}=0.","Now, \\sum _{j=0}^{q_i-1}\\omega ^{j\\beta _i}=\\frac{\\omega ^{q_i\\beta _i}-1}{\\omega ^{\\beta _i}-1}.","This is well defined because \\omega is of order m and \\beta _i\\ne 0.","The claim now follows from Claim \\ref {clm-beta} which makes the expression zero.\\end{proof}$ With the above setup in place, we can now proceed with the proof of Lemma REF .","For $\\beta \\in \\Theta $ , define $S_{\\beta }=\\lbrace x \\in \\mathbb {Z}_m^n|\\langle v_i,x\\rangle =\\beta _i, 1\\le i\\le r\\rbrace .$ Denoting $\\mu \\left(\\beta \\right)=\\mathbf {Pr}_{b \\in B}[b \\in S_{\\beta }]$ , we observe that $\\cup _{\\beta \\in \\Theta }\\left(B\\cap S_{\\beta }\\right)=B$ .","Hence, $\\sum _{\\beta \\in \\Theta }\\mu \\left(\\beta \\right)=1$ .","For $a \\in W$ , define $h\\left(a\\right)=\\mathbb {E}_{b \\in B}\\left[\\omega ^{\\langle a,b \\rangle }\\right]$ .","If $a=\\sum _{i=1}^r\\alpha _iv_i$ then $h\\left(a\\right)&=&\\mathbb {E}_{b \\in B}\\left[\\omega ^{\\langle a,b \\rangle }\\right]\\\\&=&\\mathbb {E}_{b \\in B}\\left[\\omega ^{\\langle \\sum _{i=1}^r\\alpha _iv_i,b \\rangle }\\right]\\\\&=&\\sum _{\\beta \\in \\Theta }\\mu \\left(\\beta \\right)\\omega ^{\\langle \\alpha ,\\beta \\rangle }\\\\&=&\\sum _{\\beta \\in \\Theta }\\mu \\left(\\beta \\right)\\omega ^{\\langle \\tau ^{-1}\\left(a\\right) ,\\beta \\rangle }.$ We will prove upper and lower bounds for the sum $\\sum _{a \\in A}\\left|h\\left(a\\right)\\right|^2$ .","On the one hand, $\\sum _{a \\in A}\\left|h\\left(a\\right)\\right|^2&\\ge &\\frac{1}{|A|}\\left(\\sum _{a \\in A}\\left|h\\left(a\\right)\\right|\\right)^2 \\ \\text{(Cauchy Scwartz inequality)}\\\\&\\ge &\\frac{1}{|A|}\\left(\\sum _{a \\in A}\\epsilon \\right)^2 \\ \\text{($A \\subseteq \\textnormal {Spec}_{\\epsilon }\\left(B\\right)$ implies $\\left|h\\left(a\\right)\\right|\\ge \\epsilon $ )}\\\\&\\ge & |A|\\epsilon ^2.$ On the other hand, $\\sum _{a \\in A}\\left|h\\left(a\\right)\\right|^2&\\le & \\sum _{a \\in W}\\left|h\\left(a\\right)\\right|^2\\\\&=&\\sum _{a \\in W}\\sum _{\\beta \\in \\Theta , \\beta ^{\\prime } \\in \\Theta ^{\\prime }}\\mu \\left(\\beta \\right)\\mu \\left(\\beta ^{\\prime }\\right)\\omega ^{\\langle \\tau ^{-1}\\left(a\\right),\\beta - \\beta ^{\\prime } \\rangle }\\\\&=&\\sum _{\\beta ,\\beta ^{\\prime } \\in \\Theta ,}\\mu \\left(\\beta \\right)\\mu \\left(\\beta ^{\\prime }\\right)\\sum _{a \\in W}\\omega ^{\\langle \\tau ^{-1}\\left(a\\right),\\beta - \\beta ^{\\prime } \\rangle }\\\\&=&\\sum _{\\beta \\in \\Theta }\\mu \\left(\\beta \\right)^2|W| \\qquad \\text{(Claim \\ref {clm-zero})}\\\\&\\le & |W| \\max _{\\beta \\in \\Theta }\\lbrace \\mu \\left(\\beta \\right)\\rbrace .$ Now, combining the upper and lower bounds, $\\max _{\\beta \\in \\Theta }\\lbrace \\mu \\left(\\beta \\right)\\rbrace \\ge \\epsilon ^2\\frac{|A|}{|W|}$ .","Thus, there exists a $\\beta \\in \\Theta $ such that $\\mu \\left(\\beta \\right)\\ge \\epsilon ^2\\frac{|A|}{|W|}$ .","This means that the subset $B^{\\prime }=B \\cap S_{\\beta }$ is of size at least $\\epsilon ^2\\frac{|A|}{|W|}|B|$ .","Now, any $a \\in A$ can be written as $a=\\sum _{i=1}^r \\alpha _i v_i$ , and for $b \\in B^{\\prime }$ , the inner product $\\langle a,b \\rangle =\\langle \\alpha ,\\beta \\rangle $ is independent of $b$ .","Now, for $i \\in [m]$ , let $A_i \\subseteq A$ be such that for $a \\in A_i$ , for all $b \\in B^{\\prime }$ , $\\langle a,b \\rangle =\\langle \\alpha ,\\beta \\rangle =i$ .","Now there exists some $A_i$ , call it $A^{\\prime }$ , of size at least $|A|/m$ such that $\\langle a^{\\prime },b^{\\prime } \\rangle =i$ for all $a^{\\prime }\\in A^{\\prime },b^{\\prime } \\in B^{\\prime }$ , that is, $D_{\\omega }\\left(A^{\\prime },B^{\\prime }\\right)=1$ and this proves the lemma.","We continue along the lines of [5] and prove the following lemma.","Lemma 7.9 Suppose the twin free lists $U, V \\subseteq \\mathbb {Z}_m^n$ satisfy $D_{\\omega }\\left(U, V\\right) \\ge \\epsilon $ ̨ where $\\omega $ is a primitive root of unity of order $m$ .","Also, let $\\textnormal {rank}\\left(P_{U,V}\\right)=r$ .","Then assuming Conjecture REF , for every $K > 1$ , letting $\\ell = r/\\log _m K$ , there exist lists $U^{\\prime }\\subseteq U, V^{\\prime }\\subseteq V$ such that $D_{\\omega }\\left(U^{\\prime }, V^{\\prime }\\right) = 1$ , and $|U^{\\prime }| \\ge poly_m\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell }}}{rK}\\right)\\left(2mr\\right)^{-\\ell }|U|$ , $|V^{\\prime }| \\ge poly_m\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell }}}{rK}\\right)m^{-\\ell }|V|$ .","Let $U=\\left(u_1, \\cdots u_t\\right)$ and $V=\\left(v_1, \\cdots v_t\\right)$ .","Since $P_{U,V}$ has rank $r$ there exists a $t\\times r$ matrix $U_M$ and $r \\times t$ matrix $V_M$ so that $U_MV_M=P_{U,V}$ .","Thus if we let $A$ denote the rows of $U_M$ and $B$ denote the columns of $V_M$ , then $A,B \\subseteq \\mathbb {Z}_m^r$ .","The proof does not care about the order of elements and hence we now consider $A,B$ which are sets.","Note that $|A|=|B|=t$ and if $A=\\left(a_1, \\cdots a_t\\right)$ and $B=\\left(b_1, \\cdots b_t\\right)$ then $\\langle a_i,b_j \\rangle =\\langle u_i, v_j \\rangle $ for $1\\le i,j \\le t$ .","Thus, $D_{\\omega }\\left(U, V\\right) \\ge \\epsilon $ implies $D_{\\omega }\\left(A, B\\right) \\ge \\epsilon $ .","Following [5] consider a sequence of constants $\\epsilon _1=\\epsilon /2$ , $\\epsilon _2=\\epsilon _1^2/2$ , $\\epsilon _3=\\epsilon _2^2/2$ , $\\cdots $ and a sequence of sets $A_1=A \\cap \\textnormal {Spec}_{\\epsilon _1}\\left(B\\right)$ and $A_{i} \\subseteq \\left(A_{i-1}-A_{i-1}\\right) \\cap \\textnormal {Spec}_{\\epsilon _i}\\left(B\\right)$ .","The way the subsets are chosen for $A_i$ 's will be made precise shortly.","Now by the pigeonhole principle, there exists a minimal index $\\ell \\le r/\\log _m K$ such that $|A_{\\ell +1}| \\le K|A_{\\ell }|$ .","To give a precise definition of the $A_i$ 's , we have the following.","Let $A_1=A \\cap \\textnormal {Spec}_{\\epsilon /2}\\left(B\\right)$ .","For $i\\ge 2$ , assuming $\\epsilon _{i-1}$ and $A_{i-1}$ , let $j_i$ be the the integer index which maximizes the size of $\\lbrace \\left(a,a^{\\prime }\\right) \\in A_{i-1}\\times A_{i-1}|a-a^{\\prime } \\in \\textnormal {Spec}_{\\epsilon _i}\\left(B\\right) \\ and \\ m^{j_i}\\le rep_{A_{i-1}}\\left(a-a^{\\prime }\\right) \\le m^{j_i+1}\\rbrace ,$ and let $A_i=\\lbrace a-a^{\\prime } | a,a^{\\prime } \\in A_{i-1}, a-a^{\\prime } \\in \\textnormal {Spec}_{\\epsilon _i}\\left(B\\right) \\ and \\ m^{j_i}\\le rep_{A_{i-1}}\\left(a-a^{\\prime }\\right) \\le m^{j_i+1}\\rbrace .$ Claim 7.10 For $i=1$ we have $|A_1| \\ge \\left(\\epsilon ̨/2\\right)|A|$ .","For $i>1$ we have $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{i-1}}[a-a^{\\prime } \\in A_i] \\ge \\epsilon _i/r$ and additionally $|A_i| \\ge \\frac{\\epsilon _i}{m^{j_i+1}r}|A_{i-1}|^2$ .","The case of $i=1$ follows from Markov inequality.","For larger $i$ , we show that $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{i-1}}[a-a^{\\prime } \\in \\textnormal {Spec}_{\\epsilon _i}\\left(B\\right)] \\ge \\epsilon _i.$ This follows from the fact that $\\epsilon _{i-1}^2 \\le \\left|\\mathbb {E}_{b \\in B, a \\in A_{i-1}}\\left[\\omega ^{\\langle a,b \\rangle }\\right]\\right|^2 \\le \\mathbb {E}_{b \\in B}\\left|\\mathbb {E}_{a \\in A_{i-1}}\\left[\\omega ^{\\langle a,b \\rangle }\\right]\\right|^2 = \\mathbb {E}_{a,a^{\\prime }\\in A_{i-1}}\\mathbb {E}_{b \\in B}\\left[\\omega ^{\\langle a-a^{\\prime },b \\rangle }\\right].$ Now applying Markov inequality we get that $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{i-1}}[a-a^{\\prime } \\in \\textnormal {Spec}_{\\epsilon _i}(B)] \\ge \\epsilon _i=\\epsilon _{i-1}^2/2$ .","Now selecting $j_i$ as in the construction gives that $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{i-1}}[a-a^{\\prime } \\in A_i] \\ge \\epsilon _i/r$ .","To prove the second part of the lemma, observe that by the above, we have shown that $\\left|\\lbrace \\left(a,a^{\\prime }\\right) \\in A_{i-1}\\times A_{i-1}|a-a^{\\prime } \\in A_i\\rbrace \\right| \\ge \\frac{\\epsilon _i}{r}|A_{i-1}|^2.$ Also, by construction of $A_i$ , since every $x \\in A_i$ can be represented as $x=a-a^{\\prime }$ with $a,a^{\\prime } \\in A_{i-1}$ in at most $m^{j_i+1}$ ways, we have that $|A_i| \\ge \\frac{\\epsilon _i}{m^{j_i+1}r}\\left|A_{i-1}\\right|^2$ .","This completes the proof.","Below we will use the following additive-combinatorics lemma.","Theorem 7.11 ([4], [11]) There exists an absolute constant $c>0$ such that the following holds.","Let $A$ be any arbitrary subset of an abelian group $G$ .","Let $S \\subseteq G$ be such that $|S|\\le C|A|$ .","If $\\mathbf {Pr}_{a,a^{\\prime } \\in A}[a-a^{\\prime } \\in S] \\ge 1/C$ , then there exists a subset $A^{\\prime } \\subseteq A$ such that $|A^{\\prime }|\\ge \\frac{|A|}{C^c}$ and $|A^{\\prime }-A^{\\prime }| \\le C^c|A|$ .","Now we come to the main claim.","Claim 7.12 For $i=\\ell ,\\ell -1, \\cdots 1$ there exist subsets $A_i^{\\prime } \\subseteq A_i$ , $B_i^{\\prime } \\subseteq B$ such that $D_{\\omega }\\left(A_i^{\\prime },B_i^{\\prime }\\right)=1$ and $|A_i^{\\prime }|\\ge \\alpha _i|A_i|$ and $|B_i^{\\prime }|\\ge \\beta _i|B|$ where $\\alpha _i=poly_m\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)\\left(2mr\\right)^{-\\left(\\ell -i\\right)}\\left(\\prod _{j=i}^{\\ell }\\epsilon _{j+1}\\right), \\beta _i=poly_m\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)m^{-\\left(\\ell -i\\right)}$ Base Case.","The base case of $i=\\ell $ is proved by an application of the Balog-Szemeredi-Gowers theorem followed by Conjecture REF followed by Lemma REF .","To see this, we know that $|A_{\\ell +1}| \\le K|A_{\\ell }|$ and $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{\\ell }}[a-a^{\\prime } \\in A_{\\ell +1}] \\ge \\epsilon _{\\ell +1}/r$ .","Hence by Theorem REF (with $C=\\frac{rK}{\\epsilon _{\\ell +1}}$ ), there exists a set $A_{\\ell }^{\\prime \\prime }\\subseteq A_{\\ell }$ such that $|A_{\\ell }^{\\prime \\prime }| \\ge poly\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)\\left|A_{\\ell }\\right|$ and $|A_{\\ell }^{\\prime \\prime } - A_{\\ell }^{\\prime \\prime }|\\le poly\\left(\\frac{rK}{\\epsilon _{\\ell +1}}\\right)\\left|A_{\\ell }^{\\prime \\prime }\\right|$ .","Now by Conjecture REF applied to $A_{\\ell }^{\\prime \\prime }$ , there exists a set $A_{\\ell }^{\\prime \\prime \\prime }\\subseteq A_{\\ell }^{\\prime \\prime }$ such that $\\left|A_{\\ell }^{\\prime \\prime \\prime }\\right| \\ge poly_m\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)\\left|A_{\\ell }^{\\prime \\prime }\\right|$ and $\\left|\\textnormal {span}\\left(A_{\\ell }^{\\prime \\prime \\prime }\\right)\\right|\\le m|A_{\\ell }^{\\prime \\prime }| = poly_m\\left(\\frac{rK}{\\epsilon _{\\ell +1}}\\right)\\left|A_{\\ell }^{\\prime \\prime \\prime }\\right|$ .","(Note the extra factor of $m$ in front of $|A_{\\ell }^{\\prime \\prime }|$ as we get a coset of size $|A_{\\ell }^{\\prime \\prime }|$ and its span incurs an additional factor of $m$ ) Also, as $A_{\\ell }^{\\prime \\prime \\prime } \\in \\textnormal {Spec}_{\\epsilon _{\\ell }}\\left(B\\right)$ , applying Lemma REF to $A_{\\ell }^{\\prime \\prime \\prime }$ and $B$ , we get $A_{\\ell }^{\\prime } \\subseteq A_{\\ell }^{\\prime \\prime \\prime }$ and $B_{\\ell }^{\\prime } \\subseteq B$ such that $D_{\\omega }\\left(A_{\\ell }^{\\prime },B_{\\ell }^{\\prime }\\right)=1$ , $|A_{\\ell }^{\\prime }|\\ge poly_m\\left(\\frac{\\epsilon _{{\\ell }+1}}{rK}\\right)\\left|A_{\\ell }\\right|$ and $|B_{\\ell }^{\\prime }|\\ge poly_m\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)\\left|B\\right|$ .","This completes the base case.","Let us come to the inductive case.","$\\Box $ Inductive Case.","Suppose the statement is true for $i$ and let us argue for $i-1$ .","Let $G=\\left(A_{i-1},E\\right)$ be the graph whose vertices are the elements in $A_{i-1}$ and $\\left(a,a^{\\prime }\\right)$ is an edge if $a-a^{\\prime } \\in A_i^{\\prime }$ .","Now, $|E|&\\ge &m^{j_i}|A_i^{\\prime }|\\\\&\\ge & m^{j_i}\\alpha _i|A_i| \\ \\ \\text{(inductive hypothesis)}\\\\&\\ge & m^{j_i}\\alpha _i\\frac{\\epsilon _i}{m^{j_i+1}r}|A_{i-1}|^2 \\ \\ \\text{(Claim \\ref {clm-claim24})}\\\\&=&2\\alpha _{i-1}|A_{i-1}|^2$ Now the graph has at least $2\\alpha _{i-1}|A_{i-1}|^2$ edges and $|A_{i-1}|$ vertices and therefore has a connected component of size at least $2\\alpha _{i-1}|A_{i-1}|$ vertices.","Let us call these vertices $A_{i-1}^{\\prime \\prime }$ .","Let $\\tilde{a}$ be any element of $A_{i-1}^{\\prime \\prime }$ .","Partition $B_i^{\\prime }$ into $B_{i,j}^{\\prime }$ for $0 \\le j \\le m-1$ such that all elements of $B_{i,j}^{\\prime }$ have inner product $j$ with $\\tilde{a}$ .","Let $B_{i-1}^{\\prime }=B_{i,j_1}$ be the largest of them.","Note that $|B_{i-1}^{\\prime }| \\ge |B_i^{\\prime }|/m$ .","By assumption $D_{\\omega }\\left(A_i^{\\prime },B_i^{\\prime }\\right)=1$ .","Hence, $D_{\\omega }\\left(A_i^{\\prime },B_{i-1}^{\\prime }\\right)=1$ .","Therefore, for some $j_2$ , $\\langle a,b \\rangle = j_2$ for all $a \\in A_i^{\\prime }$ and $b \\in B_{i-1}^{\\prime }$ .","Now, in the connected component obtained above, whenever $a,a^{\\prime } \\in A_{i-1}^{\\prime \\prime }$ are neighbours, $\\langle a-a^{\\prime },b \\rangle =j_2$ for $b \\in B_{i-1}^{\\prime }$ .","Thus, starting with $\\tilde{a}$ as the anchor and propagating throughout the connected component, we can classify the vertices in $A_{i-1}^{\\prime \\prime }$ based on the inner product it has with all elements in $B_{i-1}^{\\prime }$ , which is either $j_1$ or $j_2-j_1$ .","Pick the larger set and call it $A_{i-1}^{\\prime }$ .","Hence, $D_{\\omega }\\left(A_{i-1}^{\\prime },B_{i-1}^{\\prime }\\right)=1$ .","Thus, $|A_{i-1}^{\\prime }|\\ge |A_{i-1}^{\\prime \\prime }|/2 \\ge \\alpha _{i-1}|A_{i-1}|$ and $B_{i-1}^{\\prime }\\ge |B_i^{\\prime }|/m \\ge \\frac{\\beta _i}{m}|B|= \\beta _{i-1}|B|$ .","This completes the inductive case.","$\\Box $ Put $i=1$ in the above claim.","Also observe that as $\\epsilon _{j+1}=\\epsilon ^{2^j}/2^{2^j-1}\\ge \\left(\\epsilon /2\\right)^{2^j}$ .","Thus, $\\epsilon _{\\ell +1}\\ge \\left(\\epsilon /2\\right)^{2^{\\ell }}$ and $\\prod _{j=1}^{\\ell }\\epsilon _{j+1}\\ge \\left(\\epsilon /2\\right)^{2^{\\ell +1}}$ there exist $A^{\\prime } \\subseteq A_1 \\subseteq A$ , $B^{\\prime } \\subseteq B$ , such that $|A^{\\prime }| \\ge poly\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell }}}{rK}\\right)\\left(2mr\\right)^{-\\ell }\\left|A\\right|$ and $ |B^{\\prime }| \\ge poly\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell }}}{rK}\\right)m^{-\\ell }\\left|B\\right|$ .","Observing that the lower bounds grow weaker with increasing $\\ell $ ,and that $\\ell \\le \\ell ^{\\prime }=r/\\log _m K$ we get $|A^{\\prime }| \\ge poly\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell ^{\\prime }}}}{rK}\\right)\\left(2mr\\right)^{-\\ell ^{\\prime }}\\left|A\\right|$ and $|B^{\\prime }| \\ge poly\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell ^{\\prime }}}}{rK}\\right)m^{-\\ell ^{\\prime }}\\left|B\\right|$ where $\\ell ^{\\prime }=r/\\log _m K$ .","Therefore, if we take the list $U^{\\prime }\\subseteq U$ (corresponding to $A^{\\prime }\\subseteq A$ ) and $V^{\\prime }\\subseteq V$ (corresponding to $B^{\\prime }\\subseteq B$ ) then as $\\langle a_i,b_j \\rangle =\\langle u_i, v_j \\rangle $ the statement of the lemma follows.","This completes the proof of Lemma REF We can now prove the Sub-Matrix Lemma, Lemma REF ."],["Proof of Lemma ","Set $K=s^{4r/\\log r}, \\ell =\\frac{\\log r}{4}, \\epsilon =1/2m^{3/2}$ while applying Lemma REF over $\\mathbb {Z}_s$ .","We get $|A^{\\prime }|\\ge \\delta _s |A|$ , $|B^{\\prime }|\\ge \\delta _s |B|$ where $\\delta _s&=&poly_s\\left(\\frac{1}{m^{r^{1/4}}}\\right)2^{-c_1\\left(s\\right)r/\\log r}\\ \\ \\text{(for some constant $c_1\\left(s\\right)$ depending only on $s$)}\\\\&\\ge & poly_m\\left(\\frac{1}{m^{r^{1/4}}}\\right)2^{-c_1\\left(s\\right)r/\\log r}$ Now let $c_2\\left(m\\right)=\\max _{s|m, s\\ge 2}\\lbrace c_1\\left(s\\right)\\rbrace $ .","Thus, $\\delta _s \\ge poly_m\\left(\\frac{1}{m^{r^{1/4}}}\\right)2^{-c_2\\left(m\\right)r/\\log r}\\ge 2^{-c\\left(m\\right)r/\\log r}$ for some constant $c$ that depends only on $m$ .","$\\Box $"],["A Calculation","Claim A.1 Let $b>1, n \\ge 2$ be arbitrary integers.","Then $\\sum _{i=1}^{\\lfloor \\log _b n\\rfloor }\\frac{1}{b^{i-1}\\log \\left(n/b^{i-1}\\right)}\\le f(b)/\\log n$ where $f(b)=\\frac{10b}{b-1}+\\frac{10}{\\log b}+\\frac{16e}{\\log ^2b}$ .","When $b=4/3$ , $f(b)<300$ .","We divide the summation into two parts.","The first part consists of the first $\\lfloor \\log _b \\log n \\rfloor $ terms and the second part consists of the remaining terms.","In the first part, $\\frac{1}{b^{i-1}\\log n/b^{i-1}}\\le \\frac{1}{b^{i-1}0.1\\log n}$ whenever $n \\ge 2$ and hence the first part summation is bounded from above by $\\frac{10b}{(b-1)\\log n}$ .","In the second part of the summation, we use the monotonicity of $x \\log \\left(n/x\\right)$ .","The function increases with $x$ as long as $x\\le n/e$ .","Therefore, for terms with $b^{i-1}\\le n/e$ , the maximum value of each summand is given by substituting $i=\\log _b\\log n$ which gives an upper bound of $\\frac{1}{0.1\\log ^2 n}$ .","The remaining terms corresponding to $n/b \\ge b^{i-1} > n/e$ (note that these extra terms arise only if $b1, n \\ge 2$ be arbitrary integers.","Then $\\sum _{i=1}^{\\lfloor \\log _b n\\rfloor }\\frac{1}{b^{i-1}\\log \\left(n/b^{i-1}\\right)}\\le f(b)/\\log n$ where $f(b)=\\frac{10b}{b-1}+\\frac{10}{\\log b}+\\frac{16e}{\\log ^2b}$ .","When $b=4/3$ , $f(b)<300$ .","We divide the summation into two parts.","The first part consists of the first $\\lfloor \\log _b \\log n \\rfloor $ terms and the second part consists of the remaining terms.","In the first part, $\\frac{1}{b^{i-1}\\log n/b^{i-1}}\\le \\frac{1}{b^{i-1}0.1\\log n}$ whenever $n \\ge 2$ and hence the first part summation is bounded from above by $\\frac{10b}{(b-1)\\log n}$ .","In the second part of the summation, we use the monotonicity of $x \\log \\left(n/x\\right)$ .","The function increases with $x$ as long as $x\\le n/e$ .","Therefore, for terms with $b^{i-1}\\le n/e$ , the maximum value of each summand is given by substituting $i=\\log _b\\log n$ which gives an upper bound of $\\frac{1}{0.1\\log ^2 n}$ .","The remaining terms corresponding to $n/b \\ge b^{i-1} > n/e$ (note that these extra terms arise only if $b=0$, and for any $i \\\\ne j$, $< u_i,v_j> \\\\ne 0$.\",\n \"A MV family is called $q$-restricted if inner products $< u_i,v_j>$ take at most $q$ different values.\",\n \"Our interest in MV families stems from their recent application in the construction of sub-exponential locally decodable codes (LDCs).\",\n \"There, $q$-restricted MV families are used to construct LDCs with $q$ queries, and there is special interest in the regime where $q$ is constant.\",\n \"When $m$ is a prime it is known that such constructions yield codes with exponential block length.\",\n \"However, for composite $m$ the behaviour is dramatically different.\",\n \"A recent work by Efremenko [STOC 2009] (based on an approach initiated by Yekhanin [JACM 2008]) gives the first sub-exponential LDC with constant queries.\",\n \"It is based on a construction of a MV family of super-polynomial size by Grolmusz [Combinatorica 2000] modulo composite $m$.\",\n \"In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families.\",\n \"When $q$ is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length.\",\n \"When the modulus $m$ is constant (as it is in the construction of Efremenko) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over $\\\\mathbb{Z}_m$.\"\n ],\n [\n \"Introduction\",\n \"A Matching Vector Family (MV Family) is a combinatorial object that arises in several contexts including Ramsey graphs, weak representation of OR polynomials and recently in constant query locally decodable codes (LDCs).\",\n \"It is defined by two ordered lists $U=\\\\left(u_1, \\\\cdots u_t\\\\right)$ and $V=\\\\left(v_1, \\\\cdots v_t\\\\right)$ where $u_i, v_j \\\\in \\\\mathbb {Z}_m^n$ and $m$ and $n$ are integers greater than 1.\",\n \"The property that the two lists have to satisfy is the following: for all $i\\\\in [t]$ , $\\\\langle u_i, v_i \\\\rangle = 0\\\\ (mod\\\\ m)$ whereas for all $i \\\\ne j\\\\in [t]$ , $\\\\langle u_i, v_j \\\\rangle \\\\ne 0\\\\ (mod\\\\ m)$ .\",\n \"By $\\\\langle \\\\cdot , \\\\cdot \\\\rangle $ we denote the standard inner product.\",\n \"Let us call this the standard definition of a MV family.\",\n \"If in addition, all the inner products $\\\\langle u_i, v_j \\\\rangle \\\\ (mod\\\\ m)$ lie in a set of size $q$ , then it is called a $q-restricted$ MV family.\",\n \"Note that $q=m$ corresponds to the standard MV family.\",\n \"The size of the MV family is $t$ , the number of vectors in the list.\",\n \"In this paper, we shall prove upper bounds on $q-restricted$ MV families in the first part and on standard MV families in the later part.\",\n \"Let $\\\\mathbf {MV}(m,n)$ denote the largest $t$ such that there exists a MV family of size $t$ in $\\\\mathbb {Z}_m^n$ .\",\n \"Analogously, let $\\\\mathbf {MV}(m,n,q)$ denote the largest $t$ such that there exists a $q-restricted$ MV family of size $t$ in $\\\\mathbb {Z}_m^n$ .\",\n \"The question of bounding $\\\\mathbf {MV}(m,n)$ (or $\\\\mathbf {MV}(m,n,q)$ ) is closely related to the well-known combinatorial problem of set systems with restricted modular intersections [2], [22], [12], [13] (in this setting the vectors $u_i,v_i$ are required to have entries that are either 0 or 1).\",\n \"The systematic study of this more general problem, in the context of MV codes, was initiated in [8].\",\n \"The setting of prime $m$ is well understood.\",\n \"For large prime $m=p$ , it is known that $\\\\mathbf {MV}(p,n)=O\\\\left(p^{n/2}\\\\right)$ [8].\",\n \"Infact, this is almost tight.\",\n \"When $m$ is a small prime, again we have a tight upper bound of $O\\\\left(n^{p-1}\\\\right)$ [2].\",\n \"Surprisingly, the setting of small composite $m$ leads to very useful constructions of Ramsey graphs and constant query LDCs.\",\n \"This is due to a construction of MV family over $Z_6$ by Grolmusz [12] of superpolynomial size in contrast to a polynomial upper bound when $m$ is a small prime.\",\n \"Thus, it is interesting to study the behavior of MV families for small composite $m$ , and more generally arbitrary general composites.\",\n \"We will see later the connection between upper bounds on $\\\\mathbf {MV}(m,n,q)$ and lower bounds on the encoding lengths of MV Codes (a family of LDCs).\",\n \"For general $m$ , the best upper bound known [8] is $\\\\mathbf {MV}(m,n) \\\\le m^{n-1 + o_m(1)},$ with $o_m(1)$ denoting a function that goes to zero when $m$ grows.\",\n \"It was conjectured in [8] that an upper bound of $ \\\\sim m^{n/2}$ should hold for any $m$ (not just prime).\",\n \"This would be tight for large $m$ as there are constructions of MV families almost meeting this bound [28].\",\n \"However, the proof method used in [8] to prove the $O\\\\left(p^{n/2}\\\\right)$ bound does not extend to non primes.\",\n \"In this work, we prove the conjecture for $q-restricted$ MV families in $\\\\mathbb {Z}_m^n$ , for any $m$ as long as $q= \\\\frac{o(n)\\\\log m}{\\\\log \\\\left(o(n)\\\\log m\\\\right)}$ (See Theorem REF ).\",\n \"When $m=p$ is a fixed small prime, it follows from [2] that $\\\\mathbf {MV}(p,n) = O\\\\left(n^{p-1}\\\\right)$ .\",\n \"On the other hand, when $m$ is a fixed composite, say $m=6$ , there exists a MV family of superpolynomial size $\\\\Omega \\\\left(\\\\exp \\\\left(\\\\log ^2n /\\\\log \\\\log n\\\\right)\\\\right)$ ) [12].\",\n \"We prove a stronger upper bound on $\\\\mathbf {MV}(m,n)$ , compared to Theorem REF in such a case assuming a well known conjecture in additive combinatorics (see Theorem REF ).\",\n \"Table REF lists the known and new upper bounds on MV families.\",\n \"Table: List of upper bounds on 𝐌𝐕(m,n)\\\\mathbf {MV}(m,n), 𝐌𝐕(m,n,q)\\\\mathbf {MV}(m,n,q)Theorem 1 For all $m\\\\ge 2, n \\\\ge 1$ we have $ \\\\mathbf {MV}(m,n,q) \\\\le q^{O(q\\\\log q)}m^{n/2} $ Hence, Theorem REF resolves the conjecture of [8] for any $m$ and for $q = \\\\frac{o(n)\\\\log m}{\\\\log \\\\left(o(n)\\\\log m\\\\right)}$ .\",\n \"When $m >> n$ , our bound is quite close to the best known construction of MV families which gives $\\\\mathbf {MV}(m,n) \\\\ge \\\\left( \\\\frac{m+1}{n-2} \\\\right)^{n/2 - 1}$ [28].\",\n \"Our second result assumes the polynomial Freiman-Ruzsa conjecture (PFR) conjecture (discussed below) and gives a stronger upper bound on the size of MV families when $m$ is a constant and $n$ grows.\",\n \"Before we state the conjecture, we need to define what a difference set is.\",\n \"For an abelian group $G$ let $A \\\\subseteq G$ .\",\n \"Then the difference set $A-A=\\\\lbrace a_1-a_2:a_1,a_2 \\\\in A\\\\rbrace $ Conjecture 1 (PFR Conjecture in $\\\\mathbb {Z}_m^n$ ) Suppose $A \\\\subseteq \\\\mathbb {Z}_m^n$ and $|A-A|\\\\le \\\\lambda \\\\cdot |A|$ .\",\n \"Then one can find a subgroup $H$ of size at most $|A|$ such that $A$ can be covered by $\\\\lambda ^{\\\\prime } = \\\\lambda ^{c_m}$ many translates of $H$ , where $c_m$ depends only on $m$ .\",\n \"We note that the PFR conjecture has already found several applications in computer science.\",\n \"Ben-Sasson and Zewi [6] used it to construct two-source extractors from affine extractors; and Ben-Sasson, Lovett and Zewi [5] used it to bound the deterministic communication complexity of functions whose corresponding matrix has low rank.\",\n \"Our work provides another application for the PFR and demonstrates its wide-reaching applicability.\",\n \"We further note that a quasi-polynomial version of the PFR conjecture was recently proved by Sanders [21] (see also the exposition in [18]).\",\n \"Unfortunately, all the applications discussed above require the truly polynomial version of the conjecture, and so cannot apply to Sanders' result.\",\n \"We now state the second theorem.\",\n \"Theorem 2 Assuming the PFR conjecture over $\\\\mathbb {Z}_m^n$ (Conjecture REF ) we have $\\\\mathbf {MV}(m,n) \\\\le \\\\exp \\\\left( c(m)\\\\frac{n}{\\\\log n} \\\\right), $ with $c(m)$ an explicit function of $m$ .\",\n \"From a technical point of view, one of the ingredients in this work builds on the recent work of Ben-Sasson, Lovett and Zewi [5] who used the PFR conjecture to show that matrices over $\\\\mathbb {Z}_2$ with large bias (say, with many more ones than zeros) and small rank must contain a large monochromatic sub-matrix.\",\n \"An important ingredient in our proof is a generalization of their results from $\\\\mathbb {Z}_2$ to $\\\\mathbb {Z}_m$ for all $m$ , not necessarily prime.\",\n \"We note however that this is just one ingredient in our overall proof.\"\n ],\n [\n \"Lower Bounds on LDCs: Motivation for MV Family\",\n \"Locally Decodable Codes (LDCs) are a special kind of Error Correcting Codes (ECCs) that allow the receiver to decode a single symbol of the message by querying a small number of positions in a corrupted encoding.\",\n \"More formally, an $(q,\\\\delta ,\\\\epsilon )$ -LDC encodes $K$ -symbol messages $x$ to $N$ -symbol codewords $C(x),$ such that for every $i\\\\in [K],$ the symbol $x_i$ can be recovered with probability $1-\\\\epsilon ,$ by a randomized decoding procedure that makes only $q$ queries, even if the codeword $C(x)$ is corrupted in up to $\\\\delta N$ locations.\",\n \"Since the early 90's, LDC's have found exciting applications in various areas ranging from data transmission to complexity theory to cryptography/privacy.\",\n \"We refer the reader to [23], [27] for more background.\",\n \"A central research question, which is far from being solved, has to do with understanding the best possible `stretch' of an LDC with a constant number of queries.\",\n \"That is, how large $N$ has to be as a function of $K$ for constant $q$ and with constant $\\\\delta ,{\\\\varepsilon }$ (these two last parameters are not our focus here and we will generally assume them to be small fixed constants).\",\n \"For $q=1,2$ this question is completely answered.\",\n \"There are no LDC's for $r=1$ [16] and the best LDC's with $q=2$ have exponential encoding length [10], [15].\",\n \"For $q > 2$ there are huge gaps in our understanding.\",\n \"Katz and Trevisan were the first to study this problem [16] and, today, the best general lower bounds on $N$ are slightly super-linear bounds of the form $\\\\tilde{\\\\Omega }\\\\left(K^{1+1/\\\\left(\\\\lceil r/2\\\\rceil -1\\\\right)}\\\\right)$ [24].\",\n \"Notice that, when the number of queries is 3 or 4, these bounds are quadratic (see also [15], [25] for the $q=3,4$ case).\",\n \"The upper bounds were, until recently, those coming from polynomial codes and were of the order of $N \\\\le \\\\exp \\\\left(K^{\\\\frac{1}{q-1}}\\\\right)$ .\",\n \"Improved upper bounds, breaking this barrier slightly, were given in [3].\",\n \"This state of affairs changed dramatically when, in a breakthrough paper, Yekhanin [26] developed a new approach for constructing LDCs, called MV codes, that have much shorter codeword length than polynomial codes.\",\n \"Efremenko [9] was the first to show that this approach could yield codes with subexponential encoding length (Yekhanin's paper showed this under a number theoretic assumption).\",\n \"More refinements and improvements to this new framework were obtained [20], [17], [14], [19], [8], [1] to give LDC's with $q$ queries and with encoding length that grows, when $q$ is a constant, roughly like $N \\\\sim \\\\exp \\\\exp \\\\left((\\\\log K)^{O(1/\\\\log q)}(\\\\log \\\\log K)^{1-1/\\\\log q}\\\\right).$ While significantly smaller than the length of polynomial codes, the codeword length of these new codes is still super polynomial in $K$ .\",\n \"The most general setting of parameters was addressed in [8] where the authors had given a black box construction of $q$ query MV codes using $q-restricted$ MV families in $\\\\mathbb {Z}_m^n$ .\",\n \"Using the standard definition of MV families, this implied $m$ query MV codes using MV families in $Z_m^n$ .\",\n \"In this basic, yet general reduction, it was shown that upper bounds on MV families would lead to lower bounds on the encoding length of MV codes.\",\n \"With this motivation in mind, the authors in [8] made a conjecture on the upper bound on the size of MV families which would lead to lower bounds on the encoding length of MV codes under the basic framework.\",\n \"We note that Yekhanin in [26] used restricted MV families in $\\\\mathbb {Z}_p^n$ where $p$ is a very large Mersenne prime and used a specialized technique to reduce the number of queries from $p$ to 3.\",\n \"Another instance of reduction in the number of queries from what the standard construction gives, was given by Efremenko [9] where he again used restricted MV families.\",\n \"A certain gadget was discovered using computer search whereby the author worked in $\\\\mathbb {Z}_{511}$ but got down the number of queries to 3 from the basic bound of 511.\",\n \"The following is a corollary of Theorem REF .\",\n \"Corollary 3 For an arbitrary positive integer $m$ , consider an infinite family of $q$ -query Matching Vector code $C_n:{\\\\mathbb {F}}^k \\\\rightarrow {\\\\mathbb {F}}^N$ for $n \\\\in \\\\mathbb {N}$ , where $k(n)$ and $N(n)$ are growing functions of $n$ , constructed using the black box reduction from a $q$ -restricted Matching Vector Family in $\\\\mathbb {Z}_m^n$ ([8]).\",\n \"For large enough $n$ , if $q=\\\\frac{o(n)\\\\log m}{\\\\log \\\\left(o(n)\\\\log m\\\\right)}$ , then $ N \\\\ge k^{2-o(1)}$ Specifically, if $q=O(1)$ , then $N = \\\\Omega \\\\left(k^{2}\\\\right)$ .\",\n \"Next we have the following corollary from Theorem REF .\",\n \"Corollary 4 For some arbitrary positive integer $m$ , assume the PFR conjecture over $\\\\mathbb {Z}_m^n$ (Conjecture REF ).\",\n \"Consider an infinite family of $m$ -query Matching Vector code $C_n:{\\\\mathbb {F}}_q^k \\\\rightarrow {\\\\mathbb {F}}_q^N$ for $n \\\\in \\\\mathbb {N}$ , where $k(n)$ and $N(n)$ are growing functions of $n$ , constructed using the black box reduction from a standard Matching Vector Family in $\\\\mathbb {Z}_m^n$ ([8]).\",\n \"For large enough $n$ , if $m=O(1)$ , then $ N = \\\\exp \\\\left(\\\\Omega _m\\\\left(\\\\log k \\\\ \\\\log \\\\log k\\\\right)\\\\right)$ Thus Corollary REF states that, assuming Conjecture REF , MV codes with constant number of queries must have super polynomial encoding length in the basic framework.\",\n \"Note that we get the same bound in Efremenko's framework for 3 queries.\",\n \"This is because the form of the superpolynomial bound is assuming a constant $m$ and applying our bound to Efremenko's work again leads to a superpolynomial bound as $m=511$ in his setting (another constant).\",\n \"(He uses $\\\\mathbb {Z}_{511}$ to construct the MV family and further reduces the number of queries to 3.)\",\n \"This essentially means that in order to construct polynomial length codes, one needs to construct MV families in $\\\\mathbb {Z}_m^n$ for non-constant $m$ and use some specialized gadget to reduce the number of queries.\",\n \"One way is to ensure it is a $q-restricted$ (constant $q$ ) MV family.\",\n \"This automatically ensures $q$ query decoding.\",\n \"However, the quadratic lower bound continues to hold even in this scenario for constant $q$ .\",\n \"To beat the quadratic lower bound for constant query MV codes, one needs to construct $q-restricted$ MV families for growing $m$ and $q=\\\\frac{o(n)\\\\log m}{\\\\log \\\\left(o(n)\\\\log m\\\\right)}$ and then develop some special gadget to get the number of queries down further from $q$ to some constant.\"\n ],\n [\n \"Proof Overview\",\n \"The proof of Theorem REF relies on intuitions coming from the theory of two-source extractors [7], which are functions of two variables $F(X,Y)$ such that the output of $F$ is distributed in a close-to-uniform fashion whenever the two inputs are drawn, independently, from two distributions of sufficiently high entropy.\",\n \"Since our proof does not use two-source extractors explicitly we do not define them formally and just use them to explain the high level idea behind the proof.\",\n \"It is a well known fact [7] that the inner product function $F(X,Y) = \\\\langle X,Y \\\\rangle $ , say over $\\\\mathbb {Z}_2^n \\\\times \\\\mathbb {Z}_2^n$ is a good two source extractor when the two inputs $X$ and $Y$ are both drawn uniformly from sets $S_X,S_Y \\\\subseteq \\\\mathbb {Z}_2^n$ of size larger $2^{n/2}$ .\",\n \"This immediately suggests a connection to MV families, since, if we take $S_X = U$ and $S_Y = V$ for a MV family $U,V$ in $\\\\mathbb {Z}_2^n$ , we would get a completely non-uniform output (it will be zero with exponentially small probability).\",\n \"This means that the size of $U,V$ is bounded from above by approximately $2^{n/2}$ .\",\n \"If we try to use a similar argument over $\\\\mathbb {Z}_m$ we run into trouble since the inner product function modulo $m$ is not a good two source extractors for sources of size $m^{n/2}$ .\",\n \"Take, for example, $S_X = S_Y=\\\\lbrace 0,2,4\\\\rbrace ^n \\\\subseteq \\\\mathbb {Z}_6^n$ and observe that $\\\\langle X,Y \\\\rangle $ is always divisible by 2 and so is far from being uniformly distributed over $\\\\mathbb {Z}_6$ .\",\n \"It is, however, possible to show that this example is, in some sense, the only example and that, in general, we can always find a certain number of elements of either $S_X$ or $S_Y$ that `agree' modulo some factor of $m$ .\",\n \"This observation suggests proving Theorem REF by induction on the number of factors of $m$ , which is the way we proceed.\",\n \"The proof of Theorem REF uses a slightly different view of MV families as matrices with certain zero/non-zero pattern and small rank.\",\n \"Specifically, for a MV family $U,V$ of size $t$ in $\\\\mathbb {Z}_m^n$ consider the $t \\\\times t$ matrix $P$ whose $(i,j)$ 'th entry is $\\\\langle u_i,v_j \\\\rangle \\\\mod {m}$ .\",\n \"The definition of a MV family implies that $P$ has zeros on the diagonal and non-zeros everywhere else.\",\n \"If $m$ was a prime, we could think of $\\\\mathbb {Z}_m$ as a field ${\\\\mathbb {F}}$ and say that, since $P$ is the inner product matrix of vectors of length $n$ over a field, it must have rank at most $n$ .\",\n \"Conversely, every $t \\\\times t$ matrix over a field ${\\\\mathbb {F}}$ with these properties (zero on the diagonal and non-zero off the diagonal) and with rank $n$ gives a MV family of size $t$ in ${\\\\mathbb {F}}^n$ .\",\n \"We can call a matrix with this pattern of zeros/non-zeros an MV matrix.\",\n \"Thus, when $m$ is prime, the question of bounding the size of a MV family is the same as lower bounding the rank of a MV-matrixFor technical reasons, the actual proof will not be entirely using matrices and will keep the MV family in the background.\",\n \"This is because we need to keep certain invariants throughout the proof and these are easier to define for families of vectors than for matrices..\",\n \"When $m$ is composite, this whole approach should be re-examined since $\\\\mathbb {Z}_m$ is no longer a field and our familiar understanding of matrices and linear algebra over a field are no longer valid.\",\n \"We do, however, manage to carry over this correspondence between the two problems by defining the notion of rank in a careful way (more on this issue below).\",\n \"Assume for the purpose of this overview that the usual notion of rank and other intuitions from linear algebra are valid over $\\\\mathbb {Z}_m$ and let us proceed with sketching the proof of Theorem REF using the equivalent formulation as bounding (from below) the rank of a MV matrix $P$ .\",\n \"The starting point is a generalization of a result of [5], mentioned above, from $\\\\mathbb {Z}_2$ to $\\\\mathbb {Z}_m$ .\",\n \"We show that every matrix $P$ over $\\\\mathbb {Z}_m$ that is biased (i.e., its values are not distributed close to uniformly) and has low rank, contains a large monochromatic sub-matrix modulo some factor $m^{\\\\prime }$ of $m$.\",\n \"The size of the sub-matrix is bounded from below by $\\\\sim |P|\\\\exp (-r^{\\\\prime }/\\\\log (r^{\\\\prime }))$ , where $r^{\\\\prime }$ is the rank of $P$ modulo $m^{\\\\prime }$ (this factor depends on the specific way the matrix is biased).\",\n \"This generalizes the result of [5] which assumes $m=2$ and finds a large monochromatic sub-matrix (modulo 2).\",\n \"We note that the sub-matrix lemma is the only component in the proof that relies on the PFR conjecture.\",\n \"Let us refer to this result from now on as the sub-matrix lemma.\",\n \"We can apply the sub-matrix lemma to a MV matrix $P$ since its values are far from uniform (the probability of zero is much less than $1/m$ ) and since its rank is assumed (towards a contradiction) to be low.\",\n \"Suppose for the sake of simplicity that $m=p \\\\cdot q$ , with $p,q$ distinct primes (the proof for general $m$ is significantly more technical but relies on the same basic intuitions).\",\n \"Applying the sub-matrix lemma we obtain a sub-matrix $P_1$ of $P$ that is constant modulo some factor $m_1$ of $m$ (so $m_1$ is either $p$ , $q$ or $m$ ) of size at least $|P|\\\\exp (-r_1/\\\\log (r_1))$ , where $r_1\\\\le n$ is the rank of $P \\\\mod {m}_1$ .\",\n \"Using some matrix manipulations, and subtracting a rank one matrix, we can get a large sub-matrix $P_1^{\\\\prime }$ that does not intersect the diagonal of $P$ and s.t all of the entries of $P_1^{\\\\prime }$ are zero modulo $m_1$ .\",\n \"Suppose $|P_1^{\\\\prime }| = t_1$ and consider the $2t_1 \\\\times 2t_1$ sub-matrix $P_1^{\\\\prime \\\\prime }$ of $P$ that has $P_1^{\\\\prime }$ as its top-right (or bottom-left) block and s.t the top-left and bottom-right blocks are taken to have zero diagonal elements.\",\n \"Formally, if $P_1^{\\\\prime }$ is indexed by rows in $R$ and columns in $T$ with $R \\\\cap T = \\\\emptyset $ then the rows/columns of $P_1^{\\\\prime \\\\prime }$ will be indexed by $R \\\\cup T$ .\",\n \"If we consider the matrix $P_1^{\\\\prime \\\\prime }$ modulo $m_1$ then it has top-right block which is all zero and so its rank (modulo $m_1$ ) will be the sum of the ranks of the top-left and bottom right blocks.\",\n \"Thus, one of these blocks, w.l.o.g the top-left one, must have rank at most $n/2$ (over $\\\\mathbb {Z}_{m_1}$ ).\",\n \"Notice also that both of these blocks are themselves MV matrices modulo $m$ since they are sub-matrices of $P$ with the same row and column sets.\",\n \"Let $\\\\tilde{P}_1$ be the top-left block of $P_1^{\\\\prime \\\\prime }$ .\",\n \"We can now apply, again, the monochromatic sub-matrix lemma to find a large sub-matrix $P_2$ of $\\\\tilde{P}_1$ which is constant modulo some other factor $m_2$ of $m$ .\",\n \"The size of $P_2$ will be $t_1 \\\\cdot \\\\exp (-r_2/\\\\log (r_2)) = |P|\\\\cdot \\\\exp (-r_1/\\\\log (r_1) - r_2/\\\\log (r_2) ).$ The factor $m_2$ is also either $p$ or $q$ .\",\n \"If it happens to be that $m_1=m_2$ then $r_2 \\\\le n/2$ and so we gain in the size of $P_2$ in this second step (the expression $r_2/\\\\log (r_2)$ is smaller than $n/2\\\\log (n/2)$ which is smaller by roughly a factor of two than our bound on $r_1/\\\\log (r_1)$ .\",\n \"Suppose we continue with this iterative process of finding constant sub-matrices for $\\\\ell $ steps and that, by luck, all the factors $m_1,m_2, \\\\ldots $ are equal to the same factor of $m$ (say $p$ ).\",\n \"Then, after roughly $\\\\log (n)$ iteration, we will reduce the rank modulo $p$ to one and still have at least $|P|\\\\cdot \\\\exp \\\\left(-\\\\sum _{i=1}^\\\\ell \\\\frac{n}{2^i\\\\log (n/2^i)}\\\\right)$ rows, which is close to the original size of $P$ if we assume (in contradiction) that $|P| >> \\\\exp ( n/\\\\log n)$ (this calculation is given in Claim REF ).\",\n \"In this case we obtain a new large MV family $U^{\\\\prime },V^{\\\\prime }$ modulo $m$ such that all inner products $\\\\langle u_i^{\\\\prime },v_j^{\\\\prime } \\\\rangle $ of elements $u_i^{\\\\prime }\\\\in U^{\\\\prime }, v_j^{\\\\prime } \\\\in V^{\\\\prime }$ are fixed modulo $p$ .\",\n \"From this we can easily construct a MV family of roughly the same size in $\\\\mathbb {Z}_q^n$ and then use the bounds on $\\\\mathbf {MV}(q,n)$ for primes to get a contradiction.\",\n \"In the `unlucky' case we will have different factors $m_1,m_2,\\\\ldots $ in each stage, but we can adapt the analysis to consider the decrease in rank simultaneously for all factors of $m$ .\",\n \"The full proof is by induction on the number of factors of $m$ and uses the iterative sub-matrix argument above to go from a MV family modulo $m$ to a MV family of roughly the same size modulo some proper factor of $m$ (and then uses the inductive hypothesis on this new MV family).\"\n ],\n [\n \"Matrix rank over $\\\\mathbb {Z}_m$\",\n \"An important technical issue, which was already hinted at above, is in the definition of the rank of a matrix with entries in a ring $\\\\mathbb {Z}_m$ .\",\n \"There are two main properties of matrix rank over a field that we relied on in the proof sketch above.\",\n \"The first is that a rank $r$ matrix is always the inner product matrix of vectors in $r$ dimensions.\",\n \"Equivalently, a $t \\\\times t$ matrix of rank $r$ can be written as a product of a $t \\\\times r$ matrix and an $r \\\\times t$ matrix.\",\n \"This is important if we are to go back and forth between matrices and MV families.\",\n \"Another property we used is that, if we have a $2t \\\\times 2t$ matrix composed of 4 blocks of size $t \\\\times t$ and the top-right block is zero, then the rank of the matrix is the sum of the ranks of the top-left block and the bottom right block.\",\n \"Ideally, we would like to define rank over $\\\\mathbb {Z}_m$ so that both properties are satisfied.\",\n \"This is, however, impossible as the following example shows: Consider the $2 \\\\times 2$ matrix with the two rows $(4,0)$ and $(0,3)$ over $\\\\mathbb {Z}_6$ .\",\n \"This matrix can be written as the product of the two vectors $(2,3)^T$ and $(2,3)$ and so should have rank one, if we are to satisfy the first property.\",\n \"However, if we are to satisfy the second property, its rank should be the sum of the ranks of the two $1 \\\\times 1$ matrices $(4)$ and $(3)$ , which clearly cannot have rank zero!\",\n \"Our solution to this problem is to give two different definitions of rank, each satisfying one of the two properties.\",\n \"We then show that the two definitions of rank can differ from each other by a multiplicative factor of $\\\\log m$ , which our proof can handle.\",\n \"The first definition of rank is as the smallest $r$ such that our $t \\\\times t$ matrix can be written as a product of a $t \\\\times r$ matrix and an $r \\\\times t$ matrix.\",\n \"Clearly this would satisfy the first property (but not the second).\",\n \"The second definition of rank is termed column-rank and is defined as the logarithm to the base $m$ of the size of the additive subgroup of $\\\\mathbb {Z}_m^t$ generated by the columns of the matrix.\",\n \"Notice that this definition of rank can result in the rank being non-integer.\",\n \"For example, the rank of the matrix with a single column $(2,0)$ over $\\\\mathbb {Z}_6$ would be equal to $\\\\log _6(3)$ since the subgroup generated by this column is composed of the three vectors $(2,0),(4,0),(0,0)$ .\",\n \"It is not hard to show (see Claim REF ) that this definition satisfies the second property described above regarding block matrices.\",\n \"Clearly, the two definitions agree for matrices over a field.\",\n \"We show (see Claim REF ) that the two notions of rank can differ by a multiplicative factor of at most $\\\\log m$ .\",\n \"This allows us to use both definitions in different parts of the proof without losing too much in the transition.\",\n \"We finish this discussion by noting that in no part of the proof do we use the characterization of rank using determinants, which is often very useful when working over a field.\"\n ],\n [\n \"Organization\",\n \"We begin with some preliminaries in Section .\",\n \"We prove Theorem REF in Section .\",\n \"Section contains some claims about matrices over $\\\\mathbb {Z}_m$ .\",\n \"Section introduces collision free MV families.\",\n \"Both Section and Section will be used in the proof of Theorem REF in Section .\",\n \"The proof of Theorem REF also requires the sub-matrix lemma, whose proof appears in Section .\"\n ],\n [\n \"Notations:\",\n \"Throughout the paper we will be handling ordered lists of elements.\",\n \"A list $A$ of size $t$ over a finite set $\\\\Omega $ is an ordered $t$ -tuple $A=\\\\left(a_1,a_2,\\\\cdots ,a_t\\\\right)$ where each $a_i \\\\in \\\\Omega $ .\",\n \"A list can have repetitions.\",\n \"If it doesn't we say it is twin free.\",\n \"When discussing sublists $A \\\\subseteq B$ with $B = (b_1,\\\\ldots ,b_t)$ we will use the convention that, unless specified otherwise, $A$ maintains the ordering induced by $B$ .\",\n \"For a positive integer $t$ , we let $[t]$ denote the list $\\\\left(1,\\\\cdots t\\\\right)$ .\",\n \"So, for example, when we say that $T \\\\subseteq [t]$ we mean that $T$ is a list of integers in increasing order belonging to $[t]$ .\",\n \"We say that a list $A = (a_1,\\\\ldots ,a_t)$ over $\\\\Omega $ is constant if $a_i=a_j$ for all $i,j \\\\in [t]$ .\",\n \"We assume all logarithms are in base 2 unless otherwise specified.\"\n ],\n [\n \"MV Families: Basic Facts and Definitions\",\n \"We now start with some basic definition and claims regarding MV families.\",\n \"Definition 2.1 (Matching Vector Family) Let $U=\\\\left(u_1, u_2, \\\\cdots u_t\\\\right)$ and $V=\\\\left(v_1, v_2, \\\\cdots v_t\\\\right)$ be lists over $\\\\mathbb {Z}_m^n$ .\",\n \"Then $\\\\left(U,V\\\\right)$ is called a matching vector family of size $t$ in $\\\\mathbb {Z}_m^n$ if $\\\\langle u_i,v_i \\\\rangle =0\\\\ \\\\left(mod\\\\ m\\\\right), \\\\quad \\\\forall i$ .\",\n \"$\\\\langle u_i,v_j \\\\rangle \\\\ne 0\\\\ \\\\left(mod\\\\ m\\\\right), \\\\quad \\\\forall i \\\\ne j$ .\",\n \"If in addition, we $|\\\\lbrace \\\\langle u,v \\\\rangle : u \\\\in U, v \\\\in V\\\\rbrace |=q$ , we call such a MV family an $q-restricted$ MV family.\",\n \"We denote the size of $\\\\left(U,V\\\\right)$ by $\\\\left|\\\\left(U,V\\\\right)\\\\right|$ .\",\n \"For instance, $\\\\left|\\\\left(U,V\\\\right)\\\\right|=t$ above.\",\n \"Definition 2.2 (Subset of Matching Vector Family) Let $U=\\\\left(u_1, u_2, \\\\cdots u_t\\\\right), V=\\\\left(v_1, v_2, \\\\cdots v_t\\\\right)$ form a matching vector family in $\\\\mathbb {Z}_m^n$ of size $t$ .\",\n \"By $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\subseteq \\\\left(U,V\\\\right)$ , we mean there exists a sublist $T \\\\subseteq [t]$ such that $U^{\\\\prime }=\\\\left(u_i:i \\\\in T\\\\right), V^{\\\\prime }=\\\\left(v_i:i \\\\in T\\\\right)$ .\",\n \"Observe that $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)$ is a matching vector family in $\\\\mathbb {Z}_m^n$ .\",\n \"Definition 2.3 ($\\\\mathbf {MV}\\\\left(m,n\\\\right)$ ) We denote by $\\\\mathbf {MV}\\\\left(m,n\\\\right)$ the maximum size of a matching vector family $\\\\left(U,V\\\\right)$ in $\\\\mathbb {Z}_m^n$ .\",\n \"Similarly, we denote by $\\\\mathbf {MV}\\\\left(m,n,q\\\\right)$ the maximum size of an $q-restricted$ matching vector family $\\\\left(U,V\\\\right)$ in $\\\\mathbb {Z}_m^n$ .\",\n \"We shall use the following simple facts implicitly throughout the paper.\",\n \"Fact 2.4 $\\\\mathbf {MV}\\\\left(m,n\\\\right)$ is an increasing function of $n$ .\",\n \"For $n_1q$ .\",\n \"Let $v_1,\\\\cdots ,v_q \\\\in Z_m$ be the $q$ possible nonzero values that the inner products $\\\\langle u,v \\\\rangle $ attain.\",\n \"For each $v_j$ there is some prime $p_{i_j}$ where $v_j \\\\ne 0 \\\\left( \\\\mod {p}_{i_j}^{e_{i_j}}\\\\right)$ .\",\n \"So, we can replace $m$ with just $\\\\prod _{j=1}^q p_{i_j}^{e_{i_j}}$ and discard all primes other than $p_{i_1}, \\\\cdots ,p_{i_q}$ .\",\n \"Claim 3.4 If $N$ has $r$ prime factors, then $|\\\\lbrace x \\\\in Z_N: order(x) q^{10q}$.\",\n \"By Lemma REF , we know that for $s\\\\ge m/q^{10q}$ , $\\\\mathbf {MV}_{1,s}\\\\left(m,n,q\\\\right)\\\\le 12q \\\\cdot q^{24\\\\log \\\\frac{m}{s}}\\\\left(\\\\frac{m}{s}\\\\right)^{n/2}\\\\le 12q \\\\cdot q^{24\\\\log q^{10q}}\\\\left(\\\\frac{m}{s}\\\\right)^{n/2}$ .\",\n \"Similarly, we have for $s\\\\ge m/4q$ , $\\\\mathbf {MV}_{s,1}\\\\left(m,n,q\\\\right)\\\\le 12q \\\\cdot q^{24\\\\log q^{10q}}\\\\left(\\\\frac{m}{s}\\\\right)^{n/2}$ .\",\n \"Now, suppose there is a $q-restricted$ MV family $(U,V)$ in $\\\\mathbb {Z}_m^n$ of size $t>12q \\\\cdot q^{24\\\\left(1+\\\\log q^{10q} \\\\right)}m^{n/2}$ .\",\n \"Applying Lemma REF with $r_1=r_2=1$ , we get a $q-restricted$ MV family $(U^{\\\\prime },V^{\\\\prime })\\\\subseteq (U,V)$ of size $t^{\\\\prime } \\\\ge s^{-n/2}q^{-24}t >q^{24\\\\log q^{10q}}\\\\left(\\\\frac{m}{s}\\\\right)^{n/2}$ where $s\\\\ge m/q^{10q}$ such that $\\\\langle u^{\\\\prime },v^{\\\\prime } \\\\rangle =0 \\\\ \\\\left(mod\\\\ s\\\\right)$ for all $u^{\\\\prime } \\\\in U^{\\\\prime }, v^{\\\\prime } \\\\in V^{\\\\prime }$ .\",\n \"Either $U^{\\\\prime \\\\left(s\\\\right)}$ is constant or $V^{\\\\prime \\\\left(s\\\\right)}$ is constant.\",\n \"But, by the previous paragraph, we have for $s\\\\ge m/q^{10q}$ , $\\\\mathbf {MV}_{s,1}\\\\left(m,n,q\\\\right)$ and $\\\\mathbf {MV}_{1,s}\\\\left(m,n,q\\\\right)$ are at most $12q \\\\cdot q^{24\\\\log q^{10q}}\\\\left(\\\\frac{m}{s}\\\\right)^{n/2}$ .\",\n \"This leads to a contadiction.\"\n ],\n [\n \"Proof of Lemma \",\n \"By assumption we have that $\\\\langle u,v \\\\rangle =0 \\\\ \\\\left(mod\\\\ r_1r_2\\\\right)$ for all $u \\\\in U, v \\\\in V$ .\",\n \"So, we can consider $\\\\frac{\\\\langle u,v \\\\rangle }{r_1r_2} \\\\in \\\\mathbb {Z}_{r_3}$ .\",\n \"Also, by hypothesis, the inner products $\\\\frac{\\\\langle u,v \\\\rangle }{r_1r_2}$ occupy $q^{\\\\prime }\\\\le q$ residues in $\\\\mathbb {Z}_{r_3}$ .\",\n \"We have that For $1 \\\\le i \\\\le t$ , $\\\\frac{\\\\langle u_i,v_i \\\\rangle }{r_1r_2}=0 \\\\ \\\\left(mod\\\\ r_3\\\\right)$ since $\\\\langle u_i,v_i \\\\rangle =0 \\\\pmod {m}$ .\",\n \"For $1 \\\\le i,j \\\\le t$ , $i \\\\ne j$ , $\\\\frac{\\\\langle u_i,v_j \\\\rangle }{r_1r_2} \\\\ne 0 \\\\ \\\\left(mod\\\\ r_3\\\\right)$ since $\\\\langle u_i,v_j \\\\rangle \\\\ne 0 \\\\pmod {m}$ .\",\n \"Let $\\\\mu $ denote the distribution over $\\\\mathbb {Z}_{r_3}$ defined by $\\\\frac{\\\\langle u_i,v_j \\\\rangle }{r_1r_2} \\\\mod {r}_3$ where $u_i, v_j$ are drawn independently and uniformly from $U,V$ respectively.\",\n \"Case 1: $4q^{\\\\prime } \\\\ge r_3$.\",\n \"Observe that $\\\\mu $ outputs 0 only when $i=j$ .\",\n \"Therefore, $\\\\mathbf {Pr}[\\\\mu =0]=1/t \\\\le 1/12q^{\\\\prime } \\\\le 1/3r_3$ .\",\n \"On the other hand, $\\\\mathbf {Pr}[\\\\mathcal {U}_{r_3}=0]=1/r_3$ .\",\n \"This implies that $\\\\Delta \\\\left(\\\\mu ,\\\\mathcal {U}_{r_3}\\\\right) \\\\ge 1/3r_3$ .\",\n \"Thus, applying Lemma REF with $\\\\omega =exp\\\\left(2\\\\pi i/r_3\\\\right)$ , we get that for some $1 \\\\le j \\\\le r_3-1$ , $\\\\left|\\\\mathbb {E}_{x \\\\sim \\\\mu }\\\\left[\\\\left(\\\\omega ^j\\\\right)^{x} \\\\right]\\\\right|\\\\ge \\\\frac{2}{3r_3\\\\sqrt{r_3}} \\\\ge \\\\frac{1}{12q^{\\\\prime 3/2}}.$ Let $\\\\omega ^{\\\\prime }=\\\\omega ^j$ and $ord(\\\\omega ^{\\\\prime })$ (the order of $\\\\omega ^{\\\\prime }$ ) be $s=r_3/gcd\\\\left(r_3,j\\\\right)$ .\",\n \"Also, note that as $j\\\\ge 1$ , we have $s \\\\ge 2$ .\",\n \"Also, trivially, $s \\\\ge r_3/q^{\\\\prime 10q^{\\\\prime }} \\\\ge r_3/q^{10q}$ .\",\n \"Case 2: $4q^{\\\\prime } 1$ and thus $\\\\omega ^{\\\\prime }$ is not trivial.\",\n \"Now, we shall show the existence of $\\\\omega ^{\\\\prime }$ of much higher order provided $r_3>q^{\\\\prime 10q^{\\\\prime }}$ .\",\n \"By Claim REF , for $S=q^{\\\\prime 10q^{\\\\prime }}$ and $N=r_3$ , and noting that $r_3$ has atmost $q$ prime factors by Claim REF , we have $Pr_j[ord\\\\left(\\\\omega ^j\\\\right)\\\\le r_3/S]\\\\le 1/4q^{\\\\prime }$ Thus, with probabilty at least $1/2q^{\\\\prime }-1/4q^{\\\\prime }=1/4q^{\\\\prime }$ , a random $j$ satisfies $\\\\left|\\\\mathbb {E}_{x \\\\sim \\\\mu }\\\\left[\\\\left(\\\\omega ^j\\\\right)^{x} \\\\right]\\\\right| \\\\ge 1/\\\\sqrt{2q^{\\\\prime }} \\\\ge \\\\frac{1}{12q^{3/2}}$ $s=ord\\\\left(\\\\omega ^j\\\\right) \\\\ge r_3/S$ Also, as $r_3/4q^{\\\\prime }>1$ the above two conditions are true for some $j \\\\ne 0$ .\",\n \"Now, we combine the above two cases as follows.\",\n \"Let $\\\\omega ^{\\\\prime }=\\\\omega ^j$ and ${\\\\varepsilon }= \\\\frac{1}{12q^{3/2}}$ .\",\n \"We have shown by the above case-by-case analysis that $\\\\left|\\\\mathbb {E}_{x \\\\sim \\\\mu }\\\\left[\\\\left(\\\\omega ^{\\\\prime }\\\\right)^{x} \\\\right]\\\\right|\\\\ge \\\\epsilon $ $s=ord(\\\\omega ^{\\\\prime })$ is such that $s \\\\ge \\\\max \\\\lbrace 2,r_3/q^{10q}\\\\rbrace $ Using the Cauchy-Schwartz inequality twice we get $&&\\\\left|\\\\mathbb {E}_{u \\\\sim U, v \\\\sim V}\\\\left[\\\\left(\\\\omega ^{\\\\prime }\\\\right)^{\\\\langle u,v \\\\rangle /r_1r_2}\\\\right]\\\\right|\\\\ge \\\\epsilon \\\\\\\\&\\\\Rightarrow & \\\\left|\\\\mathbb {E}_{u,\\\\tilde{u} \\\\sim U, v \\\\sim V}\\\\left[\\\\left(\\\\omega ^{\\\\prime }\\\\right)^{\\\\langle u-\\\\tilde{u},v \\\\rangle /r_1r_2}\\\\right]\\\\right|\\\\ge \\\\epsilon ^2\\\\\\\\&\\\\Rightarrow & \\\\left|\\\\mathbb {E}_{u,\\\\tilde{u} \\\\sim U, v,\\\\tilde{v} \\\\sim V}\\\\left[\\\\left(\\\\omega ^{\\\\prime }\\\\right)^{\\\\langle u-\\\\tilde{u},v-\\\\tilde{v} \\\\rangle /r_1r_2}\\\\right]\\\\right|\\\\ge \\\\epsilon ^4\\\\\\\\&\\\\Rightarrow & \\\\left|\\\\mathbb {E}_{u,\\\\tilde{u} \\\\sim U, v,\\\\tilde{v} \\\\sim V}\\\\left[\\\\left(\\\\omega ^{\\\\prime }\\\\right)^{\\\\langle \\\\left(u-\\\\tilde{u}\\\\right)/r_1,\\\\left(v-\\\\tilde{v}\\\\right)/r_2 \\\\rangle }\\\\right]\\\\right|\\\\ge \\\\epsilon ^4.$ We need to explain the last expression.\",\n \"Since by assumption $U^{\\\\left(r_1\\\\right)}$ and $V^{\\\\left(r_2\\\\right)}$ are constants, $\\\\left(u-\\\\tilde{u}\\\\right)/r_1 \\\\in \\\\mathbb {Z}_{m}^n$ and $\\\\left(v-\\\\tilde{v}\\\\right)/r_2 \\\\in \\\\mathbb {Z}_{m}^n$ are well defined.\",\n \"Thus, we can fix $\\\\tilde{u}$ and $\\\\tilde{v}$ by an averaging argument such that $\\\\left|\\\\mathbb {E}_{u \\\\sim U, v \\\\sim V}\\\\left[\\\\left(\\\\omega ^{\\\\prime }\\\\right)^{\\\\langle \\\\left(u-\\\\tilde{u}\\\\right)/r_1,\\\\left(v-\\\\tilde{v}\\\\right)/r_2 \\\\rangle }\\\\right]\\\\right|\\\\ge \\\\epsilon ^4.$ Let $U^{\\\\prime }=\\\\left(u_1^{\\\\prime },u_2^{\\\\prime }, \\\\cdots u_t^{\\\\prime }\\\\right), V^{\\\\prime }=\\\\left(v_1^{\\\\prime },v_2^{\\\\prime }, \\\\cdots v_t^{\\\\prime }\\\\right)$ where $u_i^{\\\\prime }= \\\\left(u_i -\\\\tilde{u}\\\\right)/r_1$ and $v_i^{\\\\prime }= \\\\left(v_i -\\\\tilde{v}\\\\right)/r_2$ .\",\n \"Notice that $U^{\\\\prime }$ and $V^{\\\\prime }$ are not assumed to be a MV family (later we will derive from them a MV family).\",\n \"We now define two probability distributions $\\\\mu ^{U^{\\\\prime }}$ and $\\\\mu ^{V^{\\\\prime }}$ over $\\\\mathbb {Z}_s^n$ .\",\n \"For each $w \\\\in \\\\mathbb {Z}_s^n$ , let $\\\\mu ^{U^{\\\\prime }}\\\\left(w\\\\right)=\\\\left|B_s\\\\left(w,U^{\\\\prime }\\\\right)\\\\right|/\\\\left|U^{\\\\prime }\\\\right|$ and $\\\\mu ^{V^{\\\\prime }}\\\\left(w\\\\right)=\\\\left|B_s\\\\left(w,V^{\\\\prime }\\\\right)\\\\right|/\\\\left|V^{\\\\prime }\\\\right|$ .\",\n \"That is, $\\\\mu ^{U^{\\\\prime }}\\\\left(w\\\\right)$ is the probability that $u^{\\\\prime (s)}=w$ where $u^{\\\\prime }$ is chosen uniformly in $U^{\\\\prime }$ , and similarly for $\\\\mu ^{V^{\\\\prime }}\\\\left(w\\\\right)$ .\",\n \"Therefore, since the order of $w^{\\\\prime }$ is $s$ , we have that $ \\\\left|\\\\mathbb {E}_{w_1 \\\\sim \\\\mu ^{U^{\\\\prime }},w_2 \\\\sim \\\\mu ^{V^{\\\\prime }}}\\\\left[\\\\left(\\\\omega ^{\\\\prime }\\\\right)^{\\\\langle w_1,w_2 \\\\rangle }\\\\right]\\\\right|\\\\ge \\\\epsilon ^4.$ Recalling that $s$ is the order of $\\\\omega ^{\\\\prime }$ and applying Lemma REF , we get $\\\\textnormal {cp}\\\\left(\\\\mu ^{U^{\\\\prime }}\\\\right)\\\\textnormal {cp}\\\\left(\\\\mu ^{V^{\\\\prime }}\\\\right) \\\\ge \\\\epsilon ^8/s^n$ .\",\n \"Therefore, one of $\\\\textnormal {cp}\\\\left(\\\\mu ^{U^{\\\\prime }}\\\\right)$ , $\\\\textnormal {cp}\\\\left(\\\\mu ^{V^{\\\\prime }}\\\\right)$ , say $\\\\textnormal {cp}\\\\left(\\\\mu ^{U^{\\\\prime }}\\\\right)$ , is at least $\\\\epsilon ^4/s^{n/2}$ .\",\n \"Let $w^*$ be the point of maximum probability mass given by $\\\\mu ^{U^{\\\\prime }}$ .\",\n \"Then, $\\\\mu ^{U^{\\\\prime }}\\\\left(w^*\\\\right)=\\\\mu ^{U^{\\\\prime }}\\\\left(w^*\\\\right)\\\\sum _{w \\\\in \\\\mathbb {Z}_s^n}\\\\mu ^{U^{\\\\prime }}\\\\left(w\\\\right)\\\\ge \\\\sum _{w \\\\in \\\\mathbb {Z}_s^n}\\\\mu ^{U^{\\\\prime }}\\\\left(w\\\\right)^2=\\\\textnormal {cp}\\\\left(\\\\mu ^{U^{\\\\prime }}\\\\right)\\\\ge \\\\epsilon ^4/s^{n/2}.$ Now, $\\\\mu ^{U^{\\\\prime }}\\\\left(w^*\\\\right) \\\\ge \\\\epsilon ^4/s^{n/2}$ means that $\\\\left|\\\\lbrace u \\\\in U: \\\\frac{u-\\\\tilde{u}}{r_1}=w^*\\\\ \\\\left(mod\\\\ s\\\\right)\\\\rbrace \\\\right|\\\\ge t \\\\epsilon ^4/s^{n/2} $ .\",\n \"Equivalently, $\\\\bigg |\\\\big \\\\lbrace u \\\\in U: u-\\\\tilde{u}=r_1w^*\\\\ \\\\left(mod\\\\ r_1s\\\\right)\\\\big \\\\rbrace \\\\bigg |\\\\ge t \\\\epsilon ^4/s^{n/2}.$ Let $T^{\\\\prime }=\\\\left(i:u_i=\\\\tilde{u}+r_1w^*\\\\ \\\\left(mod\\\\ r_1s\\\\right)\\\\right)$ .\",\n \"Now, define $U^{\\\\prime \\\\prime }=\\\\left(u_i:i \\\\in T^{\\\\prime }\\\\right)$ and $V^{\\\\prime \\\\prime }=\\\\left(v_i:i \\\\in T^{\\\\prime }\\\\right)$ .\",\n \"Observe that $\\\\left(U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }\\\\right)$ is a matching vector family in $\\\\mathbb {Z}_m^n$ such that $U^{\\\\prime \\\\prime \\\\left(r_1s\\\\right)}$ and $V^{\\\\prime \\\\prime \\\\left(r_2\\\\right)}$ are constants.\",\n \"$\\\\left|\\\\left(U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }\\\\right)\\\\right| \\\\ge t\\\\left(\\\\epsilon ^4/s^{n/2}\\\\right)$ .\",\n \"The only thing left is to show that $\\\\langle u,v\\\\rangle =0\\\\ \\\\left(mod\\\\ r_1r_2s\\\\right)$ for all $u \\\\in U^{\\\\prime \\\\prime },v \\\\in V^{\\\\prime \\\\prime }$ .\",\n \"This may not be true in general.\",\n \"However, we can take a large subset of the matching vector family so that the resulting matching vector family satisfies this condition.\",\n \"To see this, let $u \\\\in U^{\\\\prime \\\\prime }, v \\\\in V^{\\\\prime \\\\prime }$ be arbitrary.\",\n \"Now, $u=r_1s \\\\cdot u^{\\\\prime }+u_0$ and $v=r_2 \\\\cdot v^{\\\\prime }+v_0$ where $u^{\\\\prime }, v^{\\\\prime }$ depend on $u,v$ respectively and $u_0,v_0$ are independent of $u,v$ .\",\n \"Then, $\\\\langle u,v \\\\rangle =r_1r_2s \\\\langle u^{\\\\prime },v^{\\\\prime }\\\\rangle +r_1s \\\\langle u^{\\\\prime },v_0\\\\rangle +r_2 \\\\langle u_0,v^{\\\\prime }\\\\rangle + \\\\langle u_0,v_0\\\\rangle .$ As $u$ varies over $U^{\\\\prime \\\\prime }$ , $\\\\langle u^{\\\\prime },v_0\\\\rangle $ takes at most $q$ values modulo $r_2$ .\",\n \"Hence, $r_1s\\\\langle u^{\\\\prime },v_0\\\\rangle $ takes at most $q$ values modulo $r_1r_2s$ .\",\n \"Therefore, there exist at least $\\\\left(1/q\\\\right)\\\\left|U^{\\\\prime \\\\prime }\\\\right|$ elements of $U^{\\\\prime \\\\prime }$ such that $r_1s\\\\langle u^{\\\\prime },v_0\\\\rangle $ is a constant modulo $r_1r_2s$ .\",\n \"We take the corresponding elements from $V^{\\\\prime \\\\prime }$ to form a matching vector family $\\\\left(U^{\\\\prime \\\\prime \\\\prime },V^{\\\\prime \\\\prime \\\\prime }\\\\right)\\\\subseteq \\\\left(U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }\\\\right)$ .\",\n \"We apply another round using the same idea on $U^{\\\\prime \\\\prime \\\\prime },V^{\\\\prime \\\\prime \\\\prime }$ , this time ensuring that $r_2\\\\langle u_0,v^{\\\\prime }\\\\rangle $ is constant modulo $r_1r_2s$ as $v$ varies over a large fraction of $V^{\\\\prime \\\\prime \\\\prime }$ .\",\n \"Thus, we end up with $\\\\tilde{V}$ of size at least $\\\\left(1/q\\\\right)\\\\left|V^{\\\\prime \\\\prime \\\\prime }\\\\right|$ such that $r_2\\\\langle u_0,v_i\\\\rangle $ is a constant modulo $r_1r_2s$ .\",\n \"We take the corresponding subset $\\\\tilde{U}$ from $U^{\\\\prime \\\\prime \\\\prime }$ so that $(\\\\tilde{U}, \\\\tilde{V})\\\\subseteq \\\\left(U^{\\\\prime \\\\prime \\\\prime },V^{\\\\prime \\\\prime \\\\prime }\\\\right)$ is a matching vector family.\",\n \"Denote the size of $(\\\\tilde{U}, \\\\tilde{V})$ by $\\\\tilde{t}$ .\",\n \"Note that $\\\\tilde{U}=\\\\left(\\\\tilde{u}_1,\\\\cdots ,\\\\tilde{u}_{\\\\tilde{t}}\\\\right), \\\\tilde{V}=\\\\left(\\\\tilde{v}_1,\\\\cdots ,\\\\tilde{v}_{\\\\tilde{t}}\\\\right)$ is a matching vector family in $\\\\mathbb {Z}_m^n$ of size at least $\\\\left(1/q^2\\\\right)t\\\\left(\\\\epsilon ^4/s^{n/2}\\\\right)=s^{-n/2}q^{-\\\\left(8+4\\\\log _q \\\\left(12\\\\right)\\\\right)}t\\\\ge s^{-n/2}q^{-\\\\left(8+4\\\\log _2 \\\\left(12\\\\right)\\\\right)}t \\\\ge s^{-n/2}q^{-24}t$ .\",\n \"Also, as $\\\\langle u,v\\\\rangle $ is a constant modulo $r_1r_2s$ , for $u \\\\in \\\\tilde{U}, v \\\\in \\\\tilde{V}$ , and $\\\\langle \\\\tilde{u}_i,\\\\tilde{v}_i \\\\rangle =0\\\\ \\\\left(mod \\\\ r_1r_2s\\\\right)$ , we get that $\\\\langle u,v\\\\rangle =0 \\\\ \\\\left(mod\\\\ r_1r_2s\\\\right)$ , for $u \\\\in \\\\tilde{U}, v \\\\in \\\\tilde{V}$ .\",\n \"This concludes the proof.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Lemma \",\n \"We prove the lemma by backward induction on $r_1r_2|m$ .\",\n \"That is, to prove the claim about $\\\\mathbf {MV}_{r_1,r_2}\\\\left(m,n,q\\\\right)$ , we assume the inductive hypothesis for $\\\\mathbf {MV}_{r_1^{\\\\prime },r_2^{\\\\prime }}\\\\left(m,n,q\\\\right)$ where $r_1^{\\\\prime }r_2^{\\\\prime }>r_1r_2$ and $r_1^{\\\\prime }r_2^{\\\\prime }|m$ .\",\n \"Base Case.\",\n \"The base case of $r_1r_2=m$ is trivial.\",\n \"To see this, observe that if $\\\\langle u,v \\\\rangle =0 \\\\ \\\\left(mod\\\\ m\\\\right)$ for all $u \\\\in U, v \\\\in V$ , then by the definition of a matching vector family in $\\\\mathbb {Z}_m^n$ , the size of such a family cannot exceed 1.\",\n \"Hence, for $r_1r_2=m$ , $\\\\mathbf {MV}_{r_1,r_2}\\\\left(m,n,q\\\\right)=1 \\\\le 12q\\\\cdot q^{24\\\\log \\\\frac{m}{r_1r_2}}\\\\left(\\\\frac{m}{r_1r_2}\\\\right)^{n/2}$ .\",\n \"Inductive Step.\",\n \"Let $m=r_1r_2r_3$ with $r_1r_2r_1r_2$ and $r_1^{\\\\prime }r_2^{\\\\prime }|m$ .\",\n \"We need to show that $\\\\mathbf {MV}_{r_1,r_2}\\\\left(m,n,q\\\\right)\\\\le 12q \\\\cdot q^{24\\\\log \\\\frac{m}{r_1r_2}}\\\\left(\\\\frac{m}{r_1r_2}\\\\right)^{n/2}$ .\",\n \"Suppose this is false, so that there exists a $q-restricted$ matching vector family $\\\\left(U,V\\\\right)$ in $\\\\mathbb {Z}_m^n$ with $U=\\\\left(u_1, \\\\cdots u_t\\\\right),V=\\\\left(v_1,\\\\cdots v_t\\\\right)$ where $t >12q \\\\cdot q^{24\\\\log \\\\frac{m}{r_1r_2}}\\\\left(\\\\frac{m}{r_1r_2}\\\\right)^{n/2}$ such that $U^{\\\\left(r_1\\\\right)}$ and $V^{\\\\left(r_2\\\\right)}$ are constants.\",\n \"$\\\\langle u,v \\\\rangle =0 \\\\ \\\\left(mod\\\\ r_1r_2\\\\right)$ for all $u \\\\in U, v \\\\in V$ .\",\n \"Note that $t \\\\ge 12q$ .\",\n \"Therefore, applying Lemma REF , there exists $s|r_3$ with $s \\\\ge 2$ and matching vector family $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\subseteq \\\\left(U,V\\\\right)$ such that $\\\\left|\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\right| \\\\ge s^{-n/2}q^{-24}t$ where $\\\\langle u^{\\\\prime },v^{\\\\prime } \\\\rangle =0 \\\\ \\\\left(mod\\\\ r_1r_2s\\\\right)$ for all $u^{\\\\prime } \\\\in U^{\\\\prime }, v^{\\\\prime } \\\\in V^{\\\\prime }$ .\",\n \"either $U^{\\\\prime \\\\left(r_1s\\\\right)}$ is constant or $V^{\\\\prime \\\\left(r_2s\\\\right)}$ is constant.\",\n \"Without loss of generality, we assume that $U^{\\\\prime \\\\left(r_1s\\\\right)}$ is a constant.\",\n \"Therefore, $\\\\left|\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\right|& > & s^{-n/2}q^{-24}\\\\cdot 12q\\\\cdot q^{24\\\\log \\\\frac{m}{r_1r_2}}\\\\left(\\\\frac{m}{r_1r_2}\\\\right)^{n/2}\\\\\\\\&&=12q \\\\cdot q^{24\\\\left(\\\\log \\\\frac{m}{r_1r_2}-1\\\\right)}\\\\left(\\\\frac{m}{r_1r_2s}\\\\right)^{n/2}\\\\\\\\&&\\\\ge 12q \\\\cdot q^{24\\\\log \\\\frac{m}{r_1r_2s}}\\\\left(\\\\frac{m}{r_1r_2s}\\\\right)^{n/2},$ where the last inequality used the fact that $s \\\\ge 2$ .\",\n \"This however contradicts the inductive hypothesis.\",\n \"$\\\\Box $\"\n ],\n [\n \"Notations:\",\n \"For a $t \\\\times s$ matrix $M$ over $\\\\mathbb {Z}_m$ and for lists $T \\\\subseteq [t], S \\\\subseteq [s]$ the $T \\\\times S$ submatrix of $M$ is the matrix with rows in $T$ and columns in $S$ .\",\n \"For $i\\\\in [s]$ and $j \\\\in [t]$ we denote the $i$ 'th row of $M$ by $M(i:)$ and the $j$ 'th column by $M(:j)$ .\",\n \"Definition 4.1 (Span of a set) For $A \\\\subseteq \\\\mathbb {Z}_m^n$ let $\\\\textnormal {span}\\\\left(A\\\\right)$ denote the additive subgroup generated by $A$ .\",\n \"We say that a set $A$ spans $u \\\\in \\\\mathbb {Z}_m^n$ if $u \\\\in \\\\textnormal {span}(A)$ .\",\n \"Definition 4.2 (Rank of a matrix over $\\\\mathbb {Z}_m$ ) Let $M$ be a $t \\\\times t$ matrix over $\\\\mathbb {Z}_m$ .\",\n \"Then $\\\\textnormal {rank}\\\\left(M\\\\right)$ is the smallest $r$ such that $M=AB$ where $A$ is an $t \\\\times r$ martrix over $\\\\mathbb {Z}_m$ and $B$ is an $r \\\\times t$ matrix over $\\\\mathbb {Z}_m$ .\",\n \"Definition 4.3 (Column rank of a matrix over $\\\\mathbb {Z}_m$ ) Let $M$ be a $t \\\\times t$ matrix over $\\\\mathbb {Z}_m$ .\",\n \"Let $\\\\textnormal {colspan}\\\\left(M\\\\right)$ denote the subgroup of $\\\\mathbb {Z}_m^t$ generated by the columns of $M$ .\",\n \"The column rank of $M$ over $\\\\mathbb {Z}_m$ is defined as $\\\\textnormal {colrank}\\\\left(M\\\\right)=\\\\log _m \\\\left|\\\\textnormal {colspan}\\\\left(M\\\\right)\\\\right|.$ The column rank is, in general, a real number in the range $[0,t]$ .\",\n \"Since the rank can behave in unexpected ways over $\\\\mathbb {Z}_m$ , we make sure to prove some of the basic facts that we will be using later on.\",\n \"Fact 4.4 Let $M$ be a $t \\\\times t$ matrix over $\\\\mathbb {Z}_m$ and let $M^{\\\\prime }$ be any submatrix of $M$ .\",\n \"Then $\\\\textnormal {colrank}\\\\left(M^{\\\\prime }\\\\right)\\\\le \\\\textnormal {colrank}\\\\left(M\\\\right)$ .\",\n \"Suppose $M^{\\\\prime }$ is given by the first $t^{\\\\prime }$ rows and the first $t^{\\\\prime \\\\prime }$ columns of $M$ .\",\n \"We will define an injective map $f:\\\\textnormal {colspan}\\\\left(M^{\\\\prime }\\\\right) \\\\rightarrow \\\\textnormal {colspan}\\\\left(M\\\\right)$ .\",\n \"Given any $x \\\\in \\\\textnormal {colspan}\\\\left(M^{\\\\prime }\\\\right)$ we can write $x = \\\\sum _{j=1}^{t^{\\\\prime \\\\prime }} \\\\alpha _j \\\\cdot M^{\\\\prime }(:j)$ in some fixed way (there might be several choices of $\\\\alpha _j$ ).\",\n \"Define $f(x) = \\\\sum _{j=1}^{t^{\\\\prime \\\\prime }} \\\\alpha _j \\\\cdot M(:j)$ .\",\n \"Then, $x$ is clearly the restriction of $f(x)$ to the first $t^{\\\\prime }$ indices and so the map is injective.\",\n \"Fact 4.5 Let $M$ be a $t \\\\times t$ matrix over $\\\\mathbb {Z}_m$ and let $s|m$ .\",\n \"Then $\\\\textnormal {rank}\\\\left(M^{\\\\left(s\\\\right)}\\\\right)\\\\le \\\\textnormal {rank}\\\\left(M\\\\right)$ .\",\n \"Suppose there exist an $t \\\\times r$ matrix $A$ and an $r \\\\times t$ matrix $B$ over $\\\\mathbb {Z}_m$ such that $M=AB$ .\",\n \"Then $M^{\\\\left(s\\\\right)}=A^{\\\\left(s\\\\right)}B^{\\\\left(s\\\\right)}$ and so the rank of $M^{(s)}$ is at most $r$ .\",\n \"We will need the following claims relating the rank and the column rank of matrices over $\\\\mathbb {Z}_m$ .\",\n \"Claim 4.6 Let $M$ be an $t \\\\times t$ matrix over $\\\\mathbb {Z}_m$ .\",\n \"Then, $\\\\frac{\\\\textnormal {rank}\\\\left(M\\\\right)}{\\\\log m} \\\\le \\\\textnormal {colrank}\\\\left(M\\\\right) \\\\le \\\\textnormal {rank}\\\\left(M\\\\right).$ Let $r=\\\\textnormal {rank}\\\\left(M\\\\right)$ and $r^{\\\\prime }=\\\\textnormal {colrank}\\\\left(M\\\\right)$ .\",\n \"We first prove that $r^{\\\\prime } \\\\le r$ .\",\n \"This is equivalent to proving that $\\\\left|\\\\textnormal {colspan}\\\\left(M\\\\right)\\\\right| \\\\le m^r$ .\",\n \"Let $M=AB$ where $A$ is an $t \\\\times r$ martrix over $\\\\mathbb {Z}_m$ and $B$ is an $r \\\\times t$ matrix over $\\\\mathbb {Z}_m$ .\",\n \"Since the columns of $M$ are all in the span of the columns of $A$ we have that the column span of $M$ can contain at most $m^r$ elements.\",\n \"We now prove that $r^{\\\\prime } \\\\ge r/\\\\left(\\\\log m\\\\right)$ or, equivalently, $\\\\left|\\\\textnormal {colspan}\\\\left(M\\\\right)\\\\right| \\\\ge 2^r$ .\",\n \"Suppose in contradiction that $\\\\left|\\\\textnormal {colspan}\\\\left(M\\\\right)\\\\right| < 2^r$ .\",\n \"Take a minimal spanning set $S$ of $\\\\textnormal {colspan}\\\\left(M\\\\right)$ (that is, a set that spans $\\\\textnormal {colspan}\\\\left(M\\\\right)$ and such that no proper subset of it does).\",\n \"Suppose $|S| \\\\ge r$ and consider all linear combinations (over $\\\\mathbb {Z}_m$ ) of elements of $S$ with coefficients in $\\\\lbrace 0,1\\\\rbrace \\\\subseteq \\\\mathbb {Z}_m$ .\",\n \"Since $\\\\left|\\\\textnormal {colspan}\\\\left(M\\\\right)\\\\right| < 2^r$ there are two distinct $0-1$ linear combinations that map to the same element.\",\n \"This means that there is a linear combination with coefficients in $\\\\lbrace 1,-1\\\\rbrace $ of the elements of $S$ that is equal to zero.\",\n \"Since both 1 and $-1$ are invertible modulo $m$ we can write one of the elements of $S$ as a linear combination of the other elements.\",\n \"This contradicts the minimality of $S$ and so, we must have $|S| < r$ .\",\n \"This implies that $\\\\textnormal {rank}(M) < r$ , a contradiction, since we can write $M$ as the product of the matrix with columns in $S$ with the matrix of coefficients giving the columns of $M$ .\",\n \"Claim 4.7 Let $M$ be an $t \\\\times t$ matrix over $\\\\mathbb {Z}_m$ , let $r=\\\\textnormal {rank}\\\\left(M\\\\right)$ .\",\n \"There exists $r^{\\\\prime }$ columns of $M$ that span the rest of $M^{\\\\prime }$ s columns such that $r^{\\\\prime } \\\\le r\\\\log m$ .\",\n \"Take a minimal spanning set $S$ of the columns of $M$ (that is, a set that spans all other columns and such that no proper subset of it spans all columns).\",\n \"If $2^{\\\\left|S\\\\right|} > m^r$ , then $2^{\\\\left|S\\\\right|} > \\\\left|\\\\textnormal {colspan}(M)\\\\right|$ (by Claim REF ) and we proceed as in the proof from Claim REF above.\",\n \"If we look at all the $0-1$ combinations of the columns of $S$ , then there are two distinct $0-1$ linear combinations of the columns that map to the same element of $\\\\textnormal {colspan}\\\\left(M\\\\right)$ .\",\n \"Thus, let $\\\\sum _i\\\\alpha _iS\\\\left(:i\\\\right)=\\\\sum _i\\\\beta _iS\\\\left(:i\\\\right)$ where $\\\\alpha _i \\\\ne \\\\beta _i$ for at least one $i$ , say $i_0$ .\",\n \"Therefore, we have $\\\\sum _i\\\\left(\\\\alpha _i-\\\\beta _i\\\\right)S\\\\left(:i\\\\right)=0$ .\",\n \"Note that $\\\\left(\\\\alpha _{i_0}-\\\\beta _{i_0}\\\\right)=\\\\pm 1$ and hence is invertible.\",\n \"This lets us write $S\\\\left(:i_0\\\\right)$ as a linear combinations of the remaining columns contradicting the minimality of $S$ .\",\n \"Thus, $r^{\\\\prime }=\\\\left|S\\\\right| \\\\le r\\\\log m$ .\",\n \"The following claim shows that the column rank behaves similar to rank in terms of subadditivity.\",\n \"Claim 4.8 Let $A, B$ be $t \\\\times t$ matrices over $\\\\mathbb {Z}_m$ .\",\n \"Then, $\\\\textnormal {colrank}\\\\left(A+B\\\\right)\\\\le \\\\textnormal {colrank}\\\\left(A\\\\right)+\\\\textnormal {colrank}\\\\left(B\\\\right)$ .\",\n \"We show that $\\\\left|\\\\textnormal {colspan}\\\\left(A+B\\\\right)\\\\right|\\\\le \\\\left|\\\\textnormal {colspan}\\\\left(A\\\\right)\\\\right|\\\\left|\\\\textnormal {colspan}\\\\left(B\\\\right)\\\\right|$ .\",\n \"Note that $\\\\textnormal {colspan}\\\\left(A+B\\\\right) \\\\subseteq \\\\textnormal {colspan}\\\\left(A\\\\right)+ \\\\textnormal {colspan}\\\\left(B\\\\right) \\\\stackrel{def}{=} \\\\lbrace a+b|a\\\\in \\\\textnormal {colspan}\\\\left(A\\\\right), b \\\\in \\\\textnormal {colspan}\\\\left(B\\\\right)\\\\rbrace $ .\",\n \"Therefore, $\\\\left|\\\\textnormal {colspan}\\\\left(A+B\\\\right)\\\\right| \\\\le \\\\left|\\\\textnormal {colspan}\\\\left(A\\\\right)+ \\\\textnormal {colspan}\\\\left(B\\\\right)\\\\right| \\\\le \\\\left|\\\\textnormal {colspan}\\\\left(A\\\\right)\\\\right|\\\\left|\\\\textnormal {colspan}\\\\left(B\\\\right)\\\\right|$ .\",\n \"Claim 4.9 Let $M$ be a $2t \\\\times 2t$ matrix over $\\\\mathbb {Z}_m$ , such that $M=\\\\left(\\\\begin{array}{cc}A&0\\\\\\\\\\\\star &B\\\\end{array}\\\\right)$ where $A, B $ and $\\\\star $ are $t \\\\times t$ matrices.\",\n \"Then, $\\\\textnormal {colrank}\\\\left(A\\\\right)+\\\\textnormal {colrank}\\\\left(B\\\\right) \\\\le \\\\textnormal {colrank}\\\\left(M\\\\right)$ .\",\n \"We show that $\\\\left|\\\\textnormal {colspan}\\\\left(A\\\\right)\\\\right|\\\\left|\\\\textnormal {colspan}\\\\left(B\\\\right)\\\\right|\\\\le \\\\left|\\\\textnormal {colspan}\\\\left(M\\\\right)\\\\right|$ .\",\n \"Let $\\\\textnormal {colspan}\\\\left(A\\\\right)=R_1$ , $\\\\textnormal {colspan}\\\\left(B\\\\right)=R_2$ , $\\\\textnormal {colspan}\\\\left(M\\\\right)=R$ .\",\n \"We define $f:R_1 \\\\times R_2 \\\\rightarrow R$ and show that $f$ is injective.\",\n \"Given $r_1 \\\\in R_1$ and $r_2 \\\\in R_2$ , let $\\\\alpha _1, \\\\cdots \\\\alpha _t$ and $\\\\beta _1,\\\\cdots \\\\beta _t$ denote coefficients for linear combinations of the columns of $A$ and $B$ respectively that give $r_1$ and $r_2$ .\",\n \"There might be many such linear combinations but we fix one for each $r_i$ .\",\n \"Then, $f\\\\left(r_1,r_2\\\\right)=\\\\sum _{i=1}^{t}\\\\alpha _iM\\\\left(:i\\\\right)+\\\\sum _{i=t+1}^{2t}\\\\beta _{i-t}M\\\\left(:i\\\\right)$ .\",\n \"Now, given a column vector $f\\\\left(r_1,r_2\\\\right) \\\\in R$ , we uniquely identify $r_1$ and $r_2$ as follows.\",\n \"We look at the first $t$ rows and call it $s_1$ .\",\n \"Now $s_1=r_1$ and let $\\\\alpha _1, \\\\cdots \\\\alpha _{t}$ be the linear combination fixed for $r_1$ while defining $f$ .\",\n \"Now, consider $f\\\\left(r_1,r_2\\\\right)-\\\\sum _{i=1}^{t}\\\\alpha _iM\\\\left(:i\\\\right)$ and call the last $t$ rows $s_2$ .\",\n \"Note that $s_2=r_2$ .\",\n \"Claim 4.10 Let $M$ be a $t \\\\times t$ square matrix over $\\\\mathbb {Z}_m$ with zero diagonal entries.\",\n \"If for some $s |m$ , $\\\\textnormal {colrank}\\\\left(M^{\\\\left(s\\\\right)}\\\\right)\\\\le 2$ , then there exists at least $t^{\\\\prime }=t/m^2$ indices such that $M$ restricted to those indices as rows and columns is the all zero matrix modulo $s$ .\",\n \"As $\\\\textnormal {colrank}\\\\left(M^{\\\\left(s\\\\right)}\\\\right)\\\\le 2$ , it follows that $\\\\left|\\\\textnormal {colspan}\\\\left(M^{\\\\left(s\\\\right)}\\\\right)\\\\right|\\\\le s^2 \\\\le m^2$ .\",\n \"Hence, $M^{\\\\left(s\\\\right)}$ has at most $m^2$ distinct columns.\",\n \"Therefore, there exists a set of indices $S$ of size $t^{\\\\prime }\\\\ge t/m^2$ with $S=\\\\lbrace r_1, r_2, \\\\cdots r_{t^{\\\\prime }}\\\\rbrace $ such that all the columns $M^{\\\\left(s\\\\right)}\\\\left(:r_i\\\\right)$ are identical.\",\n \"Also, as the diagonal elements are zero modulo $m$ , they are zero modulo $s$ .\",\n \"Thus, the $S \\\\times S$ submatrix is the all zero matrix modulo $s$ .\"\n ],\n [\n \"Collision-Free MV families\",\n \"In the proof of Theorem REF it will be useful to assume that the elements of the MV family do not `collide' when reduced modulo an integer $s$ dividing $m$ .\",\n \"In this section we develop the necessary machinery to allow for this assumption.\",\n \"We start by defining a collision free matching vector family.\",\n \"Definition 5.1 (Collision free MV family) A collision free matching vector family $\\\\left(U,V\\\\right)$ in $\\\\mathbb {Z}_m^n$ is a matching vector family such that for all $s|m, s \\\\ge 2$ , all elements of $U$ are distinct modulo $s$ , and all elements of $V$ are distinct modulo $s$ .\",\n \"Note that if $\\\\left(U,V\\\\right)$ is a collision free matching vector family, then so is any $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\subseteq \\\\left(U,V\\\\right)$ .\",\n \"Lemma 5.2 Let $m\\\\ge 2$ be an arbitrary integer.\",\n \"Let $s$ be a divisor of $m$ , such that $1d\\\\left(m/2\\\\right)\\\\log m$ $2^{\\\\left(1/2\\\\right)d\\\\left(m\\\\right)n/\\\\log n} \\\\ge 3m2^{d_2\\\\left(m\\\\right)n/\\\\log n}$ $d_2\\\\left(m\\\\right)n/\\\\log n \\\\ge 2\\\\log m+d_3\\\\left(m\\\\right)n/\\\\log n$ $d_3\\\\left(m\\\\right) \\\\ge d_1\\\\left(m\\\\right) \\\\cdot d_4 \\\\cdot m \\\\log m$ $d_4 \\\\ge 300$ $d_1\\\\left(m\\\\right)\\\\ge 2c\\\\left(m\\\\right)$ $d_2 \\\\ge d_3+1$ It can be verified that the following choice for the functions meets the above conditions.\",\n \"$d\\\\left(m\\\\right)=1200 \\\\cdot c\\\\left(m\\\\right) \\\\cdot m^{6\\\\log m}$ $d_1\\\\left(m\\\\right)=2 \\\\cdot c\\\\left(m\\\\right)$ $d_2\\\\left(m\\\\right)=602 \\\\cdot c\\\\left(m\\\\right) \\\\cdot m \\\\log m$ $d_3\\\\left(m\\\\right)=600 \\\\cdot c\\\\left(m\\\\right)\\\\cdot m\\\\log m$ $d_4=300$ We shall explicitly mention which conditions of the above functions are being used in different parts of the proof.\"\n ],\n [\n \"Base Case.\",\n \"The base case is where $m=p$ is prime.\",\n \"Lemma REF implies that $\\\\mathbf {MV}\\\\left(p,n\\\\right) \\\\le 1+{n+p-2 \\\\atopwithdelims ()p-1}<\\\\left(2\\\\max \\\\lbrace n,p\\\\rbrace \\\\right)^p$ .\",\n \"If we show $\\\\left(2n\\\\right)^p \\\\le 2^{d\\\\left(p\\\\right)n/\\\\log n}$ and $\\\\left(2p\\\\right)^p \\\\le 2^{d\\\\left(p\\\\right)n/\\\\log n}$ we will be done.\",\n \"Indeed, by the choice of $d\\\\left(m\\\\right)$ (Condition 2 and 3) both of the above will hold.\"\n ],\n [\n \"Inductive Case.\",\n \"Let $n \\\\ge 2, m \\\\ge 2$ be arbitrary positive integers.\",\n \"Suppose, by induction, that $\\\\mathbf {MV}\\\\left(s,n\\\\right) < 2^{d\\\\left(s\\\\right)n/\\\\log n}$ for all $s|m, s 2$ for all $s^{\\\\prime }|m, s^{\\\\prime }\\\\ge 2$ .\",\n \"Then, for some $s|m, s\\\\ge 2$ , there exists a collision free matching vector family $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\subseteq \\\\left(U,V\\\\right)$ in $\\\\mathbb {Z}_m^n$ satisfying $\\\\left|\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\right|\\\\ge 2^{-d_1\\\\left(m\\\\right)r_s/\\\\log r_s}\\\\left|\\\\left(U,V\\\\right)\\\\right|$ where $r_s=\\\\textnormal {rank}\\\\left(P_{U,V}^{\\\\left(s\\\\right)}\\\\right)$ .\",\n \"Either $\\\\textnormal {colrank}\\\\left(P_{U^{\\\\prime },V^{\\\\prime }}^{\\\\left(s\\\\right)}\\\\right) \\\\le \\\\left(3/4\\\\right)\\\\textnormal {colrank}\\\\left(P_{U,V}^{\\\\left(s\\\\right)}\\\\right)$ or $\\\\textnormal {colrank}\\\\left(P_{U^{\\\\prime },V^{\\\\prime }}^{\\\\left(s\\\\right)}\\\\right) \\\\le 2$ .\",\n \"Claim 6.6 Let $\\\\left(U,V\\\\right)$ be a collision free matching vector family in $\\\\mathbb {Z}_m^n$ such that $\\\\left|\\\\left(U,V\\\\right)\\\\right|\\\\ge 3m \\\\cdot 2^{d_2\\\\left(m\\\\right)n/\\\\log n}$ .\",\n \"Then, there exists a collision free matching vector family $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\subseteq \\\\left(U,V\\\\right)$ in $\\\\mathbb {Z}_m^n$ satisfying $\\\\left|\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\right|\\\\ge 2^{-d_2\\\\left(m\\\\right)n/\\\\log n}\\\\left|\\\\left(U,V\\\\right)\\\\right|$ .\",\n \"$P_{U,V}^{\\\\left(s\\\\right)}$ is the all zero matrix for some $s|m, s \\\\ge 2$ .\",\n \"Let us proceed with the proof of the inductive claim assuming these two claims.\",\n \"We have a collision free matching vector family $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)$ with $\\\\left|\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\right|\\\\ge 2^{\\\\left(1/2\\\\right)d\\\\left(m\\\\right)n/\\\\log n} \\\\ge 3m \\\\cdot 2^{d_2\\\\left(m\\\\right)n/\\\\log n}$ .\",\n \"(Condition 6 satisfied by the choice of $d\\\\left(m\\\\right), d_2\\\\left(m\\\\right)$ ) Applying Claim REF , there exists a collision free matching vector family $\\\\left(U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }\\\\right)\\\\subseteq \\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\subseteq \\\\left(U,V\\\\right)$ in $\\\\mathbb {Z}_m^n$ satisfying $\\\\left|\\\\left(U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }\\\\right)\\\\right|\\\\ge 2^{-d_2\\\\left(m\\\\right)n/\\\\log n}2^{\\\\left(1/2\\\\right)d\\\\left(m\\\\right)n/\\\\log n}$ .\",\n \"$P_{U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }}^{\\\\left(s\\\\right)}$ is the all zero matrix for some $s|m, s \\\\ge 2$ .\",\n \"By the choice of $d\\\\left(m\\\\right)$ , it can be verified that $-d_2\\\\left(m\\\\right)+\\\\left(1/2\\\\right)d\\\\left(m\\\\right)>d\\\\left(m/2\\\\right)\\\\log m$ (Condition 5).\",\n \"Thus, $\\\\left|\\\\left(U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }\\\\right)\\\\right| > 2^{d\\\\left(m/2\\\\right)n\\\\log m/\\\\log n}$ .\",\n \"We now show that this is enough to get a contradiction.\",\n \"If $s=m$ , we have $\\\\left|\\\\left(U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }\\\\right)\\\\right|\\\\le 1$ as $\\\\left(U^{\\\\prime \\\\prime },V^{\\\\prime \\\\prime }\\\\right)$ is a matching vector family in $\\\\mathbb {Z}_m^n$ .\",\n \"If $s|R|/2$ or $|T| \\\\le |R|/2$ .\",\n \"In both cases, we shall exhibit a matching vector family as required in the statement of the claim.\",\n \"Case 1: $|T|>|R|/2$ .\",\n \"For $U=\\\\left(u_1, u_2, \\\\cdots u_t\\\\right)$ , $V=\\\\left(v_1, v_2, \\\\cdots v_t\\\\right)$ , let $U^{\\\\prime }=\\\\left(u_j|j\\\\in T\\\\right)$ and $V^{\\\\prime }=\\\\left(v_j|j\\\\in T\\\\right)$ , and $P^{\\\\prime }=P_{U^{\\\\prime },V^{\\\\prime }}$ .\",\n \"Now, as $P^{\\\\prime \\\\left(s\\\\right)}$ is monochromatic, and $\\\\langle u_j,v_j\\\\rangle =0\\\\ \\\\left(mod\\\\ s\\\\right)$ for $j \\\\in T$ , we have $\\\\langle u^{\\\\prime },v^{\\\\prime } \\\\rangle =0 \\\\left(mod\\\\ s\\\\right)$ for all $u^{\\\\prime } \\\\in U^{\\\\prime }, v^{\\\\prime } \\\\in V^{\\\\prime }$ .\",\n \"Observe that $\\\\left|\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\right| \\\\ge 2^{-1-c\\\\left(m\\\\right)r_s/\\\\log r_s}t \\\\ge 2^{-2c\\\\left(m\\\\right)r_s/\\\\log r_s}t\\\\ge 2^{-d_1\\\\left(m\\\\right)r_s/\\\\log r_s}t$ (by the choice of $d_1\\\\left(m\\\\right)$ , Condition 10) $\\\\textnormal {colrank}\\\\left(P_{U^{\\\\prime },V^{\\\\prime }}^{\\\\left(s\\\\right)}\\\\right) =0 \\\\le 2$ This finishes Case 1.\",\n \"Case 2: $|T| \\\\le |R|/2$ .\",\n \"Let $R^{\\\\prime }=R \\\\setminus T$ and $S^{\\\\prime }=S \\\\setminus T$ .\",\n \"Note that $R^{\\\\prime } \\\\cap S^{\\\\prime }=\\\\emptyset $ and $|R^{\\\\prime }|=|S^{\\\\prime }|$ .\",\n \"Consider the $R^{\\\\prime }\\\\cup S^{\\\\prime } \\\\times R^{\\\\prime }\\\\cup S^{\\\\prime }$ submatrix of $P_{U,V}$ .\",\n \"Call it $P^{\\\\prime }$ .\",\n \"Note that $P^{\\\\prime \\\\left(s\\\\right)}=\\\\left(\\\\begin{array}{cc}P_1^{\\\\prime }&C\\\\\\\\\\\\star &P_2^{\\\\prime }\\\\end{array}\\\\right)$ where $P_1^{\\\\prime }$ and $P_2^{\\\\prime }$ are the $R^{\\\\prime } \\\\times R^{\\\\prime }$ and the $S^{\\\\prime } \\\\times S^{\\\\prime }$ submatrices of $P_{U,V}^{\\\\left(s\\\\right)}$ respectively and $C$ is monochromatic.\",\n \"We add a matrix of column rank at most 1 to $P^{\\\\prime \\\\left(s\\\\right)}$ to yield $P^{\\\\prime \\\\prime \\\\left(s\\\\right)}$ which is the same as $P^{\\\\prime \\\\left(s\\\\right)}$ except that $C$ is replaced by the all zero block matrix.\",\n \"Thus, $P^{\\\\prime \\\\prime \\\\left(s\\\\right)}=\\\\left(\\\\begin{array}{cc}P_1^{\\\\prime }&0\\\\\\\\\\\\star &P_2^{\\\\prime }\\\\end{array}\\\\right)$ Note that by Claim REF , $\\\\textnormal {colrank}\\\\left(P^{\\\\prime \\\\prime \\\\left(s\\\\right)}\\\\right) \\\\le \\\\textnormal {colrank}\\\\left(P^{\\\\prime \\\\left(s\\\\right)}\\\\right)+1$ .\",\n \"Now, using Claim REF , $\\\\textnormal {colrank}\\\\left(P_1^{\\\\prime }\\\\right)+\\\\textnormal {colrank}\\\\left(P_2^{\\\\prime }\\\\right)\\\\le \\\\textnormal {colrank}\\\\left(P^{\\\\prime \\\\left(s\\\\right)}\\\\right)+1 \\\\le \\\\textnormal {colrank}\\\\left(P_{U,V}^{\\\\left(s\\\\right)}\\\\right)+1 \\\\le \\\\left(3/2\\\\right)\\\\textnormal {colrank}\\\\left(P_{U,V}^{\\\\left(s\\\\right)}\\\\right)$ as $\\\\textnormal {colrank}\\\\left(P_{U,V}^{\\\\left(s\\\\right)}\\\\right) >2$ .\",\n \"Therefore, one of $P_1^{\\\\prime }, P_2^{\\\\prime }$ , say $P_1^{\\\\prime }$ satisfies $\\\\textnormal {colrank}\\\\left(P_1^{\\\\prime }\\\\right)\\\\le \\\\left(3/4\\\\right)\\\\textnormal {colrank}\\\\left(P_{U,V}^{\\\\left(s\\\\right)}\\\\right)$ .\",\n \"Construct the matching vector family $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)$ as follows.\",\n \"Let $U^{\\\\prime }=\\\\left(u_j|j\\\\in R^{\\\\prime }\\\\right)$ and $V^{\\\\prime }=\\\\left(v_j|j\\\\in R^{\\\\prime }\\\\right)$ .\",\n \"Again, observe that $\\\\left|\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\right| \\\\ge 2^{-1-c\\\\left(m\\\\right)r_s/\\\\log r_s}t \\\\ge 2^{-2c\\\\left(m\\\\right)r_s/\\\\log r_s}t\\\\ge 2^{-d_1\\\\left(m\\\\right)r_s/\\\\log r_s}t$ (by the choice of $d_1\\\\left(m\\\\right)$ , Condition 10).\",\n \"$\\\\textnormal {colrank}\\\\left(P_{U^{\\\\prime },V^{\\\\prime }}^{\\\\left(s\\\\right)}\\\\right) \\\\le (3/4)\\\\textnormal {colrank}\\\\left(P_{U,V}^{\\\\left(s\\\\right)}\\\\right)$ .\",\n \"This completes the proof of Case 2.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Claim \",\n \"We will use Claim REF iteratively.\",\n \"For this, we first set up some notations.\"\n ],\n [\n \"The setup.\",\n \"Define a sequence of collision free matching vector families for $i=0,\\\\ldots ,z$ .\",\n \"$\\\\left(U,V\\\\right)=\\\\left(U_0,V_0\\\\right), \\\\left(U_1,V_1\\\\right) \\\\cdots $ Let $t_i=\\\\left|\\\\left(U_i,V_i\\\\right)\\\\right|$ .\",\n \"Each step $i$ has label $s_i|m$ (this label will be given by Claim REF ).\",\n \"Let $cr_i:\\\\mathbb {Z}^+ \\\\rightarrow \\\\mathbb {R}$ be defined by $cr_i\\\\left(s\\\\right)=\\\\textnormal {colrank}\\\\left(P_{U_i,V_i}^{\\\\left(s\\\\right)}\\\\right).$ Let $r_i:\\\\mathbb {Z}^+ \\\\rightarrow \\\\mathbb {Z}$ be defined by $r_i\\\\left(s\\\\right)=\\\\textnormal {rank}\\\\left(P_{U_i,V_i}^{\\\\left(s\\\\right)}\\\\right).$\"\n ],\n [\n \"Invariants.\",\n \"We will show how to go from step $i$ to step $i+1$ .\",\n \"We stop after stage $z$ when $cr_z\\\\left(s\\\\right) \\\\le 2$ for some $s|m, s\\\\ge 2$ .\",\n \"We shall maintain the following invariants for $0 \\\\le i \\\\le z-1$ .\",\n \"$\\\\left(U_{i+1},V_{i+1}\\\\right)\\\\subseteq \\\\left(U_{i},V_{i}\\\\right)$ and hence is a collision free matching vector family in $\\\\mathbb {Z}_m^n$ .\",\n \"$t_{i+1} \\\\ge 2^{-d_1\\\\left(m\\\\right)r_i\\\\left(s_i\\\\right)/\\\\log r_i\\\\left(s_i\\\\right)}t_i$ .\",\n \"$cr_{i+1}\\\\left(s_i\\\\right)\\\\le \\\\left(3/4\\\\right)cr_i\\\\left(s_i\\\\right)$ or $cr_{i+1}\\\\left(s_i\\\\right)\\\\le 2$ .\",\n \"$cr_{i+1}\\\\left(s^{\\\\prime }\\\\right)\\\\le cr_i\\\\left(s^{\\\\prime }\\\\right)$ for all $s^{\\\\prime }|m$ .\"\n ],\n [\n \"Step $i$ {{formula:ef557492-5072-4f74-b6af-082231e35300}} Step {{formula:d641f54b-beb7-403b-a4b6-d7eea3906b1e}} .\",\n \"We state a claim that we will prove below.\",\n \"Claim 6.7 $\\\\sum _{i=0}^{z-1}d_1\\\\left(m\\\\right)r_i\\\\left(s_i\\\\right)/\\\\log r_i\\\\left(s_i\\\\right)\\\\le d_3\\\\left(m\\\\right)n/log n$ .\",\n \"In order to apply Claim REF , we need to satisfy $t_i \\\\ge 3m$ .\",\n \"Observe that by Claim REF , $t_i\\\\ge t_z &\\\\ge & t_0\\\\prod _{j=0}^{z-1}2^{-d_1\\\\left(m\\\\right)r_j\\\\left(s_j\\\\right)/\\\\log r_j\\\\left(s_j\\\\right)}\\\\\\\\ &\\\\ge & 2^{-d_3\\\\left(m\\\\right)n/\\\\log n}t_0 \\\\\\\\&\\\\ge & 3m \\\\cdot 2^{-d_3\\\\left(m\\\\right)n/\\\\log n+d_2\\\\left(m\\\\right)n/\\\\log n}\\\\ge 3m,$ (by the choice of $d_2\\\\left(m\\\\right), d_3\\\\left(m\\\\right)$ in Condition 11).\",\n \"Apply Claim REF to $\\\\left(U_{i},V_{i}\\\\right)$ to get label $s_i$ for step $i$ and $\\\\left(U_{i+1},V_{i+1}\\\\right)\\\\subseteq \\\\left(U_{i},V_{i}\\\\right)$ .\",\n \"The first three invariants are maintained by the statement of Claim REF .\",\n \"The last invariant follows from Fact REF .\",\n \"Note that by the inequality we just established, $t_z\\\\ge 2^{-d_3\\\\left(m\\\\right)n/\\\\log n}t_0$ .\",\n \"Also, by the stopping condition, $cr_z\\\\left(s^{\\\\prime }\\\\right) \\\\le 2$ for some $s^{\\\\prime }|m, s^{\\\\prime }\\\\ge 2$ .\",\n \"Thus, applying Claim REF , we get another matching vector family $\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\subseteq \\\\left(U_z,V_z\\\\right)\\\\subseteq \\\\left(U,V\\\\right)$ such that $\\\\left|\\\\left(U^{\\\\prime },V^{\\\\prime }\\\\right)\\\\right|\\\\ge t_z/m^2 \\\\ge 2^{-2\\\\log m-d_3\\\\left(m\\\\right)n/\\\\log n}\\\\left|\\\\left(U,V\\\\right)\\\\right|\\\\ge 2^{-d_2\\\\left(m\\\\right)n/\\\\log n}\\\\left|\\\\left(U,V\\\\right)\\\\right|$ (Condition 7 satisfied by the choice of $d_2\\\\left(m\\\\right)$ and $d_3\\\\left(m\\\\right)$ ).\",\n \"$P_{U^{\\\\prime },V^{\\\\prime }}^{\\\\left(s^{\\\\prime }\\\\right)}$ is the all zero matrix.\",\n \"This finishes the Proof of Claim REF .\",\n \"Proof of Claim REF : Let $t_s$ be the number of steps with label $s$ .\",\n \"Note that as the column rank modulo $s$ goes down by a factor of at least $3/4$ each time we are in a step labeled $s$ , it is easy to see that $t_s \\\\le \\\\log _{4/3}cr_0\\\\left(s\\\\right) \\\\le \\\\log _{4/3}n$ .\",\n \"We shall rely on the monotonic increasing nature of $x/\\\\log x$ when $x \\\\ge e$ .\",\n \"As $cr_i\\\\left(s\\\\right)>2$ , by Claim REF , $r_i\\\\left(s\\\\right)\\\\ge cr_i\\\\left(s\\\\right)> 2$ which means $r_i(s) \\\\ge 3 > e$ as the rank is always an integer.\",\n \"We thus have $&&\\\\sum _{i=0}^{z-1}d_1\\\\left(m\\\\right)\\\\frac{r_i\\\\left(s_i\\\\right)}{\\\\log r_i\\\\left(s_i\\\\right)}\\\\\\\\&\\\\le & d_1\\\\left(m\\\\right)\\\\log m\\\\sum _{i=0}^{z-1}\\\\frac{cr_i\\\\left(s_i\\\\right)}{\\\\log cr_i\\\\left(s_i\\\\right)}\\\\ \\\\ \\\\text{(by Claim \\\\ref {clm-rankcolrank}) and monotonicity of $x/\\\\log x$ as discussed above}\\\\\\\\&\\\\le & d_1\\\\left(m\\\\right)\\\\log m\\\\sum _{s|m, s\\\\ge 2}\\\\sum _{j=1}^{\\\\lfloor \\\\log _{4/3}n\\\\left(s\\\\right)\\\\rfloor }\\\\left(\\\\frac{cr_0\\\\left(s\\\\right)}{\\\\left(4/3\\\\right)^{j-1}\\\\log \\\\left(cr_0\\\\left(s\\\\right)/\\\\left(4/3\\\\right)^{j-1}\\\\right)}\\\\right)\\\\\\\\&\\\\le & d_1\\\\left(m\\\\right)\\\\log m\\\\sum _{s|m, s\\\\ge 2}d_4cr_0\\\\left(s\\\\right)/\\\\log cr_0\\\\left(s\\\\right)\\\\ \\\\ \\\\text{(by Claim \\\\ref {clm-sum} and Condition $9$ satisfied by $d_4$)}\\\\\\\\&\\\\le & d_1\\\\left(m\\\\right)\\\\log m\\\\sum _{s|m, s\\\\ge 2}d_4n/\\\\log n\\\\ \\\\ \\\\text{(as $cr_0\\\\left(s\\\\right)\\\\le r_0\\\\left(s\\\\right)\\\\le r_0\\\\left(m\\\\right)\\\\le n$, by Claim \\\\ref {clm-rankcolrank} and Fact \\\\ref {rankmon})}\\\\\\\\&\\\\le & d_4d_1\\\\left(m\\\\right)m\\\\left(\\\\log m\\\\right)n/\\\\log n\\\\\\\\&\\\\le & d_3\\\\left(m\\\\right)n/\\\\log n\\\\ \\\\ \\\\text{(by the choice of $d_3\\\\left(m\\\\right)$, Condition $8$)}$ This completes the proof.\",\n \"$\\\\Box $\"\n ],\n [\n \"Monochromatic rectangles from low rank matrices\",\n \"In this section we prove Lemma REF (the Sub-Matrix Lemma).\",\n \"We begin with some preliminary definitions.\",\n \"The following is a standard result in algebra and can be find in any introductory text.\",\n \"Theorem 7.1 (Fundamental Theorem of finitely generated abelian groups) Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of prime power order and an infinite cyclic group.\",\n \"More precisely, $G \\\\cong \\\\mathbb {Z}^n \\\\times \\\\mathbb {Z}_{q_1} \\\\times \\\\mathbb {Z}_{q_2} \\\\cdots \\\\times \\\\mathbb {Z}_{q_r}$ where $q_i$ 's are prime powers with $q_1 \\\\le q_2 \\\\cdots \\\\le q_r$ .\",\n \"The decomposition is unique after applying this ordering on $q_i$ 's.\",\n \"If the group $G$ is finite, then $n=0$ .\",\n \"We will use the following two definitions regarding sumsets.\",\n \"Definition 7.2 (Difference Set) For $A \\\\subseteq \\\\mathbb {Z}_m^n$ define its difference set as $A-A{=}\\\\lbrace a-a^{\\\\prime }|a,a^{\\\\prime }\\\\in A\\\\rbrace $ .\",\n \"Definition 7.3 ($rep_S\\\\left(x\\\\right)$ ) For any $S \\\\subseteq \\\\mathbb {Z}_m^n$ and $x \\\\in \\\\mathbb {Z}_m^n$ , $rep_S\\\\left(x\\\\right)$ is the number of different representations of $x$ as an expression of the form $s-s^{\\\\prime }$ where $s,s^{\\\\prime } \\\\in S$ .\",\n \"Next, we define the $\\\\epsilon $ -spectrum of $B$ with respect to a primitive root of unity of order $m$ .\",\n \"Definition 7.4 (Spectrum) For $B \\\\subseteq \\\\mathbb {Z}_m^n$ , and $\\\\epsilon \\\\in [0,1]$ , the $\\\\epsilon $ -spectrum of $B$ with respect to $\\\\omega $ , a primitive root of unity of order $m$ , is the set $\\\\textnormal {Spec}_{\\\\epsilon }\\\\left(B\\\\right)=\\\\left\\\\lbrace x \\\\in \\\\mathbb {Z}_m^n:\\\\left|\\\\mathbb {E}_{b \\\\sim B}\\\\left[\\\\omega ^{\\\\langle x,b \\\\rangle }\\\\right]\\\\right| \\\\ge \\\\epsilon \\\\right\\\\rbrace .$ When $\\\\omega $ is implicit in the context, we will drop the phrase \\\"with respect to $\\\\omega $ \\\".\",\n \"We start by proving the following lemma which is a generalization of a lemma from [5].\",\n \"Lemma 7.5 Let $A,B \\\\subseteq \\\\mathbb {Z}_m^n$ be sets.\",\n \"Let $\\\\omega $ be a primitive root of unity of order $m$ .\",\n \"If $A \\\\subseteq \\\\textnormal {Spec}_{\\\\epsilon }\\\\left(B\\\\right)$ , then there exist sets $A^{\\\\prime }\\\\subseteq A, B^{\\\\prime }\\\\subseteq B$ , such that $|A^{\\\\prime }| \\\\ge |A|/m$ and $|B^{\\\\prime }| \\\\ge \\\\epsilon ^2\\\\frac{|A|}{|span\\\\left(A\\\\right)|}|B|$ such that $D_{\\\\omega }\\\\left(A^{\\\\prime },B^{\\\\prime }\\\\right)=1$ .\",\n \"We start by setting up some notations.\",\n \"Let $W=\\\\textnormal {span}\\\\left(A\\\\right)$ be the subgroup of $\\\\mathbb {Z}_m^n$ spanned by $A$ .\",\n \"By Theorem REF , there exists an isomorphism $\\\\tau :\\\\prod _{i=1}^r\\\\mathbb {Z}_{q_i}\\\\rightarrow W$ .\",\n \"Let $\\\\prod _{i=1}^r\\\\mathbb {Z}_{q_i}$ and note that we can think of elements of $ as vectors with integer coordinates where the $ i$^{\\\\prime }th coordinate is in $ Zqi$.\",\n \"Let $ e1,e2, er where $e_i$ is the vector that has 1 in the $i$ 'th coordinate and 0 everywhere else.\",\n \"Given $x \\\\in , $ 1, r$, with $ i Zqi$ such that $$x=\\\\sum _{i=1}^r\\\\alpha _ie_i.$$Then $ (x)=i=1ri(ei)$.\",\n \"Let $ vi=(ei)$ for $ 1 i r$.\",\n \"We can think of the $ vi$^{\\\\prime }s as a basis of $ W$.\",\n \"Therefore, for $ =(1, 2, r) we have $\\\\tau \\\\left(\\\\alpha \\\\right)=\\\\sum _{i=1}^r\\\\alpha _iv_i$ .\",\n \"Let $\\\\Theta =\\\\lbrace \\\\left(\\\\beta _1, \\\\cdots \\\\beta _r\\\\right) \\\\in \\\\mathbb {Z}_m^r|\\\\exists u \\\\in \\\\mathbb {Z}_m^n \\\\text{ such that } \\\\forall i, \\\\beta _i=\\\\langle v_i,u\\\\rangle \\\\rbrace .$ Claim 7.6 For $1 \\\\le i \\\\le r$ , $q_iv_i=0^n\\\\ \\\\left(mod \\\\ m\\\\right)$ .\",\n \"Let $x =0^r \\\\in .\",\n \"Now $ (x)=0n (mod m)$.\",\n \"Note that $ x$ can also be written as $ x=qiei$.\",\n \"Applying $$ on both sides, we get $ (x)=qivi$.\",\n \"Thus, $ qivi=0n (mod m)$.$ Claim 7.7 For $\\\\beta \\\\in \\\\Theta $ , $1 \\\\le i \\\\le r$ , $q_i\\\\beta _i=0\\\\ \\\\left(mod \\\\ m\\\\right)$ .\",\n \"As $\\\\beta \\\\in \\\\Theta $ , there is a $u \\\\in \\\\mathbb {Z}_m^n$ such that $\\\\forall i$ , $\\\\beta _i=\\\\langle v_i,u\\\\rangle $ .\",\n \"Then, $q_i\\\\beta _i=q_i\\\\langle v_i,u\\\\rangle =0\\\\ \\\\left(mod\\\\ m\\\\right)$ by Claim REF .\",\n \"For $\\\\alpha \\\\in \\\\beta \\\\in \\\\Theta $ we define their inner product $\\\\langle \\\\alpha , \\\\beta \\\\rangle \\\\in \\\\mathbb {Z}_m$ by considering $\\\\alpha _i \\\\in \\\\lbrace 0,\\\\ldots ,q_i-1\\\\rbrace , \\\\beta _i \\\\in \\\\lbrace 0,\\\\ldots ,m-1\\\\rbrace $ , taking the inner product over the integers and then reducing the result modulo $m$ .\",\n \"This is indeed an inner product by Claim REF .\",\n \"Claim 7.8 Given $\\\\beta \\\\in \\\\Theta \\\\setminus \\\\lbrace 0\\\\rbrace $ , $\\\\sum _{a \\\\in W}\\\\omega ^{\\\\langle \\\\tau ^{-1}\\\\left(a\\\\right),\\\\beta \\\\rangle }=\\\\sum _{\\\\alpha \\\\in \\\\omega ^{\\\\langle \\\\alpha ,\\\\beta \\\\rangle }=0.\",\n \"}\\\\begin{proof}Let \\\\beta _i \\\\ne 0.\",\n \"Then \\\\sum _{\\\\alpha \\\\in C}\\\\omega ^{\\\\langle \\\\alpha ,\\\\beta \\\\rangle }=0 whenever \\\\sum _{j=0}^{q_i-1}\\\\omega ^{j\\\\beta _i}=0.\",\n \"Now, \\\\sum _{j=0}^{q_i-1}\\\\omega ^{j\\\\beta _i}=\\\\frac{\\\\omega ^{q_i\\\\beta _i}-1}{\\\\omega ^{\\\\beta _i}-1}.\",\n \"This is well defined because \\\\omega is of order m and \\\\beta _i\\\\ne 0.\",\n \"The claim now follows from Claim \\\\ref {clm-beta} which makes the expression zero.\\\\end{proof}$ With the above setup in place, we can now proceed with the proof of Lemma REF .\",\n \"For $\\\\beta \\\\in \\\\Theta $ , define $S_{\\\\beta }=\\\\lbrace x \\\\in \\\\mathbb {Z}_m^n|\\\\langle v_i,x\\\\rangle =\\\\beta _i, 1\\\\le i\\\\le r\\\\rbrace .$ Denoting $\\\\mu \\\\left(\\\\beta \\\\right)=\\\\mathbf {Pr}_{b \\\\in B}[b \\\\in S_{\\\\beta }]$ , we observe that $\\\\cup _{\\\\beta \\\\in \\\\Theta }\\\\left(B\\\\cap S_{\\\\beta }\\\\right)=B$ .\",\n \"Hence, $\\\\sum _{\\\\beta \\\\in \\\\Theta }\\\\mu \\\\left(\\\\beta \\\\right)=1$ .\",\n \"For $a \\\\in W$ , define $h\\\\left(a\\\\right)=\\\\mathbb {E}_{b \\\\in B}\\\\left[\\\\omega ^{\\\\langle a,b \\\\rangle }\\\\right]$ .\",\n \"If $a=\\\\sum _{i=1}^r\\\\alpha _iv_i$ then $h\\\\left(a\\\\right)&=&\\\\mathbb {E}_{b \\\\in B}\\\\left[\\\\omega ^{\\\\langle a,b \\\\rangle }\\\\right]\\\\\\\\&=&\\\\mathbb {E}_{b \\\\in B}\\\\left[\\\\omega ^{\\\\langle \\\\sum _{i=1}^r\\\\alpha _iv_i,b \\\\rangle }\\\\right]\\\\\\\\&=&\\\\sum _{\\\\beta \\\\in \\\\Theta }\\\\mu \\\\left(\\\\beta \\\\right)\\\\omega ^{\\\\langle \\\\alpha ,\\\\beta \\\\rangle }\\\\\\\\&=&\\\\sum _{\\\\beta \\\\in \\\\Theta }\\\\mu \\\\left(\\\\beta \\\\right)\\\\omega ^{\\\\langle \\\\tau ^{-1}\\\\left(a\\\\right) ,\\\\beta \\\\rangle }.$ We will prove upper and lower bounds for the sum $\\\\sum _{a \\\\in A}\\\\left|h\\\\left(a\\\\right)\\\\right|^2$ .\",\n \"On the one hand, $\\\\sum _{a \\\\in A}\\\\left|h\\\\left(a\\\\right)\\\\right|^2&\\\\ge &\\\\frac{1}{|A|}\\\\left(\\\\sum _{a \\\\in A}\\\\left|h\\\\left(a\\\\right)\\\\right|\\\\right)^2 \\\\ \\\\text{(Cauchy Scwartz inequality)}\\\\\\\\&\\\\ge &\\\\frac{1}{|A|}\\\\left(\\\\sum _{a \\\\in A}\\\\epsilon \\\\right)^2 \\\\ \\\\text{($A \\\\subseteq \\\\textnormal {Spec}_{\\\\epsilon }\\\\left(B\\\\right)$ implies $\\\\left|h\\\\left(a\\\\right)\\\\right|\\\\ge \\\\epsilon $ )}\\\\\\\\&\\\\ge & |A|\\\\epsilon ^2.$ On the other hand, $\\\\sum _{a \\\\in A}\\\\left|h\\\\left(a\\\\right)\\\\right|^2&\\\\le & \\\\sum _{a \\\\in W}\\\\left|h\\\\left(a\\\\right)\\\\right|^2\\\\\\\\&=&\\\\sum _{a \\\\in W}\\\\sum _{\\\\beta \\\\in \\\\Theta , \\\\beta ^{\\\\prime } \\\\in \\\\Theta ^{\\\\prime }}\\\\mu \\\\left(\\\\beta \\\\right)\\\\mu \\\\left(\\\\beta ^{\\\\prime }\\\\right)\\\\omega ^{\\\\langle \\\\tau ^{-1}\\\\left(a\\\\right),\\\\beta - \\\\beta ^{\\\\prime } \\\\rangle }\\\\\\\\&=&\\\\sum _{\\\\beta ,\\\\beta ^{\\\\prime } \\\\in \\\\Theta ,}\\\\mu \\\\left(\\\\beta \\\\right)\\\\mu \\\\left(\\\\beta ^{\\\\prime }\\\\right)\\\\sum _{a \\\\in W}\\\\omega ^{\\\\langle \\\\tau ^{-1}\\\\left(a\\\\right),\\\\beta - \\\\beta ^{\\\\prime } \\\\rangle }\\\\\\\\&=&\\\\sum _{\\\\beta \\\\in \\\\Theta }\\\\mu \\\\left(\\\\beta \\\\right)^2|W| \\\\qquad \\\\text{(Claim \\\\ref {clm-zero})}\\\\\\\\&\\\\le & |W| \\\\max _{\\\\beta \\\\in \\\\Theta }\\\\lbrace \\\\mu \\\\left(\\\\beta \\\\right)\\\\rbrace .$ Now, combining the upper and lower bounds, $\\\\max _{\\\\beta \\\\in \\\\Theta }\\\\lbrace \\\\mu \\\\left(\\\\beta \\\\right)\\\\rbrace \\\\ge \\\\epsilon ^2\\\\frac{|A|}{|W|}$ .\",\n \"Thus, there exists a $\\\\beta \\\\in \\\\Theta $ such that $\\\\mu \\\\left(\\\\beta \\\\right)\\\\ge \\\\epsilon ^2\\\\frac{|A|}{|W|}$ .\",\n \"This means that the subset $B^{\\\\prime }=B \\\\cap S_{\\\\beta }$ is of size at least $\\\\epsilon ^2\\\\frac{|A|}{|W|}|B|$ .\",\n \"Now, any $a \\\\in A$ can be written as $a=\\\\sum _{i=1}^r \\\\alpha _i v_i$ , and for $b \\\\in B^{\\\\prime }$ , the inner product $\\\\langle a,b \\\\rangle =\\\\langle \\\\alpha ,\\\\beta \\\\rangle $ is independent of $b$ .\",\n \"Now, for $i \\\\in [m]$ , let $A_i \\\\subseteq A$ be such that for $a \\\\in A_i$ , for all $b \\\\in B^{\\\\prime }$ , $\\\\langle a,b \\\\rangle =\\\\langle \\\\alpha ,\\\\beta \\\\rangle =i$ .\",\n \"Now there exists some $A_i$ , call it $A^{\\\\prime }$ , of size at least $|A|/m$ such that $\\\\langle a^{\\\\prime },b^{\\\\prime } \\\\rangle =i$ for all $a^{\\\\prime }\\\\in A^{\\\\prime },b^{\\\\prime } \\\\in B^{\\\\prime }$ , that is, $D_{\\\\omega }\\\\left(A^{\\\\prime },B^{\\\\prime }\\\\right)=1$ and this proves the lemma.\",\n \"We continue along the lines of [5] and prove the following lemma.\",\n \"Lemma 7.9 Suppose the twin free lists $U, V \\\\subseteq \\\\mathbb {Z}_m^n$ satisfy $D_{\\\\omega }\\\\left(U, V\\\\right) \\\\ge \\\\epsilon $ ̨ where $\\\\omega $ is a primitive root of unity of order $m$ .\",\n \"Also, let $\\\\textnormal {rank}\\\\left(P_{U,V}\\\\right)=r$ .\",\n \"Then assuming Conjecture REF , for every $K > 1$ , letting $\\\\ell = r/\\\\log _m K$ , there exist lists $U^{\\\\prime }\\\\subseteq U, V^{\\\\prime }\\\\subseteq V$ such that $D_{\\\\omega }\\\\left(U^{\\\\prime }, V^{\\\\prime }\\\\right) = 1$ , and $|U^{\\\\prime }| \\\\ge poly_m\\\\left(\\\\frac{\\\\left(\\\\epsilon /2\\\\right)^{2^{\\\\ell }}}{rK}\\\\right)\\\\left(2mr\\\\right)^{-\\\\ell }|U|$ , $|V^{\\\\prime }| \\\\ge poly_m\\\\left(\\\\frac{\\\\left(\\\\epsilon /2\\\\right)^{2^{\\\\ell }}}{rK}\\\\right)m^{-\\\\ell }|V|$ .\",\n \"Let $U=\\\\left(u_1, \\\\cdots u_t\\\\right)$ and $V=\\\\left(v_1, \\\\cdots v_t\\\\right)$ .\",\n \"Since $P_{U,V}$ has rank $r$ there exists a $t\\\\times r$ matrix $U_M$ and $r \\\\times t$ matrix $V_M$ so that $U_MV_M=P_{U,V}$ .\",\n \"Thus if we let $A$ denote the rows of $U_M$ and $B$ denote the columns of $V_M$ , then $A,B \\\\subseteq \\\\mathbb {Z}_m^r$ .\",\n \"The proof does not care about the order of elements and hence we now consider $A,B$ which are sets.\",\n \"Note that $|A|=|B|=t$ and if $A=\\\\left(a_1, \\\\cdots a_t\\\\right)$ and $B=\\\\left(b_1, \\\\cdots b_t\\\\right)$ then $\\\\langle a_i,b_j \\\\rangle =\\\\langle u_i, v_j \\\\rangle $ for $1\\\\le i,j \\\\le t$ .\",\n \"Thus, $D_{\\\\omega }\\\\left(U, V\\\\right) \\\\ge \\\\epsilon $ implies $D_{\\\\omega }\\\\left(A, B\\\\right) \\\\ge \\\\epsilon $ .\",\n \"Following [5] consider a sequence of constants $\\\\epsilon _1=\\\\epsilon /2$ , $\\\\epsilon _2=\\\\epsilon _1^2/2$ , $\\\\epsilon _3=\\\\epsilon _2^2/2$ , $\\\\cdots $ and a sequence of sets $A_1=A \\\\cap \\\\textnormal {Spec}_{\\\\epsilon _1}\\\\left(B\\\\right)$ and $A_{i} \\\\subseteq \\\\left(A_{i-1}-A_{i-1}\\\\right) \\\\cap \\\\textnormal {Spec}_{\\\\epsilon _i}\\\\left(B\\\\right)$ .\",\n \"The way the subsets are chosen for $A_i$ 's will be made precise shortly.\",\n \"Now by the pigeonhole principle, there exists a minimal index $\\\\ell \\\\le r/\\\\log _m K$ such that $|A_{\\\\ell +1}| \\\\le K|A_{\\\\ell }|$ .\",\n \"To give a precise definition of the $A_i$ 's , we have the following.\",\n \"Let $A_1=A \\\\cap \\\\textnormal {Spec}_{\\\\epsilon /2}\\\\left(B\\\\right)$ .\",\n \"For $i\\\\ge 2$ , assuming $\\\\epsilon _{i-1}$ and $A_{i-1}$ , let $j_i$ be the the integer index which maximizes the size of $\\\\lbrace \\\\left(a,a^{\\\\prime }\\\\right) \\\\in A_{i-1}\\\\times A_{i-1}|a-a^{\\\\prime } \\\\in \\\\textnormal {Spec}_{\\\\epsilon _i}\\\\left(B\\\\right) \\\\ and \\\\ m^{j_i}\\\\le rep_{A_{i-1}}\\\\left(a-a^{\\\\prime }\\\\right) \\\\le m^{j_i+1}\\\\rbrace ,$ and let $A_i=\\\\lbrace a-a^{\\\\prime } | a,a^{\\\\prime } \\\\in A_{i-1}, a-a^{\\\\prime } \\\\in \\\\textnormal {Spec}_{\\\\epsilon _i}\\\\left(B\\\\right) \\\\ and \\\\ m^{j_i}\\\\le rep_{A_{i-1}}\\\\left(a-a^{\\\\prime }\\\\right) \\\\le m^{j_i+1}\\\\rbrace .$ Claim 7.10 For $i=1$ we have $|A_1| \\\\ge \\\\left(\\\\epsilon ̨/2\\\\right)|A|$ .\",\n \"For $i>1$ we have $\\\\mathbf {Pr}_{a,a^{\\\\prime } \\\\in A_{i-1}}[a-a^{\\\\prime } \\\\in A_i] \\\\ge \\\\epsilon _i/r$ and additionally $|A_i| \\\\ge \\\\frac{\\\\epsilon _i}{m^{j_i+1}r}|A_{i-1}|^2$ .\",\n \"The case of $i=1$ follows from Markov inequality.\",\n \"For larger $i$ , we show that $\\\\mathbf {Pr}_{a,a^{\\\\prime } \\\\in A_{i-1}}[a-a^{\\\\prime } \\\\in \\\\textnormal {Spec}_{\\\\epsilon _i}\\\\left(B\\\\right)] \\\\ge \\\\epsilon _i.$ This follows from the fact that $\\\\epsilon _{i-1}^2 \\\\le \\\\left|\\\\mathbb {E}_{b \\\\in B, a \\\\in A_{i-1}}\\\\left[\\\\omega ^{\\\\langle a,b \\\\rangle }\\\\right]\\\\right|^2 \\\\le \\\\mathbb {E}_{b \\\\in B}\\\\left|\\\\mathbb {E}_{a \\\\in A_{i-1}}\\\\left[\\\\omega ^{\\\\langle a,b \\\\rangle }\\\\right]\\\\right|^2 = \\\\mathbb {E}_{a,a^{\\\\prime }\\\\in A_{i-1}}\\\\mathbb {E}_{b \\\\in B}\\\\left[\\\\omega ^{\\\\langle a-a^{\\\\prime },b \\\\rangle }\\\\right].$ Now applying Markov inequality we get that $\\\\mathbf {Pr}_{a,a^{\\\\prime } \\\\in A_{i-1}}[a-a^{\\\\prime } \\\\in \\\\textnormal {Spec}_{\\\\epsilon _i}(B)] \\\\ge \\\\epsilon _i=\\\\epsilon _{i-1}^2/2$ .\",\n \"Now selecting $j_i$ as in the construction gives that $\\\\mathbf {Pr}_{a,a^{\\\\prime } \\\\in A_{i-1}}[a-a^{\\\\prime } \\\\in A_i] \\\\ge \\\\epsilon _i/r$ .\",\n \"To prove the second part of the lemma, observe that by the above, we have shown that $\\\\left|\\\\lbrace \\\\left(a,a^{\\\\prime }\\\\right) \\\\in A_{i-1}\\\\times A_{i-1}|a-a^{\\\\prime } \\\\in A_i\\\\rbrace \\\\right| \\\\ge \\\\frac{\\\\epsilon _i}{r}|A_{i-1}|^2.$ Also, by construction of $A_i$ , since every $x \\\\in A_i$ can be represented as $x=a-a^{\\\\prime }$ with $a,a^{\\\\prime } \\\\in A_{i-1}$ in at most $m^{j_i+1}$ ways, we have that $|A_i| \\\\ge \\\\frac{\\\\epsilon _i}{m^{j_i+1}r}\\\\left|A_{i-1}\\\\right|^2$ .\",\n \"This completes the proof.\",\n \"Below we will use the following additive-combinatorics lemma.\",\n \"Theorem 7.11 ([4], [11]) There exists an absolute constant $c>0$ such that the following holds.\",\n \"Let $A$ be any arbitrary subset of an abelian group $G$ .\",\n \"Let $S \\\\subseteq G$ be such that $|S|\\\\le C|A|$ .\",\n \"If $\\\\mathbf {Pr}_{a,a^{\\\\prime } \\\\in A}[a-a^{\\\\prime } \\\\in S] \\\\ge 1/C$ , then there exists a subset $A^{\\\\prime } \\\\subseteq A$ such that $|A^{\\\\prime }|\\\\ge \\\\frac{|A|}{C^c}$ and $|A^{\\\\prime }-A^{\\\\prime }| \\\\le C^c|A|$ .\",\n \"Now we come to the main claim.\",\n \"Claim 7.12 For $i=\\\\ell ,\\\\ell -1, \\\\cdots 1$ there exist subsets $A_i^{\\\\prime } \\\\subseteq A_i$ , $B_i^{\\\\prime } \\\\subseteq B$ such that $D_{\\\\omega }\\\\left(A_i^{\\\\prime },B_i^{\\\\prime }\\\\right)=1$ and $|A_i^{\\\\prime }|\\\\ge \\\\alpha _i|A_i|$ and $|B_i^{\\\\prime }|\\\\ge \\\\beta _i|B|$ where $\\\\alpha _i=poly_m\\\\left(\\\\frac{\\\\epsilon _{\\\\ell +1}}{rK}\\\\right)\\\\left(2mr\\\\right)^{-\\\\left(\\\\ell -i\\\\right)}\\\\left(\\\\prod _{j=i}^{\\\\ell }\\\\epsilon _{j+1}\\\\right), \\\\beta _i=poly_m\\\\left(\\\\frac{\\\\epsilon _{\\\\ell +1}}{rK}\\\\right)m^{-\\\\left(\\\\ell -i\\\\right)}$ Base Case.\",\n \"The base case of $i=\\\\ell $ is proved by an application of the Balog-Szemeredi-Gowers theorem followed by Conjecture REF followed by Lemma REF .\",\n \"To see this, we know that $|A_{\\\\ell +1}| \\\\le K|A_{\\\\ell }|$ and $\\\\mathbf {Pr}_{a,a^{\\\\prime } \\\\in A_{\\\\ell }}[a-a^{\\\\prime } \\\\in A_{\\\\ell +1}] \\\\ge \\\\epsilon _{\\\\ell +1}/r$ .\",\n \"Hence by Theorem REF (with $C=\\\\frac{rK}{\\\\epsilon _{\\\\ell +1}}$ ), there exists a set $A_{\\\\ell }^{\\\\prime \\\\prime }\\\\subseteq A_{\\\\ell }$ such that $|A_{\\\\ell }^{\\\\prime \\\\prime }| \\\\ge poly\\\\left(\\\\frac{\\\\epsilon _{\\\\ell +1}}{rK}\\\\right)\\\\left|A_{\\\\ell }\\\\right|$ and $|A_{\\\\ell }^{\\\\prime \\\\prime } - A_{\\\\ell }^{\\\\prime \\\\prime }|\\\\le poly\\\\left(\\\\frac{rK}{\\\\epsilon _{\\\\ell +1}}\\\\right)\\\\left|A_{\\\\ell }^{\\\\prime \\\\prime }\\\\right|$ .\",\n \"Now by Conjecture REF applied to $A_{\\\\ell }^{\\\\prime \\\\prime }$ , there exists a set $A_{\\\\ell }^{\\\\prime \\\\prime \\\\prime }\\\\subseteq A_{\\\\ell }^{\\\\prime \\\\prime }$ such that $\\\\left|A_{\\\\ell }^{\\\\prime \\\\prime \\\\prime }\\\\right| \\\\ge poly_m\\\\left(\\\\frac{\\\\epsilon _{\\\\ell +1}}{rK}\\\\right)\\\\left|A_{\\\\ell }^{\\\\prime \\\\prime }\\\\right|$ and $\\\\left|\\\\textnormal {span}\\\\left(A_{\\\\ell }^{\\\\prime \\\\prime \\\\prime }\\\\right)\\\\right|\\\\le m|A_{\\\\ell }^{\\\\prime \\\\prime }| = poly_m\\\\left(\\\\frac{rK}{\\\\epsilon _{\\\\ell +1}}\\\\right)\\\\left|A_{\\\\ell }^{\\\\prime \\\\prime \\\\prime }\\\\right|$ .\",\n \"(Note the extra factor of $m$ in front of $|A_{\\\\ell }^{\\\\prime \\\\prime }|$ as we get a coset of size $|A_{\\\\ell }^{\\\\prime \\\\prime }|$ and its span incurs an additional factor of $m$ ) Also, as $A_{\\\\ell }^{\\\\prime \\\\prime \\\\prime } \\\\in \\\\textnormal {Spec}_{\\\\epsilon _{\\\\ell }}\\\\left(B\\\\right)$ , applying Lemma REF to $A_{\\\\ell }^{\\\\prime \\\\prime \\\\prime }$ and $B$ , we get $A_{\\\\ell }^{\\\\prime } \\\\subseteq A_{\\\\ell }^{\\\\prime \\\\prime \\\\prime }$ and $B_{\\\\ell }^{\\\\prime } \\\\subseteq B$ such that $D_{\\\\omega }\\\\left(A_{\\\\ell }^{\\\\prime },B_{\\\\ell }^{\\\\prime }\\\\right)=1$ , $|A_{\\\\ell }^{\\\\prime }|\\\\ge poly_m\\\\left(\\\\frac{\\\\epsilon _{{\\\\ell }+1}}{rK}\\\\right)\\\\left|A_{\\\\ell }\\\\right|$ and $|B_{\\\\ell }^{\\\\prime }|\\\\ge poly_m\\\\left(\\\\frac{\\\\epsilon _{\\\\ell +1}}{rK}\\\\right)\\\\left|B\\\\right|$ .\",\n \"This completes the base case.\",\n \"Let us come to the inductive case.\",\n \"$\\\\Box $ Inductive Case.\",\n \"Suppose the statement is true for $i$ and let us argue for $i-1$ .\",\n \"Let $G=\\\\left(A_{i-1},E\\\\right)$ be the graph whose vertices are the elements in $A_{i-1}$ and $\\\\left(a,a^{\\\\prime }\\\\right)$ is an edge if $a-a^{\\\\prime } \\\\in A_i^{\\\\prime }$ .\",\n \"Now, $|E|&\\\\ge &m^{j_i}|A_i^{\\\\prime }|\\\\\\\\&\\\\ge & m^{j_i}\\\\alpha _i|A_i| \\\\ \\\\ \\\\text{(inductive hypothesis)}\\\\\\\\&\\\\ge & m^{j_i}\\\\alpha _i\\\\frac{\\\\epsilon _i}{m^{j_i+1}r}|A_{i-1}|^2 \\\\ \\\\ \\\\text{(Claim \\\\ref {clm-claim24})}\\\\\\\\&=&2\\\\alpha _{i-1}|A_{i-1}|^2$ Now the graph has at least $2\\\\alpha _{i-1}|A_{i-1}|^2$ edges and $|A_{i-1}|$ vertices and therefore has a connected component of size at least $2\\\\alpha _{i-1}|A_{i-1}|$ vertices.\",\n \"Let us call these vertices $A_{i-1}^{\\\\prime \\\\prime }$ .\",\n \"Let $\\\\tilde{a}$ be any element of $A_{i-1}^{\\\\prime \\\\prime }$ .\",\n \"Partition $B_i^{\\\\prime }$ into $B_{i,j}^{\\\\prime }$ for $0 \\\\le j \\\\le m-1$ such that all elements of $B_{i,j}^{\\\\prime }$ have inner product $j$ with $\\\\tilde{a}$ .\",\n \"Let $B_{i-1}^{\\\\prime }=B_{i,j_1}$ be the largest of them.\",\n \"Note that $|B_{i-1}^{\\\\prime }| \\\\ge |B_i^{\\\\prime }|/m$ .\",\n \"By assumption $D_{\\\\omega }\\\\left(A_i^{\\\\prime },B_i^{\\\\prime }\\\\right)=1$ .\",\n \"Hence, $D_{\\\\omega }\\\\left(A_i^{\\\\prime },B_{i-1}^{\\\\prime }\\\\right)=1$ .\",\n \"Therefore, for some $j_2$ , $\\\\langle a,b \\\\rangle = j_2$ for all $a \\\\in A_i^{\\\\prime }$ and $b \\\\in B_{i-1}^{\\\\prime }$ .\",\n \"Now, in the connected component obtained above, whenever $a,a^{\\\\prime } \\\\in A_{i-1}^{\\\\prime \\\\prime }$ are neighbours, $\\\\langle a-a^{\\\\prime },b \\\\rangle =j_2$ for $b \\\\in B_{i-1}^{\\\\prime }$ .\",\n \"Thus, starting with $\\\\tilde{a}$ as the anchor and propagating throughout the connected component, we can classify the vertices in $A_{i-1}^{\\\\prime \\\\prime }$ based on the inner product it has with all elements in $B_{i-1}^{\\\\prime }$ , which is either $j_1$ or $j_2-j_1$ .\",\n \"Pick the larger set and call it $A_{i-1}^{\\\\prime }$ .\",\n \"Hence, $D_{\\\\omega }\\\\left(A_{i-1}^{\\\\prime },B_{i-1}^{\\\\prime }\\\\right)=1$ .\",\n \"Thus, $|A_{i-1}^{\\\\prime }|\\\\ge |A_{i-1}^{\\\\prime \\\\prime }|/2 \\\\ge \\\\alpha _{i-1}|A_{i-1}|$ and $B_{i-1}^{\\\\prime }\\\\ge |B_i^{\\\\prime }|/m \\\\ge \\\\frac{\\\\beta _i}{m}|B|= \\\\beta _{i-1}|B|$ .\",\n \"This completes the inductive case.\",\n \"$\\\\Box $ Put $i=1$ in the above claim.\",\n \"Also observe that as $\\\\epsilon _{j+1}=\\\\epsilon ^{2^j}/2^{2^j-1}\\\\ge \\\\left(\\\\epsilon /2\\\\right)^{2^j}$ .\",\n \"Thus, $\\\\epsilon _{\\\\ell +1}\\\\ge \\\\left(\\\\epsilon /2\\\\right)^{2^{\\\\ell }}$ and $\\\\prod _{j=1}^{\\\\ell }\\\\epsilon _{j+1}\\\\ge \\\\left(\\\\epsilon /2\\\\right)^{2^{\\\\ell +1}}$ there exist $A^{\\\\prime } \\\\subseteq A_1 \\\\subseteq A$ , $B^{\\\\prime } \\\\subseteq B$ , such that $|A^{\\\\prime }| \\\\ge poly\\\\left(\\\\frac{\\\\left(\\\\epsilon /2\\\\right)^{2^{\\\\ell }}}{rK}\\\\right)\\\\left(2mr\\\\right)^{-\\\\ell }\\\\left|A\\\\right|$ and $ |B^{\\\\prime }| \\\\ge poly\\\\left(\\\\frac{\\\\left(\\\\epsilon /2\\\\right)^{2^{\\\\ell }}}{rK}\\\\right)m^{-\\\\ell }\\\\left|B\\\\right|$ .\",\n \"Observing that the lower bounds grow weaker with increasing $\\\\ell $ ,and that $\\\\ell \\\\le \\\\ell ^{\\\\prime }=r/\\\\log _m K$ we get $|A^{\\\\prime }| \\\\ge poly\\\\left(\\\\frac{\\\\left(\\\\epsilon /2\\\\right)^{2^{\\\\ell ^{\\\\prime }}}}{rK}\\\\right)\\\\left(2mr\\\\right)^{-\\\\ell ^{\\\\prime }}\\\\left|A\\\\right|$ and $|B^{\\\\prime }| \\\\ge poly\\\\left(\\\\frac{\\\\left(\\\\epsilon /2\\\\right)^{2^{\\\\ell ^{\\\\prime }}}}{rK}\\\\right)m^{-\\\\ell ^{\\\\prime }}\\\\left|B\\\\right|$ where $\\\\ell ^{\\\\prime }=r/\\\\log _m K$ .\",\n \"Therefore, if we take the list $U^{\\\\prime }\\\\subseteq U$ (corresponding to $A^{\\\\prime }\\\\subseteq A$ ) and $V^{\\\\prime }\\\\subseteq V$ (corresponding to $B^{\\\\prime }\\\\subseteq B$ ) then as $\\\\langle a_i,b_j \\\\rangle =\\\\langle u_i, v_j \\\\rangle $ the statement of the lemma follows.\",\n \"This completes the proof of Lemma REF We can now prove the Sub-Matrix Lemma, Lemma REF .\"\n ],\n [\n \"Proof of Lemma \",\n \"Set $K=s^{4r/\\\\log r}, \\\\ell =\\\\frac{\\\\log r}{4}, \\\\epsilon =1/2m^{3/2}$ while applying Lemma REF over $\\\\mathbb {Z}_s$ .\",\n \"We get $|A^{\\\\prime }|\\\\ge \\\\delta _s |A|$ , $|B^{\\\\prime }|\\\\ge \\\\delta _s |B|$ where $\\\\delta _s&=&poly_s\\\\left(\\\\frac{1}{m^{r^{1/4}}}\\\\right)2^{-c_1\\\\left(s\\\\right)r/\\\\log r}\\\\ \\\\ \\\\text{(for some constant $c_1\\\\left(s\\\\right)$ depending only on $s$)}\\\\\\\\&\\\\ge & poly_m\\\\left(\\\\frac{1}{m^{r^{1/4}}}\\\\right)2^{-c_1\\\\left(s\\\\right)r/\\\\log r}$ Now let $c_2\\\\left(m\\\\right)=\\\\max _{s|m, s\\\\ge 2}\\\\lbrace c_1\\\\left(s\\\\right)\\\\rbrace $ .\",\n \"Thus, $\\\\delta _s \\\\ge poly_m\\\\left(\\\\frac{1}{m^{r^{1/4}}}\\\\right)2^{-c_2\\\\left(m\\\\right)r/\\\\log r}\\\\ge 2^{-c\\\\left(m\\\\right)r/\\\\log r}$ for some constant $c$ that depends only on $m$ .\",\n \"$\\\\Box $\"\n ],\n [\n \"A Calculation\",\n \"Claim A.1 Let $b>1, n \\\\ge 2$ be arbitrary integers.\",\n \"Then $\\\\sum _{i=1}^{\\\\lfloor \\\\log _b n\\\\rfloor }\\\\frac{1}{b^{i-1}\\\\log \\\\left(n/b^{i-1}\\\\right)}\\\\le f(b)/\\\\log n$ where $f(b)=\\\\frac{10b}{b-1}+\\\\frac{10}{\\\\log b}+\\\\frac{16e}{\\\\log ^2b}$ .\",\n \"When $b=4/3$ , $f(b)<300$ .\",\n \"We divide the summation into two parts.\",\n \"The first part consists of the first $\\\\lfloor \\\\log _b \\\\log n \\\\rfloor $ terms and the second part consists of the remaining terms.\",\n \"In the first part, $\\\\frac{1}{b^{i-1}\\\\log n/b^{i-1}}\\\\le \\\\frac{1}{b^{i-1}0.1\\\\log n}$ whenever $n \\\\ge 2$ and hence the first part summation is bounded from above by $\\\\frac{10b}{(b-1)\\\\log n}$ .\",\n \"In the second part of the summation, we use the monotonicity of $x \\\\log \\\\left(n/x\\\\right)$ .\",\n \"The function increases with $x$ as long as $x\\\\le n/e$ .\",\n \"Therefore, for terms with $b^{i-1}\\\\le n/e$ , the maximum value of each summand is given by substituting $i=\\\\log _b\\\\log n$ which gives an upper bound of $\\\\frac{1}{0.1\\\\log ^2 n}$ .\",\n \"The remaining terms corresponding to $n/b \\\\ge b^{i-1} > n/e$ (note that these extra terms arise only if $b1, n \\\\ge 2$ be arbitrary integers.\",\n \"Then $\\\\sum _{i=1}^{\\\\lfloor \\\\log _b n\\\\rfloor }\\\\frac{1}{b^{i-1}\\\\log \\\\left(n/b^{i-1}\\\\right)}\\\\le f(b)/\\\\log n$ where $f(b)=\\\\frac{10b}{b-1}+\\\\frac{10}{\\\\log b}+\\\\frac{16e}{\\\\log ^2b}$ .\",\n \"When $b=4/3$ , $f(b)<300$ .\",\n \"We divide the summation into two parts.\",\n \"The first part consists of the first $\\\\lfloor \\\\log _b \\\\log n \\\\rfloor $ terms and the second part consists of the remaining terms.\",\n \"In the first part, $\\\\frac{1}{b^{i-1}\\\\log n/b^{i-1}}\\\\le \\\\frac{1}{b^{i-1}0.1\\\\log n}$ whenever $n \\\\ge 2$ and hence the first part summation is bounded from above by $\\\\frac{10b}{(b-1)\\\\log n}$ .\",\n \"In the second part of the summation, we use the monotonicity of $x \\\\log \\\\left(n/x\\\\right)$ .\",\n \"The function increases with $x$ as long as $x\\\\le n/e$ .\",\n \"Therefore, for terms with $b^{i-1}\\\\le n/e$ , the maximum value of each summand is given by substituting $i=\\\\log _b\\\\log n$ which gives an upper bound of $\\\\frac{1}{0.1\\\\log ^2 n}$ .\",\n \"The remaining terms corresponding to $n/b \\\\ge b^{i-1} > n/e$ (note that these extra terms arise only if $b [dll]> 2** , where $_{{1}}$ and $_{{2}}$ have the same state space, call it $S_$ .","Likewise, $_{{1}}$ and $_{{2}}$ have the same state space, call it $S_$ .","Note that the subscripts 1 and 2 are acting like a “time\" index along a sort of timeline.","For all $a,a^{\\prime }\\in S_$ and $b,b^{\\prime }\\in S_$ , l A2|1,1(a|a',b')= aa' A2|1,1(b|a',b')= bb' .","Thus some column vectors of a marginalizer node are equal to each other.","To avoid index clutter, we will sometimes omit the indices 1 and 2 from the graph of Eq.","(), and draw instead the following graph: = =++[o][F-] ** *(,) [ull]> [dll]> ** Grounded nodes are root nodes (i.e., nodes with no incoming arrows, only outgoing ones) which have a deterministic probability amplitude (i.e., an amplitude that equals 1 for just one of the possible states of the node and zero for all the other states).","Grounded nodes will be indicated by writing a zero near them.","Here is an example of a QB net with a grounded node: =++[o][F-] **[dll] (,)** *[ull]<0 , whereWe are assuming that $0\\in S_$ .","It doesn't matter if the amplitude $A(b)$ of node $$ equals $\\delta _b^0$ , or $\\delta _b^{b_0}$ where $b_0\\in S_$ .","Either way, it's still a grounded node.","An alternative to writing a zero next to a node to indicate that it's grounded might be writing instead the letters “grd\" or the electrical symbol for a ground.","A(b)= b0 , for all $b\\in S_$ ."],["Isometries","Consider the following QB net =++[o][F-] [l] , where $A(b|a)$ satisfies bSA(b|a) c aa' = aa' , for all $a,a^{\\prime }\\in S_$ .","The node $$ in the above QB net is called an isometry node, or just an isometry for simplicity.","Eq.","() is saying that the column vectors of the transition matrix $A(b|a)$ are orthonormal.","This is only possible if $N_\\ge N_$ .","If $N_= N_$ (i.e., transition matrix is square), then the transition matrix is unitary.","If $N_> N_$ (i.e., transition matrix is rectangular with more rows than columns), then we can use the well-known, so called Gram-Schmidt procedure to add more columns to the transition matrix (“extend it\") to produce a unitary matrix.","Since $N_\\ge N_$ and the sets $S_$ , $S_$ are finite, we may assume without loss of generality that $S_\\supset S_$ .","For every $b,A\\in S_$ , letWe are using the symbol $A$ both for an element $A$ of $S_$ , and for amplitudes $A(\\cdot )$ .","It's easy to tell which usage is intended in each instance, so this should cause no confusion.","A(b|A) ={ lA(b|a) ifA=aS given by Gram-Schmidt procedureifAS-S. .","Then bS A(b|A) c AA' =AA' for all $A,A^{\\prime }\\in S_$ .","Thus, A(b|A)= bUA , where $U_{}$ is unitary.","Here is a pictorial representation, in terms of QB nets, of the procedure just outlined for extending an isometry to a unitary matrix: =++[o][F-] [l] =++[o][F-] A[l] , where $S_=S_{A}\\supset S_$ .","Consider the following QB net =++[o][F-] (,)[l] , where $A(a,b|a^{\\prime })$ satisfies aS bSA(a,b|a') c a'a” = a'a” , for all $a^{\\prime },a^{\\prime \\prime }\\in S_$ .","The node $(, )$ in the above QB net is a special case of the general isometry node presented previously.","Just as in the case of a general isometry, the transition matrix $A(a,b |a^{\\prime })$ can be extended to a unitary matrix.","Assume $0\\in S_$ .","If we define A(a, b|a', b'=0)= A(a,b|a') , then we can find a unitary matrix $U_{,}$ such that, for all $a,a^{\\prime }\\in S_$ and $b,b^{\\prime }\\in S_$ , A(a, b|a', b')= ccc aa' U, bb' .","Here is a pictorial representation, in terms of QB nets, of the procedure just outlined for extending an isometry to a unitary matrix: =++[o][F-] (,)[l] = =++[o][F-] **[dll] (,)** *[ull]<0 =++[o][F-] **[dll] (,)** *[ull]"],["Freeing a Bound Index","We've defined the QB net corresponding to the meta ket state as having for each node $$ : (1) an index $b$ that is summed over (bound), and (2) a ket ${b}_$ .","One can free an index $b$ of a node $$ of a meta ket state by multiplying that node by ${b}$ or ${b}{b}$ .","One can use QB nets to represent these two operations.","For example, if the meta state is, =++[o][F-] [l] = a,b A(b|a)A(a)ba, then =++[o][F-] * @[l]<>tr =b =++[o][F-] * @[l]<>cl =b =++[o][F-] *@[l]<<<> [dl]>>** * [ul]<0* , where A() = 0 , A(j) = wj , A(A|j) = {l A|jif AS 0if AS-S. , A(b,|A,')= ccc bAA U, ' , A(b|b',')=bb' , A(|b',')=' ."],["Complementary Channel","The channel superoperator $\\$(\\cdot )$ given by Eq.","() can be used to define a complementary channel superoperator $\\$^{\\prime }(\\cdot )$ .","If $\\$(\\cdot )$ is generated using a measurement $\\lbrace K_\\mu \\rbrace _{\\forall \\mu }$ , then we can find a unitary operator $U_{,{\\mu }}$ such that KA = lcr U, 0 .","Now define a measurement $\\lbrace L_b\\rbrace _{\\forall b}$ using the same unitary operator $U_{,{\\mu }}$ : Lb = lcr b 0A U, .","Then = $()= KK and = $'()= b Lb Lb .","We've already shown how density matrices $\\rho _$ and $\\sigma _=\\$(\\rho _)$ can be represented by QB nets (see Eqs.","() and ()).","Likewise, density matrices $\\rho _\\mu $ and $\\sigma _\\mu =\\$^{\\prime }(\\rho _\\mu )$ can be represented by QB nets as follows: = =++[o][F-] [l] = , =+++[o][F-] *A[dl]<0* (,) [ul]>> [dl]>> ** *[ul][l]"]],"string":"[\n [\n \"An Introduction to Quantum Bayesian Networks for Mixed States\"\n ],\n [\n \"Abstract This paper is intended to be a pedagogical introduction to quantum Bayesian networks (QB nets), as I personally use them to represent mixed states (i.e., density matrices, and open quantum systems).\",\n \"A special effort is made to make contact with notions used in textbooks on quantum Shannon Information Theory (quantum SIT), such as the one by Mark Wilde (arXiv:1106.1445)\"\n ],\n [\n \"Abstract\",\n \"This paper is intended to be a pedagogical introduction to quantum Bayesian networks (QB nets), as I personally use them to represent mixed states (i.e., density matrices, and open quantum systems).\",\n \"A special effort is made to make contact with notions used in textbooks on quantum Shannon Information Theory (quantum SIT), such as the one by Mark Wilde (arXiv:1106.1445)\"\n ],\n [\n \"Introduction\",\n \"This paper is intended to be a pedagogical introduction to quantum Bayesian networks (QB nets), as I personally use them to represent mixed states (i.e., density matrices, and open quantum systems).\",\n \"A special effort is made to make contact with notions used in textbooks on quantum Shannon Information Theory (quantum SIT), such as the one by Mark Wilde[1].\",\n \"QB nets are a generalization of classical Bayesian networks (CB nets) to quantum mechanics.\",\n \"CB nets have been hot topic in AI circles since the seminal work of Judea Pearl and collaborators that started in the 1980's.\",\n \"A very complete book on CB nets is the one by Koller and Friedman[2].\",\n \"Just like mankind has devised many names for the idea of God, there are other names for CB nets and variations of the idea.\",\n \"Others have called them causal probabilistic diagrams, factor graphs, probabilistic system diagrams, etc.\",\n \"To be sure, there are some differences between some of these diagrams and CB nets, but all seem to be striving to conjure up the same divine concept.\",\n \"Some of the close siblings of CB nets are discussed in Ref.\",\n \"[3], an IEEE magazine article by Loeliger.\",\n \"One variation on the CB net idea involves using graphs in which the arrows (a.k.a.\",\n \"directed edges) represent tensor indices and the boxes (a.k.a.\",\n \"nodes, vertices) represent transition matrices.\",\n \"In this approach, call it the tensor-graphs approach, each arrow coming out of a fixed node carries different stuff.\",\n \"In the CB nets approach, the nodes again represent transition matrices, but the arrows perform a very different job.\",\n \"Each arrow coming out of a fixed node carries the same stuff, namely the name of the node the arrow originates at.\",\n \"This difference might appear subtle or even insignificant to the untrained eye, but Bayesian network believers (like me) swear by it, claiming that it is clearer and more powerful than the tensor-graphs approach when dealing with probabilities.\",\n \"Bayesian network believers think that using tensor graphs to describe probability networks is like trying to fill round holes with square pegs.\",\n \"Even when the pegs fit, they don't do a very good job.\",\n \"Classical information theorists have been using tensor-graphs in their field for a long time.\",\n \"See, for example, the chapter on “network information theory\\\" in the book by Cover and Thomas, Ref.[4].\",\n \"Or look at the book by El Gamal and Kim, Ref.\",\n \"[5], which is devoted exclusively to the subject of network information theory.\",\n \"Quantum information theorists have been using tensor-graphs in their field for a long time too, at least since the seminal work by Schumacher and collaborators.\",\n \"For an early example of a paper by Schumacher that uses tensor-graphs, see, for example, Ref.\",\n \"[6], published in 1996.\",\n \"Some early quantum information papers, for example the seminal paper Ref.\",\n \"[7] by Bennett et al.\",\n \"also use a version of tensor-graphs, but they use them in a very loose, ambiguous, imprecise way.\",\n \"Ref.\",\n \"[7] even has some diagrams that sound like sacrilege to the ears of this Bayesian nets believer, such as diagrams that have nodes intended to represent buckets of sewage (Fig.13 in Ref.[7]).\",\n \"Not only do adherents to “the QB net way\\\" espouse being good parents by treating all arrows (= children) coming out of a fixed node (= parent) the same.\",\n \"We are also very strict about our nodes.\",\n \"Each node has a numerical “value\\\", and the whole graph also has a value which equals the product of the node values.\",\n \"Nodes aren't there just for decoration, or as mere labels with no numerical value assigned to them, or only to convey an abstract notion like “do this operation now\\\".\",\n \"Many quantum information papers don't use diagrams at all.\",\n \"They specify their quantum “protocols\\\" or algorithms in terms of “pseudo code\\\".\",\n \"In my opinion, those papers would be much clearer if they described their algorithms using both, pseudo-code and QB nets, whenever this is possible.\",\n \"The recent book on quantum information theory by Mark Wilde[1] earns high marks in this regard, as it uses a lot of diagrams.\",\n \"Wilde's diagrams also have a fairly precise meaning.\",\n \"However, they are tensor-graphs, not the wonderful QB nets.\",\n \"My own work on QB nets started about 15 years ago with Ref.[8].\",\n \"In that early paper, I dealt only with QB nets for pure quantum states.\",\n \"I've been using QB nets for mixed states since at least Ref.[9].\",\n \"This paper repeats some of the ideas of Ref.\",\n \"[9] for QB nets of mixed states, with (hopefully) some small improvements.\",\n \"I've also written a Mac application that does QB nets called Quantum Fog.I stopped Quantum Fog development in 2006.\",\n \"The application is still available for free at Ref.[10].\",\n \"Quantum Fog's last version (Version 2.0) is known to work with Mac OS X $\\\\le $ 10.4.\",\n \"It probably works with some higher versions of Mac OS X too.\",\n \"Quantum Fog only does QB nets for pure states.\",\n \"That's not because QB nets can't deal with mixed states as some people think.\",\n \"It's only because I stopped developing Quantum Fog before I had a chance to add to it the capability to do mixed state calculations.\",\n \"I also have a blog called Quantum Bayesian Networks (Ref.\",\n \"[11]) in which I regularly post articles about Bayesian networks and quantum computing.\",\n \"Subsequent to Refs.\",\n \"[8], [9], other workers have devised their own types of diagrams for doing quantum information theory.\",\n \"Their diagrams are very different from the QB nets in this paper.\",\n \"Leifer and Poulin in Ref.\",\n \"[12], and later Leifer and Spekkens in Ref.\",\n \"[13], postulate some directed acyclic graphs, but they assign a whole density matrix to each node of the graph.\",\n \"Furthermore, their node density matrices would be quite hard to calculate in practice, especially for complicated graphs.\",\n \"In comparison, a whole QB net is used to describe one density matrix.\",\n \"And the transition matrix that a QB net assigns to each node comes from the definition of a probability amplitude, a really basic thing that requires almost no calculation— certainly less calculation than the node density matrices of Leifer and coworkers.\",\n \"I like to think of QB nets as: A light container of data, useful as a data structure in computer programming.\",\n \"A vehicle rather than a destination.\",\n \"A transparent, tidy way of organizing a lot of data in a pictorial way, prior to intensive calculation, not as being itself the outcome of a major calculation.\",\n \"Coecke (Ref.\",\n \"[14]) and collaborators use category theory to define their diagrams.\",\n \"In comparison, QB nets are much less abstract.\",\n \"Defining them requires no category theory, just standard, run-of-the-mill quantum mechanics.\",\n \"The QB nets in this paper are much simpler than the diagrams of (a) and (b).\",\n \"Simplicity can be a virtue in mathematics (consider, for example, abstract algebra's definition of a group, which is simplicity itself).\",\n \"QB nets are, however, complicated enough to be very expressive and useful; that is, they allow one to express numerous quantum mechanical concepts in a useful, practical and enlightening way.\",\n \"QB nets are a very parsimonious extension of CB nets to quantum mechanics.\",\n \"That is, the definition of QB nets is the smallest possible modification of the definition of CB nets that I can come up with, but enough of a modification so that one can do proper quantum mechanics with them.\",\n \"Keeping QB nets close to CB nets can be very fruitful, because much is already known about CB nets.\",\n \"And CB nets use classical probability so we can sharpen our classical understanding of a problem with them and then try to extrapolate that understanding to QB nets.\",\n \"QB nets retain the same structure as CB nets and can be reduced to them very easily, simply by applying the dephasing operator “cl\\\" (defined below) to each node.\",\n \"Thus, QB nets make the connection to the classical case very direct and explicit.\"\n ],\n [\n \"Basic Notation\",\n \"As usual, $,, $ will denote the integers, real numbers, and complex numbers, respectively.\",\n \"For $a,b\\\\in $ such that $a\\\\le b$ , let $Z_{a,b}=\\\\lbrace a,a+1,a+2,\\\\ldots , b\\\\rbrace $ .\",\n \"Let $\\\\delta ^x_{y}=\\\\delta (x,y)$ denote the Kronecker delta function; it equals 1 if $x=y$ and 0 if $x\\\\ne y$ .\",\n \"For any matrix $M\\\\in ^{p\\\\times q}$ , $M^*$ will denote its complex conjugate, $M^T$ its transpose, and $M^\\\\dagger = M^{*T}$ its Hermitian conjugate.\",\n \"Random variablesWe will use the term “random variables\\\" in both classical and quantum physics.\",\n \"Normally, random variables are defined only in classical physics, where they are defined to be functions from an outcome space to a range of values.\",\n \"For technical simplicity, here we define a random variable $$ , in both classical and quantum physics, to be merely the label of a node in a graph, or an n-tuple $_K$ of such labels.\",\n \"Each node or random variable of a CB or QB net is akin to a spacetime event or a collection of them.\",\n \"will be denoted by underlined letters; e.g., $$ .\",\n \"The (finite) set of values (states) that $$ can assume will be denoted by $S_$ .\",\n \"Let $N_=|S_|$ .\",\n \"The probability that $=a$ will be denoted by $P(=a)$ or $P_(a)$ , or simply by $P(a)$ if the latter will not lead to confusion in the context it is being used.\",\n \"We will use $pd(S_)$ to denote the set of all probability distributions with domain $S_$ .\",\n \"In quantum physics, $$ has a fixed, orthonormal basis $\\\\lbrace {a}_:a\\\\in S_\\\\rbrace $ associated with it.\",\n \"The vector space spanned by this basis will be denoted by $_$ .\",\n \"Other spans of $_$ that are not necessarily orthonormal will be denoted by Greek letters with subscripts as in $\\\\lbrace {\\\\psi _j}_\\\\rbrace _{\\\\forall j}$ .\",\n \"In quantum physics, instead of probabilities $P(=a)$ , we use “probability amplitudes\\\" (or just “amplitudes\\\" for short) $A(=a)$ (also denoted by $A_(a)$ or $A(a)$ ).\",\n \"In place of $P(a)\\\\ge 0$ and $\\\\sum _a P(a)=1$ , one has $\\\\sum _a |A(a)|^2=1$ .\",\n \"Besides probability amplitudes, we also use density matrices.\",\n \"A density matrix $\\\\rho _$ is a Hermitian, non-negative, unit trace, square matrix (or the associated linear operator) acting on $_$ .\",\n \"We will use $dm(_)$ to denote the set of all density matrices acting on $_$ .\",\n \"If $\\\\rho _{,}\\\\in dm(_{,})$ , and $\\\\rho _= _(\\\\rho _{,})= \\\\sum _a {a}_\\\\rho _{,}{a}_\\\\in dm(_)$ , we will say that $\\\\rho _$ is a partial trace of $\\\\rho _{,}$ .\",\n \"Given a density matrix $\\\\rho _{_1,_2,_3,\\\\ldots }\\\\in dm(_{_1,_2,_3,\\\\ldots })$ , its partial traces will be denoted by omitting its subscripts for the random variables that have been traced over.\",\n \"For example, $\\\\rho _{_2} = tr_{_1,_3}\\\\rho _{_1,_2,_3}$ .\",\n \"Sometimes, when two random variables 1 and 2 satisfy $S_{{1}}=S_{{2}}$ , we will omit the indices 1 and 2 and refer to both random variables as $$ .\",\n \"We shall do this sometimes even if the random variables 1 and 2 are not identically distributed!\",\n \"This notation, if used with caution, does not lead to confusion and does avoid a lot of index clutter.\",\n \"When we want to make explicit that an operator $\\\\Omega $ maps states in $_$ to states in $_$ , we will indicate this with a subscript (or superscript) as $\\\\Omega _{\\\\leftarrow }$ or as $\\\\Omega _{|}$ .\",\n \"In cases where $S_=S_$ , we will sometimes write $\\\\Omega _{}$ instead of the clearer but longer $\\\\Omega _{\\\\leftarrow }$ or $\\\\Omega _{|}$ .\",\n \"The tensor product symbol $\\\\otimes $ will often be omitted.\",\n \"Sometimes, when two vectors are being tensored, we will list the two vectors vertically instead of horizontally (the latter is more common in the literature).\",\n \"For example, we might write = = c .\",\n \"This doesn't lead to confusion as long as we indicate what vector space each vector lives in.\",\n \"(In the above example, ${\\\\phi }_$ clearly lives in $_$ and ${\\\\psi }_$ in $_$ ).\",\n \"In this paper, we consider networks (graphs) with $N$ nodes.\",\n \"Each node is labeled by a random variable $_j$ , where $j\\\\in $ .\",\n \"For any $J\\\\subset $ , the ordered set of random variables $_j$ $\\\\forall j\\\\in J$ (ordered so that the integer indices $j$ increase from left to right) will be denoted by $_{J}$ .\",\n \"For example, $_{\\\\lbrace 2,4\\\\rbrace }=(_2,_4)$ .\",\n \"We will often call the values that $_{J}$ can assume $x_{J}$ .\",\n \"For example, $x_{\\\\lbrace 2,4\\\\rbrace }=(x_2,x_4)$ .\",\n \"We will often abbreviate $_{}$ by just $$ .\",\n \"We will often call the values that $$ can assume $x.$ .\"\n ],\n [\n \"The Sandbox and its Dual\",\n \"For any expressions $\\\\Omega (x)$ and $\\\\rho $ for which this makes sense, we will use the shorthand notation: (x) c xx' = (x) (x') .\",\n \"Here “h.c.\\\"\",\n \"is an abbreviation of “hermitian conjugate\\\".\",\n \"We will usually use this notation with $\\\\rho =1$ .\",\n \"This notation is especially useful when $\\\\Omega (x)$ is a long expression and we want to avoid writing it twice.\",\n \"We will refer to the space inside the set of square brackets to the left (resp., right) of $\\\\rho $ as the sandbox (resp., its dual or mirror sandbox).\"\n ],\n [\n \"The Meta State\",\n \"QB nets for pure quantum states were first defined in Ref.[8].\",\n \"A QB net for a pure state consists of a directed acyclic graph (DAG) and a transition matrix (a complex matrix) assigned to each node of the graph.\",\n \"The transition matrices must satisfy certain requirements.\",\n \"An example of such a pure state QB net is: =++[o][F-] *[dl]* *[ul][ll] If $a\\\\in S_, b\\\\in S_, c\\\\in S_$ , $A_{|,}(a|b,c)$ is the transition matrix associated with node $$ , $A_{|}(b|c)$ is the transition matrix for node $$ , and $A_(c)$ is the transition matrix for node $$ .\",\n \"We must have a |A|,(a|b,c)|2=1 , b |A|(b|c)|2=1 , c |A(c)|2 = 1 .\",\n \"Define the total probability amplitude $A_{,,}(a,b,c)$ by A,,(a,b,c) = A|,(a|b,c) A|(b|c) A(c) .\",\n \"Note that Eqs.\",\n \"() imply a,b,c |A,,(a,b,c)|2 = 1 .\",\n \"Henceforth, we will sometimes omit the node subscripts from the probability amplitudes.\",\n \"For example, we might use $A(a|b,c)$ instead of $A_{|,}(a|b,c)$ , if no confusion will arise.\",\n \"This is analogous to probability theory, where we often use $P(a|b,c)$ instead of $P_{|,}(a|b,c)$ or $P(=a|=b,=c)$ for a probability.\",\n \"More generally, suppose the graph has $N$ nodes $_1,_2, \\\\ldots ,_N$ .\",\n \"For $j\\\\in $ , a node $_j$ with possible states $x_j\\\\in {_j}$ and with parent nodes $_{pa(_j)}$ where $pa(_j)\\\\subset $ , has a transition matrix $A(x_j|x_{pa(_j)})$ which satisfies xjj |A(xj|xpa(j))|2=1 for all $x_{pa(_j)}\\\\in {_{pa(_j)}}$ .\",\n \"Let $x_.=(x_1,x_2, \\\\ldots ,x_N)$ If the total amplitude $A(x.\",\n \")$ is defined by A(x.)\",\n \"= j A(xj|xpa(j)) , then Eqs.\",\n \"() imply x.|A(x.\",\n \")|2=1 .\",\n \"Given any transition matrix of the form $A(r|\\\\vec{c})$ , call $r$ the row index and $\\\\vec{c}$ the column indices.\",\n \"Call all $A(r|\\\\vec{c})$ entries with any $r$ but fixed $\\\\vec{c}$ , a column vector of the transition matrix.\",\n \"Eqs.\",\n \"() say that each column vector of a transition matrix is normalized.\",\n \"The column vectors may also be mutually orthogonal, in which case we say that the column vectors are orthonormal.\",\n \"For example, the transition matrix for node $$ in the QB net above might also satisfy: a A|,(a|b,c) c b,cb',c' = bb'cc' .\",\n \"The isometry nodes defined below are another example of a case where the column vectors of the transition matrix are orthonormal.\",\n \"In general, it is not necessary that the column vectors be orthonormal.\",\n \"For example, for the marginalizer nodes defined below, they aren't.\",\n \"What is always necessary is that the total amplitude be normalized, as in Eq.\",\n \"(), so as to enforce the “unitarity\\\" of quantum mechanics.\",\n \"The meta state of a QB net was first defined in Ref.[9].\",\n \"A meta ket state is a pure quantum state represented as a ket or as a QB net.\",\n \"For example: meta,,= =++[o][F-] *[dl]* *[ul][ll] = a,b,c A,,(a,b,c) c a b c , where $A(a,b,c)$ is defined by Eq.().\",\n \"We assume the states $\\\\lbrace {a}_\\\\rbrace _{\\\\forall a}$ are orthonormal, and likewise for $\\\\lbrace {b}_\\\\rbrace _{\\\\forall b}$ and $\\\\lbrace {c}_\\\\rbrace _{\\\\forall c}$ .\",\n \"Note that each node of the QB net has its own ket and its own index that is summed over (i.e., bound).\",\n \"For example, node $$ has ket ${b}_$ and bound index $b$ .\",\n \"The projection operator of a meta ket defines a density matrix which we will call the meta density matrix of the protocol under consideration.\",\n \"For example, the meta density matrix of the above meta ket is given by: (meta),, = meta,, = =++[o][F-] *[dl]* *[ul][ll] .\"\n ],\n [\n \"Generic Nodes\",\n \"We find it convenient to define certain special, generic types of nodes.\",\n \"Marginalizer nodes were first defined in Ref.[8].\",\n \"In the current version of Quantum Fog, marginalizer nodes are usually denoted by black bullets, whereas non-marginalizer nodes are denoted by larger colored circles.\",\n \"In this paper, we will represent marginalizer nodes by writing a small delta near them.\",\n \"This node “decoration\\\" or subscript is easy to draw by hand and also easy for the eye to spot.\",\n \"For example: = =+++++[o][F-] 2** *(1,1) [ull]> [dll]> 2** , where $_{{1}}$ and $_{{2}}$ have the same state space, call it $S_$ .\",\n \"Likewise, $_{{1}}$ and $_{{2}}$ have the same state space, call it $S_$ .\",\n \"Note that the subscripts 1 and 2 are acting like a “time\\\" index along a sort of timeline.\",\n \"For all $a,a^{\\\\prime }\\\\in S_$ and $b,b^{\\\\prime }\\\\in S_$ , l A2|1,1(a|a',b')= aa' A2|1,1(b|a',b')= bb' .\",\n \"Thus some column vectors of a marginalizer node are equal to each other.\",\n \"To avoid index clutter, we will sometimes omit the indices 1 and 2 from the graph of Eq.\",\n \"(), and draw instead the following graph: = =++[o][F-] ** *(,) [ull]> [dll]> ** Grounded nodes are root nodes (i.e., nodes with no incoming arrows, only outgoing ones) which have a deterministic probability amplitude (i.e., an amplitude that equals 1 for just one of the possible states of the node and zero for all the other states).\",\n \"Grounded nodes will be indicated by writing a zero near them.\",\n \"Here is an example of a QB net with a grounded node: =++[o][F-] **[dll] (,)** *[ull]<0 , whereWe are assuming that $0\\\\in S_$ .\",\n \"It doesn't matter if the amplitude $A(b)$ of node $$ equals $\\\\delta _b^0$ , or $\\\\delta _b^{b_0}$ where $b_0\\\\in S_$ .\",\n \"Either way, it's still a grounded node.\",\n \"An alternative to writing a zero next to a node to indicate that it's grounded might be writing instead the letters “grd\\\" or the electrical symbol for a ground.\",\n \"A(b)= b0 , for all $b\\\\in S_$ .\"\n ],\n [\n \"Isometries\",\n \"Consider the following QB net =++[o][F-] [l] , where $A(b|a)$ satisfies bSA(b|a) c aa' = aa' , for all $a,a^{\\\\prime }\\\\in S_$ .\",\n \"The node $$ in the above QB net is called an isometry node, or just an isometry for simplicity.\",\n \"Eq.\",\n \"() is saying that the column vectors of the transition matrix $A(b|a)$ are orthonormal.\",\n \"This is only possible if $N_\\\\ge N_$ .\",\n \"If $N_= N_$ (i.e., transition matrix is square), then the transition matrix is unitary.\",\n \"If $N_> N_$ (i.e., transition matrix is rectangular with more rows than columns), then we can use the well-known, so called Gram-Schmidt procedure to add more columns to the transition matrix (“extend it\\\") to produce a unitary matrix.\",\n \"Since $N_\\\\ge N_$ and the sets $S_$ , $S_$ are finite, we may assume without loss of generality that $S_\\\\supset S_$ .\",\n \"For every $b,A\\\\in S_$ , letWe are using the symbol $A$ both for an element $A$ of $S_$ , and for amplitudes $A(\\\\cdot )$ .\",\n \"It's easy to tell which usage is intended in each instance, so this should cause no confusion.\",\n \"A(b|A) ={ lA(b|a) ifA=aS given by Gram-Schmidt procedureifAS-S. .\",\n \"Then bS A(b|A) c AA' =AA' for all $A,A^{\\\\prime }\\\\in S_$ .\",\n \"Thus, A(b|A)= bUA , where $U_{}$ is unitary.\",\n \"Here is a pictorial representation, in terms of QB nets, of the procedure just outlined for extending an isometry to a unitary matrix: =++[o][F-] [l] =++[o][F-] A[l] , where $S_=S_{A}\\\\supset S_$ .\",\n \"Consider the following QB net =++[o][F-] (,)[l] , where $A(a,b|a^{\\\\prime })$ satisfies aS bSA(a,b|a') c a'a” = a'a” , for all $a^{\\\\prime },a^{\\\\prime \\\\prime }\\\\in S_$ .\",\n \"The node $(, )$ in the above QB net is a special case of the general isometry node presented previously.\",\n \"Just as in the case of a general isometry, the transition matrix $A(a,b |a^{\\\\prime })$ can be extended to a unitary matrix.\",\n \"Assume $0\\\\in S_$ .\",\n \"If we define A(a, b|a', b'=0)= A(a,b|a') , then we can find a unitary matrix $U_{,}$ such that, for all $a,a^{\\\\prime }\\\\in S_$ and $b,b^{\\\\prime }\\\\in S_$ , A(a, b|a', b')= ccc aa' U, bb' .\",\n \"Here is a pictorial representation, in terms of QB nets, of the procedure just outlined for extending an isometry to a unitary matrix: =++[o][F-] (,)[l] = =++[o][F-] **[dll] (,)** *[ull]<0 =++[o][F-] **[dll] (,)** *[ull]\"\n ],\n [\n \"Freeing a Bound Index\",\n \"We've defined the QB net corresponding to the meta ket state as having for each node $$ : (1) an index $b$ that is summed over (bound), and (2) a ket ${b}_$ .\",\n \"One can free an index $b$ of a node $$ of a meta ket state by multiplying that node by ${b}$ or ${b}{b}$ .\",\n \"One can use QB nets to represent these two operations.\",\n \"For example, if the meta state is, =++[o][F-] [l] = a,b A(b|a)A(a)ba, then =++[o][F-] * @[l]<>tr =b =++[o][F-] * @[l]<>cl =b =++[o][F-] *@[l]<<<> [dl]>>** * [ul]<0* , where A() = 0 , A(j) = wj , A(A|j) = {l A|jif AS 0if AS-S. , A(b,|A,')= ccc bAA U, ' , A(b|b',')=bb' , A(|b',')=' .\"\n ],\n [\n \"Complementary Channel\",\n \"The channel superoperator $\\\\$(\\\\cdot )$ given by Eq.\",\n \"() can be used to define a complementary channel superoperator $\\\\$^{\\\\prime }(\\\\cdot )$ .\",\n \"If $\\\\$(\\\\cdot )$ is generated using a measurement $\\\\lbrace K_\\\\mu \\\\rbrace _{\\\\forall \\\\mu }$ , then we can find a unitary operator $U_{,{\\\\mu }}$ such that KA = lcr U, 0 .\",\n \"Now define a measurement $\\\\lbrace L_b\\\\rbrace _{\\\\forall b}$ using the same unitary operator $U_{,{\\\\mu }}$ : Lb = lcr b 0A U, .\",\n \"Then = $()= KK and = $'()= b Lb Lb .\",\n \"We've already shown how density matrices $\\\\rho _$ and $\\\\sigma _=\\\\$(\\\\rho _)$ can be represented by QB nets (see Eqs.\",\n \"() and ()).\",\n \"Likewise, density matrices $\\\\rho _\\\\mu $ and $\\\\sigma _\\\\mu =\\\\$^{\\\\prime }(\\\\rho _\\\\mu )$ can be represented by QB nets as follows: = =++[o][F-] [l] = , =+++[o][F-] *A[dl]<0* (,) [ul]>> [dl]>> ** *[ul][l]\"\n ]\n]"},"id":{"kind":"string","value":"1204.1550"}}},{"rowIdx":1866797,"cells":{"text":{"kind":"list like","value":[["Electron Distribution in the Galactic Disk - Results From a\n Non-Equilibrium Ionization Model of the ISM"],["Abstract Using three-dimensional non-equilibrium ionization (NEI) hydrodynamical simulation of the interstellar medium (ISM), we study the electron density, $n_{e}$, in the Galactic disk and compare it with the values derived from dispersion measures towards pulsars with known distances located up to 200 pc on either side of the Galactic midplane.","The simulation results, consistent with observations, can be summarized as follows: (i) the DMs in the simulated disk lie between the maximum and minimum observed values, (ii) the log derived from lines of sight crossing the simulated disk follows a Gaussian distribution centered at \\mu=-1.4 with a dispersion \\sigma=0.21, thus, the Galactic midplane =0.04\\pm 0.01$ cm$^{-3}$, (iii) the highest electron concentration by mass (up to 80%) is in the thermally unstable regime (2004$ kpc, and 1 pc elsewhere.","Periodic and outflow boundary conditions are set along the vertical faces and top/bottom ($z=\\pm 15$ kpc) of the grid, respectively.","Figure: Histogram (solid black line) and Gaussian fit (dashed black line) of log〈n e 〉\\log \\langle n_{e} \\rangle distribution as obtained with a series of 24 pulsars located at 20010^{4.2}$ K gas), respectively, with the latter representing mostly photoionized regions.","Further insight into the simulated electron distribution can be obtained directly from DMs along LOS, with length $d$ , crossing the simulated ISM, allowing the determination of the mean electron density.","The LOS have increasing lengths (with a step length of 10 pc) up to 1 kpc from a vantage point located at $(x,y,z)=(0,0,0)$ (left bottom corner of the electron density maps in Figure REF ) for $\\left|z\\right|\\le 200$ pc and spanning 90 (with a 1 separation) when projected onto the Galactic midplane.","Figure: Angular distribution of the DMs along LOS with lengths varying between 100 and 1000pc at 350 and 400 Myr (top two panels) and the time averaged 〈DM〉 t \\langle DM\\rangle _{t} (bottom panel) over theevolution time 350-400 Myr).Figure REF shows the DM along the LOS with lengths varying between 100 and 1000 pc, with a step length of 100 pc at 350 and 400 Myr (top two panels) and its time average (bottom panel) over 501 disk snapshots taken every 0.1 Myr for $350-400$ Myr of evolution.","Due to the ongoing turbulent processes in the Galactic disk the DM has variability along and between the lines of sight (Figure REF : top two panels).","The longitudinal variation is due to the inhomogeineity of the ISM (e.g., bubbles and superbubbles) as well as due to the turbulent nature of the regions crossed by the LOS.","These variations with distance and among LOS are still present in the time averaged DM, although the longitudinal variations are smoothed in the averaging process.","However, a record of any event that prevailed over a large period of time (e.g., a superbubble whose contribution to the DM is small as in the case of the observed hump in the DM between 48and 65) is kept.","Figure: Minimum (dotted line), mean (solid line) and maximum (dashed line) of the averaged DM (top panel) andn e n_{e} (bottom panel) for all LOS crossing the disk up to 4 kpc from the Sun at 310 Myr (〈X〉 310Myr \\langle X\\rangle _{310\\mbox{ Myr}}; red lines), 360 Myr (〈X〉 360Myr \\langle X\\rangle _{360\\mbox{ Myr}}; green lines), and overthe period 350-400 Myr (〈X〉 350-400Myr \\langle X\\rangle _{350-400\\mbox{ Myr}}; black lines).","The blue circles representthe DMs (top) and 〈n e 〉\\langle n_{e}\\rangle (bottom) derived from pulsars observations.We further compare the DMs and $\\langle n_{e}\\rangle $ derived from pulsars observations (§3), with those resulting from averaging over all the LOS crossing the simulated disk ($|z|<200$ pc) and up to 4 kpc from the Sun at specific times (e.g., 310 and 360 Myr) and over the period 350-400 Myr.","Top panel of Figure REF displays the observationally derived DMs and the averaged DMs at 310 Myr ($\\langle DM\\rangle _{310\\mbox{Myr}}$ ; red lines), 360 Myr ($\\langle DM\\rangle _{360\\mbox{ Myr}}$ ; green lines), and over the period 350-400 Myr ($\\langle DM\\rangle _{350-400\\mbox{ Myr}}$ ; black lines).","Due to the limitation of the box size in our simulation, we calculated the DMs for distance $d>1$ kpc taking advantage of the periodic boundary conditions used in the simulations.","The DMs were calculated from different vantage points in the midplane (including $(x=0,y=0,x=0)$ and between 500 and 1400 pc from it) using LOS crossing the disk volume at different angles, up to 4 kpc from the vantage point and up to $z=\\pm 200 $ pc.","This procedure, although inevitable here, may introduce artefacts in the simulation results, but we do not expect that our results are strongly affected though, because the simulation shows a certain pattern repetition in the ISM density and temperature distributions on scales of the order of a few correlation lengths (that is, a few times 75 pc, according to AB0507).","The averaged $\\langle DM\\rangle _{350-400\\mbox{ Myr}}$ , after a steep growth in the first 100 pc, reaches a smooth increase for $d>500$ pc, with the deviations of the minimum and maximum regarding the mean becoming constant over distance - consistent with the DMs variations in $d$ observed in Figure REF .","The averaged DMs at 310 and 360 Myr vary widely, but with values within the pulsar derived DMs.","Therefore, the terms minimum and maximum of the $\\langle DM\\rangle _{350-400\\mbox{ Myr}}$ should not be taken as strict upper and lower values for all times, as they only represent an average over a specific time window of the simulation.","After the large variation in the first few hundred parsecs, the mean of the simulated $\\langle n_{e}\\rangle $ , for the cases shown in the bottom panel of Figure REF , reach a constant value ($0.04\\pm 0.002$ cm$^{-3}$ ) and dispersion over large distances.","This reflects the clumpy nature of the electrons in the turbulent ISM as would be expected if they are predominantly found in filaments and shells (Berkhuijsen & Müller 2008).","The histogram of the $\\log \\langle n_{e}\\rangle _{350-400\\mbox{ Myr}}$ and it's best Gaussian fit are displayed in Figure REF by solid and dashed red lines, respectively.","The fit is centered at $\\log \\langle n_{e}\\rangle =-1.4\\pm 0.01$ with $\\sigma =0.21\\pm 0.01$ .","In the simulated disk, the clumpy material dominates the LOS and volume averaged PDFs which explains the similar means between the electron density of the T$<10^{4.2}$ K regime (Figure REF ) and the time averaged $\\langle n_{e}\\rangle _t$ (Figure REF ; red lines).","These fit parameters are similar to those displayed in Figure REF (black lines) for observationally derived $\\langle n_{e}\\rangle $ .","This similarity is indicative of the smoothing out of low and high density regions along the lines of sight, thus, not appearing in the PDFs, and leading to reduced Gaussian dispersion."],["Discussion and Final Remarks","In this letter we discuss the electron density distribution in the Galactic disk using the first to date 3D high resolution NEI simulations of the ISM.","The simulations trace the dynamical and thermal evolution of the interstellar gas, calculating on-the-spot the ionization structure, electron distribution and cooling function of the gas at each cell of the grid into which the computational domain is discretized.","We did not take into account stellar ionizing photons and their transport, but an averaged diffuse photon field, which should be a reasonable approximation, when describing the mesoscale ISM.","The NEI structure modifies the cooling function, which in turn enhances the number of free electrons.","Both the simulated DMs and electron density are consistent with the observations: (i)nmost DMs lie within the maximum and minimum observed values, (ii) $\\log \\langle n_{e} \\rangle $ is consistent with a gaussian distribution, (iii) the mean electron density is $0.04\\pm 0.01$ cm$^{-3}$ , and (iv) the volume filling fraction of the WIM is bracketed between 4.9 and 6%.","These results lye in between the estimates by Gaensler et al.","(2008) and Berkuijsen & Müller (2008) for the electron density in the disk and WIM.","Furthermore, the observed Gaussian distribution is present, irrespective of whether the ISM is isothermal (see discussion in Berkuijsen & Fletcher (2008) and refereces therein) or not.","This can be explained in two complementary ways: a fully developed turbulent system can be considered as a large number of independent random variables, with the logarithm of each having a certain distribution approaching a Gaussian as the number of variables goes to infinity (Central Limit Theorem), being the case of supernova-driven ISM simulations reaching a statistical equilibrium (see, e.g., Vázquez-Semadeni & Garcia 2001).","Or, alternatively, we can use the principle of maximum entropy.","The information entropy of a continuous random variable $X$ with probability density function $p(x)$ is defined as $H(X)=\\int _{-\\infty }^{\\infty } p(x) \\log p(x) dx$ .","The lognormal distribution of $X$ maximizes $H(X)$ , implying the least prior knowledge of the system (Sveshnikov, 1968).","This is exactly what is expected in homogeneous and isotropic turbulence with a large number of independent random variables.","We study deviations from lognormal distributions in a forthcoming paper."],["Acknowledgements","We thank the anonymous referee for the detailed report and valuable suggestions that allowed us to improved this letter.","The Milipeia Supercomputer (Univ.","of Coimbra) and the ISM-cluster (Univ.","of Évora) were heavily used for the calculations.","This research is supported by the FCT project PTDC/CTE-AST/70877/2006."]],"string":"[\n [\n \"Electron Distribution in the Galactic Disk - Results From a\\n Non-Equilibrium Ionization Model of the ISM\"\n ],\n [\n \"Abstract Using three-dimensional non-equilibrium ionization (NEI) hydrodynamical simulation of the interstellar medium (ISM), we study the electron density, $n_{e}$, in the Galactic disk and compare it with the values derived from dispersion measures towards pulsars with known distances located up to 200 pc on either side of the Galactic midplane.\",\n \"The simulation results, consistent with observations, can be summarized as follows: (i) the DMs in the simulated disk lie between the maximum and minimum observed values, (ii) the log derived from lines of sight crossing the simulated disk follows a Gaussian distribution centered at \\\\mu=-1.4 with a dispersion \\\\sigma=0.21, thus, the Galactic midplane =0.04\\\\pm 0.01$ cm$^{-3}$, (iii) the highest electron concentration by mass (up to 80%) is in the thermally unstable regime (2004$ kpc, and 1 pc elsewhere.\",\n \"Periodic and outflow boundary conditions are set along the vertical faces and top/bottom ($z=\\\\pm 15$ kpc) of the grid, respectively.\",\n \"Figure: Histogram (solid black line) and Gaussian fit (dashed black line) of log〈n e 〉\\\\log \\\\langle n_{e} \\\\rangle distribution as obtained with a series of 24 pulsars located at 20010^{4.2}$ K gas), respectively, with the latter representing mostly photoionized regions.\",\n \"Further insight into the simulated electron distribution can be obtained directly from DMs along LOS, with length $d$ , crossing the simulated ISM, allowing the determination of the mean electron density.\",\n \"The LOS have increasing lengths (with a step length of 10 pc) up to 1 kpc from a vantage point located at $(x,y,z)=(0,0,0)$ (left bottom corner of the electron density maps in Figure REF ) for $\\\\left|z\\\\right|\\\\le 200$ pc and spanning 90 (with a 1 separation) when projected onto the Galactic midplane.\",\n \"Figure: Angular distribution of the DMs along LOS with lengths varying between 100 and 1000pc at 350 and 400 Myr (top two panels) and the time averaged 〈DM〉 t \\\\langle DM\\\\rangle _{t} (bottom panel) over theevolution time 350-400 Myr).Figure REF shows the DM along the LOS with lengths varying between 100 and 1000 pc, with a step length of 100 pc at 350 and 400 Myr (top two panels) and its time average (bottom panel) over 501 disk snapshots taken every 0.1 Myr for $350-400$ Myr of evolution.\",\n \"Due to the ongoing turbulent processes in the Galactic disk the DM has variability along and between the lines of sight (Figure REF : top two panels).\",\n \"The longitudinal variation is due to the inhomogeineity of the ISM (e.g., bubbles and superbubbles) as well as due to the turbulent nature of the regions crossed by the LOS.\",\n \"These variations with distance and among LOS are still present in the time averaged DM, although the longitudinal variations are smoothed in the averaging process.\",\n \"However, a record of any event that prevailed over a large period of time (e.g., a superbubble whose contribution to the DM is small as in the case of the observed hump in the DM between 48and 65) is kept.\",\n \"Figure: Minimum (dotted line), mean (solid line) and maximum (dashed line) of the averaged DM (top panel) andn e n_{e} (bottom panel) for all LOS crossing the disk up to 4 kpc from the Sun at 310 Myr (〈X〉 310Myr \\\\langle X\\\\rangle _{310\\\\mbox{ Myr}}; red lines), 360 Myr (〈X〉 360Myr \\\\langle X\\\\rangle _{360\\\\mbox{ Myr}}; green lines), and overthe period 350-400 Myr (〈X〉 350-400Myr \\\\langle X\\\\rangle _{350-400\\\\mbox{ Myr}}; black lines).\",\n \"The blue circles representthe DMs (top) and 〈n e 〉\\\\langle n_{e}\\\\rangle (bottom) derived from pulsars observations.We further compare the DMs and $\\\\langle n_{e}\\\\rangle $ derived from pulsars observations (§3), with those resulting from averaging over all the LOS crossing the simulated disk ($|z|<200$ pc) and up to 4 kpc from the Sun at specific times (e.g., 310 and 360 Myr) and over the period 350-400 Myr.\",\n \"Top panel of Figure REF displays the observationally derived DMs and the averaged DMs at 310 Myr ($\\\\langle DM\\\\rangle _{310\\\\mbox{Myr}}$ ; red lines), 360 Myr ($\\\\langle DM\\\\rangle _{360\\\\mbox{ Myr}}$ ; green lines), and over the period 350-400 Myr ($\\\\langle DM\\\\rangle _{350-400\\\\mbox{ Myr}}$ ; black lines).\",\n \"Due to the limitation of the box size in our simulation, we calculated the DMs for distance $d>1$ kpc taking advantage of the periodic boundary conditions used in the simulations.\",\n \"The DMs were calculated from different vantage points in the midplane (including $(x=0,y=0,x=0)$ and between 500 and 1400 pc from it) using LOS crossing the disk volume at different angles, up to 4 kpc from the vantage point and up to $z=\\\\pm 200 $ pc.\",\n \"This procedure, although inevitable here, may introduce artefacts in the simulation results, but we do not expect that our results are strongly affected though, because the simulation shows a certain pattern repetition in the ISM density and temperature distributions on scales of the order of a few correlation lengths (that is, a few times 75 pc, according to AB0507).\",\n \"The averaged $\\\\langle DM\\\\rangle _{350-400\\\\mbox{ Myr}}$ , after a steep growth in the first 100 pc, reaches a smooth increase for $d>500$ pc, with the deviations of the minimum and maximum regarding the mean becoming constant over distance - consistent with the DMs variations in $d$ observed in Figure REF .\",\n \"The averaged DMs at 310 and 360 Myr vary widely, but with values within the pulsar derived DMs.\",\n \"Therefore, the terms minimum and maximum of the $\\\\langle DM\\\\rangle _{350-400\\\\mbox{ Myr}}$ should not be taken as strict upper and lower values for all times, as they only represent an average over a specific time window of the simulation.\",\n \"After the large variation in the first few hundred parsecs, the mean of the simulated $\\\\langle n_{e}\\\\rangle $ , for the cases shown in the bottom panel of Figure REF , reach a constant value ($0.04\\\\pm 0.002$ cm$^{-3}$ ) and dispersion over large distances.\",\n \"This reflects the clumpy nature of the electrons in the turbulent ISM as would be expected if they are predominantly found in filaments and shells (Berkhuijsen & Müller 2008).\",\n \"The histogram of the $\\\\log \\\\langle n_{e}\\\\rangle _{350-400\\\\mbox{ Myr}}$ and it's best Gaussian fit are displayed in Figure REF by solid and dashed red lines, respectively.\",\n \"The fit is centered at $\\\\log \\\\langle n_{e}\\\\rangle =-1.4\\\\pm 0.01$ with $\\\\sigma =0.21\\\\pm 0.01$ .\",\n \"In the simulated disk, the clumpy material dominates the LOS and volume averaged PDFs which explains the similar means between the electron density of the T$<10^{4.2}$ K regime (Figure REF ) and the time averaged $\\\\langle n_{e}\\\\rangle _t$ (Figure REF ; red lines).\",\n \"These fit parameters are similar to those displayed in Figure REF (black lines) for observationally derived $\\\\langle n_{e}\\\\rangle $ .\",\n \"This similarity is indicative of the smoothing out of low and high density regions along the lines of sight, thus, not appearing in the PDFs, and leading to reduced Gaussian dispersion.\"\n ],\n [\n \"Discussion and Final Remarks\",\n \"In this letter we discuss the electron density distribution in the Galactic disk using the first to date 3D high resolution NEI simulations of the ISM.\",\n \"The simulations trace the dynamical and thermal evolution of the interstellar gas, calculating on-the-spot the ionization structure, electron distribution and cooling function of the gas at each cell of the grid into which the computational domain is discretized.\",\n \"We did not take into account stellar ionizing photons and their transport, but an averaged diffuse photon field, which should be a reasonable approximation, when describing the mesoscale ISM.\",\n \"The NEI structure modifies the cooling function, which in turn enhances the number of free electrons.\",\n \"Both the simulated DMs and electron density are consistent with the observations: (i)nmost DMs lie within the maximum and minimum observed values, (ii) $\\\\log \\\\langle n_{e} \\\\rangle $ is consistent with a gaussian distribution, (iii) the mean electron density is $0.04\\\\pm 0.01$ cm$^{-3}$ , and (iv) the volume filling fraction of the WIM is bracketed between 4.9 and 6%.\",\n \"These results lye in between the estimates by Gaensler et al.\",\n \"(2008) and Berkuijsen & Müller (2008) for the electron density in the disk and WIM.\",\n \"Furthermore, the observed Gaussian distribution is present, irrespective of whether the ISM is isothermal (see discussion in Berkuijsen & Fletcher (2008) and refereces therein) or not.\",\n \"This can be explained in two complementary ways: a fully developed turbulent system can be considered as a large number of independent random variables, with the logarithm of each having a certain distribution approaching a Gaussian as the number of variables goes to infinity (Central Limit Theorem), being the case of supernova-driven ISM simulations reaching a statistical equilibrium (see, e.g., Vázquez-Semadeni & Garcia 2001).\",\n \"Or, alternatively, we can use the principle of maximum entropy.\",\n \"The information entropy of a continuous random variable $X$ with probability density function $p(x)$ is defined as $H(X)=\\\\int _{-\\\\infty }^{\\\\infty } p(x) \\\\log p(x) dx$ .\",\n \"The lognormal distribution of $X$ maximizes $H(X)$ , implying the least prior knowledge of the system (Sveshnikov, 1968).\",\n \"This is exactly what is expected in homogeneous and isotropic turbulence with a large number of independent random variables.\",\n \"We study deviations from lognormal distributions in a forthcoming paper.\"\n ],\n [\n \"Acknowledgements\",\n \"We thank the anonymous referee for the detailed report and valuable suggestions that allowed us to improved this letter.\",\n \"The Milipeia Supercomputer (Univ.\",\n \"of Coimbra) and the ISM-cluster (Univ.\",\n \"of Évora) were heavily used for the calculations.\",\n \"This research is supported by the FCT project PTDC/CTE-AST/70877/2006.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1511"}}},{"rowIdx":1866798,"cells":{"text":{"kind":"list like","value":[["Milliarcsecond structure of water maser emission in two young high-mass\n stellar objects associated with methanol masers"],["Abstract The 22.2 GHz water masers are often associated with the 6.7 GHz methanol masers but owing to the different excitation conditions they likely probe independent spatial and kinematic regions around the powering young massive star.","We compared the emission of these two maser species on milliarcsecond scales to determine in which structures the masers arise and to test a disc-outflow scenario where the methanol emission arises in a circumstellar disc while the water emission comes from an outflow.","We obtained high-angular and spectral resolution 22.2 GHz water maser observations of the two sources G31.581+00.077 and G33.641-00.228 using the EVN.","In both objects the water maser spots form complex and filamentary structures of sizes 18-160 AU.","The emission towards the source G31.581+00.077 comes from two distinct regions of which one is related to the methanol maser source of ring-like shape.","In both targets the main axis of methanol distribution is orthogonal to the water maser distribution.","Most of water masers appear to trace shocks on a working surface between an outflow/jet and a dense envelope.","Some spots are possibly related to the disc-wind interface which is as close as 100-150 AU to the regions of methanol emission."],["Introduction","High-angular resolution observations of high-mass star forming sites in the 6.7 GHz methanol maser emission have revealed a diversity of morphological structures from very simple composed of a few milliarcsecond (mas) wide spots to complex and extended clouds of arcsecond sizes (Phillips et al.","[28]; Walsh et al.","[51]; Minier et al.","[21]; Dodson et al.","[13]; Bartkiewicz et al.","[3]; Pandian et al. [27]).","Linear structures with a velocity gradient are thought to be the signature of an edge-on disc or torus around young massive protostars (Norris et al.","[26]; Minier et al.","[21]) but there is evidence that some of them are associated with outflows (De Buizer [12]) or even a shock propagating through a rotating dense molecular clump (Dodson et al. [13]).","Arc-like or ring-like morphologies of methanol emission seem to prove a disc scenario (Bartkiewicz et al.","[3]), although most of those structures do not show signs of rotation but rather inflow or/and outflow dominate (van Langevelde et al.","[50]), it is then postulated that methanol masers arise in the shock interface between the large scale accretion and a circumstellar disc.","While the 6.7 GHz methanol maser line is radiatively pumped in warm ($T\\sim 150$ K) and dense ($n\\le 10^8$ cm$^{-3}$ ) regions (Cragg et al.","[8]; Sobolev et al.","[32]), the 22 GHz water maser emission is collisionally pumped and probes denser ($n\\ge 10^8$ cm$^{-3}$ ) and hotter ($T\\sim 400$ K) gas (Elitzur et al.","[14]).","The water maser is often excited in strong shocks, driven by young low-mass and high-mass (proto)stellar objects, on a working surface between a outflow and a dense envelope.","A number of morphologies of this emission is observed from collimated jet, wide-angle flows, expanding shells to equatorial flows (Goddi et al.","[18]; Moscadelli et al.","[23], [24]; Sanna et al.","[30], [31]; Torrelles et al. [42]).","One or more types of those maser structures occur simultaneously in star forming regions on scale sizes of a few mas to several arcsec and is possibly related to the geometry of the envelopes or the mechanisms driving the outflows.","Our recent VLA survey of 22 GHz water masers in a sample of 31 methanol sources has yielded 22 detections (Bartkiewicz et al.","[4]) where both maser species are excited by the same central objects.","We noted that a distinct group of methanol sources with ring-like structure show either no associated water masers at all or water masers that are distributed orthogonally to the major axis of the ring.","It is argued that the methanol maser structure traces a circumstellar disc/torus around a high-mass young stellar object while the water masers originate in outflows.","As the VLA data are of moderate angular and spectral resolutions of $\\sim $ 1 and 0.65 km s$^{-1}$ , respectively, we undertook the VLBI observations to examine a disc-outflow scenario in the two brightest water maser targets, which show the arc-like or ring-like morphologies that are characteristic of methanol masers.","The source G31.581$+$ 00.077The names of two targets follow the Galactic coordinates of the brightest methanol maser spots derived by Bartkiewicz et al. ([3]).","has been recognized as a massive young stellar object based on 6.7 GHz methanol maser observations (Szymczak et al. [36]).","The methanol maser spots are distributed along an arc or ring of 217 mas size (Bartkiewicz et al.","[3]), which corresponds to $\\sim $ 1200 AU for the assumed near kinematic distance of 5.5 kpc (Reid et al. [29]).","The detection of an infrared source of bolometric luminosity of 3$\\times $ 10$^4$ L$_{}$ (Urquhart et al.","[48]), 22 GHz H$_2$ O maser emission (Bartkiewicz et al.","[4]), 1665 and 1667 MHz weak OH masers (Szymczak & Gérard [38]), millimeter molecular thermal lines (Szymczak et al.","[40]; Urquhart et al.","[45]), and 5 and 8.4 GHz continuum emission at location of 31.582$+$ 00.075 (i.e., 9 apart from the maser arc/ring) (Urquhart et al.","[46]; Bartkiewicz et al.","[3]) indicates that this is a cluster of recent star formation.","The source G33.641$-$ 00.228 detected in the 6.7 GHz methanol line (Szymczak et al.","[36]) has an arc spot distribution of length 630 AU (Bartkiewicz et al.","[3]) for the assumed near kinematic distance of 3.8 kpc (Reid et al.","[29]), which seems to be more likely than the far kinematic distance (Zhang priv.comm.).","The site also contains water and OH masers (Bartkiewicz et al.","[4]; Szymczak & Gérard [38]) but no 8.4 GHz continuum emission was detected with an upper limit of 0.15 Jy (Bartkiewicz et al. [3]).","The detection of radio recombination lines at about 103 km s$^{-1}$ (Anderson et al.","[1]) and a multi-feature $^{13}$ CO line spectrum (Urquhart et al.","[45]) illustrates the complexity of this molecular cloud and its clustered star formation."],["Observations and data reduction","The EVNThe European VLBI Network is a joint facility of European, Chinese, South African and other radio astronomy institutes funded by their national research councils.","observations of G31.581$+$ 00.077 and G33.641$-$ 00.228 with the antennas at Jodrell Bank, Effelsberg, Medicina, Metsähovi, Onsala, and Yebes, were carried out at 22.23508 GHz on 2010 October 30 for 8 h (the project EB047).","The tracking phase centres were estimated from the VLA survey for the water maser spots that were located nearest to the methanol emission in each target (Bartkiewicz et al.","[4]) at $\\alpha $ =18$^{\\rm h}$ 48$^{\\rm m}$ 41951, $\\delta $ =$-$ 0110'02578 and $\\alpha $ =18$^{\\rm h}$ 53$^{\\rm m}$ 32563, $\\delta $ =$+$ 0031'39130 (J2000), respectively.","A phase-referencing scheme was applied with a reference source J1851$+$ 0035 (from the VLA calibrator catalogue), using a cycle time between the maser and phase-calibrator of 60 s$+$ 90 s. This strategy yielded 2 h on-source times for each target.","The bandwidth was set to 8 MHz and data were correlated with the Mk IV Data Processor operated by JIVE with 1024 spectral channels.","The resulting spectral resolution was 0.1 km s$^{-1}$ .","The velocity was measured with respect to the local standard of rest (LSR).","The data calibration and reduction were carried out with NRAO's Astronomical Image Processing System (AIPS) using standard procedures for spectral line observations.","We used the Effelsberg antenna as a reference.","The amplitude was calibrated by performing measurements of the system temperature at each telescope and applying the antenna gain curves.","The parallactic angle corrections were subsequently added to the data.","The source 3C454.3 was used as a delay, rate, and bandpass calibrator.","The phase-calibrator was imaged and a flux density of 190 mJy was obtained.","The maser data were corrected for all Doppler effects and self-calibrated using the brightest and most compact maser spot.","Finally, naturally weighted maps of spectral channels were created in the velocity range where the emission was seen in the scalar-averaged spectrum.","For imaging, the resulting synthesized beam was 1.0$\\times $ 2.4 mas$^2$ with a position angle (PA) of $-$ 38, while the pixel separation was 0.2 mas in both coordinates.","The rms noise levels (1$\\sigma _{\\rm rms}$ ) in line-free channels was typically 10 mJy beam$^{-1}$ .","The weakest detected maser spot had a brightness of 94 mJy beam$^{-1}$ that is more than 9$\\sigma _{\\rm rms}$ .","The positions of water maser spots in all channel maps were determined by fitting two-dimensional Gaussian models.","The formal fitting errors resulting from the beamsize/signal-to-noise ratio were smaller than 0.1 mas.","To determine the position accuracy of registered maser spots, we need to consider the following factors (Diamond et al.","[10]): i) the uncertainty in the phase-reference source position of 2 masThe GSFC ICRF2 VLBI Source Position Catalog.","; ii) the antenna positions, where the claimed accuracy of $\\sim $ 1 cm corresponds to 1 mas in RA and 2 mas in Dec; iii) the phase transfer over the separation between targets and the phase-calibrator: 192 (between G31.581$+$ 00.077 and J1851$+$ 0035) and 045 (between G33.641$-$ 00.228 and J1851$+$ 0035).","These caused potential phase-solution transfer errors corresponding to 1 mas in RA and 2 mas in Dec for G31.581$+$ 00.077 and to 0.2 mas in RA and 0.4 mas in Dec for G33.641$-$ 00.228, respectively.","In total, the absolute position accuracy (1$\\sigma _{\\rm pos}$ ) is 2.5 mas in RA and 3.5 mas in Dec for both targets."],["Results","Water maser emission towards both targets was mapped, after successful phase-referencing, in the areas of 5$\\times $ 5 and the velocity range explored by Bartkiewicz et al.","([4]) using the VLA.","The EVN maps of water maser spots (Figs.","REF and REF ) are overlayed on the VLA maps and the 6.7 GHz methanol maser distributions obtained with the EVN (Bartkiewicz et al. [3]).","The Spitzer GLIMPSE mapshttp://irsa.ipac.caltech.edu/data/SPITZER/GLIMPSE/ of the 4.5 $\\mu $ m$-$ 3.6 $\\mu $ m emission excess are added.","Following the procedure described in Bartkiewicz et al.","([3]), the water maser spots are analysed to identify individual velocity-coherent maser clouds.","Their basic parameters, such as positions, $\\Delta $ RA, $\\Delta $ Dec, LSR velocities, V$_{\\rm LSR}$ , and intensities of the brightest spot, S$_{\\rm p}$ , of each cloud are listed in Tables REF and REF .","The water and methanol maser spectra of single clouds are combined in Figures REF and REF with overlays of individual Gaussians, if emission was seen in at least three consecutive channels.","A Gaussian analysis of individual clouds is carried out and the fitted flux amplitude, S$_{\\rm fit}$ , the full width at half-maximum, FWHM, the projected extent between the most separated single spot centres within a cloud, L$_{\\rm proj}$ , and the velocity gradient, V$_{\\rm grad}$ are listed in Table REF .","The lower limit to the brightness temperature, T$_{\\rm b}$ , of each cloud is also calculated according to Eq.","9$-$ 27 of Wrobel & Walker ([49]).","For comparison purposes, the same analysis is done for the 6.7 GHz methanol maser data (Table REF ) obtained in 2007 June (G31.581$+$ 00.077) or 2003 June (G33.641$-$ 00.228) with the EVN.","Below we comment in more detail on each source."],["G31.581$+$ 00.077","A total of 91 water maser spots were detected.","They are concentrated in two distinct regions (Fig.","REF ).","The south-east (SE) region of size 50$\\times $ 30 mas$^2$ containing weak ($<$ 1.35 Jy) emission in the velocity ranges from 90.1 to 93.9 km s$^{-1}$ and from 100.0 to 103.3 km s$^{-1}$ is located close to the phase centre.","The second region (NW) of middle velocity emission (96.8$-$ 100.7 km s$^{-1}$ ) and flux density of 2.6$-$ 16.5 Jy is located offset by $\\sim $ 5 to the north-west (PA$=-60$ ) of the phase centre.","The SE region is divided into the cluster of clouds 1, 2, and 3 separated by 50 mas from the cluster of clouds 6, 7, and 8 (Fig.","REF ).","The clouds in the SE region show Gaussian velocity profiles with FWHM linewidths of 0.54$-$ 0.94 km s$^{-1}$ .","Their projected sizes are 0.07$-$ 0.47 mas, which correspond to 0.9$-$ 2.6 AU for the adopted distance of 5.5 kpc.","The brightness temperature is 0.24$-$ 1.38$\\times $ 10$^9$ K. We note that clouds 3 and 6 show the velocity gradient of 1.3$-$ 1.8 km s$^{-1}$ mas$^{-1}$ oriented along PAs of 1 and $-$ 23, respectively.","It is remarkable that the two clusters of clouds (1, 2, 3 and 6, 7, 8) appear very close in space (Table REF ) forming arc-like filaments of projected length of 4 and 3.2 mas corresponding to 22 and 18 AU, respectively, and elongated at PA=$-$ 58.","The closest clouds 3 and 6 are separated by 285 AU and a velocity difference of 6.7 km s$^{-1}$ .","The mean linear separation between the two clusters along PA=65 is 310$\\pm $ 8 AU, while the velocity spread is 13.2 km s$^{-1}$ .","The velocities of the two filaments are approximately symmetric with the regard of the systemic velocity of 96.0 km s$^{-1}$ (Szymczak et al. [40]).","The two SE filaments are likely signatures of flattened shock surfaces (Torrelles et al. [41]).","We found that the water cloud 7 coincides within $\\pm $ 0.1 km s$^{-1}$ with the methanol maser cloud k, while the spatial separation is 24.4 mas which corresponds to the projected distance of 134 AU.","Although these two lines were observed within a time span of three years their separation seems to be real.","This confirms that the water and methanol masers probe different parts of the environment of young massive stars (e.g., Beuther et al.","[5]; Sanna et al.","[30], [31]) because of the different pumping mechanisms affecting both species (Elitzur et al.","[14]; Cragg et al. [8]).","The NW region of elongation of 26 mas is composed of two clouds 4 and 5 of brightness temperatures of 2.7$-$ 17.1$\\times $ 10$^9$ K (Table REF ).","Their velocity profiles are obviously asymmetric and nicely fitted by the sums of three and two Gaussian components (Fig.","REF ) of FWHM linewidth of 0.5$-$ 0.9 km s$^{-1}$ .","The velocities of the NW water maser emission largely coincide with those of the methanol masers but their linear separation is about 27000 AU.","The map of 4.5 $\\mu $ m$-$ 3.6 $\\mu $ m emission excess (Fig.","REF ), which possibly traces shocked molecular gas in outflows from massive stars (e.g., Davies et al.","[11]), shows that the SE and NW maser clusters lie in a very complex region.","Both maser clusters are likely associated with different powering sources.","The comparison of our maps with those obtained about 14 months earlier with the VLA in a CnB configuration (Bartkiewicz et al.","[4]) implies that the cloud 1 coincides to within 13 mas, the cloud 4 to within 56 mas, and the cloud 7 to within 64 mas with the matching VLA spots.","The emission seen with the VLA from three components located $\\sim $ 1$-$ 2 eastward of the clouds 4 and 5 is not detected.","We also note that the water maser components detected have very similar amplitudes in both EVN and VLA observations.","Figure: G31.581++00.077.","a) Spitzer GLIMPSE 4.5 μ\\mu m--3.6 μ\\mu m excess image overlaidwith the water maser positions (yellow asterisks) from the EVN observation and H II region(yellow cross) detected by Bartkiewicz et al.","().b) Shows the distribution of water masers observed with the EVN (filled squares)from this paper and the VLA(open squares) from Bartkiewicz et al. ().","The colours ofthe symbols relate to theLSR velocities as indicated on the right-hand side of the plot.","The origin ofthe map isthe position of the brightest 6.7 GHz methanol maser spot (Bartkiewicz et al.",")(see also Table ).","c) Shows an enlargement of the north-western (NW) water masers.d) Shows an enlargement of the south-eastern (SE) water masers together with the distributionof the 6.7 GHz methanol masers, marked by open circles, from Bartkiewicz et al. ().","In (c) and(d), thearrows represent the velocity gradients (from blue- to red-shifted LSR velocities) detectedwithin individual cloud.","The black numbers and letters correspond to the clouds withinternal velocity gradients, while the grey ones correspond to the clouds without internalvelocity gradients (Table ).Figure: Individual component spectra with Gaussian velocity profiles of 22.2 GHz water masers(red squares and lines) and 6.7 GHz methanol masers (blue circles and lines) towardsG31.581++00.077 detected using the EVN in 2010 and 2007, respectively.","The numbers andletters correspond to the cloud labels as given in Table and Fig.",".The grey lines show individual Gaussian profiles fitted to the blended features.The clouds with only two single spots are not marked to improvethe clarity of the figure (i.e., clouds b, e, g)."],["G33.641$-$ 00.228","Twenty-one water maser spots were detected in the velocity range of 54.4$-$ 57.6 km s$^{-1}$ over the north-south elongated area 50$\\times $ 280 mas$^2$ close to the phase centre (Fig.","REF ).","Using the above-mentioned procedure, we found four maser clouds where the emission is seen in at least three contiguous spectral channels.","There are also two other clouds (5 and 6) that do not obey this criterion but seem to be real (Table REF ).","All but one of the cloud are in the southward cluster at a distance $\\sim $ 270 mas from the phase centre.","This filament cluster of size 41.6 mas is aligned along a PA of 78.","For the adopted distance of 3.8 kpc, the corresponding linear scales are 1020 AU and 157 AU.","The clouds 1$-$4 in the southern cluster have Gaussian profiles with FWHM linewidths of 0.25$-$ 0.88 km s$^{-1}$ .","We note that none of the water clouds have an internal velocity gradient.","The linear size of the individual clouds is 0.3$-$ 1.0 AU and the brightness temperature is 0.12$-$ 0.27$\\times $ 10$^9$ K (Table REF ).","The strongest emission (T$_{\\rm b}=0.63\\times 10^9$ K) comes from cloud 2, which is a blend of two Gaussian components (Fig.","REF ).","Cloud 2 is $\\sim $ 20 mas (75 AU) away from the brightest methanol maser clouds a and r and differ in terms of velocity by 1.5 km s$^{-1}$ .","This is probably the first such tight association of methanol and water masers.","We note that the emission of cloud 2 is blue-shifted by 4.2 km s$^{-1}$ and that of cloud a is red-shifted by 1.2 km s$^{-1}$ relative to the systemic velocity of 61.5 km s$^{-1}$ (Szymczak et al. [40]).","It is therefore possible that these clouds signpost a shock front in which the water emission originates behind the front, while the methanol emission (cloud a) appears outside of the shock interface in the infalling gas.","The image of the 4.5 $\\mu $ m$-$ 3.6 $\\mu $ m emission excess (Fig.","REF ) shows that the water maser emission is located 12 to the south-east of single mid-infrared object.","This is likely the source powering the outflow along a PA of 165 traced by the water masers.","The positions of the clouds 2, 3, 4 and 6 differ by about 92 mas from the VLA positions of the matching clouds (Bartkiewicz et al.","[4]), which is well within the $\\sim $ 150 mas accuracy of the CnB configuration VLA data.","A weak ($<$ 0.9 Jy) red-shifted emission in the velocity range of 83.8$-$ 85.1 km s$^{-1}$ seen with the VLA towards cloud 2 (Fig.","REF ) was not recovered in the VLBI observation.","The brightness of the emission from cloud 2 determined in the EVN observation is only slightly lower than that measured with the VLA.","In contrast, the maser spots from the southern region (clouds 1, 3$-$6) are about ten times weaker in the EVN maps that may suggest the presence of diffuse and/or highly variable emission.","Figure: Same as Fig.","but for G33.641--00.228.Figure: Same as Fig.","but for G33.641--00.228.","Thescale on the left side describes the water maser intensity, while theright side corresponds to the methanol maser intensity.","The clouds with onlytwo single spots or without Gaussian characteristic are not marked for theclarity of the figure (i.e., clouds 5, 6, e, g, i, n, p).Table: List of 22.2 GHz water maser clouds as found in EVN observationstowards G31.581++00.077.","The(0,0) point corresponds to the position of the brightest methanol maser in this source(Bartkiewicz et al.","): RA=18 h ^{\\rm h}48 m ^{\\rm m}4194108,Dec=--0110025281 (J2000).Table: List of 22.2 GHz water maser clouds as observed using EVN towards G33.641--00.228.The (0,0) point correspond to the brightest methanol maser in this source(Bartkiewicz et al.","): RA=18 h ^{\\rm h}53 m ^{\\rm m}32563,Dec=++003139180 (J2000).Table: Parameters of 22.2 GHz water and 6.7 GHz methanol maser clouds."],["Maser environments","The Spitzer GLIMPSE map (Fig.","REF ) clearly indicates that the source G31.581$+$ 00.077 lies in a large (35$\\times $ 25) complex area with an 4.5 $\\mu $ m emission excess in the form of clumped and diffuse structures.","This extended emission is proposed to be a tracer of shocked gas from regions where outflowing gas interacts with the surrounding medium (Davis et al.","[11]; Cyganowski et al. [9]).","The $^{13}$ CO(1$-$ 0) spectrum obtained with a 46 beam (Urquhart et al.","[45]) towards the position 31.5808+0.0757 has two Gaussian components centred at 96.2 and 109.8 km s$^{-1}$ .","A weak emission line is clearly detected in the range of 85 to 105 km s$^{-1}$ , i.e., in the wings of the first component suggests the presence of outflows.","A signature of ordered motions also appears in the HCO$^+$ (1$-$ 0) spectrum (Szymczak et al.","[40]), where in this slightly asymmetric profile the strongest emission is blueward of the source systemic velocity and can be interpreted as the result of infall (e.g., Fuller et al. [16]).","We then find 92 east of the SE water maser an H II region of flux density 10.3 and 15 mJy at 5 GHz (White et al.","[53]) and 8.4 GHz (Bartkiewicz et al.","[3]), respectively.","This continuum source coincides to within 06 with a clump of 3.6 $\\mu $ m$-$ 4.5 $\\mu $ m emission excess (Fig.","REF ).","The NW water masers are located at the borders of two clumps of excited gas, and their powering source is unclear although it seems unlikely that the SE and NW masers are associated with the same central object.","A comparison of the present EVN maps with those obtained with the VLA 14 months earlier (Bartkiewicz et al.","[4]) implies that there is a lack of water masers eastward of the NW clouds (Fig.","REF ).","It is likely that these are diffuse and low intensity masers resolved out with the EVN beam.","The spectrum obtained with a 40 beam sometime between November 2009 and December 2010 (Urquhart et al.","[48]) indicates that the peak flux density of 125 Jy at 99.7 km s$^{-1}$ is about one order of magnitude higher than that measured with the EVN.","We suggest that the water masers of G31.581$+$ 00.077 are produced by different young stellar objects in a complex region composed of at least a few high-mass stars well-signposted by the radio continuum and the methanol and water masers.","The water maser in G33.641$-$ 00.228 lies only 12 south-eastward of the maximum of the 3.6 $\\mu $ m$-$ 4.5 $\\mu $ m emission excess of cometary-like morphology (Fig.","REF ).","The HCO$^+$ (1$-$ 0) spectrum obtained towards the position 33.648$-$ 0.224 (Szymczak et al.","[40]) shows a small dip near 61.5 km s$^{-1}$ and a slight redward asymmetry that is indicative of outflow motions.","The less optically thin H$^{13}$ CO$^+$ (1$-$ 0) transition is detected as marginal red-shifted emission, which is consistent with an outflow.","In the region where the OH 1665, 1667, and 1720 MHz masers were detected (Szymczak & Gérard [38]) and all of them peak at 60.2 km s$^{-1}$ .","The strongest 1720 MHz emission and a broad (7.4 km s$^{-1}$ ) 1667 MHz absorption profile near 56.2 km s$^{-1}$ are indicative of shock fronts and a continuum background source, respectively.","No continuum emission at 8.4 GHz was detected with a 3$\\sigma _{\\rm rms}$ limit of 0.15 mJy beam$^{-1}$ (Bartkiewicz et al. [3]).","A weak ($\\sim $ 20 mJy) 8.6 GHz hydrogen radio recombination-line near 102.9 km s$^{-1}$ detected at position G033.645$-$ 0.227 (Anderson et al.","[1]) differs so greatly in terms of velocity from the water masers that this may be a chance projection that is unassociated with the methanol and water masers.","We argue that the water masers of G33.641$-$ 00.228 as well as the associated methanol and hydroxyl masers are excited by individual high-mass stars."],["Physical parameters of circumstellar medium","Lower limits to the brightness temperature of the SE water maser components associated with G31.581$+$ 00.077 are always lower than 1.4$\\times $ 10$^9$ K, while the measured linewidths range from 0.54 to 0.94 km s$^{-1}$ are quite commonly in the star-forming sources (e.g., Goldreich & Kwan [19]; Surcis et al.","[34], [35]).","Since the kinetic temperature of the masing gas might be expected to be about 400 K (Elitzur et al.","[14]), the intrinsic thermal linewidth given by $\\Delta v_{\\rm FWHM}=2.35482\\times \\sqrt{k T_k/m}$ , where $T_k$ is a kinetic temperature, $k$ the Boltzmann constant, and $m$ is a molecular mass should be $\\sim $ 1 km s$^{-1}$ .","This value is larger than the observed linewidths and suggests that the masers are unsaturated.","In the same volume of gas where the water maser cloud 7 and the methanol maser cloud k appear to coincide (Fig.","REF ), the methanol thermal linewidth would be 0.75 km s$^{-1}$ , whereas the measured value is 0.30 km s$^{-1}$ .","The narrowing of the line profile is expected when the maser is unsaturated.","Detailed calculations have shown that in one of the strongest known methanol sources NGC7538 about 92% of the components are unsaturated (Surcis et al. [35]).","In the source G33.641$-$ 00.228, the observed linewidths of methanol components are generally narrower than in G31.581$+$ 00.077, which suggests that the maser is also unsaturated in this region.","The mas-scale velocity gradients in both lines are observed in G31.581$+$ 00.077.","We note that the velocity gradient at 22 GHz is about one order of magnitude higher than at 6.7 GHz (Table REF ).","Extensive discussion of the velocity gradients for methanol masers by Moscadelli et al.","([25]) suggests that there is a kinematical interpretation of their origin.","We note that the methanol maser gradients in both targets can reflect the ordered motions on scales of 10$-$ 60 AU.","Large velocity gradients for water masers in G31.581$+$ 00.077 suggest that they are related to the outflow motions with velocities larger than 50 km s$^{-1}$ (see Sect.","4.3).","Table: Parameters derived by fitting the kinematics of the rotatingand expanding disc model.","The signs ++ and -- of the rotation andexpansion velocities refer to the clockwise or anti-clockwise rotation andoutflow or inflow for positive i.","Both rotation and floware reversed in the case of negative i.For each source, both signs together with the sign of i could be reversedsince our model does not give the directions unambiguously.","In the case ofG33.641--00.228, two model fits are presented as described in Sect.","4.3.Table: Parameters derived from fitting the biconical outflow modelof Moscadelli et al. ().","In the case ofG31.581++00.077, two model fits are presented as described in Sect.","4.3."],["Kinematic models","The two sources investigated in this study possibly belong to a group of objects where a ring-like or arc-like methanol maser distribution traces a circumstellar disc/torus around a high-mass young stellar object, whereas the water maser distribution is orthogonal to the major axis of the methanol structure (Bartkiewicz et al. [4]).","Furthermore, these objects are not associated with detectable continuum emission at cm wavelengths (Bartkiewicz et al.","[3]) and may represent an early stage of evolution.","The new EVN observations of both sources have shown that all of the individual water maser spots detected previously with the VLA (Bartkiewicz et al.","[4]) unfold into complex and filament structures of sizes 18$-$ 160 AU.","To examine a disc-outflow scenario in the two sources, we used a model of a rotating and expanding thin disc (Uscanga et al.","[44]) for the methanol masers and a model of the outflow (Moscadelli et al.","[22]) for the water masers.","Detailed descriptions of the modelling are given in the above cited works, and elsewhere for methanol masers in discs (Bartkiewicz et al.","[3]) and outflows (Bartkiewicz et al. [2]).","We note that in the following we analyse the new water maser data with the methanol maser data obtained 3$-$ 7 years earlier (Bartkiewicz et al. [3]).","However, our inspection of a few sources published so far and systematic EVN monitoring demonstrate that the overall methanol maser morphologies are stable on a scale time of 6-8 years (Bartkiewicz et al., in prep.).","In Tables 4 and 5, we summarize the best-fit values of both models.","For the rotating and expanding thin disc, the rotation (V$_{\\rm rot}$ ), expansion (V$_{\\rm exp}$ ), and systemic (V$_{\\rm sys}$ ) velocities as well as the inclination angle, i, i.e., the angle between the line-of-sight and the normal to the ring plane defined to be i$={\\rm acos} (\\frac{\\rm b}{\\rm a})$ is given for each source.","The solutions were based on the minimization of the $\\chi _{\\rm V}^2$ function given by eq.","8 in Uscanga et al.","([44]).","The outflow model for water masers is characterized by the vertex of the cone (X$_{\\rm o}$ , Y$_{\\rm o}$ ), the x–axis coinciding with the projection of the outflow on the plane of the sky at the position angle PA, the inclination angle between the outflow axis and the line-of-sight (i.e., the z–axis) $\\Psi $ , and the opening angle of the outflow/cone 2$\\theta $ .","The systemic LSR velocities, V$_{\\rm c}$ , were assumed to be the same as given in Sect. 3.","The $\\chi ^2$ function was assumed to be expressed as in Eq.","3 of Moscadelli et al.","([22]).","The methanol emission in G31.581$+$ 00.077 is most accurately reproduced by the model where the powering source is at the centre of the best-fit ellipse and the rotation velocity of 0.9 km s$^{-1}$ is a factor of two lower than the expansion velocity (Tab.","REF , Fig.","REF ).","These values are typical of the class of ring-like methanol masers (Bartkiewicz et al.","[3]) and suggest that the methanol structure of $\\sim $ 1000 AU diameter cannot be interpreted as a Keplerian disc.","It is instead proposed that the methanol masers arise on the interface region between a large-scale accretion flow and a stellar disc (van Langevelde et al.","[50]; Torstensson et al. [43]).","We also note that the best-fit systemic velocity of 98.54 km s$^{-1}$ corresponds well to that estimated from molecular lines (Sect.3.1).","The SE water emission is reasonably well-fitted by two different models.","In the first model, O1, the vertex of the cone with 2$\\theta $ =30 coincides with the centre of the methanol ellipse within 25 mas and the velocity of masers is 77 km s$^{-1}$ (Tab.","REF , Fig.","REF ).","We argue that this fit is consistent with the outflow scenario.","As we failed to identify any mid-infrared (MIR) counterpart to the centre of the methanol ellipse as a powering source of the methanol structure and perpendicular to the water maser outflow, we propose an alternative model.","This second model, O2, assumes that the position of the cone vertex is just between the red-shifted and blue-shifted voids of SE water masers.","The best-fit velocity of the outflow is $\\sim $ 8 km s$^{-1}$ , while the projection of the axis outflow onto the plane of the sky is roughly parallel to the main axis of the methanol structure (Tab.","REF , Fig.","REF ).","In star forming regions, different centres of activity are reported on scale sizes of a few to several hundred AUs (e.g., Torrelles et al.","[41], [42]).","The proper motion studies of G31.581$+$ 00.077 can clearly provide conclusive evidence for or against this hypothesis.","In the case of methanol masers in G33.641$-$ 00.228, we present the two kinematical models for a powering source located at the centre of the best-fit ellipse ($\\Delta $ RA, $\\Delta $ Dec)=(59 mas,17 mas) and that located at the MIR position ($-$ 495 mas, 380 mas).","In the first model, D1, the infall velocity of about 1 km s$^{-1}$ is slightly higher than the rotation velocity (Tab.","REF ).","The structure of water masers is fitted by a narrow ouflow of 2$\\theta $ =18 and velocity of 15 km s$^{-1}$ (Tab.","REF , Fig.","REF ).","It is consistent with a jet-like outflow of water masers roughly perpendicular to the main axis of methanol structure.","In the second model, D2, the visible methanol structure is only a small part of the disc radius of $\\sim $ 1900 AU and the rotation velocity of 3.7 km s$^{-1}$ (Tab.","REF ), which implies that the enclosed mass is about 30M$_{}$ .","In this case, the water masers may trace the wind gas from a disc.","We note that the kinematic models are unable to accurately reproduce the observed structures of the methanol and water masers in both targets, and that only measurements of the proper motions of maser spots of the two species can unambiguously constrain the proposed scenarios.","Figure: Velocity of the methanol maser spots (open circles) in G31.581++00.077and G33.641++00.228 versus azimuth angle measured from the major axis.The sinusoidal line represents the best-fit kinematical model of a rotatingand expanding disc of infinitesimal thickness to the methanolspots with the parameters listed inTable 4.","For completeness, the water maser spots are shown as squares.In a case of G33.641++00.228 two kinematical models arepresented for a powering source located as follow: D1 – in the centre of the best-fittedellipse, D2 – at the MIR position.Figure: Outflow models fitted to water (squares)in G31.581++00.077 and G33.641++00.228 according to themodel of Moscadelli et al. ().","The relevant parameters are listed inTable 5.","The right panel presents a comparison of obtained data (squares) vs. model (crosses).V c _{\\rm c} is the systemic LSR velocity as given in Sect.","3."],["Conclusions","We have carried out high-angular resolution studies of the 22.2 GHz of water maser line towards two methanol maser sources G31.581$+$ 00.077 and G33.641$-$ 00.228 using the EVN.","Although their morphologies did not differ significantly from the previous VLA results, the astrometry at the mas level and the properties of the maser clusters could be estimated owing to the high-angular and spectral resolution.","In total, we detected eight and six water maser clouds towards G31.581$+$ 00.077 and G33.641$-$ 00.228, respectively.","In the first target, the water maser components are associated with different centres of star-forming activity, and the components associated with the methanol ring-like structure possibly trace the outflow.","In the source G33.641$-$ 00.228, southern water masers possibly trace a wind from a disc.","The kinematic models containing ring-like or arc-like methanol maser structures are able to trace a circumstellar disc/torus around a high-mass young stellar object, whereas the water maser distribution is orthogonal to the major axis of the applied methanol structure and poorly constrained by the present data.","The present studies show that the two sources are good targets for proper motion studies in order to understand more clearly the kinematics of gas in the environments of high-mass stellar objects.","They also encourage us to extend the multi-epoch EVN observations for a whole sample.","AB and MS acknowledge support by the Polish Ministry of Science and Higher Education through grant N N203 386937.","This work has also been supported by the European Community Framework Programme 7, Advanced Radio Astronomy in Europe, grant agreement nl.",": 227290."]],"string":"[\n [\n \"Milliarcsecond structure of water maser emission in two young high-mass\\n stellar objects associated with methanol masers\"\n ],\n [\n \"Abstract The 22.2 GHz water masers are often associated with the 6.7 GHz methanol masers but owing to the different excitation conditions they likely probe independent spatial and kinematic regions around the powering young massive star.\",\n \"We compared the emission of these two maser species on milliarcsecond scales to determine in which structures the masers arise and to test a disc-outflow scenario where the methanol emission arises in a circumstellar disc while the water emission comes from an outflow.\",\n \"We obtained high-angular and spectral resolution 22.2 GHz water maser observations of the two sources G31.581+00.077 and G33.641-00.228 using the EVN.\",\n \"In both objects the water maser spots form complex and filamentary structures of sizes 18-160 AU.\",\n \"The emission towards the source G31.581+00.077 comes from two distinct regions of which one is related to the methanol maser source of ring-like shape.\",\n \"In both targets the main axis of methanol distribution is orthogonal to the water maser distribution.\",\n \"Most of water masers appear to trace shocks on a working surface between an outflow/jet and a dense envelope.\",\n \"Some spots are possibly related to the disc-wind interface which is as close as 100-150 AU to the regions of methanol emission.\"\n ],\n [\n \"Introduction\",\n \"High-angular resolution observations of high-mass star forming sites in the 6.7 GHz methanol maser emission have revealed a diversity of morphological structures from very simple composed of a few milliarcsecond (mas) wide spots to complex and extended clouds of arcsecond sizes (Phillips et al.\",\n \"[28]; Walsh et al.\",\n \"[51]; Minier et al.\",\n \"[21]; Dodson et al.\",\n \"[13]; Bartkiewicz et al.\",\n \"[3]; Pandian et al. [27]).\",\n \"Linear structures with a velocity gradient are thought to be the signature of an edge-on disc or torus around young massive protostars (Norris et al.\",\n \"[26]; Minier et al.\",\n \"[21]) but there is evidence that some of them are associated with outflows (De Buizer [12]) or even a shock propagating through a rotating dense molecular clump (Dodson et al. [13]).\",\n \"Arc-like or ring-like morphologies of methanol emission seem to prove a disc scenario (Bartkiewicz et al.\",\n \"[3]), although most of those structures do not show signs of rotation but rather inflow or/and outflow dominate (van Langevelde et al.\",\n \"[50]), it is then postulated that methanol masers arise in the shock interface between the large scale accretion and a circumstellar disc.\",\n \"While the 6.7 GHz methanol maser line is radiatively pumped in warm ($T\\\\sim 150$ K) and dense ($n\\\\le 10^8$ cm$^{-3}$ ) regions (Cragg et al.\",\n \"[8]; Sobolev et al.\",\n \"[32]), the 22 GHz water maser emission is collisionally pumped and probes denser ($n\\\\ge 10^8$ cm$^{-3}$ ) and hotter ($T\\\\sim 400$ K) gas (Elitzur et al.\",\n \"[14]).\",\n \"The water maser is often excited in strong shocks, driven by young low-mass and high-mass (proto)stellar objects, on a working surface between a outflow and a dense envelope.\",\n \"A number of morphologies of this emission is observed from collimated jet, wide-angle flows, expanding shells to equatorial flows (Goddi et al.\",\n \"[18]; Moscadelli et al.\",\n \"[23], [24]; Sanna et al.\",\n \"[30], [31]; Torrelles et al. [42]).\",\n \"One or more types of those maser structures occur simultaneously in star forming regions on scale sizes of a few mas to several arcsec and is possibly related to the geometry of the envelopes or the mechanisms driving the outflows.\",\n \"Our recent VLA survey of 22 GHz water masers in a sample of 31 methanol sources has yielded 22 detections (Bartkiewicz et al.\",\n \"[4]) where both maser species are excited by the same central objects.\",\n \"We noted that a distinct group of methanol sources with ring-like structure show either no associated water masers at all or water masers that are distributed orthogonally to the major axis of the ring.\",\n \"It is argued that the methanol maser structure traces a circumstellar disc/torus around a high-mass young stellar object while the water masers originate in outflows.\",\n \"As the VLA data are of moderate angular and spectral resolutions of $\\\\sim $ 1 and 0.65 km s$^{-1}$ , respectively, we undertook the VLBI observations to examine a disc-outflow scenario in the two brightest water maser targets, which show the arc-like or ring-like morphologies that are characteristic of methanol masers.\",\n \"The source G31.581$+$ 00.077The names of two targets follow the Galactic coordinates of the brightest methanol maser spots derived by Bartkiewicz et al. ([3]).\",\n \"has been recognized as a massive young stellar object based on 6.7 GHz methanol maser observations (Szymczak et al. [36]).\",\n \"The methanol maser spots are distributed along an arc or ring of 217 mas size (Bartkiewicz et al.\",\n \"[3]), which corresponds to $\\\\sim $ 1200 AU for the assumed near kinematic distance of 5.5 kpc (Reid et al. [29]).\",\n \"The detection of an infrared source of bolometric luminosity of 3$\\\\times $ 10$^4$ L$_{}$ (Urquhart et al.\",\n \"[48]), 22 GHz H$_2$ O maser emission (Bartkiewicz et al.\",\n \"[4]), 1665 and 1667 MHz weak OH masers (Szymczak & Gérard [38]), millimeter molecular thermal lines (Szymczak et al.\",\n \"[40]; Urquhart et al.\",\n \"[45]), and 5 and 8.4 GHz continuum emission at location of 31.582$+$ 00.075 (i.e., 9 apart from the maser arc/ring) (Urquhart et al.\",\n \"[46]; Bartkiewicz et al.\",\n \"[3]) indicates that this is a cluster of recent star formation.\",\n \"The source G33.641$-$ 00.228 detected in the 6.7 GHz methanol line (Szymczak et al.\",\n \"[36]) has an arc spot distribution of length 630 AU (Bartkiewicz et al.\",\n \"[3]) for the assumed near kinematic distance of 3.8 kpc (Reid et al.\",\n \"[29]), which seems to be more likely than the far kinematic distance (Zhang priv.comm.).\",\n \"The site also contains water and OH masers (Bartkiewicz et al.\",\n \"[4]; Szymczak & Gérard [38]) but no 8.4 GHz continuum emission was detected with an upper limit of 0.15 Jy (Bartkiewicz et al. [3]).\",\n \"The detection of radio recombination lines at about 103 km s$^{-1}$ (Anderson et al.\",\n \"[1]) and a multi-feature $^{13}$ CO line spectrum (Urquhart et al.\",\n \"[45]) illustrates the complexity of this molecular cloud and its clustered star formation.\"\n ],\n [\n \"Observations and data reduction\",\n \"The EVNThe European VLBI Network is a joint facility of European, Chinese, South African and other radio astronomy institutes funded by their national research councils.\",\n \"observations of G31.581$+$ 00.077 and G33.641$-$ 00.228 with the antennas at Jodrell Bank, Effelsberg, Medicina, Metsähovi, Onsala, and Yebes, were carried out at 22.23508 GHz on 2010 October 30 for 8 h (the project EB047).\",\n \"The tracking phase centres were estimated from the VLA survey for the water maser spots that were located nearest to the methanol emission in each target (Bartkiewicz et al.\",\n \"[4]) at $\\\\alpha $ =18$^{\\\\rm h}$ 48$^{\\\\rm m}$ 41951, $\\\\delta $ =$-$ 0110'02578 and $\\\\alpha $ =18$^{\\\\rm h}$ 53$^{\\\\rm m}$ 32563, $\\\\delta $ =$+$ 0031'39130 (J2000), respectively.\",\n \"A phase-referencing scheme was applied with a reference source J1851$+$ 0035 (from the VLA calibrator catalogue), using a cycle time between the maser and phase-calibrator of 60 s$+$ 90 s. This strategy yielded 2 h on-source times for each target.\",\n \"The bandwidth was set to 8 MHz and data were correlated with the Mk IV Data Processor operated by JIVE with 1024 spectral channels.\",\n \"The resulting spectral resolution was 0.1 km s$^{-1}$ .\",\n \"The velocity was measured with respect to the local standard of rest (LSR).\",\n \"The data calibration and reduction were carried out with NRAO's Astronomical Image Processing System (AIPS) using standard procedures for spectral line observations.\",\n \"We used the Effelsberg antenna as a reference.\",\n \"The amplitude was calibrated by performing measurements of the system temperature at each telescope and applying the antenna gain curves.\",\n \"The parallactic angle corrections were subsequently added to the data.\",\n \"The source 3C454.3 was used as a delay, rate, and bandpass calibrator.\",\n \"The phase-calibrator was imaged and a flux density of 190 mJy was obtained.\",\n \"The maser data were corrected for all Doppler effects and self-calibrated using the brightest and most compact maser spot.\",\n \"Finally, naturally weighted maps of spectral channels were created in the velocity range where the emission was seen in the scalar-averaged spectrum.\",\n \"For imaging, the resulting synthesized beam was 1.0$\\\\times $ 2.4 mas$^2$ with a position angle (PA) of $-$ 38, while the pixel separation was 0.2 mas in both coordinates.\",\n \"The rms noise levels (1$\\\\sigma _{\\\\rm rms}$ ) in line-free channels was typically 10 mJy beam$^{-1}$ .\",\n \"The weakest detected maser spot had a brightness of 94 mJy beam$^{-1}$ that is more than 9$\\\\sigma _{\\\\rm rms}$ .\",\n \"The positions of water maser spots in all channel maps were determined by fitting two-dimensional Gaussian models.\",\n \"The formal fitting errors resulting from the beamsize/signal-to-noise ratio were smaller than 0.1 mas.\",\n \"To determine the position accuracy of registered maser spots, we need to consider the following factors (Diamond et al.\",\n \"[10]): i) the uncertainty in the phase-reference source position of 2 masThe GSFC ICRF2 VLBI Source Position Catalog.\",\n \"; ii) the antenna positions, where the claimed accuracy of $\\\\sim $ 1 cm corresponds to 1 mas in RA and 2 mas in Dec; iii) the phase transfer over the separation between targets and the phase-calibrator: 192 (between G31.581$+$ 00.077 and J1851$+$ 0035) and 045 (between G33.641$-$ 00.228 and J1851$+$ 0035).\",\n \"These caused potential phase-solution transfer errors corresponding to 1 mas in RA and 2 mas in Dec for G31.581$+$ 00.077 and to 0.2 mas in RA and 0.4 mas in Dec for G33.641$-$ 00.228, respectively.\",\n \"In total, the absolute position accuracy (1$\\\\sigma _{\\\\rm pos}$ ) is 2.5 mas in RA and 3.5 mas in Dec for both targets.\"\n ],\n [\n \"Results\",\n \"Water maser emission towards both targets was mapped, after successful phase-referencing, in the areas of 5$\\\\times $ 5 and the velocity range explored by Bartkiewicz et al.\",\n \"([4]) using the VLA.\",\n \"The EVN maps of water maser spots (Figs.\",\n \"REF and REF ) are overlayed on the VLA maps and the 6.7 GHz methanol maser distributions obtained with the EVN (Bartkiewicz et al. [3]).\",\n \"The Spitzer GLIMPSE mapshttp://irsa.ipac.caltech.edu/data/SPITZER/GLIMPSE/ of the 4.5 $\\\\mu $ m$-$ 3.6 $\\\\mu $ m emission excess are added.\",\n \"Following the procedure described in Bartkiewicz et al.\",\n \"([3]), the water maser spots are analysed to identify individual velocity-coherent maser clouds.\",\n \"Their basic parameters, such as positions, $\\\\Delta $ RA, $\\\\Delta $ Dec, LSR velocities, V$_{\\\\rm LSR}$ , and intensities of the brightest spot, S$_{\\\\rm p}$ , of each cloud are listed in Tables REF and REF .\",\n \"The water and methanol maser spectra of single clouds are combined in Figures REF and REF with overlays of individual Gaussians, if emission was seen in at least three consecutive channels.\",\n \"A Gaussian analysis of individual clouds is carried out and the fitted flux amplitude, S$_{\\\\rm fit}$ , the full width at half-maximum, FWHM, the projected extent between the most separated single spot centres within a cloud, L$_{\\\\rm proj}$ , and the velocity gradient, V$_{\\\\rm grad}$ are listed in Table REF .\",\n \"The lower limit to the brightness temperature, T$_{\\\\rm b}$ , of each cloud is also calculated according to Eq.\",\n \"9$-$ 27 of Wrobel & Walker ([49]).\",\n \"For comparison purposes, the same analysis is done for the 6.7 GHz methanol maser data (Table REF ) obtained in 2007 June (G31.581$+$ 00.077) or 2003 June (G33.641$-$ 00.228) with the EVN.\",\n \"Below we comment in more detail on each source.\"\n ],\n [\n \"G31.581$+$ 00.077\",\n \"A total of 91 water maser spots were detected.\",\n \"They are concentrated in two distinct regions (Fig.\",\n \"REF ).\",\n \"The south-east (SE) region of size 50$\\\\times $ 30 mas$^2$ containing weak ($<$ 1.35 Jy) emission in the velocity ranges from 90.1 to 93.9 km s$^{-1}$ and from 100.0 to 103.3 km s$^{-1}$ is located close to the phase centre.\",\n \"The second region (NW) of middle velocity emission (96.8$-$ 100.7 km s$^{-1}$ ) and flux density of 2.6$-$ 16.5 Jy is located offset by $\\\\sim $ 5 to the north-west (PA$=-60$ ) of the phase centre.\",\n \"The SE region is divided into the cluster of clouds 1, 2, and 3 separated by 50 mas from the cluster of clouds 6, 7, and 8 (Fig.\",\n \"REF ).\",\n \"The clouds in the SE region show Gaussian velocity profiles with FWHM linewidths of 0.54$-$ 0.94 km s$^{-1}$ .\",\n \"Their projected sizes are 0.07$-$ 0.47 mas, which correspond to 0.9$-$ 2.6 AU for the adopted distance of 5.5 kpc.\",\n \"The brightness temperature is 0.24$-$ 1.38$\\\\times $ 10$^9$ K. We note that clouds 3 and 6 show the velocity gradient of 1.3$-$ 1.8 km s$^{-1}$ mas$^{-1}$ oriented along PAs of 1 and $-$ 23, respectively.\",\n \"It is remarkable that the two clusters of clouds (1, 2, 3 and 6, 7, 8) appear very close in space (Table REF ) forming arc-like filaments of projected length of 4 and 3.2 mas corresponding to 22 and 18 AU, respectively, and elongated at PA=$-$ 58.\",\n \"The closest clouds 3 and 6 are separated by 285 AU and a velocity difference of 6.7 km s$^{-1}$ .\",\n \"The mean linear separation between the two clusters along PA=65 is 310$\\\\pm $ 8 AU, while the velocity spread is 13.2 km s$^{-1}$ .\",\n \"The velocities of the two filaments are approximately symmetric with the regard of the systemic velocity of 96.0 km s$^{-1}$ (Szymczak et al. [40]).\",\n \"The two SE filaments are likely signatures of flattened shock surfaces (Torrelles et al. [41]).\",\n \"We found that the water cloud 7 coincides within $\\\\pm $ 0.1 km s$^{-1}$ with the methanol maser cloud k, while the spatial separation is 24.4 mas which corresponds to the projected distance of 134 AU.\",\n \"Although these two lines were observed within a time span of three years their separation seems to be real.\",\n \"This confirms that the water and methanol masers probe different parts of the environment of young massive stars (e.g., Beuther et al.\",\n \"[5]; Sanna et al.\",\n \"[30], [31]) because of the different pumping mechanisms affecting both species (Elitzur et al.\",\n \"[14]; Cragg et al. [8]).\",\n \"The NW region of elongation of 26 mas is composed of two clouds 4 and 5 of brightness temperatures of 2.7$-$ 17.1$\\\\times $ 10$^9$ K (Table REF ).\",\n \"Their velocity profiles are obviously asymmetric and nicely fitted by the sums of three and two Gaussian components (Fig.\",\n \"REF ) of FWHM linewidth of 0.5$-$ 0.9 km s$^{-1}$ .\",\n \"The velocities of the NW water maser emission largely coincide with those of the methanol masers but their linear separation is about 27000 AU.\",\n \"The map of 4.5 $\\\\mu $ m$-$ 3.6 $\\\\mu $ m emission excess (Fig.\",\n \"REF ), which possibly traces shocked molecular gas in outflows from massive stars (e.g., Davies et al.\",\n \"[11]), shows that the SE and NW maser clusters lie in a very complex region.\",\n \"Both maser clusters are likely associated with different powering sources.\",\n \"The comparison of our maps with those obtained about 14 months earlier with the VLA in a CnB configuration (Bartkiewicz et al.\",\n \"[4]) implies that the cloud 1 coincides to within 13 mas, the cloud 4 to within 56 mas, and the cloud 7 to within 64 mas with the matching VLA spots.\",\n \"The emission seen with the VLA from three components located $\\\\sim $ 1$-$ 2 eastward of the clouds 4 and 5 is not detected.\",\n \"We also note that the water maser components detected have very similar amplitudes in both EVN and VLA observations.\",\n \"Figure: G31.581++00.077.\",\n \"a) Spitzer GLIMPSE 4.5 μ\\\\mu m--3.6 μ\\\\mu m excess image overlaidwith the water maser positions (yellow asterisks) from the EVN observation and H II region(yellow cross) detected by Bartkiewicz et al.\",\n \"().b) Shows the distribution of water masers observed with the EVN (filled squares)from this paper and the VLA(open squares) from Bartkiewicz et al. ().\",\n \"The colours ofthe symbols relate to theLSR velocities as indicated on the right-hand side of the plot.\",\n \"The origin ofthe map isthe position of the brightest 6.7 GHz methanol maser spot (Bartkiewicz et al.\",\n \")(see also Table ).\",\n \"c) Shows an enlargement of the north-western (NW) water masers.d) Shows an enlargement of the south-eastern (SE) water masers together with the distributionof the 6.7 GHz methanol masers, marked by open circles, from Bartkiewicz et al. ().\",\n \"In (c) and(d), thearrows represent the velocity gradients (from blue- to red-shifted LSR velocities) detectedwithin individual cloud.\",\n \"The black numbers and letters correspond to the clouds withinternal velocity gradients, while the grey ones correspond to the clouds without internalvelocity gradients (Table ).Figure: Individual component spectra with Gaussian velocity profiles of 22.2 GHz water masers(red squares and lines) and 6.7 GHz methanol masers (blue circles and lines) towardsG31.581++00.077 detected using the EVN in 2010 and 2007, respectively.\",\n \"The numbers andletters correspond to the cloud labels as given in Table and Fig.\",\n \".The grey lines show individual Gaussian profiles fitted to the blended features.The clouds with only two single spots are not marked to improvethe clarity of the figure (i.e., clouds b, e, g).\"\n ],\n [\n \"G33.641$-$ 00.228\",\n \"Twenty-one water maser spots were detected in the velocity range of 54.4$-$ 57.6 km s$^{-1}$ over the north-south elongated area 50$\\\\times $ 280 mas$^2$ close to the phase centre (Fig.\",\n \"REF ).\",\n \"Using the above-mentioned procedure, we found four maser clouds where the emission is seen in at least three contiguous spectral channels.\",\n \"There are also two other clouds (5 and 6) that do not obey this criterion but seem to be real (Table REF ).\",\n \"All but one of the cloud are in the southward cluster at a distance $\\\\sim $ 270 mas from the phase centre.\",\n \"This filament cluster of size 41.6 mas is aligned along a PA of 78.\",\n \"For the adopted distance of 3.8 kpc, the corresponding linear scales are 1020 AU and 157 AU.\",\n \"The clouds 1$-$4 in the southern cluster have Gaussian profiles with FWHM linewidths of 0.25$-$ 0.88 km s$^{-1}$ .\",\n \"We note that none of the water clouds have an internal velocity gradient.\",\n \"The linear size of the individual clouds is 0.3$-$ 1.0 AU and the brightness temperature is 0.12$-$ 0.27$\\\\times $ 10$^9$ K (Table REF ).\",\n \"The strongest emission (T$_{\\\\rm b}=0.63\\\\times 10^9$ K) comes from cloud 2, which is a blend of two Gaussian components (Fig.\",\n \"REF ).\",\n \"Cloud 2 is $\\\\sim $ 20 mas (75 AU) away from the brightest methanol maser clouds a and r and differ in terms of velocity by 1.5 km s$^{-1}$ .\",\n \"This is probably the first such tight association of methanol and water masers.\",\n \"We note that the emission of cloud 2 is blue-shifted by 4.2 km s$^{-1}$ and that of cloud a is red-shifted by 1.2 km s$^{-1}$ relative to the systemic velocity of 61.5 km s$^{-1}$ (Szymczak et al. [40]).\",\n \"It is therefore possible that these clouds signpost a shock front in which the water emission originates behind the front, while the methanol emission (cloud a) appears outside of the shock interface in the infalling gas.\",\n \"The image of the 4.5 $\\\\mu $ m$-$ 3.6 $\\\\mu $ m emission excess (Fig.\",\n \"REF ) shows that the water maser emission is located 12 to the south-east of single mid-infrared object.\",\n \"This is likely the source powering the outflow along a PA of 165 traced by the water masers.\",\n \"The positions of the clouds 2, 3, 4 and 6 differ by about 92 mas from the VLA positions of the matching clouds (Bartkiewicz et al.\",\n \"[4]), which is well within the $\\\\sim $ 150 mas accuracy of the CnB configuration VLA data.\",\n \"A weak ($<$ 0.9 Jy) red-shifted emission in the velocity range of 83.8$-$ 85.1 km s$^{-1}$ seen with the VLA towards cloud 2 (Fig.\",\n \"REF ) was not recovered in the VLBI observation.\",\n \"The brightness of the emission from cloud 2 determined in the EVN observation is only slightly lower than that measured with the VLA.\",\n \"In contrast, the maser spots from the southern region (clouds 1, 3$-$6) are about ten times weaker in the EVN maps that may suggest the presence of diffuse and/or highly variable emission.\",\n \"Figure: Same as Fig.\",\n \"but for G33.641--00.228.Figure: Same as Fig.\",\n \"but for G33.641--00.228.\",\n \"Thescale on the left side describes the water maser intensity, while theright side corresponds to the methanol maser intensity.\",\n \"The clouds with onlytwo single spots or without Gaussian characteristic are not marked for theclarity of the figure (i.e., clouds 5, 6, e, g, i, n, p).Table: List of 22.2 GHz water maser clouds as found in EVN observationstowards G31.581++00.077.\",\n \"The(0,0) point corresponds to the position of the brightest methanol maser in this source(Bartkiewicz et al.\",\n \"): RA=18 h ^{\\\\rm h}48 m ^{\\\\rm m}4194108,Dec=--0110025281 (J2000).Table: List of 22.2 GHz water maser clouds as observed using EVN towards G33.641--00.228.The (0,0) point correspond to the brightest methanol maser in this source(Bartkiewicz et al.\",\n \"): RA=18 h ^{\\\\rm h}53 m ^{\\\\rm m}32563,Dec=++003139180 (J2000).Table: Parameters of 22.2 GHz water and 6.7 GHz methanol maser clouds.\"\n ],\n [\n \"Maser environments\",\n \"The Spitzer GLIMPSE map (Fig.\",\n \"REF ) clearly indicates that the source G31.581$+$ 00.077 lies in a large (35$\\\\times $ 25) complex area with an 4.5 $\\\\mu $ m emission excess in the form of clumped and diffuse structures.\",\n \"This extended emission is proposed to be a tracer of shocked gas from regions where outflowing gas interacts with the surrounding medium (Davis et al.\",\n \"[11]; Cyganowski et al. [9]).\",\n \"The $^{13}$ CO(1$-$ 0) spectrum obtained with a 46 beam (Urquhart et al.\",\n \"[45]) towards the position 31.5808+0.0757 has two Gaussian components centred at 96.2 and 109.8 km s$^{-1}$ .\",\n \"A weak emission line is clearly detected in the range of 85 to 105 km s$^{-1}$ , i.e., in the wings of the first component suggests the presence of outflows.\",\n \"A signature of ordered motions also appears in the HCO$^+$ (1$-$ 0) spectrum (Szymczak et al.\",\n \"[40]), where in this slightly asymmetric profile the strongest emission is blueward of the source systemic velocity and can be interpreted as the result of infall (e.g., Fuller et al. [16]).\",\n \"We then find 92 east of the SE water maser an H II region of flux density 10.3 and 15 mJy at 5 GHz (White et al.\",\n \"[53]) and 8.4 GHz (Bartkiewicz et al.\",\n \"[3]), respectively.\",\n \"This continuum source coincides to within 06 with a clump of 3.6 $\\\\mu $ m$-$ 4.5 $\\\\mu $ m emission excess (Fig.\",\n \"REF ).\",\n \"The NW water masers are located at the borders of two clumps of excited gas, and their powering source is unclear although it seems unlikely that the SE and NW masers are associated with the same central object.\",\n \"A comparison of the present EVN maps with those obtained with the VLA 14 months earlier (Bartkiewicz et al.\",\n \"[4]) implies that there is a lack of water masers eastward of the NW clouds (Fig.\",\n \"REF ).\",\n \"It is likely that these are diffuse and low intensity masers resolved out with the EVN beam.\",\n \"The spectrum obtained with a 40 beam sometime between November 2009 and December 2010 (Urquhart et al.\",\n \"[48]) indicates that the peak flux density of 125 Jy at 99.7 km s$^{-1}$ is about one order of magnitude higher than that measured with the EVN.\",\n \"We suggest that the water masers of G31.581$+$ 00.077 are produced by different young stellar objects in a complex region composed of at least a few high-mass stars well-signposted by the radio continuum and the methanol and water masers.\",\n \"The water maser in G33.641$-$ 00.228 lies only 12 south-eastward of the maximum of the 3.6 $\\\\mu $ m$-$ 4.5 $\\\\mu $ m emission excess of cometary-like morphology (Fig.\",\n \"REF ).\",\n \"The HCO$^+$ (1$-$ 0) spectrum obtained towards the position 33.648$-$ 0.224 (Szymczak et al.\",\n \"[40]) shows a small dip near 61.5 km s$^{-1}$ and a slight redward asymmetry that is indicative of outflow motions.\",\n \"The less optically thin H$^{13}$ CO$^+$ (1$-$ 0) transition is detected as marginal red-shifted emission, which is consistent with an outflow.\",\n \"In the region where the OH 1665, 1667, and 1720 MHz masers were detected (Szymczak & Gérard [38]) and all of them peak at 60.2 km s$^{-1}$ .\",\n \"The strongest 1720 MHz emission and a broad (7.4 km s$^{-1}$ ) 1667 MHz absorption profile near 56.2 km s$^{-1}$ are indicative of shock fronts and a continuum background source, respectively.\",\n \"No continuum emission at 8.4 GHz was detected with a 3$\\\\sigma _{\\\\rm rms}$ limit of 0.15 mJy beam$^{-1}$ (Bartkiewicz et al. [3]).\",\n \"A weak ($\\\\sim $ 20 mJy) 8.6 GHz hydrogen radio recombination-line near 102.9 km s$^{-1}$ detected at position G033.645$-$ 0.227 (Anderson et al.\",\n \"[1]) differs so greatly in terms of velocity from the water masers that this may be a chance projection that is unassociated with the methanol and water masers.\",\n \"We argue that the water masers of G33.641$-$ 00.228 as well as the associated methanol and hydroxyl masers are excited by individual high-mass stars.\"\n ],\n [\n \"Physical parameters of circumstellar medium\",\n \"Lower limits to the brightness temperature of the SE water maser components associated with G31.581$+$ 00.077 are always lower than 1.4$\\\\times $ 10$^9$ K, while the measured linewidths range from 0.54 to 0.94 km s$^{-1}$ are quite commonly in the star-forming sources (e.g., Goldreich & Kwan [19]; Surcis et al.\",\n \"[34], [35]).\",\n \"Since the kinetic temperature of the masing gas might be expected to be about 400 K (Elitzur et al.\",\n \"[14]), the intrinsic thermal linewidth given by $\\\\Delta v_{\\\\rm FWHM}=2.35482\\\\times \\\\sqrt{k T_k/m}$ , where $T_k$ is a kinetic temperature, $k$ the Boltzmann constant, and $m$ is a molecular mass should be $\\\\sim $ 1 km s$^{-1}$ .\",\n \"This value is larger than the observed linewidths and suggests that the masers are unsaturated.\",\n \"In the same volume of gas where the water maser cloud 7 and the methanol maser cloud k appear to coincide (Fig.\",\n \"REF ), the methanol thermal linewidth would be 0.75 km s$^{-1}$ , whereas the measured value is 0.30 km s$^{-1}$ .\",\n \"The narrowing of the line profile is expected when the maser is unsaturated.\",\n \"Detailed calculations have shown that in one of the strongest known methanol sources NGC7538 about 92% of the components are unsaturated (Surcis et al. [35]).\",\n \"In the source G33.641$-$ 00.228, the observed linewidths of methanol components are generally narrower than in G31.581$+$ 00.077, which suggests that the maser is also unsaturated in this region.\",\n \"The mas-scale velocity gradients in both lines are observed in G31.581$+$ 00.077.\",\n \"We note that the velocity gradient at 22 GHz is about one order of magnitude higher than at 6.7 GHz (Table REF ).\",\n \"Extensive discussion of the velocity gradients for methanol masers by Moscadelli et al.\",\n \"([25]) suggests that there is a kinematical interpretation of their origin.\",\n \"We note that the methanol maser gradients in both targets can reflect the ordered motions on scales of 10$-$ 60 AU.\",\n \"Large velocity gradients for water masers in G31.581$+$ 00.077 suggest that they are related to the outflow motions with velocities larger than 50 km s$^{-1}$ (see Sect.\",\n \"4.3).\",\n \"Table: Parameters derived by fitting the kinematics of the rotatingand expanding disc model.\",\n \"The signs ++ and -- of the rotation andexpansion velocities refer to the clockwise or anti-clockwise rotation andoutflow or inflow for positive i.\",\n \"Both rotation and floware reversed in the case of negative i.For each source, both signs together with the sign of i could be reversedsince our model does not give the directions unambiguously.\",\n \"In the case ofG33.641--00.228, two model fits are presented as described in Sect.\",\n \"4.3.Table: Parameters derived from fitting the biconical outflow modelof Moscadelli et al. ().\",\n \"In the case ofG31.581++00.077, two model fits are presented as described in Sect.\",\n \"4.3.\"\n ],\n [\n \"Kinematic models\",\n \"The two sources investigated in this study possibly belong to a group of objects where a ring-like or arc-like methanol maser distribution traces a circumstellar disc/torus around a high-mass young stellar object, whereas the water maser distribution is orthogonal to the major axis of the methanol structure (Bartkiewicz et al. [4]).\",\n \"Furthermore, these objects are not associated with detectable continuum emission at cm wavelengths (Bartkiewicz et al.\",\n \"[3]) and may represent an early stage of evolution.\",\n \"The new EVN observations of both sources have shown that all of the individual water maser spots detected previously with the VLA (Bartkiewicz et al.\",\n \"[4]) unfold into complex and filament structures of sizes 18$-$ 160 AU.\",\n \"To examine a disc-outflow scenario in the two sources, we used a model of a rotating and expanding thin disc (Uscanga et al.\",\n \"[44]) for the methanol masers and a model of the outflow (Moscadelli et al.\",\n \"[22]) for the water masers.\",\n \"Detailed descriptions of the modelling are given in the above cited works, and elsewhere for methanol masers in discs (Bartkiewicz et al.\",\n \"[3]) and outflows (Bartkiewicz et al. [2]).\",\n \"We note that in the following we analyse the new water maser data with the methanol maser data obtained 3$-$ 7 years earlier (Bartkiewicz et al. [3]).\",\n \"However, our inspection of a few sources published so far and systematic EVN monitoring demonstrate that the overall methanol maser morphologies are stable on a scale time of 6-8 years (Bartkiewicz et al., in prep.).\",\n \"In Tables 4 and 5, we summarize the best-fit values of both models.\",\n \"For the rotating and expanding thin disc, the rotation (V$_{\\\\rm rot}$ ), expansion (V$_{\\\\rm exp}$ ), and systemic (V$_{\\\\rm sys}$ ) velocities as well as the inclination angle, i, i.e., the angle between the line-of-sight and the normal to the ring plane defined to be i$={\\\\rm acos} (\\\\frac{\\\\rm b}{\\\\rm a})$ is given for each source.\",\n \"The solutions were based on the minimization of the $\\\\chi _{\\\\rm V}^2$ function given by eq.\",\n \"8 in Uscanga et al.\",\n \"([44]).\",\n \"The outflow model for water masers is characterized by the vertex of the cone (X$_{\\\\rm o}$ , Y$_{\\\\rm o}$ ), the x–axis coinciding with the projection of the outflow on the plane of the sky at the position angle PA, the inclination angle between the outflow axis and the line-of-sight (i.e., the z–axis) $\\\\Psi $ , and the opening angle of the outflow/cone 2$\\\\theta $ .\",\n \"The systemic LSR velocities, V$_{\\\\rm c}$ , were assumed to be the same as given in Sect. 3.\",\n \"The $\\\\chi ^2$ function was assumed to be expressed as in Eq.\",\n \"3 of Moscadelli et al.\",\n \"([22]).\",\n \"The methanol emission in G31.581$+$ 00.077 is most accurately reproduced by the model where the powering source is at the centre of the best-fit ellipse and the rotation velocity of 0.9 km s$^{-1}$ is a factor of two lower than the expansion velocity (Tab.\",\n \"REF , Fig.\",\n \"REF ).\",\n \"These values are typical of the class of ring-like methanol masers (Bartkiewicz et al.\",\n \"[3]) and suggest that the methanol structure of $\\\\sim $ 1000 AU diameter cannot be interpreted as a Keplerian disc.\",\n \"It is instead proposed that the methanol masers arise on the interface region between a large-scale accretion flow and a stellar disc (van Langevelde et al.\",\n \"[50]; Torstensson et al. [43]).\",\n \"We also note that the best-fit systemic velocity of 98.54 km s$^{-1}$ corresponds well to that estimated from molecular lines (Sect.3.1).\",\n \"The SE water emission is reasonably well-fitted by two different models.\",\n \"In the first model, O1, the vertex of the cone with 2$\\\\theta $ =30 coincides with the centre of the methanol ellipse within 25 mas and the velocity of masers is 77 km s$^{-1}$ (Tab.\",\n \"REF , Fig.\",\n \"REF ).\",\n \"We argue that this fit is consistent with the outflow scenario.\",\n \"As we failed to identify any mid-infrared (MIR) counterpart to the centre of the methanol ellipse as a powering source of the methanol structure and perpendicular to the water maser outflow, we propose an alternative model.\",\n \"This second model, O2, assumes that the position of the cone vertex is just between the red-shifted and blue-shifted voids of SE water masers.\",\n \"The best-fit velocity of the outflow is $\\\\sim $ 8 km s$^{-1}$ , while the projection of the axis outflow onto the plane of the sky is roughly parallel to the main axis of the methanol structure (Tab.\",\n \"REF , Fig.\",\n \"REF ).\",\n \"In star forming regions, different centres of activity are reported on scale sizes of a few to several hundred AUs (e.g., Torrelles et al.\",\n \"[41], [42]).\",\n \"The proper motion studies of G31.581$+$ 00.077 can clearly provide conclusive evidence for or against this hypothesis.\",\n \"In the case of methanol masers in G33.641$-$ 00.228, we present the two kinematical models for a powering source located at the centre of the best-fit ellipse ($\\\\Delta $ RA, $\\\\Delta $ Dec)=(59 mas,17 mas) and that located at the MIR position ($-$ 495 mas, 380 mas).\",\n \"In the first model, D1, the infall velocity of about 1 km s$^{-1}$ is slightly higher than the rotation velocity (Tab.\",\n \"REF ).\",\n \"The structure of water masers is fitted by a narrow ouflow of 2$\\\\theta $ =18 and velocity of 15 km s$^{-1}$ (Tab.\",\n \"REF , Fig.\",\n \"REF ).\",\n \"It is consistent with a jet-like outflow of water masers roughly perpendicular to the main axis of methanol structure.\",\n \"In the second model, D2, the visible methanol structure is only a small part of the disc radius of $\\\\sim $ 1900 AU and the rotation velocity of 3.7 km s$^{-1}$ (Tab.\",\n \"REF ), which implies that the enclosed mass is about 30M$_{}$ .\",\n \"In this case, the water masers may trace the wind gas from a disc.\",\n \"We note that the kinematic models are unable to accurately reproduce the observed structures of the methanol and water masers in both targets, and that only measurements of the proper motions of maser spots of the two species can unambiguously constrain the proposed scenarios.\",\n \"Figure: Velocity of the methanol maser spots (open circles) in G31.581++00.077and G33.641++00.228 versus azimuth angle measured from the major axis.The sinusoidal line represents the best-fit kinematical model of a rotatingand expanding disc of infinitesimal thickness to the methanolspots with the parameters listed inTable 4.\",\n \"For completeness, the water maser spots are shown as squares.In a case of G33.641++00.228 two kinematical models arepresented for a powering source located as follow: D1 – in the centre of the best-fittedellipse, D2 – at the MIR position.Figure: Outflow models fitted to water (squares)in G31.581++00.077 and G33.641++00.228 according to themodel of Moscadelli et al. ().\",\n \"The relevant parameters are listed inTable 5.\",\n \"The right panel presents a comparison of obtained data (squares) vs. model (crosses).V c _{\\\\rm c} is the systemic LSR velocity as given in Sect.\",\n \"3.\"\n ],\n [\n \"Conclusions\",\n \"We have carried out high-angular resolution studies of the 22.2 GHz of water maser line towards two methanol maser sources G31.581$+$ 00.077 and G33.641$-$ 00.228 using the EVN.\",\n \"Although their morphologies did not differ significantly from the previous VLA results, the astrometry at the mas level and the properties of the maser clusters could be estimated owing to the high-angular and spectral resolution.\",\n \"In total, we detected eight and six water maser clouds towards G31.581$+$ 00.077 and G33.641$-$ 00.228, respectively.\",\n \"In the first target, the water maser components are associated with different centres of star-forming activity, and the components associated with the methanol ring-like structure possibly trace the outflow.\",\n \"In the source G33.641$-$ 00.228, southern water masers possibly trace a wind from a disc.\",\n \"The kinematic models containing ring-like or arc-like methanol maser structures are able to trace a circumstellar disc/torus around a high-mass young stellar object, whereas the water maser distribution is orthogonal to the major axis of the applied methanol structure and poorly constrained by the present data.\",\n \"The present studies show that the two sources are good targets for proper motion studies in order to understand more clearly the kinematics of gas in the environments of high-mass stellar objects.\",\n \"They also encourage us to extend the multi-epoch EVN observations for a whole sample.\",\n \"AB and MS acknowledge support by the Polish Ministry of Science and Higher Education through grant N N203 386937.\",\n \"This work has also been supported by the European Community Framework Programme 7, Advanced Radio Astronomy in Europe, grant agreement nl.\",\n \": 227290.\"\n ]\n]"},"id":{"kind":"string","value":"1204.0806"}}},{"rowIdx":1866799,"cells":{"text":{"kind":"list like","value":[["Bounds on the non-real spectrum of differential operators with\n indefinite weights"],["Abstract Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces.","Under the assumption that 0 and $\\infty$ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set.","The main results are quantitative estimates for this set, which are applied to Sturm-Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains."],["Introduction","We consider linear operators associated with an indefinite differential expression $\\mathcal {L}=\\frac{1}{w}\\,\\ell ,$ where $w\\ne 0$ is a real-valued, locally integrable weight function which changes its sign and $\\ell $ is an ordinary or partial differential expression of the form $\\ell =-\\frac{d}{dx} p\\frac{d}{dx}+q\\qquad \\text{or}\\qquad \\ell =-\\sum _{j,k=1}^n \\frac{\\partial }{\\partial x_j} \\,a_{jk} \\,\\frac{\\partial }{\\partial x_k} +a$ acting on a real interval or domain $\\Omega \\subset \\mathbb {R}^n$ , respectively.","In the first case $p^{-1}$ and $q$ are assumed to be real and (locally) integrable.","In the second case $a\\in L^\\infty (\\Omega )$ is real and $a_{jk}$ are $C^{\\infty }$ -coefficients such that $\\ell $ is formally symmetric and uniformly elliptic on $\\Omega \\subset \\mathbb {R}^n$ , and, additionally, the weight function $w$ and its inverse are essentially bounded.","Together with appropriate boundary conditions (if necessary) the differential expression $\\ell $ gives rise to a selfadjoint operator $T$ in an $L^2$ Hilbert space.","Multiplication with $1/w$ leads to the corresponding indefinite differential operator $A=\\frac{1}{w} T$ associated with $\\mathcal {L}$ in (REF ), which is selfadjoint in a weighted $L^2$ Krein space.","Most of the existing literature for differential operators with an indefinite weight focuses on regular or left-definite problems.","The spectral properties of the operators associated to $\\mathcal {L}$ in the case of a regular Sturm-Liouville expression $\\ell $ were investigated in detail, we refer to [20], the monograph [56] and the detailed references therein.","Also singular left-definite Sturm-Liouville problems are well studied.","Here the selfadjoint operator $T$ associated with $\\ell $ is uniformly positive and, hence, the corresponding indefinite differential operator $A$ associated with $\\mathcal {L}$ in (REF ) has real spectrum with a gap around zero, cf.","e.g., [15], [16], [17], [41], [42] and [56].","In the case $T\\ge 0$ it is of particular interest whether the operator $A$ is similar to a selfadjoint operator; necessary and sufficient similarity criteria can be found in, e.g., [37], [38].","The slightly more general situation where the indefinite Sturm-Liouville operator $A$ has finitely many negative squares or is quasi-uniformly positive is discussed in, e.g., [12], [14], [22].","The general non-left-definite case is more difficult, especially the situation where the essential spectrum of the selfadjoint operator $T$ associated with $\\ell $ is no longer contained in $\\mathbb {R}^+$ .","In this case subtle problems appear, as, e.g., accumulation of non-real eigenvalues to the real axis, see [8], [11], [13], [39].","The spectral properties of indefinite elliptic partial differential operators have been investigated in, e.g., [25], [26], [27], [49], [50], [51] on bounded domains and in [9], [21] on unbounded domains.","In the non-left-definite situation the indefinite differential operator $A$ typically possesses non-real spectrum.","General perturbation results for selfadjoint operators in Krein spaces from [4], [7], [30] ensure that the non-real spectrum is contained in a compact set.","To the best of our knowledge, explicit bounds on the size of this set do not exist in the literature.","For singular indefinite Sturm-Liouville operators it is conjectured in [56] that the lower bound of the spectrum of $T$ is related to a bound for the non-real eigenvalues; the numerical examples contained in [11], [12] support this conjecture.","It is one of our main objectives to confirm this conjecture and to provide explicit bounds on the non-real eigenvalues of indefinite Sturm-Liouville operators and indefinite elliptic partial differential operators.","For this, we develop here an abstract Krein space perturbation approach which is designed for applications to differential operators with indefinite weights.","Besides bounds on the non-real spectrum, our general perturbation results Theorem REF and Theorem REF in Section provide also quantitative estimates for intervals containing only spectrum of positive/negative type in terms of the norm of the perturbation and some resolvent integral, as well as information on the critical point $\\infty $ of the perturbed operator.","The basic idea of our approach is simple: If $T$ is a semibounded selfadjoint operator in a Hilbert space with some negative lower bound $\\gamma $ , then the operator $T+\\gamma $ is non-negative or uniformly positive.","Hence $A_0:=A+\\frac{\\gamma }{w}=\\frac{1}{w} (T+\\gamma )$ (or, more generally, $A_0=G^{-1}(T+\\gamma )$ with some Gram operator $G$ connecting the Hilbert and Krein space inner product) is a non-negative operator in a Krein space and therefore the spectrum of $A_0$ is real.","Moreover, the difference of $A_0$ and $A$ is a bounded operator.","In general, a bounded perturbation of $A_0$ may lead to unbounded non-real spectrum, but under the additional assumption that 0 and $\\infty $ are not singular critical points of $A_0$ , the influence of the perturbation on the non-real spectrum can be controlled.","The proofs of Theorem REF and REF are based on general Krein space perturbation techniques, norm estimates and local spectral theory; they are partly inspired by methods developed in [9], [48].","The abstract perturbation results from Section are applied to ordinary and partial differential operators with indefinite weights in Section .","First we investigate the singular indefinite Sturm-Liouville operator $A$ with $p=1$ , a real potential $q\\in L^\\infty (\\mathbb {R})$ and the particularly simple weight function $w(x) =\\operatorname{sgn}(x)$ .","It turns out in Theorem REF that the nonreal spectrum of $A$ is contained in the set $\\bigl \\lbrace \\lambda \\in \\operatorname{dist}\\,\\bigl (\\lambda , (-d,d)\\bigr )\\le 5 \\Vert q\\Vert _\\infty ,\\, |\\operatorname{Im}\\lambda |\\le 2 \\Vert q\\Vert _\\infty \\bigr \\rbrace ,$ where $-d= \\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x)$ is assumed to be negative, and estimates on real spectral points of positive and negative type are obtained as well.","Our second example is a second order uniformly elliptic operator defined on an unbounded domain $\\Omega \\subset \\mathbb {R}^n$ with bounded coefficients and an essentially bounded weight function $w$ having an essentially bounded inverse.","We emphasize that the estimates for the non-real spectrum of indefinite differential operators seem to be the first ones in the mathematical literature.","Notation: As usual, $\\mathbb {C}^+$ ($\\mathbb {C}^-$ ) denotes the open upper half-plane (the open lower half-plane, respectively), $\\mathbb {R}^+:=(0,\\infty )$ and $\\mathbb {R}^-:=(-\\infty , 0)$ .","The compactification $\\mathbb {R} \\cup \\lbrace \\infty \\rbrace $ of $\\mathbb {R}$ is denoted by $\\overline{\\mathbb {R}}$ , the compactification $\\mathbb {C} \\cup \\lbrace \\infty \\rbrace $ by $\\overline{\\mathbb {C}}$ .","By $L(\\mathcal {H},\\mathcal {K})$ we denote the set of all bounded and everywhere defined linear operators from a Hilbert space $\\mathcal {H}$ to a Hilbert space $\\mathcal {K}$ .","We write $L(\\mathcal {H})$ for $L(\\mathcal {H},\\mathcal {H})$ .","For a closed, densely defined linear operator $T$ in $\\mathcal {H}$ we denote the spectrum and the resolvent set by $\\sigma (T)$ and $\\rho (T)$ , respectively.","A point $\\lambda \\in belongs to the {\\it approximate point spectrum} $ ap(T)$ of $ T$ if there exists a sequence $ (fn)$ in $ domT$ with $ fn = 1$ for $ n$\\mathbb {N}$$ and $ (T - )fn0$ as $ n$.","The essentialspectrum of $ T$ is the set $ {C : A- is not Fredholm}$.$"],["Selfadjoint operators in Krein spaces","Let $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ be a Krein space, let $J$ be a fixed fundamental symmetry in $\\mathcal {H}$ and denote by $(\\cdot \\,,\\cdot )$ the Hilbert space scalar product induced by $J$ , i.e.","$(\\cdot \\,,\\cdot )= [J\\,\\cdot \\,,\\cdot ]$ .","The induced norm is denoted by $\\Vert \\cdot \\Vert $ .","For a detailed treatment of Krein spaces and operators therein we refer to the monographs [5] and [18].","For a densely defined linear operator $A$ in $\\mathcal {H}$ the adjoint with respect to the Krein space inner product $[\\cdot ,\\cdot ]$ is denoted by $A^+$ .","We mention that $A^+ = JA^*J$ , where $A^*$ denotes the adjoint of $A$ with respect to the scalar product $(\\cdot \\,,\\cdot )$ .","The operator $A$ is called selfadjoint in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ (or $[\\cdot \\,,\\cdot ]$ -selfadjoint) if $A = A^+$ .","This is equivalent to the selfadjointness of the operator $JA$ in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ .","The spectrum of a selfadjoint operator $A$ in a Krein space is in general not contained in $\\mathbb {R}$ , but its real spectral points belong to the approximate point spectrum (see, e.g., [18]), $\\sigma (A)\\cap \\mathbb {R}\\subset \\sigma _{{ap}}(A).$ A selfadjoint operator $A$ in $(\\mathcal {H}, [\\cdot \\,,\\cdot ])$ is said to be non-negative if $\\rho (A)\\ne \\varnothing $ and if $[Af,f]\\ge 0$ holds for all $f\\in \\operatorname{dom}A$ .","If, for some $\\gamma > 0$ , $[Af,f]\\ge \\gamma \\Vert f\\Vert ^2$ holds for all $f\\in \\operatorname{dom}A$ , then $A$ is called uniformly positive.","A selfadjoint operator $A$ in $(\\mathcal {H}, [\\cdot \\,,\\cdot ])$ is uniformly positive if and only if $A$ is non-negative and $0\\in \\rho (A)$ .","It is well-known that the spectrum of a non-negative operator $A$ in a Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ is real.","Moreover, $A$ possesses a spectral function $E_A$ on $\\overline{\\mathbb {R}}$ which is defined for all Borel sets $\\Delta \\subset \\overline{\\mathbb {R}}$ whose boundary does not contain the points 0 and $\\infty $ .","The corresponding spectral projection $E_A(\\Delta )$ is bounded and $[\\cdot \\,,\\cdot ]$ -selfadjoint; see [3], [31], [43], [45], [46] for further details.","The point 0 ($\\infty $ ) is called a critical point of $A$ if for each Borel set $\\Delta \\subset \\overline{\\mathbb {R}}$ with $0\\in \\Delta $ ($\\infty \\in \\Delta $ , respectively) such that $E_A(\\Delta )$ is defined the inner product $[\\cdot \\,,\\cdot ]$ is indefinite on the subspace $E_A(\\Delta )\\mathcal {H}$ .","Moreover, if the point 0 ($\\infty $ ) is a critical point of $A$ , it is called regular if there exists $C > 0$ such that $\\Vert E_A([-\\varepsilon ,\\varepsilon ])\\Vert \\le C$ ($\\Vert E_A([-\\varepsilon ^{-1},\\varepsilon ^{-1}])\\Vert \\le C$ , respectively) holds for each $\\varepsilon \\in (0,1)$ .","Otherwise, the critical point 0 or $\\infty $ is called singular.","In this paper we will investigate operators which are non-negative outside of some compact set $K$ with $0\\in K$ , see Definition REF belowDefinition REF is slightly more general than the definition used in [10].","Contrary to the definition in [10], a spectral projector for an operator non-negative over $\\overline{\\setminus K in the sense of Definition \\ref {d:lnn}corresponding to the set \\overline{\\setminus K does, in general, not exist.}}$.","This notion is a generalization of the above concept of non-negative operators in Krein spaces and will be used in the study of additive bounded perturbations of non-negative operators in Section and .","For a set $\\Delta \\subset we denote by $ *$ the set which is obtained by reflecting $$ in the real axis, that is, $ * = {: }$.$ Definition 2.1 Let $K=K^*\\subset be a compact set such that $ 0K$ and $ +K$ is simply connected.","A selfadjoint operator$ A$ in the Krein space $ (H,[,])$ is said to be {\\em non-negative over} $K$ if for any bounded openneighborhood $ U$ of $ K$ in $ there exists a bounded $[\\cdot \\,,\\cdot ]$ -selfadjoint projection $E_\\infty $ such that with respect to the decomposition $\\mathcal {H}= (I-E_\\infty )\\mathcal {H}\\,[\\dotplus ]\\,E_\\infty \\mathcal {H}$ the operator $A$ can be written as a diagonal operator matrix $A = \\begin{pmatrix}A_0 & 0\\\\0 & A_\\infty \\end{pmatrix},$ where $A_0$ is a bounded selfadjoint operator in the Krein space $((I-E_\\infty )\\mathcal {H},[\\cdot \\,,\\cdot ])$ whose spectrum is contained in $\\overline{\\mathcal {U}}$ and $A_\\infty $ is a non-negative operator in $(E_\\infty \\mathcal {H},[\\cdot \\,,\\cdot ])$ with $\\mathcal {U}\\subset \\rho (A_\\infty )$ .","Relations (REF ) and (REF ) encode the non-negativity outside of the set $K$ , which can also be seen in the following example.","Example 2.2 Let $A$ be the direct sum of a bounded selfadjoint operator $A_1$ in the Krein space $(\\mathcal {H}_1, [\\cdot \\,,\\cdot ]_1)$ and of a non-negative operator $A_2$ in the Krein space $(\\mathcal {H}_2, [\\cdot \\,,\\cdot ]_2)$ , $A := A_1\\times A_2.$ Let $K_{r_1}$ be the closed disc around zero whose radius is the spectral radius $r_1$ of $A_1$ , then for a bounded open neighborhood $\\mathcal {U}$ of $K_{r_1}$ in $ we set$ E:= E2(RU) $, where $ E2$ is the spectral function of $ A2$.","We obtain a decomposition of the form (\\ref {e:dec_lnn})and the operator $ A$ can be written as in (\\ref {e:deco}),where $ A0$ and $ A$ have the properties stated inDefinition \\ref {d:lnn}.","Therefore, the operator $ A$ isnon-negative over $Kr1$.$ Let $A$ be non-negative over $\\overline{\\setminus K. Then from the representation (\\ref {e:deco}) it is seen thatfor each bounded open neighborhood \\mathcal {U} of K the non-real spectrum of A is contained in \\overline{\\mathcal {U}}.","Therefore,\\begin{equation}\\sigma (A)\\setminus \\mathbb {R}\\subset K.\\end{equation}By (\\ref {e:deco})the spectral properties of A and A_\\infty in \\overline{\\mathbb {R}}\\setminus K are the same.","Therefore, we say that \\infty is a{\\em singular (regular) critical point} of A if \\infty is a singular (regular, respectively) critical point of A_\\infty .The following proposition is a direct consequence of \\cite [Theorem~3.2]{cu} andthe representation (\\ref {e:deco}).","}\\begin{proposition}Let A be a selfadjoint operator in the Krein space (\\mathcal {H},[\\cdot \\,,\\cdot ]) and assume that A is non-negative over \\overline{\\setminus K.Then \\infty is not a singular critical point of A if and only if there exists a uniformly positive operator W in the Kreinspace (\\mathcal {H},[\\cdot \\,,\\cdot ]) such that W\\operatorname{dom}A\\subset \\operatorname{dom}A.","}\\end{proposition}$ In order to characterize operators which are non-negative over $\\overline{\\setminus K in Theorem \\ref {t:eqthm} below we recallthe notions of the spectral points of positive and negative type of selfadjoint operators in Krein spaces.","The following definitioncan be found in, e.g., \\cite {j03,LcMM,lmm}.","}\\begin{definition}Let A be a selfadjoint operator in the Krein space (\\mathcal {H},[\\cdot \\,,\\cdot ]).","A point \\lambda \\in \\sigma _{{ap}}(A) is called a {\\em spectral point ofpositive {\\rm (}negative{\\rm )} type} of A if for every sequence (f_n) in \\operatorname{dom}A with \\Vert f_n\\Vert = 1 and (A - \\lambda )f_n\\rightarrow 0as n\\rightarrow \\infty we have\\liminf _{n\\rightarrow \\infty }\\,[f_n,f_n] > 0\\quad \\Big (\\limsup _{n\\rightarrow \\infty }\\,[f_n,f_n] < 0,\\;\\text{respectively}\\Big ).The set of all spectral points of positive {\\rm (}negative{\\rm )} type of A will be denoted by \\sigma _+(A) {\\rm (}\\sigma _-(A),respectively{\\rm )}.","A set \\Delta \\subset is said to be of {\\em positive {\\rm (}negative{\\rm )} type} with respect to Aif each spectral point of A in \\Delta is of positive type {\\rm (}negative type, respectively{\\rm )}.\\end{definition}$ The sets $\\sigma _+(A)$ and $\\sigma _-(A)$ are contained in $\\mathbb {R}$ , open in $\\sigma (A)$ and the non-real spectrum of $A$ cannot accumulate to $\\sigma _+(A)\\cup \\sigma _-(A)$ .","Moreover, at a spectral point $\\lambda _0$ of positive or negative type the growth of the resolvent is of order one in the sense of the following definition, see also [6], [36], [48].","Definition 2.3 Let $A$ be a selfadjoint operator in the Krein space $\\mathcal {H}$ and assume that $\\lambda _0\\in \\overline{\\mathbb {R}}$ is not an accumulation point of $\\sigma (A)\\setminus \\mathbb {R}$ .","We say that the growth of the resolvent of $A$ at $\\lambda _0$ is of order $m\\ge 1$ if there exist an open neighborhood $\\mathcal {U}$ of $\\lambda _0$ in $\\overline{ and M > 0 such that \\mathcal {U}\\setminus \\overline{\\mathbb {R}}\\subset \\rho (A) and\\begin{equation}\\Vert (A - \\lambda )^{-1}\\Vert \\le M\\,\\frac{(1+|\\lambda |)^{2m-2}}{|\\operatorname{Im}\\lambda |^{m}}\\end{equation}holds for all \\lambda \\in \\mathcal {U}\\setminus \\overline{\\mathbb {R}}.","}$ Clearly, if $\\lambda _0\\ne \\infty $ , Definition REF can be formulated equivalently by replacing the enumerator in () by 1.","Moreover, if the growth of the resolvent of $A$ at $\\lambda _0$ is of order $m$ , then it is of order $n$ for each $n > m$ .","The following theorem gives an equivalent characterization for non-negative operators in a neighbourhood of $\\infty $ .","Theorem 2.4 Let $A$ be a selfadjoint operator in the Krein space $\\mathcal {H}$ and let $K=K^*\\subset be a compact set such that $ 0K$ and$ +K$ is simply connected.","Then $ A$ is non-negative over $K$ if and only if the following conditions are satisfied:\\begin{enumerate}\\item [{\\rm (i)}] \\sigma (A)\\setminus \\mathbb {R}\\subset K and (\\sigma (A)\\setminus K)\\cap \\mathbb {R}^\\pm \\subset \\sigma _\\pm (A).\\item [{\\rm (ii)}] The growth of the resolvent of A at \\infty is of finite order.\\end{enumerate}$ Let $A$ be non-negative over $\\overline{\\setminus K.It is well-known that the growth of the resolvent of a non-negative operator in a Krein space is of finite orderfor all \\lambda _0 \\in \\overline{\\mathbb {R}} and that spectral points in \\mathbb {R}^+ (\\mathbb {R}^-) are ofpositive (resp.\\ negative) type, see, e.g., \\cite [Proposition II.2.1 and Proposition II.3.1]{L82} and \\cite [Section 2.1]{J88}.Therefore, the statements (i) and (ii) are consequences ofthe representation (\\ref {e:deco}); cf.","(\\ref {nrspec}).","}Conversely, assume that (i) and (ii) are satisfied.","Then the operator $ A$ is definitizable over $K$, see \\cite {j03}.In particular $ A$ possesses a local spectral function $ E$ which is defined on all Borel sets $$\\mathbb {R}$K$ forwhich neither $$ nor points of $ K$\\mathbb {R}$$ are boundary points.","For such aset $$ the following holds:\\begin{enumerate}\\item [{\\rm (a)}] E(\\Delta ) is a bounded [\\cdot \\,,\\cdot ]-selfadjoint projection and commuteswith every bounded operator which commutes with the resolvent of A;\\item [{\\rm (b)}] \\sigma (A|E(\\Delta )\\mathcal {H})\\,\\subset \\,\\sigma (A)\\cap \\overline{\\Delta };\\item [{\\rm (c)}] \\sigma (A|(I - E(\\Delta ))\\mathcal {H})\\,\\subset \\,\\sigma (A)\\setminus \\operatorname{int}(\\Delta ), where\\operatorname{int}(\\Delta ) is the interior of \\Delta with respect to the topology of\\overline{\\mathbb {R}};\\item [{\\rm (d)}] If, in addition, \\Delta is a neighbourhood of \\infty (with respect to the topology of\\overline{\\mathbb {R}}), then A|(I-E(\\Delta )) \\mathcal {H} is a bounded operator;\\end{enumerate}cf.","\\cite [Definition 3.13, Theorems 3.15 and 4.8, Remark 4.9]{j03} and note that (d) follows from the definition of the local spectral function\\cite [Definition 3.13]{j03} and the definition of the extended spectrum $ (A)$in \\cite {j03}.$ Let $\\mathcal {U}$ be a bounded open neighborhood of $K$ in $ and set $ E:= E($\\mathbb {R}$U)$.","By (a), with respect to the decomposition$$\\mathcal {H}= (I - E_\\infty )\\mathcal {H}\\,[\\dotplus ]\\,E_\\infty \\mathcal {H},$$the operator $ A$ is represented by$$A = \\begin{pmatrix}A_0 & 0\\\\0 & A_\\infty \\end{pmatrix},$$and by (b) the spectrum of the operator $ A$ is real, $ (A) R U$.","According to (i) and (ii),the operator $ A$ is definitizable over $$ and thus definitizable, see \\cite [Theorem 4.7]{j03}.Therefore, \\cite [Corollary 3 of Proposition II.5.2]{L82} implies that $ A-1$ (and thus $ A$) is non-negative in $ EH$.By (d), the operator $ A0= A|(I-E) H$ is bounded and, by (c),$ (A0) (A) int($\\mathbb {R}$U)U$.\\Box $"],["Bounded selfadjoint perturbations of non-negative operators","In this section we prove two abstract results on additive bounded perturbations of non-negative (and some closely connected class of) operators in Krein spaces that lead to perturbed operators which are non-negative over some neighborhood of infinity.","In both cases the neighborhood is given in quantitative terms.","The results will be applied in Section to singular indefinite Sturm-Liouville operators and to second order elliptic operators with indefinite weights."],["Two perturbation results","The following notation will be useful when formulating our main results below: For a set $\\Delta \\subset \\mathbb {R}$ and $r > 0$ we define $K_r(\\Delta ) := \\lbrace z\\in \\operatorname{dist}(z,\\Delta )\\,\\le \\,r\\rbrace .$ Our first main theorem concerns bounded selfadjoint perturbations of non-negative operators in Krein spaces.","Theorem 3.1 Let $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ be a Krein space with fundamental symmetry $J$ and norm $\\Vert \\cdot \\Vert = [J\\cdot ,\\cdot ]^{1/2}$ .","Let $A_0$ be a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ such that 0 and $\\infty $ are not singular critical points of $A_0$ , and $0\\notin \\sigma _p(A_0)$ .","Furthermore, let $V\\in L(\\mathcal {H})$ be a selfadjoint operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","Then the following holds: (i) If $V$ is non-negative, then $A_0+V$ is a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","(ii) If $V$ is not non-negative, then $A_0+V$ is non-negative over $\\overline{\\setminus K_r((-d,d)),where\\begin{equation}r = \\frac{1+\\tau _0}{2}\\Vert V\\Vert ,\\qquad \\quad d = -\\frac{1+\\tau _0}{2}\\,\\min \\sigma (JV),\\end{equation}and\\begin{equation}\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert <\\infty .\\end{equation}}Moreover, in both cases $$ is not a singular critical point of $ A0 + V$.$ The following simple example shows that without further assumptions a bounded selfadjoint perturbation $V$ of a non-negative or uniformly positive $A_0$ in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ may lead to an operator $A_0+V$ with unbounded non-real spectrum.","Example 3.2 Let $(\\cal K,(\\cdot ,\\cdot ))$ be a Hilbert space and let $H$ be an unbounded selfadjoint operator in $\\cal K$ such that $\\sigma (H)\\subset (0,\\infty )$ .","Equip $\\cal H:=\\cal K\\oplus \\cal K$ with the Krein space inner product $\\left[\\begin{pmatrix}k_1 \\\\ k_2\\end{pmatrix},\\begin{pmatrix}l_1 \\\\ l_2\\end{pmatrix}\\right] :=(k_1,l_2)+(k_2,l_1),\\qquad \\begin{pmatrix}k_1 \\\\ k_2\\end{pmatrix},\\begin{pmatrix}l_1 \\\\ l_2\\end{pmatrix}\\in \\cal H,$ and consider the operators $A_0=\\begin{pmatrix} 0 & I \\\\ H & 0\\end{pmatrix},\\quad V=\\begin{pmatrix} 0 & -I \\\\ 0 & 0\\end{pmatrix},\\quad \\text{and}\\quad A_0+V =\\begin{pmatrix} 0 & 0 \\\\ H & 0\\end{pmatrix}.$ It is easy to see that $A_0$ is a non-negative operator and $V$ is a bounded selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","Moreover, as $\\operatorname{dom}(A_0+V)=\\operatorname{dom}H\\oplus {\\cal H}$ we conclude $\\operatorname{ran}(A_0+V-\\lambda )\\ne \\cal H$ for every $\\lambda \\in , that is, $ (A0+V)=.","Our second main result applies to operators of the form $A = G^{-1}T$ , where $T$ is a semibounded selfadjoint operator in a Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ and $G$ is a selfadjoint bounded and boundedly invertible operator in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ .","In other words, $A$ is selfadjoint in the Krein space $(\\mathcal {H},(G\\cdot ,\\cdot ))$ and $A + \\eta G^{-1}$ is uniformly positive for suitable $\\eta $ .","Such a situation arises, e.g., when considering elliptic differential operators with an indefinite weight function; cf.","Section 4.2.","Theorem 3.3 Let $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ be a Hilbert space, $\\Vert \\cdot \\Vert = (\\cdot \\,,\\cdot )^{1/2}$ and $[\\cdot \\,,\\cdot ]=(G\\cdot ,\\cdot )$ be as above.","Let $A$ be a selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ such that for some $\\gamma > 0$ we have $[Af,f]\\ge -\\gamma \\Vert f\\Vert ^2\\quad \\text{for all }f\\in \\operatorname{dom}A.$ Assume furthermore that for some $\\eta > \\gamma $ $\\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A + \\eta G^{-1} - it)^{-1}\\,dt\\,\\right\\Vert < \\infty ,$ holds.","Then $A$ is non-negative over $\\overline{\\setminus K_r((-r,r)), where\\begin{equation*}r = \\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert G^{-1}\\Vert .\\end{equation*}Moreover, \\infty is not a singular critical point of A.","}$ Remark 3.4 Setting $A_0 := A + \\eta G^{-1}$ in Theorem REF and $V:=-\\eta G^{-1}$ we have $A = A_0 + V$ and hence Theorem REF can also be seen as a variant of Theorem REF .","Here $A_0=A + \\eta G^{-1}$ is uniformly positive so that $0\\in \\rho (A_0)$ is not a critical point of $A_0$ and the entire imaginary axis belongs to $\\rho (A + \\eta G^{-1})$ .","It follows from [55] and [32] that $\\tau _\\eta < \\infty $ is equivalent to $\\infty $ not being a singular critical point of $A_0$ .","In what follows we describe the structure of the proofs of Theorem REF (ii) and Theorem REF .","Let $E$ denote the spectral function of the non-negative operator $A_0$ .","As 0 and $\\infty $ are not singular critical points of $A_0$ , the spectral projections $E_+ := E((0,\\infty ))$ and $E_- := E((-\\infty ,0))$ , and the corresponding spectral subspaces $\\mathcal {H}_\\pm := E_\\pm \\mathcal {H}$ of $A_0$ exist.","From $0\\notin \\sigma _p(A_0)$ it follows that $\\mathcal {H}= \\mathcal {H}_+\\,[\\dotplus ]\\,\\mathcal {H}_-$ and $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ are Hilbert spaces, see, e.g., [46].","Therefore (REF ) is a fundamental decomposition of the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","With respect to this decomposition the operator $A_0$ and the fundamental symmetry $\\widetilde{J}$ corresponding to (REF ) can be written as operator matrices: $A_0 = \\begin{pmatrix}A_{0,+} & 0\\\\0 & A_{0,-}\\end{pmatrix}\\quad \\text{and}\\quad \\widetilde{J} = \\begin{pmatrix}I & 0\\\\0 & -I\\end{pmatrix}.$ Note that the operator $\\pm A_{0,\\pm }$ is a selfadjoint non-negative operator in the Hilbert space $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ .","Hence, $A_0$ is selfadjoint in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim )$ , where $(f,g)_\\sim := [\\widetilde{J}f,g] = [E_+f,g] - [E_-f,g],\\qquad f,g\\in \\mathcal {H}.$ Now, let $V\\in L(\\mathcal {H})$ be a bounded selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","Then, with respect to the decomposition (REF ) it admits an operator matrix representation $V = \\begin{pmatrix}V_+ & B\\\\C & V_-\\end{pmatrix}.$ From the selfadjointness of $V$ in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ one concludes that $V_\\pm $ is selfadjoint in the Hilbert space $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ]) = (\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim )$ and $C = -B^{\\widetilde{*}},$ where $B^{\\widetilde{*}}$ denotes the adjoint of $B$ with respect to the scalar product $(\\cdot \\,,\\cdot )_\\sim $ .","Hence, the perturbed operator $A := A_0 + V$ is represented by $A = \\begin{pmatrix}A_{0,+} + V_+ & B\\\\-B^{\\widetilde{*}} & A_{0,-} + V_-\\end{pmatrix}.$ For operators as in (REF ) we show in Theorem REF below that $\\mathbb {R}\\setminus K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,-} + V_-))$ is of positive type, $\\mathbb {R}\\setminus K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,+} + V_+))$ is of negative type, the non-real spectrum of $A$ is contained in $K:=K_{\\Vert B\\Vert _\\sim }\\bigl (\\sigma (A_{0,+} + V_+)\\bigr )\\cap K_{\\Vert B\\Vert _\\sim } \\bigl (\\sigma (A_{0,-} + V_-)\\bigr ),$ and the growth of the resolvent of $A$ at $\\infty $ is of order one.","Here $\\Vert \\cdot \\Vert _\\sim $ stands for the operator norm induced by (REF ).","Then, by Theorem REF , the operator $A$ is non-negative over $\\overline{\\setminus K, where K is as in (\\ref {BadHersfeld}).In a final step it remains to bound the quantities\\Vert B\\Vert _\\sim \\quad \\mbox{and} \\quad K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,\\pm } + V_\\pm ))by the quantitiesr and \\tau _0 as in the statement of Theorem \\ref {t:main1}.","}$ Some auxiliary statements In this section we formulate and prove some auxiliary statements which will be used in the proofs Theorems REF and REF .","For the special case of bounded operators the first part of the following theorem was already proved in [48], see also [47] and [53], [54] for similar results with unbounded entries on the diagonal.","Theorem 3.5 Let $S_+$ and $S_-$ be selfadjoint operators in some Hilbert spaces $\\mathfrak {H}_+$ and $\\mathfrak {H}_-$ , respectively, let $M\\in L(\\mathfrak {H}_-,\\mathfrak {H}_+)$ and define the operators $S := \\begin{pmatrix}S_+ & M\\\\-M^* & S_-\\end{pmatrix}\\quad \\text{and}\\quad \\mathfrak {J}:= \\begin{pmatrix}I & 0\\\\0 & -I\\end{pmatrix}$ in the Hilbert space $\\mathfrak {H}:= \\mathfrak {H}_+\\oplus \\mathfrak {H}_-$ .","Then the operator $S$ is selfadjoint in the Krein space $(\\mathfrak {H},(\\mathfrak {J}\\cdot ,\\cdot ))$ and with $\\nu := \\Vert M\\Vert $ the following statements hold: $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_+))\\cap K_\\nu (\\sigma (S_-))$ .","$\\mathbb {R}\\setminus K_\\nu (\\sigma (S_-))$ is of positive type with respect to $S$ .","$\\mathbb {R}\\setminus K_\\nu (\\sigma (S_+))$ is of negative type with respect to $S$ .","Moreover, if $S_+$ is bounded from below and $S_-$ is bounded from above (or vice versa), the non-real spectrum of $S$ is bounded and the growth of the resolvent of $S$ at $\\infty $ is of order one.","We set $[\\cdot \\,,\\cdot ]:= (\\mathfrak {J}\\cdot ,\\cdot )$ .","Let $\\lambda \\in \\mathbb {C} \\setminus K_\\nu (\\sigma (S_-))$ and $\\alpha := \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_-)) < 1$ .","We claim that for some $r(\\alpha )>0$ and all $f\\in \\operatorname{dom}S$ the following implication $\\Vert (S - \\lambda )f\\Vert \\le \\frac{1-\\alpha ^2}{4\\alpha }\\nu \\Vert f\\Vert \\quad \\Longrightarrow \\quad [f,f]\\,\\ge \\,r(\\alpha )\\Vert f\\Vert ^2$ holds.","In fact, set $\\varepsilon := \\frac{1-\\alpha ^2}{4\\alpha }$ and let $f\\in \\operatorname{dom}S$ with $\\Vert (S - \\lambda )f\\Vert \\le \\varepsilon \\nu \\Vert f\\Vert .$ With respect to the decomposition $\\mathfrak {H}= \\mathfrak {H}_+\\oplus \\mathfrak {H}_-$ we write $f$ and $g := (S - \\lambda )f$ as column vectors $f = \\begin{pmatrix}f_+\\\\f_-\\end{pmatrix}\\in \\operatorname{dom}S_+\\oplus \\operatorname{dom}S_-, \\quad g = \\begin{pmatrix}g_+\\\\g_-\\end{pmatrix}.$ Then $g_- = -M^*f_+ + (S_- - \\lambda )f_-$ , or, equivalently, $f_- = (S_- - \\lambda )^{-1}g_- + (S_- - \\lambda )^{-1}M^*f_+.$ As $\\Vert g_-\\Vert \\le \\Vert g\\Vert = \\Vert (S - \\lambda )f\\Vert \\le \\varepsilon \\nu \\Vert f\\Vert $ and $\\Vert (S_- - \\lambda )^{-1}\\Vert = \\alpha /\\nu $ this yields $\\Vert f_-\\Vert \\,\\le \\,\\alpha \\varepsilon \\Vert f\\Vert + \\alpha \\Vert f_+\\Vert $ and hence $\\Vert f_-\\Vert ^2\\,\\le \\,\\alpha ^2\\big (\\varepsilon ^2\\Vert f\\Vert ^2 + 2\\varepsilon \\Vert f\\Vert ^2 + \\Vert f_+\\Vert ^2\\big ).$ Using $\\Vert f_+\\Vert ^2 = \\Vert f\\Vert ^2 - \\Vert f_-\\Vert ^2$ we conclude $\\Vert f_-\\Vert ^2\\,\\le \\,\\frac{\\alpha ^2}{1+\\alpha ^2}(1 + \\varepsilon )^2\\Vert f\\Vert ^2.$ Hence, as $[f,f] = \\Vert f_+\\Vert ^2 - \\Vert f_-\\Vert ^2 = \\Vert f\\Vert ^2 - 2\\Vert f_-\\Vert ^2$ , we obtain $[f,f]\\,\\ge \\,r(\\alpha )\\Vert f\\Vert ^2$ , where $r(\\alpha ) := 1 - \\frac{2\\alpha ^2}{1+\\alpha ^2}(1 + \\varepsilon )^2 = \\frac{(1 - \\alpha )^2(1 + \\alpha )(7 - \\alpha )}{8(1+\\alpha ^2)} > 0,$ i.e., the implication (REF ) holds.","It follows that $(K_\\nu (\\sigma (S_-)))\\cap \\sigma _{{ap}}(S)\\subset \\sigma _{+}(S)$ .","Hence (ii) is proved and, as $\\sigma _{+}(S)$ is real, the non-real spectrum of $S$ satisfies $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_-)).$ Similarly, as in the proof of (REF ), one proves that for $\\lambda \\in \\mathbb {C} \\setminus K_\\nu (\\sigma (S_+))$ , $f\\in \\operatorname{dom}S$ , and for $\\beta := \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_+))<1$ the implication $\\Vert (S - \\lambda )f\\Vert \\le \\frac{1-\\beta ^2}{4\\beta }\\nu \\Vert f\\Vert \\quad \\Longrightarrow \\quad [f,f]\\,\\le \\,-r(\\beta )\\Vert f\\Vert ^2$ holds with $r(\\beta )>0$ as in (REF ).","This shows (iii) and $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_+))$ .","Together with (REF ) we obtain (i).","Assume now that $S_+$ is bounded from below and $S_-$ is bounded from above.","Then the non-real spectrum of $S$ is bounded by (i) and it remains to show that the growth of the resolvent of $S$ is of order one at $\\infty $ .","Note first that for $S_0=\\begin{pmatrix}S_+ & 0 \\\\ 0 & S_-\\end{pmatrix}\\qquad \\text{and}\\qquad V=\\begin{pmatrix}0 & M \\\\ -M^* & 0\\end{pmatrix},$ and $\\lambda \\in such that $ Im2V= 2$ we have $ (S0-)-1V12$ and\\begin{equation}\\Vert (S-\\lambda )^{-1}\\Vert = \\bigl \\Vert \\bigl ((S_0-\\lambda )(I+(S_0-\\lambda )^{-1}V)\\bigr )^{-1}\\bigr \\Vert \\le \\frac{2}{\\vert \\operatorname{Im}\\lambda \\vert }.\\end{equation}For $ with $\\vert \\operatorname{Im}\\lambda \\vert < 2\\nu $ and $\\operatorname{dist}(\\lambda ,\\sigma (S_-)) > \\nu + \\delta $ with some fixed $\\delta > 0$ again define the value $\\alpha = \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_-))$ .","Then $\\alpha \\in (0,\\delta ^{\\prime })$ , where $\\delta ^{\\prime } = \\nu /(\\nu + \\delta )\\in (0,1)$ , and for $f\\in \\operatorname{dom}S$ we either have $\\Vert (S - \\lambda )f\\Vert \\ge \\frac{1-\\alpha ^2}{4\\alpha }\\nu \\Vert f\\Vert > \\frac{1-\\alpha ^2}{8\\alpha }\\vert \\operatorname{Im}\\lambda \\vert \\,\\Vert f\\Vert ,$ or, by (REF ), $r(\\alpha )|\\operatorname{Im}\\lambda |\\Vert f\\Vert ^2\\le |\\operatorname{Im}[\\lambda f,f]| = |\\operatorname{Im}[(S - \\lambda )f,f]|\\le \\Vert (S - \\lambda )f\\Vert \\Vert f\\Vert ,$ and hence $\\Vert (S - \\lambda )f\\Vert \\ge r(\\alpha )|\\operatorname{Im}\\lambda |\\,\\Vert f\\Vert .$ For $\\lambda \\in with $ Im<2$ and $ dist(,(S+)) > + $ the estimates (\\ref {abs2}) and (\\ref {abs3}) hold in a similar form.","Therefore, (\\ref {abs1})-(\\ref {abs3}) imply that for all non-real $ with $\\operatorname{dist}(\\lambda ,\\sigma (S_-)) > \\nu + \\delta $ or $\\operatorname{dist}(\\lambda ,\\sigma (S_+)) > \\nu + \\delta $ we have $\\Vert (S-\\lambda )^{-1}\\Vert \\le \\frac{C}{\\vert \\operatorname{Im}\\lambda \\vert }$ with some $C > 0$ which does not depend on $\\lambda $ .","This shows that the growth of the resolvent $(S-\\lambda )^{-1}$ is of order one at $\\infty $ .$\\Box $ Remark 3.6 If $\\operatorname{dist}(\\sigma (S_+),\\sigma (S_-))\\ge 2\\Vert M\\Vert $ , then the spectrum of $S$ in Theorem REF is real.","This result can be improved in certain special cases, cf.","[1] and also [2], where sharp norm bounds on the operator angles between reducing subspaces of $S$ and $S_+\\oplus S_-$ are given.","We mention here also [34], [35] for comparable results in the study of operators of Klein-Gordon type, where bounded perturbations of operators with unbounded off-diagonal entries are investigated.","The following simple example shows that the bounds on the non-real spectrum of $S$ in Theorem REF are sharp.","Example 3.7 Let $\\mathfrak {H}_+ = \\mathfrak {H}_- = and $ S= 1$, $ M = z and hence $S = \\begin{pmatrix}1 & z\\\\-\\overline{z} & -1\\end{pmatrix}.$ Then $\\operatorname{dist}(\\sigma (S_+),\\sigma (S_-)) = 2$ .","If $|z|\\le 1$ , then $\\sigma (S) = \\lbrace \\pm \\sqrt{1 - |z|^2}\\rbrace $ is real; cf.","Remark REF .","If $|z| > 1$ , then $\\sigma (S) = \\lbrace \\pm i\\sqrt{|z|^2 - 1}\\rbrace $ and in this case the eigenvalues of $S$ lie on the boundary of $ K_{|z|}(\\lbrace 1\\rbrace )\\cap K_{|z|}(\\lbrace -1\\rbrace )$ .","Lemma 3.8 Let $T$ be a selfadjoint operator in the Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ with $0\\notin \\sigma _p(T)$ and let $E$ be its spectral function.","If $(a_n)$ and $(b_n)$ are sequences in $[0,\\infty )$ such that $a_n\\downarrow 0$ and $b_n\\uparrow \\infty $ as $n\\rightarrow \\infty $ , and $0\\in \\rho (T)$ if $a_k=0$ for some $k\\in \\mathbb {N}$ , then $E(\\mathbb {R}^+) - E(\\mathbb {R}^-) = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{a_n}^{b_n}\\,\\left((T + it)^{-1} + (T - it)^{-1}\\right)\\,dt\\,.$ First of all we observe that $(T + it)^{-1} + (T - it)^{-1} = 2T(T^2 + t^2)^{-1}.$ For all $f,g\\in \\mathfrak {H}$ we have $\\int _{a_n}^{b_n}\\,\\big (T(T^2 + t^2)^{-1}f,g\\big )\\,dt&= \\int _{a_n}^{b_n}\\,\\int _{\\mathbb {R}}\\,\\frac{s}{s^2 + t^2}\\,d(E_sf,g)\\,dt\\\\&= \\int _{\\mathbb {R}}\\,\\int _{a_n}^{b_n}\\,\\frac{s}{s^2 + t^2}\\,dt\\,d(E_sf,g)\\\\&= \\int _{\\mathbb {R}}\\,\\big (\\arctan (b_n/s) - \\arctan (a_n/s)\\big )\\,d(E_sf,g).\\\\&= \\big (\\arctan (b_nT^{-1})f - \\arctan (a_nT^{-1})f,g\\big ).$ Fubini's theorem can be applied here since on $\\mathbb {R}\\times [a_n,b_n]$ the integrand is bounded and the measure $d(E_sf,g)\\otimes dt$ considered as a measure in $\\mathbb {R} \\times [a_n,b_n]$ is finite.","Hence, $\\int _{a_n}^{b_n}\\,2T(T^2 + t^2)^{-1}f\\,dt = 2\\arctan (b_nT^{-1})f - 2\\arctan (a_nT^{-1})f.$ Now, we apply [52] and observe that $2\\arctan (a_nT^{-1})f$ tends to zero and $2\\arctan (b_nT^{-1})f$ tends to $\\pi \\operatorname{sgn}(T^{-1})f = \\pi \\operatorname{sgn}(T)f = \\pi (E(\\mathbb {R}^+) - E(\\mathbb {R}^-))f$ as $n\\rightarrow \\infty $ .$\\Box $ A subspace $\\mathfrak {L}$ of a Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ is called uniformly positive if there exists $\\delta > 0$ such that $[f,f]\\,\\ge \\,\\delta \\Vert f\\Vert ^2 \\quad \\mbox{for all } f\\in \\mathfrak {L}.$ Here $\\Vert \\cdot \\Vert $ is a Hilbert space norm on $\\mathfrak {H}$ with respect to which $[\\cdot \\,,\\cdot ]$ is continuous.","All such norms are equivalent, see [46].","Lemma 3.9 Let $G$ be a bounded and boundedly invertible selfadjoint operator in a Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ and set $[\\cdot \\,,\\cdot ]:= (G\\cdot ,\\cdot )$ .","Let $\\mathfrak {L}$ be a closed uniformly positive subspace in the Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ .","If $P$ ($E$ ) denotes the orthogonal projection in the Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ (in the Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ , respectively) onto $\\mathfrak {L}$ , then $P(G|\\mathfrak {L})$ is a uniformly positive selfadjoint operator in $(\\mathfrak {L},(\\cdot \\,,\\cdot ))$ , and its inverse is given by $\\bigl (P(G|\\mathfrak {L})\\bigr )^{-1} = E\\bigl (G^{-1}|\\mathfrak {L}\\bigr ).$ As $\\mathfrak {L}$ is uniformly positive, for all $f\\in \\mathfrak {L}$ we have $\\big (P(G|\\mathfrak {L})f,f\\big ) = (Gf,f) = [f,f]\\,\\ge \\,\\delta \\Vert f\\Vert ^2$ with some $\\delta > 0$ .","Thus, $P(G|\\mathfrak {L})$ is a uniformly positive operator in $(\\mathfrak {L},(\\cdot \\,,\\cdot ))$ and, in particular, $0\\in \\rho (P(G|\\mathfrak {L}))$ .","Let $g\\in \\mathfrak {L}$ and set $f := E(G^{-1}|\\mathfrak {L})g\\in \\mathfrak {L}$ .","Since for all $h\\in \\mathfrak {L}$ $(g - Gf,h) = \\bigl (G(I - E)G^{-1}g,h\\bigr ) = \\bigl [(I - E)G^{-1}g,h\\bigr ] = 0,$ we conclude $g - Gf\\in \\mathfrak {L}^\\perp $ and hence $g = PGf = P(G|\\mathfrak {L})f$ .","Therefore, $\\bigl (P(G|\\mathfrak {L})\\bigr )^{-1}g = f = E\\bigl (G^{-1}|\\mathfrak {L}\\bigr )g,$ which proves the lemma.$\\Box $ Proof of Theorem REF Assertion (i) in Theorem REF is simple: As both $A_0$ and $V$ are non-negative in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ we have $[(A_0+V)f,f]\\ge 0\\quad \\text{for all}\\quad f\\in \\operatorname{dom}(A_0+V)=\\operatorname{dom}A_0.$ The operator $A_0$ is selfadjoint in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim )$ (cf.","(REF ) and (REF )) and, as $V$ is bounded, $\\rho (A_0+V)$ is nonempty.","Thus $A_0+V$ is a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ and (i) in Theorem REF is proved.","Let $E$ be the spectral function of the non-negative operator $A_0$ , define the projections $E_+ := E((0,\\infty ))$ , $E_-:=E((-\\infty ,0))$ and the subspaces $\\mathcal {H}_\\pm := E_\\pm \\mathcal {H}$ .","Then with respect to the fundamental decomposition (REF ) the operators $A_0$ and $V$ have the form in (REF ) and (REF ), respectively.","The fundamental symmetry $\\widetilde{J}$ associated with the decomposition (REF ) is also given in (REF ).","The inner product $(\\cdot \\,,\\cdot )_\\sim $ is defined as in (REF ).","Moreover, $A_{0,\\pm }$ and $V_\\pm $ from (REF ) are selfadjoint in $(\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim ) = (\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ , $\\sigma (A_{0,\\pm })\\subset \\mathbb {R}^\\pm \\cup \\lbrace 0\\rbrace $ and (REF ) holds.","We apply Lemma REF to the non-negative operators $\\pm A_{0,\\pm }$ and obtain $\\pm I_{\\mathcal {H}_\\pm } = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{1/n}^n\\,\\left((A_{0,\\pm } + it)^{-1} + (A_{0,\\pm } - it)^{-1}\\right)\\,dt.$ Therefore, $\\widetilde{J} = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{1/n}^n\\,\\left((A_{0} + it)^{-1} + (A_{0} - it)^{-1}\\right)\\,dt,$ where the strong limit is with respect to the norm $\\Vert \\cdot \\Vert _\\sim := (\\cdot \\,,\\cdot )_\\sim ^{1/2}$ .","But as $\\Vert \\cdot \\Vert _\\sim $ and $\\Vert \\cdot \\Vert $ are equivalent norms, the limit on the right hand side of (REF ) also exists with respect to the norm $\\Vert \\cdot \\Vert $ , and coincides with $\\widetilde{J}$ .","The uniform boundedness principle now yields that $\\tau _0$ in () is finite and that $\\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _0<\\infty .$ Define the constants $\\nu := \\Vert B\\Vert _\\sim ,\\quad a := \\min \\sigma (V_+)\\quad \\text{and}\\quad b := \\max \\sigma (V_-).$ Then $A_{0,+} + V_+$ is bounded from below by $a$ and $\\sigma (A_{0,+} + V_+)\\subset [a,\\infty )$ , and $A_{0,-} + V_-$ is bounded from above by $b$ and $\\sigma (A_{0,-} + V_-)\\subset (-\\infty ,b]$ .","Making use of the representation (REF ) of $A = A_0 + V$ and Theorem REF we conclude: $\\sigma (A)\\backslash \\mathbb {R}\\subset K_\\nu ((-\\infty ,b])\\cap K_\\nu ([a,\\infty ))$ .","$(b + \\nu ,\\infty )$ is of positive type with respect to $A$ .","$(-\\infty ,a - \\nu )$ is of negative type with respect to $A$ .","The growth of the resolvent of $A$ at $\\infty $ of order one, and hence of finite order.","The statement in Theorem REF (ii) follows from Theorem REF if we show that the constants $\\nu $ , $a$ and $b$ in (REF ) satisfy $\\nu \\le r, \\quad a\\ge -d, \\quad b\\le d,$ where $r$ and $d$ are as in (), because in that case the properties (a)-(d) above hold with $\\nu $ , $a$ and $b$ replaced by $r$ , $-d$ and $d$ , respectively, and $K_\\nu ((-\\infty ,d])\\cap K_\\nu ([-d,\\infty ))= K_\\nu ((-d,d)).$ First we check $\\nu \\le r$ .","Denote by $(\\cdot \\,,\\cdot )$ the scalar product corresponding to the Hilbert space norm $\\Vert \\cdot \\Vert $ .","Using the well-known fact that the spectrum of a bounded operator is always a subset of the closure of its numerical range we obtain $\\begin{split}\\nu ^2&= \\Vert B\\Vert _\\sim ^2 = \\Vert B^{\\tilde{*}}B\\Vert _\\sim = \\sup \\lbrace |\\lambda | : \\lambda \\in \\sigma (B^{\\tilde{*}}B)\\rbrace \\\\&\\le \\sup \\lbrace |(B^{\\tilde{*}}Bf,f)| : \\Vert f\\Vert =1\\rbrace \\,\\le \\,\\Vert B^{\\tilde{*}}\\Vert \\,\\Vert B\\Vert \\\\&= \\Vert E_-(V|\\mathcal {H}_+)\\Vert \\,\\Vert E_+(V|\\mathcal {H}_-)\\Vert \\,\\le \\,\\Vert E_+\\Vert \\,\\Vert E_-\\Vert \\,\\Vert V\\Vert ^2.\\end{split}$ As $E_\\pm = \\frac{1}{2}(I\\pm \\widetilde{J})$ , it follows from (REF ) that $\\Vert E_\\pm \\Vert \\le \\frac{1+\\tau _0}{2}$ which shows $\\nu \\le r$ .","Next we verify $a\\ge -d$ .","Denote by $P_\\pm $ the orthogonal projection onto $\\mathcal {H}_\\pm $ with respect to $(\\cdot \\,,\\cdot )$ .","Then, according to Lemma REF , the operator $P_+(J|\\mathcal {H}_+)$ is selfadjoint and uniformly positive in the Hilbert space $(\\mathcal {H}_+,(\\cdot \\,,\\cdot ))$ .","Hence, the same holds for its inverse $E_+(J|\\mathcal {H}_+)$ ; cf.","Lemma REF .","This implies $\\min \\sigma (P_+(J|\\mathcal {H}_+))&= \\big (\\max \\sigma (E_+(J|\\mathcal {H}_+))\\big )^{-1} = \\Vert E_+(J|\\mathcal {H}_+)\\Vert ^{-1}\\\\&\\ge \\Vert E_+J\\Vert ^{-1} \\ge \\Vert E_+\\Vert ^{-1}\\,\\ge \\,\\frac{2}{1+\\tau _0}\\,.$ Hence, for $f_+\\in \\mathcal {H}_+$ we have $[f_+,f_+] = (Jf_+,f_+) = (P_+(J|\\mathcal {H}_+)f_+,f_+)\\,\\ge \\,\\frac{2}{1+\\tau _0}\\,\\Vert f_+\\Vert ^2.$ We also use $(V_+f_+,f_+)_\\sim = [V_+f_+,f_+] = [V_+f_+ - B^{\\tilde{*}}f_+,f_+] = [Vf_+,f_+] = (JVf_+,f_+)$ for $f\\in \\mathcal {H}_+$ to conclude $a=\\min \\sigma (V_+)&=\\inf \\lbrace (V_+f_+,f_+)_\\sim : \\Vert f_+\\Vert _\\sim = 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&\\ge \\inf \\lbrace (V_+f_+,f_+)_\\sim : \\Vert f_+\\Vert _\\sim \\le 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&= \\inf \\lbrace (JVf_+,f_+) : [f_+,f_+]\\le 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&\\ge \\inf \\bigl \\lbrace (JVf_+,f_+) : \\Vert f_+\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f_+\\in \\mathcal {H}_+\\bigr \\rbrace \\\\&\\ge \\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace .$ By assumption $V$ is not non-negative in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ and hence $0>\\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace =\\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2=\\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace .$ Therefore we finally obtain $a=\\min \\sigma (V_+)&\\ge \\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2=\\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace \\\\&= \\frac{1+\\tau _0}{2}\\,\\inf \\lbrace (JVf,f) : \\Vert f\\Vert = 1,\\,f\\in \\mathcal {H}\\rbrace \\\\&= \\frac{1+\\tau _0}{2}\\,\\min \\sigma (JV)=-d.$ The proof of the inequality $b\\le d$ is analogous and is left to the reader.","This completes the proof of (ii).","As $\\infty $ is not a singular critical point of $A_0$ , the last statement in Theorem REF on the critical point $\\infty $ of $A_0+V$ follows in both cases (i) and (ii) from Proposition and $\\operatorname{dom}A=\\operatorname{dom}(A_0+V)$ .$\\Box $ Proof of Theorem REF Let $\\gamma $ and $\\eta $ be as in Theorem REF .","Then the operator $A_0 := A + \\eta G^{-1}$ is uniformly positive in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ , $[\\cdot \\,,\\cdot ]= (G\\cdot ,\\cdot )$ , since for $f\\in \\operatorname{dom}A$ we have $[A_0 f,f] = [Af,f] + \\eta [G^{-1}f,f] \\ge -\\gamma \\Vert f\\Vert ^2 + \\eta \\Vert f\\Vert ^2 = (\\eta - \\gamma )\\Vert f\\Vert ^2.$ We will show that $A$ is non-negative over $\\overline{\\setminus K_r((-r,r)).In particular, 0\\in \\rho (A_0) is not a singular critical point of A_0and, as \\tau _\\eta < \\infty , the point \\infty is not aa singular critical point of A_0, see Remark \\ref {Mai2012}.With V := -\\eta G^{-1} we haveA = A_0 + V\\qquad \\text{and}\\qquad \\Vert V\\Vert =\\eta \\Vert G^{-1}\\Vert .As in the proof of Theorem~\\ref {t:main1} let E be the spectral function of the operator A_0.","Then the fundamentaldecomposition \\mathcal {H}= E_+\\mathcal {H}\\,[\\dotplus ]\\,E_-\\mathcal {H} in (\\ref {e:dec_expl}) generates matrix representationsof A_0, V and the corresponding fundamental symmetry \\widetilde{J} as in (\\ref {e:AJ}) and (\\ref {e:V}).The operator A_0 is selfadjoint in the Hilbert space(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim ), where (\\cdot \\,,\\cdot )_\\sim is defined in (\\ref {e:sim}).","With the help of Lemma \\ref {l:veselic} we obtain{\\begin{@align*}{1}{-1}\\widetilde{J}&= \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _0^n\\,\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\big )\\,dt\\\\&= \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{-n}^n\\,(A + \\eta G^{-1} - it)^{-1}\\,dt\\,.\\end{@align*}}This yields\\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _\\eta .Following the lines of the proof of Theorem \\ref {t:main1}, it suffices to show the inequalities\\nu \\le \\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert ,\\quad \\min \\sigma (V_+)\\ge -\\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert \\quad \\text{and}\\quad \\max \\sigma (V_-)\\le \\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert ,where \\nu =\\Vert E_\\mp (V\\vert \\cal H_\\pm ) \\Vert _\\sim .The first relation is proved as in (\\ref {e:rel_proof}) and the remaining two inequalities follow from the selfadjointness of V_\\pm in (\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim ) and{\\begin{@align*}{1}{-1}\\max \\lbrace |\\min \\sigma (V_\\pm )|,|\\max \\sigma (V_\\pm )|\\rbrace &= \\Vert V_\\pm \\Vert _\\sim = \\sup \\lbrace |\\lambda | : \\lambda \\in \\sigma (V_\\pm )\\rbrace \\\\&\\le \\sup \\lbrace |(V_\\pm f_\\pm ,f_\\pm )| : \\Vert f_\\pm \\Vert =1,\\,f\\in \\mathcal {H}_\\pm \\rbrace \\\\&\\le \\Vert V_\\pm \\Vert = \\Vert E_\\pm (V|\\mathcal {H}_\\pm )\\Vert \\,\\le \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert .\\end{@align*}}Hence, A is non-negative over \\overline{\\setminus K_r((-r,r)).","As\\operatorname{dom}A= \\operatorname{dom}A_0, we conclude from Proposition \\ref {p:curgus} that \\infty is also not a critical point of A.This completes the proof of Theorem~\\ref {t:main2}.\\Box }\\begin{remark}If, in addition to the assumptions in Theorem \\ref {t:main2}, the point 0 is neither an eigenvalue nor a singular critical pointof the non-negative operator A_0 := A + \\gamma G^{-1}, then the operator A is non-negative over the set \\overline{\\setminus K_r((-r,r)), where\\begin{equation}r = \\gamma \\,\\frac{1+\\tau _\\gamma }{2}\\,\\Vert G^{-1}\\Vert \\end{equation}and\\tau _\\gamma := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\big )\\,dt\\right\\Vert .Indeed, under the assumptions of Theorem \\ref {t:main2} for any \\varepsilon > 0 the operator A + (\\gamma + \\varepsilon )G^{-1}is selfadjoint in some Hilbert space, and thus the resolvent set ofA_0 is non-empty as A + (\\gamma + \\varepsilon )G^{-1} and A_0 differ just by a bounded operator.Therefore, A_0 is a non-negative operator in the Krein space(\\cal H,[\\cdot \\,,\\cdot ]).","Moreover,the point \\infty is not a singular critical point of theoperator A + (\\gamma + \\varepsilon )G^{-1}, see Remark \\ref {Mai2012},and by Proposition \\ref {p:curgus} \\infty is not a singular critical point of A_0.Therefore, the operator A_0 admits a matrix representation as in (\\ref {e:AJ}) and the corresponding fundamental symmetry \\widetilde{J}satisfies \\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _\\gamma (see the proof of Theorem \\ref {t:main2}).Then by a similar reasoning as in the proof ofTheorem \\ref {t:main2} we see that A is non-negativeover \\overline{\\setminus K_r((-r,r)), where r is as in (\\ref {Juni2012}).","}}\\end{remark}}$ Differential operators with indefinite weights In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights.","Indefinite Sturm-Liouville operators We consider Sturm-Liouville differential expressions of the form $\\mathcal {L}(f)(x) = \\operatorname{sgn}(x)\\big (-f^{\\prime \\prime }(x) + q(x)f(x)\\big ),\\quad x\\in \\mathbb {R}\\,,$ with a real-valued potential $q\\in L^\\infty (\\mathbb {R})$ and the indefinite weight function $\\operatorname{sgn}(\\cdot )$ .","The corresponding differential operator in $L^2(\\mathbb {R})$ is defined by $Af := \\mathcal {L}(f), \\qquad \\operatorname{dom}A := H^2(\\mathbb {R}).$ Here $H^2(\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.","By $(\\cdot \\,,\\cdot )$ and $\\Vert \\cdot \\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\mathbb {R})$ .","Let $J$ be the operator of multiplication with the function $\\operatorname{sgn}(x)$ .","This operator is obviously selfadjoint and unitary in $L^2(\\mathbb {R})$ .","Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\prime \\prime } + qf,\\quad \\operatorname{dom}T := H^2(\\mathbb {R}),$ is selfadjoint in $L^2(\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\mathbb {R}),[\\cdot \\,,\\cdot ])$ , where $[f,g] := (Jf,g) = \\int _\\mathbb {R}\\,f(x)\\overline{g(x)}\\operatorname{sgn}(x)\\,dx,\\quad f,g\\in L^2(\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.","We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].","For this, denote by $m_\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\pm f_\\pm := - f_\\pm ^{\\prime \\prime } + q_\\pm f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ defined on $\\operatorname{dom}T_\\pm := \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ) : f_\\pm (0) = 0\\rbrace $ , where $q_\\pm $ is the restriction of the function $q$ to $\\mathbb {R}^\\pm $ .","The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\pm \\infty $ , then $m_\\pm =\\lim _{x\\rightarrow \\pm \\infty }\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.","Furthermore the following holds.","If $m_+ > -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ is finite; If $m_+\\le -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .","To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.","One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].","The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .","Here only the nontrivial case $\\operatornamewithlimits{ess\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).","Theorem 4.2 Assume that $\\operatornamewithlimits{ess\\,inf}q<0$ holds.","Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\overline{\\setminus K_r((-d,d)), wherer := 5\\Vert q\\Vert _\\infty ,\\quad \\text{and}\\quad d := -5\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\cap \\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .","}\\begin{proof}The proof of Theorem~\\ref {t:SL2} is split into 4 parts.","The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\ref {t:main1}.","In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\end{proof}\\ \\\\{\\it 1.\\ Preparation: } The operator $ A0$,\\begin{equation*}A_0 f:=-\\operatorname{sgn}(\\cdot )f^{\\prime \\prime },\\qquad \\operatorname{dom}A_0=H^2(\\mathbb {R}),\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\mathbb {R}$ ),[,])$.","Furthermore, we have $ (A0) = $\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\cite {cn}.","Hence $ A0$ satisfies theassumptions in Theorem~\\ref {t:main1}.","Define $ V$ as\\begin{equation*}Vf:=\\operatorname{sgn}(\\cdot ) qf,\\qquad \\operatorname{dom}V=L^2(\\mathbb {R}),\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\mathbb {R}$ ),[,])$ with$$\\min \\sigma (JV) = \\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}\\,q(x)\\quad \\text{and}\\quad \\Vert V\\Vert = \\Vert q\\Vert _\\infty .$$By Theorem \\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\frac{1+\\tau _0}{2}\\,\\Vert q\\Vert _\\infty ,\\qquad d = -\\frac{1+\\tau _0}{2}\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x),$$and$$\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert .$$Therefore, we have to show\\begin{equation}\\tau _0\\le 9\\end{equation}and\\begin{equation}\\sigma (A)\\setminus \\mathbb {R}\\,\\subset \\,\\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .\\end{equation}$ 2.","Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\begin{pmatrix} B_+ & 0 \\\\ 0 & B_- \\end{pmatrix}\\quad \\text{in}\\quad L^2(\\mathbb {R})=L^2(\\mathbb {R}^+)\\times L^2(\\mathbb {R}^-),$ where $B_\\pm f_\\pm :=\\mp f_\\pm ^{\\prime \\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\mathbb {R}^\\pm )$ which are defined on $\\operatorname{dom}B_\\pm = \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ):f_\\pm (0)=0\\rbrace $ .","Here and in the following we will denote the restrictions of a function $f$ defined on $\\mathbb {R}$ to $\\mathbb {R}^\\pm $ by $f_\\pm $ .","For $\\lambda = re^{it}$ , $r > 0$ , $t\\in [0,2\\pi )$ , we set $\\sqrt{\\lambda } := \\sqrt{r}e^{it/2}$ .","For $\\lambda \\in \\mathbb {R}$ define the function $f_\\lambda (x) :={\\left\\lbrace \\begin{array}{ll}e^{i\\sqrt{\\lambda }x}, &x> 0,\\\\e^{-i\\sqrt{-\\lambda }x}, &x < 0.\\end{array}\\right.","}$ The function $f_\\lambda $ is a solution of the equations $\\mp f_\\pm ^{\\prime \\prime } = \\lambda f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ .","From $\\overline{\\sqrt{\\lambda }} = -\\sqrt{\\overline{\\lambda }}$ it follows that $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ holds for all $\\lambda \\in \\mathbb {R}$ .","Let us now prove that for all $f\\in L^2(\\mathbb {R})$ and $\\lambda \\in \\mathbb {R}$ we have $(A_0 - \\lambda )^{-1}f = (B_0 - \\lambda )^{-1}f - \\frac{[f,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda .$ Since $\\sigma (A_0)=\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\lambda \\in \\mathbb {R}$ .","In particular, for $\\lambda \\in \\mathbb {R}$ there exists $g\\in \\operatorname{dom}B_0$ such that $f=(B_0-\\lambda )g$ holds.","The right hand side in (REF ) has the form $h:=g-\\frac{[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda $ and we will show that the function $h$ belongs to $\\operatorname{dom}A_0=H^2(\\mathbb {R})$ .","As $g_\\pm $ and $f_{\\lambda ,\\pm }$ are elements of $H^2(\\mathbb {R}^\\pm )$ the same is true for $h_\\pm $ .","Moreover, $g$ and $f_\\lambda $ are continuous at 0 and so is $h$ .","Hence it remains to check that $h^\\prime $ is continuous at 0.","For this note first that $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=\\int _0^\\infty (-g_+^{\\prime \\prime }-\\lambda g_+)\\overline{f_{\\overline{\\lambda },+}}\\, dx- \\int _{-\\infty }^0 (g_-^{\\prime \\prime }-\\lambda g_-)\\overline{f_{\\overline{\\lambda },-}}\\, dx$ and since $f_{\\overline{\\lambda },\\pm }$ are solutions of $\\mp f_\\pm ^{\\prime \\prime }=\\overline{\\lambda }f_\\pm $ , integration by parts yields $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=g_+^\\prime (0)\\overline{f_{\\overline{\\lambda },+}(0)}-g_+(0) \\overline{f^\\prime _{\\overline{\\lambda },+}(0)}-g_-^\\prime (0)\\overline{f_{\\overline{\\lambda },-}(0)}+g_-(0) \\overline{f^\\prime _{\\overline{\\lambda },-}(0)}.$ As $g\\in \\operatorname{dom}B_0$ we have $g_\\pm (0)=0$ and together with $f_{\\overline{\\lambda },\\pm }(0)=1$ we find $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]= g_+^\\prime (0)-g^\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\pm ^\\prime $ on $\\mathbb {R}^\\pm $ of the function $h$ from (REF ): $h^\\prime _\\pm =g^\\prime _\\pm -\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_{\\lambda ,\\pm }^\\prime $ and as $f^\\prime _{\\lambda ,+}(0)=i\\sqrt{\\lambda }$ and $f^\\prime _{\\lambda ,-}(0)=-i\\sqrt{-\\lambda }$ we conclude $h^\\prime _+(0)-h^\\prime _-(0)= \\bigl (g_+^\\prime (0)-g^\\prime _-(0)\\bigr )-\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}\\bigl (f^\\prime _{\\lambda ,+}(0)-f^\\prime _{\\lambda ,-}(0)\\bigr )=0,$ that is, $h^\\prime $ is continuous at 0 and therefore $h\\in \\operatorname{dom}A_0$ .","Now a straightforward computation shows that $(A_0-\\lambda )h=(B_0-\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).","3.","Proof of (): Let $f\\in L^2(\\mathbb {R})$ and $t > 0$ .","Then (REF ) and $\\sqrt{it} + \\sqrt{-it} = i\\sqrt{2t}$ yield $\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\& + \\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\in \\mathbb {R}^+$ , we have $\\big |[f,f_{it}][f_{-it},f]\\big |\\,\\le \\,\\left(\\int _0^\\infty \\,e^{-x\\sqrt{t/2}}g(x)\\,dx\\right)^2$ and thus for $n\\in \\mathbb {N}$ $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt&\\le \\int _{1/n}^n\\,\\int _0^\\infty \\,\\int _0^\\infty \\,g(x)g(y)\\,\\frac{e^{-(x+y)\\sqrt{t/2}}}{\\sqrt{2t}}\\,dy\\,dx\\,dt\\\\&= 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\left(e^{-\\frac{x+y}{\\sqrt{2n}}} - e^{-\\frac{\\sqrt{n}(x+y)}{\\sqrt{2}}}\\right)\\,dy\\,dx\\\\&\\le 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\,dy\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt\\,\\le \\,2\\pi \\Vert g\\Vert _{L^2(\\mathbb {R}^+)}^2\\,\\le \\,4\\pi \\Vert f\\Vert ^2.$ and therefore we have $\\frac{1}{\\pi }\\int _{1/n}^n\\,\\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\,dt\\,\\le \\,8\\Vert f\\Vert ^2.$ Denote by $E_0$ ($E_\\pm $ ) the spectral function of the operator $A_0$ ($B_\\pm $ , respectively).","Moreover define $J_{n,\\pm } := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}\\right)\\,dt,$ and, in addition, $J_{n,0} := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt.$ As $\\pm B_\\pm $ is non-negative, it follows that $J_{n,\\pm }$ converges strongly to $\\pm I_{L^2(\\mathbb {R}^\\pm )}$ when $n\\rightarrow \\infty $ , cf.","Lemma REF .","Moreover, by $(B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}= 2B_\\pm (B_\\pm + it)^{-1} (B_\\pm - it)^{-1}$ the sequence $\\pm J_{n,\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\mathbb {R}^\\pm )$ .","This implies $\\frac{1}{\\pi }\\,\\int _{1/n}^n\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\&\\le \\Vert f_+\\Vert _{L^2(\\mathbb {R}^+)}^2 + \\Vert f_-\\Vert _{L^2(\\mathbb {R}^-)}^2 = \\Vert f\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\,\\le \\,\\Vert f\\Vert ^2 + 8\\Vert f\\Vert ^2 = 9\\Vert f\\Vert ^2,\\quad f\\in L^2(\\mathbb {R}).$ Hence, as $J = \\operatorname{sgn}(\\cdot )$ is unitary in $L^2(\\mathbb {R})$ , we finally obtain $\\Vert J_{n,0}\\Vert = \\Vert J J_{n,0}\\Vert \\le 9$ and thus $\\tau _0 = \\limsup _{n\\rightarrow \\infty }\\,\\Vert J_{n,0}\\Vert \\le 9$ .","That is, () holds.","4.","Proof of (): Let $\\lambda \\in +$ .","Using (REF ), for $f\\in L^2(\\mathbb {R})$ we obtain $\\Vert (A_0 - \\lambda )^{-1}f\\Vert \\,\\le \\,\\Vert (B_0 - \\lambda )^{-1}f\\Vert + \\frac{|[f,f_{\\overline{\\lambda }}]|\\,\\Vert f_\\lambda \\Vert }{|\\sqrt{\\lambda } + \\sqrt{-\\lambda }|}\\,\\le \\,\\frac{\\Vert f\\Vert }{|\\operatorname{Im}\\lambda |} + \\frac{\\Vert f_\\lambda \\Vert ^2\\Vert f\\Vert }{\\sqrt{2|\\lambda |}}\\,.$ Here we have used the identity $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ .","Let $r > 0$ and $t\\in (0,\\pi )$ such that $\\lambda = re^{it}$ .","Then $\\Vert f_\\lambda \\Vert ^2&= \\int _0^\\infty \\,\\left|e^{i\\sqrt{r}e^{it/2}x}\\right|^2\\,dx + \\int _{-\\infty }^0\\,\\left|e^{-i\\sqrt{r}e^{i(t+\\pi )/2}x}\\right|^2\\,dx\\\\&= \\int _0^\\infty \\,e^{-2\\sqrt{r}\\sin (t/2)x}\\,dx + \\int _{-\\infty }^0\\,e^{2\\sqrt{r}\\cos (t/2)x}\\,dx\\\\&= \\frac{\\cos (t/2) + \\sin (t/2)}{\\sqrt{r}\\,\\sin (t)}\\,\\le \\,\\frac{\\sqrt{2}}{\\sqrt{r}\\,\\operatorname{Im}(\\lambda /r)} = \\frac{\\sqrt{2|\\lambda |}}{|\\operatorname{Im}\\lambda |}.$ Therefore, $\\Vert (A_0 - \\lambda )^{-1}\\Vert \\,\\le \\,\\frac{2}{|\\operatorname{Im}\\lambda |}\\,.$ The same estimate holds for $\\lambda \\in -$ .","Now, assume that $|\\operatorname{Im}\\lambda | > 2\\Vert q\\Vert _\\infty =2\\Vert V\\Vert $ .","Then $\\Vert V(A_0 - \\lambda )^{-1}\\Vert < 1$ , and hence $A - \\lambda = A_0 - \\lambda + V = \\left(I + V(A_0 - \\lambda )^{-1}\\right)(A_0 - \\lambda )$ is boundedly invertible, which proves ().$\\Box $ Second order elliptic operators Let $\\Omega \\subset \\mathbb {R}^n$ be a domain and let $\\ell $ be the \"formally symmetric\" uniformly elliptic second order differential expression $\\ell (f)(x):=-\\sum _{j,k=1}^n \\left( \\frac{\\partial }{\\partial x_j} a_{jk}\\frac{\\partial f}{\\partial x_k}\\right)(x)+a(x)f(x),\\quad x\\in \\Omega ,$ with bounded coefficients $a_{jk}\\in C^\\infty (\\Omega )$ satisfying $a_{jk}(x)=\\overline{a_{kj}(x)}$ for all $x\\in \\Omega $ and $j,k=1,\\dots ,n$ , the function $a\\in L^\\infty (\\Omega )$ is real valued and $\\sum _{j,k=1}^n a_{jk}(x)\\xi _j\\xi _k\\ge C\\sum _{k=1}^n\\xi _k^2$ holds for some $C>0$ , all $\\xi =(\\xi _1,\\dots ,\\xi _n)^\\top \\in \\mathbb {R}^n$ and $x\\in \\Omega $ .","With the differential expression $\\ell $ we associate the elliptic differential operator $Tf:=\\ell (f),\\qquad \\operatorname{dom}T=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\ell (f) \\in L^2(\\Omega )\\bigr \\rbrace ,$ where $H^1_0(\\Omega )$ stands for the closure of $C_0^\\infty (\\Omega )$ in the Sobolev space $H^1(\\Omega )$ .","It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\Omega ),(\\cdot ,\\cdot ))$ with spectrum semibounded from below by $\\operatornamewithlimits{ess\\,inf}\\,a$ ; cf.","[24].","Let $w$ be a real valued function such that $w,w^{-1}\\in L^\\infty (\\Omega )$ and each of the sets $\\Omega _+:=\\bigl \\lbrace x\\in \\Omega : w(x)>0\\bigr \\rbrace \\quad \\text{and}\\quad \\Omega _-:=\\bigl \\lbrace x\\in \\Omega : w(x)<0\\bigr \\rbrace $ has positive Lebesgue measure.","We define a second order elliptic differential expression $\\mathcal {L}$ with the indefinite weight $w$ by $\\mathcal {L}(f)(x):=\\frac{1}{w(x)}\\,\\ell (f)(x),\\qquad x\\in \\Omega .$ The multiplication operator $G_w f=w f$ , $f\\in L^2(\\Omega )$ , is an isomorphism in $L^2(\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\in L^2(\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\int _\\Omega f(x)\\overline{g(x)}\\,w(x)\\,dx,\\qquad f,g\\in L^2(\\Omega ).$ The differential operator associated with $\\mathcal {L}$ is defined as $Af=\\mathcal {L}(f),\\qquad \\operatorname{dom}A=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\mathcal {L}(f) \\in L^2(\\Omega )\\bigr \\rbrace .$ Since for $f\\in H^1_0(\\Omega )$ we have $\\ell (f)\\in L^2(\\Omega )$ if and only if $\\mathcal {L}(f)\\in L^2(\\Omega )$ it follows that $\\operatorname{dom}A=\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.","Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ .","In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\min \\sigma _{\\rm ess}(T)\\le 0$ holds.","This also implies that the domain $\\Omega $ is unbounded as otherwise $\\sigma _{\\rm ess}(T)=\\varnothing $ .","A discussion of the cases $\\sigma _{\\rm ess}(T)=\\varnothing $ and $\\min \\sigma _{\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].","Fix some $\\eta >0$ such that $-\\eta <\\min \\sigma (T)$ and define the spaces $\\mathcal {H}_s$ , $s\\in [0,2]$ , as the domains of the $\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\eta $ in $L^2(\\Omega )$ , $\\mathcal {H}_s:=\\operatorname{dom}\\bigl ((T+\\eta )^\\frac{s}{2}\\bigr ),\\qquad s\\in [0,2].$ Note that $\\mathcal {H}=\\mathcal {H}_0$ , $\\operatorname{dom}T=\\mathcal {H}_2$ and the form domain of $T$ is $\\mathcal {H}_1$ .","The spaces $\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\eta $ ; cf.","[40].","The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.","[9].","Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ such that $W\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds for some $s\\in (0,2]$ .","Then $A$ is non-negative over $ \\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r=\\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert w^{-1}\\Vert _\\infty \\quad \\text{and}\\quad \\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A+\\eta G_w^{-1} - it)^{-1} \\,dt\\,\\right\\Vert \\,<\\,\\infty .$ Moreover, $\\infty $ is not a singular critical point of $A$ .","In the next corollary the special case $\\Omega =\\mathbb {R}^n$ is treated; cf.","[9].","From now on we assume that $\\Omega =\\mathbb {R}^n$ and $\\Omega _\\pm =\\lbrace x\\in \\mathbb {R}^n:\\pm w(x)>0\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\in C^\\infty (\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.","Note that either $\\Omega _+$ or $\\Omega _-$ is bounded.","Corollary 4.4 Suppose that for some $s\\in (0,\\frac{1}{2})$ the Sobolev spaces $H^s(\\Omega _\\pm )$ are invariant under multiplication with $w\\!\\upharpoonright _{\\Omega _\\pm }$ .","Then the indefinite elliptic operator $A$ is non-negative over $\\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .","Moreover, $\\infty $ is not a singular critical point of $A$ .","A sufficient criterion for the invariance of $H^s(\\Omega _\\pm )$ in the above corollary can be deduced from [28].","Corollary 4.5 Suppose that the weight function $w\\!\\upharpoonright _{\\Omega _\\pm }$ belongs to some Hölder space $C^{0,\\alpha }(\\Omega _\\pm )$ , $\\alpha \\in (0,\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.","Then the claim in Corollary REF holds.","of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\operatorname{dom}A=\\operatorname{dom}T=H^2(\\mathbb {R}^n)\\qquad \\text{and}\\qquad \\mathcal {H}_s=H^s(\\mathbb {R}^n),\\qquad s\\in [0,2],$ by elliptic regularity and interpolation.","Since the Sobolev spaces $H^s(\\Omega _\\pm )$ are assumed to be invariant under multiplication with the functions $w\\!\\upharpoonright _{\\Omega _\\pm }$ for some $s\\in (0,\\frac{1}{2})$ it follows from [9] that also $H^s(\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds.","Furthermore, $G_w$ is uniformly positive in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\ge \\operatornamewithlimits{ess\\,inf}w^2 \\Vert f\\Vert ^2$ and $w^{-1}\\in L^\\infty (\\mathbb {R}^n)$ .","Now the assertion follows from Theorem REF .$\\Box $ Acknowledgements The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.","Contact information Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, behrndt@tugraz.at Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.","Juni 136, 10623 Berlin, Germany, fmphilipp@gmail.com Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, carsten.trunk@tu-ilmenau.de"],["Bounded selfadjoint perturbations of non-negative operators","In this section we prove two abstract results on additive bounded perturbations of non-negative (and some closely connected class of) operators in Krein spaces that lead to perturbed operators which are non-negative over some neighborhood of infinity.","In both cases the neighborhood is given in quantitative terms.","The results will be applied in Section to singular indefinite Sturm-Liouville operators and to second order elliptic operators with indefinite weights."],["Two perturbation results","The following notation will be useful when formulating our main results below: For a set $\\Delta \\subset \\mathbb {R}$ and $r > 0$ we define $K_r(\\Delta ) := \\lbrace z\\in \\operatorname{dist}(z,\\Delta )\\,\\le \\,r\\rbrace .$ Our first main theorem concerns bounded selfadjoint perturbations of non-negative operators in Krein spaces.","Theorem 3.1 Let $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ be a Krein space with fundamental symmetry $J$ and norm $\\Vert \\cdot \\Vert = [J\\cdot ,\\cdot ]^{1/2}$ .","Let $A_0$ be a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ such that 0 and $\\infty $ are not singular critical points of $A_0$ , and $0\\notin \\sigma _p(A_0)$ .","Furthermore, let $V\\in L(\\mathcal {H})$ be a selfadjoint operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","Then the following holds: (i) If $V$ is non-negative, then $A_0+V$ is a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","(ii) If $V$ is not non-negative, then $A_0+V$ is non-negative over $\\overline{\\setminus K_r((-d,d)),where\\begin{equation}r = \\frac{1+\\tau _0}{2}\\Vert V\\Vert ,\\qquad \\quad d = -\\frac{1+\\tau _0}{2}\\,\\min \\sigma (JV),\\end{equation}and\\begin{equation}\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert <\\infty .\\end{equation}}Moreover, in both cases $$ is not a singular critical point of $ A0 + V$.$ The following simple example shows that without further assumptions a bounded selfadjoint perturbation $V$ of a non-negative or uniformly positive $A_0$ in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ may lead to an operator $A_0+V$ with unbounded non-real spectrum.","Example 3.2 Let $(\\cal K,(\\cdot ,\\cdot ))$ be a Hilbert space and let $H$ be an unbounded selfadjoint operator in $\\cal K$ such that $\\sigma (H)\\subset (0,\\infty )$ .","Equip $\\cal H:=\\cal K\\oplus \\cal K$ with the Krein space inner product $\\left[\\begin{pmatrix}k_1 \\\\ k_2\\end{pmatrix},\\begin{pmatrix}l_1 \\\\ l_2\\end{pmatrix}\\right] :=(k_1,l_2)+(k_2,l_1),\\qquad \\begin{pmatrix}k_1 \\\\ k_2\\end{pmatrix},\\begin{pmatrix}l_1 \\\\ l_2\\end{pmatrix}\\in \\cal H,$ and consider the operators $A_0=\\begin{pmatrix} 0 & I \\\\ H & 0\\end{pmatrix},\\quad V=\\begin{pmatrix} 0 & -I \\\\ 0 & 0\\end{pmatrix},\\quad \\text{and}\\quad A_0+V =\\begin{pmatrix} 0 & 0 \\\\ H & 0\\end{pmatrix}.$ It is easy to see that $A_0$ is a non-negative operator and $V$ is a bounded selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","Moreover, as $\\operatorname{dom}(A_0+V)=\\operatorname{dom}H\\oplus {\\cal H}$ we conclude $\\operatorname{ran}(A_0+V-\\lambda )\\ne \\cal H$ for every $\\lambda \\in , that is, $ (A0+V)=.","Our second main result applies to operators of the form $A = G^{-1}T$ , where $T$ is a semibounded selfadjoint operator in a Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ and $G$ is a selfadjoint bounded and boundedly invertible operator in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ .","In other words, $A$ is selfadjoint in the Krein space $(\\mathcal {H},(G\\cdot ,\\cdot ))$ and $A + \\eta G^{-1}$ is uniformly positive for suitable $\\eta $ .","Such a situation arises, e.g., when considering elliptic differential operators with an indefinite weight function; cf.","Section 4.2.","Theorem 3.3 Let $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ be a Hilbert space, $\\Vert \\cdot \\Vert = (\\cdot \\,,\\cdot )^{1/2}$ and $[\\cdot \\,,\\cdot ]=(G\\cdot ,\\cdot )$ be as above.","Let $A$ be a selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ such that for some $\\gamma > 0$ we have $[Af,f]\\ge -\\gamma \\Vert f\\Vert ^2\\quad \\text{for all }f\\in \\operatorname{dom}A.$ Assume furthermore that for some $\\eta > \\gamma $ $\\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A + \\eta G^{-1} - it)^{-1}\\,dt\\,\\right\\Vert < \\infty ,$ holds.","Then $A$ is non-negative over $\\overline{\\setminus K_r((-r,r)), where\\begin{equation*}r = \\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert G^{-1}\\Vert .\\end{equation*}Moreover, \\infty is not a singular critical point of A.","}$ Remark 3.4 Setting $A_0 := A + \\eta G^{-1}$ in Theorem REF and $V:=-\\eta G^{-1}$ we have $A = A_0 + V$ and hence Theorem REF can also be seen as a variant of Theorem REF .","Here $A_0=A + \\eta G^{-1}$ is uniformly positive so that $0\\in \\rho (A_0)$ is not a critical point of $A_0$ and the entire imaginary axis belongs to $\\rho (A + \\eta G^{-1})$ .","It follows from [55] and [32] that $\\tau _\\eta < \\infty $ is equivalent to $\\infty $ not being a singular critical point of $A_0$ .","In what follows we describe the structure of the proofs of Theorem REF (ii) and Theorem REF .","Let $E$ denote the spectral function of the non-negative operator $A_0$ .","As 0 and $\\infty $ are not singular critical points of $A_0$ , the spectral projections $E_+ := E((0,\\infty ))$ and $E_- := E((-\\infty ,0))$ , and the corresponding spectral subspaces $\\mathcal {H}_\\pm := E_\\pm \\mathcal {H}$ of $A_0$ exist.","From $0\\notin \\sigma _p(A_0)$ it follows that $\\mathcal {H}= \\mathcal {H}_+\\,[\\dotplus ]\\,\\mathcal {H}_-$ and $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ are Hilbert spaces, see, e.g., [46].","Therefore (REF ) is a fundamental decomposition of the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","With respect to this decomposition the operator $A_0$ and the fundamental symmetry $\\widetilde{J}$ corresponding to (REF ) can be written as operator matrices: $A_0 = \\begin{pmatrix}A_{0,+} & 0\\\\0 & A_{0,-}\\end{pmatrix}\\quad \\text{and}\\quad \\widetilde{J} = \\begin{pmatrix}I & 0\\\\0 & -I\\end{pmatrix}.$ Note that the operator $\\pm A_{0,\\pm }$ is a selfadjoint non-negative operator in the Hilbert space $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ .","Hence, $A_0$ is selfadjoint in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim )$ , where $(f,g)_\\sim := [\\widetilde{J}f,g] = [E_+f,g] - [E_-f,g],\\qquad f,g\\in \\mathcal {H}.$ Now, let $V\\in L(\\mathcal {H})$ be a bounded selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .","Then, with respect to the decomposition (REF ) it admits an operator matrix representation $V = \\begin{pmatrix}V_+ & B\\\\C & V_-\\end{pmatrix}.$ From the selfadjointness of $V$ in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ one concludes that $V_\\pm $ is selfadjoint in the Hilbert space $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ]) = (\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim )$ and $C = -B^{\\widetilde{*}},$ where $B^{\\widetilde{*}}$ denotes the adjoint of $B$ with respect to the scalar product $(\\cdot \\,,\\cdot )_\\sim $ .","Hence, the perturbed operator $A := A_0 + V$ is represented by $A = \\begin{pmatrix}A_{0,+} + V_+ & B\\\\-B^{\\widetilde{*}} & A_{0,-} + V_-\\end{pmatrix}.$ For operators as in (REF ) we show in Theorem REF below that $\\mathbb {R}\\setminus K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,-} + V_-))$ is of positive type, $\\mathbb {R}\\setminus K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,+} + V_+))$ is of negative type, the non-real spectrum of $A$ is contained in $K:=K_{\\Vert B\\Vert _\\sim }\\bigl (\\sigma (A_{0,+} + V_+)\\bigr )\\cap K_{\\Vert B\\Vert _\\sim } \\bigl (\\sigma (A_{0,-} + V_-)\\bigr ),$ and the growth of the resolvent of $A$ at $\\infty $ is of order one.","Here $\\Vert \\cdot \\Vert _\\sim $ stands for the operator norm induced by (REF ).","Then, by Theorem REF , the operator $A$ is non-negative over $\\overline{\\setminus K, where K is as in (\\ref {BadHersfeld}).In a final step it remains to bound the quantities\\Vert B\\Vert _\\sim \\quad \\mbox{and} \\quad K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,\\pm } + V_\\pm ))by the quantitiesr and \\tau _0 as in the statement of Theorem \\ref {t:main1}.","}$ Some auxiliary statements In this section we formulate and prove some auxiliary statements which will be used in the proofs Theorems REF and REF .","For the special case of bounded operators the first part of the following theorem was already proved in [48], see also [47] and [53], [54] for similar results with unbounded entries on the diagonal.","Theorem 3.5 Let $S_+$ and $S_-$ be selfadjoint operators in some Hilbert spaces $\\mathfrak {H}_+$ and $\\mathfrak {H}_-$ , respectively, let $M\\in L(\\mathfrak {H}_-,\\mathfrak {H}_+)$ and define the operators $S := \\begin{pmatrix}S_+ & M\\\\-M^* & S_-\\end{pmatrix}\\quad \\text{and}\\quad \\mathfrak {J}:= \\begin{pmatrix}I & 0\\\\0 & -I\\end{pmatrix}$ in the Hilbert space $\\mathfrak {H}:= \\mathfrak {H}_+\\oplus \\mathfrak {H}_-$ .","Then the operator $S$ is selfadjoint in the Krein space $(\\mathfrak {H},(\\mathfrak {J}\\cdot ,\\cdot ))$ and with $\\nu := \\Vert M\\Vert $ the following statements hold: $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_+))\\cap K_\\nu (\\sigma (S_-))$ .","$\\mathbb {R}\\setminus K_\\nu (\\sigma (S_-))$ is of positive type with respect to $S$ .","$\\mathbb {R}\\setminus K_\\nu (\\sigma (S_+))$ is of negative type with respect to $S$ .","Moreover, if $S_+$ is bounded from below and $S_-$ is bounded from above (or vice versa), the non-real spectrum of $S$ is bounded and the growth of the resolvent of $S$ at $\\infty $ is of order one.","We set $[\\cdot \\,,\\cdot ]:= (\\mathfrak {J}\\cdot ,\\cdot )$ .","Let $\\lambda \\in \\mathbb {C} \\setminus K_\\nu (\\sigma (S_-))$ and $\\alpha := \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_-)) < 1$ .","We claim that for some $r(\\alpha )>0$ and all $f\\in \\operatorname{dom}S$ the following implication $\\Vert (S - \\lambda )f\\Vert \\le \\frac{1-\\alpha ^2}{4\\alpha }\\nu \\Vert f\\Vert \\quad \\Longrightarrow \\quad [f,f]\\,\\ge \\,r(\\alpha )\\Vert f\\Vert ^2$ holds.","In fact, set $\\varepsilon := \\frac{1-\\alpha ^2}{4\\alpha }$ and let $f\\in \\operatorname{dom}S$ with $\\Vert (S - \\lambda )f\\Vert \\le \\varepsilon \\nu \\Vert f\\Vert .$ With respect to the decomposition $\\mathfrak {H}= \\mathfrak {H}_+\\oplus \\mathfrak {H}_-$ we write $f$ and $g := (S - \\lambda )f$ as column vectors $f = \\begin{pmatrix}f_+\\\\f_-\\end{pmatrix}\\in \\operatorname{dom}S_+\\oplus \\operatorname{dom}S_-, \\quad g = \\begin{pmatrix}g_+\\\\g_-\\end{pmatrix}.$ Then $g_- = -M^*f_+ + (S_- - \\lambda )f_-$ , or, equivalently, $f_- = (S_- - \\lambda )^{-1}g_- + (S_- - \\lambda )^{-1}M^*f_+.$ As $\\Vert g_-\\Vert \\le \\Vert g\\Vert = \\Vert (S - \\lambda )f\\Vert \\le \\varepsilon \\nu \\Vert f\\Vert $ and $\\Vert (S_- - \\lambda )^{-1}\\Vert = \\alpha /\\nu $ this yields $\\Vert f_-\\Vert \\,\\le \\,\\alpha \\varepsilon \\Vert f\\Vert + \\alpha \\Vert f_+\\Vert $ and hence $\\Vert f_-\\Vert ^2\\,\\le \\,\\alpha ^2\\big (\\varepsilon ^2\\Vert f\\Vert ^2 + 2\\varepsilon \\Vert f\\Vert ^2 + \\Vert f_+\\Vert ^2\\big ).$ Using $\\Vert f_+\\Vert ^2 = \\Vert f\\Vert ^2 - \\Vert f_-\\Vert ^2$ we conclude $\\Vert f_-\\Vert ^2\\,\\le \\,\\frac{\\alpha ^2}{1+\\alpha ^2}(1 + \\varepsilon )^2\\Vert f\\Vert ^2.$ Hence, as $[f,f] = \\Vert f_+\\Vert ^2 - \\Vert f_-\\Vert ^2 = \\Vert f\\Vert ^2 - 2\\Vert f_-\\Vert ^2$ , we obtain $[f,f]\\,\\ge \\,r(\\alpha )\\Vert f\\Vert ^2$ , where $r(\\alpha ) := 1 - \\frac{2\\alpha ^2}{1+\\alpha ^2}(1 + \\varepsilon )^2 = \\frac{(1 - \\alpha )^2(1 + \\alpha )(7 - \\alpha )}{8(1+\\alpha ^2)} > 0,$ i.e., the implication (REF ) holds.","It follows that $(K_\\nu (\\sigma (S_-)))\\cap \\sigma _{{ap}}(S)\\subset \\sigma _{+}(S)$ .","Hence (ii) is proved and, as $\\sigma _{+}(S)$ is real, the non-real spectrum of $S$ satisfies $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_-)).$ Similarly, as in the proof of (REF ), one proves that for $\\lambda \\in \\mathbb {C} \\setminus K_\\nu (\\sigma (S_+))$ , $f\\in \\operatorname{dom}S$ , and for $\\beta := \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_+))<1$ the implication $\\Vert (S - \\lambda )f\\Vert \\le \\frac{1-\\beta ^2}{4\\beta }\\nu \\Vert f\\Vert \\quad \\Longrightarrow \\quad [f,f]\\,\\le \\,-r(\\beta )\\Vert f\\Vert ^2$ holds with $r(\\beta )>0$ as in (REF ).","This shows (iii) and $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_+))$ .","Together with (REF ) we obtain (i).","Assume now that $S_+$ is bounded from below and $S_-$ is bounded from above.","Then the non-real spectrum of $S$ is bounded by (i) and it remains to show that the growth of the resolvent of $S$ is of order one at $\\infty $ .","Note first that for $S_0=\\begin{pmatrix}S_+ & 0 \\\\ 0 & S_-\\end{pmatrix}\\qquad \\text{and}\\qquad V=\\begin{pmatrix}0 & M \\\\ -M^* & 0\\end{pmatrix},$ and $\\lambda \\in such that $ Im2V= 2$ we have $ (S0-)-1V12$ and\\begin{equation}\\Vert (S-\\lambda )^{-1}\\Vert = \\bigl \\Vert \\bigl ((S_0-\\lambda )(I+(S_0-\\lambda )^{-1}V)\\bigr )^{-1}\\bigr \\Vert \\le \\frac{2}{\\vert \\operatorname{Im}\\lambda \\vert }.\\end{equation}For $ with $\\vert \\operatorname{Im}\\lambda \\vert < 2\\nu $ and $\\operatorname{dist}(\\lambda ,\\sigma (S_-)) > \\nu + \\delta $ with some fixed $\\delta > 0$ again define the value $\\alpha = \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_-))$ .","Then $\\alpha \\in (0,\\delta ^{\\prime })$ , where $\\delta ^{\\prime } = \\nu /(\\nu + \\delta )\\in (0,1)$ , and for $f\\in \\operatorname{dom}S$ we either have $\\Vert (S - \\lambda )f\\Vert \\ge \\frac{1-\\alpha ^2}{4\\alpha }\\nu \\Vert f\\Vert > \\frac{1-\\alpha ^2}{8\\alpha }\\vert \\operatorname{Im}\\lambda \\vert \\,\\Vert f\\Vert ,$ or, by (REF ), $r(\\alpha )|\\operatorname{Im}\\lambda |\\Vert f\\Vert ^2\\le |\\operatorname{Im}[\\lambda f,f]| = |\\operatorname{Im}[(S - \\lambda )f,f]|\\le \\Vert (S - \\lambda )f\\Vert \\Vert f\\Vert ,$ and hence $\\Vert (S - \\lambda )f\\Vert \\ge r(\\alpha )|\\operatorname{Im}\\lambda |\\,\\Vert f\\Vert .$ For $\\lambda \\in with $ Im<2$ and $ dist(,(S+)) > + $ the estimates (\\ref {abs2}) and (\\ref {abs3}) hold in a similar form.","Therefore, (\\ref {abs1})-(\\ref {abs3}) imply that for all non-real $ with $\\operatorname{dist}(\\lambda ,\\sigma (S_-)) > \\nu + \\delta $ or $\\operatorname{dist}(\\lambda ,\\sigma (S_+)) > \\nu + \\delta $ we have $\\Vert (S-\\lambda )^{-1}\\Vert \\le \\frac{C}{\\vert \\operatorname{Im}\\lambda \\vert }$ with some $C > 0$ which does not depend on $\\lambda $ .","This shows that the growth of the resolvent $(S-\\lambda )^{-1}$ is of order one at $\\infty $ .$\\Box $ Remark 3.6 If $\\operatorname{dist}(\\sigma (S_+),\\sigma (S_-))\\ge 2\\Vert M\\Vert $ , then the spectrum of $S$ in Theorem REF is real.","This result can be improved in certain special cases, cf.","[1] and also [2], where sharp norm bounds on the operator angles between reducing subspaces of $S$ and $S_+\\oplus S_-$ are given.","We mention here also [34], [35] for comparable results in the study of operators of Klein-Gordon type, where bounded perturbations of operators with unbounded off-diagonal entries are investigated.","The following simple example shows that the bounds on the non-real spectrum of $S$ in Theorem REF are sharp.","Example 3.7 Let $\\mathfrak {H}_+ = \\mathfrak {H}_- = and $ S= 1$, $ M = z and hence $S = \\begin{pmatrix}1 & z\\\\-\\overline{z} & -1\\end{pmatrix}.$ Then $\\operatorname{dist}(\\sigma (S_+),\\sigma (S_-)) = 2$ .","If $|z|\\le 1$ , then $\\sigma (S) = \\lbrace \\pm \\sqrt{1 - |z|^2}\\rbrace $ is real; cf.","Remark REF .","If $|z| > 1$ , then $\\sigma (S) = \\lbrace \\pm i\\sqrt{|z|^2 - 1}\\rbrace $ and in this case the eigenvalues of $S$ lie on the boundary of $ K_{|z|}(\\lbrace 1\\rbrace )\\cap K_{|z|}(\\lbrace -1\\rbrace )$ .","Lemma 3.8 Let $T$ be a selfadjoint operator in the Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ with $0\\notin \\sigma _p(T)$ and let $E$ be its spectral function.","If $(a_n)$ and $(b_n)$ are sequences in $[0,\\infty )$ such that $a_n\\downarrow 0$ and $b_n\\uparrow \\infty $ as $n\\rightarrow \\infty $ , and $0\\in \\rho (T)$ if $a_k=0$ for some $k\\in \\mathbb {N}$ , then $E(\\mathbb {R}^+) - E(\\mathbb {R}^-) = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{a_n}^{b_n}\\,\\left((T + it)^{-1} + (T - it)^{-1}\\right)\\,dt\\,.$ First of all we observe that $(T + it)^{-1} + (T - it)^{-1} = 2T(T^2 + t^2)^{-1}.$ For all $f,g\\in \\mathfrak {H}$ we have $\\int _{a_n}^{b_n}\\,\\big (T(T^2 + t^2)^{-1}f,g\\big )\\,dt&= \\int _{a_n}^{b_n}\\,\\int _{\\mathbb {R}}\\,\\frac{s}{s^2 + t^2}\\,d(E_sf,g)\\,dt\\\\&= \\int _{\\mathbb {R}}\\,\\int _{a_n}^{b_n}\\,\\frac{s}{s^2 + t^2}\\,dt\\,d(E_sf,g)\\\\&= \\int _{\\mathbb {R}}\\,\\big (\\arctan (b_n/s) - \\arctan (a_n/s)\\big )\\,d(E_sf,g).\\\\&= \\big (\\arctan (b_nT^{-1})f - \\arctan (a_nT^{-1})f,g\\big ).$ Fubini's theorem can be applied here since on $\\mathbb {R}\\times [a_n,b_n]$ the integrand is bounded and the measure $d(E_sf,g)\\otimes dt$ considered as a measure in $\\mathbb {R} \\times [a_n,b_n]$ is finite.","Hence, $\\int _{a_n}^{b_n}\\,2T(T^2 + t^2)^{-1}f\\,dt = 2\\arctan (b_nT^{-1})f - 2\\arctan (a_nT^{-1})f.$ Now, we apply [52] and observe that $2\\arctan (a_nT^{-1})f$ tends to zero and $2\\arctan (b_nT^{-1})f$ tends to $\\pi \\operatorname{sgn}(T^{-1})f = \\pi \\operatorname{sgn}(T)f = \\pi (E(\\mathbb {R}^+) - E(\\mathbb {R}^-))f$ as $n\\rightarrow \\infty $ .$\\Box $ A subspace $\\mathfrak {L}$ of a Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ is called uniformly positive if there exists $\\delta > 0$ such that $[f,f]\\,\\ge \\,\\delta \\Vert f\\Vert ^2 \\quad \\mbox{for all } f\\in \\mathfrak {L}.$ Here $\\Vert \\cdot \\Vert $ is a Hilbert space norm on $\\mathfrak {H}$ with respect to which $[\\cdot \\,,\\cdot ]$ is continuous.","All such norms are equivalent, see [46].","Lemma 3.9 Let $G$ be a bounded and boundedly invertible selfadjoint operator in a Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ and set $[\\cdot \\,,\\cdot ]:= (G\\cdot ,\\cdot )$ .","Let $\\mathfrak {L}$ be a closed uniformly positive subspace in the Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ .","If $P$ ($E$ ) denotes the orthogonal projection in the Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ (in the Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ , respectively) onto $\\mathfrak {L}$ , then $P(G|\\mathfrak {L})$ is a uniformly positive selfadjoint operator in $(\\mathfrak {L},(\\cdot \\,,\\cdot ))$ , and its inverse is given by $\\bigl (P(G|\\mathfrak {L})\\bigr )^{-1} = E\\bigl (G^{-1}|\\mathfrak {L}\\bigr ).$ As $\\mathfrak {L}$ is uniformly positive, for all $f\\in \\mathfrak {L}$ we have $\\big (P(G|\\mathfrak {L})f,f\\big ) = (Gf,f) = [f,f]\\,\\ge \\,\\delta \\Vert f\\Vert ^2$ with some $\\delta > 0$ .","Thus, $P(G|\\mathfrak {L})$ is a uniformly positive operator in $(\\mathfrak {L},(\\cdot \\,,\\cdot ))$ and, in particular, $0\\in \\rho (P(G|\\mathfrak {L}))$ .","Let $g\\in \\mathfrak {L}$ and set $f := E(G^{-1}|\\mathfrak {L})g\\in \\mathfrak {L}$ .","Since for all $h\\in \\mathfrak {L}$ $(g - Gf,h) = \\bigl (G(I - E)G^{-1}g,h\\bigr ) = \\bigl [(I - E)G^{-1}g,h\\bigr ] = 0,$ we conclude $g - Gf\\in \\mathfrak {L}^\\perp $ and hence $g = PGf = P(G|\\mathfrak {L})f$ .","Therefore, $\\bigl (P(G|\\mathfrak {L})\\bigr )^{-1}g = f = E\\bigl (G^{-1}|\\mathfrak {L}\\bigr )g,$ which proves the lemma.$\\Box $ Proof of Theorem REF Assertion (i) in Theorem REF is simple: As both $A_0$ and $V$ are non-negative in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ we have $[(A_0+V)f,f]\\ge 0\\quad \\text{for all}\\quad f\\in \\operatorname{dom}(A_0+V)=\\operatorname{dom}A_0.$ The operator $A_0$ is selfadjoint in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim )$ (cf.","(REF ) and (REF )) and, as $V$ is bounded, $\\rho (A_0+V)$ is nonempty.","Thus $A_0+V$ is a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ and (i) in Theorem REF is proved.","Let $E$ be the spectral function of the non-negative operator $A_0$ , define the projections $E_+ := E((0,\\infty ))$ , $E_-:=E((-\\infty ,0))$ and the subspaces $\\mathcal {H}_\\pm := E_\\pm \\mathcal {H}$ .","Then with respect to the fundamental decomposition (REF ) the operators $A_0$ and $V$ have the form in (REF ) and (REF ), respectively.","The fundamental symmetry $\\widetilde{J}$ associated with the decomposition (REF ) is also given in (REF ).","The inner product $(\\cdot \\,,\\cdot )_\\sim $ is defined as in (REF ).","Moreover, $A_{0,\\pm }$ and $V_\\pm $ from (REF ) are selfadjoint in $(\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim ) = (\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ , $\\sigma (A_{0,\\pm })\\subset \\mathbb {R}^\\pm \\cup \\lbrace 0\\rbrace $ and (REF ) holds.","We apply Lemma REF to the non-negative operators $\\pm A_{0,\\pm }$ and obtain $\\pm I_{\\mathcal {H}_\\pm } = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{1/n}^n\\,\\left((A_{0,\\pm } + it)^{-1} + (A_{0,\\pm } - it)^{-1}\\right)\\,dt.$ Therefore, $\\widetilde{J} = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{1/n}^n\\,\\left((A_{0} + it)^{-1} + (A_{0} - it)^{-1}\\right)\\,dt,$ where the strong limit is with respect to the norm $\\Vert \\cdot \\Vert _\\sim := (\\cdot \\,,\\cdot )_\\sim ^{1/2}$ .","But as $\\Vert \\cdot \\Vert _\\sim $ and $\\Vert \\cdot \\Vert $ are equivalent norms, the limit on the right hand side of (REF ) also exists with respect to the norm $\\Vert \\cdot \\Vert $ , and coincides with $\\widetilde{J}$ .","The uniform boundedness principle now yields that $\\tau _0$ in () is finite and that $\\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _0<\\infty .$ Define the constants $\\nu := \\Vert B\\Vert _\\sim ,\\quad a := \\min \\sigma (V_+)\\quad \\text{and}\\quad b := \\max \\sigma (V_-).$ Then $A_{0,+} + V_+$ is bounded from below by $a$ and $\\sigma (A_{0,+} + V_+)\\subset [a,\\infty )$ , and $A_{0,-} + V_-$ is bounded from above by $b$ and $\\sigma (A_{0,-} + V_-)\\subset (-\\infty ,b]$ .","Making use of the representation (REF ) of $A = A_0 + V$ and Theorem REF we conclude: $\\sigma (A)\\backslash \\mathbb {R}\\subset K_\\nu ((-\\infty ,b])\\cap K_\\nu ([a,\\infty ))$ .","$(b + \\nu ,\\infty )$ is of positive type with respect to $A$ .","$(-\\infty ,a - \\nu )$ is of negative type with respect to $A$ .","The growth of the resolvent of $A$ at $\\infty $ of order one, and hence of finite order.","The statement in Theorem REF (ii) follows from Theorem REF if we show that the constants $\\nu $ , $a$ and $b$ in (REF ) satisfy $\\nu \\le r, \\quad a\\ge -d, \\quad b\\le d,$ where $r$ and $d$ are as in (), because in that case the properties (a)-(d) above hold with $\\nu $ , $a$ and $b$ replaced by $r$ , $-d$ and $d$ , respectively, and $K_\\nu ((-\\infty ,d])\\cap K_\\nu ([-d,\\infty ))= K_\\nu ((-d,d)).$ First we check $\\nu \\le r$ .","Denote by $(\\cdot \\,,\\cdot )$ the scalar product corresponding to the Hilbert space norm $\\Vert \\cdot \\Vert $ .","Using the well-known fact that the spectrum of a bounded operator is always a subset of the closure of its numerical range we obtain $\\begin{split}\\nu ^2&= \\Vert B\\Vert _\\sim ^2 = \\Vert B^{\\tilde{*}}B\\Vert _\\sim = \\sup \\lbrace |\\lambda | : \\lambda \\in \\sigma (B^{\\tilde{*}}B)\\rbrace \\\\&\\le \\sup \\lbrace |(B^{\\tilde{*}}Bf,f)| : \\Vert f\\Vert =1\\rbrace \\,\\le \\,\\Vert B^{\\tilde{*}}\\Vert \\,\\Vert B\\Vert \\\\&= \\Vert E_-(V|\\mathcal {H}_+)\\Vert \\,\\Vert E_+(V|\\mathcal {H}_-)\\Vert \\,\\le \\,\\Vert E_+\\Vert \\,\\Vert E_-\\Vert \\,\\Vert V\\Vert ^2.\\end{split}$ As $E_\\pm = \\frac{1}{2}(I\\pm \\widetilde{J})$ , it follows from (REF ) that $\\Vert E_\\pm \\Vert \\le \\frac{1+\\tau _0}{2}$ which shows $\\nu \\le r$ .","Next we verify $a\\ge -d$ .","Denote by $P_\\pm $ the orthogonal projection onto $\\mathcal {H}_\\pm $ with respect to $(\\cdot \\,,\\cdot )$ .","Then, according to Lemma REF , the operator $P_+(J|\\mathcal {H}_+)$ is selfadjoint and uniformly positive in the Hilbert space $(\\mathcal {H}_+,(\\cdot \\,,\\cdot ))$ .","Hence, the same holds for its inverse $E_+(J|\\mathcal {H}_+)$ ; cf.","Lemma REF .","This implies $\\min \\sigma (P_+(J|\\mathcal {H}_+))&= \\big (\\max \\sigma (E_+(J|\\mathcal {H}_+))\\big )^{-1} = \\Vert E_+(J|\\mathcal {H}_+)\\Vert ^{-1}\\\\&\\ge \\Vert E_+J\\Vert ^{-1} \\ge \\Vert E_+\\Vert ^{-1}\\,\\ge \\,\\frac{2}{1+\\tau _0}\\,.$ Hence, for $f_+\\in \\mathcal {H}_+$ we have $[f_+,f_+] = (Jf_+,f_+) = (P_+(J|\\mathcal {H}_+)f_+,f_+)\\,\\ge \\,\\frac{2}{1+\\tau _0}\\,\\Vert f_+\\Vert ^2.$ We also use $(V_+f_+,f_+)_\\sim = [V_+f_+,f_+] = [V_+f_+ - B^{\\tilde{*}}f_+,f_+] = [Vf_+,f_+] = (JVf_+,f_+)$ for $f\\in \\mathcal {H}_+$ to conclude $a=\\min \\sigma (V_+)&=\\inf \\lbrace (V_+f_+,f_+)_\\sim : \\Vert f_+\\Vert _\\sim = 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&\\ge \\inf \\lbrace (V_+f_+,f_+)_\\sim : \\Vert f_+\\Vert _\\sim \\le 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&= \\inf \\lbrace (JVf_+,f_+) : [f_+,f_+]\\le 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&\\ge \\inf \\bigl \\lbrace (JVf_+,f_+) : \\Vert f_+\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f_+\\in \\mathcal {H}_+\\bigr \\rbrace \\\\&\\ge \\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace .$ By assumption $V$ is not non-negative in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ and hence $0>\\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace =\\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2=\\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace .$ Therefore we finally obtain $a=\\min \\sigma (V_+)&\\ge \\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2=\\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace \\\\&= \\frac{1+\\tau _0}{2}\\,\\inf \\lbrace (JVf,f) : \\Vert f\\Vert = 1,\\,f\\in \\mathcal {H}\\rbrace \\\\&= \\frac{1+\\tau _0}{2}\\,\\min \\sigma (JV)=-d.$ The proof of the inequality $b\\le d$ is analogous and is left to the reader.","This completes the proof of (ii).","As $\\infty $ is not a singular critical point of $A_0$ , the last statement in Theorem REF on the critical point $\\infty $ of $A_0+V$ follows in both cases (i) and (ii) from Proposition and $\\operatorname{dom}A=\\operatorname{dom}(A_0+V)$ .$\\Box $ Proof of Theorem REF Let $\\gamma $ and $\\eta $ be as in Theorem REF .","Then the operator $A_0 := A + \\eta G^{-1}$ is uniformly positive in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ , $[\\cdot \\,,\\cdot ]= (G\\cdot ,\\cdot )$ , since for $f\\in \\operatorname{dom}A$ we have $[A_0 f,f] = [Af,f] + \\eta [G^{-1}f,f] \\ge -\\gamma \\Vert f\\Vert ^2 + \\eta \\Vert f\\Vert ^2 = (\\eta - \\gamma )\\Vert f\\Vert ^2.$ We will show that $A$ is non-negative over $\\overline{\\setminus K_r((-r,r)).In particular, 0\\in \\rho (A_0) is not a singular critical point of A_0and, as \\tau _\\eta < \\infty , the point \\infty is not aa singular critical point of A_0, see Remark \\ref {Mai2012}.With V := -\\eta G^{-1} we haveA = A_0 + V\\qquad \\text{and}\\qquad \\Vert V\\Vert =\\eta \\Vert G^{-1}\\Vert .As in the proof of Theorem~\\ref {t:main1} let E be the spectral function of the operator A_0.","Then the fundamentaldecomposition \\mathcal {H}= E_+\\mathcal {H}\\,[\\dotplus ]\\,E_-\\mathcal {H} in (\\ref {e:dec_expl}) generates matrix representationsof A_0, V and the corresponding fundamental symmetry \\widetilde{J} as in (\\ref {e:AJ}) and (\\ref {e:V}).The operator A_0 is selfadjoint in the Hilbert space(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim ), where (\\cdot \\,,\\cdot )_\\sim is defined in (\\ref {e:sim}).","With the help of Lemma \\ref {l:veselic} we obtain{\\begin{@align*}{1}{-1}\\widetilde{J}&= \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _0^n\\,\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\big )\\,dt\\\\&= \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{-n}^n\\,(A + \\eta G^{-1} - it)^{-1}\\,dt\\,.\\end{@align*}}This yields\\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _\\eta .Following the lines of the proof of Theorem \\ref {t:main1}, it suffices to show the inequalities\\nu \\le \\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert ,\\quad \\min \\sigma (V_+)\\ge -\\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert \\quad \\text{and}\\quad \\max \\sigma (V_-)\\le \\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert ,where \\nu =\\Vert E_\\mp (V\\vert \\cal H_\\pm ) \\Vert _\\sim .The first relation is proved as in (\\ref {e:rel_proof}) and the remaining two inequalities follow from the selfadjointness of V_\\pm in (\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim ) and{\\begin{@align*}{1}{-1}\\max \\lbrace |\\min \\sigma (V_\\pm )|,|\\max \\sigma (V_\\pm )|\\rbrace &= \\Vert V_\\pm \\Vert _\\sim = \\sup \\lbrace |\\lambda | : \\lambda \\in \\sigma (V_\\pm )\\rbrace \\\\&\\le \\sup \\lbrace |(V_\\pm f_\\pm ,f_\\pm )| : \\Vert f_\\pm \\Vert =1,\\,f\\in \\mathcal {H}_\\pm \\rbrace \\\\&\\le \\Vert V_\\pm \\Vert = \\Vert E_\\pm (V|\\mathcal {H}_\\pm )\\Vert \\,\\le \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert .\\end{@align*}}Hence, A is non-negative over \\overline{\\setminus K_r((-r,r)).","As\\operatorname{dom}A= \\operatorname{dom}A_0, we conclude from Proposition \\ref {p:curgus} that \\infty is also not a critical point of A.This completes the proof of Theorem~\\ref {t:main2}.\\Box }\\begin{remark}If, in addition to the assumptions in Theorem \\ref {t:main2}, the point 0 is neither an eigenvalue nor a singular critical pointof the non-negative operator A_0 := A + \\gamma G^{-1}, then the operator A is non-negative over the set \\overline{\\setminus K_r((-r,r)), where\\begin{equation}r = \\gamma \\,\\frac{1+\\tau _\\gamma }{2}\\,\\Vert G^{-1}\\Vert \\end{equation}and\\tau _\\gamma := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\big )\\,dt\\right\\Vert .Indeed, under the assumptions of Theorem \\ref {t:main2} for any \\varepsilon > 0 the operator A + (\\gamma + \\varepsilon )G^{-1}is selfadjoint in some Hilbert space, and thus the resolvent set ofA_0 is non-empty as A + (\\gamma + \\varepsilon )G^{-1} and A_0 differ just by a bounded operator.Therefore, A_0 is a non-negative operator in the Krein space(\\cal H,[\\cdot \\,,\\cdot ]).","Moreover,the point \\infty is not a singular critical point of theoperator A + (\\gamma + \\varepsilon )G^{-1}, see Remark \\ref {Mai2012},and by Proposition \\ref {p:curgus} \\infty is not a singular critical point of A_0.Therefore, the operator A_0 admits a matrix representation as in (\\ref {e:AJ}) and the corresponding fundamental symmetry \\widetilde{J}satisfies \\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _\\gamma (see the proof of Theorem \\ref {t:main2}).Then by a similar reasoning as in the proof ofTheorem \\ref {t:main2} we see that A is non-negativeover \\overline{\\setminus K_r((-r,r)), where r is as in (\\ref {Juni2012}).","}}\\end{remark}}$ Differential operators with indefinite weights In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights.","Indefinite Sturm-Liouville operators We consider Sturm-Liouville differential expressions of the form $\\mathcal {L}(f)(x) = \\operatorname{sgn}(x)\\big (-f^{\\prime \\prime }(x) + q(x)f(x)\\big ),\\quad x\\in \\mathbb {R}\\,,$ with a real-valued potential $q\\in L^\\infty (\\mathbb {R})$ and the indefinite weight function $\\operatorname{sgn}(\\cdot )$ .","The corresponding differential operator in $L^2(\\mathbb {R})$ is defined by $Af := \\mathcal {L}(f), \\qquad \\operatorname{dom}A := H^2(\\mathbb {R}).$ Here $H^2(\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.","By $(\\cdot \\,,\\cdot )$ and $\\Vert \\cdot \\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\mathbb {R})$ .","Let $J$ be the operator of multiplication with the function $\\operatorname{sgn}(x)$ .","This operator is obviously selfadjoint and unitary in $L^2(\\mathbb {R})$ .","Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\prime \\prime } + qf,\\quad \\operatorname{dom}T := H^2(\\mathbb {R}),$ is selfadjoint in $L^2(\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\mathbb {R}),[\\cdot \\,,\\cdot ])$ , where $[f,g] := (Jf,g) = \\int _\\mathbb {R}\\,f(x)\\overline{g(x)}\\operatorname{sgn}(x)\\,dx,\\quad f,g\\in L^2(\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.","We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].","For this, denote by $m_\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\pm f_\\pm := - f_\\pm ^{\\prime \\prime } + q_\\pm f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ defined on $\\operatorname{dom}T_\\pm := \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ) : f_\\pm (0) = 0\\rbrace $ , where $q_\\pm $ is the restriction of the function $q$ to $\\mathbb {R}^\\pm $ .","The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\pm \\infty $ , then $m_\\pm =\\lim _{x\\rightarrow \\pm \\infty }\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.","Furthermore the following holds.","If $m_+ > -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ is finite; If $m_+\\le -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .","To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.","One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].","The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .","Here only the nontrivial case $\\operatornamewithlimits{ess\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).","Theorem 4.2 Assume that $\\operatornamewithlimits{ess\\,inf}q<0$ holds.","Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\overline{\\setminus K_r((-d,d)), wherer := 5\\Vert q\\Vert _\\infty ,\\quad \\text{and}\\quad d := -5\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\cap \\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .","}\\begin{proof}The proof of Theorem~\\ref {t:SL2} is split into 4 parts.","The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\ref {t:main1}.","In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\end{proof}\\ \\\\{\\it 1.\\ Preparation: } The operator $ A0$,\\begin{equation*}A_0 f:=-\\operatorname{sgn}(\\cdot )f^{\\prime \\prime },\\qquad \\operatorname{dom}A_0=H^2(\\mathbb {R}),\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\mathbb {R}$ ),[,])$.","Furthermore, we have $ (A0) = $\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\cite {cn}.","Hence $ A0$ satisfies theassumptions in Theorem~\\ref {t:main1}.","Define $ V$ as\\begin{equation*}Vf:=\\operatorname{sgn}(\\cdot ) qf,\\qquad \\operatorname{dom}V=L^2(\\mathbb {R}),\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\mathbb {R}$ ),[,])$ with$$\\min \\sigma (JV) = \\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}\\,q(x)\\quad \\text{and}\\quad \\Vert V\\Vert = \\Vert q\\Vert _\\infty .$$By Theorem \\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\frac{1+\\tau _0}{2}\\,\\Vert q\\Vert _\\infty ,\\qquad d = -\\frac{1+\\tau _0}{2}\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x),$$and$$\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert .$$Therefore, we have to show\\begin{equation}\\tau _0\\le 9\\end{equation}and\\begin{equation}\\sigma (A)\\setminus \\mathbb {R}\\,\\subset \\,\\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .\\end{equation}$ 2.","Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\begin{pmatrix} B_+ & 0 \\\\ 0 & B_- \\end{pmatrix}\\quad \\text{in}\\quad L^2(\\mathbb {R})=L^2(\\mathbb {R}^+)\\times L^2(\\mathbb {R}^-),$ where $B_\\pm f_\\pm :=\\mp f_\\pm ^{\\prime \\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\mathbb {R}^\\pm )$ which are defined on $\\operatorname{dom}B_\\pm = \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ):f_\\pm (0)=0\\rbrace $ .","Here and in the following we will denote the restrictions of a function $f$ defined on $\\mathbb {R}$ to $\\mathbb {R}^\\pm $ by $f_\\pm $ .","For $\\lambda = re^{it}$ , $r > 0$ , $t\\in [0,2\\pi )$ , we set $\\sqrt{\\lambda } := \\sqrt{r}e^{it/2}$ .","For $\\lambda \\in \\mathbb {R}$ define the function $f_\\lambda (x) :={\\left\\lbrace \\begin{array}{ll}e^{i\\sqrt{\\lambda }x}, &x> 0,\\\\e^{-i\\sqrt{-\\lambda }x}, &x < 0.\\end{array}\\right.","}$ The function $f_\\lambda $ is a solution of the equations $\\mp f_\\pm ^{\\prime \\prime } = \\lambda f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ .","From $\\overline{\\sqrt{\\lambda }} = -\\sqrt{\\overline{\\lambda }}$ it follows that $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ holds for all $\\lambda \\in \\mathbb {R}$ .","Let us now prove that for all $f\\in L^2(\\mathbb {R})$ and $\\lambda \\in \\mathbb {R}$ we have $(A_0 - \\lambda )^{-1}f = (B_0 - \\lambda )^{-1}f - \\frac{[f,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda .$ Since $\\sigma (A_0)=\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\lambda \\in \\mathbb {R}$ .","In particular, for $\\lambda \\in \\mathbb {R}$ there exists $g\\in \\operatorname{dom}B_0$ such that $f=(B_0-\\lambda )g$ holds.","The right hand side in (REF ) has the form $h:=g-\\frac{[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda $ and we will show that the function $h$ belongs to $\\operatorname{dom}A_0=H^2(\\mathbb {R})$ .","As $g_\\pm $ and $f_{\\lambda ,\\pm }$ are elements of $H^2(\\mathbb {R}^\\pm )$ the same is true for $h_\\pm $ .","Moreover, $g$ and $f_\\lambda $ are continuous at 0 and so is $h$ .","Hence it remains to check that $h^\\prime $ is continuous at 0.","For this note first that $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=\\int _0^\\infty (-g_+^{\\prime \\prime }-\\lambda g_+)\\overline{f_{\\overline{\\lambda },+}}\\, dx- \\int _{-\\infty }^0 (g_-^{\\prime \\prime }-\\lambda g_-)\\overline{f_{\\overline{\\lambda },-}}\\, dx$ and since $f_{\\overline{\\lambda },\\pm }$ are solutions of $\\mp f_\\pm ^{\\prime \\prime }=\\overline{\\lambda }f_\\pm $ , integration by parts yields $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=g_+^\\prime (0)\\overline{f_{\\overline{\\lambda },+}(0)}-g_+(0) \\overline{f^\\prime _{\\overline{\\lambda },+}(0)}-g_-^\\prime (0)\\overline{f_{\\overline{\\lambda },-}(0)}+g_-(0) \\overline{f^\\prime _{\\overline{\\lambda },-}(0)}.$ As $g\\in \\operatorname{dom}B_0$ we have $g_\\pm (0)=0$ and together with $f_{\\overline{\\lambda },\\pm }(0)=1$ we find $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]= g_+^\\prime (0)-g^\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\pm ^\\prime $ on $\\mathbb {R}^\\pm $ of the function $h$ from (REF ): $h^\\prime _\\pm =g^\\prime _\\pm -\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_{\\lambda ,\\pm }^\\prime $ and as $f^\\prime _{\\lambda ,+}(0)=i\\sqrt{\\lambda }$ and $f^\\prime _{\\lambda ,-}(0)=-i\\sqrt{-\\lambda }$ we conclude $h^\\prime _+(0)-h^\\prime _-(0)= \\bigl (g_+^\\prime (0)-g^\\prime _-(0)\\bigr )-\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}\\bigl (f^\\prime _{\\lambda ,+}(0)-f^\\prime _{\\lambda ,-}(0)\\bigr )=0,$ that is, $h^\\prime $ is continuous at 0 and therefore $h\\in \\operatorname{dom}A_0$ .","Now a straightforward computation shows that $(A_0-\\lambda )h=(B_0-\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).","3.","Proof of (): Let $f\\in L^2(\\mathbb {R})$ and $t > 0$ .","Then (REF ) and $\\sqrt{it} + \\sqrt{-it} = i\\sqrt{2t}$ yield $\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\& + \\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\in \\mathbb {R}^+$ , we have $\\big |[f,f_{it}][f_{-it},f]\\big |\\,\\le \\,\\left(\\int _0^\\infty \\,e^{-x\\sqrt{t/2}}g(x)\\,dx\\right)^2$ and thus for $n\\in \\mathbb {N}$ $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt&\\le \\int _{1/n}^n\\,\\int _0^\\infty \\,\\int _0^\\infty \\,g(x)g(y)\\,\\frac{e^{-(x+y)\\sqrt{t/2}}}{\\sqrt{2t}}\\,dy\\,dx\\,dt\\\\&= 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\left(e^{-\\frac{x+y}{\\sqrt{2n}}} - e^{-\\frac{\\sqrt{n}(x+y)}{\\sqrt{2}}}\\right)\\,dy\\,dx\\\\&\\le 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\,dy\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt\\,\\le \\,2\\pi \\Vert g\\Vert _{L^2(\\mathbb {R}^+)}^2\\,\\le \\,4\\pi \\Vert f\\Vert ^2.$ and therefore we have $\\frac{1}{\\pi }\\int _{1/n}^n\\,\\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\,dt\\,\\le \\,8\\Vert f\\Vert ^2.$ Denote by $E_0$ ($E_\\pm $ ) the spectral function of the operator $A_0$ ($B_\\pm $ , respectively).","Moreover define $J_{n,\\pm } := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}\\right)\\,dt,$ and, in addition, $J_{n,0} := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt.$ As $\\pm B_\\pm $ is non-negative, it follows that $J_{n,\\pm }$ converges strongly to $\\pm I_{L^2(\\mathbb {R}^\\pm )}$ when $n\\rightarrow \\infty $ , cf.","Lemma REF .","Moreover, by $(B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}= 2B_\\pm (B_\\pm + it)^{-1} (B_\\pm - it)^{-1}$ the sequence $\\pm J_{n,\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\mathbb {R}^\\pm )$ .","This implies $\\frac{1}{\\pi }\\,\\int _{1/n}^n\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\&\\le \\Vert f_+\\Vert _{L^2(\\mathbb {R}^+)}^2 + \\Vert f_-\\Vert _{L^2(\\mathbb {R}^-)}^2 = \\Vert f\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\,\\le \\,\\Vert f\\Vert ^2 + 8\\Vert f\\Vert ^2 = 9\\Vert f\\Vert ^2,\\quad f\\in L^2(\\mathbb {R}).$ Hence, as $J = \\operatorname{sgn}(\\cdot )$ is unitary in $L^2(\\mathbb {R})$ , we finally obtain $\\Vert J_{n,0}\\Vert = \\Vert J J_{n,0}\\Vert \\le 9$ and thus $\\tau _0 = \\limsup _{n\\rightarrow \\infty }\\,\\Vert J_{n,0}\\Vert \\le 9$ .","That is, () holds.","4.","Proof of (): Let $\\lambda \\in +$ .","Using (REF ), for $f\\in L^2(\\mathbb {R})$ we obtain $\\Vert (A_0 - \\lambda )^{-1}f\\Vert \\,\\le \\,\\Vert (B_0 - \\lambda )^{-1}f\\Vert + \\frac{|[f,f_{\\overline{\\lambda }}]|\\,\\Vert f_\\lambda \\Vert }{|\\sqrt{\\lambda } + \\sqrt{-\\lambda }|}\\,\\le \\,\\frac{\\Vert f\\Vert }{|\\operatorname{Im}\\lambda |} + \\frac{\\Vert f_\\lambda \\Vert ^2\\Vert f\\Vert }{\\sqrt{2|\\lambda |}}\\,.$ Here we have used the identity $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ .","Let $r > 0$ and $t\\in (0,\\pi )$ such that $\\lambda = re^{it}$ .","Then $\\Vert f_\\lambda \\Vert ^2&= \\int _0^\\infty \\,\\left|e^{i\\sqrt{r}e^{it/2}x}\\right|^2\\,dx + \\int _{-\\infty }^0\\,\\left|e^{-i\\sqrt{r}e^{i(t+\\pi )/2}x}\\right|^2\\,dx\\\\&= \\int _0^\\infty \\,e^{-2\\sqrt{r}\\sin (t/2)x}\\,dx + \\int _{-\\infty }^0\\,e^{2\\sqrt{r}\\cos (t/2)x}\\,dx\\\\&= \\frac{\\cos (t/2) + \\sin (t/2)}{\\sqrt{r}\\,\\sin (t)}\\,\\le \\,\\frac{\\sqrt{2}}{\\sqrt{r}\\,\\operatorname{Im}(\\lambda /r)} = \\frac{\\sqrt{2|\\lambda |}}{|\\operatorname{Im}\\lambda |}.$ Therefore, $\\Vert (A_0 - \\lambda )^{-1}\\Vert \\,\\le \\,\\frac{2}{|\\operatorname{Im}\\lambda |}\\,.$ The same estimate holds for $\\lambda \\in -$ .","Now, assume that $|\\operatorname{Im}\\lambda | > 2\\Vert q\\Vert _\\infty =2\\Vert V\\Vert $ .","Then $\\Vert V(A_0 - \\lambda )^{-1}\\Vert < 1$ , and hence $A - \\lambda = A_0 - \\lambda + V = \\left(I + V(A_0 - \\lambda )^{-1}\\right)(A_0 - \\lambda )$ is boundedly invertible, which proves ().$\\Box $ Second order elliptic operators Let $\\Omega \\subset \\mathbb {R}^n$ be a domain and let $\\ell $ be the \"formally symmetric\" uniformly elliptic second order differential expression $\\ell (f)(x):=-\\sum _{j,k=1}^n \\left( \\frac{\\partial }{\\partial x_j} a_{jk}\\frac{\\partial f}{\\partial x_k}\\right)(x)+a(x)f(x),\\quad x\\in \\Omega ,$ with bounded coefficients $a_{jk}\\in C^\\infty (\\Omega )$ satisfying $a_{jk}(x)=\\overline{a_{kj}(x)}$ for all $x\\in \\Omega $ and $j,k=1,\\dots ,n$ , the function $a\\in L^\\infty (\\Omega )$ is real valued and $\\sum _{j,k=1}^n a_{jk}(x)\\xi _j\\xi _k\\ge C\\sum _{k=1}^n\\xi _k^2$ holds for some $C>0$ , all $\\xi =(\\xi _1,\\dots ,\\xi _n)^\\top \\in \\mathbb {R}^n$ and $x\\in \\Omega $ .","With the differential expression $\\ell $ we associate the elliptic differential operator $Tf:=\\ell (f),\\qquad \\operatorname{dom}T=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\ell (f) \\in L^2(\\Omega )\\bigr \\rbrace ,$ where $H^1_0(\\Omega )$ stands for the closure of $C_0^\\infty (\\Omega )$ in the Sobolev space $H^1(\\Omega )$ .","It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\Omega ),(\\cdot ,\\cdot ))$ with spectrum semibounded from below by $\\operatornamewithlimits{ess\\,inf}\\,a$ ; cf.","[24].","Let $w$ be a real valued function such that $w,w^{-1}\\in L^\\infty (\\Omega )$ and each of the sets $\\Omega _+:=\\bigl \\lbrace x\\in \\Omega : w(x)>0\\bigr \\rbrace \\quad \\text{and}\\quad \\Omega _-:=\\bigl \\lbrace x\\in \\Omega : w(x)<0\\bigr \\rbrace $ has positive Lebesgue measure.","We define a second order elliptic differential expression $\\mathcal {L}$ with the indefinite weight $w$ by $\\mathcal {L}(f)(x):=\\frac{1}{w(x)}\\,\\ell (f)(x),\\qquad x\\in \\Omega .$ The multiplication operator $G_w f=w f$ , $f\\in L^2(\\Omega )$ , is an isomorphism in $L^2(\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\in L^2(\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\int _\\Omega f(x)\\overline{g(x)}\\,w(x)\\,dx,\\qquad f,g\\in L^2(\\Omega ).$ The differential operator associated with $\\mathcal {L}$ is defined as $Af=\\mathcal {L}(f),\\qquad \\operatorname{dom}A=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\mathcal {L}(f) \\in L^2(\\Omega )\\bigr \\rbrace .$ Since for $f\\in H^1_0(\\Omega )$ we have $\\ell (f)\\in L^2(\\Omega )$ if and only if $\\mathcal {L}(f)\\in L^2(\\Omega )$ it follows that $\\operatorname{dom}A=\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.","Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ .","In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\min \\sigma _{\\rm ess}(T)\\le 0$ holds.","This also implies that the domain $\\Omega $ is unbounded as otherwise $\\sigma _{\\rm ess}(T)=\\varnothing $ .","A discussion of the cases $\\sigma _{\\rm ess}(T)=\\varnothing $ and $\\min \\sigma _{\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].","Fix some $\\eta >0$ such that $-\\eta <\\min \\sigma (T)$ and define the spaces $\\mathcal {H}_s$ , $s\\in [0,2]$ , as the domains of the $\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\eta $ in $L^2(\\Omega )$ , $\\mathcal {H}_s:=\\operatorname{dom}\\bigl ((T+\\eta )^\\frac{s}{2}\\bigr ),\\qquad s\\in [0,2].$ Note that $\\mathcal {H}=\\mathcal {H}_0$ , $\\operatorname{dom}T=\\mathcal {H}_2$ and the form domain of $T$ is $\\mathcal {H}_1$ .","The spaces $\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\eta $ ; cf.","[40].","The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.","[9].","Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ such that $W\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds for some $s\\in (0,2]$ .","Then $A$ is non-negative over $ \\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r=\\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert w^{-1}\\Vert _\\infty \\quad \\text{and}\\quad \\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A+\\eta G_w^{-1} - it)^{-1} \\,dt\\,\\right\\Vert \\,<\\,\\infty .$ Moreover, $\\infty $ is not a singular critical point of $A$ .","In the next corollary the special case $\\Omega =\\mathbb {R}^n$ is treated; cf.","[9].","From now on we assume that $\\Omega =\\mathbb {R}^n$ and $\\Omega _\\pm =\\lbrace x\\in \\mathbb {R}^n:\\pm w(x)>0\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\in C^\\infty (\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.","Note that either $\\Omega _+$ or $\\Omega _-$ is bounded.","Corollary 4.4 Suppose that for some $s\\in (0,\\frac{1}{2})$ the Sobolev spaces $H^s(\\Omega _\\pm )$ are invariant under multiplication with $w\\!\\upharpoonright _{\\Omega _\\pm }$ .","Then the indefinite elliptic operator $A$ is non-negative over $\\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .","Moreover, $\\infty $ is not a singular critical point of $A$ .","A sufficient criterion for the invariance of $H^s(\\Omega _\\pm )$ in the above corollary can be deduced from [28].","Corollary 4.5 Suppose that the weight function $w\\!\\upharpoonright _{\\Omega _\\pm }$ belongs to some Hölder space $C^{0,\\alpha }(\\Omega _\\pm )$ , $\\alpha \\in (0,\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.","Then the claim in Corollary REF holds.","of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\operatorname{dom}A=\\operatorname{dom}T=H^2(\\mathbb {R}^n)\\qquad \\text{and}\\qquad \\mathcal {H}_s=H^s(\\mathbb {R}^n),\\qquad s\\in [0,2],$ by elliptic regularity and interpolation.","Since the Sobolev spaces $H^s(\\Omega _\\pm )$ are assumed to be invariant under multiplication with the functions $w\\!\\upharpoonright _{\\Omega _\\pm }$ for some $s\\in (0,\\frac{1}{2})$ it follows from [9] that also $H^s(\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds.","Furthermore, $G_w$ is uniformly positive in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\ge \\operatornamewithlimits{ess\\,inf}w^2 \\Vert f\\Vert ^2$ and $w^{-1}\\in L^\\infty (\\mathbb {R}^n)$ .","Now the assertion follows from Theorem REF .$\\Box $ Acknowledgements The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.","Contact information Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, behrndt@tugraz.at Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.","Juni 136, 10623 Berlin, Germany, fmphilipp@gmail.com Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, carsten.trunk@tu-ilmenau.de"],["Differential operators with indefinite weights","In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights."],["Indefinite Sturm-Liouville operators","We consider Sturm-Liouville differential expressions of the form $\\mathcal {L}(f)(x) = \\operatorname{sgn}(x)\\big (-f^{\\prime \\prime }(x) + q(x)f(x)\\big ),\\quad x\\in \\mathbb {R}\\,,$ with a real-valued potential $q\\in L^\\infty (\\mathbb {R})$ and the indefinite weight function $\\operatorname{sgn}(\\cdot )$ .","The corresponding differential operator in $L^2(\\mathbb {R})$ is defined by $Af := \\mathcal {L}(f), \\qquad \\operatorname{dom}A := H^2(\\mathbb {R}).$ Here $H^2(\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.","By $(\\cdot \\,,\\cdot )$ and $\\Vert \\cdot \\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\mathbb {R})$ .","Let $J$ be the operator of multiplication with the function $\\operatorname{sgn}(x)$ .","This operator is obviously selfadjoint and unitary in $L^2(\\mathbb {R})$ .","Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\prime \\prime } + qf,\\quad \\operatorname{dom}T := H^2(\\mathbb {R}),$ is selfadjoint in $L^2(\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\mathbb {R}),[\\cdot \\,,\\cdot ])$ , where $[f,g] := (Jf,g) = \\int _\\mathbb {R}\\,f(x)\\overline{g(x)}\\operatorname{sgn}(x)\\,dx,\\quad f,g\\in L^2(\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.","We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].","For this, denote by $m_\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\pm f_\\pm := - f_\\pm ^{\\prime \\prime } + q_\\pm f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ defined on $\\operatorname{dom}T_\\pm := \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ) : f_\\pm (0) = 0\\rbrace $ , where $q_\\pm $ is the restriction of the function $q$ to $\\mathbb {R}^\\pm $ .","The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\pm \\infty $ , then $m_\\pm =\\lim _{x\\rightarrow \\pm \\infty }\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.","Furthermore the following holds.","If $m_+ > -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ is finite; If $m_+\\le -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .","To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.","One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].","The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .","Here only the nontrivial case $\\operatornamewithlimits{ess\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).","Theorem 4.2 Assume that $\\operatornamewithlimits{ess\\,inf}q<0$ holds.","Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\overline{\\setminus K_r((-d,d)), wherer := 5\\Vert q\\Vert _\\infty ,\\quad \\text{and}\\quad d := -5\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\cap \\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .","}\\begin{proof}The proof of Theorem~\\ref {t:SL2} is split into 4 parts.","The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\ref {t:main1}.","In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\end{proof}\\ \\\\{\\it 1.\\ Preparation: } The operator $ A0$,\\begin{equation*}A_0 f:=-\\operatorname{sgn}(\\cdot )f^{\\prime \\prime },\\qquad \\operatorname{dom}A_0=H^2(\\mathbb {R}),\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\mathbb {R}$ ),[,])$.","Furthermore, we have $ (A0) = $\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\cite {cn}.","Hence $ A0$ satisfies theassumptions in Theorem~\\ref {t:main1}.","Define $ V$ as\\begin{equation*}Vf:=\\operatorname{sgn}(\\cdot ) qf,\\qquad \\operatorname{dom}V=L^2(\\mathbb {R}),\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\mathbb {R}$ ),[,])$ with$$\\min \\sigma (JV) = \\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}\\,q(x)\\quad \\text{and}\\quad \\Vert V\\Vert = \\Vert q\\Vert _\\infty .$$By Theorem \\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\frac{1+\\tau _0}{2}\\,\\Vert q\\Vert _\\infty ,\\qquad d = -\\frac{1+\\tau _0}{2}\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x),$$and$$\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert .$$Therefore, we have to show\\begin{equation}\\tau _0\\le 9\\end{equation}and\\begin{equation}\\sigma (A)\\setminus \\mathbb {R}\\,\\subset \\,\\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .\\end{equation}$ 2.","Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\begin{pmatrix} B_+ & 0 \\\\ 0 & B_- \\end{pmatrix}\\quad \\text{in}\\quad L^2(\\mathbb {R})=L^2(\\mathbb {R}^+)\\times L^2(\\mathbb {R}^-),$ where $B_\\pm f_\\pm :=\\mp f_\\pm ^{\\prime \\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\mathbb {R}^\\pm )$ which are defined on $\\operatorname{dom}B_\\pm = \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ):f_\\pm (0)=0\\rbrace $ .","Here and in the following we will denote the restrictions of a function $f$ defined on $\\mathbb {R}$ to $\\mathbb {R}^\\pm $ by $f_\\pm $ .","For $\\lambda = re^{it}$ , $r > 0$ , $t\\in [0,2\\pi )$ , we set $\\sqrt{\\lambda } := \\sqrt{r}e^{it/2}$ .","For $\\lambda \\in \\mathbb {R}$ define the function $f_\\lambda (x) :={\\left\\lbrace \\begin{array}{ll}e^{i\\sqrt{\\lambda }x}, &x> 0,\\\\e^{-i\\sqrt{-\\lambda }x}, &x < 0.\\end{array}\\right.","}$ The function $f_\\lambda $ is a solution of the equations $\\mp f_\\pm ^{\\prime \\prime } = \\lambda f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ .","From $\\overline{\\sqrt{\\lambda }} = -\\sqrt{\\overline{\\lambda }}$ it follows that $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ holds for all $\\lambda \\in \\mathbb {R}$ .","Let us now prove that for all $f\\in L^2(\\mathbb {R})$ and $\\lambda \\in \\mathbb {R}$ we have $(A_0 - \\lambda )^{-1}f = (B_0 - \\lambda )^{-1}f - \\frac{[f,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda .$ Since $\\sigma (A_0)=\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\lambda \\in \\mathbb {R}$ .","In particular, for $\\lambda \\in \\mathbb {R}$ there exists $g\\in \\operatorname{dom}B_0$ such that $f=(B_0-\\lambda )g$ holds.","The right hand side in (REF ) has the form $h:=g-\\frac{[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda $ and we will show that the function $h$ belongs to $\\operatorname{dom}A_0=H^2(\\mathbb {R})$ .","As $g_\\pm $ and $f_{\\lambda ,\\pm }$ are elements of $H^2(\\mathbb {R}^\\pm )$ the same is true for $h_\\pm $ .","Moreover, $g$ and $f_\\lambda $ are continuous at 0 and so is $h$ .","Hence it remains to check that $h^\\prime $ is continuous at 0.","For this note first that $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=\\int _0^\\infty (-g_+^{\\prime \\prime }-\\lambda g_+)\\overline{f_{\\overline{\\lambda },+}}\\, dx- \\int _{-\\infty }^0 (g_-^{\\prime \\prime }-\\lambda g_-)\\overline{f_{\\overline{\\lambda },-}}\\, dx$ and since $f_{\\overline{\\lambda },\\pm }$ are solutions of $\\mp f_\\pm ^{\\prime \\prime }=\\overline{\\lambda }f_\\pm $ , integration by parts yields $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=g_+^\\prime (0)\\overline{f_{\\overline{\\lambda },+}(0)}-g_+(0) \\overline{f^\\prime _{\\overline{\\lambda },+}(0)}-g_-^\\prime (0)\\overline{f_{\\overline{\\lambda },-}(0)}+g_-(0) \\overline{f^\\prime _{\\overline{\\lambda },-}(0)}.$ As $g\\in \\operatorname{dom}B_0$ we have $g_\\pm (0)=0$ and together with $f_{\\overline{\\lambda },\\pm }(0)=1$ we find $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]= g_+^\\prime (0)-g^\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\pm ^\\prime $ on $\\mathbb {R}^\\pm $ of the function $h$ from (REF ): $h^\\prime _\\pm =g^\\prime _\\pm -\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_{\\lambda ,\\pm }^\\prime $ and as $f^\\prime _{\\lambda ,+}(0)=i\\sqrt{\\lambda }$ and $f^\\prime _{\\lambda ,-}(0)=-i\\sqrt{-\\lambda }$ we conclude $h^\\prime _+(0)-h^\\prime _-(0)= \\bigl (g_+^\\prime (0)-g^\\prime _-(0)\\bigr )-\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}\\bigl (f^\\prime _{\\lambda ,+}(0)-f^\\prime _{\\lambda ,-}(0)\\bigr )=0,$ that is, $h^\\prime $ is continuous at 0 and therefore $h\\in \\operatorname{dom}A_0$ .","Now a straightforward computation shows that $(A_0-\\lambda )h=(B_0-\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).","3.","Proof of (): Let $f\\in L^2(\\mathbb {R})$ and $t > 0$ .","Then (REF ) and $\\sqrt{it} + \\sqrt{-it} = i\\sqrt{2t}$ yield $\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\& + \\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\in \\mathbb {R}^+$ , we have $\\big |[f,f_{it}][f_{-it},f]\\big |\\,\\le \\,\\left(\\int _0^\\infty \\,e^{-x\\sqrt{t/2}}g(x)\\,dx\\right)^2$ and thus for $n\\in \\mathbb {N}$ $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt&\\le \\int _{1/n}^n\\,\\int _0^\\infty \\,\\int _0^\\infty \\,g(x)g(y)\\,\\frac{e^{-(x+y)\\sqrt{t/2}}}{\\sqrt{2t}}\\,dy\\,dx\\,dt\\\\&= 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\left(e^{-\\frac{x+y}{\\sqrt{2n}}} - e^{-\\frac{\\sqrt{n}(x+y)}{\\sqrt{2}}}\\right)\\,dy\\,dx\\\\&\\le 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\,dy\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt\\,\\le \\,2\\pi \\Vert g\\Vert _{L^2(\\mathbb {R}^+)}^2\\,\\le \\,4\\pi \\Vert f\\Vert ^2.$ and therefore we have $\\frac{1}{\\pi }\\int _{1/n}^n\\,\\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\,dt\\,\\le \\,8\\Vert f\\Vert ^2.$ Denote by $E_0$ ($E_\\pm $ ) the spectral function of the operator $A_0$ ($B_\\pm $ , respectively).","Moreover define $J_{n,\\pm } := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}\\right)\\,dt,$ and, in addition, $J_{n,0} := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt.$ As $\\pm B_\\pm $ is non-negative, it follows that $J_{n,\\pm }$ converges strongly to $\\pm I_{L^2(\\mathbb {R}^\\pm )}$ when $n\\rightarrow \\infty $ , cf.","Lemma REF .","Moreover, by $(B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}= 2B_\\pm (B_\\pm + it)^{-1} (B_\\pm - it)^{-1}$ the sequence $\\pm J_{n,\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\mathbb {R}^\\pm )$ .","This implies $\\frac{1}{\\pi }\\,\\int _{1/n}^n\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\&\\le \\Vert f_+\\Vert _{L^2(\\mathbb {R}^+)}^2 + \\Vert f_-\\Vert _{L^2(\\mathbb {R}^-)}^2 = \\Vert f\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\,\\le \\,\\Vert f\\Vert ^2 + 8\\Vert f\\Vert ^2 = 9\\Vert f\\Vert ^2,\\quad f\\in L^2(\\mathbb {R}).$ Hence, as $J = \\operatorname{sgn}(\\cdot )$ is unitary in $L^2(\\mathbb {R})$ , we finally obtain $\\Vert J_{n,0}\\Vert = \\Vert J J_{n,0}\\Vert \\le 9$ and thus $\\tau _0 = \\limsup _{n\\rightarrow \\infty }\\,\\Vert J_{n,0}\\Vert \\le 9$ .","That is, () holds.","4.","Proof of (): Let $\\lambda \\in +$ .","Using (REF ), for $f\\in L^2(\\mathbb {R})$ we obtain $\\Vert (A_0 - \\lambda )^{-1}f\\Vert \\,\\le \\,\\Vert (B_0 - \\lambda )^{-1}f\\Vert + \\frac{|[f,f_{\\overline{\\lambda }}]|\\,\\Vert f_\\lambda \\Vert }{|\\sqrt{\\lambda } + \\sqrt{-\\lambda }|}\\,\\le \\,\\frac{\\Vert f\\Vert }{|\\operatorname{Im}\\lambda |} + \\frac{\\Vert f_\\lambda \\Vert ^2\\Vert f\\Vert }{\\sqrt{2|\\lambda |}}\\,.$ Here we have used the identity $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ .","Let $r > 0$ and $t\\in (0,\\pi )$ such that $\\lambda = re^{it}$ .","Then $\\Vert f_\\lambda \\Vert ^2&= \\int _0^\\infty \\,\\left|e^{i\\sqrt{r}e^{it/2}x}\\right|^2\\,dx + \\int _{-\\infty }^0\\,\\left|e^{-i\\sqrt{r}e^{i(t+\\pi )/2}x}\\right|^2\\,dx\\\\&= \\int _0^\\infty \\,e^{-2\\sqrt{r}\\sin (t/2)x}\\,dx + \\int _{-\\infty }^0\\,e^{2\\sqrt{r}\\cos (t/2)x}\\,dx\\\\&= \\frac{\\cos (t/2) + \\sin (t/2)}{\\sqrt{r}\\,\\sin (t)}\\,\\le \\,\\frac{\\sqrt{2}}{\\sqrt{r}\\,\\operatorname{Im}(\\lambda /r)} = \\frac{\\sqrt{2|\\lambda |}}{|\\operatorname{Im}\\lambda |}.$ Therefore, $\\Vert (A_0 - \\lambda )^{-1}\\Vert \\,\\le \\,\\frac{2}{|\\operatorname{Im}\\lambda |}\\,.$ The same estimate holds for $\\lambda \\in -$ .","Now, assume that $|\\operatorname{Im}\\lambda | > 2\\Vert q\\Vert _\\infty =2\\Vert V\\Vert $ .","Then $\\Vert V(A_0 - \\lambda )^{-1}\\Vert < 1$ , and hence $A - \\lambda = A_0 - \\lambda + V = \\left(I + V(A_0 - \\lambda )^{-1}\\right)(A_0 - \\lambda )$ is boundedly invertible, which proves ().$\\Box $"],["Second order elliptic operators","Let $\\Omega \\subset \\mathbb {R}^n$ be a domain and let $\\ell $ be the \"formally symmetric\" uniformly elliptic second order differential expression $\\ell (f)(x):=-\\sum _{j,k=1}^n \\left( \\frac{\\partial }{\\partial x_j} a_{jk}\\frac{\\partial f}{\\partial x_k}\\right)(x)+a(x)f(x),\\quad x\\in \\Omega ,$ with bounded coefficients $a_{jk}\\in C^\\infty (\\Omega )$ satisfying $a_{jk}(x)=\\overline{a_{kj}(x)}$ for all $x\\in \\Omega $ and $j,k=1,\\dots ,n$ , the function $a\\in L^\\infty (\\Omega )$ is real valued and $\\sum _{j,k=1}^n a_{jk}(x)\\xi _j\\xi _k\\ge C\\sum _{k=1}^n\\xi _k^2$ holds for some $C>0$ , all $\\xi =(\\xi _1,\\dots ,\\xi _n)^\\top \\in \\mathbb {R}^n$ and $x\\in \\Omega $ .","With the differential expression $\\ell $ we associate the elliptic differential operator $Tf:=\\ell (f),\\qquad \\operatorname{dom}T=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\ell (f) \\in L^2(\\Omega )\\bigr \\rbrace ,$ where $H^1_0(\\Omega )$ stands for the closure of $C_0^\\infty (\\Omega )$ in the Sobolev space $H^1(\\Omega )$ .","It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\Omega ),(\\cdot ,\\cdot ))$ with spectrum semibounded from below by $\\operatornamewithlimits{ess\\,inf}\\,a$ ; cf.","[24].","Let $w$ be a real valued function such that $w,w^{-1}\\in L^\\infty (\\Omega )$ and each of the sets $\\Omega _+:=\\bigl \\lbrace x\\in \\Omega : w(x)>0\\bigr \\rbrace \\quad \\text{and}\\quad \\Omega _-:=\\bigl \\lbrace x\\in \\Omega : w(x)<0\\bigr \\rbrace $ has positive Lebesgue measure.","We define a second order elliptic differential expression $\\mathcal {L}$ with the indefinite weight $w$ by $\\mathcal {L}(f)(x):=\\frac{1}{w(x)}\\,\\ell (f)(x),\\qquad x\\in \\Omega .$ The multiplication operator $G_w f=w f$ , $f\\in L^2(\\Omega )$ , is an isomorphism in $L^2(\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\in L^2(\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\int _\\Omega f(x)\\overline{g(x)}\\,w(x)\\,dx,\\qquad f,g\\in L^2(\\Omega ).$ The differential operator associated with $\\mathcal {L}$ is defined as $Af=\\mathcal {L}(f),\\qquad \\operatorname{dom}A=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\mathcal {L}(f) \\in L^2(\\Omega )\\bigr \\rbrace .$ Since for $f\\in H^1_0(\\Omega )$ we have $\\ell (f)\\in L^2(\\Omega )$ if and only if $\\mathcal {L}(f)\\in L^2(\\Omega )$ it follows that $\\operatorname{dom}A=\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.","Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ .","In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\min \\sigma _{\\rm ess}(T)\\le 0$ holds.","This also implies that the domain $\\Omega $ is unbounded as otherwise $\\sigma _{\\rm ess}(T)=\\varnothing $ .","A discussion of the cases $\\sigma _{\\rm ess}(T)=\\varnothing $ and $\\min \\sigma _{\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].","Fix some $\\eta >0$ such that $-\\eta <\\min \\sigma (T)$ and define the spaces $\\mathcal {H}_s$ , $s\\in [0,2]$ , as the domains of the $\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\eta $ in $L^2(\\Omega )$ , $\\mathcal {H}_s:=\\operatorname{dom}\\bigl ((T+\\eta )^\\frac{s}{2}\\bigr ),\\qquad s\\in [0,2].$ Note that $\\mathcal {H}=\\mathcal {H}_0$ , $\\operatorname{dom}T=\\mathcal {H}_2$ and the form domain of $T$ is $\\mathcal {H}_1$ .","The spaces $\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\eta $ ; cf.","[40].","The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.","[9].","Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ such that $W\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds for some $s\\in (0,2]$ .","Then $A$ is non-negative over $ \\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r=\\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert w^{-1}\\Vert _\\infty \\quad \\text{and}\\quad \\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A+\\eta G_w^{-1} - it)^{-1} \\,dt\\,\\right\\Vert \\,<\\,\\infty .$ Moreover, $\\infty $ is not a singular critical point of $A$ .","In the next corollary the special case $\\Omega =\\mathbb {R}^n$ is treated; cf.","[9].","From now on we assume that $\\Omega =\\mathbb {R}^n$ and $\\Omega _\\pm =\\lbrace x\\in \\mathbb {R}^n:\\pm w(x)>0\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\in C^\\infty (\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.","Note that either $\\Omega _+$ or $\\Omega _-$ is bounded.","Corollary 4.4 Suppose that for some $s\\in (0,\\frac{1}{2})$ the Sobolev spaces $H^s(\\Omega _\\pm )$ are invariant under multiplication with $w\\!\\upharpoonright _{\\Omega _\\pm }$ .","Then the indefinite elliptic operator $A$ is non-negative over $\\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .","Moreover, $\\infty $ is not a singular critical point of $A$ .","A sufficient criterion for the invariance of $H^s(\\Omega _\\pm )$ in the above corollary can be deduced from [28].","Corollary 4.5 Suppose that the weight function $w\\!\\upharpoonright _{\\Omega _\\pm }$ belongs to some Hölder space $C^{0,\\alpha }(\\Omega _\\pm )$ , $\\alpha \\in (0,\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.","Then the claim in Corollary REF holds.","of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\operatorname{dom}A=\\operatorname{dom}T=H^2(\\mathbb {R}^n)\\qquad \\text{and}\\qquad \\mathcal {H}_s=H^s(\\mathbb {R}^n),\\qquad s\\in [0,2],$ by elliptic regularity and interpolation.","Since the Sobolev spaces $H^s(\\Omega _\\pm )$ are assumed to be invariant under multiplication with the functions $w\\!\\upharpoonright _{\\Omega _\\pm }$ for some $s\\in (0,\\frac{1}{2})$ it follows from [9] that also $H^s(\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds.","Furthermore, $G_w$ is uniformly positive in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\ge \\operatornamewithlimits{ess\\,inf}w^2 \\Vert f\\Vert ^2$ and $w^{-1}\\in L^\\infty (\\mathbb {R}^n)$ .","Now the assertion follows from Theorem REF .$\\Box $"],["Acknowledgements","The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.","Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, behrndt@tugraz.at Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.","Juni 136, 10623 Berlin, Germany, fmphilipp@gmail.com Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, carsten.trunk@tu-ilmenau.de The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.","Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, behrndt@tugraz.at Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.","Juni 136, 10623 Berlin, Germany, fmphilipp@gmail.com Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, carsten.trunk@tu-ilmenau.de"]],"string":"[\n [\n \"Bounds on the non-real spectrum of differential operators with\\n indefinite weights\"\n ],\n [\n \"Abstract Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces.\",\n \"Under the assumption that 0 and $\\\\infty$ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set.\",\n \"The main results are quantitative estimates for this set, which are applied to Sturm-Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.\"\n ],\n [\n \"Introduction\",\n \"We consider linear operators associated with an indefinite differential expression $\\\\mathcal {L}=\\\\frac{1}{w}\\\\,\\\\ell ,$ where $w\\\\ne 0$ is a real-valued, locally integrable weight function which changes its sign and $\\\\ell $ is an ordinary or partial differential expression of the form $\\\\ell =-\\\\frac{d}{dx} p\\\\frac{d}{dx}+q\\\\qquad \\\\text{or}\\\\qquad \\\\ell =-\\\\sum _{j,k=1}^n \\\\frac{\\\\partial }{\\\\partial x_j} \\\\,a_{jk} \\\\,\\\\frac{\\\\partial }{\\\\partial x_k} +a$ acting on a real interval or domain $\\\\Omega \\\\subset \\\\mathbb {R}^n$ , respectively.\",\n \"In the first case $p^{-1}$ and $q$ are assumed to be real and (locally) integrable.\",\n \"In the second case $a\\\\in L^\\\\infty (\\\\Omega )$ is real and $a_{jk}$ are $C^{\\\\infty }$ -coefficients such that $\\\\ell $ is formally symmetric and uniformly elliptic on $\\\\Omega \\\\subset \\\\mathbb {R}^n$ , and, additionally, the weight function $w$ and its inverse are essentially bounded.\",\n \"Together with appropriate boundary conditions (if necessary) the differential expression $\\\\ell $ gives rise to a selfadjoint operator $T$ in an $L^2$ Hilbert space.\",\n \"Multiplication with $1/w$ leads to the corresponding indefinite differential operator $A=\\\\frac{1}{w} T$ associated with $\\\\mathcal {L}$ in (REF ), which is selfadjoint in a weighted $L^2$ Krein space.\",\n \"Most of the existing literature for differential operators with an indefinite weight focuses on regular or left-definite problems.\",\n \"The spectral properties of the operators associated to $\\\\mathcal {L}$ in the case of a regular Sturm-Liouville expression $\\\\ell $ were investigated in detail, we refer to [20], the monograph [56] and the detailed references therein.\",\n \"Also singular left-definite Sturm-Liouville problems are well studied.\",\n \"Here the selfadjoint operator $T$ associated with $\\\\ell $ is uniformly positive and, hence, the corresponding indefinite differential operator $A$ associated with $\\\\mathcal {L}$ in (REF ) has real spectrum with a gap around zero, cf.\",\n \"e.g., [15], [16], [17], [41], [42] and [56].\",\n \"In the case $T\\\\ge 0$ it is of particular interest whether the operator $A$ is similar to a selfadjoint operator; necessary and sufficient similarity criteria can be found in, e.g., [37], [38].\",\n \"The slightly more general situation where the indefinite Sturm-Liouville operator $A$ has finitely many negative squares or is quasi-uniformly positive is discussed in, e.g., [12], [14], [22].\",\n \"The general non-left-definite case is more difficult, especially the situation where the essential spectrum of the selfadjoint operator $T$ associated with $\\\\ell $ is no longer contained in $\\\\mathbb {R}^+$ .\",\n \"In this case subtle problems appear, as, e.g., accumulation of non-real eigenvalues to the real axis, see [8], [11], [13], [39].\",\n \"The spectral properties of indefinite elliptic partial differential operators have been investigated in, e.g., [25], [26], [27], [49], [50], [51] on bounded domains and in [9], [21] on unbounded domains.\",\n \"In the non-left-definite situation the indefinite differential operator $A$ typically possesses non-real spectrum.\",\n \"General perturbation results for selfadjoint operators in Krein spaces from [4], [7], [30] ensure that the non-real spectrum is contained in a compact set.\",\n \"To the best of our knowledge, explicit bounds on the size of this set do not exist in the literature.\",\n \"For singular indefinite Sturm-Liouville operators it is conjectured in [56] that the lower bound of the spectrum of $T$ is related to a bound for the non-real eigenvalues; the numerical examples contained in [11], [12] support this conjecture.\",\n \"It is one of our main objectives to confirm this conjecture and to provide explicit bounds on the non-real eigenvalues of indefinite Sturm-Liouville operators and indefinite elliptic partial differential operators.\",\n \"For this, we develop here an abstract Krein space perturbation approach which is designed for applications to differential operators with indefinite weights.\",\n \"Besides bounds on the non-real spectrum, our general perturbation results Theorem REF and Theorem REF in Section provide also quantitative estimates for intervals containing only spectrum of positive/negative type in terms of the norm of the perturbation and some resolvent integral, as well as information on the critical point $\\\\infty $ of the perturbed operator.\",\n \"The basic idea of our approach is simple: If $T$ is a semibounded selfadjoint operator in a Hilbert space with some negative lower bound $\\\\gamma $ , then the operator $T+\\\\gamma $ is non-negative or uniformly positive.\",\n \"Hence $A_0:=A+\\\\frac{\\\\gamma }{w}=\\\\frac{1}{w} (T+\\\\gamma )$ (or, more generally, $A_0=G^{-1}(T+\\\\gamma )$ with some Gram operator $G$ connecting the Hilbert and Krein space inner product) is a non-negative operator in a Krein space and therefore the spectrum of $A_0$ is real.\",\n \"Moreover, the difference of $A_0$ and $A$ is a bounded operator.\",\n \"In general, a bounded perturbation of $A_0$ may lead to unbounded non-real spectrum, but under the additional assumption that 0 and $\\\\infty $ are not singular critical points of $A_0$ , the influence of the perturbation on the non-real spectrum can be controlled.\",\n \"The proofs of Theorem REF and REF are based on general Krein space perturbation techniques, norm estimates and local spectral theory; they are partly inspired by methods developed in [9], [48].\",\n \"The abstract perturbation results from Section are applied to ordinary and partial differential operators with indefinite weights in Section .\",\n \"First we investigate the singular indefinite Sturm-Liouville operator $A$ with $p=1$ , a real potential $q\\\\in L^\\\\infty (\\\\mathbb {R})$ and the particularly simple weight function $w(x) =\\\\operatorname{sgn}(x)$ .\",\n \"It turns out in Theorem REF that the nonreal spectrum of $A$ is contained in the set $\\\\bigl \\\\lbrace \\\\lambda \\\\in \\\\operatorname{dist}\\\\,\\\\bigl (\\\\lambda , (-d,d)\\\\bigr )\\\\le 5 \\\\Vert q\\\\Vert _\\\\infty ,\\\\, |\\\\operatorname{Im}\\\\lambda |\\\\le 2 \\\\Vert q\\\\Vert _\\\\infty \\\\bigr \\\\rbrace ,$ where $-d= \\\\,\\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}q(x)$ is assumed to be negative, and estimates on real spectral points of positive and negative type are obtained as well.\",\n \"Our second example is a second order uniformly elliptic operator defined on an unbounded domain $\\\\Omega \\\\subset \\\\mathbb {R}^n$ with bounded coefficients and an essentially bounded weight function $w$ having an essentially bounded inverse.\",\n \"We emphasize that the estimates for the non-real spectrum of indefinite differential operators seem to be the first ones in the mathematical literature.\",\n \"Notation: As usual, $\\\\mathbb {C}^+$ ($\\\\mathbb {C}^-$ ) denotes the open upper half-plane (the open lower half-plane, respectively), $\\\\mathbb {R}^+:=(0,\\\\infty )$ and $\\\\mathbb {R}^-:=(-\\\\infty , 0)$ .\",\n \"The compactification $\\\\mathbb {R} \\\\cup \\\\lbrace \\\\infty \\\\rbrace $ of $\\\\mathbb {R}$ is denoted by $\\\\overline{\\\\mathbb {R}}$ , the compactification $\\\\mathbb {C} \\\\cup \\\\lbrace \\\\infty \\\\rbrace $ by $\\\\overline{\\\\mathbb {C}}$ .\",\n \"By $L(\\\\mathcal {H},\\\\mathcal {K})$ we denote the set of all bounded and everywhere defined linear operators from a Hilbert space $\\\\mathcal {H}$ to a Hilbert space $\\\\mathcal {K}$ .\",\n \"We write $L(\\\\mathcal {H})$ for $L(\\\\mathcal {H},\\\\mathcal {H})$ .\",\n \"For a closed, densely defined linear operator $T$ in $\\\\mathcal {H}$ we denote the spectrum and the resolvent set by $\\\\sigma (T)$ and $\\\\rho (T)$ , respectively.\",\n \"A point $\\\\lambda \\\\in belongs to the {\\\\it approximate point spectrum} $ ap(T)$ of $ T$ if there exists a sequence $ (fn)$ in $ domT$ with $ fn = 1$ for $ n$\\\\mathbb {N}$$ and $ (T - )fn0$ as $ n$.\",\n \"The essentialspectrum of $ T$ is the set $ {C : A- is not Fredholm}$.$\"\n ],\n [\n \"Selfadjoint operators in Krein spaces\",\n \"Let $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ be a Krein space, let $J$ be a fixed fundamental symmetry in $\\\\mathcal {H}$ and denote by $(\\\\cdot \\\\,,\\\\cdot )$ the Hilbert space scalar product induced by $J$ , i.e.\",\n \"$(\\\\cdot \\\\,,\\\\cdot )= [J\\\\,\\\\cdot \\\\,,\\\\cdot ]$ .\",\n \"The induced norm is denoted by $\\\\Vert \\\\cdot \\\\Vert $ .\",\n \"For a detailed treatment of Krein spaces and operators therein we refer to the monographs [5] and [18].\",\n \"For a densely defined linear operator $A$ in $\\\\mathcal {H}$ the adjoint with respect to the Krein space inner product $[\\\\cdot ,\\\\cdot ]$ is denoted by $A^+$ .\",\n \"We mention that $A^+ = JA^*J$ , where $A^*$ denotes the adjoint of $A$ with respect to the scalar product $(\\\\cdot \\\\,,\\\\cdot )$ .\",\n \"The operator $A$ is called selfadjoint in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ (or $[\\\\cdot \\\\,,\\\\cdot ]$ -selfadjoint) if $A = A^+$ .\",\n \"This is equivalent to the selfadjointness of the operator $JA$ in the Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot ))$ .\",\n \"The spectrum of a selfadjoint operator $A$ in a Krein space is in general not contained in $\\\\mathbb {R}$ , but its real spectral points belong to the approximate point spectrum (see, e.g., [18]), $\\\\sigma (A)\\\\cap \\\\mathbb {R}\\\\subset \\\\sigma _{{ap}}(A).$ A selfadjoint operator $A$ in $(\\\\mathcal {H}, [\\\\cdot \\\\,,\\\\cdot ])$ is said to be non-negative if $\\\\rho (A)\\\\ne \\\\varnothing $ and if $[Af,f]\\\\ge 0$ holds for all $f\\\\in \\\\operatorname{dom}A$ .\",\n \"If, for some $\\\\gamma > 0$ , $[Af,f]\\\\ge \\\\gamma \\\\Vert f\\\\Vert ^2$ holds for all $f\\\\in \\\\operatorname{dom}A$ , then $A$ is called uniformly positive.\",\n \"A selfadjoint operator $A$ in $(\\\\mathcal {H}, [\\\\cdot \\\\,,\\\\cdot ])$ is uniformly positive if and only if $A$ is non-negative and $0\\\\in \\\\rho (A)$ .\",\n \"It is well-known that the spectrum of a non-negative operator $A$ in a Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ is real.\",\n \"Moreover, $A$ possesses a spectral function $E_A$ on $\\\\overline{\\\\mathbb {R}}$ which is defined for all Borel sets $\\\\Delta \\\\subset \\\\overline{\\\\mathbb {R}}$ whose boundary does not contain the points 0 and $\\\\infty $ .\",\n \"The corresponding spectral projection $E_A(\\\\Delta )$ is bounded and $[\\\\cdot \\\\,,\\\\cdot ]$ -selfadjoint; see [3], [31], [43], [45], [46] for further details.\",\n \"The point 0 ($\\\\infty $ ) is called a critical point of $A$ if for each Borel set $\\\\Delta \\\\subset \\\\overline{\\\\mathbb {R}}$ with $0\\\\in \\\\Delta $ ($\\\\infty \\\\in \\\\Delta $ , respectively) such that $E_A(\\\\Delta )$ is defined the inner product $[\\\\cdot \\\\,,\\\\cdot ]$ is indefinite on the subspace $E_A(\\\\Delta )\\\\mathcal {H}$ .\",\n \"Moreover, if the point 0 ($\\\\infty $ ) is a critical point of $A$ , it is called regular if there exists $C > 0$ such that $\\\\Vert E_A([-\\\\varepsilon ,\\\\varepsilon ])\\\\Vert \\\\le C$ ($\\\\Vert E_A([-\\\\varepsilon ^{-1},\\\\varepsilon ^{-1}])\\\\Vert \\\\le C$ , respectively) holds for each $\\\\varepsilon \\\\in (0,1)$ .\",\n \"Otherwise, the critical point 0 or $\\\\infty $ is called singular.\",\n \"In this paper we will investigate operators which are non-negative outside of some compact set $K$ with $0\\\\in K$ , see Definition REF belowDefinition REF is slightly more general than the definition used in [10].\",\n \"Contrary to the definition in [10], a spectral projector for an operator non-negative over $\\\\overline{\\\\setminus K in the sense of Definition \\\\ref {d:lnn}corresponding to the set \\\\overline{\\\\setminus K does, in general, not exist.}}$.\",\n \"This notion is a generalization of the above concept of non-negative operators in Krein spaces and will be used in the study of additive bounded perturbations of non-negative operators in Section and .\",\n \"For a set $\\\\Delta \\\\subset we denote by $ *$ the set which is obtained by reflecting $$ in the real axis, that is, $ * = {: }$.$ Definition 2.1 Let $K=K^*\\\\subset be a compact set such that $ 0K$ and $ +K$ is simply connected.\",\n \"A selfadjoint operator$ A$ in the Krein space $ (H,[,])$ is said to be {\\\\em non-negative over} $K$ if for any bounded openneighborhood $ U$ of $ K$ in $ there exists a bounded $[\\\\cdot \\\\,,\\\\cdot ]$ -selfadjoint projection $E_\\\\infty $ such that with respect to the decomposition $\\\\mathcal {H}= (I-E_\\\\infty )\\\\mathcal {H}\\\\,[\\\\dotplus ]\\\\,E_\\\\infty \\\\mathcal {H}$ the operator $A$ can be written as a diagonal operator matrix $A = \\\\begin{pmatrix}A_0 & 0\\\\\\\\0 & A_\\\\infty \\\\end{pmatrix},$ where $A_0$ is a bounded selfadjoint operator in the Krein space $((I-E_\\\\infty )\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ whose spectrum is contained in $\\\\overline{\\\\mathcal {U}}$ and $A_\\\\infty $ is a non-negative operator in $(E_\\\\infty \\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ with $\\\\mathcal {U}\\\\subset \\\\rho (A_\\\\infty )$ .\",\n \"Relations (REF ) and (REF ) encode the non-negativity outside of the set $K$ , which can also be seen in the following example.\",\n \"Example 2.2 Let $A$ be the direct sum of a bounded selfadjoint operator $A_1$ in the Krein space $(\\\\mathcal {H}_1, [\\\\cdot \\\\,,\\\\cdot ]_1)$ and of a non-negative operator $A_2$ in the Krein space $(\\\\mathcal {H}_2, [\\\\cdot \\\\,,\\\\cdot ]_2)$ , $A := A_1\\\\times A_2.$ Let $K_{r_1}$ be the closed disc around zero whose radius is the spectral radius $r_1$ of $A_1$ , then for a bounded open neighborhood $\\\\mathcal {U}$ of $K_{r_1}$ in $ we set$ E:= E2(RU) $, where $ E2$ is the spectral function of $ A2$.\",\n \"We obtain a decomposition of the form (\\\\ref {e:dec_lnn})and the operator $ A$ can be written as in (\\\\ref {e:deco}),where $ A0$ and $ A$ have the properties stated inDefinition \\\\ref {d:lnn}.\",\n \"Therefore, the operator $ A$ isnon-negative over $Kr1$.$ Let $A$ be non-negative over $\\\\overline{\\\\setminus K. Then from the representation (\\\\ref {e:deco}) it is seen thatfor each bounded open neighborhood \\\\mathcal {U} of K the non-real spectrum of A is contained in \\\\overline{\\\\mathcal {U}}.\",\n \"Therefore,\\\\begin{equation}\\\\sigma (A)\\\\setminus \\\\mathbb {R}\\\\subset K.\\\\end{equation}By (\\\\ref {e:deco})the spectral properties of A and A_\\\\infty in \\\\overline{\\\\mathbb {R}}\\\\setminus K are the same.\",\n \"Therefore, we say that \\\\infty is a{\\\\em singular (regular) critical point} of A if \\\\infty is a singular (regular, respectively) critical point of A_\\\\infty .The following proposition is a direct consequence of \\\\cite [Theorem~3.2]{cu} andthe representation (\\\\ref {e:deco}).\",\n \"}\\\\begin{proposition}Let A be a selfadjoint operator in the Krein space (\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ]) and assume that A is non-negative over \\\\overline{\\\\setminus K.Then \\\\infty is not a singular critical point of A if and only if there exists a uniformly positive operator W in the Kreinspace (\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ]) such that W\\\\operatorname{dom}A\\\\subset \\\\operatorname{dom}A.\",\n \"}\\\\end{proposition}$ In order to characterize operators which are non-negative over $\\\\overline{\\\\setminus K in Theorem \\\\ref {t:eqthm} below we recallthe notions of the spectral points of positive and negative type of selfadjoint operators in Krein spaces.\",\n \"The following definitioncan be found in, e.g., \\\\cite {j03,LcMM,lmm}.\",\n \"}\\\\begin{definition}Let A be a selfadjoint operator in the Krein space (\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ]).\",\n \"A point \\\\lambda \\\\in \\\\sigma _{{ap}}(A) is called a {\\\\em spectral point ofpositive {\\\\rm (}negative{\\\\rm )} type} of A if for every sequence (f_n) in \\\\operatorname{dom}A with \\\\Vert f_n\\\\Vert = 1 and (A - \\\\lambda )f_n\\\\rightarrow 0as n\\\\rightarrow \\\\infty we have\\\\liminf _{n\\\\rightarrow \\\\infty }\\\\,[f_n,f_n] > 0\\\\quad \\\\Big (\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,[f_n,f_n] < 0,\\\\;\\\\text{respectively}\\\\Big ).The set of all spectral points of positive {\\\\rm (}negative{\\\\rm )} type of A will be denoted by \\\\sigma _+(A) {\\\\rm (}\\\\sigma _-(A),respectively{\\\\rm )}.\",\n \"A set \\\\Delta \\\\subset is said to be of {\\\\em positive {\\\\rm (}negative{\\\\rm )} type} with respect to Aif each spectral point of A in \\\\Delta is of positive type {\\\\rm (}negative type, respectively{\\\\rm )}.\\\\end{definition}$ The sets $\\\\sigma _+(A)$ and $\\\\sigma _-(A)$ are contained in $\\\\mathbb {R}$ , open in $\\\\sigma (A)$ and the non-real spectrum of $A$ cannot accumulate to $\\\\sigma _+(A)\\\\cup \\\\sigma _-(A)$ .\",\n \"Moreover, at a spectral point $\\\\lambda _0$ of positive or negative type the growth of the resolvent is of order one in the sense of the following definition, see also [6], [36], [48].\",\n \"Definition 2.3 Let $A$ be a selfadjoint operator in the Krein space $\\\\mathcal {H}$ and assume that $\\\\lambda _0\\\\in \\\\overline{\\\\mathbb {R}}$ is not an accumulation point of $\\\\sigma (A)\\\\setminus \\\\mathbb {R}$ .\",\n \"We say that the growth of the resolvent of $A$ at $\\\\lambda _0$ is of order $m\\\\ge 1$ if there exist an open neighborhood $\\\\mathcal {U}$ of $\\\\lambda _0$ in $\\\\overline{ and M > 0 such that \\\\mathcal {U}\\\\setminus \\\\overline{\\\\mathbb {R}}\\\\subset \\\\rho (A) and\\\\begin{equation}\\\\Vert (A - \\\\lambda )^{-1}\\\\Vert \\\\le M\\\\,\\\\frac{(1+|\\\\lambda |)^{2m-2}}{|\\\\operatorname{Im}\\\\lambda |^{m}}\\\\end{equation}holds for all \\\\lambda \\\\in \\\\mathcal {U}\\\\setminus \\\\overline{\\\\mathbb {R}}.\",\n \"}$ Clearly, if $\\\\lambda _0\\\\ne \\\\infty $ , Definition REF can be formulated equivalently by replacing the enumerator in () by 1.\",\n \"Moreover, if the growth of the resolvent of $A$ at $\\\\lambda _0$ is of order $m$ , then it is of order $n$ for each $n > m$ .\",\n \"The following theorem gives an equivalent characterization for non-negative operators in a neighbourhood of $\\\\infty $ .\",\n \"Theorem 2.4 Let $A$ be a selfadjoint operator in the Krein space $\\\\mathcal {H}$ and let $K=K^*\\\\subset be a compact set such that $ 0K$ and$ +K$ is simply connected.\",\n \"Then $ A$ is non-negative over $K$ if and only if the following conditions are satisfied:\\\\begin{enumerate}\\\\item [{\\\\rm (i)}] \\\\sigma (A)\\\\setminus \\\\mathbb {R}\\\\subset K and (\\\\sigma (A)\\\\setminus K)\\\\cap \\\\mathbb {R}^\\\\pm \\\\subset \\\\sigma _\\\\pm (A).\\\\item [{\\\\rm (ii)}] The growth of the resolvent of A at \\\\infty is of finite order.\\\\end{enumerate}$ Let $A$ be non-negative over $\\\\overline{\\\\setminus K.It is well-known that the growth of the resolvent of a non-negative operator in a Krein space is of finite orderfor all \\\\lambda _0 \\\\in \\\\overline{\\\\mathbb {R}} and that spectral points in \\\\mathbb {R}^+ (\\\\mathbb {R}^-) are ofpositive (resp.\\\\ negative) type, see, e.g., \\\\cite [Proposition II.2.1 and Proposition II.3.1]{L82} and \\\\cite [Section 2.1]{J88}.Therefore, the statements (i) and (ii) are consequences ofthe representation (\\\\ref {e:deco}); cf.\",\n \"(\\\\ref {nrspec}).\",\n \"}Conversely, assume that (i) and (ii) are satisfied.\",\n \"Then the operator $ A$ is definitizable over $K$, see \\\\cite {j03}.In particular $ A$ possesses a local spectral function $ E$ which is defined on all Borel sets $$\\\\mathbb {R}$K$ forwhich neither $$ nor points of $ K$\\\\mathbb {R}$$ are boundary points.\",\n \"For such aset $$ the following holds:\\\\begin{enumerate}\\\\item [{\\\\rm (a)}] E(\\\\Delta ) is a bounded [\\\\cdot \\\\,,\\\\cdot ]-selfadjoint projection and commuteswith every bounded operator which commutes with the resolvent of A;\\\\item [{\\\\rm (b)}] \\\\sigma (A|E(\\\\Delta )\\\\mathcal {H})\\\\,\\\\subset \\\\,\\\\sigma (A)\\\\cap \\\\overline{\\\\Delta };\\\\item [{\\\\rm (c)}] \\\\sigma (A|(I - E(\\\\Delta ))\\\\mathcal {H})\\\\,\\\\subset \\\\,\\\\sigma (A)\\\\setminus \\\\operatorname{int}(\\\\Delta ), where\\\\operatorname{int}(\\\\Delta ) is the interior of \\\\Delta with respect to the topology of\\\\overline{\\\\mathbb {R}};\\\\item [{\\\\rm (d)}] If, in addition, \\\\Delta is a neighbourhood of \\\\infty (with respect to the topology of\\\\overline{\\\\mathbb {R}}), then A|(I-E(\\\\Delta )) \\\\mathcal {H} is a bounded operator;\\\\end{enumerate}cf.\",\n \"\\\\cite [Definition 3.13, Theorems 3.15 and 4.8, Remark 4.9]{j03} and note that (d) follows from the definition of the local spectral function\\\\cite [Definition 3.13]{j03} and the definition of the extended spectrum $ (A)$in \\\\cite {j03}.$ Let $\\\\mathcal {U}$ be a bounded open neighborhood of $K$ in $ and set $ E:= E($\\\\mathbb {R}$U)$.\",\n \"By (a), with respect to the decomposition$$\\\\mathcal {H}= (I - E_\\\\infty )\\\\mathcal {H}\\\\,[\\\\dotplus ]\\\\,E_\\\\infty \\\\mathcal {H},$$the operator $ A$ is represented by$$A = \\\\begin{pmatrix}A_0 & 0\\\\\\\\0 & A_\\\\infty \\\\end{pmatrix},$$and by (b) the spectrum of the operator $ A$ is real, $ (A) R U$.\",\n \"According to (i) and (ii),the operator $ A$ is definitizable over $$ and thus definitizable, see \\\\cite [Theorem 4.7]{j03}.Therefore, \\\\cite [Corollary 3 of Proposition II.5.2]{L82} implies that $ A-1$ (and thus $ A$) is non-negative in $ EH$.By (d), the operator $ A0= A|(I-E) H$ is bounded and, by (c),$ (A0) (A) int($\\\\mathbb {R}$U)U$.\\\\Box $\"\n ],\n [\n \"Bounded selfadjoint perturbations of non-negative operators\",\n \"In this section we prove two abstract results on additive bounded perturbations of non-negative (and some closely connected class of) operators in Krein spaces that lead to perturbed operators which are non-negative over some neighborhood of infinity.\",\n \"In both cases the neighborhood is given in quantitative terms.\",\n \"The results will be applied in Section to singular indefinite Sturm-Liouville operators and to second order elliptic operators with indefinite weights.\"\n ],\n [\n \"Two perturbation results\",\n \"The following notation will be useful when formulating our main results below: For a set $\\\\Delta \\\\subset \\\\mathbb {R}$ and $r > 0$ we define $K_r(\\\\Delta ) := \\\\lbrace z\\\\in \\\\operatorname{dist}(z,\\\\Delta )\\\\,\\\\le \\\\,r\\\\rbrace .$ Our first main theorem concerns bounded selfadjoint perturbations of non-negative operators in Krein spaces.\",\n \"Theorem 3.1 Let $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ be a Krein space with fundamental symmetry $J$ and norm $\\\\Vert \\\\cdot \\\\Vert = [J\\\\cdot ,\\\\cdot ]^{1/2}$ .\",\n \"Let $A_0$ be a non-negative operator in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ such that 0 and $\\\\infty $ are not singular critical points of $A_0$ , and $0\\\\notin \\\\sigma _p(A_0)$ .\",\n \"Furthermore, let $V\\\\in L(\\\\mathcal {H})$ be a selfadjoint operator in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"Then the following holds: (i) If $V$ is non-negative, then $A_0+V$ is a non-negative operator in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"(ii) If $V$ is not non-negative, then $A_0+V$ is non-negative over $\\\\overline{\\\\setminus K_r((-d,d)),where\\\\begin{equation}r = \\\\frac{1+\\\\tau _0}{2}\\\\Vert V\\\\Vert ,\\\\qquad \\\\quad d = -\\\\frac{1+\\\\tau _0}{2}\\\\,\\\\min \\\\sigma (JV),\\\\end{equation}and\\\\begin{equation}\\\\tau _0 = \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{1/n}^n\\\\,\\\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\right)\\\\,dt\\\\,\\\\right\\\\Vert <\\\\infty .\\\\end{equation}}Moreover, in both cases $$ is not a singular critical point of $ A0 + V$.$ The following simple example shows that without further assumptions a bounded selfadjoint perturbation $V$ of a non-negative or uniformly positive $A_0$ in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ may lead to an operator $A_0+V$ with unbounded non-real spectrum.\",\n \"Example 3.2 Let $(\\\\cal K,(\\\\cdot ,\\\\cdot ))$ be a Hilbert space and let $H$ be an unbounded selfadjoint operator in $\\\\cal K$ such that $\\\\sigma (H)\\\\subset (0,\\\\infty )$ .\",\n \"Equip $\\\\cal H:=\\\\cal K\\\\oplus \\\\cal K$ with the Krein space inner product $\\\\left[\\\\begin{pmatrix}k_1 \\\\\\\\ k_2\\\\end{pmatrix},\\\\begin{pmatrix}l_1 \\\\\\\\ l_2\\\\end{pmatrix}\\\\right] :=(k_1,l_2)+(k_2,l_1),\\\\qquad \\\\begin{pmatrix}k_1 \\\\\\\\ k_2\\\\end{pmatrix},\\\\begin{pmatrix}l_1 \\\\\\\\ l_2\\\\end{pmatrix}\\\\in \\\\cal H,$ and consider the operators $A_0=\\\\begin{pmatrix} 0 & I \\\\\\\\ H & 0\\\\end{pmatrix},\\\\quad V=\\\\begin{pmatrix} 0 & -I \\\\\\\\ 0 & 0\\\\end{pmatrix},\\\\quad \\\\text{and}\\\\quad A_0+V =\\\\begin{pmatrix} 0 & 0 \\\\\\\\ H & 0\\\\end{pmatrix}.$ It is easy to see that $A_0$ is a non-negative operator and $V$ is a bounded selfadjoint operator in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"Moreover, as $\\\\operatorname{dom}(A_0+V)=\\\\operatorname{dom}H\\\\oplus {\\\\cal H}$ we conclude $\\\\operatorname{ran}(A_0+V-\\\\lambda )\\\\ne \\\\cal H$ for every $\\\\lambda \\\\in , that is, $ (A0+V)=.\",\n \"Our second main result applies to operators of the form $A = G^{-1}T$ , where $T$ is a semibounded selfadjoint operator in a Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot ))$ and $G$ is a selfadjoint bounded and boundedly invertible operator in the Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot ))$ .\",\n \"In other words, $A$ is selfadjoint in the Krein space $(\\\\mathcal {H},(G\\\\cdot ,\\\\cdot ))$ and $A + \\\\eta G^{-1}$ is uniformly positive for suitable $\\\\eta $ .\",\n \"Such a situation arises, e.g., when considering elliptic differential operators with an indefinite weight function; cf.\",\n \"Section 4.2.\",\n \"Theorem 3.3 Let $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot ))$ be a Hilbert space, $\\\\Vert \\\\cdot \\\\Vert = (\\\\cdot \\\\,,\\\\cdot )^{1/2}$ and $[\\\\cdot \\\\,,\\\\cdot ]=(G\\\\cdot ,\\\\cdot )$ be as above.\",\n \"Let $A$ be a selfadjoint operator in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ such that for some $\\\\gamma > 0$ we have $[Af,f]\\\\ge -\\\\gamma \\\\Vert f\\\\Vert ^2\\\\quad \\\\text{for all }f\\\\in \\\\operatorname{dom}A.$ Assume furthermore that for some $\\\\eta > \\\\gamma $ $\\\\tau _\\\\eta := \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{-n}^n\\\\,(A + \\\\eta G^{-1} - it)^{-1}\\\\,dt\\\\,\\\\right\\\\Vert < \\\\infty ,$ holds.\",\n \"Then $A$ is non-negative over $\\\\overline{\\\\setminus K_r((-r,r)), where\\\\begin{equation*}r = \\\\eta \\\\,\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert G^{-1}\\\\Vert .\\\\end{equation*}Moreover, \\\\infty is not a singular critical point of A.\",\n \"}$ Remark 3.4 Setting $A_0 := A + \\\\eta G^{-1}$ in Theorem REF and $V:=-\\\\eta G^{-1}$ we have $A = A_0 + V$ and hence Theorem REF can also be seen as a variant of Theorem REF .\",\n \"Here $A_0=A + \\\\eta G^{-1}$ is uniformly positive so that $0\\\\in \\\\rho (A_0)$ is not a critical point of $A_0$ and the entire imaginary axis belongs to $\\\\rho (A + \\\\eta G^{-1})$ .\",\n \"It follows from [55] and [32] that $\\\\tau _\\\\eta < \\\\infty $ is equivalent to $\\\\infty $ not being a singular critical point of $A_0$ .\",\n \"In what follows we describe the structure of the proofs of Theorem REF (ii) and Theorem REF .\",\n \"Let $E$ denote the spectral function of the non-negative operator $A_0$ .\",\n \"As 0 and $\\\\infty $ are not singular critical points of $A_0$ , the spectral projections $E_+ := E((0,\\\\infty ))$ and $E_- := E((-\\\\infty ,0))$ , and the corresponding spectral subspaces $\\\\mathcal {H}_\\\\pm := E_\\\\pm \\\\mathcal {H}$ of $A_0$ exist.\",\n \"From $0\\\\notin \\\\sigma _p(A_0)$ it follows that $\\\\mathcal {H}= \\\\mathcal {H}_+\\\\,[\\\\dotplus ]\\\\,\\\\mathcal {H}_-$ and $(\\\\mathcal {H}_\\\\pm ,\\\\pm [\\\\cdot \\\\,,\\\\cdot ])$ are Hilbert spaces, see, e.g., [46].\",\n \"Therefore (REF ) is a fundamental decomposition of the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"With respect to this decomposition the operator $A_0$ and the fundamental symmetry $\\\\widetilde{J}$ corresponding to (REF ) can be written as operator matrices: $A_0 = \\\\begin{pmatrix}A_{0,+} & 0\\\\\\\\0 & A_{0,-}\\\\end{pmatrix}\\\\quad \\\\text{and}\\\\quad \\\\widetilde{J} = \\\\begin{pmatrix}I & 0\\\\\\\\0 & -I\\\\end{pmatrix}.$ Note that the operator $\\\\pm A_{0,\\\\pm }$ is a selfadjoint non-negative operator in the Hilbert space $(\\\\mathcal {H}_\\\\pm ,\\\\pm [\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"Hence, $A_0$ is selfadjoint in the Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot )_\\\\sim )$ , where $(f,g)_\\\\sim := [\\\\widetilde{J}f,g] = [E_+f,g] - [E_-f,g],\\\\qquad f,g\\\\in \\\\mathcal {H}.$ Now, let $V\\\\in L(\\\\mathcal {H})$ be a bounded selfadjoint operator in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"Then, with respect to the decomposition (REF ) it admits an operator matrix representation $V = \\\\begin{pmatrix}V_+ & B\\\\\\\\C & V_-\\\\end{pmatrix}.$ From the selfadjointness of $V$ in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ one concludes that $V_\\\\pm $ is selfadjoint in the Hilbert space $(\\\\mathcal {H}_\\\\pm ,\\\\pm [\\\\cdot \\\\,,\\\\cdot ]) = (\\\\mathcal {H}_\\\\pm ,(\\\\cdot \\\\,,\\\\cdot )_\\\\sim )$ and $C = -B^{\\\\widetilde{*}},$ where $B^{\\\\widetilde{*}}$ denotes the adjoint of $B$ with respect to the scalar product $(\\\\cdot \\\\,,\\\\cdot )_\\\\sim $ .\",\n \"Hence, the perturbed operator $A := A_0 + V$ is represented by $A = \\\\begin{pmatrix}A_{0,+} + V_+ & B\\\\\\\\-B^{\\\\widetilde{*}} & A_{0,-} + V_-\\\\end{pmatrix}.$ For operators as in (REF ) we show in Theorem REF below that $\\\\mathbb {R}\\\\setminus K_{\\\\Vert B\\\\Vert _\\\\sim } (\\\\sigma (A_{0,-} + V_-))$ is of positive type, $\\\\mathbb {R}\\\\setminus K_{\\\\Vert B\\\\Vert _\\\\sim } (\\\\sigma (A_{0,+} + V_+))$ is of negative type, the non-real spectrum of $A$ is contained in $K:=K_{\\\\Vert B\\\\Vert _\\\\sim }\\\\bigl (\\\\sigma (A_{0,+} + V_+)\\\\bigr )\\\\cap K_{\\\\Vert B\\\\Vert _\\\\sim } \\\\bigl (\\\\sigma (A_{0,-} + V_-)\\\\bigr ),$ and the growth of the resolvent of $A$ at $\\\\infty $ is of order one.\",\n \"Here $\\\\Vert \\\\cdot \\\\Vert _\\\\sim $ stands for the operator norm induced by (REF ).\",\n \"Then, by Theorem REF , the operator $A$ is non-negative over $\\\\overline{\\\\setminus K, where K is as in (\\\\ref {BadHersfeld}).In a final step it remains to bound the quantities\\\\Vert B\\\\Vert _\\\\sim \\\\quad \\\\mbox{and} \\\\quad K_{\\\\Vert B\\\\Vert _\\\\sim } (\\\\sigma (A_{0,\\\\pm } + V_\\\\pm ))by the quantitiesr and \\\\tau _0 as in the statement of Theorem \\\\ref {t:main1}.\",\n \"}$ Some auxiliary statements In this section we formulate and prove some auxiliary statements which will be used in the proofs Theorems REF and REF .\",\n \"For the special case of bounded operators the first part of the following theorem was already proved in [48], see also [47] and [53], [54] for similar results with unbounded entries on the diagonal.\",\n \"Theorem 3.5 Let $S_+$ and $S_-$ be selfadjoint operators in some Hilbert spaces $\\\\mathfrak {H}_+$ and $\\\\mathfrak {H}_-$ , respectively, let $M\\\\in L(\\\\mathfrak {H}_-,\\\\mathfrak {H}_+)$ and define the operators $S := \\\\begin{pmatrix}S_+ & M\\\\\\\\-M^* & S_-\\\\end{pmatrix}\\\\quad \\\\text{and}\\\\quad \\\\mathfrak {J}:= \\\\begin{pmatrix}I & 0\\\\\\\\0 & -I\\\\end{pmatrix}$ in the Hilbert space $\\\\mathfrak {H}:= \\\\mathfrak {H}_+\\\\oplus \\\\mathfrak {H}_-$ .\",\n \"Then the operator $S$ is selfadjoint in the Krein space $(\\\\mathfrak {H},(\\\\mathfrak {J}\\\\cdot ,\\\\cdot ))$ and with $\\\\nu := \\\\Vert M\\\\Vert $ the following statements hold: $\\\\sigma (S)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\, K_\\\\nu (\\\\sigma (S_+))\\\\cap K_\\\\nu (\\\\sigma (S_-))$ .\",\n \"$\\\\mathbb {R}\\\\setminus K_\\\\nu (\\\\sigma (S_-))$ is of positive type with respect to $S$ .\",\n \"$\\\\mathbb {R}\\\\setminus K_\\\\nu (\\\\sigma (S_+))$ is of negative type with respect to $S$ .\",\n \"Moreover, if $S_+$ is bounded from below and $S_-$ is bounded from above (or vice versa), the non-real spectrum of $S$ is bounded and the growth of the resolvent of $S$ at $\\\\infty $ is of order one.\",\n \"We set $[\\\\cdot \\\\,,\\\\cdot ]:= (\\\\mathfrak {J}\\\\cdot ,\\\\cdot )$ .\",\n \"Let $\\\\lambda \\\\in \\\\mathbb {C} \\\\setminus K_\\\\nu (\\\\sigma (S_-))$ and $\\\\alpha := \\\\nu /\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_-)) < 1$ .\",\n \"We claim that for some $r(\\\\alpha )>0$ and all $f\\\\in \\\\operatorname{dom}S$ the following implication $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\le \\\\frac{1-\\\\alpha ^2}{4\\\\alpha }\\\\nu \\\\Vert f\\\\Vert \\\\quad \\\\Longrightarrow \\\\quad [f,f]\\\\,\\\\ge \\\\,r(\\\\alpha )\\\\Vert f\\\\Vert ^2$ holds.\",\n \"In fact, set $\\\\varepsilon := \\\\frac{1-\\\\alpha ^2}{4\\\\alpha }$ and let $f\\\\in \\\\operatorname{dom}S$ with $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\le \\\\varepsilon \\\\nu \\\\Vert f\\\\Vert .$ With respect to the decomposition $\\\\mathfrak {H}= \\\\mathfrak {H}_+\\\\oplus \\\\mathfrak {H}_-$ we write $f$ and $g := (S - \\\\lambda )f$ as column vectors $f = \\\\begin{pmatrix}f_+\\\\\\\\f_-\\\\end{pmatrix}\\\\in \\\\operatorname{dom}S_+\\\\oplus \\\\operatorname{dom}S_-, \\\\quad g = \\\\begin{pmatrix}g_+\\\\\\\\g_-\\\\end{pmatrix}.$ Then $g_- = -M^*f_+ + (S_- - \\\\lambda )f_-$ , or, equivalently, $f_- = (S_- - \\\\lambda )^{-1}g_- + (S_- - \\\\lambda )^{-1}M^*f_+.$ As $\\\\Vert g_-\\\\Vert \\\\le \\\\Vert g\\\\Vert = \\\\Vert (S - \\\\lambda )f\\\\Vert \\\\le \\\\varepsilon \\\\nu \\\\Vert f\\\\Vert $ and $\\\\Vert (S_- - \\\\lambda )^{-1}\\\\Vert = \\\\alpha /\\\\nu $ this yields $\\\\Vert f_-\\\\Vert \\\\,\\\\le \\\\,\\\\alpha \\\\varepsilon \\\\Vert f\\\\Vert + \\\\alpha \\\\Vert f_+\\\\Vert $ and hence $\\\\Vert f_-\\\\Vert ^2\\\\,\\\\le \\\\,\\\\alpha ^2\\\\big (\\\\varepsilon ^2\\\\Vert f\\\\Vert ^2 + 2\\\\varepsilon \\\\Vert f\\\\Vert ^2 + \\\\Vert f_+\\\\Vert ^2\\\\big ).$ Using $\\\\Vert f_+\\\\Vert ^2 = \\\\Vert f\\\\Vert ^2 - \\\\Vert f_-\\\\Vert ^2$ we conclude $\\\\Vert f_-\\\\Vert ^2\\\\,\\\\le \\\\,\\\\frac{\\\\alpha ^2}{1+\\\\alpha ^2}(1 + \\\\varepsilon )^2\\\\Vert f\\\\Vert ^2.$ Hence, as $[f,f] = \\\\Vert f_+\\\\Vert ^2 - \\\\Vert f_-\\\\Vert ^2 = \\\\Vert f\\\\Vert ^2 - 2\\\\Vert f_-\\\\Vert ^2$ , we obtain $[f,f]\\\\,\\\\ge \\\\,r(\\\\alpha )\\\\Vert f\\\\Vert ^2$ , where $r(\\\\alpha ) := 1 - \\\\frac{2\\\\alpha ^2}{1+\\\\alpha ^2}(1 + \\\\varepsilon )^2 = \\\\frac{(1 - \\\\alpha )^2(1 + \\\\alpha )(7 - \\\\alpha )}{8(1+\\\\alpha ^2)} > 0,$ i.e., the implication (REF ) holds.\",\n \"It follows that $(K_\\\\nu (\\\\sigma (S_-)))\\\\cap \\\\sigma _{{ap}}(S)\\\\subset \\\\sigma _{+}(S)$ .\",\n \"Hence (ii) is proved and, as $\\\\sigma _{+}(S)$ is real, the non-real spectrum of $S$ satisfies $\\\\sigma (S)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\, K_\\\\nu (\\\\sigma (S_-)).$ Similarly, as in the proof of (REF ), one proves that for $\\\\lambda \\\\in \\\\mathbb {C} \\\\setminus K_\\\\nu (\\\\sigma (S_+))$ , $f\\\\in \\\\operatorname{dom}S$ , and for $\\\\beta := \\\\nu /\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_+))<1$ the implication $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\le \\\\frac{1-\\\\beta ^2}{4\\\\beta }\\\\nu \\\\Vert f\\\\Vert \\\\quad \\\\Longrightarrow \\\\quad [f,f]\\\\,\\\\le \\\\,-r(\\\\beta )\\\\Vert f\\\\Vert ^2$ holds with $r(\\\\beta )>0$ as in (REF ).\",\n \"This shows (iii) and $\\\\sigma (S)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\, K_\\\\nu (\\\\sigma (S_+))$ .\",\n \"Together with (REF ) we obtain (i).\",\n \"Assume now that $S_+$ is bounded from below and $S_-$ is bounded from above.\",\n \"Then the non-real spectrum of $S$ is bounded by (i) and it remains to show that the growth of the resolvent of $S$ is of order one at $\\\\infty $ .\",\n \"Note first that for $S_0=\\\\begin{pmatrix}S_+ & 0 \\\\\\\\ 0 & S_-\\\\end{pmatrix}\\\\qquad \\\\text{and}\\\\qquad V=\\\\begin{pmatrix}0 & M \\\\\\\\ -M^* & 0\\\\end{pmatrix},$ and $\\\\lambda \\\\in such that $ Im2V= 2$ we have $ (S0-)-1V12$ and\\\\begin{equation}\\\\Vert (S-\\\\lambda )^{-1}\\\\Vert = \\\\bigl \\\\Vert \\\\bigl ((S_0-\\\\lambda )(I+(S_0-\\\\lambda )^{-1}V)\\\\bigr )^{-1}\\\\bigr \\\\Vert \\\\le \\\\frac{2}{\\\\vert \\\\operatorname{Im}\\\\lambda \\\\vert }.\\\\end{equation}For $ with $\\\\vert \\\\operatorname{Im}\\\\lambda \\\\vert < 2\\\\nu $ and $\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_-)) > \\\\nu + \\\\delta $ with some fixed $\\\\delta > 0$ again define the value $\\\\alpha = \\\\nu /\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_-))$ .\",\n \"Then $\\\\alpha \\\\in (0,\\\\delta ^{\\\\prime })$ , where $\\\\delta ^{\\\\prime } = \\\\nu /(\\\\nu + \\\\delta )\\\\in (0,1)$ , and for $f\\\\in \\\\operatorname{dom}S$ we either have $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\ge \\\\frac{1-\\\\alpha ^2}{4\\\\alpha }\\\\nu \\\\Vert f\\\\Vert > \\\\frac{1-\\\\alpha ^2}{8\\\\alpha }\\\\vert \\\\operatorname{Im}\\\\lambda \\\\vert \\\\,\\\\Vert f\\\\Vert ,$ or, by (REF ), $r(\\\\alpha )|\\\\operatorname{Im}\\\\lambda |\\\\Vert f\\\\Vert ^2\\\\le |\\\\operatorname{Im}[\\\\lambda f,f]| = |\\\\operatorname{Im}[(S - \\\\lambda )f,f]|\\\\le \\\\Vert (S - \\\\lambda )f\\\\Vert \\\\Vert f\\\\Vert ,$ and hence $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\ge r(\\\\alpha )|\\\\operatorname{Im}\\\\lambda |\\\\,\\\\Vert f\\\\Vert .$ For $\\\\lambda \\\\in with $ Im<2$ and $ dist(,(S+)) > + $ the estimates (\\\\ref {abs2}) and (\\\\ref {abs3}) hold in a similar form.\",\n \"Therefore, (\\\\ref {abs1})-(\\\\ref {abs3}) imply that for all non-real $ with $\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_-)) > \\\\nu + \\\\delta $ or $\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_+)) > \\\\nu + \\\\delta $ we have $\\\\Vert (S-\\\\lambda )^{-1}\\\\Vert \\\\le \\\\frac{C}{\\\\vert \\\\operatorname{Im}\\\\lambda \\\\vert }$ with some $C > 0$ which does not depend on $\\\\lambda $ .\",\n \"This shows that the growth of the resolvent $(S-\\\\lambda )^{-1}$ is of order one at $\\\\infty $ .$\\\\Box $ Remark 3.6 If $\\\\operatorname{dist}(\\\\sigma (S_+),\\\\sigma (S_-))\\\\ge 2\\\\Vert M\\\\Vert $ , then the spectrum of $S$ in Theorem REF is real.\",\n \"This result can be improved in certain special cases, cf.\",\n \"[1] and also [2], where sharp norm bounds on the operator angles between reducing subspaces of $S$ and $S_+\\\\oplus S_-$ are given.\",\n \"We mention here also [34], [35] for comparable results in the study of operators of Klein-Gordon type, where bounded perturbations of operators with unbounded off-diagonal entries are investigated.\",\n \"The following simple example shows that the bounds on the non-real spectrum of $S$ in Theorem REF are sharp.\",\n \"Example 3.7 Let $\\\\mathfrak {H}_+ = \\\\mathfrak {H}_- = and $ S= 1$, $ M = z and hence $S = \\\\begin{pmatrix}1 & z\\\\\\\\-\\\\overline{z} & -1\\\\end{pmatrix}.$ Then $\\\\operatorname{dist}(\\\\sigma (S_+),\\\\sigma (S_-)) = 2$ .\",\n \"If $|z|\\\\le 1$ , then $\\\\sigma (S) = \\\\lbrace \\\\pm \\\\sqrt{1 - |z|^2}\\\\rbrace $ is real; cf.\",\n \"Remark REF .\",\n \"If $|z| > 1$ , then $\\\\sigma (S) = \\\\lbrace \\\\pm i\\\\sqrt{|z|^2 - 1}\\\\rbrace $ and in this case the eigenvalues of $S$ lie on the boundary of $ K_{|z|}(\\\\lbrace 1\\\\rbrace )\\\\cap K_{|z|}(\\\\lbrace -1\\\\rbrace )$ .\",\n \"Lemma 3.8 Let $T$ be a selfadjoint operator in the Hilbert space $(\\\\mathfrak {H},(\\\\cdot \\\\,,\\\\cdot ))$ with $0\\\\notin \\\\sigma _p(T)$ and let $E$ be its spectral function.\",\n \"If $(a_n)$ and $(b_n)$ are sequences in $[0,\\\\infty )$ such that $a_n\\\\downarrow 0$ and $b_n\\\\uparrow \\\\infty $ as $n\\\\rightarrow \\\\infty $ , and $0\\\\in \\\\rho (T)$ if $a_k=0$ for some $k\\\\in \\\\mathbb {N}$ , then $E(\\\\mathbb {R}^+) - E(\\\\mathbb {R}^-) = \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _{a_n}^{b_n}\\\\,\\\\left((T + it)^{-1} + (T - it)^{-1}\\\\right)\\\\,dt\\\\,.$ First of all we observe that $(T + it)^{-1} + (T - it)^{-1} = 2T(T^2 + t^2)^{-1}.$ For all $f,g\\\\in \\\\mathfrak {H}$ we have $\\\\int _{a_n}^{b_n}\\\\,\\\\big (T(T^2 + t^2)^{-1}f,g\\\\big )\\\\,dt&= \\\\int _{a_n}^{b_n}\\\\,\\\\int _{\\\\mathbb {R}}\\\\,\\\\frac{s}{s^2 + t^2}\\\\,d(E_sf,g)\\\\,dt\\\\\\\\&= \\\\int _{\\\\mathbb {R}}\\\\,\\\\int _{a_n}^{b_n}\\\\,\\\\frac{s}{s^2 + t^2}\\\\,dt\\\\,d(E_sf,g)\\\\\\\\&= \\\\int _{\\\\mathbb {R}}\\\\,\\\\big (\\\\arctan (b_n/s) - \\\\arctan (a_n/s)\\\\big )\\\\,d(E_sf,g).\\\\\\\\&= \\\\big (\\\\arctan (b_nT^{-1})f - \\\\arctan (a_nT^{-1})f,g\\\\big ).$ Fubini's theorem can be applied here since on $\\\\mathbb {R}\\\\times [a_n,b_n]$ the integrand is bounded and the measure $d(E_sf,g)\\\\otimes dt$ considered as a measure in $\\\\mathbb {R} \\\\times [a_n,b_n]$ is finite.\",\n \"Hence, $\\\\int _{a_n}^{b_n}\\\\,2T(T^2 + t^2)^{-1}f\\\\,dt = 2\\\\arctan (b_nT^{-1})f - 2\\\\arctan (a_nT^{-1})f.$ Now, we apply [52] and observe that $2\\\\arctan (a_nT^{-1})f$ tends to zero and $2\\\\arctan (b_nT^{-1})f$ tends to $\\\\pi \\\\operatorname{sgn}(T^{-1})f = \\\\pi \\\\operatorname{sgn}(T)f = \\\\pi (E(\\\\mathbb {R}^+) - E(\\\\mathbb {R}^-))f$ as $n\\\\rightarrow \\\\infty $ .$\\\\Box $ A subspace $\\\\mathfrak {L}$ of a Krein space $(\\\\mathfrak {H},[\\\\cdot \\\\,,\\\\cdot ])$ is called uniformly positive if there exists $\\\\delta > 0$ such that $[f,f]\\\\,\\\\ge \\\\,\\\\delta \\\\Vert f\\\\Vert ^2 \\\\quad \\\\mbox{for all } f\\\\in \\\\mathfrak {L}.$ Here $\\\\Vert \\\\cdot \\\\Vert $ is a Hilbert space norm on $\\\\mathfrak {H}$ with respect to which $[\\\\cdot \\\\,,\\\\cdot ]$ is continuous.\",\n \"All such norms are equivalent, see [46].\",\n \"Lemma 3.9 Let $G$ be a bounded and boundedly invertible selfadjoint operator in a Hilbert space $(\\\\mathfrak {H},(\\\\cdot \\\\,,\\\\cdot ))$ and set $[\\\\cdot \\\\,,\\\\cdot ]:= (G\\\\cdot ,\\\\cdot )$ .\",\n \"Let $\\\\mathfrak {L}$ be a closed uniformly positive subspace in the Krein space $(\\\\mathfrak {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"If $P$ ($E$ ) denotes the orthogonal projection in the Hilbert space $(\\\\mathfrak {H},(\\\\cdot \\\\,,\\\\cdot ))$ (in the Krein space $(\\\\mathfrak {H},[\\\\cdot \\\\,,\\\\cdot ])$ , respectively) onto $\\\\mathfrak {L}$ , then $P(G|\\\\mathfrak {L})$ is a uniformly positive selfadjoint operator in $(\\\\mathfrak {L},(\\\\cdot \\\\,,\\\\cdot ))$ , and its inverse is given by $\\\\bigl (P(G|\\\\mathfrak {L})\\\\bigr )^{-1} = E\\\\bigl (G^{-1}|\\\\mathfrak {L}\\\\bigr ).$ As $\\\\mathfrak {L}$ is uniformly positive, for all $f\\\\in \\\\mathfrak {L}$ we have $\\\\big (P(G|\\\\mathfrak {L})f,f\\\\big ) = (Gf,f) = [f,f]\\\\,\\\\ge \\\\,\\\\delta \\\\Vert f\\\\Vert ^2$ with some $\\\\delta > 0$ .\",\n \"Thus, $P(G|\\\\mathfrak {L})$ is a uniformly positive operator in $(\\\\mathfrak {L},(\\\\cdot \\\\,,\\\\cdot ))$ and, in particular, $0\\\\in \\\\rho (P(G|\\\\mathfrak {L}))$ .\",\n \"Let $g\\\\in \\\\mathfrak {L}$ and set $f := E(G^{-1}|\\\\mathfrak {L})g\\\\in \\\\mathfrak {L}$ .\",\n \"Since for all $h\\\\in \\\\mathfrak {L}$ $(g - Gf,h) = \\\\bigl (G(I - E)G^{-1}g,h\\\\bigr ) = \\\\bigl [(I - E)G^{-1}g,h\\\\bigr ] = 0,$ we conclude $g - Gf\\\\in \\\\mathfrak {L}^\\\\perp $ and hence $g = PGf = P(G|\\\\mathfrak {L})f$ .\",\n \"Therefore, $\\\\bigl (P(G|\\\\mathfrak {L})\\\\bigr )^{-1}g = f = E\\\\bigl (G^{-1}|\\\\mathfrak {L}\\\\bigr )g,$ which proves the lemma.$\\\\Box $ Proof of Theorem REF Assertion (i) in Theorem REF is simple: As both $A_0$ and $V$ are non-negative in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ we have $[(A_0+V)f,f]\\\\ge 0\\\\quad \\\\text{for all}\\\\quad f\\\\in \\\\operatorname{dom}(A_0+V)=\\\\operatorname{dom}A_0.$ The operator $A_0$ is selfadjoint in the Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot )_\\\\sim )$ (cf.\",\n \"(REF ) and (REF )) and, as $V$ is bounded, $\\\\rho (A_0+V)$ is nonempty.\",\n \"Thus $A_0+V$ is a non-negative operator in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ and (i) in Theorem REF is proved.\",\n \"Let $E$ be the spectral function of the non-negative operator $A_0$ , define the projections $E_+ := E((0,\\\\infty ))$ , $E_-:=E((-\\\\infty ,0))$ and the subspaces $\\\\mathcal {H}_\\\\pm := E_\\\\pm \\\\mathcal {H}$ .\",\n \"Then with respect to the fundamental decomposition (REF ) the operators $A_0$ and $V$ have the form in (REF ) and (REF ), respectively.\",\n \"The fundamental symmetry $\\\\widetilde{J}$ associated with the decomposition (REF ) is also given in (REF ).\",\n \"The inner product $(\\\\cdot \\\\,,\\\\cdot )_\\\\sim $ is defined as in (REF ).\",\n \"Moreover, $A_{0,\\\\pm }$ and $V_\\\\pm $ from (REF ) are selfadjoint in $(\\\\mathcal {H}_\\\\pm ,(\\\\cdot \\\\,,\\\\cdot )_\\\\sim ) = (\\\\mathcal {H}_\\\\pm ,\\\\pm [\\\\cdot \\\\,,\\\\cdot ])$ , $\\\\sigma (A_{0,\\\\pm })\\\\subset \\\\mathbb {R}^\\\\pm \\\\cup \\\\lbrace 0\\\\rbrace $ and (REF ) holds.\",\n \"We apply Lemma REF to the non-negative operators $\\\\pm A_{0,\\\\pm }$ and obtain $\\\\pm I_{\\\\mathcal {H}_\\\\pm } = \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _{1/n}^n\\\\,\\\\left((A_{0,\\\\pm } + it)^{-1} + (A_{0,\\\\pm } - it)^{-1}\\\\right)\\\\,dt.$ Therefore, $\\\\widetilde{J} = \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _{1/n}^n\\\\,\\\\left((A_{0} + it)^{-1} + (A_{0} - it)^{-1}\\\\right)\\\\,dt,$ where the strong limit is with respect to the norm $\\\\Vert \\\\cdot \\\\Vert _\\\\sim := (\\\\cdot \\\\,,\\\\cdot )_\\\\sim ^{1/2}$ .\",\n \"But as $\\\\Vert \\\\cdot \\\\Vert _\\\\sim $ and $\\\\Vert \\\\cdot \\\\Vert $ are equivalent norms, the limit on the right hand side of (REF ) also exists with respect to the norm $\\\\Vert \\\\cdot \\\\Vert $ , and coincides with $\\\\widetilde{J}$ .\",\n \"The uniform boundedness principle now yields that $\\\\tau _0$ in () is finite and that $\\\\Vert \\\\widetilde{J}\\\\Vert \\\\,\\\\le \\\\,\\\\tau _0<\\\\infty .$ Define the constants $\\\\nu := \\\\Vert B\\\\Vert _\\\\sim ,\\\\quad a := \\\\min \\\\sigma (V_+)\\\\quad \\\\text{and}\\\\quad b := \\\\max \\\\sigma (V_-).$ Then $A_{0,+} + V_+$ is bounded from below by $a$ and $\\\\sigma (A_{0,+} + V_+)\\\\subset [a,\\\\infty )$ , and $A_{0,-} + V_-$ is bounded from above by $b$ and $\\\\sigma (A_{0,-} + V_-)\\\\subset (-\\\\infty ,b]$ .\",\n \"Making use of the representation (REF ) of $A = A_0 + V$ and Theorem REF we conclude: $\\\\sigma (A)\\\\backslash \\\\mathbb {R}\\\\subset K_\\\\nu ((-\\\\infty ,b])\\\\cap K_\\\\nu ([a,\\\\infty ))$ .\",\n \"$(b + \\\\nu ,\\\\infty )$ is of positive type with respect to $A$ .\",\n \"$(-\\\\infty ,a - \\\\nu )$ is of negative type with respect to $A$ .\",\n \"The growth of the resolvent of $A$ at $\\\\infty $ of order one, and hence of finite order.\",\n \"The statement in Theorem REF (ii) follows from Theorem REF if we show that the constants $\\\\nu $ , $a$ and $b$ in (REF ) satisfy $\\\\nu \\\\le r, \\\\quad a\\\\ge -d, \\\\quad b\\\\le d,$ where $r$ and $d$ are as in (), because in that case the properties (a)-(d) above hold with $\\\\nu $ , $a$ and $b$ replaced by $r$ , $-d$ and $d$ , respectively, and $K_\\\\nu ((-\\\\infty ,d])\\\\cap K_\\\\nu ([-d,\\\\infty ))= K_\\\\nu ((-d,d)).$ First we check $\\\\nu \\\\le r$ .\",\n \"Denote by $(\\\\cdot \\\\,,\\\\cdot )$ the scalar product corresponding to the Hilbert space norm $\\\\Vert \\\\cdot \\\\Vert $ .\",\n \"Using the well-known fact that the spectrum of a bounded operator is always a subset of the closure of its numerical range we obtain $\\\\begin{split}\\\\nu ^2&= \\\\Vert B\\\\Vert _\\\\sim ^2 = \\\\Vert B^{\\\\tilde{*}}B\\\\Vert _\\\\sim = \\\\sup \\\\lbrace |\\\\lambda | : \\\\lambda \\\\in \\\\sigma (B^{\\\\tilde{*}}B)\\\\rbrace \\\\\\\\&\\\\le \\\\sup \\\\lbrace |(B^{\\\\tilde{*}}Bf,f)| : \\\\Vert f\\\\Vert =1\\\\rbrace \\\\,\\\\le \\\\,\\\\Vert B^{\\\\tilde{*}}\\\\Vert \\\\,\\\\Vert B\\\\Vert \\\\\\\\&= \\\\Vert E_-(V|\\\\mathcal {H}_+)\\\\Vert \\\\,\\\\Vert E_+(V|\\\\mathcal {H}_-)\\\\Vert \\\\,\\\\le \\\\,\\\\Vert E_+\\\\Vert \\\\,\\\\Vert E_-\\\\Vert \\\\,\\\\Vert V\\\\Vert ^2.\\\\end{split}$ As $E_\\\\pm = \\\\frac{1}{2}(I\\\\pm \\\\widetilde{J})$ , it follows from (REF ) that $\\\\Vert E_\\\\pm \\\\Vert \\\\le \\\\frac{1+\\\\tau _0}{2}$ which shows $\\\\nu \\\\le r$ .\",\n \"Next we verify $a\\\\ge -d$ .\",\n \"Denote by $P_\\\\pm $ the orthogonal projection onto $\\\\mathcal {H}_\\\\pm $ with respect to $(\\\\cdot \\\\,,\\\\cdot )$ .\",\n \"Then, according to Lemma REF , the operator $P_+(J|\\\\mathcal {H}_+)$ is selfadjoint and uniformly positive in the Hilbert space $(\\\\mathcal {H}_+,(\\\\cdot \\\\,,\\\\cdot ))$ .\",\n \"Hence, the same holds for its inverse $E_+(J|\\\\mathcal {H}_+)$ ; cf.\",\n \"Lemma REF .\",\n \"This implies $\\\\min \\\\sigma (P_+(J|\\\\mathcal {H}_+))&= \\\\big (\\\\max \\\\sigma (E_+(J|\\\\mathcal {H}_+))\\\\big )^{-1} = \\\\Vert E_+(J|\\\\mathcal {H}_+)\\\\Vert ^{-1}\\\\\\\\&\\\\ge \\\\Vert E_+J\\\\Vert ^{-1} \\\\ge \\\\Vert E_+\\\\Vert ^{-1}\\\\,\\\\ge \\\\,\\\\frac{2}{1+\\\\tau _0}\\\\,.$ Hence, for $f_+\\\\in \\\\mathcal {H}_+$ we have $[f_+,f_+] = (Jf_+,f_+) = (P_+(J|\\\\mathcal {H}_+)f_+,f_+)\\\\,\\\\ge \\\\,\\\\frac{2}{1+\\\\tau _0}\\\\,\\\\Vert f_+\\\\Vert ^2.$ We also use $(V_+f_+,f_+)_\\\\sim = [V_+f_+,f_+] = [V_+f_+ - B^{\\\\tilde{*}}f_+,f_+] = [Vf_+,f_+] = (JVf_+,f_+)$ for $f\\\\in \\\\mathcal {H}_+$ to conclude $a=\\\\min \\\\sigma (V_+)&=\\\\inf \\\\lbrace (V_+f_+,f_+)_\\\\sim : \\\\Vert f_+\\\\Vert _\\\\sim = 1,\\\\,f_+\\\\in \\\\mathcal {H}_+\\\\rbrace \\\\\\\\&\\\\ge \\\\inf \\\\lbrace (V_+f_+,f_+)_\\\\sim : \\\\Vert f_+\\\\Vert _\\\\sim \\\\le 1,\\\\,f_+\\\\in \\\\mathcal {H}_+\\\\rbrace \\\\\\\\&= \\\\inf \\\\lbrace (JVf_+,f_+) : [f_+,f_+]\\\\le 1,\\\\,f_+\\\\in \\\\mathcal {H}_+\\\\rbrace \\\\\\\\&\\\\ge \\\\inf \\\\bigl \\\\lbrace (JVf_+,f_+) : \\\\Vert f_+\\\\Vert ^2\\\\le \\\\tfrac{1+\\\\tau _0}{2},\\\\,f_+\\\\in \\\\mathcal {H}_+\\\\bigr \\\\rbrace \\\\\\\\&\\\\ge \\\\inf \\\\bigl \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert ^2\\\\le \\\\tfrac{1+\\\\tau _0}{2},\\\\,f\\\\in \\\\mathcal {H}\\\\bigr \\\\rbrace .$ By assumption $V$ is not non-negative in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ and hence $0>\\\\inf \\\\bigl \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert ^2\\\\le \\\\tfrac{1+\\\\tau _0}{2},\\\\,f\\\\in \\\\mathcal {H}\\\\bigr \\\\rbrace =\\\\inf \\\\bigl \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert ^2=\\\\tfrac{1+\\\\tau _0}{2},\\\\,f\\\\in \\\\mathcal {H}\\\\bigr \\\\rbrace .$ Therefore we finally obtain $a=\\\\min \\\\sigma (V_+)&\\\\ge \\\\inf \\\\bigl \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert ^2=\\\\tfrac{1+\\\\tau _0}{2},\\\\,f\\\\in \\\\mathcal {H}\\\\bigr \\\\rbrace \\\\\\\\&= \\\\frac{1+\\\\tau _0}{2}\\\\,\\\\inf \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert = 1,\\\\,f\\\\in \\\\mathcal {H}\\\\rbrace \\\\\\\\&= \\\\frac{1+\\\\tau _0}{2}\\\\,\\\\min \\\\sigma (JV)=-d.$ The proof of the inequality $b\\\\le d$ is analogous and is left to the reader.\",\n \"This completes the proof of (ii).\",\n \"As $\\\\infty $ is not a singular critical point of $A_0$ , the last statement in Theorem REF on the critical point $\\\\infty $ of $A_0+V$ follows in both cases (i) and (ii) from Proposition and $\\\\operatorname{dom}A=\\\\operatorname{dom}(A_0+V)$ .$\\\\Box $ Proof of Theorem REF Let $\\\\gamma $ and $\\\\eta $ be as in Theorem REF .\",\n \"Then the operator $A_0 := A + \\\\eta G^{-1}$ is uniformly positive in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ , $[\\\\cdot \\\\,,\\\\cdot ]= (G\\\\cdot ,\\\\cdot )$ , since for $f\\\\in \\\\operatorname{dom}A$ we have $[A_0 f,f] = [Af,f] + \\\\eta [G^{-1}f,f] \\\\ge -\\\\gamma \\\\Vert f\\\\Vert ^2 + \\\\eta \\\\Vert f\\\\Vert ^2 = (\\\\eta - \\\\gamma )\\\\Vert f\\\\Vert ^2.$ We will show that $A$ is non-negative over $\\\\overline{\\\\setminus K_r((-r,r)).In particular, 0\\\\in \\\\rho (A_0) is not a singular critical point of A_0and, as \\\\tau _\\\\eta < \\\\infty , the point \\\\infty is not aa singular critical point of A_0, see Remark \\\\ref {Mai2012}.With V := -\\\\eta G^{-1} we haveA = A_0 + V\\\\qquad \\\\text{and}\\\\qquad \\\\Vert V\\\\Vert =\\\\eta \\\\Vert G^{-1}\\\\Vert .As in the proof of Theorem~\\\\ref {t:main1} let E be the spectral function of the operator A_0.\",\n \"Then the fundamentaldecomposition \\\\mathcal {H}= E_+\\\\mathcal {H}\\\\,[\\\\dotplus ]\\\\,E_-\\\\mathcal {H} in (\\\\ref {e:dec_expl}) generates matrix representationsof A_0, V and the corresponding fundamental symmetry \\\\widetilde{J} as in (\\\\ref {e:AJ}) and (\\\\ref {e:V}).The operator A_0 is selfadjoint in the Hilbert space(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot )_\\\\sim ), where (\\\\cdot \\\\,,\\\\cdot )_\\\\sim is defined in (\\\\ref {e:sim}).\",\n \"With the help of Lemma \\\\ref {l:veselic} we obtain{\\\\begin{@align*}{1}{-1}\\\\widetilde{J}&= \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _0^n\\\\,\\\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\big )\\\\,dt\\\\\\\\&= \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _{-n}^n\\\\,(A + \\\\eta G^{-1} - it)^{-1}\\\\,dt\\\\,.\\\\end{@align*}}This yields\\\\Vert \\\\widetilde{J}\\\\Vert \\\\,\\\\le \\\\,\\\\tau _\\\\eta .Following the lines of the proof of Theorem \\\\ref {t:main1}, it suffices to show the inequalities\\\\nu \\\\le \\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert V\\\\Vert ,\\\\quad \\\\min \\\\sigma (V_+)\\\\ge -\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert V\\\\Vert \\\\quad \\\\text{and}\\\\quad \\\\max \\\\sigma (V_-)\\\\le \\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert V\\\\Vert ,where \\\\nu =\\\\Vert E_\\\\mp (V\\\\vert \\\\cal H_\\\\pm ) \\\\Vert _\\\\sim .The first relation is proved as in (\\\\ref {e:rel_proof}) and the remaining two inequalities follow from the selfadjointness of V_\\\\pm in (\\\\mathcal {H}_\\\\pm ,(\\\\cdot \\\\,,\\\\cdot )_\\\\sim ) and{\\\\begin{@align*}{1}{-1}\\\\max \\\\lbrace |\\\\min \\\\sigma (V_\\\\pm )|,|\\\\max \\\\sigma (V_\\\\pm )|\\\\rbrace &= \\\\Vert V_\\\\pm \\\\Vert _\\\\sim = \\\\sup \\\\lbrace |\\\\lambda | : \\\\lambda \\\\in \\\\sigma (V_\\\\pm )\\\\rbrace \\\\\\\\&\\\\le \\\\sup \\\\lbrace |(V_\\\\pm f_\\\\pm ,f_\\\\pm )| : \\\\Vert f_\\\\pm \\\\Vert =1,\\\\,f\\\\in \\\\mathcal {H}_\\\\pm \\\\rbrace \\\\\\\\&\\\\le \\\\Vert V_\\\\pm \\\\Vert = \\\\Vert E_\\\\pm (V|\\\\mathcal {H}_\\\\pm )\\\\Vert \\\\,\\\\le \\\\,\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert V\\\\Vert .\\\\end{@align*}}Hence, A is non-negative over \\\\overline{\\\\setminus K_r((-r,r)).\",\n \"As\\\\operatorname{dom}A= \\\\operatorname{dom}A_0, we conclude from Proposition \\\\ref {p:curgus} that \\\\infty is also not a critical point of A.This completes the proof of Theorem~\\\\ref {t:main2}.\\\\Box }\\\\begin{remark}If, in addition to the assumptions in Theorem \\\\ref {t:main2}, the point 0 is neither an eigenvalue nor a singular critical pointof the non-negative operator A_0 := A + \\\\gamma G^{-1}, then the operator A is non-negative over the set \\\\overline{\\\\setminus K_r((-r,r)), where\\\\begin{equation}r = \\\\gamma \\\\,\\\\frac{1+\\\\tau _\\\\gamma }{2}\\\\,\\\\Vert G^{-1}\\\\Vert \\\\end{equation}and\\\\tau _\\\\gamma := \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{1/n}^n\\\\,\\\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\big )\\\\,dt\\\\right\\\\Vert .Indeed, under the assumptions of Theorem \\\\ref {t:main2} for any \\\\varepsilon > 0 the operator A + (\\\\gamma + \\\\varepsilon )G^{-1}is selfadjoint in some Hilbert space, and thus the resolvent set ofA_0 is non-empty as A + (\\\\gamma + \\\\varepsilon )G^{-1} and A_0 differ just by a bounded operator.Therefore, A_0 is a non-negative operator in the Krein space(\\\\cal H,[\\\\cdot \\\\,,\\\\cdot ]).\",\n \"Moreover,the point \\\\infty is not a singular critical point of theoperator A + (\\\\gamma + \\\\varepsilon )G^{-1}, see Remark \\\\ref {Mai2012},and by Proposition \\\\ref {p:curgus} \\\\infty is not a singular critical point of A_0.Therefore, the operator A_0 admits a matrix representation as in (\\\\ref {e:AJ}) and the corresponding fundamental symmetry \\\\widetilde{J}satisfies \\\\Vert \\\\widetilde{J}\\\\Vert \\\\,\\\\le \\\\,\\\\tau _\\\\gamma (see the proof of Theorem \\\\ref {t:main2}).Then by a similar reasoning as in the proof ofTheorem \\\\ref {t:main2} we see that A is non-negativeover \\\\overline{\\\\setminus K_r((-r,r)), where r is as in (\\\\ref {Juni2012}).\",\n \"}}\\\\end{remark}}$ Differential operators with indefinite weights In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights.\",\n \"Indefinite Sturm-Liouville operators We consider Sturm-Liouville differential expressions of the form $\\\\mathcal {L}(f)(x) = \\\\operatorname{sgn}(x)\\\\big (-f^{\\\\prime \\\\prime }(x) + q(x)f(x)\\\\big ),\\\\quad x\\\\in \\\\mathbb {R}\\\\,,$ with a real-valued potential $q\\\\in L^\\\\infty (\\\\mathbb {R})$ and the indefinite weight function $\\\\operatorname{sgn}(\\\\cdot )$ .\",\n \"The corresponding differential operator in $L^2(\\\\mathbb {R})$ is defined by $Af := \\\\mathcal {L}(f), \\\\qquad \\\\operatorname{dom}A := H^2(\\\\mathbb {R}).$ Here $H^2(\\\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.\",\n \"By $(\\\\cdot \\\\,,\\\\cdot )$ and $\\\\Vert \\\\cdot \\\\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\\\mathbb {R})$ .\",\n \"Let $J$ be the operator of multiplication with the function $\\\\operatorname{sgn}(x)$ .\",\n \"This operator is obviously selfadjoint and unitary in $L^2(\\\\mathbb {R})$ .\",\n \"Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\\\prime \\\\prime } + qf,\\\\quad \\\\operatorname{dom}T := H^2(\\\\mathbb {R}),$ is selfadjoint in $L^2(\\\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\\\mathbb {R}),[\\\\cdot \\\\,,\\\\cdot ])$ , where $[f,g] := (Jf,g) = \\\\int _\\\\mathbb {R}\\\\,f(x)\\\\overline{g(x)}\\\\operatorname{sgn}(x)\\\\,dx,\\\\quad f,g\\\\in L^2(\\\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.\",\n \"We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].\",\n \"For this, denote by $m_\\\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\\\pm f_\\\\pm := - f_\\\\pm ^{\\\\prime \\\\prime } + q_\\\\pm f_\\\\pm $ in $L^2(\\\\mathbb {R}^\\\\pm )$ defined on $\\\\operatorname{dom}T_\\\\pm := \\\\lbrace f_\\\\pm \\\\in H^2(\\\\mathbb {R}^\\\\pm ) : f_\\\\pm (0) = 0\\\\rbrace $ , where $q_\\\\pm $ is the restriction of the function $q$ to $\\\\mathbb {R}^\\\\pm $ .\",\n \"The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\\\pm \\\\infty $ , then $m_\\\\pm =\\\\lim _{x\\\\rightarrow \\\\pm \\\\infty }\\\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.\",\n \"Furthermore the following holds.\",\n \"If $m_+ > -m_-$ then $\\\\sigma (A)\\\\setminus \\\\mathbb {R}$ is finite; If $m_+\\\\le -m_-$ then $\\\\sigma (A)\\\\setminus \\\\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .\",\n \"To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.\",\n \"One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].\",\n \"The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .\",\n \"Here only the nontrivial case $\\\\operatornamewithlimits{ess\\\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).\",\n \"Theorem 4.2 Assume that $\\\\operatornamewithlimits{ess\\\\,inf}q<0$ holds.\",\n \"Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\\\overline{\\\\setminus K_r((-d,d)), wherer := 5\\\\Vert q\\\\Vert _\\\\infty ,\\\\quad \\\\text{and}\\\\quad d := -5\\\\,\\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\\\cap \\\\lbrace \\\\lambda \\\\in |\\\\operatorname{Im}\\\\lambda |\\\\le 2\\\\Vert q\\\\Vert _\\\\infty \\\\rbrace .\",\n \"}\\\\begin{proof}The proof of Theorem~\\\\ref {t:SL2} is split into 4 parts.\",\n \"The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\\\ref {t:main1}.\",\n \"In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\\\end{proof}\\\\ \\\\\\\\{\\\\it 1.\\\\ Preparation: } The operator $ A0$,\\\\begin{equation*}A_0 f:=-\\\\operatorname{sgn}(\\\\cdot )f^{\\\\prime \\\\prime },\\\\qquad \\\\operatorname{dom}A_0=H^2(\\\\mathbb {R}),\\\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\\\mathbb {R}$ ),[,])$.\",\n \"Furthermore, we have $ (A0) = $\\\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\\\cite {cn}.\",\n \"Hence $ A0$ satisfies theassumptions in Theorem~\\\\ref {t:main1}.\",\n \"Define $ V$ as\\\\begin{equation*}Vf:=\\\\operatorname{sgn}(\\\\cdot ) qf,\\\\qquad \\\\operatorname{dom}V=L^2(\\\\mathbb {R}),\\\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\\\mathbb {R}$ ),[,])$ with$$\\\\min \\\\sigma (JV) = \\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}\\\\,q(x)\\\\quad \\\\text{and}\\\\quad \\\\Vert V\\\\Vert = \\\\Vert q\\\\Vert _\\\\infty .$$By Theorem \\\\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\\\frac{1+\\\\tau _0}{2}\\\\,\\\\Vert q\\\\Vert _\\\\infty ,\\\\qquad d = -\\\\frac{1+\\\\tau _0}{2}\\\\,\\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}q(x),$$and$$\\\\tau _0 = \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{1/n}^n\\\\,\\\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\right)\\\\,dt\\\\,\\\\right\\\\Vert .$$Therefore, we have to show\\\\begin{equation}\\\\tau _0\\\\le 9\\\\end{equation}and\\\\begin{equation}\\\\sigma (A)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\,\\\\lbrace \\\\lambda \\\\in |\\\\operatorname{Im}\\\\lambda |\\\\le 2\\\\Vert q\\\\Vert _\\\\infty \\\\rbrace .\\\\end{equation}$ 2.\",\n \"Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\\\begin{pmatrix} B_+ & 0 \\\\\\\\ 0 & B_- \\\\end{pmatrix}\\\\quad \\\\text{in}\\\\quad L^2(\\\\mathbb {R})=L^2(\\\\mathbb {R}^+)\\\\times L^2(\\\\mathbb {R}^-),$ where $B_\\\\pm f_\\\\pm :=\\\\mp f_\\\\pm ^{\\\\prime \\\\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\\\mathbb {R}^\\\\pm )$ which are defined on $\\\\operatorname{dom}B_\\\\pm = \\\\lbrace f_\\\\pm \\\\in H^2(\\\\mathbb {R}^\\\\pm ):f_\\\\pm (0)=0\\\\rbrace $ .\",\n \"Here and in the following we will denote the restrictions of a function $f$ defined on $\\\\mathbb {R}$ to $\\\\mathbb {R}^\\\\pm $ by $f_\\\\pm $ .\",\n \"For $\\\\lambda = re^{it}$ , $r > 0$ , $t\\\\in [0,2\\\\pi )$ , we set $\\\\sqrt{\\\\lambda } := \\\\sqrt{r}e^{it/2}$ .\",\n \"For $\\\\lambda \\\\in \\\\mathbb {R}$ define the function $f_\\\\lambda (x) :={\\\\left\\\\lbrace \\\\begin{array}{ll}e^{i\\\\sqrt{\\\\lambda }x}, &x> 0,\\\\\\\\e^{-i\\\\sqrt{-\\\\lambda }x}, &x < 0.\\\\end{array}\\\\right.\",\n \"}$ The function $f_\\\\lambda $ is a solution of the equations $\\\\mp f_\\\\pm ^{\\\\prime \\\\prime } = \\\\lambda f_\\\\pm $ in $L^2(\\\\mathbb {R}^\\\\pm )$ .\",\n \"From $\\\\overline{\\\\sqrt{\\\\lambda }} = -\\\\sqrt{\\\\overline{\\\\lambda }}$ it follows that $f_{\\\\overline{\\\\lambda }} = \\\\overline{f_\\\\lambda }$ holds for all $\\\\lambda \\\\in \\\\mathbb {R}$ .\",\n \"Let us now prove that for all $f\\\\in L^2(\\\\mathbb {R})$ and $\\\\lambda \\\\in \\\\mathbb {R}$ we have $(A_0 - \\\\lambda )^{-1}f = (B_0 - \\\\lambda )^{-1}f - \\\\frac{[f,f_{\\\\overline{\\\\lambda }}]}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_\\\\lambda .$ Since $\\\\sigma (A_0)=\\\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\\\lambda \\\\in \\\\mathbb {R}$ .\",\n \"In particular, for $\\\\lambda \\\\in \\\\mathbb {R}$ there exists $g\\\\in \\\\operatorname{dom}B_0$ such that $f=(B_0-\\\\lambda )g$ holds.\",\n \"The right hand side in (REF ) has the form $h:=g-\\\\frac{[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_\\\\lambda $ and we will show that the function $h$ belongs to $\\\\operatorname{dom}A_0=H^2(\\\\mathbb {R})$ .\",\n \"As $g_\\\\pm $ and $f_{\\\\lambda ,\\\\pm }$ are elements of $H^2(\\\\mathbb {R}^\\\\pm )$ the same is true for $h_\\\\pm $ .\",\n \"Moreover, $g$ and $f_\\\\lambda $ are continuous at 0 and so is $h$ .\",\n \"Hence it remains to check that $h^\\\\prime $ is continuous at 0.\",\n \"For this note first that $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]=\\\\int _0^\\\\infty (-g_+^{\\\\prime \\\\prime }-\\\\lambda g_+)\\\\overline{f_{\\\\overline{\\\\lambda },+}}\\\\, dx- \\\\int _{-\\\\infty }^0 (g_-^{\\\\prime \\\\prime }-\\\\lambda g_-)\\\\overline{f_{\\\\overline{\\\\lambda },-}}\\\\, dx$ and since $f_{\\\\overline{\\\\lambda },\\\\pm }$ are solutions of $\\\\mp f_\\\\pm ^{\\\\prime \\\\prime }=\\\\overline{\\\\lambda }f_\\\\pm $ , integration by parts yields $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]=g_+^\\\\prime (0)\\\\overline{f_{\\\\overline{\\\\lambda },+}(0)}-g_+(0) \\\\overline{f^\\\\prime _{\\\\overline{\\\\lambda },+}(0)}-g_-^\\\\prime (0)\\\\overline{f_{\\\\overline{\\\\lambda },-}(0)}+g_-(0) \\\\overline{f^\\\\prime _{\\\\overline{\\\\lambda },-}(0)}.$ As $g\\\\in \\\\operatorname{dom}B_0$ we have $g_\\\\pm (0)=0$ and together with $f_{\\\\overline{\\\\lambda },\\\\pm }(0)=1$ we find $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]= g_+^\\\\prime (0)-g^\\\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\\\pm ^\\\\prime $ on $\\\\mathbb {R}^\\\\pm $ of the function $h$ from (REF ): $h^\\\\prime _\\\\pm =g^\\\\prime _\\\\pm -\\\\frac{g_+^\\\\prime (0)-g^\\\\prime _-(0)}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_{\\\\lambda ,\\\\pm }^\\\\prime $ and as $f^\\\\prime _{\\\\lambda ,+}(0)=i\\\\sqrt{\\\\lambda }$ and $f^\\\\prime _{\\\\lambda ,-}(0)=-i\\\\sqrt{-\\\\lambda }$ we conclude $h^\\\\prime _+(0)-h^\\\\prime _-(0)= \\\\bigl (g_+^\\\\prime (0)-g^\\\\prime _-(0)\\\\bigr )-\\\\frac{g_+^\\\\prime (0)-g^\\\\prime _-(0)}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}\\\\bigl (f^\\\\prime _{\\\\lambda ,+}(0)-f^\\\\prime _{\\\\lambda ,-}(0)\\\\bigr )=0,$ that is, $h^\\\\prime $ is continuous at 0 and therefore $h\\\\in \\\\operatorname{dom}A_0$ .\",\n \"Now a straightforward computation shows that $(A_0-\\\\lambda )h=(B_0-\\\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).\",\n \"3.\",\n \"Proof of (): Let $f\\\\in L^2(\\\\mathbb {R})$ and $t > 0$ .\",\n \"Then (REF ) and $\\\\sqrt{it} + \\\\sqrt{-it} = i\\\\sqrt{2t}$ yield $\\\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\\\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\\\\\& + \\\\frac{2}{\\\\sqrt{2t}}\\\\,\\\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\\\in \\\\mathbb {R}^+$ , we have $\\\\big |[f,f_{it}][f_{-it},f]\\\\big |\\\\,\\\\le \\\\,\\\\left(\\\\int _0^\\\\infty \\\\,e^{-x\\\\sqrt{t/2}}g(x)\\\\,dx\\\\right)^2$ and thus for $n\\\\in \\\\mathbb {N}$ $\\\\int _{1/n}^n\\\\,\\\\frac{|[f,f_{it}][f_{-it},f]|}{\\\\sqrt{2t}}\\\\,dt&\\\\le \\\\int _{1/n}^n\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,g(x)g(y)\\\\,\\\\frac{e^{-(x+y)\\\\sqrt{t/2}}}{\\\\sqrt{2t}}\\\\,dy\\\\,dx\\\\,dt\\\\\\\\&= 2\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,\\\\frac{g(x)g(y)}{x+y}\\\\left(e^{-\\\\frac{x+y}{\\\\sqrt{2n}}} - e^{-\\\\frac{\\\\sqrt{n}(x+y)}{\\\\sqrt{2}}}\\\\right)\\\\,dy\\\\,dx\\\\\\\\&\\\\le 2\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,\\\\frac{g(x)g(y)}{x+y}\\\\,dy\\\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\\\int _{1/n}^n\\\\,\\\\frac{|[f,f_{it}][f_{-it},f]|}{\\\\sqrt{2t}}\\\\,dt\\\\,\\\\le \\\\,2\\\\pi \\\\Vert g\\\\Vert _{L^2(\\\\mathbb {R}^+)}^2\\\\,\\\\le \\\\,4\\\\pi \\\\Vert f\\\\Vert ^2.$ and therefore we have $\\\\frac{1}{\\\\pi }\\\\int _{1/n}^n\\\\,\\\\frac{2}{\\\\sqrt{2t}}\\\\,\\\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\\\,dt\\\\,\\\\le \\\\,8\\\\Vert f\\\\Vert ^2.$ Denote by $E_0$ ($E_\\\\pm $ ) the spectral function of the operator $A_0$ ($B_\\\\pm $ , respectively).\",\n \"Moreover define $J_{n,\\\\pm } := \\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,\\\\left((B_\\\\pm + it)^{-1} + (B_\\\\pm - it)^{-1}\\\\right)\\\\,dt,$ and, in addition, $J_{n,0} := \\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,\\\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\right)\\\\,dt.$ As $\\\\pm B_\\\\pm $ is non-negative, it follows that $J_{n,\\\\pm }$ converges strongly to $\\\\pm I_{L^2(\\\\mathbb {R}^\\\\pm )}$ when $n\\\\rightarrow \\\\infty $ , cf.\",\n \"Lemma REF .\",\n \"Moreover, by $(B_\\\\pm + it)^{-1} + (B_\\\\pm - it)^{-1}= 2B_\\\\pm (B_\\\\pm + it)^{-1} (B_\\\\pm - it)^{-1}$ the sequence $\\\\pm J_{n,\\\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\\\mathbb {R}^\\\\pm )$ .\",\n \"This implies $\\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\\\\\&\\\\le \\\\Vert f_+\\\\Vert _{L^2(\\\\mathbb {R}^+)}^2 + \\\\Vert f_-\\\\Vert _{L^2(\\\\mathbb {R}^-)}^2 = \\\\Vert f\\\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\\\,\\\\le \\\\,\\\\Vert f\\\\Vert ^2 + 8\\\\Vert f\\\\Vert ^2 = 9\\\\Vert f\\\\Vert ^2,\\\\quad f\\\\in L^2(\\\\mathbb {R}).$ Hence, as $J = \\\\operatorname{sgn}(\\\\cdot )$ is unitary in $L^2(\\\\mathbb {R})$ , we finally obtain $\\\\Vert J_{n,0}\\\\Vert = \\\\Vert J J_{n,0}\\\\Vert \\\\le 9$ and thus $\\\\tau _0 = \\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\Vert J_{n,0}\\\\Vert \\\\le 9$ .\",\n \"That is, () holds.\",\n \"4.\",\n \"Proof of (): Let $\\\\lambda \\\\in +$ .\",\n \"Using (REF ), for $f\\\\in L^2(\\\\mathbb {R})$ we obtain $\\\\Vert (A_0 - \\\\lambda )^{-1}f\\\\Vert \\\\,\\\\le \\\\,\\\\Vert (B_0 - \\\\lambda )^{-1}f\\\\Vert + \\\\frac{|[f,f_{\\\\overline{\\\\lambda }}]|\\\\,\\\\Vert f_\\\\lambda \\\\Vert }{|\\\\sqrt{\\\\lambda } + \\\\sqrt{-\\\\lambda }|}\\\\,\\\\le \\\\,\\\\frac{\\\\Vert f\\\\Vert }{|\\\\operatorname{Im}\\\\lambda |} + \\\\frac{\\\\Vert f_\\\\lambda \\\\Vert ^2\\\\Vert f\\\\Vert }{\\\\sqrt{2|\\\\lambda |}}\\\\,.$ Here we have used the identity $f_{\\\\overline{\\\\lambda }} = \\\\overline{f_\\\\lambda }$ .\",\n \"Let $r > 0$ and $t\\\\in (0,\\\\pi )$ such that $\\\\lambda = re^{it}$ .\",\n \"Then $\\\\Vert f_\\\\lambda \\\\Vert ^2&= \\\\int _0^\\\\infty \\\\,\\\\left|e^{i\\\\sqrt{r}e^{it/2}x}\\\\right|^2\\\\,dx + \\\\int _{-\\\\infty }^0\\\\,\\\\left|e^{-i\\\\sqrt{r}e^{i(t+\\\\pi )/2}x}\\\\right|^2\\\\,dx\\\\\\\\&= \\\\int _0^\\\\infty \\\\,e^{-2\\\\sqrt{r}\\\\sin (t/2)x}\\\\,dx + \\\\int _{-\\\\infty }^0\\\\,e^{2\\\\sqrt{r}\\\\cos (t/2)x}\\\\,dx\\\\\\\\&= \\\\frac{\\\\cos (t/2) + \\\\sin (t/2)}{\\\\sqrt{r}\\\\,\\\\sin (t)}\\\\,\\\\le \\\\,\\\\frac{\\\\sqrt{2}}{\\\\sqrt{r}\\\\,\\\\operatorname{Im}(\\\\lambda /r)} = \\\\frac{\\\\sqrt{2|\\\\lambda |}}{|\\\\operatorname{Im}\\\\lambda |}.$ Therefore, $\\\\Vert (A_0 - \\\\lambda )^{-1}\\\\Vert \\\\,\\\\le \\\\,\\\\frac{2}{|\\\\operatorname{Im}\\\\lambda |}\\\\,.$ The same estimate holds for $\\\\lambda \\\\in -$ .\",\n \"Now, assume that $|\\\\operatorname{Im}\\\\lambda | > 2\\\\Vert q\\\\Vert _\\\\infty =2\\\\Vert V\\\\Vert $ .\",\n \"Then $\\\\Vert V(A_0 - \\\\lambda )^{-1}\\\\Vert < 1$ , and hence $A - \\\\lambda = A_0 - \\\\lambda + V = \\\\left(I + V(A_0 - \\\\lambda )^{-1}\\\\right)(A_0 - \\\\lambda )$ is boundedly invertible, which proves ().$\\\\Box $ Second order elliptic operators Let $\\\\Omega \\\\subset \\\\mathbb {R}^n$ be a domain and let $\\\\ell $ be the \\\"formally symmetric\\\" uniformly elliptic second order differential expression $\\\\ell (f)(x):=-\\\\sum _{j,k=1}^n \\\\left( \\\\frac{\\\\partial }{\\\\partial x_j} a_{jk}\\\\frac{\\\\partial f}{\\\\partial x_k}\\\\right)(x)+a(x)f(x),\\\\quad x\\\\in \\\\Omega ,$ with bounded coefficients $a_{jk}\\\\in C^\\\\infty (\\\\Omega )$ satisfying $a_{jk}(x)=\\\\overline{a_{kj}(x)}$ for all $x\\\\in \\\\Omega $ and $j,k=1,\\\\dots ,n$ , the function $a\\\\in L^\\\\infty (\\\\Omega )$ is real valued and $\\\\sum _{j,k=1}^n a_{jk}(x)\\\\xi _j\\\\xi _k\\\\ge C\\\\sum _{k=1}^n\\\\xi _k^2$ holds for some $C>0$ , all $\\\\xi =(\\\\xi _1,\\\\dots ,\\\\xi _n)^\\\\top \\\\in \\\\mathbb {R}^n$ and $x\\\\in \\\\Omega $ .\",\n \"With the differential expression $\\\\ell $ we associate the elliptic differential operator $Tf:=\\\\ell (f),\\\\qquad \\\\operatorname{dom}T=\\\\bigl \\\\lbrace f\\\\in H^1_0(\\\\Omega ):\\\\ell (f) \\\\in L^2(\\\\Omega )\\\\bigr \\\\rbrace ,$ where $H^1_0(\\\\Omega )$ stands for the closure of $C_0^\\\\infty (\\\\Omega )$ in the Sobolev space $H^1(\\\\Omega )$ .\",\n \"It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\\\Omega ),(\\\\cdot ,\\\\cdot ))$ with spectrum semibounded from below by $\\\\operatornamewithlimits{ess\\\\,inf}\\\\,a$ ; cf.\",\n \"[24].\",\n \"Let $w$ be a real valued function such that $w,w^{-1}\\\\in L^\\\\infty (\\\\Omega )$ and each of the sets $\\\\Omega _+:=\\\\bigl \\\\lbrace x\\\\in \\\\Omega : w(x)>0\\\\bigr \\\\rbrace \\\\quad \\\\text{and}\\\\quad \\\\Omega _-:=\\\\bigl \\\\lbrace x\\\\in \\\\Omega : w(x)<0\\\\bigr \\\\rbrace $ has positive Lebesgue measure.\",\n \"We define a second order elliptic differential expression $\\\\mathcal {L}$ with the indefinite weight $w$ by $\\\\mathcal {L}(f)(x):=\\\\frac{1}{w(x)}\\\\,\\\\ell (f)(x),\\\\qquad x\\\\in \\\\Omega .$ The multiplication operator $G_w f=w f$ , $f\\\\in L^2(\\\\Omega )$ , is an isomorphism in $L^2(\\\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\\\in L^2(\\\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\\\int _\\\\Omega f(x)\\\\overline{g(x)}\\\\,w(x)\\\\,dx,\\\\qquad f,g\\\\in L^2(\\\\Omega ).$ The differential operator associated with $\\\\mathcal {L}$ is defined as $Af=\\\\mathcal {L}(f),\\\\qquad \\\\operatorname{dom}A=\\\\bigl \\\\lbrace f\\\\in H^1_0(\\\\Omega ):\\\\mathcal {L}(f) \\\\in L^2(\\\\Omega )\\\\bigr \\\\rbrace .$ Since for $f\\\\in H^1_0(\\\\Omega )$ we have $\\\\ell (f)\\\\in L^2(\\\\Omega )$ if and only if $\\\\mathcal {L}(f)\\\\in L^2(\\\\Omega )$ it follows that $\\\\operatorname{dom}A=\\\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.\",\n \"Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ .\",\n \"In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\\\min \\\\sigma _{\\\\rm ess}(T)\\\\le 0$ holds.\",\n \"This also implies that the domain $\\\\Omega $ is unbounded as otherwise $\\\\sigma _{\\\\rm ess}(T)=\\\\varnothing $ .\",\n \"A discussion of the cases $\\\\sigma _{\\\\rm ess}(T)=\\\\varnothing $ and $\\\\min \\\\sigma _{\\\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].\",\n \"Fix some $\\\\eta >0$ such that $-\\\\eta <\\\\min \\\\sigma (T)$ and define the spaces $\\\\mathcal {H}_s$ , $s\\\\in [0,2]$ , as the domains of the $\\\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\\\eta $ in $L^2(\\\\Omega )$ , $\\\\mathcal {H}_s:=\\\\operatorname{dom}\\\\bigl ((T+\\\\eta )^\\\\frac{s}{2}\\\\bigr ),\\\\qquad s\\\\in [0,2].$ Note that $\\\\mathcal {H}=\\\\mathcal {H}_0$ , $\\\\operatorname{dom}T=\\\\mathcal {H}_2$ and the form domain of $T$ is $\\\\mathcal {H}_1$ .\",\n \"The spaces $\\\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\\\eta $ ; cf.\",\n \"[40].\",\n \"The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.\",\n \"[9].\",\n \"Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ such that $W\\\\mathcal {H}_s\\\\subset \\\\mathcal {H}_s$ holds for some $s\\\\in (0,2]$ .\",\n \"Then $A$ is non-negative over $ \\\\overline{\\\\mathbb {C}} \\\\setminus K_r((-r,r))$ , where $r=\\\\eta \\\\,\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert w^{-1}\\\\Vert _\\\\infty \\\\quad \\\\text{and}\\\\quad \\\\tau _\\\\eta := \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{-n}^n\\\\,(A+\\\\eta G_w^{-1} - it)^{-1} \\\\,dt\\\\,\\\\right\\\\Vert \\\\,<\\\\,\\\\infty .$ Moreover, $\\\\infty $ is not a singular critical point of $A$ .\",\n \"In the next corollary the special case $\\\\Omega =\\\\mathbb {R}^n$ is treated; cf.\",\n \"[9].\",\n \"From now on we assume that $\\\\Omega =\\\\mathbb {R}^n$ and $\\\\Omega _\\\\pm =\\\\lbrace x\\\\in \\\\mathbb {R}^n:\\\\pm w(x)>0\\\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\\\in C^\\\\infty (\\\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.\",\n \"Note that either $\\\\Omega _+$ or $\\\\Omega _-$ is bounded.\",\n \"Corollary 4.4 Suppose that for some $s\\\\in (0,\\\\frac{1}{2})$ the Sobolev spaces $H^s(\\\\Omega _\\\\pm )$ are invariant under multiplication with $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ .\",\n \"Then the indefinite elliptic operator $A$ is non-negative over $\\\\overline{\\\\mathbb {C}} \\\\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .\",\n \"Moreover, $\\\\infty $ is not a singular critical point of $A$ .\",\n \"A sufficient criterion for the invariance of $H^s(\\\\Omega _\\\\pm )$ in the above corollary can be deduced from [28].\",\n \"Corollary 4.5 Suppose that the weight function $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ belongs to some Hölder space $C^{0,\\\\alpha }(\\\\Omega _\\\\pm )$ , $\\\\alpha \\\\in (0,\\\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.\",\n \"Then the claim in Corollary REF holds.\",\n \"of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\\\operatorname{dom}A=\\\\operatorname{dom}T=H^2(\\\\mathbb {R}^n)\\\\qquad \\\\text{and}\\\\qquad \\\\mathcal {H}_s=H^s(\\\\mathbb {R}^n),\\\\qquad s\\\\in [0,2],$ by elliptic regularity and interpolation.\",\n \"Since the Sobolev spaces $H^s(\\\\Omega _\\\\pm )$ are assumed to be invariant under multiplication with the functions $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ for some $s\\\\in (0,\\\\frac{1}{2})$ it follows from [9] that also $H^s(\\\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\\\mathcal {H}_s\\\\subset \\\\mathcal {H}_s$ holds.\",\n \"Furthermore, $G_w$ is uniformly positive in $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\\\ge \\\\operatornamewithlimits{ess\\\\,inf}w^2 \\\\Vert f\\\\Vert ^2$ and $w^{-1}\\\\in L^\\\\infty (\\\\mathbb {R}^n)$ .\",\n \"Now the assertion follows from Theorem REF .$\\\\Box $ Acknowledgements The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.\",\n \"Contact information Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, behrndt@tugraz.at Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.\",\n \"Juni 136, 10623 Berlin, Germany, fmphilipp@gmail.com Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, carsten.trunk@tu-ilmenau.de\"\n ],\n [\n \"Bounded selfadjoint perturbations of non-negative operators\",\n \"In this section we prove two abstract results on additive bounded perturbations of non-negative (and some closely connected class of) operators in Krein spaces that lead to perturbed operators which are non-negative over some neighborhood of infinity.\",\n \"In both cases the neighborhood is given in quantitative terms.\",\n \"The results will be applied in Section to singular indefinite Sturm-Liouville operators and to second order elliptic operators with indefinite weights.\"\n ],\n [\n \"Two perturbation results\",\n \"The following notation will be useful when formulating our main results below: For a set $\\\\Delta \\\\subset \\\\mathbb {R}$ and $r > 0$ we define $K_r(\\\\Delta ) := \\\\lbrace z\\\\in \\\\operatorname{dist}(z,\\\\Delta )\\\\,\\\\le \\\\,r\\\\rbrace .$ Our first main theorem concerns bounded selfadjoint perturbations of non-negative operators in Krein spaces.\",\n \"Theorem 3.1 Let $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ be a Krein space with fundamental symmetry $J$ and norm $\\\\Vert \\\\cdot \\\\Vert = [J\\\\cdot ,\\\\cdot ]^{1/2}$ .\",\n \"Let $A_0$ be a non-negative operator in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ such that 0 and $\\\\infty $ are not singular critical points of $A_0$ , and $0\\\\notin \\\\sigma _p(A_0)$ .\",\n \"Furthermore, let $V\\\\in L(\\\\mathcal {H})$ be a selfadjoint operator in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"Then the following holds: (i) If $V$ is non-negative, then $A_0+V$ is a non-negative operator in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"(ii) If $V$ is not non-negative, then $A_0+V$ is non-negative over $\\\\overline{\\\\setminus K_r((-d,d)),where\\\\begin{equation}r = \\\\frac{1+\\\\tau _0}{2}\\\\Vert V\\\\Vert ,\\\\qquad \\\\quad d = -\\\\frac{1+\\\\tau _0}{2}\\\\,\\\\min \\\\sigma (JV),\\\\end{equation}and\\\\begin{equation}\\\\tau _0 = \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{1/n}^n\\\\,\\\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\right)\\\\,dt\\\\,\\\\right\\\\Vert <\\\\infty .\\\\end{equation}}Moreover, in both cases $$ is not a singular critical point of $ A0 + V$.$ The following simple example shows that without further assumptions a bounded selfadjoint perturbation $V$ of a non-negative or uniformly positive $A_0$ in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ may lead to an operator $A_0+V$ with unbounded non-real spectrum.\",\n \"Example 3.2 Let $(\\\\cal K,(\\\\cdot ,\\\\cdot ))$ be a Hilbert space and let $H$ be an unbounded selfadjoint operator in $\\\\cal K$ such that $\\\\sigma (H)\\\\subset (0,\\\\infty )$ .\",\n \"Equip $\\\\cal H:=\\\\cal K\\\\oplus \\\\cal K$ with the Krein space inner product $\\\\left[\\\\begin{pmatrix}k_1 \\\\\\\\ k_2\\\\end{pmatrix},\\\\begin{pmatrix}l_1 \\\\\\\\ l_2\\\\end{pmatrix}\\\\right] :=(k_1,l_2)+(k_2,l_1),\\\\qquad \\\\begin{pmatrix}k_1 \\\\\\\\ k_2\\\\end{pmatrix},\\\\begin{pmatrix}l_1 \\\\\\\\ l_2\\\\end{pmatrix}\\\\in \\\\cal H,$ and consider the operators $A_0=\\\\begin{pmatrix} 0 & I \\\\\\\\ H & 0\\\\end{pmatrix},\\\\quad V=\\\\begin{pmatrix} 0 & -I \\\\\\\\ 0 & 0\\\\end{pmatrix},\\\\quad \\\\text{and}\\\\quad A_0+V =\\\\begin{pmatrix} 0 & 0 \\\\\\\\ H & 0\\\\end{pmatrix}.$ It is easy to see that $A_0$ is a non-negative operator and $V$ is a bounded selfadjoint operator in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"Moreover, as $\\\\operatorname{dom}(A_0+V)=\\\\operatorname{dom}H\\\\oplus {\\\\cal H}$ we conclude $\\\\operatorname{ran}(A_0+V-\\\\lambda )\\\\ne \\\\cal H$ for every $\\\\lambda \\\\in , that is, $ (A0+V)=.\",\n \"Our second main result applies to operators of the form $A = G^{-1}T$ , where $T$ is a semibounded selfadjoint operator in a Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot ))$ and $G$ is a selfadjoint bounded and boundedly invertible operator in the Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot ))$ .\",\n \"In other words, $A$ is selfadjoint in the Krein space $(\\\\mathcal {H},(G\\\\cdot ,\\\\cdot ))$ and $A + \\\\eta G^{-1}$ is uniformly positive for suitable $\\\\eta $ .\",\n \"Such a situation arises, e.g., when considering elliptic differential operators with an indefinite weight function; cf.\",\n \"Section 4.2.\",\n \"Theorem 3.3 Let $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot ))$ be a Hilbert space, $\\\\Vert \\\\cdot \\\\Vert = (\\\\cdot \\\\,,\\\\cdot )^{1/2}$ and $[\\\\cdot \\\\,,\\\\cdot ]=(G\\\\cdot ,\\\\cdot )$ be as above.\",\n \"Let $A$ be a selfadjoint operator in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ such that for some $\\\\gamma > 0$ we have $[Af,f]\\\\ge -\\\\gamma \\\\Vert f\\\\Vert ^2\\\\quad \\\\text{for all }f\\\\in \\\\operatorname{dom}A.$ Assume furthermore that for some $\\\\eta > \\\\gamma $ $\\\\tau _\\\\eta := \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{-n}^n\\\\,(A + \\\\eta G^{-1} - it)^{-1}\\\\,dt\\\\,\\\\right\\\\Vert < \\\\infty ,$ holds.\",\n \"Then $A$ is non-negative over $\\\\overline{\\\\setminus K_r((-r,r)), where\\\\begin{equation*}r = \\\\eta \\\\,\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert G^{-1}\\\\Vert .\\\\end{equation*}Moreover, \\\\infty is not a singular critical point of A.\",\n \"}$ Remark 3.4 Setting $A_0 := A + \\\\eta G^{-1}$ in Theorem REF and $V:=-\\\\eta G^{-1}$ we have $A = A_0 + V$ and hence Theorem REF can also be seen as a variant of Theorem REF .\",\n \"Here $A_0=A + \\\\eta G^{-1}$ is uniformly positive so that $0\\\\in \\\\rho (A_0)$ is not a critical point of $A_0$ and the entire imaginary axis belongs to $\\\\rho (A + \\\\eta G^{-1})$ .\",\n \"It follows from [55] and [32] that $\\\\tau _\\\\eta < \\\\infty $ is equivalent to $\\\\infty $ not being a singular critical point of $A_0$ .\",\n \"In what follows we describe the structure of the proofs of Theorem REF (ii) and Theorem REF .\",\n \"Let $E$ denote the spectral function of the non-negative operator $A_0$ .\",\n \"As 0 and $\\\\infty $ are not singular critical points of $A_0$ , the spectral projections $E_+ := E((0,\\\\infty ))$ and $E_- := E((-\\\\infty ,0))$ , and the corresponding spectral subspaces $\\\\mathcal {H}_\\\\pm := E_\\\\pm \\\\mathcal {H}$ of $A_0$ exist.\",\n \"From $0\\\\notin \\\\sigma _p(A_0)$ it follows that $\\\\mathcal {H}= \\\\mathcal {H}_+\\\\,[\\\\dotplus ]\\\\,\\\\mathcal {H}_-$ and $(\\\\mathcal {H}_\\\\pm ,\\\\pm [\\\\cdot \\\\,,\\\\cdot ])$ are Hilbert spaces, see, e.g., [46].\",\n \"Therefore (REF ) is a fundamental decomposition of the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"With respect to this decomposition the operator $A_0$ and the fundamental symmetry $\\\\widetilde{J}$ corresponding to (REF ) can be written as operator matrices: $A_0 = \\\\begin{pmatrix}A_{0,+} & 0\\\\\\\\0 & A_{0,-}\\\\end{pmatrix}\\\\quad \\\\text{and}\\\\quad \\\\widetilde{J} = \\\\begin{pmatrix}I & 0\\\\\\\\0 & -I\\\\end{pmatrix}.$ Note that the operator $\\\\pm A_{0,\\\\pm }$ is a selfadjoint non-negative operator in the Hilbert space $(\\\\mathcal {H}_\\\\pm ,\\\\pm [\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"Hence, $A_0$ is selfadjoint in the Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot )_\\\\sim )$ , where $(f,g)_\\\\sim := [\\\\widetilde{J}f,g] = [E_+f,g] - [E_-f,g],\\\\qquad f,g\\\\in \\\\mathcal {H}.$ Now, let $V\\\\in L(\\\\mathcal {H})$ be a bounded selfadjoint operator in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"Then, with respect to the decomposition (REF ) it admits an operator matrix representation $V = \\\\begin{pmatrix}V_+ & B\\\\\\\\C & V_-\\\\end{pmatrix}.$ From the selfadjointness of $V$ in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ one concludes that $V_\\\\pm $ is selfadjoint in the Hilbert space $(\\\\mathcal {H}_\\\\pm ,\\\\pm [\\\\cdot \\\\,,\\\\cdot ]) = (\\\\mathcal {H}_\\\\pm ,(\\\\cdot \\\\,,\\\\cdot )_\\\\sim )$ and $C = -B^{\\\\widetilde{*}},$ where $B^{\\\\widetilde{*}}$ denotes the adjoint of $B$ with respect to the scalar product $(\\\\cdot \\\\,,\\\\cdot )_\\\\sim $ .\",\n \"Hence, the perturbed operator $A := A_0 + V$ is represented by $A = \\\\begin{pmatrix}A_{0,+} + V_+ & B\\\\\\\\-B^{\\\\widetilde{*}} & A_{0,-} + V_-\\\\end{pmatrix}.$ For operators as in (REF ) we show in Theorem REF below that $\\\\mathbb {R}\\\\setminus K_{\\\\Vert B\\\\Vert _\\\\sim } (\\\\sigma (A_{0,-} + V_-))$ is of positive type, $\\\\mathbb {R}\\\\setminus K_{\\\\Vert B\\\\Vert _\\\\sim } (\\\\sigma (A_{0,+} + V_+))$ is of negative type, the non-real spectrum of $A$ is contained in $K:=K_{\\\\Vert B\\\\Vert _\\\\sim }\\\\bigl (\\\\sigma (A_{0,+} + V_+)\\\\bigr )\\\\cap K_{\\\\Vert B\\\\Vert _\\\\sim } \\\\bigl (\\\\sigma (A_{0,-} + V_-)\\\\bigr ),$ and the growth of the resolvent of $A$ at $\\\\infty $ is of order one.\",\n \"Here $\\\\Vert \\\\cdot \\\\Vert _\\\\sim $ stands for the operator norm induced by (REF ).\",\n \"Then, by Theorem REF , the operator $A$ is non-negative over $\\\\overline{\\\\setminus K, where K is as in (\\\\ref {BadHersfeld}).In a final step it remains to bound the quantities\\\\Vert B\\\\Vert _\\\\sim \\\\quad \\\\mbox{and} \\\\quad K_{\\\\Vert B\\\\Vert _\\\\sim } (\\\\sigma (A_{0,\\\\pm } + V_\\\\pm ))by the quantitiesr and \\\\tau _0 as in the statement of Theorem \\\\ref {t:main1}.\",\n \"}$ Some auxiliary statements In this section we formulate and prove some auxiliary statements which will be used in the proofs Theorems REF and REF .\",\n \"For the special case of bounded operators the first part of the following theorem was already proved in [48], see also [47] and [53], [54] for similar results with unbounded entries on the diagonal.\",\n \"Theorem 3.5 Let $S_+$ and $S_-$ be selfadjoint operators in some Hilbert spaces $\\\\mathfrak {H}_+$ and $\\\\mathfrak {H}_-$ , respectively, let $M\\\\in L(\\\\mathfrak {H}_-,\\\\mathfrak {H}_+)$ and define the operators $S := \\\\begin{pmatrix}S_+ & M\\\\\\\\-M^* & S_-\\\\end{pmatrix}\\\\quad \\\\text{and}\\\\quad \\\\mathfrak {J}:= \\\\begin{pmatrix}I & 0\\\\\\\\0 & -I\\\\end{pmatrix}$ in the Hilbert space $\\\\mathfrak {H}:= \\\\mathfrak {H}_+\\\\oplus \\\\mathfrak {H}_-$ .\",\n \"Then the operator $S$ is selfadjoint in the Krein space $(\\\\mathfrak {H},(\\\\mathfrak {J}\\\\cdot ,\\\\cdot ))$ and with $\\\\nu := \\\\Vert M\\\\Vert $ the following statements hold: $\\\\sigma (S)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\, K_\\\\nu (\\\\sigma (S_+))\\\\cap K_\\\\nu (\\\\sigma (S_-))$ .\",\n \"$\\\\mathbb {R}\\\\setminus K_\\\\nu (\\\\sigma (S_-))$ is of positive type with respect to $S$ .\",\n \"$\\\\mathbb {R}\\\\setminus K_\\\\nu (\\\\sigma (S_+))$ is of negative type with respect to $S$ .\",\n \"Moreover, if $S_+$ is bounded from below and $S_-$ is bounded from above (or vice versa), the non-real spectrum of $S$ is bounded and the growth of the resolvent of $S$ at $\\\\infty $ is of order one.\",\n \"We set $[\\\\cdot \\\\,,\\\\cdot ]:= (\\\\mathfrak {J}\\\\cdot ,\\\\cdot )$ .\",\n \"Let $\\\\lambda \\\\in \\\\mathbb {C} \\\\setminus K_\\\\nu (\\\\sigma (S_-))$ and $\\\\alpha := \\\\nu /\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_-)) < 1$ .\",\n \"We claim that for some $r(\\\\alpha )>0$ and all $f\\\\in \\\\operatorname{dom}S$ the following implication $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\le \\\\frac{1-\\\\alpha ^2}{4\\\\alpha }\\\\nu \\\\Vert f\\\\Vert \\\\quad \\\\Longrightarrow \\\\quad [f,f]\\\\,\\\\ge \\\\,r(\\\\alpha )\\\\Vert f\\\\Vert ^2$ holds.\",\n \"In fact, set $\\\\varepsilon := \\\\frac{1-\\\\alpha ^2}{4\\\\alpha }$ and let $f\\\\in \\\\operatorname{dom}S$ with $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\le \\\\varepsilon \\\\nu \\\\Vert f\\\\Vert .$ With respect to the decomposition $\\\\mathfrak {H}= \\\\mathfrak {H}_+\\\\oplus \\\\mathfrak {H}_-$ we write $f$ and $g := (S - \\\\lambda )f$ as column vectors $f = \\\\begin{pmatrix}f_+\\\\\\\\f_-\\\\end{pmatrix}\\\\in \\\\operatorname{dom}S_+\\\\oplus \\\\operatorname{dom}S_-, \\\\quad g = \\\\begin{pmatrix}g_+\\\\\\\\g_-\\\\end{pmatrix}.$ Then $g_- = -M^*f_+ + (S_- - \\\\lambda )f_-$ , or, equivalently, $f_- = (S_- - \\\\lambda )^{-1}g_- + (S_- - \\\\lambda )^{-1}M^*f_+.$ As $\\\\Vert g_-\\\\Vert \\\\le \\\\Vert g\\\\Vert = \\\\Vert (S - \\\\lambda )f\\\\Vert \\\\le \\\\varepsilon \\\\nu \\\\Vert f\\\\Vert $ and $\\\\Vert (S_- - \\\\lambda )^{-1}\\\\Vert = \\\\alpha /\\\\nu $ this yields $\\\\Vert f_-\\\\Vert \\\\,\\\\le \\\\,\\\\alpha \\\\varepsilon \\\\Vert f\\\\Vert + \\\\alpha \\\\Vert f_+\\\\Vert $ and hence $\\\\Vert f_-\\\\Vert ^2\\\\,\\\\le \\\\,\\\\alpha ^2\\\\big (\\\\varepsilon ^2\\\\Vert f\\\\Vert ^2 + 2\\\\varepsilon \\\\Vert f\\\\Vert ^2 + \\\\Vert f_+\\\\Vert ^2\\\\big ).$ Using $\\\\Vert f_+\\\\Vert ^2 = \\\\Vert f\\\\Vert ^2 - \\\\Vert f_-\\\\Vert ^2$ we conclude $\\\\Vert f_-\\\\Vert ^2\\\\,\\\\le \\\\,\\\\frac{\\\\alpha ^2}{1+\\\\alpha ^2}(1 + \\\\varepsilon )^2\\\\Vert f\\\\Vert ^2.$ Hence, as $[f,f] = \\\\Vert f_+\\\\Vert ^2 - \\\\Vert f_-\\\\Vert ^2 = \\\\Vert f\\\\Vert ^2 - 2\\\\Vert f_-\\\\Vert ^2$ , we obtain $[f,f]\\\\,\\\\ge \\\\,r(\\\\alpha )\\\\Vert f\\\\Vert ^2$ , where $r(\\\\alpha ) := 1 - \\\\frac{2\\\\alpha ^2}{1+\\\\alpha ^2}(1 + \\\\varepsilon )^2 = \\\\frac{(1 - \\\\alpha )^2(1 + \\\\alpha )(7 - \\\\alpha )}{8(1+\\\\alpha ^2)} > 0,$ i.e., the implication (REF ) holds.\",\n \"It follows that $(K_\\\\nu (\\\\sigma (S_-)))\\\\cap \\\\sigma _{{ap}}(S)\\\\subset \\\\sigma _{+}(S)$ .\",\n \"Hence (ii) is proved and, as $\\\\sigma _{+}(S)$ is real, the non-real spectrum of $S$ satisfies $\\\\sigma (S)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\, K_\\\\nu (\\\\sigma (S_-)).$ Similarly, as in the proof of (REF ), one proves that for $\\\\lambda \\\\in \\\\mathbb {C} \\\\setminus K_\\\\nu (\\\\sigma (S_+))$ , $f\\\\in \\\\operatorname{dom}S$ , and for $\\\\beta := \\\\nu /\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_+))<1$ the implication $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\le \\\\frac{1-\\\\beta ^2}{4\\\\beta }\\\\nu \\\\Vert f\\\\Vert \\\\quad \\\\Longrightarrow \\\\quad [f,f]\\\\,\\\\le \\\\,-r(\\\\beta )\\\\Vert f\\\\Vert ^2$ holds with $r(\\\\beta )>0$ as in (REF ).\",\n \"This shows (iii) and $\\\\sigma (S)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\, K_\\\\nu (\\\\sigma (S_+))$ .\",\n \"Together with (REF ) we obtain (i).\",\n \"Assume now that $S_+$ is bounded from below and $S_-$ is bounded from above.\",\n \"Then the non-real spectrum of $S$ is bounded by (i) and it remains to show that the growth of the resolvent of $S$ is of order one at $\\\\infty $ .\",\n \"Note first that for $S_0=\\\\begin{pmatrix}S_+ & 0 \\\\\\\\ 0 & S_-\\\\end{pmatrix}\\\\qquad \\\\text{and}\\\\qquad V=\\\\begin{pmatrix}0 & M \\\\\\\\ -M^* & 0\\\\end{pmatrix},$ and $\\\\lambda \\\\in such that $ Im2V= 2$ we have $ (S0-)-1V12$ and\\\\begin{equation}\\\\Vert (S-\\\\lambda )^{-1}\\\\Vert = \\\\bigl \\\\Vert \\\\bigl ((S_0-\\\\lambda )(I+(S_0-\\\\lambda )^{-1}V)\\\\bigr )^{-1}\\\\bigr \\\\Vert \\\\le \\\\frac{2}{\\\\vert \\\\operatorname{Im}\\\\lambda \\\\vert }.\\\\end{equation}For $ with $\\\\vert \\\\operatorname{Im}\\\\lambda \\\\vert < 2\\\\nu $ and $\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_-)) > \\\\nu + \\\\delta $ with some fixed $\\\\delta > 0$ again define the value $\\\\alpha = \\\\nu /\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_-))$ .\",\n \"Then $\\\\alpha \\\\in (0,\\\\delta ^{\\\\prime })$ , where $\\\\delta ^{\\\\prime } = \\\\nu /(\\\\nu + \\\\delta )\\\\in (0,1)$ , and for $f\\\\in \\\\operatorname{dom}S$ we either have $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\ge \\\\frac{1-\\\\alpha ^2}{4\\\\alpha }\\\\nu \\\\Vert f\\\\Vert > \\\\frac{1-\\\\alpha ^2}{8\\\\alpha }\\\\vert \\\\operatorname{Im}\\\\lambda \\\\vert \\\\,\\\\Vert f\\\\Vert ,$ or, by (REF ), $r(\\\\alpha )|\\\\operatorname{Im}\\\\lambda |\\\\Vert f\\\\Vert ^2\\\\le |\\\\operatorname{Im}[\\\\lambda f,f]| = |\\\\operatorname{Im}[(S - \\\\lambda )f,f]|\\\\le \\\\Vert (S - \\\\lambda )f\\\\Vert \\\\Vert f\\\\Vert ,$ and hence $\\\\Vert (S - \\\\lambda )f\\\\Vert \\\\ge r(\\\\alpha )|\\\\operatorname{Im}\\\\lambda |\\\\,\\\\Vert f\\\\Vert .$ For $\\\\lambda \\\\in with $ Im<2$ and $ dist(,(S+)) > + $ the estimates (\\\\ref {abs2}) and (\\\\ref {abs3}) hold in a similar form.\",\n \"Therefore, (\\\\ref {abs1})-(\\\\ref {abs3}) imply that for all non-real $ with $\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_-)) > \\\\nu + \\\\delta $ or $\\\\operatorname{dist}(\\\\lambda ,\\\\sigma (S_+)) > \\\\nu + \\\\delta $ we have $\\\\Vert (S-\\\\lambda )^{-1}\\\\Vert \\\\le \\\\frac{C}{\\\\vert \\\\operatorname{Im}\\\\lambda \\\\vert }$ with some $C > 0$ which does not depend on $\\\\lambda $ .\",\n \"This shows that the growth of the resolvent $(S-\\\\lambda )^{-1}$ is of order one at $\\\\infty $ .$\\\\Box $ Remark 3.6 If $\\\\operatorname{dist}(\\\\sigma (S_+),\\\\sigma (S_-))\\\\ge 2\\\\Vert M\\\\Vert $ , then the spectrum of $S$ in Theorem REF is real.\",\n \"This result can be improved in certain special cases, cf.\",\n \"[1] and also [2], where sharp norm bounds on the operator angles between reducing subspaces of $S$ and $S_+\\\\oplus S_-$ are given.\",\n \"We mention here also [34], [35] for comparable results in the study of operators of Klein-Gordon type, where bounded perturbations of operators with unbounded off-diagonal entries are investigated.\",\n \"The following simple example shows that the bounds on the non-real spectrum of $S$ in Theorem REF are sharp.\",\n \"Example 3.7 Let $\\\\mathfrak {H}_+ = \\\\mathfrak {H}_- = and $ S= 1$, $ M = z and hence $S = \\\\begin{pmatrix}1 & z\\\\\\\\-\\\\overline{z} & -1\\\\end{pmatrix}.$ Then $\\\\operatorname{dist}(\\\\sigma (S_+),\\\\sigma (S_-)) = 2$ .\",\n \"If $|z|\\\\le 1$ , then $\\\\sigma (S) = \\\\lbrace \\\\pm \\\\sqrt{1 - |z|^2}\\\\rbrace $ is real; cf.\",\n \"Remark REF .\",\n \"If $|z| > 1$ , then $\\\\sigma (S) = \\\\lbrace \\\\pm i\\\\sqrt{|z|^2 - 1}\\\\rbrace $ and in this case the eigenvalues of $S$ lie on the boundary of $ K_{|z|}(\\\\lbrace 1\\\\rbrace )\\\\cap K_{|z|}(\\\\lbrace -1\\\\rbrace )$ .\",\n \"Lemma 3.8 Let $T$ be a selfadjoint operator in the Hilbert space $(\\\\mathfrak {H},(\\\\cdot \\\\,,\\\\cdot ))$ with $0\\\\notin \\\\sigma _p(T)$ and let $E$ be its spectral function.\",\n \"If $(a_n)$ and $(b_n)$ are sequences in $[0,\\\\infty )$ such that $a_n\\\\downarrow 0$ and $b_n\\\\uparrow \\\\infty $ as $n\\\\rightarrow \\\\infty $ , and $0\\\\in \\\\rho (T)$ if $a_k=0$ for some $k\\\\in \\\\mathbb {N}$ , then $E(\\\\mathbb {R}^+) - E(\\\\mathbb {R}^-) = \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _{a_n}^{b_n}\\\\,\\\\left((T + it)^{-1} + (T - it)^{-1}\\\\right)\\\\,dt\\\\,.$ First of all we observe that $(T + it)^{-1} + (T - it)^{-1} = 2T(T^2 + t^2)^{-1}.$ For all $f,g\\\\in \\\\mathfrak {H}$ we have $\\\\int _{a_n}^{b_n}\\\\,\\\\big (T(T^2 + t^2)^{-1}f,g\\\\big )\\\\,dt&= \\\\int _{a_n}^{b_n}\\\\,\\\\int _{\\\\mathbb {R}}\\\\,\\\\frac{s}{s^2 + t^2}\\\\,d(E_sf,g)\\\\,dt\\\\\\\\&= \\\\int _{\\\\mathbb {R}}\\\\,\\\\int _{a_n}^{b_n}\\\\,\\\\frac{s}{s^2 + t^2}\\\\,dt\\\\,d(E_sf,g)\\\\\\\\&= \\\\int _{\\\\mathbb {R}}\\\\,\\\\big (\\\\arctan (b_n/s) - \\\\arctan (a_n/s)\\\\big )\\\\,d(E_sf,g).\\\\\\\\&= \\\\big (\\\\arctan (b_nT^{-1})f - \\\\arctan (a_nT^{-1})f,g\\\\big ).$ Fubini's theorem can be applied here since on $\\\\mathbb {R}\\\\times [a_n,b_n]$ the integrand is bounded and the measure $d(E_sf,g)\\\\otimes dt$ considered as a measure in $\\\\mathbb {R} \\\\times [a_n,b_n]$ is finite.\",\n \"Hence, $\\\\int _{a_n}^{b_n}\\\\,2T(T^2 + t^2)^{-1}f\\\\,dt = 2\\\\arctan (b_nT^{-1})f - 2\\\\arctan (a_nT^{-1})f.$ Now, we apply [52] and observe that $2\\\\arctan (a_nT^{-1})f$ tends to zero and $2\\\\arctan (b_nT^{-1})f$ tends to $\\\\pi \\\\operatorname{sgn}(T^{-1})f = \\\\pi \\\\operatorname{sgn}(T)f = \\\\pi (E(\\\\mathbb {R}^+) - E(\\\\mathbb {R}^-))f$ as $n\\\\rightarrow \\\\infty $ .$\\\\Box $ A subspace $\\\\mathfrak {L}$ of a Krein space $(\\\\mathfrak {H},[\\\\cdot \\\\,,\\\\cdot ])$ is called uniformly positive if there exists $\\\\delta > 0$ such that $[f,f]\\\\,\\\\ge \\\\,\\\\delta \\\\Vert f\\\\Vert ^2 \\\\quad \\\\mbox{for all } f\\\\in \\\\mathfrak {L}.$ Here $\\\\Vert \\\\cdot \\\\Vert $ is a Hilbert space norm on $\\\\mathfrak {H}$ with respect to which $[\\\\cdot \\\\,,\\\\cdot ]$ is continuous.\",\n \"All such norms are equivalent, see [46].\",\n \"Lemma 3.9 Let $G$ be a bounded and boundedly invertible selfadjoint operator in a Hilbert space $(\\\\mathfrak {H},(\\\\cdot \\\\,,\\\\cdot ))$ and set $[\\\\cdot \\\\,,\\\\cdot ]:= (G\\\\cdot ,\\\\cdot )$ .\",\n \"Let $\\\\mathfrak {L}$ be a closed uniformly positive subspace in the Krein space $(\\\\mathfrak {H},[\\\\cdot \\\\,,\\\\cdot ])$ .\",\n \"If $P$ ($E$ ) denotes the orthogonal projection in the Hilbert space $(\\\\mathfrak {H},(\\\\cdot \\\\,,\\\\cdot ))$ (in the Krein space $(\\\\mathfrak {H},[\\\\cdot \\\\,,\\\\cdot ])$ , respectively) onto $\\\\mathfrak {L}$ , then $P(G|\\\\mathfrak {L})$ is a uniformly positive selfadjoint operator in $(\\\\mathfrak {L},(\\\\cdot \\\\,,\\\\cdot ))$ , and its inverse is given by $\\\\bigl (P(G|\\\\mathfrak {L})\\\\bigr )^{-1} = E\\\\bigl (G^{-1}|\\\\mathfrak {L}\\\\bigr ).$ As $\\\\mathfrak {L}$ is uniformly positive, for all $f\\\\in \\\\mathfrak {L}$ we have $\\\\big (P(G|\\\\mathfrak {L})f,f\\\\big ) = (Gf,f) = [f,f]\\\\,\\\\ge \\\\,\\\\delta \\\\Vert f\\\\Vert ^2$ with some $\\\\delta > 0$ .\",\n \"Thus, $P(G|\\\\mathfrak {L})$ is a uniformly positive operator in $(\\\\mathfrak {L},(\\\\cdot \\\\,,\\\\cdot ))$ and, in particular, $0\\\\in \\\\rho (P(G|\\\\mathfrak {L}))$ .\",\n \"Let $g\\\\in \\\\mathfrak {L}$ and set $f := E(G^{-1}|\\\\mathfrak {L})g\\\\in \\\\mathfrak {L}$ .\",\n \"Since for all $h\\\\in \\\\mathfrak {L}$ $(g - Gf,h) = \\\\bigl (G(I - E)G^{-1}g,h\\\\bigr ) = \\\\bigl [(I - E)G^{-1}g,h\\\\bigr ] = 0,$ we conclude $g - Gf\\\\in \\\\mathfrak {L}^\\\\perp $ and hence $g = PGf = P(G|\\\\mathfrak {L})f$ .\",\n \"Therefore, $\\\\bigl (P(G|\\\\mathfrak {L})\\\\bigr )^{-1}g = f = E\\\\bigl (G^{-1}|\\\\mathfrak {L}\\\\bigr )g,$ which proves the lemma.$\\\\Box $ Proof of Theorem REF Assertion (i) in Theorem REF is simple: As both $A_0$ and $V$ are non-negative in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ we have $[(A_0+V)f,f]\\\\ge 0\\\\quad \\\\text{for all}\\\\quad f\\\\in \\\\operatorname{dom}(A_0+V)=\\\\operatorname{dom}A_0.$ The operator $A_0$ is selfadjoint in the Hilbert space $(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot )_\\\\sim )$ (cf.\",\n \"(REF ) and (REF )) and, as $V$ is bounded, $\\\\rho (A_0+V)$ is nonempty.\",\n \"Thus $A_0+V$ is a non-negative operator in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ and (i) in Theorem REF is proved.\",\n \"Let $E$ be the spectral function of the non-negative operator $A_0$ , define the projections $E_+ := E((0,\\\\infty ))$ , $E_-:=E((-\\\\infty ,0))$ and the subspaces $\\\\mathcal {H}_\\\\pm := E_\\\\pm \\\\mathcal {H}$ .\",\n \"Then with respect to the fundamental decomposition (REF ) the operators $A_0$ and $V$ have the form in (REF ) and (REF ), respectively.\",\n \"The fundamental symmetry $\\\\widetilde{J}$ associated with the decomposition (REF ) is also given in (REF ).\",\n \"The inner product $(\\\\cdot \\\\,,\\\\cdot )_\\\\sim $ is defined as in (REF ).\",\n \"Moreover, $A_{0,\\\\pm }$ and $V_\\\\pm $ from (REF ) are selfadjoint in $(\\\\mathcal {H}_\\\\pm ,(\\\\cdot \\\\,,\\\\cdot )_\\\\sim ) = (\\\\mathcal {H}_\\\\pm ,\\\\pm [\\\\cdot \\\\,,\\\\cdot ])$ , $\\\\sigma (A_{0,\\\\pm })\\\\subset \\\\mathbb {R}^\\\\pm \\\\cup \\\\lbrace 0\\\\rbrace $ and (REF ) holds.\",\n \"We apply Lemma REF to the non-negative operators $\\\\pm A_{0,\\\\pm }$ and obtain $\\\\pm I_{\\\\mathcal {H}_\\\\pm } = \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _{1/n}^n\\\\,\\\\left((A_{0,\\\\pm } + it)^{-1} + (A_{0,\\\\pm } - it)^{-1}\\\\right)\\\\,dt.$ Therefore, $\\\\widetilde{J} = \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _{1/n}^n\\\\,\\\\left((A_{0} + it)^{-1} + (A_{0} - it)^{-1}\\\\right)\\\\,dt,$ where the strong limit is with respect to the norm $\\\\Vert \\\\cdot \\\\Vert _\\\\sim := (\\\\cdot \\\\,,\\\\cdot )_\\\\sim ^{1/2}$ .\",\n \"But as $\\\\Vert \\\\cdot \\\\Vert _\\\\sim $ and $\\\\Vert \\\\cdot \\\\Vert $ are equivalent norms, the limit on the right hand side of (REF ) also exists with respect to the norm $\\\\Vert \\\\cdot \\\\Vert $ , and coincides with $\\\\widetilde{J}$ .\",\n \"The uniform boundedness principle now yields that $\\\\tau _0$ in () is finite and that $\\\\Vert \\\\widetilde{J}\\\\Vert \\\\,\\\\le \\\\,\\\\tau _0<\\\\infty .$ Define the constants $\\\\nu := \\\\Vert B\\\\Vert _\\\\sim ,\\\\quad a := \\\\min \\\\sigma (V_+)\\\\quad \\\\text{and}\\\\quad b := \\\\max \\\\sigma (V_-).$ Then $A_{0,+} + V_+$ is bounded from below by $a$ and $\\\\sigma (A_{0,+} + V_+)\\\\subset [a,\\\\infty )$ , and $A_{0,-} + V_-$ is bounded from above by $b$ and $\\\\sigma (A_{0,-} + V_-)\\\\subset (-\\\\infty ,b]$ .\",\n \"Making use of the representation (REF ) of $A = A_0 + V$ and Theorem REF we conclude: $\\\\sigma (A)\\\\backslash \\\\mathbb {R}\\\\subset K_\\\\nu ((-\\\\infty ,b])\\\\cap K_\\\\nu ([a,\\\\infty ))$ .\",\n \"$(b + \\\\nu ,\\\\infty )$ is of positive type with respect to $A$ .\",\n \"$(-\\\\infty ,a - \\\\nu )$ is of negative type with respect to $A$ .\",\n \"The growth of the resolvent of $A$ at $\\\\infty $ of order one, and hence of finite order.\",\n \"The statement in Theorem REF (ii) follows from Theorem REF if we show that the constants $\\\\nu $ , $a$ and $b$ in (REF ) satisfy $\\\\nu \\\\le r, \\\\quad a\\\\ge -d, \\\\quad b\\\\le d,$ where $r$ and $d$ are as in (), because in that case the properties (a)-(d) above hold with $\\\\nu $ , $a$ and $b$ replaced by $r$ , $-d$ and $d$ , respectively, and $K_\\\\nu ((-\\\\infty ,d])\\\\cap K_\\\\nu ([-d,\\\\infty ))= K_\\\\nu ((-d,d)).$ First we check $\\\\nu \\\\le r$ .\",\n \"Denote by $(\\\\cdot \\\\,,\\\\cdot )$ the scalar product corresponding to the Hilbert space norm $\\\\Vert \\\\cdot \\\\Vert $ .\",\n \"Using the well-known fact that the spectrum of a bounded operator is always a subset of the closure of its numerical range we obtain $\\\\begin{split}\\\\nu ^2&= \\\\Vert B\\\\Vert _\\\\sim ^2 = \\\\Vert B^{\\\\tilde{*}}B\\\\Vert _\\\\sim = \\\\sup \\\\lbrace |\\\\lambda | : \\\\lambda \\\\in \\\\sigma (B^{\\\\tilde{*}}B)\\\\rbrace \\\\\\\\&\\\\le \\\\sup \\\\lbrace |(B^{\\\\tilde{*}}Bf,f)| : \\\\Vert f\\\\Vert =1\\\\rbrace \\\\,\\\\le \\\\,\\\\Vert B^{\\\\tilde{*}}\\\\Vert \\\\,\\\\Vert B\\\\Vert \\\\\\\\&= \\\\Vert E_-(V|\\\\mathcal {H}_+)\\\\Vert \\\\,\\\\Vert E_+(V|\\\\mathcal {H}_-)\\\\Vert \\\\,\\\\le \\\\,\\\\Vert E_+\\\\Vert \\\\,\\\\Vert E_-\\\\Vert \\\\,\\\\Vert V\\\\Vert ^2.\\\\end{split}$ As $E_\\\\pm = \\\\frac{1}{2}(I\\\\pm \\\\widetilde{J})$ , it follows from (REF ) that $\\\\Vert E_\\\\pm \\\\Vert \\\\le \\\\frac{1+\\\\tau _0}{2}$ which shows $\\\\nu \\\\le r$ .\",\n \"Next we verify $a\\\\ge -d$ .\",\n \"Denote by $P_\\\\pm $ the orthogonal projection onto $\\\\mathcal {H}_\\\\pm $ with respect to $(\\\\cdot \\\\,,\\\\cdot )$ .\",\n \"Then, according to Lemma REF , the operator $P_+(J|\\\\mathcal {H}_+)$ is selfadjoint and uniformly positive in the Hilbert space $(\\\\mathcal {H}_+,(\\\\cdot \\\\,,\\\\cdot ))$ .\",\n \"Hence, the same holds for its inverse $E_+(J|\\\\mathcal {H}_+)$ ; cf.\",\n \"Lemma REF .\",\n \"This implies $\\\\min \\\\sigma (P_+(J|\\\\mathcal {H}_+))&= \\\\big (\\\\max \\\\sigma (E_+(J|\\\\mathcal {H}_+))\\\\big )^{-1} = \\\\Vert E_+(J|\\\\mathcal {H}_+)\\\\Vert ^{-1}\\\\\\\\&\\\\ge \\\\Vert E_+J\\\\Vert ^{-1} \\\\ge \\\\Vert E_+\\\\Vert ^{-1}\\\\,\\\\ge \\\\,\\\\frac{2}{1+\\\\tau _0}\\\\,.$ Hence, for $f_+\\\\in \\\\mathcal {H}_+$ we have $[f_+,f_+] = (Jf_+,f_+) = (P_+(J|\\\\mathcal {H}_+)f_+,f_+)\\\\,\\\\ge \\\\,\\\\frac{2}{1+\\\\tau _0}\\\\,\\\\Vert f_+\\\\Vert ^2.$ We also use $(V_+f_+,f_+)_\\\\sim = [V_+f_+,f_+] = [V_+f_+ - B^{\\\\tilde{*}}f_+,f_+] = [Vf_+,f_+] = (JVf_+,f_+)$ for $f\\\\in \\\\mathcal {H}_+$ to conclude $a=\\\\min \\\\sigma (V_+)&=\\\\inf \\\\lbrace (V_+f_+,f_+)_\\\\sim : \\\\Vert f_+\\\\Vert _\\\\sim = 1,\\\\,f_+\\\\in \\\\mathcal {H}_+\\\\rbrace \\\\\\\\&\\\\ge \\\\inf \\\\lbrace (V_+f_+,f_+)_\\\\sim : \\\\Vert f_+\\\\Vert _\\\\sim \\\\le 1,\\\\,f_+\\\\in \\\\mathcal {H}_+\\\\rbrace \\\\\\\\&= \\\\inf \\\\lbrace (JVf_+,f_+) : [f_+,f_+]\\\\le 1,\\\\,f_+\\\\in \\\\mathcal {H}_+\\\\rbrace \\\\\\\\&\\\\ge \\\\inf \\\\bigl \\\\lbrace (JVf_+,f_+) : \\\\Vert f_+\\\\Vert ^2\\\\le \\\\tfrac{1+\\\\tau _0}{2},\\\\,f_+\\\\in \\\\mathcal {H}_+\\\\bigr \\\\rbrace \\\\\\\\&\\\\ge \\\\inf \\\\bigl \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert ^2\\\\le \\\\tfrac{1+\\\\tau _0}{2},\\\\,f\\\\in \\\\mathcal {H}\\\\bigr \\\\rbrace .$ By assumption $V$ is not non-negative in $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ and hence $0>\\\\inf \\\\bigl \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert ^2\\\\le \\\\tfrac{1+\\\\tau _0}{2},\\\\,f\\\\in \\\\mathcal {H}\\\\bigr \\\\rbrace =\\\\inf \\\\bigl \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert ^2=\\\\tfrac{1+\\\\tau _0}{2},\\\\,f\\\\in \\\\mathcal {H}\\\\bigr \\\\rbrace .$ Therefore we finally obtain $a=\\\\min \\\\sigma (V_+)&\\\\ge \\\\inf \\\\bigl \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert ^2=\\\\tfrac{1+\\\\tau _0}{2},\\\\,f\\\\in \\\\mathcal {H}\\\\bigr \\\\rbrace \\\\\\\\&= \\\\frac{1+\\\\tau _0}{2}\\\\,\\\\inf \\\\lbrace (JVf,f) : \\\\Vert f\\\\Vert = 1,\\\\,f\\\\in \\\\mathcal {H}\\\\rbrace \\\\\\\\&= \\\\frac{1+\\\\tau _0}{2}\\\\,\\\\min \\\\sigma (JV)=-d.$ The proof of the inequality $b\\\\le d$ is analogous and is left to the reader.\",\n \"This completes the proof of (ii).\",\n \"As $\\\\infty $ is not a singular critical point of $A_0$ , the last statement in Theorem REF on the critical point $\\\\infty $ of $A_0+V$ follows in both cases (i) and (ii) from Proposition and $\\\\operatorname{dom}A=\\\\operatorname{dom}(A_0+V)$ .$\\\\Box $ Proof of Theorem REF Let $\\\\gamma $ and $\\\\eta $ be as in Theorem REF .\",\n \"Then the operator $A_0 := A + \\\\eta G^{-1}$ is uniformly positive in the Krein space $(\\\\mathcal {H},[\\\\cdot \\\\,,\\\\cdot ])$ , $[\\\\cdot \\\\,,\\\\cdot ]= (G\\\\cdot ,\\\\cdot )$ , since for $f\\\\in \\\\operatorname{dom}A$ we have $[A_0 f,f] = [Af,f] + \\\\eta [G^{-1}f,f] \\\\ge -\\\\gamma \\\\Vert f\\\\Vert ^2 + \\\\eta \\\\Vert f\\\\Vert ^2 = (\\\\eta - \\\\gamma )\\\\Vert f\\\\Vert ^2.$ We will show that $A$ is non-negative over $\\\\overline{\\\\setminus K_r((-r,r)).In particular, 0\\\\in \\\\rho (A_0) is not a singular critical point of A_0and, as \\\\tau _\\\\eta < \\\\infty , the point \\\\infty is not aa singular critical point of A_0, see Remark \\\\ref {Mai2012}.With V := -\\\\eta G^{-1} we haveA = A_0 + V\\\\qquad \\\\text{and}\\\\qquad \\\\Vert V\\\\Vert =\\\\eta \\\\Vert G^{-1}\\\\Vert .As in the proof of Theorem~\\\\ref {t:main1} let E be the spectral function of the operator A_0.\",\n \"Then the fundamentaldecomposition \\\\mathcal {H}= E_+\\\\mathcal {H}\\\\,[\\\\dotplus ]\\\\,E_-\\\\mathcal {H} in (\\\\ref {e:dec_expl}) generates matrix representationsof A_0, V and the corresponding fundamental symmetry \\\\widetilde{J} as in (\\\\ref {e:AJ}) and (\\\\ref {e:V}).The operator A_0 is selfadjoint in the Hilbert space(\\\\mathcal {H},(\\\\cdot \\\\,,\\\\cdot )_\\\\sim ), where (\\\\cdot \\\\,,\\\\cdot )_\\\\sim is defined in (\\\\ref {e:sim}).\",\n \"With the help of Lemma \\\\ref {l:veselic} we obtain{\\\\begin{@align*}{1}{-1}\\\\widetilde{J}&= \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _0^n\\\\,\\\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\big )\\\\,dt\\\\\\\\&= \\\\frac{1}{\\\\pi }\\\\;\\\\,\\\\underset{n\\\\rightarrow \\\\infty }{{\\\\rm s}{\\\\text{-}}\\\\!\\\\lim }\\\\,\\\\!\\\\int _{-n}^n\\\\,(A + \\\\eta G^{-1} - it)^{-1}\\\\,dt\\\\,.\\\\end{@align*}}This yields\\\\Vert \\\\widetilde{J}\\\\Vert \\\\,\\\\le \\\\,\\\\tau _\\\\eta .Following the lines of the proof of Theorem \\\\ref {t:main1}, it suffices to show the inequalities\\\\nu \\\\le \\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert V\\\\Vert ,\\\\quad \\\\min \\\\sigma (V_+)\\\\ge -\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert V\\\\Vert \\\\quad \\\\text{and}\\\\quad \\\\max \\\\sigma (V_-)\\\\le \\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert V\\\\Vert ,where \\\\nu =\\\\Vert E_\\\\mp (V\\\\vert \\\\cal H_\\\\pm ) \\\\Vert _\\\\sim .The first relation is proved as in (\\\\ref {e:rel_proof}) and the remaining two inequalities follow from the selfadjointness of V_\\\\pm in (\\\\mathcal {H}_\\\\pm ,(\\\\cdot \\\\,,\\\\cdot )_\\\\sim ) and{\\\\begin{@align*}{1}{-1}\\\\max \\\\lbrace |\\\\min \\\\sigma (V_\\\\pm )|,|\\\\max \\\\sigma (V_\\\\pm )|\\\\rbrace &= \\\\Vert V_\\\\pm \\\\Vert _\\\\sim = \\\\sup \\\\lbrace |\\\\lambda | : \\\\lambda \\\\in \\\\sigma (V_\\\\pm )\\\\rbrace \\\\\\\\&\\\\le \\\\sup \\\\lbrace |(V_\\\\pm f_\\\\pm ,f_\\\\pm )| : \\\\Vert f_\\\\pm \\\\Vert =1,\\\\,f\\\\in \\\\mathcal {H}_\\\\pm \\\\rbrace \\\\\\\\&\\\\le \\\\Vert V_\\\\pm \\\\Vert = \\\\Vert E_\\\\pm (V|\\\\mathcal {H}_\\\\pm )\\\\Vert \\\\,\\\\le \\\\,\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert V\\\\Vert .\\\\end{@align*}}Hence, A is non-negative over \\\\overline{\\\\setminus K_r((-r,r)).\",\n \"As\\\\operatorname{dom}A= \\\\operatorname{dom}A_0, we conclude from Proposition \\\\ref {p:curgus} that \\\\infty is also not a critical point of A.This completes the proof of Theorem~\\\\ref {t:main2}.\\\\Box }\\\\begin{remark}If, in addition to the assumptions in Theorem \\\\ref {t:main2}, the point 0 is neither an eigenvalue nor a singular critical pointof the non-negative operator A_0 := A + \\\\gamma G^{-1}, then the operator A is non-negative over the set \\\\overline{\\\\setminus K_r((-r,r)), where\\\\begin{equation}r = \\\\gamma \\\\,\\\\frac{1+\\\\tau _\\\\gamma }{2}\\\\,\\\\Vert G^{-1}\\\\Vert \\\\end{equation}and\\\\tau _\\\\gamma := \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{1/n}^n\\\\,\\\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\big )\\\\,dt\\\\right\\\\Vert .Indeed, under the assumptions of Theorem \\\\ref {t:main2} for any \\\\varepsilon > 0 the operator A + (\\\\gamma + \\\\varepsilon )G^{-1}is selfadjoint in some Hilbert space, and thus the resolvent set ofA_0 is non-empty as A + (\\\\gamma + \\\\varepsilon )G^{-1} and A_0 differ just by a bounded operator.Therefore, A_0 is a non-negative operator in the Krein space(\\\\cal H,[\\\\cdot \\\\,,\\\\cdot ]).\",\n \"Moreover,the point \\\\infty is not a singular critical point of theoperator A + (\\\\gamma + \\\\varepsilon )G^{-1}, see Remark \\\\ref {Mai2012},and by Proposition \\\\ref {p:curgus} \\\\infty is not a singular critical point of A_0.Therefore, the operator A_0 admits a matrix representation as in (\\\\ref {e:AJ}) and the corresponding fundamental symmetry \\\\widetilde{J}satisfies \\\\Vert \\\\widetilde{J}\\\\Vert \\\\,\\\\le \\\\,\\\\tau _\\\\gamma (see the proof of Theorem \\\\ref {t:main2}).Then by a similar reasoning as in the proof ofTheorem \\\\ref {t:main2} we see that A is non-negativeover \\\\overline{\\\\setminus K_r((-r,r)), where r is as in (\\\\ref {Juni2012}).\",\n \"}}\\\\end{remark}}$ Differential operators with indefinite weights In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights.\",\n \"Indefinite Sturm-Liouville operators We consider Sturm-Liouville differential expressions of the form $\\\\mathcal {L}(f)(x) = \\\\operatorname{sgn}(x)\\\\big (-f^{\\\\prime \\\\prime }(x) + q(x)f(x)\\\\big ),\\\\quad x\\\\in \\\\mathbb {R}\\\\,,$ with a real-valued potential $q\\\\in L^\\\\infty (\\\\mathbb {R})$ and the indefinite weight function $\\\\operatorname{sgn}(\\\\cdot )$ .\",\n \"The corresponding differential operator in $L^2(\\\\mathbb {R})$ is defined by $Af := \\\\mathcal {L}(f), \\\\qquad \\\\operatorname{dom}A := H^2(\\\\mathbb {R}).$ Here $H^2(\\\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.\",\n \"By $(\\\\cdot \\\\,,\\\\cdot )$ and $\\\\Vert \\\\cdot \\\\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\\\mathbb {R})$ .\",\n \"Let $J$ be the operator of multiplication with the function $\\\\operatorname{sgn}(x)$ .\",\n \"This operator is obviously selfadjoint and unitary in $L^2(\\\\mathbb {R})$ .\",\n \"Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\\\prime \\\\prime } + qf,\\\\quad \\\\operatorname{dom}T := H^2(\\\\mathbb {R}),$ is selfadjoint in $L^2(\\\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\\\mathbb {R}),[\\\\cdot \\\\,,\\\\cdot ])$ , where $[f,g] := (Jf,g) = \\\\int _\\\\mathbb {R}\\\\,f(x)\\\\overline{g(x)}\\\\operatorname{sgn}(x)\\\\,dx,\\\\quad f,g\\\\in L^2(\\\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.\",\n \"We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].\",\n \"For this, denote by $m_\\\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\\\pm f_\\\\pm := - f_\\\\pm ^{\\\\prime \\\\prime } + q_\\\\pm f_\\\\pm $ in $L^2(\\\\mathbb {R}^\\\\pm )$ defined on $\\\\operatorname{dom}T_\\\\pm := \\\\lbrace f_\\\\pm \\\\in H^2(\\\\mathbb {R}^\\\\pm ) : f_\\\\pm (0) = 0\\\\rbrace $ , where $q_\\\\pm $ is the restriction of the function $q$ to $\\\\mathbb {R}^\\\\pm $ .\",\n \"The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\\\pm \\\\infty $ , then $m_\\\\pm =\\\\lim _{x\\\\rightarrow \\\\pm \\\\infty }\\\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.\",\n \"Furthermore the following holds.\",\n \"If $m_+ > -m_-$ then $\\\\sigma (A)\\\\setminus \\\\mathbb {R}$ is finite; If $m_+\\\\le -m_-$ then $\\\\sigma (A)\\\\setminus \\\\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .\",\n \"To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.\",\n \"One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].\",\n \"The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .\",\n \"Here only the nontrivial case $\\\\operatornamewithlimits{ess\\\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).\",\n \"Theorem 4.2 Assume that $\\\\operatornamewithlimits{ess\\\\,inf}q<0$ holds.\",\n \"Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\\\overline{\\\\setminus K_r((-d,d)), wherer := 5\\\\Vert q\\\\Vert _\\\\infty ,\\\\quad \\\\text{and}\\\\quad d := -5\\\\,\\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\\\cap \\\\lbrace \\\\lambda \\\\in |\\\\operatorname{Im}\\\\lambda |\\\\le 2\\\\Vert q\\\\Vert _\\\\infty \\\\rbrace .\",\n \"}\\\\begin{proof}The proof of Theorem~\\\\ref {t:SL2} is split into 4 parts.\",\n \"The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\\\ref {t:main1}.\",\n \"In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\\\end{proof}\\\\ \\\\\\\\{\\\\it 1.\\\\ Preparation: } The operator $ A0$,\\\\begin{equation*}A_0 f:=-\\\\operatorname{sgn}(\\\\cdot )f^{\\\\prime \\\\prime },\\\\qquad \\\\operatorname{dom}A_0=H^2(\\\\mathbb {R}),\\\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\\\mathbb {R}$ ),[,])$.\",\n \"Furthermore, we have $ (A0) = $\\\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\\\cite {cn}.\",\n \"Hence $ A0$ satisfies theassumptions in Theorem~\\\\ref {t:main1}.\",\n \"Define $ V$ as\\\\begin{equation*}Vf:=\\\\operatorname{sgn}(\\\\cdot ) qf,\\\\qquad \\\\operatorname{dom}V=L^2(\\\\mathbb {R}),\\\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\\\mathbb {R}$ ),[,])$ with$$\\\\min \\\\sigma (JV) = \\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}\\\\,q(x)\\\\quad \\\\text{and}\\\\quad \\\\Vert V\\\\Vert = \\\\Vert q\\\\Vert _\\\\infty .$$By Theorem \\\\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\\\frac{1+\\\\tau _0}{2}\\\\,\\\\Vert q\\\\Vert _\\\\infty ,\\\\qquad d = -\\\\frac{1+\\\\tau _0}{2}\\\\,\\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}q(x),$$and$$\\\\tau _0 = \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{1/n}^n\\\\,\\\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\right)\\\\,dt\\\\,\\\\right\\\\Vert .$$Therefore, we have to show\\\\begin{equation}\\\\tau _0\\\\le 9\\\\end{equation}and\\\\begin{equation}\\\\sigma (A)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\,\\\\lbrace \\\\lambda \\\\in |\\\\operatorname{Im}\\\\lambda |\\\\le 2\\\\Vert q\\\\Vert _\\\\infty \\\\rbrace .\\\\end{equation}$ 2.\",\n \"Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\\\begin{pmatrix} B_+ & 0 \\\\\\\\ 0 & B_- \\\\end{pmatrix}\\\\quad \\\\text{in}\\\\quad L^2(\\\\mathbb {R})=L^2(\\\\mathbb {R}^+)\\\\times L^2(\\\\mathbb {R}^-),$ where $B_\\\\pm f_\\\\pm :=\\\\mp f_\\\\pm ^{\\\\prime \\\\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\\\mathbb {R}^\\\\pm )$ which are defined on $\\\\operatorname{dom}B_\\\\pm = \\\\lbrace f_\\\\pm \\\\in H^2(\\\\mathbb {R}^\\\\pm ):f_\\\\pm (0)=0\\\\rbrace $ .\",\n \"Here and in the following we will denote the restrictions of a function $f$ defined on $\\\\mathbb {R}$ to $\\\\mathbb {R}^\\\\pm $ by $f_\\\\pm $ .\",\n \"For $\\\\lambda = re^{it}$ , $r > 0$ , $t\\\\in [0,2\\\\pi )$ , we set $\\\\sqrt{\\\\lambda } := \\\\sqrt{r}e^{it/2}$ .\",\n \"For $\\\\lambda \\\\in \\\\mathbb {R}$ define the function $f_\\\\lambda (x) :={\\\\left\\\\lbrace \\\\begin{array}{ll}e^{i\\\\sqrt{\\\\lambda }x}, &x> 0,\\\\\\\\e^{-i\\\\sqrt{-\\\\lambda }x}, &x < 0.\\\\end{array}\\\\right.\",\n \"}$ The function $f_\\\\lambda $ is a solution of the equations $\\\\mp f_\\\\pm ^{\\\\prime \\\\prime } = \\\\lambda f_\\\\pm $ in $L^2(\\\\mathbb {R}^\\\\pm )$ .\",\n \"From $\\\\overline{\\\\sqrt{\\\\lambda }} = -\\\\sqrt{\\\\overline{\\\\lambda }}$ it follows that $f_{\\\\overline{\\\\lambda }} = \\\\overline{f_\\\\lambda }$ holds for all $\\\\lambda \\\\in \\\\mathbb {R}$ .\",\n \"Let us now prove that for all $f\\\\in L^2(\\\\mathbb {R})$ and $\\\\lambda \\\\in \\\\mathbb {R}$ we have $(A_0 - \\\\lambda )^{-1}f = (B_0 - \\\\lambda )^{-1}f - \\\\frac{[f,f_{\\\\overline{\\\\lambda }}]}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_\\\\lambda .$ Since $\\\\sigma (A_0)=\\\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\\\lambda \\\\in \\\\mathbb {R}$ .\",\n \"In particular, for $\\\\lambda \\\\in \\\\mathbb {R}$ there exists $g\\\\in \\\\operatorname{dom}B_0$ such that $f=(B_0-\\\\lambda )g$ holds.\",\n \"The right hand side in (REF ) has the form $h:=g-\\\\frac{[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_\\\\lambda $ and we will show that the function $h$ belongs to $\\\\operatorname{dom}A_0=H^2(\\\\mathbb {R})$ .\",\n \"As $g_\\\\pm $ and $f_{\\\\lambda ,\\\\pm }$ are elements of $H^2(\\\\mathbb {R}^\\\\pm )$ the same is true for $h_\\\\pm $ .\",\n \"Moreover, $g$ and $f_\\\\lambda $ are continuous at 0 and so is $h$ .\",\n \"Hence it remains to check that $h^\\\\prime $ is continuous at 0.\",\n \"For this note first that $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]=\\\\int _0^\\\\infty (-g_+^{\\\\prime \\\\prime }-\\\\lambda g_+)\\\\overline{f_{\\\\overline{\\\\lambda },+}}\\\\, dx- \\\\int _{-\\\\infty }^0 (g_-^{\\\\prime \\\\prime }-\\\\lambda g_-)\\\\overline{f_{\\\\overline{\\\\lambda },-}}\\\\, dx$ and since $f_{\\\\overline{\\\\lambda },\\\\pm }$ are solutions of $\\\\mp f_\\\\pm ^{\\\\prime \\\\prime }=\\\\overline{\\\\lambda }f_\\\\pm $ , integration by parts yields $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]=g_+^\\\\prime (0)\\\\overline{f_{\\\\overline{\\\\lambda },+}(0)}-g_+(0) \\\\overline{f^\\\\prime _{\\\\overline{\\\\lambda },+}(0)}-g_-^\\\\prime (0)\\\\overline{f_{\\\\overline{\\\\lambda },-}(0)}+g_-(0) \\\\overline{f^\\\\prime _{\\\\overline{\\\\lambda },-}(0)}.$ As $g\\\\in \\\\operatorname{dom}B_0$ we have $g_\\\\pm (0)=0$ and together with $f_{\\\\overline{\\\\lambda },\\\\pm }(0)=1$ we find $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]= g_+^\\\\prime (0)-g^\\\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\\\pm ^\\\\prime $ on $\\\\mathbb {R}^\\\\pm $ of the function $h$ from (REF ): $h^\\\\prime _\\\\pm =g^\\\\prime _\\\\pm -\\\\frac{g_+^\\\\prime (0)-g^\\\\prime _-(0)}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_{\\\\lambda ,\\\\pm }^\\\\prime $ and as $f^\\\\prime _{\\\\lambda ,+}(0)=i\\\\sqrt{\\\\lambda }$ and $f^\\\\prime _{\\\\lambda ,-}(0)=-i\\\\sqrt{-\\\\lambda }$ we conclude $h^\\\\prime _+(0)-h^\\\\prime _-(0)= \\\\bigl (g_+^\\\\prime (0)-g^\\\\prime _-(0)\\\\bigr )-\\\\frac{g_+^\\\\prime (0)-g^\\\\prime _-(0)}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}\\\\bigl (f^\\\\prime _{\\\\lambda ,+}(0)-f^\\\\prime _{\\\\lambda ,-}(0)\\\\bigr )=0,$ that is, $h^\\\\prime $ is continuous at 0 and therefore $h\\\\in \\\\operatorname{dom}A_0$ .\",\n \"Now a straightforward computation shows that $(A_0-\\\\lambda )h=(B_0-\\\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).\",\n \"3.\",\n \"Proof of (): Let $f\\\\in L^2(\\\\mathbb {R})$ and $t > 0$ .\",\n \"Then (REF ) and $\\\\sqrt{it} + \\\\sqrt{-it} = i\\\\sqrt{2t}$ yield $\\\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\\\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\\\\\& + \\\\frac{2}{\\\\sqrt{2t}}\\\\,\\\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\\\in \\\\mathbb {R}^+$ , we have $\\\\big |[f,f_{it}][f_{-it},f]\\\\big |\\\\,\\\\le \\\\,\\\\left(\\\\int _0^\\\\infty \\\\,e^{-x\\\\sqrt{t/2}}g(x)\\\\,dx\\\\right)^2$ and thus for $n\\\\in \\\\mathbb {N}$ $\\\\int _{1/n}^n\\\\,\\\\frac{|[f,f_{it}][f_{-it},f]|}{\\\\sqrt{2t}}\\\\,dt&\\\\le \\\\int _{1/n}^n\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,g(x)g(y)\\\\,\\\\frac{e^{-(x+y)\\\\sqrt{t/2}}}{\\\\sqrt{2t}}\\\\,dy\\\\,dx\\\\,dt\\\\\\\\&= 2\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,\\\\frac{g(x)g(y)}{x+y}\\\\left(e^{-\\\\frac{x+y}{\\\\sqrt{2n}}} - e^{-\\\\frac{\\\\sqrt{n}(x+y)}{\\\\sqrt{2}}}\\\\right)\\\\,dy\\\\,dx\\\\\\\\&\\\\le 2\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,\\\\frac{g(x)g(y)}{x+y}\\\\,dy\\\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\\\int _{1/n}^n\\\\,\\\\frac{|[f,f_{it}][f_{-it},f]|}{\\\\sqrt{2t}}\\\\,dt\\\\,\\\\le \\\\,2\\\\pi \\\\Vert g\\\\Vert _{L^2(\\\\mathbb {R}^+)}^2\\\\,\\\\le \\\\,4\\\\pi \\\\Vert f\\\\Vert ^2.$ and therefore we have $\\\\frac{1}{\\\\pi }\\\\int _{1/n}^n\\\\,\\\\frac{2}{\\\\sqrt{2t}}\\\\,\\\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\\\,dt\\\\,\\\\le \\\\,8\\\\Vert f\\\\Vert ^2.$ Denote by $E_0$ ($E_\\\\pm $ ) the spectral function of the operator $A_0$ ($B_\\\\pm $ , respectively).\",\n \"Moreover define $J_{n,\\\\pm } := \\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,\\\\left((B_\\\\pm + it)^{-1} + (B_\\\\pm - it)^{-1}\\\\right)\\\\,dt,$ and, in addition, $J_{n,0} := \\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,\\\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\right)\\\\,dt.$ As $\\\\pm B_\\\\pm $ is non-negative, it follows that $J_{n,\\\\pm }$ converges strongly to $\\\\pm I_{L^2(\\\\mathbb {R}^\\\\pm )}$ when $n\\\\rightarrow \\\\infty $ , cf.\",\n \"Lemma REF .\",\n \"Moreover, by $(B_\\\\pm + it)^{-1} + (B_\\\\pm - it)^{-1}= 2B_\\\\pm (B_\\\\pm + it)^{-1} (B_\\\\pm - it)^{-1}$ the sequence $\\\\pm J_{n,\\\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\\\mathbb {R}^\\\\pm )$ .\",\n \"This implies $\\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\\\\\&\\\\le \\\\Vert f_+\\\\Vert _{L^2(\\\\mathbb {R}^+)}^2 + \\\\Vert f_-\\\\Vert _{L^2(\\\\mathbb {R}^-)}^2 = \\\\Vert f\\\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\\\,\\\\le \\\\,\\\\Vert f\\\\Vert ^2 + 8\\\\Vert f\\\\Vert ^2 = 9\\\\Vert f\\\\Vert ^2,\\\\quad f\\\\in L^2(\\\\mathbb {R}).$ Hence, as $J = \\\\operatorname{sgn}(\\\\cdot )$ is unitary in $L^2(\\\\mathbb {R})$ , we finally obtain $\\\\Vert J_{n,0}\\\\Vert = \\\\Vert J J_{n,0}\\\\Vert \\\\le 9$ and thus $\\\\tau _0 = \\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\Vert J_{n,0}\\\\Vert \\\\le 9$ .\",\n \"That is, () holds.\",\n \"4.\",\n \"Proof of (): Let $\\\\lambda \\\\in +$ .\",\n \"Using (REF ), for $f\\\\in L^2(\\\\mathbb {R})$ we obtain $\\\\Vert (A_0 - \\\\lambda )^{-1}f\\\\Vert \\\\,\\\\le \\\\,\\\\Vert (B_0 - \\\\lambda )^{-1}f\\\\Vert + \\\\frac{|[f,f_{\\\\overline{\\\\lambda }}]|\\\\,\\\\Vert f_\\\\lambda \\\\Vert }{|\\\\sqrt{\\\\lambda } + \\\\sqrt{-\\\\lambda }|}\\\\,\\\\le \\\\,\\\\frac{\\\\Vert f\\\\Vert }{|\\\\operatorname{Im}\\\\lambda |} + \\\\frac{\\\\Vert f_\\\\lambda \\\\Vert ^2\\\\Vert f\\\\Vert }{\\\\sqrt{2|\\\\lambda |}}\\\\,.$ Here we have used the identity $f_{\\\\overline{\\\\lambda }} = \\\\overline{f_\\\\lambda }$ .\",\n \"Let $r > 0$ and $t\\\\in (0,\\\\pi )$ such that $\\\\lambda = re^{it}$ .\",\n \"Then $\\\\Vert f_\\\\lambda \\\\Vert ^2&= \\\\int _0^\\\\infty \\\\,\\\\left|e^{i\\\\sqrt{r}e^{it/2}x}\\\\right|^2\\\\,dx + \\\\int _{-\\\\infty }^0\\\\,\\\\left|e^{-i\\\\sqrt{r}e^{i(t+\\\\pi )/2}x}\\\\right|^2\\\\,dx\\\\\\\\&= \\\\int _0^\\\\infty \\\\,e^{-2\\\\sqrt{r}\\\\sin (t/2)x}\\\\,dx + \\\\int _{-\\\\infty }^0\\\\,e^{2\\\\sqrt{r}\\\\cos (t/2)x}\\\\,dx\\\\\\\\&= \\\\frac{\\\\cos (t/2) + \\\\sin (t/2)}{\\\\sqrt{r}\\\\,\\\\sin (t)}\\\\,\\\\le \\\\,\\\\frac{\\\\sqrt{2}}{\\\\sqrt{r}\\\\,\\\\operatorname{Im}(\\\\lambda /r)} = \\\\frac{\\\\sqrt{2|\\\\lambda |}}{|\\\\operatorname{Im}\\\\lambda |}.$ Therefore, $\\\\Vert (A_0 - \\\\lambda )^{-1}\\\\Vert \\\\,\\\\le \\\\,\\\\frac{2}{|\\\\operatorname{Im}\\\\lambda |}\\\\,.$ The same estimate holds for $\\\\lambda \\\\in -$ .\",\n \"Now, assume that $|\\\\operatorname{Im}\\\\lambda | > 2\\\\Vert q\\\\Vert _\\\\infty =2\\\\Vert V\\\\Vert $ .\",\n \"Then $\\\\Vert V(A_0 - \\\\lambda )^{-1}\\\\Vert < 1$ , and hence $A - \\\\lambda = A_0 - \\\\lambda + V = \\\\left(I + V(A_0 - \\\\lambda )^{-1}\\\\right)(A_0 - \\\\lambda )$ is boundedly invertible, which proves ().$\\\\Box $ Second order elliptic operators Let $\\\\Omega \\\\subset \\\\mathbb {R}^n$ be a domain and let $\\\\ell $ be the \\\"formally symmetric\\\" uniformly elliptic second order differential expression $\\\\ell (f)(x):=-\\\\sum _{j,k=1}^n \\\\left( \\\\frac{\\\\partial }{\\\\partial x_j} a_{jk}\\\\frac{\\\\partial f}{\\\\partial x_k}\\\\right)(x)+a(x)f(x),\\\\quad x\\\\in \\\\Omega ,$ with bounded coefficients $a_{jk}\\\\in C^\\\\infty (\\\\Omega )$ satisfying $a_{jk}(x)=\\\\overline{a_{kj}(x)}$ for all $x\\\\in \\\\Omega $ and $j,k=1,\\\\dots ,n$ , the function $a\\\\in L^\\\\infty (\\\\Omega )$ is real valued and $\\\\sum _{j,k=1}^n a_{jk}(x)\\\\xi _j\\\\xi _k\\\\ge C\\\\sum _{k=1}^n\\\\xi _k^2$ holds for some $C>0$ , all $\\\\xi =(\\\\xi _1,\\\\dots ,\\\\xi _n)^\\\\top \\\\in \\\\mathbb {R}^n$ and $x\\\\in \\\\Omega $ .\",\n \"With the differential expression $\\\\ell $ we associate the elliptic differential operator $Tf:=\\\\ell (f),\\\\qquad \\\\operatorname{dom}T=\\\\bigl \\\\lbrace f\\\\in H^1_0(\\\\Omega ):\\\\ell (f) \\\\in L^2(\\\\Omega )\\\\bigr \\\\rbrace ,$ where $H^1_0(\\\\Omega )$ stands for the closure of $C_0^\\\\infty (\\\\Omega )$ in the Sobolev space $H^1(\\\\Omega )$ .\",\n \"It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\\\Omega ),(\\\\cdot ,\\\\cdot ))$ with spectrum semibounded from below by $\\\\operatornamewithlimits{ess\\\\,inf}\\\\,a$ ; cf.\",\n \"[24].\",\n \"Let $w$ be a real valued function such that $w,w^{-1}\\\\in L^\\\\infty (\\\\Omega )$ and each of the sets $\\\\Omega _+:=\\\\bigl \\\\lbrace x\\\\in \\\\Omega : w(x)>0\\\\bigr \\\\rbrace \\\\quad \\\\text{and}\\\\quad \\\\Omega _-:=\\\\bigl \\\\lbrace x\\\\in \\\\Omega : w(x)<0\\\\bigr \\\\rbrace $ has positive Lebesgue measure.\",\n \"We define a second order elliptic differential expression $\\\\mathcal {L}$ with the indefinite weight $w$ by $\\\\mathcal {L}(f)(x):=\\\\frac{1}{w(x)}\\\\,\\\\ell (f)(x),\\\\qquad x\\\\in \\\\Omega .$ The multiplication operator $G_w f=w f$ , $f\\\\in L^2(\\\\Omega )$ , is an isomorphism in $L^2(\\\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\\\in L^2(\\\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\\\int _\\\\Omega f(x)\\\\overline{g(x)}\\\\,w(x)\\\\,dx,\\\\qquad f,g\\\\in L^2(\\\\Omega ).$ The differential operator associated with $\\\\mathcal {L}$ is defined as $Af=\\\\mathcal {L}(f),\\\\qquad \\\\operatorname{dom}A=\\\\bigl \\\\lbrace f\\\\in H^1_0(\\\\Omega ):\\\\mathcal {L}(f) \\\\in L^2(\\\\Omega )\\\\bigr \\\\rbrace .$ Since for $f\\\\in H^1_0(\\\\Omega )$ we have $\\\\ell (f)\\\\in L^2(\\\\Omega )$ if and only if $\\\\mathcal {L}(f)\\\\in L^2(\\\\Omega )$ it follows that $\\\\operatorname{dom}A=\\\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.\",\n \"Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ .\",\n \"In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\\\min \\\\sigma _{\\\\rm ess}(T)\\\\le 0$ holds.\",\n \"This also implies that the domain $\\\\Omega $ is unbounded as otherwise $\\\\sigma _{\\\\rm ess}(T)=\\\\varnothing $ .\",\n \"A discussion of the cases $\\\\sigma _{\\\\rm ess}(T)=\\\\varnothing $ and $\\\\min \\\\sigma _{\\\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].\",\n \"Fix some $\\\\eta >0$ such that $-\\\\eta <\\\\min \\\\sigma (T)$ and define the spaces $\\\\mathcal {H}_s$ , $s\\\\in [0,2]$ , as the domains of the $\\\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\\\eta $ in $L^2(\\\\Omega )$ , $\\\\mathcal {H}_s:=\\\\operatorname{dom}\\\\bigl ((T+\\\\eta )^\\\\frac{s}{2}\\\\bigr ),\\\\qquad s\\\\in [0,2].$ Note that $\\\\mathcal {H}=\\\\mathcal {H}_0$ , $\\\\operatorname{dom}T=\\\\mathcal {H}_2$ and the form domain of $T$ is $\\\\mathcal {H}_1$ .\",\n \"The spaces $\\\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\\\eta $ ; cf.\",\n \"[40].\",\n \"The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.\",\n \"[9].\",\n \"Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ such that $W\\\\mathcal {H}_s\\\\subset \\\\mathcal {H}_s$ holds for some $s\\\\in (0,2]$ .\",\n \"Then $A$ is non-negative over $ \\\\overline{\\\\mathbb {C}} \\\\setminus K_r((-r,r))$ , where $r=\\\\eta \\\\,\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert w^{-1}\\\\Vert _\\\\infty \\\\quad \\\\text{and}\\\\quad \\\\tau _\\\\eta := \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{-n}^n\\\\,(A+\\\\eta G_w^{-1} - it)^{-1} \\\\,dt\\\\,\\\\right\\\\Vert \\\\,<\\\\,\\\\infty .$ Moreover, $\\\\infty $ is not a singular critical point of $A$ .\",\n \"In the next corollary the special case $\\\\Omega =\\\\mathbb {R}^n$ is treated; cf.\",\n \"[9].\",\n \"From now on we assume that $\\\\Omega =\\\\mathbb {R}^n$ and $\\\\Omega _\\\\pm =\\\\lbrace x\\\\in \\\\mathbb {R}^n:\\\\pm w(x)>0\\\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\\\in C^\\\\infty (\\\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.\",\n \"Note that either $\\\\Omega _+$ or $\\\\Omega _-$ is bounded.\",\n \"Corollary 4.4 Suppose that for some $s\\\\in (0,\\\\frac{1}{2})$ the Sobolev spaces $H^s(\\\\Omega _\\\\pm )$ are invariant under multiplication with $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ .\",\n \"Then the indefinite elliptic operator $A$ is non-negative over $\\\\overline{\\\\mathbb {C}} \\\\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .\",\n \"Moreover, $\\\\infty $ is not a singular critical point of $A$ .\",\n \"A sufficient criterion for the invariance of $H^s(\\\\Omega _\\\\pm )$ in the above corollary can be deduced from [28].\",\n \"Corollary 4.5 Suppose that the weight function $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ belongs to some Hölder space $C^{0,\\\\alpha }(\\\\Omega _\\\\pm )$ , $\\\\alpha \\\\in (0,\\\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.\",\n \"Then the claim in Corollary REF holds.\",\n \"of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\\\operatorname{dom}A=\\\\operatorname{dom}T=H^2(\\\\mathbb {R}^n)\\\\qquad \\\\text{and}\\\\qquad \\\\mathcal {H}_s=H^s(\\\\mathbb {R}^n),\\\\qquad s\\\\in [0,2],$ by elliptic regularity and interpolation.\",\n \"Since the Sobolev spaces $H^s(\\\\Omega _\\\\pm )$ are assumed to be invariant under multiplication with the functions $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ for some $s\\\\in (0,\\\\frac{1}{2})$ it follows from [9] that also $H^s(\\\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\\\mathcal {H}_s\\\\subset \\\\mathcal {H}_s$ holds.\",\n \"Furthermore, $G_w$ is uniformly positive in $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\\\ge \\\\operatornamewithlimits{ess\\\\,inf}w^2 \\\\Vert f\\\\Vert ^2$ and $w^{-1}\\\\in L^\\\\infty (\\\\mathbb {R}^n)$ .\",\n \"Now the assertion follows from Theorem REF .$\\\\Box $ Acknowledgements The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.\",\n \"Contact information Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, behrndt@tugraz.at Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.\",\n \"Juni 136, 10623 Berlin, Germany, fmphilipp@gmail.com Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, carsten.trunk@tu-ilmenau.de\"\n ],\n [\n \"Differential operators with indefinite weights\",\n \"In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights.\"\n ],\n [\n \"Indefinite Sturm-Liouville operators\",\n \"We consider Sturm-Liouville differential expressions of the form $\\\\mathcal {L}(f)(x) = \\\\operatorname{sgn}(x)\\\\big (-f^{\\\\prime \\\\prime }(x) + q(x)f(x)\\\\big ),\\\\quad x\\\\in \\\\mathbb {R}\\\\,,$ with a real-valued potential $q\\\\in L^\\\\infty (\\\\mathbb {R})$ and the indefinite weight function $\\\\operatorname{sgn}(\\\\cdot )$ .\",\n \"The corresponding differential operator in $L^2(\\\\mathbb {R})$ is defined by $Af := \\\\mathcal {L}(f), \\\\qquad \\\\operatorname{dom}A := H^2(\\\\mathbb {R}).$ Here $H^2(\\\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.\",\n \"By $(\\\\cdot \\\\,,\\\\cdot )$ and $\\\\Vert \\\\cdot \\\\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\\\mathbb {R})$ .\",\n \"Let $J$ be the operator of multiplication with the function $\\\\operatorname{sgn}(x)$ .\",\n \"This operator is obviously selfadjoint and unitary in $L^2(\\\\mathbb {R})$ .\",\n \"Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\\\prime \\\\prime } + qf,\\\\quad \\\\operatorname{dom}T := H^2(\\\\mathbb {R}),$ is selfadjoint in $L^2(\\\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\\\mathbb {R}),[\\\\cdot \\\\,,\\\\cdot ])$ , where $[f,g] := (Jf,g) = \\\\int _\\\\mathbb {R}\\\\,f(x)\\\\overline{g(x)}\\\\operatorname{sgn}(x)\\\\,dx,\\\\quad f,g\\\\in L^2(\\\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.\",\n \"We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].\",\n \"For this, denote by $m_\\\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\\\pm f_\\\\pm := - f_\\\\pm ^{\\\\prime \\\\prime } + q_\\\\pm f_\\\\pm $ in $L^2(\\\\mathbb {R}^\\\\pm )$ defined on $\\\\operatorname{dom}T_\\\\pm := \\\\lbrace f_\\\\pm \\\\in H^2(\\\\mathbb {R}^\\\\pm ) : f_\\\\pm (0) = 0\\\\rbrace $ , where $q_\\\\pm $ is the restriction of the function $q$ to $\\\\mathbb {R}^\\\\pm $ .\",\n \"The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\\\pm \\\\infty $ , then $m_\\\\pm =\\\\lim _{x\\\\rightarrow \\\\pm \\\\infty }\\\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.\",\n \"Furthermore the following holds.\",\n \"If $m_+ > -m_-$ then $\\\\sigma (A)\\\\setminus \\\\mathbb {R}$ is finite; If $m_+\\\\le -m_-$ then $\\\\sigma (A)\\\\setminus \\\\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .\",\n \"To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.\",\n \"One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].\",\n \"The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .\",\n \"Here only the nontrivial case $\\\\operatornamewithlimits{ess\\\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).\",\n \"Theorem 4.2 Assume that $\\\\operatornamewithlimits{ess\\\\,inf}q<0$ holds.\",\n \"Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\\\overline{\\\\setminus K_r((-d,d)), wherer := 5\\\\Vert q\\\\Vert _\\\\infty ,\\\\quad \\\\text{and}\\\\quad d := -5\\\\,\\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\\\cap \\\\lbrace \\\\lambda \\\\in |\\\\operatorname{Im}\\\\lambda |\\\\le 2\\\\Vert q\\\\Vert _\\\\infty \\\\rbrace .\",\n \"}\\\\begin{proof}The proof of Theorem~\\\\ref {t:SL2} is split into 4 parts.\",\n \"The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\\\ref {t:main1}.\",\n \"In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\\\end{proof}\\\\ \\\\\\\\{\\\\it 1.\\\\ Preparation: } The operator $ A0$,\\\\begin{equation*}A_0 f:=-\\\\operatorname{sgn}(\\\\cdot )f^{\\\\prime \\\\prime },\\\\qquad \\\\operatorname{dom}A_0=H^2(\\\\mathbb {R}),\\\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\\\mathbb {R}$ ),[,])$.\",\n \"Furthermore, we have $ (A0) = $\\\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\\\cite {cn}.\",\n \"Hence $ A0$ satisfies theassumptions in Theorem~\\\\ref {t:main1}.\",\n \"Define $ V$ as\\\\begin{equation*}Vf:=\\\\operatorname{sgn}(\\\\cdot ) qf,\\\\qquad \\\\operatorname{dom}V=L^2(\\\\mathbb {R}),\\\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\\\mathbb {R}$ ),[,])$ with$$\\\\min \\\\sigma (JV) = \\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}\\\\,q(x)\\\\quad \\\\text{and}\\\\quad \\\\Vert V\\\\Vert = \\\\Vert q\\\\Vert _\\\\infty .$$By Theorem \\\\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\\\frac{1+\\\\tau _0}{2}\\\\,\\\\Vert q\\\\Vert _\\\\infty ,\\\\qquad d = -\\\\frac{1+\\\\tau _0}{2}\\\\,\\\\operatornamewithlimits{ess\\\\,inf}_{x\\\\in \\\\mathbb {R}}q(x),$$and$$\\\\tau _0 = \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{1/n}^n\\\\,\\\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\right)\\\\,dt\\\\,\\\\right\\\\Vert .$$Therefore, we have to show\\\\begin{equation}\\\\tau _0\\\\le 9\\\\end{equation}and\\\\begin{equation}\\\\sigma (A)\\\\setminus \\\\mathbb {R}\\\\,\\\\subset \\\\,\\\\lbrace \\\\lambda \\\\in |\\\\operatorname{Im}\\\\lambda |\\\\le 2\\\\Vert q\\\\Vert _\\\\infty \\\\rbrace .\\\\end{equation}$ 2.\",\n \"Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\\\begin{pmatrix} B_+ & 0 \\\\\\\\ 0 & B_- \\\\end{pmatrix}\\\\quad \\\\text{in}\\\\quad L^2(\\\\mathbb {R})=L^2(\\\\mathbb {R}^+)\\\\times L^2(\\\\mathbb {R}^-),$ where $B_\\\\pm f_\\\\pm :=\\\\mp f_\\\\pm ^{\\\\prime \\\\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\\\mathbb {R}^\\\\pm )$ which are defined on $\\\\operatorname{dom}B_\\\\pm = \\\\lbrace f_\\\\pm \\\\in H^2(\\\\mathbb {R}^\\\\pm ):f_\\\\pm (0)=0\\\\rbrace $ .\",\n \"Here and in the following we will denote the restrictions of a function $f$ defined on $\\\\mathbb {R}$ to $\\\\mathbb {R}^\\\\pm $ by $f_\\\\pm $ .\",\n \"For $\\\\lambda = re^{it}$ , $r > 0$ , $t\\\\in [0,2\\\\pi )$ , we set $\\\\sqrt{\\\\lambda } := \\\\sqrt{r}e^{it/2}$ .\",\n \"For $\\\\lambda \\\\in \\\\mathbb {R}$ define the function $f_\\\\lambda (x) :={\\\\left\\\\lbrace \\\\begin{array}{ll}e^{i\\\\sqrt{\\\\lambda }x}, &x> 0,\\\\\\\\e^{-i\\\\sqrt{-\\\\lambda }x}, &x < 0.\\\\end{array}\\\\right.\",\n \"}$ The function $f_\\\\lambda $ is a solution of the equations $\\\\mp f_\\\\pm ^{\\\\prime \\\\prime } = \\\\lambda f_\\\\pm $ in $L^2(\\\\mathbb {R}^\\\\pm )$ .\",\n \"From $\\\\overline{\\\\sqrt{\\\\lambda }} = -\\\\sqrt{\\\\overline{\\\\lambda }}$ it follows that $f_{\\\\overline{\\\\lambda }} = \\\\overline{f_\\\\lambda }$ holds for all $\\\\lambda \\\\in \\\\mathbb {R}$ .\",\n \"Let us now prove that for all $f\\\\in L^2(\\\\mathbb {R})$ and $\\\\lambda \\\\in \\\\mathbb {R}$ we have $(A_0 - \\\\lambda )^{-1}f = (B_0 - \\\\lambda )^{-1}f - \\\\frac{[f,f_{\\\\overline{\\\\lambda }}]}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_\\\\lambda .$ Since $\\\\sigma (A_0)=\\\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\\\lambda \\\\in \\\\mathbb {R}$ .\",\n \"In particular, for $\\\\lambda \\\\in \\\\mathbb {R}$ there exists $g\\\\in \\\\operatorname{dom}B_0$ such that $f=(B_0-\\\\lambda )g$ holds.\",\n \"The right hand side in (REF ) has the form $h:=g-\\\\frac{[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_\\\\lambda $ and we will show that the function $h$ belongs to $\\\\operatorname{dom}A_0=H^2(\\\\mathbb {R})$ .\",\n \"As $g_\\\\pm $ and $f_{\\\\lambda ,\\\\pm }$ are elements of $H^2(\\\\mathbb {R}^\\\\pm )$ the same is true for $h_\\\\pm $ .\",\n \"Moreover, $g$ and $f_\\\\lambda $ are continuous at 0 and so is $h$ .\",\n \"Hence it remains to check that $h^\\\\prime $ is continuous at 0.\",\n \"For this note first that $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]=\\\\int _0^\\\\infty (-g_+^{\\\\prime \\\\prime }-\\\\lambda g_+)\\\\overline{f_{\\\\overline{\\\\lambda },+}}\\\\, dx- \\\\int _{-\\\\infty }^0 (g_-^{\\\\prime \\\\prime }-\\\\lambda g_-)\\\\overline{f_{\\\\overline{\\\\lambda },-}}\\\\, dx$ and since $f_{\\\\overline{\\\\lambda },\\\\pm }$ are solutions of $\\\\mp f_\\\\pm ^{\\\\prime \\\\prime }=\\\\overline{\\\\lambda }f_\\\\pm $ , integration by parts yields $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]=g_+^\\\\prime (0)\\\\overline{f_{\\\\overline{\\\\lambda },+}(0)}-g_+(0) \\\\overline{f^\\\\prime _{\\\\overline{\\\\lambda },+}(0)}-g_-^\\\\prime (0)\\\\overline{f_{\\\\overline{\\\\lambda },-}(0)}+g_-(0) \\\\overline{f^\\\\prime _{\\\\overline{\\\\lambda },-}(0)}.$ As $g\\\\in \\\\operatorname{dom}B_0$ we have $g_\\\\pm (0)=0$ and together with $f_{\\\\overline{\\\\lambda },\\\\pm }(0)=1$ we find $[(B_0-\\\\lambda )g,f_{\\\\overline{\\\\lambda }}]= g_+^\\\\prime (0)-g^\\\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\\\pm ^\\\\prime $ on $\\\\mathbb {R}^\\\\pm $ of the function $h$ from (REF ): $h^\\\\prime _\\\\pm =g^\\\\prime _\\\\pm -\\\\frac{g_+^\\\\prime (0)-g^\\\\prime _-(0)}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}f_{\\\\lambda ,\\\\pm }^\\\\prime $ and as $f^\\\\prime _{\\\\lambda ,+}(0)=i\\\\sqrt{\\\\lambda }$ and $f^\\\\prime _{\\\\lambda ,-}(0)=-i\\\\sqrt{-\\\\lambda }$ we conclude $h^\\\\prime _+(0)-h^\\\\prime _-(0)= \\\\bigl (g_+^\\\\prime (0)-g^\\\\prime _-(0)\\\\bigr )-\\\\frac{g_+^\\\\prime (0)-g^\\\\prime _-(0)}{i(\\\\sqrt{\\\\lambda }+\\\\sqrt{-\\\\lambda })}\\\\bigl (f^\\\\prime _{\\\\lambda ,+}(0)-f^\\\\prime _{\\\\lambda ,-}(0)\\\\bigr )=0,$ that is, $h^\\\\prime $ is continuous at 0 and therefore $h\\\\in \\\\operatorname{dom}A_0$ .\",\n \"Now a straightforward computation shows that $(A_0-\\\\lambda )h=(B_0-\\\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).\",\n \"3.\",\n \"Proof of (): Let $f\\\\in L^2(\\\\mathbb {R})$ and $t > 0$ .\",\n \"Then (REF ) and $\\\\sqrt{it} + \\\\sqrt{-it} = i\\\\sqrt{2t}$ yield $\\\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\\\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\\\\\& + \\\\frac{2}{\\\\sqrt{2t}}\\\\,\\\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\\\in \\\\mathbb {R}^+$ , we have $\\\\big |[f,f_{it}][f_{-it},f]\\\\big |\\\\,\\\\le \\\\,\\\\left(\\\\int _0^\\\\infty \\\\,e^{-x\\\\sqrt{t/2}}g(x)\\\\,dx\\\\right)^2$ and thus for $n\\\\in \\\\mathbb {N}$ $\\\\int _{1/n}^n\\\\,\\\\frac{|[f,f_{it}][f_{-it},f]|}{\\\\sqrt{2t}}\\\\,dt&\\\\le \\\\int _{1/n}^n\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,g(x)g(y)\\\\,\\\\frac{e^{-(x+y)\\\\sqrt{t/2}}}{\\\\sqrt{2t}}\\\\,dy\\\\,dx\\\\,dt\\\\\\\\&= 2\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,\\\\frac{g(x)g(y)}{x+y}\\\\left(e^{-\\\\frac{x+y}{\\\\sqrt{2n}}} - e^{-\\\\frac{\\\\sqrt{n}(x+y)}{\\\\sqrt{2}}}\\\\right)\\\\,dy\\\\,dx\\\\\\\\&\\\\le 2\\\\,\\\\int _0^\\\\infty \\\\,\\\\int _0^\\\\infty \\\\,\\\\frac{g(x)g(y)}{x+y}\\\\,dy\\\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\\\int _{1/n}^n\\\\,\\\\frac{|[f,f_{it}][f_{-it},f]|}{\\\\sqrt{2t}}\\\\,dt\\\\,\\\\le \\\\,2\\\\pi \\\\Vert g\\\\Vert _{L^2(\\\\mathbb {R}^+)}^2\\\\,\\\\le \\\\,4\\\\pi \\\\Vert f\\\\Vert ^2.$ and therefore we have $\\\\frac{1}{\\\\pi }\\\\int _{1/n}^n\\\\,\\\\frac{2}{\\\\sqrt{2t}}\\\\,\\\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\\\,dt\\\\,\\\\le \\\\,8\\\\Vert f\\\\Vert ^2.$ Denote by $E_0$ ($E_\\\\pm $ ) the spectral function of the operator $A_0$ ($B_\\\\pm $ , respectively).\",\n \"Moreover define $J_{n,\\\\pm } := \\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,\\\\left((B_\\\\pm + it)^{-1} + (B_\\\\pm - it)^{-1}\\\\right)\\\\,dt,$ and, in addition, $J_{n,0} := \\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,\\\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\\\right)\\\\,dt.$ As $\\\\pm B_\\\\pm $ is non-negative, it follows that $J_{n,\\\\pm }$ converges strongly to $\\\\pm I_{L^2(\\\\mathbb {R}^\\\\pm )}$ when $n\\\\rightarrow \\\\infty $ , cf.\",\n \"Lemma REF .\",\n \"Moreover, by $(B_\\\\pm + it)^{-1} + (B_\\\\pm - it)^{-1}= 2B_\\\\pm (B_\\\\pm + it)^{-1} (B_\\\\pm - it)^{-1}$ the sequence $\\\\pm J_{n,\\\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\\\mathbb {R}^\\\\pm )$ .\",\n \"This implies $\\\\frac{1}{\\\\pi }\\\\,\\\\int _{1/n}^n\\\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\\\\\&\\\\le \\\\Vert f_+\\\\Vert _{L^2(\\\\mathbb {R}^+)}^2 + \\\\Vert f_-\\\\Vert _{L^2(\\\\mathbb {R}^-)}^2 = \\\\Vert f\\\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\\\,\\\\le \\\\,\\\\Vert f\\\\Vert ^2 + 8\\\\Vert f\\\\Vert ^2 = 9\\\\Vert f\\\\Vert ^2,\\\\quad f\\\\in L^2(\\\\mathbb {R}).$ Hence, as $J = \\\\operatorname{sgn}(\\\\cdot )$ is unitary in $L^2(\\\\mathbb {R})$ , we finally obtain $\\\\Vert J_{n,0}\\\\Vert = \\\\Vert J J_{n,0}\\\\Vert \\\\le 9$ and thus $\\\\tau _0 = \\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\Vert J_{n,0}\\\\Vert \\\\le 9$ .\",\n \"That is, () holds.\",\n \"4.\",\n \"Proof of (): Let $\\\\lambda \\\\in +$ .\",\n \"Using (REF ), for $f\\\\in L^2(\\\\mathbb {R})$ we obtain $\\\\Vert (A_0 - \\\\lambda )^{-1}f\\\\Vert \\\\,\\\\le \\\\,\\\\Vert (B_0 - \\\\lambda )^{-1}f\\\\Vert + \\\\frac{|[f,f_{\\\\overline{\\\\lambda }}]|\\\\,\\\\Vert f_\\\\lambda \\\\Vert }{|\\\\sqrt{\\\\lambda } + \\\\sqrt{-\\\\lambda }|}\\\\,\\\\le \\\\,\\\\frac{\\\\Vert f\\\\Vert }{|\\\\operatorname{Im}\\\\lambda |} + \\\\frac{\\\\Vert f_\\\\lambda \\\\Vert ^2\\\\Vert f\\\\Vert }{\\\\sqrt{2|\\\\lambda |}}\\\\,.$ Here we have used the identity $f_{\\\\overline{\\\\lambda }} = \\\\overline{f_\\\\lambda }$ .\",\n \"Let $r > 0$ and $t\\\\in (0,\\\\pi )$ such that $\\\\lambda = re^{it}$ .\",\n \"Then $\\\\Vert f_\\\\lambda \\\\Vert ^2&= \\\\int _0^\\\\infty \\\\,\\\\left|e^{i\\\\sqrt{r}e^{it/2}x}\\\\right|^2\\\\,dx + \\\\int _{-\\\\infty }^0\\\\,\\\\left|e^{-i\\\\sqrt{r}e^{i(t+\\\\pi )/2}x}\\\\right|^2\\\\,dx\\\\\\\\&= \\\\int _0^\\\\infty \\\\,e^{-2\\\\sqrt{r}\\\\sin (t/2)x}\\\\,dx + \\\\int _{-\\\\infty }^0\\\\,e^{2\\\\sqrt{r}\\\\cos (t/2)x}\\\\,dx\\\\\\\\&= \\\\frac{\\\\cos (t/2) + \\\\sin (t/2)}{\\\\sqrt{r}\\\\,\\\\sin (t)}\\\\,\\\\le \\\\,\\\\frac{\\\\sqrt{2}}{\\\\sqrt{r}\\\\,\\\\operatorname{Im}(\\\\lambda /r)} = \\\\frac{\\\\sqrt{2|\\\\lambda |}}{|\\\\operatorname{Im}\\\\lambda |}.$ Therefore, $\\\\Vert (A_0 - \\\\lambda )^{-1}\\\\Vert \\\\,\\\\le \\\\,\\\\frac{2}{|\\\\operatorname{Im}\\\\lambda |}\\\\,.$ The same estimate holds for $\\\\lambda \\\\in -$ .\",\n \"Now, assume that $|\\\\operatorname{Im}\\\\lambda | > 2\\\\Vert q\\\\Vert _\\\\infty =2\\\\Vert V\\\\Vert $ .\",\n \"Then $\\\\Vert V(A_0 - \\\\lambda )^{-1}\\\\Vert < 1$ , and hence $A - \\\\lambda = A_0 - \\\\lambda + V = \\\\left(I + V(A_0 - \\\\lambda )^{-1}\\\\right)(A_0 - \\\\lambda )$ is boundedly invertible, which proves ().$\\\\Box $\"\n ],\n [\n \"Second order elliptic operators\",\n \"Let $\\\\Omega \\\\subset \\\\mathbb {R}^n$ be a domain and let $\\\\ell $ be the \\\"formally symmetric\\\" uniformly elliptic second order differential expression $\\\\ell (f)(x):=-\\\\sum _{j,k=1}^n \\\\left( \\\\frac{\\\\partial }{\\\\partial x_j} a_{jk}\\\\frac{\\\\partial f}{\\\\partial x_k}\\\\right)(x)+a(x)f(x),\\\\quad x\\\\in \\\\Omega ,$ with bounded coefficients $a_{jk}\\\\in C^\\\\infty (\\\\Omega )$ satisfying $a_{jk}(x)=\\\\overline{a_{kj}(x)}$ for all $x\\\\in \\\\Omega $ and $j,k=1,\\\\dots ,n$ , the function $a\\\\in L^\\\\infty (\\\\Omega )$ is real valued and $\\\\sum _{j,k=1}^n a_{jk}(x)\\\\xi _j\\\\xi _k\\\\ge C\\\\sum _{k=1}^n\\\\xi _k^2$ holds for some $C>0$ , all $\\\\xi =(\\\\xi _1,\\\\dots ,\\\\xi _n)^\\\\top \\\\in \\\\mathbb {R}^n$ and $x\\\\in \\\\Omega $ .\",\n \"With the differential expression $\\\\ell $ we associate the elliptic differential operator $Tf:=\\\\ell (f),\\\\qquad \\\\operatorname{dom}T=\\\\bigl \\\\lbrace f\\\\in H^1_0(\\\\Omega ):\\\\ell (f) \\\\in L^2(\\\\Omega )\\\\bigr \\\\rbrace ,$ where $H^1_0(\\\\Omega )$ stands for the closure of $C_0^\\\\infty (\\\\Omega )$ in the Sobolev space $H^1(\\\\Omega )$ .\",\n \"It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\\\Omega ),(\\\\cdot ,\\\\cdot ))$ with spectrum semibounded from below by $\\\\operatornamewithlimits{ess\\\\,inf}\\\\,a$ ; cf.\",\n \"[24].\",\n \"Let $w$ be a real valued function such that $w,w^{-1}\\\\in L^\\\\infty (\\\\Omega )$ and each of the sets $\\\\Omega _+:=\\\\bigl \\\\lbrace x\\\\in \\\\Omega : w(x)>0\\\\bigr \\\\rbrace \\\\quad \\\\text{and}\\\\quad \\\\Omega _-:=\\\\bigl \\\\lbrace x\\\\in \\\\Omega : w(x)<0\\\\bigr \\\\rbrace $ has positive Lebesgue measure.\",\n \"We define a second order elliptic differential expression $\\\\mathcal {L}$ with the indefinite weight $w$ by $\\\\mathcal {L}(f)(x):=\\\\frac{1}{w(x)}\\\\,\\\\ell (f)(x),\\\\qquad x\\\\in \\\\Omega .$ The multiplication operator $G_w f=w f$ , $f\\\\in L^2(\\\\Omega )$ , is an isomorphism in $L^2(\\\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\\\in L^2(\\\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\\\int _\\\\Omega f(x)\\\\overline{g(x)}\\\\,w(x)\\\\,dx,\\\\qquad f,g\\\\in L^2(\\\\Omega ).$ The differential operator associated with $\\\\mathcal {L}$ is defined as $Af=\\\\mathcal {L}(f),\\\\qquad \\\\operatorname{dom}A=\\\\bigl \\\\lbrace f\\\\in H^1_0(\\\\Omega ):\\\\mathcal {L}(f) \\\\in L^2(\\\\Omega )\\\\bigr \\\\rbrace .$ Since for $f\\\\in H^1_0(\\\\Omega )$ we have $\\\\ell (f)\\\\in L^2(\\\\Omega )$ if and only if $\\\\mathcal {L}(f)\\\\in L^2(\\\\Omega )$ it follows that $\\\\operatorname{dom}A=\\\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.\",\n \"Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ .\",\n \"In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\\\min \\\\sigma _{\\\\rm ess}(T)\\\\le 0$ holds.\",\n \"This also implies that the domain $\\\\Omega $ is unbounded as otherwise $\\\\sigma _{\\\\rm ess}(T)=\\\\varnothing $ .\",\n \"A discussion of the cases $\\\\sigma _{\\\\rm ess}(T)=\\\\varnothing $ and $\\\\min \\\\sigma _{\\\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].\",\n \"Fix some $\\\\eta >0$ such that $-\\\\eta <\\\\min \\\\sigma (T)$ and define the spaces $\\\\mathcal {H}_s$ , $s\\\\in [0,2]$ , as the domains of the $\\\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\\\eta $ in $L^2(\\\\Omega )$ , $\\\\mathcal {H}_s:=\\\\operatorname{dom}\\\\bigl ((T+\\\\eta )^\\\\frac{s}{2}\\\\bigr ),\\\\qquad s\\\\in [0,2].$ Note that $\\\\mathcal {H}=\\\\mathcal {H}_0$ , $\\\\operatorname{dom}T=\\\\mathcal {H}_2$ and the form domain of $T$ is $\\\\mathcal {H}_1$ .\",\n \"The spaces $\\\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\\\eta $ ; cf.\",\n \"[40].\",\n \"The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.\",\n \"[9].\",\n \"Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ such that $W\\\\mathcal {H}_s\\\\subset \\\\mathcal {H}_s$ holds for some $s\\\\in (0,2]$ .\",\n \"Then $A$ is non-negative over $ \\\\overline{\\\\mathbb {C}} \\\\setminus K_r((-r,r))$ , where $r=\\\\eta \\\\,\\\\frac{1+\\\\tau _\\\\eta }{2}\\\\,\\\\Vert w^{-1}\\\\Vert _\\\\infty \\\\quad \\\\text{and}\\\\quad \\\\tau _\\\\eta := \\\\frac{1}{\\\\pi }\\\\,\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\,\\\\left\\\\Vert \\\\int _{-n}^n\\\\,(A+\\\\eta G_w^{-1} - it)^{-1} \\\\,dt\\\\,\\\\right\\\\Vert \\\\,<\\\\,\\\\infty .$ Moreover, $\\\\infty $ is not a singular critical point of $A$ .\",\n \"In the next corollary the special case $\\\\Omega =\\\\mathbb {R}^n$ is treated; cf.\",\n \"[9].\",\n \"From now on we assume that $\\\\Omega =\\\\mathbb {R}^n$ and $\\\\Omega _\\\\pm =\\\\lbrace x\\\\in \\\\mathbb {R}^n:\\\\pm w(x)>0\\\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\\\in C^\\\\infty (\\\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.\",\n \"Note that either $\\\\Omega _+$ or $\\\\Omega _-$ is bounded.\",\n \"Corollary 4.4 Suppose that for some $s\\\\in (0,\\\\frac{1}{2})$ the Sobolev spaces $H^s(\\\\Omega _\\\\pm )$ are invariant under multiplication with $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ .\",\n \"Then the indefinite elliptic operator $A$ is non-negative over $\\\\overline{\\\\mathbb {C}} \\\\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .\",\n \"Moreover, $\\\\infty $ is not a singular critical point of $A$ .\",\n \"A sufficient criterion for the invariance of $H^s(\\\\Omega _\\\\pm )$ in the above corollary can be deduced from [28].\",\n \"Corollary 4.5 Suppose that the weight function $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ belongs to some Hölder space $C^{0,\\\\alpha }(\\\\Omega _\\\\pm )$ , $\\\\alpha \\\\in (0,\\\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.\",\n \"Then the claim in Corollary REF holds.\",\n \"of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\\\operatorname{dom}A=\\\\operatorname{dom}T=H^2(\\\\mathbb {R}^n)\\\\qquad \\\\text{and}\\\\qquad \\\\mathcal {H}_s=H^s(\\\\mathbb {R}^n),\\\\qquad s\\\\in [0,2],$ by elliptic regularity and interpolation.\",\n \"Since the Sobolev spaces $H^s(\\\\Omega _\\\\pm )$ are assumed to be invariant under multiplication with the functions $w\\\\!\\\\upharpoonright _{\\\\Omega _\\\\pm }$ for some $s\\\\in (0,\\\\frac{1}{2})$ it follows from [9] that also $H^s(\\\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\\\mathcal {H}_s\\\\subset \\\\mathcal {H}_s$ holds.\",\n \"Furthermore, $G_w$ is uniformly positive in $(L^2(\\\\Omega ),[\\\\cdot ,\\\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\\\ge \\\\operatornamewithlimits{ess\\\\,inf}w^2 \\\\Vert f\\\\Vert ^2$ and $w^{-1}\\\\in L^\\\\infty (\\\\mathbb {R}^n)$ .\",\n \"Now the assertion follows from Theorem REF .$\\\\Box $\"\n ],\n [\n \"Acknowledgements\",\n \"The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.\",\n \"Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, behrndt@tugraz.at Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.\",\n \"Juni 136, 10623 Berlin, Germany, fmphilipp@gmail.com Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, carsten.trunk@tu-ilmenau.de The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.\",\n \"Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, behrndt@tugraz.at Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.\",\n \"Juni 136, 10623 Berlin, Germany, fmphilipp@gmail.com Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, carsten.trunk@tu-ilmenau.de\"\n 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[
"A Framework for FFT-based Homogenization on Anisotropic Lattices"
],
[
"Abstract In order to take structural anisotropies of a given composite and different shapes of its unit cell into account, we generalize the Basic Scheme in Homogenization by Moulinec and Suquet to arbitrary sampling lattices and tilings of the d-dimensional Euclidean space.",
"Employing a Fourier transform on arbitrary lattices, which generate sampling patterns in the unit cell of interest, we derive a generalization of this scheme.",
"In several cases, this Fourier transform is of lower dimension than the space itself; for many lattices it even reduces to a one-dimensional Fourier transform having the same leading coefficient as the fastest Fourier transform implementation available.",
"We illustrate the generalized Basic Scheme on an anisotropic laminate and a generalized ellipsoidal Hashin structure.",
"For both we derive an analytical solution to the elasticity problem, in two- and three dimensions, respectively.",
"We then illustrate the possibilities of choosing a pattern.",
"Compared to classical grids this introduces both a reduction of computation time and a reduced error of the numerical method.",
"It also allows for anisotropic subsampling, i.e.",
"choosing a sub lattice of a pixel or voxel grid based on anisotropy information of the material at hand."
],
[
"Introduction",
"Modern materials are often composites of multiple components which are designed to obtain overall properties like high durability, flexibility or stiffness.",
"These inhomogeneities are usually small in comparison to the overall structure of the material or tool.",
"Therefore it is computationally beneficial and sometimes even necessary to replace the inhomogeneous material by a homogeneous one having the same macroscopic properties, called homogenization.",
"The underlying assumption is that the microstructure can be represented by a reference volume that can be repeated periodically to generate the geometry.",
"While many of these microstructures show macroscopically isotropic behavior there are also composites that have one or multiple predominant directions.",
"The classical algorithm to solve such homogenization problems of periodic microstructures on regular grids was proposed by Moulinec and Suquet [20], [21].",
"This algorithm is also called the Basic Scheme and has evoked many enhancements and modifications.",
"Amongst them are different discretization methods of the differential operator [30], [25], [24], adaptions to composites with infinite contrast, e.g.",
"porous media, [18], [24], incorporation of additional information about the geometry [12] and the solution of homogenization problems of higher order [27].",
"All of them have in common that they are formulated on regular tensor product grids, i.e.",
"they make use of the commonly known multidimensional Fast Fourier Transform (FFT).",
"In some cases it is not possible to rotate the representative volume element without violating the periodicity condition of the microstructure and then a change of the discretization grid can remedy this.",
"Galipeau and Castañeda [10], [9], for example, construct a periodic laminate structure of elastomers where each of the two phases consists of aligned elongated particles of a magnetic material.",
"In this material, the two phases differ in orientation and do not face into the direction of lamination nor orthogonal to it.",
"Lahellec, et. al.",
"[15] consider a multi-particle problem where they have an evolving computational grid.",
"The basis vectors of this grid depend on the macroscopic velocity of a Newtonian fluid and they hint that the grid for the FFT does not have to be a rectangular one without elaborating this point.",
"In both cases it might be beneficial to consider a more general sampling, i.e.",
"sampling on anisotropic lattices.",
"Besides the theory of a discrete Fourier transform (DFT) on abelian groups, also known as generalized Fourier transform [1], the DFT has also been generalized to arbitrary sampling lattices, e.g.",
"in order to derive periodic wavelets [16], [6] and a corresponding fast Fourier and fast wavelet transform [2].",
"The computational complexity on these lattices even stays the same as on the usual rectangular or pixel grid.",
"Furthermore using the theory of rank-1-lattices, Kämmerer et al.",
"[14], [13] and Potts and Volkmer [22] derive several adaptive schemes to approximate both a certain set of frequencies as well as a set of sampling points and derive approximation errors for functions of certain smoothness.",
"This also includes a constructive derivation of the vector that generates the lattice.",
"For these special lattices, the Fourier transform even in high-dimensional space reduces to a one-dimensional Fourier transform and hence reducing both the organization of the sampled data and the computational cost to compute the FFT.",
"The theory of rank-1-lattices therefore both allows for directly taking known anisotropic properties of a function into account and thereby reducing the necessary number of sample values or measurements by adapting the lattice.",
"Furthermore it also reduces the computational cost or data organization overhead due to the reduction from a high-dimensional FFT to a one-dimensional one.",
"In this paper we generalize the Basic Scheme by Moulinec and Suquet to arbitrary anisotropic periodic lattices.",
"This introduces the possibility to prefer directions other than the coordinate axes in the reference volume and hence in the solution.",
"This allows for aligning the basis functions with the dominant orientations of the geometry and controlling the refinement in these directions.",
"This generalization of the Basic Scheme to arbitrary anisotropic sampling lattices introduces the form of the grid as a algorithmic parameter without additional computational costs.",
"For a special set of rank-1-lattices, after sampling in a high-dimensional space, the computation of the fast Fourier transform even reduces to a one-dimensional FFT.",
"Therefore, additionally to the new possibility of choosing directions of preference, one can also choose these to reduce the computational efforts.",
"The remainder of the paper is organized as follows.",
"In Section we establish the preliminaries regarding the parametrization, properties of anisotropic lattices and their patterns on the unit cube.",
"Further, we introduce the FFT on such patterns, where the usual tensor product grid is a special case.",
"Exemplary for a homogenization problem we introduce the periodic equations of quasi-static elasticity in Section and elaborate on the unmodified Basic Scheme how to generalize it.",
"Based on this we explain the difference between making a coordinate transformation and choosing a lattice adapted to the geometry of the lattice.",
"In Section we generalize two known geometries to an anisotropic setting: the laminate structure and the Hashin structure that serves as the main analytical example for this work.",
"Both are anisotropic structures within isotropic material laws that provide an analytic solution for the strain field and the effective matrix.",
"This allows us to study effect of the pattern orientation on the solution and the effective properties in Section ."
],
[
"Preliminaries",
"Throughout this paper we will employ the following notation: The symbols $a\\in \\mathbb {C}$ , $\\mathbf {a}\\in \\mathbb {C}^d$ and $\\mathbf {A}\\in \\mathbb {C}^{d\\times d}$ denote scalars, vectors, and matrices, respectively.",
"The only exception from this are $f,g,h$ which are reserved for functions.",
"We denote the inner product of two vectors by $\\mathbf {a}^\\mathrm {T}\\mathbf {b} \\sum _i a_ib_i$ and reserve the symbol $\\langle \\cdot ,\\cdot \\rangle $ for inner products of two functions or two generalized sequences, respectively.",
"For a complex number $a = b+{\\mathrm {i}}c$ , $b,c\\in \\mathbb {R}$ , we denote the complex conjugate by $\\overline{a} b+{\\mathrm {i}}c$ .",
"Usually, we are concerned with $d$ -dimensional data, where $d=2,3$ , but the theory can also be written in arbitrary dimensions.",
"Sets are denoted by capital case calligraphic letters, e.g.",
"$\\mathcal {P}(\\cdot )$ or $\\mathcal {G}(\\cdot )$ and the same for the Fourier transform $\\mathcal {F}(\\cdot )$ ; all of these might depend on a scalar $n$ or matrix $\\mathbf {M}$ given in brackets.",
"We denote second-order tensors by small Greek letters as $\\lambda ,\\epsilon $ with entries $\\lambda _{ij}$ are indexed again by scalars $i,j$ and similarly we denote fourth-order tensors by capital calligraphic letters, where $\\mathcal {C}$ is the most prominent one.",
"Finally, constants like Euler's number ${\\,\\mathrm {e}}$ or the imaginary unit ${\\mathrm {i}}$ , i.e.",
"${\\mathrm {i}}^2 = -1$ , are set upright."
],
[
"Arbitrary patterns and the Fourier transform",
"The space of functions we are concerned with is the Hilbert space $L^2(d)$ of (equivalence classes of) square integrable functions on the $d$ -dimensional torus $[-\\pi ,\\pi )^d$ with inner product $\\langle f,g \\rangle =\\frac{1}{(2\\pi )^d}\\int _{d}f(\\mathbf {x})\\overline{g(\\mathbf {x})}\\,\\mathrm {d}\\mathbf {x},\\qquad f,g \\in L^2(d)\\text{.",
"}$ In several cases, the functions of interest are tensor-valued.",
"For these functions, we take the tensor product of the Hilbert space, e.g.",
"$L^2(\\mathbb {T}^d)^{n\\times n}$ for the space of functions $f\\colon \\mathbb {T}^d \\rightarrow \\mathbb {C}^{n\\times n}$ that have values being $n\\times n$ -dimensional matrices.",
"The following Fourier transform can be generalized to these tensor product spaces by performing the operations element wise.",
"We restrict the following preliminaries of this subsection therefore to the case of $L^2(\\mathbb {T}^d)$ .",
"Every function $f \\in L^2(d)$ can be written in its Fourier series representation $f(\\mathbf {x})= \\sum _{\\mathbf {k} \\in \\mathbb {Z}^d} c_{\\mathbf {k}}(f){\\,\\mathrm {e}}^{{\\mathrm {i}}\\mathbf {k}^\\mathrm {T}\\mathbf {x}},$ introducing the the multivariate Fourier coefficients $c_{\\mathbf {k}}(f) = \\langle f,{\\,\\mathrm {e}}^{{\\mathrm {i}}\\mathbf {k}^\\mathrm {T}\\circ }\\rangle $ , $\\mathbf {k}\\in \\mathbb {Z}^d$ .",
"The equality in (REF ) is meant in $L^2(d)$ sense.",
"We denote by $\\mathbf {c}(f)= \\bigl \\lbrace (c_{\\mathbf {k}}(f)\\bigr \\rbrace _{\\mathbf {k}\\in \\mathbb {Z}^d}\\in \\ell ^2(\\mathbb {Z}^d)$ the generalized sequences which form a Hilbert space with the inner product $\\langle \\mathbf {c},\\mathbf {d}\\rangle = \\sum _{\\mathbf {k} \\in \\mathbb {Z}^d}c_{\\mathbf {k}}\\overline{d_{\\mathbf {k}}},\\qquad \\mathbf {c},\\mathbf {d}\\in \\ell ^2(\\mathbb {Z}^d)\\text{.",
"}$ The Parseval equation reads $\\langle f, g \\rangle = \\langle \\mathbf {c}(f),\\mathbf {c}(g) \\rangle = \\sum _{\\mathbf {k} \\in \\mathbb {Z}^d} c_{\\mathbf {k}}(f) \\overline{c_{\\mathbf {k}}(g)}\\text{.",
"}$ For any regular matrix $\\mathbf {M} \\in \\mathbb {Z}^{d\\times d}$ , we define the congruence relation for $\\mathbf {h},\\mathbf {k} \\in \\mathbb {Z}^d$ with respect to $\\mathbf {M}$ by $\\mathbf {h} \\equiv \\mathbf {k} \\bmod \\unknown.",
"\\mathbf {M}\\Leftrightarrow \\exists \\,\\mathbf {z} \\in \\mathbb {Z}^d\\colon \\mathbf {k} = \\mathbf {h} + \\mathbf {M}\\mathbf {z}\\text{.",
"}$ We define the lattice $\\Lambda (\\mathbf {M}) \\mathbf {M}^{-1}\\mathbb {Z}^d= \\lbrace \\mathbf {y}\\in \\mathbb {R}^d : \\mathbf {M}\\mathbf {y} \\in \\mathbb {Z}^d\\rbrace ,$ and the pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ as any set of congruence representant of the lattice with respect to $\\bmod \\ 1$ , e.g.",
"$ \\Lambda (\\mathbf {M})\\cap [0,1)^d$ or $\\Lambda (\\mathbf {M})\\cap \\bigl [-\\tfrac{1}{2},\\tfrac{1}{2}\\bigr )^d$ .",
"For the rest of the paper we will refer to the set of congruence class representants in the symmetric unit cube $\\bigl [-\\tfrac{1}{2},\\tfrac{1}{2}\\bigr )^d$ .",
"The generating set $\\operatorname{\\mathcal {G}}(\\mathbf {M})$ is defined by $\\operatorname{\\mathcal {G}}(\\mathbf {M})\\mathbf {M}\\operatorname{\\mathcal {P}}(\\mathbf {M})$ for any pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ .",
"For both, the number of elements is given by ${\\operatorname{\\mathcal {P}}(\\mathbf {M})}={\\operatorname{\\mathcal {G}}(\\mathbf {M})}={\\det {\\mathbf {M}}}m,$ which follows directly from [7].",
"Finally for any factorization $\\mathbf {M} = \\mathbf {J}\\mathbf {N}$ of an integer matrix $\\mathbf {M}\\in \\mathbb {Z}^{d\\times d}$ into two integer matrices $\\mathbf {J},\\mathbf {N}\\in \\mathbb {Z}^d$ , we have $\\mathbf {x}\\in \\Lambda (\\mathbf {N}) \\Rightarrow \\mathbf {N}\\mathbf {x} \\in \\mathbb {Z}^d\\Rightarrow \\mathbf {J}\\mathbf {N}\\mathbf {x}\\in \\mathbb {Z}^d,$ and hence $\\Lambda (\\mathbf {N})\\subset \\Lambda (\\mathbf {M})$ .",
"By construction of the pattern, we directly obtain $\\operatorname{\\mathcal {P}}(\\mathbf {N}) \\subset \\operatorname{\\mathcal {P}}(\\mathbf {M})$ ; see [16] and [2] for a more general introduction.",
"We call the smaller pattern $\\operatorname{\\mathcal {P}}(\\mathbf {N})$ a subpattern of $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ .",
"Note that this is not commutative with respect to $\\mathbf {J}$ and $\\mathbf {N}$ .",
"Looking at the generating sets for the decomposition $\\mathbf {M} = \\mathbf {JN}$ we have $\\operatorname{\\mathcal {G}}(\\mathbf {N}^\\mathrm {T}) \\subset \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ .",
"A subpattern of a tensor product grid, the so called quincunx pattern, is shown in Fig.",
"REF , left."
],
[
"A fast Fourier transform on patterns",
"The discrete Fourier transform on the pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ is defined [8] by $\\mathcal {F}(\\mathbf {M})\\frac{1}{m}\\Bigl ({\\,\\mathrm {e}}^{- 2\\pi {\\mathrm {i}}\\mathbf {h}^\\mathrm {T}\\mathbf {y}}\\Bigr )_{\\mathbf {h} \\in \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T}),\\,\\mathbf {y} \\in \\operatorname{\\mathcal {P}}(\\mathbf {M})},$ where $\\mathbf {h}\\in \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ indicate the rows and $\\mathbf {y} \\in \\operatorname{\\mathcal {P}}(\\mathbf {M})$ indicate the columns of the Fourier matrix $\\mathcal {F}(\\mathbf {M})$ .",
"This discrete Fourier transform was also investigated in [2], [16].",
"For both the pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ and the generating set $\\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ , an arbitrary but fixed ordering has to be chosen.",
"The discrete Fourier transform on $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ is defined for a vector $\\mathbf {a} = (a_{\\mathbf {y}})_{\\mathbf {y}\\in \\operatorname{\\mathcal {P}}(\\mathbf {M})}\\in \\mathbb {C}^m$ arranged in the same ordering as the columns in (REF ) by $\\mathbf {\\hat{a}} = (\\hat{a}_{\\mathbf {h}})_{\\mathbf {h}\\in \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})}= \\mathcal {F}(\\mathbf {M})\\mathbf {a},$ where the resulting vector $\\mathbf {\\hat{a}}$ is ordered as the columns of $\\mathcal {F}(\\mathbf {M})$ in (REF ).",
"For a diagonal matrix $\\mathbf {M} = \\operatorname{diag}(n,n)\\in \\mathbb {N}^{2\\times 2}$ having the same entry $n\\in \\mathbb {N}$ on both diagonal entries, the pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ is the set $\\bigl (y_1,y_2)^\\mathrm {T}, y_1,y_2\\in \\frac{1}{m}\\lbrace 0,\\ldots ,m-1\\rbrace $ .",
"The generating set $\\operatorname{\\mathcal {G}}(\\mathbf {M})$ then reads $\\lbrace \\mathbf {k}\\in \\mathbb {Z}^2 : 0\\le k_j \\le m-1, j=1,2\\rbrace $ .",
"Both have $m = n^2$ elements.",
"The Fourier transform for this diagonal matrix is just the usual 2D DFT.",
"Figure: By choosing 𝐌 1 =𝐉𝐍=1-11144-44\\mathbf {M}_1 = \\mathbf {JN} = \\bigl ( {\\begin{matrix}1&-1\\\\1&1\\end{matrix}}\\bigr )\\bigl ({\\begin{matrix}4&4\\\\-4&4\\end{matrix}}\\bigr ) we obtain a usual rectangular grid pattern error(𝐌 1 )\\operatorname{\\mathcal {P}}(\\mathbf {M}_1) (left) of a diagonal matrix and its sub pattern error(𝐍\\operatorname{\\mathcal {P}}(\\mathbf {N} (dark), a quincux pattern.",
"We can prefer certain directions like for pattern error(𝐌 2 )\\operatorname{\\mathcal {P}}(\\mathbf {M}_2), 𝐌 2 =8-427\\mathbf {M}_2= \\bigl ({\\begin{matrix} 8&-4\\\\2&7\\end{matrix}}\\bigr ), (middle).",
"Certain patterns, like error(𝐌 3 )\\operatorname{\\mathcal {P}}(\\mathbf {M}_3), 𝐌 3 =8-108\\mathbf {M}_3= \\bigl ({\\begin{matrix}8&-1\\\\0&8\\end{matrix}}\\bigr ), (right) are even generated by only one generating vector.",
"Note that all matrices have the same determinant and hence the patterns have the same number of points.Fig.",
"REF illustrates that the Fourier transform (REF ) generalizes the usual discretization on a pixel grid and enables to prefer certain directions by choosing different patterns having the same number of points.",
"To efficiently implement the Fourier transform (REF ) on an arbitrary pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M)}$ for a regular integer matrix $\\mathbf {M}\\in \\mathbb {Z}^{d\\times d}$ , we have to fix a certain order of the elements therein.",
"Following the construction in [2], we use the Smith normal form $\\mathbf {M} = \\mathbf {QER}$ , where $\\mathbf {Q},\\mathbf {R}$ are of determinant 1 and $\\mathbf {E} = \\operatorname{diag}(e_1,\\ldots ,e_d)$ is the diagonal matrix of elementary divisors where $e_j$ is a divisor of $e_{j+1}$ , $j=1,\\ldots ,d-1$ .",
"We further denote by $d_{\\mathbf {M}}\\vert \\lbrace j : e_j>1\\rbrace \\vert $ the dimension of the pattern.",
"For the special case, that $d_{\\mathbf {M}}=1$ , the lattice is also called rank-1-lattice.",
"Such a lattice is shown in Fig.",
"REF (right) Introducing the pattern basis vector(s) $\\mathbf {y}_j \\frac{1}{e_{d+d_{\\mathbf {M}}+j}}\\mathbf {e}_{d-d_{\\mathbf {M}}+j},\\qquad j=1,\\ldots ,d_{\\mathbf {M}},$ where $\\mathbf {e}_j$ denotes the $j$ th unit vector, we obtain a basis for the pattern.",
"Hence we can write $\\mathbf {y} = \\sum _{j=1}^{d_{\\mathbf {M}}}\\lambda _j\\mathbf {y}_j,\\qquad \\lambda _j\\in \\lbrace 0,\\ldots ,e_j-1\\rbrace ,\\ j=1,\\ldots ,d_{\\mathbf {M}},$ where the summation is meant on the congruence classes, i.e.",
"with respect to $\\bmod \\ 1$ onto the pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ , we obtain a unique addressing for each pattern point using the coefficients $\\lambda _1,\\ldots ,\\lambda _{d_{\\mathbf {M}}}$ .",
"With the lexicographical ordering of the vectors $(0,\\ldots ,0)^\\mathrm {T},\\ldots ,$ $(\\lambda _1-1,\\ldots ,\\lambda _{d_{\\mathbf {M}}}-1)$ one not only obtains an array representation of any coefficient vector $\\mathbf {a} = \\bigl (a_{\\mathbf {y}}\\bigr )_{\\mathbf {y}\\in \\operatorname{\\mathcal {P}}(\\mathbf {M})}= \\bigl (a_{\\lambda _1,\\ldots ,\\lambda _{d_{\\mathbf {M}}}}\\bigr )_{\\lambda _1=0,\\ldots ,\\lambda _{d_{\\mathbf {M}}}=0}^{e_{d-d_{\\mathbf {M}}+1}-1,\\ldots ,e_d-1},$ but similarly also for any vector $\\hat{\\mathbf {a}}$ corresponding to the generating set using the generating set basis vector(s) $\\mathbf {h}_j \\mathbf {M}^\\mathrm {T}\\tilde{\\mathbf {y}}_j = \\mathbf {R}^\\mathrm {T}\\mathbf {e}_j,\\qquad j=1,\\ldots ,d_{\\mathbf {M}},$ where $\\tilde{\\mathbf {y}}$ denotes the basis vector(s) of $\\operatorname{\\mathcal {P}}(\\mathbf {M}^\\mathrm {T})$ constructed as above.",
"Note that the pattern dimension $d_{\\mathbf {M}}=d_{\\mathbf {M}^\\mathrm {T}}$ and the elementary divisors $e_j$ are identical for the patterns $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ and $\\operatorname{\\mathcal {P}}(\\mathbf {M}^\\mathrm {T})$ .",
"With these fixed orderings of the vector entries of $\\mathbf {a}$ and $\\hat{\\mathbf {a}}$ , the Fourier transform (REF ) can be computed using an ordinary $d_{\\mathbf {M}}$ -dimensional Fourier transform even having the same leading coefficient in its complexity of $\\mathcal {O}(m\\log m)$ [2].",
"Note that for rank-1-lattices, the Fourier transform on the pattern even reduces to a one-dimensional FFT for patterns in 2, 3 or even more dimensions.",
"Let $f\\in L_2(\\mathbb {T}^d)$ denote a square integrable function on the torus, such that its Fourier series (REF ) converges absolutely, i.e.",
"$\\sum _{\\mathbf {k}\\in \\mathbb {Z}^d} {c_{\\mathbf {k}}((f))} < \\infty .$ Sampling $f$ at the points given by pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ of a regular matrix $\\mathbf {M}\\in \\mathbb {Z}^{d\\times d}$ , i.e.",
"$a_{\\mathbf {y}} f(2\\pi \\mathbf {y})$ , $\\mathbf {y}\\in \\operatorname{\\mathcal {P}}(\\mathbf {M})$ , and performing a discrete Fourier transform we obtain the discrete Fourier coefficients $c_{\\mathbf {h}}^{\\mathbf {M}}(f) = \\hat{a}_{\\mathbf {h}}$ , $\\mathbf {h}\\in \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ , where $\\mathbf {\\hat{a}} = \\mathcal {F}(\\mathbf {M})\\mathbf {a}$ .",
"A relation between the Fourier coefficients $c_{\\mathbf {k}}(f)$ and the discrete Fourier coefficients $c_{\\mathbf {h}}^{\\mathbf {M}}(f)$ is given by the following Lemma.",
"Lemma 2.1 Let $ f\\in L^2(d) $ with absolutely convergent Fourier series and the regular matrix $\\mathbf {M}\\in \\mathbb {Z}^{d\\times d}$ be given.",
"Then the discrete Fourier coefficients $ c_{\\mathbf {h}}^{\\mathbf {M}}(f) $ , $\\mathbf {h}\\in \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ , fulfill the relation $c_{\\mathbf {h}}^{\\mathbf {M}}(f)=\\sum _{\\mathbf {z}\\in \\mathbb {Z}^d} c_{\\mathbf {h}+\\mathbf {M}^\\mathrm {T}\\mathbf {z}}(f),\\qquad \\mathbf {h}\\in \\operatorname{\\mathcal {G}}(\\mathbf {M)^\\mathrm {T}}\\text{.",
"}$ For a proof, see [5].",
"This Lemma is also called Aliasing formula and can be interpreted as follows: If the Fourier coefficients $c_{\\mathbf {k}}(f)$ of $f$ decay slowly along a certain direction $\\mathbf {h}_j$ being a basis vector of $\\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T}$ ) and the corresponding $\\epsilon _j$ is small, then the effect of the summands $\\mathbf {z}\\ne \\mathbf {0}$ is quite large or in other words the approximation $c_\\mathbf {h}^{\\mathbf {M}}(f) \\approx c_{\\mathbf {h}}(f)$ is not sufficient enough.",
"This might be e.g.",
"due to presence of an edge orthogonal to $\\mathbf {h}_j$ .",
"If on the other hand, the spectrum is bounded in this direction and $\\epsilon _j$ is large enough, then the approximation is of better quality."
],
[
"Pattern congruence classes",
"Following the pattern classification, cf.",
"[16] we notice that $\\operatorname{\\mathcal {P}}(\\mathbf {M}) = \\operatorname{\\mathcal {P}}(\\mathbf {N})$ whenever $\\mathbf {M} = \\mathbf {Q}\\mathbf {N}$ holds for a matrix $\\mathbf {Q}\\in \\mathbb {Z}^{d\\times d}$ , $\\vert \\det \\mathbf {Q}\\vert = 1$ .",
"We define the pattern congruence of two matrices by $\\mathbf {M}\\sim _{\\operatorname{\\mathcal {P}}}\\mathbf {N}$ whenever they generate the same pattern.",
"By [16] there exists a congruence class representant $\\mathbf {M}^{\\circ } = (m^{\\circ }_{i,j})_{j,i=1}^{d,d}$ in every congruence class, such that $\\mathbf {M}^{\\circ }$ is upper triangular, and $0\\le m^{\\circ }_{i,j} < m^{\\circ }_{j,j}$ for all $i<j$ .",
"An easy consequence of this is, that for a diagonal matrix $\\mathbf {M} = \\bigl ({\\begin{matrix}m_1 & 0 \\\\0 & m_2\\end{matrix}}\\bigr )$ choosing $\\mathbf {Q} = \\bigl ( {\\begin{matrix}1&n\\\\0&1\\end{matrix}})$ , $n\\in \\mathbb {N}$ , reveals, that all matrices $\\mathbf {M}_n \\mathbf {Q}\\mathbf {M} = \\bigl ({\\begin{matrix}m_1 & m_2n\\\\0&m_2\\end{matrix}}\\bigr )$ possess the same pattern or in other words the same sampling points on the torus.",
"However, their corresponding generating sets $\\operatorname{\\mathcal {G}}(\\mathbf {M}_n^\\mathrm {T})$ differ.",
"This can be interpreted as choosing different directional sine and cosine functions that can be defined on the same set of points in order to analyze given discrete data or employing the periodicity of $c_{\\mathbf {h}}^{\\mathbf {M}}(f)$ with respect to $\\mathbf {M}^\\mathrm {T}$ that is implied by (REF ).",
"This introduces the possibility of anisotropic analysis and interpretation even on a usual pixel grid.",
"From the computational point, two different matrices $\\mathbf {M}_1 \\sim _{\\operatorname{\\mathcal {P}}} \\mathbf {M}_2$ having the same pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M}_1) = \\operatorname{\\mathcal {P}}(\\mathbf {M}_2)$ only require an rearranging of the data arrays due to the change of the basis vectors, i.e.",
"using $\\mathbf {y}^{(1)}_j$ or $\\mathbf {y}^{(2)}_j$ , $j=1,\\ldots ,d_{\\mathbf {M}_1}=d_{\\mathbf {M}_2}$ , when addressing $\\mathbf {a}$ in (REF ) and similarly the ordering with respect to the generating sets $\\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T}_i)$ , $i=1,2$ .",
"This rearrangement can easily be computed, see [2].",
"Figure: The three matrices 𝐌=84n04,n=0,1,2\\mathbf {M} = \\bigl ({\\begin{matrix}8 & 4n\\\\0&4\\end{matrix}}\\bigr ), n=0,1,2 induce the same sample point set but different scaled unit cells 1 2𝐌 n -1 [-1,1] 2 \\frac{1}{2}\\mathbf {M}_n^{-1}[-1,1]^2 (solid, dashed, dotted; left) and different generating sets error(𝐌 n T )\\operatorname{\\mathcal {G}}(\\mathbf {M}_n^\\mathrm {T}) (∘\\circ , ++, ×\\times ; right, respectively) yielding different frequency sets for the Fourier transform.Example 2.2 For the simple matrix $\\mathbf {M} = \\mathbf {M}_0 = \\bigl ({\\begin{matrix}8&0\\\\0&4\\end{matrix}}\\bigr )$ the pattern is just the rectangular sampling grid, a subset of which is displayed in Fig.",
"REF (left).",
"The two matrices $\\mathbf {M}_1 = \\bigl ({\\begin{matrix}8&4\\\\0&4\\end{matrix}}\\bigr )$ and $\\mathbf {M}_2 = \\bigl ({\\begin{matrix}8&8\\\\0&4\\end{matrix}}\\bigr )$ possess the same sample points.",
"However, their scaled unit cells $\\frac{1}{2}\\mathbf {M}_i^{-1}[-1,1]^2$ , also shown in Fig.",
"REF (left), differ.",
"This illustrates the different directional preference of these matrices, though they possess the same sampling lattice.",
"While the diagonal matrix resembles the form of (stretched) pixels, the shear introduced by looking at other sine/cosine terms can be clearly seen for $i=1,2$ for the dashed and dotted parallelotopes.",
"The three different generating sets $\\operatorname{\\mathcal {G}}(\\mathbf {M}_n^\\mathrm {T})$ , $n=1,2,3$ are denoted as circles, pluses and crosses in Fig.",
"REF (right), respectively.",
"Hence both figures illustrate one way to visualize the anisotropy.",
"We want to investigate composite structures which consist of a finite number of materials composed into one material.",
"Common examples are fiber reinforced polymers [12], polycrystalline structures [26] or metall foams [17].",
"As these structures are typically small they can seldom be resolved exactly in simulations of larger structures.",
"This gives rise to the concept of homogenization where the composite material is replaced by a homogeneous one having the same relevant, i.e.",
"macroscopic, properties.",
"The basic assumption to do this is that the microstructure is periodic, i.e.",
"it is sufficient to look at a representative volume element (RVE) and that the scales of the microstructure and macroscopic structure are separated.",
"These assumptions allow to calculate the homogenized behavior of the material, the so called effective properties or effective matrix.",
"This can be inserted into a macroscopic calculation or can be used to determine the isotropy of the structure or relevant elastic properties.",
"Typical examples of such problems involve the steady-state heat equation or the quasi-static equation of linear elasticity which we want to focus on henceforth."
],
[
"The equation of quasi-static linear elasticity in homogenization",
"The partial differential equation (PDE) we consider as an example in this publication is the equation of quasi-static linear elasticity.",
"It will serve as the basis to develop a numerical algorithm solving the PDE later on.",
"Consider a periodic stiffness distribution $\\mathcal {C}\\in L^\\infty (d)^{d \\times d \\times d \\times d}$ that is essentially bounded with major and minor symmetries, i.e.",
"with $\\mathcal {C}_{ijkl}= \\mathcal {C}_{jikl} = \\mathcal {C}_{ijlk} = \\mathcal {C}_{klij}$ characterizing the microstructure in the representative volume element.",
"The entries in $\\mathcal {C}$ specify the material behavior, e.g.",
"for an isotropic material we have $\\mathcal {C}_{ijkl}= \\lambda \\delta _{ij}\\delta _{kl} + \\mu (\\delta _{ik}\\delta _{jl}+ \\delta _{il}\\delta _{jk})$ where $\\lambda \\in \\mathbb {R}$ and $\\mu \\ge 0$ are the first and second Lamé parameter, respectively.",
"For the variational formulation of the steady-state heat equation Vordřejc et.al.",
"[29] derive an equivalent integral formula.",
"This approach can be directly applied to the quasi-static linear elasticity equation as follows.",
"We define the space $\\mathcal {E}(d) \\Bigl \\lbrace \\mathbf {v} \\in L^2(d)^{d \\times d} : \\mathbf {v} = \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}\\mathbf {u}, \\mathbf {u} \\in H^1(d)^d \\Bigr \\rbrace $ where $\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}\\mathbf {u} \\frac{1}{2} (\\operatorname{\\nabla }\\mathbf {u} + (\\operatorname{\\nabla }\\mathbf {u})^\\mathrm {T})$ is the symmetric gradient operator applied to displacement $\\mathbf {u}$ and $H^1(d)$ is the Sobolev space with Sobolev index 1, i.e.",
"the space of functions from $L^2(d)$ where the first weak derivative is also in $L^2(d)$ .",
"In the following we will use Einstein's summation convention.",
"For the product between a symmetric tensor $\\mathcal {A} \\in \\mathbb {R}^{d \\times d \\times d \\times d}$ of fourth order and a symmetric second-order tensor $\\alpha \\in \\mathbb {R}^{d \\times d}$ we introduce the notation $\\mathcal {A} :\\alpha (\\mathcal {A}_{ijkl} \\alpha _{kl})_{ij}.$ Definition 3.1 The partial differential equation of quasi-static linear elasticity in homogenization reads: Find for a macroscopic strain $\\epsilon ^0 \\in \\mathbb {R}^{d \\times d}$ the strain function $\\tilde{\\epsilon } = \\tilde{\\epsilon }_{\\epsilon ^0} \\in \\mathcal {E}(d)$ such that for all $\\nu \\in \\mathcal {E}(d)$ $\\langle \\nu , \\mathcal {C}:(\\epsilon ^0 + \\tilde{\\epsilon })\\rangle = 0$ holds true.",
"We further call $\\mathcal {C}^\\mathrm {eff}\\in \\mathbb {R}^{d \\times d \\times d \\times d}$ the effective matrix which is connected with the PDE above by $\\mathcal {C}^\\mathrm {eff}:\\epsilon ^0 \\int _{d} \\mathcal {C}:\\bigl (\\tilde{\\epsilon } + \\epsilon ^0 \\bigr ) \\,\\mathrm {d}\\mathbf {x}.$ By [29] this is equivalent to solving an integral equation for the strain $\\epsilon = \\tilde{\\epsilon } + \\epsilon ^0$ introducing a constant non-zero reference stiffness $\\mathcal {C}^0$ .",
"Definition 3.2 The Lippmann-Schwinger equation is given as: Find $\\epsilon \\in L^2(d)^{d \\times d}$ such that $\\epsilon = \\epsilon ^0 - \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}(\\operatorname{div}:\\mathcal {C}^0 :\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}})^{-1} \\operatorname{div}(\\mathcal {C}- \\mathcal {C}^0) :\\epsilon $ in weak sense.",
"The divergence operator is formally the negative adjoint of the symmetric gradient operator, i.e.",
"$\\operatorname{div}= -\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}^\\mathrm {H}$ , and therefore (REF ) reduces to finding a strain $\\epsilon \\in L^2(d)^{d \\times d}$ such that $\\epsilon = \\epsilon ^0 - \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}(\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}^\\mathrm {H}:\\mathcal {C}^0:\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}})^{-1} \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}^\\mathrm {H}(\\mathcal {C}- \\mathcal {C}^0):\\epsilon ,$ see [24].",
"The strain $\\epsilon \\in L^2(d)^{d \\times d}$ can be represented by its Fourier series and we obtain for $\\mathbf {k} \\in \\mathbb {Z}^d\\setminus \\lbrace \\mathbf {0}\\rbrace $ that $c_\\mathbf {k}(\\epsilon ) =- \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {k} (\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}^\\mathrm {H}_\\mathbf {k} :\\mathcal {C}^0:\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {k})^{-1} \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}^\\mathrm {H}_\\mathbf {k}c_\\mathbf {k} \\bigl ((\\mathcal {C}- \\mathcal {C}^0) :\\epsilon \\bigr ),$ and $c_{\\mathbf {0}}(\\epsilon ) = \\epsilon ^0$ .",
"The Fourier multiplier $\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {k}$ hereby represents the action of the operator $\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}$ with respect to a Fourier coefficient index $\\mathbf {k}\\in \\mathbb {Z}^d\\backslash \\lbrace \\mathbf {0}\\rbrace $ , respectively.",
"For $u \\in H^1(d)^d$ it can be derived as $c_{\\mathbf {k}}(\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}\\mathbf {u})\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {k} c_\\mathbf {k}(\\mathbf {u})= \\frac{{\\mathrm {i}}}{2} (\\mathbf {k} c_\\mathbf {k}(\\mathbf {u})^\\mathrm {T}+ c_\\mathbf {k}(\\mathbf {u}) \\mathbf {k}^\\mathrm {T}).$"
],
[
"Homogenization on anisotropic lattices",
"The approach of Moulinec and Suquet [20], [21] to discretize (REF ) is based on collocation on a Cartesian grid, see also [31].",
"To take into account preferred directions in composite we generalize this approach to arbitrary patterns $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ introduced in the Section .",
"Let a regular integer matrix $\\mathbf {M}\\in \\mathbb {Z}^{d \\times d}$ be given.",
"Following the idea of Moulinec and Suquet we collocate (REF ) at the points $2\\pi \\mathbf {y}\\in \\mathbb {T}^d, \\mathbf {y}\\in \\operatorname{\\mathcal {P}}(\\mathbf {M})$ , of the pattern.",
"This discretization leads to the problem of finding symmetric matrices $\\epsilon _\\mathbf {y} \\in \\mathbb {R}^{d \\times d}$ for each $\\mathbf {y} \\in \\operatorname{\\mathcal {P}}(\\mathbf {M})$ such that for $\\mathbf {y}\\in \\operatorname{\\mathcal {P}}(\\mathbf {M})\\setminus \\lbrace \\mathbf {0}\\rbrace $ we have $\\begin{split}\\epsilon _\\mathbf {y} = -&\\sum _{\\mathbf {h} \\in \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})}\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {h} \\bigl ( \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}^\\mathrm {H}_\\mathbf {h} :\\mathcal {C}^0:\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {h} \\bigr )^{-1} \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {h}\\\\&\\quad \\times \\frac{1}{m}\\sum _{\\mathbf {z} \\in \\operatorname{\\mathcal {P}}(\\mathbf {M})} (\\mathcal {C}_\\mathbf {z} - \\mathcal {C}^0) :\\epsilon _\\mathbf {z} {\\,\\mathrm {e}}^{-2 \\pi {\\mathrm {i}}\\mathbf {h}^\\mathrm {T}\\mathbf {z}} {\\,\\mathrm {e}}^{2 \\pi {\\mathrm {i}}\\mathbf {h}^\\mathrm {T}\\mathbf {y}}\\end{split}$ and $\\epsilon _{\\mathbf {0}} = \\epsilon ^0$ .",
"This discretization gives rise to the generalized basic scheme for patterns based on [21] summarized in Algorithm REF .",
"[t] $\\epsilon ^{(0)}_\\mathbf {y} \\leftarrow \\epsilon ^0$ for all $\\mathbf {y} \\in \\operatorname{\\mathcal {P}}(\\mathbf {M})$ $n \\leftarrow 0$ $\\tau ^{(n+1)}_\\mathbf {y}\\leftarrow \\left(\\mathcal {C}_\\mathbf {y}- \\mathcal {C}^0\\right) :\\epsilon _\\mathbf {y}^{(n)},\\quad \\mathbf {y}\\in \\operatorname{\\mathcal {P}}(\\mathbf {M})$ $\\hat{\\tau }^{(n+1)} \\leftarrow \\operatorname{\\mathcal {F}}(\\mathbf {M}) \\tau ^{(n+1)}$ $\\hat{\\epsilon }_\\mathbf {h}^{(n+1)}\\leftarrow - \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {h}\\bigl ( \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}^\\mathrm {H}_\\mathbf {h} :\\mathcal {C}^0:\\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {h} \\bigr )^{-1} \\operatorname{\\operatorname{\\nabla }_{\\mathrm {Sym}}}_\\mathbf {h}\\hat{\\tau }_\\mathbf {h}^{(n+1)},\\quad \\mathbf {h}\\in \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})\\setminus \\lbrace \\mathbf {0}\\rbrace $ $\\hat{\\epsilon }_\\mathbf {0}^{(n+1)} \\leftarrow \\epsilon ^0 $ $\\epsilon ^{(n+1)} \\leftarrow \\operatorname{\\mathcal {F}}^{-1}(\\mathbf {M}) \\hat{\\epsilon }^{(n+1)}$ $n \\leftarrow n+1$ a convergence criterion is reached Fixed-point algorithm on patterns."
],
[
"Anisotropic unit cells",
"By [3], the necessary structure for the sampling pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ is an additive group structure.",
"All these groups can be characterized by the congruence class representants $\\mathbf {M}^{\\circ }$ .",
"Despite from that approach, another one is as follows: let $\\mathbf {L} \\in \\mathbb {R}^{d \\times d}$ be a regular matrix and $\\mathcal {N} \\subset \\mathbb {R}^d$ a set such that $(\\mathcal {N}, + \\mathbf {\\unknown.",
"}\\mod {\\mathbf {}{L}) is a group endowed with theaddition + \\mathbf {\\unknown.",
"}\\mod {\\mathbf {}{L}.",
"Then one can find a corresponding pattern\\operatorname{\\mathcal {P}}(\\mathbf {M}) such that \\mathbf {L} \\operatorname{\\mathcal {P}}(\\mathbf {M}) = \\mathcal {N}, i.e.~wecan define a discrete Fourier transform on such a group.",
"}This setting allows for a more generalized notion of a pattern with examplesgiven in Figures~\\ref {fig:unitcells}.",
"}$ Figure: While the usual of a diagonal matrix indtroduces a rectangular grid (most left), the matrix 𝐋\\mathbf {L} of the transformed pattern error 𝐋 (𝐌)\\operatorname{\\mathcal {P}}_{\\mathbf {L}}(\\mathbf {M}) introduces rotation and scaling of the unit cube (left).",
"Furthermore, in such a cell, a rank-1-lattice can be used (right), and it is even possible to use arbitrary shapes like an arbitrary hexagonal shape 𝒮\\mathcal {S} (most right).Dotted lines indicate lines of the basis vectors and all pattern consist of 72 points, whose boundary elements are repeated on the other side (the half circles each).Definition 3.3 Let $\\mathbf {M} \\in \\mathbb {Z}^{d \\times d}$ be an regular integral matrix and $\\mathbf {L} \\in \\mathbb {R}^{d \\times d}$ be a regular matrix.",
"Then the transformed pattern is defined by $\\operatorname{\\mathcal {P}}_\\mathbf {L}(\\mathbf {M}) \\mathbf {L} \\Lambda (\\mathbf {M}) \\cap \\mathbf {L}\\bigl [-\\tfrac{1}{2},\\tfrac{1}{2}\\bigr )^d.$ One can even take any set of integer points inside a certain shape $\\mathcal {S}$ , where the shifts $\\mathcal {S} + \\mathbf {L}\\mathbf {z}$ , tile the $\\mathbb {R}^d$ , i.e.",
"for all $\\mathbf {y}\\in \\mathbb {R}^d$ exists a unique $\\mathbf {z}\\in \\mathbb {Z}^d$ such that $\\mathbf {y}-\\mathbf {z}\\in \\mathcal {S}$ , e.g see Fig.",
"REF (most right) This gives rise to a huge variety of unit cells to model the microscopic periodic media.",
"The transformed patterns $\\operatorname{\\mathcal {P}}_\\mathbf {L}(\\mathbf {M})$ especially introduce the possibility to take anisotropic cell structures inside the unit cell or RVE into account.",
"Definition REF can also be interpreted in terms of a coordinate transformation.",
"Consider a regular matrix $(A_{ij})_{i,j} = \\mathbf {A} \\in \\mathbb {R}^{d\\times d}$ and transformed coordinates $\\tilde{\\mathbf {x}} = \\mathbf {A} \\mathbf {x}$ with $\\mathbf {x} \\in [-\\frac{1}{2},\\frac{1}{2})^d$ .",
"Let $\\mathbf {u}(\\mathbf {x})$ solve the PDE (REF ) with stiffness distribution $\\mathcal {C}(\\mathbf {x})$ and macroscopic strain $\\epsilon ^0$ .",
"By [19] the displacement $\\tilde{\\mathbf {u}}(\\tilde{\\mathbf {x}})$ that solves (REF ) with transformed $\\tilde{\\mathcal {C}}_{ijkl}(\\tilde{\\mathbf {x}}) &= A_{im} A_{jn}A_{ko}A_{lp} \\mathcal {C}_{mnop}(\\mathbf {x}),$ and $\\tilde{\\epsilon ^0} = \\mathbf {A}^{-T} \\epsilon ^0 \\mathbf {A}^{-1}$ is connected to $\\mathbf {u}(\\mathbf {x})$ by $\\tilde{\\mathbf {u}}(\\tilde{\\mathbf {x}}) =\\mathbf {A}^{-\\mathrm {T}} \\mathbf {u}(\\mathbf {x})$ .",
"Discretizing $\\mathbf {u}(2 \\pi \\mathbf {y})$ with $\\mathbf {y} \\in \\operatorname{\\mathcal {P}}(\\mathbf {M})$ leads to $\\mathbf {u}(2 \\pi \\mathbf {y}) = \\mathbf {A}^\\mathrm {T}\\tilde{\\mathbf {u}}(2 \\pi \\mathbf {A} \\mathbf {y})= \\mathbf {A}^\\mathrm {T}\\tilde{\\mathbf {u}}(2 \\pi \\tilde{\\mathbf {y}})$ with $\\tilde{\\mathbf {y}} = \\mathbf {A} \\mathbf {y} \\in \\operatorname{\\mathcal {P}}(\\mathbf {M})$ or equivalently $\\tilde{\\mathbf {y}} \\in \\operatorname{\\mathcal {P}}_\\mathbf {A}(\\mathbf {M})$ .",
"Compared to the usual coordinate transform, c.f.",
"[15], [28], to the unit cell, the homogenization on patterns allows for further preference of directions in the cell other than the transformed coordinate axes."
],
[
"Convergence of the discretization",
"Let $n$ be the smallest eigenvalue of $\\mathbf {M}$ being larger than 1, where $\\mathbf {M}\\in \\mathbb {Z}^{d\\times d}$ .",
"Then interpolation error on the generating set $\\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ can be bounded from above by the diagonal matrix $\\tilde{\\mathbf {M}} \\operatorname{diag}(\\lfloor n\\rfloor ,\\ldots ,\\lfloor n\\rfloor )\\in \\mathbb {Z}^{d\\times d}$ , because the hypercube $\\frac{1}{2}\\tilde{\\mathbf {M}}[-1,1]^d$ is contained in the parallelepiped $\\tfrac{1}{2}\\mathbf {M}^\\mathrm {T}[-1,1]^d$ which is used to define $\\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ and hence $\\operatorname{\\mathcal {G}}(\\tilde{\\mathbf {M}}^\\mathrm {T})\\subset \\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ .",
"From any sequence of discretizations $\\mathbf {M}_i$ whose determinants $m_i$ tend to infinity, we also obtain, that the smallest eigenvalue tends to infinity because the matrix has to span the frequency lattice $\\mathbb {Z}^d$ .",
"With this argument the convergence of the such discretization sequence is dominated by a standard case of a Cartesian grid and the convergence proof of e.g.",
"[23] the convergence in this setting follows directly.",
"To examine the effects the pattern matrix has on the solution, this section describes two problems where the effective stiffness tensor and an analytic expression for the strain field are known.",
"The first structure is a laminate which as a basically one-dimensional structure exhibiting straight interfaces and thus a unique dominant direction to analyze.",
"The second geometry we introduce is given by two confocal ellipsoids defining a coated core in a matrix material.",
"The stiffnesses of the involved materials are chosen in such a way that the inclusion acts neutrally with respect to a specific macroscopic strain.",
"This generalizes the (isotropic) Hashin structure [11].",
"The curved interfaces make it more difficult to sample the structure efficiently and the occurring effects can be expected to be more complex."
],
[
"The laminate structure",
"The probably most simple structure with a predominant direction is a periodic laminate as shown in Fig.",
"REF ( middle) .",
"This structure consists of two isotropic materials alternating in the direction of lamination $\\mathbf {n} \\in \\mathbb {R}^d$ with ${\\mathbf {n}} = 1$ and being constant perpendicular to it.",
"For this structure Milton [19] derives an analytic equation for the effective matrix $\\mathcal {C}^\\mathrm {eff}$ that is given by $\\left(\\mathcal {S}-\\mathcal {T}\\right)^{-1} &=\\int _{d} \\left(\\tilde{\\mathcal {S}}(\\mathbf {x})-\\mathcal {T}\\right)^{-1} \\,\\mathrm {d}\\mathbf {x}\\multicolumn{2}{l}{\\text{with}}\\\\\\mathcal {S} &\\sigma _0 \\left( \\sigma _0 \\operatorname{Id}- \\mathcal {C}^\\mathrm {eff}\\right)^{-1},\\qquad \\tilde{\\mathcal {S}}(\\mathbf {x}) \\sigma _0 \\left( \\sigma _0 \\operatorname{Id}-\\mathcal {C}(\\mathbf {x}) \\right)^{-1},\\\\\\mathcal {T}_{ijkl} &\\frac{1}{2} (n_i \\delta _{jk} n_l + n_i \\delta _{jl} n_k+n_j \\delta _{ik} n_l + n_j \\delta _{il} n_k ) - n_i n_j n_k n_l.$ The choice of the free parameter $\\sigma _0$ is explained in [12].",
"Let $\\sigma $ be the largest eigenvalue of the spectra of the stiffness tensors $\\mathcal {C}(\\mathbf {x})$ then a choice of $\\sigma _0 > \\sigma $ ensures that all the inversions necessary to solve for $\\mathcal {C}^\\mathrm {eff}$ can be done.",
"The resulting strain field $\\epsilon $ is, like the structure itself, piecewise constant and varies only in the direction of lamination.",
"Let the volume fractions of the two materials be $f_1$ and $f_2$ with $f_1+f_2=1$ and call the corresponding constant strain fields $\\epsilon ^1$ and $\\epsilon ^2$ .",
"The geometry is constant perpendicular to the direction of lamination.",
"Hence the problem of finding $\\epsilon ^1$ and $\\epsilon ^2$ reduces to a one-dimensional problem.",
"This gives a system of linear equations involving the macroscopic strain $\\epsilon ^0$ to be solved, namely $f_1 \\epsilon ^1 + f_2 \\epsilon ^2 = \\epsilon ^0,\\qquad f_1 \\mathcal {C}^\\mathrm {eff}\\epsilon ^1 + f_2 \\mathcal {C}^\\mathrm {eff}\\epsilon ^2 =\\mathcal {C}^\\mathrm {eff}\\epsilon ^0.$"
],
[
"The generalized Hashin structure",
"The idea of the generalized Hashin structure due to Hashin and Shtrikman [11] is based on constructing a inclusion embedded in a matrix material that acts neural to a specific macroscopic strain, i.e.",
"the inclusion does not effect the surrounding stress field.",
"An example of such an inclusion is the assemblage of coated confocal ellipsoids described by Milton [19] whose derivation we want to follow.",
"A schematic of such a structure is depicted in Fig.",
"REF , left.",
"Figure: The generalized Hashin structure, c 1 =0.05c_1 = 0.05, c 2 =0.35c_2 = 0.35, c 3 =∞c_3=\\infty , ρ c =0\\rho _{\\mathrm {c}}=0, ρ e =0.09\\rho _{\\mathrm {e}}=0.09, and 𝐧=(1 2,1) T \\mathbf {n} = (\\tfrac{1}{2},1)^\\mathrm {T} in the xyxy-plane, i.e.",
"the 2D setting is shown in a schematic visualization of the material (left) and the analytic solution ϵ(𝐱)\\epsilon (\\mathbf {x}) to the elasticity equation (right).The derivation in Milton is based on an ellipsoid that is aligned with the coordinate axes.",
"An application of [19] then allows to generalize this to arbitrary orientations resulting in the formulae stated in the following.",
"They allow to predict for a macroscopic strain $\\epsilon ^0$ that will be given below to analytically express the resulting strain field $\\epsilon $ and the action of the effective stiffness tensor on this input, $\\mathcal {C}^\\mathrm {eff}\\epsilon ^0$ .",
"To define the geometry consider confocal ellipsoidal coordinates given for $\\tilde{\\mathbf {x}} \\in \\mathbb {R}^3$ by the equation $\\frac{\\tilde{x}_1^2}{c_1^2+\\rho } + \\frac{\\tilde{x}_2^2}{c_2^2+\\rho } +\\frac{\\tilde{x}_3^2}{c_3^2+\\rho } = 1$ where w.l.o.g.",
"the constants $0 \\le c_1 \\le c_2 \\le c_3 \\le \\infty $ determine the relative lengths of the semi-axes of the ellipsoid.",
"A constant $\\rho \\ge -c_1^2$ specifies the boundary of an ellipsoid where the lengths of the semi-axes are given by $l_j(\\rho ) \\sqrt{c_j^2+\\rho }$ with $j =1,2,3$ .",
"For a fixed $\\tilde{\\mathbf {x}} \\in \\mathbb {R}^3$ the ellipsoidal radius $\\rho (\\tilde{\\mathbf {x}})$ is the uniquely determined largest of the possible three solutions of (REF ) and fulfills $\\rho (\\tilde{x}) \\ge -c_1^2$ .",
"For sake of simplicity, we want to restrict ourselves in this work to prolate spheroids with $c_1 = c_2 \\le c_3$ , oblate spheroids with $c_1 \\le c_2 =c_3$ and elliptic cylinders.",
"The latter are the limit case $c_3 \\rightarrow \\infty $ .",
"For these, according to [19], the depolarization factors are given by $d_1(\\rho ) = d_2(\\rho ),\\qquad d_2(\\rho ) = 2-2 d_3(\\rho ),\\qquad d_3(\\rho ) = \\frac{1-\\delta ^2}{\\delta ^2}\\Bigl (\\frac{1}{2 \\delta }\\log \\Bigl (\\tfrac{1+\\delta }{1-\\delta }\\Bigr ) \\Bigr ),$ for prolate spheroids, where $\\delta = \\sqrt{1-\\tfrac{l_2(\\rho )^2}{l_3(\\rho )^2}}$ .",
"Furthermore we have for oblate spheroids using $ \\delta = \\sqrt{1-\\tfrac{l_1(\\rho )^2}{l_2(\\rho )^2}}$ the factors $d_1(\\rho ) = \\frac{1}{\\delta ^2} \\Bigl (1-\\tfrac{\\sqrt{1-\\delta ^2}}{\\delta } \\sin ^{-1}\\delta \\Bigr ),\\qquad d_2(\\rho ) = 2 - 2d_1(\\rho ),\\qquad d_3(\\rho ) = d_2(\\rho ),$ and finally for elliptic cylinders $d_1(\\rho ) = \\frac{l_2(\\rho )}{l_1(\\rho )+l_2(\\rho )},\\qquad d_2(\\rho ) = \\frac{l_1(\\rho )}{l_1(\\rho )+l_2(\\rho )},\\qquad d_3(\\rho ) = 0.$ Now let $\\rho _{\\mathrm {c}}$ and $\\rho _{\\mathrm {e}}$ be the ellipsoidal radius of the core and the exterior coating, respectively, cf. Fig.",
"REF (left), with $-c_1^2 < \\rho _{\\mathrm {c}} < \\rho _{\\mathrm {e}}$ and with $l_3(\\rho _{\\mathrm {e}}) < \\frac{1}{2}$ for $c_3 < \\infty $ and $l_2(\\rho _{\\mathrm {e}}) < \\frac{1}{2}$ for $c_3 = \\infty $ , i.e.",
"the exterior ellipsoid should be contained in $[-\\frac{1}{2},\\frac{1}{2})^3$ .",
"Further, let $\\mathbf {n} \\in \\mathbb {R}^3$ with ${\\mathbf {n}} = 1$ be the direction the shortest semi-axis of the ellipsoid with length $l_1(\\rho )$ .",
"Define the rotation matrix that transforms the vector $(1,0,0)^\\mathrm {T}$ to $\\mathbf {n}$ by $\\mathbf {R} \\begin{pmatrix} 1 & -n_2 & -n_3 \\\\ n_2 & 1 & 0\\\\ n_3 & 0 & 1\\end{pmatrix}+ \\frac{1-n_1}{\\sqrt{n_2^2+n_3^2}}\\begin{pmatrix} -n_2^2-n_3^2 & 0 & 0\\\\ 0 & -n_2^2 & -n_2n_3\\\\0 & -n_2n_3 &-n_3^2 \\end{pmatrix}.$ Then the core, the exterior coating and the surrounding matrix are given by $\\Omega _{\\mathrm {c}} &\\left\\lbrace \\mathbf {x} \\in \\mathbb {R}^3 : \\rho (\\mathbf {R}^{-1}\\mathbf {x}) \\le \\rho _{\\mathrm {c}} \\right\\rbrace ,\\\\\\Omega _{\\mathrm {e}} &\\left\\lbrace \\mathbf {x} \\in \\mathbb {R}^3 : \\rho _{\\mathrm {c}} < \\rho (\\mathbf {R}^{-1}\\mathbf {x}) \\le \\rho _{\\mathrm {e}} \\right\\rbrace ,\\\\\\Omega _{\\mathrm {m}} &\\bigl [ -\\tfrac{1}{2}, \\tfrac{1}{2} \\bigr )^3 \\setminus \\left(\\Omega _{\\mathrm {c}} \\cup \\Omega _{\\mathrm {e}} \\right),$ respectively.",
"With $f(\\rho ) \\frac{\\sqrt{g(\\rho _{\\mathrm {c}})}}{\\sqrt{g(\\rho )}}\\quad \\text{ and }\\quad g(\\rho ) {\\left\\lbrace \\begin{array}{ll} (c_1^2+\\rho )(c_2^2+\\rho )(c_3^2+\\rho ),&\\text{for } c_3 < \\infty ,\\\\(c_1^2+\\rho )(c_2^2+\\rho ),&\\text{else}, \\end{array}\\right.",
"}$ the volume fraction of $\\Omega _{\\mathrm {c}}$ in the coated ellipsoid is given by $f(\\rho _{\\mathrm {e}})$ .",
"We assume that the material in the core and in the exterior coating behave isotropically, i.e.",
"they are described by stiffness matrices of the form $\\mathcal {C}_{ijkl} = \\lambda \\delta _{ij} \\delta _{kl} + \\mu (\\delta _{ik}\\delta _{jl}+\\delta _{il} \\delta _{jk})$ .",
"The parameter $\\lambda $ is Lamé's first parameter and $\\mu $ is the shear modulus.",
"We denote the parameters in the core by $\\lambda _{\\mathrm {c}}$ and $\\mu _{\\mathrm {c}}$ and in the exterior coating by $\\lambda _{\\mathrm {e}}$ and $\\mu _{\\mathrm {e}}$ , respectively.",
"Further, the bulk modulus of an isotropic material is given as $\\kappa \\lambda + \\frac{2}{3} \\mu $ .",
"Following [19] we impose a macroscopic strain of $\\epsilon ^0 = \\mathbf {R}\\left(\\frac{3 \\kappa _{\\mathrm {e}} +4 \\mu _{\\mathrm {e}}}{9(\\kappa _{\\mathrm {c}}-\\kappa _{\\mathrm {e}})} \\operatorname{Id}+(1-f(\\rho _{\\mathrm {e}}))\\mathbf {S}(\\rho _{\\mathrm {e}})\\right) \\mathbf {R}^\\mathrm {T}$ with $\\mathbf {S}(\\rho ) &(1-f(\\rho ))^{-1} (\\mathbf {D}(\\rho _{\\mathrm {c}}) - f(\\rho ) \\mathbf {D}(\\rho ))\\quad \\text{ and }\\quad \\mathbf {D}(\\rho ) \\operatorname{diag}(d_i(\\rho )_{i=1,2,3}).$ The effective matrix of the structure and likewise —to ensure the neutrality of the inclusion— the stiffness matrix of the matrix material $\\Omega _{\\mathrm {m}}$ are given by their action on the macroscopic strain as $\\mathcal {C}^\\mathrm {eff}\\epsilon ^0 \\mathbf {R}\\Bigl ( \\tfrac{\\kappa _{\\mathrm {e}}}{\\kappa _{\\mathrm {c}}-\\kappa _{\\mathrm {e}}}(\\kappa _{\\mathrm {c}}+\\frac{4}{3} \\mu _{\\mathrm {e}})+ \\frac{4}{3} \\mu _{\\mathrm {e}} f(\\rho _{\\mathrm {e}}) \\Bigr ) \\operatorname{Id}\\mathbf {R}^\\mathrm {T}+ \\mathbf {R}\\frac{2}{3} \\mu _{\\mathrm {e}} (1-f(\\rho _{\\mathrm {e}}))(3 \\mathbf {S}(\\rho _{\\mathrm {e}}) - \\operatorname{Id}) \\mathbf {R}^\\mathrm {T}.$ The resulting strain field is then for $\\mathbf {x} \\in \\Omega _{\\mathrm {c}}$ given as $\\epsilon (\\mathbf {x}) = \\mathbf {R}\\frac{3 \\kappa _{\\mathrm {e}} + 4 \\mu _{\\mathrm {e}}}{9(\\kappa _{\\mathrm {c}}-\\kappa _{\\mathrm {e}})} \\operatorname{Id}\\mathbf {R}^\\mathrm {T},$ and for $\\mathbf {x} \\in \\Omega _{\\mathrm {m}}$ as $\\epsilon (\\mathbf {x}) = \\epsilon ^0$ , respectively.",
"The strain field is constant in the core and in the matrix material.",
"In the external coating we have for $\\mathbf {x} \\in \\Omega _{\\mathrm {e}}$ $\\epsilon (\\mathbf {x})&= \\mathbf {R}\\biggl (\\tfrac{3 \\kappa _{\\mathrm {e}} + 4 \\mu _{\\mathrm {e}}}{9(\\kappa _{\\mathrm {c}}-\\kappa _{\\mathrm {e}})}\\operatorname{Id}+\\mathbf {D}(\\rho _{\\mathrm {c}}) - f(\\rho (\\tilde{\\mathbf {x}}))\\mathbf {D}(\\rho (\\tilde{\\mathbf {x}}))+\\tfrac{\\sqrt{g(\\rho _{\\mathrm {c}})}}{2} \\mathbf {q}(\\tilde{\\mathbf {x}})\\operatorname{\\nabla }_{\\tilde{\\mathbf {x}}}^T \\rho (\\tilde{\\mathbf {x}})\\biggr )\\mathbf {R}^\\mathrm {T},\\multicolumn{2}{l}{\\text{with $\\tilde{\\mathbf {x}} \\mathbf {R}^{-1} \\mathbf {x}$ and }}\\\\\\mathbf {q}(\\tilde{\\mathbf {x}})_i &= \\frac{\\tilde{x}_i}{(c_i^2+\\rho (\\tilde{\\mathbf {x}}))\\sqrt{g(\\rho (\\tilde{\\mathbf {x}}))}},\\quad \\left(\\operatorname{\\nabla }_{\\tilde{\\mathbf {x}}} \\rho (\\tilde{\\mathbf {x}}) \\right)_i = \\frac{2\\tilde{x}_i}{c_i^2+\\rho (\\tilde{\\mathbf {x}})} \\Biggl (\\sum _{j=1}^3\\frac{\\tilde{x}_i}{c_i^2+\\rho (\\tilde{\\mathbf {x}})}\\Biggr )^{-1},\\multicolumn{2}{l}{\\text{ $i=1,2,3$.",
"For the case $c_3 = \\infty $ we additionally have $\\mathbf {q}(\\tilde{\\mathbf {x}})_3 = 0$,}}\\\\\\left(\\operatorname{\\nabla }_{\\tilde{\\mathbf {x}}} \\rho (\\tilde{\\mathbf {x}}) \\right)_i&=\\frac{2\\tilde{x}_i}{c_i^2+\\rho (\\tilde{\\mathbf {x}})} \\Biggl (\\sum _{j=1}^2\\frac{\\tilde{x}_i}{c_i^2+\\rho (\\tilde{\\mathbf {x}})}\\Biggr )^{-1},\\quad i=1,2,\\text{ and }\\left(\\operatorname{\\nabla }_{\\tilde{\\mathbf {x}}} \\rho (\\tilde{\\mathbf {x}}) \\right)_3 = 0.$ Example 4.1 Let an ellipse in 2D be given, i.e.",
"we set the third dimension to be constant by $c_3=\\infty $ , by choosing $c_1 = 0.05$ , $c_2 = 0.35$ .",
"Choose further $\\rho _c = 0$ and $\\rho _e = 0.09$ and introduce a rotation of $60^\\circ $ counter clockwise by setting $\\mathbf {n} = (\\tfrac{1}{2},1,0)^\\mathrm {T}$ .",
"The resulting geometry is shown in REF (left), the analytic solution $\\epsilon (\\mathbf {x})$ to the elasticity equation in REF (right)."
],
[
"Numerics",
"The algorithms used in this paper are implemented in !MatLab!",
"R2015b in a modular and fast way using vectorization.",
"For the Fourier transform on arbitrary patterns we employ the multivariate periodic anisotropic wavelet library (MPAWL)[4], which was recently ported to Matlabsee https://github.com/kellertuer/MPAWL-Matlab.",
"It uses Matlab's internal !fftn!",
"command to apply the fast Fourier transform.",
"All tests were run on a MacBook Pro running Mac OS X 10.11.5, Core i5, 2.6 GHz, with 8 GB RAM using Matlab 2016a and the clang-700.1.76 compiler."
],
[
"Hashin",
"Consider the geometry of the coated ellipsoid as described in Section REF with the parameters in Example REF seen as a two-dimensional problem, i.e.",
"sampled with only one point in $x_3$ -direction.",
"This structure is strongly orthotropic with the dominant directions being $\\mathbf {n}$ and $\\mathbf {n}^\\perp $ .",
"To analyze the influence of the sampling matrix $\\mathbf {M}$ we compare the relative $\\ell ^2$ -error of the strain $\\epsilon $ compared with the analytic solution and the relative error in the effective stiffness tensor sampled on $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ , i.e we define $e_{\\ell ^2}(\\mathbf {M}) \\frac{{\\epsilon -\\tilde{\\epsilon }}[l^2(\\operatorname{\\mathcal {P}}(\\mathbf {M}))]}{{\\tilde{\\epsilon }}[l^2(\\operatorname{\\mathcal {P}}(\\mathbf {M}))]}\\quad \\text{ and }\\quad e_{\\mathrm {eff}}(\\mathbf {M}) \\frac{{\\mathcal {C}^\\mathrm {eff}\\epsilon ^0 - \\tilde{\\mathcal {C}}^\\mathrm {eff}\\epsilon ^0}[2]}{{\\tilde{\\mathcal {C}}^\\mathrm {eff}\\epsilon ^0}[2]},$ where $\\epsilon $ and $\\mathcal {C}^\\mathrm {eff}\\epsilon ^0$ are the numerical solutions obtained by Algorithm REF , and $\\tilde{\\epsilon }$ and $\\tilde{\\mathcal {C}}^0 \\epsilon ^0$ are the analytic solutions.",
"The pattern matrices are parametrized by $\\mathbf {M}_{j,k,\\alpha } \\begin{pmatrix}2^j & \\alpha k & 0\\\\(1-\\alpha )k & 2^{14-j} & 0\\\\0 & 0 & 1\\end{pmatrix}$ with $j \\in \\lbrace 7,\\dots ,9 \\rbrace $ , $\\alpha \\in \\lbrace 0,1 \\rbrace $ and $k\\in 16 \\cdot \\lbrace -32, \\dots , 32 \\rbrace $ .",
"For all parameters these matrices have determinant $2^{14}$ , i.e.",
"the number of sampling points stays constant.",
"The parameter $k$ shears the pattern, $\\alpha $ determines the direction of the shearing and $j$ controls the refinement in the direction of the pattern basis vectors $\\mathbf {y}_j$ , $j=1,\\ldots ,d_{\\mathbf {M}}$ , cf.",
"(REF ).",
"This induces both an anisotropy or preference of direction in the pattern $\\operatorname{\\mathcal {P}}(\\mathbf {M})$ as well as for the basis vectors $\\mathbf {h}_j$ of the corresponding generating set $\\operatorname{\\mathcal {G}}(\\mathbf {M}^\\mathrm {T})$ , cf.",
"(REF ), representing the frequencies.",
"Further, we use pattern matrices of the form $\\tilde{\\mathbf {M}}_j \\frac{1}{2}\\begin{pmatrix} 1&1&0\\\\-1&1&0\\\\0&0&2\\end{pmatrix}\\begin{pmatrix}2^j&0&0\\\\0&2^{14-j}&0\\\\0&0&1\\end{pmatrix}$ corresponding to a rotation of the grid in the direction of $\\mathbf {n}$ with $\\det (\\tilde{\\mathbf {M}}_j) = 2^{13}$ .",
"Figure: Shear parameter against relative error in the effective stiffnesstensor.",
"The corresponding matrices for the curve (j,α)(j,\\alpha ) at point kk is𝐌 j,k,α \\mathbf {M}_{j,k,\\alpha } from ().",
"Circles mark theresults for matrices 𝐌 j,0,0 \\mathbf {M}_{j,0,0} and crosses stand for matrices of theform 𝐌 ˜ j \\tilde{\\mathbf {M}}_j.In Figure REF the effect of the shearing parameter $k$ on $e_{\\mathrm {eff}}$ is depicted, neglecting curves that do not perform better than $\\mathbf {M}_{7,k,\\alpha }$ with $\\mathbf {M}_{9,k,1}$ as an example for such a curve.",
"For each value of $j$ we choose one color (brightness) and indicate $\\alpha =0$ by a solid, $\\alpha =1$ by a dotted line, and indicate $\\tilde{\\mathbf {M}}_j$ , $j=7,8,9$ , by crosses as well as the corresponding diagonal matrices $\\mathbf {M}_{j,0,0}=\\mathbf {M}_{j,0,1}$ by circles.",
"The circles correspond to the classical rectangular “pixel sampling grid”.",
"The reference point for the analysis of the results is the unsheared matrix $\\mathbf {M}_{7,0,0}$ , i.e.",
"the standard equidistant Cartesian grid, giving an error of $3.8 \\cdot 10^{-3}$ .",
"Shearing this matrix in either direction results in a larger errors, e.g.",
"with twice the error for $\\mathbf {M}_{7,-512,0}$ and an error of $5 \\cdot 10^{-3}$ for $\\mathbf {M}_{7,512,0}$ .",
"In contrast $\\mathbf {M}_{8,k,0}$ behaves similarly to $\\mathbf {M}_{7,k,0}$ with the exception of $k_{-1} = -2^8+16$ , $k_{0} = 16$ and $k_{1} = 2^8+16$ , where the error is $2.4 \\cdot 10^{-3}$ for the first two and $2.2 \\cdot 10^{-3}$ for the third point.",
"Note that the matrices possess the same pattern due to $\\mathbf {M}_{8,k_{-1},0} \\sim _{\\operatorname{\\mathcal {P}}} \\mathbf {M}_{8,k_0,0} \\sim _{\\operatorname{\\mathcal {P}}}\\mathbf {M}_{8,k_1,0}$ .",
"Figure: The strain field ϵ 11 \\epsilon _{11} (first row) using the colormap above andits corresponding e log e_{\\log }-error (second row) given bye log =log(1+|ϵ-ϵ ˜|)e_{\\log }=\\log (1+\\vert \\epsilon -\\tilde{\\epsilon }\\vert ) using the colormapbelow.",
"The matrices inducing the sampling pattern pointsare (left to right):𝐌 7,0,0 =2 7 002 7 \\mathbf {M}_{7,0,0}=\\bigl ({\\begin{matrix}2^7&0\\\\0&2^7\\end{matrix}}\\bigr ),𝐌 ˜ 7 =2 6 2 6 -2 6 2 6 \\tilde{\\mathbf {M}}_{7}=\\bigl ({\\begin{matrix}2^6&2^6\\\\-2^6&2^6\\end{matrix}}\\bigr ),𝐌 8,16,0 =2 8 02 4 2 6 \\mathbf {M}_{8,16,0}=\\bigl ({\\begin{matrix}2^8&0\\\\2^4&2^6\\end{matrix}}\\bigr ), and𝐌 9,16,0 =2 9 02 4 2 5 \\mathbf {M}_{9,16,0}=\\bigl ({\\begin{matrix}2^9&0\\\\2^4&2^5\\end{matrix}}\\bigr ).This effect is even more dominant for $\\mathbf {M}_{9,k,0}$ that gives an error of around $6 \\cdot 10^{-3}$ almost everywhere except for $k = -2^9+16$ and $k=16$ where it drops to $1.5 \\cdot 10^{-3}$ .",
"These matrices are also congruent with respect to pattern congruence $\\sim _{\\operatorname{\\mathcal {P}}}$ .",
"The strain field and the pointwise error for the $\\epsilon _{11}$ -component of $\\mathbf {M}_{7,0,0}$ are depicted in Fig.",
"REF , first column.",
"As the ellipsoid is not aligned with the pattern, the interface along the long side of the ellipsoid is not resolved well.",
"As the sine functions are not perpendicular to this interface the Gibbs phenomenon dominates the $\\ell ^2$ -error.",
"Likewise the strain peak at the tip of the ellipsoid is not captured correctly and worsens $e_\\mathrm {eff}$ tremendously.",
"If we choose a matrix with a certain shear, e.g.",
"$\\mathbf {M}_{8,16,0}$ , where the strain $\\epsilon _{11}$ and the corresponding error are shown in Fig.",
"REF (third column) , we obtain a quite small error in the effective stiffness tensor and resolve the strain peaks correctly.",
"As the pattern, however, is not aligned with the ellipsoid, the interfaces show large errors and inwards and outwards facing corners result in a very large $\\ell ^2$ -error.",
"For the matrices $\\mathbf {M}_{9,-2^9+16,0}$ and $\\mathbf {M}_{9,16,0}$ the pattern is sheared in such a way that it is aligned with the ellipsoid, c.f.",
"Fig.",
"REF , last column, refined in along the longer semi-axis.",
"The strain field $\\epsilon _{11}$ is characterized by slowly changing values in the direction of the shorter semi-axis and especially rapidly changing and high strains at the tips of the inner ellipsoid.",
"Therefore, to get small errors, we need only few points in the direction of the shorter semi-axis, in this case only 32 points.",
"The high strains at the tips of the ellipsoid, however, require a high resolution like 512 sampling points in this case.",
"This leads to a lower approximation of the edges orthogonal to the smaller semi-axis and hence the Gibbs phenomenon increases the $\\ell ^2$ -norm, cf. Fig.",
"REF , first column, for the log-error.",
"This effect concentrates around the tips of the ellipsoid resulting in a core almost shaped like a rectangle.",
"The averaging done to compute the effective stiffness tensor cancels these errors and does therefore not influence the error.",
"Table: Relative effective stiffness and ℓ 2 \\ell ^2-errors for several shearing matrices.",
"While matching the direction, 𝐌 9,16,0 \\mathbf {M}_{9,16,0} reduces the e eff e_{\\text{eff}} error, the e ℓ 2 e_{\\ell ^2}-error is reduced tremendously by e.g.",
"shearing the standard grid.",
"This can be seen by looking at 𝐌 7,2 8 +16,1 \\mathbf {M}_{7,2^8+16,1}.Table REF shows both the $\\ell ^2$ - and the $e_{\\mathrm {eff}}$ -errors for several matrices.",
"The smallest value of $e_\\mathrm {eff}$ is reached for $\\mathbf {M}_{9,-2^9+16,0}$ and $\\mathbf {M}_{9,16,0}$ .",
"The error for shear parameters around $\\mathbf {M}_{9,16,0}$ get slightly worse by changing to $k=15$ or $k=17$ , respectively.",
"The smallest $\\ell ^2$ -errors can be obtained by $\\mathbf {M}_{7,2^8,1}$ and $\\mathbf {M}_{7,2^8+16,1}$ , giving errors only half as large as in the standard case.",
"The pattern $\\mathbf {M}_{7,2^8,1}$ involves taking a standard pixel grid for sampling and then shearing the unit cell in such a way that it is aligned with the ellipsoid.",
"This gives a balance between resolving the interfaces and capturing the strain peak.",
"The alignment of the sine ansatz functions with the ellipsoid reduces the Gibbs phenomenon and the comparatively large number of frequencies used in the direction of the strain peaks allows for small errors there.",
"The matrix $\\mathbf {M}_{7,2^8+16,1}$ further shears the pattern by a small amount and resolves the strain peak better while preserving the good resolution of the ellipsoid.",
"Subsampling of the complete rectangular grid of one $\\mathbf {M}_{j,0,0}$ on the so-called quincunx pattern induced by the matrix $\\tilde{\\mathbf {M}}_j$ shown as circles for the former and crosses for the latter matrices in Fig.",
"REF , gives slightly smaller values of $e_\\mathrm {eff}$ .",
"The $\\ell ^2$ -errors like for $\\tilde{\\mathbf {M}}_7$ , cf.",
"Figs.",
"REF , second column, also decrease even especially when taking into account that only half the sampling values are used."
],
[
"Subsampling",
"We study possibilities to subsample given (large) data, when certain directional information is given, e.g.",
"the laminate from Section REF and a given normal vector of $\\mathbf {n} = (1,\\tfrac{1}{2})^\\mathrm {T}$ .",
"We choose a matrix $\\mathbf {N}$ being a factorization of a pixel grid and of the form $\\mathbf {M}=\\begin{pmatrix}a&0\\\\0&a\\end{pmatrix}=\\begin{pmatrix}1&-\\frac{a}{2}\\\\0&a\\end{pmatrix}\\begin{pmatrix}a & \\frac{a}{2}\\\\0&1\\end{pmatrix}\\mathbf {J}\\mathbf {N}$ and hence $\\operatorname{\\mathcal {P}}(\\mathbf {N}) \\subset \\operatorname{\\mathcal {P}}(\\mathbf {M})$ if $\\frac{a}{2}$ is an integer.",
"Note that in the sub pattern the most dominant direction in the Fourier domain is the direction orthogonal to the edge direction.",
"Setting $a=2^{6}$ the pixel sampling contains just $2^{12} = 4096$ data items, a size where the dominant numerical effects like the Gibbs phenomenon can still be observed.",
"We compare the following patterns: first we sample on the full grid, i.e.",
"we choose $\\mathbf {M}_{\\text{a}} = \\bigl ({\\begin{matrix}64&0\\\\0&64\\end{matrix}}\\bigr )$ .",
"We consider a directional sub lattice given by $\\mathbf {M}_\\text{b} =\\bigl ({\\begin{matrix}64&32\\\\0&1\\end{matrix}}\\bigr )$ following the construction above and resulting in 64 data points, only the square root of the number of points of the full grid.",
"By construction we have $\\operatorname{\\mathcal {P}}(\\mathbf {M}_\\text{b}) \\subset \\operatorname{\\mathcal {P}}(\\mathbf {M}_\\text{a})$ and that the spanning vector $\\mathbf {y}_1$ of this rank-1-lattice, is orthogonal to the edges present in the material.",
"This can for example be done by examining the large(r) dataset with respect to its edge directions and subsampling accordingly.",
"We study the effect of these in contrast to $\\mathbf {M}_\\text{c} = \\bigl ({\\begin{matrix}8&0\\\\0&8\\end{matrix}}\\bigr )$ having also 64 points.",
"Clearly this is also a sub lattice of $\\mathbf {M}_\\text{a}$ .",
"We employ Algorithm REF and a Chauchy criterion for stopping, i.e.",
"$\\tfrac{{\\epsilon ^{(n+1)}-\\epsilon ^{(n)}}}{{\\epsilon ^{(0)}}}$ with a threshold of $10^{-9}$ .",
"The results are shown in Figure REF .",
"The full grid based approach of $\\mathbf {M}_\\text{a}$ shown in Fig.",
"REF (left) suffers from the well known Gibbs phenomenon.",
"The subsampling on $\\operatorname{\\mathcal {P}}(\\mathbf {M}_\\text{b})$ in Fig REF (middle) reduces the number of iterations tremendously from 94 to only 9 and the computational times from $41.2$ to only $4.5$ seconds.",
"This is not the case for the pixel grid subsampling given by $\\operatorname{\\mathcal {P}}(\\mathbf {M}_\\text{c})$ shown in Fig REF (right) which still requires 89 iterations and $46.1$ seconds.",
"Looking at the $\\ell ^2$ -error depicted in the captions of Fig.",
"REF , reducing the number of points from $64^2$ in $\\operatorname{\\mathcal {P}}(\\mathbf {M}_a)$ to 64 in $\\operatorname{\\mathcal {P}}(\\mathbf {M}_b)$ also reduces the error by a factor of roughly 3.1.",
"The tensor pixel grid of $\\mathbf {M}_{\\text{c}}$ with $8\\times 8 = 64$ is by a factor of 8 worse than our new anisotropic approach given by $\\mathbf {M}_b$ .",
"Finally, we analyse the error of the effective stiffness $e_{\\text{eff}}(\\mathbf {M}_x)$ , $x\\in \\lbrace a,b,c\\rbrace $ .",
"The values are $e_{\\text{eff}}(\\mathbf {M}_a)=0.0042$ , $e_{\\text{eff}}(\\mathbf {M}_b)=0.0134$ , and $e_{\\text{eff}}(\\mathbf {M}_c)=0.0495$ .",
"Hence having only 64 times the number of points as $\\operatorname{\\mathcal {P}}(\\mathbf {M}_{\\text{b}})$ the first example is only about a factor of $3.2$ times better.",
"This pattern yields an error that is by a factor of roughly $3.6$ smaller than the tensor product grid $\\operatorname{\\mathcal {P}}(\\mathbf {M}_c)$ having the same number of points.",
"In total, such a construction is also possible for other integer values, though the factorization might not be that easy to find.",
"Other normal vector directions can be approximated, e.g.",
"applying the dyadic decomposition in frequency as in [3].",
"Then, the direction to be approximated by the factorization of the matrix $\\mathbf {M}$ has to be orthogonal to the direction, which should be sampled the most dense, i.e.",
"for the laminate this direction of interest in the Fourier domain is along the laminate."
],
[
"Summary and Conclusion",
"This article generalizes the discretization of the Lippmann-Schwinger equation and the resulting numerical algorithm to anisotropic sampling lattices.",
"This allows to refine other directions than the coordinate axes, even supporting non-orthogonal refinement.",
"This leads to smaller errors in the strain field and a better approximation of the effective stiffness tensor when taking the anisotropic properties of a material into account.",
"Furthermore, the orientation of the sine functions for the real-valued discrete Fourier transform can be chosen.",
"This allows for alignment of the ansatz functions with interfaces and increases their resolution while reducing the Gibbs phenomenon.",
"Especially regions and directions of high strain can thus be resolved better.",
"We show that these additional choices can not be reproduced by linear transformations of the problem, e.g.",
"rotations of the geometry.",
"Subsampling on suitable patterns makes these techniques also accessible for data given on standard Cartesian grids present in many applications.",
"The application of the corresponding fast Fourier transform on patterns does not increase the computational effort and might even be computationally advantageous in case of lattices of rank 1.",
"In this case the Fast Fourier transform reduces to a one-dimensional transform, greatly reducing the computational complexity required of the FFT algorithm.",
"Other modifications applied to the Basic scheme of Moulinec and Suquet can still be applied to the anisotropic lattice version introduced in this paper.",
"These modifications include adaptions of the numerical algorithm to increase both robustness and speed.",
"Furthermore, schemes stemming from finite difference or finite element methods can also be incorporated into the anisotropic setting.",
"An open problem that has to be studied in detail is an automatically performed choice of the pattern matrix $\\mathbf {M}$ .",
"While the currently chosen matrices already stem from geometric interpretation of how to choose the directions of interest in the sampling lattice, the choice is up to now manually done.",
"An analysis of the main directions of interfaces may provide a good selection of a pattern, but there may be additional restraints to take into account when selecting a sampling scheme."
],
[
"Acknowledgment",
"The authors would like to thank Bernd Simeon and Gabriele Steidl for their idea to start this collaboration."
]
] | 1605.05712 |
[
[
"One class of conservative difference schemes for solving molecular\n dynamics equations of motion"
],
[
"Abstract Simulation of many-particle system evolution by molecular dynamics takes to decrease integration step to provide numerical scheme stability on the sufficiently large time interval.",
"It leads to a significant increase of the volume of calculations.",
"An approach for constructing symmetric simplectic numerical schemes with given approximation accuracy in relation to integration step, for solving molecular dynamics Hamiltonian equations, is proposed in this paper.",
"Numerical experiments show that obtained under this approach symmetric simplectic third order scheme is more stable for integration step, time-reversible and conserves Hamiltonian of the system with more accuracy at a large integration interval then second order velocity Verlet numerical schemes."
],
[
"Introduction",
"Numerical schemes, which is using for solving systems of equations of many-particle dynamics, can have restrictions on a step and an interval of integration because if they increase, the numerical schemes become unstable and do not conserve integrals of motion.",
"As a result, when we simulate many-particle system behaviour on the sufficiently large time interval we should decrease an integration step, which leads to considerable increasing of computation quantity."
],
[
"HAMILTONIAN MOLECULAR DYNAMICS",
"A motion of $N$ -particle system in a field with potential $V(\\mathbf {q})$ can be described by the system of Hamiltonian equations [1] $\\dot{\\mathbf {p}} = -\\frac{\\partial H(\\mathbf {p},\\mathbf {q})}{\\partial \\mathbf {q}},\\,\\,\\dot{\\mathbf {q}} = \\frac{\\partial H(\\mathbf {p},\\mathbf {q})}{\\partial \\mathbf {p}},$ with initial conditions $\\mathbf {p}(0)=\\mathbf {p}^0,\\,\\mathbf {q}(0)=\\mathbf {q}^0$ .",
"Here $\\mathbf {q}=(q_1,…,q_d)^T$ - particle coordinates, $\\mathbf {p}=(p_1,… ,p_d)^T$ - particle momentums, $d=3N$ - a dimension of coordinate space and $H(\\mathbf {p},\\mathbf {q})=\\frac{1}{2}\\mathbf {p}^T M^{-1}(\\mathbf {q})\\ \\mathbf {p}+V(\\mathbf {q})$ is a separable Hamiltonian of the system with symmetric and positive definite mass matrix $M(\\mathbf {q})$ and a field potential $V(\\mathbf {q})$ ."
],
[
"DESIGN OF CONSERVATIVE DIFFERENCE SCHEMES",
"An approach to design of conservative difference scheme for solving Hamiltonian equations (REF ) is based on following stages.",
"First is a choice of appropriate type of generating function $S$ , which fixes a definite family of simplectic difference scheme [2].",
"Simplectic difference scheme corresponds to the canonical transformation of canonical variables [3].",
"Solution of Hamiltonian system (REF ) for a definite time moment can be represented as canonical transformation and that is why conserves the value of Hamiltonian [4].",
"Second is using of “forward” and “backward” Taylor expansion of coordinate and momentum on a time step for getting of a corresponding generating function and remaining values of coordinate and momentum.",
"At last, obtained at the previous stage explicit and implicit schemes are used for constructing symmetric simplectic scheme for solving Hamiltonian equations (REF ).",
"$S = \\frac{h}{2}\\left(\\frac{\\mathbf {q}^{k+1}-\\mathbf {q}^k}{h}\\right)^2 - \\frac{h}{2}\\left[\\ V(\\mathbf {q}^k) + V(\\mathbf {q}^{k+1})\\ \\right] -$ $ - \\frac{h}{12}\\left[\\ \\nabla V(\\mathbf {q}^{k+1})-\\nabla V(\\mathbf {q}^k)\\ \\right] \\cdot \\left(\\mathbf {q}^{k+1}-\\mathbf {q}^k\\right),$ $\\mathbf {p}^k = \\frac{\\mathbf {q}^{k+1}-\\mathbf {q}^k}{h}-\\frac{h}{12}\\left[\\ 5\\nabla V(\\mathbf {q}^k)+\\nabla V(\\mathbf {q}^{k+1})\\ \\right]-$ $-\\frac{h}{12}\\mathcal {H}(V)(\\mathbf {q}^k)\\cdot (\\mathbf {q}^{k+1}-\\mathbf {q}^k),$ $\\mathbf {p}^{k+1} = \\frac{\\mathbf {q}^{k+1}-\\mathbf {q}^k}{h}+\\frac{h}{12}\\left[\\ \\nabla V(\\mathbf {q}^k)+5\\nabla V(\\mathbf {q}^{k+1})\\ \\right] -$ $- \\frac{h}{12}\\mathcal {H}(V)(\\mathbf {q}^{k+1})\\cdot (\\mathbf {q}^{k+1}-\\mathbf {q}^k),$ where $\\mathcal {H}(V)(q)$ is Hessian matrix for $V(q)$ ."
],
[
"NUMERICAL EXPERIMENTS",
"Obtained by described above approach difference scheme of third order [5] was tested by solving the system of equations (REF ) with Hamiltonian $H(\\mathbf {p},\\mathbf {q})=\\frac{1}{2}(p_1^2+p_2^2)-\\frac{1}{\\sqrt{q_1^2+q_2^2}},$ so called Kepler two body problem, Results of numerical calculations are compared for the symmetric simplectic scheme of third order and well-known second order velocity Verlet scheme with a time step $h=0.2$ and a time interval $[0,T], \\,T=5000$ .",
"Approximate solution by Verlet scheme with a time step h=0.02 is used as an exact solution [5].",
"Figure: Phase trajectories.Figure: Conservation of Hamiltonian."
],
[
"Results",
"A new approach for constructing symmetric simplectic numerical schemes for solving Hamiltonian systems of equations is proposed.",
"The numerical schemes constructed by this approach produce more stable and accurate solution of Hamiltonian system (1) and better conserve the energy of a system on the large interval of numerical integration for a relatively large integration step in comparision with the Verlet method, which is usually using for solving equations of motion in molecular dynamics.",
"Acknowledgments.",
"E.G.N.",
"acknowledge Dr. B. Batgerel for computations and numerical results.",
"The work was supported in part by the Russian Foundation for Basic Research, Grant No.",
"15-01-06055."
]
] | 1605.05714 |
[
[
"The latent logarithm"
],
[
"Abstract Count or non-negative data are often log transformed to improve heteroscedasticity and scaling.",
"To avoid undefined values where the data are zeros, a small pseudocount (e.g.",
"1) is added across the dataset prior to applying the transformation.",
"This pseudocount considers neither the measured object's a priori abundance nor the confidence with which the measurement was made, making this practice convenient but statistically unfounded.",
"I introduce here the latent logarithm, or lag.",
"lag assumes that each observed measurement is a noisy realization of an unmeasured latent abundance.",
"By taking the logarithm of this learned latent abundance, which reflects both sampling confidence/depth and the object's a priori abundance, lag provides a probabilistically coherent, stable, and intuitive alternative to the questionable, but conventional \"log($x$ + pseudocount).\""
],
[
"Introduction",
"When working with count, or more generally, non-negative data one often encounters zeros.",
"Furthermore, such data usually demonstrate heteroscedasticity, with variance increasing in mean – a consequence of their governing probabilistic processes (e.g.",
"Poisson, or Negative-Binomial sampling) [1].",
"In order to render the data more homoscedastic, it is popular to log-transform the data prior to performing data analysis.",
"However, as $\\log (0)$ is undefined, applying this transformation directly across a dataset is not possible.",
"One often therefore adds a small pseudocount (e.g.",
"$+1$ when working with count data) prior to applying the log, but this is a questionable practice.",
"Samples are often unevenly explored, and it can be unclear whether a zero actually suggests zero abundance, or some minuscule abundance that was not within the resolution of sampling.",
"Indeed, the authors of [2] argue against log-transforming count data prior to applying more standard, easy-to-work with statistical approaches (e.g.",
"linear regression).",
"Instead they argue for modeling the data directly using a Poisson or Negative Binomial GLM, both of which model the logarithm of the expected value of the datalog transforming the data and applying linear regression is modeling the expected value of the log of the data, which is not the same as modeling the log of the expected value.",
"I agree with these intuitions.",
"Nevertheless, by virtue of modeling (log) expectations as a linear combination of user defined predictors, GLMs impose structure on the data and require it to be treated in a supervised manner.",
"Thus, GLMs offer us little in the way of applying unsupervised methods (e.g.",
"PCA, MDS, clustering) to non-negative data.",
"Using a Poisson-Normal hierarchical model, I propose here the latent logarithm, hereafter “lag”, that computes the logarithm of the measured object's latent, or denoised abundance in an unsupervised manner.",
"Importantly, in the limit of data lag $=$ log, and in the absence of it, lag returns a prior belief.",
"Furthermore, lag considers the level of confidence in or exploration of a sample, such that a zero for a sample that was well explored is treated differently than a zero for a sample for which we have low confidence.",
"Thus the latent logarithm provides an intuitive and more nuanced alternative to the standard psuedocounted logarithm for application to count or non-negative data."
],
[
"Model",
"Let $t \\in \\mathbb {R}_{\\ge 0}^n$ be a $n$ -samples long vector of data (e.g.",
"counts, rates).",
"Let $o \\in \\mathbb {R}_{> 0}^n$ be a $n$ -samples long vector of `offsets', `exposures', sampling depths, or `confidences' as makes conceptual sense.",
"For example, in the case where $t_i$ represents the number of times a specific species of animal was observed in a random sampling, $o_i$ would be the size of the random sample.",
"I assume that associated with each $t_i$ there is a true, but unmeasured log-latent rate $z_i$ that $t_i$ is ultimately a noisy realization of.",
"Specifically, I assume the following hierarchical model, $z_i &\\sim \\mathcal {N}\\left(\\mu , \\sigma ^2\\right) \\\\t_i & \\sim \\textrm {Continuous-Poisson}(\\exp (z_i)o_i)$ Here $\\mu $ and $\\sigma ^2$ denote the mean and variance of $z_i$ .",
"As $z_i$ is the log-latent rate, $\\exp (z_i)$ is therefore the latent rate and $\\exp (z_i)o_i$ is the latent abundance.",
"I define the Continuous-Poisson distribution to have the following density function with support on $x \\in [0, \\infty )$ : $f(x|\\lambda ) = C_{\\lambda }\\frac{ \\lambda ^x e^{-\\lambda }}{\\Gamma (x + 1)}$ where $C_\\lambda $ is a normalization constant that ensures the density integrates to unity.",
"This distribution has the same shape and moments as the Poisson, but has the added advantage of being able to consider all non-negative data – not just counts.",
"As an example, suppose we are looking for different species of birds.",
"In this case, $t_i$ would represent the number of times we saw the bird on excursion $i$ , and $o_i$ would represent the number of hours we spent looking.",
"For a rare bird, the latent rate $\\exp (z_i)$ would be close to zero sightings/hour (negative $z_i$ ), whereas for a common bird, $\\exp (z_i)$ would be a positive number (positive $z_i$ ).",
"From this example it's clear that we have, $\\textrm {units}(\\textrm {latent rate}) = \\textrm {units}(\\exp (z_i)) = \\frac{ \\textrm {units}(t_i)}{ \\textrm {units}(o_i)}$ With these intuitions in mind, I define the latent logarithm to be, $\\textrm {lag}(t_i) = \\log (\\mathbb {E}[t_i|z_i]) = z_i + \\log (o_i)$ Given we are often more directly interested in the rate of an event (e.g.",
"seeing a bird many times when looking for many hours is the same as seeing a bird only a few times when looking for only a few hours), I also define the normalized latent logarithm (nlag) to be, $\\textrm {nlag}(t_i) = \\log (\\mathbb {E}[t_i|z_i]/o_i) = z_i$ In practice, we will not know $z_i$ , $\\mu $ , and $\\sigma ^2$ and must therefore learn their value."
],
[
"Inference",
"The joint likelihood of $\\lbrace t_i\\rbrace _{i=1}^n$ and $\\lbrace z_i\\rbrace _{i=1}^n$ is given by, $p(\\lbrace t_i\\rbrace _{i=1}^n, \\lbrace z_i\\rbrace _{i=1}^n | \\mu , \\sigma ^2) & = p(\\lbrace t_i\\rbrace _{i=1}^n | \\lbrace z_i\\rbrace _{i=1}^n, \\mu , \\sigma ^2) p(\\lbrace z_i\\rbrace _{i=1}^n |\\mu , \\sigma ^2)\\\\& = \\prod _{i=1}^n p(t_i | z_i, \\mu , \\sigma ^2) p(z_i | \\mu , \\sigma ^2) \\\\& = \\prod _{i=1}^n C_{\\exp (z_i)o_i} \\frac{ [\\exp (z_i)o_i]^{t_i} e^{-\\exp (z_i)o_i}}{\\Gamma (t_i + 1)} \\\\ & \\qquad \\times \\frac{1}{\\sqrt{2\\pi \\sigma ^2}} \\exp \\left( -\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2 \\right)$ This objective is difficult to optimize jointly in $z$ and in $\\mu $ and $\\sigma ^2$ .",
"However, given $z$ the task is considerably easier.",
"Our estimates for $\\mu $ and $\\sigma ^2$ , would simply the usual sample mean and variance of $z$ .",
"This suggests an approach where we iteratively condition on some estimate or distribution over $z$ given estimates $\\hat{\\mu }$ and $\\hat{\\sigma }^2$ , and subsequently maximize $\\mu $ and $\\sigma ^2$ given our current understanding of $z$ .",
"One approach could be to use Expectation-Maximization (EM), in which we continuously maximize the expected log-likelihood.",
"Because this expectation is taken with respect to $z$ , we require a posterior distribution over $z$ , which in turn requires us to calculate the marginal distribution of $p(t_i | \\mu , \\sigma ^2) = \\int p(t_i | z_i, \\mu , \\sigma ^2)p(z_i | \\mu , \\sigma ^2) \\textrm {d}z_i$ .",
"Unfortunately, this integral is not analytically solvable, making an exact EM approach hard.",
"While we could resort to inexact sampling techniques (e.g.",
"using Metropolis-Hastings MCMC, where the marginal distribution is not needed to sample from the posterior), these approaches are slow.",
"Therefore, instead, I consider $z$ to simply be another variable in the model, and employ the iterative conditional modes (ICM) algorithm [3].",
"As mentioned before, the estimates of $\\mu $ and $\\sigma ^2$ are straightforward given an estimate of $z$ .",
"Given $\\mu $ and $\\sigma ^2$ , our goal will then be to set $z$ to be the maximizer of its posterior distribution.",
"These two steps can then be iterated until convergence of the (log) data likelihood given above."
],
[
"Maximum ",
"A reasonable estimate of $z_i$ is given by the mode value of its posterior distribution.",
"This objective is given by, $\\hat{z}_i = \\operatornamewithlimits{argmax}_{z_i} p\\left(z_i|t_i,\\mu , \\sigma ^2 \\right) & = \\operatornamewithlimits{argmax}_{z_i} \\log \\left[ \\frac{ p\\left(t_i|z_i,\\mu , \\sigma ^2 \\right)p\\left(z_i |\\mu , \\sigma ^2 \\right) }{ \\int _{-\\infty }^{\\infty } p\\left(t_i|x,\\mu , \\sigma ^2 \\right)p\\left(x |\\mu , \\sigma ^2 \\right) \\textrm {d}x } \\right] \\\\& = \\operatornamewithlimits{argmax}_{z_i} \\log p\\left(t_i|z_i,\\mu , \\sigma ^2 \\right) + \\log p\\left(z_i |\\mu , \\sigma ^2 \\right) \\\\& = \\operatornamewithlimits{argmax}_{z_i} \\log \\left\\lbrace C_{\\exp (z_i)o_i} \\frac{ [\\exp (z_i)o_i]^{t_i} e^{-\\exp (z_i)o_i}}{\\Gamma (t_i + 1)}\\right\\rbrace \\\\& \\hspace{38.41121pt} + \\log \\left\\lbrace \\frac{1}{\\sqrt{2\\pi \\sigma ^2}} \\exp \\left( -\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2 \\right) \\right\\rbrace \\\\& = \\operatornamewithlimits{argmax}_{z_i} t_i z_i - \\exp (z_i)o_i -\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2$ where I have progressively dropped constants that do not depend on $z_i$ .",
"Differentiating we get, $\\frac{\\partial }{\\partial z_i} t_i z_i - \\exp (z_i)o_i -\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2 = t_i - \\exp (z_i)o_i -\\frac{z_i - \\mu }{\\sigma ^2}.$ We cannot solve for this gradient analytically, so instead we rely on Newton-Raphson to optimize the gradient numerically.",
"The required Hessian is given by, $\\frac{\\partial }{\\partial z_i} t_i - \\exp (z_i)o_i -\\frac{z_i - \\mu }{\\sigma ^2} = - \\exp (z_i)o_i -\\frac{1}{\\sigma ^2} < 0 .$ Note that the Hessian is negative-definite, which implies there is a unique maximum for our objective."
],
[
"Properties",
"I now note some important properties of the latent logarithm.",
"Note that given some moderate $z_i$ , $t_i \\rightarrow \\infty $ as $o_i \\rightarrow \\infty $ since $\\mathbb {E}[t_i|z_i] = \\exp (z_i)o_i$ .",
"Thus in the limit of data we have, $& \\lim _{t_i \\rightarrow \\infty , o_i \\rightarrow \\infty } \\operatornamewithlimits{argmax}_{z_i} t_i z_i - \\exp (z_i)o_i -\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2 = \\operatornamewithlimits{argmax}_{z_i} t_i z_i - \\exp (z_i)o_i$ which is simply an optimization of the Poisson likelihood.",
"With one data point the maximum likelihood estimator of a Poisson mean is just the value of the datum itself.",
"Consequently, $\\lim _{t_i \\rightarrow \\infty ,o_i \\rightarrow \\infty }\\textrm {lag}(t_i) = \\log (\\mathbb {E}(t_i|z_i)) = \\log (t_i).$ Now consider the situation in which again we have ample observation ($o_i \\rightarrow \\infty $ ), but that $t_i = 0$ .",
"Intuitively, this must be because the latent rate is minuscule.",
"To confirm this intuition mathematically, notice that in this case our objective becomes, $& \\lim _{t_i \\rightarrow 0, o_i \\rightarrow \\infty } \\operatornamewithlimits{argmax}_{z_i} t_i z_i - \\exp (z_i)o_i -\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2 = \\operatornamewithlimits{argmax}_{z_i} - \\exp (z_i)o_i$ Note that $- \\exp (z_i)o_i$ has no maximum, but that it is always increasing as $z_i \\rightarrow -\\infty $ .",
"This confirms our intuitions, and we can see that when samples are deeply explored but still no events are detected, lag (or more appropriately in this case, nlag), will suggest the rate of the process is low.",
"Consider the case where we have limited data and observation (e.g.",
"$t_i = 0$ and $o_i = 0$ ).",
"Here we have, $\\lim _{t_i \\rightarrow 0, o_i \\rightarrow 0} \\operatornamewithlimits{argmax}_{z_i} t_i z_i - \\exp (z_i)o_i -\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2 = \\operatornamewithlimits{argmax}_{z_i} -\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2$ which is simply the Normal likelihood, for which $z_i = \\mu $ is the maximizer.",
"Thus, $\\lim _{t_i \\rightarrow 0,o_i \\rightarrow 0}\\textrm {nlag}(t_i) = \\log (\\mathbb {E}(t_i|z_i)/o_i) = \\mu .$ This says that in the absence of data, our best guess for the (log) latent rate of an event, is simply the mean of the distribution that encodes prior beliefs about the process rate.",
"Finally, consider the influence of the prior.",
"We can write the objective as follows: $\\operatornamewithlimits{argmax}_{z_i} \\underbrace{[t_i z_i - \\exp (z_i)o_i]}_{\\textrm {Poisson component}} + \\underbrace{\\left[-\\frac{1}{2\\sigma ^2}(z_i - \\mu )^2\\right]}_{\\textrm {Normal component}}$ As the prior variance decreases, thereby encoding stronger prior beliefs, the influence of the `Normal component' of the posterior increases.",
"Similarly, as the prior mean shifts toward extreme values (again encoding stronger prior beliefs), the `Normal component' of the posterior again dominates and accordingly “drags” along with it the estimate of $z_i$ ."
],
[
"Results",
"I first examine the latent logarithm when $\\mu $ and $\\sigma $ are fixed in order to gain a better understanding of how the Normal prior distribution interacts with the Continuous-Poisson layer.",
"I then demonstrate complete use of the latent logarithm, where $z$ , $\\mu $ , and $\\sigma ^2$ are all learned.",
"Figure: Behavior of the latent logarithm on a gene expression dataset.",
"left) Comparison of lag (t i )\\textrm {lag}(t_i) versus a conventional psuedocount approach of taking log(t i +1)\\log (t_i + 1).",
"The red line corresponds to the x=yx = y line.",
"Points in blue are those for which t i t_i equals 0. middle) Estimated log-latent TPM (xx-axis) as the sequencing depth (yy-axis) decreases for samples for which the gene was not detected.",
"right) Estimates of log-latent TPMs for synthetic data where sequencing depth is shrunk to effectively zero.",
"In the middle and right plots the gray curve illustrates the prior density."
],
[
"Fixed $\\mu $ and {{formula:c3462ad8-7a97-4200-9305-e37722434eb5}}",
"In order to examine the behavior of the latent logarithm under a fixed prior, I investigated gene expression data consisting of 2597 samples for the Arabidopsis thaliana gene AT2G43386 [4].",
"This gene is rarely expressed and in most of these samples has a measured abundance in Transcripts per Million reads (TPM) of 0.",
"However, some of these 0 TPM samples were explored more heavily$\\endcsname $Gene expression is often measured by RNA sequencing, which for the purposes of this discussion, incompletely samples a large, heterogeneous pool of transcripts (copies of expressed genes).",
"The number of detected transcripts for a specific gene is therefore proportional to the sequencing depth of that sample – or in other words, how deeply it was probed.",
"than others, and for the remaining samples a positive TPM was measured.",
"I hereafter refer to this gene as the `rare' gene.",
"To obtain $t$ , I first converted the measured TPMs to a measured number of transcripts by multiplying the measured TPM by the sequencing depth of the sample (in number of reads) divided by 1 million.",
"Intuitively then, $o$ is simply the sequencing depths of each sample (in millions of reads).",
"Note that given the original measurements were already given as rates (TPMs), we could directly model these rates in our model by setting $o_i = 1$ for all $i$ .",
"However, doing so would ignore the information encoded in the sequencing depth.",
"For example, a TPM of 0 (indicating no expression) in a lightly sequenced sample is not as believable as a TPM of 0 in a heavily sequenced sample.",
"To clearly demonstrate the influence of the prior distribution, I set the prior mean $\\mu $ to $0.25$ and the the prior variance to $0.05$ .",
"Figure 1 illustrates the results of applying the latent logarithm to the above gene expression dataset.",
"Specifically, Figure 1 (left) shows how $\\textrm {lag}(t_i)$ compares to $\\log (t_i + 1)$ .",
"The psuedocount of 1, a commonly added value, was added so that 0 values were not undefined.",
"Notice that as $t_i$ grows large $\\log (t_i + 1) \\approx \\log (t_i) \\approx \\textrm {lag}(t_i)$ , as expected, and the prior has little influence.",
"Note that the positive bias (above the red line) observed in the plot for moderate (non-zero), but not large $t_i$ is due to the effect of the pseudocount.",
"Additionally, we see some variation between $\\textrm {lag}(t_i)$ and $\\log (t_i + 1)$ for when $t_i$ is moderate, but not large (e.g.",
"around where $\\log (t_i + 1) = 1.5$ ).",
"For these samples lag estimates that the gene is actually more abundant than measured.",
"Indeed, this is because these samples were not heavily sequenced, and so lag does not know whether the low measured abundance was due to variability introduced by shallow sequencing or because the gene is truly not very expressed.",
"Consequently, it places some emphasis on the prior.",
"How lag manages the trade-off between trusting the data versus the prior is most clearly seen for samples in which the measured TPM abundance equals 0 (blue points in Figure 1 (left)).",
"Intuitively, we should be most confident in such an estimate when our sequencing depth is high.",
"This should be reflected in the fact that the estimated latent rate should be low.",
"Conversely, if we measured an abundance of 0 but did not sequence very deeply, then we do not have much information to tell us whether that 0 really represents zero abundance or a zero due to technical error.",
"Therefore, our best estimate for the estimated latent rate should simply reflect our prior belief.",
"Indeed, this is exactly what we see in Figure 2 (middle), where I have plotted the estimated log-latent TPM ($\\textrm {nlag}(t_i)$ ) along with the sequencing depth of that sample.",
"Here, as sequencing depth increases, the latent logarithm is more and more confident that a measured abundance of 0 corresponds to a small log-latent TPM.",
"Conversely, as depth decreases, the latent logarithm puts greater trust in the prior.",
"To further confirm the latent logarithm behaves as expected and is numerically stable for even the most scant data situations, I generated synthetic data points where $t_i = 0$ and $o_i$ ranged from 10 to $1 \\times 10^{-6}$ million reads on a logarithmic grid.",
"The case where 0 transcripts are detected for a rare gene when a sample is sequenced to a depth of only 1 read is functionally equivalent to never probing that sample in the first place.",
"Thus, the log-latent rate should converge to the prior mean.",
"Figure 1 (right) confirms this intuition exactly."
],
[
"Complete usage",
"In practice, we do not know $\\mu $ and $\\sigma ^2$ , and must learn it from the data.",
"We initialize the prior mean and variance for the ICM algorithm using the $+1$ pseudocount approach such that $\\hat{\\mu }_{\\textrm {init}}$ and $\\hat{\\sigma ^2}_{\\textrm {init}}$ equals the sample mean and variance of $\\log (t/o + 1)$ (here, the division is taken element wise).",
"Figure 2 illustrates the completely estimated ($z$ , $\\mu $ , $\\sigma ^2$ all learned) lag function for the `rare' gene.",
"The learned latent mean and variance are $-4.63$ and $2.30$ , which define a latent abundance distribution that is considerably left shifted and wider than the fixed-prior example given above (Figure 2, right).",
"Stated another way, lag a priori believes for this gene that an observed TPM of 0 is really a TPM more like $\\exp (-4.63) = 0.0098$ .",
"If, as depth increases to a large value, the observed TPM is still 0, then lag estimates the latent TPM to be approximately $\\exp (-6) = 0.0025$ .",
"Note that these values are a few orders of magnitude less than 1.",
"So, whether one interprets the $+1$ pseudocount as a prior belief of the transcript's abundance or applies it out of convenience, it's clear that for this rarely expressed gene (TPM equals 0 94$\\%$ of the time), a pseudocount of 1 is far too generous.",
"This is made clear in examining Figure 2 (left) where $\\log (t_i + 1)$ consistently exceeds $\\textrm {lag}(t_i)$ for small $t_i$ .",
"I repeated this analysis for a more abundant gene, AT1G14630, which I hereafter simply refer to as the `abundant gene.'",
"Unlike the rare gene, the abundant gene has non-zero TPM 97$\\%$ of the time in the dataset.",
"As expected, the learned latent abundance prior distribution is right shifted compared to the rare gene, with mean and variance equal to 0.83 and 1.80, respectively (Figure 3, right).",
"Additionally, as seen before in previous examples, for samples with zero TPM levels the estimated latent abundance decreases in increasing sampling depth.",
"However curiously, even for a shallowly sequenced sample (2.1096 million reads) in which we observe 0 TPM, the estimated log-latent TPM is still far from the prior mean.",
"This is because for this abundant gene, 2 million reads should still be enough to detect it and so it's more likely that the observed 0 is real.",
"lag hedges its bets by assigning this sample a latent TPM of 0.43.",
"Finally, note that unlike for the rare gene, $\\textrm {lag}(t_i)$ generally (though not always when sufficient sampling is available) exceeds $\\textrm {log}(t_i)$ , especially for 0 TPM samples.",
"Intuitively, this is simply because for a robustly expressed gene, with an average number of transcripts of 100.2 and a median transcript count of 49.8, a pseudocount of 1 is too stingy, especially for deeply sequenced samples.",
"Figure: Behavior of the latent logarithm for the `abundant gene.'",
"The left and right panels are presented in the same style as the left and middle panels of Figure 1."
],
[
"Discussion",
"In count or non-negative data, there are different kinds of zeros.",
"Zero measured abundances are more believable for a priori rare objects/events or deeply probed samples, than for abundant objects/events or shallowly probed samples.",
"Applying a fixed pseudocount across the dataset ignores this information.",
"lag addresses this shortcoming by taking the logarithm of the object's learned latent abundance, which considers both the object's overall abundance and sampling depth.",
"Nevertheless, unlike log, lag is a statistical routine, and like all methods of inference, it is only as good as the data it is applied to.",
"Though our algorithm converges even for two samples, the estimated latent prior distribution and the latent abundances are not necessarily stable.",
"In these situations, it may be prudent to constrain lag by placing prior distributions on the parameters of latent prior density.",
"Alternatively and perhaps ideally, the latent prior density should be learned on a larger previously assembled dataset that's representative of the dataset to be analyzed.",
"Importantly, when considering multiple intellectually linked datasets, the same latent prior distribution should be used so that lag values are globally comparable.",
"This can be done by using an already learned prior (from a previous dataset), or by applying lag to a concatenation of all datasets under consideration.",
"Finally, one may question the use of the Normal distribution as a prior density over the log of the Poisson rate parameter$\\endcsname $or equivalently, the log-normal distribution over the Poisson rate , instead of a more standard distribution, e.g.",
"Gamma, which is conjugate to the Poisson.",
"I argue the Normal actually enables greater expressivity.",
"It's known that the Poisson-Normal hierarchy allows for the same mean-variance flexibility as the Negative-BinomialLet $t_i \\sim \\textrm {Poisson}(\\lambda )$ and $\\lambda \\sim \\textrm {Gamma}(\\alpha , \\beta )$ .",
"Then $p(t_i|\\alpha ,\\beta ) = \\int _\\lambda p(t_i|\\lambda )p(\\lambda |\\alpha ,\\beta )\\textrm {d}\\lambda $ is a Negative-Binomial distribution.",
"[5].",
"However, unlike what can be done with the Gamma prior, one can encode considerable model structure into the mean and (co)variance of the Normal, as well as take advantage of its nice conditioning and marginalization properties.",
"For example, one might consider extending $\\mu $ to be a conditional mean $\\mu (X) = X\\beta $ , which depends on some user known/input set of covariates collected in the columns of $X$ .",
"Perhaps most excitingly, the univariate normal distribution may be extended to the multivariate domain, in which now the Normal's covariance matrix can couple latent abundances – and therefore ultimately the observed abundances – across multiple count measured objects [6], [5].",
"Thus, the Normal-Poisson hierarchy can enable the application of very well established and powerful multivariate methods to count data.",
"Though lag does not attempt this complexity, it provides a simple and sound transformation that can serve as an initialization or null model for these more expressive models.",
"Importantly, it also offers a more principled way to preprocess non-negative data for unsupervised learning tasks than does the usual pseudocounted logarithm."
],
[
"Conclusion",
"I introduced lag, a probabilistic alternative to the standard “$\\textrm {log}(x + \\textrm {pseudocount})$ ” that computes the logarithm of the measured object's learned latent abundance.",
"In the limit of increasing data, lag quickly converges to log.",
"However, in situations of poor sampling or object rarity, lag considers all available information to calculate the object's log-latent abundance.",
"This statistically sound and more nuanced approach is an improvement over the commonplace practice of transforming count-data-plus-pseudocount with the standard logarithm."
]
] | 1605.06064 |
[
[
"A Lagrangian view on complete integrability of the two-component\n Camassa-Holm system"
],
[
"Abstract We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa-Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem.",
"The respective coordinates correspond to different normalizations of an associated first order system.",
"In particular, we will see that the two-component Camassa-Holm system in Lagrangian variables is completely integrable as well."
],
[
"Introduction",
"The Camassa–Holm (CH) equation [8], [9] $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0,$ which serves as a model for shallow water waves [16], has been studied intensively over the last twenty years, due to its rich mathematical structure.",
"For example, it is bi-Hamiltonian [23], formally completely integrable [11], has infinitely many conserved quantities [36], and for a huge class of smooth initial data, the corresponding classical solution only exists locally in time due to wave breaking [12], [13], [14].",
"Especially the last property attracted a lot of attention and led to the construction of different types of global weak solutions via a generalized method of characteristics [6], [7], [32], [33], [26].",
"For conservative solutions, another possible approach is based on the solution of an inverse problem for an indefinite Sturm–Liouville problem [3], [18], [19], [21]; the inverse spectral method.",
"The aim of this note is to point out some connections between these two ways of describing weak conservative solutions.",
"Over the last few years various generalizations of the CH equation have been introduced.",
"A lot of them have been constructed in such a way that one or several properties of the CH equation are preserved.",
"Among them is the two-component Camassa–Holm (2CH) system [15] $u_t - u_{xxt} + 3uu_x - 2u_x u_{xx} - u u_{xxx} + \\rho \\rho _x = 0, \\\\\\rho _t + (u\\rho )_x = 0,$ that may also be written in the alternative form $u_t+uu_x+p_x& =0,\\\\\\rho _t+(u\\rho )_x& =0,$ where the auxiliary function $p$ solves the differential equation $p-p_{xx}=u^2+\\frac{1}{2} u_x^2+\\frac{1}{2} \\rho ^2.$ From our point of view this generalization is of special interest, not only because it has been derived in the context of shallow water waves [15], but also because weak solutions can be described via a generalized method of characteristics [25], [26] as well as via an underlying isospectral problem [10], [15], [34].",
"Thus we are going to study the 2CH system, which reduces to the CH equation when $\\rho $ vanishes identically.",
"As already hinted above and presented in [26], there is not only one class of weak solutions but several of them, dependent on how the energy is manipulated when wave breaking occurs.",
"This means that the spatial derivative $u_x(\\,\\cdot \\,, t)$ of the solution $(u(\\,\\cdot \\,,t), \\rho (\\,\\cdot \\,,t))$ becomes unbounded within finite time, while both $\\left\\Vert u(\\,\\cdot \\,,t)\\right\\Vert _{H^1({\\mathbb {R}})}$ and $\\left\\Vert \\rho (\\,\\cdot \\,,t)\\right\\Vert _{L^2({\\mathbb {R}})}$ remain bounded, see e.g.",
"[27], [28], [29], [30], [43], [44] and the references therein.",
"In addition, energy concentrates on sets of measure zero.",
"If one continues solutions after wave breaking in such a way that the total amount of energy, which is described by a Radon measure $\\mu $ , remains unchanged in time, one obtains the so-called weak conservative solutions [25].",
"Thus any weak conservative solution is described by a triplet $(u, \\rho ,\\mu )$ , where the connection between $u$ , $\\rho $ and $\\mu $ is given through $\\mu _{\\mathrm {ac}}=(u_x^2+\\rho ^2)dx$ , and $\\mu ({\\mathbb {R}},t)$ is independent of time.",
"The construction of these solutions by a generalized method of characteristics relies on a transformation from Eulerian to Lagrangian coordinates [25], based on [6], [32], which will be reviewed in Section (we refer to [25] for details).",
"Under this transformation, the 2CH system can be rewritten in Lagrangian variables (for conservative solutions) as $ y_t & = U,\\\\U_t & = -Q, \\\\h_t & = 2(U^2-P)U_\\xi , \\\\r_t & = 0,$ where the functions $P$ and $Q$ are given by $P(\\xi ,t) & =\\frac{1}{4} \\int _{\\mathbb {R}}e^{-\\vert y(\\xi ,t)-y(s,t)\\vert } (2U^2y_\\xi +h)(s,t)ds, \\\\ Q(\\xi ,t) & = -\\frac{1}{4} \\int _{\\mathbb {R}}\\mathrm {sign}(\\xi -s)e^{-\\vert y(\\xi ,t)-y(s,t)\\vert } (2U^2y_\\xi +h)(s,t)ds.$ Note that the three Eulerian coordinates $(u,\\rho ,\\mu )$ are mapped to four Lagrangian coordinates $(y,U,h,r)$ , which indicates that to each element $(u,\\rho ,\\mu )$ there corresponds an equivalence class in Lagrangian coordinates.",
"These equivalence classes can be identified with the help of relabeling functions.",
"The purpose of this note is to study what the change from Eulerian to Lagrangian variables means in terms of the isospectral problem underlying the 2CH system.",
"It is known [10], [15], [34] that the 2CH system can be written as the condition of compatibility for the overdetermined system $- \\psi _{xx} + \\frac{1}{4}\\psi & = z (u-u_{xx}) \\psi + z^2 \\rho ^2 \\psi , \\\\\\psi _t & = \\frac{1}{2} u_x \\psi - \\left(\\frac{1}{2z}+u\\right) \\psi _x.$ In particular, the spectrum associated with (REF ) is invariant under the 2CH flow.",
"We will see in Lemma REF that the isospectral problem, that is, the differential equation (REF ) can be rewritten as a particular equivalent first order system.",
"It then turns out that a normalizing standard transformation (which is well-known in the theory of canonical systems [42]) takes this system to another equivalent first order system that only involves the Lagrangian variables $(y, U,h,r)$ as coefficients; see Lemma REF .",
"Moreover, relabeling of Lagrangian variables simply amounts to an elementary reparametrization of the first order system; see Lemma REF .",
"From this point of view, the relation between Eulerian and Lagrangian variables can be understood as different kinds of normalizations of the same (that is, equivalent up to reparametrizations) first order system.",
"Lagrangian coordinates (in $\\mathcal {F}_0$ ) correspond to trace normalization of an associated weight matrix and Eulerian coordinates correspond to normalization of its bottom-right entry.",
"That our newly obtained first order system (REF ) indeed serves as an isospectral problem for the Lagrangian version of the 2CH system is then shown in Section .",
"More precisely, we will see that the system () turns out to be completely integrable in the sense that it can be reformulated as the compatibility condition for an overdetermined system."
],
[
"Notation",
"For integrals of a continuous function $f$ with respect to a Radon measure $\\nu $ on ${\\mathbb {R}}$ , we will employ the convenient notation $\\int _x^y f d\\nu = {\\left\\lbrace \\begin{array}{ll}\\int _{[x,y)} f d\\nu , & y>x, \\\\0, & y=x, \\\\-\\int _{[y,x)} f d\\nu , & y< x,\\end{array}\\right.}",
"\\qquad x,\\,y\\in {\\mathbb {R}},$ rendering the integral left-continuous as a function of $y$ .",
"If $f$ is even locally absolutely continuous on ${\\mathbb {R}}$ and $g$ denotes a left-continuous distribution function of $\\nu $ , then we have the integration by parts formula $\\int _{x}^y f d\\nu = \\left.",
"g f\\right|_x^y - \\int _{x}^y g(s) f^{\\prime }(s) ds, \\quad x,\\,y\\in {\\mathbb {R}},$ which can be found in [5] for example."
],
[
"From Eulerian to Lagrangian coordinates",
"In this section we will briefly outline the change from Eulerian to Lagrangian coordinates for the two-component Camassa–Holm system.",
"This has been done for the conservative case in [25], where the interested reader may find additional details.",
"For the sake of simplicity and readability, we will only consider the case of vanishing spatial asymptotics, that is, when the initial data $(u_0,\\rho _0)$ belongs to $H^1({\\mathbb {R}})\\times L^2({\\mathbb {R}})$ .",
"It is well-known that even in the case of smooth initial data, wave breaking can occur within finite time, that is, energy may concentrate on sets of Lebesgue measure zero.",
"Dependent on how the concentrated energy is manipulated, one may obtain different kinds of global weak solutions, the most prominent ones being the conservative and dissipative ones; see [26].",
"For our purposes (that is, viewing the two-component Camassa–Holm system as an integrable system), the conservative solutions are the appropriate choice.",
"In order to obtain a well-posed notion of global solutions, we need to take wave breaking into account by augmenting the Eulerian coordinates with a non-negative Radon measure $\\mu $ describing the energy of a solution.",
"Definition 2.1 (Eulerian coordinates) The set $\\mathcal {D}$ is composed of all triples $(u,\\rho ,\\mu )$ such that $u$ is a real-valued function in $H^1({\\mathbb {R}})$ , $\\rho $ is a non-negative function in $L^2({\\mathbb {R}})$ and $\\mu $ is a non-negative and finite Radon measure on ${\\mathbb {R}}$ , whose absolutely continuous part $\\mu _{\\mathrm {ac}}$ is given by $\\mu _{\\mathrm {ac}}= \\left(u_x^2 +\\rho ^2\\right) dx.$ The main benefit of the change from Eulerian to Lagrangian coordinates lies in the fact that the measure $\\mu $ turns into a function which allows to apply a generalized method of characteristics in a suitable Banach space to solve the two-component Camassa–Holm system in Lagrangian variables.",
"Before introducing the set of Lagrangian coordinates $\\mathcal {F}$ , we have to define the set of relabeling functions, which will also enable us to identify equivalence classes in Lagrangian coordinates.",
"Definition 2.2 (Relabeling functions) We denote by $\\mathcal {G}$ the subgroup of the group of homeomorphisms $\\phi $ from ${\\mathbb {R}}$ to ${\\mathbb {R}}$ such that $\\phi -{\\mathrm {id}}\\text{ and } \\phi ^{-1}-{\\mathrm {id}}& \\text{ both belong to } W^{1,\\infty }({\\mathbb {R}}),\\\\\\text{and }\\phi _\\xi -1 & \\text{ belongs to } L^2({\\mathbb {R}}).$ Definition 2.3 (Lagrangian coordinates) The set $\\mathcal {F}$ is composed of all quadruples of real-valued functions $(y, U, h, r)$ such that $& (y - {\\mathrm {id}}, U, h,r, y_\\xi - 1, U_\\xi )\\in L^\\infty ({\\mathbb {R}})\\times [L^2({\\mathbb {R}})\\cap L^\\infty ({\\mathbb {R}})]^5,\\\\& y_\\xi \\ge 0,~h\\ge 0,~y_\\xi +h>0 \\text{ almost everywhere on }{\\mathbb {R}}, \\\\ & y_\\xi h=U_\\xi ^2+ r^2 \\text{ almost everywhere on }{\\mathbb {R}}, \\\\& y+H\\in \\mathcal {G},$ where we introduce $H$ by setting $H(\\xi ) = \\int _{-\\infty }^\\xi h(s)ds$ .",
"With these definitions, we are now able to describe the transformation between the sets of Eulerian and Lagrangian coordinates.",
"Definition 2.4 For any $(u,\\rho ,\\mu )$ in $\\mathcal {D}$ we define $(y,U,h,r)$ by $y(\\xi )&=\\sup \\lbrace x\\in {\\mathbb {R}}\\mid x+\\mu ((-\\infty ,x))<\\xi \\rbrace ,\\\\ U(\\xi )& = u\\circ y(\\xi ),\\\\ h(\\xi )& = 1-y_\\xi (\\xi ),\\\\r(\\xi )& = y_\\xi (\\xi )\\,\\rho \\circ y(\\xi ).$ Then $(y,U,h,r)$ belongs to $\\mathcal {F}$ and we denote by $L:\\mathcal {D}\\mapsto \\mathcal {F}$ the mapping which to any $(u,\\rho ,\\mu )\\in \\mathcal {D}$ associates $(y,U,h,r)\\in \\mathcal {F}$ as given by (REF ).",
"In order to get back from Lagrangian to Eulerian coordinates, we also introduce the following mapping, where the quantity $y_\\#(\\nu )$ denotes the push-forward by the function $y$ of a Radon measure $\\nu $ on ${\\mathbb {R}}$ .",
"Definition 2.5 For any $ (y,U,h,r)$ in $\\mathcal {F}$ we define $(u, \\rho , \\mu )$ by $u(x)& =U(\\xi ) \\text{ for any } \\xi \\text{ such that } x=y(\\xi ),\\\\\\mu & = y_\\#(h(\\xi )d\\xi ),\\\\\\rho (x)dx& = y_\\#(r(\\xi )d\\xi ).$ Then $(u, \\rho , \\mu )$ belongs to $\\mathcal {D}$ and we denote by $M:\\mathcal {F}\\mapsto \\mathcal {D}$ the mapping which to any $(y,U,h,r)\\in \\mathcal {F}$ associates $(u, \\rho , \\mu )\\in \\mathcal {D}$ as given by (REF ).",
"We say that $X$ and $\\hat{X}\\in \\mathcal {F}$ are equivalent, if there exists a relabeling function $\\phi \\in \\mathcal {G}$ such that $\\hat{X}=X\\circ \\phi $ , where $X\\circ \\phi $ denotes $(y\\circ \\phi , U\\circ \\phi , \\phi _\\xi \\cdot h\\circ \\phi , \\phi _\\xi \\cdot r\\circ \\phi )$ .",
"Upon taking equivalence classes in $\\mathcal {F}$ , it turns out that the mappings $L$ and $M$ are inverse to each other.",
"In particular, if we introduce the class $\\mathcal {F}_0=\\lbrace X\\in \\mathcal {F}\\mid y+H={\\mathrm {id}}\\rbrace ,$ then $\\mathcal {F}_0$ contains exactly one representative of each equivalence class in $\\mathcal {F}$ .",
"Moreover, one readily sees that the range of the mapping $L$ is precisely the set $\\mathcal {F}_0$ .",
"The reformulation of the two-component Camassa–Holm system in Lagrangian coordinates (for conservative solutions) is given by () and admits a continuous semigroup of solutions.",
"Denoting by $S_t(X_0)$ the solution at time $t$ with initial data $X(0)=X_0\\in \\mathcal {F}$ , one has $S_t(X_0\\circ \\phi )=S_t(X_0)\\circ \\phi $ for all $\\phi \\in \\mathcal {G}$ (that is, the time evolution respects equivalence classes in $\\mathcal {F}$ ).",
"Upon going back to Eulerian coordinates, we obtain a continuous semigroup $M\\circ S_t\\circ L$ of solutions in $\\mathcal {D}$ that gives rise to global conservative weak solutions of the two-component Camassa–Holm system ()."
],
[
"Transformations of the isospectral problem",
"Throughout this section, we fix some $(u,\\rho ,\\mu )\\in \\mathcal {D}$ and define $\\omega $ in $H^{-1}({\\mathbb {R}})$ by $\\omega (h) = \\int _{\\mathbb {R}}u(x)h(x)dx + \\int _{\\mathbb {R}}u_x(x)h_x(x)dx, \\quad h\\in H^1({\\mathbb {R}}),$ so that $\\omega =u-u_{xx}$ in a distributional sense, as well as a non-negative and finite Radon measure $\\upsilon $ on ${\\mathbb {R}}$ such that $\\mu (B) = \\int _B u_x(x)^2 dx + \\upsilon (B)$ for every Borel set $B\\subseteq {\\mathbb {R}}$ .",
"Let us point out that it is always possible to recover the triple $(u,\\rho ,\\mu )$ from the distribution $\\omega $ and the measure $\\upsilon $ .",
"The isospectral problem for smooth solutions of the two-component Camassa–Holm system has the form $-f_{xx} + \\frac{1}{4} f = z\\, \\omega f + z^2 \\upsilon f,$ where $z$ is a complex spectral parameter.",
"Moreover, there are good reasons (see [18], [19] as well as Section ) to expect that it also serves as an isospectral problem for global conservative solutions of the two-component Camassa–Holm system [25].",
"Of course, due to the low regularity of the coefficients, the differential equation (REF ) has to be understood in a distributional sense; cf.",
"[18], [20], [24], [40].",
"Definition 3.1 A solution of (REF ) is a function $f\\in H^1_{{\\mathrm {loc}}}({\\mathbb {R}})$ such that $\\int _{{\\mathbb {R}}} f_x(x) h_x(x) dx + \\frac{1}{4} \\int _{\\mathbb {R}}f(x)h(x)dx = z\\, \\omega (fh) + z^2 \\int _{\\mathbb {R}}f h \\,d\\upsilon $ for every function $h\\in H^1_{\\mathrm {c}}({\\mathbb {R}})$ .",
"We will first show that, as long as $z$ is non-zero, the differential equation (REF ) can be transformed into an equivalent first order system of the form $\\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix} F_x = \\begin{pmatrix} u - \\frac{1}{4z} & 0 \\\\ 0 & 0 \\end{pmatrix} F + z \\begin{pmatrix} u_x^2 & u_x \\\\ u_x & 1 \\end{pmatrix} F + z \\begin{pmatrix} \\upsilon & 0 \\\\ 0 & 0 \\end{pmatrix} F.$ Since $\\upsilon $ may be a genuine measure, this system has to be understood as a measure differential equation [2], [4], [22], [38] in general: A solution of the system (REF ) is a function $F:{\\mathbb {R}}\\rightarrow 2$ which is locally of bounded variation with $\\begin{split}-F\\big |_{x_1}^{x_2} = \\int _{x_1}^{x_2} \\begin{pmatrix} 0 & 0 \\\\ u(s) - \\frac{1}{4z} & 0 \\end{pmatrix} F(s)ds & + z \\int _{x_1}^{x_2} \\begin{pmatrix} -u_x(s) & -1 \\\\ u_x(s)^2 & u_x(s) \\end{pmatrix} F(s) ds \\\\ & + z \\int _{x_1}^{x_2} \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} F d\\upsilon \\end{split}$ for all $x_1$ , $x_2\\in {\\mathbb {R}}$ .",
"In this case, the first component of $F$ is clearly locally absolutely continuous, whereas the second component is only left-continuous; cf.",
"(REF ).",
"Lemma 3.2 If the function $f$ is a solution of the differential equation (REF ), then there is a unique left-continuous function $f^{[1]}$ such that $f^{[1]}(x) = f_x(x) - z u_x(x) f(x)$ for almost all $x\\in {\\mathbb {R}}$ and the function $\\begin{pmatrix} z f \\\\ f^{[1]}\\end{pmatrix}$ is a solution of the system (REF ).",
"Conversely, if the function $F$ is a solution of the system (REF ), then its first component is a solution of the differential equation (REF ).",
"Upon integrating by parts in (REF ), we first note that a function $f\\in H^1_{\\mathrm {loc}}({\\mathbb {R}})$ is a solution of (REF ) if and only if there is a $c\\in {\\mathbb {R}}$ and a constant $d\\in such that{\\begin{@align}{1}{-1}\\begin{split}f_x(x) = d + \\frac{1}{4} \\int _c^x f(s)ds & -z\\int _c^x u(s)f(s)+u_x(s)f_x(s)\\,ds + z u_x(x)f(x) \\\\ & - z^2 \\int _c^x f \\,d\\upsilon \\end{split}\\end{@align}}for almost all $ xR$.So if $ f$ is a solution of~(\\ref {eqnDEho}), then this guarantees that there is a unique left-continuous function $ f[1]$ such that~(\\ref {eqnfqd}) holds for almost all $ xR$.It is straightforward to show that the function in~(\\ref {eqnFOSeulerVec}) is a solution of the system~(\\ref {eqnFOSeuler}).$ Now suppose that $F$ is a solution of the system (REF ) and denote the respective components with subscripts.",
"The first component of the integral equation (REF ) shows that $F_{1}$ belongs to $H^1_{\\mathrm {loc}}({\\mathbb {R}})$ with $z F_{2}(x) = F_{1,x}(x) - z u_x(x) F_{1}(x)$ for almost all $x\\in {\\mathbb {R}}$ .",
"In combination with the second component of (REF ) this shows that () holds with $f$ replaced by $F_{1}$ for some $c\\in {\\mathbb {R}}$ , $d\\in and almost every $ xR$, which shows that $ F1$ is a solution of the differential equation~(\\ref {eqnDEho}).$ Except for the potential term (that is, the first term on the right-hand side), the equivalent first order system (REF ) has the form of a canonical system; we only mention a small selection of references [1], [17], [31], [35], [37], [39], [41], [42].",
"If the measure $\\upsilon $ is absolutely continuous, then it is well known (see, for example, [42]) that the system (REF ) can be transformed (by a reparametrization) into an equivalent system with a trace normed weight matrix (that is, the matrix multiplying the spectral parameter on the right-hand side).",
"This is furthermore true in the general case upon slightly modifying the transformation; see [20].",
"In fact, upon denoting with $X=(y,U,h,r)\\in \\mathcal {F}_0$ the Lagrangian quantities corresponding to $(u,\\rho ,\\mu )$ as in Definition REF , it turns out that this transformation takes the system (REF ) into an equivalent system of the form $\\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix} G_\\xi = y_\\xi \\begin{pmatrix} U - \\frac{1}{4z} & 0 \\\\ 0 & 0 \\end{pmatrix} G + z \\begin{pmatrix} H_\\xi & U_\\xi \\\\ U_\\xi & y_\\xi \\end{pmatrix} G.$ One notes that the weight matrix is now a trace normed (that is, with trace equal to one almost everywhere) locally integrable function with determinant $H_\\xi y_\\xi - U_\\xi ^2 = y_\\xi ^2\\, \\rho ^2\\circ y = r^2$ in view of (REF ).",
"Thus, the system (REF ) can be understood in a standard sense.",
"Lemma 3.3 Let $\\sigma $ be the distribution function on ${\\mathbb {R}}$ defined by $\\sigma (x) = x + \\mu ((-\\infty ,x)), \\quad x\\in {\\mathbb {R}},$ so that $y$ is a generalized inverse of $\\sigma $ .",
"If the function $F$ is a solution of the system (REF ), then the function $G$ defined by $G(\\xi ) = \\begin{pmatrix} 1 & 0 \\\\ z(\\sigma \\circ y (\\xi ) - \\xi ) & 1 \\end{pmatrix} F\\circ y(\\xi ), \\quad \\xi \\in {\\mathbb {R}},$ is a solution of the system (REF ).",
"Conversely, if the function $G$ is a solution of the system (REF ), then (REF ) defines a function $F$ that is a solution of the system (REF ).",
"To begin with, let us note the simple identities $y \\circ \\sigma (x) & = x, \\quad x\\in {\\mathbb {R}}; & \\sigma \\circ y(\\xi ) & = \\xi , \\quad \\xi \\in \\mathrm {ran}(\\sigma ).$ In particular, since $y$ is locally constant on ${\\mathbb {R}}\\backslash \\mathrm {ran}(\\sigma )$ , this gives the equality $y_\\xi (\\xi ) G(\\xi ) = y_\\xi (\\xi ) F\\circ y(\\xi )$ for almost all $\\xi \\in {\\mathbb {R}}$ , if $F$ and $G$ are related by (REF ).",
"The remaining ingredients are two substitution formulas: Firstly, for every function $h\\in L^1_{{\\mathrm {loc}}}({\\mathbb {R}})$ , we have $\\int _{y(\\xi _1)}^{y(\\xi _2)} h(s) ds = \\int _{\\xi _1}^{\\xi _2} y_\\xi (s) h\\circ y(s) ds, \\quad \\xi _1,\\, \\xi _2\\in {\\mathbb {R}},$ according to, for example, [5].",
"Secondly, we will also use the identity $\\int _{\\sigma (x_1)}^{\\sigma (x_2)} h\\circ y(s)ds = \\int _{x_1}^{x_2} h(s) ds + \\int _{x_1}^{x_2} h(s)d\\mu (s), \\quad x_1,\\, x_2\\in {\\mathbb {R}},$ which holds for all continuous functions $h$ on ${\\mathbb {R}}$ (see, for example [5]) Now suppose that $F$ is a solution of the system (REF ) and let $\\xi _1$ , $\\xi _2\\in {\\mathbb {R}}$ .",
"Then the integral equation (REF ), identity (REF ) as well as (REF ) and (REF ) give $\\begin{split}-F\\big |_{y(\\xi _1)}^{y(\\xi _2)} = \\int _{\\xi _1}^{\\xi _2} y_\\xi (s) \\begin{pmatrix} 0 & 0 \\\\ U(s) - \\frac{1}{4z} & 0 \\end{pmatrix} G(s)ds & + z \\int _{\\xi _1}^{\\xi _2} \\begin{pmatrix} -U_\\xi (s) & -y_\\xi (s) \\\\ -y_\\xi (s) & U_\\xi (s) \\end{pmatrix} G(s) ds \\\\ & + z \\int _{\\sigma \\circ y(\\xi _1)}^{\\sigma \\circ y(\\xi _2)} \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} G(s) ds.\\end{split}$ Moreover, since $y$ is constant on $[\\sigma \\circ y(\\xi _i),\\xi _i]$ when $\\xi _i\\notin \\mathrm {ran}(\\sigma )$ we have $z (\\xi _i - \\sigma \\circ y(\\xi _i)) F_1\\circ y(\\xi _i) = z \\int _{\\sigma \\circ y(\\xi _i)}^{\\xi _i} F_1\\circ y(s)ds = z \\int _{\\sigma \\circ y(\\xi _i)}^{\\xi _i} G_1(s)ds$ for $i=1$ , 2.",
"After a straightforward calculation, all this finally gives $\\begin{split}-G\\big |_{\\xi _1}^{\\xi _2} = \\int _{\\xi _1}^{\\xi _2} y_\\xi (s) \\begin{pmatrix} 0 & 0 \\\\ U(s) - \\frac{1}{4z} & 0 \\end{pmatrix} G(s)ds & + z \\int _{\\xi _1}^{\\xi _2} \\begin{pmatrix} -U_\\xi (s) & -y_\\xi (s) \\\\ H_\\xi (s) & U_\\xi (s) \\end{pmatrix} G(s) ds,\\end{split}$ which shows that $G$ is a solution of the system (REF ).",
"For the converse, suppose that $G$ is a solution of the system (REF ).",
"In order to show that a function $F$ is well-defined by (REF ), let $\\xi _1$ , $\\xi _2\\in {\\mathbb {R}}$ such that $y(\\xi _1) = y(\\xi _2)$ and assume that $\\xi _1\\le \\xi _2$ without loss of generality.",
"Then the functions $y$ and $U$ are constant on the interval $[\\xi _1,\\xi _2]$ and we readily infer that $G_1(\\xi _1) & = G_1(\\xi _2), & -G_2\\big |_{\\xi _1}^{\\xi _2} & = z (\\xi _2-\\xi _1) G_1(\\xi _1),$ which shows that $F$ is well-defined (also note that the range of $y$ is all of ${\\mathbb {R}}$ ).",
"Now for any given $x_1$ , $x_2\\in {\\mathbb {R}}$ , we may evaluate $-F\\big |_{x_1}^{x_2} = -G\\big |_{\\sigma (x_1)}^{\\sigma (x_2)}$ using the integral equation (REF ), identity (REF ) as well as (REF ) and (REF ) to see that $F$ is a solution of the system (REF ).",
"Note that the matrix in (REF ) disappears if the measure $\\upsilon $ is absolutely continuous and the transformation in Lemma REF becomes a simple reparametrization.",
"In order to show what relabeling means in terms of the isospectral problem, let $\\phi \\in \\mathcal {G}$ be a relabeling function and consider the relabeled Lagrangian variables $\\hat{X}=X\\circ \\phi $ as in Section .",
"Then it is not hard to see that the original system (REF ) is indeed equivalent to the corresponding system with relabeled variables $\\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix} \\hat{G}_\\xi = \\hat{y}_\\xi \\begin{pmatrix} \\hat{U} - \\frac{1}{4z} & 0 \\\\ 0 & 0 \\end{pmatrix} \\hat{G} + z \\begin{pmatrix} \\hat{H}_\\xi & \\hat{U}_\\xi \\\\ \\hat{U}_\\xi & \\hat{y}_\\xi \\end{pmatrix} \\hat{G},$ by means of the following simple transformation.",
"Lemma 3.4 A function $G$ is a solution of the system (REF ) if and only if the function $\\hat{G}=G\\circ \\phi $ is a solution of the system (REF ).",
"The claim follows immediately upon applying the substitution rule [5] to the integral equation (REF ).",
"Concluding, we have seen that (REF ) and (REF ) represent the same (that is, equivalent up to reparametrizations) first order system with different kinds of normalizations.",
"The Lagrangian coordinates in $\\mathcal {F}_0$ correspond to trace normalization of the weight matrix, whereas the Eulerian coordinates in $\\mathcal {D}$ correspond to normalization of the bottom-right entry in the weight matrix.",
"Complete integrability In this final section, we will show that the Lagrangian version () of the two-component Camassa–Holm system is completely integrable in the sense that it can be formulated as the condition of compatibility for an overdetermined linear system.",
"To this end, let us first write down the corresponding reformulation of the two-component Camassa–Holm system () as a condition of compatibility: For sufficiently smooth functions $u$ , $\\rho $ and $p$ , we set $V & = \\begin{pmatrix} 0 & 0 \\\\ \\frac{1}{4z}-u & 0 \\end{pmatrix} -z \\begin{pmatrix} -u_x & -1 \\\\ u_x^2 + \\rho ^2 &u_x \\end{pmatrix}, \\\\W & = \\frac{1}{4z} \\begin{pmatrix} 0 & 0 \\\\ u -\\frac{1}{2z} & 0 \\end{pmatrix} + \\begin{pmatrix} 0 &- \\frac{1}{2} \\\\ p & 0 \\end{pmatrix} +z u \\begin{pmatrix} -u_x & -1 \\\\ u_x^2+ \\rho ^2 & u_x \\end{pmatrix}.$ Then a straightforward calculation shows that $V_t & = - \\begin{pmatrix} 0 & 0 \\\\ u_t & 0 \\end{pmatrix} - z \\begin{pmatrix} -u_{xt} & 0 \\\\ 2u_x u_{xt} + 2\\rho \\rho _t & u_{xt} \\end{pmatrix}, \\\\\\begin{split} W_x & = \\frac{1}{4z} \\begin{pmatrix} 0 & 0 \\\\ u_x & 0 \\end{pmatrix} + \\begin{pmatrix} 0 & 0 \\\\ p_x & 0 \\end{pmatrix} \\\\&\\qquad + z u_x \\begin{pmatrix} -u_x & -1 \\\\ u_x^2 + \\rho ^2 & u_x \\end{pmatrix} + z u \\begin{pmatrix} -u_{xx} & 0 \\\\ 2u_x u_{xx} + 2\\rho \\rho _x & u_{xx} \\end{pmatrix}, \\end{split} \\\\\\begin{split} [V,W] & = \\frac{1}{4z} \\begin{pmatrix} 0 & 0 \\\\ u_x & 0 \\end{pmatrix} - \\begin{pmatrix} 0 & 0 \\\\ uu_x & 0 \\end{pmatrix} \\\\&\\qquad + z \\left(u^2-p\\right) \\begin{pmatrix} -1 & 0 \\\\ 2u_x & 1 \\end{pmatrix} - \\frac{z}{2} \\begin{pmatrix} u_x^2+\\rho ^2 & 2u_x \\\\ 0 & -\\left(u_x^2 + \\rho ^2\\right) \\end{pmatrix}.",
"\\end{split}$ Upon collecting equal powers of $z$ , we see that the condition of compatibility $V_t-W_x + [V,W] = 0$ for the overdetermined system $\\Psi _x & = V \\Psi , & \\Psi _t & = W \\Psi ,$ is indeed equivalent to the two-component Camassa–Holm system ().",
"Now suppose that the functions $y$ , $U$ , $H$ and $P$ are sufficiently smooth and define $N & = y_\\xi \\begin{pmatrix} 0 & 0 \\\\ \\frac{1}{4z} -U & 0\\end{pmatrix}-z\\begin{pmatrix} -U_\\xi & -y_\\xi \\\\ H_\\xi & U_\\xi \\end{pmatrix}, \\\\M & = \\frac{1}{2z} \\begin{pmatrix} 0 & 0 \\\\ U - \\frac{1}{4z} & 0 \\end{pmatrix} - \\begin{pmatrix} 0 & \\frac{1}{2} \\\\ U^2 - P & 0\\end{pmatrix}.$ Then one readily computes that $N_t& =\\frac{1}{4z}\\begin{pmatrix} 0 & 0\\\\ y_{\\xi t}& 0\\end{pmatrix} - \\begin{pmatrix}0 & 0\\\\ U_t y_\\xi + U y_{\\xi t} & 0\\end{pmatrix} - z\\begin{pmatrix}Ê- U_{\\xi t} & - y_{\\xi t}\\\\ H_{\\xi t} & U_{\\xi t}\\end{pmatrix}, \\\\M_\\xi & = \\frac{1}{2z} \\begin{pmatrix} 0 & 0\\\\ U_\\xi & 0\\end{pmatrix} - \\begin{pmatrix} 0 & 0\\\\ 2UU_\\xi - P_\\xi & 0\\end{pmatrix}, \\\\\\begin{split}[N, M] & = \\frac{1}{4z} \\begin{pmatrix} 0 & 0\\\\ U_\\xi & 0\\end{pmatrix} - \\begin{pmatrix} 0 & 0 \\\\ UU_\\xi & 0\\end{pmatrix}Ê\\\\& \\qquad +z \\left(U^2-P\\right) \\begin{pmatrix} -y_\\xi & 0 \\\\Ê2U_\\xi & y_\\xi \\end{pmatrix} - \\frac{z}{2} \\begin{pmatrix} H_\\xi & 2U_\\xi \\\\Ê0 & -H_\\xi \\end{pmatrix}.\\end{split}$ Again, upon collecting equal powers of $z$ , the condition of compatibility $N_t - M_\\xi +[N, M]=0$ for the overdetermined system $\\Psi _\\xi & = N \\Psi , & \\Psi _t & = M \\Psi ,$ is seen to be equivalent to the system $ y_{\\xi t}& =U_\\xi , \\\\ U_{\\xi t}& = \\frac{1}{2} H_\\xi +(U^2-P)y_\\xi ,\\\\ H_{\\xi t} & = 2(U^2-P)U_\\xi , \\\\ U_ty_\\xi & = -P_\\xi .$ Note that in this case, the function $P$ is determined by $y$ , $U$ , $H$ to the extent that $y_\\xi P_{\\xi \\xi } - y_{\\xi \\xi } P_\\xi - y_\\xi ^3 P = -\\frac{1}{2}y_\\xi ^2 (2U^2y_\\xi + H_\\xi ),$ which follows upon combining () and ().",
"We finally want to show that this system indeed reduces to the system () under the additional assumption that the functions $y-{\\mathrm {id}}$ , $U$ , $H$ and $P$ (as well as their derivatives) decay spatially.",
"In fact, it is immediate that the system () implies ().",
"Conversely, given a solution of the system (), we first obtain (REF ) from (REF ) upon exploiting the decay assumption.",
"Furthermore, one sees that for every $t\\in {\\mathbb {R}}$ we have $P(\\xi ,t)=\\frac{1}{4} \\int _{\\mathbb {R}}e^{-\\vert y(\\xi ,t)-y(s,t)\\vert }(2U^2y_\\xi +h)(s,t)ds, \\quad \\xi \\in {\\mathbb {R}},$ since the function on the right-hand side in this equation is a solution of (REF ) as well (also take into account that both functions in this equation are constant whenever $y$ is constant).",
"In view of (), this shows that () holds with $P$ given by (REF ).",
"Because we clearly have $P_\\xi (\\xi ,t) = y_\\xi (\\xi ,t) Q(\\xi ,t), \\quad \\xi \\in {\\mathbb {R}},$ with $Q$ defined as in (), we see from () that () holds whenever $y_\\xi $ is non-zero.",
"If $y$ is constant on some interval, then () and () show that $U_{t\\xi } = \\frac{1}{2} H_\\xi = -Q_\\xi ,$ which implies that () holds everywhere indeed.",
"It remains to note that the time evolution of $r$ can be derived by using $r^2 = y_\\xi h - U_\\xi ^2$ , which yields $2rr_t=0$ ."
],
[
"Complete integrability",
"In this final section, we will show that the Lagrangian version () of the two-component Camassa–Holm system is completely integrable in the sense that it can be formulated as the condition of compatibility for an overdetermined linear system.",
"To this end, let us first write down the corresponding reformulation of the two-component Camassa–Holm system () as a condition of compatibility: For sufficiently smooth functions $u$ , $\\rho $ and $p$ , we set $V & = \\begin{pmatrix} 0 & 0 \\\\ \\frac{1}{4z}-u & 0 \\end{pmatrix} -z \\begin{pmatrix} -u_x & -1 \\\\ u_x^2 + \\rho ^2 &u_x \\end{pmatrix}, \\\\W & = \\frac{1}{4z} \\begin{pmatrix} 0 & 0 \\\\ u -\\frac{1}{2z} & 0 \\end{pmatrix} + \\begin{pmatrix} 0 &- \\frac{1}{2} \\\\ p & 0 \\end{pmatrix} +z u \\begin{pmatrix} -u_x & -1 \\\\ u_x^2+ \\rho ^2 & u_x \\end{pmatrix}.$ Then a straightforward calculation shows that $V_t & = - \\begin{pmatrix} 0 & 0 \\\\ u_t & 0 \\end{pmatrix} - z \\begin{pmatrix} -u_{xt} & 0 \\\\ 2u_x u_{xt} + 2\\rho \\rho _t & u_{xt} \\end{pmatrix}, \\\\\\begin{split} W_x & = \\frac{1}{4z} \\begin{pmatrix} 0 & 0 \\\\ u_x & 0 \\end{pmatrix} + \\begin{pmatrix} 0 & 0 \\\\ p_x & 0 \\end{pmatrix} \\\\&\\qquad + z u_x \\begin{pmatrix} -u_x & -1 \\\\ u_x^2 + \\rho ^2 & u_x \\end{pmatrix} + z u \\begin{pmatrix} -u_{xx} & 0 \\\\ 2u_x u_{xx} + 2\\rho \\rho _x & u_{xx} \\end{pmatrix}, \\end{split} \\\\\\begin{split} [V,W] & = \\frac{1}{4z} \\begin{pmatrix} 0 & 0 \\\\ u_x & 0 \\end{pmatrix} - \\begin{pmatrix} 0 & 0 \\\\ uu_x & 0 \\end{pmatrix} \\\\&\\qquad + z \\left(u^2-p\\right) \\begin{pmatrix} -1 & 0 \\\\ 2u_x & 1 \\end{pmatrix} - \\frac{z}{2} \\begin{pmatrix} u_x^2+\\rho ^2 & 2u_x \\\\ 0 & -\\left(u_x^2 + \\rho ^2\\right) \\end{pmatrix}.",
"\\end{split}$ Upon collecting equal powers of $z$ , we see that the condition of compatibility $V_t-W_x + [V,W] = 0$ for the overdetermined system $\\Psi _x & = V \\Psi , & \\Psi _t & = W \\Psi ,$ is indeed equivalent to the two-component Camassa–Holm system ().",
"Now suppose that the functions $y$ , $U$ , $H$ and $P$ are sufficiently smooth and define $N & = y_\\xi \\begin{pmatrix} 0 & 0 \\\\ \\frac{1}{4z} -U & 0\\end{pmatrix}-z\\begin{pmatrix} -U_\\xi & -y_\\xi \\\\ H_\\xi & U_\\xi \\end{pmatrix}, \\\\M & = \\frac{1}{2z} \\begin{pmatrix} 0 & 0 \\\\ U - \\frac{1}{4z} & 0 \\end{pmatrix} - \\begin{pmatrix} 0 & \\frac{1}{2} \\\\ U^2 - P & 0\\end{pmatrix}.$ Then one readily computes that $N_t& =\\frac{1}{4z}\\begin{pmatrix} 0 & 0\\\\ y_{\\xi t}& 0\\end{pmatrix} - \\begin{pmatrix}0 & 0\\\\ U_t y_\\xi + U y_{\\xi t} & 0\\end{pmatrix} - z\\begin{pmatrix}Ê- U_{\\xi t} & - y_{\\xi t}\\\\ H_{\\xi t} & U_{\\xi t}\\end{pmatrix}, \\\\M_\\xi & = \\frac{1}{2z} \\begin{pmatrix} 0 & 0\\\\ U_\\xi & 0\\end{pmatrix} - \\begin{pmatrix} 0 & 0\\\\ 2UU_\\xi - P_\\xi & 0\\end{pmatrix}, \\\\\\begin{split}[N, M] & = \\frac{1}{4z} \\begin{pmatrix} 0 & 0\\\\ U_\\xi & 0\\end{pmatrix} - \\begin{pmatrix} 0 & 0 \\\\ UU_\\xi & 0\\end{pmatrix}Ê\\\\& \\qquad +z \\left(U^2-P\\right) \\begin{pmatrix} -y_\\xi & 0 \\\\Ê2U_\\xi & y_\\xi \\end{pmatrix} - \\frac{z}{2} \\begin{pmatrix} H_\\xi & 2U_\\xi \\\\Ê0 & -H_\\xi \\end{pmatrix}.\\end{split}$ Again, upon collecting equal powers of $z$ , the condition of compatibility $N_t - M_\\xi +[N, M]=0$ for the overdetermined system $\\Psi _\\xi & = N \\Psi , & \\Psi _t & = M \\Psi ,$ is seen to be equivalent to the system $ y_{\\xi t}& =U_\\xi , \\\\ U_{\\xi t}& = \\frac{1}{2} H_\\xi +(U^2-P)y_\\xi ,\\\\ H_{\\xi t} & = 2(U^2-P)U_\\xi , \\\\ U_ty_\\xi & = -P_\\xi .$ Note that in this case, the function $P$ is determined by $y$ , $U$ , $H$ to the extent that $y_\\xi P_{\\xi \\xi } - y_{\\xi \\xi } P_\\xi - y_\\xi ^3 P = -\\frac{1}{2}y_\\xi ^2 (2U^2y_\\xi + H_\\xi ),$ which follows upon combining () and ().",
"We finally want to show that this system indeed reduces to the system () under the additional assumption that the functions $y-{\\mathrm {id}}$ , $U$ , $H$ and $P$ (as well as their derivatives) decay spatially.",
"In fact, it is immediate that the system () implies ().",
"Conversely, given a solution of the system (), we first obtain (REF ) from (REF ) upon exploiting the decay assumption.",
"Furthermore, one sees that for every $t\\in {\\mathbb {R}}$ we have $P(\\xi ,t)=\\frac{1}{4} \\int _{\\mathbb {R}}e^{-\\vert y(\\xi ,t)-y(s,t)\\vert }(2U^2y_\\xi +h)(s,t)ds, \\quad \\xi \\in {\\mathbb {R}},$ since the function on the right-hand side in this equation is a solution of (REF ) as well (also take into account that both functions in this equation are constant whenever $y$ is constant).",
"In view of (), this shows that () holds with $P$ given by (REF ).",
"Because we clearly have $P_\\xi (\\xi ,t) = y_\\xi (\\xi ,t) Q(\\xi ,t), \\quad \\xi \\in {\\mathbb {R}},$ with $Q$ defined as in (), we see from () that () holds whenever $y_\\xi $ is non-zero.",
"If $y$ is constant on some interval, then () and () show that $U_{t\\xi } = \\frac{1}{2} H_\\xi = -Q_\\xi ,$ which implies that () holds everywhere indeed.",
"It remains to note that the time evolution of $r$ can be derived by using $r^2 = y_\\xi h - U_\\xi ^2$ , which yields $2rr_t=0$ ."
]
] | 1605.05865 |
[
[
"Momentum transport in strongly coupled anisotropic plasmas in the\n presence of strong magnetic fields"
],
[
"Abstract We present a holographic perspective on momentum transport in strongly coupled, anisotropic non-Abelian plasmas in the presence of strong magnetic fields.",
"We compute the anisotropic heavy quark drag forces and Langevin diffusion coefficients and also the anisotropic shear viscosities for two different holographic models, namely, a top-down deformation of strongly coupled $\\mathcal{N} = 4$ Super-Yang-Mills (SYM) theory triggered by an external Abelian magnetic field, and a bottom-up Einstein-Maxwell-dilaton (EMD) model which is able to provide a quantitative description of lattice QCD thermodynamics with $(2+1)$-flavors at both zero and nonzero magnetic fields.",
"We find that, in general, energy loss and momentum diffusion through strongly coupled anisotropic plasmas are enhanced by a magnetic field being larger in transverse directions than in the direction parallel to the magnetic field.",
"Moreover, the anisotropic shear viscosity coefficient is smaller in the direction of the magnetic field than in the plane perpendicular to the field, which indicates that strongly coupled anisotropic plasmas become closer to the perfect fluid limit along the magnetic field.",
"We also present, in the context of the EMD model, holographic predictions for the entropy density and the crossover critical temperature in a wider region of the $(T,B)$ phase diagram that has not yet been covered by lattice simulations.",
"Our results for the transport coefficients in the phenomenologically realistic magnetic EMD model could be readily used as inputs in numerical codes for magnetohydrodynamics."
],
[
"Introduction",
"The study of the behavior of QCD matter under extreme conditions is a very active area of research regarding the physics of the strong interactions.",
"Ultrarelativistic heavy ion collisions [1], [2], [3], [4], [5] are currently probing matter in a region of the QCD phase diagram close to the crossover transition [6], where the system behaves as a strongly coupled quark-gluon plasma (QGP) [7] (for recent reviews, see [8], [9]).",
"One of the most striking features of this strongly coupled QGP is its nearly perfect fluid behavior characterized by a very small value (when compared to weak coupling QCD calculations [10], [11]) for the shear viscosity to entropy density ratio, which according to recent hydrodynamic simulations [12] simultaneously matching experimental data for different physical observables, is given by the value $\\eta / s\\approx 0.095$ (at least near the crossover region).",
"This small value is remarkably close to the estimate $\\eta / s=1/4\\pi $ valid for a broad class of strongly coupled holographic plasmas with spatially isotropic and translationally invariant gravity duals characterized by actions with at most two derivatives [13], [14], [15].",
"This observation suggested that the holographic gauge/gravity correspondence [16], [17], [18], [19] could be useful to obtain insight on the non-equilibrium transport properties of strongly coupled non-Abelian plasmas such as the QGP (for recent reviews on applications of the holographic correspondence to the physics of the QGP, see [20], [21]).",
"The fact that the gauge/gravity duality may be employed to calculate real time non-equilibrium observables [22], [23], [24], [25] is particularly interesting since weak coupling QCD calculations cannot reliably describe the strongly coupled region close to the crossover transition, while lattice QCD simulations, though very successful in handling calculations of equilibrium quantities such as the equation of state (at least at zero baryon density), suffer from severe technical difficulties to perform real time calculations [26].",
"Therefore, one may resort to the holographic duality as a non-perturbative tool to compute observables which are very difficult to calculate using first principle QCD techniques as it is the case of real time transport coefficients near the crossover region.",
"Indeed, the gauge/gravity duality has already been used to compute several transport coefficients of different strongly coupled non-Abelian plasmas - see for instance Refs.",
"[13], [14], [15], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38].",
"One of the many branches of applications of holographic techniques to the physics of strongly coupled systems, which is the one we are particularly interested in exploring in the present work, regards the influence of strong external Abelian magnetic fields on the equilibrium and transport properties of strongly interacting non-Abelian plasmas.",
"In fact, very intense magnetic fields ranging fromNote: $eB=1$ GeV$^2\\Rightarrow B \\simeq 1.69\\times 10^{20}$ G. $eB\\sim m_\\pi ^2\\sim 0.02 \\, \\mathrm {GeV^2}$ at the Relativistic Heavy Ion Collider (RHIC) to $eB\\sim 15 m_\\pi ^2\\sim 0.3 \\, \\mathrm {GeV^2}$ at the Large Hadron Collider (LHC) may be produced at the earliest stages of ultrarelativistic peripheral heavy ion collisions [39], [40], [41], [42], [43], [44].At first, one may expect that such strong magnetic fields rapidly decrease in intensity in the later stages when the QGP is formed (after $\\sim 1$ fm/c) due to the departure of the spectators from the collision region.",
"However, the electric conductivity of the QGP may sensitively slow down the decay of the magnetic field in the medium [45], [46] and the quantum nature of the sources [47] may delay this decay even further.",
"However, it remains unclear whether the large magnetic fields produced at the earliest stages of peripheral collisions remain strong enough to affect transport and equilibrium properties of the QGP.",
"Less intense, but still very strong magnetic fields (at least when compared to fields of terrestrial, non-astrophysical origins) up to $eB\\sim 5\\times 10^{-5} m_\\pi ^2\\sim 1\\, \\textrm {MeV}^2$ are expected to be present inside magnetars [48], while much stronger fields of order $eB\\sim 200 m_\\pi ^2\\sim 4\\, \\textrm {GeV}^2$ are believed to have been generated in the primordial Universe [49], [50], [51].",
"Due to this wide range of scenarios where strong magnetic fields may play a relevant role in the properties and the evolution of different physical systems, a large amount of research on related topics has been carried out in the last years, see for instance Refs.",
"[52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [64], [63], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87] and also [88], [89], [90], [91] for recent reviews.",
"In the holographic scenario, some calculations of physical observables in the presence of strong magnetic fields in different gauge/gravity models were discussed, for instance, in Refs.",
"[92], [93], [94], [95], [96], [97], [98], [99], [100], [101], [102], [103], [104], [105], [106], [107], [108], [109], [110], [111], [112], [113], [114], [115], [116], [117], [118].",
"In the present work, we are specifically interested in analyzing anisotropic momentum transport coefficients of strongly interacting magnetized plasmas, namely, heavy quark drag forces, Langevin diffusion coefficients, and also anisotropic shear viscosities.",
"For hard probes plowing through the strongly coupled plasma, the momentum loss of the probe to the medium may be described by its drag force [119], [120], [121], [122], [123], [124].",
"If one considers the influence of thermal fluctuations, it may be further characterized by the momentum diffusion of the probe along (and transversely) to its initial velocity via the diffusion coefficients associated to the Brownian motion of the probe described by a local Langevin equation [125], [126], [127].",
"Another important transport coefficient associated with the hydrodynamic evolution of the energy-momentum tensor of the system is the shear viscosity.",
"As we shall review in this work, the presence of an external Abelian magnetic field explicitly breaks $SO(3)$ rotational symmetry down to $SO(2)$ rotations in the plane transverse to the magnetic field direction inducing an anisotropy in the system which, in turn, implies in a splitting of these observables into several new transport coefficients.",
"For instance, while in the isotropic case at zero magnetic field there is one drag force, one shear viscosity, and two Langevin diffusion coefficients (one corresponding to fluctuations transverse and the other to fluctuations parallel to the heavy quark velocity), the anisotropy induced by a nonzero magnetic field (as in the models considered in the present work) causes the appearance of two different nontrivial shear viscosities, two different drag forces, and five different Langevin diffusion coefficients depending on the orientation of the momentum diffusion relative to the directions of the magnetic field and the velocity of the probe.",
"In the context of a top-down anisotropic deformation of a strongly coupled $\\mathcal {N} = 4$ Super-Yang-Mills (SYM) plasma driven by a nontrivial profile for a bulk axion field [128], [129], the corresponding anisotropic drag forces [130], [131], [132], Langevin diffusion coefficients [133], [134], [135], and shear viscosities [136], have been already computed but a detailed holographic study of anisotropic momentum transport driven by an external magnetic field has not yet been done.",
"Recently, in Refs.",
"[137], [98], some of the weakly coupled perturbative QCD Langevin diffusion coefficients at leading order in the strong coupling constant, $\\alpha _s$ , and strong magnetic fields were computed in the $\\alpha _s eB \\ll T^2 \\ll eB$ limit.",
"In Ref.",
"[98], the anisotropic drag forces and some of the Langevin diffusion coefficients were also computed strictly in the particular limit of strong magnetic fields, $eB/T^2\\gg 1$ , for the top-down anisotropic deformation of a strongly coupled SYM plasma driven by an external magnetic field, called the “magnetic brane model” [104], [105], [106].",
"In those works, it was found that the heavy quark diffusion for a probe moving perpendicularly to the magnetic field is larger than in the case of parallel motion suggesting that this may contribute to heavy quark elliptic flow [137].",
"In the present work, as a warm-up calculation in a top-down holographic model, we go beyond the analytical limit $eB/T^2\\gg 1$ worked out in Ref.",
"[98] and derive full numerical results for the anisotropic momentum transport coefficients of the magnetic brane model, which are valid for any value of the ratio $eB/T^2$ .",
"We compute for the first time the full results for the two drag forces, the five Langevin diffusion coefficients (two of them were not discussed in Ref.",
"[98] in any limit) and, for completeness, we also review the main result of Ref.",
"[108] concerning the calculation of the anisotropic shear viscosities for this magnetized SYM plasma.",
"However, when thinking about possible applications to real world QCD at finite temperature, it is desirable to work with a holographic model which is able to emulate at least some of the effects of the dynamical infrared breaking of conformal symmetry associated to the emergence of the dimensional transmutation scale $\\Lambda _{\\textrm {QCD}}$ .",
"This is clearly not the case of the top-down magnetic brane model proposed in Refs.",
"[104], [105], [106] since, for instance, all the observables in this model are functions of the dimensionless ratio $eB/T^2$ instead of $eB$ and $T$ , separately.",
"The reason for that is the fact that the SYM plasma is a conformal system at zero magnetic field and, in this case, if $B=0$ the temperature is the only scale of the system and one is only able to physically distinguish the zero temperature from the finite temperature case with $T\\ne 0$ being a fixed scale of the system.",
"When this theory is deformed by the introduction of an external magnetic field (which explicitly breaks conformal symmetry [110]), the value of this field is then naturally measured in terms of the fixed temperature scale.",
"The situation in QCD is completely different since due to the dynamical breaking of conformal symmetry in the infrared regime, $\\Lambda _{\\textrm {QCD}}$ emerges as the natural (quantum) scale of the theory, even in the vacuum.",
"By turning on the temperature in QCD, $T$ is in fact a variable (differently from what happens in the magnetic brane setup), which is naturally measured in terms of the fixed scale $\\Lambda _{\\textrm {QCD}}$ .",
"In the very same way, by applying an external magnetic field to QCD matter, the magnetic field is naturally measured in terms of $\\Lambda _{\\textrm {QCD}}$ and, therefore, both $T$ and $eB$ may be independently varied.",
"In order to induce a dynamical breaking of conformal symmetry in holographic settings, one may consider a bottom-up Einstein-dilaton model with a nontrivial dilaton potential responsible for emulating the effects of the $\\Lambda _{\\textrm {QCD}}$ scale, as originally proposed in [138] (see also [27], [36], [28], [29], [30], [31], [38] for further applications).",
"In Ref.",
"[139] (see also [32], [33], [34], [37] for further applications), an extension of the holographic setup proposed in [138] encompassed the construction of an Einstein-Maxwell-dilaton (EMD) model describing a QCD-like theory at finite temperature, nonzero baryon chemical potential, and zero magnetic field at the boundary of isotropic, asymptotically AdS$_5$ spaces.",
"More recently, some of us proposed a different EMD model [111] describing a QCD-like theory at finite temperature and nonzero magnetic field (at zero chemical potential) at the boundary of spatially anisotropic, asymptotically AdS$_5$ spaces.",
"The fundamental reasoning involved in these phenomenological bottom-up approaches may be properly dubbed as some type of “black hole engineering”, which consists in adequately “teaching” the dilatonic black hole model how to behave in a QCD-like manner, on phenomenologically interesting regions of the QCD phase diagram.",
"More precisely, one seeds the holographic model with adequate lattice and/or experimental/observational data, which are used to dynamically fix the free parameters of the bottom-up setup considered.",
"Once these parameters are fixed, further calculations of different observables (usually related to real time transport coefficient) provide true predictions of the holographic setup.",
"In the magnetic EMD setting proposed in Ref.",
"[111], the free parameters of the model were dynamically fixed by matching the holographic equation of state and magnetic susceptibility at zero magnetic field with the corresponding $(2+1)$ -flavor lattice QCD data with physical quark masses presented in Refs.",
"[140], [141], respectively.",
"Then, the equation of state at finite magnetic field follows as a prediction of the holographic model.",
"In [111] we obtained a reasonable agreement with the equation of state at finite $T$ and $B$ calculated recently on the lattice [142] for magnetic fields up to $eB\\sim 0.3$ GeV$^2$ .",
"In the present work, we update the model proposed in [111] by seeding it with more recent lattice data for the equation of state at $B=0$ [143] and by performing a global matching to different observables characterizing the equation of state at zero magnetic field.",
"The updated magnetic EMD model to be discussed in the present work greatly improves the quantitative agreement with the finite $T$ and $B$ lattice QCD equation of state of Ref.",
"[142], also extending it to higher values of the magnetic field.",
"As the main results of the present work, we employ this updated magnetic EMD model to compute for the first time the $T$ and $B$ dependence of the anisotropic drag forces, Langevin diffusion coefficients and shear viscosities in a realistic magnetized QCD-like holographic dual.",
"We finish this introductory section by providing an overview of the paper in order to guide the reader: The thermodynamics, drag forces, Langevin diffusion coefficients, and shear viscosities for the top-down magnetic brane model are discussed in Section ; The thermodynamics, drag forces, Langevin diffusion coefficients, and shear viscosities for the phenomenological QCD-like bottom-up magnetic EMD model are presented in Section .",
"The results presented in this section will be of more interest to the phenomenologically oriented reader; In the concluding Section we discuss the most important implications of our calculations and outline future projects that may be pursued; For completeness, we provide in Appendix a review of the general holographic formalism for the computation of drag forces, Langevin diffusion coefficients and shear viscosities, in both isotropic and anisotropic settings.",
"In Appendix , we present a comprehensive derivation of the anisotropic Kubo formulas for the several shear (and bulk) viscosities appearing in first order viscous magnetohydrodynamics.",
"In this work, we use natural units $c=\\hbar =k_B=1$ and a mostly plus (Lorentzian) metric signature.",
"Action.",
"The first class of magnetized backgrounds we will use in the present work corresponds to a top-down model of magnetic branes dual to a deformation of strongly coupled SYM theory triggered by an external magnetic field [104], [105], [106].",
"The model is described by the Einstein-Maxwell action, $S = \\frac{1}{16\\pi G_5}\\int _{\\mathcal {M}_5}d^5x\\sqrt{-g}\\left[R + \\frac{12}{L^2}- F_{\\mu \\nu }^2\\right] +S_{\\textrm {CS}}+S_{\\textrm {GHY}}+S_{\\textrm {CT}},$ where $L$ is the asymptotic $\\mathrm {AdS}_5$ radius, which we set to unity, $S_{\\textrm {CS}}$ is the topological $(4+1)$ -dimensional Abelian Chern-Simons term (which vanishes on-shell for the backgrounds considered here though it is useful [104] to fix the relation between the bulk magnetic field - which we denote in this section by $B$ - and the physically observable magnetic field at the boundary - which we denote in this section by $\\mathcal {B}$ ), $S_{\\textrm {GHY}}$ is the Gibbons-Hawking-York term [174], [175] needed in order to give a well posed initial value problem, and $S_{\\textrm {CT}}$ is the counterterm action [176], [177], [178], [179], [180] needed in order to render the complete on-shell action finite.",
"However, as we will not need to compute here the on-shell action, we do not need to specify the explicit form of $S_{\\textrm {CS}}$ , $S_{\\textrm {GHY}}$ , and $S_{\\textrm {CT}}$ .",
"Ansatz and equations of motion.",
"The magnetic brane background is described by the following ansatz in coordinates which we call the standard coordinates, denoted by a tilde, $ds^2=-\\tilde{U}(\\tilde{r})d\\tilde{t}^2+\\frac{d\\tilde{r}^2}{\\tilde{U}(\\tilde{r})} + e^{2\\tilde{V}(\\tilde{r})}(d\\tilde{x}^2 + d\\tilde{y}^2) + e^{2\\tilde{W}(\\tilde{r})} d\\tilde{z}^2, \\quad F = B d\\tilde{x} \\wedge d\\tilde{y}.$ One can check that Maxwell's equations following from Eq.",
"(REF ) are trivially satisfied by the ansatz (REF ), while Einstein's equations reduce to $\\tilde{U}(\\tilde{V}^{\\prime \\prime }-\\tilde{W}^{\\prime \\prime }) + (\\tilde{U}^{\\prime } + \\tilde{U}(2\\tilde{V}^{\\prime }+\\tilde{W}^{\\prime }))(\\tilde{V}^{\\prime }-\\tilde{W}^{\\prime }) & = -2B^2 e^{-4\\tilde{V}}, \\\\2 \\tilde{V}^{\\prime \\prime }+ \\tilde{W}^{\\prime \\prime } + 2(\\tilde{V}^{\\prime })^2 + (\\tilde{W}^{\\prime })^2 & = 0, \\\\\\frac{1}{2} \\tilde{U}^{\\prime \\prime } + \\frac{1}{2} \\tilde{U}^{\\prime } (2 \\tilde{V}^{\\prime } + \\tilde{W}^{\\prime }) & = 4 + \\frac{2}{3} B^2 e^{-4 \\tilde{V}}.$ The fourth equation follows from the above and it is a constraint on initial data: $2 \\tilde{U}^{\\prime } \\tilde{V}^{\\prime } + \\tilde{U}^{\\prime } \\tilde{W}^{\\prime } + 2 \\tilde{U} (\\tilde{V}^{\\prime })^2 + 4 \\tilde{U} \\tilde{V}^{\\prime } \\tilde{W}^{\\prime } = 12 - 2 B^2 e^{-4 \\tilde{V}}.$ Asymptotics.",
"The magnetic brane solution corresponds to a holographic renormalization group flow interpolating between a $\\mathrm {BTZ} \\times \\mathrm {R}^2$ near horizon solution given by ($r = r_H$ is the BTZ black hole [181] horizon) $ds^2 = \\left[ -3 (\\tilde{r}^2 - \\tilde{r}_H^2) dt^2 + 3 \\tilde{r}^2 d\\tilde{z}^2 + \\frac{d\\tilde{r}^2}{3(\\tilde{r}^2-\\tilde{r}_H^2)} \\right] + \\frac{B}{\\sqrt{3}} (d\\tilde{x}^2 + d\\tilde{y}^2)$ describing the deep infrared, and a near boundary $\\mathrm {AdS}_5$ asymptotic solution at $\\tilde{r} \\rightarrow \\infty $ , $ds^2 = \\tilde{r}^2 (- d\\tilde{t}^2 + d\\tilde{x}^2+d\\tilde{y}^2+ d\\tilde{z}^2) + \\frac{d\\tilde{r}^2}{\\tilde{r}^2},$ describing the ultraviolet.",
"As discussed in Ref.",
"[104], the gauge variation of the Chern-Simons term in Eq.",
"(REF ) may be used to compute a 3-point function in the gauge theory and compare it with the SYM chiral anomaly, which gives the following relation between the physically observable magnetic field at the boundary and the bulk magnetic field: $\\mathcal {B} = \\sqrt{3} B$ .",
"Numerical solutions.",
"In order to numerically solve the equations of motion we introduce new coordinates, which we call numerical coordinates, represented without the tildes $\\tilde{t} = t, \\,\\,\\, \\tilde{r} = r, \\,\\,\\,\\tilde{x} = \\frac{x}{\\sqrt{v(b)}}, \\,\\,\\,\\tilde{y} = \\frac{y}{\\sqrt{v(b)}}, \\,\\,\\,\\tilde{z} = \\frac{z}{\\sqrt{w(b)}};\\,\\,\\, \\left(\\Rightarrow B=\\frac{b}{v(b)}\\right),$ where $b$ is the rescaled magnetic field in the numerical coordinates, which is taken as an initial condition (each value of this initial condition will generate a numerical background corresponding to some definite physical state at the gauge theory), and $v(b)$ , $w(b)$ are functions extracted from the numerical solutions $V(r)$ , $W(r)$ by fitting their near-boundary behavior as $e^{2V(r\\rightarrow \\infty )}\\sim v(b)r^2$ , $e^{2W(r\\rightarrow \\infty )}\\sim w(b) r^2$ , respectively.",
"These numerical coordinates are chosen such that the horizon is located at $r_H = 1$ and that the rescaled metric functions $U(r)$ , $V(r)$ , and $W(r)$ satisfy the boundary conditions $U^{\\prime }(1) = 1\\, (\\Rightarrow T=1/4\\pi )$ , $V(1) = W(1) = 0$ .",
"Moreover, $U(1)=0$ .",
"One can then show that, on-shell, $V^{\\prime }(1) = 4-\\frac{4}{3} b^2 \\quad \\mathrm {and} \\quad W^{\\prime }(1) = 4+\\frac{2}{3} b^2,$ and one can then numerically integrate the equations of motion from the horizon to the boundary.",
"One can also compute the temperature and the entropy density normalized by the gauge theory magnetic field $\\mathcal {B}$ in terms of the scaling functions $v(b)$ and $w(b)$ [104], $\\frac{T}{\\sqrt{\\mathcal {B}}} = \\frac{1}{4\\pi \\, 3^{1/4}} \\sqrt{\\frac{v(b)}{b}} \\quad \\mathrm {and} \\quad \\frac{s}{N_c^2 \\mathcal {B}^{3/2}} = \\frac{1}{2\\pi \\, 3^{3/4}} \\sqrt{\\frac{v(b)}{b^3 w(b)}}.$ The corresponding equation of state is plotted in Fig.",
"REF , where one can notice that at large (small) magnetic fields (compared to the fixed temperature scale) the behavior of the dimensionless ratio $s/N_c^2\\mathcal {B}^{3/2}$ is linear (cubic) in the dimensionless ratio $T/\\sqrt{\\mathcal {B}}$ .",
"Therefore, one indeed recovers the AdS$_5$ -Schwarzschild result for D3-branes in the ultraviolet, $s\\sim T^3$ .",
"Figure: Normalized entropy density for the magnetic brane setup as a function of the dimensionless ratio T/ℬT/\\sqrt{\\mathcal {B}}.",
"In the inset we show the corresponding behavior for large magnetic fields."
],
[
"Drag force",
"The anisotropic drag forces described by Eqs.",
"(REF ) to (REF ) may be computed in the magnetic brane backgrounds by considering,Since the background dilaton field is zero in the magnetic brane model it follows that $g_{\\mu \\nu }^{(s)}=g_{\\mu \\nu }$ .",
"$g_{tt}^{(s)}=-\\tilde{U}(\\tilde{r})=-U(r),\\,\\,\\,g_{rr}^{(s)}=\\frac{1}{U(r)},\\,\\,\\, g_{xx}^{(s)}=g_{yy}^{(s)}=e^{2\\tilde{V}(\\tilde{r})}=\\frac{e^{2V(r)}}{v(b)},\\,\\,\\, g_{zz}^{(s)}=e^{2\\tilde{W}(\\tilde{r})}=\\frac{e^{2W(r)}}{w(b)}.$ Our numerical results for the magnetic field induced anisotropic drag forces (normalized by the isotropic SYM result at zero magnetic field given in Eq.",
"(REF )), valid for arbitrary values of the dimensionless ratio $\\mathcal {B}/T^2$ , are displayed in Figs.",
"REF and REF .",
"We see that both drag forces increase with increasing magnetic field relatively to the isotropic zero magnetic field case with the drag force being generally stronger in the transverse plane to the magnetic field direction, which indicates that probes traversing the plasma in the perpendicular plane to the magnetic field lose more energy than probes moving along the magnetic field direction.",
"Furthermore, we note that the dependence of both drag forces with the magnetic field is enhanced at higher speeds, which shows that faster/lighter probes are more affected by drag force effects in a medium with a strong magnetic field.",
"The stronger magnetic field dependence of the configuration with $\\vec{v} \\perp \\vec{\\mathcal {B}}$ is compatible with the expectation that, for strong magnetic fields, the system described in the gauge theory may be effectively spatially decomposed into a 2-dimensional system lying in the plane transverse to the magnetic field and an 1-dimensional system along the magnetic field direction, with the transverse system being more sensitive to the Landau levels induced by the magnetic field.",
"Figure: (Color online) Anisotropic drag force F drag (v∥ℬ) F_{\\textrm {drag}}^{(v\\parallel \\mathcal {B})} in the magnetic brane model normalized by the isotropic SYM result at zero magnetic field.",
"Left: 3D plot as function of ℬ/T 2 \\mathcal {B}/T^2 and vv.",
"Right: as a function of ℬ/T 2 \\mathcal {B}/T^2 for some fixed values of vv.Figure: (Color online) Anisotropic drag force F drag (v⊥ℬ) F_{\\textrm {drag}}^{(v\\perp \\mathcal {B})} in the magnetic brane model normalized by the isotropic SYM result at zero magnetic field.",
"Left: 3D plot as function of ℬ/T 2 \\mathcal {B}/T^2 and vv.",
"Right: as a function of ℬ/T 2 \\mathcal {B}/T^2 for some fixed values of vv.From Figs.",
"REF and REF , one observes that as the system approaches the ultraviolet fixed point in the limit $\\mathcal {B}/T^2\\ll 1$ the anisotropy induced by the magnetic field ceases and both drag forces converge to the SYM result given in Eq.",
"(REF ), as expected.",
"Moreover, in the opposite infrared limit, $F_{\\textrm {drag}}^{(v\\parallel B)}$ ($F_{\\textrm {drag}}^{(v\\perp B)}$ ) acquires a constant (linear) dependence on the dimensionless ratio $\\mathcal {B}/T^2$ in accordance with the analytical results obtained in Ref.",
"[98] in the limit $\\mathcal {B}/T^2\\gg 1$ .",
"Moreover, we note that $F_{\\textrm {drag}}^{(v\\parallel B)}$ reaches its asymptotic behavior in the infrared only for much larger values of the magnetic field than in the case of $F_{\\textrm {drag}}^{(v\\perp B)}$ , especially for large $v$ ."
],
[
"Langevin diffusion coefficients",
"The anisotropic Langevin diffusion coefficients described by Eqs.",
"(REF ) to (REF ) and Eqs.",
"(REF ) to () may be computed in the magnetic brane backgrounds by considering the relations in Eq.",
"(REF ).",
"Our numerical results for the magnetic field induced anisotropic Langevin coefficients, normalized by the SYM results at zero magnetic field given by the relations in Eq.",
"(REF ), which are valid for arbitrary values of the dimensionless ratio $\\mathcal {B}/T^2$ , are displayed in Figs.",
"REF to REF .",
"We see that all the Langevin coefficients are enhanced in the presence of an external magnetic field relative to the zero magnetic field case, which means that diffusion through the plasma is facilitated by the presence of a magnetic field.",
"One also notes that momentum diffusion in directions transverse to the magnetic field is generally larger than in the direction of the field.",
"This is line with what is observed in the drag force: the probe loses more energy and diffuses more momentum when moving along directions in the plane perpendicular to the magnetic field.",
"Furthermore, the coefficients $\\kappa _{(\\parallel v)}^{(v\\parallel \\mathcal {B})}$ , $\\kappa _{(\\parallel v)}^{(v\\perp \\mathcal {B})}$ , $\\kappa _{(\\perp v, \\perp \\mathcal {B})}^{(v \\perp \\mathcal {B})}$ , and $\\kappa _{(\\perp v,\\parallel \\mathcal {B})}^{(v \\perp \\mathcal {B})}$ always increase with increasing velocity, contrary to what happens with the coefficient $\\kappa _{(\\perp v)}^{(v\\parallel \\mathcal {B})}$ at large $v$ .",
"We also checked that in the infrared limit, the Langevin diffusion coefficients agree with the analytical behavior derived in Ref.",
"[98] in the limit $\\mathcal {B}/T^2\\gg 1$ .",
"Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (∥v) (v∥ℬ) \\kappa _{(\\parallel v)}^{(v\\parallel \\mathcal {B})} in the magnetic brane model normalized by the SYM result at zero magnetic field.",
"Left: 3D plot as function of ℬ/T 2 \\mathcal {B}/T^2 and vv.",
"Right: as a function of ℬ/T 2 \\mathcal {B}/T^2 for some fixed values of vv.Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (⊥v) (v∥ℬ) \\kappa _{(\\perp v)}^{(v\\parallel \\mathcal {B})} in the magnetic brane model normalized by the SYM result at zero magnetic field.",
"Left: 3D plot as function of ℬ/T 2 \\mathcal {B}/T^2 and vv.",
"Right: as a function of ℬ/T 2 \\mathcal {B}/T^2 for some fixed values of vv.Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (∥v) (v⊥ℬ) \\kappa _{(\\parallel v)}^{(v\\perp \\mathcal {B})} in the magnetic brane model normalized by the SYM result at zero magnetic field.",
"Left: 3D plot as function of ℬ/T 2 \\mathcal {B}/T^2 and vv.",
"Right: as a function of ℬ/T 2 \\mathcal {B}/T^2 for some fixed values of vv.Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (⊥v,⊥ℬ) (v⊥ℬ) \\kappa _{(\\perp v, \\perp \\mathcal {B})}^{(v\\perp \\mathcal {B})} in the magnetic brane model normalized by the SYM result at zero magnetic field.",
"Left: 3D plot as function of ℬ/T 2 \\mathcal {B}/T^2 and vv.",
"Right: as a function of ℬ/T 2 \\mathcal {B}/T^2 for some fixed values of vv.Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (⊥v,∥ℬ) (v⊥ℬ) \\kappa _{(\\perp v, \\parallel \\mathcal {B})}^{(v\\perp \\mathcal {B})} in the magnetic brane model normalized by the SYM result at zero magnetic field.",
"Left: 3D plot as function of ℬ/T 2 \\mathcal {B}/T^2 and vv.",
"Right: as a function of ℬ/T 2 \\mathcal {B}/T^2 for some fixed values of vv."
],
[
"Shear viscosity",
"The anisotropic shear viscosities described by Eqs.",
"(REF ) and () may be computed in the magnetic brane backgrounds by considering the relations in Eq.",
"(REF ).",
"In Fig.",
"REF we plot our numerical results for the ratio between the parallel and perpendicular shear viscosities, which was originally obtained in Ref.",
"[108] and it is displayed here for completeness.",
"Figure: Ratio between the parallel and perpendicular shear viscosities in the magnetic brane model as a function of the dimensionless ratio ℬ/T 2 \\mathcal {B}/T^2.One can see that the shear viscosity is reduced in the direction of the magnetic field in comparison to the value of viscosity perpendicular to the field, which indicates that the strongly coupled anisotropic plasma becomes even closer to the perfect fluid limit along the magnetic field direction."
],
[
"The magnetic Einstein-Maxwell-dilaton model",
"In this section, we begin by briefly reviewing and updating the phenomenologically constructed, QCD-like, bottom-up magnetic EMD model originally proposed in Ref.",
"[111] (we refer the interested reader to consult this reference for further details and discussions).",
"We then obtain the updated results for the equation of state at finite $B$ and also calculate the anisotropic drag forces, Langevin diffusion coefficients, and shear viscosities.",
"The update of the magnetic EMD model we shall introduce in the present work refers to the fact that this time we will dynamically fix the free parameters of the bottom-up model by performing a global matching of the holographic equation of state (including the entropy and internal energy densities, the speed of sound squared, the pressure, and the trace anomaly) and the magnetic susceptibility at zero magnetic field with the latest $(2+1)$ -flavor lattice QCD data with physical quark masses from Refs.",
"[143] and [141], respectively.",
"Previously, in Ref.",
"[111], some of us used lattice data just for the speed of sound squared and the pressure at $B=0$ from the older Ref.",
"[140] to fix the dilaton potential of the EMD model.",
"The simple improvements referred above to fix the free parameters of the model at $B=0$ will result in a much better quantitative agreement between the finite $B$ holographic and lattice QCD equations of state (as previously suggested in Ref.",
"[111]).",
"We shall also present holographic predictions for the entropy density and the crossover temperature in the presence of a magnetic field in a wider region of the $(T,B)$ phase diagram that has not yet been explored in lattice simulations.",
"It is also important to mention that even though the EMD setting worked out here does not explicitly introduce fundamental flavors in the gauge theory by means of the standard holographic dictionary (which would require the use of flavor D-branes), the dilaton potential used here is such that the holographic equation of state for the black brane at $B=0$ matches the corresponding lattice QCD results with $(2+1)$ flavors.",
"In this sense, the setup used here corresponds to an emulator which is able to mimic some of the relevant properties of QCD with dynamical flavors, such as the crossover transition.",
"Such an effective approach was originally proposed in Ref.",
"[138] (see also [182] for more recent discussions) where it was also shown that different parametrizations for the dilaton potential can mimic not only the QCD crossover but also first and second order phase transitions, which may be relevant for different applications ranging from the QGP to condensed matter physics."
],
[
"The model and its thermodynamics",
"Action.",
"The bulk action for the EMD model is given by $S&=\\frac{1}{16\\pi G_5}\\int _{\\mathcal {M}_5}d^5x\\sqrt{-g}\\left[R-\\frac{1}{2}(\\partial _\\mu \\phi )^2-V(\\phi ) -\\frac{f(\\phi )}{4}F_{\\mu \\nu }^2\\right],$ which is supplemented by boundary terms, as before.",
"However, as before these boundary terms will not be needed for the calculations carried out in the present work and, therefore, we omit them from our discussion.",
"We shall dynamically fix the dilaton potential $V(\\phi )$ and the gravitational constant $\\kappa ^2\\equiv 8\\pi G_5$ by matching the holographic equation of state at $B=0$ with the corresponding lattice data for $(2+1)$ -flavor QCD from Ref.",
"[143].",
"The dynamical dilaton field breaks conformal symmetry in the infrared (the interior of the bulk) where it acquires a nontrivial profile, while close to the boundary it vanishes and the theory goes back to the ultraviolet fixed point corresponding to the AdS$_5$ geometry.",
"The potential $V(\\phi )$ has the near-boundary ultraviolet expansionAs before, we set to unity the radius $L$ of the asymptotically AdS$_5$ geometries.",
"$V(\\phi \\rightarrow 0) \\approx -12 + m^2\\phi ^2/2$ , where the mass $m$ of the scalar field $\\phi $ defines the scaling dimension $\\Delta $ of the dual operator in the gauge theory through the relation $m^2 = -\\nu \\Delta $ , where $\\nu = 4-\\Delta $ .",
"For the dilaton potential we shall fix in what follows, the Breitenlohner-Freedman bound [183], [184] is satisfied.",
"The Maxwell-dilaton gauge coupling $f(\\phi )$ will be dynamically fixed by matching the holographic magnetic susceptibility at $B=0$ to the corresponding lattice data for $(2+1)$ -flavor QCD from Ref.",
"[141].",
"Ansatz and equations of motion.",
"The ansatz for the anisotropic magnetic backgrounds in the so-called standard coordinates, denoted as before by a tilde, is given by $d\\tilde{s}^2&=e^{2\\tilde{a}(\\tilde{r})}\\left[-\\tilde{h}(\\tilde{r})d\\tilde{t}^2+d\\tilde{z}^2\\right]+ e^{2\\tilde{c}(\\tilde{r})}(d\\tilde{x}^2+d\\tilde{y}^2)+\\frac{e^{2\\tilde{b}(\\tilde{r})} d\\tilde{r}^2}{\\tilde{h}(\\tilde{r})},\\nonumber \\\\\\tilde{\\phi }&=\\tilde{\\phi }(\\tilde{r}),\\,\\,\\, \\tilde{A}=\\tilde{A}_\\mu d\\tilde{x}^\\mu =\\hat{B}\\tilde{x}d\\tilde{y}\\Rightarrow \\tilde{F}=d\\tilde{A}=\\hat{B}d\\tilde{x}\\wedge d\\tilde{y},$ where the hat in $\\hat{B}$ is used to denote the magnetic field in units of the inverse of the asymptotically AdS$_5$ radius squared (we shall use $B$ to denote the boundary magnetic field in physical units, as we are going to discuss in a moment).",
"The boundary of the asymptotically $\\mathrm {AdS}_5$ backgrounds lies at $\\tilde{r} \\rightarrow \\infty $ while the horizon is at $\\tilde{r} = \\tilde{r}_H$ .",
"The equation of motion for the dilaton field is given by $\\tilde{\\phi }^{\\prime \\prime } + \\left(2\\tilde{a}^{\\prime }+2\\tilde{c}^{\\prime }-\\tilde{b}^{\\prime }+\\frac{\\tilde{h}^{\\prime }}{\\tilde{h}} \\right) \\tilde{\\phi }^{\\prime } - \\frac{e^{2\\tilde{b}}}{\\tilde{h}} \\left(\\frac{\\partial V}{\\partial \\tilde{\\phi }} + \\frac{\\hat{B}^2 e^{-4\\tilde{c}}}{2} \\frac{\\partial f}{\\partial \\tilde{\\phi }} \\right) = 0,$ while Einstein's equations are given by $\\tilde{a}^{\\prime \\prime }+\\left( \\frac{14}{3} \\tilde{c}^{\\prime } - \\tilde{b}^{\\prime } + \\frac{4}{3} \\frac{\\tilde{h}^{\\prime }}{h} \\right) \\tilde{a}^{\\prime } + \\frac{8}{3} \\tilde{a}^{\\prime 2} + \\frac{2}{3} \\tilde{c}^{\\prime 2} + \\frac{2}{3} \\frac{\\tilde{h}^{\\prime }}{\\tilde{h}} \\tilde{c}^{\\prime } + \\frac{2}{3} \\frac{e^{2\\tilde{b}}}{\\tilde{h}} V - \\frac{1}{6} \\tilde{\\phi }^{\\prime 2} & =0, \\\\\\tilde{c}^{\\prime \\prime } - \\left(\\frac{10}{3} \\tilde{a}^{\\prime }+\\tilde{b}^{\\prime }+\\frac{1}{3} \\frac{\\tilde{h}^{\\prime }}{\\tilde{h}} \\right) + \\frac{2}{3} \\tilde{c}^{\\prime 2} - \\frac{4}{3} \\tilde{a}^{\\prime 2} - \\frac{2}{3} \\frac{\\tilde{h}^{\\prime }}{\\tilde{h}} \\tilde{a}^{\\prime } - \\frac{1}{3}\\frac{e^{2\\tilde{b}}}{\\tilde{h}} V + \\frac{1}{3} \\tilde{\\phi }^{\\prime 2} & = 0, \\\\\\tilde{h}^{\\prime \\prime }+(2\\tilde{a}^{\\prime }+2\\tilde{c}^{\\prime }-\\tilde{b}^{\\prime })\\tilde{h}^{\\prime } & = 0.$ One can derive one last useful equation from the above, which is taken as a constraint on initial data: $\\tilde{a}^{\\prime 2} + \\tilde{c}^{\\prime 2} - \\frac{1}{4} \\tilde{\\phi }^{\\prime 2} + \\left( \\frac{\\tilde{a}^{\\prime }}{2} + \\tilde{c}^{\\prime } \\right)\\frac{\\tilde{h}^{\\prime }}{\\tilde{h}} + 4\\tilde{a}^{\\prime }\\tilde{c}^{\\prime } + \\frac{e^{2\\tilde{b}}}{2\\tilde{h}} \\left( V + \\frac{\\hat{B}^2 e^{-4\\tilde{c}}}{2} f \\right) = 0.$ Maxwell's equations are automatically satisfied by the ansatz (REF ); also, the function $\\tilde{b}(\\tilde{r})$ has no equation of motion to satisfy and, in fact, it may be gauge-fixed at will using invariance under reparametrizations of the radial coordinate.",
"In the following, we set $\\tilde{b}(\\tilde{r})=0$ .",
"Numerical coordinates.",
"As it was done before in this paper, in order to numerically solve the equations of motion we introduce numerical coordinates which are represented without the tildes.",
"The background fields are expressed in the numerical coordinates as follows (using again the $b(r)=0$ gauge), $ds^2&=e^{2a(r)}\\left[-h(r)dt^2+dz^2\\right]+e^{2c(r)}(dx^2+dy^2)+\\frac{dr^2}{h(r)},\\nonumber \\\\\\phi &=\\phi (r),\\,\\,\\,A=A_\\mu dx^\\mu =\\mathcal {B}xdy\\Rightarrow F=dA=\\mathcal {B}dx\\wedge dy.$ Let $X(r)$ be any of the background functions $a(r)$ , $c(r)$ , $h(r)$ , or $\\phi (r)$ , and take near-horizon Taylor expansions, $X(r) = X_0 + X_1(r-r_H) + X_2 (r-r_H)^2 + \\ldots $ Working with Taylor expansions up to second order results in a total of 12 coefficients to be determined in order to start the numerical integration of the equations of motion.",
"One of these infrared Taylor coefficients is the value of the dilaton field at the horizon, $\\phi _0$ , which is taken as one of the initial conditions of the system (the second initial condition corresponds to the value of the magnetic field in the numerical coordinates, which we denote in this section by $\\mathcal {B}$ ; we are going to derive later in this paper the relation between $\\hat{B}$ and $\\mathcal {B}$ ).",
"As discussed in Ref.",
"[111], one may use the freedom to rescale the bulk spacetime coordinates in order to fix the horizon location and three of the infrared Taylor coefficients as follows: $r_H = 0$ , $a_0 = c_0 = 0$ , and $h_1 = 1$ , while $h_0 = 0$ follows from the defining properties of the blackening function $h(r)$ .",
"The 7 remaining infrared Taylor coefficients are fixed on-shell as functions of the pair of initial conditions $(\\phi _0,\\mathcal {B})$ by substituting these near-horizon expansions into the equations of motion expressed in the numerical coordinates and solving the resulting coupled system of algebraic equations.",
"In order to perform the numerical integration of the equations of motion in the numerical coordinates we start a little above the horizonThis is done in order to avoid the singular point of the equations of motion corresponding to the radial location of the black hole horizon., at $r_{\\textrm {start}} = 10^{-8}$ , and integrate up to $r_{\\textrm {max}} = 10$ , which is a large value of the radial coordinate where all the geometries generated in the present work have already reached the ultraviolet fixed point corresponding to the AdS$_5$ geometry.For the set of initial conditions considered in the present work the resulting black hole geometries generally reach the ultraviolet fixed point already before $r_{\\textrm {conformal}} = 2$ but we consider integrations up to larger values of $r$ for technical reasons involving the calculation of the holographic magnetic susceptibility (see the discussion in Ref.",
"[111]) and also the fixing of the leading coefficient of the near-boundary, ultraviolet expansion for the dilaton field.",
"Then, each value of the pair of initial conditions $(\\phi _0,\\mathcal {B})$ will generate numerical black hole geometries dual to a definite physical state in the gauge theory.",
"It is also important to remark that, as discussed in Ref.",
"[111],See also a similar discussion in Ref.",
"[32] though in the different context of the EMD model at $B=0$ and finite baryon chemical potential.",
"there is an upper bound on the value of the initial condition $\\mathcal {B}$ , given some initial condition $\\phi _0$ , in order to generate asymptotically AdS$_5$ geometries.",
"This bound may be derived numerically and we display it in Fig.",
"REF .",
"Figure: Numerical bound on the maximum value for the initial condition ℬ\\mathcal {B} given some initial condition φ 0 \\phi _0: initial conditions below the curve give asymptotically AdS 5 _5 solutions.In order to determine the thermodynamic and transport observables to be presented in the following sections we generated 850,000 different anisotropic black hole geometries by constructing a rectangular grid in the plane of initial conditions $(\\phi _0,\\mathcal {B})$ , with 1000 points in the $\\phi _0$ direction varying in equally spaced steps within the interval $[0.3,7.5]$ , and for each value of $\\phi _0$ , we considered 850 points in the $\\mathcal {B}/\\mathcal {B}_\\textrm {max}(\\phi _0)$ direction varying in equally spaced steps within the interval $[0,0.99]$ .",
"These settings were enough to obtain smooth results within the phenomenologically interesting region of the physical $(T,B)$ plane studied below.",
"Asymptotics and coordinate transformations.",
"Physical observables are naturally computed in the gravity theory using the standard coordinates while numerical solutions for the background black hole geometries are obtained in the numerical coordinates defined above.",
"Therefore, we need to relate these different coordinate systems and here this is done by matching the near-boundary, ultraviolet expansions for the EMD fields in each case.",
"In the standard coordinates, the ultraviolet asymptotics attained at large $\\tilde{r}$ are given by [139], [111] $\\tilde{a} (\\tilde{r}) = \\tilde{r} + \\ldots , \\,\\,\\,\\tilde{c} (\\tilde{r}) = \\tilde{r} + \\ldots , \\,\\,\\,\\tilde{h} (\\tilde{r}) = 1 + \\ldots , \\,\\,\\,\\tilde{\\phi } (\\tilde{r}) = e^{- \\nu \\tilde{r}} + \\ldots \\,.$ On the other hand, in the numerical coordinates, the near-boundary asymptotics are given by [111] $a(r) & = \\alpha (r) + \\ldots , \\\\c(r) & = \\alpha (r) + c^{\\mathrm {far}}_0 - a^{\\mathrm {far}}_0 + \\ldots , \\\\h(r) & = h^{\\mathrm {far}}_0 + h^{\\mathrm {far}}_4 e^{-4 \\alpha (r)} + \\ldots , \\\\\\phi (r) & = \\phi _A e^{-\\nu \\alpha (r)} + \\phi _B e^{-\\Delta \\alpha (r)} + \\ldots ,$ where $\\alpha (r) = a^{\\mathrm {far}}_0 + r/\\sqrt{h^{\\mathrm {far}}_0}$ .",
"The coefficients $a^{\\mathrm {far}}_0$ , $c^{\\mathrm {far}}_0$ , $h^{\\mathrm {far}}_0$ , and $\\phi _A$ , which are required to calculate most of the physical quantities, may be found by fitting the numerical solutions close to the boundary using the ultraviolet asymptotics above.",
"We found that an accurate fitting procedure may be specified as follows: $h_0^{\\textrm {far}}=h(r_{\\textrm {conformal}})$ , while $a^{\\mathrm {far}}_0$ and $c^{\\mathrm {far}}_0$ may be extracted by employing the fitting profiles $a(r)=a^{\\mathrm {far}}_0+r/\\sqrt{h_0^{\\textrm {far}}}$ and $c(r)=c^{\\mathrm {far}}_0+r/\\sqrt{h_0^{\\textrm {far}}}$ within the interval $r\\in [r_{\\textrm {conformal}}-1,r_{\\textrm {conformal}}]$ .",
"The ultraviolet coefficient $\\phi _A$ is the most difficult to extract due to the fact that the dilaton field vanishes exponentially fast in the near-boundary region.",
"In the present work, we employed the following procedure to fix this coefficient:For the magnetic EMD model used in Ref.",
"[111] this procedure gives the same results found by the simpler procedure discussed in that reference.",
"However, for the choice of $V(\\phi )$ and $f(\\phi )$ used here, the procedure discussed in Ref.",
"[111] can only cover a very limited region of the plane of initial conditions while this new numerical procedure discussed in this paper can cover a much broader region.",
"We also checked that the present procedure works fairly well with a wide variety of different choices for $V(\\phi )$ and $f(\\phi )$ .",
"first, one defines the following adaptive variables, $r_{\\textrm {IR}}(\\phi _0,\\mathcal {B})\\equiv \\phi ^{-1}(10^{-3})$ and $r_{\\textrm {UV}}(\\phi _0,\\mathcal {B})\\equiv \\phi ^{-1}(10^{-5})$ , using next the fitting profile $\\phi (r)=\\phi _A e^{-\\nu a(r)}$ within the adaptive interval $r\\in [r_{\\textrm {IR}},r_{\\textrm {UV}}]$ to reliably extract $\\phi _A$ from the numerical solutions generated by different pairs of initial conditions $(\\phi _0,\\mathcal {B})$ .",
"By comparing both sets of asymptotic solutions in the ultraviolet, one finds a dictionary relating the standard and the numerical coordinates [111]: $\\tilde{r}&=\\frac{r}{\\sqrt{h_0^{\\textrm {far}}}}+a_0^{\\textrm {far}}-\\ln \\left(\\phi _A^{1/\\nu }\\right),\\,\\,\\,\\tilde{t}=\\phi _A^{1/\\nu }\\sqrt{h_0^{\\textrm {far}}}t,\\,\\,\\,\\tilde{x}=\\phi _A^{1/\\nu }e^{c_0^{\\textrm {far}}-a_0^{\\textrm {far}}}x,\\nonumber \\\\\\tilde{y}&=\\phi _A^{1/\\nu }e^{c_0^{\\textrm {far}}-a_0^{\\textrm {far}}}y,\\,\\,\\,\\tilde{z}=\\phi _A^{1/\\nu }z,\\,\\,\\,\\tilde{a}(\\tilde{r})=a(r)-\\ln \\left(\\phi _A^{1/\\nu }\\right),\\nonumber \\\\\\tilde{c}(\\tilde{r})&=c(r)-(c_0^{\\textrm {far}}-a_0^{\\textrm {far}})-\\ln \\left(\\phi _A^{1/\\nu }\\right),\\,\\,\\,\\tilde{h}(\\tilde{r})=\\frac{h(r)}{h_0^{\\textrm {far}}},\\,\\,\\,\\tilde{\\phi }(\\tilde{r})=\\phi (r),\\nonumber \\\\\\hat{B}&=\\frac{e^{2(a_0^{\\textrm {far}}-c_0^{\\textrm {far}})}}{\\phi _A^{2/\\nu }}\\mathcal {B}.$ Thermodynamics at zero magnetic field.",
"By employing the usual formulas for the Hawking temperature of the black hole horizon and the Bekenstein-Hawking relation for the black hole entropy density in the standard coordinates, and then going over to the numerical coordinates, one obtains that $\\hat{T} = \\frac{1}{4\\pi \\phi _A^{1/\\nu } \\sqrt{h^{\\mathrm {far}}_0}} \\quad \\textrm {and} \\quad \\hat{s} = \\frac{2\\pi e^{2(a_0^{\\textrm {far}}-c_0^{\\textrm {far}})}}{\\kappa ^2\\phi _A^{3/\\nu }}.$ As mentioned before, the hats in Eqs.",
"(REF ) and (REF ) denote the physical observables in units of powers of the inverse of the AdS$_5$ radius $L$ , which we have already set to unity.",
"In order to express observables in physical units, we introduce a scaling factor $\\Lambda $ with units of MeV such that any observable $\\hat{X}$ with mass dimension $p$ is expressed in physical units as $X=\\hat{X}\\Lambda ^p$ [MeV$^p$ ] [31], [33], [34], [111], [37].",
"Note that, by doing so, we do not introduce any new free parameter in the theory, since this procedure just exchanges the freedom in the choice of $L$ with the freedom in the choice of the scaling factor $\\Lambda $ .",
"Therefore, one has for instance, $B=\\hat{B}\\Lambda ^2$ [MeV$^2$ ], $T=\\hat{T}\\Lambda $ [MeV], and $s=\\hat{s}\\Lambda ^3$ [MeV$^3$ ].",
"As discussed in detail in Ref.",
"[111], the holographic formula for the magnetic susceptibility at zero magnetic field in the numerical coordinates is given byIdeally, one should take $T_{\\textrm {small}}=0$ , however, due to technical difficulties in subtracting numerical geometries at finite $T$ and the vacuum geometry, in practice we numerically subtracted a zero magnetic field geometry with a very small but nonzero temperature.",
"$\\chi (T,B=0)=-\\frac{1}{2\\kappa ^2}\\left[\\left(\\frac{1}{\\sqrt{h_0^{\\textrm {far}}}} \\int _{r_{\\textrm {start}}}^{r^{\\textrm {var}}_{\\textrm {max}}} dr f(\\phi (r))\\right)\\biggr |_{T,B=0}-\\left(\\textrm {same}\\right)\\biggr |_{T_{\\textrm {small}},B=0} \\right]_{\\textrm {on-shell}},$ where $r^{\\textrm {var}}_{\\textrm {max}}\\equiv \\sqrt{h_0^{\\textrm {far}}}\\left[\\tilde{r}^{\\textrm {fixed}}_{\\textrm {max}}- a_0^{\\textrm {far}}+\\ln \\left(\\phi _A^{1/\\nu }\\right)\\right]$ , with $\\tilde{r}^{\\textrm {fixed}}_{\\textrm {max}}$ being a fixed numerical ultraviolet cutoff which must be chosen in such a way that the upper limits of integration in Eq.",
"(REF ) satisfy the condition $r_{\\textrm {conformal}}\\le r^{\\textrm {var}}_{\\textrm {max}}\\le r_{\\textrm {max}}$ for all the geometries considered.",
"Now we have all the necessary ingredients to dynamically fix the dilaton potential $V(\\phi )$ , the gravitational constant $\\kappa ^2=8\\pi G_5$ , the scaling factor $\\Lambda $ , and the Maxwell-dilaton gauge coupling function $f(\\phi )$ , which are the free parameters of the bottom-up EMD model.",
"We do so by matching the holographic equation of state (represented by the temperature dependence of the entropy and internal energy densities, the speed of sound squared, the pressure, and the trace anomaly) and the holographic magnetic susceptibility at $B=0$ with the corresponding lattice results from Refs.",
"[143] and [141], respectively.",
"The results are shown in Eq.",
"(REF ) and Fig.",
"REF .",
"$V(\\phi )&=-12\\cosh (0.63\\phi )+0.65\\phi ^2-0.05\\phi ^4+0.003\\phi ^6,\\nonumber \\\\\\kappa ^2&=8\\pi G_5=8\\pi (0.46), \\quad \\Lambda =1058.83\\,\\textrm {MeV},\\nonumber \\\\f(\\phi )&=0.95\\,\\textrm {sech}(0.22\\phi ^2-0.15\\phi -0.32).$ From the dilaton potential above one can obtain that the scaling dimension of the dual operator in the gauge theory is $\\Delta \\approx 2.73$ (a relevant operator).",
"Figure: Thermodynamics of the magnetic EMD model at B=0B=0 matched to lattice data in order to fix the free parameters of the holographic model.",
"Top left: entropy density.",
"Top right: speed of sound squared.",
"Middle left: pressure.",
"Middle right: internal energy density.",
"Bottom left: trace anomaly.",
"Bottom right: magnetic susceptibility.Thermodynamics at nonzero magnetic field.",
"In Fig.",
"REF we show our numerical results for the entropy density, $s(T,B)/T^3$ , and the pressure differences, $\\Delta p(T,B)\\equiv p(T,B) - p(T_{\\textrm {ref}}=125\\textrm {MeV},B)$ , compared to the corresponding lattice data from Ref.",
"[142].",
"The pressure was obtained by integrating the entropy density with respect to temperature keeping the magnetic field fixed.",
"Therefore, it corresponds to the anisotropic pressure in the direction of the magnetic field when the magnetic flux is kept fixed under compressions, which is equal to the isotropic pressure obtained by keeping instead the magnetic field fixed under compressions [142].",
"We remark that while the results for the thermodynamics at $B=0$ displayed in Fig.",
"REF are not predictions of the EMD model, since they were dynamically fixed in order to determine the free parameters of the model in Eq.",
"(REF ), the thermodynamic results at nonzero $B$ shown in Fig.",
"REF constitute genuine predictions of the holographic model.",
"A fairly good quantitative agreement is obtained with the current lattice data up to $eB=0.6$ GeV$^2$ .",
"Figure: (Color online) Thermodynamics of the magnetic EMD model at nonzero BB compared to lattice data.",
"Left: entropy density as function of temperature for some fixed values of the magnetic field.",
"Right: pressure difference (in the direction of the magnetic field), Δp(T,B)≡p(T,B)-p(T ref =125 MeV ,B)\\Delta p(T,B)\\equiv p(T,B) - p(T_{\\textrm {ref}}=125\\textrm {MeV},B), as a function of temperature for some fixed values of the magnetic field.In Fig.",
"REF we present new holographic predictions for the entropy density and the crossover temperature extracted from its inflection point in a much wider region of the $(T,B)$ phase diagram, which has not yet been covered by lattice simulations.",
"One observes that the crossover temperature decreases with increasing magnetic field in quantitative agreement with the lattice QCD results for the values of the magnetic field already simulated on the lattice.",
"Figure: (Color online) New holographic predictions from the magnetic EMD model for the entropy density and the crossover temperature in an extended region of the (T,B)(T,B) phase diagram not yet covered on the lattice.",
"Top: entropy density.",
"Bottom: crossover temperature extracted from the inflection point of s/T 3 s/T^3 compared to the corresponding lattice data.We plan to further explore, in an upcoming work, the phase diagram of the model in the $(T,B)$ plane by accessing much larger values of magnetic fields in order to look for possible signs of real phase transitions (instead of just the smooth crossover observed here) in the magnetic medium."
],
[
"Drag force",
"The anisotropic drag forces described by Eqs.",
"(REF ) to (REF ) may be computed in the magnetic EMD backgrounds by considering $g_{tt}^{(s)}&= -\\tilde{h}(\\tilde{r})e^{2\\tilde{a}(\\tilde{r})+\\sqrt{2/3}\\,\\tilde{\\phi }(\\tilde{r})}= -\\frac{h(r)}{h_0^{\\textrm {far}}} \\frac{e^{2a(r)+\\sqrt{2/3}\\,\\phi (r)}}{\\phi _A^{2/\\nu }},\\nonumber \\\\g_{rr}^{(s)}&= \\frac{e^{\\sqrt{2/3}\\,\\tilde{\\phi }(\\tilde{r})}}{\\tilde{h}(\\tilde{r})} = \\frac{h_0^{\\textrm {far}}e^{\\sqrt{2/3}\\,\\phi (r)}}{h(r)},\\nonumber \\\\g_{xx}^{(s)}&=g_{yy}^{(s)}= e^{2\\tilde{c}(\\tilde{r})+\\sqrt{2/3}\\,\\tilde{\\phi }(\\tilde{r})}= \\frac{e^{2(c(r)-c_0^{\\textrm {far}}+a_0^{\\textrm {far}})+\\sqrt{2/3}\\,\\phi (r)}}{\\phi _A^{2/\\nu }},\\nonumber \\\\g_{zz}^{(s)}&= e^{2\\tilde{a}(\\tilde{r})+\\sqrt{2/3}\\,\\tilde{\\phi }(\\tilde{r})}= \\frac{e^{2a(r)+\\sqrt{2/3}\\,\\phi (r)}}{\\phi _A^{2/\\nu }},$ where one must be also careful in correctly applying the chain rule for radial derivatives when passing from the standard to the numerical coordinates, $^{\\prime }\\equiv \\partial _{\\tilde{r}}=\\sqrt{h_0^{\\textrm {far}}}\\partial _r$ .",
"Our numerical results for the magnetic field induced anisotropic drag forces, normalized by the isotropic SYM result at zero magnetic field given in Eq.",
"(REF ), are displayed in Figs.",
"REF and REF .",
"Note that in the EMD model we have three independent variables, $T$ , $B$ , and $v$ that may be varied independently.",
"From Fig.",
"REF we see that $F_{\\textrm {drag}}^{(v\\parallel B)}$ is not affected by the magnetic field for moderate velocities while it increases with $B$ for ultrarelativistic velocities.",
"On the other hand, its overall magnitude decreases with increasing $v$ .",
"This means that faster/lighter (charm) quarks moving in the direction of the magnetic field are more sensitive to the effects of a nonzero $B$ than slower/heavier (bottom) quarks.",
"Furthermore, we note that $F_{\\textrm {drag}}^{(v\\parallel B)}$ acquires a non-monotonic behavior as function of $T$ in the crossover region only in the case of ultrarelativistic velocities.",
"Additionally, from Fig.",
"REF one observes that $F_{\\textrm {drag}}^{(v\\perp B)}$ is affected by a nonzero magnetic field both at moderate and ultrarelativistic speeds and that it increases with $B$ .",
"Also, in the perpendicular channel the drag force is reduced with increasing $v$ .",
"In the transverse plane to the magnetic field direction, at lower temperatures, bottom quarks feel more the effects of a nonzero $B$ than charm quarks, with this tendency being inverted at higher temperatures.",
"Moreover, we find that $F_{\\textrm {drag}}^{(v\\perp B)}$ develops a non-monotonic behavior as function of $T$ in the crossover region just for large $v$ and low $B$ .",
"Figure: (Color online) Anisotropic drag force F drag (v∥B) F_{\\textrm {drag}}^{(v\\parallel B)} in the magnetic EMD model normalized by the isotropic SYM result at zero magnetic field.",
"Top: results for the heavy quark velocity v=0.50v=0.50.",
"Bottom: results in the ultrarelativistic limit v=0.99v=0.99.Figure: (Color online) Anisotropic drag force F drag (v⊥B) F_{\\textrm {drag}}^{(v\\perp B)} in the magnetic EMD model normalized by the isotropic SYM result at zero magnetic field.",
"Top: results for the heavy quark velocity v=0.50v=0.50.",
"Bottom: results in the ultrarelativistic limit v=0.99v=0.99.Figs.",
"REF and REF also show that in general, $F_{\\textrm {drag}}^{(v\\perp B)} > F_{\\textrm {drag}}^{(v\\parallel B)}$ , i.e., the energy loss is larger in the transverse plane than along the magnetic field direction as also noticed previously in the magnetic brane setup.",
"The discussion in this section illustrates how rich (and complex) heavy quark energy loss may become once spatial isotropy is broken by a magnetic field."
],
[
"Langevin diffusion coefficients",
"The anisotropic Langevin diffusion coefficients described by Eqs.",
"(REF ) to (REF ) and Eqs.",
"(REF ) to () may be computed in the magnetic EMD background by considering the relations in Eq.",
"(REF ).",
"Our numerical results for the magnetic field induced anisotropic Langevin coefficients, normalized by the SYM results at zero magnetic field given by the relations in Eq.",
"(REF ), are displayed in Figs.",
"REF to REF .",
"Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (∥v) (v∥B) \\kappa _{(\\parallel v)}^{(v\\parallel B)} in the magnetic EMD model normalized by the SYM result at zero magnetic field.",
"Top: results for the heavy quark velocity v=0.50v=0.50.",
"Bottom: results in the ultrarelativistic limit v=0.99v=0.99.Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (⊥v) (v∥B) \\kappa _{(\\perp v)}^{(v\\parallel B)} in the magnetic EMD model normalized by the SYM result at zero magnetic field.",
"Top: results for the heavy quark velocity v=0.50v=0.50.",
"Bottom: results in the ultrarelativistic limit v=0.99v=0.99.Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (∥v) (v⊥B) \\kappa _{(\\parallel v)}^{(v\\perp B)} in the magnetic EMD model normalized by the SYM result at zero magnetic field.",
"Top: results for the heavy quark velocity v=0.50v=0.50.",
"Bottom: results in the ultrarelativistic limit v=0.99v=0.99.Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (⊥v,⊥B) (v⊥B) \\kappa _{(\\perp v,\\perp B)}^{(v\\perp B)} in the magnetic EMD model normalized by the SYM result at zero magnetic field.",
"Top: results for the heavy quark velocity v=0.50v=0.50.",
"Bottom: results in the ultrarelativistic limit v=0.99v=0.99.Figure: (Color online) Anisotropic Langevin diffusion coefficient κ (⊥v,∥B) (v⊥B) \\kappa _{(\\perp v,\\parallel B)}^{(v\\perp B)} in the magnetic EMD model normalized by the SYM result at zero magnetic field.",
"Top: results for the heavy quark velocity v=0.50v=0.50.",
"Bottom: results in the ultrarelativistic limit v=0.99v=0.99.Our results show that momentum diffusion is always reduced with increasing velocityFor fixed energy, bottom quarks are slower than charm quarks and, thus, the latter should experience less momentum diffusion than the former..",
"In the $\\vec{v}\\parallel \\vec{B}$ channel, $\\kappa _{(\\parallel v)}^{(v\\parallel B)}$ always increases with $B$ while it decreases with $T$ at lower speeds (bottom quarks), whilst increasing with $T$ for larger velocities (charm quarks).",
"The same observations hold also for $\\kappa _{(\\perp v)}^{(v\\parallel B)}$ , but in general, $\\kappa _{(\\perp v)}^{(v\\parallel B)} > \\kappa _{(\\parallel v)}^{(v\\parallel B)}$ .",
"On the other hand, in the $\\vec{v}\\perp \\vec{B}$ channel, $\\kappa _{(\\parallel v)}^{(v\\perp B)}$ and $\\kappa _{(\\perp v,\\perp B)}^{(v\\perp B)}$ always increase with $B$ while decreasing with $T$ at lower velocities, whilst increasing [decreasing] with $T$ at higher velocities for lower [higher] values of the magnetic field.",
"Regarding $\\kappa _{(\\perp v,\\parallel B)}^{(v\\perp B)}$ , we see that it is not affected by the magnetic field at lower velocities (bottom quarks) while it increases with $B$ for larger velocities (charm quarks).",
"This quantity decreases with $T$ at lower velocities while it increases [decreases] with $T$ for larger velocities and lower [larger] values of the magnetic field.",
"Thus, associating slower moving probes with bottom quarks one finds that in this case $\\kappa _{(\\perp v,\\perp B)}^{(v\\perp B)} > \\kappa _{(\\parallel v)}^{(v\\perp B)} > \\kappa _{(\\perp v,\\parallel B)}^{(v\\perp B)}$ , while for more rapidly moving heavy probes (charm quarks), $\\kappa _{(\\perp v,\\perp B)}^{(v\\perp B)} > \\kappa _{(\\parallel v)}^{(v\\perp B)} \\sim \\kappa _{(\\perp v,\\parallel B)}^{(v\\perp B)}$ .",
"Consequently, one concludes that heavy quark momentum diffusion in directions transverse to the magnetic field is generally larger than in the direction of the field, as also noticed before in the magnetic brane setup."
],
[
"Shear viscosity",
"The anisotropic shear viscosities described by Eqs.",
"(REF ) and () may be computed in the magnetic EMD background by considering the relations in Eq.",
"(REF ).",
"In Fig.",
"REF we plot our numerical results for the ratio between the parallel and the perpendicular shear viscosities.",
"Figure: (Color online) Ratio between the parallel and the perpendicular shear viscosities in the magnetic EMD model as a function of TT and BB.As previously found for the magnetic brane model, also in the EMD model one observes that the shear viscosity is reduced in the direction of the magnetic field relative to the viscosity transverse to the field.",
"This shows that the anisotropic plasma becomes a more perfect fluid along the magnetic field direction.",
"Note also the strong $T$ dependence of this ratio near the crossover region, which could not be studied before in the context of the magnetic brane model discussed in the previous sections.",
"Moreover, the behavior of $\\eta _{\\parallel }$ is similar to the one found within a Boltzmann equation calculation [185] in which the anisotropic viscosities decrease with increasing magnetic field.",
"With the values of the anisotropic shear viscosities for the QCD-like model at hand, one may use them in realistic numerical viscous magnetohydrodynamics calculations to investigate the effects of a magnetic field on the hydrodynamic evolution of the QGP (assuming that the magnetic field remains strong enough after $\\sim 1$ fm/c to affect its evolution).",
"Even though the current state-of-the-art of relativistic hydrodynamics [8] has not yet incorporated viscous magnetohydrodynamic effects in a stable and causal mannerSee, for instance, Refs.",
"[186], [187], [188], [189], [190], [191], [192] for efforts towards deriving a causal an stable dissipative theory of anisotropic fluid dynamics., it must be possible to express the corresponding anisotropic viscosities in any hydrodynamic theory via Kubo formulas - and this is exactly what was done here.",
"Furthermore, it may be possible to investigate qualitatively new effects from the (anisotropic) viscosity by extending previous works that exploit simple flow patterns, such as the Bjorken flow studied already in the context of ideal magnetohydrodynamics [193], [194], [195].",
"Additionally, it would be interesting to check how the magnetic field affects the bulk viscosity of the medium (see Appendix for a discussion of the corresponding Kubo formulas), which was shown to be relevant in heavy ion collision simulations in [196], [197], [198] and, more recently, in [12].",
"In the present work, we conducted a systematic investigation of momentum transport in strongly coupled anisotropic plasmas in the presence of strong magnetic fields.",
"We studied two different holographic settings, one corresponding to a top-down deformation of SYM theory triggered by an external magnetic field, and the other one corresponding to a phenomenologically motivated bottom-up EMD model, which is able to quantitatively reproduce $(2+1)$ flavor lattice QCD thermodynamics with physical quark masses at both zero and nonzero magnetic fields.",
"The main conclusions of the present work regarding transport properties in strongly coupled magnetized media hold for both models.",
"Namely, energy loss and momentum diffusion are generally enhanced in the presence of a magnetic field being larger in transverse directions than in the direction parallel to the magnetic field.",
"Moreover, the anisotropic shear viscosity was found to be lower in the direction of the magnetic field than in the plane perpendicular to the field, which indicates that strongly coupled magnetized plasmas can become even closer to the idealized perfect fluid limit along the magnetic field direction.",
"Compared to other anisotropic models available in the literature, such as the Einstein-axion-dilaton model from Refs.",
"[128], [129], our results for the Einstein-Maxwell and EMD models have some interesting common features and some qualitative differences as well.",
"For instance, the anisotropic shear viscosity $\\eta _\\parallel /s\\le \\eta _\\perp /s=1/4\\pi $ always decreases as the magnitude of the source of anisotropy (a magnetic field or a nontrivial bulk axion profile) increases; indeed, this is the main topic of Ref.",
"[157].",
"On the other hand, the anisotropic drag force in the transverse plane is larger than in the direction of the magnetic field, which is the opposite behavior found in the Einstein-axion-dilaton model for an axion driven anisotropy [131].",
"The phenomenological consequences of our results to the current effort in the heavy ion community toward finding direct effects of strong magnetic fields on the QGP will be discussed elsewhere.",
"However, an immediate consequence of the suppression of the shear viscosity along the magnetic field direction found here seems to be that it provides a potential source of suppression for the elliptic flow coefficient in strongly magnetized media.",
"Naively, such a suppression would occur since the flow along the magnetic field direction (which should be, on average, perpendicular to the 2nd harmonic event plane [44]) is nearly dissipationless while the viscosity in the direction perpendicular to the field can be twice as large.",
"However, a realistic estimate of such an effect requires numerical magnetohydrodynamic calculationsFor a discussion of other effects of strong magnetic fields on elliptic flow in ideal hydrodynamics see [199]..",
"In this regard, our results for the phenomenologically realistic magnetic EMD model, which constitute the main outcome of the present work, could be readily used as inputs in numerical codes for magnetohydrodynamics.",
"We also presented, in the context of the QCD-like magnetic EMD model, new holographic predictions for the behavior of the entropy density and the crossover temperature extracted from its inflection point in a broader region of the $(T,B)$ phase diagram which has not yet been covered by lattice simulations.",
"In particular, the crossover temperature, which is also in quantitative agreement with the lattice results, was found to be reduced all the way up to $eB=1.5$ GeV$^2$ .",
"Very recently, we used the magnetic EMD model presented here to compute the holographic Polyakov loop and heavy quark entropy in the presence of strong magnetic fields, finding quantitative agreement with lattice results in the deconfined plasma phase [158].",
"We also intend to study in an upcoming work the QCD phase diagram in the $(T,B)$ plane for much larger values of the magnetic field looking for possible signs of real phase transitions [86] instead of the smooth crossover observed here up to $eB=1.5$ GeV$^2$ .",
"Other projects we plan to pursue in the near future include the calculation of quasinormal modes and the anisotropic bulk viscosities in the context of the phenomenological magnetic EMD setup.",
"J.N.",
"thanks G. Endrodi for discussions about the QCD phase diagram in the presence of strong magnetic fields and also for making available the corresponding lattice data.",
"S.I.F.",
"was supported by the São Paulo Research Foundation (FAPESP) under FAPESP grant number 2015/00240-7 and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).",
"R.C.",
"acknowledges financial support by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).",
"R.R.",
"acknowledges financial support by FAPESP under FAPESP grant number 2013/04036-0.",
"J.N.",
"acknowledges financial support by FAPESP and CNPq.",
"We also thank Pavel Kovtun and Juan Hernandez for pointing out inconsistencies in some of the Kubo relations derived in Appendix B of the previous version of this work, which we corrected in the present version."
],
[
"Revision of heavy quark momentum transport",
"In this Appendix we review the general holographic formalism used to calculate the heavy quark drag force, the Langevin diffusion coefficients, and the shear viscosity.",
"The discussion presented here describes both isotropic and anisotropic plasmas.",
"In the cases we are interested in, the anisotropy will be induced by an external magnetic field acting on the plasma, which explicitly breaks $SO(3)$ spatial rotation symmetry down to $SO(2)$ rotations in the plane transverse to the direction of the magnetic field, which we shall consider to be constant and uniform in the $z$ direction.",
"The derivations reviewed in this section have been treated extensively in the literature and are included here for completeness and also to set a clear notation for our purposes.",
"In the isotropic cases, the formalism for calculating the heavy quark drag force was originally discussed in Refs.",
"[119], [120], [121], [122], [123], [124], while the formalism for the Langevin diffusion coefficients was presented in Refs.",
"[125], [126], [127].",
"The method for calculating the holographic shear viscosity in isotropic gravity duals with actions containing at most two derivatives was treated in Refs.",
"[13], [14], [15].",
"In anisotropic environments, the approach used to obtain the drag forces was discussed in Refs.",
"[130], [131], [132], while the Langevin coefficients were studied in Refs.",
"[133], [134], [135].",
"The calculation of anisotropic shear viscosities in holography was done, for instance, in Refs.",
"[136], [108], [156], [157].",
"See also Ref.",
"[33] for the calculation of the isotropic drag force and the Langevin diffusion coefficients in a phenomenologically realistic EMD model at finite baryon chemical potential and zero magnetic field, and Ref.",
"[144] for the calculation of the drag forces in an anisotropic holographic plasma at finite chemical potential and zero magnetic field.",
"Before proceeding, let us fix the notation used in this paper.",
"We deal with five dimensional black hole backgrounds which may be written, in the Einstein frame, as followsSince we use a mostly plus Lorentzian metric signature, $g_{tt}<0$ .",
"$ds^2 = g_{tt}(r)dt^2+g_{rr}(r)dr^2+g_{xx}(r)(dx^2+dy^2)+g_{zz}(z)dz^2,$ where $r$ is the holographic radial coordinate.",
"In the different coordinate systems we use in this work, the boundary of the asymptotically $\\mathrm {AdS}_5$ geometries is always at $r\\rightarrow \\infty $ .",
"The geometries we consider have a black hole horizon at $r = r_H$ , where $g_{tt}(r)$ and $g_{rr}^{-1}(r)$ have a simple zero.",
"For nonzero external magnetic fields, these backgrounds display an anisotropy that differentiates the $(x,y)$ plane transverse to the magnetic field from the $z$ direction, that is, $g_{xx} \\ne g_{zz}$ for $B\\ne 0$ , while $g_{xx} = g_{zz}$ in the isotropic $B=0$ case.",
"The starting point to determine the holographic drag forces and Langevin diffusion coefficients is to consider the Nambu-Goto action for a classical relativistic bosonic string which, in the string frame denoted here by a superscript $(s)$ , is written as follows $S_{\\textrm {NG}} = -\\frac{1}{2\\pi \\alpha ^{\\prime }} \\int d\\tau d\\sigma \\sqrt{-\\gamma ^{(s)}},$ where $\\alpha ^{\\prime }=l_s^2$ is the square of the string length, which may be related to the effective t' Hooft coupling in the dual gauge theory as (we set in the present work the radius of the asymptotically $\\mathrm {AdS}_5$ spaces to unity) $\\alpha ^{\\prime }=\\lambda _t^{-1/2}$ , $\\gamma ^{(s)}\\equiv \\det \\gamma _{ab}^{(s)}$ , with $\\gamma ^{(s)}_{ab} = g_{\\mu \\nu }^{(s)}\\partial _{a} \\mathcal {X}^\\mu \\partial _{b} \\mathcal {X}^\\nu $ being the induced metric on the string worldsheet (i.e., the pullback).",
"The string worldsheet is parametrized by the internal coordinates $a, b \\in \\lbrace \\tau , \\sigma \\rbrace $ , $\\mathcal {X}^\\mu (\\sigma ,\\tau )$ is the worldsheet embedding on the target background spacetime, and $g_{\\mu \\nu }^{(s)}$ is the string frame metric of the background.",
"The relation between the string frame metric, $g_{\\mu \\nu }^{(s)}$ , and the Einstein frame metric, $g_{\\mu \\nu }$ , is given by $g_{\\mu \\nu }^{(s)}=e^{\\sqrt{\\frac{2}{3}}\\phi }g_{\\mu \\nu },$ where $\\phi $ is the background dilaton field (following the general philosophy of Improved Holographic QCD [145], [146], [147]).",
"The general idea that comes from the holographic dictionary is that, following the original proposals to calculate holographic Wilson loops [148], [149], any string endpoint attached to an ultraviolet brane near the boundary of the background spacetime is to be regarded as a probe quark living in the gauge theory.",
"The motion of this string endpoint at the boundary is influenced by the bulk dynamics of the string and, therefore, by studying its dynamics one can extract information about the behavior of the probe heavy quark in the gauge theory."
],
[
"Isotropic case",
"We start by reviewing the holographic formalism to calculate the drag force in isotropic media.",
"A heavy probe moving with a constant velocity $\\vec{v}$ is inserted at the boundary and one computes the force that must be applied on the probe in order to equilibrate the effect of the drag force exerted by the medium.",
"We will consider a heavy probe moving along the $x$ axis, i.e, $\\vec{v}=v\\hat{x}$ .",
"We work in the so-called static gauge for the string worldsheet where one fixes $\\sigma = r$ and $\\tau = t$ .",
"The trailing string ansatz for the embedding function describing this situation is given by $\\mathcal {X}^\\mu =(\\mathcal {X}^t,\\mathcal {X}^r,\\mathcal {X}^x,\\mathcal {X}^y,\\mathcal {X}^z)= (t,r,x(t,r)=vt+\\xi (r),0,0),$ where $\\xi (r)$ describes the bulk string profile in the $(r,x)$ plane, which is dynamically fixed by extremizing the Nambu-Goto action (REF ) on the black brane backgrounds (REF ).",
"The Nambu-Goto action with the trailing string ansatz becomesFor heavy quarks with finite masses one should consider the minimal coupling of the background Maxwell field describing the external magnetic field at the boundary with the string worldsheet, as done in Ref.",
"[112].",
"However, in the present work we consider only infinitely heavy probes and in this case the minimal coupling is $\\alpha ^{\\prime }$ suppressed relatively to the Nambu-Goto action and may be neglected at leading order in the t' Hooft coupling.",
"$S_{\\textrm {NG}} = -\\frac{1}{2\\pi \\alpha ^{\\prime }} \\int dr dt \\sqrt{-g_{tt}^{(s)} g_{rr}^{(s)} -g_{tt}^{(s)} g_{xx}^{(s)} \\xi ^{\\prime }(r)^2 - g_{rr}^{(s)} g_{xx}^{(s)} v^2},$ where $\\xi ^{\\prime }(r)\\equiv \\partial _r\\xi (r)$ .",
"This action does not depend explicitly on $\\xi (r)$ and, therefore, the corresponding radial conjugate momentum, $\\pi _{\\xi } = \\frac{\\delta S_{\\textrm {NG}}}{\\delta \\xi ^{\\prime }} = - \\frac{1}{2 \\pi \\alpha ^{\\prime }} \\frac{g_{tt}^{(s)} g_{xx}^{(s)} \\xi ^{\\prime }(r)}{\\sqrt{-g^{(s)}_{tt} g^{(s)}_{rr} -g_{tt}^{(s)} g_{xx}^{(s)} \\xi ^{\\prime }(r)^2 - g_{rr}^{(s)} g_{xx}^{(s)} v^2}},$ is conserved in the radial direction and we may compute it at any value of $r$ .",
"We can do it by first solving the above relation for $\\xi ^{\\prime }(r)$ , $\\xi ^{\\prime }(r) = \\sqrt{\\frac{-g_{tt}^{(s)} g_{rr}^{(s)} - g_{xx}^{(s)} g_{rr}^{(s)} v^2}{g_{tt}^{(s)} g_{xx}^{(s)} \\left(1 + \\frac{g_{tt}^{(s)} g_{xx}^{(s)}}{(2\\pi \\alpha ^{\\prime } \\pi _{\\xi })^2} \\right)}},$ which must be realOtherwise the probe's trajectory at the boundary, $x(t,r\\rightarrow \\infty )=vt+\\xi (r\\rightarrow \\infty )$ , would not be real-valued..",
"Therefore, in order to render the expression inside the square root nonnegative for every $r$ , when the numerator changes sign, the denominator must also change sign which happens at a common value of the radial coordinate $r=r_\\star $ obtained by solving the equation $g_{tt}^{(s)} (r_\\star ) + g_{xx}^{(s)} (r_\\star ) v^2 = 0.$ As discussed in detail in Refs.",
"[125], [126], [127], the induced worldsheet metric has the form of a two dimensional black hole with a horizon precisely at $r=r_\\star $ .",
"Evaluating (REF ) at the worldsheet horizon, one obtains $\\pi _{\\xi } = - \\frac{\\sqrt{g_{tt}^{(s)}(r_\\star ) g_{xx}^{(s)} (r_\\star )}}{2\\pi \\alpha ^{\\prime }}.$ The flux of momentum from the string endpoint at the boundary to the worldsheet horizon in the interior of the bulk is equal to the drag force exerted by the plasma on the infinitely heavy probe at the boundary, which is given by $\\pi _{\\xi }$ .",
"Therefore, the drag force exerted by the medium on the probe reads [124] $F^{\\textrm {(iso)}}_{\\textrm {drag}} = \\frac{dp_x}{dt} = \\pi _{\\xi } = -\\frac{1}{2\\pi \\alpha ^{\\prime }} \\sqrt{-g_{tt}^{(s)} (r_\\star ) g_{xx}^{(s)} (r_\\star )} = -\\frac{\\sqrt{\\lambda _t}}{2\\pi } g_{xx}^{(s)}(r_\\star ) v.$ Specializing the isotropic background to the AdS$_5$ -Schwarzschild case, which is the gravity dual of a strongly coupled SYM plasma at $B=0$ , we obtain the well-known result [119], [120] $\\frac{F_{\\textrm {drag}}^{\\textrm {(SYM)}}}{\\sqrt{\\lambda _t} T^2} = -\\frac{\\pi \\gamma v}{2},$ where $\\gamma =1/\\sqrt{1-v^2}$ is the Lorentz factor."
],
[
"Anisotropic case",
"In the anisotropic case, one needs to consider the angle $\\varphi $ formed between the velocity $\\vec{v}$ of the heavy probe and the anisotropic direction $\\hat{z}$ , which in our case is the direction of the magnetic field.",
"A general discussion which yields the drag force in terms of an arbitrary angle $\\varphi $ was presented in Appendix A of [132], generalizing the original derivation for transverse and longitudinal angles first presented in Appendix B of [131].",
"We consider below an alternative derivation based in Appendix A of Ref.",
"[132], but here we make a rotation of the $xz$ axes before computing the drag force.",
"We then particularize our results to the transverse and longitudinal orientations, obtaining the same results as in Refs.",
"[131], [132].",
"Let us reorient our old $xz$ axes into new $XZ$ axes such that now it is the new $X$ direction which coincides with the probe's velocity direction.",
"That is, $dx = \\cos (\\theta ) dX - \\sin (\\theta ) dZ \\quad \\mathrm {and} \\quad dz = \\sin (\\theta ) dX + \\cos (\\theta ) dZ,$ where $\\theta \\equiv \\pi /2 - \\varphi $ is the angle between the $\\hat{x}$ direction and the velocity of the probe $\\vec{v}\\parallel \\hat{X}$ .",
"The (Einstein frame) rotated metric becomes $ds^2 = g_{tt} dt^2 + g_{rr} dr^2 + g_{XX} dX^2 + g_{yy} dy^2 + g_{ZZ} dZ^2 + 2 g_{XZ} dX dZ,$ where $g_{XX} = g_{xx} \\cos ^2 \\theta + g_{zz} \\sin ^2 \\theta ,\\,\\,\\, g_{ZZ} = g_{xx} \\sin ^2 \\theta + g_{zz} \\cos ^2 \\theta ,\\,\\,\\,g_{XZ} = \\cos \\theta \\sin \\theta (g_{zz} - g_{xx}).$ The derivation of the drag force carried out in the isotropic case can be repeated here with almost no modifications, except for the substitution $x \\rightarrow X$ , in which case one obtains that $F^{\\textrm {(aniso)}}_{\\textrm {drag}}(\\theta ) = -\\frac{\\sqrt{\\lambda _t}}{2\\pi } g_{XX}^{(s)}(\\theta , r_\\star (\\theta )) v,$ where $r_\\star (\\theta )$ is the root of $g_{tt}(r_\\star (\\theta ))+g_{XX}(r_\\star (\\theta ),\\theta )v^2=0.$ If one takes $\\theta = \\pi /2$ , such that the quark's velocity is parallel to the magnetic field, one obtains that [131], $F_{\\textrm {drag}}^{(v\\parallel B)} \\equiv F^{\\textrm {(aniso)}}_{\\textrm {drag}}(\\theta =\\pi /2) = - \\frac{\\sqrt{\\lambda _t}}{2\\pi } g_{zz}^{(s)} (r_\\star ^{\\parallel }) v,$ where $r_\\star ^{\\parallel }$ is the root of $g_{tt}^{(s)} (r_\\star ^{\\parallel }) + g_{zz}^{(s)} (r_\\star ^{\\parallel }) v^2 = 0.$ The other limiting case we shall consider here corresponds to a heavy quark moving perpendiculary to the magnetic field direction, which is obtained by taking $\\theta = 0$ [131], $F_{\\textrm {drag}}^{(v\\perp B)} \\equiv F^{\\textrm {(aniso)}}_{\\textrm {drag}}(\\theta = 0) = - \\frac{\\sqrt{\\lambda _t}}{2\\pi } g_{xx}^{(s)} (r_\\star ^{\\perp }) v,$ where $r_\\star ^{\\perp }$ is the root of $g_{tt}^{(s)} (r_\\star ^{\\perp }) + g_{xx}^{(s)} (r_\\star ^{\\perp }) v^2 = 0.$"
],
[
"Diffusion from worldsheet membrane paradigm",
"The drag force derived previously in the trailing string scenario defines a matrix of friction coefficients, $\\eta ^D_{ij}$ , $i,j\\in \\lbrace x,y,z\\rbrace $ , according to the following equation, $F_i^{\\textrm {drag}}=\\frac{dp_i}{dt}=-\\eta ^D_{ij}p^j, \\quad \\eta ^D_{ij}\\equiv \\frac{\\sqrt{\\lambda _t}}{2\\pi } \\frac{g_{ij}^{(s)}(r_\\star )}{m_q\\gamma },$ where $m_q$ is the quark mass and $p_i=m_q\\gamma v_i$ is its 4-momentum.",
"Note that for a diagonal background metric, the friction matrix $\\eta ^D_{ij}$ in Eq.",
"(REF ) is also diagonal.",
"In order to calculate the Langevin diffusion coefficients associated with the Brownian motion of the heavy quark under the influence of thermal fluctuations one needs to add a small perturbation $\\delta \\mathcal {X}^\\mu $ for the worldsheet embedding function on top of the trailing string ansatz in the static gauge given by Eq.",
"(REF ), $\\bar{\\mathcal {X}}^\\mu =\\mathcal {X}^\\mu +\\delta \\mathcal {X}^\\mu = (t,r,x^\\ell (t,r),x^i(t,r),x^j(t,r))=(t,r, vt+\\xi (r)+\\delta \\mathcal {X}^\\ell (t,r),\\delta \\mathcal {X}^i(t,r),\\delta \\mathcal {X}^j(t,r)),$ where we employ the spatial index $\\ell $ to denote the component in the direction of the quark's velocity, $\\vec{v}=v\\hat{x}_\\ell $ .",
"The inclusion of thermal fluctuations modifies the equation of motion (REF ) for the probe quark in the gauge theory.",
"In fact, the boundary value of the worldsheet embedding fluctuation, $\\delta \\mathcal {X}_i(t,r\\rightarrow \\infty )$ , serves as a source for an operator $\\mathcal {F}_i(t)$ playing the role of a random force acting on the heavy quark in the gauge theory.",
"Assuming that for long times the temporal correlation functions of the operator $\\mathcal {F}_i(t)$ are proportional to Dirac delta distributions, with the proportionality factors defining the Langevin diffusion coefficients $\\kappa _{ij}$ , one may derive in this limit an effective equation of motion for the heavy quark taking into account thermal fluctuations, which has the form of a local Langevin equation [127], [133], $\\frac{dp_i}{dt}=-\\eta ^D_{ij}p^j+\\mathcal {F}_i(t),$ and one can also show that the Langevin diffusion coefficients satisfy the following modified Einstein's relationEq.",
"(REF ) is a modification of the usual Einstein's relation since it relates the friction and the diffusion coefficients through the worldsheet temperature $T_\\star $ instead of the heat bath temperature $T$ .",
"According to Ref.",
"[127], $T_\\star $ is the temperature measured by a probe quark moving at speed $v$ inside a strongly coupled plasma.",
"[127], [133], $\\kappa _{ij} = 2 T_\\star \\eta ^D_{ij} = -2 T_\\star \\lim _{\\omega \\rightarrow 0} \\frac{\\text{Im}\\,G^{R}_{ij}(\\omega )}{\\omega },$ where $G^{R}_{ij}(\\omega )$ is the retarded propagator of the random force operator $\\mathcal {F}_i(t)$ and $T_\\star $ is the worldsheet Hawking's temperature, to be discussed in the sequel.",
"The friction coefficients $\\eta ^D_{ij}=-\\lim _{\\omega \\rightarrow 0}\\text{Im}\\,G^{R}_{ij}(\\omega )/\\omega $ may be directly extracted from the quadratic action for the worldsheet embedding fluctuations to be derived next using the membrane paradigm [150] applied to the worldsheet horizon, as we are going to discuss below."
],
[
"Thermal fluctuations on the trailing string",
"In this subsection we review the results of Refs.",
"[133], [134] concerning the derivation of the quadratic fluctuated action for the string worldsheet.",
"The disturbed metric induced on the string worldsheet in the string frame, associated with the perturbed ansatz (REF ), reads $\\bar{\\gamma }^{(s)}_{ab}= \\gamma ^{(s)}_{ab}+\\delta \\gamma _{ab}^{(s)\\,(1)}+\\delta \\gamma _{ab}^{(s)\\,(2)},$ where $\\gamma _{ab}^{(s)}$ is the unperturbed pullback and we also have linear and quadratic terms in the perturbations, $\\delta \\gamma _{ab}^{(s)\\,(1)} &=g_{\\ell \\ell }^{(s)}\\left( \\partial _{a}\\mathcal {X}^{\\ell }\\partial _{b}\\delta \\mathcal {X}^{\\ell }+\\partial _{b}\\mathcal {X}^{\\ell }\\partial _{a}\\delta \\mathcal {X}^{\\ell } \\right), \\\\\\delta \\gamma _{ab}^{(s)\\,(2)}& = \\sum _{i}g_{ii}^{(s)} \\partial _{a}\\delta \\mathcal {X}^{i}\\partial _{b}\\delta \\mathcal {X}^{i}.$ The task now is to obtain an expression for $\\sqrt{-\\bar{\\gamma }}$ where $\\bar{\\gamma }$ is the determinant of the perturbed pullback.",
"Since the fluctuations of the worldsheet embedding are small, we retain only terms up to second order in the perturbations, $\\sqrt{-\\bar{\\gamma }^{(s)}}=\\sqrt{-\\gamma ^{(s)}}\\left[1 +\\frac{1}{2}\\delta \\gamma ^{(s)(1)a}_{a}-\\frac{1}{4}\\delta \\gamma ^{(s)(1)ab}\\delta \\gamma ^{(s)(1)}_{ab}+\\frac{1}{8}(\\delta \\gamma ^{(s)(1)a}_{a})^2+\\frac{1}{2}\\delta \\gamma ^{(s)(2)a}_{a} \\right].$ In the above expression, we are not interested in the first two terms since they are not quadratic in the perturbations and, therefore, they do not contribute to the 2-point Green's function of the stochastic force operator $\\mathcal {F}_i(t)$ sourced by the boundary value of the worldsheet embedding fluctuations.",
"The explicit corrections contributing to this propagator are given by the last three terms in Eq.",
"(REF ), which can be written as $\\frac{1}{4}\\delta \\gamma ^{(s)(1)ab}\\delta \\gamma _{ab}^{(s)(1)} &= \\frac{1}{2}g_{\\ell \\ell }^{(s)2}\\left[(\\gamma ^{(s)cd}\\partial _{c}\\mathcal {X}^{\\ell }\\partial _{d}\\mathcal {X}^{\\ell })\\gamma ^{(s)ab}\\partial _{a}\\delta \\mathcal {X}^{\\ell }\\partial _{b}\\delta \\mathcal {X}^{\\ell } +(\\gamma ^{(s)cd}\\partial _{c}\\mathcal {X}^{\\ell }\\partial _{d}\\delta \\mathcal {X}^{\\ell })^2 \\right], \\\\\\frac{1}{8}(\\delta \\gamma ^{(s)(1)a}_{a})^2 &= \\frac{1}{2}g_{\\ell \\ell }^{(s)2}(\\gamma ^{(s)cd}\\partial _{c}\\mathcal {X}^{\\ell }\\partial _{d}\\delta \\mathcal {X}^{\\ell })^2, \\\\\\frac{1}{2}\\delta \\gamma ^{(s)(2)a}_{a} &= \\gamma ^{(s)ab}g_{ij}^{(s)}\\partial _{a}\\delta \\mathcal {X}^{i}\\partial _{b}\\delta \\mathcal {X}^{j}.$ Plugging these corrections into the Nambu-Goto action, we obtain the following quadratic action for the worldsheet embedding fluctuations $S_2=-\\frac{1}{2\\pi \\alpha ^{\\prime }}\\int dt dr \\sqrt{-\\gamma ^{(s)}}\\frac{\\gamma ^{(s)ab}}{2}\\left[ (1-g_{\\ell \\ell }^{(s)}\\gamma ^{(s)cd}\\partial _{c}\\mathcal {X}^{\\ell }\\partial _{d}\\mathcal {X}^{\\ell })g_{\\ell \\ell }^{(s)}\\partial _a \\delta \\mathcal {X}^{\\ell }\\partial _b \\delta X^{\\ell }+\\sum _{i\\ne l}g_{ii}^{(s)}\\partial _a \\delta \\mathcal {X}^{i}\\partial _b \\delta \\mathcal {X}^{i} \\right].$ We can still make one further simplification in the kinetic term of the $\\ell $ -fluctuation.",
"First, we note that $\\gamma ^{(s)ab}\\underbrace{g_{\\mu \\nu }^{(s)}\\partial _{a}\\mathcal {X}^{\\mu }\\partial _{b}\\mathcal {X}^{\\nu }}_{=\\gamma ^{(s)}_{ab}} = \\delta _a^a = 2 \\Rightarrow 1- g_{\\ell \\ell }^{(s)} \\gamma ^{(s)ab}\\partial _{a}\\mathcal {X}^{\\ell }\\partial _{b}\\mathcal {X}^{\\ell } = -1 + \\gamma ^{(s)tt}g_{tt}^{(s)}+\\gamma ^{(s)rr}g_{rr}^{(s)},$ and since, from the holographic drag force calculation, $\\gamma ^{(s)tt}=\\frac{\\left(g_{tt}^{(s)}\\right)^2-v^2(2\\pi \\alpha ^{\\prime }\\pi _\\xi )^2}{\\left(g_{tt}^{(s)}\\right)^2(g_{tt}^{(s)}+v^2g_{\\ell \\ell })}, \\ \\ \\ \\gamma ^{(s)rr} = \\frac{g_{tt}^{(s)}g_{\\ell \\ell }^{(s)}+(2\\pi \\alpha ^{\\prime }\\pi _\\xi )^2}{g_{tt}^{(s)}g_{rr}^{(s)}g_{\\ell \\ell }^{(s)}},$ we have that $(1- g_{\\ell \\ell }^{(s)} \\gamma ^{(s)ab}\\partial _{a}\\mathcal {X}^{\\ell }\\partial _{b}\\mathcal {X}^{\\ell })g_{\\ell \\ell }^{(s)} = \\frac{g_{tt}^{(s)}g_{\\ell \\ell }^{(s)}+(2\\pi \\alpha ^{\\prime }\\pi _\\xi )^2}{g_{tt}^{(s)}+v^2g_{\\ell \\ell }^{(s)}}\\equiv N(r).$ Therefore, a simple expression for the quadratic action in the worldsheet embedding fluctuations can be found [133], [134] $S_2 = -\\frac{1}{2\\pi \\alpha ^{\\prime }}\\int dt dr \\sqrt{-\\gamma ^{(s)}}\\frac{\\gamma ^{(s)ab}}{2}\\left[ N(r)\\partial _a \\delta \\mathcal {X}^{\\ell }\\partial _b \\delta \\mathcal {X}^{\\ell }+\\sum _{i\\ne \\ell }g_{ii}^{(s)}\\partial _a \\delta \\mathcal {X}^{i}\\partial _b \\delta \\mathcal {X}^{i} \\right],$ which reduces to Eq.",
"(3.33) of Ref.",
"[127] in the isotropic limit.",
"Now we diagonalize the pullback by making a worldsheet coordinate redefinition: $dt\\rightarrow dt-dr\\,\\gamma _{tr}/\\gamma _{tt}$ .",
"The quadratic action for the fluctuations now takes the form [133], [134] $S_2 = -\\frac{1}{2\\pi \\alpha ^{\\prime }}\\int dt dr \\sqrt{-h^{(s)}}\\frac{h^{(s)ab}}{2}\\left[ N(r)\\partial _a \\delta \\mathcal {X}^{\\ell }\\partial _b \\delta \\mathcal {X}^{\\ell }+\\sum _{i\\ne \\ell }g_{ii}^{(s)}\\partial _a \\delta \\mathcal {X}^{i}\\partial _b \\delta \\mathcal {X}^{i} \\right],$ where $h_{ab}^{(s)}$ is the diagonalized pullback, whose explicit form in terms of the background metric is given by $h_{ab}^{(s)} = \\text{diag}\\left\\lbrace g_{tt}^{(s)}+v^2g_{\\ell \\ell }^{(s)}, \\frac{g_{tt}^{(s)}g_{rr}^{(s)}g_{\\ell \\ell }^{(s)}}{g_{tt}^{(s)}g_{\\ell \\ell }^{(s)}+v^2(g_{\\ell \\ell }^{(s)}(r_\\star ))^2} \\right\\rbrace .$ This metric has a worldsheet black hole horizon and the associated worldsheet Hawking's temperature reads [133], [134] $T_\\star &=\\frac{1}{4\\pi }\\sqrt{\\left|h^{\\prime (s)}_{tt}\\left(h^{(s)rr}\\right)^{\\prime }\\right|_{r_\\star }}, \\\\&= \\frac{1}{4\\pi }\\sqrt{\\left|(g_{tt}^{(s)}+ v^2g_{\\ell \\ell }^{(s)})^{\\prime }\\left(\\frac{g_{tt}^{(s)}g_{\\ell \\ell }^{(s)}+ v^2(g_{\\ell \\ell }^{(s)}(r_\\star ))^2}{g_{tt}^{(s)}g_{\\ell \\ell }^{(s)}g_{rr}^{(s)}} \\right)^{\\prime }\\right|_{r_\\star }}, \\\\&= \\frac{1}{4\\pi }\\sqrt{\\left|\\frac{(g^{\\prime (s)}_{tt})^2-v^4(g^{\\prime (s)}_{\\ell \\ell })^2}{g_{tt}^{(s)}g_{rr}^{(s)}} \\right|_{r_\\star }}, \\\\&= \\frac{1}{4\\pi \\sqrt{-g_{tt}^{(s)}(r_\\star )g_{rr}^{(s)}(r_\\star )}}\\sqrt{\\left|(g_{tt}^{(s)}g_{\\ell \\ell }^{(s)})^{\\prime }\\left(\\frac{g_{tt}^{(s)}}{g_{\\ell \\ell }^{(s)}}\\right)^{\\prime }\\right|_{r_\\star }}.$ Note, from the expression above, that in the anisotropic case the worldsheet temperature will depend on which direction the string propagates."
],
[
"Isotropic case",
"In the case of zero magnetic field $g_{xx}=g_{zz}$ .",
"If we choose, without loss of generality, $\\ell =x$ , the action (REF ) takes the form $S_2 =- \\frac{1}{2\\pi \\alpha ^{\\prime }}\\int dt dr \\sqrt{-h^{(s)}}\\frac{h^{(s)ab}}{2}\\left[ N(r)\\partial _a \\delta \\mathcal {X}^{x}\\partial _b \\delta \\mathcal {X}^{x}+\\sum _{i=y,z}g_{ii}^{(s)}\\partial _a \\delta \\mathcal {X}^{i}\\partial _b \\delta \\mathcal {X}^{i} \\right].$ The diffusion coefficients parallel and transverse to the probe's velocity are then related to the 2-point functions obtained from the fluctuations $\\delta \\mathcal {X}^x$ and $\\delta \\mathcal {X}^y$ (or $\\delta \\mathcal {X}^z$ ), respectively.",
"The easiest way to extract them is via the membrane paradigm [150], which implies that the transport coefficient associated to the propagator of a massless fluctuation via linear response theory, in the limit of zero spatial momentum and zero frequency, can be directly extracted from the coefficient in front of the effective kinetic term for the fluctuation evaluated at the black hole horizon.",
"More specifically, considering a quadratic action for some generic bulk massless perturbation $\\phi $ in arbitrary $(d+1)$ -dimensions, $S_2^\\phi = -\\frac{1}{2}\\int drd^{d}x \\sqrt{-g}\\frac{g^{\\mu \\nu }}{q(r)}\\partial _\\mu \\phi \\partial _\\nu \\phi ,$ the transport coefficient $\\chi $ extracted from (REF ) is given by [150] $\\chi =- \\lim _{\\omega \\rightarrow 0} \\frac{\\text{Im}\\,G^{R}(\\omega ,\\vec{k}=\\vec{0})}{\\omega } =\\lim _{r\\rightarrow r_H}\\frac{1}{q(r)}\\sqrt{\\frac{-g(r)}{g_{tt}(r)g_{rr}(r)}}.$ In the case of (REF ), one first applies the membrane paradigm to the two dimensional worldsheet black hole horizon (i.e., take $g_{\\mu \\nu }\\rightarrow h_{ab}^{(s)}$ and $r_H\\rightarrow r_\\star $ in Eq.",
"(REF )) to obtain the friction coefficients $\\eta ^D_{ij}=-\\lim _{\\omega \\rightarrow 0}\\text{Im}\\,G^{R}_{ij}(\\omega )/\\omega $ associated to the worldsheet embedding fluctuations, and then uses the modified Einstein's relation (REF ) to obtain the corresponding Langevin coefficients [133], [134],Note from Eqs.",
"(REF ) and (REF ) that $N(r_\\star )\\rightarrow 0/0$ .",
"Therefore, one needs to use the L'Hopital's rule to calculate the limit $\\lim _{r\\rightarrow r_\\star }N(r)$ .",
"In order to do this, one first rewrites (REF ) as $N(r)=\\frac{1}{g_{\\ell \\ell }}\\times \\frac{g^{(s)}_{tt}g^{(s)}_{\\ell \\ell }+(2\\pi \\alpha ^{\\prime }\\pi _\\xi )^2}{(g^{(s)}_{tt}/g^{(s)}_{\\ell \\ell })+v^2}$ , which gives $\\lim _{r\\rightarrow r_\\star }N(r)=\\frac{1}{g^{(s)}_{\\ell \\ell }}\\times \\frac{(g^{(s)}_{tt}g^{(s)}_{\\ell \\ell })^{\\prime }}{(g^{(s)}_{tt}/g^{(s)}_{\\ell \\ell })^{\\prime }}\\biggr |_{r_\\star }$ .",
"$\\kappa _{(\\parallel v)}^{\\textrm {(iso)}}&=\\frac{T_\\star }{\\pi \\alpha ^{\\prime }}\\lim _{r\\rightarrow r_\\star }N(r) = \\frac{T_\\star }{\\pi \\alpha ^{\\prime }}\\frac{1}{g_{xx}^{(s)}(r_\\star )} \\frac{(g_{tt}^{(s)}g_{xx}^{(s)})^{\\prime }}{(g^{(s)}_{tt}/g_{xx}^{(s)})^{\\prime }}\\biggr |_{r_\\star },\\\\\\kappa _{(\\perp v)}^{\\textrm {(iso)}}&=\\frac{T_\\star }{\\pi \\alpha ^{\\prime }} g_{xx}^{(s)}(r_\\star ),$ where the subscript $(\\parallel v)\\,[(\\perp v)]$ denotes the momentum diffusion parallel [perpendicular] to the initial probe's velocity.",
"Then, for a SYM plasma, one obtains the well-known results [125], [126] $\\frac{\\kappa _{(\\parallel v)}^{\\textrm {(SYM)}}}{\\sqrt{\\lambda _t}T^3}=\\pi \\gamma ^{5/2} \\quad \\textrm {and} \\quad \\frac{\\kappa _{(\\perp v)}^{\\textrm {(SYM)}}}{\\sqrt{\\lambda _t}T^3}&=\\pi \\gamma ^{1/2}.$"
],
[
"Anisotropic case",
"To deal with the general anisotropic case for the diffusion coefficients using the generic diagonal background (REF ), one would need to consider three angles: $\\textrm {ang}(\\vec{v},\\vec{\\kappa })$ , $\\textrm {ang}(\\vec{B},\\vec{\\kappa })$ , and $\\textrm {ang}(\\vec{v},\\vec{B})$ , where $\\vec{\\kappa }$ is the direction of momentum diffusion.",
"For this work, though, we shall investigate the following cases: (i) $\\vec{v}$ parallel to $\\vec{B}$ , which is characterized by two diffusion coefficients; and (ii) $\\vec{v}$ perpendicular to $\\vec{B}$ , which is characterized by three diffusion coefficients.",
"Thus, our goal here is to calculate the momentum diffusion along the spatial directions $\\hat{x}$ , $\\hat{y}$ , and $\\hat{z}$ for each case, as illustrated in Fig.",
"REF Figure: (Color online).",
"Schematic representation of the anisotropic Langevin diffusion coefficients computed in this work.Starting with the case (i), $\\vec{v}\\parallel \\vec{B}$ , with $\\vec{B}=B\\hat{z}$ , one may write the disturbed action (REF ) as follows [133], [134], $S_2^{(v\\parallel B)} = -\\frac{1}{2\\pi \\alpha ^{\\prime }}\\int dt dr \\sqrt{-h^{(s)}}\\frac{h^{(s)ab}}{2}\\left[ N(r)\\partial _a \\delta \\mathcal {X}^{z}\\partial _b \\delta \\mathcal {X}^{z}+\\sum _{i=x,y}g_{ii}^{(s)}\\partial _a \\delta \\mathcal {X}^{i}\\partial _b \\delta \\mathcal {X}^{i} \\right].$ Proceeding with the membrane paradigm calculation, just as done in the previous subsection, one obtains the following diffusion coefficients [133], [134], $\\frac{\\kappa _{(\\parallel v)}^{(v\\parallel B)}}{\\sqrt{\\lambda _t}} &= \\frac{T_\\star ^{\\parallel }}{\\pi g_{zz}^{(s)}(r_{\\star }^{\\parallel })}\\left.\\frac{(g_{tt}^{(s)}g_{zz}^{(s)})^{\\prime }}{(g_{tt}^{(s)}/g_{zz}^{(s)})^{\\prime }}\\right|_{r_{\\star }^{\\parallel }}, \\\\\\frac{\\kappa _{(\\perp v)}^{(v\\parallel B)}}{\\sqrt{\\lambda _t}} &= \\frac{T_\\star ^{\\parallel }}{\\pi }g_{xx}^{(s)}(r_{\\star }^{\\parallel }),$ where [133], [134], $T_\\star ^{\\parallel } &= \\frac{1}{4\\pi \\sqrt{-g_{tt}^{(s)}(r_{\\star }^{\\parallel })g_{rr}^{(s)}(r_{\\star }^{\\parallel })}}\\sqrt{\\left|(g_{tt}^{(s)}g_{zz}^{(s)})^{\\prime }\\left(\\frac{g_{tt}^{(s)}}{g_{zz}^{(s)}}\\right)^{\\prime }\\right|_{r_{\\star }^{\\parallel }}},\\\\0 &= g_{tt}^{(s)}(r_{\\star }^{\\parallel })+v^2g_{zz}^{(s)}(r_\\star ^{\\parallel }).$ Considering now the case (ii), $\\vec{v}\\perp \\vec{B}$ , with $\\vec{v}=v\\hat{x}$ , the fluctuated action (REF ) becomes [133], [134], $S_2^{(v\\perp B)} = -\\frac{1}{2\\pi \\alpha ^{\\prime }}\\int dt dr \\sqrt{-h^{(s)}}\\frac{h^{(s)ab}}{2}\\left[ N(r)\\partial _a \\delta \\mathcal {X}^{x}\\partial _b \\delta \\mathcal {X}^{x}+g_{xx}^{(s)}\\partial _a \\delta \\mathcal {X}^{y}\\partial _b \\delta \\mathcal {X}^{y} +g_{zz}^{(s)}\\partial _a \\delta \\mathcal {X}^{z}\\partial _b \\delta \\mathcal {X}^{z}\\right].$ From the membrane paradigm, one obtains the following diffusion coefficients [133], [134], $\\frac{\\kappa _{(\\parallel v)}^{(v\\perp B)}}{\\sqrt{\\lambda _t}} &= \\frac{T_\\star ^{\\perp }}{\\pi g_{xx}^{(s)}(r_{\\star }^{\\perp })}\\left.\\frac{(g_{tt}^{(s)}g_{xx}^{(s)})^{\\prime }}{(g_{tt}^{(s)}/g_{xx}^{(s)})^{\\prime }}\\right|_{r_{\\star }^{\\perp }}, \\\\\\frac{\\kappa _{(\\perp v, \\perp B)}^{(v\\perp B)}}{\\sqrt{\\lambda _t}} &= \\frac{T_\\star ^{\\perp }}{\\pi }g_{xx}^{(s)}(r_{\\star }^{\\perp }),\\\\\\frac{\\kappa _{(\\perp v, \\parallel B)}^{(v\\perp B)}}{\\sqrt{\\lambda _t}} &= \\frac{T_\\star ^{\\perp }}{\\pi } g_{zz}^{(s)}(r_{\\star }^{\\perp }),$ where [133], [134], $T_\\star ^{\\perp } &= \\frac{1}{4\\pi \\sqrt{-g_{tt}^{(s)}(r_\\star ^{\\perp })g_{rr}^{(s)}(r_\\star ^{\\perp })}}\\sqrt{\\left|(g_{tt}^{(s)}g_{xx}^{(s)})^{\\prime }\\left(\\frac{g_{tt}^{(s)}}{g_{xx}^{(s)}}\\right)^{\\prime }\\right|_{r_\\star ^{\\perp }}},\\\\0 &= g_{tt}^{(s)}(r_{\\star }^{\\perp })+v^2g_{xx}^{(s)}(r_\\star ^{\\perp }).$"
],
[
"Shear viscosity",
"Along with the drag force and the Langevin diffusion coefficients, we will also compute how the shear viscosity coefficient $\\eta $ changes with the inclusion of a magnetic field.",
"Any quantum field theory that possess an isotropic gravity dual whose action contains terms up to two derivatives has the following value for the shear viscosity to entropy density ratio, $\\frac{\\eta }{s}=\\frac{1}{4\\pi },$ which was previously conjectured [15] to be a lower bound for the value of the ratio $\\eta /s$ in Nature, which is known in the literature as the Kovtun-Son-Starinets (KSS) bound.",
"In order to obtain departures and possible violations of the KSS bound (REF ) in holographic settings, one may include higher order curvature terms in the gravity action [153], [151], [152].",
"Furthermore, these higher order derivative corrections may be also employed in conjunction to a nontrivial dilaton potential breaking conformal symmetry in the infrared to provide a non-constant temperature profile for the ratio $\\eta /s$ [154].",
"Another way to violate the bound (REF ) in holographic setups is to break rotational or translational symmetries.",
"The first calculation of anisotropic shear viscosities was done in Ref.",
"[136] for the case of an anisotropic plasma created by a spatially dependent bulk axion profile originally proposed in Refs.",
"[128], [129].",
"The result is similar to the one obtained in Ref.",
"[108] in the context of the magnetic brane model originally proposed in Refs.",
"[104], [105], [106] - see also Ref.",
"[155] for results concerning $p$ -form magnetically charged branes.",
"One may also find in the literature models with a dilaton driven anisotropy [156], [157], anisotropic $SU(2)$ Einstein-Yang-Mills models used as gravity duals of holographic superfluids [159], [160], [161], and a black brane model whose temperature is modulated in the spatial directions [162], [163].",
"Recently, violations of the KSS bound were also found in isotropic theories dual to massive gravity models which break translational invariance [164], [165], and in the context of Horndeski gravity duals [166].",
"The shear viscosity is obtained from the imaginary part of the retarded 2-point function associated with the stress-energy tensor, $\\eta _{ijkl} = - \\lim _{\\omega \\rightarrow 0} \\frac{1}{\\omega } \\text{Im} \\ G_{T_{ij}T_{kl}}^{R} (\\omega , \\vec{k} = \\vec{0}) \\ \\text{with} \\ i, j,k, l = x, y, z,$ where $G^{R}_{T_{ij}T_{kl}}(\\omega ,\\vec{k}) \\equiv - i\\int d^4x e^{i(\\omega t-\\vec{k}\\cdot \\vec{x})}\\theta (t) \\langle \\left[T_{ij}(t,\\vec{x}), T_{kl}(0,\\vec{0}) \\right] \\rangle .$ In isotropic and homogeneous theories there is only one shear viscosity coefficient which is obtained from the non-diagonal part of the Green's function (REF ), i.e.",
"$\\eta = \\eta _{xyxy}$ .",
"However, since the backgrounds to be considered here will be anisotropic one has that $\\eta _{xyxy}\\ne \\eta _{xzxz}$ , for instance.",
"When the anisotropy in the fluid is induced by a magnetic field one has now to consider up to seven different viscosity coefficients, with five shear viscosities and two bulk viscosities [167], [168], [169], [170], [171].",
"In order to clarify as much as possible how one may obtain these different viscosity coefficients in an anisotropic theory induced by a magnetic field, we provide in Appendix a detailed analysis of the Kubo's formulas for all the viscosities that may appear in such case.",
"Although the general viscous magnetohydrodynamics constructed with the viscous tensor described in Eq.",
"(REF ) of Appendix may contain seven different nontrivial viscosity coefficients, for the magnetized gravity backgrounds considered here, where $SO(3)$ rotational invariance is broken down to $SO(2)$ , but do not take into account, for instance, the contribution of a nontrivial angular momentum, the number of nontrivial independent viscosity coefficients will be less since we have only four independent fluctuations of the metric field which are important to compute the viscositiesNote that $h_{\\mu \\nu }$ in this section denotes the metric fluctuations and has nothing to do with the diagonal disturbed pullback $h^{(s)}_{ab}$ discussed in the previous section.",
"(cf.",
"Eq.",
"(REF )): $h_{xy}$ , $h_{xz}$ , $h_{xx}+h_{zz}$ , and $h_{zz}$ .",
"Moreover, the diagonal fluctuations are related to the bulk viscosities (they also couple to the dilaton fluctuation $\\delta \\phi $ as discussed in Ref.",
"[27]), whose calculations are much more involved and deferred for a future work.",
"Consequently, in the present we focus on the two different nontrivial shear viscosities appearing in our anisotropic magnetized backgrounds.",
"Note that we will also neglect the Abelian field fluctuations $\\delta A_i$ since they only couple to vector fluctuations, e.g.",
"$h_{ti}$ , which are important, for example, in the calculation of the electric conductivity.",
"The two nontrivial shear viscosities we calculate in the present work are given by the Kubo's formulas $\\eta _{\\perp } \\equiv \\eta _{xyxy} = - \\lim _{\\omega \\rightarrow 0} \\frac{1}{\\omega } \\text{Im} \\ G_{T_{xy}T_{xy}}^{R} (\\omega , \\vec{k} = \\vec{0}), \\\\\\eta _{\\parallel } \\equiv \\eta _{xzxz} = - \\lim _{\\omega \\rightarrow 0} \\frac{1}{\\omega } \\text{Im} \\ G_{T_{xz}T_{xz}}^{R} (\\omega , \\vec{k} = \\vec{0}).",
"$ We compute $\\eta _{\\perp }$ and $\\eta _{\\parallel }$ via holography using the membrane paradigm approach, as it was done with the Langevin diffusion coefficients in the previous subsection.",
"For such a task, one needs to obtain the second order disturbed action for the metric fluctuations $h_{xy}\\equiv \\delta g_{xy}$ and $h_{xz}\\equiv \\delta g_{xz}$ coupling to $T^{xy}$ and $T^{xz}$ , respectively.",
"It was shown in Ref.",
"[108] that the quadratic part of the disturbed actions with respect to $h_{x}^{y}=g^{xx}h_{xy}$ and $h_{x}^{z}=g^{zz}h_{zx}$ are given by the following expressions $S_{2}^{(\\perp )}&=-\\frac{1}{16\\pi G_5}\\int d^5x \\sqrt{-g}\\frac{1}{2}(\\partial h_{x}^{y})^2, \\\\S_{2}^{(\\parallel )}&=-\\frac{1}{16\\pi G_5}\\int d^5x \\sqrt{-g}\\left(\\frac{g_{zz}}{2g_{xx}}(\\partial h_{x}^{z})^2\\right).",
"$ Notice also that the presence of a nontrivial dilaton field $\\phi $ in the background does not alter the above expressions since its fluctuation $\\delta \\phi $ couples only to the diagonal part of the disturbed metric field (defining the so-called “scalar channel”), which is needed to calculate the bulk viscosities but not the shear viscosities as aforementioned.",
"We obtain the shear viscosity transport coefficients, $\\eta _{\\perp }$ and $\\eta _{\\parallel }$ , by employing the membrane paradigm directly to the quadratic fluctuated actions (REF ) and (), respectively, $\\eta _{\\perp } &= \\frac{1}{16\\pi G_5}\\sqrt{g_{xx}^{2}(r_H)g_{zz}(r_H)}, \\\\\\eta _{\\parallel } &= \\frac{1}{16\\pi G_5}\\sqrt{g_{zz}^{3}(r_H)}.$ On the other hand, the entropy density $s$ obtained from the background (REF ) by using the Bekenstein-Hawking's relation [172], [173] is given by $s = \\frac{\\sqrt{g_{xx}^{2}(r_H)g_{zz}(r_H)}}{4G_5}.$ Therefore, the ratios $\\eta _{\\perp }/s$ and $\\eta _{\\parallel }/s$ become $\\frac{\\eta _{\\perp }}{s}&=\\frac{1}{4\\pi }, \\\\\\frac{\\eta _{\\parallel }}{s}&=\\frac{1}{4\\pi }\\frac{g_{zz}(r_H)}{g_{xx}(r_H)}.",
"$ These formulas were first obtained in the context of the Einstein-axion-dilaton model [136], and also in the particular cases of the EMD model given by the anisotropic Einstein-dilaton model [156] (see also [157])We warn the reader that the notation for $\\eta _\\perp $ and $\\eta _\\parallel $ followed in Refs.",
"[156], [157] are reversed compared to the notation adopted here and in Refs.",
"[136], [108].",
"Moreover, we also remark that here and also in Ref.",
"[108] we considered the fluctuation $h_{x}^{z}$ instead of the fluctuation $h_{z}^{x}$ considered in Refs.",
"[136], [156], [157].",
"and the Einstein-Maxwell model [108].",
"The perpendicular shear viscosity to entropy density ratio $\\eta _{\\perp }/s$ associated with the fluctuation $h^{x}_{y}$ of the metric (REF ), which has the residual $SO(2)$ rotational symmetry in the plane transverse to the magnetic field, does not deviate from the KSS result (REF ).",
"Consequently, the goal of sections REF and REF will be to unveil how the parallel shear viscosity to entropy density ratio $\\eta _{\\perp }/s$ is modified relatively to the KSS formula in the presence of an external magnetic field for the magnetic brane and magnetic EMD models, respectively."
],
[
"Derivation of anisotropic Kubo formulas for viscosity from linear response theory",
"In this Appendix we investigate how the breaking of the $SO(3)$ rotation symmetry down to $SO(2)$ affects the dissipative properties of relativistic fluids, i.e., we shall discuss how one may generalize the viscosity coefficients (shear and bulk) in order to accommodate the anisotropic nature of the magnetized fluid in the presence of a magnetic field.",
"Historically, calculations of anisotropic transport coefficients in Abelian plasmas were carried out in the 1950's, mainly by Braginskii [167], [168] - see also the more recent Ref.",
"[185].",
"Recently, there has been an increasing interest in the effects of strong fields on high energy relativistic systems, such as neutron stars [169], [170], where the anisotropic nature of the plasma may play an important role.",
"Although our discussion will be restricted to anisotropic viscosities in a plasma whose anisotropy is driven by an external magnetic field, we emphasize that this phenomenon occurs in various systems such as plastics and superfluids [200]; see Refs.",
"[159], [160], [161] for holographic approaches to the latter.",
"Ultimately, we are interested in relativistic viscous plasmas and, consequently, we want a causal and stable theory of relativistic magnetohydrodynamics.",
"One approach to viscous magnetohydrodynamics corresponds to the Navier-Stokes-Fourier-Ohm theory [169], which is an extension of the acausal and unstable [201], [202] relativistic Navier-Stokes theory - we shall not dwell into this approach here.",
"Relativistic effects in magnetohydrodynamics for weakly collisional (Abelian) plasmas were studied in [171], which may be important to study black hole accretion flows where the magnetic field is intense.",
"Recently, though, Ref.",
"[192] extended the Israel-Stewart formalism [203] to derive the equations of motion of an anisotropic dissipative fluid obtained from the Boltzmann equation using the moments method developed in [204]The usefulness of the moments method in dealing with the relativistic Boltzmann equation goes beyond the context of heavy ion collision applications.",
"In fact, the moments method may be used to obtain the equations of motion describing magnetohydrodynamics directly from the Boltzmann-Vlasov equations [205] and also the out-of-equilibrium dynamics of gases in an expanding universe [206]..",
"The task now is to derive the form of the viscous stress tensor $\\Pi _{\\mu \\nu }$ in order to obtain Kubo formulas for the anisotropic viscosities; for completeness we revisit Appendix A of Ref.",
"[108] and some aspects of Ref.",
"[169].",
"For instance, in isotropic theories, one hasWe assume a 4D spacetime from now on.",
"$\\Pi _{\\mu \\nu }=-2\\eta \\left(w_{\\mu \\nu }-\\Delta _{\\mu \\nu }\\frac{\\theta }{3} \\right)-\\zeta \\theta ,$ where $w_{\\mu \\nu }=\\frac{1}{2}\\left(D_{\\mu }u_{\\nu }+D_{\\nu }u_{\\mu }\\right)$ , $u^\\mu $ is the four-velocity with normalization $u^\\mu u_\\mu =-1$ , $D_{\\mu }=\\Delta _{\\mu \\alpha }\\partial ^{\\alpha }$ , $\\Delta _{\\mu \\nu }=g_{\\mu \\nu }+u_{\\mu }u_{\\nu }$ (orthogonal projector), and $\\theta =\\partial _\\mu u^\\mu $ .",
"The expression (REF ) cannot hold for a highly magnetized plasma since it has a reduced axial symmetry around the magnetic vector.",
"From the gravity side of the holographic correspondence the anisotropic metric (REF ) tells us the same.",
"Therefore, we need a rank-4 viscosity tensor $\\eta ^{\\alpha \\beta \\mu \\nu }$ obeying the following relation $\\Pi ^{\\mu \\nu }= \\eta ^{\\mu \\nu \\alpha \\beta }w_{\\alpha \\beta }.$ The tensorial structure of the viscosity tensor $\\eta ^{\\mu \\nu \\alpha \\beta }$ depends solely on $\\Delta ^{\\mu \\nu }$ , $b^{\\mu }$ (a unit spacelike vector normal to the magnetic field), and $b^{\\mu \\nu }=\\epsilon ^{\\mu \\nu \\alpha \\beta }b^\\alpha u^\\beta $ .",
"Furthermore, the viscosity tensor must satisfy the following symmetry relations (where $B$ is the magnetic field) $\\eta ^{\\mu \\nu \\alpha \\beta }(B)&= \\eta ^{\\nu \\mu \\alpha \\beta }(B) =\\eta ^{\\mu \\nu \\beta \\alpha }(B), \\\\\\eta ^{\\mu \\nu \\alpha \\beta }(B) &= \\eta ^{\\alpha \\beta \\mu \\nu }(-B).",
"\\ \\ \\ \\text{(Onsager principle)} $ The linearly independent structures which may be constructed using $\\Delta ^{\\mu \\nu }$ , $b^{\\mu }$ , and $b^{\\mu \\nu }$ , which respect the symmetries (REF ) and (), are given by $\\text{(i)}& \\ \\Delta ^{\\mu \\nu }\\Delta ^{\\alpha \\beta }, \\\\\\text{(ii)}& \\ \\Delta ^{\\mu \\alpha }\\Delta ^{\\nu \\beta }+\\Delta ^{\\mu \\beta }\\Delta ^{\\nu \\alpha }, \\\\\\text{(iii)}& \\ \\Delta ^{\\mu \\nu }b^{\\alpha }b^{\\beta }+\\Delta ^{\\alpha \\beta }b^{\\mu }b^{\\nu }, \\\\\\text{(iv)}& \\ b^{\\mu }b^{\\nu }b^{\\alpha }b^{\\beta }, \\\\\\text{(v)} & \\ \\Delta ^{\\mu \\alpha }b^{\\nu }b^{\\beta }+\\Delta ^{\\mu \\beta }b^{\\nu }b^{\\alpha }+\\Delta ^{\\nu \\alpha }b^{\\mu }b^{\\beta }+\\Delta ^{\\nu \\beta }b^{\\mu }b^{\\alpha }, \\\\\\text{(vi)}& \\ \\Delta ^{\\mu \\alpha }b^{\\nu \\beta }+\\Delta ^{\\mu \\beta }b^{\\nu \\alpha }+\\Delta ^{\\nu \\alpha }b^{\\mu \\beta }+\\Delta ^{\\nu \\beta }b^{\\mu \\alpha }, \\\\\\text{(vii)}& \\ b^{\\mu \\alpha }b^{\\nu }b^{\\beta }+b^{\\mu \\beta }b^{\\nu }b^{\\alpha }+b^{\\nu \\alpha }b^{\\mu }b^{\\beta }+b^{\\nu \\beta }b^{\\mu }b^{\\alpha }.$ The viscosity tensor will be composed by linear combinations of the above relations with the viscosity coefficients being the factors in front of each structure.",
"Consequently, one concludes that there must be seven viscosity coefficients for this theory of viscous magnetohydrodynamics, divided into five shear viscosities and two bulk viscosities.",
"We adopt a similar convention of viscosity coefficients to the one followed in Ref.",
"[170] (HSR)This is a different convention from the one followed in Ref.",
"[167] and in Chapter 13 of Ref.",
"[168]., except that in the present work (FCRN), $\\eta _{2}^{\\textrm {(FCRN)}}=-\\eta _{2}^{\\textrm {(HSR)}}$ and $\\eta _{3}^{\\textrm {(FCRN)}}=-2\\eta _{3}^{\\textrm {(HSR)}}$ .",
"In this case, the viscosity tensor assumes the form $\\eta ^{\\mu \\nu \\alpha \\beta } =& (-2/3\\eta _0 +1/4\\eta _1 +3/2\\zeta _\\perp )\\text{(i)} + (\\eta _0 )\\text{(ii)} +(3/4\\eta _1+3/2\\zeta _\\perp )\\text{(iii)}\\\\& +(9/4\\eta _1 -4\\eta _2 +3/2\\zeta _\\perp +3\\zeta _\\parallel )\\text{(iv)} +(-\\eta _2 )\\text{(v)} +(-\\eta _4)\\text{(vi)} \\\\& +(-\\eta _3+\\eta _4)\\text{(vii)},$ with the $\\eta ^{\\prime }$ s being the shear viscosities and the $\\zeta ^{\\prime }$ s being the bulk viscosities.",
"Substituting (REF ) into (REF ) we find the following viscous tensor $\\Pi _{\\mu \\nu } &= -2\\eta _0\\left(w_{\\mu \\nu }-\\Delta _{\\mu \\nu }\\frac{\\theta }{3} \\right)-\\eta _1\\left(\\Delta _{\\mu \\nu }-\\frac{3}{2}\\Xi _{\\mu \\nu } \\right)\\left(\\theta -\\frac{3}{2}\\phi \\right)+2\\eta _2\\left(b_\\mu \\Xi _{\\nu \\alpha }b_{\\beta }+b_\\nu \\Xi _{\\mu \\alpha }b_{\\beta }\\right)w^{\\alpha \\beta } \\\\& +\\eta _3\\left(\\Xi _{\\mu \\alpha }b_{\\nu \\beta }+\\Xi _{\\nu \\alpha }b_{\\mu \\beta }\\right)w^{\\alpha \\beta }-2\\eta _4\\left(b_{\\mu \\alpha }b_{\\nu }b_{\\beta }+b_{\\nu \\alpha }b_{\\mu }b_{\\beta }\\right)w^{\\alpha \\beta }-\\frac{3}{2}\\zeta _{\\perp }\\Xi _{\\mu \\nu }\\phi - 3\\zeta _{\\parallel }b_{\\mu }b_{\\nu }\\varphi ,$ where $w_{\\mu \\nu }=\\frac{1}{2}\\left(D_{\\mu }u_{\\nu }+D_{\\nu }u_{\\mu }\\right)$ , $D_{\\mu }=\\Delta _{\\mu \\alpha }\\nabla ^{\\alpha }$ , $\\Xi _{\\mu \\nu }\\equiv \\Delta _{\\mu \\nu }-b_\\mu b_\\nu $ (orthogonal projector), $\\theta =\\nabla _\\mu u^\\mu $ , $\\phi \\equiv \\Xi _{\\mu \\nu }w^{\\mu \\nu }$ and $\\varphi \\equiv b_{\\mu }b_{\\nu }w^{\\mu \\nu }$ .",
"Note that the differential operator $D_\\mu $ is given in terms of the covariant derivative, i.e., we are generalizing the viscous tensor to a curved spacetime; this will be essential to extract the Kubo formulas since they are obtained here by considering gravity fluctuations."
],
[
"Kubo formulas for viscous magnetohydrodynamics",
"With the expression for the viscous tensor $\\Pi _{\\mu \\nu }$ at hand, it is time to derive the Kubo formulas that relate the viscosity coefficients to the retarded Green's functions.",
"We remark that Ref.",
"[170] also derived the Kubo formulas although using the Zubarev formalism.",
"Let us summarize the usual procedure to obtain the Kubo formulas for the viscosity: adopting a Minkowski background, we perform small gravity perturbations in $\\Pi _{\\mu \\nu }$ assuming that they are all homogeneous, which means that we can work only with the spatial indices, i.e., $g_{ij}= \\eta _{ij}+h_{ij}(t)$ , with $h_{00}=h_{0i}=0$ .",
"Also, we work in the rest frame of the fluid where $u^{\\mu }=(1,0,0,0)$ .In other words, we will work in the Landau-Lifshitz frame where $u_\\mu \\Pi ^{\\mu \\nu }=0$ and all the information about the viscosities are in the components $\\lbrace i, j, k, l \\rbrace $ of the retarded Green's function (REF ).",
"Finally, we equate the fluctuated form of (REF ) to $h_{kl}G^{R, \\, kl}_{ij}(\\omega )$ in order to extract the Kubo formulas.",
"The novelty here is the presence of a magnetic field which is assumed to be constant and uniform along the $z$ direction, i.e., $b^{\\mu }=(0,0,0,1)$ .",
"Thus, we have the variation for the viscous tensorNote that: $\\delta \\Xi _{\\mu \\nu }=h_{\\mu \\nu }, \\ \\ \\ \\delta \\theta = \\frac{1}{2}\\partial _t h^{\\lambda }_{\\lambda }, \\ \\ \\ \\delta \\varphi =\\frac{1}{2}\\partial _t h_{zz}.",
"$ $\\delta \\Pi _{ij}=\\delta (i)+\\delta (ii)+\\delta (iii)+\\delta (iv)+\\delta (v)+\\delta (vi)+\\delta (vii),$ where $\\delta (i) &= -\\eta _0\\left(\\partial _t h_{ij}-\\frac{1}{3}\\delta _{ij}\\partial _t h^{k}_{k}\\right),$ $\\delta (ii) &=-\\frac{1}{4} \\eta _1 \\left[ -(\\delta _{ij}-3b_i b_j)\\partial _t h^{k}_{k} +\\frac{3}{2}(\\delta ^{kl}-b^{k}b^{l})(\\delta _{ij}-3b_i b_j)\\partial _t h_{kl}\\right],$ $\\delta (iii) &= \\eta _2 \\left[ b_i b^k\\partial _t h_{jk}+b_j b^k\\partial _t h_{ik} - 2b_i b_j b^k b^l\\partial _t h_{kl} \\right],$ $\\delta (iv) &=\\frac{1}{2}\\eta _3 \\partial _t h_{kl} \\left(\\delta _{i}^{k}\\epsilon _{j}^{\\ lz}+\\delta ^{jl}\\epsilon _{i}^{\\ kz} - b_{i}b^{k}\\epsilon _{j}^{\\ lz} - b_{j}b^{k}\\epsilon _{i}^{\\ lz} \\right),$ $\\delta (v) &= -\\eta _4 \\left(b_{ik}b_{j}b_{l}+b_{jk}b_{i}b_{l} \\right)\\partial _t h^{kl},$ $\\delta (vi) &=-\\frac{3}{4}\\zeta _\\perp \\left(\\delta _{ij}-b_ib_j \\right) \\left(\\partial _t h^{k}_{k}+\\partial _t h_{zz}\\right),$ $\\delta (vii) &= -\\frac{3}{2}\\zeta _\\parallel b_ib_j\\partial _t h_{zz}.$ The next step is to write the variations above in Fourier space using a plane-wave Ansatz for the perturbations, which gives the following expressions $\\delta (i) &= \\frac{i\\omega }{2}h_{kl}(\\omega )\\left[ \\eta _0 \\left( \\delta ^{k}_{i}\\delta ^{l}_{j} +\\delta ^{l}_{i}\\delta ^{k}_{j}-\\frac{2}{3}\\delta _{ij}\\delta ^{kl}\\right) \\right],$ $\\delta (ii) &= \\frac{i\\omega }{2}h_{kl}(\\omega )\\frac{1}{4} \\eta _1 \\left[ -2\\delta ^{kl}(\\delta _{ij}-3b_i b_j) +3(\\delta ^{kl}-b^{k}b^{l})(\\delta _{ij}-3b_i b_j)\\right],$ $\\delta (iii) &= -\\frac{i\\omega }{2}h_{kl}(\\omega )\\eta _2 \\left( b_{i}b^{k}\\delta ^{l}_{j}+b_{i}b^{l}\\delta ^{k}_{j}+b_{j}b^{k}\\delta ^{l}_{i}+b_{j}b^{l}\\delta ^{k}_{i}-4b_{i}b_{j}b^{k}b^l \\right),$ $\\delta (iv) &= -\\frac{i\\omega }{2}h_{kl}(\\omega )\\eta _3 \\left(\\delta _{i}^{k}\\epsilon _{j}^{\\ lz}+\\delta ^{jl}\\epsilon _{i}^{\\ kz} - b_{i}b^{k}\\epsilon _{j}^{\\ lz} - b_{j}b^{k}\\epsilon _{i}^{\\ lz} \\right),$ $\\delta (iv) &= \\frac{i\\omega }{2}h_{kl}(\\omega ) \\eta _4 \\left(b_{i}^{\\ k}b_{j}b^{l} +b_{i}^{\\ l}b_{j}b^{k}+ b_{j}^{\\ k}b_{i}b^{\\ l}+b_{j}^{\\ l}b_{i}b^{k} \\right) ,$ $\\delta (v) &= \\frac{i\\omega }{2}h_{kl}(\\omega )\\left[ \\frac{3}{2}\\zeta _\\perp \\left(\\delta _{ij}\\delta ^{kl}+\\delta _{ij}\\delta _{z}^{k}\\delta _{z}^{l}-b_ib_j\\delta ^{kl}-b_ib_j\\delta _{z}^{k}\\delta _{z}^{l} \\right) \\right],$ $\\delta (vii) &=\\frac{i\\omega }{2}h_{kl}(\\omega )\\left[ 3\\zeta _\\parallel b_ib_j\\delta _{z}^{k}\\delta _{z}^{l} \\right].$ Collecting all the variations in Fourier space, we write $\\delta \\Pi _{ij}(\\omega )&=\\frac{i\\omega }{2}h_{kl}(\\omega )\\left[ \\eta _0\\left(\\delta ^{k}_{i}\\delta ^{l}_{j}+\\delta ^{l}_{i}\\delta ^{k}_{j}-\\frac{2}{3}\\delta _{ij}\\delta ^{kl}\\right) +\\frac{1}{4} \\eta _1 \\left[ -2\\delta ^{kl}(\\delta _{ij}-3b_i b_j) +3(\\delta ^{kl}-b^{k}b^{l})(\\delta _{ij}-3b_i b_j)\\right] \\right.",
"\\\\&\\left.",
"- \\eta _2 \\left( b_{i}b^{k}\\delta ^{l}_{j}+b_{i}b^{l}\\delta ^{k}_{j}+b_{j}b^{k}\\delta ^{l}_{i}+b_{j}b^{l}\\delta ^{k}_{i}-4b_{i}b_{j}b^{k}b^l \\right) -\\eta _3 \\left(\\delta _{i}^{k}\\epsilon _{j}^{\\ lz}+\\delta ^{jl}\\epsilon _{i}^{\\ kz} - b_{i}b^{k}\\epsilon _{j}^{\\ lz} - b_{j}b^{k}\\epsilon _{i}^{\\ lz} \\right) \\right.\\\\&\\left.",
"+ \\eta _4 \\left(b_{i}^{\\ k}b_{j}b^{l} +b_{i}^{\\ l}b_{j}b^{k}+ b_{j}^{\\ k}b_{i}b^{\\ l}+b_{j}^{\\ l}b_{i}b^{k} \\right) +\\frac{3}{2}\\zeta _\\perp \\left(\\delta _{ij}\\delta ^{kl}+\\delta _{ij}\\delta _{z}^{k}\\delta _{z}^{l}-b_ib_j\\delta ^{kl}-b_ib_j\\delta _{z}^{k}\\delta _{z}^{l} \\right) \\right.\\\\&\\left.",
"+3\\zeta _\\parallel b_ib_j\\delta _{z}^{k}\\delta _{z}^{l} \\right],$ which allows us to express the retarded Green's function in terms of the viscosities, $- \\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }\\text{Im}\\,G^{R, \\, kl}_{ij}(\\omega ) = & \\eta _0\\left(\\delta ^{k}_{i}\\delta ^{l}_{j}+\\delta ^{l}_{i}\\delta ^{k}_{j}-\\frac{2}{3}\\delta _{ij}\\delta ^{kl}\\right) +\\frac{1}{4} \\eta _1 \\left[ -2\\delta ^{kl}(\\delta _{ij}-3b_i b_j) +3(\\delta ^{kl}-b^{k}b^{l})(\\delta _{ij}-3b_i b_j)\\right] \\\\& -\\eta _2 \\left( b_{i}b^{k}\\delta ^{l}_{j}+b_{i}b^{l}\\delta ^{k}_{j}+b_{j}b^{k}\\delta ^{l}_{i}+b_{j}b^{l}\\delta ^{k}_{i}-4b_{i}b_{j}b^{k}b^l \\right) \\\\& -\\eta _3 \\left(\\delta _{i}^{k}\\epsilon _{j}^{\\ lz}+\\delta ^{jl}\\epsilon _{i}^{\\ kz} - b_{i}b^{k}\\epsilon _{j}^{\\ lz} - b_{j}b^{k}\\epsilon _{i}^{\\ lz} \\right) \\\\& + \\eta _4 \\left(b_{i}^{\\ k}b_{j}b^{l} +b_{i}^{\\ l}b_{j}b^{k}+ b_{j}^{\\ k}b_{i}b^{\\ l}+b_{j}^{\\ l}b_{i}b^{k} \\right)\\\\& +\\frac{3}{2}\\zeta _\\perp \\left(\\delta _{ij}\\delta ^{kl}+\\delta _{ij}\\delta _{z}^{k}\\delta _{z}^{l}-b_ib_j\\delta ^{kl}-b_ib_j\\delta _{z}^{k}\\delta _{z}^{l} \\right) +3\\zeta _\\parallel b_ib_j\\delta _{z}^{k}\\delta _{z}^{l}.$ The final step is to isolate the viscosities and obtain the corresponding Kubo formulas.",
"For such a task, we only need to select specific components of $G^{R}_{ij,kl}$ .",
"For instance, if we take $i=k=x$ and $j=l=y$ in (REF ), we have $\\eta _0 = - \\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }G^{R}_{T_{xy}T_{xy}}(\\omega ),$ and so forth.",
"Finally, we obtain the following Kubo formulas $\\eta _0 &= - \\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }\\text{Im}\\, G^{R}_{T_{xy}T_{xy}}(\\omega ) , \\\\\\eta _1 &= -\\frac{4}{3}\\eta _0+ 4\\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }\\text{Im}\\, G^{R}_{P_{\\parallel }P_{\\perp }}(\\omega ), \\\\\\eta _2 &= \\eta _0+\\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }\\text{Im}\\, G^{R}_{T_{xz}T_{xz}}(\\omega ), \\\\\\eta _3 &= \\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }G^{R}_{P_{\\perp }T_{12}}(\\omega ), \\\\\\eta _4 &= - \\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }\\text{Im}\\, G^{R}_{T_{xz}T_{yz}}(\\omega ), \\\\\\zeta _\\perp &= -\\frac{2}{3}\\lim _{\\omega \\rightarrow 0} \\frac{1}{\\omega }\\left[ \\text{Im}G^{R}_{P_\\perp P_\\perp }(\\omega ) + \\text{Im}G^{R}_{P_\\parallel , P_\\perp }(\\omega ) \\right], \\\\\\zeta _\\parallel &= -\\frac{4}{3}\\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }\\left[ \\text{Im}G^{R}_{P_\\perp P_\\parallel }(\\omega ) + \\text{Im}G^{R}_{P_\\parallel , P_\\parallel }(\\omega ) \\right],$ where $P_{\\perp } \\equiv \\frac{1}{2}T^{a}_{\\ a}=\\frac{1}{2}(T^{x}_{\\ x}+T^{y}_{\\ y}), \\ \\ P_{\\parallel } \\equiv \\frac{1}{2} T^{z}_{\\ z}.$ The results in Eqs.",
"(REF ) — () agree with the ones obtained in Ref.",
"[207].",
"At first sight, the Kubo formulas obtained here seem different from the ones obtained in Ref.",
"[170].",
"The reason is that the formulas written in [170] are in a fully covariant form.",
"However, if we use the following identity $\\left\\langle \\left[\\int d^3x T^{00}, A \\right]\\right\\rangle = \\langle \\left[H, A \\right]\\rangle = i\\left\\langle \\frac{\\partial A}{\\partial t}\\right\\rangle = 0,$ where $A$ is a generic operator and $H$ is the Hamiltonian, we get rid of the term $\\hat{\\epsilon } \\sim T^{00}$ - recall that the mean values $\\langle \\cdots \\rangle $ for the Kubo formulas are always related to the equilibrium state.",
"Furthermore, when we recover isotropy, i.e., when $B=0$ , the formulas for both bulk viscosities, $\\zeta _{\\perp }$ and $\\zeta _{\\parallel }$ , return to the well-known isotropic formula.",
"Moreover, due to the structure of the Kubo formulas for the bulk viscosity, we have the relation $\\zeta =\\frac{2}{3}\\zeta _\\parallel +\\frac{1}{3}\\zeta _\\perp ,$ where $\\zeta $ is the isotropic bulk viscosity obtained by the Kubo formula, $\\zeta =-\\frac{4}{9}\\lim _{\\omega \\rightarrow 0}\\frac{1}{\\omega }\\textrm {Im}\\ G_{PP}^{R}(\\omega ,\\vec{k}=\\vec{0}),$ where $P\\equiv P_\\perp +P_\\parallel $ .",
"Following the usual convention, we defineDue to the different sign conventions for $\\eta _2$ , the definition for $\\eta _\\parallel $ differs from the one adopted in Ref.",
"[108].",
"$\\eta _{\\perp }\\equiv \\eta _0, \\ \\ \\eta _{\\parallel }\\equiv \\eta _0-\\eta _2.$ Another common way to write the formulas for the shear viscosities is $\\eta _{ijkl} = - \\lim _{\\omega \\rightarrow 0} \\frac{1}{\\omega } \\text{Im} \\ G_{T_{ij}T_{kl}}^{R} (\\omega , \\vec{k} = \\vec{0}) \\ \\ \\text{with} \\ i, j,k, l = x, y, z.$ For instance, in the above notation the isotropic shear viscosity $\\eta _0$ reads $\\eta _0=\\eta _{xyxy}=\\eta _{\\perp }.$ We finish this appendix emphasizing that the Kubo formulas for $\\eta _1$ , $\\eta _3$ and $\\eta _4$ trivially vanish in the backgrounds considered here, which are of the form (REF ).",
"For instance, the Kubo formula for $\\eta _3$ () depends on the operators $P_\\perp $ and $T^{xy}$ ; however, the dual bulk fields of these operators, $h_{xx}$ and $h_{xy}$ , respectively, are decoupled in the disturbed on-shell action, which makes the corresponding 2-point Green's function vanish.",
"Therefore, for the magnetic brane setup and the magnetic EMD model, one has to compute only two shear viscosities, $\\eta _{\\perp }$ and $\\eta _{\\parallel }$ , as we have done in the previous sections."
]
] | 1605.06061 |
[
[
"A Search for Highly Dispersed Fast Radio Bursts in Three Parkes\n Multibeam Surveys"
],
[
"Abstract We have searched three Parkes multibeam 1.4 GHz surveys for the presence of fast radio bursts (FRBs) out to a dispersion measure (DM) of 5000 pc cm$^{-3}$.",
"These surveys originally targeted the Magellanic Clouds (in two cases) and unidentified gamma-ray sources at mid-Galactic latitudes (in the third case) for new radio pulsars.",
"In previous processing, none of these surveys were searched to such a high DM limit.",
"The surveys had a combined total of 719 hr of Parkes multibeam on-sky time.",
"One known FRB, 010724, was present in our data and was detected in our analysis but no new FRBs were found.",
"After adding in the on-sky Parkes time from these three surveys to the on-sky time (7512 hr) from the five Parkes surveys analysed by Rane et al., all of which have now been searched to high DM limits, we improve the constraint on the all-sky rate of FRBs above a fluence level of 3.8 Jy ms at 1.4 GHz to $R = 3.3^{+3.7}_{-2.2} \\times 10^{3}$ events per day per sky (at the 99% confidence level).",
"Future Parkes surveys that accumulate additional multibeam on-sky time (such as the ongoing high-resolution Parkes survey of the LMC) can be combined with these results to further constrain the all-sky FRB rate."
],
[
"INTRODUCTION",
"In recent years a number of short-duration (millisecond) radio bursts (“fast radio bursts”, or FRBs) have been detected by the Parkes, Arecibo, and Green Bank radio telescopes in large-scale pulsar surveys.",
"These bursts have characteristics which indicate that they are not of terrestrial origin and are likely of extragalactic origin.",
"The broadband dispersion characteristics observed for FRBs very closely obey the cold plasma dispersion law in which the signal delay is proportional to the inverse square of the observing frequency (e.g., Lorimer & Kramer 2005).",
"This is expected if the signal originates from an astrophysical source (unlike, e.g., similar signals detected in some surveys, such as Perytons, which have been identified as terrestrial microwave interference; Petroff et al.",
"2015b).",
"The FRBs detected to date also have dispersion measures (DMs) significantly larger than what the Galactic plasma content along the line of sight is likely to account for [8].",
"This fact, along with the recently proposed association of FRB 150418 with an elliptical galaxy at redshift 0.5 [17], suggests that these bursts originate from very large (cosmological) distances.",
"Note, however, that [36] and [35] have called into question this association, and other models have been proposed in which the high DM can be accounted for more locally [7].",
"To date, only one of these bursts, FRB 121102 [32], has been observed to repeat [33], [31], despite numerous efforts and many hours spent trying to redetect FRBs in the same sky location using the same observing system.",
"With the possible exception of FRB 150418, no FRBs have yet been localized to the point where associations with known objects can be established.",
"Thus, the physical origin of FRBs remains uncertain, though the repeating nature of at least a subset of FRBs indicates that some of them do not originate from a cataclysmic event that destroys the source object.",
"Models for FRBs such as supergiant pulses emanating from magnetars in other galaxies (e.g., Cordes & Wasserman 2016) are currently favored.",
"For a recent overview and list of references to a variety of proposed models for FRBs, see [29] and [16].",
"The current tally of FRBs that have been detected and published is presented in the Swinburne FRB Catalogue [28].http://www.astronomy.swin.edu.au/pulsar/frbcat/ All but two of these FRBs were detected with the Parkes 64-m telescope, and all but one have been detected at or near an observing frequency of 1400 MHz.",
"Efforts are now underway both to comb existing pulsar survey data for FRBs that may have been missed in previous analyses of the data and to detect FRBs as they occur using real-time observing and detection systems.",
"Examples of the latter include the Parkes telescope at 1400 MHz [26], the ARTEMIS backend and LOFAR array operating at a much lower frequency of 145 MHz [15], and the BURST project with the Molonglo Observatory Synthesis Telescope which operates at an intermediate frequency of 843 MHz [5].",
"When the expected DM contribution from the Galaxy is removed using the [8] Galactic electron model, none of the FRBs detected to date has an extragalactic DM contribution (DM excess) larger than $\\sim 1550$ pc cm$^{-3}$ [6].",
"[37] have shown that there is a complicated non-linear relationship between the DM contribution from the intergalactic medium (IGM) and redshift.",
"However, as seen in their Fig.",
"1, for small redshifts ($z3$ ), a linear approximation can be made in which $\\sim 900$ -1100 pc cm$^{-3}$ of DM is contributed per redshift unit.",
"A DM excess of $\\sim 1550$ would then correspond to a redshift of $z \\sim 1.5$ assuming that the IGM is the primary source of the dispersion.",
"However, significant local dispersion near the source or contributions from the host galaxy could further boost the measured DM.",
"Thus, if FRBs beyond this redshift range are to be discovered, larger DMs must be searched for burst signals.",
"We have searched three Parkes radio pulsar surveys to try and detect putative bursts at very large DMs (up to a DM of 5000 pc cm$^{-3}$ ).",
"These three surveys were previously searched for FRBs in the single-pulse search analysis done during the original data processing, and in fact one of the surveys contains the first FRB found, FRB 010724 [22].",
"One of the other surveys has a known Peryton present [3].",
"However, none of the three surveys have yet been searched out to high DMs ($> 1000$ pc cm$^{-3}$ ).",
"All three of the surveys targeted sky regions away from the Galactic plane (all beams had Galactic latitudes $|b| > 5^{\\circ }$ ; see Table REF ).",
"This avoids foreground effects from the Galaxy that can negatively affect the detectability of FRBs through increased pulse scattering and sky temperature [4], [25].",
"Below we describe each of the three surveys we searched and outline our FRB search procedure.",
"We then describe our results and the subsequent constraints on the all-sky rate of FRBs."
],
[
"OVERVIEW OF THE SURVEYS",
"The three surveys we have analysed were all conducted with the Parkes radio telescope using the 13-beam multibeam receiver [34].",
"Note that all but two of the FRBs detected and listed to date in the Swinburne FRB Catalogue [28] were detected with this same observing system.",
"Table REF describes the observing parameters for each of these three surveys, which have a cumulative multibeam on-sky time of 719 hr.",
"The first survey (“SMC”) was a deep search for pulsars in the Magellanic Clouds.",
"Both the Small (SMC) and Large (LMC) Magellanic Clouds were searched with the same analog filterbank system as was used in the highly successful Parkes Multibeam Pulsar Survey [23].",
"A total of 22 new pulsars were discovered in this survey during several processing passes through the data [11], [24], [30].",
"The first FRB ever discovered, FRB 010724 [22], was also detected in this survey.",
"Prior to the work described here, this survey had only been searched for periodic signals and single pulses out to a maximum DM of 800 pc cm$^{-3}$ .",
"A total of 488 hr of on-sky time was recorded in the survey.",
"The second survey (“EGU”) targeted 56 unidentified mid-Galactic-latitude gamma-ray sources from the third EGRET catalog [14].",
"The same observing system was used for this survey as for the SMC survey described above, but with different integration and sampling times (see Table REF ).",
"The results of the survey were presented by [12].",
"The data were previously searched out to DM = 1000 pc cm$^{-3}$ , and six new pulsars were discovered.",
"One of these was PSR J1614$-$ 2230, a binary system with a pulsar mass of $1.97 \\pm 0.04~M_{\\odot }$ [13].",
"A Peryton RFI burst signal was also discovered in this survey [3].",
"This survey recorded a total of 135 hr of on-sky time.",
"The third survey (“PLMC”) is a new pulsar and transient survey of the LMC which is sensitive to millisecond pulsars in the LMC for the first time.",
"Like the two surveys above, it uses the Parkes telescope and the multibeam receiver, but with the Berkeley-Parkes-Swinburne Recorder (BPSR) digital backend [20].",
"This has a fast sampling capability and narrow frequency channels (see Table REF for details), and 20% of the total survey data has been collected and processed so far (corresponding to 96 hr of on-sky time).",
"The initial results from this work were described by [30], where 3 new pulsars were discovered.",
"In this processing, the data were searched for pulsations and single bursts out to DM = 500 pc cm$^{-3}$ , but no new FRBs have yet been detected in this survey."
],
[
"ANALYSIS",
"In our re-analysis of these surveys, we searched the data for impulsive signals at a much larger range of DMs than previously searched.",
"We searched DMs ranging from 0 to 5000 pc cm$^{-3}$ with a variable DM trial spacing that had a wider spacing at larger DMs.",
"The spacing was chosen so that the smearing introduced from a DM offset would not significantly increase the DM smearing already present within the finite frequency channels.",
"Table REF lists the number of DM trials used in each survey analysis.",
"Each dedispersed time series was searched for signals using a widely-used single-pulse detection algorithm in the SIGPROChttp://sigproc.sourceforge.net/ pulsar analysis package.",
"This algorithm is described in detail by [9] (see also [29] for a discussion) and uses a boxcar smoothing technique to maintain sensitivity to pulses at a wide range of time-scales.",
"The boxcar filters were produced by averaging adjacent time samples in 10 successive groups of two, yielding boxcar widths ranging from 1 to 1024 time samples (see Table REF for the sampling times used for the different surveys).",
"The boxcar sample aggregation was successively applied to each dedispersed time series, with the highest resulting signal-to-noise ratio (S/N) signal from the passes through the data being kept.",
"Only signals with a S/N $\\ge 5$ were recorded.",
"Note that this technique has been shown by [18] to reduce sensitivity to events which are offset from the boxcar centers by as much as a factor of $\\sqrt{2}$ (as compared to a convolution of the time series with comparable boxcar filters).",
"This sensitivity reduction was taken into account in our estimate of the all-sky FRB event rate.",
"A single-pulse diagnostic plot was produced for each beam (see Fig.",
"REF ).",
"In this plot, a short, dispersed impulse would appear in the DM vs. time plot as a signal at a non-zero DM that is localized in both dimensions.",
"The size of the circles indicates the S/N.",
"No radio frequency interference (RFI) excision was performed prior to the dedispersion and pulse search.",
"However, RFI was cleaned in the resulting single-pulse plots.",
"If an RFI signal appeared at a DM of zero (indicating a terrestrial signal), then samples at all DMs corresponding to the time of that event were removed.",
"This technique efficiently removed both non-dispersed broadband RFI and sporadic narrowband RFI while maintaining the detectability of dispersed broadband pulses (see Fig.",
"REF ).",
"After cleaning, both the cleaned and uncleaned plots were checked by eye for indications of dispersed pulse events."
],
[
"RESULTS AND DISCUSSION",
"We discovered no new FRBs in the three surveys we searched out to a DM of 5000 pc cm$^{-3}$ .",
"We did clearly detect both FRB 010724 [22] in an SMC survey beam and the Peryton interference signal present in several EGU survey beams [3].",
"This signal has been identified as a source of RFI [27], but since it mimics some of the characteristics of FRBs it is a good test of our single-pulse detection algorithm.",
"Both of these detections were made blindly (i.e., during the routine analysis of the survey data).",
"These detections are shown in Fig.",
"REF , and these are the only FRB-type signals known to be present in these surveys.",
"Note that in addition to these two single-burst events, a number of known pulsars were also detected during the processing as single-pulse sources.",
"We used the null detection of any new FRBs in these three surveys we analysed plus the results from five other Parkes multibeam 1.4 GHz surveys that have been searched for FRBs (see Rane et al.",
"2016) to determine a new constraint on the all-sky FRB rate.",
"These other large-scale Parkes surveys have all been searched out to high DMs (at least a DM of 3000 pc cm$^{-3}$ ; see Table 2 of Burke-Spolaor & Bannister 2014).",
"The five additional surveys we included in our FRB rate estimate are listed below (see also Table 2 of [29] which gives additional details of each survey): The high-time resolution universe south (HTRU-S) high-latitude survey [6].",
"A total of 2812 hr of on-sky time was recorded in this survey and 9 new FRBs were discovered.",
"The HTRU-S intermediate-latitude survey [25].",
"A total of 1154 hr of on-sky time was recorded.",
"No new FRBs were found.",
"The Swinburne Multibeam (SWMB) Survey [4].",
"925 hr total was recorded and one new FRB was discovered.",
"The Parkes Multibeam Pulsar Survey (PMPS) [23].",
"This survey targeted low Galactic latitudes and had an on-sky integration of 2115 hr.",
"One new FRB was detected here.",
"The Parkes High-latitude (PH) Survey [1].",
"506 hr of total on-sky time was recorded with no new FRBs found.",
"We added the 7512 hr of time from the surveys above to the 719 hr from our three surveys, and following the method of [29], we ran a likelihood analysis to determine a statistically likely all-sky rate of detectable FRBs.",
"From this combined survey set, we find a rate of $R = 3.3^{+3.7}_{-2.2} \\times 10^{3}$ events per day per sky above a fluence limit of 3.8 Jy ms at the 99% confidence level.",
"This is an improvement over the [29] limit of $R = 4.4^{+5.2}_{-3.1}\\times 10^{3}$ events per day above a 4 Jy ms fluence limit (99% confidence).",
"Fig.",
"REF shows the likelihood function for both the old and new all-sky rates.",
"Our derived FRB event rate above a uniform fluence threshold combines the results of the rates determined individually from the 8 different Parkes surveys, while also accounting for the different single-pulse search processing methods and different telescope backends used in these surveys (see Rane et al.",
"2016 for further details).",
"Other rate estimates that have been published from Parkes observations have used only a single survey or subset of surveys (e.g., the HTRU-S survey analysed by Champion et al.",
"2016 and Keane & Petroff 2015) which have a smaller total on-sky time than the combined set of surveys that we used.",
"Given the large uncertainties in all of these rates (including ours), they are all compatible with each other.",
"However, our rate is an improvement on the recent [29] rate estimate since we have included the on-sky time from 3 more surveys to their analysis and detected no additional FRBs.",
"We note that the PMPS was conducted at low Galactic latitudes ($|b| < 5^{\\circ }$ ), and Galactic-plane effects may significantly influence detectability of any FRBs present and hence can affect underlying FRB rate estimates.",
"The inclusion of this information in the future analysis of other Parkes multibeam surveys (such as the complete PLMC survey, of which only 20% has been observed and processed; Ridley et al.",
"2013) will help further constrain the all-sky FRB rate."
],
[
"CONCLUSIONS",
"We have analysed three Parkes multibeam surveys for FRBs at a range of DMs extending out to 5000 pc cm$^{-3}$ , a much higher DM limit than what was previously searched in these surveys.",
"We detected one known FRB and one known Peryton interference signal that were present in these surveys, but found no new FRBs.",
"We used the 719 hr of multibeam on-sky time from our three surveys and the 7512 hr of on-sky time from five other large-scale Parkes multibeam surveys searched out to high DMs (at least 3000 pc cm$^{-3}$ ) to improve the constraint on the FRB all-sky rate.",
"We determine a rate of $R = 3.3^{+3.7}_{-2.2} \\times 10^{3}$ events per day per sky above a fluence limit of 3.8 Jy ms at the 99% confidence level.",
"Results from future Parkes surveys will be able to be combined with these results to further constrain the underlying FRB rate."
],
[
"ACKNOWLEDGMENTS",
"The Parkes radio telescope is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.",
"Work at Franklin & Marshall College was partially supported by the Hackman scholarship fund.",
"FC thanks the McGill Space Institute for hospitality during the completion of this manuscript.",
"Figure: Single-pulse detections of two known burst signals present inthe surveys we analysed: FRB 010724 from the SMC survey(shown in the top two panels), and a Peryton RFI signal from the EGU survey (shown in the bottom two panels).",
"Each pair ofpanels shows DM vs. time prior to RFI cleaning (top panel) and aftercleaning (bottom panel) for that particular source.",
"The symbol sizeindicates signal strength.",
"The cleaning process removes undispersedbroadband terrestrial RFI (clustered at DM = 0 throughout theintegration) and narrowband RFI (occasional thin vertical signals)while preserving broadband dispersed signals.",
"Both signals areclearly detected in the diagnostic plots.Figure: The posterior probability density function of the event rateof Parkes-detectable FRBs, determined from the 5 Parkes surveys(totaling 7512 hr) analysed by (dashed curve) and fromthe addition of the 3 Parkes surveys (totaling 719 hr) described inthis paper (solid curve).",
"See also Table 2 of .",
"Allsurveys were searched to high DMs (at least 3000 pc cm -3 ^{-3}).",
"Therate analysis considered the different single-pulse search processingmethods and observing backends used in the different surveys.",
"Theresulting new all-sky FRB rate is R=3.3 -2.2 +3.7 ×10 3 R = 3.3^{+3.7}_{-2.2} \\times 10^{3} events per day per sky above a fluence limit of 3.8 Jy ms atthe 99% confidence level.lccc Summary of Three Parkes Surveys Searched 0pt Survey SMC EGU PLMC Galactic Latitude Range $|b| \\sim 45^{\\circ }$ (SMC); $5^{\\circ } < |b| < 73^{\\circ }$ $|b| \\sim 53^{\\circ }$ (LMC) $|b| \\sim 53^{\\circ }$ (LMC) Total Number of Survey Beams 2717a 3016 520b Integration Time Per Pointing (s) 8400 2100 8600 Total On-sky Survey Time (hr) 488 135 96b Sampling Time (ms) 1.000 0.125 0.512c Number of Frequency Channels 96 96 870d Observing Bandwidth (MHz) 288 288 340d Center Observing Frequency (MHz) 1374 1374 1352d Max.",
"Galactic DM Contribution (pc cm$^{-3}$ )e $\\sim 50$ $\\sim 500$ f $\\sim 50$ Original Max.",
"DM Searched (pc cm$^{-3}$ )g 800 1000 500 New Max.",
"DM Searched (pc cm$^{-3}$ ) 5000 5000 5000 Number of Trial DMs in New Search 256 371 1431 Known Burst Signals Detected 1 FRBh 1 Perytoni $-$ Survey References [24]; [12] [30] [30] All three surveys used the 13-beam multibeam receiver [34] on the Parkes 64-m telescope, and all surveys were conducted at 1.4 GHz.",
"All three surveys therefore had the same beam size and instantaneous sensitivity as other large-scale Parkes surveys recently searched for FRBs.",
"aOur analysis used 2756 beams (2717 original survey beams plus 39 unique extra beams that were not used in the [24] survey grid).",
"bThis corresponds to the first 20% of the total survey coverage, which is the fraction of the survey that has been observed (and processed) to date.",
"cFor the analysis here, the raw time samples were aggregated into groups of 8 to create an effective sampling time of 0.512 ms from the native 0.064 ms sampling at the telescope recorder.",
"dThe BPSR data recorder used at the Parkes telescope has 400 MHz of bandwidth split into 1024 channels with a 1382 MHz center observing frequency [20].",
"However, the receiver is not sensitive to the top 60 MHz of the band, which is blanked during the data analysis.",
"The table therefore shows the effective values with this 60 MHz band removed.",
"eMaximum Galactic DM contribution estimated from the NE2001 Galactic electron model [8] for all survey lines of sight.",
"fThe expected maximum Galactic DM along the line of sight is 100 pc cm$^{-3}$ for more than half of the target sources in this survey, and no lines of sight have an expected maximum Galactic DM greater than 500 pc cm$^{-3}$ .",
"gMaximum DM searched for pulsars and impulsive signals in the original survey analysis.",
"hFRB 010724 was discovered by [22] in this survey with DM = 375 pc cm$^{-3}$ (see Fig.",
"REF ).",
"This signal was detected in our analysis.",
"iOne Peryton was discovered by [3] which had a fitted DM $\\sim 375$ pc cm$^{-3}$ .",
"This signal was detected in our analysis (see Fig.",
"REF ), but it has been determined to be terrestrial in origin [27]."
]
] | 1605.06074 |
[
[
"CP-even scalar boson production via gluon fusion at the LHC"
],
[
"Abstract In view of the searches at the LHC for scalar particle resonances in addition to the 125 GeV Higgs boson, we present the cross section for a CP-even scalar produced via gluon fusion at N3LO in perturbative QCD assuming that it couples directly to gluons in an effective theory approach.",
"We refine our prediction by taking into account the possibility that the scalar couples to the top-quark and computing the corresponding contributions through NLO in perturbative QCD.",
"We assess the theoretical uncertainties of the cross section due to missing higher-order QCD effects and we provide the necessary information for obtaining the cross section value and uncertainty from our results in specific scenarios beyond the Standard Model.",
"We also give detailed results for the case of a 750 GeV scalar, which will be the subject of intense experimental studies."
],
[
"Introduction",
"The Higgs boson is the only elementary scalar particle of the Standard Model (SM) and the first such particle whose existence has been confirmed experimentally [1], [2].",
"Many well-motivated extensions of the SM predict additional electrically neutral and colourless scalar particles (elementary or not).",
"Experimental searches to hunt for these hypothetical new particles are thus of extreme importance.",
"The LHC experiments have the potential to discover new scalar resonances by using similar techniques and signatures as in the search for the SM Higgs boson.",
"In particular, in this context both the ATLAS an CMS experiments have recently reported an excess in the diphoton mass spectrum at an invariant mass of $\\sim 750$ GeV [3], [4].",
"Even if it is still premature to speculate if the source of this excess is indeed a new resonance or merely a statistical fluctuation, this excess is exemplary for the kind of signals that may arise from the production of heavy singlet scalars in the current run of the LHC.",
"While specific models with colour-neutral scalars may differ drastically in the principles which motivate them as well as in their spectra and interactions, the computation of QCD radiative corrections to their production cross section could share many common features that are independent of the underlying model.",
"In particular, the production of hypothetical Higgs-like scalars at the LHC may be favourable in the gluon-fusion process due to the large value of the gluon-gluon luminosity.",
"In this case, the model-dependence of the production cross section may only enter through the coupling describing the (effective) interaction of the Higgs-like scalar with the gluons and quarks.",
"The purpose of this paper is to provide benchmark cross sections for a colorless CP-even scalar in an effective theory which is similar to the one obtained for the Higgs boson in the Standard Model after integrating out the top-quark.",
"The cross section for the scalar can be obtained from the cross section of the Higgs boson for a wide range of models, by appropriately taking into account the corresponding Wilson coefficient.",
"Our computation is valid through N$^3$ LO in perturbative QCD and is based on recent results for the SM Higgs boson gluon-fusion cross section to that order [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].",
"We consider mass values for the new scalar particle ranging from 10 GeV to 3 TeV, which is the same range as the one targeted by the searches of ATLAS and CMS in the Run 2 of the LHC [21].",
"In some extensions of the Standard Model (see, for example, ref.",
"[22] and references therein) the new Higgs-like scalar has a non-negligible coupling to the Standard Model top-quark.",
"For scalar masses below the top-pair threshold the Wilson coefficient in our effective field theory (EFT) can be made to account for a coupling of the scalar to the top-quark.",
"However, for scalar masses around and above threshold, the effective theory approach is not accurate.",
"For this purpose, we extend the effective theory by adding a Yukawa-type interaction of the top-quark with the scalar and compute the additional contributions generated by this coupling.",
"While we can correct our N$^3$ LO cross-section in the effective theory for contributions of light Standard Model particles through NLO in QCD, it is not possible to do so in a model-independent way for contributions from new relatively light particles with a mass smaller than about half the mass of the scalar.",
"The presence of such particles will invalidate our effective-theory computation.",
"To estimate this effect, we consider the difference between the effective theory NLO cross-section and the exact NLO cross-section for a 750 GeV scalar which couples to a new top-like quark, as a function of the mass of the quark.",
"We find that the two cross-sections can differ by more than the QCD scale uncertainty for top-like quark masses as heavy as twice the mass of the scalar.",
"Nevertheless, we observe that one can still make use of the effective theory computation for such a model also for lower values of the quark mass in order to estimate the relative size of the QCD corrections ($K-$ factor).",
"This article is organised as follows.",
"In section we present the effective theory setup of our calculation and discuss how the results can be adapted to suite a large class of models through the Wilson coefficient.",
"We also discuss the theoretical uncertainty that is associated with the N$^3$ LO QCD corrections in the effective theory.",
"In section , we discuss the inclusion of finite width effects in the calculation and provide a fit of the zero-width cross section to enable the adaptation of our results to models where the scalar resonance is expected to have a non-vanishing but narrow width.",
"In section , we investigate the validity of our effective theory approach under the assumption that the coupling between the scalar and the gluons is mediated by a heavy top partner of varying mass.",
"In section we go beyond a purely effective description of the scalar interaction and allow for a non-vanishing coupling between the Standard Model top and the scalar, for which we present the NLO QCD corrections.",
"We conclude in section and provide appendices with tables of cross section values for a wide range of masses of the scalar."
],
[
"N$^3$ LO QCD corrections in the effective theory approach",
"Let us consider a model where the SM is extended by a colourless singlet scalar $S$ of mass $m_S$ and width $\\Gamma _S$ , which only couples to the SM in a minimal way.",
"For now, we assume that the only SM fields that the new scalar $S$ couples to are the gluons, through an effective operator of dimension five.",
"We will discuss Yukawa couplings to heavy quarks in Section .",
"Such a model can be described by a Lagrangian of the form $\\mathcal {L}_{\\text{eff}}=\\mathcal {L}_{\\textrm {SM}}+\\mathcal {L}_S-\\frac{1}{4 v}C_{S} \\, S\\, G_{\\mu \\nu }^a G_a^{\\mu \\nu },$ where $\\mathcal {L}_S$ collects the kinetic term and the potential of the scalar $S$ .",
"The vacuum expectation value $v$ is [23], $v = \\frac{1}{\\sqrt{\\sqrt{2} G_f}} = 246.22~\\textrm {GeV},$ and $C_S$ is a Wilson coefficient parametrizing the strength of the coupling of the particle $S$ with the gluons.",
"Except for the value of the Wilson coefficient, this is the same dimension-five operator as the one coupling the SM Higgs boson to the gluons, after the top-quark has been integrated out.",
"Since the scalar $S$ couples to gluons, it can be produced at hadron colliders.",
"Its hadronic production cross section can be cast in the form $\\sigma _S(m_S,\\Gamma _S,\\Lambda _{\\rm UV})= \\left| C_S(\\mu , \\Lambda _{\\rm UV}) \\right|^2 \\eta (\\mu ,m_S,\\Gamma _S)\\,,$ where $\\mu $ is the mass scale introduced by dimensional regularisation and $\\Lambda _{\\rm UV}$ is the scale of new physics, representing collectively the masses of the heavy particles in some ultraviolet (UV) completion of the theory.",
"At all orders in perturbation theory, the cross section is independent of the arbitrary scale $\\mu $ , with the scale variation of the square of the Wilson coefficient cancelling the one of the matrix-elements $\\eta $ .",
"Since $\\eta $ does not depend on $\\Lambda _{\\rm UV}$ , the scale dependence of the Wilson coefficient itself is universal, and it does not depend on the (renormalizable) UV completion of the effective theory.",
"In particular, the renormalization group evolution equation that relates the Wilson coefficient at two different scales reads [24]: $\\frac{C_S(\\mu ,\\Lambda _{\\rm UV})}{C_S(\\mu _0,\\Lambda _{\\rm UV})} = \\frac{\\beta (\\mu )}{\\beta (\\mu _0)}\\frac{\\alpha (\\mu _0)}{\\alpha (\\mu )}\\,,$ with $\\beta (\\mu )$ the QCD $\\beta $ -function.",
"We see that, as expected, the evolution equation only depends on the low-energy effective theory, and it is independent of the details of the UV completion.",
"As a consequence, if we divide the production cross section by the value of the Wilson coefficient at some reference scale $\\mu _0$ , $\\frac{\\sigma _S(m_S,\\Gamma _S,\\Lambda _{\\rm UV})}{\\left| C_S(\\mu _0, \\Lambda _{\\rm UV})\\right|^2} =\\left[\\frac{\\beta (\\mu )}{\\beta (\\mu _0)} \\frac{\\alpha (\\mu _0)}{\\alpha (\\mu )}\\right]^2 \\eta (\\mu ,m_S,\\Gamma _S)\\,,$ we obtain a universal ratio that does not depend on the UV completion (up to corrections due to differences that the various high-energy theories may induce on the width of the scalar particle $S$ ).",
"In particular, since the dimension-five operator in eq.",
"(REF ) is the same as the dimension-five operator mediating Higgs production in gluon fusion in the SM, we can extract the right-hand side of eq.",
"(REF ) from Higgs production in gluon-fusion, $\\sigma _S(m_S,\\Gamma _S,\\Lambda _{\\rm UV}) =\\left|\\frac{ C_S(\\mu _0, \\Lambda _{\\rm UV})}{ C(\\mu _0, m_{t})}\\right|^2\\,\\sigma _H(m_S,\\Gamma _S,m_t) \\; .$ Here $\\sigma _H$ is the gluon-fusion production cross section of a Higgs boson of mass $m_S$ and width $\\Gamma _S$ , in a variant of the SM with $N_f=5$ massless quarks and the top-quark integrated out, $m_t$ is the top-quark mass and $C(\\mu _0,m_{t})$ is the Wilson coefficient multiplying the SM gluon-fusion operator evaluated at our reference scale $\\mu _0$ .",
"In the $\\overline{\\rm MS}$ scheme it reads [25], [26] $C(\\mu , m_t) & = & - \\frac{a_s}{3} \\Bigg \\lbrace 1 +a_s\\,\\frac{11}{4}+ a_s^2 \\left[\\frac{2777}{288} - \\frac{19}{16} \\log \\left(\\frac{m_t^2}{\\mu ^2}\\right) -N_f\\left(\\frac{67}{96}+\\frac{1}{3}\\log \\left(\\frac{m_t^2}{\\mu ^2}\\right)\\right)\\right]\\nonumber \\\\&&+ a_s^3\\Bigg [-\\left(\\frac{6865}{31104}+ \\frac{77}{1728} \\log \\left(\\frac{m_t^2}{\\mu ^2}\\right)+ \\frac{1}{18}\\log ^2\\left(\\frac{m_t^2}{\\mu ^2}\\right)\\right) N_f^2 \\\\&& + \\left(\\frac{23}{32} \\log ^2\\left(\\frac{m_t^2}{\\mu ^2}\\right) -\\frac{55}{54} \\log \\left(\\frac{m_t^2}{\\mu ^2}\\right)+\\frac{40291}{20736}- \\frac{110779}{13824} \\zeta _3 \\right) N_f \\nonumber \\\\&& -\\frac{2892659}{41472}+\\frac{897943}{9216}\\zeta _3+ \\frac{209}{64} \\log ^2\\left(\\frac{m_t^2}{\\mu ^2}\\right)- \\frac{1733}{288}\\log \\left(\\frac{m_t^2}{\\mu ^2}\\right)\\Bigg ] +\\begin{cal}O\\end{cal}(a_s^4) \\Bigg \\rbrace \\, ,\\nonumber $ with $a_s = \\alpha _s(\\mu )/\\pi $ , $\\alpha _s$ being the strong coupling constant in QCD with $N_f=5$ flavors.",
"We note at this point that eq.",
"(REF ) is valid to all orders in perturbation theory, and so neither $\\sigma _S$ nor $\\sigma _H$ depend on the arbitrary scales $\\mu _0$ and $\\mu $ .",
"They do, however, acquire a dependence on the scale $\\mu $ once the perturbative series is truncated at some finite order.",
"It is of course possible to further separate $\\mu $ into a factorization and a renormalization scale, $\\mu _f$ and $\\mu _r$ , as it is customary, and we will assume in the following that this is done implicitly.",
"In addition, $\\sigma _S$ also acquires a dependence on our reference scale $\\mu _0$ order by order in perturbation theory, resulting from the truncation of the perturbative expansion of the ratio of Wilson coefficients.",
"Equation (REF ) is the basic formula which allows us to predict the hadronic production cross section of a generic CP-even singlet scalar through gluon-fusion to high order in perturbative QCD.",
"Indeed, the gluon-fusion cross section is known through N$^3$ LO in QCD in the large-$m_t$ limit [19] (for $\\Gamma _S=0$ ) as an expansion around the Higgs threshold, and eq.",
"(REF ) easily allows us to convert the result for the Higgs boson into the corresponding N$^3$ LO cross section for a generic CP-even scalar.",
"At this point we need to make a small technical comment: both the ratio of Wilson coefficient and the Higgs production cross section on the right-hand side of eq.",
"(REF ) admit a perturbative expansion up to N$^3$ LO.",
"Multiplying these two perturbative expansions, as suggested by eq.",
"(REF ), introduces additional terms into the perturbative expansion which are of N$^4$ LO accuracy or higher, thus beyond the reach and the control of our N$^3$ LO computation.",
"In ref.",
"[19] it was shown that the numerical impact of these terms is captured by the scale variation at N$^3$ LO if the central scale is chosen as $\\mu =m_S/2$ , and the effects of such terms are therefore accounted for by the scale variation uncertainty assigned to our prediction.",
"A second comment regards the dependence of the Standard Model Wilson coefficient on the various scales.",
"The SM Wilson coefficient $C(\\mu _0, m_{t})$ depends on the reference scale $\\mu _0$ and the top-quark mass $m_t$ .",
"While the value of $\\mu _0$ is arbitrary and can be chosen freely, in practise we choose the value of $\\mu _0$ to be the mass of the scalar, $\\mu _0=m_S$ , and we set the value of the mass of the top quark to its physical value.",
"In doing so, we may introduce large logarithms into the computation, order by order in perturbation theory.",
"For example, we see that at NNLO and N$^3$ LO the Wilson coefficient contains a logarithm of the form $\\log (m_t/\\mu _0)$ .",
"Similarly, the Wilson coefficient expression for the scalar $C_S(\\mu _0,\\Lambda _{\\rm UV})$ will contain logarithms of the type $\\log (\\Lambda _{\\rm UV}/\\mu _0)$ .",
"In order to avoid large logarithms due to widely disparate scales in the ratio of the two Wilson coefficients, it may be preferable to evaluate the SM Wilson coefficient with a top-quark mass value $m_t \\sim \\Lambda _{\\rm UV}$ rather than its physical value.",
"The value of the top mass entering the SM Higgs cross-section can easily be changed with a simple rescaling, $\\sigma _H(m_S,\\Gamma _S,\\Lambda _{\\rm UV}) = \\left|\\frac{ C(\\mu _0, \\Lambda _{\\rm UV})}{ C(\\mu _0,m_t)}\\right|^2\\,\\sigma _H(m_S,\\Gamma _S,m_t)\\; .$ Figure: Effective theory production cross section of a scalar particle as a function of the particle massm S ∈10,50m_S\\in \\left[10,50 \\right] GeV through increasing orders in perturbationtheory, at a 13 TeV proton-proton collider.",
"The bands enveloping the respective orders represent the variation of the cross section due to variations of the scale μ∈m S /4,m S \\mu \\in \\left[{m_S}/{4},m_S\\right].The value of the top mass is set to m t (m t )=162.7m_t(m_t)=162.7 GeV.Figure: Effective theory production cross section of a scalar particle of massm S ∈50,150m_S\\in \\left[50,150\\right] GeV through increasing orders in perturbationtheory.",
"For further details see the caption of Fig.",
".Figure: Effective theory production cross section of a scalar particle with massm S ∈150,500m_S\\in \\left[150,500\\right] GeV through increasing orders inperturbation theory.",
"For further details see the caption of Fig.",
".Figure: Effective theory production cross section of a scalar particle as a function of the particle mass m S ∈500,3000m_S\\in \\left[500,3000\\right] GeV through increasing orders in perturbation theory.",
"For further details see the caption of Fig.",
".Let us now discuss our predictions for the production cross section $\\sigma _S$ .",
"In Figs.",
"REF - REF we show the effective theory cross section $\\sigma _H(m_S,\\Gamma _S=0,m_t)$ as a function of $m_S$ through N$^3$ LO, at a proton-proton collider with a center-of-mass energy of 13 TeV.",
"The theory uncertainty bands correspond to a variation of the scale $\\mu =\\mu _r=\\mu _f$ in the range $\\mu \\in \\left[{m_S}/{4}, m_S\\right]$ .",
"The value of the top mass is set to $m_t(m_t)=162.7$ GeV and we use the PDF4LHC15 [27] parton distribution function (PDF) set.",
"We observe that for the range of scalar masses between 10 GeV and 3 TeV the N$^3$ LO scale variation band is always contained inside the NNLO band.",
"This is also true for the lowest values of this mass range, $m_S\\sim 10-50 \\textrm { GeV}$ (Fig.",
"REF ), where one observes especially large corrections at NLO and NNLO.",
"For higher masses, the scale variation is reduced.",
"As an illustration, we show in Fig.",
"REF the scale dependence for the cross section of a CP-even scalar with a mass of 750 GeV.",
"Figure: Effective theory production cross section of a scalar particle with a massm S =m_S= 750 GeV as a function of the common renormalization and factorization scales μ\\mu through increasing orders in perturbation theory.Sources of theoretical uncertainty affecting the gluon-fusion cross section at N$^3$ LO other than scale variation have been considered in detail in ref. [19].",
"In particular, these sources of uncertainty are due to the lack of N$^3$ LO parton densities and to the truncation of the threshold expansion for the N$^3$ LO correction.",
"In order to estimate the uncertainty on our computation, we follow faithfully the uncertainty estimation prescription of ref.",
"[19] with one modification: the uncertainty due to the lack of N$^3$ LO parton densities is estimated here as the envelope of the corresponding uncertainties formed by using the CT14 [28], NNPDF30 [29] and PDF4LHC sets, which are themselves computed according to the prescription of ref. [19].",
"The reason for modifying the prescription in this case is that the PDF4LHC set leads to very small uncertainties for scalars in the region $500-1000$ GeV and, in particular, to a vanishing uncertainty estimate for a scalar with mass around 770 GeV.",
"This feature is not shared with individual PDF sets.",
"We therefore use, conservatively, the envelope of CT14, NNPDF30 and PDF4LHC, which leads to an uncertainty due to the lack of N$^3$ LO parton densities at the level of $0.9\\%-3\\%$ for scalars in the range 50 GeV$-3$ TeV.",
"This uncertainty remains of the order of a few percent also at lower masses, but it increases rapidly to ${\\cal O}(10\\%)$ for $m_S \\lesssim 20$ GeV.",
"We present the cross section values and uncertainties for this range of scalar masses in Appendix .",
"In particular, in Tab.",
"REF we focus on the range between 730 and 770 GeV."
],
[
"Finite width effects and the line-shape",
"The results of the previous section hold formally only when the width of the scalar is set to zero.",
"In many beyond the Standard Model (BSM) scenarios, however, finite-width effects cannot be neglected.",
"In this section we present a way to include leading finite-width effects into our results, in the case where the width is not too large compared to the mass.",
"The total cross section for the production of a scalar boson of total width $\\Gamma _S$ can be obtained from the cross section in the zero-width approximation via a convolution $\\sigma _S(m_S,\\Gamma _S,\\Lambda _{\\rm UV})= \\int dQ^2 \\frac{Q \\Gamma _S(Q)}{\\pi }\\frac{\\sigma _S(Q,\\Gamma _S=0,\\Lambda _{\\rm UV})}{(Q^2-m_S^2)^2+ m_S^2 \\Gamma ^2(m_S)} + \\mathcal {O}\\left(\\Gamma _S(m_S)/m_S\\right)\\,,$ where $Q$ is the virtuality of the scalar particle.",
"This expression is accurate at leading order in $\\Gamma _S(m_S)/m_S$ .",
"For large values of the width relative to the mass, subleading corrections and signal-background interference effects are important and are not captured by eq.",
"(REF ).",
"Let us also note that in order to use eq.",
"(REF ) faithfully, one needs the width as a function of the virtuality of the scalar, which may bear a substantial model dependence.",
"However, it is often the case that the width can be approximated as $\\Gamma _S(Q \\approx m_S) \\equiv \\Gamma _S\\,.$ The invariant mass distribution in this approximation for the production of a CP-even scalar with a mass of $m_S= 750$ GeV and total width from 2 to $10\\%$ of the scalar mass is shown in Fig.",
"REF .",
"This result has been obtained from an interpolation of the zero-width cross section values of Tab.",
"REF in Appendix .",
"We caution that if the results of Appendix are used to derive cross section values with non-zero width effects following the strategy outlined in this section, an additional uncertainty of the order ${\\cal O}\\left({\\Gamma _S}/{m_S}\\right)$ should be assigned.",
"Figure: Line-shape of a 750 GeV CP-even scalar boson at the LHC for differentvalues of its total width.In order to facilitate the computation of the line-shape we perform a parametric fit of the production cross section of a zero-width scalar boson as a function of its virtuality (with $\\Lambda _{\\textrm {UV}}=m_S$ ) for a center-of-mass energy of 13TeV.",
"To guarantee agreement of the fitted cross section with the actual cross section at a level better than $1\\%$ , we split the range of interesting scalar boson mass values into three intervals: in the range $m_S\\in [10\\textrm { GeV},150\\textrm { GeV}]$ , cf.",
"Tab.",
"REF in Appendix , we find $\\sigma _S(x)&\\approx & \\left(3.89881\\times 10^6\\, x^2 -1.90274\\times 10^6\\, x -202261\\,x\\, \\log ^2x+1623.77\\, \\log ^2x \\right.",
"\\nonumber \\\\&-&\\left.",
"923052 \\,x\\, \\log x+24108.2\\, \\log x+95652.2\\right) \\textrm { pb}\\,,$ in the range $m_S\\in [150\\textrm { GeV},500\\textrm { GeV}]$ , cf.",
"Tab.",
"REF , we find $\\sigma _S(x) \\approx \\left(1-\\@root 3 \\of {x}\\right)^{9.52798}\\, x^{-0.0415044 \\,\\log x-1.50381}\\,\\, \\textrm { pb}\\,,$ in the range $m_S\\in [500\\textrm { GeV},3000\\textrm { GeV}]$ , cf.",
"Tabs.",
"REF -REF , we find $\\sigma _S(x) \\approx \\left(1-\\@root 3 \\of {x}\\right)^{9.71562} \\,x^{-0.0040194\\,\\log ^3x-0.0474683\\, \\log ^2x-0.240878 \\,\\log x-1.81243} \\textrm { pb}\\,,$ where $x \\equiv \\frac{Q/\\textrm {GeV}}{13 \\textrm { TeV}}$ .",
"The fits of Eqs.",
"(REF )-(REF ) can be used as the kernel of the convolution in eq.",
"(REF )."
],
[
"Validity of the EFT approach",
"So far, we have assumed that the effective theory of eq.",
"(REF ) furnishes an accurate description of the gluon-scalar interaction.",
"However, this assumption may be challenged if the scalar couples to light coloured particles.",
"To investigate this effect, we will consider a scalar of mass $m_S= 750$ GeV which couples to a top-like quark of mass $m_T$ .",
"In this scenario, we can compute the cross-section exactly through NLO in perturbative QCD.",
"We can then compare this prediction with the cross-section derived with the effective theory of eq.",
"(REF ).",
"The red line in Fig.",
"REF shows the percent difference of the two predictions, $\\delta _{\\rm EFT} = \\frac{\\sigma ^{\\rm NLO}_{\\rm exact} -\\sigma ^{\\rm NLO}_{\\rm EFT}}{\\sigma ^{\\rm NLO}_{\\rm exact}} \\times 100\\% \\; ,$ as a function of $m_T$ for a scalar with a mass of $m_S = 750$ GeV.",
"While in the region $m_T \\sim m_S/2$ the effective field theory is inadequate, for larger values of $m_T$ the EFT description becomes accurate very quickly.",
"Indeed, for $m_T \\sim 1.5 m_S$ the discrepancy with respect to the exact result is already below the theoretical uncertainty of QCD origin (of about $5\\%$ ).",
"Figure: Percent difference () between the exact and rescaled EFT (rEFT) cross sectionat NLO (red line/green line) as a function of the quark mass for the production of a750 GeV CP-even scalar.",
"The vertical line corresponds to m S /2m_S/2.Let us now define a rescaled effective theory cross-section, $\\sigma ^{\\rm NLO}_{\\rm rEFT}= \\frac{\\sigma ^{{\\rm LO}}_{\\rm exact}}{\\sigma ^{{\\rm LO}}_{\\rm EFT}}\\, \\sigma ^{{\\rm NLO}}_{\\rm EFT} \\;.$ The green line of Fig.",
"REF shows the relative difference $\\delta _{\\rm rEFT} = 100 \\times \\, \\frac{\\sigma ^{\\rm NLO}_{\\rm exact}-\\sigma ^{\\rm NLO}_{\\rm rEFT}}{\\sigma ^{\\rm NLO}_{\\rm exact}}$ between the rescaled effective theory prediction for the cross-section and the exact NLO cross section.",
"We notice that the rescaled effective theory cross-section reproduces the exact cross section very well (as it has also been the case for the Standard Model Higgs boson [30], [31]).",
"We conclude that, while for the study of scalars which couple to relatively light particles one should resort to a calculation within a specific model, also in such situations the effective theory computation can be utilised as a means to compute with a good accuracy the corresponding QCD $K-$ factor with respect to the exact LO cross section."
],
[
"Top-quark contributions",
"So far we have considered the case where the interaction between the heavy scalar $S$ and the Standard Model is mediated exclusively by the dimension-five operator in eq.",
"(REF ).",
"In many extensions of the Standard Model new scalars may also couple directly to the quarks of the third generation as well as the $W$ and $Z$ bosons.",
"However, the absence of resonances in top-pair production and four-lepton production suggests that these couplings should be small.",
"It may be nonetheless useful to estimate the contribution to the inclusive $S$ -scalar cross section due to the heavy SM quarks and gauge bosons for the purpose of setting precise experimental constraints on their couplings to $S$ .",
"In this section, we study the effects due to the top quark, which in many scenarios are expected to give the most important contribution.",
"For light scalar masses, $m_S < {\\cal O} ( m_t)$ , contributions due to the coupling of the top quark can be simply taken into account through N$^3$ LO in perturbative QCD by using an appropriate Wilson coefficient $C_S$ .",
"For heavier scalars with masses around and above the top-pair threshold, however, it is not justified theoretically to integrate out the top-quark.",
"For this reason we study in the following the effect of modifying the Largangian in eq.",
"(REF ) by including a direct Yukawa interaction between the scalar $S$ and the top quark, i.e., we consider the Lagrangian $\\mathcal {L}_{\\text{eff}}=\\mathcal {L}_{\\textrm {SM}}+\\mathcal {L}_S-\\frac{\\lambda _{\\rm wc}}{4 v}C \\,S\\, G_{\\mu \\nu }^a G_a^{\\mu \\nu } - \\lambda _t \\frac{m_t}{v}\\, S\\, \\bar{t} t\\,,$ where $\\lambda _t$ is the ratio of the Yukawa coupling of the scalar $S$ and the Yukawa coupling of the Higgs boson to the top quark and $\\lambda _{\\rm wc}= \\frac{C_S}{C}$ denotes the ratio of the Wilson coefficients in this theory and the SM with the top-quark integrated out.",
"Figure: NLO contributions to the total cross section for a scalar particle of mass m S m_S as described in eqs.",
"(), ().",
"The bands correspond to the theory uncertainty (based on eq. ()).",
"In the lower pane we present the relative theory uncertainty (%\\%) for each of the contributions.The production cross section now depends on the values of $\\lambda _{\\rm wc}$ and $\\lambda _t$ : $\\sigma _S \\equiv \\sigma _S[\\lambda _{\\rm wc}, \\lambda _t]\\,.$ It does not take too much effort to show that the NLO cross section can be cast in the form $\\begin{split}\\sigma _S^{\\rm NLO}[\\lambda _{\\rm wc}, \\lambda _t] &\\,=\\lambda _{\\rm wc} (\\lambda _{\\rm wc} - \\lambda _t ) \\sigma _S^{\\rm NLO}[1, 0]+ \\lambda _t (\\lambda _t - \\lambda _{\\rm wc} ) \\sigma _S^{\\rm NLO}[0,1] \\\\&\\,+ \\lambda _{\\rm wc} \\lambda _t \\, \\sigma _S^{\\rm NLO}[1, 1] \\,.\\end{split}$ The cross section with no Yukawa coupling, $\\sigma _S^{\\rm NLO}[1, 0]$ , is known through N$^3$ LO.",
"Hence, we can improve the previous expression by including all N$^3$ LO corrections to the terms proportional to $\\lambda _{\\rm wc}^2$ , $\\begin{split}\\sigma _S[\\lambda _{\\rm wc}, \\lambda _t] &\\,=\\lambda _{\\rm wc}^2 \\sigma _S^{\\textrm {N}^3\\textrm {LO}}[1,0]-\\lambda _{\\rm wc} \\lambda _t \\sigma _S^{\\rm NLO}[1, 0]+ \\lambda _t (\\lambda _t - \\lambda _{\\rm wc}) \\sigma _S^{\\rm NLO}[0,1] \\\\&\\,+ \\lambda _{\\rm wc} \\lambda _t \\, \\sigma _S^{\\rm NLO}[1, 1] \\,.\\end{split}$ Table: Contributions to the cross section () for theproduction of a CP-even scalar with mass 750 GeV at a proton-proton collider.Although we have taken into account QCD corrections within the EFT framework through N$^3$ LO , the result for the cross section is formally only NLO-accurate because we are missing finite-mass effects beyond NLO.",
"We therefore estimate the uncertainties on the NLO cross-section as $\\frac{ \\delta \\sigma ^{\\rm NLO}[n_1,n_2]}{\\sigma ^{\\rm NLO}[n_1,n_2]} = \\pm \\delta _{>{\\rm NLO}} \\, (1 + \\delta _{\\rm scheme}[n_1,n_2])\\,,\\quad n_i\\in \\lbrace 0,1\\rbrace \\,,$ where $\\delta _{>{\\rm NLO}}= \\left( \\frac{\\sigma ^{\\rm N^3LO}[1,0]-\\sigma ^{\\rm NLO}[1,0]}{\\sigma ^{\\rm NLO}[1,0]} \\right)_{\\rm EFT}$ is the relative change of the gluon-fusion cross-section in the effective theory from NLO to N$^3$ LO.",
"Note that in this way make the assumption that the cross-section components that are not known beyond NLO will not have a worse perturbative convergence than in the effective theory.",
"Finally, we enlarge this uncertainty further by $\\delta _{\\rm scheme}[n_1,n_2] =\\frac{\\left|\\sigma ^{\\rm NLO, \\overline{MS}}_{\\textrm {exact}}[n_1,n_2] -\\sigma ^{\\rm NLO,OS}_{\\textrm {exact}}[n_1,n_2]\\right|}{\\sigma ^{\\rm NLO, \\overline{MS}}_{\\textrm {exact}}[n_1,n_2]}\\;,$ which measures the scheme dependence of the top-quark contribution at NLO.",
"Equation (REF ) is our best prediction for the gluon-fusion production cross section of a generic scalar $S$ .",
"We recall that the values of $\\sigma _S^{\\textrm {N}^3\\textrm {LO}}[1,0]$ as a function of the scalar boson mass can be read off from Tabs.",
"REF -REF in Appendix .",
"The NLO cross sections $\\sigma _S^{\\rm NLO}[n_1,n_2]$ ($n_1,n_2 \\in \\lbrace 0,1\\rbrace $ ) are reported in Tabs.",
"REF -REF .",
"As an illustration, we present the complete set of terms entering in eq.",
"(REF ) for a scalar of mass $m_S=750\\textrm { GeV}$ in Tab.",
"REF .",
"The following fits are a good approximation to the central values of the cross sections introduced in this section.",
"The fits are functions of $x=\\frac{m_S/\\textrm {GeV}}{13 \\textrm {TeV}}$ valid for a center-of-mass energy of 13TeV and PDF4LHC15 parton distribution functions.",
"In the range $m_S\\in [50\\textrm { GeV},150\\textrm { GeV}]$ , we find $\\begin{aligned}\\sigma _S^{\\textrm {NLO}}[1,1]/\\textrm {pb} &= -4.79097\\times 10^9 x^2-\\frac{454.08}{x^2}-\\frac{48.2912 \\log x}{x^2}+1.4105\\times 10^9x\\\\&-\\frac{1.22684\\times 10^6}{x}+1.79376\\times 10^7 \\log ^2x+1.11478\\times 10^9 x \\log x\\\\&+1.59613\\times 10^8 \\log x-\\frac{168047 \\log x}{x}+4.39955\\times 10^8\\,,\\end{aligned}$ ${\\begin{@align}{1}{-1}\\nonumber \\sigma _S^{\\textrm {NLO}}[1,0]/\\textrm {pb} &= 2.71176\\times 10^9 x^2+\\frac{302.023}{x^2}+\\frac{32.2322 \\log x}{x^2}-8.07812\\times 10^8 x\\\\\\nonumber &+\\frac{787193}{x}-1.11305\\times 10^7 \\log ^2x-6.62451\\times 10^8 x \\log x\\\\&-9.8134\\times 10^7 \\log x+\\frac{108365 \\log x}{x}-2.68924\\times 10^8\\,,\\end{@align}}$ $\\begin{aligned}\\sigma _S^{\\textrm {NLO}}[0,1]/\\textrm {pb} &= 910970 -3.05552\\times 10^7 x^2+7.62007\\times 10^6 x-\\frac{544.886}{x}\\\\&+26407.1 \\log ^2x+3.86595\\times 10^6 x \\log x+290640 \\log x\\\\&-\\frac{58.3182 \\log x}{x}\\,.\\end{aligned}$ In the range $m_S\\in [150\\textrm { GeV},350\\textrm { GeV}]$ , we find ${\\begin{@align}{1}{-1}\\nonumber \\sigma _S^{\\textrm {NLO}}[1,1]/\\textrm {pb} &=5.78391\\times 10^{11} x^3-6.04905\\times 10^{12} x^3 \\log x-3.97065\\times 10^{12}x^2\\\\\\nonumber &-1.60239\\times 10^{12} x^2 \\log x-2.28312\\times 10^{11} x-\\frac{7.35167\\times 10^6}{x}\\\\&+1.05353\\times 10^8 \\log ^2x-4.9669\\times 10^{10} x \\log x\\\\\\nonumber &+6.11874\\times 10^8 \\log x-\\frac{865250 \\log x}{x}+5.46682\\times 10^8\\,,\\end{@align}}$ ${\\begin{@align}{1}{-1}\\nonumber \\sigma _S^{\\textrm {NLO}}[1,0]/\\textrm {pb} &=-1.18403\\times 10^{10} x^3+8.55496\\times 10^{10} x^3 \\log x+5.7597\\times 10^{10}x^2\\\\\\nonumber &+2.3659\\times 10^{10} x^2 \\log x+3.48892\\times 10^9x+\\frac{121488.",
"}{x}\\\\&-1.66608\\times 10^6 \\log ^2x+7.67582\\times 10^8 x \\log x\\\\\\nonumber &-9.42982\\times 10^6 \\log x+\\frac{14369.3 \\log x}{x}-7.62306\\times 10^6\\,,\\end{@align}}$ ${\\begin{@align}{1}{-1}\\nonumber \\sigma _S^{\\textrm {NLO}}[0,1]/\\textrm {pb} &=3.58507\\times 10^{11} x^3-3.67114\\times 10^{12} x^3 \\log x-2.41271\\times 10^{12}x^2\\\\\\nonumber &-9.74512\\times 10^{11} x^2 \\log x-1.3909\\times 10^{11} x-\\frac{4.49756\\times 10^6}{x}\\\\&+6.4297\\times 10^7 \\log ^2x-3.02763\\times 10^{10} x \\log x\\\\\\nonumber &+3.72922\\times 10^8 \\log x-\\frac{529486 \\log x}{x}+3.31551\\times 10^8\\,.\\end{@align}}$ In the range $m_S\\in [350\\textrm { GeV},500\\textrm { GeV}]$ , we find ${\\begin{@align}{1}{-1}\\nonumber \\sigma _S^{\\textrm {NLO}}[1,1]/\\textrm {pb} &=-1.55048\\times 10^{10} x^2+\\frac{20602.3}{x^2}+4.12759\\times 10^{10} x^2 \\log x\\\\\\nonumber &+3.27683\\times 10^{10} x+\\frac{3.90175\\times 10^6}{x}-1.00487\\times 10^8 \\log ^2x\\\\&-\\frac{1.21312\\times 10^6 \\log ^2x}{x}+1.28157\\times 10^{10} x \\log x\\\\\\nonumber &-1.37351\\times 10^8 \\log x-\\frac{6.26022\\times 10^6 \\log x}{x}+8.75837\\times 10^8\\,,\\end{@align}}$ ${\\begin{@align}{1}{-1}\\nonumber \\sigma _S^{\\textrm {NLO}}[1,0]/\\textrm {pb} &=-1.74667\\times 10^9 x^2+\\frac{2345.03}{x^2}+4.57918\\times 10^9 x^2 \\log x\\\\&+3.64833\\times 10^9 x+\\frac{450670}{x}-1.12934\\times 10^7 \\log ^2x\\\\\\nonumber &-\\frac{137023 \\log ^2x}{x}+1.43013\\times 10^9 x \\log x-1.53273\\times 10^7 \\log x\\\\\\nonumber &-\\frac{704422 \\log x}{x}+9.83324\\times 10^7\\,,\\end{@align}}$ ${\\begin{@align}{1}{-1}\\nonumber \\sigma _S^{\\textrm {NLO}}[0,1]/\\textrm {pb} &=1.59986\\times 10^9 x^2-\\frac{2171.87}{x^2}-4.2072\\times 10^9 x^2 \\log x\\\\&-3.34991\\times 10^9 x-\\frac{417719}{x}+1.03789\\times 10^7 \\log ^2x\\\\\\nonumber &+\\frac{126226 \\log ^2x}{x}-1.3131\\times 10^9 x \\log x+1.40646\\times 10^7 \\log x\\\\\\nonumber &+\\frac{647893 \\log x}{x}-9.03229\\times 10^7\\,.\\end{@align}}$ In the range $m_S\\in [500\\textrm { GeV},1000\\textrm { GeV}]$ , we find $\\begin{aligned}\\sigma _S^{\\textrm {NLO}}[1,1]/\\textrm {pb} &=1.0459\\times 10^8 x^2+\\frac{478.474}{x^2}-7.72699\\times 10^7 x^2 \\log x\\\\&-8.14486\\times 10^7 x-\\frac{2.50557\\times 10^6}{x}-95661.5 \\log ^2x\\\\&-\\frac{71863.4 \\log ^2x}{x}-7.67177\\times 10^7 x \\log x\\\\&-1.27372\\times 10^7 \\log x-\\frac{802665 \\log x}{x}-3.08306\\times 10^7\\,,\\end{aligned}$ $\\begin{aligned}\\sigma _S^{\\textrm {NLO}}[1,0]/\\textrm {pb} &=1.76181\\times 10^7 x^2+\\frac{81.1265}{x^2}-1.3039\\times 10^7 x^2 \\log x\\\\&-1.37328\\times 10^7 x-\\frac{421944}{x}-16421.7 \\log ^2x\\\\&-\\frac{12116.3 \\log ^2x}{x}-1.29224\\times 10^7 x \\log x\\\\&-2.14537\\times 10^6 \\log x-\\frac{135235 \\log x}{x}-5.19164\\times 10^6\\,,\\end{aligned}$ $\\begin{aligned}\\sigma _S^{\\textrm {NLO}}[0,1]/\\textrm {pb} &=2.43245\\times 10^7 x^2+\\frac{110.92}{x^2}-1.78774\\times 10^7 x^2 \\log x\\\\&-1.88708\\times 10^7 x-\\frac{585093}{x}-22465.3 \\log ^2x\\\\&-\\frac{16766.8 \\log ^2x}{x}-1.78573\\times 10^7 x \\log x\\\\&-2.97483\\times 10^6 \\log x-\\frac{187382 \\log x}{x}-7.19613\\times 10^6\\,.\\end{aligned}$ In the range $m_S\\in [1000\\textrm { GeV},3000\\textrm { GeV}]$ , we find $\\begin{aligned}\\sigma _S^{\\textrm {NLO}}[1,1]/\\textrm {fb} &= x^{-0.407849 \\log ^3x-0.841896 \\log ^2x+77.9612 x \\log x+14.6265 \\log x+49.7825}\\\\&\\times \\left(1-\\@root 3 \\of {x}\\right)^{-1.74158}\\,,\\end{aligned}$ $\\begin{aligned}\\sigma _S^{\\textrm {NLO}}[1,0]/\\textrm {fb} &= x^{-0.0682707 \\log ^3x-0.812125 \\log ^2x-2.68171 x \\log x-3.91644 \\log x-10.0547}\\\\&\\times \\left(1-\\@root 3 \\of {x}\\right)^{7.95188}\\,,\\end{aligned}$ $\\begin{aligned}\\sigma _S^{\\textrm {NLO}}[0,1]/\\textrm {fb} &= x^{-2.40603 x+0.812624 \\log ^2x+18.7175 x \\log x+6.90869 \\log x+12.7991}\\\\&\\times \\left(1-\\@root 3 \\of {x}\\right)^{8.05859}\\,.\\end{aligned}$"
],
[
"Conclusion",
"In this paper we have presented the most precise predictions for the production of a CP-even scalar produced in gluon fusion at the LHC.",
"We assume that the coupling of the scalar to the gluons can be described in an effective field theory approach.",
"This enables us to provide precision QCD corrections at N$^3$ LO to the production cross section at the LHC.",
"We find that for a scalar with a mass of 750 GeV the N$^3$ LO corrections yield a theoretical uncertainty of $\\sim 2\\%$ when we choose the central renormalization and factorization scales to be $m_S/2$ and follow the prescriptions layed out in ref. [19].",
"If we assume that the effective theory Wilson coefficient coupling of the CP-even scalar and the gluons is generated by a heavy top-partner, we can estimate the validity of the effective description as a function of the heavy fermion mass.",
"We find that in the threshold region, when the mass of the top-partner is about half the mass of the scalar, the corrections can be as large as $\\sim 60\\%$ .",
"However, from direct searches the low mass ranges for top-partners are tightly constrained, so that at least for the discussion of a 750 GeV scalar this configuration is unlikely.",
"For heavier fermion masses the effective theory description performs very well and is already accurate at $\\sim 5\\%$ when then fermion mass is about $~1.5$ times the scalar mass.",
"Additionally, we have considered the case that the new scalar couples directly to the SM top quark.",
"In this case we need to take into account the top corrections to the cross section, which are unfortunately only available through NLO in QCD, and they thus come with uncertainties of up to $\\sim 20\\%$ .",
"We incorporate the top corrections to the scalar production cross section as a function of the Yukawa coupling of the top quark to the scalar, allowing for a model-dependent choice of the parameters.",
"In summary, we have provided the ingredients to endow predictions for the production of a CP-even scalar at the LHC with the most precise QCD corrections available.",
"The only assumptions that have been made in the calculation are that the scalar couples to the gluons through an effective theory like operator.",
"By determining the Wilson coefficient for the scalar coupling to the gluons as well as the Yukawa coupling to the top within a concrete model, it is now possible, using the numbers provided here, to easily incorporate QCD corrections through N$^3$ LO into the cross section predictions for the LHC.",
"While the results presented in this article concern strictly the production of a CP-even scalar, we can use the same techniques to compute the cross-section for resonance production of different CP/spin types which may be phenomenologically relevant (see, for example, ref. [32]).",
"This will be the subject of future works."
],
[
"Acknowledgements",
"CD, EF and TG are grateful to the KITP, Santa Barbara, for the hospitality during the final stages of this work.",
"We are grateful to Riccardo Barbieri and Giuliano Panico for useful dicussions.",
"This research was supported in part by the National Science Foundation under Grant No.",
"NSF PHY11-25915, by the Swiss National Science Foundation (SNF) under contracts 200021-165772 and 200020-162487 and by the European Commission through the ERC grants “pertQCD”, “HEPGAME” (320651), “HICCUP”, “MathAm” and “MC@NNLO” (340983)."
],
[
"Reference tables for the production cross section of a CP-even scalar through top-quark loops",
"Here we report the values of the N$^3$ LO production cross section $\\sigma _H(m_S, \\Gamma _S=0, m_t)$ for a scalar of mass $m_S \\in [10\\textrm { GeV},3\\textrm { TeV}]$ at the 13 TeV LHC, in an effective theory where the top quark has been integrated out.",
"The $\\overline{\\textrm {MS}}$ -mass of the top quark is chosen to be 162.7 GeV.",
"We also show the two components that enter this result (cf.",
"eq.",
"(REF ), with $\\sigma _S \\rightarrow \\sigma _H$ , $\\Lambda _{\\rm UV} \\rightarrow m_t$ , $C_S(\\mu , \\Lambda _{\\rm UV}) \\rightarrow C(\\mu _0, m_t) $ ), i.e.",
"the squared SM Wilson coefficient $\\left| C(\\mu _0 = m_S/2, m_t)\\right|^2$ and the matrix-element $\\eta $ .",
"The theory error is computed from the variation of the common renormalization and factorization scale $\\mu $ in the range $\\left[{m_S}/{4}, m_S \\right]$ .",
"To it we add linearly the uncertainty from missing N$^3$ LO PDFs, evaluated following the method of sec.",
", and from the truncation of the threshold expansion.",
"We refer to ref.",
"[19] for an explanation of how the latter is computed.",
"We use the PDF set PDF4LHC14 [27] and derive the combined PDF+$\\alpha _s$ error following the indications of the PDF4LHC working group.",
"Table: Production cross section for a scalar particle with a mass in the range10 GeV to 150 GeV.Table: Production cross section for a scalar particle with a mass in the range 150 GeV to 500 GeV.Table: Production cross section for a scalar particle with a mass in the range 500 GeV to 1700 GeV.Table: Production cross section for a scalar particle with a mass in the range 1750 GeV to 3000 GeV.Table: Production cross section for a scalar particle with a mass in the range 730 GeV to 770 GeV.Table: NLO contributions to the production cross section for ascalar particle, as they are defined in eq. ().",
"The theory uncertaintyis computed as in eq.",
"()Table: NLO contributions to the production cross sectionfor a scalar particle, as they are defined in eq. ().",
"The theoryuncertainty is computed as in eq.",
"().Table: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. ().",
"The theory uncertainty is computed as in eq.",
"().Table: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. ().",
"The theory uncertainty is computed as in eq.",
"().Table: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. ().",
"The theory uncertainty is computed as in eq.",
"().Table: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. ().",
"The theory uncertainty is computed as in eq.",
"().Table: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. ().",
"The theory uncertainty is computed as in eq.",
"().Table: NLO contributions to the production cross section for a scalar particle with a mass around 750 GeV, as they are defined in eq. ().",
"The theory uncertainty is computed as in eq.",
"()."
]
] | 1605.05761 |
[
[
"Studies of the Lennard-Jones fluid in 2, 3, and 4 dimensions highlight\n the need for a liquid-state 1/d expansion"
],
[
"Abstract The recent theoretical prediction by Maimbourg and Kurchan [arXiv:1603.05023] that for regular pair-potential systems the virial potential-energy correlation coefficient increases towards unity as the dimension $d$ goes to infinity is investigated for the standard 12-6 Lennard-Jones fluid.",
"This is done by computer simulations for $d=2,3,4$ going from the critical point along the critical isotherm/isochore to higher density/temperature.",
"In all cases the virial potential-energy correlation coefficient increases significantly.",
"For a given density and temperature relative to the critical point, with increasing number of dimension the Lennard-Jones system conforms better to the hidden-scale-invariance property characterized by high virial potential-energy correlations (a property that leads to the existence of isomorphs in the thermodynamic phase diagram, implying that it becomes effectively one-dimensional in regard to structure and dynamics).",
"The present paper also gives the first numerical demonstration of isomorph invariance of structure and dynamics in four dimensions.",
"Our findings emphasize the need for a universally applicable $1/d$ expansion in liquid-state theory; we conjecture that the systems known to obey hidden scale invariance in three dimensions are those for which the yet-to-be-developed $1/d$ expansion converges rapidly."
],
[
"Studies of the Lennard-Jones fluid in 2, 3, and 4 dimensions highlight the need for a liquid-state $1/d$ expansion Lorenzo Costigliola, Thomas B. Schrøder, and Jeppe C. Dyre \"Glass and Time\", IMFUFA, Dept.",
"of Science and Environment, Roskilde University, P. O.",
"Box 260, DK-4000 Roskilde, Denmark The recent theoretical prediction by Maimbourg and Kurchan [arXiv:1603.05023] that for regular pair-potential systems the virial potential-energy correlation coefficient increases towards unity as the dimension $d$ goes to infinity is investigated for the standard 12-6 Lennard-Jones fluid.",
"This is done by computer simulations for $d=2,3,4$ going from the critical point along the critical isotherm/isochore to higher density/temperature.",
"In all cases the virial potential-energy correlation coefficient increases significantly.",
"For a given density and temperature relative to the critical point, with increasing number of dimension the Lennard-Jones system conforms better to the hidden-scale-invariance property characterized by high virial potential-energy correlations (a property that leads to the existence of isomorphs in the thermodynamic phase diagram, implying that it becomes effectively one-dimensional in regard to structure and dynamics).",
"The present paper also gives the first numerical demonstration of isomorph invariance of structure and dynamics in four dimensions.",
"Our findings emphasize the need for a universally applicable $1/d$ expansion in liquid-state theory; we conjecture that the systems known to obey hidden scale invariance in three dimensions are those for which the yet-to-be-developed $1/d$ expansion converges rapidly.",
"Recent years have brought notable progress in the understanding of the liquid state coming from studies of the high-dimensional limit.",
"With roots back in time [1], [2], [3], [4], [5] and in a continuation of recent progress [6], [7], [8], [9], [10], Charbonneau and collaborators in 2014 in a tour de force replica symmetry breaking calculation solved the glass problem in high dimensions for the prototypical hard-sphere (HS) model [11].",
"This was followed by a proof by Maimbourg, Kurchan, and Zamponi that the dynamics satisfies a universal equation in high dimensions for the general case of a system of particles interacting via pairwise additive forces [12].",
"This is how a “simple” liquid is traditionally defined [13], [14], [15], [16], [17], although during the last 20 years it has gradually become clear that some such systems – like the Gaussian core model, the Lennard-Jones Gaussian model, and the Jagla model – exhibit quite complex behavior (see, e.g., Ref.",
"ing12 and its references).",
"Very recently, Maimbourg and Kurchan showed that in the condensed phase, i.e., for states dominated by hard repulsions, any well-behaved pair-potential system has strong virial potential-energy correlations in sufficiently high dimensions [19], [20].",
"Specifically, it was shown that the Pearson correlation coefficient $R$ of the constant-volume canonical-ensemble equilibrium fluctuations of virial $W$ and potential energy $U$ , $R=\\frac{\\langle \\Delta W\\Delta U\\rangle }{\\sqrt{\\langle (\\Delta W)^2\\rangle \\langle (\\Delta U)^2\\rangle }}\\,,$ converges to unity as the number of dimensions $d$ goes to infinity.",
"The analysis presented in Ref.",
"mai16 also showed that the ${\\rm EXP}$ pair potential (a simple exponential decay in space) plays the role as a building block of all pair potentials [21], [22].",
"Note that, in contrast to the inverse-power-law pair potentials $\\propto r^{-n}$ ($r$ being the pair distance), due to its rapid spatial decay the ${\\rm EXP}$ pair potential has a thermodynamic limit in all dimensions.",
"Systems with $R$ close to unity are characterized by “hidden scale invariance”, an approximate symmetry that has been studied in several publications since its introduction in 2008; there are now also experimental verifications of the concept for van der Waals liquids [23], [24], [25], [26].",
"Systems with hidden scale invariance are simple because they have so-called isomorphs in the thermodynamic phase diagram, which are lines along which structure and dynamics in suitably reduced units are invariant to a good approximation.",
"The isomorph theory has been applied to atomic and molecular liquid and crystalline models in thermal equilibrium, as well as to non-equilibrium phenomena like shear flows of liquids and glasses (see, e.g., Ref.",
"dyr14 and its references).",
"Recently, it was shown from state-of-the-art DFT ab initio simulations of 58 liquid elements at their triple points that most metals possess hidden scale invariance [28].",
"An overview of the isomorph theory was given in Ref.",
"dyr14 from 2014.",
"After that paper was written, it became clear that Roskilde (R) simple systems [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42] – those with $R>0.9$ – are characterized by approximately obeying the following condition [43]: $U(R_a)=U(R_b)\\Leftrightarrow U(\\lambda R_a)= U(\\lambda R_b)$ in which $R$ specifies all particle positions and $U(R)$ is the potential-energy function.",
"Thus hidden scale invariance is equivalent to an approximate conformal invariance property.",
"The non-trivial finding of the above-mentioned works is that many realistic model systems – as well as many real-world liquids and solids – obey hidden scale invariance.",
"It appears that most metals and van der Waals bonded liquids and solids exhibit hidden scale invariance, whereas systems with strong directional bonds like covalently or hydrogen-bonded systems do not and are generally more complex [27].",
"This paper presents computer simulations of the standard 12-6 Lennard-Jones (LJ) system in two, three, and four dimensions consisting of particles interacting via the pair potential $v_{\\rm LJ}(r)\\,=\\,4\\varepsilon \\left[ \\left(\\frac{r}{\\sigma }\\right)^{-12} - \\left(\\frac{r}{\\sigma }\\right)^{-6} \\right] \\,.$ Here $\\varepsilon $ and $\\sigma $ define the characteristic energy and length scales of the pair potential.",
"The LJ system does not have a proper thermodynamic limit in more than five dimensions, and one may argue what is the correct generalization of this system to arbitrary dimension $d$ (for instance, $v(r)\\propto (r/\\sigma )^{-(d+9)}-(r/\\sigma )^{-(d+3)}$ or $v(r)\\propto (r/\\sigma )^{-4d}-(r/\\sigma )^{-2d}$ or a third option).",
"We avoided this problem by staying at low dimensions.",
"It is not obvious how to compare results for different thermodynamic state points in different dimensions.",
"In high dimensions one may compare different state points by scaling the density such that the HS packing fraction remains invariant [44].",
"In our case of relatively low dimensions this is too crude; in any case, we also need a scaling of the temperature in order to be able to compare results obtained in different dimensions.",
"The critical point of the LJ system is known for $d=2,3,4$ [45], [46], [47], and we used this as reference state point.",
"This choice has the further advantage that in three dimensions the virial potential-energy correlations are weak in the vicinity of the critical point, which allows one to monitor how $R$ increases when the condensed “strongly correlating” liquid phase is approached upon increasing density or temperature.",
"Molecular dynamics simulations have been performed before in four spatial dimensions [48], [49], [50].",
"The simulations reported below used a homemade code applicable in arbitrary dimensions [51].",
"The code implements $NVT$ dynamics with periodic boundary conditions [52] based on the leap-frog algorithm coupled with a Nose-Hoover thermostat.",
"A shifted-forces cutoff at $2.5 \\sigma $ was used in all simulations [53].",
"The time step $t_s$ varied with state point such that the reduced time step, $\\tilde{t}_s\\equiv t_s \\rho ^{1/d} \\sqrt{k_B T /m}$ , was 0.001 (here $\\rho \\equiv N/V$ is the particle density and $m$ the particle mass).",
"After melting and equilibrating from a simple cubic configuration, the LJ system was simulated at every liquid state point for $2\\cdot 10^7$ time steps.",
"In two dimensions the system crystallized at the three highest-density state points; simulations at these state points were performed with a reduced time step of $\\tilde{t}_s=0.0005$ and the number of time steps was doubled.",
"In all cases the thermostat relaxation time was 80 time steps.",
"The system size was $N=1225$ in two, $N=1728$ in three, and $N=2401$ in four dimensions.",
"Figure: Radial distribution function of the Lennard-Jones (LJ) fluid along the critical isotherm in two, three, and four dimensions (black, red, and green colors, respectively; r ˜≡ρ 1/d r\\tilde{r}\\equiv \\rho ^{1/d}r where ρ\\rho is the particle density and rr the interparticle distance).",
"(a) shows results at 1.41.4 times the critical density ρ c \\rho _c, (b) shows results at twice the critical density.",
"In both cases the fluid's long-range structure becomes markedly less pronounced as the number of dimensions increases.Our focus is on what happens in the fluid region of phase space in which the correlation coefficient $R$ of Eq.",
"(REF ) is far from unity in 3d.",
"This number is close to unity in the “ordinary” 3d condensed liquid phase not too far from the melting line, as well as in the entire crystalline phase [54], [55], [56], but approaching the gas phase and, in particular, the critical point in 3d, $R$ drops quickly and the system is no more R simple [55].",
"Figure REF reports the reduced-unit radial distribution function $g(r)$ at the critical temperature $T_c$ at $1.4$ and 2 times the critical density; the black symbols mark $g(r)$ in two dimensions, the red curves in three dimensions, and the green curves in four dimensions.",
"Figure REF nicely confirms the argument of Maimbourg and Kurchan that in higher dimensions the nearest-neighbor distance increasingly dominates the physics [19].",
"Thus beyond the first coordination shell $g(r)$ converges quickly to unity in high dimensions.",
"In the words of Ref.",
"mai16, what happens in high dimensions is that a single pair distance dominates the physics because “particles that are too close are exponentially few in numbers, while those that are too far interact exponentially weakly”.",
"This argument presupposes, of course, that the pair potential in question has been generalized to any number of dimensions in a way ensuring a proper thermodynamic limit, i.e., such that it decays more rapidly than $r^{-d}$ at long distances.",
"A system for which a single pair distance dominates the physics even in three dimensions is the hard-sphere (HS) system for which the radial distribution function at contact determines the equation of state [57], [58].",
"The above suggests that one may regard the $3d$ HS system as a poor man's version of the $d\\rightarrow \\infty $ limit; indeed, it has been known for some time that the pair correlations of the HS system becomes increasingly trivial as $d$ increases [59], [44], [60].",
"We note, however, that the HS system is not the only possibility of a $3d$ poor man's $d\\rightarrow \\infty $ limit; alternatives are the Gaussian core model [61] or the Mari-Kurchan model [62].",
"Figure: The LJ fluid's virial potential-energy correlation coefficient in 2, 3, and 4 dimensions.",
"(a) shows a sketch of the temperature-density phase diagram in which both variables are normalized to their values at the critical point, T c T_c and ρ c \\rho _c , , .",
"The black symbols and full curves represent the phase limits of the LJ system in 3d (see, e.g., Refs.",
"hey15a,cos16 and their references).",
"The orange crosses mark the state points simulated in 2d, 3d, and 4d, whose virial potential-energy correlation coefficients are reported in (b) and (c), the green symbols indicate the three isomorphic state points simulated in 4d (Fig.",
").",
"(b) The virial potential-energy correlation coefficient RR (Eq.",
"()) along the critical isochore.",
"There is generally a convergence to the hidden-scale-invariance property characterizing R simple systems defined by R>0.9R>0.9 (dashed horizontal line), a situation that is reached much earlier in four than in three dimensions, where it is reached much earlier than in two dimensions.",
"In two dimensions the system developed “holes” close to the critical point (see the main text), which is indicated by the two open symbols.",
"(c) The virial potential-energy correlation coefficient along the critical isotherm.",
"There is convergence to the hidden-scale-invariance scenario characterizing R simple systems, a situation that is reached much earlier in four than in three dimensions, where it is reached much earlier than in two dimensions.",
"The two-dimensional system crystallized at the highest densities (ρ/ρ c >2.5\\rho /\\rho _c>2.5), which is indicated by black square symbols; as in (b) the four open symbols at lower densities indicate that the sample developed “holes” close to the critical point.In order to systematically compare what happens in different dimensions we studied the variation of the virial potential-energy correlation coefficient $R$ of Eq.",
"(REF ) as one moves away from the critical point along the critical isochore and isotherm, respectively.",
"In units of $\\varepsilon /k_B$ for temperature and $1/\\sigma ^d$ for density the critical point is given by $(\\rho ,T)=(0.355,0.515)$ in two dimensions [45], by $(\\rho ,T)=(0.316,1.312)$ in three dimensions [63], and by $(\\rho ,T)=(0.34,3.404)$ in four dimensions [47] (the 2d and 3d critical point data were calculated by Monte Carlo (MC) simulations with the LJ potential truncated at the half box length, the 4d critical point was determined by MC simulations with the potential truncated at $2.5\\sigma $ ).",
"Figure REF (a) gives an overview of the density-temperature thermodynamic phase diagram in which both variables in the standard van der Waals way have been normalized to unity at the critical point.",
"The full black curves indicate the freezing and melting lines for the 3d case, and the orange crosses mark the state points simulated.",
"The results for $R$ are shown in Fig.",
"REF (b) for the critical isochore and in Fig.",
"REF (c) for the critical isotherm (in both figures the horizontal dashed lines mark the (a bit arbitrary) threshold $R=0.9$ defining R simple systems [54]).",
"In two dimensions the system developed visible “holes” close to the critical point deriving from large density fluctuations [66]; the corresponding simulations are marked by open (black) symbols.",
"In all cases, along both the isochore and the isotherm the correlations increase significantly as one moves away from the critical point.",
"Note that in four dimensions $R$ is fairly large already at the critical point.",
"When contemplating these findings one should keep in mind that $R$ is close to unity for the LJ system in three dimensions in the “ordinary” condensed liquid phase not too far from the melting line.",
"Our conclusions based on Fig.",
"REF are: 1) The simulations confirm the prediction of Maimbourg and Kurchan that all systems in their condensed-matter (“hard”) regime have strong correlations in high dimensions.",
"2) There is a striking difference between two, three, and four dimensions, and already in four dimensions the correlations are strong whenever density and temperature are above their critical values.",
"Before proceeding to discuss the implications of these findings for liquid-state theory we take the opportunity to demonstrate the existence of isomorphs in four dimensions.",
"The most general method for mapping out an isomorph in the thermodynamic phase diagram makes use of the fact that isomorphs are configurational adiabats [67], [43] in conjunction with the following standard fluctuation identity [67] (in which ${S}_{\\rm ex}$ is the entropy minus that of an ideal gas at the same density and temperature): $\\left(\\frac{\\partial \\ln T}{\\partial \\ln \\rho }\\right)_{{S}_{\\rm ex}}\\,=\\,\\frac{\\langle \\Delta W\\Delta U\\rangle }{\\langle (\\Delta U)^2\\rangle }\\,.$ We changed density in steps of 1%, 2%, and 5%, respectively, in each step calculating from Eq.",
"(REF ) the temperature change needed to keep ${S}_{\\rm ex}$ constant.",
"An alternative way of generating isomorphic state points, which is limited to LJ-type systems, utilizes the fact that due to invariance of the structure in reduced units, the quantity $h(\\rho )/T$ is isomorph invariant where $h(\\rho )=A\\rho ^{12/d}-B\\rho ^{6/d}$ (the two constants $A$ and $B$ , which are (slightly) isomorph dependent, are determined from simulations at a reference state point specifying the isomorph in question; see Refs.",
"cos16,boh14 for justification and more details of this procedure).",
"Figure REF (a) demonstrates consistency between the two different ways of generating an isomorph in 4d, although for the largest density step (5%) there is a small disagreement.",
"The isomorph invariance of $h(\\rho )/T$ signals a breakdown of the theory at low densities at which the above expression for $h(\\rho )$ becomes negative.",
"This means that along any isomorph the virial potential-energy correlations must eventually weaken at low densities, which is also observed [54], [69].",
"Since $R\\rightarrow 1$ in high dimensions for the state points not too far away from the melting line [19], one may speculate that in the $d\\rightarrow \\infty $ limit there is a phase transition between a phase of increasingly perfect hidden scale invariance and one of poor virial potential-energy correlations [19], [51].",
"Starting from the 4d state point $(\\rho \\sigma ^4,k_BT/\\varepsilon )=(0.9,1.0)$ two isomorphic state points were found.",
"The first one $(\\rho \\sigma ^4,k_BT/\\varepsilon )=(0.945,1.23)$ was identified using Eq.",
"(REF ) as described above, the second one $(\\rho \\sigma ^4,k_BT/\\varepsilon )=(1.5,6.56)$ was determined using the isomorph invariance of $h(\\rho )/T$ .",
"Figure REF (b) shows the pair distribution function as a function of the reduced radius for the three state points.",
"The collapse validates structural invariance along the 4d isomorph.",
"Figure REF (c) shows the mean-square displacement as a function of time in reduced units for the same three state points, demonstrating isomorph invariance also of the dynamics.",
"Figure: Validation of isomorph invariance in four dimensions based on simulations at three state points on the same isomorph.",
"Starting from the reference state point (ρσ 4 ,k B T/ε)=(0.9,1.0)(\\rho \\sigma ^4,k_BT/\\varepsilon )=(0.9,1.0) two isomorphic state points were generated as described in the text.",
"(a) Consistency check of the two different ways of generating isomorphs detailed in the text.",
"The yellow, purple, and green symbols give the results of using Eq.",
"() repeatedly for, respectively, a 1%, 2%, and 5% density increase starting from the reference state point; the blue point was calculated by the h(ρ)h(\\rho ) method described in the text.",
"(b) The pair distribution function at the three isomorphic state points plotted as a function of the reduced pair distance.",
"The collapse demonstrates structural invariance along the isomorph.",
"(c) Reduced mean-square displacement as a function of reduced time for the same three isomorphic state points, demonstrating isomorph invariance of the dynamics (reduced units are defined in Ref.",
"IV).Turning back to the dimensionality dependence of the virial potential-energy correlations, our findings may be summarized as follows.",
"Above the critical point as the number of dimensions increases the LJ system converges rapidly to the state of perfect hidden scale invariance shown by Maimbourg and Kurchan to characterize the high-dimensional limit.",
"Assuming that the challenge of generalizing arbitrary systems to arbitrary dimensions has been addressed, we conjecture the following: 1) All systems (also molecular ones) obey hidden scale invariance in sufficiently high dimensions in their condensed phases; 2) the rate with which this property translates into lower dimensions depends on the system in question.",
"In other words, if one defines the van der Waals scaled density $\\tilde{\\rho }\\equiv \\rho /\\rho _c$ and temperature $\\tilde{T}\\equiv T/T_c$ , we conjecture that $R(\\tilde{\\rho },\\tilde{T})\\rightarrow 1$ as $d\\rightarrow \\infty $ for all systems, at least whenever $\\tilde{\\rho }>1$ and $\\tilde{T}>1$ .",
"The rate of convergence determines whether or not the system is R simple in three dimensions.",
"If the above conjecture is correct, any system at any given condensed-matter state point has a “transition region” of dimensionalities above which it becomes R simple.",
"This range of dimensions is located below three dimensions for systems that are R simple in three dimensions (at the state point in question) and above three for those that are not.",
"An important task for the future will be to construct a systematic $1/d$ expansion taking one from the case of guaranteed R simple behavior as $d\\rightarrow \\infty $ to three dimensions.",
"Hints of how this may be done were given in Ref.",
"cha14 for the HS case, but a more general approach is needed.",
"We find it conceivable that future textbooks in liquid-state theory start by deriving a simple and general theory for the limit of high dimensions and subsequently translates this into three dimensions via a $1/d$ expansion, but clearly much remains to be done before this becomes reality.",
"We are indebted to Thibaud Maimbourg for his comments to an early draft of this paper.",
"This work was supported in part by the Danish National Research Foundation via grant DNRF61."
]
] | 1605.06007 |
[
[
"Heuristics for Planning, Plan Recognition and Parsing"
],
[
"Abstract In a recent paper, we have shown that Plan Recognition over STRIPS can be formulated and solved using Classical Planning heuristics and algorithms.",
"In this work, we show that this formulation subsumes the standard formulation of Plan Recognition over libraries through a compilation of libraries into STRIPS theories.",
"The libraries correspond to AND/OR graphs that may be cyclic and where children of AND nodes may be partially ordered.",
"These libraries include Context-Free Grammars as a special case, where the Plan Recognition problem becomes a parsing with missing tokens problem.",
"Plan Recognition over the standard libraries become Planning problems that can be easily solved by any modern planner, while recognition over more complex libraries, including Context-Free Grammars (CFGs), illustrate limitations of current Planning heuristics and suggest improvements that may be relevant in other Planning problems too."
],
[
"Introduction",
"Plan Recognition is a common task in a number of areas where the goal and plan of an agent must be inferred from observations of its behavior [12], [4], [10].",
"Plan Recognition is a form of Planning in reverse: while in Planning, we seek the actions that achieve a goal, in Plan Recognition, we seek the goals that explain the observed actions.",
"Work in Plan Recognition, however, has proceeded independently of the work in Planning, using mostly handcrafted libraries or algorithms not related to Planning [8], [14], [3], [9], [7], [1].",
"Recently, we have shown that Plan Recognition can be formulated and solved using Classical Planning algorithms [11].",
"This is important since Classical Planning algorithms have become quite powerful in recent years.",
"This formulation does not work over libraries but over Strips theories where a set ${\\cal G}$ of possible goals is given.",
"The Plan Recognition task is defined as the problem of identifying the goals $G \\in {\\cal G}$ that have some optimal plan compatible with the observations $O$ .",
"Such goals are grouped into the optimal goal set ${\\cal G}^*$ , ${\\cal G}^* \\subseteq {\\cal G}$ .",
"The reason for focusing on the optimal plans is that they represent the possible behaviors of a perfectly rational agent pursuing the goal $G$ [2].",
"By suitable transformation, it is then shown in [11] that this optimal set ${\\cal G}^*$ can be computed exactly by means of optimal Planning algorithms and approximately by efficient suboptimal Planning algorithms and polynomial heuristics.",
"In this work, we show that this formulation subsumes the standard formulation of Plan Recognition over libraries through a compilation of libraries into Strips.",
"The libraries correspond AND/OR graphs that may be cyclic and where children of AND nodes may be partially ordered.",
"This libraries include Context-Free Grammars as a special case, where the Plan Recognition problem becomes a parsing problem.",
"Plan Recognition over the standard Plan Libraries become simple Planning problems that can be easily solved by any modern planner, while recognition over more complex libraries, including CFGs, illustrate limitations of current Planning heuristics and improvements that may be relevant in other Planning problems as well.Parsing in CFGs is polynomial while Planning is known to be NP–hard.",
"This worst complexity bounds, however, do not imply that the reduction of parsing to Planning is necessarily a bad idea.",
"First, many Planning problems – like manySAT problems – can be solved quite efficiently; second, parsing with constraints, as required in Natural Language Processing, is also intractable, yet many of these constraints can be handled naturally in Planning.",
"In addition, the mapping handles missing tokens in the input sentence and yields interesting lessons for Planning heuristics.",
"The paper is organized as follows.",
"First we review the formulation of Plan Recognition over Strips theories in [11], then we consider Plan Recognition over libraries, present some experimental results, and draw some conclusions."
],
[
"Plan Recognition as Planning",
"A Strips Planning problem is a tuple $P= \\langle F,I,A,G \\rangle $ where $F$ is the set of fluents, $I \\subseteq F$ and $G \\subseteq F$ are the initial and goal situations, and $A$ is a set of actions $a$ with precondition, add, and delete lists $Pre(a)$ , $Add(a)$ , and $Del(a)$ respectively, all of which are subsets of $F$ .",
"For each action $a \\in A$ , we assume that there is a non-negative cost $c(a)$ so that the cost of a sequential plan $\\pi = a_1, \\ldots , a_n$ is $c(\\pi ) = \\sum c(a_i)$ .",
"A plan $\\pi $ is optimal if it has minimum cost.",
"For unit costs, i.e., $c(a)=1$ for all $a \\in A$ , plan cost is plan length, and the optimal plans are the shortest ones.",
"Unless stated otherwise, action costs are assumed to be 1."
],
[
"Definition",
"The Plan Recognition problem given a plan library $L$ for a set $\\cal G$ of possible goals $G$ can be understood, at an abstract level, as the problem of finding a goal $G$ with a plan $\\pi $ in the library, written $\\pi \\in \\Pi _L(G)$ , such that $\\pi $ satisfies the observations.",
"We define the Plan Recognition problem over a domain theory in a similar way just changing the set $\\Pi _L(G)$ of plans for $G$ in the library by the set $\\Pi ^*_P(G)$ of optimal plans for $G$ given the domain $P$ .",
"We use $P =\\langle F,I,O\\rangle $ to represent Planning domains so that a Planning problem $P(G)$ is obtained by concatenating a Planning domain with a goal $G$ , which is a set of fluents.",
"We define a Plan Recognition problem or theory as follows: Definition 1 A Plan Recognition problem or theory is a triplet $T = \\langle P, {\\cal G}, O \\rangle $ where $P= \\langle F,I,A\\rangle $ is a Planning domain, $\\cal G$ is the set of possible goals $G$ , $G \\subseteq F$ , and $O = o_1, \\ldots , o_m$ is an observation sequence with each $o_i$ being an action in $A$ .",
"We also need to make precise what it means for an action sequence to satisfy an observation sequence made up of actions.",
"E.g., the action sequence $\\pi =\\lbrace a,b,c,d,e,a\\rbrace $ satisfies the observation sequences $O_1 = \\lbrace b,d,a\\rbrace $ and $O_2=\\lbrace a,c,a\\rbrace $ , but not $O_3 = \\lbrace b,d,c\\rbrace $ .",
"This can be formalized with the help of a function that maps observation indices in $O$ into action indices in $A$ : Definition 2 An action sequence $\\pi = a_1, \\ldots , a_n$ satisfies the observation sequence $O = o_1, \\ldots , o_m$ if there is a monotonic function $f$ mapping the observation indices $j=1, \\ldots , m$ into action indices $i=1, \\ldots , n$ , such that $a_{f(j)}=o_j$ .",
"The solution to a Plan Recognition theory $T=\\langle P,{\\cal G}, O\\rangle $ is given by the goals $G$ that admit an optimal plan that is compatible with the observations: Definition 3 The exact solution to a theory $T=\\langle P, {\\cal G}, O\\rangle $ is given by the optimal goal set ${\\cal G}_T^*$ which comprises the goals $G \\in {\\cal G}$ such that for some $\\pi \\in \\Pi _P^*(G)$ , $\\pi $ satisfies $O$ .",
"Figure: Plan Recognition: Is the agent headed to CC, II, or KK?",
"The observations are the transitionsfrom AA to BB and FF to GG in that order.Figure REF shows a simple Plan Recognition problem.",
"Room A (marked with a circle) is the initial position of the agent, while Rooms C, I and K (marked with a square) are its possible destinations.",
"Arrows between Rooms A and B, and F and G, are the observed agent movements in that order.",
"In the resulting theory $T$ , the only possible goals that have optimal plans compatible with the observation sequence are I and K. In the terminology above, the set of possible goals ${\\cal G}$ is given by the atoms $at(C)$ , $at(I)$ , and $at(K)$ , while the optimal goal set ${\\cal G}_T^*$ comprises $at(I)$ and $at(K)$ , leaving out the possible goal $at(C)$ ."
],
[
"Computation",
"In order to solve the Plan Recognition problem using Planning algorithms, we get rid of the observations.",
"For simplicity, we assume that no pair of observations $o_i$ and $o_j$ refer to the same action $a$ in $P$ .",
"When this is not so, we create a copy $a^{\\prime }$ of the action $a$ in $P$ so that $o_i$ refers to $a^{\\prime }$ and $o_j$ refers to $a$ .",
"We will eliminate observations by mapping the theory $T=\\langle P,{\\cal G}, O\\rangle $ into an slightly different theory $T^{\\prime }=\\langle P^{\\prime },{\\cal G}^{\\prime }, O^{\\prime }\\rangle $ with an empty set $O^{\\prime }$ of observations, such that the solution set ${\\cal G}^*_T$ for $T$ can be read off from the solution set ${\\cal G}^*_{T^{\\prime }}$ for $T^{\\prime }$ .",
"Definition 4 For a theory $T=\\langle P,{\\cal G}, O\\rangle $ , the transformed theory is $T^{\\prime }=\\langle P^{\\prime },{\\cal G}^{\\prime }, O^{\\prime }\\rangle $ with $P^{\\prime }= \\langle F^{\\prime }, I^{\\prime }, A^{\\prime }\\rangle $ has fluents $F^{\\prime } = F \\cup F_o$ , initial situation $I^{\\prime } = I$ , and actions $A^{\\prime } = A \\cup A_o$ , where $P= \\langle F, I, A\\rangle $ , $F_0 = \\lbrace p_a \\ | \\ a \\in O\\rbrace $ , and $A_o = \\lbrace o_a \\ | \\ a \\in O\\rbrace $ , ${\\cal G}^{\\prime }$ contains the goal $G^{\\prime } = G \\cup G_o$ for each goal $G$ in $\\cal G$ , where $G_o = F_o$ , $O^{\\prime }$ is empty The new actions $o_a$ in $P^{\\prime }$ have the same precondition, add, and delete lists as the actions $a$ in $P$ except for the new fluent $p_a$ that is added to $Add(o_a)$ , and the fluent $p_b$ , for the action $b$ that immediately precedes $a$ in $O$ , if any, that is added to $Pre(o_a)$ .",
"In the transformed theory $T^{\\prime }$ , the observations $a \\in O$ are encoded as extra fluents $p_a \\in F_o$ , extra actions $o_a \\in A_o$ , and extra goals $p_a \\in G_o$ .",
"Moreover, these extra goals $p_a$ can only be achieved by the new actions $o_a$ , that due to the precondition $p_b$ for the action $b$ that precedes $a$ in $O$ , can be applied only after all the actions preceding $a$ in $O$ , have been executed.",
"The result is that the plans that achieve the goal $G^{\\prime } = G \\cup G_o$ in $P^{\\prime }$ are in correspondence with the plans that achieve the goal $G$ in $P$ that satisfy the observations $O$ : Proposition 5 $\\pi = a_1, \\ldots , a_n$ is a plan for $G$ in $P$ that satisfies the observations $O=o_1, \\ldots ,o_m$ under the function $f$ iff $\\pi ^{\\prime }=b_1, \\ldots , b_n$ is a plan for $G^{\\prime }$ in $P^{\\prime }$ with $b_i=o_{a_i}$ , if $i=f(j)$ for some $j \\in [1,m]$ , and $b_i=a_i$ otherwise.",
"It follows from this that $\\pi $ is an optimal plan for $G$ in $P$ that satisfies the observations iff $\\pi ^{\\prime }$ is an optimal plan in $P^{\\prime }$ for two different goals: $G$ , on the one hand, and $G^{\\prime }=G \\cup G_o$ on the other.",
"If we let $\\Pi _P^*(G)$ stand for the set of optimal plans for $G$ in $P$ , we can thus test whether a goal $G$ in $\\cal G$ accounts for the observation as follows:Note that while a plan for $G^{\\prime } = G \\cup G_o$ is always a plan for $G$ , it is not true that an optimal plan for $G^{\\prime }$ is an optimal plan for $G$ , or even that a good plan for $G^{\\prime }$ is a good plan for $G$ .",
"Theorem 6 $G \\in {\\cal G}_T^*$ iff there is an action sequence $\\pi $ in $\\Pi _{P^{\\prime }}^*(G) \\cap \\Pi ^*_{P^{\\prime }}(G^{\\prime })$ .",
"Moreover, since $G \\subseteq G^{\\prime }$ , if we let $c^*_{P^{\\prime }}(G)$ stand for the optimal cost of achieving $G$ in $P^{\\prime }$ , we can state this result in a simpler form: Theorem 7 $G \\in {\\cal G}_T^*$ iff $c^*_{P^{\\prime }}(G) = c^*_{P^{\\prime }}(G^{\\prime })$ The optimal goal set ${\\cal G}^*$ can be computed, using this result, by solving two optimal Planning problems for each possible goal $G$ : one extending the domain $P^{\\prime }$ with the goal $G$ , the other extending $P^{\\prime }$ with the goal $G^{\\prime }$ made up of $G$ and the dummy goals $G_o$ encoding the observations.",
"The goal $G$ explains the observations and thus belongs to ${\\cal G}^*_T$ iff the solutions to these two optimal Planning problems have the same cost.",
"In [11], a more efficient method for computing this set exactly is introduced, where the cost of the first problem is used as the upper bound in the solution of the second.",
"In addition, two methods that approximate ${\\cal G}_T^*$ and scale up much better are presented.",
"For Plan Recognition over libraries, the situation is simpler, as the resulting Planning problems have zero action costs, and hence all plans are optimal."
],
[
"Plan Recognition over Libraries",
"As mentioned above, the Plan Recognition problem given a plan library $L$ for a set $\\cal G$ of possible goals $G$ can be understood, at an abstract level, as the problem of finding a goal $G$ with a plan $\\pi \\in L$ , written $\\pi \\in \\Pi _L(G)$ , such that $\\pi $ satisfies the observations $O$ .",
"We show now that a library $L$ for a goal $G$ can be compiled intro a Strips Planning problem $P_L(G)$ so that $\\pi $ is in $\\Pi _L(G)$ iff $\\pi $ is a plan for $P_L(G)$ .",
"Provided with this correspondence and by setting the cost of all the actions in $P_L$ to zero, so that no plan in the library is ruled out due to their cost, a plan in the library for $G$ will satisfy the observations $O$ iff $G$ is in the optimal goal set ${\\cal G}_{T}^*$ of the theory $T = \\langle P_L, {\\cal G}, O\\rangle $ , a set that can be computed by using an off-the-shelf classical planner upon the Planning problems $P^{\\prime }_L(G^{\\prime })$ obtained from the transformation that compiles the observation $O$ in $T$ away."
],
[
"Plan Libraries",
"As it is standard, we take a library $L$ for a goal $G$ to be a rooted, ordered AND/OR graph where each node is a AND node, an OR node, or a leaf.",
"Leaves represent primitive task (actions), OR nodes represent non-primitive tasks, and AND nodes represent methods for decomposing non-primitive task.",
"The children of OR nodes are AND nodes or leaves, while the children of AND nodes are OR nodes or leaves.",
"The children of an AND node $n$ can be ordered partially; we write $n^{\\prime } <_n n^{\\prime \\prime }$ to express that child $n^{\\prime }$ of $n$ must come before child $n^{\\prime \\prime }$ .",
"The root of the library is a task (OR node) that represents the goal $G$ to be achieved.",
"We will allow libraries to be cyclic, and thus, CFGs will be an special case where the OR nodes stand for the non-terminal symbols in the grammar, the AND nodes stands for the grammar rules, and the leaves stand for the grammar terminals.",
"The children of the AND/OR graphs that represent CFGs are normally cyclic and the children of AND nodes (rules) are ordered linearly.",
"The set of solutions to one such AND/OR graph can be defined by means of derivations as it is common in parsing, with the only difference that a partially ordered rule $X \\rightarrow Y_1, \\ldots , Y_m$ represented by an AND node, stands for the set of all totally ordered rules $X \\rightarrow Y_{i_1}, \\ldots , Y_{i_m}$ compatible with the partial order.",
"The set of plans $\\Pi _L(G)$ in the library for $G$ denotes the set of 'strings' (sequences of terminal tasks or actions) that can be derived from the root node corresponding to $G$ ."
],
[
"Compilation",
"The compilation of the library $L$ for a goal $G$ into a Strips Planning problem $P_L(G)$ depends on a depth parameter $N$ , and it ensures that the plans in $P^N_L(G)$ are in correspondence with the set of plans (strings of primitive tasks) $\\Pi _L(G)$ that can be derived from the library by bounding the depth of the derivation to $N$ .",
"If the library is acyclic, it suffices to set $N$ to the depth of the graph to ensure completeness; otherwise, the parameter $N$ puts a bound on the number of derivations.",
"For simplicity, we often drop the index $N$ from the notation.",
"The Planning problem $P_L(G) = \\langle F_L, I_L, G_L, A_L\\rangle $ have a set of fluents $F_L$ , initial and goal situations $I_L$ and $G_L$ , and actions $A_L$ .",
"For simplicity, we will describe the problem assuming a Strips language with negation.",
"Negation, however, can be easily compiled away [5].",
"The fluents $F_L$ in $P_L$ are $started(n,i)$ , $\\lnot started(n,i)$ , $finished(n,i)$ , $\\lnot finished(n,i)$ , and $top(i)$ , where $n$ corresponds to the nodes in the AND/OR graph representing the library $L$ , and $i=[0 \\ldots N]$ .",
"The integers $i$ aim to capture the possible levels of the stack, with the true level captured by the fluent $top(i)$ that is mutex with $top(k)$ for $k\\ne i$ .",
"In a state, where $top(i)$ is true, the fluents $started(n,k)$ and $finished(n,k)$ for $k \\le i$ express the contents of the stack.",
"In any such a state, all fluents $started(n,k)$ and $finished(n,k)$ for $k > i$ will be false.",
"The initial and goal situations of $P_L$ are $I_L = \\lbrace top(0)\\rbrace $ and $G_L = \\lbrace finished(n,0)\\rbrace $ , where $n$ is the single (OR) root node of the library $L$ .",
"That is, the stack starts at level 0 empty with no node started, and the goal is to finish with the root node executed at the same level.",
"For doing this, the stack will expand and contract, while the execution of a node will allow the execution of its children.",
"Roughly the $started(n,i)$ fluents flow downward in the graph, and the fluents $finished(n,i)$ flow upward, with the actions $start(n,i)$ and $end(n,i)$ emulating the start and ending of the primitive and non-primitive tasks in the AND/OR graph.",
"As a convenient abbreviation, we write $i \\!",
"+ \\!",
"1$ and $i \\!",
"- \\!",
"1$ to denote constants $i^{\\prime }$ defined as the successor and predecessor of the constant $i$ in the encoding.",
"The actions in $P_{L}^{N}(G)$ are: Calls from And nodes $n$ to non–terminal children $n^{\\prime }$ are represented by actions $start(n,n^{\\prime },i)$ with preconditions $\\begin{split}Pre &= \\lbrace top(i), started(n,i), \\lnot finished(n^{\\prime }) \\rbrace \\\\& \\cup \\lbrace finished(n^{\\prime \\prime }, i) \\, | \\, n^{\\prime \\prime } <_{n} n^{\\prime }\\rbrace \\\\& \\cup \\lbrace \\lnot started(n^{\\prime \\prime }, i) \\, | \\, n^{\\prime \\prime } \\in children(n) \\rbrace \\end{split}$ add list $Add = \\lbrace top(i\\!",
"+ \\!1), started(n^{\\prime },i\\!",
"+ \\!1\\rbrace $ and delete list $Del = \\lbrace top(i)\\rbrace $ .",
"For calls to terminal children $n^{\\prime }$ , the precondition of $start(n,n^{\\prime },i)$ is the same as the one described above but $Add = \\lbrace finished(n^{\\prime },i) \\rbrace $ and $Del = \\emptyset $ .",
"Termination of calls made from And nodes $n$ are encoded with actions $end(n,i)$ , with preconditions $\\begin{split}Pre &= \\lbrace top(i), started(n,i) \\rbrace \\\\& \\cup \\lbrace finished(n^{\\prime }) \\, | \\, n^{\\prime } \\in children(n) \\rbrace \\end{split}$ and add list $Add = \\lbrace finished(n,i-1), top(i-1) \\rbrace $ and delete list $Del = Pre$ .",
"Calls from internal Or nodes $n$ to children $n^{\\prime }$ are represented by actions $start(n,n^{\\prime },i)$ with precondition $\\begin{split}Pre &= \\lbrace top(i), started(n,i) \\rbrace \\\\&\\cup \\lbrace \\lnot finished(n^{\\prime \\prime }, i\\!",
"+ \\!1) \\, | \\, n^{\\prime \\prime } \\in children(n) \\rbrace \\\\&\\cup \\lbrace \\lnot started(n^{\\prime \\prime }, i\\!",
"+ \\!1) \\, | \\, n^{\\prime \\prime } \\in children(n) \\rbrace \\end{split}$ add list $Add = \\lbrace top(i\\!",
"+ \\!1), started(n^{\\prime }, i\\!",
"+ \\!",
"1) \\rbrace $ and delete list $Del = \\lbrace top(i) \\rbrace $ .",
"Termination of calls from internal Or nodes are represented by actions $end(n,n^{\\prime },i)$ , where $n^{\\prime }$ is a child of $n$ , with precondition $Pre = \\lbrace top(i), started(n,i), finished(n^{\\prime },i) \\rbrace $ , add list $Add = \\lbrace finished(n, i\\!",
"- \\!1), top(i\\!",
"- \\!1) \\rbrace $ and delete list $Del = Pre$ .",
"Root Or nodes are handled like other OR nodes, except that the action $end(n,i=0)$ adds $finished(n,0)$ rather than adding $finished(n,i\\!",
"- \\!1)$ .",
"For a plan $\\pi $ for $P_L(G)$ , let us keep only the sequence of $start(n,n^{\\prime },k)$ actions where $n$ is a leaf node of $L$ , and let us set $f_L(\\pi )$ to the corresponding sequence with the $start(n,n^{\\prime },k)$ actions replaced by the primitive actions associated with the nodes $n$ .",
"The first result is about the correspondence between the set of plans in the library $L$ for $G$ with depth bounded by $N$ , $\\Pi _L^N(G)$ , and the sequences of primitive actions $f(\\pi )$ for plans $\\pi $ for $P_{L}^{N}(G)$ : Theorem 8 (Correspondence) For a library $L$ for goal $G$ and a positive integer $N$ , $\\pi \\in \\Pi _L^N(G)$ iff there is a plan $\\pi ^{\\prime }$ for $P_L(G)$ such that $\\pi = f_L(\\pi )$ .",
"The second result exploits this correspondence for computing the plans in the library that comply with a set of observations $O$ using an off-the-shelf classical planner, suboptimal or not, over the problem $P^{\\prime }_L(G^{\\prime })$ obtained from $P_L(G)$ by compiling the observations $O$ away (Definition REF ): Theorem 9 (Computation) For a library $L$ for a goal $G$ , and a positive integer $N$ , $G$ has a plan in $\\Pi _L^N(G)$ compatible with the observations $O$ iff there is a plan for the Planning problem ${P^{\\prime }}_L^N(G^{\\prime })$ obtained from $P_L(G)$ by compiling the observations $O$ away.",
"The third result is semantic, and shows that this computational method follows from the general formulation for Plan Recognition from Strips theories when action costs are taken to be zero: Theorem 10 (Subsumption) Let ${\\cal G}$ be a set of possible goals, and let $L$ be the library for $G \\in {\\cal G}$ .",
"Then $G$ has a plan in the library that satisfies the observations $O$ with depth no greater than $N$ iff there is an optimal plan for the problem $P^N_L(G)$ that satisfies $O$ , assuming action costs to be zero.",
"Indeed, this result follows from the one above, as when all action costs are zero, any plan for ${P^{\\prime }}^N_L(G^{\\prime })$ is an optimal plan for ${P^{\\prime }}^N_L(G)$ , which in turn from Proposition REF , represents a plan for the problem $P_L(G)$ that satisfies the observations."
],
[
"Experimental Results",
"We test below the Plan Recognition framework laid out above over plan libraries and Context-Free Grammars."
],
[
"The Soccer Plan Library",
"From the descriptions found [13] on plan hierarchies for controlling simulated RoboSoccer teams, we have defined ourselves a set plan libraries for recognizing the intentions of the opposing soccer team.",
"Each library considers one of the following four root tasks, namely, Frontal–Attack, Flank–Attack, Fight–Back and Fall–Back Names of tasks loosely correspond with those of top–level goals featured by the plan hierarchy found in the ISIS source distribution.. Plans in the first two libraries share a substantial amount of activities, e.g.",
"kicking the ball, or running towards the general direction of the opposing team, which do not or hardly take place in plans conveyed by the latter two libraries.",
"In Figure REF we show the plan library for the task Frontal–Attack.",
"Figure: Plan library for Frontal–Attack.",
"Nodes with elliptic shape are OR nodes or leaves (primitive tasks), box–shaped nodes are AND nodes.",
"Precedence constraints between children of AND nodes are not shown.In the experiments, we test which plan libraries are compatible with a sequence of observations drawn from a plan obtained from one of them.",
"The planner we used to search for such plans is the satisfying classical planner FFv2.3.",
"Table: Average time, number of nodes expanded for determining that a library was compatible with the observations, length of resulting explanations (plans) and algorithm used by FF – either Ehc or Bfs.",
"|O||O| is the size of the observation sequence.",
"Derivation depth NN was set to 5.",
"Sequence #1 is {\\lbrace run-forward, kick}\\rbrace , sequence #2 is {\\lbrace run-forward, turn-to-player, kick}\\rbrace , sequence #3 is {\\lbrace run-forward, turn-away, kick-short, run-to-ball,turn-to-goal, kick }\\rbrace and sequence #4 consists in repeating five times the activity kick.",
"None of the plan libraries account for the observation sequences #2, #3 and #4.In Table REF we can see that the Planning problems we obtain from our compilation are handled easily by FF.",
"It is important to note that the size of the observation sequence $|O|$ does not seem to be related with the running time.",
"While the plan library depicted in Figure REF might be very simple, it is not simpler than the plan libraries typically found in the Plan Recognition literature."
],
[
"Context-Free Grammars",
"Context-free grammars (CFGs) appear to present more interesting Planning challenges than the common plan libraries.",
"First, most, if not all, CFGs of interest in Natural Language Processing (NLP) are cyclic, though in languages like English, the depth of the derivation is not big.",
"On the other hand, CFGs used as benchmarks for parsers, like ATIS–3 or CommandTalk These grammars can be found in the NLTK (http://www.nltk.org) Natural Language Processing toolkit corpora., feature thousands of rules.",
"Compiling such grammars into Plan Libraries results in graphs with several thousand nodes.",
"Until recently [6], there has not been any serious attempt at developing a set of challenging benchmarks for Plan Recognition algorithms.",
"We have tested our compilation in a toy CFG of the English language, described below: $S \\rightarrow NP\\,\\,VP$ $VP \\rightarrow V\\,\\,NP | V | VP\\,\\,PP$ $NP \\rightarrow Det\\,\\,N | Name | NP\\,\\,PP$ $PP \\rightarrow P\\,\\,NP$ $V \\rightarrow saw|ate|ran$ $N \\rightarrow boy|cookie|table|telescope|hill$ $Name \\rightarrow Jack|Bob$ $P \\rightarrow with|under$ $Det \\rightarrow the|a|my$ Compiling this simple CFG yields a Plan Library with 85 nodes, which in turn yields a Planning problem with about 800 actions after having fixed the maximum derivation depth to 10.",
"Table: Time needed by FF to accept or reject input sentences, number of nodes expanded in each case, length of explanations (plans) and algorithm used by FF to find the solution.",
"Note that sentences #2 and #3 are incomplete.",
"In sentence #2 the sentence subject is missing, and in sentence #3 the verb is missing.",
"There is no plan accounting for sentence #4.Table REF confirms our intuition that even very simple CFGs yield significantly more challenging Planning problems than Plan Libraries do.",
"In general more search is required to find a parse tree for the input token sequence.",
"One very interesting property inherent to our approach is its ability to “interpolate” missing tokens from the input sentence, as is the case of sentence #3.",
"In that sentence there is no verb, and the planner introduces one of the available productions for non–terminal $V$ in order to obtain a correct parse tree.",
"In sentence #2 the subject is missing, and in this case the planner introduces a noun–phrase.",
"Encouraged by these results, we wanted to conduct an experiment with a “real grammar”.",
"We aimed at obtaining a parse for sentences using the ATIS–3 benchmark CFG.",
"Yet this grammar contains over 3,000 different production rules, which resulted in an AND/OR graph with over 6,000 nodes.",
"The Planning problem resulting from compiling that graph featured over 300,000 actions and a disk footprint of about 2 Gigabytes.",
"We have thus tested our Plan Recognition framework over a CFG not as complex as ATIS–3 but a bit more complex than the toy CFG above.",
"This second grammar features a much richer lexicon: 7 verbs with tenses and number, over twenty nouns, pronouns, auxiliary verbs and all of English prepositions.",
"It also features rules for modeling pragmatics – statements, questions and commands – and taking as well into account applicable syntactic cases – declarative, imperative and interrogation – for each pragmatic.",
"This second grammar, when compiled, resulted in an AND/OR graph with 251 nodes, which, after fixing the derivation depth $N$ to 30 to ensure solubility, resulted in a Planning problem with over 10,000 actions.",
"Table: Average time, number of nodes expanded – for timeouts an educated guess is provided – and plan lengths obtained with the second grammar.",
"TO stands for timeout (time limit was set to 600 seconds).",
"Twelve sentences were divided into three sets.",
"The Covered set contained full sentences covered by the grammar.",
"The Incomplete set contained covered sentences with missing tokens.",
"The final set, Not Covered, refers to non–English sequences of tokens.",
"The results of applying our scheme to this second grammar are shown in Table REF , where three types of sentences are considered.",
"Interestingly FF, solved pretty well the sentences in the Covered set, but had trouble processing the non–English token sequences in the Not Covered set.",
"The timeout we get in the Covered set corresponds to the sentence “why did you take the book”, while the sentence “take the book” was solved after having to expand just 441 nodes.",
"This observation and the fact that incomplete sentences are much smaller than the average sentence in the Complete set, leads us to conclude that in the context of parsing as Planning, the length of the sentence to parse seems to be relevant for the hardness of the problem.",
"We can also see that the “interpolating” behavior of our scheme is biased towards providing a reasonably sized parse tree.",
"It is also worthy to note that none of the problems was solved with the incomplete EHC procedure.",
"The result confirms that the search for plans in the resulting theories becomes much more expensive due to the limitations of current heuristics that make planners like FF get lost in much larger search spaces.",
"Moreover, we have found that it is possible to incorporate some ideas from parsing algorithms like CYK [15] into relaxed–plan graph heuristics, while keeping the heuristic itself computable in polynomial time.",
"We think that such heuristics will help the search to become more focused.",
"Interestingly, the new heuristic is general and thus applies to Planning problems that are completely unrelated to parsing.",
"Unfortunately, we haven't had the time to test these ideas yet, but would like to do that for the camera–ready version if the paper is accepted for the workshop."
],
[
"Discussion",
"We have shown that the framework for plan recognition over Strips theories, formulated recently in [11], subsumes the Plan Recognition problem over libraries, as they can be compiled into Strips.",
"We have also shown that recognition over standard libraries become Planning problems that can be easily solved by modern planners, while recognition over more complex libraries, including CFGs, illustrate limitations of current Planning heuristics and suggest improvements that may be relevant in other Planning problems as well (to be worked out and shown)."
]
] | 1605.05807 |
[
[
"Experimental observation of $\\beta$-delayed neutrons from $^{9}$Li as a\n way to study short-pulse laser-driven deuteron production"
],
[
"Abstract A short-pulse laser-driven deuteron beam is generated in the relativistic transparency regime and aimed at a beryllium converter to generate neutrons at the TRIDENT laser facility.",
"These prompt neutrons have been used for active interrogation to detect nuclear materials, the first such demonstration of a laser-driven neutron source.",
"During the experiments, delayed neutrons from $^9$Li decay was observed.",
"It was identified by its characteristic half-life of 178.3 ms. Production is attributed to the nuclear reactions $^9$Be(d,2p)$^9$Li and $^9$Be(n,p)$^9$Li inside the beryllium converter itself.",
"These reactions have energy thresholds of 18.42 and 14.26 MeV respectively, and we estimate the (d,2p) reaction to be the dominant source of $^9$Li production.",
"Therefore, only the higher-energy portion of the deuteron spectrum contributes to the production of the delayed neutrons.",
"It was observed that the delayed-neutron yield decreases with increasing distance between the converter and the deuteron source.",
"This behavior is consistent with deuteron production with energy greater than $\\sim$20 MeV within a cone with a half-angle greater than 40$^{\\circ}$.",
"Prompt-neutron time-of-flight measurements at varying separation between the converter and the laser target indicate that the fast deuteron population above threshold is severely depleted on axis out to $\\sim$20$^{\\circ}$.",
"These measurements are consistent with emission of the fast deuterons (i.e., above 10 MeV/nucleon) in a ring-like fashion around the central axis.",
"Such an inferred ring-like structure is qualitatively consistent with a documented signature of the breakout afterburner (BOA) laser-plasma ion acceleration mechanism.",
"The measurement of $\\beta$-delayed neutrons from $^9$Li decay could provide an important new diagnostic tool for the study of the features of the deuteron production mechanism in a non-intrusive way."
],
[
"Introduction",
"Intense laser-driven ion beams have been the subject of considerable study for over a decade [1], [2].",
"Based on advanced mechanisms of laser-driven ion acceleration, a new intense and short-duration neutron source with record flux ($> 10^{10}$ n/sr) [3], [4] has been pioneered at Los Alamos National Laboratory (LANL).",
"The neutrons are generated from a multistep process starting with the interaction of a short-pulse laser with a deuterated-plastic nanofoil target to make an intense beam of protons and deuterons.",
"The ion beam subsequently impinges on a suitable converter material to drive the neutron beam.",
"This source has the particularly useful properties of high intensity, short-duration and a forward peaked distribution.",
"Laser-driven neutron sources offer an alternative path for the development of compact, bright and penetrating sources for many applications [5].",
"One of the motivations at LANL for such a source is the capability to perform an assay of special nuclear materials for nuclear materials accountancy, safeguards and national security applications.",
"This source is particularly suitable for the latter two applications.",
"A penetreting intense neutron burst offers the potential of achieving a high signal-to-noise ratio in difficult environments (e.g., with high neutron background emitted by the interrogated item) and promises a short assay time, which translates to a high interrogated item throughput.",
"This application, also known as active interrogation, is based on the measurements of induced neutron signatures to identify/assay nuclear materials during (prompt fission neutron) and after (delayed neutrons from fission products) an interrogation with an external neutron pulse [6], [7], such as the laser driven neutron source at Trident.",
"Laser-driven ion acceleration relies on very intense ($I > 10^{18}$ W/cm$^2$ ) laser fields on plasma targets to create collective effects that drive large accelerating electric fields with field strengths of tens of TV/m over very short distances (microns).",
"The LANL Trident laser facility [8] provides a very high-contrast laser pulse of energy $\\sim $ 80 J, wavelength $\\lambda _0 = $ 1053 nm, 600 fs FWHM duration, and peak on-target laser intensities up to 10$^{21}$ W/cm$^2$ .",
"The high-contrast enables fielding plastic-foil targets ($\\sim $ 1 g.cm$^{-3}$ ) with thicknesses typically in the range of 300$-$ 700 nm.",
"The target heats up rapidly during the main pulse rise and becomes relativistically transparent [9] to the laser by the time of peak power.",
"After transparency, the laser interacts volumetrically with the plasma and accelerates ions in the interaction region to high energies [10], [11], [12], [13], [14].",
"Recent experiments have demonstrated that high-energy deuterons produced in the relativistically transparent regime can create a forward directed neutron emission primarily from deuteron break-up mechanisms in the converter, in addition to the prompt isotropic component from reactions such as $^9$ Be(d,n)$^{10}$ B [4].",
"Figure: Experimental setup at Trident facility (not in scale).We show for the first time that $\\beta $ -delayed neutron production from certain nuclear reactions is useful to probe the deuterium beam.",
"We consider two distinct such reactions that result in $^9$ Li: $^9$ Be(d,2p)$^9$ Li [15] and $^9$ Be(n,p)$^9$ Li [16].",
"$^9$ Li subsequently $\\beta $ -decays, with one branch producing delayed neutrons with the 1/e-value of $\\tau =$ 257.2 millisecond (half-life of 178.3 ms) [17].",
"As discussed below, only high-energy deuterons are implicated in generating $^9$ Li because of the energy thresholds for these reactions.",
"In this letter, we discuss the angular distribution of high-energy deuteron production in the relativistic transparency regime that are revealed through the analysis of the delayed neutron production from $^9$ Li decay and of the prompt neutron spectra along the axis."
],
[
"Setup",
"Our experimental setup is shown in Fig.",
"REF .",
"The target chamber houses the laser focusing optics (in this experiment a F/3 parabolic mirror), the target (for this run, primarily $\\sim $ 350 nm thick deuterated polyethylene (CD$_2$ ) foils) and the converter (a Be cylinder 20 mm diameter and 40 mm depth).",
"All the diagnostic equipment, described below, was positioned outside the chamber.",
"The chamber radius is 1 m and its wall is 20 mm thick stainless steel, with 25 mm thick Al flanges located around the chamber.",
"The neutron diagnostic arrangment was motivated to utilize laser-driven neutron beams for active interrogation.",
"Two $^3$ He thermal neutron coincidence well counters were placed outside the chamber for the detection of $\\beta $ -delayed neutrons from active interrogation of nuclear material.",
"These arise from neutron rich nuclei created following induced fission.",
"One of the well counters contained a sample of nuclear material for active interrogation, while the other was kept empty to serve as a reference for background comparison.",
"The counters used were high-level neutron coincidence counters (HLNCC-II) [18] composed of a single ring of 18 $^3$ He-filled proportional detectors embedded in high-density polyethylene.",
"Neutrons impingings on the HLNCC-II detectors are counted in pulse mode being spread out in time by the thermalization and diffusion property of the moderator assembly.",
"Both counters were located on the equator, closely straddling the central beam axis, defined by the laser propagation direction.",
"It is important to emphasize that for the purpose of this letter, only data acquired in the reference (empty) detector are considered for the analysis, where no fissionable material that could lead to delayed neutron production is present.",
"In addition, a single moderated $^3$ He-filled proportional detector, was located $\\sim $ 90$^{\\circ }$ off-axis.",
"Furthermore, 5 neutron time-of-flight (nTOF) plastic scintillator detectors were positioned around the target chamber.",
"One nTOF detector (nTOF #5) was located 6.2 m away from the converter along the central beam axis (the laser-propagation direction and the symmetry axis of the converter).",
"This detector was used to measure the neutron energy distribution in the forward direction.",
"It was located outside the building to detect the high-energy portion of the neutron spectrum with better resolution.",
"Several bubble detectors were distributed around the chamber to measure laser generated neutron flux in multiple directions [3], [4], [5].",
"The moderated $^3$ He detector, as well as the HLNCC-II well counter, were covered in a thin ($\\sim $ 1 mm thick) Cd foil to block the contribution of slow neutrons returning to the detectors after scattering in the room [19].",
"Prompt beam-neutrons produced by the laser that strike these detectors exhibit a characteristic 1/e die-away time of several tens of microseconds due to the thermalization and diffusion in the high-density polyethylene moderator [19].",
"The die-away times of HLNCC-II and the single moderated $^3$ He detector correspond to $\\sim $ 43 and $\\sim $ 20 $\\mu $ s, respectively.",
"These time intervals are short compared to the delayed neutron production which typically extends over several milliseconds or more.",
"Thus, the delayed neutron signal can be clearly identified in these thermal-neutron detection systems.",
"The single $^3$ He detector positioned off the beam-axis was used as a neutron flux monitor, since its die-away time characteristics enabled data collection over an extended period of time beyond the initial time window, when the well counter was recovering from the initial neutron burst.",
"The count rates from both detectors were acquired using a list mode (time stamp)card with 10 ns time resolution and analysed in the form of time-interval distributions of the neutron detection times following the laser trigger pulse."
],
[
"Experiment",
"In the initial phase of these experiments, various measurements were made to empirically optimize the neutron production and its directionality for active interrogation.",
"The characteristics of the neutron beam were investigated by changing target/converter configurations and varying the distance between them in a series of laser shots.",
"The distance of the front face of the Be converter to the target was systematically increased to 3.6, 6.0, 8.0 and 12.0 mm over several shots and the results are reported here.",
"During the measurements in which the Be converter was placed at a separation of either 3.6, 6.0 or 8.0 mm, an unexpected delayed-neutron tail following the prompt neutron peak was observed in the $^3$ He detectors not containing any nuclear material.",
"So the origin of these neutrons cannot be from induced fission.",
"The tail was observed in the reference HLNCC-II well counter located in the forward direction as well as in the single $^3$ He detector located at $\\sim $ 90$^{\\circ }$ off-axis, as shown in Figs.",
"REF and REF , respectively.",
"These Figures show the time-interval distribution of the neutron detection recorded following the laser pulse.",
"It can be seen that the tail decays completely in $\\sim $ 1 second.",
"The 1/e-value ($\\tau $ ) extracted from these experiments corresponds to 260 $\\pm $ 20 ms.",
"This value was obtained by summing-up the time-interval distributions measured in the HLNCC-II well counter for target-to-converter distances of 3.6, 6.0 and 8.0 mm and performing an exponential fit to determine the slope of the decay (see Fig.",
"REF ).",
"The extracted 1/e-value ($\\tau $ ) corresponds to the delayed neutron production via decay of $^9$ Li with the 1/e-value of $\\tau =$ 257.2 ms [17].",
"This delayed neutron production is expected to be isotropic, as confirmed by the comparison of the signals from the HLNCC-II and the $^3$ He flux monitor, located at $\\sim $ 0$^{\\circ }$ and $\\sim $ 90$^{\\circ }$ , respectively.",
"Figure: Delayed neutron production in the forward direction from 9 ^9Li decay is shown for four different shots in which Be ion-to-neutron converter was placed at different distances from the target as measured with the HLNCC-II detector.",
"As the converter gets closer to the target, more high energy deuterons interact with the converter, hence increasing the 9 ^9Li production, and the subsequent delayed neutron decaying signal.",
"These are the raw data before normalization for neutron yield.",
"Note that zero on the x-axis corresponds to the time of laser pulse and negative values represent background data acquired immediately before the shot.Figure: Summed time-interval distributions measured in the HLNCC counter for target/converter distances of 3.6, 6.0 and 8.0 mm.",
"The decay constant (260 ±\\pm 20 ms ) is calculated by a weighted least squares fit and the uncertainty (1σ\\sigma ) by varying the fit region.Because of the die-away time characteristics of the $^3$ He detectors used, the temporal signal produced directly by the laser-driven prompt neutron pulse dissipates approximately within a ms, since 1 ms corresponds to more than 20 times the 1/e-die-away period.",
"In addition, return of the scattered thermal neutrons from the room is suppressed by the external Cd shield.",
"Contribution of these effects to the delayed neutron signature, observed over the period of 50$-$ 1000 ms, can therefore be excluded.",
"The energy thresholds of the nuclear reactions $^9$ Be(d,2p)$^9$ Li and $^9$ Be(n,p)$^9$ Li are 18.42 and 14.26 MeV, respectively [19].",
"Therefore, the delayed neutrons can only be produced from the high-energy deuterons or neutrons impinging on the $^9$ Be converter.",
"The high-energy deuterons with energies $\\ge $ 18.42 MeV can lead to $^9$ Li production directly via $^9$ Be(d,2p)$^9$ Li.",
"In addition, the deuterons with energies greater than about 30 MeV can also contribute, by break-up reactions, to the production of the high-energy neutrons, that lead to $^9$ Li production by the (n,p) reaction.",
"The conversion efficiency of deuterons-to-neutrons for the deuterons above 15 MeV has been measured to be around 0.1% [4] for the same size Be converter.",
"The low conversion efficiency combined makes the $^9$ Be(d,2p)$^9$ Li branch the dominant process.",
"Figure: 9 ^9Li delayed neutron spectra in the 3 ^3He neutron monitor located at ∼\\sim 90 ∘ ^{\\circ } from the forward direction shown for two shots with different converter-target distances.",
"The 9 ^9Be converter catches high energy deuterons when at 3.6 mm, while at 12 mm, it misses most them.",
"Delayed neutron from 9 ^9Li decay is expected isotropic, thus consistently the 9 ^9Li delayed neutron signal is measured by detectors located at ∼\\sim 90 ∘ ^{\\circ }."
],
[
"Results",
"We observed that as the distance of the converter to the CD$_2$ -target increases, the yield of the delayed neutrons, proportional to $^9$ Li production, decreases.",
"Figs.",
"REF and REF illustrates this trend.",
"Therefore, at a large enough target-to-converter distance, the flux of the high-energy deuterons on the converter, that govern the $^9$ Li production, approaches zero.",
"The integral of the delayed neutron counts normalized to the laser neutron yield are shown in Fig.",
"REF for all the target-converter distances.",
"The delayed neutron counts displayed were integrated over an interval of 50$-$ 1000 ms after the laser pulse.",
"It can be anticipated that at a sufficiently close distance between the converter and the deuteron source most of the deuterons would hit the Be and the dependence versus separation would level off.",
"Such distances (less than 3.6 mm) were not investigated in this experiment in order to avoid Be damage and contamination inside the chamber.",
"These observations and their scaling imply a large divergence angle for the high-energy deuteron production responsible for the $^9$ Li (see Discussion).",
"Although these data enable us to estimate a bound for the cone angle of the fast deuterons, they don't provide information about the energy distribution within that cone, except that it exceeds the $\\sim $ 20 MeV threshold.",
"Figure: Neutron time-of-flight spectra (nTOF #5) for two shots with the Be converter placed at 3.6 and 12.0 mm from the target, respectively.",
"The vertical lines mark the highest observed energy of the neutron distribution of ∼\\sim 55 MeV and ∼\\sim 18 MeV for 3.6 mm and 12 mm distances, respectively.",
"A further vertical line marks 10 MeV neutron energy.",
"Higher energy neutrons (to the left of this line) can only be produced by breakup of high energy (>>20 MeV) deuterons.Independent data from the nTOF detector located on the central beam axis (nTOF #5) provides insight on the energy distribution of the fast deuterons within the cone discussed above.",
"nTOF #5 data in Fig.",
"REF show the dependence of the neutron beam energy on the target-to-converter distance.",
"Specifically, it shows time-of-flight spectra of the prompt neutron beam arising from the deuterium disintegration in the converter for target-to-converter distances of 3.6 and 12 mm.",
"The deuterons of interest ($>$ 20 MeV), when undergoing break-up, produce neutrons with energies of $\\sim $ 10 MeV or more.",
"Therefore in Fig.",
"REF , we compare the neutron spectra for times 143 ns (corresponding to 10 MeV) and lower (corresponding to higher energies).",
"If the deuteron spectra were the same at all angles we would expect the prompt neutron spectra in nTOF#5 to be self-similar for any separation.",
"Instead, Fig.",
"REF shows a spectrum severely depleted of neutrons above 10 MeV when the converter separation is 12 mm, measured relative to the spectrum at the minimum separation of 3.6 mm.",
"This suggests an angular distribution of the fast deuterons that is hollow along the central axis (see discussion below.)"
],
[
"Discussion",
"If the observed delayed neutron production from $^9$ Li decay was caused by a mechanism involving a quasi-parallel beam of forward directed deuterons above the $^9$ Li production threshold striking the Be converter, the delayed neutron yield would be comparatively insensitive to the distance between the CD$_2$ laser target and the Be converter.",
"This is inconsistent with our observations.",
"Instead, our observations and interpretations imply a large divergence angle for the high-energy deuterons responsible for the $^9$ Li production.",
"There must be a large number of high energy deuterons that impinge on the Be converter when it is close to the target but miss the converter completely when it is farther away.",
"The minimum cone half-angle relative to the central axis (the axis of the converter cylinder as well as the laser propagation direction) of these fast deuterons can be estimated in two ways.",
"First, we simply take the $\\arctan $ of the angle defined by the ratio of the converter radius (1 cm), and the closest separation (0.4 cm), which yields 68$^{\\circ }$ .",
"To refine this estimate, we note that a representative energetic deuteron ($\\sim $ 50 MeV) [3], [4] would penetrate $\\sim $ 0.85 cm into the Be converter.",
"So we could take instead the closest separation $+$ 0.85 cm, which yields an angle of $\\sim $ 40$^{\\circ }$ .",
"Figure: 9 ^9Li delayed neutron (DN) production, normalized for the neutron yield (HLNCC-II at 0 ∘ ^{\\circ }), as a function of the distance between the Be back face and the target.",
"The uncertainties are statistical only.",
"The uncertainty on the 3.6 mm datum is larger then the others because it contains a correction factor for the orientation axis of the HLNCC-II detector.As mentioned above, if the spectra of the deuterons driving the prompt neutrons in Fig.",
"REF were the same at any angle, we would expect the prompt neutron spectra to be self-similar for any separation between the laser target and neutron converter.",
"Instead, Fig.",
"REF shows that the fast deuteron angular distribution cannot be uniform.",
"The lack of neutrons with energies greater than 10 MeV at the larger target-converter distance implies that most high energy deuterons are missing the converter.",
"Using a similar estimation as above, the fast deuteron angular distribution is depleted along the central axis up to an angle $\\sim $ 20$^{\\circ }$ .",
"All these observations are consistent with the bulk of the fast deuterons coming out in a ring-like fashion.",
"The observation of a ring-like morphology for the fast deuteron population is consistent in light of prior simulations and experiments on laser-driven C$^{6+}$ beams in the BOA regime.",
"We note that deuterons have the same charge to mass ratio of fully ionized C. 3D simulations of the interaction of an intense laser with diamond nanofoils resulting in BOA acceleration discussed in Ref.",
"[20] show such a ring-like structure for the high-energy portion of the C$^{6+}$ spectrum.",
"The experiments in Ref.",
"[20] carried out on the Trident laser with diamond nanofoils also showed an increasingly ring-like emission versus angle for increased C$^{6+}$ ion energy, albeit at a smaller angle $\\sim $ 10$^{\\circ }$ .",
"Although, deuterated plastic and diamond targets are different in their chemical structure, with proper optimization of the target thickness to yield a similar time during the laser pulse when the laser target becomes relativistically transparent, it is not unreasonable to expect similar behavior.",
"However, the response of the two materials to the laser pre-pulse and the laser hydrodynamic disassembly would not be identical, so quantitative differences are expected.",
"Our results demonstrate the power of selected nuclear reactions as a highly specific diagnostic for deuteron beam spectra and morphology.",
"In the case of Be converter $\\beta $ -delayed neutron measurements from $^9$ Li decay offer an attractive alternative relative to other diagnostics tools, such as radio-chromic films, because they can be measured continuously throughout any experimental campaign, and the converter obstructs and thus complicates the use of other beam diagnostics.",
"As demonstrated in Fig.",
"REF , a simple moderated $^3$ He-filled proportional counter may be used to detect this delayed neutron signature while providing a flux monitoring capability.",
"The Be converter does not need to be segmented, as the use of radio-chromic films would require, and the data are acquired automatically as part of regular flux monitoring.",
"Nuclear diagnostic methods such as this one, could also be very useful to resolve ambiguous results with particle spectrometers when ions with similar charge to mass ratios are involved.",
"The measured integral delayed neutron counts together with known efficiency of the $^3$ He detector used and the $^9$ Li production cross section, could be used to determine the fraction of high energy deuterons in every shot.",
"In general, the quantitative power of the technique would be greatly increased if the reaction cross section was known as a function of energy.",
"Unfortunately, there is only a theoretical calculation available [21].",
"To the extent of our knowledge, the $^9$ Li production from $^9$ Be(d,2p)$^9$ Li reaction has not been reported in a peer reviewed publication since 1951 [15].",
"Therefore, our work provides an impetus for further work to measure the cross section experimentally.",
"In fact, a modified version of the experimental set-up utilized in the current work could be used for cross section measurements.",
"In addition, the observation of $\\beta $ -delayed neutrons from sources other than nuclear fission has important implications for the correct understanding and interpretation of laser-driven active interrogation measurements, when $\\beta $ -delayed neutrons from fissions are being used as the signature of nuclear material.",
"In this case, delayed neutron sources other than nuclear fission have to be understood and accounted for to prevent biases in the estimates of the amounts of nuclear material [5]."
],
[
"Conclusions",
"An intense and energetic deuterium beam is generated by irradiating a sub-micron thick polyethylene target with the intense Trident short-pulse laser.",
"The ion acceleration mechanism has been identified as BOA, which operates in the relativistic transparency regime of laser-plasmas.",
"The deuterons produce neutrons by impinging on a beryllium cylinder placed in the beam path.",
"This technique produces a higher forward-directed neutrons flux per unit laser energy than any alternative available so far.",
"Our neutron detectors, which were designed for active interrogation of special nuclear material, unexpectedly detected delayed neutrons from $^9$ Li decay with its characteristic half-life of 178.3 ms. We attribute the $^9$ Li production primarily to $^9$ Be(d,2p) reactions driven by energetic deuterons above the reaction threshold of 18.42 MeV.",
"The delayed neutron yield gives a direct on-line measure of the number of high energy deuterons impinging on the beryllium collector.",
"The measured delayed neutron yield from the $^9$ Li decay is observed to decrease steeply as the separation between the laser target and neutron converter is increased.",
"That dependence implies that the high energy portion of the deuteron beam is emitted from the laser target in a cone with a half angle (relative to the central axis) of at least $\\sim $ 40$^{\\circ } - 70^{\\circ }$ .",
"Moreover, data from the neutron time of flight diagnostic along the central axis have been used to measure the energy spectrum of the prompt beam neutrons created by deuterium disintegration in the converter, for various laser-target to converter separations.",
"Those neutron spectra have been used to estimate the relative abundance of the fast deuterons, i.e., above the $^9$ Li production threshold.",
"It is found that the flux of those fast deuterons is severely depleted along the axis, up to an angle $\\sim $ 20$^{\\circ }$ .",
"These observations are consistent with the hypothesis that the fast deuteron population is being emitted in a ring-like fashion around the central axis.",
"This hypothesis is qualitatively consistent with one of the documented unique signatures of the BOA mechanism.",
"Observation of the delayed neutron production from $^9$ Li decay can be a useful diagnostic for neutron production experiments using deuterons on beryllium converters.",
"The utility of this nuclear diagnostic to determine the beam morphology motivates measurements of the $^9$ Be(d,2p) cross-section to compare with current model calculations.",
"Indeed, it is possible to use a modified version of our experimental setup to measure those cross sections.",
"In addition, as outlined above, this unanticipated source of delayed neutrons must be considered in the interpretation of laser-driven active neutron interrogation measurements."
],
[
"Acknowledgment",
"Research reported in this publication was supported by the LANL Laboratory Directed Research and Development (LDRD) program at Los Alamos National Laboratory.",
"The Trident Laser Facility is supported by the NNSA Science and ICF campaigns."
]
] | 1605.05702 |
[
[
"The b --> s l^+ l^- anomalies and their implications for new physics"
],
[
"Abstract Recently, the LHC has found several anomalies in exclusive semileptonic b --> s l^+ l^- decays.",
"In this proceeding, we summarize the most important results of our global analysis of the relevant decay modes.",
"After a discussion of the hadronic uncertainties entering the theoretical predictions, we present an interpretation of the data in terms of generic new physics scenarios.",
"To this end, we have performed model-independent fits of the corresponding Wilson coefficients to the data and have found that in certain scenarios the best fit point is prefererred over the Standard Model by a global significance of more than 4{\\sigma}.",
"Based on the results, the discrimination between high-scale new physics and low-energy QCD effects as well as the possibility of lepton-flavour universality violation are discussed."
],
[
"Introduction",
"The flavour-changing neutral current (FCNC) transition $b\\rightarrow s\\ell ^+\\ell ^-$ can be probed through various decay channels, currently studied in detail at the LHC in the LHCb, CMS and ATLAS experiments, as well as at Belle.",
"Recent experimental results have shown interesting deviations from the SM: The LHCb analysis [1] of the 3 fb$^{-1}$ data on $B\\rightarrow K^*\\mu ^+\\mu ^-$ in particular confirms a $\\sim 3\\sigma $ anomaly in two large $K^*$ -recoil bins of the angular observable $P_5^\\prime $ [2], [3] that was already present in the 1 fb$^{-1}$ results presented in 2013 [4].",
"The observable $R_K=Br(B\\rightarrow K\\mu ^+\\mu ^-)/Br(B\\rightarrow Ke^+e^-)$ was measured by LHCb [5] in the dilepton mass range from 1 to 6 GeV$^2$ as $0.745^{+0.090}_{-0.074}\\pm 0.036$ , corresponding to a $2.6\\sigma $ tension with its SM value predicted to be equal to 1 (to a very good accuracy).",
"Finally, also the LHCb results [6] on the branching ratio of $B_s\\rightarrow \\phi \\mu ^+\\mu ^-$ exhibit deviations at the $\\sim 3\\sigma $ level in two large-recoil bins.",
"Figure: Effective couplings (') ^{(\\prime )} contributing to b→sℓ + ℓ - b\\rightarrow s\\ell ^+\\ell ^- transitions andsensitivity of the various radiative and (semi-)leptonic B (s) B_{(s)} decays to them.The appearance of several tensions in different $b\\rightarrow s\\ell ^+\\ell ^-$ channels is quite intriguing because all these observables are sensitive to the same effective couplings $^{(\\prime )}$ illustrated in Fig.",
"REF and induced by the operators $&&\\mathcal {O}_9^{(\\prime )}=\\frac{\\alpha }{4\\pi }[\\bar{s}\\gamma ^\\mu P_{L(R)}b][\\bar{\\mu }\\gamma _\\mu \\mu ],\\hspace{28.45274pt}\\mathcal {O}_{10}^{(\\prime )}=\\frac{\\alpha }{4\\pi }[\\bar{s}\\gamma ^\\mu P_{L(R)}b][\\bar{\\mu }\\gamma _\\mu \\gamma _5\\mu ],\\nonumber \\\\&&\\mathcal {O}_7^{(\\prime )}=\\frac{\\alpha }{4\\pi }m_b[\\bar{s}\\sigma _{\\mu \\nu }P_{R(L)}b]F^{\\mu \\nu },$ where $P_{L,R}=(1 \\mp \\gamma _5)/2$ and $m_b$ denotes the $b$ quark mass.",
"It is thus natural to ask whether a new physics contribution to these couplings could simultaneously account for the various tensions in the data.",
"Beyond the SM, contributions to $^{(\\prime )}$ are for instance generated at tree level in scenarios with $Z^\\prime $ bosons or lepto–quarks.",
"Note that additional scalar or pseudoscalar couplings $$ cannot address the above-mentioned anomalies since their contributions are suppressed by small lepton masses.",
"Therefore we will not discuss this possibility in the following.",
"The parameter space spanned by the couplings $^{(\\prime )}$ is probed through various observables in radiative and (semi-)leptonic $B_{(s)}$ decays, each of them sensitive to a different subset of coefficients (see Fig.",
"REF ).",
"A complete investigation of potential new physics effects thus requires a combined study of these observables including correlations among them.",
"The first analysis in this spirit, performed in Ref.",
"[7] with the data of 2013, pointed to a large negative contribution to the Wilson coefficient $$ .",
"This general picture was confirmed later on by other groups, using different/additional observables, different theoretical input for the form factors etc.",
"(e.g.",
"Refs.",
"[8], [9]).",
"In this proceeding, we report the most important results of our analysis in Ref.",
"[10] which can be compared to other recent global analyses [11], [12], [13] and which improves the original study in Ref.",
"[7] in many aspects: It includes the latest experimental results of all relevant decays (using the LHCb data for the exclusive), uses refined techniques to estimate uncertainties originating from power corrections to the hadronic form factors and from non-perturbative charm loops, and consistently takes into account experimental and theoretical correlations.",
"Before presenting the results from our fits in Sec.",
", with a special emphasis on the possibility of discriminating between high-scale new physics and low-energy QCD effects as well as on the possibility of lepton-flavour universality violation, we discuss in Sec.",
"the hadronic uncertainties entering the theoretical predictions of the relevant observables.",
"Our conclusions are given in Sec.",
"."
],
[
"Hadronic uncertainties",
"Predictions for exclusive semileptonic $B$ decays are plagued by QCD effects of perturbative and non-perturbative nature.",
"At leading order (LO) in the effective theory, predictions involve tree-level diagrams with insertions of the operators $\\mathcal {O}_{7,9,10}$ (generated at one loop in the SM), as well as one-loop diagrams with an insertion of the charged-current operator $\\mathcal {O}_2=[\\bar{s}\\gamma ^\\mu P_Lc][\\bar{c}\\gamma _\\mu P_Lb]$ (generated at tree level in the SM).",
"In contributions of the first type, the leptonic and the hadronic currents factorize, and QCD corrections are constrained to the hadronic $B\\rightarrow M$ current (first two diagrams in Fig.",
"REF ).",
"This class of factorizable QCD corrections thus forms part of the hadronic form factors parametrizing the $B\\rightarrow M$ transition.",
"Contributions of the second type, on the other hand, receive non-factorizable QCD corrections (third diagram in Fig.",
"REF ) that cannot be absorbed into form factors.",
"In the following we discuss the uncertainties stemming from the two types of corrections and their implementation in our analysis."
],
[
"Form factor uncertainties",
"The form factors are available from lattice as well as from light-cone sum rule (LCSR) calculations, with the former being suited for the region of high $q^2>15$ GeV$^2$ and the latter for the region of low $q^2<8$ GeV$^2$ .",
"Since the form factors introduce a dominant source of uncertainties into the theory predictions, it is desirable to reduce the sensitivity to them as much as possible.",
"For $B\\rightarrow V\\ell ^+\\ell ^-$ decays, with $V$ being a vector meson, this can be achieved in the low-$q^2$ region by exploiting large-recoil symmetries of QCD.",
"At LO in $\\alpha _s$ and $\\Lambda /m_b$ , these symmetries enforce certain relations among the seven hadronic form factors $V$ , $A_1$ , $A_2$ , $A_0$ , $T_1$ , $T_2$ , $T_3$ , like e.g.",
"$\\frac{m_B(m_B+m_{K^*})A_1-2E(m_B-m_{K^*})A_2}{m_B^2T_2-2Em_BT_3}\\,=\\,1+\\mathcal {O}(\\alpha _s,\\Lambda /m_b),$ where $m_B$ denotes the mass of the $B$ meson, and $m_{K^*}$ and $E$ the mass and the energy of the $K^*$ meson.",
"From the experimentally measured coefficients of the differential angular distribution of $B\\rightarrow V\\ell ^+\\ell ^-$ , one can construct observables that involve ratios like the one in eq.",
"(REF ).",
"The resulting observables $P_i^{(\\prime )}$ then only exhibit a mild form factor dependence, suppressed by powers of $\\alpha _s$ and $\\Lambda /m_b$ .",
"For the cancellation of the form factor uncertainties in ratios like the one in eq.",
"(REF ), it is crucial to have control of the correlations among the errors of the different form factors.",
"These correlations can be taken into account via two orthogonal approaches: Either they can be assessed directly from the LCSR calculation (Ref.",
"[14] provides LCSR form factors with correlation matrices), or they can be implemented resorting to the large-recoil symmetry relations.",
"Whereas the former method is limited to the particular set of LCSR form factors from Ref.",
"[14] and hence sensitive to details of the corresponding calculation, the latter method determines the correlations in a model-independent way from first principles and can thus also be applied to different sets of form factors like the ones from Ref.",
"[15], where no correlations were provided.",
"As a drawback, correlations are obtained from large-recoil symmetries only up to $\\Lambda /m_b$ corrections which have to be estimated.",
"For the estimate of these factorizable power corrections, we follow the strategy that was developed in Ref.",
"[16] based on and further refining a method first proposed in Ref. [17].",
"We assume a generic size of $10\\%$ factorizable power corrections to the form factors, which is consistent with the results that are obtained from a fit to the particular LCSR form factors from Refs.",
"[15], [14]."
],
[
"Uncertainties from $c\\bar{c}$ loops",
"Long-distance charm-loop effects (third diagram in Fig.",
"REF ) can mimic the effect of an effective coupling $^{c\\bar{c}}$ and have been suggested as a solution of the anomaly in $B\\rightarrow K^*\\mu ^+\\mu ^-$ [20], [21].",
"Due to the non-local structure of these corrections, their contribution is expected to have a non-constant $q^2$ -dependence, where $q^2$ is the squared invariant masses of the lepton pair.",
"Together with the perturbative SM contribution $^{\\rm eff}$ and a potential constant new physics coupling $^{\\rm SM}$ , it can be cast into an effective Wilson coefficient $^{\\rm eff\\;i}(q^2)\\,=\\,^{\\rm eff}(q^2)\\,+\\,^{\\rm NP}\\,+\\,^{c\\bar{c}\\;i}(q^2),$ with a different $^{c\\bar{c}\\;i}$ and hence also a different $^{\\rm eff\\;i}$ for the three transversity amplitudes $i=0,\\Vert ,\\perp $ .",
"Currently, only a partial calculation [15] exists, yielding values $^{c\\bar{c}\\;i}$ that tend to enhance the anomalies.",
"In our analysis, we assume that this partial result is representative for the order of magnitude of the total charm-loop contribution and we assign an error to unknown charm-loop effects varying $^{c\\bar{c}\\;i}(q^2)\\;=\\;s_i\\;^{c\\bar{c}\\;i}(q^2),\\hspace{14.22636pt}\\textrm {for }-1\\le s_i\\le 1.$"
],
[
"Results of the global fit",
"Our reference fits are obtained using the following experimental input: branching ratios and angular observables of the decays $B\\rightarrow K^*\\mu ^+\\mu ^-$ and $B_s\\rightarrow \\phi \\mu ^+\\mu ^-$ , branching ratios of the charged and neutral modes $B\\rightarrow K\\mu ^+\\mu ^-$ , the branching ratios of $B\\rightarrow X_s\\mu ^+\\mu ^-$ , $B_s\\rightarrow \\mu ^+\\mu ^-$ and $B\\rightarrow X_s\\gamma $ , as well as the isospin asymmetry $A_I$ and the time-dependent CP asymmetry $S_{K^*\\gamma }$ of $B\\rightarrow K^*\\gamma $ .",
"For the theoretical predictions, we use lattice form factors from Refs.",
"[18], [19] in the low-recoil region, and LCSR form factors from Ref.",
"[15] (except for $B_s\\rightarrow \\phi $ where Ref.",
"[14] is used), with correlations assessed from the large-recoil symmetries.",
"Table: Results of various one-parameter fits for the Wilson coefficients {ȉ}\\lbrace ȉ\\rbrace .Starting from a model hypothesis with $n$ free parameters for the Wilson coefficients $\\lbrace ȉ^{\\rm NP}\\rbrace $ , we then perform a frequentist $\\Delta \\chi ^2$ -fit, including experimental and theoretical correlation matrices.",
"In Tab.",
"REF we show our results for various one-parameter scenarios.",
"In the last column we give the SM-pull for each scenario, i.e.",
"we quantify by how many sigmas the best fit point is preferred over the SM point $\\lbrace ȉ^{\\rm NP}\\rbrace =0$ in the given scenario.",
"A scenario with a large SM-pull thus allows for a big improvement over the SM and a better description of the data.",
"From the results in Tab.",
"REF we infer that a large negative $^{\\rm NP}$ is required to explain the data.",
"In a scenario where only this coefficient is generated a fairly good goodness-of-fit is yielded for $^{\\rm NP}\\sim -1.1$ .",
"A decomposition into the different exclusive decay channels, as well as into low- and large-recoil regions, shows that each of these individual contributions points to the same solution, i.e.",
"a negative $^{\\rm NP}$ , albeit with varying significance.",
"We refer the reader to Ref.",
"[10] for further results, e.g.",
"for fits in various 2-parameter scenarios as well as for the full 6-parameter fit of $^{(\\prime )\\rm NP}$ resulting in a SM-pull of $3.6\\sigma $ .",
"Figure: Left: Bin-by-bin fit of the one-parameter scenario with a single coefficient NP ^{\\rm NP}.Right: Fit with independent coefficients NP ^{\\rm NP} and NP ^{\\rm NP}."
],
[
"New physics vs. non-perturbative charm-contribution",
"According to Eq.",
"(REF ), a potential new physics contribution $^{\\rm NP}$ enters amplitudes always together with a charm-loop contribution $^{c\\bar{c}\\;i}(q^2)$ , spoiling an unambiguous interpretation of the fit result from the previous section in terms of new physics.",
"However, whereas $^{\\rm NP}$ does not depend on the squared invariant mass $q^2$ of the lepton pair, $^{c\\bar{c}\\;i}(q^2)$ is expected to exhibit a non-trivial $q^2$ -dependence.",
"Following Ref.",
"[12], we show in Fig.",
"REF on the left a bin-by-bin fit for the one-parameter scenario with a single coefficient $^{\\rm NP}$ .",
"The results obtained in the individual bins are consistent with each other, allowing thus for a solution $^{\\rm NP}$ that is constant in the whole $q^2$ region, as required for an interpretation in terms of new physics, though the situation is not conclusive due to the large uncertainties in the single bins.",
"An alternative strategy to address this question has been followed recently in Ref.",
"[21] where a direct fit of the $q^2$ -dependent charm contribution $^{c\\bar{c}\\;i}(q^2)$ to the data on $B\\rightarrow K^*\\mu ^+\\mu ^-$ (at low $q^2$ ) has been performed under the hypothesis of the absence of new physics.",
"The fact that only one out of the 12 parameters encoding a non-constant $q^2$ -dependence in $^{c\\bar{c}\\;i}(q^2)$ differs from zero by $\\gtrsim 2\\sigma $ makes the result obtained in Ref.",
"[21] compatible The probability for a fluctuation by $\\gtrsim 2\\sigma $ in at least one out of 12 parameters is given by $1-0.954^{12}\\approx 43\\%$ , well within $1\\sigma $ .",
"with a $q^2$ -independent new physics solution $^{\\rm NP}$ , in agreement with our findings from Fig.",
"REF .",
"Note further that the results in Ref.",
"[21] do not allow to draw any conclusions on whether a $q^2$ -dependent solution of the anomalies via $^{c\\bar{c}\\;i}(q^2)$ is preferred compared to a solution via a constant $^{\\rm NP}$ since this would require a comparison of the goodness of the fit taking into account the different number of free parameters of the two parametrizations.",
"Moreover, we like to stress that the results for the observables presented in Ref.",
"[21] should not be interpreted as SM predictions, as they are based on a fit to the experimental data."
],
[
"Lepton-flavour universality violation",
"Since the measurement of $R_K$ suggests the violation of lepton-flavour universality, we also studied the situation where the muon- and the electron-components of the operators $^{(\\prime )}$ receive independent new physics contributions $^{\\rm NP}$ and $^{\\rm NP}$ , respectively.",
"The electron-couplings $^{\\rm NP}$ are constrained by adding the decays $B\\rightarrow K^{(*)}e^+e^-$ to the global fit.",
"Note that the correlated fit to $B\\rightarrow K\\mu ^+\\mu ^-$ and $B\\rightarrow Ke^+e^-$ simultaneously is equivalent to a direct inclusion of the observable $R_K$ .",
"In Fig.",
"REF on the right we display the result for the two-parameter fit to the coefficients $^{\\rm NP}$ and $^{\\rm NP}$ .",
"The fit prefers an electron-phobic scenario with new physics coupling to $\\mu ^+\\mu ^-$ but not to $e^+e^-$ .",
"Under this hypothesis, that should be tested by measuring $R_{K^*}$ and $R_\\phi $ , the SM-pull increases by $\\sim 0.5\\sigma $ compared to the value in Tab.",
"REF for the lepton-flavour universal scenario."
],
[
"Conclusions",
"LHCb data on $b\\rightarrow s\\ell ^+\\ell ^-$ decays shows several tensions with SM predictions, in particular in the angular observable $P_5^\\prime $ of $B\\rightarrow K^*\\mu ^+\\mu ^-$ , in the branching ratio of $B_s\\rightarrow \\phi \\mu ^+\\mu ^-$ , and in the ratio $R_K=Br(B\\rightarrow K\\mu ^+\\mu ^-)/Br(B\\rightarrow Ke^+e^-)$ (all of them at the $\\sim 3\\sigma $ level).",
"In global fits of the Wilson coefficients to the data, scenarios with a large negative $^{\\rm NP}$ are preferred over the SM by typically more than $4\\sigma $ .",
"A bin-by-bin analysis demonstrates that the fit is compatible with a $q^2$ -indepedent effect generated by high-scale new physics, though a $q^2$ -dependent QCD effect cannot be excluded with the current precision.",
"Note, however, that a QCD effect could not explain the tension in $R_K$ .",
"The latter observable further favours a lepton-flavour violating scenario with new physics coupling only to $\\mu ^+\\mu ^-$ but not to $e^+e^-$ , a scenario to be probed by a measurement of the analogous ratios $R_{K^*}$ and $R_\\phi $ to probe this hypothesis."
],
[
"Acknowledgments",
"L.H.",
"is grateful to the organizers for the invitation to the workshop and thanks the participants for stimulating discussions.",
"The work of L.H.",
"was supported by the grants FPA2013-46570-C2-1-P and 2014-SGR-104, and partially by the Spanish MINECO under the project MDM-2014-0369 of ICCUB (Unidad de Excelencia “María de Maeztu”).",
"JV is funded by the DFG within research unit FOR 1873 (QFET), and acknowledges financial support from CNRS.",
"SDG, JM and JV acknowledge financial support from FPA2014-61478-EXP.",
"This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreements No 690575, No 674896 and No.",
"692194."
]
] | 1605.06059 |
[
[
"Interaction induced topological protection in one-dimensional conductors"
],
[
"Abstract We discuss two one-dimensional model systems -- the first is a single channel quantum wire with Ising anisotropy, while the second is two coupled helical edge states.",
"We show that the two models are governed by the same low energy effective field theory, and interactions drive both systems to exhibit phases which are metallic, but with all single particle excitations gapped.",
"We show that such states may be either topological or trivial; in the former case, the system demonstrates gapless end states, and insensitivity to disorder."
],
[
"Introduction",
"The importance of topology in different aspects of condensed matter physics has long been known, from topological excitations [1], [2] through topological order [3] to topological quantum computing [4].",
"Perhaps one of the most surprising roles of topology however was the discovery of topological insulators (TIs) about a decade ago [5], [6], [7].",
"The central idea is that gapped states of non-interacting electrons can be characterised by a topological index [8].",
"The original topological index for the integer quantum Hall states, the TKNN index [9] has now been extended to all possible dimensions and underlying symmetries of the system in question, leading to a `periodic table' of possible topological insulators [10], [11].",
"While the value of the appropriate topological index is a technical way of defining a topological insulator, the defining feature is the presence of edge states – gapless excitations that live only at the edge of the sample, while the bulk of the system remains gapped.",
"These edge states also have the feature that they can not be localised by disorder.",
"They remain conducting – perfectly so in two-dimensional topological states – no matter how rough or disordered the edge is so long as none of the protecting symmetries are broken.",
"Specialising now to two dimensional systems with time reversal symmetry, we have the quantum spin Hall (QSH) effect which has a $\\mathbb {Z}_2$ topological index.",
"Typically, this state appears in semiconductors which have band inversion due to strong spin-orbit coupling [6].",
"The inverted band may be viewed as a negative band gap, and the defining topological index can be reduced to the sign of the gap – a positive gap means a normal insulator, while a negative gap is the topological state.",
"This leads immediately to one of the main questions discussed in this work: Can one obtain states akin to a topological insulator when the gap is not present in the band structure, but dynamically generated by interactions?",
"It is worthwhile noting that topological superconductors [12] are one example of such a mechanism occurring – after a mean field decoupling, these reduce to a non-interacting Hamiltonian and a classification as before [11].",
"We will discuss other examples which have no local symmetry breaking and no local order parameter, and so they cannot be so obviously put into the existing framework.",
"The role of interactions in topological insulators is currently a vibrant field .",
"One of the questions often asked is how interactions affect the metallic states at the edge of the topological insulator – it is well understood now [14], [13], [15], [16], [17], [18] that interactions destroy the perfect conduction predicted by the non-interacting theory due to inelastic scattering processes.",
"For weak to moderate interactions, this gives a temperature dependent reduction to the conductance, the perfect conductance still being realised at zero temperature.",
"For strong interactions however, the topological protection may be broken completely, and the edge state may localise.",
"The second question we will be concentrating on in this work is the opposite of this: Can interactions enhance the topological protection of the state, i.e.",
"make it less sensitive to impurities?",
"We will discuss two models that have recently been proposed that show emergent topological properties as discussed above.",
"The first is a model of a single (spinful) channel quantum wire with Ising anisotropy [19], [20].",
"The second is two coupled quantum spin Hall helical edge states [21], [22].",
"In this case, it is important to note that the $\\mathbb {Z}_2$ topological classification of the quantum spin Hall effect means that an even number of coupled edge states (i.e.",
"two) are not topologically protected (in the absence of interactions).",
"The common feature of these two models is that they are each interacting one-dimensional systems with two channels – in the former case, these two channels are the two spin projections; in the latter, the two distinct helical edge states.",
"The low-energy effective Hamiltonian is thus equivalent in both cases.",
"When the interactions are treated [23], [24], the channels rearrange themselves into two independent propagating collective modes – one carrying charge, and the other carrying the remaining degree of freedom (spin in the former case, relative distribution of charge in the latter).",
"We will describe the conditions under which this charge mode remains gapless, but the other mode acquires a gap.",
"In line with our previous discussion, we will see in both cases that this gap can have two possible signs – one of which is topological and exhibits edge states.",
"The system is however not an insulator, as there are gapless modes in the charge sector throughout the entire one-dimensional bulk.",
"The metallic charge sector does not escape the influence of topology however – we show that due to the gap in the spin sector, the charge sector is robust (or even topologically protected) against backscattering from impurities in the same way as a single (topologically protected) helical edge in the quantum spin Hall effect.",
"This is in strong contrast to conventional wisdom of impurities in a one-dimensional Luttinger liquid, whereby the states are localised for all but the strongest attractive interactions [25], while even a single impurity drives the conduction to zero for repulsive interactions [26].",
"The emergent topology therefore manifests itself in two ways – firstly in the presence of edge states in the spin/relative charge sector, and secondly in the protection against backscattering of the gapless charge modes.",
"We will show at the end that these two properties are related, in that one implies the other.",
"We will also discuss how to potentially classify the emergent topological phase, which although inspired from the ideas of topological insulators cannot be manifestly written as a non-interacting theory."
],
[
"Quantum wire with Ising anisotropy",
"The first system we want to discuss is a single channel quantum wire.",
"It is thereby important that the spin-rotation invariance of the electrons in the wire is broken, such that the low energy model of the system contains Ising (easy-axis) anisotropy in the spin degrees of freedom; the reason will become clear shortly.",
"We begin by introducing the low energy model that describes the electrons in the wire.",
"Thereby, it is convenient to define the vector of fermionic fields $ {c}(k) = (c_{\\uparrow ,R} c_{\\downarrow ,R},c_{\\uparrow ,L},c_{\\downarrow ,L}) $ , where $c_{\\sigma ,\\eta }$ (k) destroys a fermion with momentum $k$ , spin $\\sigma =\\uparrow ,\\downarrow $ and chirality $\\eta = R,L$ .",
"In this notation the electron Hamiltonian takes the form $H = H_0 + H_{\\rm int}$ , with the kinetic term H0 = k c(k) h0(k) c(k) .",
"Here, $h_0 = v_F k \\,\\sigma ^0_{\\sigma ,\\sigma ^{\\prime }} \\tau ^z_{\\eta ,\\eta ^{\\prime }}$ is a hermitian matrix, where $v_F$ is the Fermi velocity and $\\sigma ^a,\\tau ^a$ denote the Pauli matrices in spin- and chiral space, respectively.",
"The interaction term is given by Hint = U x n n + V x ( cR, cL, cL, cR, + H.c. ) , with the fermionic density $n_{\\sigma } = c_{\\sigma ,R}^{\\dagger } c_{\\sigma ,R}^{\\phantom{\\dagger }} + c_{\\sigma ,L}^{\\dagger } c_{\\sigma ,L}^{\\phantom{\\dagger }}$ .",
"The electrons in a single channel quantum wire are a good realization of a Tomonaga Luttinger liquid (TLL).",
"The distinctive feature of this state is that the elementary excitations are not single electrons but collective modes: charge plasmons and spinons.",
"Technically, the collective nature of excitations becomes apparent under the bosonization mapping: c,R = R2 a0 ei 4 ( - ) , c,L = L2 a0 ei 4 (+ ) .",
"Here, $\\kappa _{\\eta }$ denote Klein factors and $a_0$ is the short distance cutoff of the field theory.",
"The bosonic fields $\\varphi _{\\sigma }$ and $\\theta _{\\sigma }$ obey the equal time commutation relations [(x), y (y)] = -i (x-y) .",
"Introducing the charge and spin components $\\varphi _{c,s} = ( \\varphi _{\\uparrow } \\pm \\varphi _{\\downarrow }) / \\sqrt{2}$ the Hamiltonian density in real space decouples into charge and spin parts $\\mathcal {H} = \\mathcal {H}_c +\\mathcal {H}_s$ , where $\\mathcal {H}_c =& \\frac{v_c}{2} \\big [ K_c (\\partial _x \\theta _{c})^2 + K_c^{-1}(\\partial _x \\varphi _{c})^2 \\big ]\\, ,\\\\\\mathcal {H}_s =& \\frac{v_F}{2} \\Bigl [ (\\partial _x \\theta _{s})^2 + \\Big ( 1-\\frac{g_{\\parallel }}{\\pi v_F}\\Big ) (\\partial _x \\varphi _{s})^2\\Bigr ] \\\\& +\\frac{g_{\\perp }}{2 (\\pi a_0)^2} \\cos \\big ( \\sqrt{8 \\pi } \\varphi _s \\big ) \\, .$ Here, $K_c \\simeq 1 - a_0 U /2 \\pi v_F$ , $g_{\\parallel } = a_0 U$ , $g_{\\perp } = a_0 V$ and $v_{c,s} = v_F/ K_{c,s}$ .",
"The charge sector describes a Luttinger liquid with plasmon velocity $v_c$ and Luttinger parameter $K_c$ .",
"The spin sector consists of two terms, the kinetic energy, with coupling constant $g_{\\parallel }$ and the potential energy with coupling constant $g_{\\perp }$ .",
"The competition between these two terms determines the phase diagram of the model depicted in Fig.",
"REF .",
"If the potential energy is large the system develops a gap in the spin sector, but is still gapless in the charge sector.",
"The resulting strong coupling phases are thermodynamically equivalent, but have opposite signs of the spin-gap, $\\Delta _s = g_{\\perp }/2\\pi a_0$ .",
"They can be characterised by looking at potential local order parameters.",
"As the charge mode always remains gapless, the order parameters are never nonzero in the thermodynamic limit.",
"Rather the phase of the system is determined by the order parameter with the slowest decaying correlations.",
"For definiteness we consider repulsive interactions in the charge sector, $K_c <1$ .",
"In this case we have to study two possibilities for the order parameter.",
"For $g_{\\perp } <0$ the potential energy $\\sim g_{\\perp } \\cos (\\sqrt{8 \\pi } \\varphi _s)$ is minimized by $\\varphi _{s,n}^{\\text{CDW}} \\equiv \\sqrt{\\pi /2} n$ .",
"For these mean-field configurations, $\\langle \\cos (\\sqrt{2 \\pi } \\varphi _s)\\rangle \\ne 0$ and the dominant correlations are of the CDW type, with the order parameter $\\mathcal {O}_{\\text{CDW}} =& \\sum _{\\sigma ,\\sigma ^{\\prime }} \\sum _{\\eta ,\\eta ^{\\prime }} c_{\\sigma ,\\eta }^{\\dagger } \\sigma ^0_{\\sigma ,\\sigma ^{\\prime }} \\tau ^x_{\\eta ,\\eta ^{\\prime }}c_{\\sigma ^{\\prime },\\eta ^{\\prime }}^{\\phantom{\\dagger }} \\\\=& \\frac{2}{\\pi a_0} \\sin (\\sqrt{2 \\pi } \\varphi _{c}) \\cos (\\sqrt{2 \\pi } \\varphi _{s}) \\, .$ On the other hand if $g_{\\perp } >0$ the potential energy is minimized by $\\varphi _{s,n}^{\\text{SDW}} = \\sqrt{\\pi /2} (n+1/2)$ and the order parameter $\\mathcal {O}_{\\text{SDW}} =& \\sum _{\\sigma ,\\sigma ^{\\prime }} \\sum _{\\eta ,\\eta ^{\\prime }}c_{\\sigma ,\\eta }^{\\dagger } \\sigma ^z_{\\sigma ,\\sigma ^{\\prime }} \\tau ^x_{\\eta ,\\eta ^{\\prime }} c_{\\sigma ^{\\prime },\\eta ^{\\prime }}^{\\phantom{\\dagger }} \\\\=& \\frac{2}{\\pi a_0} \\cos (\\sqrt{2 \\pi } \\varphi _{c}) \\sin (\\sqrt{2 \\pi } \\varphi _{s}) \\, ,$ Figure: Strong coupling phases of the model in Eq.",
"() with K c <1K_c<1.",
"Spin degrees of freedom order while charge degrees of freedom (red solid circles) fluctuate around their average position.describing the z-component of a spin density wave, becomes dominant since $\\langle \\sin (\\sqrt{2 \\pi } \\varphi _s)\\rangle \\ne 0$ .",
"A cartoon picture of the two types of quasi-longrange order is depicted in Fig.",
"REF .",
"The analysis for attractive interactions, $K_c>1$ , is analogous and in that case the dominant order parameters is triplet superconductivity (TS) for $g_{\\perp }>0$ and singlet superconductivity (SS) for $g_{\\perp }<0$ .",
"We will show that one of the gapped phases (SDW or triplet SC) is topological, while the other (CDW or singlet SC) is topologically trivial.",
"We will continue to use terminology for the $K_c<1$ phases, but since the topological properties are solely determined by the spin sector, the results hold also for the corresponding superconducting phases.",
"A subtle point, that is often overlooked in the literature is, that the topological SDW phase can only be realized in systems with broken spin-rotational symmetry.",
"Technically, the SU(2) symmetry in the spin sector manifests itself by the condition that the coupling constants, $g_{\\perp }=g_{\\parallel } =g$ , are equal and thus the system is either in the TLL phase if $g>0$ or in the CDW phase if $g<0$ .",
"Furthermore, we will show that the protection of the system against Anderson localization crucially depends on conserved time reversal symmetry (TRS), so that breaking the SU(2) symmetry by application of a magnetic field is not desired.",
"In the context of quantum wires with broken SU(2) symmetry in the spin sector several experimental realizations of the SDW or TS phase have been proposed.",
"In Ref.",
"[19] the authors considered a quasi-one-dimensional semiconducting quantum wire with strong Rashba spin-orbit coupling (SOC) and identified a parameter regime where the SDW phase forms.",
"A possible realization of the TS phase was proposed in Ref.",
"[20], where the authors studied a semiconducting quantum wire proximity coupled to a superconducting wire with SOC in a certain regime of parameters.",
"Another realization of the TS phase may also be possible in quasi-one dimensional organic conductors where spin anisotropic interactions are believed to be present.",
"[25] Next, we discuss coupled edge states of 2D TIs as another system, where the SDW phase can emerge."
],
[
"Two coupled helical modes",
"The second system where we see the appearance of emergent topological properties are the edge states of the QSH insulator.",
"These edge states have a helical structure, meaning that each consists of two counterpropagating modes with opposite spin orientation.",
"A single helical edge mode is protected against Anderson localization by time reversal symmetry, which forbids elastic scattering between Kramers partners.",
"On the other hand, when two sets of Kramers pairs are coupled, scattering between the states is expected to localize the edge modes.",
"We will show that the above situation changes drastically, when we include interaction between the edge states.",
"The Hamiltonian of the system $H=H_0+H_{\\rm int}$ then consists of the noninteracting part $H_0$ and the interaction Hamiltonian $H_{\\rm int}$ .",
"Defining the vector of fermionic fields ${c}(k)=(c_{\\uparrow ,1}(k),c_{\\downarrow ,1}(k),c_{\\uparrow ,2}(k),c_{\\downarrow ,2}(k))^T$ , where $c_{\\sigma ,a}(k)$ destroys a fermion in the helical mode ($a=1,2$ ) with a spin ($\\sigma =\\uparrow ,\\downarrow $ ) and momentum $k$ , the non-interacting part can be written as $H_0=\\sum _{k}{c}^\\dagger (k)h_0(k){c}(k),$ with the hermitian matrix $h_0=\\delta _{aa^{\\prime }}(v_F\\sigma ^z_{\\sigma \\sigma ^{\\prime }}+\\alpha _{SO}\\sigma ^x_{\\sigma \\sigma ^{\\prime }})k-t_\\perp \\tau ^x_{aa^{\\prime }}\\delta _{\\sigma \\sigma ^{\\prime }}.$ Here, $\\sigma ^{x,y,z},\\tau ^{x,y,z}$ are the Pauli matrices in spin and mode space, respectively.",
"The Hamiltonian $H_0$ accounts for the kinetic energy (with dispersion $\\epsilon _{\\uparrow /\\downarrow }(k)=\\pm v_F k$ ), spin orbit coupling ($\\alpha _{SO}$ ), and tunneling ($t_\\perp $ ) between the helical modes.",
"The interaction Hamiltonian is given by $H_{\\rm int}=U_0\\sum _{x,a}n_a(x)n_a(x)+2U\\sum _{x}n_1(x)n_2(x).$ The interaction constants within the same mode and between different modes are $U_0$ and $U$ respectively.",
"Under generic conditions these two constants are different ($U_0\\ne U$ ).",
"The fermion densities are defined similar to before as $n_a(x)=c^\\dagger _{\\uparrow ,a}(x)c_{\\uparrow ,a}(x)+c^\\dagger _{\\downarrow ,a}(x)c_{\\downarrow ,a}(x)$ .",
"Going to a diagonal basis and bosonizing the system [22], the full Hamiltonian density $\\mathcal {H}_0+\\mathcal {H}_{\\rm int}$ splits into two commuting parts $\\mathcal {H}=\\mathcal {H}_+ + \\mathcal {H}_-$ .",
"The Hamiltonian density $\\mathcal {H}_+$ is given by $\\mathcal {H}_+=\\frac{u_+}{2} \\left[\\frac{(\\partial _x\\varphi _+)^2}{K}+(\\partial _x\\theta _+)^2K\\right],$ with Luttinger parameter $K=\\sqrt{(v+g^{\\prime })/(v+g^{\\prime }+4g)}$ , renormalized velocity $u_+=\\sqrt{(v+g^{\\prime })(v+g^{\\prime }+4g)}$ and $g^{\\prime }=a_0(U_0-U)/2\\pi $ .",
"For energies above $t_\\perp $ , the Hamiltonian density $\\mathcal {H}_-$ is $\\mathcal {H}_- &=&\\frac{u_-}{2}\\left[(\\partial _x\\varphi _-)^2+(\\partial _x\\theta _-)^2\\right]-\\frac{g^{\\prime }}{\\pi a_0^2} \\cos (\\sqrt{8\\pi }\\theta _-),$ with $u_-=v-g^{\\prime }$ .",
"For $g^{\\prime }>0$ , interactions within each helical mode are stronger than between them and the cosine potential in (REF ) has a minimum at $\\theta _-=\\sqrt{\\pi /2}n$ where $n\\in \\mathbb {Z}$ .",
"In this case, the order parameter $\\nonumber \\mathcal {O}_{I}&=&i\\sum _{\\sigma \\sigma ^{\\prime } aa^{\\prime }}\\tilde{c}^\\dagger _{\\sigma a}(\\tau ^y)_{aa^{\\prime }}[\\cos 2\\bar{k}_F x\\;\\sigma ^z-\\sin 2\\bar{k}_F x\\;\\sigma ^y]_{\\sigma \\sigma ^{\\prime }}\\tilde{c}_{\\sigma ^{\\prime } a^{\\prime }}\\\\&=&\\frac{2}{\\pi a_0}\\cos (\\sqrt{2\\pi }\\theta _-)\\cos (\\sqrt{2\\pi }\\varphi _+), $ becomes dominant as $\\langle \\cos \\sqrt{2\\pi }\\theta _-\\rangle \\ne 0$ .",
"Here, $\\bar{k}_F= (k_F^1+k_F^2)/2 \\equiv \\epsilon _F/v$ and we introduced the fermion operators $\\tilde{c}_{\\sigma ,a}=\\sum _{\\sigma ^{\\prime }}(e^{i(\\beta -\\pi /4)\\sigma ^y})_{\\sigma \\sigma ^{\\prime }}c_{\\sigma ^{\\prime },a}$ , with the rotation angle $2 \\beta ={\\rm tan}^{-1}\\left(\\alpha _{SO}/v_F\\right)$ .",
"The presence of $\\sigma ^x$ and $\\sigma ^y$ (rather than $\\sigma ^0$ ) in this order parameter means that these are spin currents.",
"The matrix $\\tau ^y$ in the mode space indicates that a spin current flows between the two spin edges.",
"The spatially dependent part in brackets describes a spiral for the axis of quantization of these currents.",
"We therefore interpret this order as a spin-nematic phase [27] in the spirally varying tilted spin basis, depicted pictorially in Fig.",
"REF Figure: (color online) Patern of spin currents in the spin nematic (g ' >0g^{\\prime }>0) phase, i.e.",
"for inter-helical modes interactionweaker than intra mode interaction.",
"Each spin projection guide traces a spiral along the edge, whilealso moving between the two helical modes (represented by black guides).",
"However within these guides, the order parameter is not the spin itself, but the spin current, represented by the arrows.For $g^{\\prime }<0$ the minimum of $-g^{\\prime }\\cos (\\sqrt{8\\pi }\\theta _-)$ occurs at $\\theta _-=\\sqrt{\\pi /2}\\left(n+1/2\\right)$ with $n$ integer.",
"In this case, the order parameter $\\nonumber \\mathcal {O}_{II}&=&\\sum _{\\sigma \\sigma ^{\\prime } aa^{\\prime }}\\tilde{c}^\\dagger _{\\sigma a}(\\tau ^z)_{aa^{\\prime }}[\\cos 2\\bar{k}_F x\\,\\sigma _z-\\sin 2\\bar{k}_F x\\,\\sigma ^y]_{\\sigma \\sigma ^{\\prime }}\\tilde{c}_{\\sigma ^{\\prime },a^{\\prime }}\\\\&=&\\frac{2}{\\pi a_0}\\sin (\\sqrt{2\\pi }\\theta _-)\\cos (\\sqrt{2\\pi }\\varphi _+), $ is dominant since $\\langle \\sin \\sqrt{2\\pi }\\theta _-\\rangle \\ne 0$ .",
"The only and main difference between this order parameter and that in Eq.",
"(REF ) is the presence of $\\tau ^z$ instead of $\\tau ^y$ .",
"This implies that one has a pattern of spins instead of spin currents, with the two different helical edges antiferromagnetically connected.",
"This order parameter corresponds to a spin density wave, where as before the axis of quantization traces a spiral pattern along the edge of the sample.",
"We illustrate this order parameter in Fig.",
"REF Figure: (color online) A pattern of spins in the SDW phase g ' <0g^{\\prime }<0.",
"Each spin projectiontraces a spiral but edge modes do not mix.Putting these two results together, the entire phase diagram of the problem in the absence of the disorder is depicted in Fig.",
"REF .",
"Figure: (color online).- Dominant order parameter for two interacting helical modes,as a function of the ratio between the interaction strenghts U 0 U_0 (same helical mode) vs UU (different helical modes)for strong tunneling.",
"Above one, dominant correlations are of the spin-nematic type; below one, the dominant correlationsare of spin-density wave type.",
"H - H_- remain gapless at U 0 =UU_0=U.We point out that the order parameters of the strong coupling phases in Eq.",
"() and (REF ) are completely analogous to the ones discussed in the context of quantum wires in Eq.",
"(REF ) and (REF ).",
"To connect with the results of previous sections, we include a table relating the similar operators that appear in the two physical situations.",
"In the following we will discuss the \"topological\" properties of the strong coupling phases in the context of the quantum wire setup.",
"However, all results can be mapped to the model of helical edge modes by means of the substitutions outlined in the table.",
"Table: Comparison between the quantum wire model and the edge of a TRTI with two helical modes."
],
[
"Disorder",
"In this section we study the transport properties of the topological phases in the presence of disorder.",
"We consider both the effect of a single impurity and random disorder and show that one strong coupling phase (CDW) is very susceptible to disorder scattering and will become localized, while the other (SDW) remains a ballistic conductor, even when disorder is added.",
"We model disorder by the Hamiltonian Hdis = d x U(x) c(x) c(x) +H.c.",
".",
"Here, $\\mathcal {U}$ denotes the matrix of the disorder potential in spin and chiral space, whose entries are in general complex.",
"For a single impurity the potential is a delta function $\\mathcal {U}(x) = \\mathcal {U}_{\\rm imp} \\delta (x)$ at the position of the impurity, say $x=0$ .",
"In the case of disorder the potential is a random matrix $\\mathcal {U}_{\\rm dis}(x)$ , which we assume to be gaussian correlated, i.e.",
"$\\overline{\\mathcal {U}^{\\ast }_{dis}(x) \\mathcal {U}_{dis}(y)} = \\mathcal {D} \\delta (x-y)$ .",
"The allowed matrix elements of the disorder potential are severely constricted by the symmetries of the system.",
"First, due to time-reversal symmetry the matrix must be diagonal in spin space.",
"Second, chiral symmetry allows for the decomposition $\\mathcal {U}_{\\eta ,\\eta ^{\\prime }} = \\mathcal {U}^{\\parallel } \\delta _{\\eta ,\\eta ^{\\prime }} +\\mathcal {U}^{\\perp } \\delta _{\\eta ,\\bar{\\eta }}$ , where $\\bar{R} = L$ and vice versa.",
"The Hamiltonian containing the forward scattering component $\\mathcal {U}^{\\parallel }$ can be removed by a unitary transformation.",
"[23] The physical reason is, that forward scattering does not relax current or that impurity scattering inside the same helical edge is forbidden by time reversal symmetry, in the case of helical edges.",
"Summarizing, we study the disorder Hamiltonian.",
"Hdis = d x U(x) c,R(x) c,L(x) +H.c.",
".",
"In the bosonized form the Hamiltonian takes the form Hdis =- i4 d x U(x) e- 2 c (2 s) +H.c.",
".",
"Figure: (a) Schematic showing pinning of a classical density wave by an impurity forspinless electrons.",
"(b) In the SDW case for spinful electrons, the density waves of spin up and spin down electrons are locked out of phase.",
"If the impurity acts equally on the two spin projections (time reversal symmetry), then it can no longer pin the density waves.In the case of single impurity, we can directly analyze the scaling dimension of ().",
"For $g_{\\perp } <0$ the system is in the CDW phase and the expectation value of $\\cos (\\sqrt{2 \\pi } \\varphi _{s})$ is finite.",
"The scaling of the impurity operator is then determined by $\\sin (\\sqrt{2 \\pi } \\varphi _{c})$ and it becomes relevant for $K_c<2$ .",
"We conclude that a single impurity in the CDW phase is a relevant perturbation that will drive the system to an insulating phase.",
"On the other hand if $g_{\\perp } >0$ the system is in the SDW phase, where the expectation value of $\\cos (\\sqrt{2 \\pi } \\varphi _{s})$ vanishes.",
"Corrections to conductance then arise from higher order scattering processes that are generated from the impurity term under the renormalization group flow [28], [29].",
"The leading perturbation $\\sim \\mathcal {U}_{\\rm imp}^2 \\cos (\\sqrt{8 \\pi } \\varphi _c)$ , which describes coherent scattering of two electrons with opposite spin off the impurity, becomes relevant at $K_c<1/2$ .",
"Consequently, the SDW phase can be regarded as a ballistic conductor, even in the presence of an impurity, as long as interactions are not too strong.",
"We emphasize that time reversal symmetry is crucial for the above analysis.",
"If TRS were broken, the impurity Hamiltonian in Eq.",
"() would contain an additional term proportional to $\\cos (\\sqrt{2 \\pi } \\varphi _c) \\sin (\\sqrt{2 \\pi } \\varphi _s)$ which becomes relevant already for $K_c<2$ and renders the SDW phase insulating.",
"To analyze the effect of random disorder, we average over the randomness.",
"This yields the replicated action $\\nonumber S_{\\rm dis}^{AV}&=&\\frac{\\mathcal {D}}{(\\pi a_0)^2}\\sum _{\\alpha \\beta }\\int dx d\\tau _1d\\tau _2\\cos (\\sqrt{2\\pi }\\varphi _s^\\alpha )\\cos (\\sqrt{2\\pi }\\varphi _s^\\beta )\\\\&\\times &\\cos (\\sqrt{2\\pi }[\\varphi _c^\\alpha -\\varphi _c^\\beta ]).$ Deep in the gapped SDW phase we can expand $\\cos (\\sqrt{2\\pi }\\varphi _s)$ around its minimum $\\varphi _s=\\varphi _{s,n}^{\\rm SDW}+\\delta \\varphi _s$ .",
"Integrating out the massive $\\delta \\varphi _s$ mode, the model for the charge field $\\varphi _c$ maps to a Giamarchi-Schultz [25] model with Luttinger parameter $K^{\\rm GS}=2K_c$ .",
"Therefore the random disorder is a relevant perturbation for $K_c<3/4$ .",
"Lastly, we present a semiclassical argument for the protection of the SDW phase against impurity scattering.",
"In a spinless Luttinger liquid the excitations are charge density waves that are pinned, such that the electron density at the position of the impurity is minimized as depicted in Fig.REF (a).",
"In the SDW phase the relative displacement between density waves of opposite spins, $\\sqrt{2 \\pi } \\varphi _s$ , is pinned to $\\pi $ .",
"The total charge density is therefore uniformally distributed and can not be pinned by the impurity, as depicted in Fig.REF (b)."
],
[
"Topological properties",
"Besides the protection against Anderson localization in the charge sector, there is another topological property of the SDW phase: the spin sector hosts zero-energy boundary modes with fractional spin.",
"Let us first consider the spin sector at the exactly solvable Luther-Emery point $K_s \\equiv 1+ g_{\\parallel }/2 \\pi v_F = 1/2$ .",
"At this special point the wave function of the edge states can be explicitly calculated by mapping the sine Gordon model in Eq.",
"(REF ) to a model of spinless fermions with mass $\\Delta _s = g_s / 2 \\pi a$ .",
"On a semi-infinite line with open boundary conditions the fermionic Hamiltonian has zero energy solutions at the boundary.",
"[30] The wave function of the boundary mode 0(x) = svs e-svs |x| decays exponentially into the bulk on the scale of the correlation length $\\xi \\sim v_s/\\Delta _s$ .",
"A crucial point, that is not appreciated in the original publication, is that above solution is only normalizable if $\\Delta _s >0$ , i.e.",
"the edge state only exists if the bulk is in the SDW phase.",
"Next, we consider a boundary between a topologically trivial phase with $\\Delta _s<0$ (e.g., the vacuum) and the topologically nontrivial phase with $\\Delta _s>0$ at the point $x=0$ .",
"Since the field $\\varphi _s$ is pinned to $\\varphi ^{\\text{CDW}}_{s,n_1} = \\sqrt{\\pi /2} n_1$ for $\\Delta _s<0$ and to $\\varphi ^{\\text{SDW}}_{s,n_2} = \\sqrt{\\pi /2} (n_2+1/2)$ for $Delta_s>0$ where $n_1$ , $n_2$ are integers, there must be a kink of minimal magnitude $\\sqrt{\\pi /8}$ in $\\varphi _s$ across the boundary.",
"Such a kink in $\\varphi _s$ corresponds to an accumulation of half of the electron spin at the boundary: $S_z =& \\int \\!",
"\\mathrm {d} x \\, \\rho _s(x) = \\frac{1}{\\sqrt{2 \\pi }} \\int \\!",
"\\mathrm {d} x \\, \\partial _x \\varphi _s(x) = \\pm \\frac{1}{4} \\, .$ It is instructive to compare this behavior with the excitation spectrum in the bulk.",
"The minimal excitation in the bulk is a soliton which corresponds to a transition from one minimum of the cosine potential to the next, $\\varphi _{s,n}^{\\text{SDW}} \\rightarrow \\varphi _{s,n\\pm 1}^{\\text{SDW}}$ with spin $S_z =\\Delta \\varphi _s / \\sqrt{2 \\pi } = \\pm 1/2$ .",
"Since the edge state describes a transition from a minimum of the bulk SDW phase to the minimum of the trivial CDW phase they describe \"half\" of a soliton with spin $S_z = \\pm 1/4$ , as we found above.",
"The soliton excitations, contrasted with a phase boundary between SDW and CDW phase, are depicted pictorially in Fig REF .",
"So far we have discussed the protection against disorder and the edge states of the topologically nontrivial phase separately.",
"However, the protection against impurity scattering and the zero energy bound states at the edge are intricately connected, as we will show now.",
"We consider an infinite system in the topological phase with two impurities at sites $x_1=0$ and $x_2=L$ .",
"As discussed before their Hamiltonian reads as $\\nonumber U_{\\rm well}&=& \\sum _{\\sigma ,i} \\int \\!",
"\\mathrm {d} x \\, h_w \\left( c_{\\sigma ,R}^{\\dagger }(x) c^{\\phantom{\\dagger }}_{\\sigma ,L}(x) + {\\rm H.c.} \\right) \\delta (x-x_i) \\\\&=&\\left.\\frac{2 h_w}{ \\pi a_0} \\sin (\\sqrt{2 \\pi } \\varphi _{c})\\cos (\\sqrt{2 \\pi } \\varphi _{s}) \\right|_{x=0}^{x=L} \\, .$ The energy scale of the impurities $h_w/a_0$ is assumed to be much larger than any other scale in the problem.",
"The potential well (REF ) then pins the field $\\varphi _s$ to the value $\\sqrt{\\pi /2}m $ with $m\\in \\mathbb {Z}$ , close to the boundary.",
"In the bulk the field $\\varphi _s$ is pinned to either $\\varphi ^{\\rm CDW}_{s,n} = \\sqrt{\\pi /2}n$ for $g_{\\perp }<0$ or $\\varphi ^{\\rm SDW}_{s,n} = \\sqrt{\\pi /2}(n+1/2)$ for $g_{\\perp }>0$ .",
"This implies that for $g_{\\perp }>0$ the field $\\varphi _s$ has to change by $\\pm \\sqrt{\\pi /8}$ close to the boundary (see Fig.",
"REF ).",
"As we already discussed, this kink in the $\\varphi _s(x)$ field corresponds to a spin $1/4$ excitation near the edge.",
"The two different ground states, shown in Fig.",
"REF , correspond to configurations with kink and anti-kink pairs.",
"Both configurations have the same energy.",
"This degeneracy of the $\\varphi _s$ field at the edge of the samples allows particles to tunnel in or out at the edges without paying the energy cost of the gap.",
"These modes therefore describe the same topologically protected localized zero-mode at the boundaries of the sample that we discussed before.",
"Figure: (color online).",
"Spatial profile of the ϕ s (x)\\varphi _s(x) field in the topological phase (g ⊥ >0)(g_{\\perp }>0).The two different groundstates in a finite helical system correspond to thetwo choices of kink anti-kink in the boundary, where the field has to minimizethe backscattering potential.",
"Different colors represent different ground state profiles for ϕ s (x)\\varphi _s(x)."
],
[
"Discussion",
"In this work we studied the emergence of topological phases in systems that in the absence of interactions are in a topologically trivial state.",
"We focused on the case of repulsive interactions, and analyzed two physical realizations: a single channel wire with a broken $\\text{SU}(2)$ symmetry and a pair of coupled helical edge states.",
"Although these models may look completely different a first sight, we have shown that they are described by the same low energy effective theory.",
"The low energy physics is characterised by two collective modes – one gapless, carrying charge; and the other gapped, carrying other quantum numbers which is now system specific.",
"In both cases, we have analyzed the possible low energy fixed points and identified “order parameters”, defined in this context as the correlation function that decays slowest with distance.",
"In one phase (a spin density wave in both cases), the ground state has emergent topological properties; while in the other phase, the system remains topologically trivial.",
"It is of crucial importance here that the gapless charge mode means that the local order parameters do not acquire a non-zero expectation value – there is no long range order and no spontaneous symmetry breaking.",
"This means that there is no non-interacting mean-field description of the quasi-ordered state, which makes them rather distinct from the topological superconductors within the Bogoliubov-de-Genne universality classes [12].",
"Although these topological states were motivated by the non-interacting topological insulators, this last statement means that it is not obvious at all how to place them into the classification of Ref. [11].",
"In this framework, one looks at the properties of the Hamiltonian under certain symmetries.",
"While this is relatively straightforward for non-interacting topological insulators, there is an important ambiguity for the present system.",
"One can look at the symmetries of the original interacting Hamiltonian (which has gapless modes so is not strictly speaking an insulator), or one can exploit the spin-charge separation and study only the gapped sector of the model.",
"The two strategies are not equivalent: the former was applied in Ref.",
"[20] putting the model in the class DIII, which is a $\\mathbb {Z}_2$ topological insulator in one dimension; the latter was applied in Ref.",
"[19] putting the model in the class BDI which has a $\\mathbb {Z}$ topological index.",
"It is easy to find potential problems with both methods.",
"Studying the symmetries as a whole, one can imagine, at least in principle, modifying the symmetry in a way that affects only the gapless charge sector.",
"This, by definition, will do nothing to the edge states in the gapped sector, and therefore should not change the symmetry classification.",
"On the other hand, looking only at the gapped spin sector may miss important global properties.",
"To begin with, the spin fields are related to the original fermionic operators in a non-local way.",
"This means that properties that are seemingly topological with respect to the spin field are actually local with respect to the original fermions.",
"More importantly, if another interaction (e.g.",
"Umklapp terms) is added that gaps the charge sector, one is left with a fully ordered spin density wave (Néel) state that has long range order, spontaneously breaks time reversal symmetry, and is clearly not topological.",
"To the best of our knowledge, the question of topological classification of these states is unresolved at present.",
"However, the physical properties resulting from this topology are unambiguous.",
"Edge states exist at the zero-dimensional ends of the one-dimensional systems; and the system is protected against backscattering by impurities (and hence Anderson localization) in the full one-dimensional metallic bulk.",
"Although the underlying physics is identical, the interpretation now depends on the system in question.",
"In the case of coupled helical edge states, it is well known that the topological protection may be destroyed by interactions [17], [18].",
"Here, we have demonstrated the opposite – that the lack of topological protection in two coupled non-interacting edges can be reinstated by interactions.",
"The case of the spin-anisotropic quantum wire is even more intriguing.",
"In this case, we have shown that the topological protection against impurity scattering normally associated with edge states can be achieved, even without the wire being the edge of anything.",
"If this state could be realised in practice, it has many potential applications in quantum electronics due to its close-to-perfect conduction properties."
]
] | 1605.06058 |
[
[
"Search for supersymmetry in the multijet and missing transverse momentum\n final state: an analysis performed on 2.3 fb$^{-1}$ of 13 TeV pp collision\n data collected with the CMS detector"
],
[
"Abstract We present results from a generic search for strongly produced supersymmetric particles in pp collisions in the multijet + missing transverse momentum final state.",
"The data sample corresponds to 2.3 fb$^{-1}$ recorded by the CMS experiment at $\\sqrt{s}=13$ TeV.",
"This search is motivated by supersymmetry (SUSY) models that avoid fine-tuning of the higgs mass.",
"In such models, certain strongly produced SUSY particles, including the gluino and top squark, are predicted to have masses on the order of a TeV.",
"These particles also have some of the highest production cross sections in SUSY and give rise to final states with distinct, high jet multiplicity event signatures.",
"To make the analysis sensitive to a wide range of such final states, events are classified by the number of jets, the scalar sum of the transverse momenta of the jets, the vector sum of the transverse momenta of the jets, and the number of b-tagged jets.",
"No significant excess is observed beyond the standard model (SM) expectation.",
"The results are interpreted as limits on simplified SUSY models.",
"In these models, gluino masses as high as 1600 GeV are excluded at 95\\% CL for scenarios with low $\\tilde{\\chi}_{1}^{0}$ mass, exceeding the most stringent limits set in by CMS at $\\sqrt{s}=8$ TeV by more than 200 GeV in several simplified models."
],
[
"Introduction",
"Strongly produced supersymmetric particles, including gluinos and third generation squarks, are among the most attractive search targets in early $\\sqrt{s}=13$ TeV LHC data.",
"At the mass limits set on simplified models [1] with low $\\tilde{\\chi }_{1}^{0}$ mass at $\\sqrt{s}=8$ TeV, the cross section for gluino pair production, for example, is enhanced by a factor of thirty with respect to $\\sqrt{s}=8$ TeV due to the mass-dependent increase in parton luminosity at $\\sqrt{s}=13$ TeV [2].",
"Consequently, a search for strongly produced SUSY performed at $\\sqrt{s}=13$ TeV needs only a fraction of the data taken at $\\sqrt{s}=8$ TeV to exceed the sensitivity obtained in Run I for many models.",
"This search is also highly motivated by models of natural SUSY.",
"In these models, the stop, left-handed sbottom, and gluino must have masses near the electroweak scale to compensate for large radiative corrections to the mass of the higgs boson from SM particles without fine tuning.",
"Although there is no way to rigorously define what degree of fine tuning is acceptable, a convention among post-Run I natural SUSY models requires the masses of the gluino and top squarks to be less than 2 TeV and 1 TeV, respectively [3], [4].",
"With sufficient data, these models will be probed over much of the natural SUSY parameter space in Run II.",
"In $R$ -parity conserving SUSY models, in which the LSP is the $\\tilde{\\chi }_{1}^{0}$ , gluinos and squarks are expected to decay to multiple hadrons, as well as weakly-interacting SUSY particles that escape detection.",
"To target this signature, we conduct our search in a sample of events with multiple jets, large missing transverse momentum ($H_T^{\\rm miss}$ ), a large scalar sum of jet transverse momenta ($H_T$ ), and no charged electrons or muons.",
"Diagrams for three of the simplified SUSY models we target are shown in Figure REF .",
"This analysis [5] expands and combines strategies from two searches performed by CMS at $\\sqrt{s}=8$ TeV [6], [7].",
"Figure: From left to right, diagrams for the simplified modelspp→g ˜g ˜→bb ¯bb ¯χ ˜ 1 0 χ ˜ 1 0 pp\\rightarrow \\tilde{g}\\tilde{g}\\rightarrow b\\bar{b}b\\bar{b}\\;\\tilde{\\chi }_1^0\\tilde{\\chi }_1^0, pp→g ˜g ˜→tt ¯tt ¯χ ˜ 1 0 χ ˜ 1 0 pp\\rightarrow \\tilde{g}\\tilde{g}\\rightarrow t\\bar{t}t\\bar{t}\\;\\tilde{\\chi }_1^0\\tilde{\\chi }_1^0, pp→g ˜g ˜→qq ¯qq ¯χ ˜ 1 0 χ ˜ 1 0 pp\\rightarrow \\tilde{g}\\tilde{g}\\rightarrow q\\bar{q}q\\bar{q}\\;\\tilde{\\chi }_1^0\\tilde{\\chi }_1^0, andpp→t ˜t ˜ ¯→tt ¯χ ˜ 1 0 χ ˜ 1 0 pp\\rightarrow \\tilde{t}\\bar{\\tilde{t}}\\rightarrow t\\bar{t}\\;\\tilde{\\chi }_1^0\\tilde{\\chi }_1^0."
],
[
"Search strategy and event selection",
"Since SUSY offers an enormous variety of models and final states, we design our search to be inclusive and generic so that we maintain sensitivity to a diverse array of new physics scenarios.",
"We consider events with at least four jets with transverse momenta ($p_T$ ) of at least 30 GeV, at least 200 GeV of $H_T^{\\rm miss}$ , at least 500 GeV of $H_T$ , and no charged leptons.",
"To increase sensitivity to different signal topologies, and to help characterize a potential excess beyond the measured SM backgrounds, we divide our search region into 72 independent search regions defined by the jet multiplicity, the $H_T^{\\rm miss}$ , the $H_T$ , and the b-tagged jet multiplicity of each event.",
"The bins of $H_T^{\\rm miss}$ and $H_T$ that define the search regions are illustrated in Figure REF (left).",
"The dominant SM backgrounds for the all-hadronic multijet + missing momentum final state include production of top quarks and $W$ and $Z$ bosons in association with jets, as well as strongly produced (QCD) multijets.",
"To suppress the multijet background, which arises primarily from mismeasurement of jet momenta resulting in a fake-$H_T^{\\rm miss}$ signature, we reject events in which any of the four jets of largest $p_T$ is aligned with the $H_T^{\\rm miss}$ , specifically, we require $\\Delta \\phi (j_i\\;,\\;H_T^{\\rm miss}) > 0.5,\\;\\;\\;\\;\\Delta \\phi (j_j\\;,\\;H_T^{\\rm miss}) > 0.3$ where $i=1,2$ (i.e., the two leading jets) and $j=3,4$ .",
"Top and $W$ events enter the search region if a $W$ decays either to an electron or muon that is out of kinematic or geometric detector acceptance; if it decays to an electron or muon that fails lepton reconstruction, identification, or isolation requirements; or if it decays to a tau lepton that decays hadronically.",
"To further suppress this background, we reject events with an isolated charged track identified by the particle flow algorithm [8], [9] as an electron, muon, or charged hadron."
],
[
"Background estimation",
"Following the event selection described in Section , the dominant background in bins with large jet and b-tagged jet multiplicity is top quark pair production, while in bins with lower multiplicity, the composition is distributed among $W$ , $Z$ , and QCD multijet events.",
"The background composition, as determined in simulation for select search regions, is shown in Figure REF (center, right).",
"Figure: Distributions from simulated event samples showing kinematic shape comparisons of the number of jets, H T miss H_T^{\\rm miss}, H T H_T, and the number of b-tagged jets for the main background processes and six representative gluino production signal models.",
"The full baseline selection is applied in each plot.As the cross sections of these backgrounds are generally larger than those of our target SUSY models by multiple orders of magnitude, many of the search regions with highest signal sensitivity lie on extreme tails of kinematic distributions (Figure REF ).",
"The tails of these distributions are difficult to model, so we measure their shapes using dedicated techniques and control regions (CRs) in data for each background process.",
"The CRs are designed to capture the kinematic properties of the backgrounds in the search region up to differences understood sufficiently well that they can be corrected by simulation."
],
[
"Results",
"No significant excess is observed beyond the measured SM backgrounds.",
"Figure REF shows the observed data are compared to the summed estimated backgrounds in each of the 72 search regions.",
"The results are interpreted as 95% CL upper limits on the production cross sections for simplified SUSY models.",
"Three of these interpretations are shown in Figure REF .",
"The strongest mass limits on gluino pair production are obtained for the model in which the gluino decays exclusively to $\\tilde{b}\\bar{b}$ , yielding the striking final state $b\\bar{b}b\\bar{b}\\;\\tilde{\\chi }_1^0\\tilde{\\chi }_1^0$ .",
"For this model, we can exclude gluino masses below 1600 GeV for light $\\tilde{\\chi }_1^0$ at the 95% CL, an improvement of over 200 GeV with respect to the strongest limits set by CMS at $\\sqrt{s}=8$ TeV.",
"For simplified models in which the gluino decays to $\\tilde{t}\\bar{t}$ and $\\tilde{q}\\bar{q}$ , we exclude gluino masses below 1550 and 1440 GeV, respectively, for light $\\tilde{\\chi }_1^0$ , also significantly extending previous limits."
],
[
"Acknowledgments",
"I acknowledge support from the US Department of Energy Office of Science and the Graduate Division of the University of California, Santa Barbara."
]
] | 1605.05762 |
[
[
"Space-Time Codes Based on Rank-Metric Codes and Their Decoding"
],
[
"Abstract We propose a new class of space-time block codes based on finite-field rank-metric codes in combination with a rank-metric-preserving mapping to the set of Eisenstein integers.",
"It is shown that these codes achieve maximum diversity order and improve upon certain existing constructions.",
"Moreover, we present a new decoding algorithm for these codes which utilizes the algebraic structure of the underlying finite-field rank-metric codes and employs lattice-reduction-aided equalization.",
"This decoder does not achieve the same performance as the classical maximum-likelihood decoding methods, but has polynomial complexity in the matrix dimension, making it usable for large field sizes and numbers of antennas."
],
[
"Introduction",
"Space-time (ST) codes were introduced in [1] for multiple-input/multiple-output (MIMO) fading channels in point-to-point single-user (multi-antenna) scenarios.",
"Several code constructions have been proposed so far, both ST convolutional and block codes.",
"ST codes are usually maximum-likelihood (ML) decoded, yielding an exponential decoding complexity.",
"An important design criterion for ST codes is that the rank distance of two codewords must be as large as possible [1].",
"In [2], [3], finite-field rank-metric codes were used to construct ST block codes by mapping the finite-field elements to a modulation alphabet in the complex plane.",
"It was shown that this mapping preserves the minimum rank distance of the finite-field code in case of binary phase-shift keying and subsets of the Gaussian integers $\\mathbb {G}$ [4], as well as for other important constellations [5].",
"In this paper, we prove that there is a rank-metric-preserving mapping in the case of Eisenstein integers $\\mathbb {E}$ [6].",
"The use of this modulation alphabet promises to improve upon other modulation alphabets in $\\mathbb {C}$ , since Eisenstein integers form the hexagonal lattice in $\\mathbb {C}$ , the densest possible lattice in a 2-dimensional real vector space.",
"Furthermore, we present an alternative decoding method for these ST codes, using lattice-reduction-aided (LRA) equalization techniques in combination with a decoding algorithm of the underlying finite-field rank-metric code.",
"This decoder is sub-optimal in terms of failure probability compared to the classical ML decoding methods, but has polynomial complexity and therefore can be used for a larger set of parameters.",
"The paper is organized as follows.",
"In Section , we describe the channel model and provide basics on Eisenstein integers and rank-metric codes.",
"We propose a new ST code construction in Section and present alternative decoding methods in Section .",
"Section concludes the paper.",
"We assume a flat-fading MIMO channel with additive white Gaussian noise, i.e.",
"${\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}= {\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}{\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}+ {\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}},$ and ${N_{\\mathrm {tx}}},{N_{\\mathrm {rx}}},{N_{\\mathrm {time}}}\\in \\mathbb {N}$ denote the numbers of transmit antennas, receive antennas and time steps, respectively, ${\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}\\in \\mathbb {E}^{{N_{\\mathrm {tx}}}\\times {N_{\\mathrm {time}}}}$ is the sent codeword and ${\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}\\in \\mathbb {C}^{{N_{\\mathrm {rx}}}\\times {N_{\\mathrm {time}}}}$ is the received word (both over space (rows) and time (columns)).",
"${\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}\\in \\mathbb {C}^{{N_{\\mathrm {rx}}}\\times {N_{\\mathrm {tx}}}}$ is the channel matrix, which is known at the receiver (perfect channel state information) and whose entries are drawn i.i.d.",
"from the zero-mean unit-variance complex Gaussian distribution.",
"Also, ${\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}\\in \\mathbb {C}^{{N_{\\mathrm {rx}}}\\times {N_{\\mathrm {time}}}}$ is the noise matrix, which is unknown at the receiver and whose entries are sampled i.i.d.",
"from a zero-mean complex Gaussian distribution [1].",
"The signal-to-noise (SNR) ratio is given by the transmit energy per information bit $E_\\mathrm {b,TX}$ in relation to the noise power spectral density $N_0$ ."
],
[
"Eisenstein Integers",
"Let $\\omega = \\mathrm {e}^{\\mathrm {j}\\frac{2 \\pi }{3}}$ .",
"Then the ring $\\mathbb {E}:= \\mathbb {Z}[\\omega ] = \\lbrace a+\\omega b : a,b \\in \\mathbb {Z}\\rbrace \\subseteq \\mathbb {C}$ is called Eisenstein integers [6].",
"$\\mathbb {E}$ is a principal ideal domain (PID), a Euclidean domain, and a lattice.",
"The units of $\\mathbb {E}$ are the sixth roots of unity $\\mathbb {E}^{\\times }= \\lbrace \\mathrm {e}^{\\mathrm {j}\\frac{\\ell \\pi }{6}} : \\ell =1,\\dots ,6\\rbrace $ .",
"Let $\\Theta \\in \\mathbb {E}\\setminus \\lbrace 0\\rbrace $ .",
"Then $\\Theta \\mathbb {E}$ is a sub-lattice of $\\mathbb {E}$ and for any $z \\in \\mathbb {C}$ we can define a quantization function $\\mathcal {Q}_{\\Theta \\mathbb {E}}(z) = \\mathop {\\mathrm {argmin}}\\limits _{y \\in \\Theta \\mathbb {E}} |z-y|$ and a modulo function $\\mathrm {mod}_{\\Theta \\mathbb {E}}(z) = z - \\mathcal {Q}_{\\Theta \\mathbb {E}}(z).$ Both $\\mathcal {Q}_{\\Theta \\mathbb {E}}(\\cdot )$ and $\\mathrm {mod}_{\\Theta \\mathbb {E}}(\\cdot )$ can be extended to vector or matrix inputs by applying them component-wise.",
"The Eisenstein integer constellation of $\\Theta \\in \\mathbb {E}\\setminus \\lbrace 0\\rbrace $ is the set $\\mathcal {E}_{\\Theta } = \\lbrace \\mathrm {mod}_{\\Theta \\mathbb {E}}(z) : z \\in \\mathbb {E}\\rbrace .$ Note that $\\mathcal {E}_{\\Theta } = \\mathcal {R}_\\mathrm {V}(\\Theta \\mathbb {E}) \\cap \\mathbb {E}$ , where $\\mathcal {R}_\\mathrm {V}(\\Theta \\mathbb {E})$ is the Voronoi region of the lattice $\\Theta \\mathbb {E}$ [7], [8].",
"$\\mathcal {E}_{\\Theta }$ contains $|\\Theta |^2$ elements.",
"The resulting signal constellation has a hexagonal boundary region and is more densely packed than a signal constellation of the same cardinality over the Gaussian integers or quadrature amplitude modulation, cf. [8].",
"Besides its high packing density, Eisenstein integer constellations have another major advantage compared to classical signal constellations: they possess algebraic structure.",
"In order to use this fact, we need the following lemma.",
"Lemma 1 ([6]) $\\Theta $ is a prime in $\\mathbb {E}$ if one of the following conditions is true.",
"(i) $\\Theta = u \\cdot p$ for some $u \\in \\mathbb {E}^{\\times }$ and $p$ is a prime in $\\mathbb {N}$ with $p \\equiv 2 \\mod {3}$ (Type $\\mathrm {I}$ ).",
"(ii) $|\\Theta |^2 = p$ is a prime in $\\mathbb {N}$ with $p \\equiv 1 \\mod {3}$ or $p=3$ (Type $\\mathrm {II}$ ).",
"We define multiplication and addition of $a,b \\in \\mathcal {E}_{\\Theta }$ as $a \\oplus b = \\mathrm {mod}_{\\Theta \\mathbb {E}}(a + b) \\quad \\text{and} \\quad a \\otimes b = \\mathrm {mod}_{\\Theta \\mathbb {E}}(a \\cdot b),$ where $+$ and $\\cdot $ are the ordinary operations in $\\mathbb {C}$ .",
"Then the set $\\mathcal {E}_{\\Theta }$ with these operations $(\\mathcal {E}_{\\Theta },\\oplus ,\\otimes )$ is a ring and—even stronger—the following theorem holds.",
"Theorem 1 ([6]) Let $\\Theta $ be a prime in $\\mathbb {E}$ .",
"Then $(\\mathcal {E}_{\\Theta },\\oplus ,\\otimes )$ is a finite field.",
"More precisely, the following isomorphisms hold.",
"$(\\mathcal {E}_{\\Theta },\\oplus ,\\otimes ) \\cong {\\left\\lbrace \\begin{array}{ll}\\mathbb {F}_{p^2}, &\\text{if $\\Theta $ is of Type $\\mathrm {I}$}, \\\\\\mathbb {F}_{p}, &\\text{if $\\Theta $ is of Type $\\mathrm {II}$}.\\end{array}\\right.",
"}$ A table of suitable Eisenstein integer constellations of size up to 127 can be found in [8], along with a list of constellations which are subsets of the Gaussian integers.",
"The table also states their resulting average power (mean squared absolute value of a constellation)."
],
[
"Rank-Metric and Gabidulin Codes",
"Rank-metric codes are sets of matrices where the distance of two elements is measured by the rank metric instead of the classical Hamming metric.",
"The most famous class of rank-metric codes are Gabidulin codes, which were independently introduced in [9], [10], [11] and are used in many applications such as random linear network coding [12] and cryptography [13].",
"In general, a rank-metric code $\\mathcal {C}$ over a field $\\mathbb {K}$ is a subset of $\\mathbb {K}^{m \\times n}$ , along with the rank metric $\\operatorname{d_R}: \\mathbb {K}^{m \\times n} \\times \\mathbb {K}^{m \\times n} &\\rightarrow \\lbrace 0,\\dots ,\\min \\lbrace n,m\\rbrace \\rbrace , \\\\({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}},{\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}) &\\mapsto \\operatorname{rank}({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}-{\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}).$ It has minimum rank distance $d := \\min \\limits _{\\begin{array}{c}1,2 \\in \\mathcal {C}\\\\ 1 \\ne 2\\end{array}} \\operatorname{d_R}(1,2).$ Let $q$ be a prime power and $m \\in \\mathbb {N}$ .",
"Thus, $\\mathbb {F}_{q^m}$ can be seen as a vector space of dimension $m$ over $\\mathbb {F}_{q}$ and for some $n \\in \\mathbb {N}$ , there is a mapping $\\mathrm {ext}: \\mathbb {F}_{q^m}^n \\rightarrow \\mathbb {F}_{q}^{m \\times n}, \\; {\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}\\mapsto $ where each component of the vector ${\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}$ is extended into a fixed basisE.g., $\\mathcal {B}=\\left(1,\\alpha , \\alpha ^2, \\dots , \\alpha ^{m-1}\\right)$ , where $\\alpha $ is a primitive element of $\\mathbb {F}_{q^m}$ .",
"$\\mathcal {B}$ of $\\mathbb {F}_{q^m}$ over $\\mathbb {F}_{q}$ .",
"The expansion of the $i$ th component of ${\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}$ is then the $i$ th column of $.A \\emph {linearized polynomial} over $$\\mathbb {F}_{q^m}$$ of $ q$-degree $ df $\\mathbb {N}$ 0$ is a polynomial of the form{\\begin{@align}{1}{-1}f(X) = \\sum \\limits _{i=0}^{d_f} f_i X^{q^i}, \\quad f_i \\in \\mathbb {F}_{q^m}, \\quad f_{d_f} \\ne 0.\\end{@align}}The zero polynomial $ f(X)=0$ is also a linearized polynomial and has $ q$-degree $ df = -$.The set of linearized polynomials over $$\\mathbb {F}_{q^m}$$ is denoted by $ Lqm$.$ Let $k,n \\in \\mathbb {N}$ be such that $k < n \\le m$ .",
"We choose $g_1,\\dots ,g_n \\in \\mathbb {F}_{q^m}$ to be linearly independent over $\\mathbb {F}_{q}$ .",
"A Gabidulin code of length $n$ and dimension $k$ is given by $\\mathcal {C}_\\mathrm {G}[n,k] = \\lbrace [f(g_1),\\dots ,f(g_n)] : f \\in \\mathcal {L}_{q^m}, d_f < k\\rbrace .$ The codewords ${\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}= [f(g_1),\\dots ,f(g_n)] \\in \\mathbb {F}_{q^m}^n$ can be interpreted as matrices $\\mathbb {F}_{q}^{m \\times n}$ using the $\\mathrm {ext}$ mapping and thus, the rank metric is well-defined.",
"The minimum rank distance of $\\mathcal {C}_\\mathrm {G}[n,k]$ is $d = n-k+1$ and therefore fulfills the rank-metric Singleton bound with equality [9], [10], [11].",
"It is shown in [14]In the published version of this paper, a wrong reference was given.",
"that we can reconstruct $\\mathcal {C}_\\mathrm {G}$ from $ {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}+ {\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}_\\mathrm {R}{\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}_\\mathrm {R}+ {\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}_\\mathrm {C}{\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}_\\mathrm {C},$ where $\\operatorname{rank}({\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}) = \\tau $ , ${\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}_\\mathrm {R}\\in \\mathbb {F}_{q}^{m \\times \\rho }, {\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}_\\mathrm {R}\\in \\mathbb {F}_{q}^{\\rho \\times n}$ , ${\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}_\\mathrm {C}\\in \\mathbb {F}_{q}^{m \\times \\delta }, {\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}_\\mathrm {C}\\in \\mathbb {F}_{q}^{\\delta \\times n}$ , whenever $2 \\tau + \\rho + \\delta < d $ and ${\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}_\\mathrm {R}$ and ${\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}_\\mathrm {C}$ are known at the receiver (error and erasure decoder).",
"The decoding complexity is $\\mathcal {O}(m^3)$ operations in $\\mathbb {F}_{q}$ [14], or $\\mathcal {O}^{\\sim }(n^{1.69} m)$ using the algorithms in [15], where $\\mathcal {O}^{\\sim }$ is the asymptotic complexity neglecting $\\log (nm)$ factors.",
"A criss-cross error is a matrix that contains non-zero entries only in a limited number of rows and columns, cf. [11].",
"In general, if such a matrix can be covered with $\\tau _\\mathrm {r}$ rows and $\\tau _\\mathrm {c}$ columns such that outside the cover, there is no error, the matrix has rank $\\le \\tau _\\mathrm {r} + \\tau _\\mathrm {c}$ .",
"Therefore, criss-cross and rank errors are closely related.",
"In this section, we present a new construction method for ST codes based on finite-field rank-metric codes in combination with Eisenstein integers.",
"The construction is similar to the one in [4], but uses a different embedding of the finite-field elements into the complex numbers.",
"We give a proof that this mapping is rank-distance-preserving, which implies that the spacial diversity order of the ST code is lower-bounded by the minimum rank distance of the finite-field code.",
"Furthermore, we present simulation results that show a coding gain compared to the codes constructed in [4]."
],
[
"Code Construction",
"Let $\\mathbb {F}_{q}$ be a finite field which is isomorphic to an Eisenstein integer constellation $\\mathcal {E}_{\\Theta } \\subseteq \\mathbb {C}$ with modulo arithmetic $\\oplus $ and $\\otimes $ , cf.",
"Theorem REF .",
"We choose an isomorphismE.g., if $\\Theta $ is of Type $\\mathrm {II}$ , $q$ is a prime, we can write $\\mathbb {F}_{q}= \\lbrace 0,\\dots ,q-1\\rbrace $ , and $\\varphi (x) = \\mathrm {mod}_{\\Theta \\mathbb {E}}(x)$ for all $x \\in \\mathbb {F}_{q}$ is an automorphism.",
"$\\varphi : \\mathbb {F}_{q}\\rightarrow \\mathcal {E}_{\\Theta }$ and extend the mapping to matrices by applying it entry-wise $\\Phi : \\mathbb {F}_{q}^{m \\times n} &\\rightarrow \\mathbb {C}^{m \\times n}, \\\\[x_{ij}]_{i,j} &\\mapsto [\\varphi (x_{ij})]_{i,j}.$ We can also define a generalized inverse $\\Phi ^{-1} : \\mathbb {E}&\\rightarrow \\mathbb {F}_{q}^{m \\times n}, \\\\[x_{ij}]_{i,j} &\\mapsto [\\varphi ^{-1}(\\mathrm {mod}_{\\Theta \\mathbb {E}}(x_{ij}))]_{i,j}.$ The following theorem lays the foundation for a new class of ST codes based on Eisenstein integers.",
"Theorem 2 The mapping $\\Phi $ is minimum rank-distance-preserving, i.e., for any rank-metric code $\\mathcal {C}\\subseteq \\mathbb {F}_{q}^{m \\times n}$ of minimum distance $d$ the code $\\mathcal {C}_\\mathbb {E}= \\Phi (\\mathcal {C}) \\subseteq \\mathbb {C}^{m \\times n}$ has minimum distance $d$ .",
"W.l.o.g., $n \\le m$ ; otherwise transpose all matrices.",
"Let ${(1)},{(2)}\\in \\mathcal {C}$ , ${(1)}\\ne {(2)}$ .",
"Then, $\\operatorname{rank}({(1)}-{(2)}) \\ge d$ .",
"Take $d$ linearly independent columns of ${(1)}-{(2)}$ , w.l.o.g.",
"${\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(1)}_1-{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(2)}_1,\\dots ,{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(1)}_d-{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(2)}_d \\in \\mathbb {F}_{q}^m$ .",
"We can expand this set of vectors to a basis ${\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(1)}_1-{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(2)}_1,\\dots ,{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(1)}_d-{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(2)}_d,\\tilde{{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}}_{d+1},\\dots ,\\tilde{{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}}_m$ of $\\mathbb {F}_{q}^m$ and define the matrices $\\tilde{^{(1)}&= [{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(1)}_1,\\dots ,{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(1)}_d,\\tilde{{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}}_{d+1},\\dots ,\\tilde{{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}}_m] \\in \\mathbb {F}_{q}^{m \\times m}, \\\\\\tilde{^{(2)}&= [{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(2)}_1,\\dots ,{\\mathchoice{\\mbox{$\\displaystyle c$}}{\\mbox{$\\textstyle c$}}{\\mbox{$\\scriptstyle c$}}{\\mbox{$\\scriptscriptstyle c$}}}^{(2)}_d,{\\mathchoice{\\mbox{$\\displaystyle 0$}}{\\mbox{$\\textstyle 0$}}{\\mbox{$\\scriptstyle 0$}}{\\mbox{$\\scriptscriptstyle 0$}}},\\dots ,{\\mathchoice{\\mbox{$\\displaystyle 0$}}{\\mbox{$\\textstyle 0$}}{\\mbox{$\\scriptstyle 0$}}{\\mbox{$\\scriptscriptstyle 0$}}}] \\in \\mathbb {F}_{q}^{m \\times m}.",
"}Thus, \\operatorname{rank}(\\tilde{^{(1)}-\\tilde{^{(2)}) = m and \\det (\\tilde{^{(1)}-\\tilde{^{(2)}) \\ne 0.",
"}Since \\varphi : \\mathbb {F}_{q}\\rightarrow \\mathbb {E} is an isomorphism, we know that \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\Phi (\\tilde{^{(1)})-\\Phi (\\tilde{^{(2)})) = \\Phi (\\tilde{^{(1)}-\\tilde{^{(2)}), implying that there is an {\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}\\in (\\Theta \\mathbb {E})^{m \\times m} such that \\Phi (\\tilde{^{(1)})-\\Phi (\\tilde{^{(2)}) = {\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}+ \\Phi (\\tilde{^{(1)}-\\tilde{^{(2)}).It follows from Lemmas~\\ref {lem:mod_det} \\mathrm {(*)} and \\ref {lem:det_invariant_under_phi} \\mathrm {(**)} (in the appendix) that{\\begin{@align}{1}{-1}a &:= \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det (\\Phi (\\tilde{^{(1)})-\\Phi (\\tilde{^{(2)}))) \\\\&= \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}+ \\Phi (\\tilde{^{(1)}-\\tilde{^{(2)}))) \\\\&\\overset{\\mathrm {(*)}}{=} \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det (\\Phi (\\tilde{^{(1)}-\\tilde{^{(2)}))) \\\\&\\overset{\\mathrm {(**)}}{=} \\varphi (\\underset{\\ne 0}{\\underbrace{\\det (\\tilde{^{(1)}-\\tilde{^{(2)})}}) \\ne 0.",
"}}Thus, \\det (\\Phi (\\tilde{^{(1)})-\\Phi (\\tilde{^{(2)})) = a + b \\ne 0, for some b \\in \\Theta \\mathbb {E} (\\Theta \\nmid a but \\Theta \\mid b), and \\operatorname{rank}(\\Phi (\\tilde{^{(1)})-\\Phi (\\tilde{^{(2)}))=m (full rank).Hence, the first d columns of \\Phi ({(1)})-\\Phi ({(1)}) are linearly independent and{\\begin{@align}{1}{-1}\\operatorname{rank}(\\Phi ({(1)})-\\Phi ({(2)})) \\ge d,\\end{@align}}proving the claim.",
"}}It can be shown (one of the two design criteria in \\cite {tarokh1998space}) that the diversity order of an ST code is lower-bounded by its rank distance.Since the mapping \\Phi is rank-distance-preserving, we can design the diversity order of the ST code by choosing the finite-field rank-metric code accordingly.\\begin{example}We can take a Gabidulin code \\mathcal {C}_\\mathrm {G}[n,k] over the field \\mathbb {F}_{q^m} with minimum distance d = n-k+1 and obtain an ST code \\mathcal {C}_\\mathrm {ST}= \\Phi (\\mathcal {C}_\\mathrm {G}) with spatial diversity d.The resulting codewords {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}\\in \\mathcal {C}_\\mathrm {ST} are complex matrices of dimension m \\times n (we must choose m = {N_{\\mathrm {tx}}} and n = {N_{\\mathrm {time}}}.",
"If {N_{\\mathrm {tx}}}>{N_{\\mathrm {time}}}, we transpose the codewords and set m = {N_{\\mathrm {time}}} and n = {N_{\\mathrm {tx}}}).Since k can be chosen 1\\le k < n, we are flexible in the tradeoff between code rate \\tfrac{k}{n} and diversity d=n-k+1.In the special case of k=1 (rank-metric repetition code equivalent), the resulting code \\mathcal {C}_\\mathrm {ST} has maximum diversity n.\\end{example}}\\subsection {Numerical Results}}Figure~\\ref {fig:simulation_ML} shows simulation results (\\emph {frame error rate} (FER) over SNR) of a comparison of Gaussian integer ST codes from finite-field Gabidulin codes \\cite {bossert2002space} and our construction presented in Section~\\ref {subsec:new_ST_construcion}.We use the channel model described in Section~\\ref {subsec:channel_model} with {N_{\\mathrm {tx}}}={N_{\\mathrm {rx}}}={N_{\\mathrm {time}}}=4.The ST codes are defined using a \\mathcal {C}_\\mathrm {G}[4,1] code of minimum distance d=4.As usual, we ML-decode ST codes by determining{\\begin{@align}{1}{-1}\\hat{{\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}} = \\mathop {\\mathrm {argmin}}\\limits _{{\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}^{\\prime } \\in \\mathcal {C}} \\Vert {\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}{\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}^{\\prime } - {\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}\\Vert _\\mathrm {F}, \\end{@align}}where \\Vert \\cdot \\Vert _\\mathrm {F} is the Frobenius norm, using exhaustive search.",
"}We use q=13 and q=37 as the field size since for both, there are Gaussian and Eisenstein primes whose constellations are isomorphic to \\mathbb {F}_{13} and \\mathbb {F}_{37} respectively, cf.~\\cite {stern2015lattice}.",
"}\\begin{figure}[b]\\clearpage {}\\begin{tikzpicture}\\end{tikzpicture}\\begin{axis}[width=0.4height=2.2in,at={(0.758in,0.481in)},scale only axis,xmin=-5,xmax=13,xlabel={10 \\log _{10}(E_\\mathrm {b,TX}/N_0) \\mathrm {[dB]} \\longrightarrow },xmajorgrids,ymode=log,ymin=1e-04,ymax=1,yminorticks=true,ylabel={\\mathrm {FER} \\longrightarrow },ymajorgrids,yminorgrids,axis background/.style={fill=white},title style={font=\\bfseries },legend style={at={(0.02,0.02)},anchor=south west, legend cell align=left,align=left,draw=white!15!black}][color=red,solid,mark=+]table[row sep=crcr]{-8 0.908521127709156\\\\-7 0.853307560716066\\\\-6 0.774463556890583\\\\-5 0.672100917307615\\\\-4 0.551023047185442\\\\-3 0.422381766995784\\\\-2 0.301369899917833\\\\-1 0.200781260523445\\\\0 0.124869677662684\\\\1 0.0727824997642748\\\\2 0.0400274788184108\\\\3 0.0208865959940193\\\\4 0.010111935774997\\\\5 0.00433734290601975\\\\6 0.00163660609652608\\\\7 0.000523983351068845\\\\8 0.000150864101078948\\\\9 4.04100270747181e-05\\\\10 4.04100270747181e-06\\\\11 1.3470009024906e-06\\\\12 0\\\\13 0\\\\14 0\\\\15 0\\\\16 0\\\\17 0\\\\18 0\\\\19 0\\\\20 0\\\\21 0\\\\22 0\\\\23 0\\\\24 0\\\\25 0\\\\26 0\\\\27 0\\\\};{Gaussian (q=13)};\\end{axis}[color=blue,solid,mark=x]table[row sep=crcr]{-8 0.903494120341061\\\\-7 0.844508950820997\\\\-6 0.761462304179744\\\\-5 0.653563490887539\\\\-4 0.528101132827759\\\\-3 0.398305472864667\\\\-2 0.279623917348025\\\\-1 0.182431067228815\\\\0 0.111659639811959\\\\1 0.0643772141327335\\\\2 0.0348603833564569\\\\3 0.0176551408289444\\\\4 0.0079715513409394\\\\5 0.0033553792481041\\\\6 0.0012042188068266\\\\7 0.000359649240964991\\\\8 0.000102372068589286\\\\9 2.69400180498121e-05\\\\10 1.3470009024906e-06\\\\11 0\\\\12 0\\\\13 0\\\\14 0\\\\15 0\\\\16 0\\\\17 0\\\\18 0\\\\19 0\\\\20 0\\\\21 0\\\\22 0\\\\23 0\\\\24 0\\\\25 0\\\\26 0\\\\27 0\\\\};{Eisenstein (q=13)};\\end{figure}[color=red,solid,mark=o]table[row sep=crcr]{-6 0.963157512153014\\\\-5 0.934157466721185\\\\-4 0.889223570033165\\\\-3 0.824658579801009\\\\-2 0.739559311253464\\\\-1 0.637769297169597\\\\0 0.526732997137795\\\\1 0.415377765662623\\\\2 0.311622370632865\\\\3 0.221691881332061\\\\4 0.148552996229158\\\\5 0.0926836581709145\\\\6 0.0527118259052292\\\\7 0.0271091726863841\\\\8 0.012317477624824\\\\9 0.00488664758529826\\\\10 0.00169915042478761\\\\11 0.000496115578574349\\\\12 0.000121757303166599\\\\13 1.99900049975012e-05\\\\14 3.63454636318205e-06\\\\15 0\\\\};{Gaussian (q=37)};}[color=blue,solid,mark=asterisk]table[row sep=crcr]{-6 0.959330334832584\\\\-5 0.927665258279951\\\\-4 0.878694289219027\\\\-3 0.808789241742765\\\\-2 0.718396256417246\\\\-1 0.612039434828041\\\\0 0.497678433510517\\\\1 0.384103402844033\\\\2 0.281303893507792\\\\3 0.194164735813911\\\\4 0.124624051610558\\\\5 0.0737058743355595\\\\6 0.0392485575394121\\\\7 0.0185034755349598\\\\8 0.00754350097678433\\\\9 0.00265867066466767\\\\10 0.000775067011948571\\\\11 0.000188996410885466\\\\12 3.81627368134115e-05\\\\13 8.1777293171596e-06\\\\14 9.08636590795511e-07\\\\15 0\\\\};{Eisenstein (q=37)};}}}\\caption {ML decoding results for {N_{\\mathrm {tx}}}={N_{\\mathrm {rx}}}={N_{\\mathrm {time}}}=4, using a Gabidulin code \\mathcal {C}_\\mathrm {G}[4,1] over \\mathbb {F}_{q^4}, mapped to Gaussian and Eisenstein integer constellations with q \\in \\lbrace 13,37\\rbrace .I.i.d.\\ unit-variance complex-Gaussian channel matrix {\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}, additive i.i.d.\\ complex-Gaussian noise matrix {\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}.",
"}\\end{@align}}In both scenarios, our construcion provides a coding gain compared to the Gaussian integer ST codes from~\\cite {bossert2002space}.At \\mathrm {FER}= 10^{-3}, for q=13, the gain is approximately 0.3 dB and in the q=37 case, we are more than 0.6 dB better.This gain is expected since Eisenstein integers are more densely packed in the complex plane than Gaussian integers, cf.~\\cite {conway2013sphere}.",
"}\\section {Alternative Decoding}}The complexity of the ML-decoding method used above is proportional to the number of ST codewords.For instance, the ST code constructed in Example~\\ref {ex:full_diversity_gabidulin_ST_code} has q^{N_{\\mathrm {tx}}} codewords and ML decoding is not possible in sufficiently short time already for small field sizes q or transmit antenna numbers {N_{\\mathrm {tx}}}.",
"}It is interesting to note that although rank-metric codes have been used before to construct new ST codes, to the best of our knowledge, their decoding has not yet been employed.\\footnote {In \\cite {robert2015new} a decoder of a generalized Gabidulin code is used.",
"In their channel model, {\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}} is always the identity matrix and {\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}} naturally contains criss-cross error patterns.",
"Hence, it differs significantly from the channel model for which ST codes were originally designed \\cite {tarokh1998space}.",
"}In this section, we propose a new decoding scheme which utilizes the decoding capabilities of Gabidulin codes in combination with a channel transformation based on LRA equalization.For simplicity, we assume {N_{\\mathrm {tx}}}={N_{\\mathrm {rx}}}, implying that {\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}} is invertible with probability 1 (see, e.g., \\cite {fischer2016factorization} on how LRA equalization works if {N_{\\mathrm {tx}}}\\ne {N_{\\mathrm {rx}}}).",
"}}\\subsection {Channel Transformation using LRA Techniques}}}In LRA zero-forcing linear equalization \\cite {windpassinger2003low,fischer2016factorization}, the inverse channel matrix {\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}^{-1} is decomposed\\footnote {See \\cite {fischer2016factorization} for an overview of different factorization criteria.}",
"into{\\begin{@align}{1}{-1}{\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}^{-1} = {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}{\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}\\end{@align}}such that {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}\\in \\mathbb {E}, \\det {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}\\in \\mathbb {E}^{\\times } (implying {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1} \\in \\mathbb {E}), and the maximum of the row norms{\\begin{@align}{1}{-1}\\max _{i} \\Vert i\\Vert _2 \\quad (i \\text{ is the $i$th row of ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}$})\\end{@align}}is minimal among all decompositions.The problem of finding such a decomposition is equivalent to solving the \\emph {shortest basis problem} (SBP) in an \\mathbb {E}-lattice\\footnote {The same decomposition is possible for Gaussian integers.",
"However, it performs better (in terms of \\max _{i} \\Vert i\\Vert _2) for Eisenstein integers, cf.\\ \\cite {stern2015lattice}.}",
"(with the rows of {\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}^{-1} forming a basis of the lattice).The SBP is NP-hard.However, we can find an approximate solution using the LLL algorithm\\footnote {For Eisenstein integers, the LLL algorithm has to be adapted, cf.~\\cite {stern2015lattice,stern2016advanced}.}",
"in time \\mathcal {O}(m^4).Since we know {\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}, we can compute the alternative receive matrix{\\begin{@align}{1}{-1}\\tilde{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}} = {\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}= {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}+ {\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}.\\end{@align}}Due to {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}\\in \\mathbb {E}^{m \\times n}, we can make a component-wise decision of the entries of \\tilde{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}} to the closest point in \\mathbb {E} using the quantization function and obtain{\\begin{@align}{1}{-1}\\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}} = \\mathcal {Q}_{\\mathbb {E}}(\\tilde{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}) = {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}+ \\mathcal {Q}_{\\mathbb {E}}({\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}) =: {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}+ {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}.\\end{@align}}Since \\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}} \\in \\mathbb {E}^{m \\times n}, we can use the generalized inverse of \\Phi to get back to finite fields{\\begin{@align}{1}{-1}\\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}_\\mathbb {F}&= \\Phi ^{-1}(\\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}) = \\Phi ^{-1}({\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1}) \\Phi ^{-1}({\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}) + \\Phi ^{-1}({\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}) \\\\&=: {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{-1} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}_\\mathbb {F}^{\\vphantom{-1}} + {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\vphantom{-1}}.\\end{@align}}Also, \\det ({\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{-1}) = \\det (\\Phi ^{-1}({\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1})) = \\varphi ^{-1}(\\det ({\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}})^{-1}) \\ne 0 (since \\det ({\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}) is a unit in \\mathbb {E}) and thus, {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{-1} is invertible and we can compute{\\begin{@align}{1}{-1}\\overline{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}_\\mathbb {F}= {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}\\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}_\\mathbb {F}= {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}_\\mathbb {F}+ {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}{\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}.\\end{@align}}We have transformed the MIMO fading channel, which can be seen as a \\emph {multiplicative additive matrix channel} over \\mathbb {C}, into an \\emph {additive matrix channel} over \\mathbb {F}_{q}.Figure~\\ref {fig:illustration_channel_trafo} illustrates the channel transformation procedure.",
"}\\begin{figure}[ht]\\begin{tikzpicture}\\node (Z) at (0,1.5) {{\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}_\\mathbb {F}\\in \\mathcal {C}_\\mathrm {G}};\\node [right] (A1) at (0,0.5*1.5) {\\leavevmode {\\color {darkgreen}\\Phi ({\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}_\\mathbb {F})}};\\node (A) at (0,0) {{\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}\\in \\mathcal {E}_{\\Theta }^{{N_{\\mathrm {tx}}}\\times {N_{\\mathrm {time}}}}};\\node (B) at (0,-1.5) {{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}= {\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}{\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}+ {\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}\\in \\mathbb {C}^{{N_{\\mathrm {rx}}}\\times {N_{\\mathrm {time}}}}};\\node [right] (C1) at (0,-1.5*1.5) {\\leavevmode {\\color {darkgreen}{\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}};\\node (C) at (0,-2*1.5) {\\tilde{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}} = {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}+ {\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}\\in \\mathbb {C}^{{N_{\\mathrm {tx}}}\\times {N_{\\mathrm {time}}}}};\\node [right] (D1) at (0,-2.5*1.5) {\\leavevmode {\\color {darkgreen}\\mathcal {Q}_{\\mathbb {E}}(\\tilde{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}})}};\\node (D) at (0,-3*1.5) {\\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}} = {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}^{-1} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}+ {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}\\in \\mathbb {E}^{{N_{\\mathrm {tx}}}\\times {N_{\\mathrm {time}}}}};\\node [right] (E1) at (0,-3.5*1.5) {\\leavevmode {\\color {darkgreen}\\Phi ^{-1}(\\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}})}};\\node (E) at (0,-4*1.5) {\\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}_\\mathbb {F}^{\\vphantom{-1}} = {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{-1} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}_\\mathbb {F}^{\\vphantom{-1}} + {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\vphantom{-1}} \\in \\mathbb {F}_{q}^{{N_{\\mathrm {tx}}}\\times {N_{\\mathrm {time}}}}};\\node [right] (F1) at (0,-4.5*1.5) {\\leavevmode {\\color {darkgreen}{\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}\\hat{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}_\\mathbb {F}}};\\node (F) at (0,-5*1.5) {\\overline{{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}}_\\mathbb {F}= {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}_\\mathbb {F}+ {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}{\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}\\in \\mathbb {F}_{q}^{{N_{\\mathrm {tx}}}\\times {N_{\\mathrm {time}}}}};\\end{tikzpicture}\\end{figure}\\node (A2) at (2*,-0.5*) {{\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}};\\node [right] (B3) at (2*,-1*) {\\leavevmode {\\color {darkgreen}LLL alg.",
"}};\\node (B2) at (2*,-1.5*) {{\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}},{\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}};\\node [right] (F3) at (2*,-3*) {\\leavevmode {\\color {darkgreen}\\Phi ^{-1}({\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}})}};\\node (F2) at (2*,-4.5*) {{\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}};}[->,>=latex] (Z) -- (A);[->,>=latex] (A) -- (B);[->,>=latex] (B) -- (C);[->,>=latex] (C) -- (D);[->,>=latex] (D) -- (E);[->,>=latex] (E) -- (F);}[->,>=latex] (A) -- (A2);[->,>=latex] (A2) -- (B2);[->,>=latex] (B2) -- (F2);}[->,>=latex,dashed] (B2) -- (C1);[->,>=latex,dashed] (F2) -- (F1);}[rotate=8,dashed,darkgreen] (1.1,-0.75) ellipse (1.7 and 0.15);\\node [right] (A1) at (1.7,0) {\\leavevmode {\\color {darkgreen}MIMO channel}};$ Illustration of the channel transformation."
],
[
"Decoding Using Rank-Metric Decoder",
"In order to see how rank-metric codes can be used to correct errors of the form ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}{\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{}$ , we have a closer look at the error matrix ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}$ .",
"An entry of ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}$ is non-zero if the corresponding entry in ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}$ is large enough (by absolute value) to be closer to some element of $\\mathbb {E}\\setminus \\lbrace 0\\rbrace $ than to 0.",
"It can be observed that the rows of ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}$ have different norms $\\Vert i\\Vert _2$ .",
"Since the entries of ${\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}$ are i.i.d.",
"$\\mathcal {N}(0,\\sigma _n^2)$ distributed for some noise variance $\\sigma _n^2$ , an entry in the $i$ th row of ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}$ is $\\mathcal {N}(0,\\Vert i\\Vert _2^2 \\sigma _n^2)$ distributed (and i.i.d.",
"to other entries in that row).",
"Thus, those rows of ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}$ with larger $\\Vert i\\Vert _2$ tend to contain more errors than others.",
"Since the $\\Vert i\\Vert _2$ 's might differ a lot,Finding the distribution of the row norms of ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}$ is an open problem and is beyond the scope of this paper since it would involve a detailed analysis of the numerical properties of the LLL algorithm.",
"in general, non-zero entries of ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}$ tend to occur row-wise.",
"Also, entries in columns are no longer independent and thus, if there is a relatively large entry in ${\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}$ , this value might influence the entries of the entire column in ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}$ , or ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}$ .",
"We can thus conclude that ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}$ tends to contain criss-cross error patterns and therefore has low rank.",
"We cannot use arbitrary criss-cross error correcting codes because the multiplication by ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}$ in the final error matrix destroys the criss-cross pattern.",
"However, the rank is preserved, meaning that the matrix ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}{\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}$ tends to have low rank and can be corrected using a rank-metric code.",
"Example 1 Let ${N_{\\mathrm {tx}}}={N_{\\mathrm {rx}}}={N_{\\mathrm {time}}}=7$ and $6~\\mathrm {dB}$ SNR.",
"A realistic output of the channel matrix decomposition is ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}\\in \\mathbb {C}^{7 \\times 7}$ with squared row norms: Table: NO_CAPTIONFor instance, the error matrix ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{}$ in the channel transformation procedure can have the form (here, $*$ means that this entry is non-zero, all other entries are zero) ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}=\\begin{bmatrix}*&*& &*&*&*&*\\\\& & & & &*& \\\\*&*&*& &*&*& \\\\& & & & & & \\\\& & & & &*& \\\\*& &*& &*&*&*\\\\& & & & &*& \\\\\\end{bmatrix}\\; \\Rightarrow \\; \\operatorname{rank}({\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}) \\le 4.$ The rows which contain many errors ($i=1$ , 3 and 6) are due to large values of $\\Vert i\\Vert _2^2$ and the corrupted column ($j=6$ ) results from a large value in the $j$ th column of the original noise matrix ${\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}$ , which spreads through the entire column due to the matrix multiplication ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}$ ."
],
[
"Improved Decoding Using GMD",
"Since we know ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}$ , the squared row norms $\\Vert i\\Vert _2^2$ provide reliability information of the rows of ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}$ .",
"Thus, we can use generalized minimum distance (GMD) decoding [20] in combination with an error-and-erasure decoding algorithm for Gabidulin codes (cf.",
"Section REF ) to obtain better results.",
"More exactly, we can start by trying to decode without erasures.",
"Then, incrementally from $\\ell =1$ to $d-1$ , we estimate the likeliest $\\ell $ rows of ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}$ which are in error, using the soft information given by the $\\Vert i\\Vert _2$ 's, say $\\mathcal {E}_\\ell \\subseteq \\lbrace 1,\\dots ,m\\rbrace $ , $|\\mathcal {E}_\\ell | = \\ell $ (e.g., $\\mathcal {E}_2 = \\lbrace 1,6\\rbrace $ in Example REF ).",
"Then we can decompose the error into ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{} {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{} = {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{} {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime } + {\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{} {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime \\prime },$ where ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime \\prime }$ contains non-zero values only in the rows $\\mathcal {E}_\\ell $ and ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime }$ has zero rows in $\\mathcal {E}_\\ell $ .",
"We can re-write ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}{\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime \\prime } = [{\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}]_{\\mathcal {E}_\\ell } [{\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime \\prime }]_{\\mathcal {E}_\\ell }$ , where $[{\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}]_{\\mathcal {E}_\\ell }\\in \\mathbb {F}_{q}^{m \\times \\ell }$ consists of the columns of ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}$ with indices in $\\mathcal {E}_\\ell $ and the rows of $[{\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime \\prime }]_{\\mathcal {E}_\\ell }\\in \\mathbb {F}_{q}^{\\ell \\times m}$ are the non-zero rows of ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime \\prime }$ .",
"The procedure is illustrated in the following example.",
"Example 2 Let ${\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{}$ be as in Example REF and $k=1$ .",
"Thus, our finite-field Gabidulin code has parameters $[7,1]$ , minimum rank distance 7, and we cannot correct the rank error with a half-the minimum rank distance decoder since $\\operatorname{rank}({\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}) = 4 > 3 = \\frac{d-1}{2}$ .",
"Using GMD, we can, e.g., declare $\\ell =2$ erasures as follows (recall that $\\mathcal {E}_2 = \\lbrace 1,6\\rbrace $ ): ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}\\begin{bmatrix}*&*& &*&*&*&*\\\\& & & & &*& \\\\*&*&*& &*&*& \\\\& & & & & & \\\\& & & & &*& \\\\*&\\phantom{*}&*&\\phantom{*}&*&*&*\\\\& & & & &*& \\\\\\end{bmatrix}&={\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}\\left(\\begin{bmatrix}& & & & &\\phantom{*}&\\phantom{*}\\\\& & & & &*& \\\\*&*&*&\\phantom{*}&*&*& \\\\& & & & &\\phantom{*}& \\\\& & & & &*& \\\\& & & & &\\phantom{*} \\\\& & & & &*& \\\\\\end{bmatrix}+\\begin{bmatrix}*&*& &*&*&*&*\\\\& & & & & & \\\\& & & & & & \\\\& & & & & & \\\\& & & & & & \\\\*& &*& &*&*&*\\\\& & & & & & \\\\\\end{bmatrix}\\right) \\\\&={\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{} {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime }+[{\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}]_{\\mathcal {E}_2}\\begin{bmatrix}*&*& &*&*&*&*\\\\*& &*& &*&*&*\\\\\\end{bmatrix},$ where $\\operatorname{rank}({\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{} {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime }) = 2$ (unknown) and $[{\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}]_{\\mathcal {E}_2} \\in \\mathbb {F}_{q}^{7 \\times 2}$ (known) consists of the columns of ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}$ with indices $\\mathcal {E}_2$ .",
"Thus, we can correctly decode due to (REF ) and $2 \\cdot \\operatorname{rank}({\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{} {\\mathchoice{\\mbox{$\\displaystyle E$}}{\\mbox{$\\textstyle E$}}{\\mbox{$\\scriptstyle E$}}{\\mbox{$\\scriptscriptstyle E$}}}_\\mathbb {F}^{\\prime }) + \\operatorname{rank}([{\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{}]_{\\mathcal {E}_\\ell }) = 4 + 2 < 7 = d.$ If we use a Gabidulin code of dimension 1 as in [4] or Example REF , we need to know only one row in ${\\mathchoice{\\mbox{$\\displaystyle Z$}}{\\mbox{$\\textstyle Z$}}{\\mbox{$\\scriptstyle Z$}}{\\mbox{$\\scriptscriptstyle Z$}}}_\\mathbb {F}^{} {\\mathchoice{\\mbox{$\\displaystyle X$}}{\\mbox{$\\textstyle X$}}{\\mbox{$\\scriptstyle X$}}{\\mbox{$\\scriptscriptstyle X$}}}_\\mathbb {F}^{}$ which does not contain an error for decoding successfully.",
"Since there are only as many possibilities as there are rows, we can simply “try” all rows, meaning that iteratively for each row $i$ we declare an erasure in all other rows than the $i$ th one, decode and obtain a candidate codeword.",
"Among these candidates, we then find the one with minimum Frobenius norm difference to the received word as in ().",
"We call this method multi-trial (MT) GMD decoding here."
],
[
"Numerical Results",
"Figure REF shows simulation results.",
"We use ST codes based on a $\\mathcal {C}_\\mathrm {G}[4,1]$ code of minimum distance $d=4$ with an Eisenstein integer constellation of size $q=13$ , and the channel model described in Section REF with ${N_{\\mathrm {tx}}}={N_{\\mathrm {rx}}}={N_{\\mathrm {time}}}=4$ .",
"We compare ML decoding to the alternative decoding methods described in this section; BMD as in Section REF and both GMD and MT GMD as in Section REF .",
"For comparison, we perform factorization and equalization based on both zero-forcing (ZF) linear equalization (as described in Section ) and the minimum mean-squared error (MMSE) criterion.",
"The latter is not described here in detail for reasons of clarity, but can, e.g., be found in [17].",
"Figure: Comparison of ML decoding and alternative decoders (BMD, GMD, MT GMD) based on LRA equalization in the case q=13q=13, N tx =N rx =N time =4{N_{\\mathrm {tx}}}={N_{\\mathrm {rx}}}={N_{\\mathrm {time}}}=4, and 𝒞 G [4,1]\\mathcal {C}_\\mathrm {G}[4,1].",
"ZF and MMSE indicates that the ZF or the MMSE criterion, respectively, was used for both factorization and equalization.I.i.d.",
"unit-variance complex-Gaussian channel matrix 𝐇{\\mathchoice{\\mbox{$\\displaystyle H$}}{\\mbox{$\\textstyle H$}}{\\mbox{$\\scriptstyle H$}}{\\mbox{$\\scriptscriptstyle H$}}}, additive i.i.d.",
"complex-Gaussian noise matrix 𝐍{\\mathchoice{\\mbox{$\\displaystyle N$}}{\\mbox{$\\textstyle N$}}{\\mbox{$\\scriptstyle N$}}{\\mbox{$\\scriptscriptstyle N$}}}.It can be seen that all alternative decoding methods are suboptimal compared to the ML case.",
"The best of the alternatives, multi-trial GMD with MMSE factorization and equalization, is approximately 7 dB worse than ML decoding at FER $10^{-3}$ .",
"This effect can be expected due to the following reasons.",
"The row norms of ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}$ do not provide actual soft information.",
"They merely describe a statistical tendency of the errors in ${\\mathchoice{\\mbox{$\\displaystyle F$}}{\\mbox{$\\textstyle F$}}{\\mbox{$\\scriptstyle F$}}{\\mbox{$\\scriptscriptstyle F$}}}{\\mathchoice{\\mbox{$\\displaystyle Y$}}{\\mbox{$\\textstyle Y$}}{\\mbox{$\\scriptstyle Y$}}{\\mbox{$\\scriptscriptstyle Y$}}}$ .",
"GMD decoding of Gabidulin codes cannot fully utilize soft information.",
"To our knowledge, there is no soft-information decoding algorithm for Gabidulin codes, yet.",
"The LLL algorithm only finds an approximate solution to the shortest basis problem.",
"However, all alternative decoding methods share the advantage that their decoding has polynomial decoding complexity in the parameters ${N_{\\mathrm {tx}}}$ , ${N_{\\mathrm {rx}}}$ , and ${N_{\\mathrm {time}}}$ of the code.",
"It can therefore be used for larger parameter sets.",
"We have presented a new class of space-time codes based on finite-field rank-metric codes and Eisenstein integers.",
"These codes achieve maximum diversity order and improve upon existing ST codes based on Gaussian integers.",
"We have also shown how to decode the new code class in polynomial time using a channel transformation based on lattice-reduction-aided equalization.",
"In future work, the problems causing the sub-optimality of the alternative decoder, as discussed in Section REF , should be solved in order to reduce the gap to ML decoding.",
"Alternatively, a modification of the code construction using concatenation with Hamming-error-correcting codes in the rows can be considered, which could shift all curves (including ML decoding) to lower SNR values since the probability of a row being in error decreases.",
"However, if long Hamming-error correcting codes of large dimension are used, ML decoding becomes impractical due to the large number of codewords, resulting in an advantage for our alternative decoding method."
],
[
"Technical Proofs",
"We choose $\\Theta $ and $\\varphi $ as in Section .",
"Lemma 2 Let ${\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}\\in (\\Theta \\mathbb {E})^{m \\times n}$ , ${\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}\\in \\mathbb {E}^{m \\times n}$ .",
"Then, $\\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}+ {\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}})) = \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det ({\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}})).$ For $a,b \\in \\Theta $ it holds that $\\mathrm {mod}_{\\Theta \\mathbb {E}}(a+b) &= \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\mathrm {mod}_{\\Theta \\mathbb {E}}(a)+\\mathrm {mod}_{\\Theta \\mathbb {E}}(b)), \\\\\\mathrm {mod}_{\\Theta \\mathbb {E}}(a \\cdot b) &= \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\mathrm {mod}_{\\Theta \\mathbb {E}}(a) \\cdot \\mathrm {mod}_{\\Theta \\mathbb {E}}(b)).",
"$ The determinant is a finite sum of finitely many multiplications of matrix elements, so this relation extends to $\\det $ as follows: $&\\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}+{\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}})) \\\\&= \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det (\\mathrm {mod}_{\\Theta \\mathbb {E}}({\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}}))) = \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det ({\\mathchoice{\\mbox{$\\displaystyle B$}}{\\mbox{$\\textstyle B$}}{\\mbox{$\\scriptstyle B$}}{\\mbox{$\\scriptscriptstyle B$}}})),$ which proves the claim (note that $\\mathrm {mod}_{\\Theta \\mathbb {E}}({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}})={{\\mathchoice{\\mbox{$\\displaystyle 0$}}{\\mbox{$\\textstyle 0$}}{\\mbox{$\\scriptstyle 0$}}{\\mbox{$\\scriptscriptstyle 0$}}}}$ ).",
"Lemma 3 For any ${\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}\\in \\mathbb {F}_{q}^{m \\times n}$ , $\\varphi (\\det ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}})) = \\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det (\\Phi ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}))).$ Since $\\varphi : \\mathbb {F}_{q}\\rightarrow (\\mathbb {E},\\oplus ,\\otimes )$ is an isomorphism, $\\varphi (\\det ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}})) = \\det _{\\oplus ,\\otimes }(\\Phi ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}))$ , where $\\det _{\\oplus ,\\otimes }$ is the determinant under modulo addition $\\oplus $ and multiplication $\\otimes $ .",
"We obtain $\\mathrm {mod}_{\\Theta \\mathbb {E}}(\\det (\\Phi ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}))) = \\det \\nolimits _{\\oplus ,\\otimes }(\\Phi ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}})) = \\varphi (\\det ({\\mathchoice{\\mbox{$\\displaystyle A$}}{\\mbox{$\\textstyle A$}}{\\mbox{$\\scriptstyle A$}}{\\mbox{$\\scriptscriptstyle A$}}}))$ , where the first equality follows by (REF ) and ()."
]
] | 1605.05716 |
[
[
"Longitudinal spin-relaxation of donor-bound electrons in direct bandgap\n semiconductors"
],
[
"Abstract We measure the donor-bound electron longitudinal spin-relaxation time ($T_1$) as a function of magnetic field ($B$) in three high-purity direct-bandgap semiconductors: GaAs, InP, and CdTe, observing a maximum $T_1$ of $1.4~\\text{ms}$, $0.4~\\text{ms}$ and $1.2~\\text{ms}$, respectively.",
"In GaAs and InP at low magnetic field, up to $\\sim2~\\text{T}$, the spin-relaxation mechanism is strongly density and temperature dependent and is attributed to the random precession of the electron spin in hyperfine fields caused by the lattice nuclear spins.",
"In all three semiconductors at high magnetic field, we observe a power-law dependence ${T_1 \\propto B^{-\\nu}}$ with ${3\\lesssim \\nu \\lesssim 4}$.",
"Our theory predicts that the direct spin-phonon interaction is important in all three materials in this regime in contrast to quantum dot structures.",
"In addition, the \"admixture\" mechanism caused by Dresselhaus spin-orbit coupling combined with single-phonon processes has a comparable contribution in GaAs.",
"We find excellent agreement between high-field theory and experiment for GaAs and CdTe with no free parameters, however a significant discrepancy exists for InP."
],
[
"Introduction",
"In the last decade, the prospects for spin-based quantum information have spurred renewed interest in the fundamental mechanisms for spin relaxation in semiconductors [1], [2], [3], [4].",
"Shallow impurities in direct-bandgap materials are promising candidates for quantum applications relying on spin-photon interfaces [5], [6], [7], as these systems boast high optical homogeneity [8], strong spin-photon coupling, and the potential in II-VI materials [9] to enhance spin coherence times with isotope purification [10], [11].",
"While electron spin relaxation is now relatively well understood in III-V semiconductor quantum dots both theoretically and experimentally [4], [12], [13], [14], [15], [16], [17], it is still an open question whether the same processes dominate in the similar direct band-gap donor system.",
"In contrast to quantum dots, in which the size, shape, composition, and strain field for each dot are to a large extent unknown, the physical properties relevant to spin relaxation for the homogeneous donor system have been measured.",
"This enables quantitative comparison of spin-relaxation rates between theory and experiment which should help predict which donor systems are most promising for future applications.",
"Here we measure the longitudinal spin-flip time $T_1$ , the fundamental limit for the storage time for quantum information, in three semiconductors: GaAs, InP, and CdTe.",
"All three are direct bandgap materials with similar band structure allowing for the optical pumping of the donor-bound electron spins under resonant exciton excitation.",
"We show that at low magnetic fields, ${T_1}$ is proportional to ${B^2}$ with a proportionality constant highly dependent on temperature and donor density.",
"At high magnetic fields, we find that $T_1$ is proportional to $B^{-\\nu }$ , with the power $\\nu $ in the range ${3\\lesssim \\nu \\lesssim 4}$ .",
"The competition of these two dependencies leads to a maximum of $T_1$ in GaAs and InP at relatively high magnetic field: $(1.4 \\pm 0.1)~\\mathrm {ms} \\mathrm {~at~} 4~\\mathrm {T}$ for GaAs and $(0.40 \\pm 0.01)~\\mathrm {ms} \\mathrm {~at~} 1.9~\\mathrm {T}$ for InP.",
"Due to technical issues, we are unable to observe this maximum for CdTe; however, the highest $T_1$ measured is $(1.23\\pm 0.07)~\\mathrm {ms} \\mathrm {~at~} 1.1~\\mathrm {T}$ with $T_1$ expected to rapidly increase at lower fields.",
"The low magnetic-field $T_1$ behavior for GaAs and InP is consistent with a spin relaxation mechanism controlled by the hyperfine coupling of the electron spin with static fluctuations of the host-lattice nuclear spins.",
"In this situation, spin precession is randomized due to the finite electron correlation time at each donor site [18], [19].",
"Although the mechanism for the extremely-short correlation time $\\tau _c$ ($\\tau _{c,\\mathrm {GaAs}} \\simeq 25~\\mathrm {ns}$ , $\\tau _{c,\\mathrm {InP}} \\simeq 40~\\mathrm {ns}$ ) is not completely clear, our measurement is consistent with prior works [19], [20].",
"Our results show that the nuclear-spin environment, known to be the dominant factor in spin dephasing [12], [2], plays an important role in longitudinal relaxation even at low doping densities ($\\sim $ 10$^{14}$ cm$^{-3}$ ) and moderate magnetic fields (up to several tesla).",
"On the high-field side, the similar magnetic-field dependence observed in all three semiconductors is suggestive of a universal mechanism.",
"We theoretically investigate the dominant spin-relaxation mechanisms and find that two mechanisms, (i) the direct spin-phonon interaction and, (ii) the admixture mechanism caused by Dresselhaus spin-orbit coupling combined with the piezoelectric electron-phonon interaction, can account for the magnitude of the observed relaxation in GaAs and CdTe.",
"The strength of the direct spin-phonon interaction is surprising because it was found to be negligible in the similar quantum dot system [4].",
"We find, however, that both interactions are too weak to account for the observed relaxation in InP.",
"The paper is organized as follows: Section presents the studied samples and experimental technique to measure $T_1$ , the experimental results are summarized in Sec. .",
"Section presents the theory and comparison with experiment.",
"The paper is summarized by a short conclusion .",
"Appendices include additional experimental and theoretical details."
],
[
"Samples and experimental technique",
"We study two GaAs, three InP, and two CdTe $n$ -doped samples with the parameters given in Table REF .",
"Spin-relaxation is measured optically in the Voigt geometry (photon wave vector $\\mathbf {k}\\perp \\mathbf {B}$ ) with the magnetic field aligned parallel to the sample surface.",
"Magneto-photoluminescene spectra exhibiting optically resolved Zeeman transitions for all three semiconductors are shown in Appendix .",
"$\\Lambda $ -transitions suitable for optically pumping the electron spin are found by resonantly exciting one of the Zeeman sublevels of the neutral donor (D$^0$ ) to the lowest neutral donor-bound exciton (D$^0$ X) transition and observing the corresponding Raman transition.",
"The optically excited and collected transitions for InP (GaAs, CdTe) are labelled in the energy diagram and photoluminescence spectra in Figs.",
"REF (a),(b) [Appendix , Figs.",
"REF (a),(b),(e),(f)].",
"Table: Sample parameters.",
"N e =N D -N A {N_e= N_D - N_A} is the electron density, ℓ\\ell is the sample thickness.",
"Metal organic vapour phase epitaxy and molecular beam epitaxy are abbreviated by MOCVD and MBE respectively.",
"The InP epilayer is grown directly on an InP substrate.",
"The GaAs epilayer is grown on 4 microns of Al 0.3 _{0.3}Ga 0.7 _{0.7}As on a GaAs substrate.",
"Further details on sample growth are given in the references.Figure: (a) Energy level diagram for the InP donor system.",
"(b) Photoluminescence spectrum of InP.",
"Excitation at 1.549 eV with 50 μ\\mu W power, for the two above-bandgap excitation spectra (red and blue).",
"σ\\sigma (π\\pi ) denote linear collection polarization perpendicular (parallel) to the magnetic field.",
"Resonant excitation spectrum (black) uses excitation at 1.417 eV with 100 μ\\mu W π\\pi -polarized light, with σ\\sigma -polarized light collected.",
"(c) Pulse sequence for optical pumping.",
"The Ti:Sapphire laser is pulsed on and off repetitively on the π\\pi transition, while PL from the σ\\sigma transition is detected.",
"The time between pulses significantly exceeds T 1 T_1.",
"(d) Optical pumping trace for InP with laser power 10 μ\\mu W. The inset sketches the population transfer process during optical pumping.",
"The amplitude of the exponential curve is proportional to the population in ↑\\uparrow .",
"(e) Pulse sequence for T 1 T_1 measurement.",
"The detector gate-on time is 2 μ\\mu s and the laser pulse length is 50 μ\\mu s.(f) T 1 T_1 measurement for InP with laser power 10 μ\\mu W. The data is fit with an exponential plus a background yielding the time constant T 1 =(0.23±0.1)ms{T_1 = (0.23\\pm 0.1)~\\text{ms}}.",
"Error bars denote the standard deviation of the recovery signal in each time bin over the many repetitions of the pulse sequence.",
"The corresponding representative data for GaAs and CdTe are given in Appendix .",
"All experiments used ∼\\sim 30 μm\\mu \\mathrm {m} laser spot size.To measure the spin relaxation time in the magnetic field, we optically deplete one of the Zeeman spin sublevels and monitor the recovery of its thermal population in the course of spin relaxation.",
"At high magnetic fields, the optically-resolved spin Raman transitions enable frequency-selective optical pumping of the donor electron state.",
"At low fields, while the transitions cannot be spectrally resolved, optical pumping is still obtained by utilizing the optical polarization selection rules.",
"Optical pumping is confirmed by monitoring the time-dependence of the collected transition intensity during optical excitation after the system has reached thermal equilibrium.",
"A typical high-field optical pumping pulse sequence and photoluminescence trace are depicted in Figs.",
"REF (c),(d).",
"The decrease in photoluminescence intensity is only observed with resonant spin excitation.",
"Two-laser experiments in GaAs have also confirmed that this decrease is due to spin-pumping and not, for example, due to photo-induced ionization [25].",
"A clear optical pumping signal cannot be observed in the highest purity InP sample, InP-1.",
"The cause is attributed to surface depletion effects discussed further in Appendix .",
"For the remainder of the paper we will restrict ourselves to the remaining six samples, where reliable signals are detected.",
"Spin-relaxation measurements are performed by varying the recovery time between optical pumping pulses which are produced by an acousto-optic modulator (AOM) from the output of a narrow-band continuous-wave Ti:Sapphire laser.",
"The AOM extinction ratio, $r_e$ was measured to be ${>}10^4$ giving an upper-bound of the maximum measurable $T_1$ of $r_e\\tau _{op}$ , in which $\\tau _{op}$ is the characteristic timescale of optical pumping.",
"Given the several microsecond $\\tau _{op}$ [Fig.",
"REF (d)], we have the ability to measure $T_1$ exceeding 10 ms.",
"The “Raman” photoluminescence is collected during the first part of the optical pumping pulse, see Fig.",
"REF (e).",
"As the recovery time increases, we observe an increase in the collected signal as the system returns to thermal equilibrium.",
"At each magnetic field, the recovery is fitted to a weighted exponential with time constant $T_1$ [26], as shown in Fig.",
"REF (e).",
"Measurements are performed for fields up to 7.0 T. Reduced visibility of the optical pumping signal places a technical limit on the minimum magnetic field measurement for each sample."
],
[
"Experimental Results",
"The longitudinal spin relaxation times $T_1$ as a function of the electron Zeeman splitting ${\\Delta E = |g\\mu B|}$ for InP, GaAs and CdTe are shown in Fig REF .",
"Here, $g$ is the effective electron $g$ -factor and $\\mu $ is the Bohr magneton.",
"The data show several notable features.",
"First, all samples approach a ${T_1 \\sim B^{-\\nu }}$ dependence, with ${3\\lesssim \\nu \\lesssim 4}$ , at high magnetic fields.",
"The proportionality constant depends on the semiconductor sample.",
"A $B^{-3}$ dependence, included in Fig.",
"REF , fits all curves well, however we note that higher-field data would be desirable for GaAs because the small electron $g$ -factor prevents us from accessing the high-Zeeman-splitting limit, where ${|g\\mu B| \\gg k_B T}$ .",
"Also, a $B^{-4}$ power-law is reasonable for CdTe, as the magnetic field dependence becomes steeper in CdTe with decreasing field, see also Fig.",
"REF .",
"The high-field $T_1$ process appears to be independent of donor concentration.",
"Even the $T_1$ curve for the high-density InP-2 sample approaches the InP-1 curve at the highest fields.",
"At low fields, $T_1$ in InP and GaAs approaches a $B^2$ dependence with a donor-concentration-dependent pre-factor.",
"This is extremely pronounced for the InP samples in which the donor-bound electron density $N_e$ , the difference between the donor and acceptor densities in the sample, ${N_D-N_A}$ , differs by a factor of 4.",
"The effect is also present in GaAs in which $N_e$ differs by a factor of 1.7.",
"Finally, the maximum $T_1$ observed in all three materials is similar: $T_1=1.4$ , 0.4, and 1.2 ms for GaAs, InP, and CdTe respectively.",
"Figure: (a) T 1 T_1 as a function of the Zeeman splitting for InP-2 at 1.5 K. The arrows show the magnetic field values at which the temperature dependence study was performed.",
"(b-d) Temperature dependence of T 1 T_1 at (b) B=0.48T{B = 0.48~\\text{T}}, (c) 1.9 T, and (d) 5.7 T. The dotted line denotes |gμB|/k B {|g\\mu B|/k_B}.Figure: T 1 T_1 as a function of Zeeman splitting for CdTe-1 at T=1.5K{T = 1.5~\\text{K}} and T=5K{T = 5~\\text{K}}.",
"The red and blue lines are mutually fitted by an empirical formula T 1 =bB 4 /F ph {T_1 = b B^4 / F_{ph}}, where b=2000μs/T 4 {b = 2000 ~\\mu s/\\mathrm {T}^4}.",
"The red and blue dashed lines denote the energy at 1.5 K and 5 K.Measurements of the temperature $T$ effect on $T_1$ are also performed.",
"In InP-2, the sample in which $T_1$ can be obtained for the largest range of Zeeman energies, $T_1(T)$ was measured at 0.5 T (low field regime), 1.9 T (peak $T_1$ ), and 5.7 T (high-field regime) with the results depicted in Fig.",
"REF .",
"In the low-field regime, an extremely-steep inverse dependence of $T_1$ on temperature is observed indicative of a strong phonon-assisted process.",
"In the high-field regime, the relaxation time is almost independent of temperature at the lowest temperatures in our experiments, and drops with an increase in $T$ .",
"This high-field behavior is consistent with a model in which $T_1$ is inversely dependent on the phonon factor $F_{ph} = 2N_{ph} + 1$ , in which ${N_{ph} = [\\exp (|g\\mu B|/k_BT)-1]^{-1}}$ is the phonon occupation number.",
"A comparison of magnetic-field-dependent measurements at 1.5 K and 5 K for CdTe-1 also support a high-field single-phonon mechanism.",
"The ratio of the two curves in Fig.",
"REF is given by $F_{ph}(5~{\\rm K})/F_{ph}(1.5~\\rm K)$ ."
],
[
"Theory",
"Here we consider the mechanisms resulting in spin relaxation of donor-bound electrons.",
"We start with the limit of relatively-low magnetic fields, where spin relaxation is controlled by the hyperfine coupling of the electron and nuclear spins.",
"Next, we turn to the regime of high enough magnetic fields where the nuclei-induced spin relaxation is unimportant and the spin-flip processes caused by the joint effects of the electron-phonon and the spin-orbit interactions play the major role."
],
[
"Low-field spin-relaxation",
"At low temperatures and low donor densities, the electrons in bulk semiconductors are localized.",
"At low and moderate magnetic fields, the electron spin relaxation is controlled by the hyperfine interaction with the host lattice nuclei [18], [27].",
"The spin dynamics of the electron in the ensemble of donors obey the set of kinetic equations [28], [29] $\\frac{d \\mathbf {S}_i}{dt} + \\mathbf {S}_i \\times \\mathbf {\\Omega }_i = \\mathbf { \\mathcal {Q}}_i,$ where $\\mathbf {S}_i$ is the electron spin at the site $i$ , $\\mathbf {\\Omega }_i = \\mathbf {\\Omega }_{i,{\\rm nucl}} + \\mathbf {\\Omega }_B$ is the electron spin precession frequency caused by the hyperfine interaction with nuclear spins, $\\mathbf {\\Omega }_{i,{\\rm nucl}}$ , and by the Larmor precession in the external field, $\\mathbf {\\Omega }_B$ .",
"The collision integral $\\mathbf { \\mathcal {Q}}_i$ describes the variations of the spins due to the electron hopping between sites, processes of ionization and recombination, exchange diffusion, etc.",
"[18], [30].",
"The schematic illustration of the spin dynamics of localized electrons is presented in Fig.",
"REF (a).",
"Here we employ the simplest model of the collision integral by introducing a single correlation time $\\tau _c$ , disregarding the spread of the transition probabilities [18], [28].",
"We assume that the nuclear fluctuations are frozen on the timescale of $\\tau _c$ and that the Zeeman splitting in the external field is negligible as compared with the thermal energy.",
"Hence, we obtain a simple analytical formula for the relaxation time of the spin component parallel to the magnetic field ${\\mathbf {B}\\parallel z}$ [28]: $T_{1,hf} = \\frac{\\tau _c \\mathcal {A}}{1-\\mathcal {A}},$ where $\\mathcal {A} = \\left\\langle \\frac{1+ \\Omega _{i,z}^2\\tau _c^2}{1+\\Omega _i^2\\tau _c^2} \\right\\rangle ,$ and the angular brackets denote the averaging over the distribution of random nuclear fields.",
"Equation (REF ) is valid for an arbitrary relationship between the spin precession frequency and $\\tau _c$ .",
"In the experimentally relevant range of magnetic field, ${\\Omega _B = |g\\mu B|/\\hbar }$ exceeds by far the spin precession frequency in the field of nuclear fluctuations and the inverse correlation time.",
"It follows then from Eqs.",
"(REF ), (REF ) that $T_{1,hf}= \\frac{3\\tau _c\\Omega _B^2}{2\\langle \\Omega _{\\rm nucl}^2\\rangle }\\propto \\tau _c B^2,$ where $\\langle \\Omega _{\\rm nucl}^2\\rangle $ is the mean square fluctuation of the nuclear field averaged over the ensemble of donors.",
"This expression shows the $B^2$ power law which is observed in experiment, Fig.",
"REF .",
"This increase in spin-relaxation time with increasing field is related to the suppression of the relaxation by the magnetic field: At ${\\Omega _B \\gg \\tau _c^{-1},\\langle \\Omega _{\\rm nucl}^2\\rangle ^{1/2}}$ , the electron spin precesses around the total field ${\\mathbf {\\Omega }_{B} + \\mathbf {\\Omega }_{i,{\\rm nucl}}}$ during the correlation time.",
"Its precession axis is almost parallel to $\\mathbf {\\Omega }_B$ and its orientation changes by a small random angle $\\sim \\Omega _{i,{\\rm nucl}}/\\Omega _B$ when the electron hops between the localization sites.",
"Such a random process results in the spin relaxation rate $\\sim \\tau _c^{-1}(\\Omega _{i,{\\rm nucl}}/\\Omega _B)^2 \\propto 1/(\\tau _cB^{2})$ in agreement with Eq.",
"(REF ).",
"For known mechanisms of electron correlation time at a donor, such as electron hopping and the exchange diffusion, see Ref.",
"[30] for review, an exponential sensitivity to the donor density (and, in the former case, to the temperature) is expected [30], [31].",
"Correspondingly, for these mechanisms $T_1$ should be strongly affected by these parameters.",
"Such trends are clearly seen in the experiment, Fig.",
"REF and Fig.",
"REF (b).",
"The developed model enables quantitative comparison with the experiment.",
"To that end, we evaluate the mean square of the donor-bound electron spin precession frequency in the nuclear field as [32] $\\langle \\Omega _{\\rm nucl}^2 \\rangle = \\frac{{V}_0}{8\\pi (a_B^*)^3 \\hbar ^2}\\sum _\\alpha {\\left(A_\\alpha ^{hf}\\right)^2} I_\\alpha (I_\\alpha +1),$ where ${a_B^*{=\\varepsilon \\hbar ^2/(m^*e^2)}}$ is the donor Bohr radius, ${V_0 = a_0^3}$ is the unit lattice volume, $I_\\alpha $ is the spin of $\\alpha ^{\\rm th}$ nucleus in a unit cell, $A_\\alpha $ is the hyperfine interaction constant.",
"Taking for GaAs ${A_{^{69}\\rm Ga} = 38.2~\\mu \\text{eV}}$ , ${A_{^{71}\\rm Ga} = 48.5~\\mu \\text{eV}}$ and ${A_{^{75}\\rm As} = 46~\\mu \\text{eV}}$ [33] we obtain $\\sqrt{\\langle \\Omega _{\\rm nucl}^2 \\rangle } = 0.47\\times 10^8~\\mbox{s}^{-1}$ .",
"Fitting the experimental data with Eq.",
"(REF ), we determine a correlation time $\\tau _c \\approx 25$ ns for the GaAs-2 sample.",
"Such a value of the correlation time is consistent with previous studies of GaAs samples with similar donor densities [19], [20].",
"A somewhat longer $\\tau _c$ of $\\sim 40$ ns is determined for the InP-2 sample, where the hyperfine interaction is dominated by $^{115}$ In isotopes with ${I_{\\rm In} = 9/2}$ .",
"The estimate for $A_{\\rm In}$ comes from Ref.",
"[34] where the Overhauser effect for InSb was measured.",
"The literature reports a spread of $A_{\\rm In}$ : 47 $\\mu $ eV [35], 56 $\\mu $ eV [36] and 84 $\\mu $ eV [33].",
"Here we use the middle value of ${A_{\\rm In} = 56~\\mu \\text{eV}}$ , which yields $\\sqrt{\\langle \\Omega _{\\rm nucl}^2 \\rangle } = 1.6\\times 10^9~\\mbox{s}^{-1}$ .",
"Although the experimental sensitivity of $T_1$ to temperature and carrier density are consistent with the known mechanisms contributing to the donor electron correlation time, the magnitude of $\\tau _c$ is orders of magnitude shorter than these mechanisms predict for the low donor densities used in this study.",
"Our result is consistent with prior works [19], [20] and suggests additional, unknown mechanisms may be at play, such as an inhomogeneous donor distribution resulting in the formation of clusters with a relatively high donor density, and short $\\tau _c$ .",
"According to Eq.",
"(REF ), the electron spin relaxation time associated with the hyperfine interaction strongly increases with an increase in field.",
"Hence, at sufficiently strong magnetic fields this mechanism becomes inefficient as compared with mechanisms caused by the combination of the electron-phonon and spin-orbit interactions described below.",
"By contrast, $T_1$ due to these processes decreases with an increase in $B$ .",
"Figure: Schematic of spin-relaxation mechanisms.",
"(a) At low magnetic fields, spin-relaxation is dominated by the interaction of the electron spin with lattice nuclear spins.",
"Panels (b,c) are relevant for the high-field spin relaxation mechanism.",
"(b) Energy level structure for unperturbed donor-bound electron in magnetic field, described by zero-field quantum numbers.",
"(c) Dresselhaus spin-orbit coupling mixes states with opposite spin and different angular momentum components.",
"In the admixture mechanism, phonons cause relaxation between the two eigenstates via the components with like spin.",
"The direct spin-phonon interaction causes spin-relaxation via the components with opposite spin."
],
[
"High-field spin-relaxation",
"While the spin-orbit interaction alone is not sufficient to cause a spin-flip of a localized charge carrier, a combination of the electron-phonon interaction and spin-orbit coupling serves as a main source of localized electron spin relaxation at high magnetic fields [37], [38], [39].",
"Phonons can also modulate the hyperfine coupling of the electron and the lattice-nuclei spins giving rise to ${T_1\\propto B^{-3}}$ dependence [37].",
"Similar to the quantum dot case, this effect is negligible for donor-bound electrons.",
"Two-phonon processes [38] are also very weak for the range of temperatures and fields studied here.",
"An exhaustive theoretical investigation of the spin-flip mechanisms has been performed for the related GaAs quantum dot system [4], [40], [41].",
"In GaAs quantum dots, all reported spin-orbit related mechanisms exhibit a ${T_1 \\propto B^{-\\nu }}$ dependence with $\\nu \\ge 5$ .",
"For bulk GaAs-like semiconductors, such a study has not been performed before to the best of our knowledge.",
"The orbitals for the donor-bound electron differ from those for quantum dots, leading to the use of a different approximation for the Dresselhaus spin-orbit Hamiltonian and different selection rules.",
"Experimentally we observe that the high-field spin relaxation is consistent with a single phonon process.",
"This limits us to mechanisms that combine Dresselhaus spin-orbit coupling and spin-conserving phonon-induced relaxation, and direct spin-phonon mechanisms.",
"In this section, we present the detailed calculation for the high-field $T_1$ due to both mechanisms and compare our theoretical results to the experimental data."
],
[
"Admixture mechanism caused by Dresselhaus spin-orbit coupling",
"We are first interested in the spin relaxation between the Zeeman sublevels of the donor-bound electron ground state mediated by spin-orbit and electron-phonon coupling (admixture mechanism).",
"This is the dominant relaxation mechanism for III-V quantum dots [4], [40] and naively may also be expected to play the dominant role in the similar donor system.",
"For this mechanism, the spin-orbit interaction modifies the ground-state Zeeman sublevels by the admixture of the excited sublevels with the opposite spin component.",
"Hence, the spin-independent electron-phonon coupling causes spin-relaxation through the components of the states with the same spin, as depicted in Fig.",
"REF (b)-(c).",
"The interaction Hamiltonian for the admixture mechanism is $H_{adm} = U_{ph} + H_{so},$ where $U_{ph}$ is the spin-conserving electron-phonon interaction Hamiltonian and $H_{so}$ is the spin-orbit Hamiltonian.",
"In the high-field limit, the Zeeman splitting can be comparable or even exceed the thermal energy.",
"In such a case, the transition rates from the Zeeman sublevel $\\downarrow $ to $\\uparrow $ , $\\Gamma _{\\uparrow \\downarrow }$ , and back, $\\Gamma _{\\downarrow \\uparrow }$ , differ.",
"The observed longitudinal spin relaxation time satisfies ${T_1 = (\\Gamma _{\\uparrow \\downarrow } + \\Gamma _{\\downarrow \\uparrow } )^{-1} }.$ The individual rates are found using Fermi's golden rule, e.g., $\\Gamma _{\\downarrow \\uparrow } = \\frac{2 \\pi }{\\hbar } \\sum _{\\mathbf {q},\\alpha } |M_{\\downarrow \\uparrow }|^2 \\delta (\\hbar q s_\\alpha - |g \\mu B|),$ where $\\mathbf {q}$ is the phonon wavevector, $s_\\alpha $ is the speed of sound in phonon branch $\\alpha $ and $\\alpha =t,l$ for the transverse and longitudinal modes, respectively.",
"Hereafter we assume for convenience that the spin-up state has higher energy than the spin-down one, hence, ${g\\mu B>0}$ , as illustrated in Fig.",
"REF , so that the rate in Eq.",
"(REF ) corresponds to the phonon emission process.",
"Electron spin-relaxation occurs via a second order process due to the quantum interference of $U_{ph}$ and $H_{so}$ in the Hamiltonian (REF ), see Ref.",
"[4] for details, $\\nonumber M_{\\downarrow \\uparrow , adm} = -\\sum _{e} \\bigg [ \\frac{\\langle \\text{1s}, \\downarrow |U_{ph}|e, \\downarrow \\rangle \\langle e, \\downarrow |H_{so}|\\text{1s}, \\uparrow \\rangle }{E_e - E_{\\text{1s}} + g \\mu B} \\\\+\\frac{\\langle \\text{1s}, \\downarrow |H_{so}|e, \\uparrow \\rangle \\langle e, \\uparrow |U_{ph}|\\text{1s}, \\uparrow \\rangle }{E_e - E_{\\text{1s}} - g \\mu B} \\bigg ],$ where $|\\text{1s}\\rangle $ is the ground orbital state of the donor-bound electron, $|e\\rangle $ denotes the excited orbital states, and $E_e$ , $E_{\\text{1s}}$ are the energies of the corresponding orbitals.",
"Due to the small localization energy of the donor-bound electron ($\\lesssim 10$ meV), the electron wave function in a magnetic field is well described with effective mass theory using the hydrogenic Hamiltonian $H_{0} = \\frac{\\hbar ^2}{2m^*} \\left(\\mathbf {k} - \\frac{e}{\\hbar }\\mathbf {A}\\right)^2 - \\frac{1}{4 \\pi \\varepsilon _0 }\\frac{e^2}{ \\varepsilon r} + \\frac{1}{2} g \\mu \\mathbf {\\sigma }\\cdot {\\mathbf {B}},$ where $m^*$ is the electron effective mass, $e$ is the electron charge, $\\mathbf {A}$ is the vector potential of the magnetic field $\\mathbf {B}$ , $\\mathbf {r}$ is the position vector, ${r=|\\mathbf {r}|}$ , ${\\mathbf {k} = - i \\partial /\\partial \\mathbf {r}}$ is the wavevector, $\\varepsilon $ is the relative dielectric constant of the material, and ${\\mathbf {\\sigma }}$ is the vector composed of the Pauli matrices.",
"In the presence of the magnetic field, the Hamiltonian, Eq.",
"(REF ), possesses an axial symmetry and its eigenstates are characterized by four quantum numbers: principal quantum number $\\nu $ , angular momentum $z$ -projection $m$ , $z$ -parity $\\pi _z$ and spin $z$ -projection $m_s$ .",
"To establish a link with the hydrogen-like series of donor-bound electron states at $B=0$ , we will label the orbitals by their zero-field quantum numbers $nlm$ , where $n$ is the principal quantum number and $l$ is the angular momentum quantum number, when appropriate.",
"The energy of a phonon involved in the spin-flip transition is the Zeeman splitting between the spin sublevels.",
"Therefore, the phonon wavevector ${q_{\\alpha }=g\\mu B/(\\hbar s_{\\alpha })\\rightarrow 0}$ as ${B\\rightarrow 0}$ .",
"Thus, at moderate magnetic fields in piezoelectric crystals such as GaAs, InP and CdTe studied here, we found that the piezoelectric electron-phonon interaction with ${U_{ph}^{(pz)} \\propto q^{-1/2}}$ dominates over the deformation potential interaction, where ${U_{ph}^{(dp)} \\propto q^{1/2}}$ [50], see Appendix .",
"The piezoelectric electron-phonon interaction reads $U_{ph}^{(pz)} = \\sqrt{\\frac{\\hbar }{2\\rho \\omega _{\\mathbf {q},\\alpha }}}e^{i (\\mathbf {q r} - \\omega _{\\mathbf {q},\\alpha } t)} (e A_{\\mathbf {q},\\alpha }) b^{\\dagger }_{\\mathbf {q},\\alpha } + {\\rm c.c.",
"},$ where $A_{\\mathbf {q}, \\alpha } = h_{14}\\sum _{ijk} \\beta _{ijk}\\xi _i \\xi _j \\hat{e}_k^{(\\mathbf {q},\\alpha )},$ $\\rho $ is the mass density of the material, $\\omega _{\\mathbf {q}, \\alpha }$ is the phonon frequency, $b^\\dagger _{\\mathbf {q},\\alpha }$ is the creation operator for a phonon, $\\mathbf {\\xi }= \\mathbf {q}/q$ is the unit vector along the phonon wavevector, $\\hat{\\mathbf {e}}$ is the phonon polarization vector, the only nonzero components of $\\beta _{ijk}$ are those with different subscripts, $\\beta _{xyz} = \\ldots = \\beta _{zyx}=1$ , and $h_{14}$ is the piezoelectric constant [50].",
"Since all the samples studied here are bulk semiconductors characterized by the $T_d$ point symmetry group, the only relevant spin-orbit coupling comes from the cubic-in-the-electron-wavevector Dresselhaus spin-orbit term, $H_{so}$ .",
"It arises from the lack of inversion symmetry in zinc-blende crystals and has the form $H_{so} = \\gamma \\sum _i \\sigma _i k_i(k_{i+1}^2-k_{i+2}^2),$ where $\\gamma $ is the Dresselhaus spin-orbit coupling constant and the subscript $i$ cycles through $x$ , $y$ , $z$ .",
"Depending on the relation between the magnetic length, ${l_b=\\sqrt{\\hbar /|eB|}}$ , and the effective Bohr radius, ${a_B^*}$ , various regimes of the spin-flip can be realized.",
"At sufficiently weak magnetic fields, where ${l_b \\gg a_B^*}$ , the magnetic field does not affect the hydrogen-like states of the donor-bound electron.",
"In this case, the Dresselhaus spin-orbit interaction admixes $n$ f-shell states with principal quantum numbers ${n=4,5, \\ldots }$ and orbital momentum ${l=3}$ to the 1s-shell state.",
"With the long wavelength approximation (LWA) for the phonons, where ${|g \\mu B| a_B^*/(\\hbar s_\\alpha ) \\ll 1}$ , we obtain the longitudinal spin relaxation time $T_{1,adm}^{(low)} \\propto B^{-9} F^{-1}_{ph},$ see Appendix for details.",
"Hence, at low temperatures, $T_1$ is inversely proportional to $B^9$ , while for $k_B T \\gtrsim g\\mu B$ , $T_1 \\propto B^{-8}$ .",
"We do not observe this regime in experiments due to the dominating low-field nuclear-electron hyperfine mechanism.",
"In the opposite limit, where ${l_b \\ll a_B^*}$ and, moreover, ${\\hbar \\omega _c \\gg \\mathcal {E}_{Ry}^*}$ , where ${\\omega _c = |eB/m^*|}$ is the cyclotron frequency and ${\\mathcal {E}_{Ry}^*= {m^*e^4/[2(4\\pi \\varepsilon \\varepsilon _0)^2\\hbar ^2]}}$ is the donor-bound-electron binding energy, the magnetic field shrinks the wave functions of the ground and excited states of the donor-bound electron.",
"This situation is similar to the case of an electron localized in the $(xy)$ plane by a parabolic potential, like in the quantum dot system studied in Ref. [4].",
"Here the excited states with $|m|=1$ (in addition to those with $|m|=3$ ) are admixed and, in the LWA, we obtain for the spin flip time $T_{1, adm}^{(high)} \\propto B^{-3} F^{-1}_{ph},$ see Appendix .",
"This high-field limit is not realized for the studied samples and magnetic fields accessible in our experiments.",
"Moreover, in this limit, the LWA in our system is no longer valid.",
"Therefore, we have performed the full numerical evaluation of the spin relaxation time according to Eqs.",
"(REF ) and (REF ) using the numerical solutions to Eq.",
"(REF ) [51] and the material parameters from Table REF .",
"Additional details on the numerical calculation can be found in Appendix .",
"These results, which include 18 excited state orbitals, are given by the black curves in Fig.",
"REF .",
"We numerically find that the first excited state which evolves from 2p$_-$ makes the dominant contribution to the spin relaxation rate, as shown by the dashed green curves in Fig.",
"REF .",
"This numerical result, together with the analysis of the wavefunctions in Appendix , motivates using Gaussian shapes of the ground and excited state wave functions, Eqs.",
"(REF ), to obtain an analytic solution for further insight into the intermediate field behavior.",
"The magnetic field induced shrinking is taken into account by assuming different characteristic lengths ${l_{z,\\text{1s}} = a_B^*}$ and ${l_{\\rho ,\\text{1s}}=[1/(a_B^*)^2+1/(2l_b^2)]^{-1/2}}$ for the motion along and perpendicular to the field.",
"After some transformations, we obtain (see Appendix for details): $\\frac{1}{T_{1, adm}} = {\\frac{256 \\chi ^{10}}{35(1+\\chi ^2)^{12}}}\\frac{\\gamma ^2 e^4 h_{14}^2 {|}g \\mu {|}^3 B^5}{ \\pi \\rho \\hbar ^6} \\times \\\\\\times \\left(\\frac{1}{\\Delta E}-\\frac{1}{\\Delta E + \\hbar \\omega _c}\\right)^2\\left( \\frac{f_l}{s_l^5} + \\frac{4f_t}{3s_t^5} \\right) F_{ph}\\:.$ Here, $\\Delta E = E_{\\text{2p}_-} - E_{\\text{1s}}$ is the energy difference between the hydrogen-like ground 1s and excited $\\text{2p}_-$ state.",
"The factors $f_{\\alpha } = \\exp {\\lbrace -(\\chi g\\mu B{l_\\rho })^2/[(1+\\chi ^2)\\hbar ^2s_{\\alpha }^2]\\rbrace }$ take into account that the phonon wavelength can be comparable with the donor-bound electron state size.",
"These factors are particularly sensitive to the wavefunction shape.",
"Finally, the parameter $\\chi $ is a parameter of the wave functions which characterizes the ratio of the effective radii for the excited and the ground states, see Eqs.",
"(REF ).",
"By comparing the trial wavefunction to the numerical 2p wavefunction, we find reasonable choices for $\\chi $ of 1.5, 1.7 and 2.2 for GaAs, InP and CdTe over the experimental range of magnetic field, as shown in Fig.",
"REF .",
"The magnitude of $T_{1,adm}$ calculated according to Eq.",
"(REF ) is quite sensitive to the choice of $\\chi $ .",
"A comparison between the experimental, numerical, and analytic results for $T_1$ is shown in Fig.",
"REF .",
"We stress that these calculations contain no fitting parameters.",
"Qualitatively we observe similar behavior between the analytic and numerical calculations.",
"At sufficiently strong magnetic fields, we find the LWA fails for InP and CdTe due to their relatively large electron $g$ -factors as compared with GaAs.",
"This effect is taken into account by factors $f_l$ and $f_t$ in Eq.",
"(REF ).",
"It softens the exponent in $B$ -dependence giving approximately $3 \\lesssim \\nu \\lesssim 4$ in the accessible field range.",
"Further increase in $B$ results in a minimum in $T_1(B)$ .",
"It is noteworthy that at such magnetic fields, the deformation potential interaction may become important, see [52] and Appendix for details; moreover, in such fields the result could be quite sensitive to the shape of the wave functions.",
"Hence, for sufficiently high fields, Eq.",
"(REF ) provides only an indication of the trend.",
"We find that the numerically calculated values of $T_1$ for InP and CdTe are orders of magnitude longer than the experimentally observed spin relaxation times in these samples, which indicates the importance of other spin-flip mechanisms in the materials, see below.",
"By contrast, in GaAs the calculated magnitude of $T_1$ is quite close to the experimental values, demonstrating that the admixture mechanism is significant in this material."
],
[
"Direct spin-phonon interaction",
"Although the direct spin-phonon interaction was not found to be a dominant relaxation mechanism for electrons in semiconductor quantum dots [4], we demonstrate here that it contributes significantly to donor-bound electron spin relaxation.",
"To some extent, this is because the role of the admixture mechanism is diminished due to the cubic-in-the-wavevector spin-orbit splitting in the bulk material, as compared with $\\mathbf {k}$ -linear terms used for quantum dot systems [4].",
"The direct spin-phonon interaction Hamiltonian is [48] $U_{dir} = \\frac{\\hbar v_0}{2} [\\sigma _x (u_{xy}k_y - u_{xz}k_z) + \\\\+ \\sigma _y (u_{yz} k_z - u_{yx} k_x) + \\sigma _z (u_{zx} k_x - u_{zy} k_y)],$ Here $u_{ij}=u_{ji}$ is the deformation tensor, and, as above, ${\\mathbf {k} = -i \\mathbf {\\nabla }- (e/{\\hbar }) \\mathbf {A}}$ .",
"The coupling constant $v_0$ has the dimension of velocity.",
"It has been determined by experiment for GaAs and InP but is unknown for CdTe (see Table REF ).",
"For numerical evaluation for CdTe, we use a spread of values, ${8\\times 10^5~\\mathrm {m/s}}<{ v_0^{\\rm CdTe}}< {3\\times 10^6~\\mathrm {m/s}}$ with the lower (upper) bound corresponding to $v_0^{\\rm GaAs}$ $\\left(v_0^{\\rm InSb}\\right)$ [48].",
"The relaxation rates $\\Gamma _{\\uparrow \\downarrow }$ are calculated using Eq.",
"(REF ) with the first-order matrix element ${M_{\\uparrow \\downarrow }= \\langle \\text{1s},\\uparrow | U_{dir} | \\text{1s},\\downarrow \\rangle }$ , as depicted in Fig.",
"REF (c).",
"We use an approximate exponential wave function with a characteristic length ${l = [{(a_B^*)}^{-2} + 1/(2l_b^2)]^{-1/2}}$ to obtain analytic expressions for $\\Gamma _{\\uparrow \\downarrow }$ and, correspondingly, for the associated longitudinal spin relaxation time $T_{1,dir}$ .",
"The choice of the wave function is motivated by the fact that only the $\\text{1s}$ orbital state is involved, which is not significantly perturbed at the experimentally accessible magnetic fields.",
"Moreover, the precise symmetry of the wave function for the direct phonon mechanism is not critical.",
"The evaluation of Eq.",
"(REF ) yields (see Appendix ): $\\begin{aligned}& \\frac{1}{T_{1,dir}} = \\frac{(e v_0 l^2)^2 |g \\mu |^5 |B|^7}{560 \\pi \\rho \\hbar ^6}F_{ph} \\times \\\\& \\left( \\frac{1}{s_l^7}\\frac{1}{(1+Q_l^2)^6}+\\frac{4}{3 s_t^7}\\frac{1}{(1+Q_t^2)^6} \\right) ,\\end{aligned}$ where ${Q_{\\alpha }= |g\\mu B| l/(2\\hbar s_{{\\alpha }})}$ .",
"Equation (REF ) demonstrates that the spin-flip time is proportional to $B^{-7}$ at weak magnetic fields.",
"An increase in the field results in a softening of the $B$ -field dependence due to decrease of the efficiency of the electron-phonon interaction (breakdown of the LWA) described by the factors $(1+Q_\\alpha ^2)^{-6}$ .",
"In addition to the analytic approximation, we performed the full calculation using the numerically-obtained ground-state donor wave function.",
"The very good agreement between the analytic and numerical calculations, seen in Fig.",
"REF , can be attributed to the minor effect of the magnetic field on the ground-orbital-state wave functions at the experimental fields.",
"A comparison between the theoretical calculations with no fit parameters and the experimental data is also provided in Fig.",
"REF .",
"For GaAs, we find that the magnitude of the direct-phonon mechanism is approximately the same as the admixture mechanism.",
"Also included in Fig.",
"REF (a) is the sum of these two mechanisms.",
"Accounting for both mechanisms results in a difference between the theory and the data of approximately a factor of 2, which can be easily attributed to the uncertainties in the system parameters in Table REF .",
"For InP and CdTe, the direct spin-phonon mechanism is found to be significantly stronger than the admixture mechanism.",
"For CdTe, the agreement between theory and experiment is extremely good if the direct spin-phonon interaction strength in CdTe is similar to that of InSb.",
"This may be reasonable given the similar valence band spin-orbit splitting in the two materials, 0.8 eV in InSb [53] and 0.9 eV in CdTe [54], [55], [56].",
"Here, an independent measurement of $v_0$ , like those performed in Ref.",
"[48] for GaAs and InP, or its independent first-principles calculation, is needed to corroborate our result.",
"There is still a significant discrepancy between theory and experiment for InP, where the experimental spin-relaxation time is 15 to 30 times shorter than the predicted value from the direct spin-phonon coupling.",
"Its origin is not clear and further studies, both experimental and theoretical, are needed to resolve this discrepancy."
],
[
"Conclusion",
"In this work we measure the longitudinal spin relaxation time as a function of magnetic field for electrons bound to donors in three different high-purity direct bandgap semiconductors.",
"We observe for the first time the crossover between low-field spin relaxation resulting from a hyperfine coupling of the electron and lattice nuclear spins and high-field single-phonon-mediated spin relaxation.",
"From a fundamental perspective, the existence of both regimes is expected.",
"However, the comparison of the data with the developed theory in terms of the magnitude of the relaxation raises new questions.",
"Low field measurements indicate a tens of nanoseconds electron spin correlation time of so far unknown origin.",
"High-field measurements strongly suggest the admixture mechanism is important in GaAs, while the direct spin-phonon interaction is important in both CdTe and GaAs.",
"However for InP, the discrepancy between theory and experiment calls for further investigation.",
"In the context of possible applications, the high-field $B^{-\\nu }$ dependence of $T_1$ , combined with the density and temperature dependent low-field $B^{2}$ behavior, has practical implications.",
"If the crossover point can be pushed to lower fields, extremely-long spin-relaxation times may be possible.",
"This could be realized with lower impurity density, lower temperature, larger binding energies, and a nuclear-spin-free matrix.",
"In support of this, we note that no crossover is observed in CdTe even when ${k_BT >|g\\mu B|}$ .",
"This may reflect the role of the higher donor binding energy and/or the reduced nuclear-spin environment in CdTe.",
"In this context, isotope purification, which is known to significantly affect spin dephasing, may also significantly increase the maximum achievable $T_1$ for electrons bound to shallow donors."
],
[
"Acknowledgements",
"The authors thank P. Rivera, P. Wilhelm, and B. Ebinger for assistance with building the measurement apparatus and E.L. Ivchenko for valuable discussions.",
"We thank Colin Stanley for the MBE GaAs samples provided via the University of Glasgow.",
"This material is based upon work supported by the National Science Foundation under Grant Number 1150647 and the National Science Foundation Graduate Research Fellowship under grant number DGE-1256082.",
"E.Y.S.",
"acknowledges support of the University of the Basque Country UPV/EHU under program UFI 11/55, Spanish MEC (FIS2012-36673-C03-01 and FIS2015-67161-P) and Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-472-10).",
"M.M.G.",
"was partially supported by RFBR project No.",
"14-02-00168, the Russian Federation President Grant MD-5726.2015.2, Dynasty foundation.",
"M.V.D.",
"was partially supported by RFBR project No.",
"16-32-60175 and the Dynasty foundation."
],
[
"Magneto-photoluminescence spectra for GaAs, InP, and CdTe",
"Representative magneto-photoluminescence spectra for GaAs-2, InP-2, and CdTe-2 are shown in Fig.",
"REF .",
"In all three samples we can observe the free exciton (labeled X), donor-bound exciton $\\text{D}^0\\text{X} \\rightarrow \\text{D}^0\\text{,1s}$ transition (labeled D$^0$ X), ionized donor-bound exciton transition $\\text{D}^+\\text{X} \\rightarrow \\text{D}^+$ (labeled D$^+$ X), and acceptor-bound exciton $\\text{A}^0\\text{X} \\rightarrow \\text{A}^0\\text{,1s}$ transition (labeled A$^0$ X).",
"Also observed in GaAs and InP are the D$^0$ X two-electron satellite (TES) transitions which correspond to the $\\text{D}^0\\text{X} \\rightarrow \\text{D}^0,nl^{m}$ transition, where $n,l,m$ specify the quantum numbers of the excited D$^0$ orbital at $B=0$ .",
"For GaAs and InP, the fine-structure of the D$^0$ X spectra is well resolved due to the hole spin and spin-orbit interaction as well as the nearby D$^0$ X excited orbital states.",
"In the CdTe samples, which are bulk crystals, this structure is unresolved, limiting our ability to optically pump the system to electron Zeeman splittings greater than 0.1 meV.",
"Figure: Magneto-photoluminescence spectra in the Voigt geometry.",
"(a) GaAs-2.",
"The oscillations in photoluminescence intensity with field areattributed to oscillations in magneto-absorption due to the diamagnetic exciton effect , T=2K{T=2~\\text{K}}, excitation and collection are performed in linear polarizations oriented at ±45 ∘ \\pm 45^\\circ with respect to the magnetic field direction, 1 mW excitation power at 810 nm.",
"(b) InP-2, T=2.3K{T=2.3~\\text{K}}, σ\\sigma -polarization excitation, all polarizations collected, 40 μ\\mu W above band-gap excitation power.",
"(c) CdTe-2.",
"T=1.6K{T=1.6~\\text{K}}.",
"π\\pi -polarization excitation, σ\\sigma -polarization collection, 20 μ\\mu W above band-gap excitation power."
],
[
"GaAs and CdTe $T_1$ measurements",
"Representative energy diagrams, spectra, optical pumping traces, and $T_1$ recovery traces for CdTe and GaAs are shown in Fig.",
"REF .",
"For GaAs and InP, the lower energy Zeeman pair transition was used for optical pumping.",
"Although this results in a weaker signal due to the lower thermal population in the higher electron spin level, the lower energy transition is clearly resolved from all other D$^0$ X transitions enabling efficient optical pumping.",
"For CdTe, there is significant inhomogeneous optical broadening of the D$^0$ X lines.",
"This can be observed by comparing the non-resonant and resonant excitation spectral linewidths in Fig.",
"REF (b).",
"Optical pumping visibility is thus significantly smaller in this sample relative to GaAs and InP.",
"Empirically we find the best signal-to-noise is obtained by pumping the high-energy Zeeman pair transition due to the significantly larger thermal population in the lower energy spin state.",
"Due to the large $g$ -factor in CdTe, the thermal population in the high energy state at 7 T and 1.5 K is only 0.6%."
],
[
"Surface depletion effects",
"In the GaAs and InP samples, a $\\mu $ s-scale time-dependent increase in luminescence was observed in all band edge PL after the start of an optical excitation pulse.",
"The magnitude of this effect varied significantly between samples and depended on both the wavelength and intensity of the optical excitation.",
"The effect was greater in InP than in GaAs and was greater in lower doped samples.",
"It did not significantly depend on emission wavelength.",
"Free exciton, D$^+$ X, D$^0$ X, and A$^0$ X transitions all behaved similarly.",
"Figure REF a depicts a representative example of this effect in sample InP-2.",
"In two experiments at 0 T and 1.6 K, the D$^0$ X emission is detected during an excitation pulse.",
"In the first experiment, the sample is excited with a 5 $\\mu $ W excitation pulse resonant with the D$^0$ X transition.",
"During the application of this pulse a small increase in optical emission at the beginning of the pulse can be observed on the microsecond time-scale (blue trace).",
"This effect decreases with increasing field and increases with excitation power intensity.",
"In the second experiment, we use a 5 $\\mu $ W excitation pulse with energy greater than the bandgap.",
"A significant emission increase is observed.",
"Using a pulse sequence similar to the $T_1$ sequence [Fig.",
"REF (e)], we find the sample relaxes to its initial state on the timescale of 50 microseconds.",
"For all $T_1$ measurements reported in the main manuscript, the resonant excitation power is always kept low enough so that this emission enhancement effect is negligible.",
"Due to this effect, we are unable to obtain a $T_1$ measurement for donors in InP-1, the InP sample with the lowest donor concentration.",
"Example optical pumping curves for this sample are shown in Fig.",
"REF (b) at 4 T and 1.6 K. The blue trace shows data corresponding to the standard optical pumping pulse sequence depicted in Fig.",
"REF (c).",
"The visibility is poor and in addition to the small optical pumping feature, we see an increase in the PL intensity after the initial optical pumping phase.",
"For InP-1, this “brightening” effect is observed at all reasonable powers (i.e.",
"powers for which we can obtain enough signal to reliably obtain a $T_1$ measurement) and the decay of this signal is the dominant contribution in $T_1$ pulse sequence measurements.",
"Additionally, we performed a two-pulse experiment where a 50 $\\mu $ s pulse with energy above the bandgap is applied to the sample until 5 $\\mu $ s before the the optical pumping pulse begins.",
"The effect of the pre-pulse is dramatic (Fig.",
"REF ): the larger visibility can be attributed to the pre-pulse depolarizing bound electron spins.",
"However we also note that the intensity in the optically pumped steady-state, near the end of the pulse, is flat and significantly larger in the pre-pulse case.",
"This indicates that in terms of emission intensity, the sample has reached steady-state during the application of the pre-pulse.",
"We attribute the brightening effect to the elimination of near-surface fields in the GaAs and InP samples under optical illumination [58].",
"There is evidence that very small fields, on the order of V/cm, can substantially quench fluorescence [59].",
"We do not observe this effect in CdTe, which is a true bulk sample rather than a few-micron-thick film.",
"Figure: Time-resolved photoluminescence during optical pulse.",
"(a) Collection of D 0 ^0X emission during resonant D 0 ^0X excitation (blue curve) and above-bandgap excitation (red curve).",
"A significant increase in emission intensity is observed for above-bandgap excitation.",
"(b) Optical pumping traces for InP-1 at 4 T. Blue curve: Standard optical pumping experiment.",
"Red curve: Prior to the optical pumping pulse, a 50 μ\\mu s long above-bandgap pre-pulse is applied.",
"The end of the pre-pulse is 5 μ\\mu s before the start of the optical pumping pulse."
],
[
"Numerical solution of donor-bound electron in magnetic field",
"The numerical solution of the hydrogen atom in a magnetic field is a nontrivial problem [60].",
"Of particular difficulty is the transition from the low-field to high-field regime, where the solutions cannot be conveniently expanded in hydrogen or Landau orbitals [61].",
"We have used a readily available finite element solver to find the energies and wave functions of hydrogen in a magnetic field of arbitrary strength [51].",
"These solutions can be mapped onto the donor-bound electron problem by replacing the electron mass, Bohr radius, $g$ -factor and other parameters by their effective values for the donor-bound electron.",
"The magnetic field is measured by a dimensionless quantity $\\beta =B/B_0$ , where the reference magnetic field ${B_0=2\\hbar /[|e|(a_B^*)^2]}$ is found by considering when the Larmor radius $\\sqrt{2\\hbar /|eB|}$ is equal to the donor Bohr radius.",
"For GaAs, InP, and CdTe, $B_0$ is, 13.4 T, 19.3 T and 49.8 T respectively.",
"In our experiment, the maximum applied field is 7 T, implying that the 1s wave function is a good approximation for the ground state.",
"However, we note that for higher energy orbitals $n$ , the magnetic field at which magnetic effects begin to dominate Coulomb ones occurs at $B_0/n^3$ [51].",
"Thus, higher energy orbitals are significantly perturbed even at small $\\beta $ .",
"The energy difference between the excited states and the ground state are shown in Fig.",
"REF .",
"The energy is scaled by the effective binding energy $\\mathcal {E}_{Ry}^*$ , which is 5.8, 7.0 and 13.6 meV for GaAs, InP and CdTe respectively."
],
[
"Theory of spin-relaxation via the admixture mechanism",
"In this appendix, we calculate the spin-relaxation rate due to the admixture mechanism in several different ways.",
"First we provide general simplifications that are common to all calculations.",
"We then evaluate the expression for $T_1$ numerically at all fields and analytically at low and moderate fields."
],
[
"General expression for the admixture spin-relaxation rate",
"It is convenient to represent the spin-relaxation rate, Eq.",
"(REF ), in a simplified form.",
"First, we investigate which excited states may contribute to spin relaxation by symmetry.",
"The Dresselhaus spin-orbit interaction Hamiltonian (REF ) is cubic in the electron wavevector.",
"It can be conveniently decomposed in the spherical angular harmonics, $Y_{l}^{m}({\\theta }_{ k}, {\\phi }_{k})$ , where the subscripts $l=0,1,2,\\ldots $ , $m=-l,-l+1, \\ldots ,l-1,l$ , and ${\\theta }_{ k}$ , ${\\phi }_{ k}$ are the polar and azimuthal angles of the wavevector $\\mathbf {k}$ in the spherical coordinate system with $z$ being the polar axis.",
"Corresponding summands in Eq.",
"(REF ) take the form: $&\\frac{k_x(k_y^2 - k_z^2)}{k^3} = \\sqrt{\\pi }\\left(\\frac{Y_3^{3} - Y_3^{-3} }{\\sqrt{35}} + \\frac{Y_3^1 - Y_3^{-1}}{\\sqrt{21}} \\right), \\\\&\\frac{k_y(k_z^2 - k_x^2)}{k^3} = \\sqrt{\\pi } \\left(\\frac{Y_3^{3} + Y_3^{-3} }{i\\sqrt{35}} - \\frac{Y_3^1 + Y_3^{-1}}{i\\sqrt{21}} \\right),\\\\&\\frac{k_z(k_x^2-k_y^2)}{k^3}= 2\\sqrt{\\frac{2\\pi }{105}}\\left(Y_3^2+Y_3^{-2}\\right).$ Here the arguments of the spherical harmonics are omitted for brevity.",
"In our frame of axes where ${\\mathbf {B} \\parallel z}$ , the eigenstates of the donor-bound electron in the magnetic field are characterized by the angular momentum component $m$ onto the $z$ axis.",
"Note that the term $ \\sigma _zk_z(k_x^2-k_y^2)$ in Eq.",
"(REF ) does not play a role in the spin flip process.",
"Hence, the intermediate states for the admixture mechanism, in agreement with the first two lines of Eq.",
"(REF ), are those with ${m=\\pm 1}$ , ${m=\\pm 3}$ .",
"In relatively weak fields where the magnetic field does not perturb the ground and excited stated wave functions, the donor-bound electron has spherical symmetry and Eq.",
"(REF ) imposes a strict selection rule for the excited states: only ${l=3}$ (and ${m=\\pm 1}$ , ${m=\\pm 3}$ ) can cause spin-relaxation.",
"As such, in the sum over excited states in Eq.",
"(REF ), we only need to include ${m=\\pm 1}$ and ${m=\\pm 3}$ states.",
"We also note that due to the azimuthal symmetry, $\\langle \\nu ,\\pi _z,m| k_x (k_y^2-k_z^2) | {\\text{1s}} \\rangle = e^{-i m \\frac{\\pi }{2}} \\langle \\nu ,\\pi _z,m| k_y (k_x^2-k_z^2) | {\\text{1s}} \\rangle ,$ it is sufficient to calculate the contribution due to the $\\sigma _x k_x(k_y^2-k_z^2)$ term in the Dresselhaus Hamiltonian.",
"By combining positive and negative $m$ terms and simplifying, we find $& |M_{\\uparrow \\downarrow }|^2 = \\frac{2\\gamma ^2 \\hbar }{\\rho \\omega _{\\mathbf {q},\\alpha }} |e A_{\\mathbf {q},\\alpha } |^2 \\cdot \\\\& \\nonumber \\Biggl | \\hspace{8.5359pt}\\sum _{{\\begin{array}{c}m=1,-3\\\\ \\nu =1,2,\\dots \\\\ \\pi _z=1\\end{array}}}\\langle {\\text{1s}} |e^{i {\\mathbf {q}} \\mathbf {r}} |\\nu ,\\pi _z,m \\rangle \\langle \\nu ,\\pi _z,m| k_x (k_y^2-k_z^2) | {\\text{1s}} \\rangle G_{\\nu ;m}\\Biggr |^2$ where $G_{\\nu ;m} = (\\Delta E_{\\nu ,\\pi _z,m} - g \\mu B)^{-1} - (\\Delta E_{\\nu ,\\pi _z,-m} + g \\mu B)^{-1}, $ and $\\pi _z=1$ by symmetry.",
"By integrating over phonon modes, we find the general expression $\\frac{1}{T_1}= F_{ph} \\frac{\\gamma ^2}{2 \\pi ^2 \\hbar ^2 \\rho } \\sum _\\alpha \\sum _{m=1,-3} \\frac{|g \\mu B|}{s_\\alpha ^3} \\int d \\Omega _q |e A_{\\mathbf {q}, \\alpha }|^2\\Bigg |\\sum _{\\nu }\\left[ \\langle {\\text{1s}} |e^{i \\mathbf {q} \\mathbf {r}} |\\nu ,\\pi _z,m \\rangle \\right]_{q=q_\\alpha } \\langle \\nu ,\\pi _z,m| k_x (k_y^2-k_z^2) | {\\text{1s}} \\rangle G_{\\nu ;m}\\Bigg |^2\\:,$ where the phonon matrix element is evaluated at a wavevector $q$ magnitude corresponding to the Zeeman energy ${q=q_\\alpha \\equiv |g\\mu B|/\\hbar s_\\alpha }$ .",
"In the following sections, we will evaluate Eq.",
"(REF ) using numerically calculated functions and an analytic approximation for the hydrogenic wavefunctions in a magnetic field."
],
[
"Numerical calculation of admixture spin-relaxation rate",
"For the two matrix elements in Eq.",
"(REF ), the integrals over the azimuthal angle of the position vector $\\mathbf {r}$ can be performed analytically.",
"This greatly speeds the evaluation time and improves the accuracy of the numerical calculation.",
"The wave functions are written in cylindrical coordinates as $ \\langle \\mathbf {r}|\\nu ,\\pi _z,m\\rangle = \\Phi _{\\nu ,\\pi _z,m}(\\rho ,z) e^{i m \\phi }$ , where $\\rho $ is the radial coordinate, $z$ the axial coordinate and $\\phi $ the azimuthal angle.",
"By transforming the differential operators into cylindrical coordinates and integrating over $\\phi $ , we find $\\left\\langle \\nu ,\\pi _z,\\pm 1\\right| k_x (k_y^2-k_z^2) & \\left| \\text{1s} \\right\\rangle = \\nonumber \\\\-\\frac{\\pi }{4} \\int \\rho \\, d\\rho \\, dz \\, \\Phi _{\\nu ,\\pi _z,\\pm 1}& \\left[ \\frac{1}{\\rho ^2} \\partial _\\rho - \\frac{1}{\\rho } \\partial _\\rho ^2 - \\partial _\\rho ^3 + 4 \\partial _\\rho \\partial _z^2 \\right] \\Phi _\\text{1s}, \\nonumber \\\\\\left\\langle \\nu ,\\pi _z,\\pm 3\\right| k_x (k_y^2-k_z^2) & \\left| \\text{1s} \\right\\rangle = \\nonumber \\\\-\\frac{\\pi }{4} \\int \\rho \\, d\\rho \\, dz \\, \\Phi _{\\nu ,\\pi _z,\\pm 3}& \\left[ \\frac{3}{\\rho ^2} \\partial _\\rho - \\frac{3}{\\rho } \\partial _\\rho ^2 + \\partial _\\rho ^3 \\right] \\Phi _\\text{1s}, $ and that the matrix element is zero for any other excited state magnetic quantum number, as it must be by symmetry.",
"Similarly, for the matrix element ${\\langle \\nu ,\\pi _z,m | e^{i \\mathbf {q} \\mathbf {r}} | \\text{1s} \\rangle }$ , we note that aligning the ${\\phi =0}$ plane along $\\mathbf {q}$ results in multiplying the integrand by a phase $\\exp {(- i m \\phi _{q})}$ , where $ \\phi _q$ is the azimuthal angle of the phonon wavevector $\\mathbf {q}$ .",
"The $\\phi $ integral can then be performed analytically with the help of a Bessel function identity.",
"Using a few additional simplifications involving the $z$ -parity of the wave functions, the matrix element becomes $\\langle \\nu , \\pi _z, m | & e^{i \\mathbf {q} \\mathbf {r}} | \\text{1s} \\rangle = \\, 4\\pi \\,e^{-i m \\phi _q} \\int _0^\\infty \\rho \\, d\\rho \\int _{0}^\\infty \\, dz \\, \\cdot \\\\\\nonumber & \\Phi _{\\nu ,\\pi _z,m} \\Phi _\\text{1s} \\cos (z\\, q \\cos \\theta _q) J_m(\\rho \\,q \\sin \\theta _q),$ where $\\theta _q$ is the polar angle of the wavevector $\\mathbf {q}$ , and $J_m$ is the $m^\\text{th}$ Bessel function of the first kind.",
"Lastly, we evaluate the integral over the phonon azimuthal angle $\\phi _q$ in Eq.",
"(REF ).",
"We additionally note that ${m=\\pm 1}$ states cannot interfere with ${m=\\pm 3}$ states due to the $e^{-i m \\phi _q}$ factor in Eq.",
"(REF ).",
"We thus arrive at the expression for $T_1$ via the admixture mechanism, $\\frac{1}{T_{1}} = F_{ph} \\frac{\\gamma ^2 |g \\mu B|}{2 \\pi \\hbar ^2 \\rho } \\int \\sin \\theta _q d \\theta _q \\sum _\\alpha \\sum _{m=1,-3} \\left|\\sum _{\\nu =1}^\\infty \\left[ \\langle \\text{1s} | e^{i \\mathbf {q} \\mathbf {r}} | \\nu , \\pi _z, m \\rangle \\right]_{\\phi _q=0}^{q=q_\\alpha } \\left\\langle \\nu ,\\pi _z,m\\right| k_x (k_y^2-k_z^2) \\left|\\text{1s} \\right\\rangle \\right|^2 P_\\alpha (\\theta _q)\\:,$ where the phonon matrix element is evaluated at the wavevector corresponding to the Zeeman energy and at ${\\phi _q=0}$ Since $\\phi _q$ just adds a phase, the expression is valid for any choice of $\\phi _q$ ..",
"The functions $P_\\alpha (\\theta _q)$ describe the contributions of different phonon modes and different electron-phonon interaction mechanisms.",
"We take into account both the piezoelectric interaction with longitudinal and transverse modes, Eq.",
"(REF ), as well as the deformation potential interaction.",
"The latter is described by the Hamiltonian [50], [52] $U_{ph}^{(dp)}(\\mathbf {r}) = \\sqrt{\\frac{\\hbar }{2\\rho \\omega _{\\mathbf {q}\\alpha }}} e^{i \\mathbf {q} \\mathbf {r}} q (\\xi _i \\hat{e}_i) D b^\\dag _{\\mathbf {q}\\alpha } + {\\rm c.c.",
"},$ where $D$ is the deformation potential constant and involves longitudinal phonons only.",
"As a result, disregarding the interference of piezo and deformation potential interactions, we have $\\begin{aligned}P_1(\\theta _q) & = \\frac{9e^2 h_{14}^2}{s_l^3} \\cos ^2 \\theta _q \\sin ^4 \\theta _q\\:, \\\\P_2(\\theta _q) & =\\frac{e^2 h_{14}^2}{8s_t^3} (27 + 28 \\cos 2 \\theta _q + 9 \\cos 4 \\theta _q )\\:, \\\\P_3(\\theta _q) & = \\frac{ 2(g \\mu B)^2 D^2}{\\hbar ^2 s_l^5} .\\end{aligned}$ The simplified matrix elements Eq.",
"(REF ), (REF ) are calculated numerically using standard procedures.",
"The scripts have been made readily available [63]."
],
[
"Admixture mechanism in moderate magnetic fields",
"In the regime of moderate fields, when ${l_b \\sim a_B^*}$ , we take into account the modification of the excited state wavefunctions by the magnetic field.",
"In this regime, admixture with the lowest energy excited states (p-shell states) is allowed and we will further take into account only these two states assuming a Gaussian form of the wavefunctions: $\\psi _{\\text{1s}} &=& \\frac{1}{\\pi ^{3/4} l_{\\rho } \\sqrt{l_z}} \\ \\exp {\\left(-\\frac{x^2+y^2}{2 (l_{\\rho ,\\text{1s}})^2} -\\frac{z^2}{2 (l_{z,\\text{1s}})^2}\\right)}, \\\\\\psi _{\\text{2p}_\\pm } &=& \\frac{ (x \\pm iy)}{\\pi ^{3/4} \\chi ^{5/2} l_\\rho ^2 \\sqrt{l_z}} \\ \\exp {\\left(-\\frac{x^2+y^2}{2 (l_{\\rho ,\\text{2p}})^2}-\\frac{z^2}{2 (l_{z,\\text{2p}})^2}\\right)}\\:.",
"\\nonumber $ Further, for simplicity, we will assume a proportionality ${l_{\\rho ,\\text{2p}} = \\chi l_{\\rho ,\\text{1s}} }$ and ${l_{z,\\text{2p}} = \\chi l_{z,\\text{1s}} }$ for the analytic wavefunctions.",
"Here ${l_{\\rho ,\\text{1s}}=[1/(a_B^*)^2+1/{(2l_b^2)}]^{-1/2}}$ and ${l_{z,\\text{1s}} = a_B^*}$ are the wave function effective sizes in the $(xy)$ -plane and the $z$ -direction, respectively.",
"At zero field, $l_{\\rho ,\\text{1s}}$ is just the Bohr radius and both lengths coincide.",
"A non-zero magnetic field shrinks the wave function in the $(xy)$ -plane leading to anisotropy of the $\\text{1s}$ -state with ${l_{\\rho ,\\text{1s}} < l_{z,\\text{1s}}}$ .",
"In the limit of very strong fields ${l_{\\rho ,\\text{1s}}=\\sqrt{2}l_b}$ , in agreement with the free-electron wave function in a magnetic field in the symmetric gauge.",
"The exact numerical wavefunctions (Appendix ) are used to determine the value of $\\chi $ for the analytic wavefunctions, Eq.",
"(REF ).",
"To determine the best value of $\\chi $ , we numerically optimized the overlap integral between the analytic wavefunctions, Eq.",
"(REF ), and the numerical ones.",
"The ratio of the wavefunction size for the 2p$_{-1}$ and 1s states are shown in Fig.",
"REF .",
"In the figure we have also shown the limiting values of $\\beta $ for which the experimental high-field dependence is observed.",
"By averaging over the ratio of lengths for $\\rho $ and $z$ directions, we obtain a best-choice $\\chi $ of 1.5, 1.7 and 2.2 for GaAs, InP, and CdTe respectively.",
"Figure: The overlap integral between the Gaussian approximation for the wavefunction, Eq.",
"(), and the numerical solution was maximized as a function of β\\beta .",
"The ratio of the radial and axial Gaussian sizes is plotted as a function of BB.",
"Also shown are the experimental limits of the “high-field\" regime where T 1 T_1 goes as B -ν B^{-\\nu } for the three different materials.",
"Using these limits, reasonable choices of χ\\chi are 1.5, 1.7 and 2.2 for GaAs, InP, and CdTe respectively.To obtain the matrix element $M_{\\downarrow \\uparrow }$ given by Eq.",
"(REF ), we need to calculate ${\\langle 2 p_\\pm | \\partial _x(\\partial _y^2-\\partial _z^2)| \\text{1s}\\rangle }$ and ${\\langle \\text{1s} | e^{i \\mathbf {q} \\mathbf {r}}| \\text{2p}_\\pm \\rangle }$ .",
"Utilizing the Gaussian wave functions we assumed, Eq.",
"(REF ), the results for the integrals are $\\langle \\text{2p}_\\pm | \\partial _x(\\partial _y^2-\\partial _z^2)| \\text{1s}\\rangle &=& \\frac{{\\sqrt{2}\\chi ^{5/2}}}{{(1+\\chi ^2)^{7/2}}l_b^2 l_{\\rho }}\\:, \\\\\\langle \\text{2p}_\\pm | \\partial _y(\\partial _z^2-\\partial _x^2)| \\text{1s}\\rangle &=& \\pm \\frac{i{\\sqrt{2}\\chi ^{5/2}}}{{(1+\\chi ^2)^{7/2}} l_b^2 l_{\\rho }}\\:{.",
"}$ and $ \\langle \\text{1s} | e^{i \\mathbf {q} \\mathbf {r}}| 2 p_\\pm \\rangle = { i 2\\sqrt{2} \\left( \\frac{\\chi }{1+\\chi ^2} \\right)^{5/2}} (q_x \\pm i q_y) l_\\rho \\times \\\\\\times {\\exp {\\left(-\\frac{\\chi ^2}{2(1 + \\chi ^2)} [(q_x^2+q_y^2) l_\\rho ^2+ q_z^2 l_{z}^2]\\right)}} \\:.$ Note that nonzero matrix elements in Eq.",
"(REF ) are proportional to $B$ , so that they vanish in the limit of low fields (this regime is considered below in Appendix REF ).",
"As the magnetic fields in our experiments are not very strong, the change of the characteristic length is small and we can neglect the difference between $l_\\rho $ and $l_z$ in the exponential part of Eq.",
"(REF ) by setting ${l_{z,\\text{1s}} = l_{\\rho ,\\text{1s}} = l}$ .",
"In fact, the wavefunction size $l$ is important only when the long-wavelength approximation for the electron-acoustic phonon interaction fails, in which case the result may be quite sensitive to the overall shape of the wave function, and, additionally, the deformation-potential interaction may be important.",
"Substituting these into Eqs.",
"(REF ),(REF ), the relaxation rate is $\\frac{1}{T_1} = {\\frac{32}{\\pi \\hbar \\rho } \\frac{\\chi ^{10}}{(1 + \\chi ^2)^{12}}} \\frac{\\gamma ^2 (e h_{14})^2}{l_b^4} \\left(\\frac{1}{\\Delta E_{R}}-\\frac{1}{\\Delta E_{R}+ \\hbar \\omega _c}\\right)^2 \\\\ \\times F_{ph}\\sum \\limits _\\alpha \\frac{q_\\alpha ^3}{s_\\alpha ^2} {\\exp {\\left(-{\\frac{\\chi ^2}{1 + \\chi ^2}} q_\\alpha ^2 l^2\\right)}} \\left\\langle \\sin ^2 \\theta _{ q} a_{\\mathbf {q}, \\alpha }^2 \\right\\rangle _{\\Omega }\\:{.",
"}$ The quantity $\\Delta E = E_{\\text{2p}_-} - E_{\\text{1s}}$ .",
"Numerically we find that $\\Delta E = 3/4 \\mathcal {E}_{Ry}^*$ is a good approximation across the entire experimental range of fields (see Fig.",
"REF ).",
"For longitudinal phonons we have $\\left\\langle \\sin ^2 \\theta _{ q} a_{ q, l}^2 \\right\\rangle _{\\Omega } = \\frac{8}{35}\\:,$ while for transverse modes $\\left\\langle \\sin ^2 \\theta _{ q} a_{ q, t}^2 \\right\\rangle _{\\Omega } = \\frac{32}{105}\\:.$ Substituting these integrals into the Eq.",
"(REF ), the final result for the spin relaxation rate is $\\frac{1}{T_1} = {\\frac{256 \\chi ^{10}}{35(1+\\chi ^2)^{12}}}\\frac{\\gamma ^2 e^4 h_{14}^2 (g \\mu )^3 B^5}{ \\pi \\rho \\hbar ^6} \\times \\\\\\times \\left(\\frac{1}{\\Delta E_{R}}-\\frac{1}{\\Delta E_{R}+ \\hbar \\omega _c}\\right)^2 F_{ph} \\times \\\\\\times \\left[ \\frac{1}{s_l^5} {\\exp {\\left(-{\\frac{\\chi ^2q_l^2 l^2}{1 + \\chi ^2}} \\right)}} + \\frac{4}{3s_t^5} {\\exp {\\left(-{\\frac{\\chi ^2 q_t^2 l^2}{1 + \\chi ^2}} \\right)}} \\right] \\:,$ in agreement with Eq.",
"(REF ) of the main text.",
"We note that in the limit of very strong magnetic fields where ${l_b \\ll a_B^*}$ and ${\\hbar \\omega _c \\gg \\Delta E_R}$ one has to take into account the modification of the separation between the ground and the excited states by the magnetic field.",
"As a rough estimate one may replace $\\Delta E_R$ in Eq.",
"(REF ) by $\\hbar \\omega _c$ , in which case, within the LWA, one has ${T_1 \\propto B^3}$ .",
"Our estimates show that this limit is not fulfilled in any sample for the magnetic fields under study.",
"Finally, we briefly analyze the deformation potential interaction in which case instead of Eq.",
"(REF ) one has Eq.",
"(REF ).",
"It follows from Eq.",
"(REF ) that transition can be assisted by the logitudinal acoustic phonons only.",
"Making use of the analytical form of the wavefunctions, Eq.",
"(REF ), and the matrix elements of the Dresselhaus spin-orbit interaction, Eq.",
"(REF ), as well as Eq.",
"(REF ) we obtain $\\frac{1}{T_1} = \\frac{{64} \\chi ^{10}}{3\\pi {\\hbar } \\rho (1+\\chi ^2)^{12}} \\left(\\frac{g\\mu B}{\\hbar s_l}\\right)^5 \\frac{{\\gamma ^2}D^2}{{s_l^2}l_b^4} F_{ph}\\times \\\\\\left(\\frac{1}{\\Delta E_{R}}-\\frac{1}{\\Delta E_{R}+ \\hbar \\omega _c}\\right)^2 \\exp {\\left(-\\frac{\\chi ^2 q_l^2l^2}{1+\\chi ^2}\\right)}.$ The angular integrations over the phonon wavevectors has been carried out, as before, neglecting the difference between $l_{\\rho ,\\text{1s}}$ and $l_{z,\\text{1s}}$ in the exponent and using the expression ${\\langle (\\xi _i \\hat{e}_i)^2 (\\xi _x^2+\\xi _y^2) \\rangle _\\Omega =2/3}$ .",
"The analysis shows that the deformation potential contribution is much smaller than the piezo-interaction, Eq.",
"(REF ) for the relevant magnetic fields.",
"The contribution from the deformation potential interaction is more important than the piezo-interaction at high fields due to its stronger $B$ -field dependence.",
"The crossover field for GaAs is about 40 T, which is much larger than the magnetic fields in this study.",
"The crossover fields for InP and CdTe are 9.1 T and 3.9 T. Although deformation potential interaction for these two materials is comparable to the piezo-interaction at the fields achievable in our experiment, it is still much weaker than the direct spin-phonon interaction."
],
[
"Admixture mechanism in low magnetic fields",
"In the low-field limit we take the wave functions of the ground 1s and excited ($|e \\rangle = |n\\text{f},m \\rangle $ , where f denotes f-orbitals) states in a hydrogen-like form: $\\psi _{\\text{1s}} &=& \\frac{1}{\\sqrt{\\pi a^3}} e^{-r/a} \\nonumber \\:, \\\\\\psi _{n\\text{f},m} &=& R_{n3}(r) Y_3^m ({\\theta }, {\\phi })\\:.$ Here and $R_{n3}$ are the radial functions of the f-orbitals ($l = 3$ ).",
"To calculate the spin-flip rate we use the matrix element $M_{\\downarrow \\uparrow }$ given by Eq.",
"(REF ).",
"We note that in the absence of a magnetic field the following relation for the matrix elements of $H_{so}$ holds $\\left\\langle \\text{1s} \\uparrow \\left| H_{so} \\right| n\\text{f}, m \\downarrow \\right\\rangle = - \\left\\langle n\\text{f}, -m \\uparrow \\left| H_{so} \\right| \\text{1s} \\downarrow \\right\\rangle \\:.$ Using this relation and keeping in mind that the energies of $m$ and $-m$ states are the same at zero magnetic field the second order matrix element, Eq.",
"(REF ), vanishes as ${B \\rightarrow 0}$ [4].",
"At nonzero magnetic field $M_{\\downarrow \\uparrow }$ becomes nonzero due to (i) Zeeman splitting of spin sublevels and (ii) orbital splitting of ${m = \\pm 3}$ and ${m = \\pm 1}$ states.",
"Taking into account only the Zeeman splitting we obtain the following expression for the matrix element $M_{\\downarrow \\uparrow }$ in the low-field limit (the effect of the field-induced orbital splitting is briefly addressed in the end of this subsection) $M_{\\downarrow \\uparrow } = 2g\\mu B \\sum _{n,m} \\frac{\\langle \\text{1s}, \\downarrow |U_{ph}|n\\text{f},m, \\downarrow \\rangle \\langle n\\text{f},m,\\downarrow |H_{so}|\\text{1s}, \\uparrow \\rangle }{(E_{n\\text{f},m} - E_{\\text{1s}})^2}\\:.$ Here the wavefunctions and energies are taken at $B = 0$ .",
"The sum over $m$ in Eq.",
"(REF ) can be evaluated using the fact that at ${B = 0}$ the energy spectrum is degenerate with respect to $m$ and the following formula: $\\sum \\limits _{m} \\langle \\text{1s}, \\downarrow |e^{i \\mathbf {q} \\mathbf {r}}|n\\text{f},m, \\downarrow \\rangle \\langle n\\text{f},m,\\downarrow |H_{so}|\\text{1s}, \\uparrow \\rangle = \\\\= -4 \\pi \\frac{\\gamma }{a^3} \\langle R_{n3}(r) \\Phi (r) \\rangle _r \\langle \\psi _{\\text{1s}}(r) R_{n3}(r) j_3(qr) \\rangle _r F(\\mathbf {\\xi })\\:.$ Here ${\\Phi = (1 + 3a/r + 3a^2/r^2) \\psi _{\\text{1s}}}$ , $j_3$ is the spherical Bessel function of the third order, $F(\\mathbf {\\xi }) = \\xi _x(\\xi _y^2 - \\xi _z^2) + i \\xi _y(\\xi _z^2 - \\xi _x^2)$ , and ${\\mathbf {\\xi }= \\mathbf {q} /q}$ .",
"Angular brackets with the subscript $r$ denote integration over $r$ , i.e.",
"$\\langle f_1(r) f_2(r) \\rangle _r = \\int r^2 f_1 f_2 dr$ .",
"Equation (REF ) is obtained using the decomposition of $e^{i \\mathbf {q} \\mathbf {r}}$ over the spherical harmonics, Eqs.",
"(REF ) and orthogonality of $Y_l^m$ with respect to $m$ .",
"The spin-flip rate calculated according to Eqs.",
"(REF ) and (REF ) has a form: $\\frac{1}{T_1} = \\frac{32 \\pi }{105^2} \\frac{(\\gamma e h_{14})^2}{\\rho \\hbar ^8 (\\mathcal {E}_{Ry}^*)^4} S^2 (g \\mu B)^9 \\sum \\limits _\\alpha \\frac{1}{s_\\alpha ^9} \\left\\langle |F(\\mathbf {\\xi })|^2 a_{\\mathbf {q}, \\alpha }^2 \\right\\rangle _{\\Omega }\\:,$ where $a_{\\mathbf {q}, \\alpha } = A_{\\mathbf {q}, \\alpha }/h_{14}$ , $\\langle \\dots \\rangle _\\Omega $ denotes an average over the angles ${\\theta }_{ q}$ and ${\\phi }_{ q}$ of $\\mathbf {q}$ , and $S = \\sum \\limits _{n \\ge 4} \\frac{1}{\\left( 1 - E_{n\\text{f}}/\\mathcal {E}_{Ry}^* \\right)^2} \\left\\langle R_{n3} \\Phi \\right\\rangle _r \\left\\langle \\psi _{\\text{1s}} \\bar{r}^3 R_{n3} \\right\\rangle _r\\:$ is the dimensionless sum over all excited states with different $n$ (including the states of the continuous spectrum) and ${\\bar{r} = r/a_B^*}$ .",
"In the evaluation of Eq.",
"(REF ) we used the long wavelength approximation for phonons by using the asymptotic $j_3(qr) \\approx (qr)^3/105$ at $qr \\ll 1$ .",
"For longitudinal acoustic phonons, ${\\hat{\\mathbf {e}}^{(\\mathbf {q},l)} = \\mathbf {\\xi }}$ , and hence $\\left\\langle |F(\\mathbf {\\xi })|^2 a_{\\mathbf {q}, \\alpha }^2 \\right\\rangle _{\\Omega } = \\frac{96}{5005}\\:.$ For transverse acoustic modes, the polarization vectors satisfy ${\\langle \\hat{e}_i^{(\\mathbf {q},t)} \\hat{e}_j^{(\\mathbf {q},t)} \\rangle = (\\delta _{ij}-\\xi _i \\xi _j)/2}$ .",
"Considering that there are two transverse modes, we obtain $\\left(\\sum _{i, j, k} \\beta _{i j k} \\xi _i \\xi _j \\hat{e}_k^{(\\mathbf {q},t)}\\right)^2 = 4(\\xi _x^2\\xi _y^2+\\xi _x^2\\xi _z^2+\\xi _y^2\\xi _z^2-9\\xi _x^2\\xi _y^2\\xi _z^2)\\:,$ and $\\left\\langle |F(\\mathbf {\\xi })|^2 a_{\\mathbf {q}, \\alpha }^2 \\right\\rangle _{\\Omega } = \\frac{2048}{45045}\\:.$ Using these averages one finally obtains for the spin relaxation rate $\\frac{1}{T_1} = \\zeta \\frac{(\\gamma e h_{14})^2}{\\rho \\hbar ^8 (\\mathcal {E}_{Ry}^*)^4} S^2 (g \\mu B)^9 \\left[ \\frac{1}{s_l^9} + \\left( \\frac{4}{3} \\right)^3 \\frac{1}{s_t^9} \\right]\\:,$ with $\\zeta = 1024 \\pi /(1287 \\times 35^3) \\approx 0.000175$ .",
"Let us now turn to the evaluation of the $S$ parameter.",
"It comprises the sum over the discrete spectrum (index $n$ ) and the integral over the continuum spectrum (index $\\eta $ ): $S = \\sum \\limits _{n = 4}^{+\\infty } \\frac{1}{(1 - 1/n^2)^2} \\left\\langle R_{n3} \\Phi \\right\\rangle _r \\left\\langle \\psi _{\\text{1s}} \\bar{r}^3 R_{n3} \\right\\rangle _r + \\\\+ \\int \\limits _0^{+\\infty } \\frac{\\left\\langle R_{\\eta 3} \\Phi \\right\\rangle _r \\left\\langle \\psi _{\\text{1s}} \\bar{r}^3 R_{\\eta 3} \\right\\rangle _r}{(1 + \\eta ^2)^2} d\\eta \\:.$ Matrix elements entering Eq.",
"(REF ) are calculated analytically using formulas (f,1) and (f,2) of Ref.",
"[64]: $\\left\\langle \\psi _{\\text{1s}} \\bar{r}^3 R_{n3} \\right\\rangle _r = \\frac{96}{\\sqrt{\\pi } n^5} \\sqrt{\\frac{(n+3)!}{(n-4)!}}",
"\\frac{(1-1/n)^{n-5}}{(1+1/n)^{n+5}}\\:, \\nonumber \\\\\\left\\langle \\psi _{\\text{1s}} \\bar{r}^3 R_{\\eta 3} \\right\\rangle _r = \\frac{96}{\\sqrt{\\pi }} \\left( \\frac{\\eta }{1 - e^{-2\\pi /\\eta }} \\right)^{1/2} \\times \\nonumber \\\\\\sqrt{\\prod \\limits _{s=1}^3 (s^2 \\eta ^2 +1)} \\frac{(1- i \\eta )^{-i/\\eta -5}}{(1+ i \\eta )^{-i/\\eta +5}}\\:,$ and $\\left\\langle R_{n3} \\Phi \\right\\rangle _r = \\frac{16}{7!",
"\\sqrt{\\pi } n^5} \\sqrt{\\frac{(n+3)!}{(n-4)!}}",
"\\times \\nonumber \\\\\\sum \\limits _{\\nu = 4,5,6} c_\\nu \\frac{(\\nu - 1)!",
"}{(1 + 1/n)^\\nu } {}_2{F}_1 \\left(4-n, \\nu , 8, \\frac{2}{n+1}\\right)\\:, \\nonumber \\\\\\left\\langle R_{\\eta 3} \\Phi \\right\\rangle _r = \\frac{16}{7!",
"\\sqrt{\\pi }} \\left( \\frac{\\eta }{1 - e^{-2\\pi /\\eta }} \\right)^{1/2} \\sqrt{\\prod \\limits _{s=1}^3 (s^2 \\eta ^2 +1)} \\times \\nonumber \\\\\\sum \\limits _{\\nu = 4,5,6} c_\\nu \\frac{(\\nu - 1)!",
"}{(1 + i \\eta )^\\nu } {}_2{F}_1\\left(\\frac{i}{\\eta } + 4, \\nu , 8, \\frac{2 i \\eta }{1+ i \\eta }\\right)\\:.$ Here ${c_4 = 3}$ , ${c_5 = 3}$ , ${c_6 = 1}$ , and ${}_2{F}_1$ is the ordinary hypergeometric function.",
"Using these matrix elements, a numerical summation in Eq.",
"(REF ) is performed yielding ${S \\approx 0.487}$ .",
"To analyze the contribution of the excited states we provide two estimates for $S$ : ${S_{\\mathrm {low}}< S < S_{\\mathrm {up}}}$ .",
"Here the lower limit ($S_{\\mathrm {low}}$ ) is the first term in the sum with $n = 4$ , and the upper limit ($S_{\\mathrm {up}}$ ) is the sum over a complete set of functions $\\lbrace R_{n3}, R_{\\eta 3} \\rbrace $ with a fixed denominator equal to an energy distance between the 4f and 1s state: $S_{\\mathrm {low}} = \\frac{256}{225} \\left\\langle \\mathrm {R}_{43} \\Phi \\right\\rangle _r \\left\\langle \\psi _{\\text{1s}} \\bar{r}^3 \\mathrm {R}_{43} \\right\\rangle _r \\approx 0.006 \\:,$ $S_{\\mathrm {up}} = \\frac{256\\left\\langle \\Phi \\bar{r}^3 \\psi _{\\text{1s}} \\right\\rangle _r }{225} \\approx 1.9\\:.$ Noteworthy, $S$ exceeds $S_{\\mathrm {low}}$ by more than two orders of magnitude, demonstrating the importance of accounting for all excited states of the spectrum.",
"However, for highly excited states the LWA breaks down which somehow reduces the estimate of $S$ .",
"Moreover, for the experimental donor densities the overlap of states with large ${n \\gtrsim 5}$ belonging to different donors is not negligible.",
"The account for such an overalap is beyond the scope of the present paper.",
"Equation (REF ) was derived assuming that the Zeeman splitting dominates the orbital $B$ -linear splitting of the excited states with opposite values of $m$ .",
"Such an approximation works well in quantum dot systems where the dot anisotropy lifts the degeneracy in $m$ .",
"For the donor-bound electron this is not the case, since the problem (at ${B=0}$ ) has a spherical symmetry.",
"In this situation, the splitting of the $n\\text{f}$ states with $m$ and $-m$ is $|m|\\hbar \\omega _c \\gg g\\mu B$ .",
"To estimate $T_1$ in this case one should replace $(g\\mu B)^9$ in Eq.",
"(REF ) by ${(g\\mu B)^7 (\\hbar \\omega _c)^2}$ in agreement with Eq.",
"(REF ) in the main text."
],
[
"General expression for the direct spin-phonon spin-relaxation rate",
"The direct spin-phonon interaction Hamiltonian [48], [65], [39] is $U_{dir} = \\frac{\\hbar v_0}{2} [\\sigma _x (u_{xy}k_y - u_{xz}k_z) + \\sigma _y (u_{yz}k_z - u_{xy}k_x)],$ where we ignore the $\\sigma _z$ term because it does not contribute to spin relaxation, ${\\mathbf {k} = -i \\mathbf {\\nabla }- (e/\\hbar ) \\mathbf {A}}$ , and we use the symmetric gauge ${\\mathbf {A} = (-By/2, Bx/2, 0)}$ .",
"The deformation tensor $u_{ij}$ due to phonon $\\mathbf {q}, \\alpha $ is $u_{ij}^{\\mathbf {q}, \\alpha } = \\sqrt{\\frac{\\hbar }{2\\rho \\omega _{\\mathbf {q},\\alpha }}} e^{i (\\mathbf {q} \\mathbf {r} - \\omega _{\\mathbf {q}, \\alpha } t)} \\frac{i (\\hat{e}_i^{(\\mathbf {q}, \\alpha )} q_j + \\hat{e}_j^{(\\mathbf {q}, \\alpha )} q_i)}{2} b^\\dag _{\\mathbf {q}, \\alpha } + {\\rm c.c.",
"}$ where $\\mathbf {e}_{\\alpha }$ is the polarization of phonon mode $\\alpha $ : $\\begin{aligned}\\mathbf {e}_{l} &= q^{-1} [q_x,q_y,q_z],\\\\\\mathbf {e}_{t_1} &= (q_x^2+q_y^2)^{-1/2} [q_y,-q_x,0], \\\\\\mathbf {e}_{t_2} &= q^{-1}(q_x^2+q_y^2)^{-1/2} [q_xq_z,q_yq_z,-(q_x^2+q_y^2)]{.",
"}\\end{aligned}$ Subscripts $t_1$ and $t_2$ denote two degenerate transverse modes.",
"The relaxation rate $\\Gamma _{\\downarrow \\uparrow }$ is found from Eq.",
"(REF ) using $M_{\\downarrow \\uparrow } = \\left\\langle \\text{1s}, \\downarrow \\left| U_{dir} \\right| \\text{1s}, \\uparrow \\right\\rangle \\:.$ According to the general principles of quantum mechanics, the momentum operator and deformation tensor in Eq.",
"(REF ) must be symmetrized, i.e.",
"${u_{ij} k_l \\rightarrow \\lbrace u_{ij},k_l\\rbrace }$ , where ${\\lbrace a,b\\rbrace = (ab+ba)/2}$ [4], [66], [67].",
"Due to this symmetrization and the fact that the ground state is a localized state, all terms with $k_z$ integrate to zero.",
"Further simplifications yield $\\nonumber \\langle \\text{1s},\\downarrow | U_{dir} | \\text{1s},\\uparrow \\rangle = i \\frac{\\hbar v_0}{4} \\sqrt{\\frac{\\hbar }{2 \\rho \\omega _{\\mathbf {q}, \\alpha }}} \\left( \\hat{e}_{x}^{(\\mathbf {q}, \\alpha )} q_y + \\hat{e}_{y}^{(\\mathbf {q}, \\alpha )} q_x \\right) \\times \\\\\\nonumber \\left[ \\langle \\text{1s} | \\lbrace \\exp (i \\mathbf {q r}), k_y \\rbrace | \\text{1s} \\rangle -i \\langle \\text{1s} | \\lbrace \\exp (i \\mathbf {q r}), k_x \\rbrace | \\text{1s} \\rangle \\right],$ $\\langle \\text{1s} | \\lbrace \\exp (i \\mathbf {q r}), k_x\\rbrace | \\text{1s} \\rangle &=& \\frac{e B}{2 \\hbar } \\langle \\text{1s} | \\exp (i \\mathbf {q r}) y | \\text{1s} \\rangle \\:, \\\\\\langle \\text{1s} | \\lbrace \\exp (i \\mathbf {q r}), k_y\\rbrace | \\text{1s} \\rangle &=& -\\frac{e B}{2 \\hbar } \\langle \\text{1s} | \\exp (i \\mathbf {q r}) x | \\text{1s} \\rangle \\: \\nonumber .$ Substituting Eq.",
"(REF ) into Eq.",
"(REF ) and taking the phonon factor $F_{ph}$ into consideration, we obtain the general expression for the spin-relaxation rate $\\frac{1}{T_1} = F_{ph} \\frac{v_0^2 e^2 B^2}{2^{9} \\pi ^2 \\hbar \\rho } \\sum _{\\alpha } \\frac{q_{\\alpha }^3}{s_{\\alpha }^2} \\int \\, d\\Omega _{q}\\, \\big |\\left( \\hat{e}_{x}^{\\mathbf {q}, \\alpha } \\xi _y + \\hat{e}_{y}^{\\mathbf {q}, \\alpha } \\xi _x \\right)\\cdot \\\\\\langle \\text{1s} | \\exp (i \\mathbf {q}_{\\alpha } \\mathbf {r}) (x-iy) | \\text{1s} \\rangle \\big |^2\\:,$ which can be evaluated either numerically or using an analytic approximation."
],
[
"Numerical calculation of direct spin-phonon spin-relaxation rate",
"Similar to Appendix REF , the azimuthal part of the integral $\\langle \\text{1s} | \\exp (i \\mathbf {q}_{\\alpha } \\mathbf {r}) (x-iy) | \\text{1s} \\rangle $ can be calculated analytically to simplify the numerical calculation.",
"We introduce the notation for this matrix element: ${\\kappa }_\\alpha (\\theta _q) = e^{i \\phi _q}\\langle \\text{1s}| e^{i \\mathbf {q}_{\\alpha } \\mathbf {r}} (x - iy) | \\text{1s} \\rangle ,$ and obtain $\\kappa _\\alpha (\\theta _{ q}) = \\, 4\\pi \\int _0^\\infty \\rho ^2\\, d\\rho \\int _{0}^\\infty \\, dz \\, \\Phi _{\\text{1s}}^2(\\rho ,z) \\cdot \\\\\\cos (z\\, q_{\\alpha } \\cos \\theta _q) J_1(\\rho \\,q_{\\alpha } \\sin \\theta _q)\\:.$ The simplified expression for the spin-relaxation rate is $\\begin{aligned}\\frac{1}{T_1}&=F_{ph} \\frac{\\nu _0^2 e^2 B^2}{2^{9} \\pi \\hbar \\rho } \\int _0^\\pi \\, d\\theta {_{ q}} \\, \\sin ^3 \\theta {_q} \\cdot \\\\& \\left[ \\sin ^2 \\theta {_q} \\frac{q_l^3}{s_l^2} |\\kappa _l|^2+(1+\\cos ^2 \\theta {_q}) \\frac{q_t^3}{s_t^2} |\\kappa _t|^2 \\right]\\:,\\end{aligned}$ which can be calculated numerically using standard procedures."
],
[
"Analytic calculation of direct spin-phonon spin-relaxation rate",
"To derive an analytical result we use trial wavefunctions of a Gaussian or exponential form.",
"First, we approximate the ground state wave function by a Gaussian $\\psi _{{\\text{1s}}} = \\frac{1}{(\\sqrt{\\pi }l)^{3/2}} e^{-{r^2}/{(2 l^2)}},$ with $l=[1/(a_B^*)^2+1/{(2l_b^2)}]^{-1/2}$ .",
"The matrix element can be found analytically, $\\langle \\text{1s} | \\exp (i \\mathbf {q r}) r_j | \\text{1s} \\rangle = \\frac{1}{2}i q_j l^2 e^{-q^2 l^2/4}\\:,$ where $j = x,y$ .",
"Using Eqs.",
"(REF ), (REF ) we obtain the relaxation rate assuming that the spin-up state has a higher energy as compared with the spin-down one $\\Gamma _{\\downarrow \\uparrow } &=& \\frac{ (N_{ph}+1) v_0^2}{256 \\pi \\rho \\hbar } (e B l^2)^2 \\sum _\\alpha \\frac{(g \\mu B)^5}{\\hbar ^5 s_\\alpha ^7} I_\\alpha e^{-{q_{\\alpha }^2 l^2}/{2}}, \\qquad \\\\I_\\alpha &=& \\left\\langle \\left( \\hat{e}_{x}^{\\mathbf {q}, \\alpha } \\xi _y + \\hat{e}_{y}^{\\mathbf {q}, \\alpha } \\xi _x \\right)^2 \\xi _x^2 \\right\\rangle _\\Omega \\:.$ The integrals over phonon angle for the longitudinal and both transverse modes are ${I_l = 4/35}$ and ${I_t = 16/105}$ .",
"Taking into account the phonon factor $F_{ph}$ , the final result for the relaxation rate by the direct spin-phonon process is $\\frac{1}{T_1} =\\frac{1}{2240 \\pi } \\frac{(e v_0 l^2)^2 (g \\mu )^5 B^7}{\\rho \\hbar ^6} \\times \\\\\\left( \\frac{e^{-{q_l^2 l^2}/{2}}}{s_l^7}+\\frac{4 e^{-{q_t^2 l^2}/{2}}}{3 s_t^7} \\right) F_{ph}\\:.$ Another possible choice of wave function for the donor-bound electron is an exponential $\\psi _{\\text{1s}} = \\frac{1}{\\sqrt{\\pi l^3}} e^{-r/l}.$ For this wave function, $\\langle \\text{1s} | \\exp (i \\mathbf {q r}) r_j | \\text{1s} \\rangle = \\frac{i\\, l^2 q_j}{(1+q^2 l^2/4)^3}\\:.$ The relaxation rate using an exponential wave function is the same as Eq.",
"(REF ) with $\\exp {(-q_\\alpha ^2 l^2/2)}$ replaced by ${4/(1+ q_\\alpha ^2 l^2/4)^6}$ , in agreement with Eq.",
"(REF ) of the main text.",
"We note that in the presence of a magnetic field the form of the donor-bound electron functions depends on the gauge, which calls for special care in evaluating the matrix elements in Eq.",
"(REF ).",
"Particularly, Eqs.",
"(REF ) and (REF ) are valid in the symmetric gauge.",
"For instance, in the Landau gauge, where ${\\mathbf {A}=(0,Bx,0)}$ , Eqs.",
"(REF ) and (REF ) acquire extra phase factors $\\exp {[i eB xy/(2\\hbar )]}$ .",
"Taking these phase factors into account one can readily check that Eqs.",
"(REF ) and, correspondingly, Eqs.",
"(REF ), (REF ) and (REF ) are gauge invariant."
]
] | 1605.05978 |
[
[
"On the Equivalence of Maximum SNR and MMSE Estimation: Applications to\n Additive Non-Gaussian Channels and Quantized Observations"
],
[
"Abstract The minimum mean-squared error (MMSE) is one of the most popular criteria for Bayesian estimation.",
"Conversely, the signal-to-noise ratio (SNR) is a typical performance criterion in communications, radar, and generally detection theory.",
"In this paper we first formalize an SNR criterion to design an estimator, and then we prove that there exists an equivalence between MMSE and maximum-SNR estimators, for any statistics.",
"We also extend this equivalence to specific classes of suboptimal estimators, which are expressed by a basis expansion model (BEM).",
"Then, by exploiting an orthogonal BEM for the estimator, we derive the MMSE estimator constrained to a given quantization resolution of the noisy observations, and we prove that this suboptimal MMSE estimator tends to the optimal MMSE estimator that uses an infinite resolution of the observation.",
"Besides, we derive closed-form expressions for the mean-squared error (MSE) and for the SNR of the proposed suboptimal estimators, and we show that these expressions constitute tight, asymptotically exact, bounds for the optimal MMSE and maximum SNR."
],
[
"Introduction",
"Bayesian estimation of a parameter, a source, or a signal, from noisy observations, is a general framework in statistical inference, with widespread applications in signal processing, communications, controls, machine learning, etc.",
"[1].",
"The minimum mean-squared error (MMSE) is the most popular criterion in this framework, intuitively connected to the maximum signal-to-noise ratio (MSNR) criterion, mostly used for communication and detection applications [1], [2].",
"After the first seminal work in [3], the connections between the MMSE and the signal-to-noise ratio (SNR) have attracted several research interests, and there is a quite abundant literature to establish links among them and the mutual information (see [4]–[8] and the references therein).",
"In the context of signal classification (i.e., detection), [9] has shown the interdependencies between the mean-squared error (MSE) and other second-order measures of quality, including many definitions of SNR.",
"However, a thorough investigation of the links between MSE and SNR, in the context of estimation, is still lacking.",
"Some connections between MMSE and SNR have been explored in [3], which proves that the MMSE in the additive noise channel is inversely proportional to the SNR.",
"However, the SNR of [3] is defined at the input of the estimator, while we are interested in the SNR at the output of the estimator.",
"Motivated to further explore the links between SNR and MSE, in this paper we first define the SNR for the output of a generic estimator, and then we prove the equivalence between the MMSE and MSNR criteria in the context of estimation design.",
"Actually, when the parameter to be estimated and the observations are jointly Gaussian, it is well known that the MMSE estimator, the maximum likelihood (ML) estimator, and the maximum a posteriori (MAP) estimator, are linear in the observation and are equivalent to the MSNR estimator (up to a scalar multiplicative coefficient) [11], [12]: indeed, in this simple Gaussian case, all these estimators produce the same output SNR, which is both maximum and identical to the input SNR.",
"Differently, this paper considers a more general case, where the parameter to be estimated and the observations can be non-Gaussian.",
"In this general case, to the best of our knowledge, the natural question if the MMSE and MSNR estimation criteria are equivalent or not, is still unansweredWe believe that this question has never been addressed in detail in the context of estimation problems: the investigation done in [9] for detection cannot be extended to estimation, since the SNR definitions used in [9] are quite different from the output SNR considered in this paper..",
"While classical estimation typically deals with the MMSE criterion, some authors have been looking for an MSNR solution, such as [10], ignoring if this solution has anything to do with the MMSE solution.",
"Specifically, this paper proves that the equivalence between MMSE and MSNR estimators always holds true, even when the parameter to be estimated and the observations are non-Gaussian: in this case, both the MMSE and the MSNR estimators are usually nonlinear in the observations.",
"This equivalence establishes a strong theoretical link between MMSE and MSNR criteria, traditionally used in different contexts, i.e., estimation and detection, respectively.",
"Then, we prove that the equivalence between the MSNR and MMSE criteria holds true also for any suboptimal estimator that is expressed by a linear combination of fixed basis functions, according to a basis expansion model (BEM) [13].",
"Within this framework, we derive the suboptimal MMSE estimator, and other equivalent MSNR estimators, constrained to a given quantization resolution of the noisy observations.",
"Notheworthy, each quantization-constrained estimator corresponds to a specific choice of the set of BEM functions.",
"These quantization-constrained estimators may have practical interest in low-complexity applications that use analog-to-digital (A/D) converters with limited number of bits, such as low-power wireless sensor applications.",
"Specifically, we prove that the suboptimal quantization-constrained MMSE (Q-MMSE) estimator tends to the optimal (unquantized) MMSE estimator that uses an infinite resolution of the observation.",
"In addition, we derive closed-form expressions for the SNR and for the MSE of the proposed suboptimal estimators.",
"Note that these closed-form expressions can be used as lower bounds on the SNR of the MSNR estimators, or as upper bounds on the MSE of the optimal MMSE estimator: indeed, in case of non-Gaussian statistics, analytical expressions for the MMSE value are difficult to obtain [14]; anyway, we also provide some analytical expressions for the MMSE and MSNR values.",
"To provide an example for practical applications, we apply the derived suboptimal estimators to an additive non-Gaussian noise model, where the noisy observation is simply a signal-plus-noise random variable.",
"We include a numerical example where the signal has a Laplacian statistic, while the noise distribution is a Laplacian mixture, bearing in mind that the results in this paper are valid for any signal and noise statistics.",
"The obtained results show that the proposed suboptimal Q-MMSE and quantization-constrained MSNR (Q-MSNR) estimators outperform other alternative estimators discussed in Section V. The numerical results also confirm and that, when the size of the quantization intervals tends to zero, the MSE (and SNR) of the Q-MMSE estimator tends to the optimal MMSE (and MSNR) value, as expected by design.",
"The rest of this paper is organized as follows.",
"Section II proves the equivalence between the MSNR and MMSE criteria and discusses several theoretical links.",
"In Section III, we derive the equivalence results for BEM-based estimators, such as the Q-MMSE.",
"Section IV considers the special case of additive non-Gaussian noise channel, while Section V illustrates a numerical example.",
"Section VI concludes the paper."
],
[
"Maximum SNR and MMSE Estimators",
"For real-valued scalar observation and parameters, Bayesian estimation deals with statistical inference of a random parameter of interest $x$ from a possibly noisy observation $y$ , assuming that the joint probability density function (pdf) $f_{\\rm {XY}}(x,y)$ is known.",
"The estimator of the scalar parameter $x$ is a function $g(\\cdot )$ that produces the estimated parameter $\\hat{x} = g(y)$ .",
"By a linear regression analysis, for any zero-mean $x$ and $y$ and any estimator $g(\\cdot )$ , it is possible to express the estimator output as $\\hat{x} = g(y) = K_{g}x + w_g,$ where $K_{g} = \\frac{E_{\\rm {XY}}\\lbrace x g(y)\\rbrace }{\\sigma ^2_x},$ $\\sigma ^2_x = E_{\\rm {X}}\\lbrace x^2\\rbrace $ , and $w_g$ is the zero-mean output noise, which is orthogonal to the parameter of interest $x$ and characterized by $\\sigma ^2_{W_g} = E_{{\\rm {W}}_g}\\lbrace w_g^2\\rbrace $ .",
"It is well known that the estimator $g_{\\textrm {MMSE}}(\\cdot )$ that minimizes the Bayesian MSE $J_{g} = E_{\\rm {XY}}\\lbrace (g(y)-x)^2\\rbrace $ is expressed by [1], [2], [14], [15] $g_{\\textrm {MMSE}}(y) = E_{\\rm {X|Y}}\\lbrace x|y\\rbrace = \\int _{-\\infty }^{\\infty }{x f_{\\rm {X|Y}}(x|y)dx}.$ However, other Bayesian criteria are possible, such as the MAP, the minimum mean-absolute error, etc.",
"[2].",
"Actually we may choose $g(\\cdot )$ that maximizes the SNR at the estimator output in (REF ), as done for detection in [10], [16].",
"In this sense, the definition of $K_g$ in (REF ) leads to the output SNR $\\gamma _{g} = \\frac{K_{g}^2 \\sigma ^2_x}{\\sigma ^2_{w_g}},$ defined as the power ratio of the noise-free signal and the uncorrelated noise in (REF ).",
"Alternatively, we may maximize the gain $K_{g}$ in (REF ) (instead of the SNR), under a power constraint.",
"Using the orthogonality in (REF ), the output power is $E_{\\rm {Y}}\\lbrace g^2(y)\\rbrace = K^2_{g} \\sigma ^2_x + \\sigma ^2_{w_g},$ and hence, using (REF ) and (REF ), we obtain $J_{g} =& E_{\\rm {Y}}\\lbrace g^2(y)\\rbrace + (1-2K_{g})\\sigma ^2_x \\\\=& (1-K_{g})^2 \\sigma ^2_x + \\sigma ^2_{w_g}.$ From (REF ) and (), it is straightforward that the MSE $J_{g}$ and the SNR $\\gamma _g$ are linked by $J_{g} = (1-K_{g})^2 \\sigma _x^2 + \\frac{K_{g}^2\\sigma _x^2}{\\gamma _{g}}.$"
],
[
"Equivalence of MSNR and MMSE Estimators",
"While for jointly Gaussian statistics the equivalence between MSNR and MMSE is easy to establish (since the MMSE estimator is linear in $y$ ), herein we consider the most general case, without any assumption on the statistics of $x$ and $y$ .",
"Theorem 1: Among all the possible estimators $g(\\cdot )$ , the MMSE estimator (REF ) maximizes the SNR (REF ) at the estimator output, for any pdf $f_{\\rm {XY}}(x,y)$ .",
"Let us denote with $g_{\\textrm {MMSE}}(y)$ the MMSE estimator (REF ), and with $K_{g_{\\textrm {MMSE}}}$ its associated gain (REF ).",
"In addition, let us denote with $g_{\\textrm {MSNR}}(y)$ an estimator that maximizes the SNR (REF ), as expressed by $g_{\\textrm {MSNR}}(y)= \\mathop {\\arg \\max }\\limits _{g(\\cdot )}\\left[\\frac{K_{g}^2\\sigma _x^2}{E_{\\rm {Y}} \\lbrace g^2(y)\\rbrace -K_{g}^2\\sigma _x^2}\\right],$ and by $K_{g_{\\textrm {MSNR}}}$ its associated gain in (REF ).",
"This MSNR estimator is not unique, since also any other estimator $g_{a,{\\textrm {MSNR}}}(y) = a g_{\\textrm {MSNR}}(y),$ with $a \\in \\mathbb {R} \\setminus \\lbrace 0\\rbrace $ , maximizes the SNR.",
"Indeed, due to the scaling factor $a$ , by means of (REF ) both the noise-free power $K_g^2\\sigma _x^2$ and the noise power $\\sigma _{w_g}^2 = E_{\\rm {Y}} \\lbrace g^2(y)\\rbrace -K_{g}^2\\sigma _x^2$ are multiplied by the same quantity $a^2$ , hence the SNR in (REF ) is invariant with $a$ .",
"By (REF ) and (REF ), the gain $K_{g_{a,{\\textrm {MSNR}}}}$ of $g_{a,{\\textrm {MSNR}}}(y)$ is equal to $K_{g_{a,\\textrm {MSNR}}} = a K_{g_{\\textrm {MSNR}}}.$ Conversely, the MMSE estimator is unique and has a unique gain $K_{g_{\\textrm {MMSE}}}$ .",
"Thus, we have to prove the equivalence of the MMSE estimator $g_{\\textrm {MMSE}}(y)$ with the specific $g_{a,{\\textrm {MSNR}}}(y)$ characterized by $K_{g_{a,\\textrm {MSNR}}} = K_{g_{\\textrm {MMSE}}}$ .",
"Therefore, by (REF ), we have to choose the MSNR estimator with the specific value $a=\\tilde{a}$ expressed by $\\tilde{a} = \\frac{K_{g_{\\textrm {MMSE}}}}{K_{g_{\\textrm {MSNR}}}}.$ The MSNR estimator $g_{\\tilde{a},{\\textrm {MSNR}}}(y)$ is actually the MSNR estimator that corresponds to an optimization problem restricted to the subclass of all the estimators $g(\\cdot )$ characterized by the same gain $K_g=K_{g_{\\textrm {MMSE}}}$ , as expressed by $g_{\\tilde{a},\\textrm {MSNR}}(y)= \\mathop {\\arg \\max }\\limits _{g(\\cdot ), K_g = K_{g_{\\textrm {MMSE}}}}\\left[\\frac{{K^2_g}\\sigma _x^2}{E_{\\rm {Y}} \\lbrace g^2(y)\\rbrace -{K^2_g}\\sigma _x^2}\\right].$ Note that, despite the constraint $K_g = K_{g_{\\textrm {MMSE}}}$ , we still obtain the unconstrained MMSE estimator (REF ), which by definition belongs to the subclass of estimators being characterized by $K_g=K_{g_{\\textrm {MMSE}}}$ .",
"Using the constraint $K_g = K_{g_{\\textrm {MMSE}}}$ , it is clear in (REF ) that the dependence of the MSE functional $J_g$ on $g(\\cdot )$ is only through $\\gamma _g $ , and no longer also through $K_g$ as in the general case: consequently, the MMSE estimator is $g_{\\textrm {MMSE}}(y)= & \\mathop {\\arg \\min }\\limits _{g(\\cdot ), K_g = K_{g_{\\textrm {MMSE}}}}\\left[J_{g}\\right] = \\mathop {\\arg \\min }\\limits _{g(\\cdot ), K_g = K_{g_{\\textrm {MMSE}}}}\\left[\\frac{\\sigma _x^2}{\\gamma _g}\\right] \\nonumber \\\\=& \\mathop {\\arg \\max }\\limits _{g(\\cdot ), K_g = K_{g_{\\textrm {MMSE}}}}\\left[{\\gamma _g}\\right] = g_{\\tilde{a},\\textrm {MSNR}}(y).$ Thus, (REF ) shows that the estimator that maximizes the SNR with a fixed $K_g=K_{g_{\\textrm {MMSE}}}$ is equivalent to the estimator that minimizes the MSE, i.e., $g_{\\tilde{a},\\textrm {MSNR}}(y)$ = $g_{\\textrm {MMSE}}(y)$ .",
"Basically, Theorem 1 explains that $\\lbrace g_{a,{\\textrm {MSNR}}}(y)\\rbrace $ are all scaled versions of $g_{\\textrm {MMSE}}(y)$ .",
"In other words, each scaled version of the MSNR produces the same SNR, but a different MSE: only a unique MSNR estimator is the MMSE estimator, and, in this sense, the two estimation criteria are equivalent."
],
[
"Theoretical Properties of MSNR and MMSE Estimators",
"Property 1: The output power $E_{\\rm {Y}} \\lbrace g_{\\textrm {MMSE}}^2(y)\\rbrace $ of the MMSE estimator (REF ) is equal to $K_{g_{\\textrm {MMSE}}} \\sigma _x^2$ .",
"Indeed, from (REF ) and (REF ), we obtain $K_{g_{\\textrm {MMSE}}} \\sigma _x^2 =& E_{\\rm {XY}}\\lbrace x g_{\\textrm {MMSE}}(y)\\rbrace = E_{\\rm {XY}}\\lbrace x E_{\\rm {X|Y}}\\lbrace x|y\\rbrace \\rbrace \\nonumber \\\\=& \\int _{-\\infty }^{\\infty }{ E_{\\rm {X|Y}}\\lbrace x E_{\\rm {X|Y}}\\lbrace x|y\\rbrace | y \\rbrace f_{\\rm {Y}}(y) dy} \\nonumber \\\\=& \\int _{-\\infty }^{\\infty }{ \\left[ E_{\\rm {X|Y}}\\lbrace x|y\\rbrace \\right] ^2 f_{\\rm {Y}}(y) dy} \\nonumber \\\\=& E_{\\rm {Y}} \\lbrace g_{\\textrm {MMSE}}^2(y)\\rbrace .$ Property 2: The MMSE $J_{g_{\\textrm {MMSE}}}$ is equal to $(1 - K_{g_{\\textrm {MMSE}}}) \\sigma _x^2$ .",
"Indeed, from (REF ) and (REF ), we obtain $J_{g_{\\textrm {MMSE}}} =& E_{\\rm {Y}}\\lbrace g_{\\textrm {MMSE}}^2(y)\\rbrace + (1-2K_{g_{\\textrm {MMSE}}})\\sigma ^2_x \\nonumber \\\\=& (1-K_{g_{\\textrm {MMSE}}})\\sigma ^2_x.$ Property 3: The power of the uncorrelated noise term $w_g$ at the output of the MMSE estimator is equal to $ K_{g_{\\textrm {MMSE}}}(1 - K_{g_{\\textrm {MMSE}}})\\sigma ^2_x $ .",
"Indeed, from (REF ), (REF ), and (REF ), we obtain $\\sigma ^2_{w_{g_{\\textrm {MMSE}}}} =& E_{\\rm {Y}}\\lbrace g_{\\textrm {MMSE}}^2(y)\\rbrace - K_{g_{\\textrm {MMSE}}}^2 \\sigma ^2_x \\nonumber \\\\=& K_{g_{\\textrm {MMSE}}} ( 1 - K_{g_{\\textrm {MMSE}}}) \\sigma _x^2 \\\\=& K_{g_{\\textrm {MMSE}}} J_{g_{\\textrm {MMSE}}}.",
"\\nonumber $ Equation (REF ) confirms that $K_{g_{\\textrm {MMSE}}} \\in [0,1]$ .",
"Property 4: The MSNR $\\gamma _{g_{\\textrm {MSNR}}}$ is equal to $ K_{g_{\\textrm {MMSE}}} / (1 - K_{g_{\\textrm {MMSE}}})$ .",
"Indeed, from (REF ) and (REF ), we obtain $\\gamma _{g_{\\textrm {MSNR}}} = \\gamma _{g_{\\textrm {MMSE}}} = \\frac{K_{g_{\\textrm {MMSE}}}^2 \\sigma _x^2}{\\sigma ^2_{w_{g_{\\textrm {MMSE}}}}} = \\frac{K_{g_{\\textrm {MMSE}}}}{1 - K_{g_{\\textrm {MMSE}}}}.$ By (REF )–(REF ), the MSNR is related to the MMSE by $\\gamma _{g_{\\textrm {MSNR}}} = \\gamma _{g_{\\textrm {MMSE}}} = \\frac{E_{\\rm {Y}} \\lbrace g_{\\textrm {MMSE}}^2(y)\\rbrace }{J_{g_{\\textrm {MMSE}}}} = \\frac{\\sigma ^2_x - J_{g_{\\textrm {MMSE}}}}{J_{g_{\\textrm {MMSE}}}}.$ Property 5: The unbiased MMSE (UMMSE) estimator $g_{\\textrm {UMMSE}}(y)$ maximizes the SNR: therefore, the UMMSE estimator is a scaled version of the MMSE estimator, i.e., $g_{\\textrm {UMMSE}}(y) = \\frac{g_{\\textrm {MMSE}}(y)}{K_{g_{\\textrm {MMSE}}}}.$ Indeed, for any estimator $g(y)$ , we can make it unbiased by dividing $g(y)$ by $K_g$ , as expressed by $\\hat{x} = h(y) = \\frac{g(y)}{K_g} = x + \\frac{w_g}{K_g}.$ By (REF ), $ h(y) = K_h x + w_h$ , therefore $K_h = 1$ and $w_h = {w_g}/{K_g}$ .",
"Hence, for unbiased estimators, the minimization over $h(\\cdot )$ of the MSE $\\sigma _{w_h}^2$ is equivalent to the minimization over $g(\\cdot )$ of $\\sigma _{w_g}^2 / {K_g^2}$ , which coincides with the maximization over $g(\\cdot )$ of the SNR (REF ).",
"As a consequence, the UMMSE estimator is the unique MSNR estimator characterized by $K_{g_{\\textrm {MSNR}}} = 1$ .",
"Since all MSNR estimators are scaled versions of $g_{\\textrm {MMSE}}(y)$ , the unique UMMSE estimator coincides with (REF ).",
"Property 6: The MSE $J_{g_{\\textrm {UMMSE}}}$ of the UMMSE estimator is equal to $J_{g_{\\textrm {MMSE}}} /K_{g_{\\textrm {MMSE}}}$ .",
"Indeed, from (REF ), (REF ), and (REF ), it is easy to show that $J_{g_{\\textrm {UMMSE}}} = \\frac{\\sigma _{w_{g_{\\textrm {MMSE}}}}^2}{K_{g_{\\textrm {MMSE}}}^2} = \\frac{1-K_{g_{\\textrm {MMSE}}}}{K_{g_{\\textrm {MMSE}}}}\\sigma _x^2 = \\frac{J_{g_{\\textrm {MMSE}}}}{K_{g_{\\textrm {MMSE}}}}.$ Since $K_{g_{\\textrm {MMSE}}} \\le 1$ , then $J_{g_{\\textrm {UMMSE}}} \\ge J_{g_{\\textrm {MMSE}}}$ .",
"The Properties 1-6, summarized in Table REF , show that all the theoretical expressions for both MMSE and MSNR basically depend on $K_{g_{\\textrm {MMSE}}}$ .",
"Since the definition of $K_{g_{\\textrm {MMSE}}}$ in (REF ) involves a double integration over the joint pdf $f_{\\rm {XY}}(x,y)$ , in general the exact value of $K_{g_{\\textrm {MMSE}}}$ is difficult to obtain analytically.",
"Hence, we introduce some suboptimal estimators that allow for an analytical evaluation of their MSE and SNR.",
"Table: Summary of Theoretical Properties"
],
[
"Suboptimal Estimators",
"Suboptimal MMSE and MSNR estimators for non-Gaussian statistics are interesting for several reasons.",
"For instance, closed-form computation of the MMSE estimator $g_{\\textrm {MMSE}}(y)$ in (REF ) may be cumbersome.",
"Furthermore, the optimal MMSE nonlinear function $g_{\\textrm {MMSE}}(y)$ may be too complicated to be implemented by low-cost hardware, such as wireless sensors.",
"Additionally, the MMSE $J_{g_{\\textrm {MMSE}}}$ is difficult to compute in closed form.",
"Consequently, a simpler analytical expression for a suboptimal estimator $g(\\cdot )$ may permit to compute the associated MSE and SNR, which provide an upper bound on the MMSE and a lower bound on the MSNR, respectively.",
"Considering a wide class of suboptimal estimators, we assume that $g(\\cdot )$ is expressed by a BEM of $N$ known functions $u_i(\\cdot )$ and $N$ unknown coefficients $g_i$ : $g(y) = \\sum _{i=1}^{N}{g_i u_i(y)}.$ Each function $u_i(y)$ can be interpreted as a specific (possibly highly suboptimal) estimator, and $g(y)$ in (REF ) as a linear combination of simpler estimators.",
"We are not interested in the optimization of the basis functions $\\lbrace u_i(\\cdot )\\rbrace $ : therefore, the design of $g(\\cdot )$ becomes the design of the coefficients $\\lbrace g_i\\rbrace $ .",
"Actually, we have no constraints on the choice of $\\lbrace u_i(\\cdot )\\rbrace $ ; for instance, saturating or blanking functions, or a mix of them, are typically beneficial to contrast impulsive noise [10], [16].",
"However, in Section III.C, we will show that an orthogonal design simplifies the computation of $\\lbrace g_i\\rbrace $ , and that the proposed design is general enough for any context.",
"In the following two subsections, we show that, for BEM-constrained suboptimal estimators (REF ), the MSNR and MMSE design criteria still continue to be equivalent."
],
[
"B-MSNR Estimators",
"Herein we derive the MSNR estimators constrained to the BEM (REF ), denoted as BEM-MSNR (B-MSNR) estimators.",
"By (REF ) and (REF ), the SNR $\\gamma _{g}$ in (REF ) can be expressed by $\\gamma _{g} = \\frac{{K_{g}^2} \\sigma ^2_x}{E_{\\rm {Y}}\\lbrace g^2(y)\\rbrace - {K_{g}^2} \\sigma ^2_x} =\\frac{\\mathbf {g}^T \\theta \\theta ^T\\mathbf {g}}{\\mathbf {g}^T (\\sigma ^2_x \\mathbf {R} - \\theta \\theta ^T) \\mathbf {g}}.$ where $\\mathbf {g} =& [g_1, g_2, ..., g_N]^T, \\\\\\theta =& [\\theta _1, \\theta _2, ..., \\theta _N]^T, \\\\\\theta _i =& E_{\\rm {XY}}\\lbrace x u_i(y)\\rbrace , \\\\\\mathbf {R} =& \\begin{bmatrix} R_{11} & \\cdots & R_{1N} \\\\ \\vdots &\\ddots & \\vdots \\\\ R_{N1} & \\cdots & R_{NN} \\end{bmatrix}, \\\\R_{ij} =& E_{\\rm {Y}}\\lbrace u_i(y) u_j(y)\\rbrace .$ In order to maximize (REF ), we take the eigenvalue decomposition of the symmetric matrix $\\sigma ^2_x \\mathbf {R} - \\theta \\theta ^T = \\mathbf {U} \\Lambda \\mathbf {U}^T$ , which is assumed to be full rank.",
"Note that $\\mathbf {U}$ is orthogonal and $\\Lambda $ is diagonal.",
"Then, we express the SNR in (REF ) as $\\gamma _{g} = \\frac{\\mathbf {v}^T \\mathbf {b} \\mathbf {b}^T \\mathbf {v}}{\\mathbf {v}^T \\mathbf {v}},$ where $\\mathbf {v} = \\Lambda ^{1/2} \\mathbf {U}^T \\mathbf {g}$ and $\\mathbf {b} = \\Lambda ^{-1/2} \\mathbf {U}^T \\theta $ .",
"The ratio in (REF ) is maximum [17] when $\\mathbf {v} = \\mathbf {v}_{\\textrm {B-MSNR}} = c \\mathbf {b} = c \\Lambda ^{-1/2} \\mathbf {U}^T \\theta $ , where $c \\in \\mathbb {R} \\setminus \\lbrace 0\\rbrace $ is an arbitrary constant, and therefore the SNR in (REF ) is maximum when the estimator is $\\mathbf {g}_{\\textrm {B-MSNR}} = \\mathbf {U} \\Lambda ^{-1/2} \\mathbf {v}_{\\textrm {B-MSNR}} = c (\\sigma ^2_x \\mathbf {R} - \\theta \\theta ^T)^{-1} \\theta .$ By (REF ) and (REF ), using the Sherman-Morrison formula [17], the SNR of B-MSNR estimators is expressed by $\\gamma _{\\textrm {B-MSNR}} = \\theta ^T (\\sigma ^2_x \\mathbf {R} -\\theta \\theta ^T)^{-1} \\theta =\\frac{\\theta ^T \\mathbf {R}^{-1} \\theta }{\\sigma ^2_x-\\theta ^T \\mathbf {R}^{-1} \\theta }.$"
],
[
"B-MMSE Estimator",
"Now we derive the MMSE estimator constrained to the BEM (REF ), denoted as BEM-MMSE (B-MMSE) estimator.",
"By (REF ) and (REF )–(), the MSE $J_{g}$ in () becomes $J_g = \\sigma ^2_x -2 \\mathbf {g}^T \\theta + \\mathbf {g}^T \\mathbf {R}\\mathbf {g}.$ By taking the derivative of (REF ) with respect to $\\mathbf {g}$ and setting it to zero, we obtain the B-MMSE estimator, expressed by $\\mathbf {g}_{\\textrm {B-MMSE}} = \\mathbf {R}^{-1} \\theta .$ By (REF ) and (REF ), the MSE of the B-MMSE estimator is $J_{g_{\\textrm {B-MMSE}}} = \\sigma ^2_x - \\mathbf {g}_{\\textrm {B-MMSE}}^T \\mathbf {R} \\mathbf {g}_{\\textrm {B-MMSE}} = \\sigma ^2_x - \\theta ^T \\mathbf {R}^{-1}\\theta .$ Using (REF ), the SNR (REF ) can be expressed by $\\gamma _{\\textrm {B-MSNR}} = \\frac{\\theta ^T \\mathbf {R}^{-1} \\theta }{J_{g_{\\textrm {B-MMSE}}}} = \\frac{\\sigma ^2_x - J_{g_{\\textrm {B-MMSE}}}}{J_{g_{\\textrm {B-MMSE}}}}.$ The similarity of (REF ) and (REF ) suggests a link between B-MMSE and B-MSNR estimators, as shown in Theorem 2.",
"Theorem 2: The B-MSNR estimator (REF ) coincides with the B-MMSE estimator (REF ), when $c = \\sigma ^2_x - \\theta ^T \\mathbf {R}^{-1} \\theta $ .",
"Using the Sherman-Morrison formula [17], (REF ) becomes $\\mathbf {g}_{\\textrm {B-MSNR}} = \\frac{c }{\\sigma ^2_x - \\theta ^T\\mathbf {R}^{-1} \\theta } \\mathbf {R}^{-1} \\theta .$ When $c = \\sigma ^2_x - \\theta ^T \\mathbf {R}^{-1}\\theta $ , $\\mathbf {g}_{\\textrm {B-MSNR}}$ in (REF ) coincides with $\\mathbf {g}_{\\textrm {B-MMSE}}$ in (REF ).",
"Theorem 2 proves that the B-MMSE estimator maximizes the SNR (REF ) among all the BEM-based estimators: therefore, each B-MSNR estimator is a scaled version of the B-MMSE estimator.",
"Also in this BEM-constrained case the equivalence between B-MMSE and B-MSNR estimators is valid for any statistic of the signal and of the noisy observation.",
"Note that in Theorem 2 the functions $\\lbrace u_i(\\cdot )\\rbrace $ are arbitrary, but fixed.",
"Differently, if we fix the coefficients $\\lbrace g_i\\rbrace $ in (REF ), and perform the optimization over a subset of functions, the equivalence between MMSE and MSNR solutions may not hold true.",
"Indeed, in case of impulsive noise mitigation by means of a soft limiter (SL), expressed by $g_{\\textrm {SL}}(y) = -\\beta $ if $y \\le -\\beta $ , $g_{\\textrm {SL}}(y) = y$ if $-\\beta < y < \\beta $ , and $g_{\\textrm {SL}}(y) = \\beta $ if $y \\ge \\beta $ , the optimization over $\\beta > 0$ generally produces an MMSE solution [15] that is different from the MSNR solution [16].",
"Therefore, the equivalence between MMSE and MSNR estimators can be invalid for non-BEM-based suboptimal estimators.",
"In addition to MMSE, there exist other criteria that maximize the SNR: as shown in Appendix A, the BEM-based unbiased MMSE estimator and a BEM-based estimator that maximizes the gain (REF ) (subject to a power constraint) both produce the same SNR of B-MMSE and B-MSNR estimators."
],
[
"Q-MMSE Estimator",
"Herein we prove that, by choosing convenient basis functions $\\lbrace u_i(\\cdot ) \\rbrace $ in (REF ), the B-MMSE estimator (REF ) converges to the optimal MMSE estimator (REF ).",
"Indeed, the rectangular disjoint (orthogonal) basis functions $u_i(y) = {\\left\\lbrace \\begin{array}{ll} 1, & \\mbox{if } y_{i-1} < y \\le y_i,\\\\ 0, & \\mbox{otherwise,} \\end{array}\\right.",
"}$ for $i=1,...,N$ , with $y_0 = -\\infty $ and $y_N = \\infty $ , greatly simplify the computation of the coefficients $\\lbrace g_i\\rbrace $ .",
"Basically, we are approximating the estimator $g(y)$ by a piecewise-constant function.",
"Using (REF ), $R_{ij}$ in () becomes $R_{ij} = {\\left\\lbrace \\begin{array}{ll}F_{\\rm {Y}}(y_i) - F_{\\rm {Y}}(y_{i-1}), & \\mbox{if } i=j,\\\\ 0, & \\mbox{if } i \\ne j, \\end{array}\\right.",
"}$ where $F_{\\rm {Y}}(y)$ is the cumulative distribution function (cdf) of the observation $y$ .",
"In this case, the matrix $\\mathbf {R}$ in () is diagonal.",
"Therefore, the coefficients of this specific B-MMSE estimator (REF ), which we refer to as Q-MMSE estimator, simply become $g_{i,\\textrm {Q-MMSE}} = \\frac{\\theta _i}{R_{ii}},$ while the associated MSE (REF ) is expressed by $J_{g_{\\textrm {Q-MMSE}}} = \\sigma ^2_x - \\sum _{i=1}^{N} {\\frac{\\theta _i^2}{R_{ii}}}.$ Note that the Q-MMSE estimator (REF ) can also be interpreted as the MMSE estimator when the observation $y$ has been discretized using $N$ quantization intervals $( y_{i-1} , y_{i} ]$ , for $i = 1,...,N$ .",
"Moreover, we should bear in mind that the number $N$ of quantization levels, as well as the edges of the quantization intervals, are fixed but arbitrary.",
"Thus, the proposed framework finds a natural application when the observed signal undergoes an A/D conversion stage.",
"However, it is important to prove that, in case of infinite number of quantization levels, the Q-MMSE estimator (REF ) tends to the optimal MMSE estimator (REF ) for unquantized observations: hence, the number $N$ of quantization levels enables a tradeoff between performance and complexity.",
"Theorem 3: When the interval size $\\Delta y_i = y_i - y_{i-1}$ tends to zero for $i=2,...,N-1$ , and when $y_1$ and $y_{N-1}$ tend to $y_0 = - \\infty $ and $y_N = \\infty $ , respectively, then the Q-MMSE estimator (REF ) tends to the MMSE estimator (REF ).",
"When $\\Delta y_i \\rightarrow 0$ , for $i=2,...,N-1$ , from (REF ) it is easy to show that $f_{\\rm {X|Y}}(x|y) u_i(y) \\rightarrow f_{\\rm {X|Y}}(x|y_i) u_i(y)$ ; hence, for $i=2,...,N-1$ , () gives $\\theta _i =& \\int _{-\\infty }^{\\infty }{x \\int _{y_{i-1}}^{y_i} {f_{\\rm {X|Y}}(x|y)f_{\\rm {Y}}(y) dy} dx} \\xrightarrow{} \\nonumber \\\\\\rightarrow & f_{\\rm {Y}}(y_i) \\Delta y_i \\int _{-\\infty }^{\\infty }{xf_{\\rm {X|Y}}(x|y_i)dx}.$ In addition, from (REF ) and (REF ), we have $R_{ii} = \\int _{y_{i-1}}^{y_i} {f_{\\rm {Y}}(y) dy} \\xrightarrow{}f_{\\rm {Y}}(y_i)\\Delta y_i.$ By taking the ratio between (REF ) and (REF ), $g_{i,\\textrm {Q-MMSE}}$ in (REF ) tends to $g_{\\textrm {MMSE}}(y_i)= E_{\\rm {X|Y}}\\lbrace x|y_i\\rbrace $ in (REF ), for $i=2,...,N-1$ .",
"This result can be extended in order to include $i=1$ and $i=N$ by noting that, when $y_1 \\rightarrow y_0 = - \\infty $ and $y_{N-1} \\rightarrow y_N = \\infty $ , then $f_{\\rm {X|Y}}(x|y) \\rightarrow f_{\\rm {X|Y}}(x|y_1)$ for $y \\in (y_0,y_1]$ and $f_{\\rm {X|Y}}(x|y) \\rightarrow f_{\\rm {X|Y}}(x|y_{N-1})$ for $y \\in (y_{N-1},y_N)$ .",
"Theorem 3 proves that, when the size of the quantization intervals tends to zero, the Q-MMSE estimator converges to the MMSE estimator, regardless of the statistics of the signal of interest $x$ and of the noisy observation $y$ .",
"In particular, the SNR of the Q-MMSE estimator converges to the SNR of the MMSE estimator.",
"Moreover, since a Q-MMSE estimator is a particular B-MMSE estimator, by Theorem 2, the Q-MMSE estimator is also a Q-MSNR estimator, for the same set of quantization thresholds.",
"Noteworthy, if we would optimize the quantization intervals $\\lbrace (y_{i-1},y_i] \\rbrace $ [i.e., the functions $\\lbrace u_i(\\cdot )\\rbrace $ in (REF )] by keeping the coefficients $g_i$ as fixed, we could end up with different quantization thresholds in an MMSE and MSNR sense."
],
[
"Q-MMSE in Additive Noise Channels",
"Herein we provide further insights on the coefficients (REF ) of the Q-MMSE estimator, when the observations are impaired by an additive noise $n$ , independent from $x$ , as expressed by $y = x + n$ and depicted in Fig.",
"REF .",
"The additive noise model (REF ) occurs in several applications, especially if the data are obtained by quantized measurements.",
"Indeed, Q-MMSE estimators are particularly useful in realistic scenarios where either the source, or the noise, or both, depart from the standard Gaussian assumption.",
"These scenarios include: (a) additive noise with a high level of impulsiveness [18]–[24]; (b) additive noise whose pdf is a mixture of statistics caused by the random occurrence of different noise sources [25]–[29]; (c) source represented by a pdf mixture, such as in applications (e.g., audio, medical, etc.)",
"that involve effective denoising of sounds or images [30], [31].",
"The optimal coefficients $\\lbrace g_i\\rbrace $ obviously depend on the specific pdfs of source and noise, and the numerical results reported in Section V give some evidence of the usefulness of Q-MMSE estimation in an additive non-Gaussian observation model.",
"According to the BEM model, we assume that the quantization thresholds have been fixed by some criterion.",
"Despite possible criteria for threshold optimization are beyond the scope of this work, in Section V we give some insights about this issue and consider some heuristic solutions.",
"To specialize the results of Section III to the additive noise model in (REF ), we observe that the pdf $f_{\\rm {Y}}(y)$ is the convolution between $f_{\\rm {X}}(x)$ and $f_{\\rm {N}}(n)$ .",
"Thus, the coefficients $\\theta _i$ and $R_{ii}$ defined in () and () can be calculated from the first-order statistics of $x$ and $n$ .",
"Using (REF ), () and (REF ), we obtain $\\theta _i = \\int _{-\\infty }^{\\infty }{x f_{\\rm {X}}(x) \\int _{y_{i-1}-x}^{y_i-x}{f_{\\rm {N}}(n) dn} dx} = D(y_i) - D(y_{i-1}),$ where $D(y) = \\int _{-\\infty }^{\\infty }{x f_{\\rm {X}}(x) F_{\\rm {N}}(y-x) dx}.$ An alternative expression can be obtained by exchanging the integration order, which leads to $\\theta _i = \\int _{-\\infty }^{\\infty }{f_{\\rm {N}}(n) \\int _{y_{i-1}-n}^{y_i-n} {xf_{\\rm {X}}(x) dx} dn} = D(y_i) - D(y_{i-1}),$ where $D(y) =& \\int _{-\\infty }^{\\infty } { \\scalebox {0.9}{ f_{\\rm {N}}(n) \\left[ (y-n)F_{\\rm {X}}(y-n) - I_{\\rm {X}}(y-n) \\right] } dn}, \\\\ I_{\\rm {X}}(y) =&\\int _{-\\infty }^{y}{F_{\\rm {X}}(x)dx}.$ Which expression is preferable, between (REF ) and (REF ), depends on the expressions of $f_{\\rm {X}}(x)$ and $f_{\\rm {N}}(n)$ .",
"Using (REF ) and (REF ), we obtain $R_{ii} =& F_{\\rm {Y}}(y_i) - F_{\\rm {Y}}(y_{i-1}), \\\\F_{\\rm {Y}}(y) =& \\int _{-\\infty }^{\\infty }{f_{\\rm {X}}(x) F_{\\rm {N}}(y-x) dx} \\\\=& \\int _{-\\infty }^{\\infty }{f_{\\rm {N}}(n) F_{\\rm {X}}(y-x) dn}.$ Thus, using (REF ), either (REF ) or (REF ), and (REF ), the Q-MMSE estimator for the additive noise model $(\\ref {eq:noise_model})$ is expressed by $g_{i,\\textrm {Q-MMSE}} = \\frac{ D(y_i) - D(y_{i-1}) }{F(y_i) - F(y_{i-1})}.$"
],
[
"A Numerical Example",
"In this section, we want to numerically compare the MSE and the SNR performances of the Q-MMSE estimator with those of the optimal MMSE estimator, in order to show the usefulness of Q-MMSE estimators with a limited number of quantization levels.",
"Therefore, first we derive the mathematical expressions of the optimal MMSE estimator and of the Q-MMSE estimator, assuming a non-trivial additive noise model (REF ) where both the signal and the noise are non-Gaussian.",
"Specifically, we model the signal $x$ with a Laplace pdf $f_{\\rm {X}}(x) = \\frac{\\alpha }{2} e^{-\\alpha |x|},$ with $\\alpha = \\sqrt{2}/\\sigma _x$ , and the noise $n$ with a Laplace mixture pdf $f_{\\rm {N}}(n) = p_0 \\frac{\\beta _0}{2} e^{-\\beta _0 |n|} + p_1 \\frac{\\beta _1}{2}e^{-\\beta _1 |n|},$ with $\\lbrace \\beta _m = \\sqrt{2}/\\sigma _{n,m}\\rbrace _{m=0,1}$ , $R =\\sigma ^2_{n,0} / \\sigma ^2_{n,1}$ , $\\sigma ^2_n = p_0 \\sigma ^2_{n,0} + p_1\\sigma ^2_{n,1}$ , $p_0 + p_1 = 1$ and $0 \\le p_0 \\le 1$ .",
"Basically, (REF ) models a noise generated by two independent sources: each noise source, characterized by a Laplace pdf with average power $\\sigma ^2_{n,m}$ , occurs with probability $p_m$ .",
"Similar results can be obtained by modeling either the noise, or the signal, or both, as a Gaussian mixture, thus covering a wide range of practical applications of non-Gaussian denoising.",
"As detailed in Appendix B, direct computation of (REF ) with (REF ) and (REF ) yields the optimal MMSE estimator $& g_{\\textrm {MMSE}}(y) = \\operatorname{sgn}{(y)} \\times \\nonumber \\\\& \\times \\frac{ \\sum \\limits _{m=0}^{1} p_m\\left[ C_{1,m} (e^{-\\beta _m |y|} - e^{-\\alpha |y|}) - C_{2,m} \\beta _m |y| e^{-\\alpha |y|} \\right]}{\\sum \\limits _{m=0}^{1} { p_m C_{2,m} \\left( \\alpha e^{-\\beta _m |y|} - \\beta _m e^{-\\alpha |y|} \\right) }}, \\\\& C_{1,m} = \\frac{\\alpha ^2 \\beta _m^2}{(\\alpha ^2 - \\beta _m^2)^2}, \\qquad C_{2,m} = \\frac{\\alpha \\beta _m}{2(\\alpha ^2 - \\beta _m^2)}.$ The Q-MMSE estimator can be calculated by solving (REF ) and () using the pdf in (REF ) and (REF ): as detailed in Appendix C, when $y>0$ , this calculation leads to $D(y) = & \\sum \\limits _{m=0}^{1} p_m \\frac{ \\beta _m^2 (3 \\alpha ^2 - \\beta _m^2) e^{-\\alpha y} }{2 \\alpha (\\alpha ^2 - \\beta _m^2)^2} \\nonumber \\\\- & \\sum \\limits _{m=0}^{1} p_m{\\frac{ \\alpha ^2 \\beta _m e^{-\\beta _m y} }{ (\\alpha ^2 - \\beta _m^2)^2} } +\\sum \\limits _{m=0}^{1} p_m \\frac{ \\beta _m^2 y e^{-\\alpha y} }{2 (\\alpha ^2 - \\beta _m^2) }, \\\\F_{\\rm {Y}}(y) = & 1 - \\sum \\limits _{m=0}^{1} {p_m \\frac{ \\alpha ^2 e^{-\\beta _m y} - \\beta _m^2 e^{-\\alpha y}}{2 (\\alpha ^2 - \\beta _m^2)} },$ which inserted into (REF ) give the final result.",
"In addition to MMSE and Q-MMSE, other two alternative estimators are included in this comparison: (a) the sampled MMSE (S-MMSE) estimator $g_{i,\\textrm {S-MMSE}}$ , obtained by sampling the optimal MMSE estimator $g_{\\textrm {MMSE}}(\\cdot )$ at the midpoint of each quantization interval, e.g., $ g_{i,\\textrm {S-MMSE}} = g_{\\textrm {MMSE}}((y_{i-1}+y_{i})/2) $ ; and (b) the optimal quantizer (OQ) obtained by applying the Lloyd-Max algorithm [32] to the signal pdf $f_{\\rm {X}}(x)$ .",
"Note that the Lloyd-Max OQ exploits the statistical knowledge of the parameter of interest $x$ only, and neglects the noise, while the Q-MMSE estimator-quantizer also exploits the knowledge of the pdf of noise $n$ : hence, the Q-MMSE estimator is expected to give better performance.",
"With reference to the choice of the $N-1$ thresholds $\\lbrace y_i\\rbrace $ of the Q-MMSE estimators, a heuristic approach chooses all the $N-1$ thresholds equispaced, such that the overload probability $P_{\\textrm {ol}}=P\\lbrace y \\in [-\\infty ,y_1)\\cup [y_{N-1},\\infty )\\rbrace $ of the quantizer is fixed: this limits the amount of saturating distortion.",
"Another option is to choose the non-uniform thresholds $\\lbrace y_i\\rbrace $ given by the Lloyd-Max algorithm [32] applied to the signal pdf $f_{\\rm {X}}(x)$ in (REF ).",
"For all the quantized estimators, we use the acronym NU for non-uniform quantization and U for uniform quantization.",
"Fig.",
"REF compares the shape of the Q-MMSE estimator $g_{\\textrm {Q-MMSE}}(\\cdot )$ with the shape of the optimal (unquantized) MMSE estimator $g_{\\textrm {MMSE}}(\\cdot )$ , when $\\sigma _x = 1$ , $\\sigma _n = 4$ , $R = 0.001$ , $p_0 = 0.9$ , and the $N-1$ thresholds are equispaced between $y_1=-10$ and $y_{N-1}=10$ , which induce an overload probability $P_{\\textrm {ol}} \\approx 0.0327$ .",
"Since all the considered MMSE estimators are odd functions of the input $y$ , Fig.",
"REF only displays the positive half.",
"Fig.",
"REF confirms that, when the number $N$ of quantization levels increases, the Q-MMSE estimator tends to the optimal MMSE estimator.",
"Note also that the Q-MMSE estimator $g_{i,\\textrm {Q-MMSE}}$ is different from the staircase curve of the S-MMSE estimator $g_{i,\\textrm {S-MMSE}}$ .",
"Figure: Comparison between the optimal (unquantized) MMSE estimator and Q-MMSE estimators with uniform quantization (NN is the number of intervals).Figure: SNR gain G g G_g of different estimators as a function of the input SNR σ x 2 /σ n 2 \\sigma _x^2 / \\sigma _n^2.Figure: Comparison between different estimators with non-uniform quantization(NN is the number of intervals).Figure: MSE J g J_g of different estimators as a function of the input SNR σ x 2 /σ n 2 \\sigma _x^2 / \\sigma _n^2.Fig.",
"REF shows the SNR gain $G_g$ provided by different estimators $g(\\cdot )$ .",
"The SNR gain $G_g$ is defined as $G_g = \\frac{\\gamma _g}{\\sigma _x^2 / \\sigma _n^2},$ where $\\gamma _g$ is the SNR at the output of the estimator, and $\\sigma _x^2 / \\sigma _n^2$ is the SNR at the input of the estimator.",
"The signal and noise parameters are the same of Fig.",
"REF , except for the variable $\\sigma _n$ .",
"Fig.",
"REF compares the SNR performance of Q-MMSE, S-MMSE, and OQ estimators, assuming uniform and non-uniform quantization versions (with labels U and NU in the legend of Fig.",
"REF ): the overload regions are the same for both versions and have been selected by the Lloyd-Max algorithm, which ends up with an overload probability $P_{\\textrm {ol}} \\approx 0.0093$ when $\\sigma _x^2 / \\sigma _n^2 = 0$ dB and $P_{\\textrm {ol}} \\approx 0.0805$ when $\\sigma _x^2 / \\sigma _n^2 = -12$ dB.",
"As a reference, Fig.",
"REF also includes an optimal Q-MMSE (with $N=127$ ) with uniform quantization obtained by an exhaustive maximization of the SNR gain over all the possible choices for the overload regions (i.e., for all the possible choices of $y_1=-y_{N-1}$ ): this is equivalent to an optimization of the interval size $\\Delta y = (y_{N-1}-y_1)/(N-2)$ of the uniform quantization intervals.",
"When the number of quantization intervals $N$ is sufficiently high, the SNR of this optimal Q-MMSE estimator basically coincides with the SNR of the optimal (unquantized) MMSE, whose simulated SNR gain is included in Fig.",
"REF as well.",
"Fig.",
"REF confirms that the SNR gain of the Q-MMSE estimator is larger than for the other quantized estimators, provided that the quantization intervals are the same.",
"The SNR of the Q-MMSE estimator can be further improved by increasing the number of intervals and by optimizing the (uniform) interval sizes, as shown in Fig.",
"REF by the curve with $N = 127$ with optimized overload regions.",
"In addition, Fig.",
"REF shows that the SNR of the optimal Q-MMSE estimator is very close to the simulated SNR of the optimal (unquantized) MMSE estimator.",
"Therefore, the proposed Q-MMSE approach permits to obtain analytical tight lower bounds on the SNR of the optimal (unquantized) MMSE estimator.",
"Fig.",
"REF compares the function $g(y)$ for the estimators of Fig.",
"REF with non-uniform quantization, when $\\sigma _x^2 / \\sigma _n^2 =-15$ dB.",
"Fig.",
"REF highlightsthat the function $g(y)$ of the Lloyd-Max OQ is nondecreasing, because the noise is neglected; differently, the function $g(y)$ of the (Q-) MMSE estimators can be non-monotonic, like in this specific example.",
"Fig.",
"REF displays the MSE of different estimators, in the same scenario of Fig.",
"REF .",
"It is evident that the Q-MMSE estimator provides the lowest MSE among all the quantized estimators that use the same quantization intervals.",
"Note that the analytical MSE of the Q-MMSE estimator can be used as an upper bound of the minimum value $J_\\textrm {MMSE}$ (obtained in Fig.",
"REF by simulation).",
"Similarly to the SNR analysis of Fig.",
"REF , tighter upper bounds on the MMSE $J_{g_\\textrm {MMSE}}$ can be obtained by increasing the number of intervals $N$ and by further optimization over all the possible overload regions."
],
[
"Conclusion",
"In this paper, we have studied a meaningful definition of the MSNR estimator, and we established its equivalence with the MMSE estimator, regardless of the statistics of the noise and of the parameter of interest.",
"We have also extended this equivalence to a specific class of suboptimal estimators expressed as a linear combination of arbitrary (fixed) functions; conversely, we have explained that the same equivalence does not hold true in general for non-BEM suboptimal estimators.",
"The developed theoretical framework has been instrumental to study Bayesian estimators whose input is a quantized observation of a parameter of interest corrupted by an additive noise.",
"We have shown that, when the size of the quantization intervals goes to zero, the Q-MMSE (Q-MSNR) estimator exactly tends to the MMSE (MSNR) estimator for unquantized observations.",
"Furthermore, by a practical example, we have shown that, using a fairly limited number of quantization levels, the Q-MMSE estimator can easily approach the performance of the optimal (unquantized) MMSE estimator: the designed Q-MMSE estimator, clearly, outperforms in SNR (and in MSE) other suboptimal estimators."
],
[
"Appendix A - Other BEM-Based Estimators",
"We detail BEM-based estimators that produce the maximum SNR, similarly to B-MMSE and B-MSNR estimators: unbiased estimators and a maximum-gain estimator.",
"Unbiased estimators are defined by $E_{\\rm {Y|X}} \\lbrace g(y)|x \\rbrace =x $ and hence are characterized by $K_{g} = 1$ in (REF ).",
"By (REF ), (REF ), (REF )–(), for the BEM-based estimators we have $K_{g} = \\frac{\\mathbf {g}^T\\theta }{\\sigma ^2_x}.$ Therefore, the BEM-based unbiased MSNR (B-UMSNR) estimator is obtained by maximizing (REF ) subject to the constraint $\\mathbf {g}^T\\theta = \\sigma ^2_x$ , while the BEM-based unbiased MMSE (B-UMMSE) estimator is obtained by minimizing (REF ) subject to the same constraint.",
"By inserting the constraint $\\mathbf {g}^T\\theta = \\sigma ^2_x$ into (REF ) and (REF ), both optimizations are equivalent to the minimization of the output power $E_{\\rm {Y}}\\lbrace g^2(y)\\rbrace = \\mathbf {g}^T \\mathbf {R} \\mathbf {g}$ subject to $\\mathbf {g}^T\\theta = \\sigma ^2_x$ , which leads to $\\mathbf {g}_{\\textrm {B-UMSNR}} = \\mathbf {g}_{\\textrm {B-UMMSE}} = \\frac{\\sigma ^2_x}{\\theta ^T \\mathbf {R}^{-1} \\theta }\\mathbf {R}^{-1} \\theta .$ The solution (REF ) is equivalent to (REF ) with $c = \\frac{\\sigma ^2_x (\\sigma ^2_x - \\theta ^T \\mathbf {R}^{-1}\\theta )}{\\theta ^T \\mathbf {R}^{-1}\\theta }.$ Hence, the B-UMMSE estimator gives the maximum SNR achievable by BEM-based estimators, and is a scaled version of the B-MMSE estimator (REF ).",
"An alternative Bayesian criterion is the maximization of the gain $K_{g}$ (REF ) or (REF ), subject to a power constraint.",
"Using the output power constraint $E_{\\rm {Y}}\\lbrace g^2(y)\\rbrace = \\mathbf {g}^T \\mathbf {R} \\mathbf {g} = P$ , the BEM-based maximum-gain (B-MG) estimator is expressed by $\\mathbf {g}_{\\textrm {B-MG}} = \\sqrt{\\frac{P}{\\theta ^T\\mathbf {R}^{-1} \\theta }} \\mathbf {R}^{-1}\\theta ,$ which is a scaled version of the B-MMSE estimator and hence an MSNR estimator among the BEM-based estimators.",
"Here we show that the computation of (REF ), when the signal pdf is (REF ) and the noise pdf is (REF ), leads to (REF ).",
"First, using the Bayes' theorem, the MMSE estimator (REF ) is rewritten as $g_{\\textrm {MMSE}}(y) = \\frac{\\int _{-\\infty }^{\\infty }{x f_{\\rm {Y|X}}(y|x) f_{\\rm {X}}(x) dx}}{f_{\\rm {Y}}(y)};$ in addition, the noise pdf (REF ) can be rewritten as $f_{\\rm {N}}(n) =& \\sum \\limits _{m=0}^{1} p_m f_{{\\rm {N}},m}(n) \\\\f_{{\\rm {N}},m}(n) =& \\frac{\\beta _m}{2} e^{-\\beta _m |n|}.$ Using (REF ), (REF ), (REF ), (REF ), and (), the numerator of (REF ), for $y>0$ , can be rewritten as $& \\int _{-\\infty }^{\\infty }{x f_{\\rm {Y|X}}(y|x) f_{\\rm {X}}(x) dx} \\\\= & \\int _{-\\infty }^{\\infty }{x f_{\\rm {N}}(y - x) f_{\\rm {X}}(x) dx} \\\\= & \\sum \\limits _{m=0}^{1} p_m \\int _{-\\infty }^{\\infty }{x f_{{\\rm {N}},m} (y - x) f_{\\rm {X}}(x) dx} \\\\= & I_1(y) + I_2(y) + I_3(y),$ where $I_1(y) = & \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha \\beta _m}{4} e^{- \\beta _m y} \\int _{-\\infty }^{0}{ x e^{( \\alpha + \\beta _m) x} dx}, \\\\I_2(y) = & \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha \\beta _m}{4} e^{- \\beta _m y} \\int _{0}^{y}{ x e^{(-\\alpha + \\beta _m) x} dx}, \\\\I_3(y) = & \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha \\beta _m}{4} e^{ \\beta _m y} \\int _{y}^{\\infty }{ x e^{(-\\alpha - \\beta _m) x} dx}.$ The three integrals (REF ), (), and (), can be solved using $\\int { x e^{a x} dx} = {\\left\\lbrace \\begin{array}{ll} \\frac{1}{a^2}[(ax-1) e^{a x}] + C, & \\mbox{if } a \\ne 0,\\\\\\frac{1}{2}x^2 + C, & \\mbox{if } a = 0, \\end{array}\\right.",
"}$ where $C$ is an arbitrary constant.",
"If we assume that $\\alpha \\ne \\beta _m$ , for $m=0,1$ , then (REF )–() become $I_1(y) = & - \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha \\beta _m e^{- \\beta _m y}}{4(\\alpha + \\beta _m)^2}, \\\\I_2(y) = & \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha \\beta _m \\lbrace [ (\\beta _m - \\alpha ) y -1 ] e^{- \\alpha y} + e^{- \\beta _m y} \\rbrace }{4(\\beta _m - \\alpha )^2}, \\\\I_3(y) = & \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha \\beta _m [ (\\alpha + \\beta _m) y +1 ] e^{- \\alpha y} }{4(\\alpha + \\beta _m)^2} .$ Hence, the numerator of (REF ), for $y>0$ , is equal to $& I_1(y) + I_2(y) + I_3(y) = \\\\& = \\sum \\limits _{m=0}^{1} p_m \\left[ C_{1,m} (e^{-\\beta _m y} - e^{-\\alpha y}) - C_{2,m} \\beta _m y e^{-\\alpha y} \\right],$ where $C_{1,m}$ and $C_{2,m}$ are expressed by ().",
"If we repeat the same procedure for $y<0$ , we obtain a similar equation.",
"On the other hand, using (REF ), (REF ), (REF ) and (), the denominator of (REF ) is equal to $f_{\\rm {Y}}(y) = & f_{\\rm {X}}(y) \\ast f_{\\rm {N}}(y) = f_{\\rm {X}}(y) \\ast \\sum \\limits _{m=0}^{1} p_m f_{{\\rm {N}},m}(y) \\\\= & \\sum \\limits _{m=0}^{1} p_m [ f_{\\rm {X}}(y) \\ast f_{{\\rm {N}},m}(y) ] = \\sum \\limits _{m=0}^{1} p_m f_{{\\rm {Y}},m}(y),$ where $\\ast $ denotes convolution and $f_{{\\rm {Y}},m}(y) = f_{\\rm {X}}(y) \\ast f_{{\\rm {N}},m}(y).$ By denoting with $C_{\\rm {X}}(u)$ the characteristic function associated with the pdf $f_{\\rm {X}}(x)$ , (REF ) translates into $C_{{\\rm {Y}},m}(u) = C_{\\rm {X}}(u) C_{{\\rm {N}},m}(u) = \\frac{\\alpha ^2}{\\alpha ^2 + 4 \\pi ^2 u^2} \\frac{\\beta _m^2}{\\beta _m^2 + 4 \\pi ^2 u^2}.$ If we assume that $\\alpha \\ne \\beta _m$ , for $m=0,1$ , then (REF ) can be decomposed in partial fractions as $C_{{\\rm {Y}},m}(u) =& \\frac{\\beta _m^2}{\\beta _m^2 - \\alpha ^2} \\frac{\\alpha ^2}{\\alpha ^2 + 4 \\pi ^2 u^2} + \\frac{\\alpha ^2}{\\alpha ^2 - \\beta _m^2} \\frac{\\beta _m^2}{\\beta _m^2 + 4 \\pi ^2 u^2}\\\\=& \\frac{\\beta _m^2}{\\beta _m^2 - \\alpha ^2} C_{\\rm {X}}(u) + \\frac{\\alpha ^2}{\\alpha ^2 - \\beta _m^2} C_{{\\rm {N}},m}(u),$ which, by means of (REF ) and (), leads to $& f_{{\\rm {Y}},m}(y) = \\frac{\\beta _m^2}{\\beta _m^2 - \\alpha ^2} f_{\\rm {X}}(y) + \\frac{\\alpha ^2}{\\alpha ^2 - \\beta _m^2} f_{{\\rm {N}},m}(y),\\\\& f_{{\\rm {Y}}}(y) = \\sum \\limits _{m=0}^{1} p_m \\left[ \\frac{\\beta _m^2}{\\beta _m^2 - \\alpha ^2} f_{\\rm {X}}(y) + \\frac{\\alpha ^2}{\\alpha ^2 - \\beta _m^2} f_{{\\rm {N}},m}(y) \\right].$ Therefore, by (), (REF ), (REF ), and (REF ), the denominator of (REF ) is equal to $f_{{\\rm {Y}}}(y) =& \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha \\beta _m^2 e^{-\\alpha |y|}}{2(\\beta _m^2 - \\alpha ^2)} + \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha ^2 \\beta _m e^{-\\beta _m |y|}}{2(\\alpha ^2 - \\beta _m^2)} \\\\=& \\sum \\limits _{m=0}^{1} p_m C_{2,m} (\\alpha e^{-\\beta _m |y|} - \\beta _m e^{-\\alpha |y|}),$ where $C_{2,m}$ is expressed by ().",
"By inserting (REF )–(), (REF ), and (REF )–() into (REF ), we obtain the mathematical expression of $g_{\\textrm {MMSE}}(y)$ for $y>0$ , and, by repeating the same procedure for negative values of $y$ , we obtain the final expression of $g_{\\textrm {MMSE}}(y)$ reported in (REF )–(), which is valid for all values of $y$ .",
"Note that, since the signal pdf and the noise pdf are both symmetric, the MMSE estimator is and odd function of $y$ , and therefore $g_{\\textrm {MMSE}}(-y) = -g_{\\textrm {MMSE}}(y)$ .",
"Herein we detail the computation of $D(y)$ in (REF ) and of $F_{\\rm {Y}}(y)$ in (): these two quantities are derived by calculating (REF ) and (), respectively, for the additive noise model (REF ), when the signal pdf is expressed by (REF ) and the noise pdf is expressed by (REF ).",
"Indeed, (REF ) and () are necessary in order to compute the Q-MMSE estimator, expressed by (REF ), via (REF )–(REF ) and (REF )–().",
"The derivations of $D(y)$ and $F_{\\rm {Y}}(y)$ are performed only for $y>0$ (those for $y<0$ are similar).",
"By (REF ) and (), the noise cdf can be expressed as $F_{\\rm {N}}(n) =& \\sum \\limits _{m=0}^{1} p_m F_{{\\rm {N}},m}(n) \\\\F_{{\\rm {N}},m}(n) =&{\\left\\lbrace \\begin{array}{ll} \\frac{1}{2}e^{\\beta _m n}, & \\mbox{if } n<0,\\\\1 - \\frac{1}{2}e^{-\\beta _m n}, & \\mbox{if } n \\ge 0, \\end{array}\\right.",
"}$ and therefore, by (REF ), $D(y)$ in (REF ) becomes $D(y) =& \\sum \\limits _{m=0}^{1} p_m \\int _{-\\infty }^{\\infty }{x f_{\\rm {X}}(x) F_{{\\rm {N}},m}(y-x) dx} \\\\=& I_4(y) + I_5(y) + I_6(y) + I_7(y) + I_8(y),$ where $I_4(y) =& \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha }{2} \\int _{-\\infty }^{ 0}{x e^{ \\alpha x} dx}, \\\\I_5(y) =& - \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha }{4} e^{-\\beta _m y} \\int _{-\\infty }^{ 0}{x e^{ (\\alpha + \\beta _m ) x} dx}, \\\\I_6(y) =& \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha }{2} \\int _{ 0}^{ y}{x e^{-\\alpha x} dx}, \\\\I_7(y) =& - \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha }{4} e^{-\\beta _m y} \\int _{ 0}^{ y}{x e^{(-\\alpha + \\beta _m ) x} dx}, \\\\I_8(y) =& \\sum \\limits _{m=0}^{1} p_m \\frac{\\alpha }{4} e^{ \\beta _m y} \\int _{ y}^{\\infty }{x e^{-(\\alpha + \\beta _m ) x} dx}.$ By assuming $\\alpha \\ne \\beta _m$ , for $m=0,1$ , and by solving the five integrals in (REF )–() using (REF ), it is easy to show that $D(y)$ in (REF ) becomes equal to (REF ).",
"The cdf $F_{\\rm {Y}}(y)$ can be easily calculated from (REF ) and (), which lead to $F_{{\\rm {Y}},m}(y) =& \\frac{\\beta _m^2}{\\beta _m^2 - \\alpha ^2} F_{\\rm {X}}(y) + \\frac{\\alpha ^2}{\\alpha ^2 - \\beta _m^2} F_{{\\rm {N}},m}(y)\\\\=& 1 + \\frac{\\beta _m^2 e^{-\\alpha y} - \\alpha ^2 e^{-\\beta _m y}}{2(\\alpha ^2 - \\beta _m^2)},$ where we have used $F_{\\rm {X}}(y) = 1 - \\frac{1}{2} e^{-\\alpha y}$ for $y>0$ .",
"Using () with (REF )–(), we obtain the final expression ()."
]
] | 1605.06044 |
[
[
"Why are flare ribbons associated with the spines of magnetic null points\n generically elongated?"
],
[
"Abstract Coronal magnetic null points exist in abundance as demonstrated by extrapolations of the coronal field, and have been inferred to be important for a broad range of energetic events.",
"These null points and their associated separatrix and spine field lines represent discontinuities of the field line mapping, making them preferential locations for reconnection.",
"This field line mapping also exhibits strong gradients adjacent to the separatrix (fan) and spine field lines, that can be analysed using the `squashing factor', $Q$.",
"In this paper we make a detailed analysis of the distribution of $Q$ in the presence of magnetic nulls.",
"While $Q$ is formally infinite on both the spine and fan of the null, the decay of $Q$ away from these structures is shown in general to depend strongly on the null-point structure.",
"For the generic case of a non-radially-symmetric null, $Q$ decays most slowly away from the spine/fan in the direction in which $|{\\bf B}|$ increases most slowly.",
"In particular, this demonstrates that the extended, elliptical high-$Q$ halo around the spine footpoints observed by Masson et al.",
"(Astrophys.",
"J., 700, 559, 2009) is a generic feature.",
"This extension of the $Q$ halos around the spine/fan footpoints is important for diagnosing the regions of the photosphere that are magnetically connected to any current layer that forms at the null.",
"In light of this, we discuss how our results can be used to interpret the geometry of observed flare ribbons in `circular ribbon flares', in which typically a coronal null is implicated.",
"We conclude that both the physics in the vicinity of the null and how this is related to the extension of $Q$ away from the spine/fan can be used in tandem to understand observational signatures of reconnection at coronal null points."
],
[
"Introduction",
"As new generations of solar telescopes allow ever more detailed views of the Sun's atmosphere, the link between magnetic topological structures and observed sites of energy release becomes increasingly apparent.",
"The magnetic structure of the corona is highly complex over a broad range of scales, as a result of the complex array of magnetic polarities that appear in a continually evolving pattern on the photosphere.",
"The magnetic flux from each polarity region on the photosphere generically connects to many other flux patches of opposite polarity.",
"The structure of the associated coronal magnetic field can appear bewilderingly complex, but advances in theory, modelling and observations have allowed a characterisation of the key features of the 3D structure – such as likely sites for dynamic events to take place.",
"One particular tool for analysing the coronal field structure is the magnetic field line mapping between positive and negative polarity regions of the photosphere.",
"In particular, field lines along which this mapping is discontinuous – usually separatrix surfaces associated with magnetic null points – or has strong gradients – at quasi-separatrix layers (QSLs) – are now known to be likely sites for current accumulation and energy dissipation.",
"These structures are defined and discussed in the following section.",
"In the last 20 years or so, a wealth of observational evidence has accumulated for energy release at both magnetic null points and QSLs in the form of flares, jets, and bright points [12], [18], [32], [31], [34], [30], [62].",
"In each of these cases, the magnetic field structure in the corona must be inferred by employing some extrapolation method that uses the observed photospheric field as a boundary condition.",
"One particular recent focus has been to understand in flaring regions how the observed $H_\\alpha $ flare ribbons map to the coronal magnetic field structure – and what one can subsequently deduce about the flare energy release process.",
"In configurations containing QSLs, the footprints of these QSLs have been shown to be co-located with observed ribbons [11], [8], [50], [51].",
"In specific cases a magnetic null point is also present.",
"It has a fan separatrix and spines which define the topology of the magnetic configuration, and since the vicinity of the null is a preferential site for current accumulation and reconnection, so the footpoints of the spine and fan structures are often where the flare ribbons are located.",
"While the fan surface footprint naturally defines elongated ribbons, the spine lines define locally compact regions, so that one would expect compact ribbons – however they are also observed to be elongated.",
"A link is observed between this elongation and the squashing degree $Q$ surrounding the fan and spines [32] – the reason for this link is explored herein.",
"[34] observed a so-called circular flare ribbon associated with the footprint of the separatrix surface of a coronal null point – and they noted again the elongation of the spine footpoint ribbons.",
"Since this observation, a number of further studies have confirmed that these findings are generic [59], [61], [29].",
"In this paper we make more concrete the link between the null point magnetic field structure, the geometries of associated features in the field line mapping, and the expected locations of flare ribbons.",
"This is done by analysing the field line mapping in the vicinity of generic 3D null point structures, and relating this to the known properties of current sheet formation and magnetic reconnection around these nulls.",
"The paper is organised as follows.",
"We start in Section by discussing some necessary background on 3D magnetic topology and reconnection.",
"In Sections and the field line mapping in a linear null configuration and coronal separatrix dome configuration are analysed, respectively.",
"In Section the field line mapping is studied in the context of magnetic reconnection around the null point in MHD simulations.",
"We end in Section with a discussion.",
"In this paper we deal with magnetic null points in the solar corona.",
"Such coronal null points have been demonstrated to exist in abundance by various surveys of coronal magnetic field extrapolations, both potential and force-free [49], [15], [19].",
"A magnetic null is simply a location in space at which the magnetic field strength is exactly zero, ${\\bf {B}}={\\bf 0}$ , and in three dimensions (3D) this condition is met generically only at isolated points.",
"The magnetic field in the vicinity of the null is characterised by a pair of spine field lines that asymptotically approach (or recede from) the null and a fan surface within which field lines radiate away from (or approach) the null point – see [27], [36] for a full description, and Figure REF for a visualisation.",
"The fan surface forms a separatrix surface in the field, distinguishing two volumes of magnetic flux within which the field line connectivity is topologically distinct.",
"The simplest generic coronal structure involving a magnetic null point occurs when a `parasitic' polarity region on the photosphere is surrounded by polarity region(s) of the opposite sign (and greater total flux).",
"In this configuration a magnetic null is located where the field contributions from the two polarities cancel.",
"The fan separatrix surface then forms a dome structure and separates flux connecting the dominant polarity to the parasitic polarity (beneath the dome) from that which connects from the dominant polarity to locations further away on the photosphere.",
"The effect of reconnection in such a separatrix dome configuration has been considered by [14], [46].",
"Null points and their associated separatrix surfaces also occur in many more complicated topological configurations involving multiple null points, separatrix surfaces, and separators (separatrix surface intersections) – see for example [39].",
"However, here we restrict our analysis to a single null point, considering a separatrix dome configuration in Section .",
"3D null points are one of the preferential sites for reconnection in the corona.",
"This is because in the perfectly-conducting limit, singular current layers are known to form at the null when rather general perturbations are applied [41].",
"Therefore no matter how small the dissipation, non-ideal processes will eventually become important as the field around the null point collapses.",
"Note that in an equilibrium there should be zero current at the null, since in general a pressure gradient cannot balance the Lorentz force in the vicinity of the null [37].",
"Any non-zero current at the null will in general lead to a Lorentz force that drives the null point to collapse to form a current sheet [26], [41], [20], [7].",
"The resulting reconnection may take different forms, the most general mode of reconnection being spine-fan reconnection that is associated with transfer of magnetic flux across the separatrix surface – this process permitting in principle the release of significant stored magnetic energy [2], [43], [35], [46].",
"There are also two other modes of reconnection at 3D nulls – torsional-spine and torsional-fan reconnection, that involve a rotational slippage of field lines around the spine but involve no flux transfer across the separatrix [40]."
],
[
"The squashing factor and quasi-separatrix layers",
"The principal reason why magnetic null points were first proposed as sites of current accumulation and therefore magnetic reconnection in 3D is that at the null the field line mapping is discontinuous.",
"It is now well established that 3D reconnection may also occur in the absence of null points or separatrices, and in particular natural sites for the formation of intense current layers are regions in which the field line mapping exhibits strong gradients.",
"Analysis of these gradients is typically performed by evaluating the (covariant) squashing degree, defined for planar boundaries by $Q=\\frac{(\\partial U/\\partial u)^2+(\\partial U/\\partial v)^2+(\\partial V/\\partial u)^2+(\\partial V/\\partial v)^2}{|(\\partial U/\\partial u)(\\partial V/\\partial v)-(\\partial U/\\partial v)(\\partial V/\\partial u)|},$ where $u$ and $v$ are field line footpoints on the `launch' boundary, and $U$ and $V$ are the footpoint locations on the `target' boundary, see [57].",
"The general expression for non-planar boundaries can be found in Equations (11-14) of [55].",
"Note that the denominator in Eq.",
"(REF ) can also be represented by $B_n/B_n^\\star $ where $B_n$ and $B_n^\\star $ are the field components normal to the boundaries at the launch and target footpoints, respectively.",
"Numerically it is usually more stable to use this expression in practice.",
"One potential weakness of $Q$ as defined in Eq.",
"(REF ) is that the values obtained depend on the orientation at which field lines intersect the launch and target boundaries.",
"An alternative formulation is the perpendicular covariant squashing factor, $Q_\\perp $ , as defined in equations (30-36) of [55], which removes such projection effects by evaluating the mapping deformation for infinitesimal perpendicular planes at the locations of the launch and target surfaces.",
"Bundles of magnetic field lines along which $Q$ or $Q_\\perp $ have large values ($\\gg 2$ , the minimum value, obtained for a uniform field) are known as quasi-separatrix layers, or QSLs.",
"It has been demonstrated that these are natural locations for accumulation of intense currents, using both modelling approaches [56], [22], [3], [17] and solar observations [12], [32].",
"The term QSL comes from the fact that true separatrices can be thought of as a limiting case of a QSL .",
"In particular, due to the discontinuity in the field line mapping at a separatrix, $Q$ is by definition infinite there.",
"In addition, $Q$ must also be large but finite in the region adjacent to the separatrix – and on this we focus in Section ."
],
[
"The nature of 3D reconnection",
"In order to understand energy release mediated by magnetic reconnection in 3D, it is important to understand a key property of 3D reconnection.",
"Specifically, in contrast to the 2D case reconnection in 3D always occurs in a finite volume.",
"That is, rather than field lines breaking and rejoining at a single point (the X-point) as in 2D, field lines change connectivity continuously throughout the (finite-sized) non-ideal region [48].",
"This non-ideal region is in general any region within which the electric field component parallel to the magnetic field ($E_\\Vert $ ) is non-zero, and for which $\\int E_\\Vert \\, ds\\ne 0$ , the integral being evaluated along field lines [52].",
"A consequence of the breaking and rejoining of field lines throughout the non-ideal region is as follows.",
"For reconnection in the absence of separatrices, for example at QSLs, there is an everywhere continuous `flipping' or `slipping' of reconnecting field lines [47].",
"One can also distinguish this flipping motion further, to slipping or slip-running, depending on whether the velocity of the apparent field line motion is sub- or super-Alfvénic, respectively [4].",
"Such slipping motions are now observed during energy release in the corona [5], [54], [28], [13].",
"All of the above statements hold true for 3D null point reconnection.",
"In particular, the reconnection happens not only at the null point itself, but throughout the non-ideal region (current sheet) surrounding the null [44], [45].",
"Therefore what we call `null point reconnection' is more precisely reconnection that occurs within a finite region (the current layer) surrounding a null – the importance of the null being that it is a favourable site for intense currents to develop (as described above).",
"There is still a continuous change of field line connectivity within the current sheet, and an apparent flipping motion of field lines.",
"However, in the presence of a null there is also one discontinuous jump of connectivity, for every field line that is reconnected through either the spine of the null or the separatrix (fan) surface.",
"For a given reconnection event, if there is a null point in the non-ideal region the field line velocity must by necessity be `slip-running' near the null, since it is infinite at the spine/fan.",
"For the case of a single null point in a separatrix dome configuration, that we examine below, the expected patterns of the field line slippage motions were described in detail by [46]."
],
[
"Preliminaries",
"In the following section we examine the distribution of the squashing degree, $Q$ , in the vicinity of null points.",
"First we consider the simplest case of a linear null point.",
"Note that any generic 3D null point can be represented locally (i.e.",
"sufficiently close to the null) by this linearisation – and conversely for the null point to be topologically stable, the linear term in the Taylor expansion of ${\\bf {B}}$ about the null point must be non-zero [24].",
"(Topological stability implies that an arbitrary perturbation does not destroy the topology, in contrast to the case where the first non-zero term is the quadratic term – those higher order nulls are topologically unstable since an arbitrary perturbation of ${\\bf {B}}$ changes the topology.)",
"Note that there are two factors that influence the variation of $Q$ in the vicinity of a null point and associated separatrix: one is that $Q$ is formally infinite for spine and fan field lines due to the discontinuity in the field line mapping.",
"There is then a characteristic decay of $Q$ away from these field lines.",
"The other factor is that field lines in the fan surface typically become oriented parallel to one of the fan eigenvectors (corresponding to the largest fan eigenvalue) at larger distances.",
"This naturally means there is a stronger divergence of field lines away from the fan eigenvector direction corresponding to the smaller fan eigenvalue.",
"This leads to a rotational asymmetry of the decay of $Q$ away from the spine and fan – as demonstrated below.",
"Figure: Boundary surfaces used for the calculation of the squashing factor QQ or Q ⊥ Q_\\perp in (a) Section and (b) Section .",
"Field lines are plotted for k=0.4k=0.4 (kk defines the asymmetry of the null-point field as defined by Eq.",
").Evaluation of the squashing factor requires that we select two surfaces that each field line intersects once and only once.",
"We consider two cases: in the first case we take both boundaries to be planar (Figure REF a), and in the second case we take a plane of constant $z$ and a circular cylinder surface (see Figure REF b).",
"When both surfaces are planar we can obtain exact expressions for $Q$ and $Q_\\perp $ , whereas for the cylindrical boundary we must evaluate them numerically."
],
[
"Squashing factor between two planar boundaries",
"Consider an equilibrium magnetic null point (zero current).",
"The field can be represented by ${\\bf {B}}=B_0\\left(\\begin{array}{ccc}k & 0 & 0 \\\\ 0 & 1-k & 0 \\\\ 0 & 0 & -1\\end{array}\\right)\\left(\\begin{array}{c}x\\\\ y\\\\ z\\end{array}\\right)+ \\mathcal {O}(r^2),$ with $0<k<1$ , where we have chosen to orient the coordinate system such that the spine lies along the $z$ -axis, the fan surface is coincident with the $z=0$ plane, and the two eigenvectors of $\\nabla {\\bf {B}}$ are parallel to the $x$ and $y$ axes.",
"The corresponding eigenvalues are unequal when $k\\ne 1/2$ , and in the $xy$ -plane the field strength increases most quickly away from the origin along the direction of the eigenvector associated with the largest eigenvalue – the $x$ -direction for $1/2<k<1$ .",
"We refer below to this direction as the strong field direction in the fan, and correspondingly to the orthogonal direction as the weak field direction.",
"In the corona, one would in general expect field lines in the strong field direction to connect to the strongest nearby photospheric flux concentrations (though this need not necessarily be the case).",
"Taking two planar boundaries as shown in Figure REF (a), we can obtain a mathematical expression for how $Q$ decays away from (say) the spine along the strong and weak field directions in the fan.",
"Here we take the `launch plane' for field lines to be $z=a$ and the target plane to be $x=b$ ($a$ and $b$ constant).",
"That is, we consider the mapping $(x,y,a) \\rightarrow (b, Y, Z)$ .",
"For simplicity we only examine how $Q$ decays from the spine along the $x$ -axis (setting $Y=y=0$ ).",
"This corresponds to the strong field direction in the fan if $1/2<k<1$ , and the weak field direction for $0<k<1/2$ .",
"We examine the decay in arbitrary directions in the next section.",
"Setting $y=Y=0$ , we obtain the following expression for $Q$ as a function of $x$ , the distance of the launch footpoint from the spine $Q(y=0)=\\frac{bk}{a}\\left(\\frac{x}{b}\\right)^{(2-{2}/{k})} + \\frac{a}{bk} \\left(\\frac{x}{b}\\right)^{-(2-{2}/{k})},$ see Appendix .",
"We see that, as expected, $Q\\rightarrow \\infty $ as $x\\rightarrow 0$ (the spine footpoint).",
"We are interested in the behaviour for small $x$ , and since $0<k<1$ , the relevant term close to the spine is the first one, so that $Q(y=0) \\approx \\frac{bk}{a}\\left(\\frac{x}{b}\\right)^{(2-{2}/{k})}.$ This implies that in the weak field region ($0<k<1/2$ ) $Q$ should be larger (for fixed $x$ ) since we have a larger negative exponent.",
"This is to be expected given the strong field line divergence in the $xy$ -plane in this region.",
"Hence a level curve of $Q$ would be expected to be elongated along the weak field direction.",
"Note that for the rotationally symmetric case ($k=1/2$ ) we have that $Q$ decays like $1/r^2$ .",
"Consider now the distribution of $Q_\\perp $ (see Section REF ).",
"Evaluating $Q_\\perp $ on the $x$ -axis as before we obtain the following expression $Q_\\perp (y=0)=\\frac{x^4k^2[(x/b)^{-2/k}b^2+a^2]+(x/b)^{2/k}b^4[a^2-k^2x^2]}{k^2b^2x^2\\sqrt{k^2x^2+a^2}\\sqrt{b^2+a^2(x/b)^{2/k}}}$ (Appendix ).",
"For $x\\ll b$ , we have the same dominant scaling in $x$ as before, specifically $Q_\\perp (y=0) \\approx \\frac{bk}{a}\\left(\\frac{x}{b}\\right)^{2-2/k}.$ That the behaviour of $Q$ and $Q_\\perp $ is identical in this plane is expected since in this plane as we get close to the spine and fan the field lines intersect the boundaries approximately perpendicular.",
"We now examine the dimensions of contours of $Q$ on the spine boundary ($z=\\pm a$ ).",
"Specifically, we rearrange Equation (REF ) to find the radius $x=r_x(Q_0)$ at which $Q_\\perp =Q_0$ .",
"Assuming that $x/b\\ll 1$ we can directly invert Equation (REF ) to obtain $r_x(Q_0)=b \\left( \\frac{aQ_0}{b} \\right)^{k/(2k-2)}$ One can obtain from this the same information along the $y$ -axis by making the replacement $k\\rightarrow 1-k$ to give $r_y(Q_0)$ .",
"Figure: (a) For fixed k=0.4k=0.4, r x (Q 0 )r_x(Q_0) (red) and r y (Q 0 )r_y(Q_0) (black dashed), as defined by Eq.",
"();(b) Log-log plot of the aspect ratio r x /r y r_x/r_y (see Eq.",
"), for a=ba=b and fixed k=0.45k=0.45 (red), k=0.35k=0.35 (black, dashed) and k=0.25k=0.25 (blue, dot-dashed).As shown in Figure REF (a), for $k=0.4$ the spacing of $Q_\\perp $ contours drops off more slowly in the weak field direction, consistent with above.",
"Now let us analyse the asymmetry in the $Q_\\perp $ contour produced by the asymmetry in null eigenvalues in more detail, by examining the relative decay of $Q_\\perp $ from the spine footpoint along the two axes.",
"Using Eq.",
"(REF ) we have that $\\frac{r_x(Q_0)}{r_y(Q_0)}=\\left(\\frac{aQ_0}{b}\\right)^{(k-1/2)/(k(k-1))}.$ In Figure REF (b) we plot the ratio $r_x/r_y$ for $a=b$ and three particular values of $k$ .",
"We see that, as expected, this aspect ratio increases as the magnetic field asymmetry increases.",
"Furthermore, we observe that for high $Q_\\perp $ values the $Q_\\perp $ distribution is highly `non-circular' even for moderate values of $k$ , while lower level contours of $Q_\\perp $ only show high asymmetry when $k$ is far from 0.5 (note that Figure REF (b) is a log-log plot).",
"Note also that a full range of degrees of null point eigenvalue ratios (corresponding to $k\\in (0,1)$ ) is obtained in solar extrapolations, and even for a moderate asymmetry (values of $k$ close to 0.5) the eccentricity is significant.",
"For example [10] analysed 6 nulls with $k=0.08$ – $0.33$ , the configuration analysed by [34] contained a null point with $k\\approx 0.11$ , and [19] carried out an extensive survey of potential field extrapolations from three years' worth of magnetogram data, identifying 1924 coronal null points – choosing the orientation such that $0\\le k\\le 0.5$ their data shows a relatively uniform distribution of $k$ values between 0.1 and 0.5, with a mean value of $k$ of 0.26 and a standard deviation of 0.11."
],
[
"Squashing factor for a surface encircling the fan",
"The calculations of the previous section allow us to visualise the $Q$ distribution along the coordinate directions.",
"However, since the vertical planar boundary intersects only a subset of the fan field lines, they do not allow us to visualise the $Q$ distribution all around the spine or fan footpoints.",
"In order to do this we must choose a target plane that intersects all fan field lines (such that all field lines passing close to the null intersect both the launch and target surfaces).",
"Figure: Contours of log 10 (Q ⊥ )\\log _{10} (Q_\\perp ) as calculated between surfaces z=a=1z=a=1 and r=b=1r=b=1 – visualised on z=1z=1 for different kk.",
"Contour levels are log 10 (Q ⊥ )={2,3,4,5}\\log _{10} (Q_\\perp )=\\lbrace 2,3,4,5\\rbrace , and are coloured {\\lbrace black, blue, red, green}\\rbrace .As such, we now calculate $Q$ between a planar `launch' surface at $z=a$ intersecting the spine, and a cylindrical `target' surface at $r=b$ encircling the fan, $a,b$ constant (Figure REF b).",
"That is, we study the mapping generated by the field lines $(x,y,a)\\rightarrow (b,\\Theta , Z)$ .",
"In this case we are unable to obtain a full analytical expression for the field line mapping and its inverse, and thus we evaluate $Q$ numerically.",
"We integrate between $10^5$ and $10^6$ field lines from a rectangular grid of starting points at $z=a$ to obtain their intersections with $r=b$ , then perform derivatives of the mapping using a fourth-order-accurate centred difference over that grid.",
"In this case we evaluate $Q_\\perp $ , since especially in the regions of strongly diverging field lines (the weak field region, around the $x$ -axis for $k<1/2$ ) the field lines intersect the circular cylinder surface far from perpendicular.",
"The $Q_\\perp $ maps at $z=a$ for different $k$ are presented in Figure REF .",
"As predicted by the planar boundary analysis, we see a stretching of the $Q_\\perp $ contours along the weak field direction in the fan ($x$ for $k<1/2$ ).",
"We also see that the contours do not form simple ellipses, but are pinched in the middle so that their maximum extension is at finite $x$ values.",
"Therefore in order to measure the asymmetry of the $Q_\\perp $ distribution around the spine, it is arguably most useful to determine the largest extent of these contours along the $x$ and $y$ directions, rather than simply examining the profile along the coordinate axes.",
"In Figure REF we plot (with circles and solid lines) the contour aspect ratio defined as the maximum contour extent along $x$ divided by the maximum extent along $y$ (over all $x$ ).",
"Figure: Aspect ratio of Q ⊥ Q_\\perp contours on the spine boundary, defined as the maximum contour extent in the long direction (xx for the linear null) divided by the maximum contour extent perpendicular to that (yy for the linear null).",
"Circles: Q ⊥ Q_\\perp calculated between the surfaces z=a=1z=a=1 and r=b=1r=b=1 for the linear null point in Eq. ().",
"Crosses, squares: Q ⊥ Q_\\perp around the inner and outer spine footpoints, respectively, for the separatrix dome configuration in Eq. ().",
"Blue, k=0.5k=0.5; black, k=0.45k=0.45; red, k=0.35k=0.35; green, k=0.2k=0.2.The plot demonstrates the same trends as observed in the previous section, specifically that the aspect ratio is greater for both increased null point asymmetry (smaller $k$ for $k< 0.5$ ) and for higher $Q_\\perp $ levels.",
"Note that the qualitative features discussed above are present if one considers $Q$ instead of $Q_\\perp $ .",
"Figure: Contours of log 10 (Q ⊥ )\\log _{10} (Q_\\perp ) as calculated between surfaces z=a=1z=a=1 and r=b=1r=b=1 – visualised on r=1r=1 for different kk.",
"Contour levels are log 10 (Q ⊥ )={2,3,4,5}\\log _{10} (Q_\\perp )=\\lbrace 2,3,4,5\\rbrace , and are coloured {\\lbrace black, blue, red, green}\\rbrace .We can now perform the same $Q_\\perp $ calculation procedure but with the launch and target boundaries reversed, in order to find the pattern of $Q_\\perp $ in the vicinity of the fan surface – shown in Figure REF .",
"We observe that $Q_\\perp $ decreases in some locations and increases in others as we increase the asymmetry of the null.",
"The weak field region is along the $x$ -axis, which corresponds to $\\theta =0,\\pi $ , by the usual convention.",
"Again the widest $Q_\\perp $ contours are located in the vicinity of these weak field directions.",
"That is, for a given distance from the separatrix (height $z$ ), $Q_\\perp $ is largest along the weak field direction.",
"The results above are entirely consistent with those of e.g. [34].",
"In particular, we see that one obtains extended $Q_\\perp $ (or $Q$ ) contours around the spine for even a moderate degree of null point asymmetry.",
"Of course other global features of the field could well also contribute, but the figures show that the high $Q$ region in which the null is embedded is not a special additional feature of the particular field studied by [34], but is rather a natural consequence of having a coronal null that is not rotationally symmetric."
],
[
"Simulation setup and results",
"In this section we ask the question: what is the relation between current layers that form at 3D nulls and the distribution of the squashing factor identified in the previous section, and to what extent can we expect the $Q$ (or $Q_\\perp $ ) profile to predict where flare ribbons might be observed?",
"We consider resistive MHD simulations similar to those of [21].",
"Specifically, at $t=0$ in our simulations we have a linear magnetic null point of the form ${\\bf {B}}&=& B_0 [y(2k-1)\\cos \\phi \\sin \\phi + x(k\\cos ^2\\phi + (1-k)\\sin ^2\\phi ),\\\\&&y((1-k) \\cos ^2\\phi + k\\sin ^2\\phi ) + x(2k-1)\\cos \\phi \\sin \\phi , -z], \\nonumber $ which reduces to Eq.",
"(REF ) for $\\phi =0$ .",
"For $\\phi \\ne 0$ , the fan plane eigenvectors are rotated by an angle $\\phi $ with respect to the coordinate axes.",
"The simulation domain is $x,y\\in [-3,3],\\, z\\in [-0.5,0.5]$ and we set $B_0=1$ .",
"We apply a driving velocity on the domain boundaries that advects the spine footpoints in opposite directions on opposite $z$ -boundaries, in the $y$ direction.",
"Specifically, we let ${\\bf {v}}(z=\\pm 0.5) & = & \\pm {\\bf {e}}_y \\,v_0 \\tanh \\left(\\frac{t}{T}\\right)\\,\\left(\\tanh \\left(\\frac{x-x_0}{x_h}\\right)-\\tanh \\left(\\frac{x+x_0}{x_h}\\right)\\right) \\\\& & ~~~~~~ \\times ~ \\left(\\tanh \\left(\\frac{y-y_0}{y_h}\\right)-\\tanh \\left(\\frac{y+y_0}{y_h}\\right)\\,\\right) , \\nonumber $ with $v_0=0.02, T=0.1, x_0=0.3,y_0=0.6,x_h=y_h=0.2$ .",
"Outside these regions ${\\bf {v}}={\\bf 0}$ on all boundaries and ${\\bf {B}}$ is line-tied.",
"The plasma density and pressure are initially uniform, $\\rho _0=1$ , $p_0=0.0333$ , so that the driving velocity is highly sub-sonic and sub-Alfvénic.",
"In addition, the resistivity $\\eta =2\\times 10^{-3}$ , also spatially uniform, throughout.",
"As the simulation proceeds the stress injected by the boundary driving focusses around the null point – the null point collapses and a current sheet forms around it [43].",
"We examine three different simulations; in the first the null point is rotationally symmetric, while in the second and third we take an asymmetric null with $k=0.35$ corresponding to frame (c) in Figures REF and REF .",
"Specifically, we use parameters in Eq.",
"(REF ) as follows: simulation 1: $k=0.5,\\phi =0$ ; simulation 2: $k=0.35, \\phi =0$ ; simulation 3: $k=0.35, \\phi =\\pi /4$ .",
"Hereafter we analyse the state reached in the simulations at time $t=3$ , this constituting a representative time by which the current layer has formed and reconnection is underway.",
"Figure: Current density isosurface at level |𝐉|=2.5|{\\bf {J}}|=2.5 at t=3.0t= 3.0 from the MHD simulations (Section ).",
"The view is down onto the fan surface; overlayed are fan field lines.",
"The dashed red curve indicates the dimensions of the β=1\\beta =1 curve in the z=0z=0 plane.The current layer formed at $t=3$ is shown for each simulation in Fig.",
"REF .",
"In each case the current is maximum at the null point.",
"Note that the current layer extends from the high-$\\beta $ region into the low-$\\beta $ region in all simulations.",
"In cases (a) and (b) the current extends perpendicular to the direction of the driving motion, which is also the weak field direction for case (b), while in case (c) there is a competition between the driving and weak field directions.",
"However for this moderate value of $k = 0.35$ , the extension is not very pronounced.",
"It is already known that when the null point is asymmetric, the current tends to spread preferentially along the weak field direction in the fan plane, at least when the driving has a non-zero component perpendicular to this direction [1], [21], since the weak field in this region is less able to withstand the field collapse.",
"Note that one can in general expect currents to extend along this weak field direction also due to the strong $Q$ values further from the separatrix there, which reflect the variety of field line connectivities present nearby.",
"Such field lines are anchored in distant locations at the boundary, so they typically experience different magnetic stress from the boundary motions, so different perturbed ${\\bf {B}}$ .",
"However, at this time in our simulations there has not been time for any communication with the $x$ and $y$ boundaries, and so the current accumulation is associated only with stresses being applied from the spine boundaries and the local collapse dynamics."
],
[
"Predicting flare ribbon locations",
"A proper diagnosis of expected locations of flare ribbons would require a self-consistent modelling of particle acceleration in a null point current sheet.",
"This is yet to be done – most existing studies use simplified analytical models that do not properly represent the structure of the magnetic field and current layer.",
"We emphasise then that the following analysis based on the MHD approximation is a crude first step toward predicting the expected location of energetic particles.",
"First, we should understand the mechanisms by which the acceleration could occur.",
"Null points have been proposed as efficient particle acceleration sites first because the geometry of the field around the null naturally allows for magnetic mirroring, and particle acceleration by gradient-${\\bf B}$ and curvature drifts [58], [38], [23], [53].",
"The particle dynamics in the vicinity of the null may indeed be inherently chaotic [33].",
"In addition, when a current layer is present at the null during reconnection, there can be direct acceleration by the associated electric field.",
"This was observed to be the dominant acceleration mechanism in the PIC simulations of [6], who studied null point reconnection in the corona using a configuration similar to that of [34].",
"To understand the resulting particle deposition patterns, one must first understand the structure of the electric current layer at the null.",
"This electric current distribution is determined in general by a combination of factors.",
"During spine-fan reconnection, the spine and fan of the null point locally collapse towards one another (see Figure REF ) as in our simulations.",
"This collapse of the null occurs in general when a shear perturbation of either the spine or the fan occurs [43].",
"The plane in which this collapse occurs (plane that contains the deformed spine line – see Figure REF ) is determined both by the perturbation that drives the collapse and the null point structure.",
"The associated current sheet that forms has a current vector that at the null is oriented perpendicular to the plane of collapse, see Figure REF .",
"Thus (in resistive MHD) the parallel electric field is oriented along the fan surface, perpendicular to the spine and the plane of the null point collapse [43].",
"Hence, we expect a strong acceleration layer near the null in the fan plane, and thus particle deposition in the vicinity of the fan surface footprint (of oppositely charged particles on opposite sides).",
"It is also possible that particles accelerated towards the null in this layer may follow the field lines out along the spines.",
"However, when they reach the vicinity of the null point they become effectively de-magnetised, and most particles are simply accelerated `across' the current layer, and out along the fan (rather than being deflected up the spine).",
"Exactly this effect was observed by [6], who noticed very few particles accelerated out along the spines.",
"Figure: Schematic of the structure around a null point at which spine-fan reconnection is taking place.",
"Grey and black lines are magnetic field lines that show the local field collapse.",
"Black arrows indicate the plasma flow.",
"The shaded grey surface outlines the current layer locally along the fan (as in Figure ), while the red arrows within it indicate the dominant current vector orientation.",
"Modified from .We should note that there are at least three factors that could cause enhanced acceleration along the spine structures as well, to create the spine footpoint ribbons observed by [34] and others.",
"First, at solar parameters the reconnection process around the site of the original null is likely to be significantly more complex than in the simple models where a single laminar current layer is present.",
"Indeed this current sheet is susceptible to a tearing-type instability that leads to a fragmented current layer containing many nulls, as described by [60].",
"In such a configuration, the vicinity of the original null becomes highly turbulent, and one would expect efficient particle scattering along both the large-scale spine and fan directions.",
"Second, if there is some large-scale rotational external motion, this can drive torsional spine reconnection, associated with a component of current parallel to the spine [42] which can accelerate particles along the spine [25].",
"Finally, one could expect strong mirroring of particles close to the fan footpoints to lead to a distribution of particles also around the spine footpoints (note that the PIC simulations of [6] did not cover the domain all the way to the photosphere).",
"Based on the above considerations, there exists no unequivocal way of diagnosing general expected particle deposition footprints.",
"However, independent of the details of the acceleration mechanism, if the acceleration happens during reconnection at the null point, then as particles move away from the null they will become magnetised, and would be expected to be observed in the vicinity of either the spine or fan footpoints.",
"For our simulations we make the following basic assumption: Particles will be accelerated in some manner in the vicinity of the current sheet around the null.",
"Thus, we estimate expected deposition patterns by tracing field lines from the current sheet to find their intersections with the boundaries.",
"To predict particle deposition locations on the spine boundaries ($z=\\pm 0.5$ ), we therefore perform the following procedure: We select an array of points that lie on a given current contour level within the domain, and trace field lines from each of these points to the spine boundaries.",
"We then compare the intersections of these field lines with the $Q_\\perp $ distribution on the boundary to determine whether they match, i.e.",
"whether the $Q_\\perp $ distribution can be expected to give a good prediction of the geometry of particle deposition signatures.",
"We first evaluate $Q_\\perp $ in our simulations using launch boundary $z=0.5$ and target boundary a circular cylinder of radius 2.8, as in Section REF .",
"These contours are plotted, together with footpoint locations for field lines threading the current layer, in Fig.",
"REF .",
"Consider first simulation 1 where the field is initially rotationally symmetric (Figs.",
"REF a, REF a).",
"We see that there is some $x$ -$y$ asymmetry in the 3D current distribution (Fig.",
"REF a) that is due to the orientation of the boundary driver.",
"Here the null point field was initially rotationally symmetric, and when we map field lines from the current layer to the boundary we see a similar degree of asymmetry in the 3D current layer and its 2D projection on the boundary (Fig.",
"REF a).",
"Figure: z=0.5z=0.5 boundary.Coloured crosses show intersection points with this plane of field lines that are traced from initial points lying on a 3D contour of |𝐉||{\\bf {J}}| at t=3.0t=3.0.Selected contours are at 25%, 50% and 75% of the domain maximum (green, red, and yellow crosses, respectively) – for (a) k=0.5,φ=0k=0.5, \\phi =0, (b) k=0.35,φ=0k=0.35, \\phi =0, (c) k=0.35,φ=π/4k=0.35,\\phi =\\pi /4.",
"Black contours denote log 10 Q ⊥ \\log _{10}Q_\\perp as calculated between this plane and a circular cylinder of radius 2.8.As a contrast, now compare Figs.",
"REF (b), REF (b).",
"There is little difference in the asymmetry of the 3D current density distribution from Fig.",
"REF (a).",
"However, the field geometry means that field lines approach the spine more slowly along the weak field direction, so that the projection of $|{\\bf {J}}|$ is elongated along this direction.",
"This reinforces the fact that the current preferentially spreads along that direction – but it is the 3D field line geometry rather than the current layer geometry that has the major effect on the projected $|{\\bf {J}}|$ map.",
"This projection of $|{\\bf {J}}|$ along the field lines now has a comparable geometry to the $Q_\\perp $ contours, although the projected $|{\\bf {J}}|$ map does not exhibit such high eccentricity/asymmetry.",
"We observe a similar behaviour when we examine Figure REF (c), except that in this case (where the boundary driving is at a finite angle to the null point eigenvectors) the two distributions appear to be rotated with respect to one another – more pronounced for low current contour levels.",
"These results indicate that a complicated combination of the driving geometry, the field geometry, and the current intensity in the current sheet (itself dependent on plasma parameters and the driving of the system) will influence the expected particle precipitation locations (even using this simple estimate for these locations).",
"Now consider the boundaries intersected by fan field lines.",
"The same method as applied before using field line mapping is not useful in determining the angular distribution of expected particle locations, since by definition every field line of the fan connects back to the null and therefore the maximum current region.",
"What is more important for this angular distribution is the orientation of the electric field in the acceleration region together with the global field structure.",
"Let us make the following simple considerations.",
"Particle acceleration along fan field lines will occur through direct acceleration by the DC electric field (or other mechanisms as mentioned above).",
"It is expected that at the null this electric field is directed predominantly towards the weak field direction (perpendicular to the plane of null collapse, see above).",
"However, the field lines diverge away from this direction, and converge towards the strong field region.",
"Thus the particles – which becomes re-magnetised as they are accelerated away from the null – are naturally channelled along the field lines into the neighbourhood of the strong field direction.",
"This is the region in which $Q$ (or $Q_\\perp $ ) falls off more quickly away from the separatrix – and thus we expect that particles accumulate around the strong field regions of the fan footprint, corresponding to the locations where the $Q$ contours are narrowest about the fan."
],
[
"Effect of the global field",
"The above sections showed that the asymmetry of the field in the local vicinity of the null can have a profound effect on $Q$ , and on the mapping of field lines from the current layer.",
"However, other features of the global field can clearly distort this picture.",
"In this section we return to an equilibrium field and examine the effect of the global coronal geometry, to determine whether the above results regarding the $Q_\\perp $ -distribution asymmetry carry through beyond the linear null point field.",
"We consider a null point in a separatrix dome configuration, as shown in Figure REF (a).",
"Figure: (a) Magnetic field lines (red) outlining the spine and fan structure of the coronal null point in the model field of Equation () with S=0.109S=0.109 (corresponding to k=0.35k=0.35).",
"The shading on z=0z=0 shows B z B_z there, saturated to values -1-1 (black) and +1 (white).",
"(b) B z B_z distribution on z=0z=0 together with contours of the Q ⊥ Q_\\perp distribution at levels log 10 Q ⊥ ={1.5,4}\\log _{10}Q_\\perp =\\lbrace 1.5, 4\\rbrace , coloured green and orange, respectively.The magnetic field is potential, and on the photosphere corresponds to a magnetic dipole, one polarity of which contains an embedded `parasitic polarity'.",
"This field is constructed by placing four magnetic point charges at locations outwith our domain of interest.",
"Specifically, we restrict our studies to the half-space $z>0$ , where $z=0$ represents the photosphere, and place all point charges at $z<0$ .",
"The magnetic field is given by ${\\bf {B}}=\\sum _{i=1}^4 \\epsilon _i \\frac{{\\bf {x}}-{\\bf {x}}_i}{|{\\bf {x}}-{\\bf {x}}_i|^3}$ where ${\\bf {x}}_i$ are the locations and $\\epsilon _i$ are the strengths of the point charges.",
"Here we take $\\lbrace \\epsilon _1,\\epsilon _2,\\epsilon _3,\\epsilon _4\\rbrace =\\lbrace 1,-1,-0.2,-0.2\\rbrace $ and ${\\bf {x}}_1=(0,1,-0.5)$ , ${\\bf {x}}_2=(0,-1,-0.5)$ , ${\\bf {x}}_3=(S,1+S/2,-0.2)$ , ${\\bf {x}}_4=(-S,1-S/2,-0.2)$ .",
"The charges located at ${\\bf {x}}_3$ and ${\\bf {x}}_4$ are associated with the parasitic polarity around $(x,y)=(0,1)$ .",
"The parameter $S$ controls the separation of these two charges.",
"When $S=0$ they are coincident, and the parasitic polarity is approximately circular.",
"However, as $S$ is increased, the parasitic polarity becomes increasingly stretched (see Figure REF ).",
"This elongation of the parasitic polarity means that the field strength becomes less homogeneous in the vicinity of the fan footprint, and also that the field asymmetry at the null – as measured by the fan plane eigenvalues – increases.",
"Here we analyse the magnetic topology for three different values of $S$ : $S=0$ leads to an approximately symmetric field in the local vicinity of the null that corresponds closely to the symmetric case $k=0.5$ for the linear field, $S=0.067$ that gives fan eigenvalues that correspond to $k=0.45$ for the linear null, and $S=0.109$ that corresponds to $k=0.35$ .",
"Figure: Distribution of Q ⊥ Q_\\perp around the inner (left) and outer (right) spine footprints, for the magnetic field of equation () with S=0S=0 (top), S=0.067S=0.067 (middle), and S=0.109S=0.109 (bottom).As shown in Figure REF (b), when the null point is asymmetric the level curves of $Q_\\perp $ on the photosphere form extended structures.",
"As $S$ is increased, so too the asymmetry of these level curves increases – both around the inner and outer spine footpoints – see Figure REF .",
"To determine whether the same scaling with the null asymmetry as above is observed we measure the aspect ratio of these ribbons, defined as follows.",
"The `length' of a given $Q_\\perp $ contour is defined as the maximum extent along any line passing through the spine line footpoint (location at which $Q_\\perp \\rightarrow \\infty $ ).",
"The width is then defined as the maximum extent of the contour along any line perpendicular to this, analogous to section REF .",
"The aspect ratio, being the ratio of the length over the width, is calculated for different $Q_\\perp $ contours for each value of $S$ , and the results are plotted in Figure REF .",
"What we see is that there are clear differences in the values of the calculated aspect ratio, both between the inner and outer spine footpoints, and to the results for the linear field (crosses, squares, and circles in Figure REF , respectively).",
"However, as shown in Eq.",
"(REF ) the exact value of the aspect ratio depends on the locations of the launch and target footpoints (for the linear null $a$ and $b$ ), thus we would not expect an exact agreement.",
"Moreover, we note that the overall scaling of the aspect ratio with the $Q_\\perp $ level is rather well reproduced between the linear field and coronal null point field, and that for the higher $Q_\\perp $ contour levels considered the aspect ratios are in rather good agreement."
],
[
"Discussion and conclusions",
"Coronal magnetic null points exist in abundance, as demonstrated by extrapolations of the coronal field, and have been inferred to be important for a broad range of energetic events.",
"These null points and their associated separatrix and spine field lines are preferential locations for reconnection due to the discontinuity of the field line mapping.",
"This field line mapping also exhibits strong gradients adjacent to the separatrix and spine field lines, that we have analysed here using the squashing factor $Q$ (and $Q_\\perp $ ).",
"Understanding the distribution of $Q$ in the presence of separatrices is of timely importance due to the increasing use of calculation of $Q$ maps in analysing the coronal field topology.",
"While a map of the $Q$ distribution shows the presence of both true separatrices and finite-$Q$ QSLs, one should note that the physics of current layer formation / energy storage is critically different between a high-$Q$ region containing a separatrix and one that does not.",
"In particular, current singularities are known to form in the ideal limit in the presence of separatrices [41].",
"Thus reconnection onset is inevitable irrespective of the dissipation (though may be `slow' in an energy storage phase).",
"By contrast, the current layers that form at QSLs are probably finite [7], [16], with the onset of reconnection at coronal parameters then requiring a thinning of the QSL and current layer during the energy storage phase [3], [9].",
"What is clear is that in the case of both null points (separatrices) and QSLs, the current layer formation and eventual dynamics are crucially dependent on the driving of the system, for example from the photosphere.",
"In this paper, we have made a detailed analysis of the distribution of $Q$ in the presence of magnetic nulls and their associated separatrices.",
"The main results can be summarised as follows.",
"It is generically the case that $Q$ is not uniformly distributed around the spine and fan footpoints.",
"Specifically, a generic null point is not rotationally symmetric, and while $Q$ is infinite formally on both the spine and fan of the null, it decays most rapidly away from the spine/fan in the direction in which $|{\\bf {B}}|$ increases most rapidly.",
"When a linearisation of the null is performed (this linearisation characterising the local topology of the field for any topologically stable null [24]), this direction corresponds to the eigenvector of the largest fan eigenvalue.",
"The result of the above is that contours of $Q$ are broadest along the direction of the eigenvalue with smallest fan eigenvalue – denoted herein as the `weak field direction'.",
"In particular, this demonstrates that the extended, elliptical-like high-$Q$ halo around the spine footpoints observed by, e.g., [34], [54] is not a special feature of the particular observations, but is a generic feature when a coronal null is present whose fan eigenvalues are not equal (i.e.",
"when the field strength is not homogeneous around the fan footprint).",
"The asymmetry of the halo of $Q$ contours around the spine/fan increases as the null point asymmetry (measured by the ratio of the eigenvalues) increases.",
"Furthermore, for a given null point asymmetry, the stretching of the $Q$ contours is most extreme for the highest contour levels.",
"When the global field geometry (beyond the linear field region) is considered, the exact aspect ratios of the $Q$ contours are modified from the simple linear null case, but the core of the distribution of $Q$ still reflects the conditions around the null.",
"This is especially true for high $Q$ -contour levels.",
"As a first approximation for understanding why the geometry of flare ribbons is observed to agree well with the geometry of the $Q$ halo in circular ribbon flares [34], we analysed MHD simulations of null point reconnection.",
"We traced field lines through the current layer, and analysed the relationship between their intersections with the boundary and the $Q$ contours on the boundary.",
"While no on-to-one relation was found, we showed that field lines traced from the core of the current layer match rather well with the highest $Q$ contours.",
"Thus, particularly for the kernels of the flare ribbons, the $Q$ distribution should in general be expected to predict well the location and orientation of the ribbons.",
"It is well established that an understanding of the null point structure and its relation to the driving of the system is crucial for determining the current layer formation at the null and associated dynamics.",
"We have shown here that this null point structure, defined by its local eigenvectors and eigenvalues, is intrinsically linked to the distribution of $Q$ away from the spine/fan.",
"Furthermore, this extension of the $Q$ halos around the spine/fan footpoints is in general important for diagnosing the regions of the photosphere that are magnetically connected to any current layer that forms at the null.",
"If we hypothesise this current layer to be a primary site of particle acceleration, this provides predictive properties for e.g.",
"flare ribbon formation.",
"We conclude that the physics in the vicinity of the null and how this is related to the extension of $Q$ away from the spine/fan can be used in tandem to understand observational signatures of reconnection at coronal null points.",
"The authors acknowledge fruitful discussions with G. Valori, E. Pariat, P. Wyper, E. Priest and M. Janvier.",
"D.P.",
"is grateful for financial support from the Leverhulme Trust and the UK's STFC (grant number ST/K000993).",
"The work by K.G.",
"was supported by a research grant (VKR023406) from VILLUM FONDEN."
],
[
"Calculation of $Q$ and {{formula:15cda0e2-aa5f-468c-bd6d-918f6249a911}} for the linear null",
"For the linear null point magnetic field of Equation (REF ), the field line equations ${\\rm d}{\\bf {X}}(s)/{\\rm d}s={\\bf {B}}({\\bf {X}}(s))$ may be solved to obtain parametric equations ${\\bf {X}}(s)$ for the field lines; $X(s)=X_0\\,{\\rm e}^{k s}, \\quad Y(s)=Y_0\\,{\\rm e}^{(1-k) s}, \\quad Z(s)=Z_0\\,{\\rm e}^{-s},$ where ${\\bf {X}}(s)=(X(s),Y(s),Z(s))=(X_0,Y_0,Z_0)$ at $s=0$ .",
"Now, set $s=0$ on the plane $z=Z_0=a$ , intersecting the spine, so that $(X_0,Y_0,Z_0)=(x,y,a)$ .",
"Then we can eliminate $s$ in the above equations, to obtain $Y=y\\left(\\frac{x}{X}\\right)^{(k-1)/k},\\quad Z=a\\left(\\frac{x}{X}\\right)^{1/k}.$ We now choose the `target plane' to be $(X,Y,Z)=(b,Y,Z)$ , $b$ constant.",
"Finally, identifying $\\lbrace U,V,u,v\\rbrace $ in Eq.",
"(REF ) with $\\lbrace Y,Z,x,y\\rbrace $ , the required derivatives may be obtained.",
"A little algebra leads to the following expression for $Q$ : $Q(x,y)=\\left( {\\frac{{y}^{2} \\left( k-1 \\right) ^{2}}{{k}^{2}{x}^{2}} \\left( {\\frac{x}{b}} \\right) ^{2-2/k}}+ \\left( {\\frac{x}{b}} \\right) ^{2-2/k}+{\\frac{{a}^{2}}{{k}^{2}{x}^{2}} \\left( {\\frac{x}{b}} \\right) ^{2/k}} \\right) {\\frac{bk}{a}},$ which reduces to Equation (REF ) for $y=0$ .",
"Evaluation of $Q_\\perp $ for the same planar boundaries as above requires that we calculate ${Q_\\perp }^{2}={\\frac{{{\\partial U}}^{i}}{{{\\partial u}}^{k}} \\left( \\delta _{{{\\it ij}}}-{\\frac{B_{{i}}^\\star B_{{j}}^\\star }{ \\left|{\\bf {B}}^\\star \\right| ^{2}}} \\right)\\frac{{\\partial U}^j}{{\\partial u}^l}\\left( {\\delta }^{{\\it lk}}+{\\frac{{B}^{l}{B}^{k}}{{B_{{n}}}^{2}}} \\right) } \\cdot \\frac{|{\\bf {B}}^\\star |}{|{\\bf {B}}|}$ where $\\delta $ is the Kronecker delta, and summation over repeated indices is assumed [55].",
"In addition, $\\lbrace U^1,U^2\\rbrace =\\lbrace Y,Z\\rbrace $ , $\\lbrace u^1,u^2\\rbrace =\\lbrace x,y\\rbrace $ , $B_1^\\star ,B_2^\\star ,|{\\bf {B}}^\\star |$ are $B_y, B_z, |{\\bf {B}}|$ evaluated at the target boundary $X=b$ , and $B^1,B^2,B_n$ are $B_x,B_y,B_z$ evaluated at the launch plane $z=a$ .",
"The resulting expression is too lengthy to reproduce here, but reduces to Equation (REF ) for $y=0$ ."
]
] | 1605.05704 |
[
[
"An intense, cold, velocity-controlled molecular beam by\n frequency-chirped laser slowing"
],
[
"Abstract Using frequency-chirped radiation pressure slowing, we precisely control the velocity of a pulsed CaF molecular beam down to a few m/s, compressing its velocity spread by a factor of 10 while retaining high intensity: at a velocity of 15~m/s the flux, measured 1.3~m from the source, is 7$\\times$10$^{5}$ molecules per cm$^{2}$ per shot in a single rovibrational state.",
"The beam is suitable for loading a magneto-optical trap or, when combined with transverse laser cooling, improving the precision of spectroscopic measurements that test fundamental physics.",
"We compare the frequency-chirped slowing method with the more commonly used frequency-broadened slowing method."
],
[
"Introduction",
"Molecular beams with controllable forward velocity have been at the forefront of cold ($T\\sim $ 1–1000 mK) molecule research for many years [1].",
"Such beams are increasingly being used for precise measurements that test fundamental physics, including measurements of the electron's electric dipole moment [2], [3], parity violation in nuclei [4] and chiral molecules [5], [6], changes to the fundamental constants [7], [8], [9] and tests of QED [10].",
"The precision of these measurements could be greatly improved using colder and slower molecular beams, preferably in the ultracold regime ($T\\le 1$ mK).",
"Traditional techniques for controlling the forward velocity, such as Stark deceleration and its variants [11], [12], [13], [14], as well as recently-developed alternatives [15], [16], do not provide cooling.",
"In some cases, molecules have been trapped and then cooled to lower temperatures by adiabatic [17], evaporative [18] or Sisyphus [19], [20], [21] cooling.",
"Sympathetic cooling may also be possible [22], [23].",
"Recently, a few molecular species have been directly laser cooled, either by compressing the transverse velocity distribution of a molecular beam [24], [25], or in a magneto-optical trap (MOT) which provides simultaneous trapping and cooling [26], [27], [28].",
"An important current challenge is to increase the number of molecules in the MOT by increasing the fraction delivered below the capture velocity, which is typically 10–20 m/s [29].",
"At present, radiation pressure slowing is used [30], with the laser linewidth broadened to address a wide velocity range [30], [31], [32].",
"This approach yields limited control of the final velocity and typically slows the beam without compressing the velocity distribution, delivering only a tiny fraction of the molecules at the desired position and speed.",
"Here, we present an alternative approach, using frequency-chirped laser slowing of CaF to both compress the velocity distribution into a narrow range and slow to the desired final velocity.",
"We find this approach superior to the frequency-broadened technique, realizing finer velocity control, decreased temperature, and greatly increased molecular flux, all of which are essential for making dense molecular MOTs and intense molecular beams for precise measurements."
],
[
"Experiment Setup",
"Figure REF (a) shows the relevant energy levels of CaF and the vibrational branching ratios between them, along with our notation.",
"The main cooling transition is B(0)–X(0) with wavelength $\\lambda _{\\rm {main}}$ =531 nm, linewidth $\\Gamma $ =2$\\pi \\times $ 6.3 MHz [36] and single-photon recoil velocity 1.3 cm/s.",
"Population that leaks into X(1) is returned to the cooling cycle via the A(0)–X(1) transition at $\\lambda _{\\rm {repump}}$ =628 nm.",
"From an experimental study of potential loss channels (see Sec.",
"REF ), we conclude that with only these two wavelengths, $\\sim $ 3$\\times $ 10$^{4}$ photons per molecule can be scattered, corresponding to a velocity change of 390 m/s, before half are lost from the cooling cycle.",
"Using separate upper states for the main cooling and repump lasers almost doubles the scattering rate [37] relative to all previous work [38], [24], [30], [25], [39], [31], [32] where X(0) and X(1) were both driven to A(0).",
"Figure REF (b) illustrates the apparatus.",
"A pulsed beam of CaF is produced by a cryogenic buffer gas source [40], [41], [42].",
"At $t$ =0, a pulsed laser (5 mJ, 4 ns, 1064 nm) ablates Ca into a 4 K copper cell, through which flow 1 sccm of 4 K helium and 0.01 sccm of 270 K SF$_6$ .",
"The resulting CaF molecules are cooled by the He and entrained in the flow.",
"They exit the cell at $z$ =0 via a 3.5 mm diameter aperture, and are collimated by an 8 mm diameter aperture at $z$ =15 cm that separates the source from the main chamber, where the pressure is 3$\\times $ 10$^{-7}$ mbar.",
"Within a factor of 2, the flux is 1.9$\\times $ 10$^{11}$ molecules per steradian per shot in X(0), and the pulse duration at $z$ =2.5 cm is 280 $\\mu $ s (FWHM).",
"At $z$ =130 cm the molecules are detected by driving the A(0)$\\leftarrow $ X(0) transition, imaging the resulting laser-induced fluorescence (LIF) onto a photomultiplier tube (PMT), and recording the signal with a time resolution of 5 $\\mu $ s, yielding a time-of-flight (ToF) profile.",
"The 5 mW probe beam crosses the molecular beam at 60$^{\\circ }$ or 90$^{\\circ }$ to the molecular beam propagation direction for velocity-sensitive or insensitive measurements, respectively.",
"Radio frequency sidebands applied to the probe [39] address the four hyperfine components of the transition.",
"The cooling light counter-propagates to the molecular beam and consists of 110 mW at $\\lambda _{\\rm {main}}$ applied for times between $t_{\\rm {start}}$ and $t_{\\rm {end}}$ , and 100 mW at $\\lambda _{\\rm {repump}}$ , which is applied continuously.",
"The two wavelength components have orthogonal linear polarizations, both at 45$^\\circ $ to a uniform 0.5 mT magnetic field directed along $y$ , which prevents optical pumping into dark Zeeman sub-levels [43], [44], [38].",
"For most experiments, the cooling light is collimated and has a gaussian intensity distributions with $1/e^2$ diameter of 6 mm.",
"For the experiments described in Sec.",
"REF , the light converges towards the molecular source.",
"The main cooling light is blocked on alternate experimental shots so that measurements with and without cooling can be compared.",
"To address all hyperfine components, we generate the spectrum shown in Fig.",
"REF (c,ii) by passing both lasers through electro-optic modulators (EOMs) driven at 24 MHz with a modulation index of 3.1.",
"We find the frequencies, $f_{\\rm {main}}$ and $f_{\\rm {repump}}$ , that maximize the LIF when each laser in turn is used as an orthogonal probe.",
"Then we detune the two cooling lasers so that, when counter-propagating to the molecules, they are resonant with those travelling with speed $v_{\\rm {start}}$ .",
"To compensate the changing Doppler shift as the molecules slow down, we apply linear frequency chirps with rates $\\beta $ and $\\beta \\lambda _{\\rm {main}}/\\lambda _{\\rm {repump}}$ to the main and repump lasers, respectively.",
"To compare this frequency-chirped method with the frequency-broadened method used in previous work [30], [31], [32], we fix the centre frequencies at $f_{\\rm {main}} - f_{\\rm {offset}}$ and $f_{\\rm {repump}} - f_{\\rm {offset}}\\lambda _{\\rm {main}}/\\lambda _{\\rm {repump}}$ , and produce the broadened spectrum shown in Fig.",
"REF (c,i) by sending the light through three consecutive EOMs driven at 72, 24, and 8 MHz."
],
[
"Method for determining velocity distributions",
"To determine a velocity distribution, we compare the Doppler-shifted spectrum recorded using the 60$^{\\circ }$ probe laser with the unshifted spectrum recorded using the 90$^\\circ $ probe.",
"In principle, the velocity distribution could be extracted directly from a comparison of these spectra.",
"There are three disadvantages to this direct method.",
"First, the spectrum has hyperfine structure that spans roughly the same frequency interval as the Doppler shifts, and this complicates the conversion of the spectrum into a velocity distribution.",
"Second, the spectral resolution limits the velocity resolution to about 20 m/s.",
"While this can be improved upon by deconvolving the spectral profile recorded using the 90$^\\circ $ probe, that introduces additional noise.",
"Third, the method does not make use of all the available information, in particular the fact that there is a strong correspondence between velocity and arrival time.",
"Instead, we employ a novel analysis method where we first determine that correspondence, and then use it to convert the ToF profile to a velocity distribution.",
"Figure REF illustrates the analysis method using data with $\\beta = 21$ MHz/ms, $t_{\\rm {start}}=1$ ms, $t_{\\rm {end}}=7$ ms, and $v_{\\rm {start}}=178$ m/s.",
"Data with the cooling light off (on) is referred to as “control” (“cooled”).",
"Figure REF (a) shows the control and cooled ToF profiles recorded using the 90$^\\circ $ probe, each averaged over 50 shots.",
"To measure the velocity profile we first record a Doppler-free reference spectrum using the 90$^{\\circ }$ probe.",
"The peak fluorescence signal in this spectrum defines the zero of frequency.",
"We then measure a velocity-sensitive spectrum using the 60$^{\\circ }$ probe.",
"We partition this data by arrival time, using 0.5 ms-wide time windows, so that the range of velocities is small and the spectrum is similar to the reference spectrum, but shifted according to the mean velocity.",
"Figure REF (b) shows the control and cooled spectra for molecules arriving between 7.5 and 8 ms, the time window indicated by the dashed lines in (a).",
"Because there are four hyperfine components, and the light has four rf sidebands, there are several peaks in the spectrum, three of which are clear in the data.",
"The largest peak is obtained when the four hyperfine components are simultaneously resonant.",
"We fit the data to a sum of three gaussians and use the fitted centre frequency of the largest peak to determine the mean velocity.",
"The uncertainty in this mean velocity is also obtained from this fit.",
"Applying this procedure to all time windows gives graphs of arrival time versus mean velocity, as in Fig.",
"REF (c).",
"We use these measured correlations between velocity and arrival time to turn the ToF profiles into velocity distributions.",
"To do that we need to join the points, and we have experimented with three different ways of doing this, all of which produce very similar velocity distributions.",
"The simplest is linear interpolation.",
"This works well but is not ideal because the gradient is discontinuous at each data point and the conversion between distributions is proportional to this gradient.",
"It is preferable to represent the data by a smooth curve, and we find that construction of a B-spline function can achieve that and also works well.",
"The third method, and the one we favour, is to fit the model $t = \\sum _{n=0}^{m}{a_n/v^n}$ to the data, where $a_{n}$ are free parameters and we choose $m$ appropriately.",
"We choose to use this method for all our data, since it works well and allows us to use standard fitting algorithms and goodness-of-fit measures.",
"The control data fits well with $m=1$ , as expected for zero deceleration.",
"For the cooled data, we take $m=5$ since this gives an adequate fit for all the datasets.",
"For the data in Fig.",
"REF (c) this is the smallest value of $m$ where $\\chi ^{2}$ is smaller than the median of the chi-squared distribution.",
"To find the number of molecules with velocities in the range $v\\pm \\Delta v$ , we use the curves of Fig.",
"REF (c) to find the times $t_{1,2}$ , corresponding to $v\\pm \\Delta v$ with $\\Delta v=2$ m/s, then integrate the ToF profile between $t_{1}$ and $t_{2}$ .",
"Doing this for all velocities gives the control and cooled velocity distributions such as those shown in Fig.",
"REF (d).",
"Figure: Method for determining the velocity distribution, illustrated for data with β=21\\beta = 21 MHz/ms, t start =1t_{\\rm {start}}=1 ms, t end =7t_{\\rm {end}}=7 ms, v start =178v_{\\rm {start}}=178 m/s.",
"Throughout, blue and grey data have cooling light on and off respectively.",
"(a) Control and cooled ToF profiles recorded using the 90 ∘ ^\\circ probe.",
"(b) Spectrum recorded using the 60 ∘ ^\\circ probe, for molecules arriving in the 7.5–8 ms time window [the region between the dashed lines in (a)].",
"The Doppler shift determines the mean velocity of molecules arriving in this time window.",
"Dots: data.",
"Lines: fit to sum of three gaussians.",
"(c) Dots: arrival time versus mean velocity determined this way.",
"The error bars are obtained from the fit to the spectrum.",
"Lines: fits to the model described in the text.",
"The number of molecules in a velocity bin, such as the one between the dashed lines, is found by reading off the corresponding time bin and then integrating the ToF profile within that time bin.",
"(d) Velocity distributions obtained by this method.",
"The coloured bands around the solid lines indicate the 68% confidence limits determined using the method described in the text.To determine a statistical confidence interval, we proceed as follows.",
"For each data point in Fig.",
"REF (c) we generate 400 new velocity values drawn at random from a normal distribution with mean and standard deviation given by the central value and error of that data point.",
"From these, we construct 400 new time-versus-velocity curves and associated velocity distributions using exactly the same method as described above.",
"From this large set of velocity profiles, we find the mean value at each point, along with the upper and lower limits that bound 68% of the values above and below the mean.",
"Finally, all the profiles are divided by the maximum value of the control profile, so that the peak of every control profile is set to 1.",
"The solid lines in Fig.",
"REF (d) show the mean profiles, and the bands around them represent the 68% confidence interval.",
"The accuracy of our analysis method is discussed in detail in ."
],
[
"Frequency-chirped slowing",
"The solid curves in Fig.",
"REF are experimental control and cooled ToF profiles and velocity distributions for various chirp rates, with $t_{\\rm {start}}$ =1 ms, $t_{\\rm {end}}$ =7 ms, and $v_{\\rm {start}}$ =178 m/s.",
"When $\\beta $ =0, the molecules are slowed to about 100 m/s and their velocity distribution is compressed.",
"This is reflected in the ToF profile as a depletion at early times and an enhancement at later times.",
"As $\\beta $ increases, the molecules are pushed to lower velocities, and while they arrive at the detector over a broad range of times, they always have a narrow velocity distribution.",
"The widths of the slow peaks correspond to a temperature of about 100 mK.",
"The final velocity is always lower than $v_{\\rm {end}}$ , indicating that the molecules follow the changing frequency up to the highest $\\beta $ used.",
"The dashed curves in Fig.",
"REF are simulation results.",
"For each simulation, we use a rate model [45] to determine the scattering rate versus detuning and power, and then calculate the resulting trajectories of many molecules using the experimental parameters and measured initial velocity distributions as inputs.",
"The randomness of the momentum kicks is included.",
"For all $\\beta $ , the simulations accurately predict the observed ToF profiles and velocity distributions, including the overall loss of detected molecules (see below).",
"Some predicted structure in the slowed peak is not observed experimentally, but all other features agree well, showing that the scattering rate is as expected and the experiment is well understood.",
"Supplementary simulations of a ten times longer molecular pulse, typical of most buffer-gas sources [40], [41], indicate there is no difference in the velocity distribution or the tail of the ToF profile where the slow molecules arrive, provided the light is turned on once the majority of molecules have left the source.",
"This shows that similar slowing performance can be expected for sources with more typical properties.",
"We find that the slowing depends critically on the applied magnetic field that remixes dark states.",
"In the absence of this field the slowing light has no effect.",
"The deceleration increases with applied field up to 0.5 mT, corresponding to an average Zeeman shift of 3 MHz, where the effect saturates.",
"Switching the polarization of the light [43], [25] at 5 MHz, with no applied magnetic field, gives the same results as a static polarization and a 0.5 mT magnetic field.",
"Increasing the laser intensity increases the deceleration and the number of molecules decelerated, until the intensity reaches $\\approx $ 350 mW/cm$^2$ where the effect saturates."
],
[
"Frequency-broadened slowing",
"For comparison with our frequency-chirped results, Fig.",
"REF shows ToF profiles and velocity distributions obtained using frequency-broadened light for three values of $f_{\\rm {offset}}$ .",
"Again, we address most of the molecules and slow them efficiently.",
"The velocity distribution is not as narrow as in the chirped case, but it is compressed.",
"Though not seen in previous work, this is expected [46] because all molecules are slowed until their Doppler shift is slightly below the low frequency cut-off of the broadened laser spectrum.",
"The simulations (dashed lines) agree very well with the measured ToF and velocity distributions, showing that this case is also well understood.",
"Just as for the chirped case, for the slowing to work it is essential to apply a magnetic field or to modulate the polarization of the light.",
"Once again, we found that the deceleration increases with applied field up to 0.5 mT, and that switching the polarization of the light at 5 MHz has the same effect as a 0.5 mT magnetic field.",
"The slowing saturates at a laser intensity of $\\approx $ 750 mW/cm$^2$ , about double the intensity needed for the chirped method."
],
[
"Losses",
"Both slowing techniques show a decrease in the number of detected molecules as the velocity is reduced.",
"To understand the reason, we first investigate the loss channels that might take population out of the cooling cycle.",
"The laser slowing experiments themselves provide a very sensitive way to do this.",
"To determine the fraction that leaks to state $q$ , we scan the probe laser over a transition from $q$ and measure the increase in fluorescence when the cooling light is applied.",
"Here, we use all the same parameters as in the $\\beta = 21$ MHz/ms data shown in Fig.",
"REF .",
"We determine the fraction $f(q)=\\Delta P(q)/P_{0}$ where $P_{0}$ is the initial population in X(0) and $\\Delta P(q)$ is the change in the population of $q$ induced by the slowing lasers.",
"Using the A(2) $\\leftarrow $ X(2) transition we find $f(v=2) = 3.7(1)\\%$ .",
"The simulations reproduce this result when the B(0)–X(2) branching ratio is $1.5(3)\\times 10^{-5}$ .",
"Using the Q(0) and Q(2) lines of the $A^2\\Pi _{1/2}(v=0)\\leftarrow X^2\\Sigma ^+(v=0)$ transition, we find $f(N=0) = 1.6(2)\\%$ and $f(N=2) = 0.4(2)\\%$ , corresponding to branching ratios of $7(1)\\times 10^{-6}$ to $N=0$ and $1.6(3)\\times 10^{-6}$ to $N=2$ .",
"The most obvious route to these even-parity states is the decay chain B–A–X, though there are other possibilities, including magnetic dipole transitions which are sometimes surprisingly intense for molecules [47].",
"With similar sensitivity, we searched for possible loss to $N=3$ induced by a term in the hyperfine Hamiltonian that couples states with $\\Delta N = 2$ , but found nothing.",
"From all these measurements we conclude that $\\sim 3\\times 10^{4}$ photons per molecule can be scattered before half are lost from the cooling cycle, and that very little of the loss observed in Figs.",
"REF and REF is due to leaks out of the cooling cycle.",
"Instead, the loss is due to the increased divergence of the slower molecules, compounded by stochastic transverse heating, as observed previously [30].",
"This increased divergence reduces the fraction of slow molecules that pass through the detection volume.",
"The excellent agreement between experiment and simulation confirms this, since there are no other loss mechanisms in the simulations.",
"Repeating the simulation for $\\beta $ =21 MHz/ms with transverse heating turned off, we find that the transverse heating is responsible for only 8% of the total loss.",
"Therefore, the dominant loss mechanism is the natural increase in divergence when the molecules are slowed down without any change to their transverse velocity distribution."
],
[
"Slowing to velocities below the capture velocity of a MOT",
"With the loss mechanisms understood, we increase the number of slow molecules in three ways.",
"First, we add a small transverse force by converging the cooling beam with a full angle of 8.2 mrad to a $1/e^2$ diameter of 3 mm at $z$ =0.",
"This increases the number of detected molecules by 60% relative to a collimated beam of the same power, using the same parameters as in Fig.",
"REF and $\\beta $ = 21 MHz/ms.",
"Second, we reduce the free flight time for slowed molecules by increasing $t_{\\rm {end}}$ .",
"Third, we change the chirp ramp so that the frequency is constant between $t_{\\rm {start}}$ and $t_{\\rm {chirp}}$ , then linearly chirped between $t_{\\rm {chirp}}$ and $t_{\\rm {end}}$ .",
"This slows molecules with speeds greater than $v_{\\rm {start}}$ before the chirp begins, so that they are no longer left behind, and increases the number of detected slow molecules by about 50% when $t_{\\rm {chirp}} - t_{\\rm {start}}$ =1 ms.",
"Figure REF shows the ToF profile and velocity distribution measured with these improvements.",
"Molecules arriving between 12–16 ms all have mean speeds in the narrow range 15$\\pm $ 2.5 m/s.",
"Within this range, the absolute number of molecules is 1$\\times $ 10$^{6}$ , the flux is 7$\\times $ 10$^5$ molecules per cm$^{2}$ per shot, the intensity is 2$\\times $ 10$^8$ cm$^{-2}$ s$^{-1}$ and the brightness is 5$\\times $ 10$^{9}$ cm$^{-2}$ s$^{-1}$ sr$^{-1}$ , all to within a factor of 2.",
"The velocity of these molecules is below the expected capture velocity of a MOT with $1/e^{2}$ beam diameters of 24 mm and readily available powers [29], indicating that $\\approx $ 10$^{6}$ molecules per pulse could be loaded into a MOT.",
"The corresponding simulation agrees well with the data, being just 4 m/s faster and containing about 50% more molecules.",
"Figure: Comparing slowing methods: simulated number of slow molecules at the detector in a 10 m/s-wide interval centred on the peak velocity, as a function of that velocity.",
"The number of slow molecules is expressed as a percentage of the total number of detected molecules in the control distribution.",
"The velocity is controlled via β\\beta (chirped case) and f offset f_{\\rm {offset}} (broadened case), with the slowing light on between t start t_{\\rm {start}}=4 ms and t end =t_{\\rm {end}}=12 ms. All other parameters are the same as those for Figs.",
"and ."
],
[
"Comparing the two slowing methods",
"Figure REF summarizes information from simulations where $\\beta $ and $f_{\\rm {offset}}$ are varied for the frequency-chirped and broadened cases respectively.",
"We count the number of slow molecules at the detector in a 10 m/s-wide interval centred on the peak velocity, and plot this number versus that velocity.",
"There is little difference between the two methods at higher velocities, but below 50 m/s the chirp method gives more slow molecules, e.g.",
"about ten times more at 20 m/s.",
"With broadened light, all molecules start slowing as soon as the light is turned on, those with high initial speeds never reach the final velocity, while those with low initial speeds reach it too early and then have a long way to travel with high divergence.",
"For very low final speeds, these molecules may even come to rest before reaching the detector.",
"The chirp method is more efficient because the slower molecules join the slowing process later on, and so a larger fraction of the initial distribution reaches the final velocity at a point close to the detector.",
"Figure REF also compares the effectiveness of the converging and collimated slowing beams.",
"For frequency-broadened light converging the beam reduces the molecule number.",
"This is because the slowing force has a low-velocity cut-off that shifts to higher velocities as $z$ increases, due to the falling light intensity, resulting in a much wider final velocity distribution: those that reach the cut-off early on have lower velocities than those that reach it later.",
"Thus, while there are more molecules overall, there are fewer per unit velocity range.",
"This does not happen in the chirped case, and so the converging beam yields an increase.",
"For the comparison shown in Fig.",
"REF , the slowing light turn-on and turn-off times were chosen to be $t_{\\rm {start}}$ =4 ms and $t_{\\rm {end}}$ =12 ms, respectively.",
"While useful for comparing the various methods, this choice of parameters is generally not optimum for either of the slowing techniques.",
"In simulations of frequency-broadened slowing, molecules reach their final velocity within 3–4 ms of the slowing light turning on.",
"After reaching a low enough velocity to fall out of resonance with the slowing light the molecules freely propagate to the detector at the slow final velocity and hence with a large divergence.",
"In contrast, when using frequency-chirped slowing, the forward velocity of the molecules tracks that of the chirp, decreasing linearly until the chirp ends.",
"In this case, molecules reach the final velocity at 12 ms and hence diverge less before reaching the detector.",
"A complete numerical optimisation of the laser power, convergence, turn-on time, turn-off time, initial frequency offset, and chirp rate (in the frequency-chirped case) involves too large a parameter space to be practical.",
"Instead, we fix the laser power and turn-off times at 100 mW and $t_{\\rm {end}}$ =12 ms, and vary the turn-on time $t_{\\rm {start}}$ .",
"The beam convergence is fixed to one of two values, either “collimated” or “converging”.",
"We also vary the offset frequency $f_{\\rm {offset}}$ for frequency-broadened slowing, and the chirp rate $\\beta $ for frequency-chirped slowing.",
"The initial frequency offset in the latter case is fixed at 335 MHz ($v_{\\rm {start}}$ =178 m/s).",
"For a metric to compare the simulation results over this limited parameter space, we choose the number of molecules that arrive at the MOT location with forward velocities below the expected capture velocity of $v_{c}=20$ m/s.",
"Figure: Result from simulations optimizing the number of molecules arriving at the MOT location below the expected capture velocity of 20 m/s.",
"Here, the vertical scales are arbitrary.",
"In all cases the slowing light is turned off at t end =12t_{\\rm {end}}=12 ms and the chosen deceleration method is optimized by varying t start t_{\\rm {start}}.",
"(a) Chirped laser slowing using a converging cooling beam and an initial detuning of -335 MHz (v start =178v_{\\rm {start}}=178 m/s) and various chirp rates.",
"(b) Frequency-broadened laser slowing using a collimated cooling laser and various overall detunings.",
"(c) Comparison of the best parameter settings for the four cases of chirped, frequency-broadened, collimated-beam, and converging-beam laser slowing.Figure REF shows the results of simulations aimed at optimising the number of molecules satisfying this MOT-loading metric.",
"The five curves in Figure REF (a) compare chirped-frequency slowing using a converging beam with various values of $t_{\\rm {start}}$ and $\\beta $ .",
"The best result is obtained with $\\beta = 30$ MHz/ms and $t_{\\rm {start}}=3.5$ ms.",
"The results are very sensitive to $t_{\\rm {start}}$ , as might be expected.",
"If the slowing light is turned on too late then, for a fixed chirp rate, no molecules are decelerated below the capture velocity.",
"If the slowing light is turned on too early, molecules decelerate too much and either diverge or are turned around before they reach the MOT location.",
"Figure REF (b) shows the optimization results using a collimated frequency-broadened slowing laser.",
"The results are a much weaker function of $t_{\\rm {start}}$ than in Fig.",
"REF (a) and are optimized at slightly later turn-on times.",
"The best result is obtained using $f_{\\rm {offset}}= 180$ MHz and $t_{\\rm {start}} = 6$ ms.",
"Figure REF (c) compares the best results of these optimization procedures for four cases: collimated-chirped, converging-chirped, collimated-broadened, and converging-broadened.",
"After this optimization, it is clear that chirped slowing outperforms frequency-broadened slowing in producing molecules at the MOT location and below the expected capture velocity.",
"Furthermore, this conclusion becomes even stronger if the MOT capture velocity is reduced.",
"The optimized chirp method gives 4.5 times more molecules below $v_{c}$ than the optimized broadening method when $v_{c}=$ 20 m/s, and $>$ 20 times more when $v_{c}$ =5 m/s."
],
[
"Conclusions",
"We have shown that a beam of CaF molecules can be slowed down either using the frequency-chirped method or the frequency-broadened method.",
"By driving the B-X transition, which has exceptionally favourable branching ratios, the deceleration is rapid and efficient, requiring only two laser wavelengths, each with rf sidebands.",
"Our study of losses to unaddressed states shows that $\\sim 3\\times 10^{4}$ photons per molecule can be scattered before half are lost from the cooling cycle.",
"Molecules scattering this many photons would be slowed by 390 m/s, which is far greater than needed to bring molecules to rest from a typical buffer-gas-cooled source.",
"For both slowing methods the dominant loss mechanism is the increased divergence of the slowed molecules.",
"Hence, it is best to minimize the distance that the molecules have to travel at low speed, and so they should reach their final velocity as late as possible, i.e.",
"when they reach the detector or the MOT volume.",
"The frequency-broadened method is not good at achieving this because all molecules start slowing as soon as the light is turned on, and many reach low velocity too early.",
"The chirped method is more efficient because the the slower molecules join the slowing process later on.",
"For this reason, while the two methods produce a similar number of slow molecules down to about 50 m/s, the chirped method gives far more molecules at lower speeds, e.g.",
"about ten times more at 20 m/s.",
"This advantage is especially important for loading a MOT where the capture velocity is likely to be 20 m/s or less.",
"We find that the chirped method yields more slow molecules when the slowing light converges towards the molecular source, especially for the lower velocities.",
"Using this method, we produce approximately $10^{6}$ molecules with speeds in the narrow range 15$\\pm $ 2.5 m/s.",
"Thus, our method appears very well suited for loading a MOT.",
"The chirped method also greatly compresses the velocity distribution, and it provides very precise velocity control.",
"When combined with a short region of transverse laser cooling [24] near the source, our method will produce an intense, collimated, slow and velocity-controlled beam that could improve the precision of measurements that test fundamental physics.",
"The research leading to these results has received funding from EPSRC under grants EP/I012044 and EP/M027716, and from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 320789."
],
[
"Accuracy of the method for determining velocity distributions",
"Our method for determining velocity distributions is described in Sec. .",
"In this Appendix, we discuss in detail the accuracy of this method.",
"The method must work perfectly if there is a unique correspondence between arrival time and velocity so that it is valid to assign all molecules arriving in any small time window to the mean velocity measured in that time window.",
"However, molecules with different velocities may arrive at the same time if their journeys from source to detector differ in some way, so we wish to analyse the effect of that.",
"We distinguish two ways that this can happen.",
"The first is that molecules exit the source over a range of times.",
"The second is that the force that acts may depend on other parameters such as the transverse position or transverse velocity of the molecule when it leaves the source.",
"We consider first the case where molecules leave the source over a range of times.",
"Let us define the exit time from the source, $t_0$ , the transit time from source to detector, $\\tau $ , and the arrival time $t=\\tau + t_{0}$ .",
"For now, we let the laser parameters be independent of time, so that a given initial velocity $u$ results in a specific final velocity $v$ and flight time $\\tau $ .",
"Let these be related by $v = f(\\tau )$ and the inverse, $\\tau = g(v)$ .",
"The probability density function for a variable $x$ is $P_{x}(x)$ .",
"The time-of-flight profile measured 1.3 m from the source is $P_{t}(t)$ and the one measured 2.5 cm from the source is a good approximation to $P_{t_0}(t_0)$ .",
"The time-of-flight profile is $P_t(t) = \\int P_\\tau (t-t_0) P_{t_0}(t_0) d t_0 = (P_{t_0} \\ast P_\\tau )(t),$ where $\\ast $ is the convolution operator.",
"Thus, the distribution of transit times, $P_\\tau (\\tau )$ , can be obtained from the data by the deconvolution of $P_t$ with $P_{t_0}$ .",
"The velocity distribution is related to $P_\\tau (\\tau )$ through a change of variables: $P_v(v) = P_\\tau (g(v))\\left|\\frac{d g}{d v}\\right|.$ We do not measure $g(v)$ directly.",
"Instead, we measure the mean velocity of molecules that arrive in a small time window centred at $t$ , $\\bar{v} = p(t)$ .",
"This can be expressed as $p(t) = \\frac{\\int f(t-t_0)P_\\tau (t-t_0)P_{t_0}(t_0) d t_0}{\\int P_\\tau (t-t_0)P_{t_0}(t_0) d t_0}.$ Thus, we can write $p(t) P_t(t) = (P_{t_0} \\ast f P_{\\tau })(t).$ We now have the algorithm for determining the velocity distribution from the measured data: (i) Calculate $P_\\tau $ by a deconvolution of $P_t$ with $P_{t_0}$ ; (ii) Calculate $f(t)$ by taking a deconvolution of the product $p P_t$ with $P_{t_0}$ , and then dividing by $P_\\tau $ ; (iii) Invert $f(t)$ to obtain $g(v)$ ; (iv) Take the derivative of $g(v)$ ; (v) Use Eq.",
"(REF ).",
"In our experiment, $P_{t_0}$ has a very narrow width - the distribution we measure at 2.5 cm has a FWHM of 280 $\\mu $ s, and the distribution at the source must be even narrower.",
"Using the measured velocity distribution of the unslowed beam, we infer a FWHM at the source of 240 $\\mu $ s. This width is very small compared to any of the times $t$ where $P_t(t)$ is significant, and is also very small compared to the widths of any features in $P_t(t)$ .",
"As a result, the deconvolution steps have a negligible effect.",
"In this limit, $P_v(v) \\approx P_t(q(v))\\left|\\frac{d q}{d v}\\right|,$ where $t = q(\\bar{v})$ is the inverse function to $p(t)$ , and the approximation is exact in the limit that $P_{t_0}(t_0) = \\delta (t_0)$ .",
"This is the result we use for all our data and, as we shall see below, it is very accurate for our experiment.",
"Our source emits a narrower temporal distribution than is typical of most buffer gas sources.",
"To evaluate the accuracy of our analysis method when the source emits a longer pulse, we test it on synthetic data.",
"To generate this data, we first create molecules at the source with initial velocities drawn at random from a normal distribution whose mean and width are equal to those we measure in the experiment, and with exit times drawn from a normal distribution with zero mean and FWHM $\\Delta t$ .",
"The molecules are then subject to an acceleration function $a = a_0/(1+(v-v_0)^{2}/w^{2})$ , where we choose $a_0 = -10^{4}$ m/s$^{2}$ , $v_{0} = 80$ m/s and $w = 10$ m/s.",
"We solve the equation of motion for each molecule to generate the exact arrival time and velocity distributions in a plane 1 m from the source.",
"We also determine the mean velocity in a set of time windows, just as in the experiment.",
"We then apply the same analysis routine to the synthetic data as used for the real data, and compare the velocity distribution determined this way to the exact distribution.",
"Figure REF (a) shows this comparison in the case where we set $\\Delta t = 240$ $\\mu $ s, as in the experiment.",
"The histogram is the exact velocity distribution, and the line shows the distribution from Eq.",
"(REF ).",
"As expected from the argument above, there is no noticeable difference between the two.",
"The largest difference in any velocity bin is 1.9% of the amplitude of the undecelerated distribution, and the deviations in most bins are much smaller than this.",
"Figure REF (b) shows the same comparison in the case where $\\Delta t$ is 10 times larger.",
"In this case, the distribution from Eq.",
"(REF ) deviates considerably from the true one, especially for high velocities.",
"This is to be expected since the arrival time is comparable to $\\Delta t$ for these faster molecules.",
"Interestingly, the analysis method still works well for the narrow distribution of slowed molecules which are the ones of most interest.",
"This is because these molecules take a long time to reach the detector, and because the narrow peak in the velocity distribution does not correspond to any narrow features in the time-of-flight profile.",
"On the contrary, the sharp feature in the velocity distribution arises because molecules arriving over a wide range of times all have very similiar velocities.",
"The result of applying the full algorithm described above is shown by the dashed line in Fig.",
"REF (b) and does indeed give a better approximation to the true distribution in this case where the range of exit times is broad.",
"We note that deconvolution algorithms often generate artificial oscillations in the result, especially where there are sudden changes in gradient, and that the analysis algorithm can become unstable when that occurs.",
"We find that this happens at the low velocities where the sharp peak occurs, and so we only plot the result over the range where the algorithm is stable.",
"Fortunately, the algorithm works well over the whole velocity range where the approximate method is inaccurate.",
"Figure: Accuracy of using Eq.",
"() to determine velocity distributions, assessed using model data.",
"Histograms: exact distributions.",
"Red lines: results using Eq. ().",
"Black dashed line in (b): distribution obtained using Eq.",
"(), the full analysis method.",
"The parameters used in each case are described in the text.We have also compared the exact velocity distribution with the one determined from Eq.",
"(REF ) for the case where the acceleration function is time-dependent.",
"For this comparison, the acceleration acts only for times between 1 and 7 ms, and the resonant velocity $v_{0}$ is chirped downwards in time from 180 to 60 m/s, similar to the experiment.",
"We use the narrow temporal source distribution of the experiment.",
"Once again, we find that our analysis method reproduces the correct velocity distribution to very high accuracy.",
"We turn now to the possibility that molecules arriving in a small time window may have a spread of velocities because the integrated force depends on a parameter that differs between molecules, such as the transverse position or transverse velocity at the source.",
"We use again our numerical model of the analysis method to examine the effect of this.",
"We consider the case where molecules have a range of transverse positions $x$ , but no transverse velocity.",
"We modify the acceleration function so that it drops off with transverse displacement: $a = a_{0} \\exp (-x^{2})/(1+(v-v_0)^{2}/w^{2})$ .",
"We produce the initial set of molecules as before, with $\\Delta t = 240$ $\\mu $ s, and draw the dimensionless transverse displacement $x$ at random from a normal distribution with a full width at half maximum of 2.",
"This samples a wider range of decelerations than the molecules experience in the experiment.",
"There, the molecules that we detect travel close to the centre of the laser beam, where the intensity is high and the force is strongly saturated.",
"Figure REF (c) shows the result for this case.",
"We see that the range of forces broadens the peak of slow molecules, and that the analysis method accurately recovers the correct velocity distribution.",
"Figure REF (d) shows the result when we choose instead an initial distribution which is uniform in the range $0<x<3$ .",
"This broadens and flattens the slow peak even further, and our analysis method still recovers the correct distribution.",
"We have experimented with a range of different models for how the force and the initial distribution might vary, always finding that the analysis method is accurate."
]
] | 1605.06055 |
[
[
"A Frequency Domain Test for Propriety of Complex-Valued Vector Time\n Series"
],
[
"Abstract This paper proposes a frequency domain approach to test the hypothesis that a complex-valued vector time series is proper, i.e., for testing whether the vector time series is uncorrelated with its complex conjugate.",
"If the hypothesis is rejected, frequency bands causing the rejection will be identified and might usefully be related to known properties of the physical processes.",
"The test needs the associated spectral matrix which can be estimated by multitaper methods using, say, $K$ tapers.",
"Standard asymptotic distributions for the test statistic are of no use since they would require $K \\rightarrow \\infty,$ but, as $K$ increases so does resolution bandwidth which causes spectral blurring.",
"In many analyses $K$ is necessarily kept small, and hence our efforts are directed at practical and accurate methodology for hypothesis testing for small $K.$ Our generalized likelihood ratio statistic combined with exact cumulant matching gives very accurate rejection percentages and outperforms other methods.",
"We also prove that the statistic on which the test is based is comprised of canonical coherencies arising from our complex-valued vector time series.Our methodology is demonstrated on ocean current data collected at different depths in the Labrador Sea.",
"Overall this work extends results on propriety testing for complex-valued vectors to the complex-valued vector time series setting."
],
[
"to deal with impropriety,” IEEE Transactions on Signal Processing, vol.",
"59, pp.",
"5101–5125, 2011.",
"J. C. Ruiz-Molina, “Estimation of improper complex-valued random signals in colored noise by using the Hilbert space theory,” IEEE Trans.",
"Inf.",
"Theory, vol.",
"55, pp.",
"2859–2867, 2009."
]
] | 1605.05910 |
[
[
"Pulsed quantum interaction between two distant mechanical oscillators"
],
[
"Abstract Feasible setup for pulsed quantum non-demolition interaction between two distant mechanical oscillators through optical or microwave mediator is proposed.",
"The proposal uses homodyne measurement of the mediator and feedforward control of the mechanical oscillators to reach the interaction.",
"To verify quantum nature of the interaction, we investigate the Gaussian entanglement generated in the mechanical modes.",
"We evaluate it under influence of mechanical bath and propagation loss for the mediator and propose ways to optimize the interaction.",
"Finally, both currently available optomechanical and electromechanical platforms are numerically analyzed.",
"The analysis shows that implementation is already feasible with current technology."
],
[
"Introduction",
"Quantum optomechanics and electromechanics connecting light and microwaves with mechanical motion at the quantum level is an emerging field of quantum physics and technology [1], [2], [3].",
"Recently, Gaussian quantum entanglement between mechanical oscillator and microwave field [4], and nonclassical photon-phonon correlation of mechanical membrane and optical pulse [5] have been experimentally demonstrated.",
"Both experiments used modern pulsed optomechanics [6], [7], [8], [9], [10].",
"They open new possibilities to experimentally connect other physical platforms with mechanical oscillator, like continuous-variable cold atom ensembles [11], [12], [13], and further many discrete systems like individual atoms [11], [14], superconducting qubits [15], [16], solid-state systems [17], [18], [19], [20] and semiconductor systems [21], [22].",
"Together with these interesting and challenging interdisciplinary experiments, state-of-the-art of laboratory techniques could currently allow to let interact two mechanical oscillators mediated by light or microwave field.",
"It is another interesting step forward, two similar mechanical oscillators coupled at quantum level have not been demonstrated yet.",
"It can be very stimulating especially because the connection between two mechanical systems can physically connect quantum optomechanics to classical thermodynamics.",
"If two similar quantum mechanical oscillators will be interfaced by the quantum version of the coupling typically used in classical mechanics, they can naturally generate entanglement.",
"It is a simple witness that they were coupled quantum mechanically.",
"Additionally, the mechanical-mechanical interaction can be quantum non-demolition type, which is required for basic continuous-variable quantum gate [23] , useful for its specific features, for both gate-based [24] and cluster-state-based [25] quantum computing.",
"Recently, the nonlocal optical QND gate was demonstrated [26] following the theoretical proposals in [27], [28].",
"Such the QND coupling was already broadly exploited between two atomic ensembles [29].",
"It is therefore much more important for the future to achieve such the well-defined quantum interaction of mechanical oscillators, not only the generation of entangled state of two mechanical systems.",
"Generation of entanglement between two mechanical systems have been already proposed in three different configurations.",
"In the first type of proposed setups, two mechanical oscillators have been placed in a single optical cavity [30], [31], [32], [33], [34], [35], [36], [37], [38].",
"In this case, the continuous generation of steady-state entanglement appears because the mechanical oscillators interact with join optical intra-cavity field.",
"This configuration has been extensively used to discuss continuous-time quantum synchronization [39], [40], [41].",
"In the second kind of proposals, two entangled beams of light were used to entangle two mechanical systems without necessity to measure them [42], [43], [44].",
"In the third kind of proposed setups, two continuous-wave beams of light, leaving two continuously pumped optomechanical cavities, are jointly detected in Bell measurement and photocurrent is used to correct the mechanical oscillators [45], [46], [47] .",
"These schemes can generate entanglement at a distance, however, it is very limited because of instabilities in the blue-detuned continuous-wave regime.",
"Advanced time-continuous quantum measurement and control has been suggested to prepare mechanical entanglement [48].",
"Recently, theoretical investigation of optomechanical crystals has offered many other ways how to obtain mechanical entanglement [49], [50].",
"Our goal is to propose currently feasible scheme with potential to use power of quantum optics tools to complement recent experimental test of coupled quantized mechanical oscillations of trapped ions [51].",
"In this paper, we propose currently feasible way to build basic pulsed quantum non-demolition (QND) interaction between two mechanical oscillators at a distance, connected by light or microwave field.",
"The scheme is depicted in Fig. 1.",
"Using homodyne detection of light or microwave field and feedforward control, means of both mechanical oscillators precisely follow the QND interaction.",
"To generate significant entanglement of mechanical oscillators, coherent light is sufficient, and the entanglement can be very well estimated when intra-cavity field can be adiabatically eliminated.",
"On the other hand, squeezed light is advantageous to approach ideal QND interaction between two mechanical systems.",
"Feasible squeezing of light is capable to enhance entangling power of the QND interaction.",
"However, for larger optomechanical coupling strength and larger squeezing, non-adiabatic methods taking the intracavity field fully into account are required.",
"Importantly, the non-adiabatic calculations predict a decrease of the entanglement power for larger squeezing.",
"It is due to presence of the intra-cavity field and the squeezing has to be therefore optimized to get maximum of entangling power.",
"We prove sufficient stability of the QND interaction under influence of mechanical bath and transmission loss between two separated cavities.",
"Finally, we verified that it is feasible to build the mechanical QND interaction for both current optomechanical [52] and electromechanical [53] setups.",
"The paper is organized in the following way.",
"We begin by mathematical definition of quantum-nondemolition interaction and principal description of the experimental setup.",
"First, in Sec.",
"we carry out a simple principal analysis of the physics of the setup.",
"To do so we start from a brief derivation of equations of motion for an optomechanical system in Sec.",
"REF and solve those in Sec.",
"REF ignoring for a while the decoherence and eliminating intracavity modes.",
"We quantify the interaction between the mechanical modes analyzing the transfer of first moments of quadratures, and for a figure of merit of the strength of the interaction we employ the entanglement between the modes.",
"We use logarithmic negativity [54] as a measure of entanglement.",
"We show principal possibility of the protocol performance and derive the simplest conditions on the experimental parameters.",
"Second, in Sec.",
"we perform a full numerical analysis of the system allowing for the imperfections.",
"Those include impact of the intracavity modes that mediate the interaction between the travelling light pulse and the mechanical modes and the thermal bath causing decoherence of the latter.",
"We as well investigate the impact of the optical loss between the cavities.",
"We show that with currently available parameters the protocol can establish a QND interface between the two distant mechanical modes."
],
[
"Setup for pulsed QND interaction between mechanical oscillator",
"In this paper, we propose a feasible way of implementation of quantum non-demolition (QND) interaction between mechanical modes of two distant optomechanical cavities.",
"The QND interaction of two harmonic oscillators may be described by Hamiltonian $\\mathcal {H}_{_\\mathrm {int}} = \\hbar g Q_1 Q_2$ with $Q_{1,2}$ being the position or momentum of the corresponding oscillator and $g$ , interaction strength.",
"After the interaction both the variables $Q_1$ and $Q_2$ remain unperturbed (not demolished) whereas the complementary ones to $Q_1$ ($Q_2$ ) become linearly displaced by a value proportional to $gQ_2$ ($gQ_1$ ).",
"The nondemolition interaction has been demonstrated in a few electromechanical experiments recently [55], [56].",
"The proposed scheme is presented in the Fig.",
"REF .",
"It is the simplest setup for generation of QND coupling between two mechanical systems.",
"It is basically a serial scheme which does not require multiple pass of optical pulse through single optomechanical cavity.",
"Moreover, it exploits advantage of squeezed light, homodyne detection, which are very efficient resources of quantum optics.",
"The feedforward correction on mechanics can be done simply at any time by classical pulse of laser light.",
"The modes of two mechanical oscillators M$_{1}$ and M$_{2}$ interact by turns with an optical (or microwave) pulse $L$ via opto(electro-)mechanical coupling.",
"The pulse is then detected and the result of the detection is used to linearly displace the mechanical mode of the first cavity (if needed, in the second one is displaced as well).",
"In principle this feedforward is not necessary to achieve entanglement, as the latter could be created contional on the results of the detection.",
"A similar method was used recently for conditional state preparation in optomechanics [57].",
"The optical pulse can be prepared in a squeezed intensive coherent state and sent into the optomechanical cavity.",
"The latter in essence comprises an optical mode coupled via radiation pressure to a mechanical harmonic oscillator [58].",
"We follow the standard approach [59], [60], [1] and assume that the optical pulse is displaced with a strong classical component that is modulated at mechanical frequency.",
"This ensures that the effective interaction within the cavity is the non-demolition type.",
"The QND interaction allows a partial exchange of the variables between the mechanical mode M$_{1}$ and the travelling pulse (see Fig.",
"REF (a)).",
"The latter is then redirected to the second cavity with mechanical oscillator M$_{2}$ , which we assume to be identical to the M$_{1}$ .",
"The QND interaction within the second cavity allows to transmit a variable of the mode M$_{1}$ carried by the pulse to the mode M$_{2}$ and in turn to transmit a variable of the mode M$_{2}$ to the light.",
"The pulse is then detected and the result of detection is used to displace the mode M$_{1}$ to complete transfer of the M$_{2}$ variable.",
"A proper strong presqueezing of the light pulse and the feed-forward correction allow to eliminate all variables from the final transformation of the mechanical modes that consequently approach an ideal QND interaction between them."
],
[
"Optomechanical quantum non-demolition interaction",
"Let us first consider a single optomechanical cavity that in essence embodies an optical mode and a mechanical one.",
"The two modes are coupled by radiation pressure with the Hamiltonian [58] $H_\\mathrm {rp} = - \\hbar g_0 n_\\mathrm {cav} x/x_\\mathrm {zp}$ , where $n_\\mathrm {cav}$ stands for intracavity photon number, $x$ , mechanical displacement from equilibrium, $g_0$ , so-called single-photon coupling strength.",
"The mechanical zero-point fluctuation amplitude, denoted by $x_\\mathrm {zp}$ , for a mechanical oscillator with mass $m$ and eigen frequency $\\omega _m$ equals $x_\\mathrm {zp}= \\sqrt{ \\hbar / 2 m \\omega _m}$ .",
"In order to enhance the radiation pressure coupling, strong coherent field is used as the pump.",
"This allows to linearise the dynamics around a steady classical state and solve for quantum corrections.",
"Moreover, we assume this strong classical field to be resonant with the cavity and modulated at the frequency of the mechanical oscillator [59].",
"In this case if the mechanical frequency $\\omega _m$ exceeds all other characteristic frequencies of the system, one can perform averaging to get rid of the terms at $2 \\omega _m$ (i.e., adopt the Rotating Wave Approximation, RWA) to obtain the non-demolition coupling.",
"The latter condition is usually equivalent to the requirement that the optical decay rate $\\kappa $ of the cavity be smaller with respect to $\\omega _m$ ., known as resolved-sideband regime.",
"After the linearization and averaging out the rapid oscillating terms we arrive to the QND coupling within the optomechanical cavity with Hamiltonian that reads (depending on the phase of the pump) $\\mathcal {H} = \\hbar g X p\\quad \\text{ or }\\quad \\mathcal {H} = \\hbar g Y q,$ where $g = g_0 \\sqrt{ \\left\\langle n_\\mathrm {cav} \\right\\rangle }$ is the enhanced optomechanical coupling strength, $X$ and $Y$ , and $q$ and $p$ are quadratures of, respectively, the optical and mechanical modes which obey usual commutation relations ($[X,Y] = i; \\ [q,p] = i$ ).",
"The mechanical displacement $x$ can be expressed in terms of quadratures as $x/ x_\\mathrm {zp}= q \\cos \\omega _m t + p \\sin \\omega _m t$ and a similar expression holds for the optical quadratures.",
"The counter-rotating terms at $2 \\omega _m$ could provide additional back-action.",
"In Appendix we analyze this back-action and prove that for typical experimental parameters it is sufficient to consider the system within RWA.",
"To describe the interaction of the propagating light pulse with the optomechanical cavity we complement the Hamiltonian of the optomechanical interaction $\\mathcal {H}_1 = -\\hbar g_1 X_1 p_1$ with input-output relations [61] (henceforth we denote with index “1” or “2” quantities corresponding to the respective cavity).",
"The system is thus described by the following set of equations: $\\begin{aligned}\\dot{q}_1 & = - \\tfrac{\\gamma }{2} q_1 -g_1 X_1 + \\xi _{q1},&\\dot{X}_1 & = - \\kappa X_1 + \\sqrt{2 \\kappa } X^\\text{in},\\\\\\dot{p}_1 & = - \\tfrac{\\gamma }{2} p_1 + \\xi _{p1},&\\dot{Y}_1 & = - \\kappa Y_1 + \\sqrt{2 \\kappa } Y^\\text{in} + g_1 p_1\\end{aligned}\\\\Q^\\text{out} = \\sqrt{ 2 \\kappa } Q - Q^\\text{in},\\quad Q = X,Y.$ Here $X^\\text{in},Y^\\text{in}$ are the quadratures of the pulse with commutator $[X^\\text{in} , Y^\\text{in} (t^{\\prime })] = i \\delta (t - t^{\\prime })$ , $\\xi _{{q,p}}$ are the quadratures of mechanical noise.",
"$\\kappa $ and $\\gamma $ are respectively optical and viscous mechanical damping coefficients.",
"Figure: Entanglement between the two mechanical modes as a function of optical presqueezing.",
"Thick lines correspond to adiabatic solution; thin lines with markers to solution with cavity mode.",
"Different colors and dashings are used for different ratio of the gains K 1 K_1 and K 2 K_2.",
"Losses are absent: η=1\\eta =1.",
"Highlighted is the region of squeezing magnitudes not exceeding the value of 12.7{12.7}{} reported in Ref.",
"."
],
[
"Adiabatic regime",
"As a first approximation we consider the system in adiabatic regime.",
"Given that optical decay rate exceeds the other rates in (REF ) (which is typically the case in experiment), one can assume that the optical mode reacts to any changes instantaneously, which is equivalent to putting $\\dot{X} = \\dot{Y} = 0$ in Eqs (REF ).",
"Formally this corresponds to replacement of all the functions of time with their own versions averaged over the interval with duration $\\tau _*$ such that $\\kappa \\gg 1/\\tau _* \\gg \\gamma ,g$ .",
"Lastly, in this section we leave out the mechanical decoherence, setting $\\gamma = 0,\\ \\xi _{q1} = \\xi _{p1} =0$ .",
"With these assumptions the solution of Eqs.",
"(REF ) reads $q_1(\\tau ) & = q_1(0) - S K_1 \\mathcal {X}^\\text{in}, & \\mathcal {X}^\\text{out}_1 & = S \\mathcal {X}^\\text{in}, \\\\p_1(\\tau ) & = p_1(0) , & \\mathcal {Y}^\\text{out}_1 & = \\frac{1}{S} \\mathcal {Y}^\\text{in} + K_1 p_1(0).$ We have introduced the squeezing magnitude $S$ and the effective interaction strength $K_1 = g_1 \\sqrt{2 \\tau /\\kappa }$ .",
"We also have defined the input and output quadratures of the cavity as $\\mathcal {Q}^\\text{k} = \\frac{ 1 }{ \\sqrt{\\tau }} \\int _0^\\tau Q^\\text{k} (s) ds,\\quad Q = X,Y,\\quad \\text{k}=\\text{in,out}.$ The quadratures are normalized to obey $[\\mathcal {X}^\\text{k}, \\mathcal {Y}^\\text{k}] = i$ .",
"The output field from the first cavity is then delivered to the input of the second one through a purely lossy channel that performs an admixture of vacuum to the signal, therefore $Q^\\text{in}_2 = \\sqrt{ \\eta } Q^\\text{out}_1 + \\sqrt{ 1 - \\eta } Q_\\mathrm {ls},\\quad Q = X,Y.$ Here $Q_\\mathrm {ls}$ are the quadratures of vacuum mode.",
"The optomechanical interaction within the second cavity is described by the Hamiltonian $\\mathcal {H}_2 = \\hbar g_2 Y_2 q_2$ and starts at time $t = \\tau $ .",
"One can obtain the input-output relations for the second cavity in a similar fashion.",
"For simplicity we assume the parameters of the second cavity (except the coupling $g_2$ ) to replicate the parameters of the first one.",
"The optical output quadrature $\\mathcal {X}^\\text{out}_2$ is measured and the position of the mechanical mode of the first cavity is displaced so that the final value equals $q_1 = q_1 (\\tau ) + K_f \\mathcal {X}^\\text{out} _2$ .",
"$q_1 & = q_1(0) + K_2 K_f q_2(\\tau )\\\\ & - S \\mathcal {X}^\\text{in}\\left(K_1 - K_f \\sqrt{\\eta } \\right)+ \\mathcal {X}_\\mathrm {ls} K_f \\sqrt{1 - \\eta }, \\\\ p_1 & = p_1(0), \\\\ q_2 & = q_2(\\tau ), \\\\ p_2 & = p_2(\\tau ) - K_1 K_2 p_1(0) \\sqrt{\\eta }\\\\ & - \\mathcal {Y}^\\text{in} \\frac{K_2 \\sqrt{\\eta }}{S}- K_2 \\sqrt{1 - \\eta }\\mathcal {Y}_\\mathrm {ls}.$ Similarly, we have introduced $K_{2} = g_{2} \\sqrt{\\frac{2 \\tau }{\\kappa }}$ here.",
"To approach the ideal QND interaction of the two mechanical modes with Hamiltonian $\\mathcal {H}_\\mathrm {QND} = \\hbar K_1 K_2 \\tau ^{-1} p_1 q_2$ one needs to fulfill a few conditions.",
"First, ensure low loss ($\\eta \\rightarrow 1$ ) to get rid of the noisy mode $Q_\\mathrm {ls}$ .",
"Second, pick a proper feed-forward gain $K_f = K_1 / \\sqrt{ \\eta }$ and provide high squeezing $S \\gg 1$ to suppress the optical mode $\\mathcal {Q}^\\text{in}$ .",
"To quantify the strength of the interaction we estimate the entanglement between the two mechanical modes, namely the logarithmic negativity [54] (see Appendix for details).",
"In the lossless case the optimal value of squeezing yielding maximum of entanglement is given by $S = \\left| K_2 / ( K_1 - K_f ) \\right|$ .",
"Therefore for the feedforward $K_f = K_1$ the entanglement increases with squeezing infinitely.",
"In the limit of moderately strong coupling ($K_{1,2} \\gtrsim 1$ ) the following approximation holds: $E_\\eta \\approx - \\ln \\frac{ 1 }{ 2 K_1 K_2 } \\sqrt{ 1 + \\frac{ K_2^2 }{ S^2 }}.$ From this expression follows that although increase of both $S$ and $K_{1,2}$ leads to stronger entanglement, it is more efficient to increase $K_1$ .",
"This can be seen from the latter equation in (REF ), where the noisy mediator quadrature $\\mathcal {Y}^\\text{in}$ enters with a multiplier $\\propto K_2$ .",
"The LN for this simple model is presented as a function of the presqueezing $S$ in Fig.",
"REF (solid lines).",
"The parameters used for simulation are $\\kappa / 2 \\pi = {221.5}{}$ , $\\gamma / 2 \\pi = {328}{}$ , $\\tau = {4.5}{}$ that correspond to a recent optomechanical experiment [52] with increased pulse duration $\\tau $ .",
"From the Fig.",
"REF is is clear that for low squeezing the LN is mostly defined by the interaction strength $K_1$ in the first cavity as it follows from (REF ).",
"In the limit of high squeezing the LN saturates to the value that is defined by the product of gains $K_1 K_2$ , again in agreement with (REF )."
],
[
"Robustness to imperfections",
"There are two sources of hindrance that we left out for the previous section.",
"First is the intracavity modes that mediate interaction between the propagating pulse and the mechanical modes of interest.",
"As well the intracavity modes produce unwanted memory effects that disturb the desired QND interaction.",
"Second is the interaction of mechanical modes with the thermal environment.",
"In this section we first study these two sources independently and finally provide a full solution taking both into account simultaneously."
],
[
"Impact of the intracavity modes",
"To consider the effect of the intracavity modes on the QND interface, we solve the set of dynamical equations (REF ) without the mechanical decoherence ($\\gamma = 0$ , $\\xi _{{q,p}} = 0$ ).",
"The solution reads (for compactness we write the solution for the lossless case, $\\eta =1$ ) $q_1 = & q_1(0) + q_2(\\tau ) K_2 K_f \\left( 1 - \\frac{1 - e^{-\\kappa \\tau }}{\\kappa \\tau } \\right) - S (K_1 - K_f )\\frac{1}{\\sqrt{\\tau }} \\int _0^{\\tau } X^\\text{in}_1(s) ds\\\\ & + S K_1 \\int _0^{\\tau } X^\\text{in}_1(s)\\left(e^{-\\kappa (\\tau - s)}\\left[1 - 4\\kappa (\\tau - s)\\frac{K_f}{K_1} \\right] \\right) ds+ X_1(0)\\left( \\frac{ 2 g_f}{\\kappa } \\left[ (1 - e^{-\\kappa \\tau }) - 2 \\kappa \\tau e^{-\\kappa \\tau } \\right] - \\frac{g_1}{\\kappa } \\left[ 1 - e^{-\\kappa \\tau } \\right] \\right)\\\\ & + X_2(0) \\frac{2 g_f}{\\kappa } \\left( 1 - e^{-\\kappa \\tau } \\right),\\\\ p_1 = & p_1(0),\\\\ q_2 = & q_2(\\tau ),\\\\ p_2 = & p_2(\\tau )- p_1(0) K_1 K_2 \\left( 1 + e^{-\\kappa \\tau } - \\frac{2}{\\kappa \\tau }(1 - e^{-\\kappa \\tau }) \\right)- \\frac{K_2}{S} \\frac{1}{\\sqrt{\\tau }} \\int _0^{\\tau } \\left( 1 - e^{-\\kappa (\\tau - s)}[2 \\kappa ( \\tau - s) + 1] \\right) Y_1^\\text{in}(s) ds\\\\ & - Y_1(0) \\frac{ 2 g_2}{ \\kappa } \\left( 1 - e^{-\\kappa \\tau } (1 + \\kappa \\tau ) \\right)- Y_2(0) \\frac{g_2}{\\kappa }(1 - e^{-\\kappa \\tau }),$ where we defined $g_f \\equiv K_f \\sqrt{ \\kappa / 2 \\tau }$ .",
"These equations deviate from the idealized set (REF ) by presence of the initial intracavity quadratures $Q_{1,2} (0)$ .",
"As well the pulse quadratures $Q^\\text{in}$ can no longer be eliminated completely by a proper choice of $K_f$ and high squeezing $S$ .",
"Moreover, in this case high squeezing apmlifies the noisy summand with $X^\\text{in}$ degrading the interface.",
"The impact of this summand can be reduced by redefining the temporal mode of the output pulse (applying optimal time filter at the detection).",
"This, however, cannot cancel the noisy summand completely as the optical quadratures that are written during the first pass ($X^\\text{in}$ ) and second pass ($Y^\\text{in}$ ) are distorted in different manner, see Eq.",
"(REF ).",
"From the Eqs.",
"(REF ) follows that in the limit $\\kappa \\gg g_{1,2,f}$ and $\\kappa \\tau \\gg 1$ these equations reduce to the pure QND transformations (REF ).",
"Furthermore, from the first equation it follows that the effect of the unwanted summand $\\propto S X^\\text{in}$ can be reduced by decreasing $K_1$ .",
"This is illustrated in Fig.",
"REF where we plot the LN for solution including the cavity modes as a function of squeezing for different couplings.",
"At high squeezing the full solution deviates from the adiabatic one, however the curves with lower $K_1$ show this deviation at higher squeezing than the curves with higher $K_1$ .",
"The proper choice of the coupling thus allows to approach the performance of the idealized adiabatic regime.",
"Note that in order to increase the LN it is more efficient to increase $K_1$ than $K_2$ .",
"To increase the LN staying close to the preferred adiabatic regime (and therefore a pure QND interface between the mechanical modes) on the contrary it is preferable to increase $K_2$ ."
],
[
"Mechanical thermal bath",
"Finally we consider the system in presence of the thermal mechanical environment.",
"We assume that each of the mechanical modes is coupled at rate $\\gamma $ to its own environment that is in a thermal state with occupation $n_\\mathrm {th}$ (see Fig.",
"REF ).",
"The coupling for both modes takes place during the interaction with the pulse.",
"Moreover, the first mode remains coupled to the environment during the interaction of the second system with the pulse.",
"Before the interaction with the pulse the mechanical modes are in the ground state (the possibility to precool mechanical oscillator close to the ground state has been demonstrated for a number of setups [63], [64], [5]).",
"The thermal bath is represented in the equations (REF ) by Langevin force quadratures $\\xi _\\mathrm {q,p}$ .",
"These quadratures are assumed Markovian so that $\\mathinner {\\langle { \\xi _a (t) \\xi _a (t^{\\prime }) + \\xi _a (t^{\\prime }) \\xi _a (t) }\\rangle } = \\gamma ( 2 n_\\mathrm {th} + 1 ) \\delta (t - t^{\\prime }),\\ a=q,p;\\\\\\mathinner {\\langle { \\xi _q (t) \\xi _p (t^{\\prime }) + \\xi _p (t^{\\prime }) \\xi _q (t) }\\rangle } = 0.$ The LN in adiabatic regime with intracavity modes eliminated is approximately given by (here $K_1 = K_2 = K$ ) $E_\\eta \\approx - \\ln \\frac{ 1 }{ 2 K^2 } \\sqrt{ 1 + \\Gamma K^4 + \\frac{ K^2 }{ S^2 } ( 1 + \\Gamma K^4 ) },\\quad \\Gamma = 2 \\gamma \\tau n_\\mathrm {th}.$ In case of zero mechanical damping the expression is reduced to (REF ).",
"The LN corresponding to the full solution with all the imperfections is plotted as a function of the squeezing $S$ in Fig.",
"REF for a set of different parameters.",
"The main means how the mechanical environment affects the entanglement is adding the thermal noise to the mechanical quadratures.",
"Besides this the environment also creates small imbalance that prohibits the perfect cancellation of the optical mediator mode in $q_1$ by feedforward.",
"The magnitude of this imbalance is however almost negligible.",
"We as well plot the LN as a function of the squeezing for nonzero loss ($1 - \\eta \\ne 0$ ).",
"The Fig.",
"REF shows that at higher squeezing the entanglement between the mechanical modes is more tolerant to the mechanical bath than to the optical loss.",
"Nevertheless, even with realistic loss parameters the entanglement does not vanish.",
"We observe that adiabatic elimination is capable to very well fit the results for wide range of feasible squeezing of radiation.",
"Numerical analysis shows that the nonzero occupation of the mechanical bath creates a threshold for the coupling that allows the entanglement.",
"At the same time, nonzero optical loss only decreases the value of the LN, so in case of zero occupation of the bath, the entanglement can tolerate any finite loss."
],
[
"Coupling optimization for experiments {{cite:04bcae92ca0a7fd60b8beb6069faf2a470f87c2b}}, {{cite:bd107ae709f7fbd43768d89dd836ff0923f4c767}}",
"In prior sections we focused on approaching a pure QND interaction between the two mechanical modes.",
"Therefore we assumed the feedforward to be adjusted in a way that helps to cancel most of the optical mediator quadrature $X^\\text{in}$ , i.e.",
"$K_f = K_1 /\\sqrt{ \\eta }$ .",
"Now we aim for maximization of the entanglement between the two modes.",
"We waive the constraint on $K_f$ and numerically optimize the logarithmic negativity with respect to the optomechanical gains $K_{1,2}$ , feedforward strength $K_f$ , and the pulse duration $\\tau $ given a limitation on the coupling strength.",
"The results of the numerical optimization are presented in Fig.",
"REF .",
"The optimal regime to achieve maximal entanglement appears to be very close to the regime of pure QND between the mechanical oscillators with long pulses $\\kappa \\tau \\gg 1$ and $K_f = K_1/\\sqrt{\\eta }$ .",
"Squeezed source of radiation apparently helps to improve entanglement in both opto- and electromechanical scheme for large $\\eta $ close to perfect transmission and smaller $n_\\mathrm {th}$ .",
"Simultaneously, the threshold for $g/\\kappa $ to observe entanglement is lowered as well for higher $\\eta $ and lower $n_\\mathrm {th}$ .",
"On the other hand, for larger $n_\\mathrm {th}$ and lower $\\eta $ , the squeezing of radiation is not important, however, we can still observe entanglement of mechanical systems if $\\gamma /\\kappa $ is not too large and $g/\\kappa $ is sufficiently large.",
"Our analysis (see Appendix ) shows that under these conditions and for moderate squeezing the rotating wave approximation standardly employed in theory of optomechanics is well justified.",
"It is therefore fully feasible to generate entanglement with state-of-the-art systems.",
"The optomechanical setup noticeably outperforms the electromechanical one due to higher eigenfrequency of the mechanical oscillator and consequently lower bath occupation.",
"The high occupation of the mechanical thermal bath in the electromechanical setup places a constraint on the available pulse durations which in turn limits the QND gain $K$ ."
],
[
"Conclusion",
"We have proposed feasible way of the simplest pulsed implementation of entangling quantum non-demolition coupling between two distant but very similar mechanical oscillators, implementable with both current electromechanical and optomechanical setups.",
"The method exploits squeezed light and microwave radiation and highly efficient homodyne detection to induce maximal entanglement for this purely mechanical coupling.",
"We verified robustness of the procedure under small transmission loss between the oscillators and under mechanical thermal baths.",
"We realized that both current optomechanical [52] and electromechanical [53] setups are sufficient for the implementation of an extended version of multiple QND interaction.",
"It will allow pulsed studies of quantum synchronization of mechanical objects [39], [40], [41].",
"Afterwards, a detailed study of quantum interaction of possibly very different mechanical systems is important for development of physical connection with quantum thermodynamics [65], [66], [67], [68].",
"The method can be further extended to controllably couple more mechanical systems in future by different type of Gaussian interactions and possibly challenging non-Gaussian transformations.",
"We acknowledge Project No.",
"GB14-36681G of the Czech Science Foundation.",
"A.A.R.",
"acknowledges support by the Development Project of Faculty of Science, Palacky University.",
"N.V. acknowledges the support of Palacky University (IGA-PrF-2016-005)."
],
[
"Logarithmic negativity",
"The mechanical modes in our system are initially in thermal states and the optical modes are all in vacuum, and the linear dynamic preserves the Gaussianity of the states of mechanical modes.",
"A Gaussian state of a two-mode system with quadratures $f = [ q_1 , p_1 , q_2 , p_2 ]^T$ is fully determined by a vector of means $\\left\\langle f \\right\\rangle $ and a covariance matrix (CM) with elements defined as $V_{i j}= \\frac{1}{2} \\mathinner {\\langle { \\Delta f_i \\Delta f_j + \\Delta f_j \\Delta f_i }\\rangle }.$ Here angular brackets denote the averaging over the quantum state, and $\\Delta f_i \\equiv f_i - \\left\\langle f_i \\right\\rangle $ .",
"Covariance matrix may be divided into $2 \\times 2$ blocks such that: $V =\\begin{bmatrix}\\mathcal {V}_1 & \\mathcal {V}_c \\\\\\mathcal {V}_c^T & \\mathcal {V}_2\\end{bmatrix},$ where $\\mathcal {V}_1$ and $\\mathcal {V}_2$ characterize internal correlations in mechanical subsystems.",
"The matrix $\\mathcal {V}_c$ stands for the correlations between the first and second mechanical modes.",
"The diagonalisation of the CM leads to symplectic eigenvalues $\\nu _{\\pm }$ [54]: $\\nu _{\\pm } = \\sqrt{\\frac{1}{2} \\left(\\Sigma (V) \\pm \\sqrt{\\Sigma (V)^2 - 4 \\det {V}} \\right)},$ with $\\Sigma (V) = \\det {\\mathcal {V}_1} + \\det {\\mathcal {V}_2} - 2 \\det {\\mathcal {V}_c}.$ Logarithmic negativity is defined then as $E_{{\\eta }} = \\max [0, -\\ln {2 \\nu _-}]$ and we use it as the measure of the entanglement of the system under the consideration."
],
[
"Beyond Rotating Wave Approximation",
"The Rotating Wave Approximation is usually adopted for considerations of the optomechanical systems working in the resolved-sideband regime ($\\kappa \\ll \\omega _m$ ).",
"In this Appendix we consider our protocol without this approximation.",
"We outline here the main steps that lead to an analytical expression for the covariance matrix of the mechanical modes.",
"The covariance matrix contains additional terms from back-action compared to the case of RWA.",
"We show that these terms do not impact the entanglement of the modes much.",
"For the sake of simplicity we do not consider in this Appendix thermal environments of mechanics and optical losses between the cavities.",
"Both these effects can be easily taken into account.",
"The equations of motion for the first system read $\\dot{q}_1 & = g_1 X_1 (\\cos 2 \\omega _m t -1 ),\\\\\\dot{p}_1 & = g_1 X_1 \\sin 2 \\omega _m t,\\\\\\dot{X}_1 & = \\sqrt{2 \\kappa } X_1^{\\text{in}}-\\kappa X_1,\\\\\\dot{Y}_1 & = \\sqrt{2 \\kappa } Y_1^{\\text{in}}-\\kappa Y_1\\\\&+ g_1 p_1 ( 1 - \\cos 2 \\omega _m t ) + g_1 q_1 \\sin 2 \\omega _m t.$ As is easily seen, this system of equations allows an analytical solution.",
"First, the Eq.",
"() has the solution $X_1 (t) = e^{ - \\kappa t } \\Big [ X_1 (0) + \\sqrt{ 2 \\kappa } \\int _0^t ds \\: e^{ \\kappa s } X_1^\\text{in} (s) \\Big ].$ We then plug this expression into Eqs.",
"(REF ,) to solve for $q_1$ and $p_1$ .",
"The solution for $p_1$ reads $p_1 (\\tau ) - p_1 (0) = X_1 (0) g_1 \\int _0^\\tau dt \\: e^{ - \\kappa t } \\sin 2 \\omega _m t\\\\+ g_1 \\sqrt{ 2 \\kappa } \\int _0^\\tau dt \\: e^{ - \\kappa t } \\sin 2 \\omega _m t \\int _0^t ds \\: e^{\\kappa s } X^\\text{in} (s)\\\\= X_1(0) g_1 \\mathcal {I}(0) + g_1 \\sqrt{ 2 \\kappa } \\int _0^\\tau ds e^{ \\kappa s } X^\\text{in} (s) \\mathcal {I}(s ),$ where $\\mathcal {I}(s ) \\equiv \\int _s^\\tau dt \\: e^{ - \\kappa t } \\sin 2 \\omega _m t.$ Notice that we swapped the order of integration when going to the last line in (REF ) in order to have $X^\\text{in}$ in the outermost integration.",
"We do a similar swap with the consequent expressions.",
"The solution for $q_1$ can be written in a similar fashion.",
"This with (REF ) can then be substituted into Eq.",
"() to obtain the solution for $Y_1$ .",
"The very same procedure repeatedly applied to the equations of motion for the second cavity and input-output relations allows to obtain a full analytical solution for the vector of quadratures of the mechanical modes.",
"The solution itself is rather cumbersome so we do not present it here.",
"Having the solution we proceed to compute the covariance matrix.",
"To demonstrate the method of calculation we use the Eq.",
"(REF ) to compute the element $V_{2,2} = \\left\\langle p_1 (\\tau )^2 \\right\\rangle $ .",
"$V_{2,2} = \\left\\langle p_1^2 (0) \\right\\rangle + \\left\\langle X_1^2 (0) \\right\\rangle g_1^2 \\mathcal {I}^2 (0)\\\\+ 2 \\kappa g_1^2 \\iint _0^\\tau \\!\\!\\!\\!",
"ds ds^{\\prime } \\: \\left\\langle X_1^\\text{in} (s) \\circ X_1^\\text{in} (s^{\\prime }) \\right\\rangle e^{ \\kappa ( s + s^{\\prime } ) } \\mathcal {I}(s) \\mathcal {I}(s^{\\prime })\\\\= \\left\\langle p_1^2 (0) \\right\\rangle + \\left\\langle X_1^2 (0) \\right\\rangle g_1^2 \\mathcal {I}^2 (0)+ V_\\mathrm {X} 2 \\kappa g_1^2 \\int _0^\\tau \\!\\!\\!\\!",
"ds \\: e^{ 2 \\kappa s } \\mathcal {I}^2 (s),$ where we used $\\left\\langle X_1^\\text{in} (s) \\circ X_1^\\text{in} (s^{\\prime }) \\right\\rangle = V_\\mathrm {X} \\delta ( s - s^{\\prime }).$ It is illustrative to estimate the difference between the full solution (REF ) and the straightforward solution $V_{2,2}^\\text{RWA} = \\left\\langle p_1^2 (0) \\right\\rangle $ obtained with advantage of RWA.",
"The quantity $\\mathcal {I}$ defined above serves as a measure of this divergence.",
"One can make estimations $& (g_1 \\mathcal {I}(0) )^2 \\sim \\left( \\frac{ g_1 }{ 2 \\omega _m } \\right)^2 = \\left( \\frac{ g_1 }{ \\kappa } \\right)^2 \\left( \\frac{ \\kappa }{ 2 \\omega _m }\\right)^2 \\ll 1,\\\\& 2 \\kappa g_1^2 \\int _0^\\tau ds \\: e^{ 2 \\kappa s } \\mathcal {I}^2 (s) \\sim \\cos ^2 2 \\omega _m \\tau \\left( \\frac{ g_1 }{ 2 \\omega _m }\\right)^2 \\ll 1.$ Besides this simple estimates we present the computed logarithmic negativity of the mechanical modes in Fig.",
"REF .",
"One can see that the adoption of RWA leads to an overestimation of entanglement due to the back-action that comes from the counterrotating terms in the Hamiltonian.",
"However, for appropriate parameters the full solution without RWA still approaches rather closely the idealized adiabatic one provided that the optomechanical coupling is not too strong (cf.",
"blue dot-dashed lines in Fig.",
"REF and REF ).",
"We use the sideband-resolution parameter $ \\kappa / \\omega _m = 0.04$ which is a conservative estimate for a number of current experimental setups [4], [52].",
"We became aware recently of another publication [69] that deals with a QND interaction beyond RWA."
]
] | 1605.05932 |
[
[
"A First Look at Ad-block Detection: A New Arms Race on the Web"
],
[
"Abstract The rise of ad-blockers is viewed as an economic threat by online publishers, especially those who primarily rely on ad- vertising to support their services.",
"To address this threat, publishers have started retaliating by employing ad-block detectors, which scout for ad-blocker users and react to them by restricting their content access and pushing them to whitelist the website or disabling ad-blockers altogether.",
"The clash between ad-blockers and ad-block detectors has resulted in a new arms race on the web.",
"In this paper, we present the first systematic measurement and analysis of ad-block detection on the web.",
"We have designed and implemented a machine learning based tech- nique to automatically detect ad-block detection, and use it to study the deployment of ad-block detectors on Alexa top- 100K websites.",
"The approach is promising with precision of 94.8% and recall of 93.1%.",
"We characterize the spectrum of different strategies used by websites for ad-block detection.",
"We find that most of publishers use fairly simple passive ap- proaches for ad-block detection.",
"However, we also note that a few websites use third-party services, e.g.",
"PageFair, for ad-block detection and response.",
"The third-party services use active deception and other sophisticated tactics to de- tect ad-blockers.",
"We also find that the third-party services can successfully circumvent ad-blockers and display ads on publisher websites."
],
[
"Introduction",
"The online advertising industry has been largely fueling the World Wide Web for the past many years.",
"According to the Interactive Advertising Bureau (IAB), the annual online ad revenues for 2014 totaled $49.5 billion in 2014, which is 15.6% higher than in 2013 [13].",
"Online advertising plays a critical role in allowing web content to be offered free of charge to end-users, with the implicit assumption that end-users agree to watch ads to support these “free” services.",
"However, online advertising is not without its problems.",
"The economic magnetism of online advertising industry has made ads an attractive target for various types of abuses, which are driven by incentives for higher monetary benefits.",
"Since publishers are paid on a per-impression or per-click basis, many publishers choose to place ads such that they interfere with the organic content and cause annoyance to end-users [21].",
"They include anything from autoplay video ads, rollovers, pop-ups, and flash animation ads to the ever-popular homepage takeover with sidebars that follow user scrolling.",
"Another major issue with online advertising is the widespread tracking of users across websites raising privacy and corporate surveillance concerns.",
"Several recent studies have shown that ad exchanges aggressively profile users and invade user privacy [23].",
"Malvertising (using ads to spread malware) is also on the rise [26], [34].",
"In addition to the above problems, many users simply desire an ad-free web experience which is much cleaner and smoother.",
"Therefore, ad-blockers have become popular in recent years and they can block ads seamlessly without requiring any user input.",
"A wide range of ad-blocking extensions are available for popular web browsers such as Chrome and Firefox [20].",
"Adblock Plus is most prominent among all these extensions [1].",
"According to a recent academic study, 22% of the most active residential broadband users of a major European ISP use Adblock Plus [30].",
"In addition, it is estimated in a recent report [32] that $22 billion will be lost due to ad-blocking in 2015, almost twice the amount estimated in 2014.",
"To the advertisement industry and content publishers, ad-blockers are becoming a growing threat to their business model.",
"To combat this, two strategies have emerged: (1) companies such as Google and Microsoft have begun to pay ad-blockers to have their ads whitelisted; and (2) websites have begun to detect the presence of ad-blockers and may refuse to serve any user with ad-blocker turned on, e.g.",
"Yahoo mail reportedly did so recently [18].",
"As not every website is willing or capable of paying ad-blockers, the 2nd strategy becomes a low-cost solution that can be easily deployed.",
"Even though anecdotes exist about websites starting to detect ad-blockers, the scale at which this occurs remains largely unknown.",
"To fill this gap, in this paper we perform the first systematic characterization of the ad-block detection phenomenon.",
"Specifically, we are interested in understanding: (1) how many websites are performing ad-block detection; (2) what type of technical approaches are used; and (3) how can ad-blockers counter or circumvent such detection.",
"Key Contributions.",
"The key contributions of the paper are the following: • We conduct a measurement study of Alexa top-100K websites using a machine learning based approach to identify the websites that use ad-block detection.",
"The approach is promising with precision of 94.8% and recall of 93.1%.",
"The results show that around 300–1100 websites are currently performing ad-block detection (details in §).",
"• We cluster different ad-block detection approaches based on the JavaScripts that are inserted in the websites.",
"The results indicate that there is a spectrum of detection solutions ranging from fairly simple (passive detection) to complex (active deception).",
"We conduct several case studies to illustrate the strengths and limitations of different approaches."
],
[
"Background",
"In this section, we provide an overview of ad-blockers and ad-block detectors.",
"The rise of ad-blockers.",
"The issues with online ads has resulted in a proliferation of ad-blocking software.",
"Ad-blocking software (or ad-blocker) is an effective tool that blocks ads seamlessly, primarily published as extensions in web browsers such as Chrome and Firefox [20].",
"More recently, Apple has also allowed content blocking plugins for Safari on iOS devices [25].",
"Other popular relevant tools include Ghostery [7] and DisconnectMe [3]; however, they are primarily focused on protecting user privacy.",
"With respect to functionality, these ad-blockers (1) block ads on websites and (2) protect user privacy by filtering network requests that profile browsing behaviors.",
"Recent reports have shown that the number of users using ad-blocking software has rapidly increased worldwide.",
"According to PageFair, up to 198 million users around the world now use ad-blocking software [32].",
"According a recent academic study, 22% of the most active residential broadband users of a major European ISP use Adblock Plus [30].",
"These ad-blocking users have been estimated to cost publishers more than $22 billion in lost revenue in 2015 [32].",
"Figure: Typical ad-block detection responsesHow do ad-blockers work?",
"Ad-blockers eliminate ads by either page element removal or web request blocking.",
"For page element removal, ad-blockers use various CSS selectors to access the elements and remove them.",
"Similarly, for web requests, ad-blocker looks for particular URLs and remove the ones which belong to advertisers.",
"For both of these actions, ad-blockers are dependent on filter lists that contain the set of rules (as regular expressions) specifying the domains and element selectors to remove.",
"There are various kinds of filter lists available which can be included in ad-blockers.",
"Each of these lists serves a different purpose.",
"For example, Adblock Plus by default includes EasyList [4], which provides rules for removing ads from English websites.",
"Similarly, Fanboy [6] is another popular list that removes only annoying ads from websites.",
"Additionally, EasyPrivacy [5] helps ad-blockers to remove spy-wares.",
"Figure: Web page load evolution for http://www.vipleague.tv.",
"Left: The original website content is loaded.",
"Middle: Ad-blocker removes ads from the page.",
"Right: Ad-block detector blocks the content and shows a pop-up notification asking the user to disable ad-blocking software.The rise of ad-block detection.",
"The widespread use of ad-blockers has prompted a cat-and-mouse game between publishers and ad-blocking software.",
"More specifically, publishers have started to detect whether users are visiting their websites while using ad-blocking software.",
"Once detected, publishers notify users to turn off their ad-blocking software.",
"These notifications can range from a mild non-intrusive message which is integrated inside website content to more aggressive blocking of website content and/or functionality.",
"Figure REF shows examples of both cases.",
"We note that the aggressive approach refrains users from accessing any website content.",
"To detect the use of ad-blocking software, publishers include scripts in the code of their web pages.",
"When a user with the ad-blocking software opens such a website, these scripts typically monitor the visibility of ads on the page to identify the use of ad-blockers.",
"If ads are found hidden or removed by the scripts, publishers take countermeasures according to their policies.",
"It is noteworthy that the strategies used by publishers to detect ad-blockers is evolving.",
"Figure: Ad-block detection JavaScript extracted from http://www.vipleague.tvIllustration of ad-block detection.",
"To understand how ad-block detection scripts operate, let's analyze the complete cycle of ad-block detection.",
"Figure REF shows the web page loading process of http://www.vipleague.tv, which employs ad-block detection, on a web browser with Adblock Plus.",
"Figure REF shows the JavaScript that is used for ad-block detection by the website.At the time of writing, this script can detect users with Adblock Plus.",
"The functionality of the JavaScript can be divided into three parts: timeout, condition check, and response.",
"In Figure REF , we note that the web browser starts loading the HTML and other page content included in the HTML code (➊).",
"While the content is loading, ad-block extension kicks in and starts evaluating the HTML code and page content to remove potential ads (➋).",
"Since the ad-blocking software starts working after a small delay, the ad-block detection script has to wait some time before monitoring the ads.",
"In Figure REF , the timeout is set at 2000 milliseconds.",
"Once the timeout expires, the condition check is executed to verify the presence/absence of ads.",
"This step is typically carried out by accessing various elements and their css properties.",
"In Figure REF , the script first checks the visibility property of the div with identifier XUinXYCfBvqpyDHOrOAVClxoWJemrlPpfYCdWfiyAzNY, as will be discuss later, this div is specifically designed for ad-block detection.",
"Ad-blockers sometimes also make the ad invisible by decreasing its dimensions; therefore, the script verifies the height and length properties of the ad related div elements for classes vip_052x003 and vip_09x827.",
"If the script detects that ads were removed or hidden, then the response step is executed.",
"As discussed earlier, the implementation details of this step varies across publishers.",
"A few publishers gently request users to remove/disable their ad-blockers, while others aggressively show a page-wide notification and/or block content (➌).",
"For example, in Figure REF , the publisher responds by first changing css properties of the div, which it verifies in a conditional check.",
"More specifically, the publisher sets the z-index of the div to make it a pop-up message."
],
[
"Measuring Ad-block Detection",
" In this section, we design and implement our approach for automatically identifying websites that employ ad-block detection.",
"The main premise of our approach is that websites conducting ad-block detection make distinct changes to their web page content for ad-block users as compared to users without ad-block.",
"Our goal is to identify, quantify, and extract such distinct features that can be leveraged for training machine learning models to automatically detect websites that employ ad-block detection."
],
[
"Overview",
"We want to identify distinct features that capture the changes made by ad-block detectors to the HTML structure of web pages.",
"To this end, we first conducted some pilot studies to test the behavior of websites that employ ad-block detection.",
"Based on our pilot studies, we found that the changes made by ad-block detectors can be categorized into: (1) addition of extra DOM nodes, (2) change in the style of existing DOM nodes, and (3) changes in the textual content.",
"We also found a few cases when the websites completely changed the web page content.",
"In addition, a few websites with ad-block detectors reacted by redirecting users to warning pages.",
"Note that the Adblock Plus is installed with the default configuration which allows acceptable ads [11].",
"This will likely suppress many ad-block detections and result in underestimating their prevalence.",
"However, since most regular users would choose the default configuration, we believe our study represents what most users would observe regarding to ad-block detection.",
"Below, we provide an overview of our proposed features and also discuss how they capture the changes by ad-block detectors.",
"Node additions.",
"We found that in order to show notification to users with ad-blockers, websites dynamically create and add new DOM nodes.",
"Thus, node additions in the DOM can potentially indicate ad-block detection.",
"We can log the total number of DOM elements inserted in a web page.",
"Style changes.",
"We found that a few websites include ad-block detection notifications which are in their page content but hidden.",
"If these websites detect the use of ad-blockers, they change the visibility of their notification.",
"To cover such cases, we can log attribute changes to DOM elements of a web page.",
"Text changes.",
"Other then structural changes, we found that some websites change the textual content (i.e., text-related nodes) in response to ad-blockers.",
"Therefore, we can log changes in the textual content of a web page and addition of text-related nodes in a web page.",
"Miscellaneous features.",
"In addition to the above-mentioned features, we also consider other features like innerHTML to detect whether the structure is completely changed and URL to detect redirection.",
"Figure: Overview of our methodology for measuring ad-block detection."
],
[
"Methodology",
"Figure REF provides an overview of our methodology to automatically measure ad-block detection on the web.",
"We conduct A/B testing to compare the contents of a web page with and without ad-blocking software.",
"To automate this process, we use the Selenium Web Driver [8] to open two separate instances of the Chrome web browser, with and without Adblock Plus (➊).",
"We implemented a custom Chrome browser extension to record changes in the content of web pages during the page load process.",
"Our extension records the structure of the DOM tree, all textual content, and HTML code of the web page (➋).",
"We implemented a feature extraction script to process the collected data and generate a feature vector for each website (➌).",
"We feed the extracted features to a supervised classification algorithm for training and testing (➍).",
"We train the machine learning model using a labeled set of websites with and without ad-block detectors.",
"Below we describe these steps in detail.",
"Web automation for A/B testing.",
"Using the Selenium Web Driver [8], we implemented a web automation tool to conduct automated measurements.",
"For A/B testing, our tool first loads a website without Adblock Plus, and then opens it with Adblock Plus in a separate browser instance.",
"However, we found that many websites host dynamic content that changes at a very small timescales.",
"For example, some websites include dynamic images (e.g., logos), which can introduce noise in our A/B testing.",
"Similarly, most news websites update their content frequently which can also add noise.",
"Thus, we may incorrectly attribute these changes to the ad-blocker or ad-block detector used by the publisher.",
"To mitigate the impact of such noise, our tool opens multiple instances of each website in parallel and excludes content that changes across multiple instances.",
"Data collection using a custom Chrome extension.",
"To collect data while a web page is loading, we use DOM Mutation Observers [9] to track changes in a DOM (e.g.",
"DOMNodeAdded, DOMAttrModified, etc.).",
"The changes we track include addition of new DOM nodes or scripts, node attribute changes like class change or style change, removal of nodes, changes in text etc.",
"We implemented the data collection module as a Chrome extension.",
"The extension is preloaded in the browser instances that are launched by our web automation tool.",
"As soon as a web page starts loading, the extension attaches an observer listener with it.",
"Whenever an event occurs, the listener fires and we record the information.",
"For example, we record the identifier, type, value, name, parent nodes, and attributes of the corresponding node.",
"For each attribute change, in addition to above-mentioned information, we record the name of attribute which changes like style or class and its old and new value.",
"We also log page level data such as the complete DOM tree, innerText, and innerHTML as well.",
"Table: Features used to identify ad-block detectorsFeature extraction.",
"We then process the output of data collector to extract a set of informative features which can distinguish between changes due to ad-block detection.",
"Recall that we load each page multiple times to mitigate noise.",
"Let A denote the data collected with ad-blocker, and let B & B' denote the data collected by loading a web page twice without an ad-blocker.",
"We provide details of the feature extraction process below.",
"Table REF includes the list of all features used in our study.",
"$\\bullet $ Node features.",
"For each instance, we extract DOM related nodes because our pilot experiments revealed that websites using ad-block detection add only DOM related nodes.",
"More specifically, we extract the list of anchor, div, h1, h2, h3, img, table, p, and iframe nodes for each instance.",
"Once we have a list of DOM nodes for each instance, we compare A vs. B' and B vs. B' to obtain the list of differences between these nodes.",
"We denote these lists as AB' and BB' lists.",
"As explained earlier, to remove number of node differences due to dynamic content of websites, we cross-validate nodes in AB' with BB' using their properties.",
"Our key idea is that if a publisher ads random nodes to a web page, they may have different identifiers but most the other properties will be almost similar.",
"Thus, we remove the nodes from AB' that also appear in BB'.",
"$\\bullet $ Attribute features.",
"For each instance, we extract changes in the style of DOM related nodes.",
"More specifically, we focus on changes to the display-related property of nodes.",
"For instance, we log whether the visibility property of a node changes from hidden to non-hidden.",
"We also log changes to the display property of a node, e.g., the number of changes in height, width, and opacity of nodes.",
"Similar to node features, we compare A, B, and B' to eliminate attribute changes from AB' that also appear in BB'.",
"$\\bullet $ Text features.",
"We get the list of all text nodes in A, B and B'.",
"Using the lists, we identify pairs of nodes with differences texts.",
"We particularly focus on line differences rather than character-level differences to mitigate noise (e.g., difference in clock time).",
"We again compare A, B, and B' to eliminate changes in textual features from AB' that also appear in BB'.",
"$\\bullet $ Structural features.",
"We compare differences in the overall page HTML using the cosine similarity metric.",
"If the cosine similarity between A and B/B' is very low, it indicates significant content change.",
"To check for potential URL redirections, we also track changes in URL.",
"Classification model training and testing.",
"We feed the extracted features to a machine learning classifier to automatically detect websites that employ ad-block detection.",
"However, in order to train the classification algorithm, we need a sufficient number of labeled examples of websites that detect ad-blockers (i.e., positive samples) and websites that do not detect ad-blockers (i.e., negative samples).",
"To get positive samples, we first use a crowd-sourced list of such websites [2].",
"We manually validated the websites in this list, and excluded websites that did not detect and respond to ad-blockers.",
"We also manually opened Alexa top 1000 websites and identified four websites that use ad-block detection.During the manual verification, we found that the response of websites after ad-block detection varies.",
"Most websites detect and respond to ad-blockers on the homepage without waiting for any input from users.",
"In contrast, some websites respond to ad-blockers only when a particular content type is requested (e.g., video is played) or when the user navigates to other pages.",
"Since it is not practical to automatically identify such requirements, we restrict ourselves to the former category of websites.",
"Also note that some websites include ad-block detection logic but they do not respond to ad-blockers.",
"We excluded these websites from the list as well.",
"Overall, we identified a total of 200 positive training samples.",
"Since a vast majority of Alexa top 1000 websites do not deploy ad-block detection, we use them as negative training samples.",
"Table: Feature ranking based on information gainFigure: Distribution of features used to identify ad-block detection"
],
[
"Feature Analysis",
"In this section, we analyze the extracted features to quantitatively understand their usefulness in identifying ad-block detection.",
"We first visualize the distributions of a few features.",
"Figure REF plots the cumulative distribution functions (CDF) of two features.",
"We observe that websites which employ ad-block detection tend to changes more lines and add div elements than other websites.",
"These distributions confirm our intuition that ad-block detectors make changes in the web content that are distinguishable.",
"To systematically study the usefulness of different features, we employ the concept of information gain [28], which uses entropy to quantify how our knowledge of a feature reduces the uncertainty in the class variable.",
"The key benefit of information gain over other correlation-based analysis methods is that it can capture non-monotone dependencies.",
"Let $H(X)$ denote the entropy (i.e., uncertainty) of feature $X$ .",
"H is defined as: $H=-\\sum \\limits _{i}{p_{i}{\\log {p_{i}}}}$ Let $H(Y)$ denote the entropy (i.e., uncertainty) of the binary class variable $Y$ .",
"Information gain is computed as: $IG(Y|X) = H(Y) - H(Y|X).$ We can normalize information gain, also called relative information gain, as: $\\frac{H(Y) - H(Y|X)}{H(Y)}.$ Using this, we can quantify what an input feature informs us about the use of ad-block detection.",
"Table REF ranks the top 10 features based on their information gain.",
"We note that text-based features (number of words changed and number of text nodes added) have the highest information gain, both exceeding 25%.",
"They are followed by node and style based features (e.g., number of div elements added, number of nodes for which height property is changed, etc.).",
"Table: Effectiveness of different classifiers"
],
[
"Classifier Evaluation",
"We train machine learning classification models using the labeled set of 1000 negative samples and 200 positives samples.",
"We use the standard $k$ -fold cross validation methodology to verify the accuracy of the trained models.",
"For this purpose we select $k=5$ , divide the data into 5 folds where one fold is used as training set while rest of folds are used for verification.",
"To quantify the classification accuracy of the trained models, we use the standard ROC metrics such as precision, recall, and area under ROC curve (AUC).",
"$\\text{Precision} = \\frac{\\text{True Positives}}{\\text{True Positives} + \\text{False Positives}}$ $\\text{Recall} = \\frac{\\text{True Positives}}{\\text{True Positives} + \\text{False Negatives}}$ We test multiple machine learning models on our data set.",
"We tuned various parameters of each of these models to optimize their classification performance.",
"Table REF summarizes the classification accuracy of these classifiers.",
"We note that the random forest classifier, which is a combination of tree classifiers, clearly outperforms the C4.5 decision tree and the naive Bayes classifiers.",
"The random forest classifier achieves 93.1% recall, 94.8% precision, and 96.0% AUC.",
"To further evaluate the effectiveness of different feature sets in identifying ad-block detection, we conduct experiments using stand alone feature sets and then evaluate their all possible combinations.",
"We divide the features into node features, attribute features, and text features.",
"Among stand alone feature sets, text-based features provide the best classification accuracy.",
"We also observe that using combinations of feature sets does improve the classification accuracy.",
"The best classification performance is achieved when all feature sets are combined.",
"Figure: Visualization of decision tree model for ad-block detectionTo further gain some intuition from the trained machine learning models, we visualize a pruned version of the decision tree model trained on labeled data in Figure REF .",
"As expected from the information gain analysis, we note that a text feature (words difference) is the root node of the decision tree.",
"If there is a positive word difference, the model detects ad-block detection.",
"Similarly, if node visibility is changed, the model detects ad-block detection.",
"It is interesting to note that the top three features in the decision tree belong to different feature categories.",
"This indicates that different feature sets complement each other, rather than capturing similar information, which we also observed earlier when evaluating different combinations of features.",
"We want to analyze the strategies and methods used by publishers for ad-block detection.",
"To this end, we first use the random forest model on Alexa top 100K websites to identify ad-block detectors.",
"Our machine learning model found a total of 292 websites that detect and respond to ad-blockers.",
"Table (in Appendix) lists these 292 ad-block detecting websites along with their Alexa rank.",
"We note that a vast majority of the websites in Table have low Alexa ranks, likely due to 1) the top web websites have paid ad-blockers to be whitelisted or 2) the top websites are worried about losing users if they take an aggressive stance against ad-blocker users.",
"Using additional string based features (e.g., “Adblock”, “Adblock Plus”), we also found a total of 797 websites that have ad-block detection scripts but do not exhibit visible behaviors, likely due to default-on acceptable ads in our Adblock Plus extension.",
"It is also possible that such websites are currently tracking the usage of ad-blockers but not necessarily ready to go aggressively against users.",
"Overall, we found 1,089 ad-block detecting websites in the Alexa top-100K list.",
"In this section, we focus our attention on the ad-block detecting websites that not only detect ad-blockers but also respond to them.",
"Our goal here is to characterize how different ad-block detection strategies operate under the hood.",
"We cluster ad-block detection strategies based on their JavaScript code similarity.",
"Our analysis allows us to measure the popularity of specific strategies and third-party ad-block detection services, e.g.",
"PageFair.",
"The result of the analysis will also help us design countermeasures against the state-of-the-art ad-block detectors."
],
[
"JavaScript Collection",
"As a first step, we collect the JavaScript code of all websites that employ ad-block detection.",
"Analyzing the functionality of JavaScript code is non-trivial because the code can be packed inside functions such as eval.",
"To overcome these issues, we leverage the fact that the code needs to unpack itself before execution.",
"We attach a debugger between the Chrome V8 JavaScript engine [15] and the web pages.",
"Specifically, we observe script.parsed function, which is invoked when eval is called or new code is added with <iframe> or <script> tags.",
"We implement the debugger as a Chrome extension and collect all JavaScript snippets parsed on a web page and identify the snippet responsible for ad-block detection.",
"Figure: Visualization of ASTs of two ad-block detector JavaScript snippets.",
"We note that although the code snippets appear to be different, the structure of their ASTs are similar."
],
[
"Clustering",
"Given these ad-block detector JavaScript snippets, we aim to cluster them into a few groups to identify their families.",
"To this end, we first compute “similarity” between JavaScript snippets and then use clustering.",
"Methodology.",
"To analyze and quantify the similarity between JavaScript snippets, we parse them to produce abstract syntax trees (ASTs).",
"ASTs have been used in prior literature for JavaScript malware detection [19], [24].",
"ASTs allow us to retain the structural and logical properties of the code while ignoring fine details like variable names, which are not useful for our analysis.",
"Figure REF shows two ad-block detection JavaScript snippets and their corresponding AST visualizations.",
"We use the Esprima JavaScript parser [14] to visualize ASTs for each JavaScript snippet.",
"We note that although the JavaScript snippets look fairly different but their ASTs have similar logical structure except minor differences near the leaf nodes.",
"We transform ASTs of all ad-block detection JavaScript snippets to normalized node sequences by performing the pre-order traversal on each tree.",
"Each variable length sequence is composed of node types that appear in the tree.",
"Note that there are 88 distinct node types in the JavaScript language.",
"To transform the variable length normalized node sequences to a fixed number of dimensions, we convert each sequence into a 88-dimensional summary vector.",
"Each JavaScript snippet is represented as an 88-dimensional point, where each dimension corresponds to a node type.",
"The value of each dimension is the node type frequency.",
"Figure: Cluster visualization using PCA.",
"The websites in the dense central cluster use simple scripts for ad-block detection.",
"The outliers represent websites that use more sophisticated third-party scripts for ad-block detection.Results.",
"We use the Principal Component Analysis (PCA) to reduce the dimensionality of the summary vector for visualization.",
"Figure REF plots the a 3-dimensional visualization of 292 ad-block detection JavaScript snippets.",
"We note that the center of the plot contains a dense cluster of instances.",
"Other outlier instances are spread out far from the central cluster.",
"We surmise that the central cluster represents websites that use a similar approach towards ad-block detection.",
"However, there are a number of outliers that represent customized and potentially more sophisticated approaches.",
"In the next section, we conduct an in-depth analysis of various ad-block detection approaches."
],
[
"Case Studies",
"Next, we analyze the ad-block detection strategies used by different clusters.",
"We first study the ad-block detection strategies of websites in the dense central cluster.",
"We refer to these websites as the “common family”.",
"We then study the ad-block detection strategies of outlier websites.",
"Our manual inspection of outliers revealed that these approaches use third-party ad-block detection scripts, including PageFair [16], FuckAdBlock [12], and Sourcepoint [17].",
"Below, we provide an in-depth analysis of both types of websites."
],
[
"The Common Family",
"The most distinct feature of ad-block detection JavaScript snippets in the common family is their simplicity.",
"Most of them are between 5-10 lines of code, yet they can successfully detect state-of-the-art ad-blockers.",
"Specifically, as discussed in §, ad-blockers tend to aggressively block ads by removing the ad frames entirely, without the intention of hiding their operation whatsoever.",
"The obvious nature of ad-blockers allows simple scripts, such as those in the common family, to easily identify ad-block users.",
"Detection timing.",
"We note that all websites in the common family launch their ad-block detection logic in the beginning of the page load process.",
"Since it may take a few seconds before an ad-blocker can remove the ads, some websites delay the execution of their logic by standard setTimeout() or setTimeIntervel().",
"In Figure REF , we show two example ad-block detection scripts, one inserting delay and another without delay.",
"Since the ad-block detection logic is a one-time check (i.e., it is not invoked periodically), ad-block detectors include the delay to ensure that ad-blockers have ample time conduct their operation.",
"Detection logic.",
"The ad-block detection JavaScript snippets in the common family typically check different ad elements to detect ad-blockers.",
"In Figure REF , we show the detection logic implemented by several websites in the common family.",
"We again note that the detection checks are fairly intuitive and simple.",
"For example, consider urlchecker.org, which checks whether the height of adcheker div is less then 10pxs or not.",
"Our further analysis revealed that the ad-blocker blocks the adsbygoogle.js script due to which the adcheker div is empty and its height is equal to 1px.",
"Other websites in Figure REF also check the properties of different div elements.",
"Since the filter lists used by ad-blockers, e.g., EasyList [4] and Fanboy [6], are publicly available, ad-block detectors can successfully setup these detection rules.",
"For instance, EasyList [4] used by ad-blockers has adsbyggole.js in its block list.",
"Response.",
"Although the detection logic used by websites in the common family is similar, their response to ad-blockers vary widely.",
"Figure REF lists a few of the responses.",
"For hentai.to, a <p> element requests users to disable the ad-blocker.",
"Since the original content is preserved, this approach is not aggressive.",
"However, for knowlet3389.blogspot.hk, the #Blog1 div is removed upon ad-block detection, which indicates that the website hides its content from ad-block users.",
"elahmad.com also aggressive responds by redirecting ad-block users to a warning page.",
"Overall, we find a wide spectrum of responses to ad-block detection, ranging from gentle request messages to more aggressive redirection.",
"Figure: Examples of detection logic used by different ad-block detectors"
],
[
"PageFair",
"PageFair is a service that allows publishers to detect ad-block usage and take mitigation actions such as display tailored non-intrusive ads to ad-block users.",
"In the list of 292 ad-block detection websites, we found 12 websites that use PageFair.",
"We note that PageFair uses dynamic JavaScript and code obfuscation.",
"For example, it involves assembling URLs on the fly to retrieve additional JavaScript codes.",
"Below, we shed light on PageFair's strategy for ad-block detection and its response.",
"Figure: Examples of responses by different ad-block detectors.Detection timing.",
"PageFair performs multiple periodic checks at various stages of the web page load process to detect ad-blockers.",
"This approach is much more sophisticated than the common family and makes it harder for ad-blockers to evade detection by simply delaying their activity.",
"Detection logic.",
"PageFair's detection logic attempts to actively trap ad-blockers by injecting different “baits” on web pages.",
"This is in stark contrast to simple passive detectors used by most websites in the common family.",
"In addition, PageFair attempts to check whether any ad-block plug-in is installed by looking for various browser resources exposed by ad-blocking extensions.",
"The use of these methods makes PageFair's detection logic difficult to evade.",
"We separately discuss both of these methods below.",
"$\\bullet $ Baiting: Figure REF shows different types of baits used by PageFair.",
"The first example shows an injected div element that is not visible on the page, i.e.",
"1x1 in size and negative values of the top and left properties.",
"The other two examples show img and script baits.",
"PageFair's detection logic injects these baits with keywords such as “ad” in the element name or URL.",
"For example, the identifier of the div element is set to influads_block.",
"Similarly, the source of script tag is set to adsense.js, which is a common script used by Google Ads.",
"$\\bullet $ Extension resources: For Chrome browser, we note that PageFair attempts to detect the presence of ad-blockers by accessing extension resources exposed by various ad-blockers at chrome-extension://.",
"Figure REF shows how PageFair accesses extension resources to identify 8 popular ad-blockers including AdBlock, Adblock Plus, AdBlock Pro, AdBlock Premium, Adblock Super, Adguard, Ad Remover, and uBlock.",
"For each type of ad-blocker, it includes a unique extension identifier, e.g.",
"gighmmpiobklfepjocnamgkkbiglidom for AdBlock, and the resource file path.",
"Note that Chrome generally does not allow web pages to directly access extension resources unless an extension specifies resources as web_accessible_ resources in the manifest file and makes them publicly accessible.",
"We find that these resources of various ad-blockers requested by PageFair are indeed publicly accessible.",
"Thus, ad-blockers are leaking the proof of their presence to ad-block detectors.",
"In addition, Figure REF shows that critical resources such as whitelisted pages are also accessible.",
"For example, Adblock Plus exposes block.html, which allows websites to get a list of blocked URLs.",
"Figure: PageFair uses different baits to detect ad-blockers.Figure: PageFair accesses extension resources to detect ad-blockers.Response.",
"PageFair provides a whitelist ad service under the acceptable ads manifesto [11].",
"To understand PageFair's service, we installed PageFair on a test website that uses Google Ads.",
"With ad-blocker, as expected, we find that the original ad is not available.",
"PageFair includes a replacement ad, which is not hosted on Google's domain.",
"Instead, the replacement ad is hosted on PageFair's domain — adsfeed.pagefair.com.",
"We found that PageFair's domain is whitelisted by EasyList [4], which is a default list on popular ad-blockers.",
"The following snippet is from EasyList [4]: \"##.pagefair-acceptable\" \"||pagefair.net⌃$third-party\" The rule clearly indicates that PageFair is allowed to show acceptable ads on partner websites.",
"PageFair's response is reflective of the growing adoption of acceptable ads by many publishers and ad-blockers [33].",
"It is worth mentioning that when we rechecked some websites several weeks after we finished the initial experiments, the behaviors of websites changed.",
"For instance, http://www.vipleague.tv used to detect ad-blockers and simply refuse to serve users via a large pop-up window.",
"It is correctly classified as the common family.",
"However, in its recent version, it has evolved to behave similarly to PageFair.",
"Instead of refusing the service, the website switches the origin of the ads from adsrvmedia, which is on the filter list of ad-blockers, to the same local domain.",
"Since the new ad URL is not on the filter list, the ad-blocker simply fails to block it.",
"This shows yet another new response that is similar in nature to PageFair."
],
[
"Related Work",
"To the best of our knowledge, this paper presents the first large-scale measurement study of ad-block detection.",
"Since the arms race between publishers and ad-blockers is a recent phenomenon, prior work has mainly focused on understanding the dynamics of ads and ad-blockers.",
"Below, we discuss prior literature on online advertising and ad-blockers.",
"Online Advertising.",
"A large number of web publishers rely on online advertising.",
"However, online advertising has recently become more intrusive and annoying to end-uses.",
"Researchers have recently focused on various security and privacy aspects of online advertising.",
"Li et al.",
"conducted the first large-scale study of malicious advertising (called malvertising) on the web [26].",
"Their analysis of 90,000 websites showed that not only malicious ads effect top websites but they also evade detection by various cloaking techniques.",
"Zarras et.",
"al.",
"also conducted a large-scale study to determine the extent at which user are exposed to malicious advertisements [34].",
"Their measurement study of more then 60,000 ads showed that around 1% ads exhibit malicious behavior.",
"They also showed that a few ad networks are more prone to supply malicious advertisements than others.",
"Other than malvertising, researchers have also analyzed privacy implications of online advertising.",
"Online advertising relies on sophisticated tracking of users across the web to target personalized ads.",
"Roesner et al.",
"conducted a comprehensive active measurement study of third-party tracking on the web [31].",
"They found more than 500 unique trackers on 1000 websites, with a few web trackers covering a large fraction of users' browsing activity.",
"Metwalley et al.",
"conducted a passive measurement study to determine the extent of tracking on the web.",
"They found that more than 400 tracking services are contacted by the users and also 80% of the users are tracked by at least one tracking service within a second after starting their surfing sessions [27].",
"Nath performed a measurement of 500K ad requests from 150K Android apps [29].",
"The analysis showed that although most ad networks collect targeting data on mobile apps, it does not significantly alter the way ads are chosen for many users.",
"Ad-blockers.",
"Due to the rapid increase in the use of ad-blockers, researchers are interested in measuring the use of ad-blockers.",
"Pujol et al.",
"conducted a measurement study using passive network traces of thousands of users from a European ISP to quantify ad-block usage [30].",
"Their results show that 22% of users use AdBlock Plus.",
"They also found that ad-blocker users still generate significant ad traffic due their enrollment in the acceptable ads program.",
"Walls et al.",
"conducted a study of the whitelists used by ad-blockers for allowing acceptable ads [33].",
"They analyzed the evolution of ad-blocker whitelists and performed measurements on 8K websites.",
"Their analysis showed that whitelists contains around 5,936 filters and 3,545 unique publisher domains.",
"Their findings highlight that whitelists are inclined towards top ranked Alexa websites (59% filters are for top 5000 websites).",
"Gugelmann et al.",
"proposed a methodology to compliment manual filter lists of ad-blockers by automatically blacklisting intrusive ads [22].",
"They train a classifier on HTTP traffic statistics and identify around 200 new advertising and tracking services.",
"Since ad-block detection is a recent phenomenon, to the best of our knowledge, no prior work has studied the mechanisms used by ad-block detectors.",
"Our work aims to fill this gap by conducting a large-scale measurement of ad-block detection on the web."
],
[
"Conclusion",
"We present the first large-scale study of ad-block detection on the web.",
"Our main observation is that about 1100 websites in the Alexa top-100K are currently performing ad-block detection.",
"Out of these, about 300 websites respond to ad-block detection by requesting users to disable ad-blockers through notifications to more aggressive blocking of website content.",
"We also find the increasing use of third party ad-block detectors such as PageFair.",
"We envision the arms race to continue in the coming years as we expect both ad-blockers and ad-block detectors to adapt in the future."
]
] | 1605.05841 |
[
[
"Double electron capture searches in $^{74}$Se"
],
[
"Abstract A search for various double electron capture modes of $^{74}$Se has been performed using an ultralow background Ge-detector in the Felsenkeller laboratory, Germany.",
"Especially for the potentially resonant transition into the 1204.2 keV excited state of $^{74}$Ge a lower half-life limit of $0.70\\cdot 10^{19}$ yr (90% credibility) has been obtained.",
"Serious concerns are raised about the validity of obtained $^{74}$Se limits in some recent publications."
],
[
"Introduction",
"The search for physics beyond the standard model is a wide spread activity in accelerator and non-accelerator physics.",
"Among all the searches for new physics, total lepton number violation plays an important role.",
"The golden channel to search for total lepton number violation is neutrinoless double beta decay $(Z,A) \\rightarrow (Z+2,A) + 2 e^- \\quad (0\\mbox{$\\nu \\beta \\beta $ decay}).$ For recent reviews see , , .",
"Any Beyond Standard Model (BSM) physics allowing $\\Delta L =2$ processes can contribute to the decay rate.",
"It has been shown that its observation would imply that neutrinos are their own antiparticles (Majorana neutrinos) which is an essential ingredient for leptogenesis, explaining the baryon asymmetry in the universe with the help of Majorana neutrinos (see for example ).",
"However, it is unknown how much individual BSM processes contribute to the double beta decay rate.",
"To observe $0\\nu \\beta \\beta $ decay, single beta decay has to be forbidden by energy conservation or at least strongly suppressed.",
"For this reason only 35 potential double beta minus emitters exist.",
"As the phase space for these decays scales strongly with the Q-value, searches are using only those nuclides with a Q-value above [2]MeV, reducing the list to 11 candidates.",
"Lower limits on half-lives beyond $10^{25}$ years of the neutrino less mode have been measured for the isotopes $^{76}$ Ge and $^{136}$ Xe , , .",
"In addition, the allowed process of neutrino accompanied double beta decay $(Z,A) \\rightarrow (Z+2,A) + 2 e^- + 2 \\mbox{$\\bar{\\nu }_e$} \\quad (2\\mbox{$\\nu \\beta \\beta $ decay})$ will occur.",
"It is the rarest decay measured in nature and has been observed in more than ten isotopes.",
"However, in both cases the measured quantity is a half-life, which is linked to the phase space $G$ and nuclear transition matrix elements $M$ .",
"In case of 2$\\nu \\beta \\beta $ decay the matrix element is purely Gamow-Teller (GT) and the relation is $\\left(\\mbox{T$_{1/2}^{2\\nu }$} \\right)^{-1} = G \\times \\mid M_{GT} \\mid ^2\\,,$ which does not require any BSM particle physics, as opposed to 0$\\nu \\beta \\beta $ decay.",
"The half-life measurements of 2$\\nu \\beta \\beta $ decay is important for understanding the nuclear structure since it will provide valuable information on the nuclear matrix elements which can be directly compared with theory.",
"Equivalent processes to double beta minus decay with the emission of two electrons could occur on the right side of the mass parabola of even-even isobars.",
"34 nuclides are candidates for this process.",
"Three different decay modes are possible involving $\\beta ^+$ decay and electron capture (EC) $(Z,A) &\\rightarrow (Z-2,A) + 2 e^+ \\; (+ 2 \\mbox{$\\nu _e$}) \\quad & \\mbox{(\\mbox{$\\beta ^+\\beta ^+$} )} \\\\e^- + (Z,A) &\\rightarrow (Z-2,A) + e^+ \\; (+ 2 \\mbox{$\\nu _e$}) \\quad & \\mbox{(\\mbox{$\\beta ^+EC$} )} \\\\2 e^- + (Z,A) &\\rightarrow (Z-2,A) \\; (+ 2 \\mbox{$\\nu _e$}) \\quad & \\mbox{(\\mbox{$ECEC$} )} $ Decay modes containing an EC emit X-rays or Auger electrons created by the atomic shell vacancy in the daughter nuclide.",
"Decay modes containing a positron have a reduced Q-value as each generated positron accounts for a reduction of 2 $m_ec^2$ in the phase space.",
"Thus, the largest phase space is available in the $ECEC$ mode and makes it the most probable one.",
"However, the $ECEC$ is also the most difficult to detect, only producing X-rays (or Auger electrons) and two neutrinos in the final state instead of [511]keV $\\gamma $ -rays resulting from the decay modes involving positrons.",
"Furthermore, it has been shown that $\\beta ^+EC$ transitions have an enhanced sensitivity to right-handed weak currents (V+A interactions) and thus would help to disentangle the physics mechanism of 0$\\nu \\beta \\beta $ decay, if observed.",
"In the 0$\\nu $$ECEC$ mode there are only X-rays (or Auger electrons) and if there is no other particle in the final state this would violate energy and momentum conservation.",
"It was suggested that the energy is released radiatively as a single internal radiative bremsstrahlung $\\gamma $ -ray, two $\\gamma $ -rays, an $e^-e^+$ -pair or an internal conversion electron , .",
"In this case the rate is reduced by several orders of magnitude due to the additional radiative coupling and half-lives in the order of $10^{29}$ to [$10^{32}$ ]yr have been predicted for various isotopes assuming an effective neutrino mass of [1]eV .",
"However, a potential resonance enhancement for 0$\\nu $$ECEC$ is expected for isotopes in which the final state of the daughter nuclide is energetically degenerated with the ground state of the mother within a few [100]eV.",
"This may lead to an up to $10^6$ times faster rate , , .",
"This paper includes searches for the (radiative) 0$\\nu $$ECEC$ of $^{74}$ Se into the ground state of $^{74}$ Ge and into the $2_1^+$ excited state, as well as for the 2$\\nu $$ECEC$ decay into the two excited states ($2_1^+$ and $2_2^+$ ).",
"An illustration of the different decay modes is shown in Fig.",
"REF .",
"Depending on the shells the electrons are captured from, the energy of the $\\gamma $ -ray emitted in the radiative 0$\\nu $$ECEC$ decay modes can vary by a few keV.",
"The dominant orbital is always the s-orbital in a given shell due to its finite probability of presence at the nucleus.",
"This work investigates several decay modes with captures from the s-orbital of the K and L shells.",
"The energy of the X-rays (or Auger electrons) is 11.10 keV due to a vacancy in the K shell and 1.41 keV due to a vacancy in the L shell .",
"The Coulomb interaction between the two holes slightly changes the energy of the system compared to two individual single electron capture cases.",
"This has been quantitative estimated in Ref.",
"for a variety of double electron capture candidates and is typically well below [1]keV.",
"This effect is well included in the systematic uncertainty of the energy calibration of the Ge-detector.",
"Additionally, the $2^+_2$ state at [1204.205(7)]keV has been considered for a possible resonance enhancement.",
"This decay mode was investigated in the past in Ref.",
", and most recently in Ref. .",
"Unfortunately, a more precise Q-value measurement of [1209.240(7)]keV , seems to disfavor a non-radiative resonant transition and a new half-life of [$5\\cdot 10^{43}$ ]yr with [1]eV effective neutrino mass has been calculated .",
"Recent measurements have set limits of $0.55\\cdot 10^{19}$ yr and $1.5\\cdot 10^{19}$ yr , , though the latter limit seems unrealistic when comparing it to the sensitivity of the experiment, as shown later.",
"The de-excitation of the $2_2^+$ state can follow two branches (Fig.",
"REF ).",
"In the first branch with (68.5$\\pm $ 1.4)% probability two $\\gamma $ -rays with energies of [595.9]keV and [608.4]keV are emitted.",
"In the second branch only one $\\gamma $ -ray with an energy of [1204.3]keV is emitted .",
"Hence, the potential signal from the non-radiative resonant 0$\\nu $$ECEC$ transition is expected to produce three peaks in the energy spectrum that correspond to the three $\\gamma $ -rays.",
"The third peak corresponding to [1204.2]keV $\\gamma $ -ray additionally includes a small contribution of less than 10% from the summation of the [595.9]keV and [608.4]keV $\\gamma $ -rays.",
"Figure: ECECECEC decay scheme of 74 ^{74}Se.",
"The radiative decays are separated into captures from the K and L atomic shells and their combination KK, KL and LL."
],
[
"Experimental Setup and Data",
"The measurement was performed in the Felsenkeller Underground Laboratory in Dresden, Germany, with a shielding of [110]m.w.e.",
"rock overburden reducing the muon flux to [${0.6}\\cdot 10^{-3}$ ]cm$^{-2}$ s$^{-1}$ .",
"A sample of [2503.6]g selenium grains was used which is a large subset of the sample used in the measurement of Ref.",
", .",
"$^{74}$ Se has a natural abundance of 0.89% translating into an isotopic mass of [20.9]g of $^{74}$ Se within the sample.",
"The sample was filled into a standard Marinelli beaker with an inner recess fitting onto the end cap of an ultra low background HPGe detector with a relative efficiency of [90]% routinely used for gamma spectroscopy measurements.",
"A schematic drawing of the arrangement can be seen in Fig.",
"REF .",
"The detector is surrounded by a [5]cm copper shielding embedded in another shielding of [15]cm of low activity lead.",
"The inner [5]cm of the lead shielding has a specific activity of [($2.7 \\pm 0.6$ )]Bq/kg $^{210}$ Pb while the outer [10]cm has [($33 \\pm 0.4$ )]Bq/kg.",
"The spectrometer is located in a measuring chamber which is an additional shielding against radiation from the ambient rock.",
"Furthermore, the detector is constantly held in a nitrogen atmosphere to avoid radon.",
"The data is collected with a 16384 channel MCA from ORTEC recording energies up to [2.8]MeV resulting in a calibration slope of about [0.17]keV/channel.",
"The energy resolution is 0.24% FWHM@[595]keV and 0.15% FWHM@[1204]keV.",
"More details can be found in , .",
"The sample was measured for 35.29 days corresponding to an exposure of [88.35]kg$\\cdot $ d. The efficiency calibration was performed by a mixture of analytically pure SiO$_2$ , the reference materials RGU and RGTh from IAEA and KCl.",
"Those activity standards contain specific activities of [($107.0\\pm 0.3$ )]Bq/kg $^{238}$ U, [($113.0\\pm 1.2$ )]Bq/kg $^{232}$ Th and [($106.9\\pm 2.1$ )]Bq/kg $^{40}$ K respectively.",
"The calibration sources were filled in an identical container as used for the selenium sample.",
"Thus, calibration source and measuring sample differ only in their self absorption behaviour.",
"The full energy detection efficiencies for $\\gamma $ -rays from the selenium sample was determined with Monte Carlo (MC) simulations based on Geant4.",
"The detector geometry was implemented in the code framework MaGe developed for particle propagation at low energies.",
"Events of the signal process were generated using a modified version of Decay0 with updated branching ratios for the $2^+_2$ state and including the angular correlation between the emitted $\\gamma $ -rays.",
"The code was validated with the calibration source in the same geometry as the selenium sample.",
"For $\\gamma $ -lines in the range of [$0.5-1$ ]MeV, the relative difference of detection efficiency between MC and calibration measurement is on average less than 10%.",
"For the resonant decay to the $2^+_2$ state the efficiencies for $\\gamma $ -rays with energies of [595.9]keV and [1204.2]keV were determined as [($1.85 \\pm 0.19$ )]% and [($0.75 \\pm 0.08$ )]% relative to one decay of a source nuclide including the branching ratio and summation effects.",
"Contributions to the uncertainty come from the MC statistics (1%), systematic uncertainties of MC processes and geometries (10%) and the packing density of the selenium sample (3%).",
"The measured energy spectrum can be seen in Fig.",
"REF .",
"The background $\\gamma $ -lines in the regions of interest (ROI) around [595.9]keV and [1204.2]keV are at [609.3]keV and [1238.1]keV from $^{214}$ Bi and at [583.2]keV $^{208}$ Tl.",
"Especially the [609.3]keV background $\\gamma $ -line is very close to the [608.4]keV signal $\\gamma $ -line and can not be separated with the given energy resolution.",
"Hence, this $\\gamma $ -line was not considered for analysis.",
"Figure: Energy spectrum after 35.29 days measuring time.The upper figure shows the spectrum in the full range up to 2.8 MeV.",
"Background γ\\gamma -lines from e + ^+e - ^--annihilation, 60 ^{60}Co, 40 ^{40}K and 208 ^{208}Tl can be identified.",
"In the bottom figures the spectrum is zoomed into the two regions of interest.",
"Background γ\\gamma -lines expected from 214 ^{214}Bi, 208 ^{208}Tl and 60 ^{60}Co are marked with arrows.",
"The dashed lines indicate the position of γ\\gamma -lines expected from the various ECECECEC decay modes of 74 ^{74}Se."
],
[
"Analysis",
"A Bayesian analysis with the help of the Bayesian Analysis Toolkit (BAT) is used to extract information about the signal count expectation for the various decay modes from the spectrum.",
"The analysis is described exemplary for the resonant 0$\\nu $$ECEC$ into the 2$^+_2$ state.",
"Results for the other decay modes are obtained analogously."
],
[
"Fit procedure",
"A combined spectral fit of the energy spectrum in the two regions of interest (ROI) around the [595.9]keV and [1204.3]keV $\\gamma $ -lines is performed.",
"The data is binned with a bin width of about [0.17]keV, which corresponds to the width of the MCA channels.",
"Both fit regions are chosen so that no background peaks are included.",
"The first ROI is defined as [585, 607] keV, limited below by the [583.2]keV peak from $^{208}$ Tl and above by the [609.3]keV peak from $^{214}$ Bi.",
"The second ROI is defined as [1176, 1235] keV, limited below by the [1173.2]keV peak from $^{60}$ Co and above by the [1238.1]keV peak from $^{214}$ Bi.",
"The same ranges are used for the other decay modes.",
"Those choices allow that the spectrum in each ROI can be described by an extended p.d.f.",
"$P(E|\\mathbf {p})$ consisting of a Gaussian signal peak and a linear background component: $P(E|\\mathbf {p}) =\\frac{\\mu _S}{\\sqrt{2\\pi }\\sigma }\\cdot e^{\\left(-\\frac{(E-E_S)^2}{2\\sigma ^2}\\right)}+ b_0 + b_1\\cdot E\\ , $ where $E$ is the energy and $\\mathbf {p}$ is the set of parameters consisting of $E_S$ being the signal peak position, $\\mu _S$ the signal count expectation, $\\sigma $ the energy resolution and $b_0$ and $b_1$ the parameters describing the linear background polynomial.",
"The expectation of the signal counts is connected to the inverse half-life via $\\mu _S = \\ln {2}\\cdot \\frac{N_A\\cdot a_{74}\\cdot M}{m_{\\rm Se}}\\cdot t\\cdot \\epsilon \\cdot T_{1/2}^{-1}\\ ,$ where $N_A$ is Avogadro's constant, $a_{74}$ the abundance of $^{74}$ Se, $m_{\\rm Se}$ the molar mass of natural selenium, $M$ the mass of the selenium sample and $\\epsilon $ the full $\\gamma $ -ray detection efficiency.",
"The probability of the data when combining both ROI is written as the product of Poisson probabilities of each bin: $P(\\mathbf {n}|\\mathbf {p}) =\\prod \\limits _{r=1}^{2} \\prod \\limits _{i=1}^{N_r} \\frac{\\lambda _{r,i}(\\mathbf {p})^{n_{r,i}}}{n_{r,i}!}",
"e^{-\\lambda _{r,i}(\\mathbf {p})}&&$ where $\\mathbf {n}$ denotes the data, $r$ the ROI, $N_r$ the number of bins in ROI $r$ and $n_{r,i}$ the number of counts in bin $i$ of ROI $r$ .",
"The expected number of counts $\\lambda _{r,i}$ in bin $i$ of ROI $r$ is obtained by integrating $P(E|\\mathbf {p})$ over the energy range of the bin.",
"By applying Bayes theorem the posterior probability $P(\\mathbf {p}|\\mathbf {n})$ is obtained which can be reduced to $P(T_{1/2}^{-1}|\\mathbf {n})$ by integrating over the nuisance parameters.",
"Gaussian shaped prior probabilities are used for the efficiency, resolution and the peak positions to account for the systematic uncertainties of those parameters.",
"A flat prior is used for $b_0$ , $b_1$ and $T_{1/2}^{-1}$ whereas the linear slope $b_1$ is restricted to negative values and the inverse half-life $T_{1/2}^{-1}$ to positive i.e.",
"physical values.",
"In case of no signal, 90% credibility lower half-life limits are obtained by the 90% quantile of the marginalized posterior probability density of $T_{1/2}^{-1}$ ."
],
[
"Results",
"The results for all the considered decay modes are compiled in Tab.",
"REF .",
"Additionally, they are discussed in the following."
],
[
"0$\\nu $",
"Three $\\gamma $ -lines are expected from the two de-excitation branches of the $2^+_2$ state (Fig.",
"REF ).",
"The $\\gamma $ -line at [608.4]keV is not considered for the analysis because of the $^{214}$ Bi background $\\gamma $ -line in its proximity.",
"The full energy detection efficiencies including the branching ratio are 1.9% for the [595.9]keV $\\gamma $ -ray and 0.7% for the [1204.2]keV $\\gamma $ -ray.",
"The data shows a slight upward fluctuation of events at the position of the [591.2]keV peak and a downward fluctuation at the position of the [1204.2]keV peak, when compared to the background level.",
"The maximum of the combined posterior probability $P(\\mathbf {p}|\\mathbf {n})$ is found at a value for $T_{1/2}^{-1}$ which corresponds to a signal count expectation of about 8 counts in the first ROI and about 3 counts in the second ROI with background expectations of [$88.3^{+2.3}_{-2.1}$ ]cts/keV and [$31.2^{+0.8}_{-0.7}$ ]cts/keV respectively.",
"However, the zero signal case is included in the smallest 68% interval of $P(T_{1/2}^{-1}|\\mathbf {n})$ which is shown for this decay mode as well as for all other decay modes in Fig.",
"REF .",
"The 90% quantile used to calculate the lower half-life limit is indicated with vertical arrows.",
"A lower half-life limit of [$T_{1/2} > 0.70 \\cdot 10^{19}$ ]yr (90% credibility) is extracted for the 0$\\nu $$ECEC$ resonant decay of $^{74}$ Se.",
"The influence of the systematic uncertainties reduces the limit by about 1.3%.",
"The energy spectrum around the ROI including the best fit and the limit are shown in Fig.",
"REF .",
"Table: Resulting half-life limits for the analyzed decay modes.",
"Also given are the final state and the capture shells as well as the gamma energies expected from each decay.",
"The energies in brackets are not included in the analysis, because they are inseparable from a background line.Figure: Marginalized posterior probability density of the inverse half-life for all decay modes.",
"The 90% quantile of the distribution is indicated with vertical arrows and used to set the lower half-life limit.Figure: Energy spectrum in the both regions of interest for the resonant transition into the 2 2 + 2^+_2 state.",
"Shown is the combined best fit (blue line) and the 90% credibility limit (red line).",
"The color bands show the central 68%, 95% and 99.7% intervals for the Poisson probability of the counts in each bin with the expectation value given by the best fit."
],
[
"0$\\nu $",
"One signal peak in the second ROI at [1196.7]keV for the KL capture and [1206.4]keV for the LL capture is expected from this decay mode.",
"The detection efficiency for this energy is 2.2%.",
"The posterior probability $P(\\mathbf {p}|\\mathbf {n})$ peaks at a value corresponding to about 7 counts and 21 counts respectively.",
"For the LL capture, the zero signal case is outside of the smallest 95% interval, but still inside the 99.7% interval.",
"The probability for the zero signal case is still reasonably high enough so that it can not be rejected.",
"Lower half-life limits of [$0.96\\cdot 10^{19}$ ]yr (KL capture) and [$0.58\\cdot 10^{19}$ ]yr (LL capture) are obtained with 90% credibility.",
"The latter limit is considerably lower because it is negatively affected by the upward fluctuation.",
"Both limits are about a factor 1.5 higher than previous limits of [$0.64\\cdot 10^{19}$ ]yr (KL) and [$0.41\\cdot 10^{19}$ ]yr (LL) ."
],
[
"0$\\nu $",
"This decay mode features a signal $\\gamma $ -line at [595.9]keV from the de-excitation $\\gamma $ -ray of the $2^+_1$ state.",
"Additionally, a second peak from the internal bremsstrahlung $\\gamma $ -ray at [591.2]keV (KK), [600.9]keV (KL capture) and [610.6]keV (LL capture) is expected.",
"The [610.6]keV $\\gamma $ -line is inseparable from the $^{214}$ Bi background $\\gamma $ -line and is not used for the analysis.",
"The detection efficiency for these $\\gamma $ -rays is 2.7%, which is higher than for the $2^+_2$ transitions, mainly because there is no branching from the $2^+_1$ state.",
"The maximum of $P(\\mathbf {p}|\\mathbf {n})$ corresponds to 0 counts for the signal expectation for the KK capture and about 11 counts per peak for both the KL and LL capture.",
"Lower half-life limits of [$1.43\\cdot 10^{19}$ ]yr (KK capture), [$1.03\\cdot 10^{19}$ ]yr (KL) and [$0.82\\cdot 10^{19}$ ]yr (LL) are obtained.",
"They are comparable to the previous limits of [$1.57\\cdot 10^{19}$ ]yr (KK), [$1.12\\cdot 10^{19}$ ]yr (KL) and [$1.30\\cdot 10^{19}$ ]yr (LL) reported in ."
],
[
"2$\\nu $",
"Only one [591.2]keV $\\gamma $ -ray from the de-excitation of the $2^+_1$ state is expected, with a relatively high detection efficiency of 3% due to no branching and no summation effects.",
"The maximum of $P(\\mathbf {p}|\\mathbf {n})$ corresponds to an expectation of about 11 counts in the signal peak hinting at a slight upward fluctuation compared to the background.",
"A lower limit of [$0.92\\cdot 10^{19}$ ]yr could be extracted, which improves the previous best limit of [$0.77\\cdot 10^{19}$ ]yr by 16%."
],
[
"2$\\nu $",
"The results for this decay mode are the same as for the resonant transition to the $2^+_2$ state, due to the identical signature of the de-excitation $\\gamma $ -ray cascade.",
"This also applies for the best previous limit."
],
[
"Discussion",
"The half-life limit for the 0$\\nu $$ECEC$ resonant transition found in this work is lower than the quoted limit of [$1.5\\cdot 10^{19}$ ]yr in Ref. .",
"However, when comparing the experimental parameters, the current work has a substantially lower background level and a higher detection efficiency which indicates a significant improvement of sensitivity compared to Ref. .",
"Even so the statistical extraction of the half-life limit $T_{1/2}$ differs between the two analyses, a simple sensitivity estimates $T_{1/2}^{\\rm sens.",
"}$ of the measurements can be quantitatively compared based on only few experimental parameters.",
"Those parameters are the sample mass $m$ , the measuring time $t$ , the detection efficiency $\\epsilon $ and the background rate $B$ (in units of counts per keV, day and kg sample mass) which are shown in Tab.",
"REF for this work and the recent measurements in Ref.",
", , for the case of the [595.9]keV $\\gamma $ -line.",
"We re-evaluated the $T_{1/2}^{\\rm sens.",
"}$ of these measurements using the definition $T_{1/2}^{\\rm sens.}",
"= R\\cdot \\alpha \\cdot \\epsilon \\cdot \\sqrt{m\\cdot t}/\\sqrt{B\\cdot \\Delta E}$ which is designed to compare different experiments .",
"$R$ is the $\\gamma $ -line branching ratio and $\\Delta E$ is the energy window of the peak window.",
"The isotope specific factor $\\alpha $ is taken as [$\\ln {2}\\cdot N_A \\cdot a_{\\rm Se74} \\cdot /(1.6\\cdot m_{\\rm Se}) = 2.9\\cdot 10^{22}$ ]kg$^{-1}$ with $N_A$ as Avogadro's constant, $a_{\\rm Se74}$ the isotopic abundance of $^{74}$ Se and $m_{\\rm Se}$ the molar mass of natural selenium.",
"The factor of 1.6 is necessary for a confidence level of 90%.",
"$T_{1/2}^{\\rm sens.",
"}$ and the quoted half-life limits are compared in the last two rows of Tab.",
"REF .",
"In Ref.",
"a Bayesian approach is adopted including all three $\\gamma $ -lines for the 0$\\nu $$ECEC$ resonant transition.",
"The re-evaluated $T_{1/2}^{\\rm sens.",
"}$ is in agreement with the published limit.",
"In Ref.",
"the method of setting a limit is not given.",
"The published value is 3 orders of magnitude larger than $T_{1/2}^{\\rm sens.",
"}$ and thus unphysical.",
"The value was corrected in Ref.",
"but remains overestimated by a factor of 30 compared to $T_{1/2}^{\\rm sens.",
"}$ .",
"In Ref.",
"a similar definition of the sensitivity as $T_{1/2}^{\\rm sens.",
"}$ is interpreted as the half-life limit.",
"We find again a factor of 20 discrepancy between the published value and the re-evaluated sensitivity.",
"The published limits cannot be explained by the statistical treatment of the data and are thus doubtful.",
"For this reason we reject the results from Ref.",
", and consider the obtained limit in the present work as experimentally more robust and trustworthy.",
"It is thus the currently most stringent limit for the 0$\\nu $$ECEC$ resonant transition of $^{74}$ Se.",
"Table: References"
]
] | 1605.05976 |
[
[
"Quantum enhanced feedback cooling of a mechanical oscillator"
],
[
"Abstract Laser cooling is a fundamental technique used in primary atomic frequency standards, quantum computers, quantum condensed matter physics and tests of fundamental physics, among other areas.",
"It has been known since the early 1990s that laser cooling can, in principle, be improved by using squeezed light as an electromagnetic reservoir; while quantum feedback control using a squeezed light probe is also predicted to allow improved cooling.",
"Here, we implement quantum feedback control of a micro-mechanical oscillator for the first time with a squeezed probe field.",
"This allows quantum-enhanced feedback cooling with a measurement rate greater than it is possible with classical light, and a consequent reduction in the final oscillator temperature.",
"Our results have significance for future applications in areas ranging from quantum information networks, to quantum-enhanced force and displacement measurements and fundamental tests of macroscopic quantum mechanics."
],
[
"Methods",
"Bright squeezed light at 1064 for tracking the mechanical motion was provided by the nonlinear process of cavity-enhanced parametric down-conversion and injected into the toroidal optical cavity with decay rate $\\kappa / 2 \\pi = {94}{}$ via evanescent coupling using a tapered single-mode fibre.",
"The total on-resonance transmission through the optical fibre and toroid was measured to be 50, mainly limited by scattering losses at the tapered region of the fibre.",
"After interaction with the mechanical mode of the toroid, the phase quadrature of the probe beam was measured by a high efficiency homodyne detector (98), and the output photocurrent was split to provide signals for both characterization of the mechanical motion and feedback.",
"The feedback loop, consisting of bandpass filtering, delay, and amplification, was designed to exert a viscous damping force on a selected mechanical mode.",
"The required velocity dependence of the applied force was controlled by a tunable signal delay.",
"To actuate the mechanical motion of the toroid, an electric field was generated between a grounded aluminium plate supporting the toroid and a sharp electrode with a tip diameter of about 3 [28].",
"A bias voltage of 295 was applied to the electrode to polarize the micro-toroid, in order to enhance the electrical transduction.",
"Following the standard approach [31], the measured and calibrated Lorentzian spectra in Figs.",
"REF a were fitted to a model including the noise squashing effect of in-loop measurements (see Supplementary information).",
"For each measurement, the mechanical resonance frequency, feedback gain, and measurement noise level were used as free parameters.",
"Substituting the fitted values into equation (REF ) the effective mode temperature was calculated."
],
[
"Acknowledgements",
"The work was supported by the Lundbeck Foundation, the Danish Council for Independent Research (Sapere Aude 4184-00338B and 0602-01686B), the Australian Research Council Centre of Excellence CE110001013, and by the Air Force Office of Scientific Research and the Asian Office of Aerospace Research and Development.",
"W.P.B.",
"is supported by the Australian Research Council Future Fellowship FT140100650.",
"Micro-toroid fabrication was undertaken within the Queensland Node of the Australian Nanofabrication Facility."
],
[
"Author contributions",
"The authors declare no competing financial interests.",
"U.L.A.",
"conceived the idea while C.S., H.K., T.G.",
"and U.L.A.",
"devised the experiment.",
"C.S.",
"performed the experiment with help from H.K., H.F., U.B.H., G.I.H., T.G.",
"and J.B. C.S.",
"and U.B.H.",
"analysed the data with input from W.P.B.",
"and U.L.A.",
"C.S., U.B.H., T.G., W.P.B.",
"and U.L.A.",
"wrote the paper.",
"C.S.",
"and U.B.H.",
"wrote the supplemental material with input from H.K.",
"T.G., A.H. and U.L.A.",
"supervised the work."
]
] | 1605.05949 |
[
[
"Observation of $\\pi^- K^+$ and $\\pi^+ K^-$ atoms"
],
[
"Abstract The observation of hydrogen-like $\\pi K$ atoms, consisting of $\\pi^- K^+$ or $\\pi^+ K^-$ mesons, is presented.",
"The atoms have been produced by 24 GeV/$c$ protons from the CERN PS accelerator, interacting with platinum or nickel foil targets.",
"The breakup (ionisation) of $\\pi K$ atoms in the same targets yields characteristic $\\pi K$ pairs, called \"atomic pairs\", with small relative momenta in the pair centre-of-mass system.",
"The upgraded DIRAC experiment has observed $349\\pm62$ such atomic $\\pi K$ pairs, corresponding to a signal of 5.6 standard deviations."
],
[
"Introduction",
"Up to now, the DIRAC collaboration has published indications about the production of $\\pi K$ atomsThe term $\\pi K$ atom or $A_{K \\pi }$ refer to $\\pi ^- K^+$ and $\\pi ^+ K^-$ atoms.",
"[1], [2], [3].",
"This time, DIRAC reports the first statistically significant observation of the strange dimesonic $\\pi K$ atom.",
"Meson-meson interactions at low energy are the simplest hadron-hadron processes and allow to test low-energy QCD, specifically Chiral Perturbation Theory (ChPT) [4], [5], [6], [7].",
"The observation and lifetime measurement of $\\pi ^+\\pi ^-$ atoms (pionium) have been reported in [8], [9], [10].",
"Going one step further, the observation and lifetime measurement of the $\\pi K$ atom involving strangeness provides a direct determination of a basic S-wave $\\pi K$ scattering length difference [11].",
"This atom is an electromagnetically bound $\\pi K$ state with a Bohr radius of $a_{B}$ = 249 fm and a ground state binding energy of $E_{B}$ = 2.9 keV.",
"It decays predominantly by strong interaction into two neutral mesons $\\pi ^0 K^0$ or $\\pi ^0 \\bar{K^0}$ .",
"The atom decay width $\\Gamma _{\\pi K}$ in the ground state (1S) is given by the relation [11], [12]: $\\Gamma _{\\pi K} = \\frac{1}{\\tau } = R (a_{0}^-)^2$ , where $a_{0}^-=\\frac{1}{3}(a_{1/2}-a_{3/2})$ is the S-wave isospin-odd $\\pi K$ scattering length ($a_I$ is the $\\pi K$ scattering length for isospin $I$ ) and $R$ a precisely known factor (relative precision 2%).",
"The scattering length $a_{0}^-$ has been studied in ChPT [13], [14], [15], in the dispersive framework [16] and in lattice QCD (see e.g.",
"[17]).",
"Using $a_{0}^-$ from [16], one predicts for the $\\pi K$ atom lifetime $\\tau = (3.5 \\pm 0.4)\\cdot 10^{-15}~\\text{s}$ .",
"A method to produce and observe hadronic atoms has been developed [18].",
"In the DIRAC experiment, relativistic dimesonic bound states, formed by Coulomb final state interaction (FSI), are moving inside the target and can break up.",
"Particle pairs from breakup (atomic pair in Fig.",
"REF ) are characterised by a small relative momentum $Q <$ 3 MeV/c in the centre-of-mass (c.m.)",
"system of the pairThe quantity $Q$ denotes the experimental c.m.",
"relative momentum.",
"The longitudinal ($Q_L$ ) and transverse $(Q_T~=~\\sqrt{Q^2_X + Q^2_Y})$ components of the vector $\\vec{Q}$ are defined with respect to the direction of the total laboratory pair momentum..",
"Figure: Inclusive πK\\pi K production in24 GeV/c p-Ni interaction:p + Ni →\\rightarrow π - K + \\pi ^- K^+ + X.The ionisation or breakup of A Kπ A_{K \\pi } leads toso-called atomic pairs.",
"(More details, see text in section .",
")A first $\\pi K$ atom investigation has been performed with a platinum target at the CERN PS with 24 GeV/$c$ protons in 2007 [1], [2].",
"An enhancement of $\\pi K$ pairs at low relative momentum has been observed, corresponding to $173 \\pm 54$ $\\pi K$ atomic pairs or a significance of 3.2 standard deviations ($\\sigma $ ).",
"In the experiment from 2008 to 2010, DIRAC has detected in a Ni target an excess of $178\\pm 49$ $\\pi K$ pairs, an effect of only 3.6 $\\sigma $ [3].",
"In the present paper, experimental data obtained in Ni and Pt targets have been analysed, using recorded informations from all detectors (see Fig.",
"REF ) and enhanced background description based on Monte Carlo (MC) simulations.",
"Setup geometry correction, detector response simulation, background suppression and admixture evaluation have been significantly improved for all runs.",
"The above mentioned improvements allow a statistically reliable observation of $\\pi K$ atoms."
],
[
"Experimental setup",
"The setup [19], sketched in Fig.",
"REF , detects and identifies $\\pi ^+ \\pi ^-$ , $\\pi ^- K^+$ and $\\pi ^+ K^-$ pairs with small $Q$ .",
"The structure of these pairs after the magnet is approximately symmetric for $\\pi ^+ \\pi ^-$ and asymmetric for $\\pi K$ .",
"Originating from a bound system, these particles travel with the nearly same velocity, and hence for $\\pi K$ atomic pairs, the kaon momentum is by a factor of about $\\frac{M_{K}}{M_{\\pi }} \\approx 3.5$ larger than the pion momentum ($M_{K}$ is the charged kaon mass and $M_{\\pi }$ the charged $\\pi $ mass).",
"The 2-arm vacuum magnetic spectrometer presented is optimized for simultaneous detection of these pairs [20], [21].",
"The 24 GeV/c primary proton beam, extracted from the CERN PS, hits a ($26\\pm 1$ ) $\\mu $ m thick Pt target in 2007The Pt target maximizes production of atomic pairs.",
"and Ni targets with thicknesses ($98\\pm 1$ ) $\\mu $ m in 2008 and ($108\\pm 1$ ) $\\mu $ m in 2009 and 2010The Ni targets are optimal for lifetime measurement..",
"The radiation thickness of the 98 (108) $\\mu $ m Ni target amounts to about $7\\cdot 10^{-3}$ $X_{0}$ (radiation length).",
"Figure: General view of the DIRAC setup:1 – target station;2 – first shielding wall;3 – microdrift chambers;4 – scintillating fiber detector;5 – ionisation hodoscope;6 – second shielding wall;7 – vacuum tube;8 – spectrometer magnet;9 – vacuum chamber;10 – drift chambers;11 – vertical hodoscope;12 – horizontal hodoscope;13 – aerogel Cherenkov;14 – heavy gas Cherenkov;15 – nitrogen Cherenkov;16 – preshower;17 – muon detector.The secondary channel (solid angle $\\Omega = 1.2 \\cdot 10^{-3}$ sr) together with the whole setup is vertically inclined relative to the proton beam by $5.7^\\circ $ upward.",
"Secondary particles are confined by the rectangular beam collimator inside of the second steel shielding wall, and the angular divergence in the horizontal (X) and vertical (Y) planes is $\\pm 1^\\circ $ .",
"With a spill duration of 450 ms, the beam intensity has been (10.5–12)$\\cdot 10^{10}$ protons/spill and, correspondingly, the single counting rate in one plane of the ionisation hodoscope (IH) (5–6)$\\cdot 10^6$ particles/spill.",
"Secondary particles propagate mainly in vacuum up to the Al foil at the exit of the vacuum chamber, which is located between the poles of the dipole magnet ($B_{max}$ = 1.65 T and $BL$ = 2.2 T$\\cdot $ m).",
"In the vacuum gap, MicroDrift Chambers (MDC) with 18 planes and a Scintillating Fiber Detector (SFD) with 3 planes X, Y and U, inclined by $45^\\circ $ , have been installed to measure particle coordinates ($\\sigma _{SFDx} = \\sigma _{SFDy} = 60~\\mu $ m, $\\sigma _{SFDu} = 120~\\mu $ m) and particle time ($\\sigma _{tSFDx} = 380$ ps, $\\sigma _{tSFDy} = \\sigma _{tSFDu} = 520$ ps).",
"The four IH planes serve to identify unresolved double track events with only one hit in SFD.",
"Each spectrometer arm is equipped with the following subdetectors: drift chambers (DC) to measure particle coordinates with about 85 $\\mu $ m precision; vertical hodoscope (VH) to measure time with 110 ps accuracy for particle identification via time-of-flight determination; horizontal hodoscope (HH) to select pairs with vertical separation less than 75 mm between the arms ($Q_{Y}$ less than 15 MeV/c); aerogel Cherenkov counter (ChA) to distinguish kaons from protons; heavy gas ($C_{4}F_{10}$ ) Cherenkov counter (ChF) to distinguish pions from kaons; nitrogen Cherenkov (ChN) and preshower (PSh) detector to identify $\\mathrm {e}^+\\mathrm {e}^-$ pairs; iron absorber and two-layer scintillation counter (Mu) to identify muons.",
"In the “negative” arm, an aerogel counter has not been installed, because the number of antiprotons is small compared to $K^{-}$ .",
"Pairs of oppositely charged time-correlated particles (prompt pairs) and accidentals in the time interval $\\pm 20$ ns are selected by requiring a 2-arm coincidence (ChN in anticoincidence) with a coplanarity restriction (HH) in the first-level trigger.",
"The second-level trigger selects events with at least one track in each arm by exploiting DC-wire information (track finder).",
"Using track information, the online trigger selects $\\pi \\pi $ and $\\pi K$ pairs with relative momenta $|Q_X| < 12~\\rm {MeV}/c$ and $|Q_L| < 30~\\rm {MeV}/c$ .",
"The trigger efficiency is about 98% for pairs with $|Q_X| < 6~\\rm {MeV}/c$ , $|Q_Y| < 4~\\rm {MeV}/c$ and $|Q_L| < 28~\\rm {MeV}/c$ .",
"Particle pairs $\\pi ^-{\\mathrm {p}}$ ($\\pi ^+\\bar{\\mathrm {p}}$ ) from $\\Lambda $ ($\\bar{\\Lambda }$ ) decay have been used for spectrometer calibration and $\\mathrm {e}^+\\mathrm {e}^-$ pairs for general detector calibration."
],
[
"Production of bound and free $\\pi ^- K^+$ and {{formula:27315296-61e8-4759-8051-9fe7fe0e4e6e}} pairs",
"Prompt $\\pi ^{\\mp }K^{\\pm }$ pairs from proton-nucleus collisions are produced either directly or originate from short-lived (e.g.",
"$\\Delta $ , $\\rho $ ), medium-lived (e.g.",
"$\\omega $ , $\\phi $ ) or long-lived (e.g.",
"$\\eta ^{\\prime }$ , $\\eta $ ) sources.",
"Pion-kaon pairs produced directly, from short- or medium-lived sources, undergo Coulomb FSI resulting in unbound states (Coulomb pair in Fig.",
"REF ) or forming bound states ($A_{K \\pi }$ in Fig.",
"REF ).",
"Pairs from long-lived sources are practically not affected by Coulomb interaction (non-Coulomb pair in Fig.",
"REF ).",
"The accidental pairs are generated via different proton-nucleus interactions.",
"The cross-section of $\\pi K$ atom production is given by the expression [18]: $\\frac{{\\rm d}\\sigma ^{n}_A}{{\\rm d}\\vec{p}_A}=(2\\pi )^3\\frac{E_A}{M_A}\\left.\\frac{{\\rm d}^2\\sigma ^0_s}{{\\rm d}\\vec{p}_K {\\rm d}\\vec{p}_\\pi }\\right|_{\\frac{\\vec{p}_K}{M_{K}} \\approx \\frac{\\vec{p}_\\pi }{M_{\\pi }}}\\hspace{-2.84526pt} \\cdot \\left|\\psi _{n}(0)\\right|^2 \\:,$ where $\\vec{p}_{A}$ , $E_{A}$ and $M_{A}$ are the momentum, total energy and mass of the $\\pi K$ atom in the laboratory (lab) system, respectively, and $\\vec{p}_K$ and $\\vec{p}_\\pi $ the momenta of the charged kaon and pion with equal velocities.",
"Therefore, these momenta obey in good approximation the relations $\\vec{p}_{K}=\\frac{M_{K}}{M_{A}} \\vec{p}_{A}$ and $\\vec{p}_{\\pi }=\\frac{M_{\\pi }}{M_{A}} \\vec{p}_{A}$ .",
"The inclusive production cross-section of $\\pi K$ pairs from short-lived sources without FSI is denoted by $\\sigma _s^0$ , and $\\psi _{n}(0)$ is the $S$ -state Coulomb atom wave function at the origin with principal quantum number $n$ .",
"According to (REF ), $\\pi K$ atoms are only produced in $S$ -states with probabilities $W_n=\\frac{W_1}{n^3}$ : $W_1=83.2\\%$ , $W_2=10.4\\%$ , $W_3=3.1\\%$ , $W_{n>3}=3.3\\%$ .",
"In complete analogy, the $\\pi ^{\\mp }K^{\\pm }$ Coulomb pair production is described in the point-like production approximation, depending on the relative momentum $q$ in the production pointThe quantity $q$ denotes the original c.m.",
"relative momentum.",
": $\\frac{{\\rm d}^2\\sigma _C}{{\\rm d}\\vec{p}_K {\\rm d}\\vec{p}_\\pi } =\\frac{{\\rm d}^2\\sigma ^0_s}{{\\rm d}\\vec{p}_K {\\rm d}\\vec{p}_\\pi }\\hspace{-5.69054pt} \\cdot A_C(q)\\quad \\mbox{with} \\quad A_C(q) = \\frac{4\\pi \\mu \\alpha /q}{1-\\exp \\left(-4\\pi \\mu \\alpha /q\\right) } \\;.$ The Coulomb enhancement function $A_C(q)$ is the well-known Sommerfeld-Gamov-Sakharov factor [22], [23], [24], $\\mu $ = 109 $\\rm {MeV/c^2}$ the reduced mass of the $\\pi ^{\\mp }K^{\\pm }$ system and $\\alpha $ the fine structure constant.",
"The relative production yield of atoms to Coulomb pairs [25] is calculated from the ratio (REF ) to (REF ).",
"For $\\pi $ and $K$ production from non-pointlike medium-lived sources, corrections at the percent level have been applied to the production cross-sections [26].",
"Strong final state elastic and inelastic interactions are negligible [26]."
],
[
"Data processing",
"Recorded events have been reconstructed with the DIRAC $\\pi \\pi $ analysis software (ARIANE) modified for analysing $\\pi K$ data."
],
[
"Tracking and setup tuning",
"Only events with one or two particle tracks in the DC detector of each arm are processed.",
"Event reconstruction is performed according to the following steps: 1) One or two hadron tracks are identified in the DC of each arm with hits in VH, HH and PSh slabs and no signal in ChN and Mu (Fig.",
"REF and related text).",
"The earliest track in each arm is used for further analysis, because these tracks induce the trigger signal starting the readout procedure.",
"2) Track segments, reconstructed in DC, are extrapolated backward to the incident proton beam position in the target, using the transfer function of the DIRAC dipole magnet.",
"This procedure provides approximate particle momenta and corresponding intersection points in MDC, SFD and IH.",
"3) Hits are searched for around the expected SFD coordinates in the region $\\pm 1$ cm, corresponding to 3–5 $\\sigma $ defined by the position accuracy, taking into account particle momenta.",
"This way, events are selected with low and medium background defined by the following criteria: the number of hits around the two tracks is $\\le 4$ in each SFD plane and $\\le 9$ in all three SFD planes.",
"The case of only one hit in the region $\\pm 1$ cm can occur because of detector inefficiency (two crossing particles, but one is not detected) or if two particles cross the same SFD column.",
"The latter event type can be regained by double ionisation selection in the corresponding slab of the IH.",
"For data collected in 2007 with the Pt target, criteria are different: the number of hits is two in the $Y$ - and $U$ -plane (SFD $X$ -plane and IH, which may resolve crossing of only one SFD column by two particles, have not been used in 2007).",
"The momentum of the positively or negatively charged particle is refined to match the $X$ -coordinates of the tracks in DC as well as the SFD hits in the $X$ - or $U$ -plane, depending on presence of hits.",
"In order to find the best two-track combination, the two tracks may not use a common SFD hit in case of more than one hit in the proper region.",
"In the final analysis, the combination with the best $\\chi ^2$ in the other SFD planes is kept.",
"In order to improve the mechanical alignment and general description of the setup geometry, the $\\Lambda $ and $\\bar{\\Lambda }$ particle decays into ${\\rm p}\\pi ^-$ and $\\pi ^+\\bar{\\rm p}$ are exploited [28], [27], [29].",
"By requiring the mass equality $M^{exp}_{\\Lambda } = M^{exp}_{\\bar{\\Lambda }}$ , the angles of the DC axes are modified.",
"In the next step, the obtained angle between the DC axes is tuned to get the PDG (Particle Data group) reference $\\Lambda $ mass: the survey value of this angle needs to be increased by a few $10^{-4}$ rad.",
"For the data set 2007–2010, the weighted average of the experimental $\\Lambda $ mass values is $M^{exp}_{\\Lambda } = (1.115680 \\pm 2.9 \\cdot 10^{-6}) \\rm {GeV}/c^2$ , in agreement with the PDG value $M^{PDG}_{\\Lambda } = (1.115683 \\pm 6 \\cdot 10^{-6}) \\rm {GeV}/c^2$ [30].",
"This confirms consistency of the setup alignment.",
"The $\\Lambda $ mass width in the simulated distribution tests how well the MC simulation reproduces the momentum and angle resolution of the setup.",
"Data of each year has been investigated which simulated distribution – with different widths – fits best the experimental $\\Lambda $ distribution.",
"Simulated $\\Lambda $ distributions providing a better $\\chi ^2$ fit to the data are created with a width increased by the following factors: $1.027 \\pm 0.003$ in 2007 (two SFD planes), while this increase in the subsequent years (three SFD planes) is not significant: $1.002 \\pm 0.004$ (2008), $1.001 \\pm 0.003$ (2009) and $1.003 \\pm 0.003$ (2010).",
"The difference between data and MC width could be the consequence of an imperfect description of the setup downstream part and can be removed by introducing a Gaussian smearing of the reconstructed momenta [3].",
"This technique is also used to evaluate the systematic error connected with reconstructed momentum smearing.",
"Taking into account momentum smearing, the momentum resolution has been evaluated as $ \\frac{dp}{p} = \\frac{p_{gen} -p_{rec} }{p_{gen}}$ with $p_{gen}$ and $p_{rec}$ the generated and reconstructed momenta, respectively.",
"Between 1.5 and 8 GeV/c, particle momenta are reconstructed with a relative precision from $2.8\\cdot 10^{-3}$ to $4.4\\cdot 10^{-3}$ [27].",
"Relative momentum resolutions after the target are: $\\sigma _{QX} \\approx \\sigma _{QY} \\approx 0.36~\\rm {MeV}/c$ , $\\sigma _{QL} \\approx 0.94~{\\rm MeV}/c$ for $p_{\\pi K} = p_{\\pi }+p_K = 5$ GeV/c and about 6% worse values for $p_{\\pi K} = 7.5$ GeV/c."
],
[
"Event selection",
"Selected events are classified into three categories: $\\pi ^-K^+$ , $\\pi ^+K^-$ and $\\pi ^-\\pi ^+$ .",
"The last category is used for calibration purposes.",
"Pairs of $\\pi K$ are cleaned of $\\pi ^-\\pi ^+$ and $\\pi ^-{\\rm p}$ background by the Cherenkov counters ChF and ChA.",
"In the momentum range from 3.8 to 7 GeV/c, pions are detected by ChF with (95–97)% efficiency [31], whereas kaons and protons (antiprotons) do not produce a signal.",
"The admixture of $\\pi ^-{\\rm p}$ pairs is suppressed by the aerogel Cherenkov detector (ChA), which records kaons but not protons [32].",
"By requiring a signal in ChA and selecting compatible time-of-flights (TOF) between the target and VH, $\\pi ^-{\\rm p}$ and $\\pi ^-\\pi ^+$ pairs, contaminating $\\pi ^-K^+$ , can be substantially suppressed.",
"Correspondingly, the admixture of $\\pi ^+\\pi ^-$ pairs to $\\pi ^+K^-$ has also been taken into account.",
"Fig.",
"REF shows, after applying the selection criteria, the well-defined $\\pi ^-K^+$ Coulomb peak at $Q_L=0$ and the strongly suppressed peak from $\\Lambda $ decays at $Q_L=-30$ MeV/c.",
"The $Q_L$ distribution of $\\pi ^+K^-$ pairs is similar [3].",
"Figure: Q L Q_L distributions of potential π - K + \\pi ^-K^+ pairsbefore (a) and after (b) applying the selection described in the text.",
"Events withpositive Q L Q_L are suppressed compared to those with negative Q L Q_Ldue to lower acceptance and lower production cross-section.",
"The final analysis sample contains only events which fulfil the following criteria: $Q_T < 4~{\\rm MeV/c} \\, , \\, |Q_L| < 20~{\\rm MeV/c} \\, .$ Due to finite detector efficiency, a certain admixture of misidentified pairs still remains in the experimental distributions.",
"Their contribution has been estimated by TOF investigation and accordingly been subtracted [33].",
"Under the assumption that all positively charged particles are $K^+$ , Fig.",
"REF compares the experimental with the simulated TOF difference distribution for $\\pi ^-K^+$ , $\\pi ^-\\pi ^+$ and $\\pi ^-{\\rm p}$ pairs.",
"Two ranges for positively charged particle momenta, (4.4–4.5) and (5.4–5.5) GeV/$c$ , have been investigated.",
"Figure: Distributions over time-of-flight difference forevents with positively charged particle momenta in the intervals:a) (4.4–4.5) GeV/cc;b) (5.4–5.5) GeV/cc.Experimental data (histogram) are fitted by the sum of the distributions:K + π - K^+\\pi ^- (red, dashed),π + π - \\pi ^+\\pi ^- (blue, dotted),pπ - {\\mathrm {p}}\\pi ^- (magenta, dotted-dashed) andaccidental pairs (green, constant).The sum of all the fractions is shown as black solid line."
],
[
"Data simulation",
"Since the $\\pi K$ data samples consist of Coulomb, non-Coulomb and atomic pairs, these event types have been generated by MC (DIPGEN [34], GEANT-DIRAC (setup simulator)).",
"The MC sample exceeds ten times the number of experimental events.",
"The events are characterised by different $q$ distributions: the non-Coulomb pairs are distributed in accordance with phase space, while the $q$ distribution of Coulomb pairs is modified by the factor $A_C(q)$ (REF ).",
"For atomic pairs, one needs to know the breakup position and the lab momentum of each pair.",
"In practice, lab momenta for MC events are generated in accordance with analytic formulae, resembling the experimental momentum distributions of such pairs [34], [35].",
"After comparing experimental momentum spectra [33] with MC distributions reconstructed by the analysis software, their ratio is used as event-by-event weight function for MC events in order to provide the same lab momentum spectra for simulated as for experimental data.",
"The breakup point, from which the ionisation occurred, the quantum numbers of the atomic state and the corresponding $q$ distribution of the atomic pair are obtained by solving numerically transport equations [36] using total and transition cross-sections [37].",
"The lab momenta of the atoms are assumed, in accordance with equation (REF ), to be the same as for Coulomb pairs.",
"The description of the charged particle propagation through the setup takes into account: a) multiple scattering in the target, detector planes and setup partitions, b) response of all detectors, c) additional momentum smearing and d) results of the SFD response analysis [33], [38], [39] with influence on the $Q_T$ resolution."
],
[
"Data analysis",
"In the analysis of $\\pi K$ data, the experimental 1-dimensional distributions of relative momentum $Q$ and $|Q_L|$ and the 2-dimensional distributions ($|Q_L|$ , $Q_T$ ) have been fitted for each year and each $\\pi K$ charge combination by simulated distributions of atomic, Coulomb and non-Coulomb pairs.",
"Their corresponding numbers $n_A$ , $N_C$ and $N_{nC}$ are free fit parameters.",
"The sum of these parameters is equal to the number of analysed events.",
"The experimental and simulated $Q$ distributions of $\\pi ^- K^+$ and $\\pi ^+ K^-$ pairs are shown in Fig.",
"REF (top) for all events with $Q_T<4$ MeV/$c$ and $|Q_L|<20$ MeV/$c$ .",
"One observes an excess of events above the sum of Coulomb and non-Coulomb pairs in the low $Q$ region, where atomic pairs are expected.",
"After background subtraction there is a signal at the level of 5.7 standard deviations, shown in Fig.",
"REF (bottom): $n_A = 349 \\pm 61$ ($\\chi ^2/n = 41/37$ , $n=$ number of degrees of freedom), see Table REF .",
"The signal shape is described by the simulated distribution of atomic pairs.",
"The numbers of atomic pairs, produced in the Ni and Pt targets, are $n_A(\\mathrm {Ni})=275\\pm 57$ ($\\chi ^2/n = 40/37$ ) and $n_A(\\mathrm {Pt})=73\\pm 22$ ($\\chi ^2/n = 40/36$ ), respectively.",
"The same analysis has been performed for all $\\pi ^- K^+$ and $\\pi ^+ K^-$ pairs separately as presented in Fig.",
"REF and Fig.",
"REF .",
"The $\\pi ^- K^+$ and $\\pi ^+ K^-$ atomic pair numbers are $n_A=243\\pm 51$ ($\\chi ^2/n = 36/37$ ) and $n_A=106\\pm 32$ ($\\chi ^2/n = 42/37$ ), respectively.",
"The experimental ratio, $2.3\\pm 0.9$ , between the two types of atom production is compatible with the ratio $2.4$ as calculated using FRITIOF [40].",
"Figure: Top: QQ distribution of experimentalπ - K + \\pi ^-K^+ and π + K - \\pi ^+K^- pairs fitted by the sum of simulated distributionsof atomic, Coulomb and non-Coulomb pairs.Atomic pairs are shown in red (dotted-dashed) andfree pairs (Coulomb in blue (dashed) and non-Coulomb in magenta (dotted))in black (solid).Bottom: Difference distribution between experimental andsimulated free pair distributions compared with simulated atomic pairs.The number of observed atomic pairs is denoted by n A n_A.Figure: Same distributions as in Fig.",
",but only for π - K + \\pi ^-K^+ pairs.Figure: Same distributions as in Fig.",
",but only for π + K - \\pi ^+K^- pairs.In the 2-dimensional ($|Q_L|$ ,$Q_T$ ) analysis, all experimental data in the same $|Q_L|$ and $Q_T$ intervals have been analysed using simulated 2-dimensional distributions.",
"The evaluated atomic pair number, $n_A=314\\pm 59$ ($\\chi ^2/n = 237/157$ ), corresponds to 5.3 standard deviations and coincides with the previous analysis result.",
"In Table REF , the results of the three analysis types (Ni and Pt target together) are presented for each atom type and combined.",
"There is a good agreement between the results of the $Q$ and ($|Q_L|$ ,$Q_T$ ) analyses.",
"The 1-dimensional $|Q_L|$ analysis for all $\\pi K$ data yields $n_A=230\\pm 92$ ($\\chi ^2/n = 52/37$ ), which does not contradict the values obtained in the other two statistically more precise analyses.",
"Compared to the previous investigation [1], in the present work the Pt data has been analysed including upstream detectors.",
"The consequence is a decrease of the statistics, but on the other hand an increase of the $Q_T$ resolution.",
"This better resolution improves the data quality.",
"Concerning the Ni target, the increase of $n_A$ , compared to [3], is caused by optimizing the time-of-flight criteria, which decreases atomic pair losses for the same fraction of background in the final distributions.",
"Table: Atomic pair numbers n A n_A by analysingthe 1-dimensional QQ and |Q L ||Q_L| distributions andthe 2-dimensional (|Q L ||Q_L|,Q T Q_T) distribution.Only statistical errors are given."
],
[
"Systematic errors",
"The evaluation of the atomic pair number $n_A$ is affected by several sources of systematic errors [29], [33].",
"Most of them are induced by imperfections in the simulation of the different $\\pi K$ pairs (atomic, Coulomb, non-Coulomb) and misidentified pairs.",
"Shape differences of experimental and simulated distributions in the fit procedure (section ) lead to biases of parameters, including atomic pair contribution, and finally on $n_A$ .",
"The influence of systematic error sources is different for the analyses of $Q$ , ($|Q_L|,Q_T$ ) and $Q_L$ distributions.",
"Table REF shows systematic errors induced by different sources.",
"Table: Systematic errors in the number n A n_A of πK\\pi K atomic pairs."
],
[
"Conclusion",
"In the dedicated experiment DIRAC at CERN, the dimesonic Coulomb bound states involving strangeness, $\\pi ^- K^+$ and $\\pi ^+ K^-$ atoms, have been observed for the first time with reliable statistics.",
"These atoms are generated by a 24 GeV/$c$ proton beam, hitting Pt and Ni targets.",
"In the same targets, a fraction of the produced atoms breaks up, leading to $\\pi ^- K^+$ and $\\pi ^+ K^-$ atomic pairs with small relative c.m.",
"momenta $Q$ .",
"The 1-dimensional $\\pi ^{\\mp }K^{\\pm }$ analysis in $Q$ yields $349\\pm 61(stat)\\pm 9(syst)=349\\pm 62(tot)$ atomic pairs (5.6 standard deviations) for both charge combinations.",
"Analogously, a 2-dimensional analysis in ($|Q_L|$ ,$Q_T$ ) has been performed with the result of $314\\pm 59(stat)\\pm 10(syst)=314\\pm 60(tot)$ atomic pairs (5.2 standard deviations), in agreement with the former number.",
"The resulting $\\pi K$ atom lifetime and $\\pi K$ scattering length from the ongoing analysis will be presented in a separate paper."
],
[
"Acknowledgements",
"We are grateful to R. Steerenberg and the CERN-PS crew for the delivery of a high quality proton beam and the permanent effort to improve the beam characteristics.",
"The project DIRAC has been supported by the CERN and JINR administration, Ministry of Education and Youth of the Czech Republic by project LG130131, the Istituto Nazionale di Fisica Nucleare and the University of Messina (Italy), the Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science, the Ministry of Education and Research (Romania), the Ministry of Education and Science of the Russian Federation and Russian Foundation for Basic Research, the Dirección Xeral de Investigación, Desenvolvemento e Innovación, Xunta de Galicia (Spain) and the Swiss National Science Foundation."
]
] | 1605.06103 |
[
[
"Automatic TM Cleaning through MT and POS Tagging: Autodesk's Submission\n to the NLP4TM 2016 Shared Task"
],
[
"Abstract We describe a machine learning based method to identify incorrect entries in translation memories.",
"It extends previous work by Barbu (2015) through incorporating recall-based machine translation and part-of-speech-tagging features.",
"Our system ranked first in the Binary Classification (II) task for two out of three language pairs: English-Italian and English-Spanish."
],
[
"Introduction",
"Autodesk has accumulated more than 40 million professionally translated segments over the past 17 years.",
"These translation units (TUs) mainly stem from user interfaces and documentation of software products localized into 32 languages.",
"As we are now unifying and centralizing all translations in a single repository, it is high time to sort out duplicate, outdated, and erroneous TUs.",
"Exploring methods to handle the latter – clearly more challenging than removing duplicate and outdated material – motivated us to participate in the First Shared Task on Translation Memory Cleaning [2].",
"Going forward, we strive to make human translation more efficient (by showing translators less erroneous fuzzy matches) and machine translation more accurate (by reducing noise in training data).",
"In this paper, we describe our submitted system for distinguishing correct from incorrect TUs.",
"Rather than tailoring it to individual languages, we aimed at a language-independent solution to cover all of the language pairs in this shared task or, looking to the future, Autodesk's production environments.",
"The system is based on previous work by [1] and uses language-independent features with language-specific plug-ins, such as machine translation, part-of-speech tagging, and language classification.",
"Specifics about previous work are given in the next section.",
"In Section [sec:Method]sec:Method1, we describe our method and, in Section [sec:Results]sec:Results1, show how it compares to [1]'s ([1]) approach as well as other submissions to this shared task.",
"Lastly, we offer preliminary conclusions and outline future work in Section [sec:Conclusions]sec:Conclusions1."
],
[
"Background",
"TM cleaning functionality in commercial tools is mostly rule-based, centering around the removal of duplicate entries, ensuring markup validity (e.g., no unclosed tags), or controlling for client or project specific terminology .",
"Although helpful, these methods fall short of identifying spurious entries that contain language errors or partial translations.",
"With crowd-sourced and automatically constructed TMs in particular, it is also necessary to identify translation units with source and target segments that do not correspond at all [11], [10].",
"[1] has proposed to cast the identification of such incorrect translations as a supervised classification problem.",
"In his work, 1,243 labelled TUs were used to train binary classifiers based on 17 features.",
"The most important of them, according to the author, were bisegment_similarity and lang_diff: the former is defined as the cosine similarity between a target segment and its machine translated source segment, while the latter denotes whether the language codes declared in a translation unit correspond with the codes detected by a language detector.",
"The best classifier, a support vector machine with linear kernel, achieved 82% precision and 81% recall on a held-out test set of 309 TUs.",
"To the best of our knowledge, [1] provided the first and so far only research contribution on automatic TM cleaning, which the author himself described as a neglected research area [1].",
"With our participation to this shared task, we seek to extend his work by examining new features based on statistical MT and POS tagging.",
"As outlined above, comparing machine translated source segments to their actual target segments has proven effective in [1]'s ([1]) experiments.",
"We propose to complement or replace the similarity function used for this comparison (cosine similarity) by two automatic MT evaluation metrics, Bleu [5] and character-based Levenshtein distance, in order to reward higher-order $n$ -gram ($n > 1$ ) and partial word overlaps, respectively.",
"Furthermore, we introduce a recall-based MT feature that takes multiple MT hypotheses ($n$ -best translations) of a given source segment into account, based on the assumption that alternative translations of words (such as buy and purchase) or phrases (such as despite and in spite of) should not be punished.",
"We also experiment with part-of-speech information to identify spurious translation units.",
"With closely related languages in particular, the rationale would be that adjectives (to name an example) in a source segment are likely to be reflected in the corresponding target segment in case of a valid translation.",
"The comparison of POS tags from language-specific tagsets will be based on a mapping to eleven coarse-grained, language-independent grammatical groups [7].",
"We acknowledge that the use of MT is discouraged by the organizers of this shared task to foster contributions that require less compute power.",
"However, as MT was found to be valuable in previous work (see above) and computational resources are hardly a limiting factor in corporate environments (see Section [sec:Resources]sec:Resources1), we decided not to refrain from including MT-based features in our submissions."
],
[
"Method",
"Our system uses labelled TUs to train classifiers based on language-independent features (see Section [sec:Features]sec:Features1) with language-specific plug-ins (see Section [sec:Resources]sec:Resources1).",
"The feature extraction pipeline is implemented in Scala (see Section [sec:Classification]sec:Classification1), and our final submission – geared to distinguish correct or almost correct from incorrect TUs – is based on a selection of nine features (see Section [sec:FeatureSelection]sec:FeatureSelection1)."
],
[
"Features",
"We re-implemented the 17 features proposed by [1].",
"In addition, we explore mt_coverage the percentage of target words contained in the $n$ -best machine translations of the source segment.",
"We use $n=20$ in our experiments.",
"mt_cfs the character-based Levenshtein distance between target segment and machine translated source segment.",
"We normalise this score such that identical and completely dissimilar segments result in scores of 1.0 and 0.0 respectively, i.e., $\\nonumber \\textit {cfs} = 1 - \\frac{\\text{Levensthein distance in characters}}{\\text{number of characters in longer segment.",
"}}$ This score is computed individually for each of the 20-best translation options; the best of these scores instantiates the feature value.",
"mt_bleu the BLEU score [5] between target segment and machine translated source segment.",
"We employ the sentence-level version of the metric as implemented in Phrasal [3].",
"As with mt_cfs, individual scores are computed for each of the 20-best translation options; the best score instantiates the feature value.",
"pos_sim_all the cosine similarity between the part-of-speech (POS) tags found in the source and target segment.",
"pos_sim_some the cosine similarity between source and target segment in terms of nouns (NOUN), verbs (VERB), adjectives (ADJ), and pronouns (PRON).",
"pos_exact whether or not the POS tag sequence in source and target segment is identical.",
"language_detection whether or not a state-of-the-art language classifier confirms the target segment's language declared in the translation unit.",
"ratio_words the ratio between number of words in source and target segment.",
"ratio_chars the ratio between number of characters in source and target segment."
],
[
"Resources",
"Some of the features described in the previous section require natural language processing (NLP) facilities.",
"For machine translation, we use our in-house systems [8], [13] based on the Moses SMT framework [4].",
"They are trained on translated software and user manuals from Autodesk products only and chosen for the sake of convenience; we would expect better performance of our MT-based features in conjunction with MT engines geared to the text domains used in this shared task (listed in Table REF ).",
"Our engines are integrated into a scalable infrastructure deployed on an elastic compute cloud, allowing high throughput even with large translation memories to be cleaned.",
"For POS tagging, we rely on [9]'s ([9]) TreeTagger and its readily available modelshttp://www.cis.uni-muenchen.de/~schmid/tools/TreeTagger/ for English, German, Italian, and Spanish.",
"To make POS tags comparable across these languages, they are mappedhttps://github.com/slavpetrov/universal-pos-tags to the Universal Tagset proposed by [7].",
"Lastly, we use the publicly available Xerox Language Identifier APIhttps://open.xerox.com/Services/LanguageIdentifier for language detection."
],
[
"Classification",
"Our feature extraction pipeline, including [1]'s ([1]) as well as our own features (see Section [sec:Features]sec:Features1), is implemented in Scala.",
"This pipeline is used to transform translation units into feature vectors and train classifiers using the scikit-learn framework [6].",
"From the various classification algorithms we tested, Random Forests performed best with our selection of features (see below)."
],
[
"Feature Selection",
"For the reasons mentioned in Section [sec:Introduction]sec:Introduction1, we aimed at finding a combination of features that would perform well with all language pairs rather than tailoring solutions to individual languages.",
"We focused on gearing our classifiers to distinguish correct or almost correct (classes 1, 2) from incorrect TUs (class 3) – i.e., the Binary Classification (II) task – by optimising the weighted F$_1$ -score (F$_1$ ) on training data (see Tables REF and REF ).",
"From the various feature combinations we tested, we found the following to be most successful: ratio_words, pos_sim_all, language_detection, mt_cfs, mt_bleu, ratio_chars (as described in Section [sec:Features]sec:Features1), alongside cg_score, only_capletters_dif, and punctuation_similarity [1].",
"Evaluation results are given in the next section."
],
[
"Results",
"We tested our final submission – a Random Forests classifier based on the nine features described in Section [sec:FeatureSelection]sec:FeatureSelection1 – on three language pairs (en–de, en–es, en–it) and two tasks: Binary II and Fine-Grained Classification (see Sections [sec:BinaryClassification]sec:BinaryClassification1 and [sec:FineGrainedClassification]sec:FineGrainedClassification1, respectively).",
"The classifier was trained solely on data provided by the organizers of this shared task for each of the language $\\times $ task conditions.",
"Each TU in this data was annotated with one of three labels: correct, almost correct, and incorrect (see Table REF )."
],
[
"Binary Classification (II)",
"Our rationale for focusing on telling apart correct or almost correct from incorrect TUs was that a first application of our method, if successful, would most likely be the filtering of TM data for MT training.",
"While eliminating almost correct TUs might decrease rather than increase MT quality, filtering out incorrect segments can have a positive impact [12].",
"Prior to submission, we benchmarked our system against the two baselines provided by the organizers: a dummy classifier assigning random classes according to the overall class distribution in the training data (Baseline 1), and a classifier based on the Church-Gale algorithm as adapted by [1] (Baseline 2).",
"More importantly, however, we compared our system to [1]'s ([1]) approach, using the classification algorithms which reportedly worked best with the 17 features in his work.",
"Our system performed well in this comparison, surpassing [1]'s approach in all language pairs except en–de, where both systems were en par.",
"Details are shown in Table REF , where we report weighted precision (P), recall (R), and F$_1$ -scores averaged over 5-fold cross-validation with 23–13 splits of the training data.",
"The final evaluation and ranking produced by the organizers, shown in Table REF , confirms our findings from experimenting with training data: our system performs well on the en–es and en–it test sets (best in class), while performance is substantially lower on the en–de test set.",
"The reasons for this are yet to be ascertained (see also Section [sec:Conclusions]sec:Conclusions1)."
],
[
"Fine-Grained Classification",
"Although geared to the Binary Classification (II) task (see above), we also assessed our system on the Fine-Grained Classification task.",
"Here, the goal was to distinguish between all of the three classes, i.e., determine whether a TU is correct, almost correct, or incorrect.",
"Again, we compared our system's performance to [1]'s ([1]) method, using 23–13 splits of the training data (5-fold cross-validation).",
"The results, shown in Table REF , implied that the nine features we selected would not suffice for a more fine-grained classification of TUs.",
"This was confirmed in the official evaluation and ranking: our system scored low on en–de and mediocre on en–es and en–it.",
"Further work will be needed to analyse these results in more detail."
],
[
"Conclusions",
"We have proposed a machine learning based method to identify incorrect entries in translation memories.",
"It is applicable to any language pair for which an MT system, a POS tagger, and a language identifier are available.",
"Implemented using off-the-shelf tools, our system achieved the best classification results for two out of three language pairs (English–Italian and English–Spanish) in the Binary Classification (II) task.",
"In future work, we would like to assess the impact of gearing NLP components to target domains on classification accuracy.",
"The training data in this shared task stems from news (German) and medical texts (Italian, Spanish) which our MT systems, for example, were not optimized for.",
"This domain mismatch might partially explain why our system did not perform well on the English–German test set.",
"More importantly, however, we would like to test our implementation as-is in Autodesk's production environments for software localization.",
"Removing incorrect segments from TMs could ultimately help make professional translation more efficient by providing better MT (through filtered training data) and more accurate fuzzy matches."
],
[
"Acknowledgements",
"We would like to thank Valéry Jacot for his vital support and guidance."
]
] | 1605.05906 |
[
[
"Rokhlin Property for Group Actions on Hilbert $C^*$-modules"
],
[
"Abstract We introduce Rokhlin properties for certain discrete group actions on $C^*$-correspondences as well as on Hilbert bimodules and analyze them.",
"It turns out that the group actions on any $C^*$-correspondence $E$ with Rokhlin property induces group actions on the associated $C^*$-algebra $\\mathcal O_E$ with Rokhlin property and the group actions on any Hilbert bimodule with Rokhlin property induces group actions on the linking algebra with Rokhlin property.",
"Permanence properties of several notions such as nuclear dimension and $\\mathcal D$-absorbing property with respect to crossed product of Hilbert $C^*$-modules with groups, where group actions have Rokhlin property, are studied.",
"We also investigate a notion of outerness for Hilbert bimodules."
],
[
"Introduction",
"Elliott in [6] initiated the classification program for nuclear $C^*$ -algebras based on the K-theory of $C^*$ -algebras.",
"Many aspects of the modern approach to this program are described in the monograph [32] by Rørdam.",
"In [7], Elliott and Toms discussed the importance of the following regularity properties in the modern study of the classification program: strict comparison, finite decomposition rank, and $Z$ -absorbing where $Z$ is the Jiang-Su algebra which is the first object in the category of strongly self-absorbing $C^*$ -algebras.",
"Toms and Winter gave a plausible conjecture: If $A$ is a unital simple nuclear separable infinite dimensional $C^*$ -algebra, then the following statements are equivalent: (1) $A$ has finite nuclear dimension; (2) $A$ is $Z$ -stable; (3) $A$ has the strict comparison.",
"The implications $(1)\\Rightarrow (2)$ and $(2)\\Rightarrow (3)$ have been proved by Winter [36] and Rørdam [33], respectively.",
"The implication $(3)\\Rightarrow (2)$ has been proved by several authors under certain assumption, see [35].",
"The classification of the crossed products of $C^*$ -algebras is a very challenging problem, i.e., if we start with a $C^*$ -algebra of a particular type and a group action on it, then it is difficult to predict the properties of their crossed product.",
"The Rokhlin property for group actions on $C^*$ -algebras has significantly influenced the approaches taken in the classification theory of $C^*$ -algebras.",
"Kishimoto [21] showed that the reduced crossed product of a simple $C^*$ -algebra, with respect to an outer action of a discrete group, is simple.",
"Herman and Ocneanu [10] defined the Rokhlin property of group actions on $C^*$ -algebras in terms of projections and this notion is stronger than the outerness.",
"The modern definition of the Rokhlin property is due to Izumi [13].",
"Several classes including AF-algebras, AI-algebras, AT-algebras are closed under finite group actions with the Rokhlin property (cf.",
"[25]).",
"An approach to the Rokhlin property, introduced by Santiago [34], for finite group actions on not necessarily unital $C^*$ -algebras (cf.",
"[12], [24]) use positive contractions instead of projections.",
"Note that when $A$ is unital, using [34], we can replace orthogonal positive contractions by orthogonal projections and get Izumi's definition of Rokhlin property.",
"We begin Section with Santiago's definition and list there some classes that are preserved under crossed products by finite group actions with this Rokhlin property.",
"In [29], from $C^*$ -correspondences Pimsner constructed $C^*$ -algebras which are known as Cuntz-Pimsner algebras.",
"The class of Cuntz-Pimsner algebras includes Cuntz algebras, Cuntz-Krieger algebras and the crossed products by $\\mathbb {Z}$ .",
"For every $C^*$ -correspondence $(E,A,\\phi )$ , Katsura in [17] defined a $C^*$ -algebra $O_{E}$ .",
"The algebra $O_{E}$ is the same as the Cuntz-Pimsner algebra when $\\phi $ is injective.",
"The algebras $O_{E}$ also generalize the crossed product, as defined in [1], by Hilbert bimodules and graph algebras (cf.",
"[8]).",
"The graph $C^*$ -algebras of topological graphs can also be realized as $O_{E}$ for some $C^*$ -correspondence $(E,A,\\phi )$ (see [18] for details).",
"In Subsection REF we define compatible action, $(\\eta ,\\alpha )$ , of a locally compact group on a $C^*$ -correspondence.",
"Hao and Ng in [9] proved that each compatible action of a locally compact group $G$ on a $C^*$ -correspondence $({E},A,\\phi )$ induces an action of $G$ on the associated $C^*$ -algebra $O_{{E}}$ .",
"It is of interest to determine at least the sufficient conditions under which for a compatible action $(\\eta , \\alpha )$ of $G,$ permanence property is exhibited by several notions related to this $C^*$ -algebra, associated to the Hilbert module with respect to the crossed product.",
"We define the Rokhlin property for finite group actions on $C^*$ -correspondences in Subsection REF and provide an answer of the above question regarding sufficient conditions in Subsection REF when $G$ is finite.",
"For any class of unital and separable $C^*$ -algebras $\\mathcal {C},$ Osaka and Phillips [25] introduced the notion of local $\\mathcal {C}$ -algebra.",
"Santiago [34] extended this approach by considering non-unital $C^*$ -algebras.",
"A notion of closed under local approximation is defined (see page 104) in terms of local $\\mathcal {C}$ -algebra.",
"If $\\mathcal {C}$ denotes certain class of $C^*$ -algebras such as purely infinite $C^*$ -algebras, simple stably projectionless $C^*$ -algebras etc.",
"listed in Theorem REF , then $\\mathcal {C}$ is closed under local approximation and under crossed product with a finite group action with the Rokhlin property.",
"As an application of the observation made in Section we show that if an action $(\\eta ,\\alpha )$ of a finite group $G$ on $(E,A,\\phi )$ has the Rokhlin property and $O_E$ belongs to one of the classes mentioned above, then $O_{E\\times _{\\eta } G}$ belongs to the same class (see Corollary REF ).",
"At the end of Section , we point out that the gauge action on the graph $C^*$ -algebra is saturated by [15], but the corresponding action on the $C^*$ -correspondence does not have the Rokhlin property.",
"We introduce, in Section 4, the Rokhlin property for compatible finite group actions on Hilbert bimodules and prove the following: If we realize a Hilbert $A$ -module $E$ as a Hilbert $\\mathcal {K}(E)$ -$A$ bimodule, and if $A$ belongs to a class $\\mathcal {C}$ in the previous paragraph and if a group action $\\eta $ of a finite group $G$ on the Hilbert $A$ -module $E$ has the Rokhlin property as a certain compatible action on the bimodule, then the linking algebra of the crossed product Hilbert $A\\times _{\\alpha } G$ -module $E\\times _{\\eta } G$ belongs to the class $\\mathcal {C}$ .",
"To obtain this result we first prove that any compatible action of a finite group $G$ with the Rokhlin property on a Hilbert bimodule induces an action of $G$ with the Rokhlin property on the linking algebra."
],
[
"Rokhlin property for finite group actions on $C^*$ -algebras",
"In this section, first we recall the definition of the Rokhlin property for finite group actions on a $C^*$ -algebra, from [34], which involves positive contractions.",
"The Rokhlin property for finite group actions on $C^*$ -algebras was studied by several authors [13], [24], [25], [28], [34], and it was proved that several classes of $C^*$ -algebras are preserved under the crossed product when the action of the group has the Rokhlin property.",
"We list here some such classes from [34].",
"Definition 2.1 Let $\\alpha : G\\rightarrow Aut(A)$ be an action of a finite group $G$ on a $C^*$ -algebra $A$ .",
"We say that $\\alpha $ has the Rokhlin property if for any $\\epsilon > 0$ and every finite subset $S$ of $A$ there exist orthogonal positive contractions $(f_g)_{g\\in G}\\subset A$ satisfying $\\Vert (\\sum _{g\\in G} f_g)a -a\\Vert <\\epsilon $ for all $a\\in S$ , $\\Vert \\alpha _h(f_g)-f_{hg}\\Vert <\\epsilon $ for $h,g\\in G$ , $\\Vert [f_g ,a]\\Vert <\\epsilon $ for $g\\in G$ and $a\\in S$ .",
"The elements $(f_g)_{g\\in G}$ are called Rokhlin elements for $\\alpha $ .",
"Remark 2.2 When $A$ is unital, $\\alpha $ has the Rokhlin property in the sense of Izumi [13].",
"That is, we can take a partition of unity $(e_g)_{g\\in G}$ consisting of projections in place of $(f_g)_{g\\in G}$ (see [34]).",
"The following notions are borrowed from [25], [34]: Let ${\\mathcal {C}}$ be a class of $C^*$ -algebras.",
"A local ${\\mathcal {C}}$ -algebra is a $C^*$ -algebra $A$ such that for every finite set $S \\subset A$ and every $\\epsilon > 0,$ there exists a $C^*$ -algebra $B$ in ${\\mathcal {C}}$ and a $*$ -homomorphism $\\pi \\colon B \\rightarrow A$ such that $dist (a, \\pi (B)) < \\epsilon $ for all $a \\in S.$ We say that ${\\mathcal {C}}$ is closed under local approximation if every local ${\\mathcal {C}}$ -algebra belongs to ${\\mathcal {C}}$ .",
"We recall the following result from [34] (cf.",
"[26], [25], [28], [11], [20], [14]).",
"Theorem 2.3 [34] The following classes are closed under local approximation and under crossed product with finite group actions with the Rokhlin property, respectively: (i) purely infinite $C^*$ -algebras, (ii) $C^*$ -algebras having stable rank one, (iii) $C^*$ -algebras with real rank zero, (iv) $C^*$ -algebras of nuclear dimension at most $n,$ where $n\\in \\mathbb {Z}_+,$ (v) separable $D$ -absorbing $C^*$ -algebras where $D$ is a strongly self-absorbing $C^*$ -algebra, (vi) simple $C^*$ -algebras, (vii) simple $C^*$ -algebras that are stably isomorphic to direct limits of sequences of $C^*$ -algebras, in a class $\\mathfrak {S}$ , where $\\mathfrak {S}$ is a class of finitely generated semiprojective $C^*$ -algebras that is closed under taking tensor products by matrix algebras over $\\mathbb {C}$ , (viii) separable AF-algebras, (ix) separable simple $C^*$ -algebras that are stably isomorphic to AI-algebras, (x) separable simple $C^*$ -algebras that are stably isomorphic to AT-algebras, (xi) $C^*$ -algebras that are stably isomorphic to sequential direct limits of one-dimensi- onal noncommutative CW-complexes, (xii) separable $C^*$ -algebras whose quotients are stably projectionless, (xiii) simple stably projectionless $C^*$ -algebras, (xiv) separable $C^*$ -algebras with almost unperforated Cuntz semigroup, (xv) simple $C^*$ -algebras with strict comparison of positive elements, (xvi) separable $C^*$ -algebras whose closed two-sided ideals are nuclear and satisfy the Universal Coefficient Theorem."
],
[
"Rokhlin property for finite group actions on $C^*$ -correspondences",
"In this section we define and explore the Rokhlin property for a compatible group action on a $C^*$ -correspondence when the group is finite."
],
[
"$C^*$ -correspondence",
"Let ${E}$ be a vector space which is a right module over a $C^*$ -algebra $A$ and satisfying $\\alpha (xa)=(\\alpha x)a=x(\\alpha a)$ for $x\\in {E},a\\in A,\\alpha \\in \\mathbb {C}$ .",
"The module ${E}$ is called an (right) inner-product $A$ -module if there exists a map $\\langle \\cdot ,\\cdot \\rangle \\!_{_{A}} : E \\times E \\rightarrow {A}$ such that $\\langle x,x \\rangle \\!_{_{A}} \\ge 0 ~\\mbox{for}~ x \\in {E} $ and $\\langle x,x \\rangle \\!_{_{A}} = 0$ only if $x = 0 ,$ $\\langle x,ya \\rangle \\!_{_{A}}= \\langle x,y \\rangle \\!_{_{A}} a ~\\mbox{for}~ x,y \\in {E}$ and $\\mbox{for}~ a\\in A, $ $\\langle x,y \\rangle \\!_{_{A}}=\\langle y,x \\rangle \\!_{_{A}}^*~\\mbox{for}~ x ,y\\in {E} ,$ $\\langle x,\\mu y+\\nu z \\rangle \\!_{_{A}}= \\mu \\langle x,y \\rangle \\!_{_{A}} +\\nu \\langle x,z \\rangle \\!_{_{A}}~\\mbox{for}~ x,y,z \\in {E} $ and $\\mbox{for}~ \\mu ,\\nu \\in \\mathbb {C}$ .",
"An (right) inner-product $A$ -module ${E}$ is called (right) Hilbert $A$ -module or (right) Hilbert $C^{*}$ -module over $A$ if it is complete with respect to the norm $\\Vert x\\Vert :=\\Vert \\langle x,x\\rangle \\!_{_{A}}\\Vert ^{1/2} ~\\mbox{for}~ x \\in E.$ If there is no ambiguity, we simply write $\\langle \\cdot ,\\cdot \\rangle $ instead of $\\langle \\cdot ,\\cdot \\rangle \\!_{_{A}}$ .",
"The notion of Hilbert $C^*$ -module was introduced independently by Paschke [27] and Rieffel [31].",
"In [16], Kasparov used Hilbert $C^*$ -modules as a tool to study a bivariate K-theory for $C^*$ -algebras.",
"Below we define the notion of $C^*$ -correspondences which will play an important role in this article.",
"Definition 3.1 Let $A$ be a $C^*$ -algebra.",
"A right Hilbert $A$ -module $E$ is called a $C^*$ -correspondence over $A$ if there exists a $*$ -homomorphism $\\phi :A\\rightarrow \\mathcal {B}^a(E)$ where $\\mathcal {B}^a(E)$ is the set of all adjointable operators on $E$ , which gives a left action of $A$ on $E$ as $ay:=\\phi (a)y~\\mbox{for all}~a\\in A,y\\in E.$ We use notation $(E,A,\\phi )$ for the $C^*$ -correspondence and denote by $\\mathcal {K}(E)$ the $C^*$ -algebra generated by maps $\\lbrace \\theta _{x,y}:x,y\\in E\\rbrace $ defined by $\\theta _{x,y}(z):=x\\langle y,z\\rangle ~\\mbox{for}~x,y,z\\in E.$ In this article we work with a certain type of action, which we define below, of a locally compact group on a $C^*$ -correspondence: Definition 3.2 Let $(G,\\alpha ,A)$ be a $C^*$ -dynamical system of a locally compact group $G$ and let $(E,A,\\phi )$ be a $C^*$ -correspondence over $A$ .",
"An $\\alpha $ -compatible action $\\eta $ of $G$ on ${E}$ is defined as a group homomorphism from $G$ into the group of invertible linear transformations on ${E}$ such that $\\eta _g (\\phi (a)x)=\\alpha _g (a)\\eta _g(x)~\\mbox{for $a\\in A,~x\\in {E},~g\\in G$,}~$ $\\langle \\eta _g (x) ,\\eta _g (y)\\rangle =\\alpha _g(\\langle x,y \\rangle )~\\mbox{for $~x,y\\in {E},~g\\in G$,}~$ and $g\\mapsto \\eta _g (x)$ is continuous from $G$ into ${E}$ for each $x\\in {E}$ .",
"We denote an $\\alpha $ -compatible action $\\eta $ by $(\\eta ,\\alpha )$ .",
"In this case, we define a $*$ -isomorphism $Ad \\eta _s:\\mathcal {B}^a(E)\\rightarrow \\mathcal {B}^a(E)$ for each $s\\in G$ by $Ad \\eta _s(T)(x):=\\eta _s(T(\\eta _{s^{-1}}(x))~\\mbox{for}~T\\in \\mathcal {B}^a(E),~x\\in E.$ Let $G$ be a locally compact group and $\\bigtriangleup $ be the modular function of $G$ .",
"Let $\\eta $ be an $\\alpha $ -compatible action of $G$ on the $C^*$ -correspondence $(E,A,\\phi )$ .",
"Then the crossed product $E\\times _{\\eta }G$ (cf.",
"[16], [5], [9]) is a Hilbert $A\\times _{\\alpha } G$ -module and is defined as the completion of an inner-product $C_c (G,A)$ -module $C_c(G,{E})$ where the module action and the $C_c (G,A)$ -valued inner-product are given by $ l\\cdot g(s)&=\\int _{G}l(t)\\alpha _{t}(g(t^{-1}s))dt,\\\\\\langle l,m\\rangle \\!_{_{C_c (G,A)}}(s)&=\\int _{G}\\alpha _{t^{-1}}(\\langle l(t),m(ts)\\rangle \\!_{_{A}})dt,$ respectively for $g\\in C_c (G,A)$ and $l,m\\in C_c (G,{E})$ .",
"For each $s\\in G$ the $*$ -isomorphism $Ad \\eta _s$ defined by Equation REF satisfies $Ad \\eta _s(\\theta _{x,y})=\\theta _{{\\eta _s (x)},{\\eta _s (y)}}$ for $x,y\\in E$ , and $(G,Ad \\eta ,\\mathcal {K}(E))$ becomes a $C^*$ -dynamical system.",
"From Definition REF (1) it follows that $\\phi :A\\rightarrow M(\\mathcal {K}(E))$ is equivariant, i.e., $\\phi (\\alpha _s(a))=Ad\\eta _s(\\phi (a))~\\mbox{for all }~a\\in A, s\\in G.$ Indeed, using $Ad \\eta _s$ we get another $*$ -isomorphism $\\Xi : \\mathcal {K}(E\\times _{\\eta } G)\\rightarrow \\mathcal {K}(E)\\times _{Ad \\eta } G$ (cf.",
"Section 3.11 of [16] and Section 2 of [9]) defined by $\\Xi (\\theta _{l,m})(s):=\\int _G \\theta _{l(r),Ad \\eta _s(m(s^{-1}r))}\\bigtriangleup (s^{-1}r) dr~\\mbox{where}~l,m\\in C_c (G,E),~s\\in G.$ From the fact that $\\phi :A\\rightarrow M(\\mathcal {K}(E))$ is equivariant we get an equivariant $*$ -homomorphism $\\chi :A\\times _{\\alpha } G\\rightarrow M(\\mathcal {K}(E)\\times _{Ad\\eta }G)$ satisfying $\\chi (f\\otimes a)=f\\otimes \\phi (a)$ for $f\\in C_c(G),~a\\in A$ .",
"We identify $\\mathcal {K}(E\\times _{\\eta } G)$ with $\\mathcal {K}(E)\\times _{Ad \\eta } G$ and treat $\\chi $ and $\\Xi ^{-1}\\circ \\chi $ as same."
],
[
"Rokhlin property for compatible finite group actions on $C^*$ -correspondences",
"Definition 3.3 Let $(G,\\alpha ,A)$ be a $C^*$ -dynamical system of a finite group $G$ on $A$ and let $({E},A,\\phi )$ be a $C^*$ -correspondence.",
"Let $(\\eta ,\\alpha )$ be an $\\alpha $ -compatible action of $G$ on ${E}$ .",
"Then we say that $\\eta $ has the Rokhlin property if for each $\\epsilon >0$ , and finite subsets $S_1$ and $S_2$ of ${E}$ and $A$ respectively, there exists $(a_g)_{g\\in G}\\subset A$ consisting of mutually orthogonal positive contractions such that $\\Vert \\sum _{g\\in G} \\phi (a_g)x -x\\Vert <\\epsilon $ , $\\Vert \\sum _{g\\in G} xa_g -x\\Vert <\\epsilon $ , $\\Vert \\sum _{g\\in G} a_g a -a\\Vert <\\epsilon $ and $\\Vert \\sum _{g\\in G} aa_g -a\\Vert <\\epsilon $ for all $x\\in S_1$ , $a\\in S_2$ , $\\Vert \\alpha _h(a_g)-a_{hg}\\Vert <\\epsilon $ for $h,g\\in G$ , $\\Vert xa_g-\\phi (a_g) x\\Vert <\\epsilon $ and $\\Vert a_g a-aa_g \\Vert <\\epsilon $ for all $x\\in S_1$ , $a\\in S_2$ and $g\\in G$ .",
"The following example is based on the construction of an action of $\\mathbb {Z}_2$ on $C_0(X)$ where $X$ is equipped with a homeomorphism of order 2 defined on it: Example 3.4 Let $X=\\lbrace \\frac{1}{n}: n\\in \\mathbb {N}\\rbrace $ and the topology on $X$ be discrete.",
"Define a map $\\psi : X\\rightarrow X$ by $\\psi (1/(2n-1)):=1/(2n),~ \\psi (1/(2n)):=1/(2n-1)~\\mbox{ for all}~n\\in \\mathbb {N}.$ Observe that $\\psi $ is a homeomorphism of order 2.",
"Thus we obtain an automorphism $\\alpha : C_0(X)\\rightarrow C_0(X)$ such that $\\alpha (g)(x):=g(\\psi ^{-1}(x))~\\mbox{for each}~x\\in X,~g\\in C_0(X).$ Indeed, $\\alpha ^2=id_{C_0(X)}$ and this provides an action of $\\mathbb {Z}_2$ on $ C_0(X)$ which we denote by $\\alpha $ .",
"Let $H$ be a Hilbert space and let $C_0(X, H)$ be the space of continuous $H$ -valued functions on $X$ vanishing at infinity.",
"The space $C_0(X, H)$ becomes a Hilbert $C_0(X)$ -module where module action and inner product are defined as follow: $~f\\cdot g(x):=g(x)f(x);~\\langle f,f^{\\prime }\\rangle (x):=\\langle f(x),f^{\\prime }(x)\\rangle $ $\\mbox{for all}~f,f^{\\prime }\\in C_0(X, H),~g\\in C_0(X).$ In fact, $C_0(X, H)$ becomes a $C^*$ -corresponden- ce over $C_0(X)$ with the left action $\\phi $ defined by $\\phi (g)f:=f\\cdot g~\\mbox{for all}~f\\in C_0(X, H),~g\\in C_0(X).$ Define $\\eta :C_0(X, H)\\rightarrow C_0(X, H)$ by $\\eta (f)(x):=f(\\psi ^{-1}(x))~\\mbox{for all}~x\\in X,~f\\in C_0(X, H).$ It follows that $\\langle \\eta (f),\\eta (f^{\\prime })\\rangle =\\alpha (\\langle f,f^{\\prime }\\rangle )$ for all $f,f^{\\prime }\\in C_0(X, H)$ .",
"Moreover, $\\eta ^2=id_{C_0(X, H)}$ and hence we get an induced $\\alpha $ -compatible $\\mathbb {Z}_2$ action on $(C_0(X, H), C_0(X),\\phi )$ say $(\\eta ,\\alpha )$ .",
"The action $(\\eta ,\\alpha )$ has the Rokhlin property in the sense of Definition REF : Let $a_n^{(0)}, a_n^{(1)}\\in C_0(X)$ be the characteristic functions of the sets $\\lbrace 1/(2k-1) : 1\\le k\\le n\\rbrace $ and $\\lbrace 1/(2k) : 1\\le k\\le n\\rbrace $ , respectively for each $n\\in \\mathbb {N}$ .",
"Note that these functions are continuous (because the given sets are open), $\\alpha (a_n^{(0)})=a_n^{(1)}$ , and $(a_n^{(0)}+a_n^{(1)})_{n\\in \\mathbb {N}}$ is an approximate unit for $\\mathrm {C}_0(X)$ .",
"It is clear that $a_n^{(0)}$ and $a_n^{(1)}$ are orthogonal.",
"If $(e_n)_{n\\in \\mathbb {N}}$ is an approximate unit for a $C^*$ -algebra $A$ and $E$ is a Hilbert $A$ -module, then $(xe_n)_{n\\in \\mathbb {N}}$ converges to $x$ for each $x\\in E$ .",
"Hence $(\\eta ,\\alpha )$ has the Rokhlin property, for $ C_0(X)$ is commutative.",
"For any subset $S$ of a $C^*$ -algebra, we use symbol $S^*$ to denote the set $\\lbrace x^*:x\\in S\\rbrace $ .",
"Example 3.5 Let $l^2(A)$ be the direct sum of a countable number of copies of a $C^*$ -algebra $A$ .",
"The vector space $l^2(A)$ is known as the standard Hilbert $C^*$ -module where the right $A$ -module action and the $A$ -valued inner-product is given by $(a_1,a_2,\\ldots ,a_n,\\ldots )a&:=(a_1a,a_2a,\\ldots ,a_na,\\ldots )~\\mbox{and}~\\\\\\langle (a_1,a_2,\\ldots ,a_n,\\ldots ),(a^{\\prime }_1,a^{\\prime }_2,\\ldots ,a^{\\prime }_n,\\ldots )\\rangle &:=\\sum ^\\infty _{i=1}a^*_ia^{\\prime }_i~\\mbox{for all}~a,a_1,a^{\\prime }_1,a_2,a^{\\prime }_2\\ldots \\in A.$ It is easy to note that $(l^2(A),A,\\phi )$ is a $C^*$ -correspondence where the adjointable left action $\\phi :A\\rightarrow \\mathcal {B}^a (l^2(A))$ is defined as $\\phi (a)(a_1,a_2,\\ldots ,a_n,\\ldots )=(aa_1,aa_2,\\ldots ,aa_n,\\ldots )~\\mbox{for all}~a,a_1,a_2,\\ldots \\in A.$ Let $(G,\\alpha ,A)$ be a finite group action.",
"Define $\\eta :G\\rightarrow Aut~ l^2(A)$ by $\\eta _t (a_1,a_2,\\ldots ,a_n,\\ldots ):=(\\alpha _t(a_1),\\alpha _t(a_2),\\ldots ,\\alpha _t(a_n),\\ldots )$ where $t\\in G$ and $(a_1,a_2,\\ldots ,a_n,\\ldots )\\in l^2(A).$ It is clear that $\\eta $ is an $\\alpha $ -compatible action of the group $G$ on $(l^2(A),A,\\phi )$ .",
"Next we show that if $\\alpha $ has the Rokhlin property, then $\\eta $ has the Rokhlin property as an $\\alpha $ -compatible action of the group $G$ on the $C^*$ -correspondence $(l^2(A),A,\\phi ).$ Let $\\epsilon >0$ and let $S_1=\\lbrace (a^j_1,a^j_2,\\ldots ,a^j_n,\\ldots ):j=1,2,\\ldots ,N\\rbrace $ and $S_2$ be finite subsets of $l^2(A)$ and $A$ respectively.",
"Thus for each $j,$ there exist positive integers $N^j$ such that $\\Vert \\sum _{n>N^j} a^{j*}_na^j_n\\Vert ^{\\frac{1}{2}}<\\frac{\\epsilon }{2(|G|^2+2|G|+1)}.$ Fix $S^{\\prime }_1:=\\lbrace a^j_n:n\\le N^j,1\\le j\\le N\\rbrace $ and let $K=(\\mbox{max}_j N^j) +1.$ Assume that $\\alpha $ has the Rokhlin property for the finite set $S=S^{\\prime }_1\\cup S^{\\prime *}_1\\cup S_2\\cup S^{*}_2,$ i.e., we get Rokhlin elements $\\lbrace f_g:g\\in G\\rbrace $ consist of mutually orthogonal positive contractions in $A$ satisfying the following: $\\Vert (\\sum _{g\\in G} f_g)a -a\\Vert <\\frac{\\epsilon }{2K}$ for all $a\\in S$ , $\\Vert \\alpha _h(f_g)-f_{hg}\\Vert <\\frac{\\epsilon }{2K}$ for $h,g\\in G$ , $\\Vert [f_g ,a]\\Vert <\\frac{\\epsilon }{2K}$ for $g\\in G$ and $a\\in S$ .",
"Now we check that the action $(\\eta ,\\alpha )$ has the Rokhlin property with Rokhlin elements $\\lbrace a_g:g\\in G\\rbrace $ where $a_g:=f_g$ for each $g\\in G$ : Note that $&\\left\\Vert \\sum _{g\\in G} \\phi (a_g)(a^j_1,a^j_2,\\ldots ,a^j_n,\\ldots ) -(a^j_1,a^j_2,\\ldots ,a^j_n,\\ldots )\\right\\Vert \\\\=&\\left\\Vert \\left(\\sum _{g\\in G} f_g a^j_1-a^j_1,\\sum _{g\\in G} f_g a^j_2-a^j_2,\\ldots ,\\sum _{g\\in G} f_g a^j_n-a^j_n,\\ldots \\right)\\right\\Vert \\\\=&\\left\\Vert \\sum ^\\infty _{n=1}\\left[\\sum _{g\\in G} f_g a^j_n-a^j_n\\right]^*\\left[\\sum _{g\\in G} f_g a^j_n-a^j_n\\right]\\right\\Vert ^\\frac{1}{2}\\\\<&\\sum ^{N^j}_{n=1}\\left\\Vert \\sum _{g\\in G} f_g a^j_n-a^j_n\\right\\Vert +\\left\\Vert \\sum _{n>{N^j}}\\left[\\sum _{g\\in G} f_g a^j_n\\right]^*\\left[\\sum _{g\\in G} f_g a^j_n\\right]\\right\\Vert ^\\frac{1}{2}\\\\& ~~~+\\left\\Vert \\sum _{n>{N^j}}\\left[\\sum _{g\\in G} f_g a^j_n\\right]^*a^j_n\\right\\Vert ^\\frac{1}{2}+\\left\\Vert \\sum _{n>{N^j}}a^{j*}_n\\left[\\sum _{g\\in G} f_g a^j_n\\right]\\right\\Vert ^\\frac{1}{2}+\\left\\Vert \\sum _{n>{N^j}}a^{j*}_na^j_n\\right\\Vert ^\\frac{1}{2}\\\\&<\\sum ^{N^j}_{n=1}\\frac{\\epsilon }{2K}+\\frac{\\epsilon }{2}<\\epsilon ,~\\mbox{and}~$ $&~~~~\\Vert (a^j_1,a^j_2,\\ldots ,a^j_n,\\ldots )a_g-\\phi (a_g)(a^j_1,a^j_2,\\ldots ,a^j_n,\\ldots )\\Vert \\\\&= \\Vert (a^j_1a_g-a_ga^j_1,a^j_2a_g-a_ga^j_2,\\ldots ,a^j_na_g-a_ga^j_n,\\ldots )\\Vert \\\\&< \\sum ^{N^j}_{n=1}\\Vert a^j_nf_g-f_ga^j_n\\Vert +\\frac{\\epsilon }{2}\\\\&<\\sum ^{N^j}_{n=1}\\frac{\\epsilon }{2K}+\\frac{\\epsilon }{2}<\\epsilon ~\\mbox{ for all $(a^j_1,a^j_2,\\ldots ,a^j_n,\\ldots )\\in S_1$ and $g\\in G$.",
"}~$ It is easy to check other conditions of the definition of Rokhlin property and hence $(\\eta ,\\alpha )$ has the Rokhlin property."
],
[
"Rokhlin property for induced actions on Cuntz-Pimsner algebras",
"For a $C^*$ -correspondence, Katsura [17] introduced the following associated $C^*$ -algebra: Definition 3.6 Let $({E},A,\\phi )$ be a $C^*$ -correspondence over a $C^*$ -algebra $A$ and $B$ be a $C^*$ -algebra.",
"A pair $(\\pi ,\\Psi )$ is called covariant representation of $({E},A,\\phi )$ on $B$ if $\\pi :A\\rightarrow B$ is a $*$ -homomorphism and $\\Psi :{E}\\rightarrow B$ is a bounded linear map satisfying $\\Psi (x)^* \\Psi (y)=\\pi (\\langle x,y\\rangle )$ for all $x,y\\in E$ , $\\pi (a) \\Psi (x)=\\Psi (\\phi (a)x)$ for all $a\\in A$ and $x\\in {E}$ , $\\pi (b)=\\Pi _{\\Psi } (\\phi (b))$ for all $b\\in J_{E}$ where $J_{{E}}:=\\phi ^{-1}(\\mathcal {K}({E}))\\cap (ker \\phi )^\\perp $ and $\\Pi _{\\Psi }:\\mathcal {K}({E})\\rightarrow B $ is a $*$ -homomorphism defined by $\\Pi _{\\Psi } (\\theta _{x,y}):=\\Psi (x)\\Psi (y)^*~\\mbox{for}~x,y\\in E.$ The notation $C^*(\\pi ,\\Psi )$ denotes the $C^*$ -algebra generated by the images of mappings $\\pi $ and $\\Psi $ in $B$ .",
"A covariant representation $(\\pi _U,\\Psi _U)$ of a $C^*$ -correspondence $({E},A,\\phi )$ is said to be universal if for any covariant representation $(\\pi ,\\Psi )$ of $({E},A,\\phi )$ on $B$ , there exists a natural surjection $\\psi :C^*(\\pi _U,\\Psi _U)\\rightarrow C^*(\\pi ,\\Psi )$ such that $\\pi =\\psi \\circ \\pi _U$ and $\\Psi =\\psi \\circ \\Psi _U$ .",
"We denote the $C^*$ -algebra $C^*(\\pi _U,\\Psi _U)$ by $O_{E}$ .",
"In Lemma 2.6 of [9] Hao and Ng proved that each action $(\\eta ,\\alpha )$ of a locally compact group $G$ on $({E},A,\\phi )$ induces a $C^*$ -dynamical system $(G, \\gamma ,O_{{E}})$ such that $\\gamma _s(\\pi _U(a))=\\pi _U(\\alpha _s (a))$ and $\\gamma _s(\\Psi _U(x))=\\Psi _U(\\eta _s (x))$ for all $a\\in A$ , $x\\in {E}$ and $s\\in G$ .",
"The theorem below shows that Definition REF is the natural choice for Rokhlin property.",
"Theorem 3.7 Let $(\\eta ,\\alpha )$ be an action of a finite group $G$ on a $C^*$ -correspondence $({E},A,\\phi )$ .",
"The following statements are equivalent: (a) The action $(\\eta ,\\alpha )$ has the Rokhlin property.",
"(b) The induced action $\\gamma $ of $G$ on $O_{{E}}$ as mentioned above has the Rokhlin property with Rokhlin elements from $\\pi _{U}(A)$ .",
"Let $\\epsilon >0$ and let $S=\\lbrace b_1,b_2,\\ldots ,b_n\\rbrace $ be any finite subset of $O_{{E}}$ .",
"For each $1\\le j\\le n$ there exist finite sets $\\lbrace x^l_j\\rbrace _{1\\le l\\le l_j}\\subset {E}$ and $\\lbrace a^m_j\\rbrace _{1\\le m\\le m_j}\\subset A$ such that $\\Vert b_j-p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))\\Vert <\\frac{\\epsilon }{3|G|}~\\mbox{where}~$ $p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))=\\sum ^{n_j}_{i=1}\\lambda _{j,i} u_{j,i,1} u_{j,i,2}\\ldots u_{j,i,{k_{j,i}}}$ is a finite linear combination of words $u_{j,i,1} u_{j,i,2}\\ldots u_{j,i,{k_{j,i}}}$ in the set $\\lbrace \\Psi _U(x^l_j),\\Psi _U(x^l_j)^*,\\pi _U(a^m_j),\\pi _U(a^m_j)^*: 1\\le l\\le l_j,1\\le m\\le m_j,1\\le j\\le n\\rbrace .$ Let $S_1=\\lbrace x^l_j\\rbrace _{l,j}$ , $S_2=\\lbrace a^m_j\\rbrace _{m,j}$ and $K_j=\\sum ^{n_j}_{i=1}|\\lambda _{j,i}| \\Vert u_{j,i,1}\\Vert \\Vert u_{j,i,2}\\Vert \\ldots \\Vert u_{j,i,{k_{j,i}}}\\Vert $ .",
"Set $K:=3 \\left(\\displaystyle \\max _{1\\le j\\le n}\\displaystyle \\max _{1\\le i\\le n_j}k_{j,i}\\right)\\left(\\displaystyle \\max _{1\\le j\\le n} K_j\\right)$ .",
"Since $(\\eta ,\\alpha )$ has the Rokhlin property, there exists $(a_g)_{g\\in G}\\subset A$ consists of mutually orthogonal positive contractions such that $\\Vert \\sum _{g\\in G} \\phi (a_g)x -x\\Vert <\\frac{\\epsilon }{K}$ , $\\Vert \\sum _{g\\in G} xa_g -x\\Vert <\\frac{\\epsilon }{K}$ , $\\Vert \\sum _{g\\in G} a_g a -a\\Vert <\\frac{\\epsilon }{K}$ and $\\Vert \\sum _{g\\in G} aa_g -a\\Vert <\\frac{\\epsilon }{K}$ for all $x\\in S_1$ , $a\\in S_2$ , $\\Vert \\alpha _h(a_g)-a_{hg}\\Vert <\\frac{\\epsilon }{K}$ for $h,g\\in G$ , $\\Vert xa_g-\\phi (a_g) x\\Vert <\\frac{\\epsilon }{K}$ and $\\Vert a_g a-aa_g \\Vert <\\frac{\\epsilon }{K}$ for all $x\\in S_1$ , $a\\in S_2$ and $g\\in G$ .",
"We show that $\\gamma $ has the Rokhlin property with respect to $(f_g)_{g\\in G}$ where for each $g\\in G$ , $f_g:=\\pi _U(a_g)$ .",
"For each $g\\in G$ , we have $\\Vert f_g\\Vert \\le 1$ , and $f_g$ 's are mutually orthogonal positive contractions.",
"Further For each $1\\le j\\le n$ , $\\Vert \\mbox{$\\sum _{g\\in G}$} f_g b_j-b_j\\Vert &\\le \\Vert \\mbox{$\\sum _{g\\in G}$} \\pi _U(a_g)b_j -\\mbox{$\\sum _{g\\in G}$}\\pi _U(a_g)p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))\\Vert \\\\&+\\Vert \\mbox{$\\sum _{g\\in G}$} \\pi _U(a_g)p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))-p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))\\Vert \\\\ &+\\Vert p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))-b_j\\Vert <\\epsilon .$ For $h,g\\in G$ we have $\\Vert \\gamma _h(\\pi _U(a_g))-\\pi _U(a_{hg})\\Vert &=\\Vert \\pi _U(\\alpha _h(a_g))-\\pi _U(a_{hg}) \\Vert <\\epsilon .$ For $1\\le j\\le n$ we get $&\\Vert \\pi _U (a_g)b_j -b_j\\pi _U(a_g)\\Vert \\\\ &=\\Vert \\pi _U (a_g)b_j -\\pi _U (a_g)p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))\\Vert \\\\ &+\\Vert \\pi _U (a_g)p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))-p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))\\pi _U (a_g)\\Vert \\\\ &+\\Vert p_j(\\Psi _U(x^l_j),\\pi _U(a^m_j))\\pi _U (a_g)-b_j \\pi _U(a_g)\\Vert <\\epsilon .$ Thus $\\gamma $ has the Rokhlin property with Rokhlin elements from the $C^*$ -algebra $\\pi _{U}(A)$ .",
"Conversely, let $S_1$ and $S_2$ be finite subsets of ${E}$ and $A$ respectively.",
"Fix $\\epsilon >0$ and $S:=\\lbrace \\Psi _U (x),\\pi _U(\\langle x,x\\rangle ),\\pi _U(a),\\pi _U(a^*):x\\in S_1,a\\in S_2\\rbrace .$ Since $\\gamma $ has the Rokhlin property with Rokhlin elements from the $C^*$ -algebra $\\pi _{U}(A)$ , there exist mutually orthogonal positive contractions $(\\pi _U(f_g))_{g\\in G}\\subset \\pi _{U}(A)$ satisfying $\\Vert (\\sum _{g\\in G} \\pi _U(f_g))a -a\\Vert <\\epsilon ^{\\prime }$ for all $a\\in S$ , $\\Vert \\alpha _h(\\pi _U(f_g))-\\pi _U(f_{hg})\\Vert <\\epsilon ^{\\prime }$ for $h,g\\in G$ , $\\Vert [\\pi _U(f_g) ,a]\\Vert <\\epsilon ^{\\prime }$ for $g\\in G$ and $a\\in S$ , where $\\epsilon ^{\\prime }=min\\lbrace \\epsilon ,\\frac{\\epsilon ^2}{|G|+1}\\rbrace $ .",
"We show below that $(\\eta ,\\alpha )$ has the Rokhlin property with Rokhlin elements $(f_g)_{g\\in G}$ : For each $x\\in S_1$ , $a\\in S_2$ we get $\\left\\Vert \\sum _{g\\in G} \\phi (f_g)x -x\\right\\Vert =&\\left\\Vert \\Psi _U\\left(\\sum _{g\\in G} \\phi (f_g)x -x\\right)\\right\\Vert \\\\=&\\left\\Vert \\sum _{g\\in G} \\pi _U(f_g)\\Psi _U(x) -\\Psi _U(x)\\right\\Vert <\\epsilon , \\\\\\left\\Vert \\sum _{g\\in G} xf_g -x\\right\\Vert ^2 =&\\left\\Vert \\sum _{g,g^{\\prime }\\in G} \\langle xf_g -x,xf_{g^{\\prime }} -x\\rangle \\right\\Vert \\\\\\le & \\sum _{g^{\\prime }\\in G}\\left\\Vert \\sum _{g\\in G} f_g \\langle x,x\\rangle -\\langle x,x\\rangle \\right\\Vert \\Vert f_{g^{\\prime }}\\Vert +\\left\\Vert \\sum _{g\\in G} f_g \\langle x,x\\rangle -\\langle x,x\\rangle \\right\\Vert \\\\ <&(|G|+1)\\left(\\frac{\\epsilon ^2}{|G|+1}\\right)=\\epsilon ^2, \\\\\\Vert \\sum _{g\\in G} f_g a -a\\Vert =&\\Vert \\pi _U(\\sum _{g\\in G} f_g a -a)\\Vert <\\epsilon \\mbox{ and}~ \\\\\\Vert \\sum _{g\\in G} af_g -a\\Vert =&\\Vert \\pi _U(\\sum _{g\\in G} af_g -a)\\Vert <\\epsilon ;$ $\\Vert \\alpha _h(f_g)-f_{hg}\\Vert =\\Vert \\pi _U(\\alpha _h(f_g)-f_{hg})\\Vert =\\Vert \\gamma _h(\\pi _U(f_g))-\\pi _U(f_{hg})\\Vert <\\epsilon $ for $h,g\\in G$ ; For all $x\\in S_1$ , $a\\in S_2$ and $g\\in G$ we obtain $\\Vert xa_g-\\phi (f_g) x\\Vert &=\\Vert \\Psi _U(xf_g-\\phi (f_g) x)\\Vert \\\\&=\\Vert \\Psi _U(x)\\pi _U(f_g)-\\pi _U(f_g) \\Psi _U(x)\\Vert <\\epsilon $ and $\\Vert f_g a-af_g \\Vert =\\Vert \\pi _U(f_g a-af_g) \\Vert =\\Vert \\pi _U(f_g) \\pi _U(a)-\\pi _U(a)\\pi _U(f_g) \\Vert <\\epsilon .$ Hence $(\\eta ,\\alpha )$ has the Rokhlin property with Rokhlin elements $(f_g)_{g\\in G}$ ."
],
[
"Applications of our characterization",
"Katsura obtained several results about the nuclearity of the $C^*$ -algebra $O_{E}$ associated to a $C^*$ -correspondence $E$ in [19].",
"We discuss permanence properties of this notion and several other notions for the $C^*$ -algebra, associated to a $C^*$ -correspondence, with respect to the crossed product $E\\times _{\\eta }G$ of a $C^*$ -correspondence $E$ for some action $(\\eta ,\\alpha )$ of a finite group $G$ with Rokhlin property.",
"The nuclear dimension of $O_{E}$ is estimated recently in [4].",
"Corollary 3.8 Assume $(\\eta ,\\alpha )$ to be an action of a finite group $G$ on a $C^*$ -correspondence $({E},A,\\phi )$ .",
"If $(\\eta ,\\alpha )$ has the Rokhlin property and if $O_{E}$ belongs to any one of the classes, say $\\mathcal {C}$ , listed in Theorem REF , then $O_{{E}\\times _{\\eta } G}$ also belongs to the same class $\\mathcal {C}$ .",
"Suppose action $(\\eta ,\\alpha )$ has the Rokhlin property.",
"By Theorem REF the induced action $\\gamma $ of $G$ on $O_{{E}}$ has the Rokhlin property.",
"Since $O_{{E}}$ belongs to the class $\\mathcal {C}$ from Theorem REF , from the remarks made just before this corollary it follows that $O_{{E}} \\times _{\\gamma } G$ also belongs to the same class.",
"The $C^*$ -algebras $O_{{E}} \\times _{\\gamma }G$ and $O_{{E}\\times _{\\eta } G}$ has been shown to be isomorphic in [9].",
"Hence $O_{{E}\\times _{\\eta } G}$ also belongs to the same class.",
"A directed graph $E=(E^0,E^1,r,s)$ consists of a countable vertex set $E^0$ , and a countable edge set $E^1$ , along with maps $r,s: E^1\\rightarrow E^0$ describing the range and the source of edges.",
"We also assume that the directed graph is always row finite, i.e., for every vertex $v\\in E_0,$ the set $s^{-1}(v)$ is a finite subset of $E_1$ .",
"Let $A$ denote the $C^*$ -algebra $C_0(E^0)$ .",
"A graph $C^*$ -algebra of the directed graph $E$ (cf.",
"[22]) is a universal $C^*$ -algebra generated by partial isometries $\\lbrace S_e:e\\in E^1\\rbrace $ and projections $\\lbrace P_v:v\\in E^0\\rbrace $ such that ${S_e}^* S_e=P_{r(e)}=\\sum _{s(f)=r(e)} S_f {S_f}^*~\\mbox{for all}~e\\in E^1.",
"$ Since the graph is row finite, the summation is finite.",
"We use the symbol $C^*(E)$ to denote the graph $C^*$ -algebra of a directed graph $E$ .",
"The vector space $C_c (E^1)$ becomes an inner-product $A$ -module with the following inner-product and module action: $\\langle f,g\\rangle (v)&:=&\\sum _{e\\in r^{-1}(v)}\\overline{f(e)}g(e)~\\mbox{for each}~v\\in E^0;\\\\(fh)(e)&:=&f(e)h(r(e))~\\mbox{for all}~e\\in E^1;$ where $f,g\\in C_c(E^1)$ and $h\\in A$ .",
"Let $E(E)$ denote the completion of the inner-product module $C_c (E^1)$ .",
"Define $\\phi :A\\rightarrow \\mathcal {B}^a(E(E))$ by $\\phi (h)f(e):=h(s(e))f(e)~\\mbox{for each}~e\\in E^1;f\\in C_c(E^1);h\\in A.$ Thus $(E(E),A,\\phi )$ is a $C^*$ -correspondence and the graph $C^*$ -algebra $C^*(E)$ of a directed graph $E$ is always isomorphic to $O_{E(E)}$ (cf.",
"[17]).",
"Definition 3.9 Let $E=(E^0,E^1,r,s)$ be a directed graph and let $c$ from $E^1$ to a countable group $G$ be a mapping.",
"The skew-product graph is the graph $E(c)=(G\\times E^0,G\\times E^1,r^{\\prime },s^{\\prime })$ where $r^{\\prime }(g,e):=(gc(e),r(e))$ and $s^{\\prime }(g,e):=(g,s(e))$ for all $g\\in G$ ; $e\\in E^1$ .",
"For a given countable abelian group $G$ and a function $c:E^1\\rightarrow G$ , we can define an action $\\gamma ^c$ of $\\widehat{G}$ on $C^*(E)$ (cf.",
"[22]) by $ \\gamma ^c_{\\chi }(S_e):=\\langle \\chi ,c(e)\\rangle S_e~\\mbox{for each}~\\chi \\in \\widehat{G},~e\\in E^1.$ Let $\\alpha $ be the trivial action of $\\widehat{G}$ on $A$ and let $\\eta $ be an action of $\\widehat{G}$ on $E(E)$ defined by $\\eta _{\\chi }(f)(e):= \\langle \\chi ,c(e)\\rangle f(e) ~\\mbox{for each}~\\chi \\in \\widehat{G},e\\in E^1,f\\in C_c(E^1).$ From [9] it follows that $\\gamma ^c$ coincides with the action of $G$ on $O_{E(E)}$ induced by the action $(\\eta ,\\alpha )$ .",
"Proposition 3.10 Let $E=(E^0,E^1,r,s)$ be a directed graph, $G$ be a finite abelian group and $c:E^1\\rightarrow G$ be a function.",
"Let $(\\eta ,\\alpha )$ be an action of $\\widehat{G}$ on the $C^*$ -correspondence $E(E)$ defined in the previous paragraph.",
"Then $(\\eta ,\\alpha )$ does not have the Rokhlin property.",
"Since $\\alpha $ is the trivial action of $\\widehat{G}$ on $A,$ we have $\\Vert \\alpha _g(f_s)-f_{gs}\\Vert &=&\\Vert f_s-f_{gs}\\Vert =\\Vert f_s+f_{gs}\\Vert \\\\&\\ge & ~\\mbox{max}~\\lbrace \\Vert f_s\\Vert ,\\Vert f_{gs}\\Vert \\rbrace ~\\mbox{for all}~ s,g\\in \\widehat{G};\\\\1&=&\\left\\Vert \\displaystyle \\sum _{s\\in \\widehat{G}} f_s\\right\\Vert \\le \\displaystyle \\sum _{s\\in \\widehat{G}}\\left\\Vert f_s\\right\\Vert .$ Thus $\\alpha $ does not have the Rokhlin property.",
"It follows that the condition $(2)$ in Definition REF is not satisfied, and hence the $\\alpha $ -compatible action $(\\eta ,\\alpha )$ does not have the Rokhlin property.",
"Corollary 3.11 Let $E=(E^0,E^1,r,s)$ be a directed graph, $G$ be a finite abelian group and $c:E^1\\rightarrow G$ be a function.",
"Let $(\\eta ,\\alpha )$ be an action of $\\widehat{G}$ on the $C^*$ -correspondence $E(E).$ Then the induced action $\\gamma ^c$ of $\\widehat{G}$ on $C^*(E)$ does not have the Rokhlin property with Rokhlin elements from $\\pi _{U}(A).$ Suppose that the induced action $\\gamma ^c$ of $\\widehat{G}$ on $C^*(E)$ has the Rokhlin property with Rokhlin elements from $\\pi _U(A).$ From [9] we have $\\gamma ^c_{\\chi }(\\pi _U (\\delta _v))&=&\\pi _U(\\alpha _\\chi (\\delta _v))=\\pi _U(\\delta _v)~\\mbox{for}~v\\in E^0,\\\\\\gamma ^c_{\\chi }(\\Psi _U (\\delta _e))&=& \\langle \\chi ,c(e)\\rangle (\\Psi _U (\\delta _e))=\\Psi _U (\\eta _\\chi (\\delta _e))~\\mbox{for}~e\\in E^1,$ for $\\chi \\in \\widehat{G}.$ Then Rokhlin elements belong to the fixed point algebra $C^*(E(c))^{\\gamma ^c}.$ Therefore the action $\\gamma ^c$ does not satisfy the condition $(2)$ in Definition REF , and we have a contradiction."
],
[
"Rokhlin property for group actions on Hilbert bimodules",
"Analogous to a right Hilbert $A$ -module, a left Hilbert $A$ -module is defined as a left $A$ -module with the positive definite form ${_{_{A}}}\\!\\langle \\cdot , \\cdot \\rangle :{E}\\times {E}\\rightarrow A$ which is, conjugate-linear in the second variable, linear in the first variable and we have ${_{_{A}}}\\!\\langle a x ,y\\rangle =a {_{_{A}}}\\!\\langle x,y\\rangle ~\\mbox{for}~x,y\\in {E},a\\in A.$ Definition 4.1 Let $A$ and $B$ be two $C^*$ -algebras.",
"A left Hilbert $B$ -module ${E}$ is called Hilbert $B$ -$A$ bimodule if it is also a right Hilbert $A$ -module satisfying ${_{_{B}}}\\!\\langle x,y \\rangle z=x \\langle y,z\\rangle \\!_{_{A}}~\\mbox{for}~x,y,z\\in E.$ On a Hilbert bimodule we consider actions of a locally compact group similar to those introduced in Definition REF .",
"Definition 4.2 Let $(G,\\alpha ,A)$ and $(G,\\beta ,B)$ be $C^*$ -dynamical systems of a locally compact group $G$ and let ${E}$ be a $B$ -$A$ Hilbert bimodule.",
"A $\\beta $ -compatible action (respectively an $\\alpha $ -compatible action) $\\eta $ of $G$ on ${E}$ is defined as a group homomorphism from $G$ into the group of invertible linear transformations on ${E}$ such that $\\eta _g (bx)=\\beta _g (b)\\eta _g(x)~(respectively~\\eta _g(xa)=\\eta _g(x)\\alpha _g (a)$ ), ${_{_{B}}}\\!\\langle \\eta _g (x) ,\\eta _g (y)\\rangle =\\beta _g({_{_{B}}}\\!\\langle x,y \\rangle )~(respectively~\\langle \\eta _g ({x}) ,\\eta _g (y)\\rangle \\!_{_{A}}=\\alpha _g(\\langle x,y \\rangle \\!_{_{A}})$ ) for $a\\in A,~b\\in B,~x,y\\in {E}$ ,$~g\\in G$ ; and $g\\mapsto \\eta _g(x)$ is continuous from $G$ into ${E}$ for each $x\\in {E}$ .",
"The combination of these two compatibility conditions will be simply called $(\\beta ,\\alpha )$ -compatibility.",
"Consider a $(\\beta ,\\alpha )$ -compatible action $\\eta $ of a locally compact group $G$ on the $B$ -$A$ Hilbert bimodule ${E}$ .",
"The crossed product bimodule $E\\times _{\\eta }G$ (cf.",
"[16], [5]) is an $B\\times _{\\beta }G$ -$A\\times _{\\alpha }G$ Hilbert bimodule obtained by completion of $C_c(G,{E})$ such that $ (l g)(s)=\\int _{G}l(t)\\alpha _{t}(g(t^{-1}s))dt&,~~~~~(f m)(s)=\\int _G f(t)\\eta _t (m(t^{-1}s))dt,\\\\\\langle l,m\\rangle \\!_{_{A\\times _{\\alpha }G}}(s)&=\\int _{G}\\alpha _{t^{-1}}(\\langle l(t),m(ts)\\rangle \\!_{_{A}})dt,\\\\{_{_{B\\times _{\\beta }G}}}\\!\\langle l,m\\rangle (s)&=\\int _{G}{_{_{B}}}\\!\\langle l(st^{-1}),\\eta _s (m(t^{-1}))\\rangle dt$ for all $f\\in C_c(G,B)$ , $g\\in C_c (G,A)$ and $l,m\\in C_c (G,{E})$ .",
"If ${E}$ is a (right) Hilbert $A$ -module, then ${E}$ is a $\\mathcal {K}({E})$ -$A$ Hilbert bimodule with respect to ${_{_{\\mathcal {K}({E})}}}\\!\\langle x,y\\rangle =\\theta _{x,y}$ for all $x,y\\in E$ .",
"Moreover, we can associate a $C^*$ -algebra called the linking algebra, defined by ${\\mathfrak {L}}_E:=\\begin{pmatrix}\\mathcal {K} ({E}) & {{E}} \\\\{{{E}}^*} & {A}\\end{pmatrix} \\subset \\mathcal {K} ({E}\\oplus A )$ (cf.",
"[30]), to each (right) Hilbert $A$ -module $E$ .",
"Let $(G,\\alpha ,A)$ be a $C^*$ -dynamical system and $\\eta $ be an $\\alpha $ -compatible action of $G$ on ${E}$ .",
"For each $s\\in G$ , let us define $Ad\\eta _s (t):=\\eta _s t \\eta _{s^{-1}}$ for $t\\in \\mathcal {K}({E})$ where ${\\eta ^*_s}({x}^*):={\\eta _s(x)}^*$ for $x\\in {E}$ .",
"Indeed, $\\eta $ is also an $(Ad \\eta , \\alpha )$ -compatible action and we get the induced action $\\theta $ of $G$ on $\\mathfrak {L}_{{E}}$ (cf.",
"[5], [23]) defined by $\\theta _s \\begin{pmatrix}{t} & {{x}} \\\\{y^*} & a\\end{pmatrix}:=\\begin{pmatrix}{Ad\\eta _s t} & {{\\eta _s}({x})} \\\\{\\eta ^*_s(y^*)} & \\alpha _s(a)\\end{pmatrix}.$ for all $s\\in G$ , $t\\in \\mathcal {K}({E})$ , $a\\in A$ and $x,y\\in {E}$ .",
"We denote this $C^*$ -dynamical system by $(G,\\theta ,{\\mathfrak {L}}_{{E}})$ ."
],
[
"Rokhlin property for induced finite group actions on linking algebras",
"Analogous to Definition REF the Rokhlin property for finite group actions on Hilbert bimodules is defined as follows: Definition 4.3 Let $(G,\\alpha ,A)$ and $(G,\\beta ,B)$ be $C^*$ -dynamical systems of a finite group $G$ and let ${E}$ be a $B$ -$A$ Hilbert bimodule.",
"Assume $\\eta $ to be a $(\\beta ,\\alpha )$ -compatible action of $G$ on ${E}$ .",
"We say that $\\eta $ has the Rokhlin property if for each $\\epsilon >0$ finite subsets $S_1$ and $S_2$ of $E$ , and finite subsets $S_3$ and $S_4$ of $B$ and $A$ respectively, there are sets $(a_g)_{g\\in G}\\subset A$ and $(b_g)_{g\\in G}\\subset B$ consisting of mutually orthogonal positive contractions such that $\\Vert \\sum _{g\\in G} a_g u -u\\Vert <\\epsilon $ for all $u\\in S^*_2\\cup S_4$ , $\\Vert \\sum _{g\\in G} b_g v -v\\Vert <\\epsilon $ for all $v\\in S_1\\cup S_3$ .",
"$\\Vert \\alpha _h(a_g)-a_{hg}\\Vert <\\epsilon $ and $\\Vert \\beta _h(b_g)-b_{hg}\\Vert <\\epsilon $ for $h,g\\in G$ , $\\Vert x a_g-b_g x\\Vert <\\epsilon $ , $\\Vert t b_g-b_g t\\Vert <\\epsilon $ and $\\Vert a_g a-a a_g\\Vert <\\epsilon $ , $\\Vert a_g y^*-y^*b_g \\Vert <\\epsilon $ for all $x\\in S_1$ , $y\\in S_2$ , $t\\in S_3$ , $a\\in S_4$ and $g\\in G$ .",
"Following theorem justifies the choice of this version of Rokhlin property for group actions on a bimodule: Theorem 4.4 Let ${E}$ be a Hilbert $A$ -module where $A$ is $C^*$ -algebra.",
"Suppose $\\alpha :G\\rightarrow Aut(A)$ is an action of a finite group $G$ and $\\eta $ is an $\\alpha $ -compatible action of $G$ on ${E}$ .",
"Then the following statements are equivalent: (a) $\\eta $ has the Rokhlin property as an (Ad$\\eta ,\\alpha $ )-compatible action.",
"(b) The action $\\theta $ of $G$ on ${\\mathfrak {L}}_{{E}}$ induced by $\\eta $ has the Rokhlin property with Rokhlin elements coming from the $C^*$ -subalgebra $\\begin{pmatrix}\\mathcal {K}({E})& {0} \\\\{0} &{A}\\end{pmatrix}$ of ${\\mathfrak {L}}_{{E}}$ .",
"Let $\\epsilon >0$ be given and $S=\\left\\lbrace \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a^{\\prime }_i\\end{pmatrix}:i=1,2,\\ldots ,n\\right\\rbrace $ be any finite subset of ${\\mathfrak {L}}_{{E}}$ .",
"Consider $S_1 =\\lbrace x_1,x_2,\\ldots ,x_n\\rbrace $ , $S_2 =\\lbrace y_1,y_2,\\ldots ,y_n \\rbrace $ , $S_3=\\lbrace t_1,t_2,\\ldots ,t_n\\rbrace $ and $S_4=\\lbrace a^{\\prime }_1,a^{\\prime }_2,\\ldots ,a^{\\prime }_n\\rbrace $ .",
"Suppose $\\eta $ has the Rokhlin property as an $(Ad\\eta ,\\alpha )$ -compatible action, there are sets $(a_g)_{g\\in G}\\subset A$ and $(b_g)_{g\\in G}\\subset \\mathcal {K}({E})$ consisting of mutually orthogonal positive contractions such that $\\Vert \\sum _{g\\in G} a_g u -u\\Vert <\\frac{\\epsilon }{4}$ for all $u\\in S^*_2\\cup S_4$ , $\\Vert \\sum _{g\\in G} b_g v -v\\Vert <\\frac{\\epsilon }{4}$ for all $v\\in S_1\\cup S_3$ .",
"$\\Vert \\alpha _h(a_g)-a_{hg}\\Vert <\\frac{\\epsilon }{4}$ and $\\Vert \\beta _h(b_g)-b_{hg}\\Vert <\\frac{\\epsilon }{4}$ for $h,g\\in G$ , $\\Vert x a_g-b_g x\\Vert <\\frac{\\epsilon }{4}$ , $\\Vert t b_g-b_g t\\Vert <\\frac{\\epsilon }{4}$ and $\\Vert a_g a-a a_g\\Vert <\\frac{\\epsilon }{4}$ , $\\Vert a_g y^*-y^*b_g \\Vert <\\frac{\\epsilon }{4}$ for all $x\\in S_1$ , $y\\in S_2$ , $t\\in S_3$ , $a\\in S_4$ and $g\\in G$ .",
"We prove that the action $\\theta $ of $G$ on ${\\mathfrak {L}}_E$ induced by $\\eta $ has the Rokhlin property with respect to $(f_g)_{g\\in G}$ where $f_g:=\\begin{pmatrix}{b_g} & {0} \\\\{0} & {a_g}\\end{pmatrix}$ .",
"For each $g\\in G$ , $ \\Vert f_g\\Vert =\\sup _{\\Vert (x,a)\\Vert \\le 1} \\left\\Vert \\begin{pmatrix}{b_g} & {0} \\\\{0} & {a_g}\\end{pmatrix}\\begin{pmatrix}{x} \\\\{a}\\end{pmatrix}\\right\\Vert \\le 1.",
"$ Further for $g,h\\in G$ with $g\\ne h$ we get $f_g f_h&= \\begin{pmatrix}{b_g} & {0} \\\\{0} & a_g\\end{pmatrix}\\begin{pmatrix}{b_h} & {0} \\\\{0} & a_h\\end{pmatrix} =0.$ Now we verify conditions (1)-(3) of Definition REF : $\\left\\Vert \\sum _{g\\in G} f_g\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a^{\\prime }_i\\end{pmatrix} -\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a^{\\prime }_i\\end{pmatrix}\\right\\Vert $ $ =\\left\\Vert \\sum _{g\\in G} \\begin{pmatrix}{b_g} & {0} \\\\{0} & a_g\\end{pmatrix}\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a^{\\prime }_i\\end{pmatrix} -\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a^{\\prime }_i\\end{pmatrix}\\right\\Vert <\\epsilon .$ For $t,g\\in G$ , we have $\\left\\Vert \\theta _h(f_g)-f_{hg}\\right\\Vert &=\\left\\Vert \\theta _h \\begin{pmatrix}{b_g} & {0} \\\\{0} & a_g\\end{pmatrix}-\\begin{pmatrix}{b_{hg}} & {0} \\\\{0} & a_{hg}\\end{pmatrix}\\right\\Vert \\\\ &=\\left\\Vert \\begin{pmatrix}{\\eta _h b_g}\\eta ^*_h & {0} \\\\{0} & \\alpha _g (a_g)\\end{pmatrix}-\\begin{pmatrix}{b_{hg}} & {0} \\\\{0} & a_{hg}\\end{pmatrix}\\right\\Vert <\\epsilon .$ For each $g\\in G$ and $1\\le i\\le n$ , we get $\\left\\Vert \\left[f_g ,\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a^{\\prime }_i\\end{pmatrix}\\right]\\right\\Vert &=\\left\\Vert \\begin{pmatrix}{b_g} & {0} \\\\{0} & a_g\\end{pmatrix}\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a^{\\prime }_i\\end{pmatrix}-\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a^{\\prime }_i\\end{pmatrix}\\begin{pmatrix}{b_g} & {0} \\\\{0} & a_g\\end{pmatrix}\\right\\Vert \\\\&=\\left\\Vert \\begin{pmatrix}{b_g t_i} & {a_g x_i} \\\\{a_g {y}^*_i} & a_g a^{\\prime }_i\\end{pmatrix}-\\begin{pmatrix}{t_i b_g} & {x_i a_g} \\\\{{y}^*_i b_g} & a^{\\prime }_i a_g\\end{pmatrix}\\right\\Vert <4(\\epsilon /4)=\\epsilon .$ Conversely, let $\\epsilon >0$ and let $S_1,~ S_2$ be finite subsets of $E$ .",
"Let $S_3$ and $S_4$ be finite subsets of $\\mathcal {K}({E})$ and $A$ , respectively.",
"Choose $S=\\left\\lbrace \\begin{pmatrix}{t} & {0} \\\\{0} & {0}\\end{pmatrix},\\begin{pmatrix}{0} & {x} \\\\{0} & 0\\end{pmatrix},\\begin{pmatrix}{0} & {0} \\\\{y^*} & 0\\end{pmatrix},\\begin{pmatrix}{0} & {0} \\\\{0} & a\\end{pmatrix}:x\\in S_1, y\\in S_2, t\\in \\mathcal {K}({E}),a\\in A\\right\\rbrace .$ Suppose $\\theta $ has the Rokhlin property with Rokhlin elements $\\left\\lbrace \\begin{pmatrix}{b_g} & {0} \\\\{0} & a_g\\end{pmatrix}:g\\in G\\right\\rbrace $ $\\subset \\begin{pmatrix}\\mathcal {K}({E})& {0} \\\\{0} &{A}\\end{pmatrix}\\subset {\\mathfrak {L}}_{{E}}$ with respect to the set $S$ and $\\epsilon >0$ .",
"Then it is easy to verify that $\\eta $ has the Rokhlin property as an $(Ad\\eta ,\\alpha )$ -compatible action with the positive contractions $(a_g)_{g\\in G}\\subset A$ and $(b_g)_{g\\in G}\\subset \\mathcal {K}({E})$ with respect to the sets $S_1,~S_2$ , $S_3$ and $S_4$ on noting that the steps and arguments of the first part of this proof can be carried out in the reverse order.",
"Let $E$ be a Hilbert $A$ -module.",
"There are several examples of $C^*$ -algebras $A$ for which nuclear dimension of ${\\mathfrak {L}}_{{E}}$ is at most $n$ .",
"One way to obtain it is by considering $A$ with nuclear dimension at most $n$ , because from the fact that $A$ is a full hereditary $C^*$ -subalgebra of ${\\mathfrak {L}}_{{E}}$ it follows that the nuclear dimension of ${\\mathfrak {L}}_{{E}}$ is at most $n$ (see [37]).",
"Corollary 4.5 Let $(G,\\alpha ,A)$ be a $C^*$ -dynamical system where $G$ is finite group and $n\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .",
"Let ${E}$ be a Hilbert $A$ -module.",
"Assume $\\eta $ to be an action of $G$ on ${E}$ which has the Rokhlin property as an $(Ad\\eta ,\\alpha )$ -compatible action and $A$ belongs to any one of the classes, say $\\mathcal {C}$ , listed in Theorem REF , then ${\\mathfrak {L}}_{{E}\\times _{\\eta } G}$ also belongs to the same class $\\mathcal {C}$ .",
"It follows from Theorem REF that the induced action $\\theta $ on ${\\mathfrak {L}}_{{E}}$ has the Rokhlin property.",
"Since $A$ is a full hereditary subalgebra of ${\\mathfrak {L}}_{{E}}$ , the linking algebra ${\\mathfrak {L}}_{{E}}$ belongs to the class $\\mathcal {C}$ from Theorem REF , then ${\\mathfrak {L}}_{{E}} \\times _{\\theta } G$ also belongs to the same class.",
"We identify ${\\mathfrak {L}}_{{E}} \\times _{\\theta } G$ and ${\\mathfrak {L}}_{{E}\\times _{\\eta } G}$ (cf.",
"the proof of [5]).",
"Hence ${\\mathfrak {L}}_{{E}\\times _{\\eta } G}$ also belongs to the same class."
],
[
"Rokhlin property for induced actions of $\\mathbb {Z}$ on linking algebras",
"Next we investigate the case where action is of the infinite discrete group $\\mathbb {Z}$ over a Hilbert bimodule $E$ instead of action of finite group on $E$ .",
"Indeed, we use notation $\\eta $ for $\\eta _1$ and $\\alpha $ for $\\alpha _1$ .",
"We first recall the definition of the Rokhlin property for the automorphisms on $C^*$ -algebras from [2].",
"Definition 4.6 Let $A$ be a $C^*$ -algebra and $\\alpha \\in Aut(A)$ .",
"We say that $\\alpha $ has the Rokhlin property if for any positive integer $p$ , any finite set $S \\subset A$ , and any $\\epsilon >0$ , there are mutually orthogonal positive contractions $e_{0,0},\\ldots ,e_{0,p-1},$ $ e_{1,0},\\ldots ,e_{1,p}$ such that $\\left\\Vert \\left( \\sum _{r=0}^1\\sum _{j=0}^{p-1+r} e_{r,j} \\right) a - a \\right\\Vert <\\epsilon $ for all $a \\in S$ , $\\left\\Vert [e_{r,j},a]\\right\\Vert < \\epsilon $ for all $r,j$ and $a \\in S$ , $\\left\\Vert \\alpha (e_{r,j})a - e_{r,j+1}a\\right\\Vert <\\epsilon $ for all $a\\in S$ , $r=0,1$ and $j=0,1,\\ldots ,p-2+r$ , $\\left\\Vert \\alpha (e_{0,p-1} + e_{1,p})a - (e_{0,0} + e_{1,0})a\\right\\Vert <\\epsilon $ for all $a \\in S$ .",
"We call elements $e_{0,0},\\ldots ,e_{0,p-1}, e_{1,0},\\ldots ,e_{1,p}$ the Rokhlin elements for $\\alpha $ .",
"Definition 4.7 Let $(\\mathbb {Z},\\alpha ,A)$ and $(\\mathbb {Z},\\beta ,B)$ be $C^*$ -dynamical systems.",
"Suppose ${E}$ is a Hilbert $B$ -$A$ bimodule and $\\eta $ is a ($\\beta $ ,$\\alpha $ )-compatible automorphism of $E$ .",
"We say that $\\eta $ has the Rokhlin property if for any $\\epsilon >0,$ any positive integer $p$ , any finite subsets $S_1$ and $S_2$ of $E$ , and any finite subsets $S_3$ and $S_4$ of $B$ and $A$ respectively, there are sets consisting of mutually orthogonal positive contractions $\\lbrace a_{0,0},\\ldots ,a_{0,p-1},a_{1,0},\\ldots ,a_{1,p}\\rbrace \\subset A$ and $\\lbrace b_{0,0},\\ldots ,b_{0,p-1},b_{1,0},\\ldots ,b_{1,p}\\rbrace \\subset B$ such that $b_{i,j}b_{i^{\\prime },j^{\\prime }}=0$ and $a_{i,j}a_{i^{\\prime },j^{\\prime }}=0$ if $(i,j)\\ne (i^{\\prime },j^{\\prime })$ .",
"$\\left\\Vert \\sum _{r=0}^1\\sum _{j=0}^{p-1+r} b_{r,j} v-v\\right\\Vert <\\epsilon $ , $\\left\\Vert \\sum _{r=0}^1\\sum _{j=0}^{p-1+r} a_{r,j} u-u\\right\\Vert <\\epsilon $ for all $v\\in S_1 \\cup S_3$ , $u \\in S^*_2 \\cup S_4$ .",
"$\\Vert x a_{r,j}-b_{r,j} x\\Vert <\\epsilon $ , $\\Vert a_{r,j}y^*-y^*b_{r,j} \\Vert <\\epsilon $ , $\\Vert b b_{r,j}-b_{r,j} b\\Vert <\\epsilon $ , $\\Vert a_{r,j} a-aa_{r,j} \\Vert <\\epsilon $ for all $x\\in S_1,y\\in S_2,b\\in S_3,a\\in S_4$ and for all $r,j$ .",
"$\\Vert \\beta (b_{r,j}) v-b_{r,j+1} v\\Vert <\\epsilon ~,~\\Vert \\alpha (a_{r,j}) u-a_{r,j+1} u\\Vert <\\epsilon $ for all $v\\in S_1 \\cup S_3$ , $u \\in S^*_2 \\cup S_4$ ; $r=0,1$ and $j=0,\\ldots ,p-2+r$ .",
"$\\Vert \\beta (b_{0,p-1}+b_{1,p})v-(b_{0,0}+b_{1,0}) v\\Vert <\\epsilon $ , $\\Vert \\alpha (a_{0,p-1}+a_{1,p}) u-(a_{0,0}+a_{1,0}) u\\Vert <\\epsilon $ for all $v\\in S_1 \\cup S_3$ , $u \\in S^*_2 \\cup S_4$ .",
"The following observation is a justification for the choice of the above definition of Rokhlin property for actions of $\\mathbb {Z}$ on a bimodule: Theorem 4.8 Suppose $(\\mathbb {Z},\\alpha ,A)$ is a $C^*$ -dynamical system.",
"Assume ${E}$ to be a Hilbert $A$ -module and $\\eta $ to be an automorphism on $E$ .",
"The following statements are equivalent: (a) $\\eta $ has the Rokhlin property as an (Ad$\\eta $ ,$\\alpha $ )-compatible automorphism.",
"(b) The automorphism $\\theta $ in $Aut({\\mathfrak {L}}_{{E}})$ induced by $\\eta $ has the Rokhlin property with Rokhlin elements coming from the $C^*$ -subalgebra $\\begin{pmatrix}\\mathcal {K}({E})& {0} \\\\{0} &{A}\\end{pmatrix}$ of ${\\mathfrak {L}}_{{E}}$ .",
"Let $\\epsilon >0$ be given and $S=\\left\\lbrace \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}:i=1,2,\\ldots ,n\\right\\rbrace $ be any finite subset of ${\\mathfrak {L}}_{{E}}$ .",
"Consider $S_1 =\\lbrace x_1,x_2,\\ldots ,x_n\\rbrace $ , $S_2 =\\lbrace y_1,y_2,\\ldots ,y_n\\rbrace $ , $S_3=\\lbrace t_1,t_2,\\ldots ,t_n\\rbrace $ and $S_4=\\lbrace a_1,a_2,\\ldots ,a_n\\rbrace $ .",
"Suppose $\\eta $ has the Rokhlin property, there are sets consist of mutually orthogonal positive contractions $\\lbrace a_{0,0},\\ldots ,$ $a_{0,p-1},a_{1,0},\\ldots ,a_{1,p}\\rbrace \\subset A$ and $\\lbrace b_{0,0},\\ldots ,b_{0,p-1},b_{1,0},\\ldots ,b_{1,p}\\rbrace \\subset \\mathcal {K}({E})$ such that $b_{i,j}b_{i^{\\prime },j^{\\prime }}=0$ and $a_{i,j}a_{i^{\\prime },j^{\\prime }}=0$ if $(i,j)\\ne (i^{\\prime },j^{\\prime })$ .",
"$\\left\\Vert \\sum _{r=0}^1\\sum _{j=0}^{p-1+r} b_{r,j} v-v\\right\\Vert <\\frac{\\epsilon }{4}$ , $\\left\\Vert \\sum _{r=0}^1\\sum _{j=0}^{p-1+r} a_{r,j} u-u\\right\\Vert <\\frac{\\epsilon }{4}$ for all $v\\in S_1 \\cup S_3$ , $u \\in S^*_2 \\cup S_4$ .",
"$\\Vert x a_{r,j}-b_{r,j} x\\Vert <\\frac{\\epsilon }{4} $ , $\\Vert a_{r,j}y^*-y^*b_{r,j} \\Vert <\\frac{\\epsilon }{4} $ , $\\Vert b b_{r,j}-b_{r,j} b\\Vert <\\frac{\\epsilon }{4} $ , $\\Vert a_{r,j} a-aa_{r,j} \\Vert <\\frac{\\epsilon }{4} $ for all $x\\in S_1,y\\in S_2,b\\in S_3,a\\in S_4$ and for all $r,j$ .",
"$\\Vert \\beta (b_{r,j}) v-b_{r,j+1} v\\Vert <\\frac{\\epsilon }{4}~,~\\Vert \\alpha (a_{r,j}) u-a_{r,j+1} u\\Vert <\\frac{\\epsilon }{4}$ for all $v\\in S_1 \\cup S_3$ , $u \\in S^*_2 \\cup S_4$ ; $r=0,1$ and $j=0,\\ldots ,p-2+r$ .",
"$\\Vert \\beta (b_{0,p-1}+b_{1,p})v-(b_{0,0}+b_{1,0}) v\\Vert <\\frac{\\epsilon }{4}$ , $\\Vert \\alpha (a_{0,p-1}+a_{1,p}) u-(a_{0,0}+a_{1,0}) u\\Vert <\\frac{\\epsilon }{4}$ for all $v\\in S_1 \\cup S_3$ , $u \\in S^*_2 \\cup S_4$ .",
"We verify that the action $\\theta $ of $G$ on ${\\mathfrak {L}}_E$ induced by $\\eta $ has the Rokhlin property as an (Ad$\\eta ,\\alpha $ )-compatible automorphism with respect to $e_{0,0},\\ldots ,e_{0,p-1},e_{1,0},\\ldots ,$ $e_{1,p}$ where $e_{i,j}:=\\begin{pmatrix}b_{i,j} & {0} \\\\{0} & a_{i,j}\\end{pmatrix}$ .",
"For each $(i,j)$ , $ \\Vert e_{i,j}\\Vert =\\sup _{\\Vert (x,a)\\Vert \\le 1} \\left\\Vert \\begin{pmatrix}{b_{i,j}} & {0} \\\\{0} & a_{i,j}\\end{pmatrix}\\begin{pmatrix}{x} \\\\{a}\\end{pmatrix}\\right\\Vert \\le 1.$ For $(i,j)\\ne (i^{\\prime },j^{\\prime })$ we have $e_{i,j} e_{i^{\\prime },j^{\\prime }}&= \\begin{pmatrix}{b_{i,j}} & {0} \\\\{0} & a_{i,j}\\end{pmatrix}\\begin{pmatrix}{b_{i^{\\prime },j^{\\prime }}} & {0} \\\\{0} & a_{i^{\\prime },j^{\\prime }}\\end{pmatrix} = \\begin{pmatrix}{b_{i,j} b_{i^{\\prime },j^{\\prime }}} & {0} \\\\{0} & a_{i,j}a_{i^{\\prime },j^{\\prime }}\\end{pmatrix}=0.$ We verify conditions (1)-(4) of Definition REF below: For $1\\le i\\le n$ we get $&\\left\\Vert \\left(\\mbox{$\\sum _{r=0}^1\\sum _{j=0}^{p-1+r}$} e_{r,j} \\right) \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix} - \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix} \\right\\Vert \\\\&=\\left\\Vert \\left( \\mbox{$\\sum _{r=0}^1\\sum _{j=0}^{p-1+r}$} \\begin{pmatrix}{b_{r,j}} & {0} \\\\{0} & a_{r,j}\\end{pmatrix} \\right) \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix} - \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\right\\Vert <4\\times \\epsilon /4=\\epsilon .$ For all $r,j$ we have $&\\left\\Vert \\left[e_{r,j},\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\right]\\right\\Vert \\\\&=\\left\\Vert \\begin{pmatrix}{b_{r,j}} & {0} \\\\{0} & a_{r,j}\\end{pmatrix} \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\right.\\left.-\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\begin{pmatrix}{b_{r,j}} & {0} \\\\{0} & a_{r,j}\\end{pmatrix} \\right\\Vert <4\\times \\epsilon /4=\\epsilon .$ For $r=0,1$ and $j=0,\\ldots ,p-2+r$ , we have $&\\left\\Vert \\theta (e_{r,j})\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix} - e_{r,j+1}\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\right\\Vert \\\\&=\\left\\Vert \\begin{pmatrix}{\\eta (b_{r,j})\\eta ^*} & {0} \\\\{0} & \\alpha (a_{r,j})\\end{pmatrix}\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix} \\right.-\\left.\\begin{pmatrix}{b_{r,j+1}} & {0} \\\\{0} & a_{r,j+1}\\end{pmatrix}\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\right\\Vert <\\epsilon .$ For $1\\le i\\le n$ , we get $ &\\left\\Vert \\theta (e_{0,p-1} +e_{1,p})\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}- (e_{0,0} + e_{1,0})\\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\right\\Vert \\\\=&\\left\\Vert \\begin{pmatrix}{\\eta (b_{0,p-1}+b_{1,p})\\eta ^*} & {0} \\\\{0} & \\alpha (a_{0,p-1}+a_{1,p})\\end{pmatrix} \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\right.\\\\ &\\left.-\\begin{pmatrix}{b_{0,0}+b_{1,0}} & {0} \\\\{0} & a_{0,0}+a_{1,0}\\end{pmatrix} \\begin{pmatrix}{t_i} & {{x_i}} \\\\{y^*_i} & a_i\\end{pmatrix}\\right\\Vert <\\epsilon .$ Conversely, let $S_1 \\cup S_2$ , $S_3$ and $S_4$ be finite subsets of $E$ , $\\mathcal {K}({E})$ and $A$ , respectively.",
"Let $S=\\left\\lbrace \\begin{pmatrix}{t} & {0} \\\\{0} & {0}\\end{pmatrix},\\begin{pmatrix}{0} & {x} \\\\{0} & 0\\end{pmatrix},\\begin{pmatrix}{0} & {0} \\\\{y^*} & 0\\end{pmatrix},\\begin{pmatrix}{0} & {0} \\\\{0} & a\\end{pmatrix}:x\\in S_1, y\\in S_2, t\\in \\mathcal {K}({E}),a\\in A\\right\\rbrace .$ Suppose $\\theta $ has the Rokhlin property with Rokhlin elements $\\begin{pmatrix}{b_{0,0}} & {0} \\\\{0} & a_{0,0}\\end{pmatrix},$ $\\begin{pmatrix}{b_{0,1}} & {0} \\\\{0} & a_{0,1}\\end{pmatrix},\\ldots ,\\begin{pmatrix}{b_{0,p-1}} & {0} \\\\{0} & a_{0,p-1}\\end{pmatrix},$ $\\begin{pmatrix}{b_{1,0}} & {0} \\\\{0} & a_{1,0}\\end{pmatrix},\\begin{pmatrix}{b_{1,1}} & {0} \\\\{0} & a_{1,1}\\end{pmatrix},\\ldots ,$ $\\begin{pmatrix}{b_{1,p}} & {0} \\\\{0} & a_{1,p}\\end{pmatrix}$ coming from the $C^*$ -algebra $\\begin{pmatrix}\\mathcal {K}({E})& {0} \\\\{0} &{A}\\end{pmatrix}\\subset {\\mathfrak {L}}_{{E}}$ with respect to the set $S$ and any $\\epsilon >0$ .",
"Then it is easy to check that $\\eta $ has the Rokhlin property as an $(Ad\\eta ,\\alpha )$ -compatible action with the positive contractions $\\lbrace a_{0,0},\\ldots ,a_{0,p-1},$ $a_{1,0},\\ldots ,a_{1,p}\\rbrace \\subset A$ and $\\lbrace b_{0,0},\\ldots ,b_{0,p-1},$ $b_{1,0},\\ldots ,b_{1,p}\\rbrace \\subset \\mathcal {K}({E})$ with respect to the sets $S_1 \\cup S_2$ , $S_3$ and $S_4$ on observing that the steps and arguments of the first part of this proof can be carried out in the reverse order.",
"We recall the definition of $D$ -absorbing (cf.",
"[11]).",
"Definition 4.9 A separable, unital $C^*$ -algebra $ {D} \\ncong \\mathbb {C}$ is strongly self-absorbing if there exists an isomorphism $ \\varphi :{D} \\rightarrow {D} \\otimes {D}$ such that $ \\varphi $ and ${id}_{{D}} \\otimes {1}_{{D}}$ are approximately unitarily equivalent $*$ -homomorphisms.",
"If $D$ is a strongly self-absorbing $C^*$ -algebra, we say that a $C^*$ -algebra $A$ is $D$ -absorbing if $A\\cong A\\otimes D$ .",
"In the following we observe a permanence property of the $D$ -absorbing property with respect to the crossed product ${E}\\times _{\\eta } \\mathbb {Z}$ of a bimodule $E$ for an (Ad$\\eta ,\\alpha $ )-compatible action with Rokhlin property: Corollary 4.10 Assume $(\\mathbb {Z},\\alpha ,A)$ to be a $C^*$ -dynamical system.",
"Let $\\eta $ be an $\\alpha $ -compatible automorphism on a Hilbert $A$ -module ${E}$ and let $D$ be a strongly self-absorbing $C^*$ -algebra.",
"If $\\eta \\in Aut({E})$ has the Rokhlin property as an (Ad$\\eta ,\\alpha $ )-compatible action and let $A$ be separable and $D$ -absorbing, then ${\\mathfrak {L}}_{{E}\\times _{\\eta } \\mathbb {Z}}$ is $D$ -absorbing.",
"By Theorem REF the induced automorphism $\\theta $ on ${\\mathfrak {L}}_{{E}}$ has the Rokhlin property.",
"Since ${A}$ is a full hereditary $C^*$ -subalgebra of ${\\mathfrak {L}}_{{E}}$ , ${\\mathfrak {L}}_{{E}}$ is separable and $D$ -absorbing, ${\\mathfrak {L}}_{{E}} \\times _{\\theta } \\mathbb {Z}$ is $D$ -absorbing (cf.",
"[2]).",
"We identify ${\\mathfrak {L}}_{{E}} \\times _{\\theta } \\mathbb {Z}$ and ${\\mathfrak {L}}_{{E}\\times _{\\eta } \\mathbb {Z}}$ (cf.",
"the proof of [5]) and hence ${\\mathfrak {L}}_{{E}\\times _{\\eta } \\mathbb {Z}}$ is $D$ -absorbing."
],
[
"Outerness for group actions on Hilbert bimodules",
"In this section we define and explore outer actions of a locally compact group on a Hilbert bimodule.",
"Definition 5.1 Let $(G,\\alpha ,A)$ and $(G,\\beta ,B)$ be unital $C^*$ -dynamical systems of a locally compact group $G$ and let ${E}$ be a $B$ -$A$ Hilbert bimodule.",
"Let $u$ and $u^{\\prime }$ be two unitaries in $A$ and $B$ , respectively.",
"Define an $(Ad(u^{\\prime }),Ad(u))$ -compatible automorphism $Ad(u^{\\prime },u):E\\rightarrow E$ by $Ad(u^{\\prime },u)(x)=u^{\\prime *} x u~\\mbox{ for each}~x\\in E.$ Let $\\eta $ be an $(\\beta ,\\alpha )$ -compatible action of $G$ on ${E}$ .",
"We say that $\\eta $ is outer if for each $t\\in G\\setminus \\lbrace e\\rbrace $ we have $\\eta _t\\ne Ad(u^{\\prime },u)$ for any unitaries $u\\in A$ and $u^{\\prime }\\in B$ , where $e$ denotes the identity of $G$ .",
"In a $B$ -$A$ Hilbert bimodule, it was pointed out in [3] that ${_{_{B}}}\\!\\langle xa,y\\rangle ={_{_{B}}}\\!\\langle x,ya^*\\rangle ~\\mbox{and}~\\langle bx,y\\rangle \\!_{_{A}}=\\langle x,b^*y\\rangle \\!_{_{A}}~\\mbox{for all}~x,y\\in E;~ a\\in A;~b\\in B.$ Indeed, using these conditions in the following computations we show that $Ad(u^{\\prime },u)$ is an $(Ad(u^{\\prime }),Ad(u))$ -compatible automorphism: For $x\\in E$ , $a\\in A$ and $b\\in B$ we get $Ad(u^{\\prime },u)(x)Ad (u)(a)=u^{\\prime *} x u u^*au=u^{\\prime *} x au=Ad(u^{\\prime },u)(xa),$ $Ad (u^{\\prime })(b)Ad(u^{\\prime },u)(x)=u^{\\prime *}bu^{\\prime }u^{\\prime *} x u =u^{\\prime *}b x u=Ad(u^{\\prime },u)(bx).$ For each $x,y\\in E$ we have $\\langle Ad(u^{\\prime },u)(x), Ad(u^{\\prime },u)(y) \\rangle \\!_{_{A}} &=\\langle u^{\\prime *} x u, u^{\\prime *} y u \\rangle \\!_{_{A}}=\\langle x u, u^{\\prime }u^{\\prime *} y u \\rangle \\!_{_{A}}=\\langle x u, y u \\rangle \\!_{_{A}}\\\\ &=u^*\\langle x , y \\rangle \\!_{_{A}} u=Ad (u)(\\langle x , y \\rangle \\!_{_{A}}),$ ${_{_{B}}}\\!\\langle Ad(u^{\\prime },u)(x), Ad(u^{\\prime },u)(y) \\rangle &={_{_{B}}}\\!\\langle u^{\\prime *} x u, u^{\\prime *} y u \\rangle ={_{_{B}}}\\!\\langle u^{\\prime *}x , u^{\\prime *} y uu^* \\rangle ={_{_{B}}}\\!\\langle u^{\\prime *} x , u^{\\prime *}y \\rangle \\\\ &=u^{\\prime *}{_{_{B}}}\\!\\langle x , y \\rangle u^{\\prime }=Ad (u^{\\prime })({_{_{B}}}\\!\\langle x , y \\rangle ).$ We check our definition of outerness for group actions on Hilbert bimodules is compatible with the definition of outerness for group actions on $C^*$ -algebras as follows: If $A=B$ , then $A$ naturally becomes an $A$ -$A$ Hilbert bimodule.",
"In this case we fix $(G,\\alpha ,A)$ and take $\\eta =\\beta =\\alpha $ .",
"If $\\eta $ is not outer, then $\\eta _s=Ad(u^{\\prime },u)$ for some $s\\in G,~u\\in A,~u^{\\prime }\\in B$ .",
"Further from the above computations, this gives $u=u^{\\prime }$ and $\\beta _s=\\alpha _s=Ad(u)$ .",
"Thus $\\alpha $ can not be outer.",
"Hence $\\alpha $ is outer implies $\\eta $ is outer.",
"The following proposition says that outerness for group actions on Hilbert bimodules is a weaker notion than outerness for group actions on $C^*$ -algebras.",
"Proposition 5.2 Suppose $(G,\\alpha ,A)$ and $(G,\\beta ,B)$ are unital $C^*$ -dynamical systems of a locally compact group $G$ and ${E}$ is a $B$ -$A$ Hilbert bimodule.",
"Let $\\eta $ be an $(\\beta ,\\alpha )$ -compatible action of $G$ on ${E}$ .",
"If $E$ is full with respect to both the inner products, and if $\\alpha $ or $\\beta $ is outer, then $\\eta $ is outer.",
"Let $s\\in G\\setminus \\lbrace e\\rbrace $ and let $u_s$ and $u^{\\prime }_s$ be unitaries in $A$ .",
"If $\\eta _s=Ad(u^{\\prime }_s,u_s)$ , then for each $x,y\\in E$ we get $\\Vert \\alpha _s(\\langle x,y\\rangle \\!_{_{A}})-u^{*}_s\\langle x,y\\rangle \\!_{_{A}} u_s\\Vert &=& \\Vert \\langle \\eta _s (x),\\eta _s (y)\\rangle \\!_{_{A}}-u^{*}_s\\langle x,y\\rangle \\!_{_{A}} u_s\\Vert \\\\&=& \\Vert \\langle \\eta _s (x),\\eta _s (y)\\rangle \\!_{_{A}}-\\langle xu_s,yu_s\\rangle \\!_{_{A}} \\Vert \\\\&=& \\Vert \\langle \\eta _s (x),\\eta _s (y)\\rangle \\!_{_{A}}-\\langle xu_s,u^{\\prime }_su^{\\prime *}_syu_s\\rangle \\!_{_{A}} \\Vert \\\\&=& \\Vert \\langle \\eta _s (x),\\eta _s (y)\\rangle \\!_{_{A}}-\\langle u^{\\prime *}_s xu_s,u^{\\prime *}_syu_s\\rangle \\!_{_{A}} \\Vert =0.$ Similarly if $\\eta _s=Ad(u^{\\prime }_s,u_s)$ , then we obtain $\\beta _s ({_{_{B}}}\\!\\langle x,y\\rangle )=u^{\\prime *}_s({_{_{B}}}\\!\\langle x,y\\rangle )u^{\\prime }_s$ for each $x,y\\in E$ .",
"Corollary 5.3 Suppose $(G,\\alpha ,A)$ and $(G,\\beta ,B)$ are unital $C^*$ -dynamical systems of a finite group $G$ and ${E}$ is a $B$ -$A$ Hilbert bimodule.",
"Let $\\eta $ be an $(\\beta ,\\alpha )$ -compatible action of $G$ on ${E}$ .",
"If $E$ is full with respect to both the inner products, and if $\\eta $ has the Rokhlin property, then $\\eta $ is outer.",
"Since $\\eta $ is an $(\\beta ,\\alpha )$ -compatible action of the finite group $G$ on ${E}$ with the Rokhlin property, it follows from Definition REF that $\\beta $ and $\\alpha $ have the Rokhlin property.",
"This implies that $\\beta $ and $\\alpha $ are outer (cf.",
"[34]).",
"Therefore $\\eta $ is outer by Proposition REF .",
"Acknowledgements: The first author was supported by Seed Grant from IRCC, IIT Bombay, the second author was supported by JSPS KAKENHI Grant Number 26400125, and the third author was supported by CSIR, India.",
"Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India Email address: [email protected] Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan Email address: [email protected] Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India Email address: [email protected]"
]
] | 1605.06050 |
[
[
"Twitter as a Lifeline: Human-annotated Twitter Corpora for NLP of\n Crisis-related Messages"
],
[
"Abstract Microblogging platforms such as Twitter provide active communication channels during mass convergence and emergency events such as earthquakes, typhoons.",
"During the sudden onset of a crisis situation, affected people post useful information on Twitter that can be used for situational awareness and other humanitarian disaster response efforts, if processed timely and effectively.",
"Processing social media information pose multiple challenges such as parsing noisy, brief and informal messages, learning information categories from the incoming stream of messages and classifying them into different classes among others.",
"One of the basic necessities of many of these tasks is the availability of data, in particular human-annotated data.",
"In this paper, we present human-annotated Twitter corpora collected during 19 different crises that took place between 2013 and 2015.",
"To demonstrate the utility of the annotations, we train machine learning classifiers.",
"Moreover, we publish first largest word2vec word embeddings trained on 52 million crisis-related tweets.",
"To deal with tweets language issues, we present human-annotated normalized lexical resources for different lexical variations."
],
[
"Introduction",
"Twitter has been extensively used as an active communication channel, especially during mass convergence events such as natural disasters like earthquakes, floods, typhoons [9], [6].",
"During the onset of a crisis, a variety of information is posted in real-time by affected people; by people who are in need of help (e.g., food, shelter, medical assistance, etc.)",
"or by people who are willing to donate or offer volunteering services.",
"Moreover, humanitarian and formal crisis response organizations such as government agencies, public health care NGOs, and military are tasked with responsibilities to save lives, reach people who need help, etc. [17].",
"Situation-sensitive requirements arise during such events and formal disaster response agencies look for actionable and tactical information in real-time to effectively estimate early damage assessment, and to launch relief efforts accordingly.",
"Recent studies have shown the importance of social media messages to enhance situational awareness and also indicate that these messages contain significant actionable and tactical information [2], [7], [14].",
"Many Natural-Language-Processing (NLP) techniques such as automatic summarization, information classification, named-entity recognition, information extraction can be used to process such social media messages [1], [9].",
"However, many social media messages are very brief, informal, and often contain slangs, typograpical errors, abbreviations, and incorrect grammar [5].",
"These issues degrade the performance of many NLP techniques when used down the processing pipeline [15], [4].",
"We present Twitter corpora consisting of more than 52 million crisis-related messages collected during 19 different crises.",
"We provide human annotations (volunteers and crowd-sourced workers) of two types.",
"First, the tweets are annotated with a set of categories such as displaced people, financial needs, infrastructure, etc.",
"These annotation schemes were built using input taken from formal crisis response agencies such as United Nations Office for the Coordination of Humanitarian Affairs (UN OCHA).",
"Second, the tweets are annotated to identify out-of-vocabulary(OOV) terms, such as slangs, places names, abbreviations, misspellings, etc.",
"and their corrections and normalized forms.",
"This dataset can form the basis for research in text classification for short messages and for research on normalizing informal language.",
"Creating large corpora for training supervised machine-learning models is hard because it requires time and money that may not be available.",
"However, since our dataset was used for disaster relief efforts, volunteers were willing to annotate it; this work can now be leveraged to improve text classification and language processing tasks.",
"Our work provides annotations for around 50,000 thousand messages, which is a significant corpus, that will enable research into applied machine learning and consequently benefit the disaster relief (and other) research communities.",
"Our dataset has been collected from various countries and during various times of the year.",
"This diversity would make it an interesting dataset that if used would be a foil to solutions that only work for specific language “dialects\", e.g., American English and would fail or suffer from degradation of quality if applied to variations, such as Indian English.",
"Our work shows that when a dataset is used for a real application, we could obtain larger number of annotations than otherwise.",
"These can then be used to improve text processing as a by-product.",
"The annotated data is also used to train machine-learning classifiers.",
"In this case, we use three well-known learning algorithms: Naive Bayes, Random Forest, and Support Vector Machines (SVM).",
"We remark that these classifiers are useful for formal crisis response organizations as well as for the research community to build more effective computational methods [13], [9] on top.",
"We also train word2vec word embeddings from all 52 million messages and make them available to research community."
],
[
"Contributions",
"The contributions of this paper are as follows: We present human-annotated crisis-related messages collected during 19 different crises We use human-annotations to built machine-learning classifiers in a multiclass classification setting to classify messages that are useful for humanitarian efforts We provide first largest word2vec word embeddings trained using 52 million crisis-related messages We use the collected data to identify OOV (out-of-vocabulary) words and provide human-annotated normalized lexical resources for different lexical variations"
],
[
"Paper organization",
"The rest of the paper is organized as follows.",
"In the next section, we describe datasets details and annotation schemes.",
"Section 3 describes supervised classification task and word2vec word embeddings.",
"Section 4 provides details of text normalization and we present related work in section 5.",
"We conclude the paper in section 6."
],
[
"Data collection",
"We collected crisis-related messages from Twitter posted during 19 different crises that took place from 2013 to 2015.",
"Table REF shows the list of crisis events along with their names, crisis type (e.g.",
"earthquake, flood), countries where they took place, and the number of tweets each crisis contains.",
"We collected these messages using our AIDR (Artificial Intelligence for Disaster Response) platform [8].",
"AIDR is an open source platform to collect and classify Twitter messages during the onset of a humanitarian crisis.",
"AIDR has been used by UN OCHA during many major disasters such as Nepal Earthquake, Typhoon Hagupit.",
"AIDR provides different convenient ways to collect messages from Twitter using the Twitter's streaming API.",
"One can use different data collection strategies.",
"For example, collecting tweets that contain some keywords and are specifically from a particular geographical area/region/city (e.g.",
"New York).",
"The detailed data collection strategies used to collect the datasets shown in Table REF are included in each dataset folder.",
"Table: Crises datasets details including crisis type, name, year, language of messages, country, # of tweets."
],
[
"Data annotation",
"Messages posted on social media vary greatly in terms of information they contain.",
"For example, users post messages of personal nature, messages useful for situational awareness (e.g.",
"infrastructure damage, causalities, individual needs), or not related to the crisis at all.",
"Depending on their information needs, different humanitarian organizations use different annotation schemes to categories these messages.",
"In this work, we use a subset of the annotations used by the United Nations Office for the Coordination of Humanitarian Affairs (UN OCHA).",
"The 9 category types (including two catch-all classes: “Other Useful Information\" and “Irrelevant\") used by the UN OCHA are shown in the below-presented annotation scheme.",
"For most of the datasets we have performed annotations by employing volunteers and paid workers.",
"To perform volunteered-based annotations, messages were collected from Twitter in real-time and passed through a de-duplication process.",
"Only unique messages were considered for human-annotation.",
"We use Stand-By-Task-Force (SBTF)http://blog.standbytaskforce.com/ volunteers to annotate messages using our MicroMappers platform.http://micromappers.org/ The real-time annotation process helps train machine learning classifiers rapidly, which are then used to classify new incoming messages.",
"This process helps address time-critical information needs requirement of many humanitarian organizations.",
"After the first round of annotations, we found that some categories are small in terms of number of labels thus showing high class-imbalance.",
"A dataset is said to be imbalanced if at least one of the classes has significantly fewer annotated instances than the others.",
"The class imbalance problem has been known to hinder the learning performance of classification algorithms.",
"In this case, we performed another round of annotations for datasets that have high class imbalance using the paid crowdsourcing platform CrowdFlower.http://crowdflower.com/ In both annotation processes, an annotation task consists of a tweet and the list of categories listed below.",
"A paid worker or volunteer reads the message and selects one of the categories most suitable for the message.",
"Messages that do not belong to any category but contain some important information are categorized as “Other Useful Information\".",
"A task is finalized (i.e.",
"a category is assigned) when three different volunteers/paid workers agree on a category.",
"According to the Twitter's data distribution policy, we are not allowed to publish actual contents of more than 50k tweets.",
"For this reason, we publish all annotated tweets, which are less than 50k, along with tweet-ids of all the unannotated messages at http://CrisisNLP.qcri.org/.",
"We also provide a tweets retrieval tool implemented in Java, which can be used to get full tweets content from Twitter.",
"In below we show the annotation scheme used for crisis events caused by natural disasters.",
"For other events, details regarding their annotations are available with the published data.",
"Annotation scheme: Categorizing messages by information types Injured or dead people: Reports of casualties and/or injured people due to the crisis Missing, trapped, or found people: Reports and/or questions about missing or found people Displaced people and evacuations: People who have relocated due to the crisis, even for a short time (includes evacuations) Infrastructure and utilities damage: Reports of damaged buildings, roads, bridges, or utilities/services interrupted or restored Donation needs or offers or volunteering services: Reports of urgent needs or donations of shelter and/or supplies such as food, water, clothing, money, medical supplies or blood; and volunteering services Caution and advice: Reports of warnings issued or lifted, guidance and tips Sympathy and emotional support: Prayers, thoughts, and emotional support Other useful information: Other useful information that helps understand the situation Not related or irrelevant: Unrelated to the situation or irrelevant"
],
[
"Classification of Messages",
"To make sense of huge amounts of Twitter messages posted during crises, we consider a basic operation, that is, the automatic categorization of messages into the categories of interest.",
"This is a multiclass categorization problem in which instances are categorized into one of several classes.",
"Specifically, we aim at learning a predictor $h : \\mathcal {X} \\rightarrow \\mathcal {Y}$ , where $\\mathcal {X}$ is the set of messages and $\\mathcal {Y}$ is a finite set of categories.",
"For this purpose, we use three well-known learning algorithms i.e.",
"Naive Bayes (NB), Support Vector Machines (SVM), and Random Forest (RF)."
],
[
"Preprocessing and feature extraction",
"Prior to learning a classifier, we perform the following preprocessing steps.",
"First, stop-words, URLs, and user-mentions are removed from the Twitter messages.",
"We perform stemming using the Lovins stemmer.",
"We use Uni-grams and bi-grams as our features.",
"Previous studies found these two features outperform when used for similar classification tasks [7].",
"Finally, we used the information gain, a well-know feature selection method to select top 1k features.",
"The labeled data we used in this task was annotated by the paid workers."
],
[
"Evaluation and Results",
"We trained all three different kinds of classifiers using the preprocessed data.",
"For the evaluation of the trained models, we used 10-folds cross-validation technique.",
"Table REF shows the results of the classification task in terms of Area Under ROC curvehttps://en.wikipedia.org/wiki/Receiver_operating_characteristic for all classes of the 8 different disaster datasets.",
"We also show the proportion of each class in each dataset.",
"Given the complexity of the task i.e.",
"multiclass classification of short messages, we can see that all three classifiers have pretty decedent results.",
"In this case, a random classifier represents an AUC = 0.50 and higher values are preferable.",
"Other than the “missing trapped or found people\" class, which is the smallest class in term of proportion across all the datasets, results for most of the other classes are at the acceptable level (i.e.",
"$\\ge $ 0.80).",
"Table: Classification results in terms of Area Under ROC Curve for selected datasets across all classes using Support Vector Machines (SVM), Naive Bayes (NB), and Random Forest (RF)."
],
[
"Crisis word embeddings",
"Many applications of machine learning and computational linguistics rely on semantic representations and relationships between words of a text document.",
"Many different types of methods have been proposed that use continuous representations of words such as Latent Semantic Analysis (LSA) and Latent Dirichlet Allocation (LDA).",
"However, recently models based on distributional representations of words become more famous.",
"In this work, we train word embeddings (i.e.",
"distributed word representations) using the 52 million Twitter messages in our datasets and make it available to research community.",
"To the best of our knowledge this is the first largest word embeddings that are trained on crisis-related tweets.",
"We use word2vec, a very popular software to train word embedding [11].",
"As preprocessing, we replaced URLs, digits, and usernames with fixed constants and removed special characters.",
"Finally, the word embeddings are generated using Continuous Bag Of Words (CBOW) architecture with negative sampling along with 300 word representation dimensionality."
],
[
"Language issues in Twitter messages",
"The quality—in terms of readability, grammar, sentence structure etc.—of Twitter messages vary significantly.",
"Typically, Twitter messages are brief, informal, noisy, unstructured, and often contain misspellings and grammatical mistakes.",
"Moreover, due to Twitter's 140 character limit restriction, Twitter users intentionally shorten words by using abbreviations, acronyms, slangs, and sometimes words without spaces.",
"The accuracy of natural language processing techniques would improve if we can identify the informal nature of the language in tweets and normalize OOV terms [5].",
"We divide these lexical variations into the following five categories: Typos/misspellings: e.g.",
"earthquak (earthquake), missin (missing), ovrcme (overcome) Single-word abbreviation/slangs: e.g.",
"pls (please), srsly (seriously), govt (government), msg (message) Multi-word abbreviation/slangs: e.g.",
"imo (in my opinion), im (i am), brb (be right back) Phonetics substitutions: e.g.",
"2morrow (tomorrow), 4ever (forever), 4g8 (forget), w8 (wait) Words without spaces: e.g.",
"prayfornepal (pray for nepal), wehelp (we help), weneedshelter (we need shelter) Figure: Crowdsourcing task for Twitter out-of-vocabulary words normalization"
],
[
"Identification of candidate OOV words",
"To identify candidate OOV words that require normalization, we first build initial vocabularies consisting of lexical variations mentioned in the previous section.",
"We use a dictionary available on the web to normalize abbreviations, chat shortcuts, and slang.http://www.innocentenglish.com/news/texting-abbreviations-collection-texting-slang.html We also use the SCOWL (Spell Checker Oriented Word Lists) aspell English dictionary http://wordlist.aspell.net/ that consists of 349,554 English words.",
"The SCOWL dictionary is suitable for English spell checkers for most of English dialects.",
"Although, the SCOWL dictionary contains places names (e.g.",
"names of countries and famous cities), after testing it on Nepal Earthquake data, we found that its coverage is not complete and a large number of cities/towns of Nepal are missing.",
"To overcome this issue, we use the MaxMind https://www.maxmind.com/en/free-world-cities-database world cities database that consists of 3,173,959 cities.",
"Using the above resources, we try to find OOV words in the dataset.",
"However, we observed that a large number of OOVs consist of misspelled words for which a correct form can be obtained using one edit-distance change (i.e.",
"by performing one insertion, deletion, or substitution operation).",
"For this purpose, we train a language model using lists of most frequent words from Wiktionary,http://en.wiktionary.org/wiki/Wiktionary the British National Corpus,http://www.kilgarriff.co.uk/bnc-readme.html and words in our SCOWL dictionary.",
"For a given misspelled word $w$ , we aim to find a correction $c$ out of all possible corrections where the probability of $c$ given $w$ is maximum, i.e., $argmax_c P(c|w)$ By Bayes Theorem this is equivalent to: $argmax_c P(c|w) = argmax_c P(w|c) P(c)/P(w)$ or it can be written as: $argmax_c P(c|w) = argmax_c P(w|c) P(c)$ where $P(c)$ is the probability that $c$ is the correct word and $P(w|c)$ is the probability that the author typed $w$ when $c$ was intended.",
"We then restrict the language model to predict corrections within one edit-distance range and from those choose the one with highest probability.",
"Misspellings for which more than one change is required, we consider them as OOVs to be corrected by human workers."
],
[
"Normalization of OOV words",
"To normalize the identified OOV words, we used the CrowdFlower crowdsourcing platform.",
"A crowdsourcing task in this case consists of a Twitter message that contains one or more OOV words and a set of instructions shown in Figure REF .",
"The workers were asked to read the instructions and examples carefully before providing an answer.",
"A worker reads the given message and provides a correct OOV tag (i.e.",
"slang/abbreviation/acronym, a location name, an organization name, a misspelled word, or a person name).",
"If an OOV is a misspelled word, the worker also provides its corrected form.",
"We provide all the resources and the results of crowdsoucing to research community."
],
[
"Related Work",
"The use of microblogging platforms such as Twitter during the sudden onset of a crisis situation has been increased in the last few years.",
"Thousands of crisis-related messages that are posted online contain important information that can also be useful to humanitarian organizations for disaster response efforts, if processed timely and effectively [6], [9].",
"Many different types of processing techniques ranging from machine learning to natural language processing to computational linguistics have been developed [3] for different purposes [10].",
"Despite there exists some resources e.g.",
"[16], [12], however, due to the scarcity of relevant data, in particular human-annotated data, crisis informatics researchers still cannot fully utilize the capabilities of different computational methods.",
"To overcome these issues, we present to research community a corpora consisting of labeled and unlabeled crisis-related Twitter messages.",
"Moreover, we also provide normalized lexical resources useful for linguistic analysis of Twitter messages."
],
[
"Conclusions",
"We present Twitter corpora consisting of over 52 million crisis-related tweets collected during 19 crisis events.",
"We provide two sets of annotations related to topic-categorization of the tweets and tagging out-of-vocabulary words and their normalizations.",
"We build machine-learning classifiers to empirically validate the effectiveness of the annotated datasets.",
"We also provide word2vec word embeddings trained on 52 million messages.",
"We believe that these resources and the tools built using them will help improve automatic natural language processing of crisis-related messages and eventually be useful for humanitarian organizations."
]
] | 1605.05894 |
[
[
"Multiplicity density at mid-rapidity in $AA$ collisions: effect of meson\n cloud"
],
[
"Abstract We study the influence of the meson cloud of the nucleon on predictions of the Monte Carlo Glauber model for the charged particle multiplicity density at mid-rapidity in $AA$ collisions.",
"We find that for central $AA$ collisions the meson cloud can increase the multiplicity density by $\\sim 16-18$\\%.",
"The meson-baryon Fock component reduces the required fraction of the binary collisions by a factor of $\\sim 2$ for Au+Au collisions at $\\sqrt{s}=0.2$ TeV and $\\sim 1.5$ for Pb+Pb collisions at $\\sqrt{s}=2.76$ TeV."
],
[
"Introduction",
"The understanding of the initial entropy/energy distribution is crucial for hydrodynamical simulation of the evolution of the hot quark-gluon plasma (QGP) in high-energy $AA$ collisions.",
"A rigorous determination of the initial conditions for the plasma fireball in $AA$ collisions is presently impossible.",
"The most popular methods in use for this purpose at the present time are the IP-Glasma model [1], [2] and the wounded nucleon Glauber model [3], [4].",
"The IP-Glasma approach is based on the pQCD color-glass condensate model [5].",
"Unfortunately the applicability of the pQCD in the IP-Glasma model is questionable since it assumes that gluon fields can be treated perturbatively down to an infrared scale $m\\sim \\Lambda _{QCD}$ [1], [2].",
"It is several times smaller than the inverse gluon correlation radius in the QCD vacuum $1/R_c\\sim 0.75$ GeV [6], which is the natural lower limit for the virtuality scale of the perturbative gluons.",
"In the dipole approach [7] for $m \\sim 0.75$ GeV the perturbative contribution to the hadronic cross sections turns out to be smaller than the nonperturbative one up to $\\sqrt{s}\\sim 10^3$ GeV [8].",
"For this reason even at the LHC energies a purely perturbative treatment of the hadron cross sections is questionable.",
"The wounded nucleon Glauber model [3], [4] is a phenomenological scheme.",
"Originally [3] it was assumed that in $AA$ collisions each nucleon undergoing inelastic soft interaction (participant) produces a fixed contribution to the multiplicity rapidity density.",
"At mid-rapidity ($\\eta =0$ ) in the c.m.",
"frame this contribution equals half of the $pp$ multiplicity rapidity density.",
"It gives for $AA$ collisions the multiplicity density $\\propto N_{part}$ , where $N_{part}$ is the number of participants in both the colliding nuclei.",
"Later, in [4] it was proposed to include in the model the contribution from hard processes that gives the particle density proportional to the number of the binary collisions $N_{coll}$ .",
"In this two component version the charged particle multiplicity density in $AA$ collisions takes the form $\\frac{dN_{ch}(AA)}{d\\eta }=\\frac{(1-\\alpha )}{2}n_{pp}N_{part}+\\alpha n_{pp}N_{coll}\\,,$ where $n_{pp}=dN_{ch}/d\\eta $ is the multiplicity density in $pp$ collisions, and $\\alpha $ characterizes the magnitude of hard processes to multiparticle production.",
"In the Glauber model model $N_{part}$ and $N_{coll}$ can be expressed via the inelastic $pp$ cross section and the nuclear density.",
"Fitting the data on the centrality dependence of the charged particle multiplicity in Au+Au collisions at $\\sqrt{s}=0.2$ TeV and in Pb+Pb collisions at 2.76 TeV gives $\\alpha \\approx 0.13-0.15$ [9], [10], [11].",
"For such a value of $\\alpha $ the hard contribution to the particle production in $AA$ collisions turns out to be rather large ($\\sim 40-50$ % for central collisions).",
"It is important that the two component Glauber model allows the Monte Carlo formulation [12], [13], [14].",
"The Monte Carlo Glauber (MCG) model has proved to be a useful tool for analysis of the event-by-event fluctuations of observables in $AA$ collisions.",
"The model of wounded nucleons has been also formulated at quark level [15], [16] when inelastic interaction of the nucleon is treated as a combination of inelastic interactions of its constituent quarks.",
"However, in this picture the quark contribution to the multiplicity required for description of data on $AA$ collisions may differ substantially from the one that is necessary for $pp$ collisions [17].",
"Say, the data on Au+Au collisions at $\\sqrt{s}=0.2$ TeV require the quark contribution suppressed by a factor $\\sim 1.4$ as compared to $pp$ interaction [17].",
"However, the situation with consistency between $AA$ and $pp$ collisions becomes better if the nucleon is treated as a quark-diquark system [18].",
"The common feature of the wounded nucleon models with internal nucleon structure is the nonlinear increase of $dN_{ch}(AA)/d\\eta $ with the number of wounded nucleons even without the hard contribution [19], [18], [17], [20].",
"This is due to the growth of the fraction of the wounded constituents in each nucleon in $AA$ collisions as compared to that in $pp$ collisions.",
"It is clear that a similar effect should arise from the meson cloud of the nucleon.",
"The total weight of the meson-baryon Fock states in the nucleon may be as large as $\\sim 40$ % [21] (with the dominant contribution from the $\\pi N$ component).",
"The purpose of the present work is to study within the MCG approach the possible effect of the meson-baryon component of the nucleon on the multiplicity rapidity density in $AA$ collisions.",
"We will analyze within the MCG model with the meson cloud data on Au+Au collisions at $\\sqrt{s}=0.2$ [22] and Pb+Pb collisions at $2.76$ TeV [23]."
],
[
"Theoretical framework",
"At high energies the wave function of the physical nucleon becomes identical to that in the infinite momentum frame (IMF).",
"It can be written in the form [21], [24] $|N\\rangle _{phys}=\\sqrt{1-n_{MB}}|N\\rangle +\\sum _{MB}\\int dxd{{\\bf k}}\\Psi _{MB}(x,{{\\bf k}})|MB\\rangle \\,,$ where $N$ , $B$ , and $M$ denote the bare baryon and meson states, $x$ is the fractional longitudinal meson momentum in the physical nucleon, $\\Psi _{MB}$ is the probability amplitude for the $MB$ Fock state, and $n_{MB} =\\sum _{MB}\\int dxd{{\\bf k}}|\\Psi _{MB}(x,{{\\bf k}})|^2$ is the total weight of the $MB$ Fock components.",
"The energy denominator of time-ordered perturbation theory in the IMF for the $MB$ component reads $E_N-E_M-E_B\\approx [m_N^2-M^{2}_{MB}(x,{{\\bf k}}^2)]/2E_N$ , where $M^{2}_{MB}(x,{{\\bf k}}^2)=\\frac{m_M^2+{{\\bf k}}^2}{x}+\\frac{m^2_{B}+{{\\bf k}}^2}{1-x}$ is the squared invariant mass of the $MB$ system.",
"For the normalization corresponding to (REF ) the IMF wave function (for point-like particles) may be written as $\\Psi _{MB}(x,{{\\bf k}})=\\frac{\\langle MB|V|N\\rangle }{4\\pi ^{3/2}\\sqrt{x(1-x)}\\left[m_N^2-M^2_{MB}(x,{{\\bf k}})\\right]}\\,.$ Here $\\langle MB|V|N\\rangle $ is the vertex factor in the IMF-limit, which depends on the form of the Lagrangian.",
"For the dominant $\\pi N$ state $\\langle \\pi N^{\\prime }|V|N\\rangle =g_{\\pi NN}\\bar{u}_{N^{\\prime }}\\gamma _5u_{N}$ (the helicity dependent vertex functions for different $MB$ states can be found in [21]).",
"In phenomenological applications the internal structure of the hadrons is accounted for by multiplying the vertex factor for point-like particles by a form factor, $F$ , which in the IMF scheme depends on $x$ and ${{\\bf k}}$ only via $M_{MB}(x,{{\\bf k}})$ [24], [21], [25].",
"To a good approximation on can account for in (REF ) only $\\pi N$ , $\\pi \\Delta $ , $\\rho N$ and $\\rho \\Delta $ two-body states [21], [25].",
"Since the bare $\\Delta $ and $\\rho $ -meson states have the same quark content as $N$ and $\\pi $ , it is reasonable to assume that their inelastic interactions are similar to that for $N$ and $\\pi $ states.",
"Then, from the point of view of the MCG model, each physical nucleon interacts with the probability $1-n_{MB}$ as the bare $N$ and with the probability $n_{MB}$ as the two-body $\\pi N$ system.",
"For $\\pi $ -meson the $x$ -distribution is peaked at $x\\sim 0.3$ and for $\\rho $ -meson at $x\\sim 0.5$ [21].",
"For simplicity in our MCG calculations we neglect fluctuations of $x$ and take $x=0.3$ .",
"For the transverse spacial distribution of the $MB$ state we use the distribution of the dominant $\\pi N$ component.",
"It was renormalized to match the total weight of the $MB$ component $n_{MB}=0.4$ [21].",
"We calculated the transverse spacial distribution using the dipole formfactor [25] $F=\\left(\\frac{\\Lambda ^2+m_N^2}{\\Lambda ^2+M^{2}_{\\pi N}(x,{{\\bf k}})}\\right)^2\\,.$ We take $\\Lambda =1.3$ GeV, such a value is supported by the data on $pp\\rightarrow nX$ [26].",
"It gives for the mean squared transverse radius of the $MB$ component $\\langle \\rho ^2_{MB}\\rangle ^{1/2}\\approx 0.87$ fm.",
"However, the results of the MCG simulation depends weakly on the value of $\\Lambda $ .",
"This is due to the fact that in the Glauber model there is no shadowing effect for inelastic interactions.",
"In our model inelastic interaction of the physical nucleons from the colliding objects is a combination of $N+N$ , $N+MB$ , $MB+N$ and $MB+MB$ interactions.",
"We assume that the inelastic cross sections for the bare states obey the constituent quark counting rule $4\\sigma ^{NN}_{in}=6\\sigma ^{MB}_{in}=9\\sigma ^{MM}_{in}$ .",
"For the profiles of the probability of $ab$ inelastic interaction in the impact parameter we use a Gaussian form $P_{ab}(\\rho )=\\exp \\left(-\\pi \\rho ^2/\\sigma _{in}^{ab}\\right)\\,.$ The value of $\\sigma _{in}^{NN}$ has been adjusted to reproduce the experimental inelastic $pp$ cross section $\\sigma _{in}^{pp}$ (see below).",
"We consider the multiplicity density at $\\eta =0\\,\\,(y=0)$ .",
"The direct data on $dN_{ch}/d\\eta $ for pion-proton and pion-pion collisions for RHIC-LHC energies are absent.",
"Calculations within the quark-gluon string scheme [27], [28] show that the charged particle multiplicity density in the central rapidity region for pion-proton and pion-pion collisions is somewhat bigger than for proton-proton collisions.",
"To good accuracy this excess compensates a possible reduction of the multiplicity density in $\\pi N$ and $\\pi \\pi $ interactions due to somewhat smaller c.m.",
"energy in our model.",
"For this reason we assume that all the wounded bare particles produce the same amount of entropy per unit pseudorapidity $\\eta $ in the c.m.",
"frame of colliding objects ($pp$ or $AA$ ).",
"We ignore the effect of a small rapidity shift ($\\sim 0.5$ ) of the c.m.",
"frame for pairs with different energies (as occurs for $\\pi N$ interactions) on the entropy rapidity density since it is flat at mid-rapidity.",
"The total rapidity density for $AA$ collisions is the sum of the contributions from the sources corresponding to the wounded constituents and to the binary collisions $\\frac{dS}{dy}=\\sum _{i=1}^{N_w} \\frac{dS_w^{i}}{dy}+\\sum _{i=1}^{N_{bin}} \\frac{dS_{bin}^{i}}{dy}\\,.$ We write the contribution of each source from the wounded constituents as $dS_w^i/dy=\\frac{(1-\\alpha )}{2}S$ .",
"The contribution of each binary collision is $dS_{bin}^i/dy=S$ , and for each pair of wounded particles the probability of a hard binary collision is $\\alpha $ .",
"We assume an isentropic expansion of the QGP.",
"In this case the initial entropy rapidity density is proportional the charged particle pseudorapidity density $dS/dy=C dN_{ch}/d\\eta $ , where $C\\approx 7.67$ [29].",
"In this approximation one can replace in (REF ) the entropy density by the pseudorapidity charged particle density.",
"And the fluctuating entropy density $S$ for each source is replaced by the fluctuating pseudorapidity charged particle density $n=S/C$ .",
"We describe the fluctuations of $n$ by the Gamma distribution $P(n,\\langle n\\rangle )=\\left(\\frac{n}{\\langle n\\rangle }\\right)^{k-1}\\frac{k^k\\exp \\left[-nk/\\langle n\\rangle \\right]}{\\langle n\\rangle \\Gamma (k)}\\,.$ The parameters $\\langle n\\rangle $ and $k$ have been fitted from $pp$ data on $dN_{ch}/d\\eta $ (see below).",
"Note however that for $AA$ collisions the results are only weakly sensitive to fluctuations of $n$ (except for the region of very high multiplicities).",
"For calculation of the centrality dependence of the charged particle multiplicity in $AA$ collisions the distribution of the entropy rapidity density in the transverse coordinates, $\\rho _s=dS/dyd\\mbox{{$\\rho $}}$ , is not important.",
"However, it is necessary for calculation of geometric quantities such as the initial anisotropy $\\epsilon _n$ [30] $\\epsilon _n=\\frac{\\int d\\mbox{{$\\rho $}}\\rho ^n e^{in\\phi }\\rho _s(\\mbox{{$\\rho $}})}{\\int d\\mbox{{$\\rho $}}\\rho ^n\\rho _s(\\mbox{{$\\rho $}})}\\,.$ In the approximation of the point-like sources we have $\\rho _s(\\mbox{{$\\rho $}})=\\sum _{i=1}^{N_w} \\delta (\\mbox{{$\\rho $}}-\\mbox{{$\\rho $}}_i)\\frac{dS_w^{i}}{dy}+\\sum _{i=1}^{N_{bin}} \\delta (\\mbox{{$\\rho $}}-\\mbox{{$\\rho $}}_i)\\frac{dS_{bin}^{i}}{dy}\\,.$ We assume that for each binary collision the source is located in the middle between colliding constituents.",
"Physically the approximation of the point-like sources is clearly unreasonable.",
"To account for qualitatively the finite size of the sources we replaced in our MCG code the $\\delta $ functions in (REF ) by a Gaussian distribution $\\exp {\\left(-\\mbox{{$\\rho $}}^2/a_s^2\\right)}/\\pi a_s^2$ with $a_s=0.7$ fm.",
"We observed that the results for the anisotropy coefficients $\\epsilon _n$ becomes sensitive to the smearing of the sources only for very peripheral collisions.",
"We perform calculations using the Woods-Saxon nuclear distribution $\\rho _{A}(r)=\\frac{c}{1+\\exp [(r-R_A)/a]}\\,,$ where $c$ is the normalization constant, $R_{A}=(1.12A^{1/3}-0.86/A^{1/3})$ fm, $a=0.54$ fm [14]."
],
[
"Numerical results",
"In numerical calculations we take $n_{pp}=2.65$ at $\\sqrt{s}=0.2$ TeV obtained by the UA1 collaboration [31].",
"The direct $pp$ data on $n_{pp}$ at $\\sqrt{s}=2.76$ TeV are absent.",
"We obtained it with the help of the power law interpolation between the CMS data at $\\sqrt{s}=2.36$ TeV [32] ($n_{pp}=4.47\\pm 0.04(\\mbox{stat.",
"})\\pm 0.16 (\\mbox{syst.",
"})$ ) and at $\\sqrt{s}=7$ TeV [33] ($n_{pp}=5.78\\pm 0.01(\\mbox{stat.})",
"\\pm 0.23 (\\mbox{syst.",
"})$ ).",
"It gives $n_{pp}\\approx 4.65$ at $\\sqrt{s}=2.76$ TeV.",
"The multiplicity densities measured in [31], [32], [33] correspond to the non-single-diffractive (NSD) events.",
"For this reason in the MCG simulation one should also take for $\\sigma _{in}^{pp}$ the cross section corresponding to the NSD event class.",
"The exclusion of the diffractive contribution to the inelastic cross section is reasonable since the diffractive events do not contribute to the mid-rapidity multiplicity density.",
"We use for the NSD $pp$ cross section at $\\sqrt{s}=0.2$ TeV the value 35 mb measured by the UA1 collaboration [31], and at $\\sqrt{s}=2.76$ TeV the value $50.24$ mb obtained by the ALICE collaboration [34].",
"Making use of the above values of the NSD $\\sigma _{in}^{pp}$ we fitted $\\sigma _{in}^{NN}$ .",
"For the scenario with meson cloud we obtained $\\sigma _{in}^{NN}[\\sqrt{s}=0.2\\,,2.76\\,\\mbox{TeV}] \\approx [26.15,\\,38.4]\\,\\,\\mbox{mb}\\,.$ In the scenario without meson cloud $\\sigma _{in}^{NN}$ is equal simply to the experimental NSD $pp$ cross section.",
"The parameters $\\langle n\\rangle $ and $k$ in the Gamma distribution (REF ) have been fitted to reproduce the experimental $n_{pp}$ and to satisfy the relation $n_{pp}/D=1$ ($D^2$ is a variance of $dN_{ch}/d\\eta $ ) which is well satisfied for the experimental multiplicity distribution in the pseudorapidity window $|\\eta |<0.5$ at $\\sqrt{s}=0.2$ TeV [31] and at $\\sqrt{s}=2.36$ TeV [35].",
"For the scenario without meson cloud $\\langle n\\rangle $ equals the experimental $n_{pp}$ for any fraction of the binary collisions.",
"For $\\alpha =0$ the relation $n_{pp}/D=1$ gives $k=0.5$ .",
"For nonzero $\\alpha $ the value of $k$ grows weakly with $\\alpha $ , but the deviation from $0.5$ is small.",
"For the scenario with meson cloud the required value of $\\langle n\\rangle $ is smaller than $n_{pp}$ , and $k$ is close to $0.5$ .",
"We first fitted the parameters $\\langle n\\rangle $ and $k$ to the $pp$ data for set of $\\alpha $ .",
"Then we used them to fit the parameter $\\alpha $ to best reproduce the data on the centrality dependence of $dN_{ch}/d\\eta $ in Au+Au collisions at $\\sqrt{s}=0.2$ TeV from STAR [22] and in Pb+Pb collisions at $\\sqrt{s}=2.76$ TeV from ALICE [23].",
"For Au+Au collisions at $\\sqrt{s}=0.2$ TeV we obtained $\\alpha \\approx 0.06$ and $\\alpha \\approx 0.135$ for the scenarios with and without meson cloud, respectively.",
"And for Pb+Pb collisions at $\\sqrt{s}=2.76$ TeV for these two scenarios our fits give $\\alpha \\approx 0.09$ and $\\alpha \\approx 0.14$ .",
"For the above values of $\\alpha $ the parameters of the Gamma distribution (REF ) obtained from the fit with meson cloud to the $pp$ data read $\\langle n\\rangle [\\sqrt{s}=0.2,\\,2.76\\,\\text{TeV}]\\approx [2.39,\\,4.13]\\,,$ $k[\\sqrt{s}=0.2,\\,2.76\\,\\text{TeV}]\\approx [0.506,\\,0.52]\\,.$ For the scenario without meson cloud $\\langle n\\rangle =n_{pp}$ , and $k[\\sqrt{s}=0.2,\\,2.76\\,\\text{TeV}]\\approx [0.57,\\,0.57]\\,.$ As expected, accounting for the meson cloud leads to a reduction of the required fraction of the binary collisions.",
"The effect becomes smaller at the LHC energy.",
"It is due to an increase of the interaction radius from RHIC to LHC, resulting in the lower sensitivity to the internal nucleon structure at the LHC energy.",
"In Figs.",
"1, 2 we compare our calculations for the fitted values of $\\alpha $ with STAR [22] and ALICE [23] data.",
"The theoretical histograms have been obtained by Monte Carlo generation of $\\sim 2\\times 10^6$ events.",
"To illustrate the magnitude of the effect of the meson cloud in Figs.",
"1a and 2a we show the results for the scenario without meson cloud obtained with $\\alpha $ for the scenario with meson cloud.",
"One can see that at small centrality the meson cloud increases the multiplicity by $\\sim 16-18$ %.",
"Note that our calculations do not assume a certain internal structure of the bare baryon and meson states.",
"For this reason one can expect that the long range meson-baryon fluctuations in the nucleon wave function should increase the multiplicity in $AA$ collisions in any scheme.",
"We also studied the effect of meson cloud on the initial anisotropy coefficients $\\epsilon _n$ ($n=2-5$ ).",
"We found that the effect of the meson cloud is small (except for very peripheral collisions where the results are not robust due to their sensitivity to the entropy distribution for the wounded constituents and the binary collisions).",
"Recently there was interest in the multiplicity dependence of the ellipticity $\\epsilon _2$ for U+U collisions [36], [39], [40].",
"In [36] it was predicted that due to prolate shape of the $^{238}$ U nucleus the initial $\\epsilon _2$ should has a knee structure at multiplicities in the top 1% U+U collisions related to the growth of the contribution of the binary collisions for the tip-tip configurations of the colliding nuclei.",
"But the elliptic flow $v_2$ measured by STAR [37], [38] in U+U collisions at $\\sqrt{s}=193$ GeV shows no indication of a knee structure.",
"This challenged the picture with a significant contribution of the binary collisions, and stimulated study of alternative ansatze for the entropy deposition in the Glauber picture [39], [40].",
"However, the Glauber calculations of [36] have been performed neglecting the fluctuations of the multiplicity in $NN$ collisions.",
"Later in [42] it was demonstrated that the knee structure vanishes when the fluctuations are taken into account.",
"Our calculations for U+U collisions also show that the knee structure in $\\epsilon _2$ is swept out (both with and without meson cloud) when the fluctuations of the sources are taken into account.",
"Figure: Centrality dependence of dN ch /dηdN_{ch}/d\\eta forAu+Au collisions at s=0.2\\sqrt{s}=0.2 TeV.",
"Left:MCG simulation for the scenario with (solid) and without (dotted) meson cloudfor α=0.06\\alpha =0.06.Right: MCG simulation for the scenario without meson cloudfor α=0.135\\alpha =0.135.",
"Data are from STAR .Figure: Centrality dependence of dN ch /dηdN_{ch}/d\\eta forPb+Pb collisions at s=2.76\\sqrt{s}=2.76 TeV.",
"Left:MCG simulation for the scenario with (solid) and without (dotted) meson cloudfor α=0.09\\alpha =0.09.Right: MCG simulation for the scenario without meson cloudfor α=0.14\\alpha =0.14.",
"Data are from ALICE ."
],
[
"Summary",
"We have studied the influence of the meson cloud on predictions of the MCG model for $AA$ collisions.",
"We find that for central $AA$ collisions the meson cloud can increase the multiplicity density in the central rapidity region by $\\sim 16-18$ %.",
"Accounting for the meson-baryon Fock components of the nucleon reduces the required fraction of the binary collisions by a factor of $\\sim 2$ for Au+Au collisions at $\\sqrt{s}=0.2$ TeV and $\\sim 1.5$ for Pb+Pb collisions at $\\sqrt{s}=2.76$ TeV.",
"One can expect that the observed increase of the multiplicity in $AA$ collisions due to the virtual meson-baryon states in the physical nucleon should exist in other models for the initial conditions in $AA$ collisions.",
"I thank W. Broniowski and S.A. Voloshin for communications.",
"This work is supported in part by the grant RFBR 15-02-00668-a."
]
] | 1605.06012 |
[
[
"Dynamical estimate of post main sequence stellar masses in 47 Tucanae"
],
[
"Abstract We use the effects of mass segregation on the radial distribution of different stellar populations in the core of 47 Tucanae to find estimates for the masses of stars at different post main sequence evolutionary stages.",
"We take samples of main sequence (MS) stars from the core of 47 Tucanae, at different magnitudes (i.e.",
"different masses), and use the effects of this dynamical process to develop a relation between the radial distance (RD) at which the cumulative distribution reaches the 20th and 50th percentile, and stellar mass.",
"From these relations we estimate the masses of different post MS populations.",
"We find that mass remains constant for stars going through the evolutionary stages between the upper MS up to the horizontal branch (HB).",
"By comparing RDs of the HB stars with stars of lower masses, we can exclude a mass loss greater than 0.09M during the red giant branch (RGB) stage at nearly the 3{\\sigma} level.",
"The slightly higher mass estimates for the asymptotic giant branch (AGB) are consistent with the AGB having evolved from somewhat more massive stars.",
"The AGB also exhibits evidence of contamination by more massive stars, possibly blue stragglers (BSS), going through the RGB phase.",
"We do not include the BSS in this paper due to the complexity of these objects, instead, the complete analysis of this population is left for a companion paper.",
"The process to estimate the masses described in this paper are exclusive to the core of 47 Tuc."
],
[
"Introduction",
"With a large sample of 157 globular clusters (GC) in the galaxy [14], these roughly spherical agglomerations of stars are one of the most widely studied systems in astronomy.",
"The high number of stars residing in a single GC and the long time they had to evolve, makes these objects an ideal place to study the evolution of stars and the dynamics of stellar systems.",
"An important process in GC dynamics is mass segregation that happens on a relaxation timescale, $t_{relax}$ .",
"Essentially, mass segregation means that more massive stars move towards the center of the cluster while less massive ones tend towards larger radii, completely changing the original mass distribution of the cluster [33].",
"This process is the result of two different mechanisms: relaxation and equipartition.",
"The first one comes from the fact that each star wanders away from its initial orbit, increasing the entropy of the system, leading it to a new configuration with a small, dense core and and a large, low-density halo [6].",
"The second one comes from kinetic theory which states that particle encounters will make those particles with large kinetic energy lose energy to those with lower energies, leading to a state where the mean-square velocity is inversely proportional to a particle's mass.",
"Massive stars transfer kinetic energy to the less massive stars, so the massive stars end up with less energy per unit mass and are restricted to the central, most-bound regions of the cluster.",
"Meanwhile the less massive stars gain energy per unit mass and inhabit the outer, less-bound regions of the cluster [20], [6].",
"Mass segregation has been quantified for different GC [12].",
"A good example of a target with multiple investigations is NGC 104 (47 Tucanae, 47 Tuc), the second largest and brightest GC in the sky.",
"One of the first detailed studies of mass segregation in 47 Tuc was carried out by [1].",
"Using images of the core of 47 Tuc, he was able to measure the luminosity function to which he fitted King-Michie models obtaining the best agreement with those models that included mass segregation.",
"But not only can mass segregation be analysed through luminosity functions, if the core of a cluster is indeed relaxed, the radial distribution of different groups of stars should also exhibit indications of this phenomenon.",
"Also, because mass segregation sorts stars by mass, when picking stars from a specific region of the CMD (if they have lived there for a period longer than a $t_{relax}$ ), their radial distribution should reflect the mass of the selected sample.",
"For 47 Tuc, $t_{relax}$ in the core was measured to be 30 Myr [16].",
"Main sequence stars (MS) last much longer than $t_{relax}$ , thus, if we can pick different ranges of masses along the MS, the radial distributions for the different masses should reflect the effects of mass segregation.",
"We will show how the high quality ultraviolet (UV) data allows us to reach a MS mass difference of $\\sim 0.2M_{\\odot }$ between the turn-off point (TO) and the faint MS stars, sufficient mass difference to show clear evidence of mass segregation.",
"This will lead to a relation between the cumulative radial distribution and mass of the stars that will subsequently be use to estimate the masses for stars in different post MS evolutionary stages.",
"For the horizontal branch (HB), the debate has been centered on how much mass loss occurs during the red giant branch (RGB) phase.",
"[23] argued that the mass loss in the RGB phase is $\\sim 0.25M_{\\odot }$ , for stars with a TO mass of $0.9M_{\\odot }$ .",
"For a TO mass of $0.85M_{\\odot }$ values of mass loss have been estimated to be between $\\sim 0.1-0.2M_{\\odot }$ [29], [23], [24], [22].",
"In [31], [31] found a mean HB mass for 47 Tuc of $0.65-0.66M_{\\odot }$ , while [13] reported the masses of the HB stars in the same cluster to be between $0.6-0.7M_{\\odot }$ for a TO mass of $0.85M_{\\odot }$ considering the mass loss rates mentioned before.",
"Recently, [15] came to the conclusion that the bulk of the mass loss occurs at the tip of the asymptotic giant branch (AGB) with a mass loss of $\\sim 0.34M_{\\odot }$ during this phase, and only $\\sim 0.02M_{\\odot }$ while the star is on the RGB.",
"In the case of white dwarfs (WD), there seems to be a better agreement in the reported masses with values around $\\sim 0.54M_{\\odot }$ for a TO mass of $0.9M_{\\odot }$ [29], [18], [15]."
],
[
"Observations",
"The data come from observations made with the Hubble Space Telescope (HST) using Wide Field Camera 3 (WFC3) with two of the most UV filters, F225W and F336W, whose central wavelengths are 235.9 nm and 335.9 nm respectively.",
"Ten fields in the core of 47 Tuc were obtained between November 2012 and August 2013 during cycle 20 of the HST program GO-12971 (PI: H. Richer).",
"The observations were planned so that each visit included two exposures in each filter, 380s and 700s for F225W and 485s and 720s for F336W.",
"Each field was offset from the previous one in order to map the entire central region of the cluster.",
"When all the images are stacked together the final field has a star-like shape; we reduced this field to a circular region to avoid biases on the radial distributions.",
"The final field of view covers a radius of $\\sim 160$ arcseconds from the center of the cluster.",
"Photometry was performed following the procedure described in [19].",
"We will also make use of data available from the core of 47 Tuc in the visible range, specifically with the HST ACS (Advanced Camera for Surveys) F606W and F814W filters.",
"This data set is part of the ACS Survey of Galactic Globular Cluster [32].",
"The survey used the ACS Wide Field Channel to obtain photometric data of 65 of the nearest globular clusters and is publicly available at: http://www.astro.ufl.edu/~ata/public_hstgc/databases.html.",
"A description of the data reduction and photometry can be found in [2]."
],
[
"Artificial Star Test: Correcting for Incompleteness",
"To estimate the number of stars lost in the photometry process, we ran artificial star tests.",
"The procedure, explained in detail in [16], consists in inserting artificial stars into the images in both F225W and F336W filters and calculating how effectively these are recovered when run through the same photometry process as the real stars.",
"The completeness rate is a function of the magnitudes of the star in each filter as well as its distance from the center of the cluster, and so, artificial stars were given a range of values covering the observed magnitudes and distances to the center of 47 Tuc.",
"The completeness rate is strongly dependent on both radius and magnitude, with only the brightest stars close to unity.",
"To test our corrections for incompleteness, we compared the cumulative radial distribution of the Small Magellanic Cloud (SMC) stars to that of $R^{2}$ , which one would expect because the SMC stars are nearly uniformly distributed on the scale of the core of 47 Tuc.",
"The SMC is a dwarf galaxy orbiting the Milky Way that happens to lie in the background of 47 Tuc.",
"The two objects are completely unrelated and very far apart (47 Tuc is 4.5 kpc [14], away from the Sun, while the SMC is at $\\sim 60$ kpc, [17]), but since they share the same region of the sky, stars from the SMC contaminate the color magnitude diagram (CMD) of 47 Tuc.",
"Figure REF shows where the MS of the SMC lies on our CMD.",
"Because the SMC is not related to 47 Tuc, the radial distribution of its stars should be proportional to the area of the field of our observations.",
"Looking at the right panel of Figure REF , we can see the comparison between the incompleteness corrected cumulative radial distributions of the SMC and $R^{2}$ (as we are only counting the stars within a circular area).",
"If our completeness rates were properly obtained then these distributions should be approximately equal.",
"In fact the Kolmogorov-Smirnov test (KS-test) yields a p-value of 0.60 telling that we cannot reject the hypothesis that the distributions are in fact the same.",
"The mean completeness fraction of the SMC sample is less than 70%, so the completeness corrections are crucial to obtaining the estimate of the underlying distribution.",
"Figure: Left: Selection of SMC stars on the UV CMD.",
"Right: Cumulative radial distribution of the SMC compared to R 2 R^2.",
"The legend on the CMD indicates the number of stars before correcting for incompleteness, while the legend in the right plot gives the size of the sample after correcting for incompleteness.",
"The agreement between both distributions allow us check the validity of our completeness rates."
],
[
"Stellar Population Selection",
"Due to the high quality of the data, each population is easily identified and can be separated one from another.",
"Each population is defined to be within a region shown by the different color boxes on the CMDs.",
"The boundaries of each region were chosen with the help of a MESA (Modules for Experiments in Stellar Astrophysics; [26] [26], [27], [28]) evolutionary model, slight modifications on the limits of these regions would only make including stars with higher photometry errors or not real members of the different branches more likely.",
"Also, including the stars surrounding the highlighted regions does not change the number of stars in each box by more than a few percent and tests done including these stars show no effect on the shapes of the cumulative radial distributions.",
"The MESA evolutionary model was created using the pre-built 1M_pre_ms_to_wd model in the test suite.",
"The initial parameters were set to the appropriate values for 47 Tuc, with a mass of $0.9M_{\\odot }$ , a metallicity of $3.36 \\times 10^{-3}$ and helium abundance of $0.256$ [4].",
"A detailed description of the construction of the model can be found in [15].",
"On the WFC3 CMD, we selected stars going through the RGB and two samples of WDs, bright (BWD) and faint (FWD) WDs.",
"The initial WD sample extends between $F225W$ magnitudes of 24 and 16.5, and is divided at magnitude 21.",
"This division gives median ages of 6 Myr for the BWD and 127 Myr for the FWD [16].",
"For the RGB we keep a distance of a few tenths of magnitudes from the sub-giant branch (SGB) which in the ACS CMD translates into a magnitude difference of $\\sim 0.5$ between the RGB selection and the TO.",
"On the UV CMD, we also select a sample of MS binaries (MSBn) that we expect to be mostly nearly equal mass binaries.",
"Using the ACS data we can extend the number of selected evolutionary stages, including now the HB and AGB.",
"To select stars on the ACS CMD, the WFC3 data set is reduced to stars within the ACS field of 105 arcseconds.",
"Both catalogues were matched so that every star taken from the ACS data has a counterpart in the WFC3 data, this could not be done the other way around as the ACS data does not go to faint magnitudes on the lower MS and WDs.",
"Figure REF , shows the location of the HB and AGB on the visible range CMD and the positions of these stars stars on the UV CMD.",
"For this data set no completeness corrections were applied, as we are only using the brightest end of the CMD and at those magnitudes the completeness is very close to unity.",
"Figure: F606W,F606W-F814WF606W,F606W-F814W (left) and F225W,F225W-F336WF225W,F225W-F336W (right) CMD with the selection of the stellar populations from the ACS data (triangles) and WFC3 data (circles), and where they fall on the opposite CMD.",
"The insets shows the number of stars selected in each of the populations within the ACS field.",
"We can see how hard it would be to identify AGB star purely based on the WFC3 photometry.",
"Also, selecting the HB stars from the UV CMD would lead to a contaminated sample of stars.Figure REF shows the location on the UV CMD of the different regions selected.",
"We point out the difference between the number of AGB stars on this figure compare to Figure REF .",
"In [25] (hereafter paper2), we find that the contamination of the AGB happens mainly at the so called AGB bump, as a consequence of our inability to separate the RGB and AGB of evolved blue straggler stars (BSS) from the normal evolution AGB.",
"The dissimilarity in the number of AGB stars is then due to the removal of the AGB bump (between $F606W$ magnitudes of $\\sim 12.5-13$ ).",
"A prominent feature not highlighted but visible in both the optical and UV CMDs is the presence of BSS.",
"We do not include the BSS in the current analysis due to the fact that we need to demonstrate that this population lasts longer than a relaxation time, which goes beyond the scope of this paper.",
"A complete inspection of the BSS and evolved BSS in the core of 47 Tuc is presented in paper2, where we also report the results for the estimate of the masses of these populations using the relations found in this paper."
],
[
"The Mass Relation",
"For the WFC3 data, the high-quality data and photometry have allowed us to go as faint as six magnitudes below the TO reaching a significant enough mass difference along the MS to be able to show mass segregation.",
"Figure REF shows the CMD of 47 Tuc in the UV, where we have highlighted three MS regions with the corresponding median mass for each box.",
"Starting from the bright end of the MS, the masses are $0.83\\pm 0.01 M_{\\odot }$ for the upper MS (UMS), $0.75\\pm 0.01 M_{\\odot }$ for the middle MS (MMS) and $0.68\\pm 0.02 M_{\\odot }$ for the lower MS (LMS).",
"The masses are calculated based on an 11 Gyr PARSEC isochrone ([7], available at http://stev.oapd.inaf.it/cmd).",
"The isochrone was constructed using the metallicity of 47 Tuc and the bolometric corrections of [8].",
"In order to fit the isochrone to the data, besides the distance modulus ($(m-M)_{0} = 13.36$ , [36]) and reddening ($E(B-V) = 0.04$ , [31]) it was necessary to add 0.4 and 0.3 magnitudes of extinction to F225W and F336W respectively.",
"The isochrone fits the CMD in F606W and F814W without any additional corrections.",
"To the right of the CMD we display the radial distributions of the different MS regions.",
"We can see from this how the brightest and more massive MS stars are significantly more centrally concentrated than the faintest sample, with the intermediate mass sample sitting in between.",
"The observable difference between the distributions can be confirmed with a KS-test which yields p-values of the order of $10^{-21}$ or lower.",
"Figure: Left: UV CMD of the core of 47 Tuc displaying the selection of three MS regions, upper (UMS), middle (MMS) and lower (LMS) MS, with the green arrows showing the corresponding masses at the center of each box based on an 11 Gyr PARSEC isochrone.",
"Right: Radial distribution of the regions pointed out on the CMD following the same colour pattern.",
"The legend on the CMD has the number of stars before correcting for incompleteness, while the legend in the right plot gives the size of the sample after correcting for incompleteness.",
"The greater central concentration of the radial distributions of the more massive MS stars is evidence of mass segregation in the core of 47 Tuc.We now use the radial distributions of the MS to find a relationship that will allow us to estimate the masses for different groups of stars in 47 Tuc.",
"For each of the three MS regions we take the value of the distance from the center of the cluster where the cumulative distributions reach 20 and 50 percent, we call these distances $R_{20}$ and $R_{50}$ respectively.",
"Plotting the logarithmic values of each mass against their corresponding $R_{20}$ and $R_{50}$ , we find a relationship for each R. The logarithmic values of mass ($M$ ) and $R$ follow a linear relation like the one in equation REF : $log(M) = A \\times log(R) + B$ We chose it to be a power law as, after many relaxation times, gravity dominates and only mass ratios matter.",
"According to [34], power laws were found to described the velocity dispersion-mass profile in GC.",
"Additionally, [5] showed (in simulations) that this is true at least for low mass stars ($\\lesssim 1M_{\\odot }$ ).",
"Because velocity dispersion maps onto radial distance (i.e.",
"larger velocity dispersions mean larger radial distance), we can then find a similar power law function to obtain masses from radial distributions.",
"By fitting a linear function to the three points retrieved from the MS, represented by the green dots in Figure REF , we get the following relationships: $log(M_{R_{20}}) = -0.631^{+0.003}_{-0.002} \\times log(R_{20}) + 0.844^{+0.005}_{-0.003}$ $log(M_{R_{50}}) = -0.757^{+0.003}_{-0.001} \\times log(R_{50}) + 1.282^{+0.006}_{-0.003}$ for $R_{20}$ and $R_{50}$ respectively.",
"Because we are counting stars, the errors in the estimated masses will be dominated by Poissonian errors.",
"To calculate the errors in our estimated masses, we need the error in $R$ ($R_{20}$ or $R_{50}$ ), using $R_{20}$ as the example we calculate the errors using the following equation: $error(log(R_{20})) = \\pm log(r[N_{R_{20}} \\pm \\sqrt{N_{R_{20}}}]) \\mp log(r[N_{R_{20}}])$ where $r[N]$ is the radius at the Nth star, for example $r[N_{R_{20}}]$ means the radius at the star where the cumulative distribution reaches 20%.",
"Then the error in the mass is just: $error(log(M_{R_{20}})) = A_{R_{20}} \\times error(log(R_{20}))$ where $A_{R_{20}}$ is the slope of the fit for $R_{20}$ .",
"Here we have neglected the errors in the determination of the slope and the determination of the masses for the MS as these are low compared to the statistical errors and have almost no effect on the final error values.",
"Doing the same exercise but limiting the UV field to 105 arcseconds to match the ACS field and taking data from the ACS CMD where possible (normally we would use the matched stars between the two catalogues but as the MS does not extend to faint magnitudes in the ACS data we do not restrict the selection of MS stars to stars measured in all four filters but we do restrain it to the same size field.",
"This is also true for the WDs).",
"The median masses were recalculated for this smaller field yielding similar results.",
"The final fits for the reduced field are: $log(M_{R_{20}}) = -0.994^{+0.004}_{-0.004} \\times log(R_{20}) + 1.303^{+0.006}_{-0.006}$ $log(M_{R_{50}}) = -1.325^{+0.004}_{-0.003} \\times log(R_{50}) + 2.160^{+0.007}_{-0.005}$ which are shown graphically in Figure REF .",
"Figure: CMD (left panel) and radial distribution (right panel) including 5 evolutionary stages: UMS, RGB, HB, AGB and WD, this last one divided into faint and bright WDs, plus the MSBn.",
"The selection of the stars for the UMS, RGB, WDs, and MSBn are taken directly from the UV CMD, while the samples for the HB and AGB have been done through the ACS data.",
"Not all the stars in F225W,F225W-F336WF225W,F225W-F336W have a counterpart on the F606W,F606W-F814WF606W,F606W-F814W CMD as the WDs were not detected with the filters in the visible range.",
"Instead the data was reduced to the same field in order to compare their radial distribution.",
"The colors for the regions on the CMD are the same as in the left plot and specified on the legend.Table: KS-test p-value results between the populations selected on Figure .",
"The numbers demonstrated that we cannot exclude the hypothesis that every stellar sample (except for the MSBn and FWD) could have been drawn from the same population."
],
[
"RESULTS",
"Going back to Figure REF , we now focus on the right panel, which displays the radial distribution of the selected regions.",
"The KS-test results between the cumulative radial distributions (see Table REF ) indicate that all the populations could have come from the same distribution with p-values over 0.40 among any combination excluding those combinations including the FWD and MSBn.",
"The FWD and MSBn show no relation to any of the other chosen populations with p-values $\\lesssim 10^{-3}$ , except between the MSBn and the AGB with a p-value of 0.24 that is probably explained by the low number of AGB stars and possible contamination to the AGB region of the CMD to be discussed in more detail in section .",
"We can also see that the MSBn have the most centrally concentrated distribution.",
"The similarity between the radial distributions of the UMS, RGB, HB and AGB point to mass loss happening late in the AGB.",
"To test the idea of mass loss during the RGB phase, we compare the HB distribution with that of stars of known lower masses.",
"Using the radial distributions of the three MS regions used to build the fits for the masses, we confirm that the HB is only related to the UMS, while the difference gets bigger as we go to lower masses with p-values of $\\sim 10^{-4}$ for MMS ($0.74M_{\\odot }$ ) and $\\sim 10^{-11}$ for the LMS ($0.65M_{\\odot }$ ) region.",
"We also compare the HB to the stars in-between the UMS and MMS, which have a median mass of $0.79M_{\\odot }$ .",
"In this case the p-value is $6.5 \\times 10^{-3}$ , nearly three sigma.",
"Having the radial distributions, we take the value of $R_{20}$ and $R_{50}$ for the different evolutionary stages.",
"We can use equations (REF ) and (REF ), or (REF ) and (REF ), depending on the size of the field, to estimate the masses of the selected stellar populations.",
"For the WFC3 complete list of stars, Figure REF shows the linear fit for the MS data points and the estimated masses for the RGB, FWD, BWD and MSBn, for both $R_{20}$ and $R_{50}$ .",
"The errors in the mass values can be found in Table REF .",
"As expected from the radial distributions, the most massive stars are the MSBn, then the RGB stars with a mass value close to the turn off mass, and finally the WDs which show evidence of mass loss.",
"Both the mass values and errors are very similar for $R_{20}$ and $R_{50}$ , with small variations for the FWD and MSBn.",
"When using the data reduced to the ACS 105 arcseconds field, we are able to include stars in the HB and AGB evolutionary stages.",
"A plot similar to Figure REF but for this smaller field is shown in Figure REF .",
"All the mass values with the corresponding errors are presented in Table REF .",
"Again, the most massive stars are the MSBn followed by the RGB, HB and AGB that show similar masses within the error bars.",
"The least massive stars are once more the WDs.",
"Comparing the results obtained for $R_{20}$ and $R_{50}$ , the largest differences are, in this case, for the AGB and FWD.",
"Figure: Relationship between log(M)log(M) and log(R)log(R) for R 20 R_{20} (red) and R 50 R_{50} (blue).",
"The equations have been obtained through a linear fit using the known masses for the three regions of the MS (green dots).",
"The inset shows the estimated masses for the MSBn, RGB and bright and faint WDs stars at both R 20 R_{20} and R 50 R_{50}, the error values can be found in table .Figure: Relationship between log(M)log(M) and log(R)log(R).",
"Similar to Figure but with the field reduced to a radius of 105 arcseconds which is the limit for the ACS field.",
"The error values for the masses can also be found in table .Table: Results for the mass estimates in both the WFC3 complete 160 arcseconds field and the reduced ACS field (105 arcseconds).",
"The masses were calculated using equations (), (), () and (), and the errors with equations () and ().Another interesting result is the difference between the masses for the MSBn, when comparing the WFC3 ($R \\le 160$ ) and ACS ($R \\le 105$ ) results.",
"To try to explain this discrepancy, we plot the frequency distribution on a straightened CMD of the MSBn, compared to a box of MS stars parallel to the MSBn in the same magnitude range.",
"Furthermore we include the complete sample of MS+MSBn stars within the $F225W$ magnitudes of 24 and 20.7 (corresponding to the limits of the MSBn box).",
"This is shown in Figure REF where we can see an excess of red stars, especially outside of 105 arcseconds.",
"These stars fall in the same color range as the MSBn.",
"This points to a contamination of lower mass stars to the MSBn box, which could explain the lower mass estimates for the lager radius sample.",
"We will expand this discussion in section .",
"Figure: Frequency distribution (divided by the width of the bins) of stars in a straightened CMD represented by dots and crosses while the continuous and dashed lines are the best fitted Gaussian fitted just to the stars blueward of the median.",
"Dots and continuous lines refer to results for stars within 105 arcseconds from the center of the cluster.",
"Crosses and dashed lines for stars between 105 and 160 arcseconds.",
"The red and orange colors are the results for the sample of MS stars parallel to the MSBn box shown in Figure .",
"The green and yellow colors show the results for the MSBn.",
"A gap of ∼0.2\\sim 0.2 magnitudes is kept between the MSBn and MS boxes.",
"Finally, blue and cyan show the complete sample of MSBn and MS (without a gap)."
],
[
"Conclusion",
"The stars in a GC will arrange themselves with the more massive stars moving towards the core.",
"If the stars in 47 Tuc have had enough time to relax, their radial distributions will show the effects of this phenomenon (i.e.",
"more massive stars should show a more centrally concentrated radial distribution), reflecting their masses.",
"We used MS stars of different magnitudes (thus different masses) to develop a relation between the radial distance at 20 and 50% of the cumulative radial distribution and stellar mass.",
"We then used this relation to estimates the masses of the MSBn; and post MS stars in the RGB, HB, AGB and WD evolutionary stages.",
"According to [15], mass loss in 47 Tuc happens when the star is close to the tip of the AGB.",
"As we mention before, the masses for the RGB and WD show evidence of mass loss between these stages of evolution.",
"When we include the masses for the HB and AGB we see that the mass loss happens between the AGB and WDs.",
"We also notice a small increase in the masses between the RGB and AGB that is consistent with the AGBs having evolved from slightly more massive stars that ran out of hydrogen earlier than those forming the current RGB.",
"Two other things might explain this behaviour; (1), the errors for the masses at this evolved stage are very large and might account for the extra mass; and (2), it is possible that even after our efforts to make the AGB as clean as possible, there is still some contamination by evolved blue straggler stars as suggested by [3] and paper2.",
"Unlike the AGB, the contamination to the HB is expected to be very low.",
"Studies show that the HB from stars more massive than the TO mass is brighter than the normal HB [29], [11], [3], [10].",
"In paper2 we arrived at the same conclusion.",
"Additionally, in paper2, we also compare the number of observed versus expected stars which, for the HB, are practically the same.",
"The agreement between the cumulative radial distributions for the UMS, RGB, HB, AGB, and BWD also point towards the bulk of the mass loss happening late in the AGB phase.",
"KS-test results confirm the similarities between these distributions.",
"When we compare the radial distribution of the HB stars to those of the MS at different magnitudes, we find that the HB radial distribution is only related to the UMS stars, and shows no relation to stars with a mass of $0.79M_{\\odot }$ or lower.",
"This allows us to exclude a mass loss greater than $\\sim 0.09M_{\\odot }$ during the RGB stage at the $3\\sigma $ level.",
"The difference between the mass estimates for the BWD and FWD, is mainly due to the fact the BWD have a median age of only 6 Myr, five times lower than the relaxation time in the core of the cluster, while the FWD are typically much older than the relaxation time.",
"In the case of the MSBn, the reported lower values when considering the WFC3 field could be explain by the presence of multiple stellar generations.",
"There is photometric [9], [21] and dynamical [30] evidence of multiple stellar populations in 47 Tuc.",
"Results show that the multiple generations formed within a period of 1 to 2 Gyr [35] and have a helium dispersion of $\\Delta Y\\sim 0.02$ [21].",
"The difference in the helium abundance between the stellar generations, is reflected in the the broadening and color dispersion of the sequences on the CMD.",
"To construct the MSBn region we draw a box about 0.75 ($2.5 \\times \\log (2)$ ) magnitudes above the MS ridge line (i.e.",
"a pair of two equal-mass MS stars will fall within it).",
"Where the main sequence is more narrow, $R<105$ , this works well.",
"Outside $R>105$ , the MS is broader, probably not due to binaries, but possibly also due to the multiple stellar generations present in the cluster.",
"In this case, the MSBn box gets contaminated with pairs of stars from the red side of the MS with a large mass ratio.",
"In consequence, the mass estimate in the outer region ends up being lower because the pairs of stars are indeed less massive.",
"The process described in this paper has only been tested for 47 Tuc.",
"The relations to estimates the masses are exclusive to the core of this cluster, where the relaxation time is short compared to the lifetime of the stars in each evolutionary stage."
]
] | 1605.05740 |
[
[
"The Information-Collecting Vehicle Routing Problem: Stochastic\n Optimization for Emergency Storm Response"
],
[
"Abstract Utilities face the challenge of responding to power outages due to storms and ice damage, but most power grids are not equipped with sensors to pinpoint the precise location of the faults causing the outage.",
"Instead, utilities have to depend primarily on phone calls (trouble calls) from customers who have lost power to guide the dispatching of utility trucks.",
"In this paper, we develop a policy that routes a utility truck to restore outages in the power grid as quickly as possible, using phone calls to create beliefs about outages, but also using utility trucks as a mechanism for collecting additional information.",
"This means that routing decisions change not only the physical state of the truck (as it moves from one location to another) and the grid (as the truck performs repairs), but also our belief about the network, creating the first stochastic vehicle routing problem that explicitly models information collection and belief modeling.",
"We address the problem of managing a single utility truck, which we start by formulating as a sequential stochastic optimization model which captures our belief about the state of the grid.",
"We propose a stochastic lookahead policy, and use Monte Carlo tree search (MCTS) to produce a practical policy that is asymptotically optimal.",
"Simulation results show that the developed policy restores the power grid much faster compared to standard industry heuristics."
],
[
"Introduction",
"Climate change is producing more powerful storms, increasing the frequency and severity of outages in our power grid.",
"Despite the importance of electricity in every aspect of our lives, we often do not know the location of the fault causing an outage, which complicates the task of restoring power.",
"Instead, utilities depend heavily on phone calls (known as “trouble calls\") from customers who have lost their power, in addition to information from the tripping of some circuit breakers/protective devices.",
"Complicating the situation is that as few as one percent of customers call when they lose their power, creating tremendous uncertainty in the knowledge of the state of the grid.",
"This in turn complicates the task of dispatching utility trucks to the location of faults (which are unknown to the utility center) to restore the grid as quickly as possible.",
"According to the Edison Electric Institute ([18]), on average 55% of power outages in the U.S. are due to weather and it can reach up to 80% in some years ([10]).",
"Wind (primarily) and ice frequently bring trees and branches down on power lines creating sporadic outages that quickly spread through the grid due to the limited number of protective devices that are triggered due to a short circuit.",
"The remaining outages are due to equipment faults, vehicle or construction accidents, maintenance or human errors, and animals.",
"It is estimated that 90% of customer outage-minutes are due to faults affecting the local distribution systems ([10]).",
"However, the remaining 10% stem from generation and transmission problems, which can cause wider-scale outages affecting larger numbers of customers.",
"Whenever a power line is damaged, the closest protective device through which the power passes to the damaged power line disconnects resulting in power outage to all customers served through that protective device.",
"Protective devices, such as fuses or circuit breakers, are usually responsible for monitoring the power flow through all the attached downstream components that include power lines, substations, transformers and other protective devices which can form a major part of the circuit.",
"The protective device disconnects the damaged circuit to avoid overloading of the network components which can severely damage them.",
"Whenever a power outage occurs, the electric utility center (EUC) is typically notified by the phone calls of the customers.",
"However, a fault in one location trips the first upstream protective device but it can create a much wider set of outages, with random phone calls coming from locations far from the fault.",
"Complicating the situation is that only a few customers will call.",
"Consequently, identifying the locations of the faults across the power system is a major dilemma for the EUC which needs to restore the network quickly.",
"Even if the EUC knows the location of the faults, routing the truck to restore the grid to minimize the number of customers in outage at any point in time is a challenge.",
"Currently, EUCs use a simple policy to route the utility trucks to restore power such as first-call, first-serve where the customers that call first to report outages are served first by searching around the call location for the fault.",
"This policy might be far from optimal since the actual fault could be a few kilometers away from the location of call.",
"We believe that managing the truck dispatching problem based on a model that properly captures both the physical system as well as the uncertainty offers the potential of significantly reducing the amount of time that customers have lost power.",
"In ([1]), we have developed a probability model that estimates the likelihood of faults occurring across the power grid using two sources of information: 1) the prior probability of power line faults depending on the storm pattern, the structure of the grid and the EUC knowledge of the power line conditions and environment, e.g., existing trees that could potentially fall on the power lines and 2) the phone calls of the customers.",
"In this paper, a stochastic optimization model is designed that makes use of the information provided by the probability model about the likelihood of the location of faults to route the utility efficiently across the power grid with the objective of restoring power as fast as possible.",
"This paper makes the following contributions: 1) it provides the first formal model of a stochastic vehicle routing problem that explicitly captures the physical state (the location of the truck and known state of the grid), information state (other information such as the history of the truck and phone calls), and belief state (capturing probabilistic knowledge of potential outages).",
"2) It proposes a stochastic lookahead policy that is solved using a specialized implementation of Monte Carlo tree search, a method that offers an asymptotically optimal policy, which is also a first for this problem class.",
"3) The performance of the developed policy is tested using a simulator that simulates the power grid of the state of New Jersey by using real data provided by PSE&G which is the main EUC in New Jersey.",
"Simulations show that a) the lookahead learning policy closely matches the performance of the optimal solution and b) significantly outperforms standard industry heuristics when tested on realistic stochastic models.",
"We also demonstrate that the industry heuristic does not make good use of better information, while our method provide information-consistent behavior, with better results as information improves (as one would expect).",
"The paper is organized as follows.",
"Section summarizes the literature of fault identification, utility crew dispatching for restoring power distribution systems and stochastic vehicle routing.",
"Section gives a mathematical model of the flow of information for a grid as some event such as a storm evolves producing faults and loss of power.",
"Section derives the sequential stochastic optimization problem that routes the utility truck across the grid and discusses several dispatch policies.",
"Section introduces MCTS as the lookahead policy that approximates the optimal one with good computational efficiency.",
"Section compares the proposed learning policy against an industry-standard escalation policy."
],
[
"Literature Review",
"We first review the literature from the power community on outage prediction and the dispatching of utility crews, followed by a review of the relevant articles from the mainstream literature on dynamic vehicle routing."
],
[
"Outage Prediction and Utility Crew Dispatching Literature",
"Utilities face two problems in responding to storm damage: 1) identifying the location of outages and 2) dispatching the utility crews.",
"The power grid contains protective devices that detect power flow interruption upon which they shut down power flow to the affected components to avoid further damage.",
"Thus, in case one of the components of the circuit faults, the closest upstream protective device shuts down causing power outage to all the downstream components.",
"Different utilities use different methodologies for management and power restoration during an outage; while most utilities install measurement units on most of the long-distance, high voltage transmission system, distribution systems are not equally automated due to the cost of the equipments.",
"Some utilities, such as Carolina Power & Light utility ([22]), have advanced automated systems that include SCADA (supervisory control and data acquisition) which is a system formed of sensors across the grid operating with coded signals over communication channels so as to provide control of remote equipment.",
"However, the majority of the utilities do not have large scale installed sensors.",
"Instead, utilities depend primarily on phone calls from customers who have lost their power.",
"Complicating the situation is that as few as one percent of customers call when they lose their power, creating tremendous uncertainty in the knowledge of the state of the grid.",
"Distribution systems have different configurations; the most studied configuration in the literature is the radial distribution system which has a tree structure starting from the substation.",
"In this model, there is a single path for power flow from the substation to the consumers.",
"This enables utilities and researchers to propose escalation algorithms for fault identification based on customer calls and grid topology ([37], [19]).",
"Escalation algorithms gather the set of calls and then from the location of each customer call, the tree is followed back to the substation until the first common location for all calls is located; this location would be identified as the faulted one.",
"Escalation algorithms suit single fault scenarios but cannot capture the case of multiple fault scenarios.",
"An improved escalation algorithm for a heat storm is presented in ([25]) where the escalation from the locations of calls depends on the type of the upstream devices.",
"Artificial intelligence techniques that make use of customer calls to identify the fault locations are also investigated to provide better performance than the simple heuristics described above.",
"For example a neural network is presented in ([27]) but the main limitation is in determining the sample training set.",
"An approach based on fuzzy set theory and tabu search is proposed in ([12]) but the limitation is in the computational complexity that becomes intractable for large networks.",
"Knowledge-based outage identification that make use of SCADA and automated meter reading to provide the EUC with knowledge about the status of the distribution system on top of customer calls is proposed in ([26]).",
"But, most of the utilities do not have automated distribution systems and thus, primarily rely on the grid topology, the phone calls of the customers and the experience of the utility personnel to estimate the locations of outages.",
"Moreover, since only a few customers will call, identifying the locations of the faults across the power system is still a major dilemma for the EUC which needs to restore the network as fast as possible.",
"Managing utility crews to restore outages attracted a modest level of attention in the research literature ([38], [42], [41]).",
"In ([42]), a utility truck is routed to the location of a customer that called to report a power outage; then, after restoring the fault, the truck is routed to the next calling customer that has the shortest travel time.",
"In ([41]), the authors develop a model based on data mining and machine learning techniques to predict outages in the grid using collected data from past six storms as well as asset information (framing, pole age, etc.",
"), in addition to environmental information.",
"Then, given the predicted outages, deterministic optimization models are developed to route a given number of utility trucks to perform a predefined number of repair jobs required at each damaged location in order to minimize the grid restoration time.",
"The authors in ([38]) propose an outage prediction model at the level of areas served by a substation, then a mechanism is proposed to plan hourly crew staffing levels across different organizations (service centers, local contractors, mutual aid crews) and different crew types in order to minimize the overall grid restoration time but they do not solve the problem of utility crew routing."
],
[
"Stochastic Vehicle Routing Literature",
"The study of stochastic and dynamic vehicle routing has a long history, with survey articles appearing as early as ([40]).",
"The problem area has attracted so much attention that there has been a steady flow of reviews and survey articles ([39], [35], [32], [16], [5], [30], [24]).",
"Most of this literature focuses on uncertainty in pickups or deliveries, although some authors address random travel times ([20]).",
"Algorithmic strategies range from scenario-based lookahead policies (e.g.",
"([23], [4]), to a wide variety of rolling horizon heuristics that involve solving deterministic approximations modified to handle uncertainty.",
"Most of the academic literature has focused on solving stochastic lookahead models ([23], [4]).",
"Since these problems combine uncertainty with the complexity of these difficult integer programs, these lookahead models are themselves quite hard to solve.",
"Often overlooked is that they are simply rolling horizon heuristics to solve a fully-sequential problem, which is typically not modeled (but is often represented in a simulator).",
"[33] provides a general framework for modeling stochastic, dynamic problems in transportation and logistics, identifying four classes of policies.",
"Focusing specifically on vehicle routing, ([17]) provides a proper model of the stochastic vehicle routing problem, and then proposes a broad class of practical roll-out policies.",
"Our paper addresses the problem of routing a single utility truck that has to find and repair outages over a power grid.",
"Since this problem is completely new, we continue a long tradition of beginning by addressing the single-vehicle version of the problem (see, for example, ([34]) and ([36])).",
"Our problem includes a unique dimension which has never been addressed in the literature: probabilistic knowledge about the state of the network, and the dimension that utility trucks are not only repairing the network, but as the vehicle progresses, it collects information that is used to update the belief knowledge.",
"This “state of knowledge� has to be represented along with the physical state (location) of the network, and the history of the truck.",
"Our solution strategy uses a technique familiar to the computer science community known as Monte Carlo Tree Search (MCTS).",
"While this is most commonly used for deterministic problems ([9]), it has been extended to stochastic problems and shown to be asymptotically optimal for certain variations of the algorithm ([3]).",
"MCTS has been popular in the computer science community (although primarily for deterministic problems), but has only recently seen applications in transportation and logistics ([28], [15]).",
"However, we are unaware of any prior application of MCTS in the setting of information collection, where the state variable includes a belief state."
],
[
"Problem Description",
" Figure: Illustration of a circuit in a distribution grid, with feeders from a single substation.In this section, we describe the parameters and variables that govern the evolution of the problem to be solved over time.",
"First, we assume an overhead power system consisting of substations, poles that carry the protective devices, power lines and transformers through which customers are fed with power as shown in Figure REF .",
"In ([1]), we have already developed a probability model that estimates the likelihood of fault occurring across the power lines of the grid based on the prior probability of power line fault and the phone calls of the customers.",
"It is assumed that a fault across a power line includes the fault across its connection ends which could be a transformer or a protective device.",
"This can be justified by the fact that a fault across a power line or one of the components at its end connections results in power outage to the same set of customers in the power system.",
"Let $\\mathcal {U}$ be the set of circuits in the power system, $\\mathcal {U}=\\lbrace u: u=1,\\ldots ,U\\rbrace $ where each circuit can be graphically represented by a tree that is rooted at the substation.",
"We also define $\\mathcal {I}=\\lbrace i: i=1,\\ldots ,I\\rbrace $ as the set of poles, $\\mathcal {I}^u=\\lbrace i: i=1,\\ldots ,I^u\\rbrace $ set of nodes of circuit $u$ which are mounted on the poles and $\\mathfrak {L}^u$ as the set of power lines on circuit $u$ where power line $i\\in \\mathfrak {L}^u$ feeds node $i\\in \\mathcal {I}^u$ with power.",
"Using the configuration of the grid, we define $d^u_i$ to be the first upstream protective device of node $i$ and power lines $i$ on circuit $u$ .",
"For each circuit $u$ , let $\\mathcal {D}^u$ be the set of protective devices and $\\mathcal {S}^u$ be the set of segments where each segment contains the power lines that trigger the same protective device.",
"We define ${\\cal S}^u_i$ as the set of power lines belonging to the same segment of node $i$ on circuit $u$ , i.e., triggering the same protective device, $\\mathcal {S}^u_i=\\lbrace j, d^u_j = d^u_i \\rbrace $ .",
"We also introduce $\\mathcal {Q}^u_{i}=\\lbrace j, \\mbox{ node } i\\in \\mathcal {I}^u \\hbox{~becomes in outage if power line~} j \\in \\mathfrak {L}^u \\hbox{~faults}\\rbrace $ as the set of power lines that if faulted result in an outage to node $i$ on circuit $u$ and $\\mathcal {W}^u_i$ as the set of downstream segments of node $i$ but with different upstream protective devices, i.e., $\\mathcal {W}^u_i=\\lbrace S^u_j, \\hbox{~segment~}S^u_j \\hbox{~is downstream of} \\linebreak \\hbox{node~}i ~\\hbox{~\\&~} d^u_i\\ne d^u_j\\rbrace $ .",
"We assume that $n_i^u$ customers are attached to node $i$ on circuit $u$ .",
"If the node is a transformer, then $n_i^u>0$ , otherwise $n^u_i=0$ because no customers are attached to the power generator or protective devices.",
"Let $G(\\mathcal {V},\\mathcal {E})$ be the graph representing the road network through which the utility crew can travel to check the power grid where $\\mathcal {V}=\\lbrace i: i=1,\\ldots ,N\\rbrace $ is the set of nodes of the road network where each node represents either a begin/end of a road segment or a pole of the power grid, and $\\mathcal {E}$ is the set containing the arcs/roads in the graph, i.e., $\\mathcal {E}=\\lbrace (i,j): \\mbox{~if there is an arc between nodes~} i \\mathrm {~and~ } j \\mathrm {~in~} \\mathcal {V}\\rbrace $ .",
"Thus, some of the arcs/roads in $\\mathcal {E}$ are parallel to the power lines of the power grid as shown in Figure REF .",
"In this paper, we assume that an important source of information that the EUCs rely on to identify power outages is the calls of the customers that lost power.",
"Let $H_t=\\lbrace H^u_{ti}: \\forall i\\in \\mathcal {I}^u, \\forall u\\in \\mathcal {U}\\rbrace $ be a random vector representing the possible realizations of received calls from the nodes of the power system by time $t$ where $H^u_{ti}$ is a random variable representing the number of received phone calls from node $i$ on circuit $u$ .",
"Customer phone calls arrive over time, thus, $H_t=H_{t-1}+\\hat{H}_t$ where $\\hat{H}_t$ represents the new set of received calls in the time interval between $t-1$ and $t$ .",
"Based on the set of received phone calls by time $t$ , we have to figure out the power lines that faulted in the grid.",
"For each circuit $u$ , let $L^u_t=\\lbrace L^u_{ti}: \\forall i\\in \\mathfrak {L}^u\\rbrace $ be a random vector representing the possible realizations of power lines that faulted by time $t$ on circuit $u$ where $L^u_{ti}$ is a random variable indicating whether power line $i$ on circuit $u$ has faulted; we assume $L^u_{ti}=1$ if the power line faulted and $L^u_{ti}=0$ otherwise.",
"Let $T_{tij}$ be a random variable representing the required travel time from node $i$ to node $j$ which depends on the traffic condition of the road, at time $t$ , and $R^u_{i}$ be a random variable representing the required repair time for power line $i$ on circuit $u$ which depends on the fault type.",
"Now, we can define $\\tau _{tij}=T_{tij} + \\sum _{u\\in \\mathcal {U}}R^u_{j}$ as the total time required to go from node $i$ to $j$ which accounts for the travel and repair times of all power lines that feed pole $j$ with power.",
"Let $\\omega $ be a sample realization of the random variables.",
"At time $t$ and according to sample path $\\omega $ , let $H_t(\\omega )$ be an outcome of customer calls, $L^u_t(\\hspace{-0.85355pt}\\omega \\hspace{-1.42271pt})$ be an outcome of the power line faults on circuit $u$ , $R^u(\\omega )$ be an outcome of the repair times on circuit $u$ and $T_{t}(\\omega )$ be an outcome of the travel conditions.",
"Then, we can define at time $t$ and according to sample path $\\omega $ , $p(H_t=H_t(\\omega ))$ as the probability of the set of received calls, $p(L^u_t=L^u_t(\\hspace{-0.85355pt}\\omega \\hspace{-1.42271pt}))$ as the prior probability of power line faults on circuit $u$ , $p(R^u=R^u(\\omega ))$ as the probability of the repair time on circuit $u$ and $p(T_t=T_t(\\omega ))$ as the probability of the travel time.",
"Define $\\Omega $ as the set of all outcomes, $\\cal F$ as the set of events and $\\cal P$ as the probability measure on $(\\Omega ,\\cal F)$ , so that $(\\Omega ,\\cal F,\\cal P)$ is the probability space.",
"$\\Omega \\subseteq \\mathcal {F} $ is formed by a set of scenarios where each scenario $\\omega $ indicates a specific set of calls, a set of power line faults, fault types and travel conditions (traffic).",
"Let $p(\\omega )$ be the probability of scenario $\\omega $ , where $\\sum _{\\omega \\in \\Omega }p(\\omega ) = 1$ .",
"Second, the EUC should estimate the prior probability, $p(L^u_{ti})$ , of fault of power line $i$ on circuit $u$ at time $t$ based on the storm pattern, the structure of the grid and the EUC knowledge of the power line conditions and environment as explained in ([1]).",
"In the worst case, the EUC can assume a uniform probability of fault for all power lines, and then the set of received of calls will identify the ones that have most likely faulted.",
"In ([1]), for any realization of $L^u_t$ and $H_t$ , the posterior probability of fault of the power lines on circuit $u$ given the phone calls is calculated using Bayes' theorem as follows: $p(L^u_t|H_t) = \\frac{p(H_t|L_t^u)p(L_t^u)}{p(H_t)},$ where $p(H_t|L_t^u)$ is the likelihood of the calls given the power line faults on circuit $u$ and the expressions have been derived in equation (5) in ([1]); it is a function of the locations of the calling customers, the calling probability which refers to the percentage of customers calling to report an outage and the grid structure.",
"The posterior probability of fault of power line $i$ on circuit $u$ , given the phone calls can be expressed as: $p(L^u_{ti}=1|H_t)=\\frac{\\sum _{L^u_t\\in \\lbrace \\mathcal {L}^u\\rbrace _{L^u_{ti}=1}}p(H_t,L^u_t)}{p(H_t)}=\\frac{\\sum _{L^u_t\\in \\lbrace \\mathcal {L}^u\\rbrace _{L^u_{ti}=1}}p(H_t|L^u_t)p(L^u_t)}{\\sum _{L^u_t\\in \\mathcal {L}^u} p(H_t|L^u_t)p(L^u_t)}, $ where $\\mathcal {L}^u$ is the set containing all power line fault combinations on circuit $u$ and $\\lbrace \\mathcal {L}^u\\rbrace _{L^u_{ti}=1}$ is the set containing a subset of vectors of $\\mathcal {L}^u$ where the variable corresponding to power line $i$ , i.e., $L^u_{ti}$ , is equal to 1.",
"Thus, $\\lbrace \\mathcal {L}^u\\rbrace _{L^u_{ti}=1}$ is the set containing all the combinations of power lines that can fault with power line $i$ on circuit $u$ .",
"In this work, another factor plays a major role in identifying the fault probability of a power line by time $t$ which is the trajectory of the truck that is going across the power grid to fix faults.",
"For example, if power line $i$ on circuit $u$ has been fixed by time $t-1$ , then its prior probability of fault, at time $t$ , is 0.",
"Let $x_{tij}$ be a binary variable representing whether the utility truck travels from node $i$ to node $j$ using roadway graph $G(\\mathcal {V},\\mathcal {E})$ at time $t$ .",
"It is assumed that if a truck travels from node $i$ to node $j$ at time $t$ then it repairs all the power lines that are attached to pole $j$ if there are faults across them.",
"Let $x_t$ be a matrix capturing the vector of decisions $(x_{tij})_{i,j\\in \\mathcal {V}}$ where $x_{tij} = 1$ if the truck is dispatched to $i$ from $j$ at time $t$ .",
"Also, let $X_t$ be the trajectory of the truck up to time $t$ , i.e., $X_t = (x_{t^{\\prime }})_{t^{\\prime }=0}^t$ .",
"The information we are looking for, at time $t$ , is the posterior probability, $p(L^u_{ti}|H_t,X_{t-1})$ , of power line $i$ being in fault given the phone calls and the trajectory of the truck up to time $t-1$ which is given by: $&&\\hspace{-19.91684pt}p(L^u_{ti}=1|H_t,X_{t-1})=\\frac{\\sum _{L^u_t\\in \\lbrace \\mathcal {L}^u\\rbrace _{L^u_{ti}=1}}p(H_t|L^u_t)p(L^u_t|X_{t-1})}{\\sum _{L^u_t\\in \\mathcal {L}^u}p(H_t|L^u_t)p(L^u_t|X_{t-1})},$ where $p(L^u_t|X_{t-1})$ is the prior probability of vector $L^u_t$ being in fault given the route of the truck; the prior probability of the power lines is updated by setting $p(L^u_{ti}|X_{t-1})=0$ if $\\sum _j x_{t^{\\prime }ji}=1, t^{\\prime }\\le t-1$ .",
"However, the likelihood $p(H_t|L^u_t)$ is independent of the route of the truck.",
"Accordingly, at each time $t$ , the prior probabilities of power line faults should be computed and the set of received calls should account for any new incoming calls between $t-1$ and $t$ .",
"After collecting the updated priors and customer calls, the posterior probability of faults are computed using the model described in ([1]) upon which the route of the truck is determined, i.e., the value of $x_{t}$ .",
"The aim of this work is to develop a policy that routes a utility truck to restore outages in the power grid as quickly as possible while accounting for all the components described above."
],
[
"Sequential Stochastic Optimization Problem",
" To develop a near-optimal policy for managing a utility truck across the grid, the problem should be formulated first as a sequential stochastic optimization problem after which we can derive the optimal policy to solve it.",
"However, this introduces several challenges as explained below.",
"First, the number of customers whose power is restored after a truck visits a location (even if a repair occurs) is a random variable since the actual value depends also on upstream and downstream outages which are uncertain.",
"Second, each time a truck travels a segment, the information collected (e.g.",
"that there is an outage on that segment, or not) is used to update the probability of outages on all lines.",
"Third, new phone calls are arriving over time, which allows us to update probabilities of outages.",
"At the same time, as trucks identify and fix outages, other phone calls may become irrelevant.",
"Finally, the time required for a truck to traverse a segment depends to a large extent on whether it finds an outage, and the time required to repair the outage.",
"In order to develop the sequential stochastic optimization problem that restores power to the maximum number of customers in outage over time, we need to define its five fundamental elements: State $S_t$ - The information capturing what we know at time $t$ ; in this work, $S_t=\\bigg (R_t,P_t^L,H_t\\bigg )$ where $R_t$ represents the physical state that indicates the location of the truck at time $t$ , $P_t^L$ is a vector containing the prior probability of all power line faults (i.e., its entries are $p(L_{ti}^u=1|X_{t-1})$ and $H_t$ is the set of received calls by time $t$ ).",
"Thus, given the prior probabilities of fault and the set of calls, we can calculate the posterior probabilities of faults which also represent the state of the network at time $t$ .",
"Decision $x_t$ - The vector $x_t = (x_{tij})_{i,j}$ captures the decision made at time $t$ about the next hop of the truck, where $x_{tij} = 1$ if the truck is sent from node $i$ to node $j$ at time $t$ .",
"Let $X^\\pi (S_t)$ be the policy that determines $x_t \\in \\mathcal {X}_t$ given $S_t$ .",
"Exogenous information $W_t$ - The new information that arrives between $t-1$ and $t$ .",
"This includes new phone calls ${\\hat{H}}_t$ , as well as information about outages (discovered by the utility trucks) and travel (or repair) times.",
"We denote the exogenous information process by $W_1, W_2, ..., W_T$ .",
"Let $\\omega $ be a sample realization of the information, which we note depends on the policy (if a truck fixes an outage, then this will produce fewer phone calls).",
"For this reason, we let $\\Omega ^\\pi $ be the set of outcomes which depends on the policy $\\pi $ that we use to dispatch trucks.",
"Let ${\\cal F}^\\pi $ be the sigma-algebra on $\\Omega ^\\pi $ , and let ${\\cal P}^\\pi $ be the probability measure on $(\\Omega ^\\pi , {\\cal F}^\\pi )$ , giving us a probability space $(\\Omega ^\\pi , {\\cal F}^\\pi , {\\cal P}^\\pi )$ .",
"We let ${\\cal F}^\\pi _t$ be the sigma-algebra generated by $W_1, ..., W_t$ , giving us the filtrations ${\\cal F}^\\pi _t \\subseteq {\\cal F}^\\pi _{t+1}$ .",
"This notation means that all variables indexed by $t$ are ${\\cal F}^\\pi _t$ -measurable.",
"The transition function $S_{t+1}=S^M(S_t,x_t,W_{t+1})$ which represents the evolution of the physical, informational and belief states; for example, if power line $j$ on circuit $u$ was repaired at time $t$ then its prior probability of fault is set to 0 at time $t+1$ , i.e., $p(L_{{t+1}j}=1|\\sum _ix_{tij}=1)=0$ .",
"This in turn updates the probabilities of faults over the power grid.",
"The function also represents the exogenous evolution of calls, i.e., $H_{t+1}=H_t+\\hat{H}_{t+1}$ and any additional information contained by $W_t$ as explained earlier.",
"The objective function - Our objective is to find the policy that maximizes the number of customers with restored power over time which is equivalent to minimizing the number of customers in outage while minimizing the truck's operating cost.",
"Let $C(S_t,x_t) = C^c(S_t,x_t)+\\gamma C^o(S_t,x_t)$ be the total cost of taking decision $x_t$ given state $S_t$ where $C^c(S_t,x_t)$ and $C^o(S_t,x_t)$ represent the customer outage-minutes and the truck's operating costs, respectively, and $\\gamma $ is a tunable parameter that balances the weight between the two costs.",
"The cost function $C^c(S_t,x_t)$ represents the number of customers in outage at time $t$ as shown in Figure REF and the sum over time, $\\sum _{t=0}^TC^c(S_t,x_t)$ , evaluates the shaded area under the curve which we refer to as “customer outage-minutes.\"",
"The total objective can be represented as $\\sum _{t=0}^TC(S_t,x_t)$ and we have to find the policy that solves: $\\min _\\pi \\mathbb {E}^\\pi \\left[\\sum _{t=0}^TC(S_t,X^{\\pi }(S_t))|S_0\\right]$ where the expectation is over all possible sequences $W_1,W_2,\\ldots ,W_T$ , which depend on the decisions taken.",
"The initial state $S_0$ captures all deterministic parameters, and priors on any uncertain information (such as outages).",
"Our goal is to find the best policy for making a decision.",
"Figure: Objective function; outage-minute is represented by the shaded area under the curve.So far, we have defined the fundamental elements influencing truck routing.",
"In this work, the flow of information and state transition occurs in the following sequence: $(S_0,x_0,S^x_0,W_1,S_1,x_1,S^x_1,W_2,S_2,\\ldots ,S_t,x_t,S^x_t,W_t,\\ldots ,S_T)$ where $S^x_t$ represents the state after taking decision $x_t$ , known as the post-decision state.",
"Based on $x_t$ , we see a specific exogenous information $W_{t+1}$ indicating that it directly depends on $x_t$ .",
"The distinguishing feature of this problem relative to the classical stochastic vehicle routing literature is the dimension that the utility truck is collecting information, and that our state variable includes the physical state of the truck, our current state of knowledge concerning the probability of outages, and the history of phone calls (there is a reason we have to retain the history).",
"A decision to dispatch a truck from $i$ to $j$ has to consider not only the change in the physical state of the truck but also the value of the information that is collected while traversing from $i$ to $j$ .",
"The contribution of this paper relative to the dynamic vehicle routing literature is that we capture the collection of information (phone calls, observations of outages) to update our beliefs about where outages may be.",
"In contrast with classical models where the state $S_t$ captures the status of trucks and unserved demands, our model uses a state variable with three types of information: the physical state $R_t$ , which includes the location of the truck, information $I_t$ which includes the history of phone calls $H_t$ and observations of actual outages, and the probabilistic knowledge $P^L_t$ which captures the probability of outages on observed lines.",
"Thus we write the state of our system as $S_t = (R_t, H_t, P^L_t)$ .",
"We update our probabilistic knowledge $P^L_t$ by applying Bayes' theorem that combines the prior $P^L_t$ with exogenous information of phone calls and grid observations to produce the posterior $P^L_{t+1}$ .",
"[31] describes four classes of policies for solving the optimization in equation (REF ): policy function approximations (PFAs), parametric cost function approximations (CFAs), policies based on value function approximations (VFAs), and lookahead policies.",
"The utility currently uses a rule-based PFA (send a truck to the next caller reporting an outage).",
"In this paper, we are going to use a policy based on a stochastic lookahead model."
],
[
"Multistage Lookahead Policy",
"A lookahead policy is particularly useful for our problem since it is time-dependent obviously because the grid has a radial structure which affects the computation of the objective.",
"First, we begin our discussion by characterizing an optimal policy using the function $X^*_t(S_t)=\\arg \\min _{x_t\\in {\\mathcal {X}_t(S_t)}}\\left(C(S_t,x_t)+\\min _\\pi \\mathbb {E}^\\pi \\left\\lbrace \\sum _{t^{\\prime }=t+1}^T C(S_{t^{\\prime }},X^\\pi _{t^{\\prime }}(S_{t^{\\prime }}))|S^x_t\\right\\rbrace \\right), $ where $S_{t+1}=S^M\\left(S_t, x_t,W_{t+1}\\right)$ .",
"Generally, (REF ) represents a lookahead policy solving a multistage stochastic program , i.e., after implementing the first-period decision and stepping forward in time, the problem has to be solved again with a new set of scenario trees.",
"In scenario trees, we choose $x_t$ , then see the information $W_{t+1}$ , then we would choose $x_{t+1}$ , after which we see $W_{t+2}$ , and so on.",
"Thus, (REF ) is equivalent to: $X^*_t(S_t)&=&\\arg \\min _{x_t\\in {\\mathcal {X}_t(S_t)}}\\bigg (C(S_t,x_t)+\\mathbb {E}_{W_{t+1}\\in \\Omega _{t+1}}\\bigg [\\min _{x_{t+1}\\in {\\mathcal {X}_{t+1}(S_{t+1})}}C(S_{t+1},x_{t+1})+\\mathbb {E}_{W_{t+2}\\in \\Omega _{t+2}}\\bigg [ \\ldots + \\nonumber \\\\&&\\hspace{156.49014pt}\\mathbb {E}_{W_{T}\\in \\Omega _{T}}\\bigg [C(S_{T})|S^x_{T-1}\\bigg ]\\ldots \\bigg ]|S^x_{t+1}\\bigg ]|S^x_t \\bigg ]\\bigg ).$ Scenario trees capture the entire history; this is particularly important for our problem since the history of the route of the truck which indicates the power lines that have been fixed affects the computation of the customers in outage due to the radial structure of the grid.",
"In practice, the policies in (REF ) or (REF ) are computationally intractable, requiring that we introduce an approximation.",
"There are several strategies that are typically used to simplify lookahead models: 1) Limiting the horizon by reducing it from $(t,T)$ to $(t,t+H)$ .",
"2) Discretizing the time, states and decisions to make the model computationally tractable.",
"3) Aggregating the outcome or sampling by using Monte Carlo sampling to choose a small set, $\\tilde{\\Omega }_t$ , of possible outcomes between $t$ and $t+H$ .",
"4) Stage aggregation which represents the process of revealing information before making another decision; a common approximation is a two-stage formulation, where we make a decision $x_t$ , then observe all future events (until $t+H$ ), and then make all remaining decisions.",
"A more accurate formulation is a multistage model but these can be computationally expensive if not strategically designed.",
"5) Dimensionality reduction where we ignore some variables in our lookahead model as a form of simplification.",
"For example, a forecast of future incoming phone calls can add a number of dimensions to the state variable.",
"While we have to track these in the original model, we can hold them fixed in the lookahead model, and then ignore them in the state variable (these become latent variables).",
"The most common lookahead strategy is a deterministic model, although even these can be difficult for vehicle routing problems.",
"For this reason ([17]) explores the use of a rollout policy based on ideas from ([6]) which uses a simple heuristic to approximate the downstream value of a decision.",
"Deterministic lookahead models are unable to capture the uncertainty of our belief state, while rollout heuristics, aside from offering no theoretical guarantees, are typically built from myopic policies which are unlikely to work well.",
"All variables in the lookahead model are indexed by $(t,t^{\\prime })$ where $t$ represents when the decision is being made (which fixes the information content) while $t^{\\prime }$ is the time within the lookahead model.",
"We also use tilde's to avoid confusion between the lookahead model (which often uses a variety of approximations) and the real model.",
"So, the stochastic lookahead model becomes $\\hspace{-14.22636pt}X^*_t(S_t)&=&\\arg \\min _{x_t\\in {\\mathcal {X}_t(S_t)}}\\bigg (C(S_t,x_t)+\\tilde{\\mathbb {E}}_{\\tilde{W}_{t,t+1}\\in \\tilde{\\Omega }_{t,t+1}}\\bigg [\\min _{\\tilde{x}_{t,t+1}\\in {\\mathcal {\\tilde{X}}_{t,t+1}(\\tilde{S}_{t,t+1})}}\\tilde{C}(\\tilde{S}_{t,t+1},\\tilde{x}_{t,t+1})+ \\nonumber \\\\&&\\hspace{-19.91684pt}\\mathbb {\\tilde{E}}_{\\tilde{W}_{t,t+2}\\in \\tilde{\\Omega }_{t,t+2}}\\bigg [ \\ldots \\mathbb {\\tilde{E}}_{\\tilde{W}_{t,t+H}\\in \\tilde{\\Omega }_{t,t+H}}\\bigg [\\tilde{C}(\\tilde{S}_{t,t+H})|\\tilde{S}^x_{t,t+H-1}\\bigg ]\\ldots \\bigg ]|\\tilde{S}^x_{t,t+1}\\bigg ]|S^x_t \\bigg ]\\bigg ),$ where the expectation $\\tilde{\\mathbb {E}}\\lbrace .|S^x_t\\rbrace $ is over the sample space in $\\tilde{\\Omega }_{t,t+1}$ which is constructed given that we are in state $S_t$ at time $t$ .",
"When computing this policy, we start in a particular state $S_t$ , but then step forward in time using: $\\tilde{S}_{t,t^{\\prime }+1}=S^M(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }},\\tilde{W}_{t,t^{\\prime }+1}), t^{\\prime }=t,\\ldots ,t+H-1.$ Alternatively, we can formulate (REF ) as: $\\hspace{-14.22636pt}X^*_t(S_t)&=&\\arg \\min _{x_t\\in {\\mathcal {X}_t(S_t)}}\\bigg (C(S_t,x_t)+\\tilde{V}^x_t(S^x_t)\\bigg ),$ where $\\tilde{V}^x_t(S^x_t)$ is the approximate average value of the post-decision state that is returned by the lookahead policy.",
"In this work, we fix the set of received calls at time $t$ while the other sources of randomness such as the fault locations and the travel/repair times are included in the tree.",
"This lookahead model could be solved via scenario trees but the size of the tree can grow exponentially if we take into account all possible random realizations since taking decision $\\tilde{x}_{tt^{\\prime }}$ from state $\\tilde{S}_{tt^{\\prime }}$ can result in many different successor states based on the observed exogenous information.",
"To simplify the problem, we can assume that the only randomness that is modeled in the tree is whether a fault is found at a certain location or not.",
"Also, one fault type can be considered which results in a deterministic repair time and the travel times can be also considered deterministic but they definitely depend on the distances between the nodes in the grid.",
"In this case, there are only two possible successor states by taking decision $x_{tt^{\\prime }}$ from state $S_{tt^{\\prime }}$ which depends on whether a fault is observed or not.",
"The number of customers whose power is restored due to decision $x_t$ from state $S_t$ is a random variable given the uncertainty about the location of outages due to the radial structure of the grid.",
"The actual number of served customers depends on whether there is a fault upstream to the visited location, which is uncertain.",
"In addition, if a location faults, it causes outage to the customers attached to its segment and all downstream segments.",
"Thus, if a fault is fixed, then all the downstream customers might restore power unless there are other downstream segments that faulted which is uncertain.",
"The actual “customer outage-minutes\" objective is represented in Figure REF , but since we cannot figure out the exact number of customers with restored power at each state $S_t$ , we evaluate the expected customer outage-minutes.",
"At state $S_t$ , the fault probabilities are updated according to (REF ) and the expected number of customers in outage is: $C^c(S_t,x_t)= \\sum _{u\\in \\mathcal {U}}\\sum _{s\\in S^u} \\left(1-\\prod _{k\\in Q_s}p(L^u_{tk}=0)\\right)\\sum _{k\\in s} n^u_k,$ where $Q_s$ represents the set of power lines that if fault result in outage to segment $s$ , the term in parenthesis is the probability of at least one fault across $Q_s$ that causes $s$ to be in outage and $\\sum _{k\\in s} n^u_k$ is the number of customers across segment $s$ .",
"Generating the entire scenario tree is computationally intractable.",
"For this reason, we turn to a popular strategy developed in the computer science community known as Monte Carlo tree search (MCTS) ([29], [9], [13], [21]) which uses an intelligent sampling procedure to create a partial tree that can be solved, and which asymptotically ensures that we would find the optimal decision given a large-enough computing budget.",
"Given that we are solving the one-truck problem, we can formulate the lookahead model as a decision tree, taking advantage of the property that the number of possible decisions for a truck at any point in time is quite small, since it is limited to paths over a road network.",
"Further, if we limit the random information to whether the truck finds a fault or not, then the random variables are binomial.",
"However, even with these restrictions, a decision tree will still grow exponentially.",
"We propose MCTS as the look-ahead policy to solve the original problem in the following way.",
"Given a current state, $S_t$ , which depends on the location of the truck and the probability of faults at time $t$ , MCTS should decide where to move the truck next in order to minimize the objective represented by the customer outage-minutes.",
"So, starting from the current state $S_t$ as the root node, sampled MCTS successively builds the look-ahead tree over the state-space.",
"Finally, the move that corresponds to the highest value from the root node will be taken.",
"At this stage, the probability space is updated taking into account any incoming exogenous information such as whether there was a fault or not by taking the move that was the outcome of the previous step, the newly arrived phone calls of customers and the consumed travel/repair times.",
"Then, the whole process is repeated until power is reconstructed to the whole power grid based on the values of the probability model; that is, when the values of the posterior probabilities drop below a certain threshold $\\epsilon ^{thr}$ which is typically a small value.",
"The pseudo-code of the proposed lookahead policy to solve the utility truck routing problem is presented in Algorithm REF .",
"[t!",
"]Lookahead Policy for Utility Truck Routing [t!]",
"Step 0.",
"Initialization: Initialize the state $S_0=(R_0,P_0^L,H_0)$ .",
"Set $t\\leftarrow 0$ .",
"Step 1.",
"While $\\left(p(L^u_{t{i}}|H_t,X_{t-1})\\ge \\epsilon ^{thr}, \\forall i,\\forall u\\right)$ do: Step 1a.",
"At time $t$ , fix the set of calls $H_t$ and determine $\\tilde{\\Omega }_t$ .",
"Step 1b.",
"Call $MCTS(S_t)$ to solve (REF ) that determines $x^*_t$ .",
"Step 1c.",
"Set $x_t\\leftarrow x^*_t$ and move the truck according to $x_t$ .",
"Step 1d.",
"Update $S_{t+1}=S^M(S_t,x_t,W_{t+1})$ until the truck reaches the destination node set by $x^*_t$ , say at time $t^{\\prime }$ .",
"Step 1e.",
"$t\\leftarrow t^{\\prime }$ End While"
],
[
"Monte Carlo Tree Search",
"Even if we are solving a stochastic lookahead for a single truck, the resulting decision tree still grows exponentially, and is too large to enumerate.",
"For this reason, we turn to a technique known as Monte Carlo tree search (MCTS), widely used in computer science, which dynamically creates a decision tree using a combination of Monte Carlo sampling and heuristic rollout policies to identify the most promising branches.",
"MCTS builds successively a search tree until some predefined computational budget is reached such as number of iterations or time constraints.",
"In deterministic MCTS, each node corresponds only to the physical state of the network and each edge corresponds to a decision which results in another unique physical state.",
"In stochastic MCTS, the exogenous information is included in the tree and thus each node corresponds to a state and each edge can represent either a decision (move) or an exogenous event both of which result in another state.",
"Each iteration of MCTS is formed of four basic steps which are Selection, Expansion, Simulation and Backpropagation ([13]).",
"The Selection step involves selecting a decision successively starting from the initial state till an expandable state is reached.",
"The Expansion step adds one or more states to the tree.",
"The Simulation step referred to as “Simulation Policy\" is used to evaluate the value of the newly added state.",
"Finally, in the Backpropagation step, the value of the newly added state is backpropagated to update the value functions of all predecessor states.",
"MCTS has evolved in the literature primarily in the context of deterministic problems.",
"Deterministic MCTS has been used extensively in the literature, as summarized in ([9]).",
"In this case, the exogenous information is handled in the simulation policy by averaging over the possible outcomes.",
"Stochastic outcomes (which characterizes our application) has been handled using a process known in the computer science community as “Determinization\" ([7], [8], [11]), or by explicitly representing exogenous information in the tree directly ([14]), which is the approach that we take.",
"It is shown in ([3]) that stochastic MCTS, in which the exogenous information is included in the tree, is asymptotically optimal if every possible action can be uniformly chosen to be included in the tree no matter how good or bad it is.",
"The same rule applies for the exogenous events.",
"In this work, we choose a more strategic way in choosing the actions to expand in order to converge faster to the optimal solution.",
"In stochastic MCTS, which is also referred to as sampled MCTS, the states in the tree are associated with the following data: 1) The pre-decision value function, $\\tilde{V}_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }})$ , the post-decision value function, $\\tilde{V}^x_{tt^{\\prime }}(\\tilde{S}^x_{tt^{\\prime }})$ , and cost, $\\tilde{C}(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }})$ , which represent the value function and cost of being in state $\\tilde{S}_{tt^{\\prime }}$ and taking decision $\\tilde{x}_{tt^{\\prime }}$ , respectively.",
"2) The visit count, $N(\\tilde{S}_{tt^{\\prime }})$ , which represents the number of rollouts that included state $\\tilde{S}_{tt^{\\prime }}$ .",
"3) The count of the state-decision, $N(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }})$ , which represents the number of times decision $\\tilde{x}_{tt^{\\prime }}$ was taken from state $\\tilde{S}_{tt^{\\prime }}$ .",
"4) The set of decisions, $\\tilde{\\mathcal {X}}_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }})$ , and set of possible random outcomes, $\\tilde{\\Omega }_{t,t^{\\prime }+1}(\\tilde{S}_{tt^{\\prime }}^{x})$ ; $\\tilde{\\mathcal {X}}_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }})$ is the set of decisions that the truck would face moving over a road network given that it is at state $\\tilde{S}_{tt^{\\prime }}$ .",
"For state $\\tilde{S}_{tt^{\\prime }}$ , let $\\tilde{\\mathcal {X}}_{tt^{\\prime }}^e(\\tilde{S}_{tt^{\\prime }})$ be the set of decisions that has been explored by the truck (i.e., expanded in the tree) by time $t^{\\prime }$ and let $\\tilde{\\mathcal {X}}_{tt^{\\prime }}^u(\\tilde{S}_{tt^{\\prime }})$ be its complement set which represents the set of unexplored decisions in the tree by time $t^{\\prime }$ .",
"Similarly, $\\tilde{\\Omega }_{t,t^{\\prime }+1}(\\tilde{S}_{tt^{\\prime }}^{x})$ is the set of all possible random events that can take place at time $t^{\\prime }+1$ given state $\\tilde{S}_{tt^{\\prime }}^{x}$ , $\\tilde{\\Omega }_{t,t^{\\prime }+1}^e(\\tilde{S}_{tt^{\\prime }}^{x})$ is the set of explored events and $\\tilde{\\Omega }_{t,t^{\\prime }+1}^u(\\tilde{S}_{tt^{\\prime }}^{x})$ is its complement.",
"In sampled MCTS, assume that we are at a pre-decision state $\\tilde{S}_{tt^{\\prime }}$ and decide to take decision $\\tilde{x}_{tt^{\\prime }}$ which takes us to the post-decision state $\\tilde{S}_{tt^{\\prime }}^{x}$ .",
"In real life, while the truck is moving to the location specified by $x_{t}$ , it encounters first the travel time which depends on traffic.",
"Second, it discovers the fault type at the intended location which determines the repair time and affects the number of customers with restored power.",
"In the lookahead model, upon determining $\\tilde{x}_{tt^{\\prime }}$ , we sample one of the possible exogenous realizations $\\tilde{W}_{t,t^{\\prime }+1}$ which immediately informs us about the expected travel and repair times and thus we can immediately know the time, $\\tau (\\tilde{x}_{tt^{\\prime }},\\tilde{W}_{t,t^{\\prime }+1})$ , required by the truck to arrive to the destination node specified by $\\tilde{x}_{tt^{\\prime }}$ .",
"Thus, in MCTS, we define the stochastic transition function as $\\tilde{S}_{t,t^{\\prime }+\\tau (\\tilde{x}_{tt^{\\prime }},\\tilde{W}_{t,t^{\\prime }+1})}=\\tilde{S}^{M,x}(\\tilde{S}_{tt^{\\prime }}^{x},\\tilde{W}_{t,t^{\\prime }+1})$ .",
"It is obvious now that taking the same decision $\\tilde{x}_{tt^{\\prime }}$ from the same state $\\tilde{S}_{tt^{\\prime }}$ results in a different outcome state based on the exogenous information $\\tilde{W}_{t,t^{\\prime }+1}$ .",
"We assume that MCTS has a computational budget of $n^{thr}$ iterations.",
"The steps of MCTS can be grouped into two main policies: a Tree Policy (formed of the Selection and Expansion steps) and a Simulation Policy (formed of the Simulation step) as shown in Figure REF .",
"After terminating the tree search, then the decision that corresponds to best value from the root node is chosen.",
"The pseudo-code for sampled MCTS is presented in Algorithm .",
"Figure: One iteration of the proposed sampled MCTS.[t!]",
"Sampled MCTS Algorithm [t!]",
"function $MCTS(S_t)$ Create root node $\\tilde{S}_{tt}$ with state $S_t$ ; set iteration counter $n=0$ while $n< n^{thr}$ $\\tilde{S}_{tt^{\\prime }}\\leftarrow TreePolicy(\\tilde{S}_{tt})$ $\\tilde{V}_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }})\\leftarrow SimPolicy(\\tilde{S}_{tt^{\\prime }})$ $Backup(\\tilde{S}_{tt^{\\prime }},\\tilde{V}_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }}))$ $n\\leftarrow n+1$ end while return $x_{t}^*=\\arg \\min _{\\tilde{x}_{tt}\\in \\tilde{\\mathcal {X}}_{tt}^e(\\tilde{S}_{tt})}\\tilde{C}(\\tilde{S}_{tt},\\tilde{x}_{tt})+\\tilde{V}^x_{tt}(\\tilde{S}^x_{tt})$ function $TreePolicy(\\tilde{S}_{tt})$ $t^{\\prime }\\leftarrow t$ while $\\tilde{S}_{tt^{\\prime }}$ is non-terminal do if $|\\tilde{\\mathcal {X}}_{tt^{\\prime }}^e(\\tilde{S}_{tt^{\\prime }})|<d^{thr}$ do(Expanding a decision out of a pre-decision state) choose decision ${\\tilde{x}_{tt^{\\prime }}}^*$ optimistically by using a two-stage lookahead model $\\tilde{S}_{tt^{\\prime }}^{x}=S^M(\\tilde{S}_{tt^{\\prime }},{\\tilde{x}_{tt^{\\prime }}}^*)$ (Expansion step) $\\tilde{\\mathcal {X}}_{tt^{\\prime }}^e(\\tilde{S}_{tt^{\\prime }})\\leftarrow \\tilde{\\mathcal {X}}_{tt^{\\prime }}^e(\\tilde{S}_{tt^{\\prime }})\\bigcup \\lbrace \\tilde{x}_{tt^{\\prime }}^*\\rbrace $ $\\tilde{\\mathcal {X}}_{tt^{\\prime }}^u(\\tilde{S}_{tt^{\\prime }})\\leftarrow \\tilde{\\mathcal {X}}_{tt^{\\prime }}^u(\\tilde{S}_{tt^{\\prime }})-\\lbrace \\tilde{x}_{tt^{\\prime }}^*\\rbrace $ else $\\tilde{x}_{tt^{\\prime }}^*=\\arg \\max _{\\tilde{x}_{tt^{\\prime }}\\in \\tilde{\\mathcal {X}}_{tt^{\\prime }}^e(\\tilde{S}_{tt^{\\prime }})}\\left(-\\left(\\tilde{C}(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }})+\\tilde{V}^x_{tt^{\\prime }}(\\tilde{S}^x_{tt^{\\prime }})\\right)+\\alpha \\sqrt{\\frac{\\ln N(\\tilde{S}_{tt^{\\prime }})}{N(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }})}}\\right)$ $\\tilde{S}_{tt^{\\prime }}^{x}=S^M(\\tilde{S}_{tt^{\\prime }},{\\tilde{x}_{tt^{\\prime }}}^*)$ end if if $|\\tilde{\\Omega }_{t,t^{\\prime }+1}^e(\\tilde{S}_{tt^{\\prime }}^{x})|<e^{thr}$ do (Expanding an exogenous outcome out of a post-decision state) choose exogenous event ${\\tilde{W}_{t,t^{\\prime }+1}}$ according to importance sampling with uniform distribution $g(\\tilde{W}_{t,t^{\\prime }+1})$ , $\\forall \\tilde{\\omega }\\in \\tilde{\\Omega }_{t,t^{\\prime }+1}^u(\\tilde{S}_{tt^{\\prime }}^{x})$ $\\tilde{S}_{t,t^{\\prime }+\\tau (\\tilde{x}_{tt^{\\prime }},\\tilde{W}_{t,t^{\\prime }+1})}=S^{M,x}(\\tilde{S}_{tt^{\\prime }}^{x},\\tilde{W}_{t,t^{\\prime }+1})$ (Expansion step) $\\tilde{\\Omega }_{t,t^{\\prime }+1}^e(\\tilde{S}_{tt^{\\prime }}^{x}) \\leftarrow \\tilde{\\Omega }_{t,t^{\\prime }+1}^e(\\tilde{S}_{tt^{\\prime }}^{x})\\bigcup \\lbrace \\tilde{W}_{t,t^{\\prime }+1}\\rbrace $ $\\tilde{\\Omega }_{t,t^{\\prime }+1}^u(\\tilde{S}_{tt^{\\prime }}^{x}) \\leftarrow \\tilde{\\Omega }_{t,t^{\\prime }+1}^u(\\tilde{S}_{tt^{\\prime }}^{x})-\\lbrace \\tilde{W}_{t,t^{\\prime }+1}\\rbrace $ $t^{\\prime }\\leftarrow t^{\\prime }+\\tau (\\tilde{x}_{tt^{\\prime }},\\tilde{W}_{t,t^{\\prime }+1})$ return $\\tilde{S}_{tt^{\\prime }}$ (stops execution of while loop) else choose exogenous event ${\\tilde{W}_{t,t^{\\prime }+1}}$ according to importance sampling with uniform distribution $g(\\tilde{W}_{t,t^{\\prime }+1})$ , $\\forall \\tilde{\\omega }\\in \\tilde{\\Omega }_{t,t^{\\prime }+1}^e(\\tilde{S}_{tt^{\\prime }}^{x})$ $\\tilde{S}_{t,t^{\\prime }+\\tau (\\tilde{x}_{tt^{\\prime }},\\tilde{W}_{t,t^{\\prime }+1})}=S^{M,x}(\\tilde{S}_{tt^{\\prime }}^{x},\\tilde{W}_{t,t^{\\prime }+1})$ $t^{\\prime }\\leftarrow t^{\\prime }+\\tau (\\tilde{x}_{tt^{\\prime }},\\tilde{W}_{t,t^{\\prime }+1})$ end if end while function $SimPolicy(\\tilde{S}_{tt^{\\prime }})$ Choose a sample path $\\tilde{\\omega }\\in \\tilde{\\Omega }_{tt^{\\prime }}$ while $\\tilde{S}_{tt^{\\prime }}$ is non-terminal Choose $\\tilde{x}_{tt^{\\prime }}\\leftarrow \\pi _0({\\tilde{S}_{tt^{\\prime }}})$ [t!]",
"[t!]",
"${\\tilde{S}_{t,t^{\\prime }+\\tau (\\tilde{x}_{tt^{\\prime }}(\\tilde{\\omega }))}}\\leftarrow S^M(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }}(\\tilde{\\omega }))$ $t^{\\prime }\\leftarrow t^{\\prime }+\\tau (\\tilde{x}_{tt^{\\prime }}(\\tilde{\\omega }))$ end while return $\\tilde{V}_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }})$ (Value function of $\\tilde{S}_{tt^{\\prime }}$ ) function $Backup(\\tilde{S}_{tt^{\\prime }},\\tilde{V}_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }}))$ while $\\tilde{S}_{tt^{\\prime }}$ is not null do $N(\\tilde{S}_{tt^{\\prime }})\\leftarrow N(\\tilde{S}_{tt^{\\prime }}) + 1$ $t^*\\leftarrow $ time when the truck was at predecessor node, i.e., $(\\tilde{S}_{tt^{\\prime }}=S^{M,x}(\\tilde{S}_{tt^*}^x,{\\tilde{W}_{t,t^*+1}}))$ where $(\\tilde{S}_{tt^*}^x=S^{M}(\\tilde{S}_{tt^*},{\\tilde{x}_{tt^*}}))$ $\\tilde{S}_{tt^*}^{x}\\leftarrow \\mathrm {~predecessor~of~} \\tilde{S}_{tt^{\\prime }}$ $N(\\tilde{S}_{tt^*},{\\tilde{x}_{tt^*}})\\leftarrow N(\\tilde{S}_{tt^*},{\\tilde{x}_{tt^*}}) + 1$ $\\tilde{V}^x_{tt^*}(\\tilde{S}^x_{tt^*})\\leftarrow \\frac{1}{\\sum _{\\tilde{W}_{t,t^*+1}\\in \\tilde{\\Omega }_{t,t^*+1}^e(\\tilde{S}_{tt^*}^{x})} p(\\tilde{W}_{t,t^*+1})} \\cdot E_{g}[p(\\tilde{W}_{t,t^*+1})/g(\\tilde{W}_{t,t^*+1})\\tilde{V}_{tt^*}(S^{M,x}(\\tilde{S}_{tt^*}^x,{\\tilde{W}_{t,t^*+1}}))]$ $\\tilde{S}_{tt^*}\\leftarrow \\mathrm {~predecessor~of~} \\tilde{S}_{tt^*}^{x}$ $\\Delta \\leftarrow \\tilde{C}(\\tilde{S}_{tt^*},{\\tilde{x}_{tt^*}}) + \\tilde{V}^x_{tt^*}(\\tilde{S}^x_{tt^*}) $ $\\tilde{V}_{tt^*}(\\tilde{S}_{tt^*})\\leftarrow \\tilde{V}_{tt^*}(\\tilde{S}_{tt^*})+\\frac{\\Delta - \\tilde{V}_{tt^*}(\\tilde{S}_{tt^*})}{N(\\tilde{S}_{tt^*})}$ $t^{\\prime }\\leftarrow t^*$ end while The choice of the parameters of MCTS is problem dependent; in this work, we choose the following parameters for the four steps of MCTS to find the optimal route of the truck.",
"1.",
"Selection: In sampled MCTS, there are two selection strategies which are applied based on the domain of selection; one is for the decision space while the other is for the exogenous event space.",
"Starting from the root node, selection chooses a decision based on previous gained information while controlling a balance between exploration and exploitation.",
"The most popular method used in the computer science literature is Upper Confidence Bounding applied to Trees (UCT) ([9], [21]).",
"However, upper confidence bounds are used for maximization problems.",
"In this work, the aim is to minimize the objective function which is equivalent to maximizing its negative value.",
"UCT builds on an extensive literature in computer science on upper confidence bounding (UCB) policies for multiarmed bandit problems ([2]).",
"UCT selects the decision that maximizes the following equation: $\\tilde{x}_{tt^{\\prime }}^*=\\arg \\max _{\\tilde{x}_{tt^{\\prime }}\\in \\tilde{\\mathcal {X}}_{tt^{\\prime }}^e(\\tilde{S}_{tt^{\\prime }})}\\left(-\\left(\\tilde{C}(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }})+\\tilde{V}^x_{tt^{\\prime }}(\\tilde{S}^x_{tt^{\\prime }})\\right)+\\alpha \\sqrt{\\frac{\\ln N(\\tilde{S}_{tt^{\\prime }})}{N(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }})}}\\right),$ where $\\alpha $ is a tunable parameter that balances exploration and exploitation.",
"The choice of decision $\\tilde{x}_{tt^{\\prime }}^*$ that maximizes the UCT equation depends on a weighted average of two terms; the first term of the UCT equation represents the average value of the state-decision after $N(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }})$ iterations.",
"So, the higher the average value of the state-decision, the more it contributes to exploiting the decision further since its reward is high.",
"The second term gives a higher weight to the decision that has been less explored since its value decreases as $N(\\tilde{S}_{tt^{\\prime }},\\tilde{x}_{tt^{\\prime }})$ increases which contributes to exploring the decisions with lower number of visits.",
"Upon choosing $\\tilde{x}_{tt^{\\prime }}^*$ , the state of the network becomes $\\tilde{S}^{x}_{tt^{\\prime }}$ after which we sample an exogenous realization $\\tilde{W}_{t,t^{\\prime }+1}$ from the set of explored exogenous events $\\tilde{\\Omega }^e_{t,t^{\\prime }+1}(\\tilde{S}_{tt^{\\prime }})$ for state $\\tilde{S}_{tt^{\\prime }}$ .",
"In the developed model, the exogenous events may have very different probability density functions where some events can lie on the tail of the probability density function.",
"Thus, in order to avoid too many iterations to catch the rare events, we propose importance sampling to choose $\\tilde{W}_{tt^{\\prime }}^*$ from the set of available samples.",
"Importance sampling yields the same expected value of the outcome of a random variable with a much lower number of iterations compared to sampling using the random variable's initial probability density function.",
"Assume that $\\tilde{\\omega }$ represents an outcome of the exogenous random variable $\\tilde{W}_{tt^{\\prime }}$ .",
"Since there are only a few outcomes for $\\tilde{W}_{tt^{\\prime }}$ , let $p(\\tilde{W}_{tt^{\\prime }}=\\tilde{\\omega })$ be the probability mass function for outcome $\\tilde{\\omega }$ and $E_{p}[\\tilde{W}_{tt^{\\prime }}]$ be its expected value.",
"Also, define a new probability mass function $g(\\tilde{W}_{tt^{\\prime }}^{\\prime }=\\tilde{\\omega })$ which is designed to balance the selection of all outcome events.",
"According to importance sampling, $E_{p}[\\tilde{W}_{tt^{\\prime }}]\\approx E_{g}[p(\\tilde{W}_{tt^{\\prime }}^{\\prime })/g(\\tilde{W}_{tt^{\\prime }}^{\\prime })\\tilde{W}_{tt^{\\prime }}^{\\prime }]$ for a much lower number of realizations of $\\tilde{W}_{tt^{\\prime }}^{\\prime }$ .",
"For simplicity, let $\\tilde{\\omega }_{tt^{\\prime }}$ be an abbreviation of the event $\\tilde{W}_{tt^{\\prime }}=\\tilde{\\omega }$ .",
"In this work, we choose $g(\\tilde{W}_{tt^{\\prime }})$ to have a uniform distribution for all random events that can take place from state $\\tilde{S}_{t,t^{\\prime }-1}^{x}$ .",
"Then, one of the outcomes $\\tilde{\\omega }$ is chosen according to $g(\\tilde{W}_{tt^{\\prime }})$ and later in the backpropagation step, the value function of $\\tilde{S}^{M,x}(\\tilde{S}_{t,t^{\\prime }-1}^{x},\\tilde{\\omega }_{tt^{\\prime }}^*)$ is weighted by $p(\\tilde{\\omega }_{tt^{\\prime }}^*)/g(\\tilde{\\omega }_{tt^{\\prime }}^*)$ in order to maintain the same expected value of the exogenous events.",
"2.",
"Expansion: This is the process of adding a child node to the tree to expand it.",
"Upon visiting a state, one can either expand an unexplored state (via a decision or exogenous event) or exploit existing states.",
"For example, one can set a threshold, $d^{thr}$ , for the number of decisions and another threshold, $e^{thr}$ , for the number of exogenous events to be expanded first before starting the exploitation process.",
"If a state has several unexplored decisions and exogenous events represented by the sets $\\tilde{\\mathcal {X}}_{tt^{\\prime }}^u(\\tilde{S}_{tt^{\\prime }})$ and $\\tilde{\\Omega }_{tt^{\\prime }}^u(\\tilde{S}_{tt^{\\prime }}^{x})$ , respectively, then an unexplored decision is chosen optimistically by using a two-stage lookahead model.",
"That is, for each unexplored decision $x_{tt^{\\prime }}\\in \\tilde{\\mathcal {X}}_{tt^{\\prime }}^u(\\tilde{S}_{tt^{\\prime }})$ , we generate a random exogenous event $\\tilde{\\omega }\\in \\tilde{\\Omega }_{t,t^{\\prime }+1}^u(\\tilde{S}_{tt^{\\prime }}^{x})$ and evaluate the value of the obtained state $\\tilde{S}^{M,x}(\\tilde{S}_{tt^{\\prime }}^{x},\\tilde{\\omega }_{t,t^{\\prime }+1})$ by calling the simulation policy.",
"Then, the decision and its corresponding exogenous event that corresponded to the highest obtained value are chosen to be expanded in the tree.",
"Finally, if the selection phase reaches a state, say $\\tilde{S}_{tt^{\\prime }}^{x}$ , that is already part of the tree but for which the threshold value of the number of exogenous events to be explores is not met, then a random exogenous sample $\\tilde{\\omega }\\in \\tilde{\\Omega }_{t,t^{\\prime }+1}^u(\\tilde{S}_{tt^{\\prime }}^{x})$ is created resulting in state $\\tilde{S}^{M,x}(\\tilde{S}_{tt^{\\prime }}^{x},\\tilde{\\omega }_{t,t^{\\prime }+1})$ and evaluated as discussed above.",
"3.",
"Simulation: The simulation policy is a heuristic policy to provide an initial estimate of the value of the state that has just been added to the tree.",
"Adding a node to the search tree at time $t^{\\prime }$ , results in state $\\tilde{S}_{tt^{\\prime }}$ and updates the probability space $\\tilde{\\Omega }_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }})$ .",
"The simulation policy selects states from the newly added state until the end of the simulation; this is a roll-out simulation starting from the expanded state.",
"To get an efficient simulation policy, we propose one that it is not too stochastic (for efficiency of evaluation) nor too deterministic (because this will bias the search tree) ([9]).",
"The proposed simulation policy is based on a lookahead model in which we evaluate the value of the newly added state at time $t^{\\prime }$ by generating a sample path $\\tilde{\\omega }\\in \\tilde{\\Omega }_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }})$ .",
"The sample path determines the set of power lines that have faulted along with the fault types, and required travel and repair times for each arc in the graph.",
"We elaborate on the simulation policy in Appendix where we formulate the problem as a sequential optimization problem; the resulting problem is a mixed integer non-linear program (MINLP) where the nonlinearity arises from the radial structure of the grid.",
"In general, MINLP problems are difficult to solve for large network sizes but Appendix shows that the problem reduces to a travelling salesman problem (TSP) since the objective is to find the tour of the truck that visits each location with a fault exactly once to repair it in order to minimize the customer outage-minutes.",
"So, Appendix shows that the optimal route of the truck given a sample path can be optimally solved via dynamic programming for a small number of generated faults whereas a heuristic TSP solution becomes necessary if the number of generated faults is large.",
"4.",
"Backpropagation: At the end of the simulation, a value, $\\tilde{V}_{tt^{\\prime }}(\\tilde{S}_{tt^{\\prime }})$ , of the newly created state, $\\tilde{S}_{tt^{\\prime }}$ , is obtained.",
"Starting from the last added node in the tree, its simulated value is backpropagated through all ancestors of state $\\tilde{S}_{tt^{\\prime }}$ until the root state to update their statistics.",
"The number of visit counts of all ancestor states of state $\\tilde{S}_{tt^{\\prime }}$ are increased by one and their values are modified according to a chosen criteria where we choose the average value of all rollouts through a state.",
"While backpropagating, assume that we have $\\tilde{S}_{tt^{\\prime }}=S^{M,x}(\\tilde{S}_{tt^*}^x,{\\tilde{\\omega }_{t,t^*+1}})$ , then the value of the ancestor post-decision state, say $\\tilde{S}_{tt^*}^x$ , should be updated as $\\tilde{V}^x_{tt^*}(\\tilde{S}^x_{tt^*})\\leftarrow \\frac{1}{\\sum _{\\tilde{W}_{t,t^*+1}\\in \\tilde{\\Omega }_{t,t^*+1}^e(\\tilde{S}_{tt^*}^{x})} p(\\tilde{W}_{t,t^*+1})} \\cdot E_{g}[p(\\tilde{W}_{t,t^*+1})/g(\\tilde{W}_{t,t^*+1})\\tilde{V}_{tt^*}(S^{M,x}(\\tilde{S}_{tt^*}^x,{\\tilde{W}_{t,t^*+1}}))]$ where the expectation is over all the explored exogenous events from post-decision state $\\tilde{S}_{tt^*}^x$ .",
"Also, the value function of the ancestor pre-decision state, $\\tilde{S}_{tt^*}$ should be updated with a value of $\\Delta \\leftarrow \\tilde{C}(\\tilde{S}_{tt^*},{\\tilde{x}_{tt^*}}) + \\tilde{V}^x_{tt^*}(\\tilde{S}^x_{tt^*})$ which accounts for the link cost and the updated post-decision state value function so that we get $\\tilde{V}_{tt^*}(\\tilde{S}_{tt^*})\\leftarrow \\tilde{V}_{tt^*}(\\tilde{S}_{tt^*})+\\frac{\\Delta - \\tilde{V}_{tt^*}(\\tilde{S}_{tt^*})}{N(\\tilde{S}_{tt^*})}$ (note that we assume that the weight of all decisions from the same state is equal)."
],
[
"Performance Results",
"To assess the performance of the proposed approaches, the simulated power grid is constructed using real data provided by PSE&G which describes the structure of circuits in their electrical distribution network.",
"The data corresponds to the northeastern portion of PSE&G's power grid in New Jersey and it is formed of 319 circuits.",
"There are an average of 41 protective devices and 724 power lines per circuit.",
"The data identifies the type and location of each component in the circuits such as substations, protective devices, power lines and transformers.",
"The simulator is programmed to generate storms that pass across the grid generating power line faults causing total or partial circuit power outages ([1]).",
"The obtained outages trigger some of the affected customers to call to report the outage.",
"When a power line faults due to storm damage, the simulator finds the nearest upstream protective device, opens it (shutting off power), and then identifies all the customers who subsequently lose power.",
"For each customer experiencing a power outage, a Bernoulli random variable is generated, with a probability of success which equals to the calling probability across the customer's segment, to determine whether the customer will call or not.",
"Thus, the higher the calling probability is, the higher the number of trouble calls will be.",
"In order to reconstruct the grid due to storm damage, a truck uses a roadway defined by the graph $G(\\mathcal {V},\\mathcal {E})$ where $\\mathcal {E}$ is the set containing the arcs/roads between two consecutive nodes of the graph.",
"We represent the network by aggregating lines and poles that are protected by the same protective device into a single segment.",
"The idea is that if a truck visits one pole or line of the circuit, it will be able to quickly see the status of nearby lines and poles, which we represent as a segment.",
"We then assume that we are guiding trucks from one segment to another (where the truck has to find the best path over the road network).",
"When a truck visits a segment, it is assumed to learn (and fix, if necessary) all outages anywhere in a segment.",
"Basically, the network $G(\\mathcal {V},\\mathcal {E})$ is formed of the poles that carries the protective devices where $\\mathcal {E}$ corresponds to the connection matrix in the form of the minimum distance between any two nodes in the network according to the real roadway.",
"As the storm passes across the grid, each power line $i$ along its way is associated with a prior probability of power line fault as explained in detail in ([1]).",
"In the simulator, the prior probability of fault of the power lines are generated based on several parameters such as the severity of the storm, its diameter and the distance of the power line from its center (refer to ([1]) for more details).",
"Since this paper addresses a single truck, we tuned the priors to create storms that generated one to as many as tens of outages.",
"In this case, the segment fault probabilities range between 0 and $0.765$ where the segments that faulted have posteriors ranging between $0.032$ and $0.765$ ."
],
[
"Lookahead Policy",
"After collecting the priors for power line faults, the obtained faults and the customers that called, the simulator executes the proposed power line fault probability model presented in (REF ) upon which the utility truck is routed to restore the grid.",
"According to the simulations, the average number of segments affected by the storm path is 1558; however, the probability model sets the posterior probability of a major number of them to a very low number, e.g., below $0.01$ .",
"Based on the statistics of 1000 networks, the minimum posterior fault of a segment that faulted is $0.032$ , so we choose the segments with posterior probability of faults greater or equal to $0.01$ as candidate segments that have faulted.",
"Figure: Customer outage-hours vs. MCTS budget for one network; twenty simulations for various calling probabilities where each simulation corresponds to the route of the truck starting from the depot till the stopping condition is met.Figures REF shows the customer outage-hours vs. MCTS budget for twenty simulations for one chosen network which has 5 faults.",
"By MCTS budget, it is meant the number of iterations, $n^{thr}$ , executed to return the decision of where to route the truck next.",
"A simulation represents the route of the truck from the depot until the stopping condition of truck routing presented in Algorithm REF is met.",
"There are two stopping conditions; one to stop MCTS search when the number of iterations performed, to return a final decision, reaches $n^{thr}$ .",
"The other is for stopping truck routing when the posterior probabilities of faults of all nodes in the network drops below a threshold $\\epsilon ^{thr}$ which is set to $0.01$ since the posterior probabilities of the segments that faulted are much higher.",
"In MCTS, there are several steps that depend on random outcomes such as which exogenous event to choose or the generated sample path to return a value function for a newly generated state; this results in different solutions for different calls of MCTS especially for a low budget; however, as the MCTS budget increases, the averaging over the random events becomes more accurate and MCTS returns solutions with nearly similar objective values.",
"Figure REF also shows that the rate of convergence to a similar objective value depends on the calling probability; that is, as the calling probability increases, then more information is revealed in the network which reduces the uncertainty and consequently MCTS returns a good solution with a lower budget.",
"According to Figure REF , a low MCTS budget is not enough to decide on the route of the truck but also a very high MCTS budget does not provide any improvement at the cost of higher computational complexity.",
"The simulations are also compared to the posterior optimal solution which corresponds to the optimal solution after revealing the locations of faults in the network obtained using the dynamic program presented in Algorithm .",
"Also, one of the important tunable parameters in the MCTS algorithm is the parameter that balances the weight between exploitation and exploration (referred to as $\\alpha $ in Algorithm ).",
"For the same network considered above, a brute force search over the best value of $\\alpha $ is conducted ranging from $0.1$ till 7 with increments of $0.1$ .",
"Starting from $0.1$ , the average objective value decreases reaching its minimum at $\\alpha =2.2$ and then it increases again.",
"So, for the rest of the simulations, we set $\\alpha = 2.2$ .",
"Figure: Average customer outage-hours vs. MCTS budget for ten networks.Figure REF shows the average customer outage-hours vs. MCTS budget of ten networks for various calling probabilities.",
"The number of faults in the networks ranges from 4 to 12 with an average of $6.09$ faults.",
"First, as the calling probability increases, the lookahead policy provides a solution that is closer to the posterior optimal since more information is provided.",
"Also, the required MCTS budget decreases as the calling probability increases; that is, for a calling probability of $0.01$ and $1.0$ , around 4000 and 1000 MCTS iterations are required, respectively, to converge to a good solution.",
"The lookahead policy provides a solution that is $18.7$ %, $35.3$ % and $58.5$ % higher than the posterior optimal for a calling probability of $1.0$ , $0.1$ and $0.01$ respectively.",
"Second, it is revealed that a high gain is obtained when the calling probability increases from $0.01$ to $0.1$ since the information provided can be from different locations which help in detecting the location of outages.",
"As the calling probability rises above $0.1$ , the benefits are more modest than the increase from $0.01$ to $0.1$ since we only need one customer out of a group across the same segment to make the phone call.",
"Figure: Average computational time of the lookahead policy to decide on the truck's next hop vs. MCTS budget.Figure REF shows the average computational time of the lookahead policy to decide on the next truck route.",
"Obviously, the computational time increases as the MCTS budget increases; however, no additional gain in terms of the objective value is obtained.",
"It is also shown that the computational time decreases as the calling probability increases since more information is provided about the locations that could have faulted which reduces the network size after considering all nodes with posterior probabilities above a certain threshold.",
"However, in the worst case, for a calling probability of $0.01$ and a good computational budget of MCTS, which is shown to be 4000 in Figure REF , the computational time of the lookahead policy is $2.5$ minutes which is suitable for online problems.",
"Figure: Average truck routing time to restore the grid and to stop routing vs. MCTS budget; the truck is stopped when the stopping condition of the lookahead policy is met.Figure REF shows the time required to restore the grid and the time required to stop truck routing compared to the posterior optimal solution.",
"Even when the grid is completely restored, the utility center cannot detect that unless the distribution system is fully equipped with SCADA which is not the case.",
"In this work, the utility center relies on the probability model to decide when the grid is restored and consequently, stop the truck routing process.",
"The posterior optimal solution indicates that 6 hours are required to restore the grid if full information is revealed.",
"For a calling probability of $1.0$ , the lookahead policy can restore the grid in $7.2$ hours on average but, the truck is stopped after 11 hours.",
"For a lower calling probability, both the restore and stop times increase for the same reasons mentioned previously but still have a good performance with respect to the posterior optimal solution."
],
[
"Industrial Heuristics",
"We also compare the performance of the proposed lookahead policy to industrial heuristics which typically rely on escalation algorithms.",
"An escalation algorithm back-traces from each trouble call location to find the first common point for all the calls.",
"It is obvious that escalation algorithms are good to locate a single fault which is assumed to be an upstream fault that triggers all downstream calls.",
"But, often this is not the case as there can be more than one fault triggering the calls.",
"Practically, escalation is performed at the control center along with other intelligence techniques as explained in the literature review in Section .",
"However, in this work, only the trouble calls and the grid structure are used as the sources of information.",
"Thus, the proposed escalation algorithm gives priority to visit the common locations that would trigger all calls but after that searches for other downstream faults as explained in Algorithm REF .",
"[t!]",
"Escalation Algorithm for Grid Restoration [t!]",
"Step 1.",
"For each circuit do Step 1a.",
"Collect all calls and back trace to find the first node that is common to all calls say node $x$ .",
"Step 1b.",
"Send the truck to node $x$ and then back trace to the substation to cover all upstream faults (when a truck visits a node, it fixes an existing fault and this applies to all steps of the algorithm).",
"Step 1c.",
"From node $x$ , perform down tracing to reach the first segment from which a call was initiated and place it in set $\\mathcal {D}$ .",
"Step 2.",
"For each segment in $\\mathcal {D}$ do Step 2a.",
"Perform down tracing to cover all nodes that called.",
"Table: Average Statistics for Escalation AlgorithmTable REF shows the average statistics of the same ten networks used earlier.",
"In the escalation algorithm, searching for a fault is just triggered by the customer calls because there is no clue for the utility center to predict the locations of faults except by visiting the locations that trigger the calls if faulted.",
"So, if there is a segment with a low number of customers where no one called to report an outage then, there is no way to tell that a fault exists unlike the proposed probability model which would recognize that there is a nonzero probability of an outage on these segments due to the storm path.",
"In Table REF , we see that for a calling probability of $0.01$ , on average 1 fault could not be identified since a total of 14 customers are only affected and non of them called at such a low calling probability.",
"So, the utility center will stop routing the truck assuming that it has recovered the grid, because it visited every location from which a call was received.",
"The average number of faults is $6.09$ with an average of 1 unrepaired fault.",
"So, the time it to took the truck to restore an average of $5.09$ faults that triggered calls is around $21.83$ hours and the customer outage-hours is $2.34*10^4$ .",
"We set the maximum truck routing time to stop to 48 hours unless it restored all the grid in a smaller amount of time.",
"The escalation algorithm was fast in locating an upstream fault whenever there is one that triggered all downstream calls but it took it a long time to locate downstream faults because there is no clue from the common point and on except by following the locations that could trigger the calls.",
"Whereas for the same calling probability, the lookahead policy is able to repair all faults in $9.15$ hours with a customer outage-hours of $1.58*10^4$ which means that is much better because it was able to reconstruct all faults with a smaller objective function and truck routing time.",
"The lookahead policy outperforms the escalation algorithm by orders of magnitude as the calling probability increases; for example, for a calling probability of $1.0$ , the escalation algorithm is able to identify all fault locations because basically every customer that lost power is calling.",
"But, the restore and stop times are $20.34$ and $26.46$ hours, respectively, whereas the customer outage-hours is $2.17*10^4$ .",
"More calls can help identify upstream faults extremely fast but there is no strategy for identifying the downstream faults because everyone is calling due to the upstream faults.",
"So, increasing the calling probability is not as beneficial for the escalation policy as it is for the lookahead policy which makes a much clever use of the provided information by decreasing the restore and repair times to $7.2$ and 9 hours with an objective of $1.17 * 10^4$ .",
"Based on the results, we see that the current rules followed in industry can work well for locating upstream faults but it does not provide accurate guidance for finding cascaded downstream faults.",
"Moreover, if no calls are received from a location with an outage, then the escalation algorithm cannot find it.",
"Not only that, when increasing the calling probability, the escalation does not make good use of the additional information except for locating upstream faults quickly."
],
[
"Conclusion",
"In this work, a lookahead policy is proposed to route the utility truck across the power grid to restore it as quickly as possible after a storm.",
"In addition to the trouble calls, the utility truck was used as a mechanism for collecting additional information to update the belief about faults in the grid.",
"Performance results have shown that even with a small percentage of customers calling to report an outage, the lookahead policy can restore the power grid efficiently.",
"It is also shown that the lookahead policy outperforms current techniques used in industry that are based on escalation to locate the faults and in making use of the additional provided information.",
"Lookahead Simulation Policy In the lookahead simulation policy, we evaluate the value of the newly added node at time $t^{\\prime }$ by using a lookahead model in which a sample path $\\tilde{\\omega }\\in \\tilde{\\Omega }_{t,t^{\\prime }+1}(\\tilde{S}_{tt^{\\prime }})$ is generated.",
"The sample path determines the set of power lines that have faulted along with the fault types, required repair and travel times for each arc in the graph.",
"For example, using the probability of fault of each power line in the system, we generate a Bernoulli random variable with a probability of success equals to the probability of fault.",
"In this case, each power line in the power system has a posterior probability of fault equal to either 1 or 0.",
"Thus, we define the following indicator function: ${1}_{\\tilde{L}^u_{t^{\\prime }i}}=\\left\\lbrace \\begin{array}{c l}1 & \\mbox{,~if~} \\tilde{L}^u_{tt^{\\prime }i}(\\tilde{\\omega })=1,\\\\0 & \\mbox{, otherwise},\\end{array}\\right.$ where ${1}_{\\tilde{L}^u_{t^{\\prime }i}}=1$ if power line $i$ on circuit $u$ is in fault at time $t^{\\prime }$ and it is equal to zero, otherwise.",
"In the simulation policy, we use the time index $t^{\\prime \\prime }$ for each state, decision and random variable.",
"Recall, at time $t$ , we are in the base model where we fix the set of calls and call MCTS to find the truck's next hop.",
"For each node included in the MCTS tree, we index it with time $tt^{\\prime }$ .",
"In the fourth step of MCTS, to evaluate the value of a newly added state $\\tilde{S}_{tt^{\\prime }}$ , we use another lookahead model to generate a sample path $\\tilde{\\omega }\\in \\tilde{\\Omega }_{t,t^{\\prime }+1}(\\tilde{S}_{tt^{\\prime }})$ at $t^{\\prime }$ .",
"While, routing the truck according to the generated path, we index the nodes by $t^{\\prime \\prime }$ ; for example, $\\tilde{S}_{t^{\\prime \\prime }}$ indicates the state at time $t^{\\prime \\prime }$ in the simulation step which is used to evaluate the value of the generated state in MCTS at time $t^{\\prime }$ .",
"In the simulation step, if at time $t^{\\prime \\prime }$ , the truck visits a location that was identified as a location with fault at time $t^{\\prime }$ , its indicator function is set to 0 from time $t^{\\prime \\prime }+1$ and on.",
"Thus, for each power line that faulted the following relation holds: ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=1-\\sum _{i\\in \\mathcal {V}}\\sum _{\\hat{t}=t^{\\prime }}^{t^{\\prime \\prime }-1}\\tilde{x}_{\\hat{t}ij}, \\mbox{~if~} \\tilde{L}^u_{tt^{\\prime }j}(\\tilde{\\omega } )=1.$ Whereas, if power line $i$ on circuit $u$ did not fault in the considered scenario then, ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }i}}=0$ , $\\forall t^{\\prime \\prime }\\ge t^{\\prime }$ .",
"Given, a sample path $\\tilde{\\omega }$ , the aim is to find the optimal truck's route that minimizes the customer outage-minutes.",
"Let $\\tilde{C}_{t^{\\prime \\prime }j}$ be a random variable representing the number of customers that regain power by visiting node $j$ at time $t^{\\prime \\prime }$ according to scenario $\\tilde{\\omega }$ .",
"Then, the value of the newly added node is obtained by solving the following optimization problem: $&&\\hspace{-28.45274pt} \\min _{\\tilde{x}_{t^{\\prime \\prime }}} \\sum _{\\hat{t}=t^{\\prime }}^{t^{\\prime }+H}\\left(N-\\sum _{t^{\\prime \\prime }=t^{\\prime }}^{\\hat{t}}\\sum _{u\\in \\mathcal {U}}\\sum _{j=1}^N\\tilde{C}^u_{t^{\\prime \\prime }j}\\right)\\\\&&\\hspace{-28.45274pt} \\mbox{subject to}\\nonumber \\\\&&\\hspace{-28.45274pt}\\tilde{C}^u_{t^{\\prime \\prime }j}=\\sum _i \\left(\\prod _{k\\in \\mathcal {Q}^u_j\\backslash j}\\hspace{-5.69046pt}1-{1}_{\\tilde{L}^u_{t^{\\prime \\prime }k}}\\right){1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}\\left(\\sum _{k\\in \\mathcal {S}^u_j} n^u_k + \\sum _{s\\in \\mathcal {W}^u_{j}}\\left(\\prod _{w=\\min \\lbrace \\mathcal {W}^u_{j}\\rbrace }^s \\prod _{k\\in w}1-{1}_{\\tilde{L}^u_{t^{\\prime \\prime }k}}\\right)\\sum _{k\\in s}n^u_k\\right)\\tilde{x}_{t^{\\prime \\prime }ij}, \\nonumber \\\\ &&\\hspace{355.65944pt}\\forall j\\in \\mathcal {V}, \\forall t^{\\prime \\prime }\\\\ &&\\hspace{-28.45274pt} {1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=1-\\sum _i\\sum _{\\hat{t}=t^{\\prime }}^{t^{\\prime \\prime }-1}\\tilde{x}_{\\hat{t}ij},\\mbox{~such~that~} \\tilde{L}^u_{tt^{\\prime }j}(\\tilde{\\omega })=1 ,\\forall j \\in \\mathcal {I}^u,\\forall u\\in \\mathcal {U}, \\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt}\\tilde{\\Delta }_{t^{\\prime \\prime }ij}\\ge T_{ij}(\\tilde{\\omega })\\tilde{x}_{t^{\\prime \\prime }ij} + \\sum _u R^u_{j}(\\tilde{\\omega }) \\left(\\tilde{x}_{t^{\\prime \\prime }ij} -\\sum _{\\hat{t}=t^{\\prime }}^{t^{\\prime \\prime }-1}\\tilde{x}_{\\hat{t}ji} \\right),\\forall (i,j)\\in \\mathcal {E},\\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt}\\tilde{\\xi }_{t^{\\prime \\prime }j}\\ge \\tilde{\\xi }_{t^{\\prime \\prime }-1i}+ \\sum _i \\tilde{\\Delta }_{t^{\\prime \\prime }ij} ,\\forall (i,j)\\in \\mathcal {E}, \\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt}\\tilde{\\xi }_{t^{\\prime \\prime }j}\\le t^{\\prime \\prime }\\sum _i\\tilde{x}_{t^{\\prime \\prime }ij}+\\zeta \\left(1-\\sum _i\\tilde{x}_{t^{\\prime \\prime }ij}\\right),\\forall j\\in \\mathcal {V}, \\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt}\\sum _{t^{\\prime \\prime }=t^{\\prime }}^{t^{\\prime }+H}\\tilde{x}_{t^{\\prime \\prime }ij}\\le 1,\\forall (i,j)\\in \\mathcal {E}\\\\&&\\hspace{-28.45274pt} \\sum _k \\tilde{x}_{(t^{\\prime \\prime }+T_{jk}(\\tilde{\\omega }))jk}+\\sum _k \\tilde{x}_{(t^{\\prime \\prime }+T_{jk}(\\tilde{\\omega })+\\sum _uR^u_{k}(\\tilde{\\omega }))jk}\\le \\sum _{i}\\tilde{x}_{t^{\\prime \\prime }ij}\\le 1,\\forall j\\in \\mathcal {V},\\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt} \\tilde{C}^u_{t^{\\prime \\prime }j}\\ge 0, \\tilde{\\xi }_{t^{\\prime \\prime }j}\\ge 0, \\tilde{\\Delta }_{t^{\\prime \\prime }ij}\\ge 0, \\tilde{x}_{t^{\\prime \\prime }ij}\\in \\lbrace 0,1\\rbrace $ This problem is a mixed integer non-linear programming (MINLP) problem.",
"The objective (REF ) minimizes the customer outage-minute which is equivalent to maximizing the number of customers with restored power (also referred to as served customers) up to time $t$ represented by the inner sum in the objective.",
"Constraint () determines the number of served customers when the truck goes from its current location, say node $i$ , to node $j$ at time $t^{\\prime \\prime }$ .",
"The number of served customers depends on whether there is a fault upstream to node $j$ or on its segment, i.e., in set $\\mathcal {Q}_j^u$ .",
"Note that, node $j$ will be favored to be visited if there is a fault across power line $j$ , i.e., if ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=1$ .Depending on the structure of the power grid, if a location faults, it causes outage to the customers attached to its segment and all downstream segments.",
"Thus, if a fault is fixed, then all these customers will be affected.",
"But, this also depends on whether there is a fault on any downstream location as shown in ().",
"Constraint () also reveals that the number of customers by visiting power line $j$ is positive if ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=1$ ; however, after visiting this location, say at time $t^*$ , ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=0, \\mbox{~for~} t^{\\prime \\prime }> t^*$ .",
"Thus, if the truck will come across the same location for the second time, then the gain will be 0 which favors the truck not to visit the same location more than once unless there is no other route for it.",
"Constraint () is the same as (REF ) and it has been explained in details above.",
"Constraint () determines the required time to traverse arc $(i,j)\\in \\mathcal {E}$ .",
"If there is no power line to be investigated at node $j$ , then ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=0$ which means that the required traversal time is equal to the travel time which depends on the traffic conditions only.",
"However, if there is a power line across arc $(i,j)$ , then one of two rules apply; if there is a fault on power line $j$ according to sample path $\\tilde{\\omega }$ , then the required traversal time accounts for the travel and repair times for power line $j$ .",
"However, if arc $(i,j)$ is traversed for the second time at time $t$ then, the traversal time is just equal to the travel time since the fault was repaired when the arc was traversed for the first time.",
"Constraints ()-() guarantee that the truck is at node $j$ at time $t^{\\prime \\prime }$ , only if $\\tilde{\\xi }_{t^{\\prime \\prime }j}=t^{\\prime \\prime }$ which sets $\\tilde{x}_{t^{\\prime \\prime }ij}=1$ .",
"If $\\tilde{x}_{t^{\\prime \\prime }ij}=1$ , then $\\tilde{\\xi }_{t^{\\prime \\prime }j}=t^{\\prime \\prime }$ , otherwise $\\tilde{\\xi }_{t^{\\prime \\prime }j}$ is less than a large positive number $\\zeta $ as shown in () but larger than the time where the truck was lastly as indicated by ().",
"But, since the objective is maximizing the number of served customers over time, the optimization problem will set the time to the least possible value that satisfies all constraints.",
"In (), $\\tilde{\\xi }_{t^{\\prime \\prime }j}$ can be equal to $t^{\\prime \\prime }$ only if it satisfies the required traverse times; the required time to reach node $j$ depends on the elapsed time to reach its direct predecessor, say node $i$ , in addition to the required time to traverse node $j$ from node $i$ , i.e., $\\tilde{\\Delta }_{t^{\\prime \\prime }ij}$ .",
"Note that, since the objective minimizes the customer outage-minutes, then the optimization problem keeps on routing the truck to cover all power lines that faulted, as favored by (), until all faults are fixed.",
"Constraint () guarantees that all arcs in the graph can be visited once in one direction and consequently, at most twice (forward and backward) which is a sufficient condition to have an Eulerian path where each power line and node with positive fault probability can be visited once.",
"Constraint () indicates that the truck can go from node $j$ to node $k$ at time $t^{\\prime \\prime }+\\tilde{\\Delta }_{t^{\\prime \\prime }jk}$ only if it was at node $j$ at time $t^{\\prime \\prime }$ and only if the necessary traverse time, $\\tilde{\\Delta }_{t^{\\prime \\prime }jk}$ , has elapsed which depends on ().",
"Moreover, this constraint removes the sub-tours in the network since the truck must have visited a node before it can travel from it.",
"Finally, constraint () shows that all variables are positive except $\\tilde{x}_{t^{\\prime \\prime }ij}$ which is binary.",
"The formulated optimization problem is very complex mainly because it is an MINLP problem which combines the complexity of non-linear programming and integer programming both of which lie in the class of NP-hard problems.",
"Thus, achieving the optimal global solution is most probably never attainable for large network sizes.",
"While there has been a tremendous achievements in solving integer programming problems given that their continuous relaxation is convex, solving non-linear optimization problems is still a non mature area that gets stuck at local optimums.",
"The only constraint that cannot be linearized is (); it can be seen that the order of non-linearity depends on the number of faults upstream and downstream of a node which is scenario dependent.",
"Thus, the radial structure of the grid is the main complicating factor in the optimization problem.",
"Though the problem is non-linear, the optimal solution can be attained by using dynamic programming.",
"For each sample path, we can transform the problem into a complete graph with nodes $\\mathcal {V}^f$ which contains all the power lines that have faulted and the location of the truck indexed with 0.",
"The connection cost between the nodes of $\\mathcal {V}^f$ are calculated by summing the shortest travel time between the nodes according to $\\tilde{\\omega }$ .",
"Let $S$ be the set of the nodes visited by the truck and $f(S)$ a function returning the number of customers still in outage after visiting the nodes of $S$ .",
"The aim of the problem is to find the optimal sequence of the truck route that visits each node exactly once (to repair it) in order to minimize the customer outage-minutes.",
"This problem is equivalent to a travelling salesman problem (TSP) which is NP-Complete; however the solution can be obtained optimally using dynamic programming with complexity $O(n^22^n)$ where $n$ is the number of nodes in the TSP graph which corresponds to the number of generated faults.",
"However, since the number of generated faults is relatively small (less than 20 faults), obtaining the optimal solution using dynamic programming is computationally feasible.",
"Let $C^f(S,i)$ be the customer outage-minutes of going from vertex 0 through the nodes of $S$ ending at node $i$ .",
"Then, the recurrence relation of the dynamic program by going from node $i$ to node $j$ can be defined as $C^f(S,j) = C^f(S-\\lbrace j\\rbrace ,i)+f(\\lbrace S-\\lbrace j\\rbrace \\rbrace )\\cdot (T_{ij}+\\sum _uR^u_j),$ where the first term of the summation accounts for the customer outage-minutes up to node $i$ whereas the second term accounts for the cumulative customer outage-minutes by going from node $i$ to node $j$ .",
"The detailed steps of the dynamic program to obtain the value of the objective function are presented in Algorithm .",
"[t!]",
"Dynamic program for optimal customer outage-minutes of a given sample path [t!]",
"Step 0.",
"Initialization For all $j\\in \\mathcal {V}^f, j\\ne 0$ do $C^f(\\lbrace 0,j\\rbrace ,j) = f(\\lbrace 0,j\\rbrace )\\cdot (T_{0j}+\\sum _uR^u_j)$ Step 1.",
"Compute customer outage-minutes for all subsets For $s=3$ to $|\\mathcal {V}^f|$ For all subset of $\\mathcal {V}^f$ of size $s$ do For all $j\\in S, j\\ne 0$ $C^f(S,j) = \\min _{i\\in S, i\\ne j} C^f(S-\\lbrace j\\rbrace ,i)+f(\\lbrace S-\\lbrace j\\rbrace \\rbrace )\\cdot (T_{ij}+\\sum _uR^u_j)$ Step 2.",
"Optimal solution $\\min _{j\\in \\mathcal {V}^f}C^f(\\mathcal {V}^f,j)$"
],
[
"Lookahead Simulation Policy",
"In the lookahead simulation policy, we evaluate the value of the newly added node at time $t^{\\prime }$ by using a lookahead model in which a sample path $\\tilde{\\omega }\\in \\tilde{\\Omega }_{t,t^{\\prime }+1}(\\tilde{S}_{tt^{\\prime }})$ is generated.",
"The sample path determines the set of power lines that have faulted along with the fault types, required repair and travel times for each arc in the graph.",
"For example, using the probability of fault of each power line in the system, we generate a Bernoulli random variable with a probability of success equals to the probability of fault.",
"In this case, each power line in the power system has a posterior probability of fault equal to either 1 or 0.",
"Thus, we define the following indicator function: ${1}_{\\tilde{L}^u_{t^{\\prime }i}}=\\left\\lbrace \\begin{array}{c l}1 & \\mbox{,~if~} \\tilde{L}^u_{tt^{\\prime }i}(\\tilde{\\omega })=1,\\\\0 & \\mbox{, otherwise},\\end{array}\\right.$ where ${1}_{\\tilde{L}^u_{t^{\\prime }i}}=1$ if power line $i$ on circuit $u$ is in fault at time $t^{\\prime }$ and it is equal to zero, otherwise.",
"In the simulation policy, we use the time index $t^{\\prime \\prime }$ for each state, decision and random variable.",
"Recall, at time $t$ , we are in the base model where we fix the set of calls and call MCTS to find the truck's next hop.",
"For each node included in the MCTS tree, we index it with time $tt^{\\prime }$ .",
"In the fourth step of MCTS, to evaluate the value of a newly added state $\\tilde{S}_{tt^{\\prime }}$ , we use another lookahead model to generate a sample path $\\tilde{\\omega }\\in \\tilde{\\Omega }_{t,t^{\\prime }+1}(\\tilde{S}_{tt^{\\prime }})$ at $t^{\\prime }$ .",
"While, routing the truck according to the generated path, we index the nodes by $t^{\\prime \\prime }$ ; for example, $\\tilde{S}_{t^{\\prime \\prime }}$ indicates the state at time $t^{\\prime \\prime }$ in the simulation step which is used to evaluate the value of the generated state in MCTS at time $t^{\\prime }$ .",
"In the simulation step, if at time $t^{\\prime \\prime }$ , the truck visits a location that was identified as a location with fault at time $t^{\\prime }$ , its indicator function is set to 0 from time $t^{\\prime \\prime }+1$ and on.",
"Thus, for each power line that faulted the following relation holds: ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=1-\\sum _{i\\in \\mathcal {V}}\\sum _{\\hat{t}=t^{\\prime }}^{t^{\\prime \\prime }-1}\\tilde{x}_{\\hat{t}ij}, \\mbox{~if~} \\tilde{L}^u_{tt^{\\prime }j}(\\tilde{\\omega } )=1.$ Whereas, if power line $i$ on circuit $u$ did not fault in the considered scenario then, ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }i}}=0$ , $\\forall t^{\\prime \\prime }\\ge t^{\\prime }$ .",
"Given, a sample path $\\tilde{\\omega }$ , the aim is to find the optimal truck's route that minimizes the customer outage-minutes.",
"Let $\\tilde{C}_{t^{\\prime \\prime }j}$ be a random variable representing the number of customers that regain power by visiting node $j$ at time $t^{\\prime \\prime }$ according to scenario $\\tilde{\\omega }$ .",
"Then, the value of the newly added node is obtained by solving the following optimization problem: $&&\\hspace{-28.45274pt} \\min _{\\tilde{x}_{t^{\\prime \\prime }}} \\sum _{\\hat{t}=t^{\\prime }}^{t^{\\prime }+H}\\left(N-\\sum _{t^{\\prime \\prime }=t^{\\prime }}^{\\hat{t}}\\sum _{u\\in \\mathcal {U}}\\sum _{j=1}^N\\tilde{C}^u_{t^{\\prime \\prime }j}\\right)\\\\&&\\hspace{-28.45274pt} \\mbox{subject to}\\nonumber \\\\&&\\hspace{-28.45274pt}\\tilde{C}^u_{t^{\\prime \\prime }j}=\\sum _i \\left(\\prod _{k\\in \\mathcal {Q}^u_j\\backslash j}\\hspace{-5.69046pt}1-{1}_{\\tilde{L}^u_{t^{\\prime \\prime }k}}\\right){1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}\\left(\\sum _{k\\in \\mathcal {S}^u_j} n^u_k + \\sum _{s\\in \\mathcal {W}^u_{j}}\\left(\\prod _{w=\\min \\lbrace \\mathcal {W}^u_{j}\\rbrace }^s \\prod _{k\\in w}1-{1}_{\\tilde{L}^u_{t^{\\prime \\prime }k}}\\right)\\sum _{k\\in s}n^u_k\\right)\\tilde{x}_{t^{\\prime \\prime }ij}, \\nonumber \\\\ &&\\hspace{355.65944pt}\\forall j\\in \\mathcal {V}, \\forall t^{\\prime \\prime }\\\\ &&\\hspace{-28.45274pt} {1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=1-\\sum _i\\sum _{\\hat{t}=t^{\\prime }}^{t^{\\prime \\prime }-1}\\tilde{x}_{\\hat{t}ij},\\mbox{~such~that~} \\tilde{L}^u_{tt^{\\prime }j}(\\tilde{\\omega })=1 ,\\forall j \\in \\mathcal {I}^u,\\forall u\\in \\mathcal {U}, \\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt}\\tilde{\\Delta }_{t^{\\prime \\prime }ij}\\ge T_{ij}(\\tilde{\\omega })\\tilde{x}_{t^{\\prime \\prime }ij} + \\sum _u R^u_{j}(\\tilde{\\omega }) \\left(\\tilde{x}_{t^{\\prime \\prime }ij} -\\sum _{\\hat{t}=t^{\\prime }}^{t^{\\prime \\prime }-1}\\tilde{x}_{\\hat{t}ji} \\right),\\forall (i,j)\\in \\mathcal {E},\\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt}\\tilde{\\xi }_{t^{\\prime \\prime }j}\\ge \\tilde{\\xi }_{t^{\\prime \\prime }-1i}+ \\sum _i \\tilde{\\Delta }_{t^{\\prime \\prime }ij} ,\\forall (i,j)\\in \\mathcal {E}, \\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt}\\tilde{\\xi }_{t^{\\prime \\prime }j}\\le t^{\\prime \\prime }\\sum _i\\tilde{x}_{t^{\\prime \\prime }ij}+\\zeta \\left(1-\\sum _i\\tilde{x}_{t^{\\prime \\prime }ij}\\right),\\forall j\\in \\mathcal {V}, \\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt}\\sum _{t^{\\prime \\prime }=t^{\\prime }}^{t^{\\prime }+H}\\tilde{x}_{t^{\\prime \\prime }ij}\\le 1,\\forall (i,j)\\in \\mathcal {E}\\\\&&\\hspace{-28.45274pt} \\sum _k \\tilde{x}_{(t^{\\prime \\prime }+T_{jk}(\\tilde{\\omega }))jk}+\\sum _k \\tilde{x}_{(t^{\\prime \\prime }+T_{jk}(\\tilde{\\omega })+\\sum _uR^u_{k}(\\tilde{\\omega }))jk}\\le \\sum _{i}\\tilde{x}_{t^{\\prime \\prime }ij}\\le 1,\\forall j\\in \\mathcal {V},\\forall t^{\\prime \\prime }\\\\&&\\hspace{-28.45274pt} \\tilde{C}^u_{t^{\\prime \\prime }j}\\ge 0, \\tilde{\\xi }_{t^{\\prime \\prime }j}\\ge 0, \\tilde{\\Delta }_{t^{\\prime \\prime }ij}\\ge 0, \\tilde{x}_{t^{\\prime \\prime }ij}\\in \\lbrace 0,1\\rbrace $ This problem is a mixed integer non-linear programming (MINLP) problem.",
"The objective (REF ) minimizes the customer outage-minute which is equivalent to maximizing the number of customers with restored power (also referred to as served customers) up to time $t$ represented by the inner sum in the objective.",
"Constraint () determines the number of served customers when the truck goes from its current location, say node $i$ , to node $j$ at time $t^{\\prime \\prime }$ .",
"The number of served customers depends on whether there is a fault upstream to node $j$ or on its segment, i.e., in set $\\mathcal {Q}_j^u$ .",
"Note that, node $j$ will be favored to be visited if there is a fault across power line $j$ , i.e., if ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=1$ .Depending on the structure of the power grid, if a location faults, it causes outage to the customers attached to its segment and all downstream segments.",
"Thus, if a fault is fixed, then all these customers will be affected.",
"But, this also depends on whether there is a fault on any downstream location as shown in ().",
"Constraint () also reveals that the number of customers by visiting power line $j$ is positive if ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=1$ ; however, after visiting this location, say at time $t^*$ , ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=0, \\mbox{~for~} t^{\\prime \\prime }> t^*$ .",
"Thus, if the truck will come across the same location for the second time, then the gain will be 0 which favors the truck not to visit the same location more than once unless there is no other route for it.",
"Constraint () is the same as (REF ) and it has been explained in details above.",
"Constraint () determines the required time to traverse arc $(i,j)\\in \\mathcal {E}$ .",
"If there is no power line to be investigated at node $j$ , then ${1}_{\\tilde{L}^u_{t^{\\prime \\prime }j}}=0$ which means that the required traversal time is equal to the travel time which depends on the traffic conditions only.",
"However, if there is a power line across arc $(i,j)$ , then one of two rules apply; if there is a fault on power line $j$ according to sample path $\\tilde{\\omega }$ , then the required traversal time accounts for the travel and repair times for power line $j$ .",
"However, if arc $(i,j)$ is traversed for the second time at time $t$ then, the traversal time is just equal to the travel time since the fault was repaired when the arc was traversed for the first time.",
"Constraints ()-() guarantee that the truck is at node $j$ at time $t^{\\prime \\prime }$ , only if $\\tilde{\\xi }_{t^{\\prime \\prime }j}=t^{\\prime \\prime }$ which sets $\\tilde{x}_{t^{\\prime \\prime }ij}=1$ .",
"If $\\tilde{x}_{t^{\\prime \\prime }ij}=1$ , then $\\tilde{\\xi }_{t^{\\prime \\prime }j}=t^{\\prime \\prime }$ , otherwise $\\tilde{\\xi }_{t^{\\prime \\prime }j}$ is less than a large positive number $\\zeta $ as shown in () but larger than the time where the truck was lastly as indicated by ().",
"But, since the objective is maximizing the number of served customers over time, the optimization problem will set the time to the least possible value that satisfies all constraints.",
"In (), $\\tilde{\\xi }_{t^{\\prime \\prime }j}$ can be equal to $t^{\\prime \\prime }$ only if it satisfies the required traverse times; the required time to reach node $j$ depends on the elapsed time to reach its direct predecessor, say node $i$ , in addition to the required time to traverse node $j$ from node $i$ , i.e., $\\tilde{\\Delta }_{t^{\\prime \\prime }ij}$ .",
"Note that, since the objective minimizes the customer outage-minutes, then the optimization problem keeps on routing the truck to cover all power lines that faulted, as favored by (), until all faults are fixed.",
"Constraint () guarantees that all arcs in the graph can be visited once in one direction and consequently, at most twice (forward and backward) which is a sufficient condition to have an Eulerian path where each power line and node with positive fault probability can be visited once.",
"Constraint () indicates that the truck can go from node $j$ to node $k$ at time $t^{\\prime \\prime }+\\tilde{\\Delta }_{t^{\\prime \\prime }jk}$ only if it was at node $j$ at time $t^{\\prime \\prime }$ and only if the necessary traverse time, $\\tilde{\\Delta }_{t^{\\prime \\prime }jk}$ , has elapsed which depends on ().",
"Moreover, this constraint removes the sub-tours in the network since the truck must have visited a node before it can travel from it.",
"Finally, constraint () shows that all variables are positive except $\\tilde{x}_{t^{\\prime \\prime }ij}$ which is binary.",
"The formulated optimization problem is very complex mainly because it is an MINLP problem which combines the complexity of non-linear programming and integer programming both of which lie in the class of NP-hard problems.",
"Thus, achieving the optimal global solution is most probably never attainable for large network sizes.",
"While there has been a tremendous achievements in solving integer programming problems given that their continuous relaxation is convex, solving non-linear optimization problems is still a non mature area that gets stuck at local optimums.",
"The only constraint that cannot be linearized is (); it can be seen that the order of non-linearity depends on the number of faults upstream and downstream of a node which is scenario dependent.",
"Thus, the radial structure of the grid is the main complicating factor in the optimization problem.",
"Though the problem is non-linear, the optimal solution can be attained by using dynamic programming.",
"For each sample path, we can transform the problem into a complete graph with nodes $\\mathcal {V}^f$ which contains all the power lines that have faulted and the location of the truck indexed with 0.",
"The connection cost between the nodes of $\\mathcal {V}^f$ are calculated by summing the shortest travel time between the nodes according to $\\tilde{\\omega }$ .",
"Let $S$ be the set of the nodes visited by the truck and $f(S)$ a function returning the number of customers still in outage after visiting the nodes of $S$ .",
"The aim of the problem is to find the optimal sequence of the truck route that visits each node exactly once (to repair it) in order to minimize the customer outage-minutes.",
"This problem is equivalent to a travelling salesman problem (TSP) which is NP-Complete; however the solution can be obtained optimally using dynamic programming with complexity $O(n^22^n)$ where $n$ is the number of nodes in the TSP graph which corresponds to the number of generated faults.",
"However, since the number of generated faults is relatively small (less than 20 faults), obtaining the optimal solution using dynamic programming is computationally feasible.",
"Let $C^f(S,i)$ be the customer outage-minutes of going from vertex 0 through the nodes of $S$ ending at node $i$ .",
"Then, the recurrence relation of the dynamic program by going from node $i$ to node $j$ can be defined as $C^f(S,j) = C^f(S-\\lbrace j\\rbrace ,i)+f(\\lbrace S-\\lbrace j\\rbrace \\rbrace )\\cdot (T_{ij}+\\sum _uR^u_j),$ where the first term of the summation accounts for the customer outage-minutes up to node $i$ whereas the second term accounts for the cumulative customer outage-minutes by going from node $i$ to node $j$ .",
"The detailed steps of the dynamic program to obtain the value of the objective function are presented in Algorithm .",
"[t!]",
"Dynamic program for optimal customer outage-minutes of a given sample path [t!]",
"Step 0.",
"Initialization For all $j\\in \\mathcal {V}^f, j\\ne 0$ do $C^f(\\lbrace 0,j\\rbrace ,j) = f(\\lbrace 0,j\\rbrace )\\cdot (T_{0j}+\\sum _uR^u_j)$ Step 1.",
"Compute customer outage-minutes for all subsets For $s=3$ to $|\\mathcal {V}^f|$ For all subset of $\\mathcal {V}^f$ of size $s$ do For all $j\\in S, j\\ne 0$ $C^f(S,j) = \\min _{i\\in S, i\\ne j} C^f(S-\\lbrace j\\rbrace ,i)+f(\\lbrace S-\\lbrace j\\rbrace \\rbrace )\\cdot (T_{ij}+\\sum _uR^u_j)$ Step 2.",
"Optimal solution $\\min _{j\\in \\mathcal {V}^f}C^f(\\mathcal {V}^f,j)$"
]
] | 1605.05711 |
[
[
"Proof of a Limited Version of Mao's Partition Rank Inequality using a\n Theta Function Identity"
],
[
"Abstract Ramanujan's congruence $p(5k+4) \\equiv 0 \\pmod 5$ led Dyson \\cite{dyson} to conjecture the existence of a measure \"rank\" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct proof of Ramanujan's congruence.",
"The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer \\cite{atkin}, who proved Dyson's conjecture.",
"More recently, Mao proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 \\cite{mao10} and for $M_2$ rank differences for partitions with no repeated odd parts modulo $6$ and $10$ \\cite{maom2}.",
"Alwaise et.",
"al.",
"proved four of Mao's conjectured inequalities \\cite{swisher}, while leaving three open.",
"Here, we prove a limited version of one of the inequalities conjectured by Mao."
],
[
"Introduction and Results",
"A partition of a positive integer $n$ is a way of writing $n$ as a sum of positive integers, usually written in non-increasing order of the summands or parts of the partition.",
"The number of partitions of $n$ is denoted by $p(n)$ .",
"For a partition $\\lambda $ , we denote the number of parts in the partition as $n(\\lambda )$ and the largest part as $l(\\lambda )$ .",
"The celebrated Ramanujan's congruences for the partition function begged for a combinatorial interpretation: $p(5k+4) &\\equiv 0 \\pmod {5}, \\\\p(7k+5) &\\equiv 0 \\pmod {7}, \\\\p(11k+6) &\\equiv 0 \\pmod {11}.$ Dyson [4] defined the rank of a partition $\\lambda $ to be $l(\\lambda ) - n(\\lambda )$ and conjectured that partitions for $5k+4$ and $7k+5$ can be divided into five and seven equal sub-classes respectively based on their rank.",
"Specifically, he claimed that $N(s, 5, 5n+4) &= \\frac{p(5n+4)}{5}, \\\\N(t, 7, 7n+4) &= \\frac{p(7n+6)}{7},$ where $N(s, m, n)$ denotes the number of partitions of $n$ with rank $s$ modulo $m$ .",
"Atkin and Swinnerton-Dyer [2] proved Dyson's conjecture by finding the generating functions for the rank differences $N(s, m, mk+d) - N(s, m, mk+d)$ for $k = 5, 7$ .",
"They obtained several other interesting identities apart from Ramanujan's congruences.",
"Lovejoy and Osburn [5] expanded on the work by Atkin and Swinnerton-Dyer to find rank differences for overpartitions and $M_2$ rank differences for partitions without repeated odd parts, which is defined for such a partition $\\lambda $ by $ \\left\\lceil \\frac{l(\\lambda )}{2} \\right\\rceil - n(\\lambda ) .$ The corresponding count for number of partitions of $n$ with no repeated odd parts having its $M_2$ rank congruent to $s$ modulo $m$ is given by $N_2(s, m, n)$ .",
"They obtained all the rank difference formulas corresponding to $m = 3, 5$ .",
"Continuing on their work, Mao [6], [7] extended the results for Dyson rank differences modulo 10 and $M_2$ rank differences modulo 6 and 10.",
"He obtained several interesting inequalities based on his results such as $N(1,10,5n+1) & > N(5,10,5n+1), \\\\N_2(0,6,3n+1) + N_2(1,6,3n+1) & > N_2(2,6,3n+1) + N_2(3,6,3n+1).$ Mao also gave some conjectures in [6], [7] based on computational evidence, both for the Dyson rank and $M_2$ rank for partitions with unique odd parts.",
"Conjecture 1.1 Computation evidence suggests that $N(0, 10, 5n) + N(1, 10, 5n) &> N(4, 10, 5n) + N(5, 10, 5n),\\\\N(1, 10, 5n) + N(2, 10, 5n) &\\ge N(3, 10, 5n) + N(4, 10, 5n), \\\\N_2(0, 10, 5n) + N_2(1, 10, 5n) &> N_2(4, 10, 5n) + N_2(5, 10, 5n),\\\\N_2(0, 10, 5n+4) + N_2(1, 10, 5n+4) &> N_2(4, 10, 5n+4) + N_2(5, 10, 5n+4),\\\\N_2(1, 10, 5n) + N_2(2, 10, 5n) &> N_2(3, 10, 5n) + N_2(4, 10, 5n),\\\\N_2(1, 10, 5n+2) + N_2(2, 10, 5n+2) &> N_2(3, 10, 5n+2) + N_2(4, 10, 5n+2), \\\\N_2(0, 6, 3n+2) + N_2(1, 6, 3n+2) &> N_2(2, 6, 3n+2) + N_2(3, 6, 3n+2).$ In (), (), and (), $n \\ge 1$ , whilst in the rest $n \\ge 0$ .",
"Alwaise et.",
"al.",
"[1] proved four of these seven inequalities conjectured by Mao, namely (REF ), (), (), and () by using elementary methods based on the number of solutions of Diophantine equations solving for the exponents in the generating functions in the corresponding rank differences.",
"They also observed that in (), the strict inequality holds.",
"However, their methods weren't strong enough to prove the remaining three conjectures, which are still open.",
"Here, we prove a limited version of ().",
"Theorem 1.2 Mao's conjecture () is true when $3 \\nmid n + 1$ .",
"Specifically, we have that the following inequalities are true for all $n \\ge 0$ : $N_2(0, 6, 9n+2) + N_2(1, 6, 9n+2) &> N_2(2, 6, 9n+2) + N_2(3, 6, 9n+2), \\\\N_2(0, 6, 9n+5) + N_2(1, 6, 9n+5) &> N_2(2, 6, 9n+5) + N_2(3, 6, 9n+5).$"
],
[
"Preliminaries",
"The standard $q$ -series notation is employed which is defined as $({a}; q)_{n} &:= \\prod _{i=0}^{n-1}(1-aq^i), \\\\({a}; q)_{\\infty } &:= \\prod _{i=0}^\\infty (1-aq^i),$ where $n \\in \\mathbb {N}$ and $a \\in \\mathbb {C}$ .",
"The empty product $({a}; q)_{0}$ is defined to be 1.",
"The following elementary identities are used in manipulation of $q$ -series to prove equalities between expressions.",
"For $a, b \\in \\mathbb {Z}$ , $c \\in \\mathbb {C}$ , and for $k \\in \\mathbb {N}$ , we have $({-q}; q)_{\\infty } \\cdot ({q}; q^{2})_{\\infty } &= 1, \\\\({q^a}; q^{b})_{\\infty } ({-q^a}; q^{b})_{\\infty } &= ({q^{2a}}; q^{2b})_{\\infty }, \\\\({cq^a}; q^{2b})_{\\infty }({cq^{a+b}}; q^{2b})_{\\infty } &= ({cq^a}; q^{b})_{\\infty }, \\\\({cq^a}; q^{kb})_{\\infty }\\cdots ({cq^{a+(k-1)b}}; q^{kb})_{\\infty } &= ({cq^a}; q^{b})_{\\infty }.$ Further, we make use of the shorthand notation as employed by both Mao [6], [7] and Alwaise et.",
"al.",
"[1].",
"$({a_1, \\dots , a_k}; q)_{n} &:= ({a_1}; q)_{n} \\cdots ({a_k}; q)_{n}, \\\\({a_1, \\dots , a_k}; q)_{\\infty } &:= ({a_1}; q)_{\\infty } \\cdots ({a_k}; q)_{\\infty }, \\\\J_b &:= ({q^b}; q^{b})_{\\infty }, \\\\J_{a, b} &:= ({q^a, q^{b-a}, q^b}; q^{b})_{\\infty }.",
"\\\\$ We will also use Mao's $M_2$ rank difference generating function to prove our result Theorem REF .",
"Mao proved the following theorem which encapsulates the pertinent rank differences.",
"Theorem 2.1 (Mao [7]) We have $&\\sum _{n \\ge 0} \\left( N_2(0, 6, n) + N_2(1, 6, n) - N_2(2, 6, n) - N_2(3, 6, n) \\right)q^n \\\\&= \\frac{1}{J_{9, 36}} \\sum _{n = - \\infty }^{\\infty } \\frac{(-1)^nq^{18n^2+9n}}{1-q^{18n+3}} + q \\frac{J_{6, 36}^2J_{18, 36}J_{36}^3}{J_{3, 36}^2J_{9, 36}J_{15, 36}^2} \\\\&+ \\frac{J_{6, 36}J_{18, 36}^2J_{36}^3}{2qJ_{3, 36}^2J_{9, 36}J_{15, 36}^2} - \\frac{1}{J_{9, 36}} \\sum _{n=-\\infty }^{\\infty } \\frac{(-1)^nq^{18n^2+9n-1}}{1+q^{18n}}.$ Apart from this, an identity of Ramanujan theta function is also used.",
"The Ramanujan's general theta function $f(a, b)$ is defined as $f(a, b) &:= \\sum _{n = -\\infty }^{\\infty } a^{\\frac{n(n+1)}{2}}b^{\\frac{n(n-1)}{2}}= (-a, -b, ab; ab)_\\infty $ with $|ab| < 1$ where the equality following through (and being equivalent to) Jacobi triple product identity.",
"We will use the following two special cases of the theta function and the function $\\chi (q)$ which are defined as $\\varphi (q) &:= f(q, q) = ({-q, -q, q^2}; q^{2})_{\\infty }, \\\\\\psi (q) &:= f(q, q^3) = \\frac{({q^2}; q^{2})_{\\infty }}{({q}; q^{2})_{\\infty }}, \\\\\\chi (q) &:= ({-q}; q^{2})_{\\infty }.$ The following theta function identity is used in the proof of our main result.",
"Theorem 2.2 (Baruah and Barman [3]) We have $ \\varphi ^2(q) + \\varphi ^2(q^3) = 2\\varphi ^2(-q^6)\\frac{\\chi (q)\\psi (-q^3)}{\\chi (-q)\\psi (q^3)}.",
"$"
],
[
"Proof of Theorem ",
"We denote $d(n) := N_2(0, 6, n) + N_2(1, 6, n) - N_2(2, 6, n) - N_2(3, 6, n)$ for simplicity.",
"We will show that the generating function $\\sum _{n\\ge 0}d(3n+2)q^n$ has strictly positive coefficients for all $n \\lnot \\equiv 2 \\pmod {3}$ .",
"We first compute the generating function $\\sum _{n\\ge 0}d(3n+2)q^n$ using Theorem REF .",
"Proposition 3.1 We have $\\sum _{n \\ge 0} d(3n+2) q^{n} = \\frac{1}{qJ_{3,12}} \\left(\\frac{J_{2, 12}J_{6, 12}^2J_{12}^3}{2J_{1, 12}^2J_{5, 12}^2} - \\sum _{n=-\\infty }^{\\infty } \\frac{(-1)^nq^{6n^2+3n}}{1+q^{6n}} \\right).$ The proof is straightforward manipulation by including only exponents congruent to 2 modulo 3 in the original generating function, and then letting $q \\mapsto q^{\\frac{1}{3}}$ as follows: $&\\sum _{n \\ge 0} d(3n+2) q^{3n+2} = \\frac{J_{6, 36}J_{18, 36}^2J_{36}^3}{2qJ_{3, 36}^2J_{9, 36}J_{15, 36}^2} - \\frac{1}{J_{9, 36}} \\sum _{n=-\\infty }^{\\infty } \\frac{(-1)^nq^{18n^2+9n-1}}{1+q^{18n}} \\\\\\Rightarrow &\\sum _{n \\ge 0} d(3n+2) q^{3n} = \\frac{J_{6, 36}J_{18, 36}^2J_{36}^3}{2q^3J_{3, 36}^2J_{9, 36}J_{15, 36}^2} - \\frac{1}{J_{9, 36}} \\sum _{n=-\\infty }^{\\infty } \\frac{(-1)^nq^{18n^2+9n-3}}{1+q^{18n}} \\\\\\Rightarrow &\\sum _{n \\ge 0} d(3n+2) q^{n} = \\frac{J_{2, 12}J_{6, 12}^2J_{12}^3}{2qJ_{1, 12}^2J_{3, 12}J_{5, 12}^2} - \\frac{1}{J_{3, 12}} \\sum _{n=-\\infty }^{\\infty } \\frac{(-1)^nq^{6n^2+3n-1}}{1+q^{6n}}.$ Remark 3.2 Note that the while there is a $q$ in the denominator of the common factor above, it is canceled because the constant term of the expression inside the parentheses is zero.",
"We will also need the following lemma which will tie together the proof: Lemma 3.3 We have $ \\frac{J_{2, 12}J_{6, 12}^2J_{12}^3}{J_{1, 12}^2J_{5, 12}^2} = \\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{2}.",
"$ We first write the expression in its constituent $q$ -series and then use () to cancel common factors in both numerator and denominator.",
"We find that $\\frac{J_{2, 12}J_{6, 12}^2J_{12}^3}{J_{1, 12}^2J_{5, 12}^2} &= \\frac{({q^2, q^{10}, q^{12}}; q^{12})_{\\infty }({q^6, q^6, q^{12}}; q^{12})_{\\infty }^2({q^{12}}; q^{12})_{\\infty }^3}{({q,q^{11}, q^{12}}; q^{12})_{\\infty }^2({q^5,q^7, q^{12}}; q^{12})_{\\infty }^2} \\\\&= \\frac{({q^2, q^{10}}; q^{12})_{\\infty }({q^6, q^6, q^{12}}; q^{12})_{\\infty }^2}{({q,q^{7}}; q^{12})_{\\infty }^2({q^5,q^{11}}; q^{12})_{\\infty }^2} \\\\&= \\varphi ^2(-q^6)\\frac{({q, q^{5}}; q^{6})_{\\infty }({-q, -q^{5}}; q^{6})_{\\infty }}{({q}; q^{6})_{\\infty }^2({q^5}; q^{6})_{\\infty }^2} \\\\&= \\varphi ^2(-q^6)\\frac{({-q, -q^5}; q^{6})_{\\infty }}{({q, q^5}; q^{6})_{\\infty }}.",
"$ We next use () to reduce the $q$ -series by multiplying the missing factors in both numerator and denominator, and simplify the expression based on () which is based on (REF ), to finally recognize the identity in Theorem REF as follows: $\\frac{J_{2, 12}J_{6, 12}^2J_{12}^3}{J_{1, 12}^2J_{5, 12}^2} &= \\varphi ^2(-q^6)\\frac{({-q, -q^5}; q^{6})_{\\infty }}{({q, q^5}; q^{6})_{\\infty }} \\\\&= \\varphi ^2(-q^6)\\frac{({-q, -q^5}; q^{6})_{\\infty }({-q^3}; q^{6})_{\\infty }({q^3}; q^{6})_{\\infty }}{({q, q^5}; q^{6})_{\\infty }({q^3}; q^{6})_{\\infty }({-q^3}; q^{6})_{\\infty }} \\\\&= \\varphi ^2(-q^6)\\frac{({-q}; q^{2})_{\\infty }({q^6}; q^{6})_{\\infty }({q^3}; q^{6})_{\\infty }}{({q}; q^{2})_{\\infty }({-q^3}; q^{6})_{\\infty }({q^6}; q^{6})_{\\infty }} \\\\&= \\varphi ^2(-q^6)\\frac{\\chi (q)\\psi (-q^3)}{\\chi (-q)\\psi (q^3)} \\\\&= \\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{2} .$ We now prove our result Theorem REF .",
"We use Lemma REF and note that all the exponents of the infinite summation inside the parentheses are $0 \\pmod {3}$ .",
"Hence, $\\sum _{n \\ge 0} d(3n+2) q^{n} &= \\frac{1}{qJ_{3,12}} \\left(\\frac{J_{2, 12}J_{6, 12}^2J_{12}^3}{2J_{1, 12}^2J_{5, 12}^2} - \\sum _{n=-\\infty }^{\\infty } \\frac{(-1)^nq^{6n^2+3n}}{1+q^{6n}} \\right) \\\\&= \\frac{1}{qJ_{3,12}} \\left( \\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{4} - \\frac{1}{2} + \\sum _{n \\ge 1} a_{3n}q^{3n} \\right),$ where $a_{3n} \\in \\mathbb {Z}$ .",
"Now let $3 \\nmid n+1$ , then $d(3n+2) &= [q^{n}] \\frac{1}{qJ_{3,12}} \\left( \\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{4} - \\frac{1}{2} + \\sum _{n \\ge 1} a_{3n}q^{3n} \\right) \\\\&= [q^{n+1}] \\left( \\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{4{J_{3,12}}} - \\frac{1}{2} + \\frac{1}{J_{3,12}} \\sum _{n \\ge 1} a_{3n}q^{3n} \\right) \\\\&= [q^{n+1}] \\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{4{J_{3,12}}} - [q^{n+1}] \\frac{1}{2} + [q^{n+1}] \\frac{1}{J_{3,12}} \\sum _{n \\ge 1} a_{3n}q^{3n} \\\\&= [q^{n+1}] \\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{4{J_{3,12}}}$ where $[x^k]f(x)$ denotes the coefficient of $x^k$ in the generating function $f(x)$ .",
"It now suffices to show that all coefficients of $\\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{J_{3,12}}$ are positive.",
"This follows as $\\frac{\\varphi ^2(q) + \\varphi ^2(q^3)}{J_{3,12}} &= \\frac{2 + 4q + 4q^2 + \\sum _{n\\ge 3}b_nq^n}{(1-q^3)({q^9, q^9, q^{12}}; q^{12})_{\\infty }} \\\\&= \\left(2 + 4q + 4q^2 + \\sum _{n\\ge 3}b_nq^n\\right)\\left(\\sum _{n \\ge 0}q^{3n}\\right)\\left(1 + \\sum _{n\\ge 0}c_nq^n\\right)$ where $b_i$ and $c_i$ are non-negative.",
"We can generate $q^{3n+k}$ using the above factors by $q^k$ from first, $q^{3n}$ from second, and 1 from the last, where $k = 0, 1, 2$ .",
"This completes our proof for Theorem REF"
],
[
"Conclusion and Remarks",
"The method employed by Alwaise et.",
"al.",
"doesn't work for this inequality because the expression inside the parentheses in Proposition REF does seem to have negative coefficients for an infinite number of coefficients.",
"This result is limited to $3n+2$ when $3 \\nmid n+1$ , but computational evidence suggests that $\\displaystyle \\frac{1}{1-q^{12}} \\left(\\frac{J_{2, 12}J_{6, 12}^2J_{12}^3}{2J_{1, 12}^2J_{5, 12}^2} - \\sum _{n=-\\infty }^{\\infty } \\frac{(-1)^nq^{6n^2+3n}}{1+q^{6n}} \\right)$ has non-negative coefficients, and given the simplification with help of Lemma REF , a stronger version of the method used in [1] along with using properties of $\\varphi ^2(q)$ , in which coefficient of $q^n$ counts number of Diophantine solutions to $a^2 + b^2 = n$ might aid in proving the inequality when $3 \\mid n + 1$ ."
]
] | 1605.06037 |
[
[
"On a Certain Function with Negative Coefficients"
],
[
"Abstract In what follows we improve an inequality related to matrix theory.",
"T. Laffey proved (2013) a weaker form of this inequality [2]."
],
[
"snakes On a Certain Function with Negative Coefficients Dov Aharonov In what follows we improve an inequality related to matrix theory.",
"T. Laffey proved (2013) a weaker form of this inequality [2].",
"Theorem 1 Given $\\rho = \\sum \\limits _{k=1}^{n}\\mu _k \\le 1$ , $\\mu _k > 0$ , $k=1,2,\\ldots ,n$ , $c_1 >c_2>\\cdots >c_n>0$ Then for $n \\ge 2$ , or $n=1$ $\\mu _1<1$ : $\\sum \\limits _{j=1}^{\\infty } \\pi (1-c_jt)^{\\mu _j} = 1-\\sum \\limits _{j=1}^{\\infty }D_j t^j\\ \\mbox{we have} \\ D_j >0 \\ , \\ j=1,2,\\ldots $ Lemma 1 In order to prove the theorem it is enough to prove it for $\\rho =1$ .",
"Proof of Lemma 1 Assume that $\\sum \\limits _{j=1}^{n}\\mu _j=1$ and that for this case the theorem is proved.",
"Then for $0 < \\rho <1$ we have $(1-\\sum D_jt^j)^{\\rho }=1-\\rho \\sum D_jt^j + \\rho (\\rho -1)\\left(\\sum D_jt^j\\right)^{2}-\\frac{1}{6}\\rho (\\rho -1)(\\rho -2)\\left(\\sum D_jt^j\\right)^{3}+ \\cdots $ $=1-\\sum \\limits _{j=1}^{\\infty }\\tilde{D}_j t^j$ Hence $\\tilde{D}_j > 0$ $\\blacksquare $ From now on we may assume, without loss of generality, $\\sum \\limits _{j=1}^{n}\\mu _j =1$ Lemma 2 Assuming $(1)$ and $(2)$ we have $\\sum \\limits _{j=1}^{n}c_j \\mu _j= D_1 \\ , \\ D_1 > c_n$ Proof of Lemma 2 $\\prod \\limits _{j=1}^{n}(1-c_jt)^{\\mu _j} =\\prod \\limits _{j=1}^{n}(1-\\mu _jc_jt+O(t^2)) =1-\\left(\\sum \\limits _{j=1}^{n}\\mu _jc_j\\right)t+ O(t^2)=1-D_1t+O(t^2)$ Hence, the first part is confirmed.",
"For the second part recall the monotonicity of $c_j$ .",
"We have $D_1 = \\sum \\limits _{j=1}^{n}c_j\\mu _j > c_n\\sum \\limits _{j=1}^{n} \\mu _j = c_n$ $\\blacksquare $ The proof of our theorem will be by induction.",
"For $n=1,2$ the proof follows very simply.",
"Details are omitted.",
"Now assume that the theorem is proved for $n-1$ ($n \\ge 3$ ).",
"We want to prove it for $n$ .",
"Suppose that this is not the case.",
"It will be convenient to use the notation $L_k = \\prod \\limits _{j=1}^{k}(1-c_jt)^{\\mu _j}$ Obviously, $L_k = L_{k-1}(1-c_kt)^{\\mu _k}$ .",
"In particular, $L_n = L_{n-1}(1-c_nt)^{\\mu _n}=1-\\sum \\limits _{j=1}^{\\infty }D_jt^j$ By our assumption, there exists among the coefficients at least one which is not negative.",
"Take the smallest index, say $m$ , for which this is true.",
"Hence $-D_j < 0 \\ , \\ 1 \\le j \\le m-1 \\ , \\ -D_m \\ge 0$ Time has come to use a simple idea, but useful.",
"From $(4)$ $L_n = L_{n-2}(1-c_{n-1}t)^{\\mu _{n-1}}(1-c_nt)^{\\mu _n}$ Then for $c_n=0$ $L_n=L_{n-1}$ and we are back to the case of $n-1$ factors, i.e.",
"the theorem is assumed to be correct by the induction assumption.",
"The same follows if we assume $c_{n-1}=c_n$ .",
"Indeed $(1-c_{n-1}t)^{\\mu _{n-1}}(1-c_nt)^{\\mu _n}=(1-c_{n-1}t)^{\\mu _{n-1}+\\mu _n}$ , and again we have $n-1$ factors.",
"(Note that if we want to prove a weaker result for $\\mu _1=\\mu _2=\\cdots =\\mu _n = \\frac{1}{n}$ the proof does not work!!)",
"Figure 1 [scale=0.5] [->] (0,0)–(0,8); [->] (0,4)–(10,4); (8,0)–(8,4.25); (4,3.75)–(4,4.25); [snake,segment amplitude = 1mm, segment length = 5mm,rounded corners] (0,2)–(8,2); above] at (4,4.25) $c_n$ ; above] at (8,4.25) $c_{n-1}$ ; left] at (0,2) $-D_j$ ; t (0,-1) $j=1,2,\\ldots , n-1$ ; By the induction assumption.",
"Figure 1 [scale=0.5] [->] (0,0)–(0,8); [->] (0,4)–(10,4); (8,0)–(8,4.25); (4,3.75)–(4,4.25); [snake,segment amplitude = 1mm, segment length = 5mm,rounded corners] (0,2)–(8,2); above] at (4,4.25) $c_n$ ; above] at (8,4.25) $c_{n-1}$ ; left] at (0,2) $-D_j$ ; t (0,-1) $j=1,2,\\ldots , n-1$ ; By the induction assumption.",
"Figure 2 [scale=0.5] [->] (0,0)–(0,8); [->] (0,4)–(10,4); below] at (4,3.8) $\\bar{c}_n$ ; (4,3.75)–(4,4.25); above] at (6,4.25) $c_{n-1}$ ; (6,3.75)–(6,4.25); (3,4) arc (180:0:.5); (2,4) arc (-180:0:.5); (1,4) arc (180:0:.5); (4,4) arc (-180:-90:.5); (4.5,3.5)–(6,3); (0,3)–(.5,3.5); (.5,3.5)–(1,4); $\\begin{array}{l}-D_m(0) <0\\\\-D_m(c_{n-1}) <0\\end{array}$ Figure 2 [scale=0.5] [->] (0,0)–(0,8); [->] (0,4)–(10,4); below] at (4,3.8) $\\bar{c}_n$ ; (4,3.75)–(4,4.25); above] at (6,4.25) $c_{n-1}$ ; (6,3.75)–(6,4.25); (3,4) arc (180:0:.5); (2,4) arc (-180:0:.5); (1,4) arc (180:0:.5); (4,4) arc (-180:-90:.5); (4.5,3.5)–(6,3); (0,3)–(.5,3.5); (.5,3.5)–(1,4); Consider now $c_n$ as a variable and $c_1, c_2, \\ldots , c_{n-1}$ as fixed.",
"$0 \\le c_n \\le c_{n-1}$ Since $-D_m \\ge 0$ at some point, say $\\bar{c}_n$ , for this interval and also $-D_m(0) < 0$ , $-D_m(c_{n-1}) < 0$ it follows by the mean value theorem that for some point in this interval, say $c^*_n$ : $-D_m(c^*_n)=0$ .",
"But there are only finite number of zeros of $-D_m$ .",
"This is true due to the fact that it is a polynomial function of the variable $c_n$ .",
"Thus, without loss of generality, $c^*_n$ may be taken as the largest zero in the interval $(0,c_{n-1})$ .",
"Figure 3 [scale=0.5] [->] (0,0)–(0,8); [->] (0,4)–(10,4); (7,6)–(7,2); left] at (0,7) $-D_m$ ; (0,2)– (1,4); (7,3) arc (270:180:1); [fill] (6,4) circle (0.07); [snake,segment amplitude = 3mm, segment length = 10mm,rounded corners] (1,4)–(6,4); t (6,5) $c^*_n$ ; below] at (7,2) $c_{n-1}$ ; Figure 4 [scale=0.5] [->] (0,0)–(0,8); [->] (0,0)–(10,0); (8,-4)–(8,4); (8,-3) arc (270:180:3); left] at (0,7) $-D_m$ ; right] at (8,-4) $c_{n-1}$ ; t (4,-4) $c_n^*$ ; Thus (due to the fact that there are finite number of zeros as explained above) we have that the interval $(c^*_n,c_{n-1})$ is free of zeros.",
"In what follows we show a contradiction.",
"this will be done by considering a sufficiently small interval $(c^*_n,c_{n}^*+ \\varepsilon )$ for which $-D_m$ is changed from zero at $c_n^*$ to a positive value.",
"We now present the calculations leading to this assertion.",
"Indeed $L_n(c_1,c_2,\\ldots ,c_{n-1},c^*_{n}+\\varepsilon )=L_{n-1}(c_1,c_2,\\ldots ,c_{n-1})(1-(c^*_{n}+\\varepsilon )t)^{\\mu _n}=$ $= L_{n-1}(1-c^*_nt-\\varepsilon t)^{\\mu _n}$ Also: $(1-c_n^*t-\\varepsilon t)^{\\mu _n}=(1-c_n^*t)^{\\mu _n}\\left(1-\\frac{\\varepsilon t}{1-c^*_nt}\\right)^{\\mu _n}=(1-c_n^*t)^{\\mu _n}\\left(1-\\frac{\\varepsilon t}{1-c^*_nt} \\mu _n +O(\\varepsilon ^2)\\right)$ Thus $L_n(c_1,c_2, \\ldots ,c_{n-1},c^*_{n}+\\varepsilon )=[L_{n-1}(c_1,c_2, \\ldots ,c_{n-1})(1-c_n^*t)^{\\mu _n}]\\left(1-\\frac{\\varepsilon t}{1-c^*_nt} \\mu _n +O(\\varepsilon ^2)\\right)$ $=(1-D_1 t - D_2t^2-\\cdots -D_{m-1}t^{m-1}+0-D_{m+1}t^{m+1}+\\cdots )\\left(1-\\frac{\\varepsilon t\\mu _n}{1-c^*_nt}+O(\\varepsilon ^2)\\right)=R \\ \\mbox{(notation)}$ To continue it will be convenient to use the notation $\\lbrace F\\rbrace _K$ for the $k$ -th coefficient of $F$ .",
"Also denote $\\mu _n\\varepsilon $ by $\\varepsilon ^*$ .",
"$\\lbrace R\\rbrace _m = \\left\\lbrace \\left(1-\\sum \\limits _{j=1}^{m-1}D_j t^j\\right)\\left(1-\\frac{\\varepsilon ^* t}{1-c^*_nt}\\right)\\right\\rbrace _m +O(\\varepsilon ^2)$ Positive terms do not destroy positivity if ignored, hence $\\lbrace R\\rbrace _m = \\left\\lbrace -\\frac{\\varepsilon ^*t}{1-c^*_nt}\\right\\rbrace _m+\\left\\lbrace (D_1t)\\left(\\frac{\\varepsilon ^*t}{1-c^*_nt}\\right)\\right\\rbrace _m+ \\ \\mbox{positive terms} \\ +O(\\varepsilon ^2)$ $\\begin{array}{rl}\\lbrace R\\rbrace _m \\ge & -\\varepsilon (c_n^*)^{m-1}+(\\varepsilon ^*)\\left\\lbrace D_1t^2\\left(1+\\sum \\limits _{k=1}^{\\infty }(c_n^*t)^k\\right)\\right\\rbrace _m+O(\\varepsilon ^2)\\\\\\lbrace R\\rbrace _m \\ge & -\\varepsilon (c_n^*)^{m-1}+\\varepsilon ^*D_1(c_n^*)^{m-2}+O(\\varepsilon ^*)\\\\\\vspace{-10.0pt}\\\\\\lbrace R\\rbrace _m \\ge & \\varepsilon ^*(c_n^*)^{m-2}[-c_n^*+D_1]+O(\\varepsilon ^*)\\end{array}$ For $\\varepsilon $ small enough this is positive by $(3)$ applied for the special value $c_n^*$ of $c_n$ .",
"Hence, positivity of the $n$ -th coefficient is established provided $\\varepsilon $ is small enough.",
"Thus we arrived at a contradiction, which ends the proof.",
"$\\blacksquare $ Figure 5 [scale=0.5] [->] (0,0)–(0,8); [->] (0,4)–(10,4); (4,5)–(4,3); below] at (4,3) $c^*_n$ ; [snake,segment amplitude = 4mm, segment length = 30mm,rounded corners] (4,4)–(8,3); (8,5)–(8,3); right] at (8,3) $c_{n-1}$ ; [fill] (5.76,4.3) circle (.07); above] at (7,5) $c^*_n+\\varepsilon $ ; Contradiction Compare with Figure 4 Figure 5 [scale=0.5] [->] (0,0)–(0,8); [->] (0,4)–(10,4); (4,5)–(4,3); below] at (4,3) $c^*_n$ ; [snake,segment amplitude = 4mm, segment length = 30mm,rounded corners] (4,4)–(8,3); (8,5)–(8,3); right] at (8,3) $c_{n-1}$ ; [fill] (5.76,4.3) circle (.07); above] at (7,5) $c^*_n+\\varepsilon $ ; Contradiction Compare with Figure 4 Acknowledgment I want to thank Prof Raphi Loewy for bringing to my attention the papers [1] and [2].",
"Department of mathematics-I.I.T,Haifa 32000.Israel.",
"E-mail address: [email protected]"
]
] | 1605.05824 |
[
[
"Searching for topological defect dark matter with optical atomic clocks"
],
[
"Abstract The total mass density of the Universe appears to be dominated by dark matter.",
"However, beyond its gravitational interactions at the galactic scale, little is known about its nature.",
"Extensions of the quantum electrodynamics Lagrangian with dark-matter coupling terms may result in changes to Standard Model parameters.",
"Recently, it was proposed that a network of atomic clocks could be used to search for transient signals of a hypothetical dark matter in the form of stable topological defects.",
"The clocks become desynchronized when a dark-matter object sweeps through the network.",
"This pioneering approach, which is applicable only for distant clocks, is limited by the quality of the fibre links.",
"Here, we present an alternative experimental approach that is applicable to both closely spaced and distant optical atomic clocks and benefits from their individual susceptibilities to dark matter, hence not requiring fibre links.",
"We explore a new dimension of astrophysical observations by constraining the strength of atomic coupling to the hypothetical dark-matter cosmic objects.",
"Our experimental constraint exceeds the previous limits; in fact, it not only reaches the ultimate level expected to be achievable with a constellation of GPS atomic clocks but also has a large potential for improvement."
],
[
"Acknowledgements",
"We are very grateful to Victor Flambaum and Yevgeny Stadnik for discussions and crucial remarks concerning the response of an atomic clock transition and an optical cavity to variations in the fine-structure constant, which helped us to properly evaluate our constraints.",
"We also thank Wim Ubachs and Szymon Pustelny for inspiring discussions.",
"The reported measurements were performed at the National Laboratory FAMO in Toruń, Poland, and were supported by a subsidy from the Ministry of Science and Higher Education.",
"Support has also been received from the project EMPIR 15SIB03 OC18.",
"This project has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme.",
"The individual contributors were partially supported by the National Science Centre of Poland through Projects Nos.",
"2015/19/D/ST2/02195, DEC-2013/09/N/ST4/00327, 2012/07/B/ST2/00235, DEC-2013/11/D/ST2/02663, 2015/17/B/ST2/02115, and 2014/15/D/ST2/05281.",
"This research was partially supported by the TEAM Programme of the Foundation for Polish Science, co-financed by the EU European Regional Development Fund and by the COST Action CM1405 MOLIM.",
"PW is supported by the Foundation for Polish Science's START Programme.",
"P.W.",
"developed the concept, performed the calculations and data analysis, and prepared the manuscript.",
"P.M. and M.B.",
"performed the experiment.",
"M.B., P.M., M.Z., D.L., A.C., and R.C.",
"contributed to the development of the experimental set-up.",
"P.W., R.C., M.Z., M.B.",
"and P.M. contributed to the interpretation and discussion of the results.",
"R.C., M.Z., D.L., P.M. and M.B.",
"contributed to the preparation of the manuscript.",
"M. Z. leads the experimental group.",
"The authors declare no competing financial interests.",
"Figure: Set-up for a dark-matter (DM) search using two co-located optical atomic clocks.",
"In each clock, the frequency of the probe laser (Laser 1 and Laser 2) is tightly locked to the ultra-stable optical cavity (Cavity 1 and Cavity 2).",
"The frequency of the light is subsequently tuned by the frequency shifter (FS1 and FS2).",
"The beam probes the clock transition in the trapped atoms (Atomic Trap 1 and Atomic Trap 2).",
"The observed transition probability is used to generate feedback to actively control the shifter to keep the frequency of the probe beams locked to the clock transition.",
"The changes to the frequency correction from each shifter (readout 1 and readout 2) reflect the changes to the frequency of the clock transition with respect to each cavity.When the Earth traverses through a DM topological defect, the presence of the DM will perturb, given a non-zero DM-SM coupling, the values of certain Standard Model (SM) parameters.",
"In particular, we may expect a transient variation in the fine-structure constant α\\alpha .",
"This will shift the frequency of the electronic clock transition and hence will directly manifest as a common component s(t)s(t) in both readouts, r 1 (t)r_1(t) and r 2 (t)r_2(t).",
"In the simplest case, s(t)s(t) can be modelled as a cascade of NN pulses of the same amplitude s 0 s_0 and durations of T 1 T_1, ..., T N T_N.",
"The amplitude of the peak of the cross-correlation of r 1 (t)r_1(t) and r 2 (t)r_2(t) gives a constraint on the strength of the DM-SM coupling (t 2 -t 1 t_2-t_1 is the length of the cross-correlated readouts).Figure: Extraction of the common signal from the clock's readouts.",
"The magnitude of the common component is retrieved by cross-correlating the two readouts.",
"First, a high-pass filter is applied to both readouts to eliminate the low-frequency common signal, which originates primarily from cavity instability.",
"The top left panel shows a 1000 s interval of the original (red) and filtered (purple) readouts from one of the clocks.",
"The corresponding power spectrum density is presented in the top right panel.",
"The dimmed area indicates the frequency range that is removed by the high-pass filter.",
"The bottom left panel shows the entire readout for each clock after filtering (one of them is artificially shifted for readability).",
"The cross-correlation of the filtered readouts is shown in the bottom right panel.Figure: Constraint on the energy scale Λ α \\Lambda _\\alpha .",
"The grey and green lines represent the constraints we infer from the present measurement by cross-correlating the readouts in their entirety and by cross-correlating only a small portion of the readouts, respectively (see the text).",
"The dashed blue and red lines, which are taken directly from Ref.",
", represent limits on the experimental constraints that could be achieved with a trans-continental network of Sr optical lattice clocks and a GPS constellation, respectively.",
"However, these limits are underestimated.",
"The dotted blue and red lines represent the corrected constraints (see the Methods section for details).",
"The dashed green line represents the limit achievable using our approach under the same conditions considered in Ref.",
"."
]
] | 1605.05763 |
[
[
"Functional renormalization group approach to the Yang-Lee edge\n singularity"
],
[
"Abstract We determine the scaling properties of the Yang-Lee edge singularity as described by a one-component scalar field theory with imaginary cubic coupling, using the nonperturbative functional renormalization group in $3 \\leq d\\leq 6$ Euclidean dimensions.",
"We find very good agreement with high-temperature series data in $d = 3$ dimensions and compare our results to recent estimates of critical exponents obtained with the four-loop $\\epsilon = 6-d$ expansion and the conformal bootstrap.",
"The relevance of operator insertions at the corresponding fixed point of the RG $\\beta$ functions is discussed and we estimate the error associated with $\\mathcal{O}(\\partial^4)$ truncations of the scale-dependent effective action."
],
[
"Introduction",
"With the pioneering work of Yang and Lee a new perspective on the properties of statistical systems was established by pointing out the importance of the distribution of zeros of the partition function [1], [2].",
"Expressed in terms of an external parameter, which we shall denote by $z$ , the partition function $Z = Z(z)$ of a finite system can in general be expressed in terms of its roots $z_{\\alpha }$ in the complex plane, i.e., we may write $Z = \\prod _{\\alpha } (z - z_{\\alpha })$ .",
"Their significance appears in the thermodynamic limit, $V\\rightarrow \\infty $ , when they coalesce along one-dimensional curves that separate different infinite volume behaviors of the partition function.In principle, the zeros may accumulate on a dense set in parameter space, which must not necessarily be one dimensional.",
"However, such a scenario is not relevant to this work.",
"These curves can be viewed as cuts that distinguish different branches of the free energy (or grand canonical potential) $\\Omega = - \\beta \\ln Z = - \\beta V\\int \\textrm {d}\\theta \\hspace{1.0pt} g(\\theta ) \\ln \\left[z - z(\\theta )\\right]$ , where $g(\\theta )$ corresponds to the normalized density of zeros ($\\int \\textrm {d}\\theta \\hspace{1.0pt} g(\\theta ) = 1$ ) on a curve parametrized as $z(\\theta )$ and $\\beta = 1/T$ is the inverse temperature ($k_B = 1$ ).",
"Clearly, once the location of the zeros, or cuts they coalesce into, $z(\\theta )$ , and the distribution $g(\\theta )$ is known, in principle, all thermodynamic properties of the system can be calculated.",
"This has led to numerous efforts to determine $g(\\theta )$ for a wide range of lattice models via numerical methods [3], [4], [5], [6] and also experimentally [7], [8], [9].",
"Besides providing a rigorous basis to study the thermodynamic properties of finite lattice systems, such attempts have also helped to elucidate features of fundamental theories.",
"Drawing on the principle of universality they have led to important insights into the phase diagram of strongly-interacting matter at nonvanishing baryon densities [10], [11], [12].",
"Typically, for lattice spin models at temperature $T$ and external field $H$ the natural variable in terms of which the partition function is a polynomial is $z = \\exp \\left(- 2 \\beta H \\right)$ .",
"The zeros of $Z(z)$ are commonly referred to as Yang-Lee or Lee-Yang zeros.",
"In particular, for the ferromagnetic Ising model one finds these zeros distributed along the unit circle $z = \\exp ( i \\theta )$ , where $\\theta = 2 i \\beta H$ and $H$ is imaginary.",
"This has been proven rigorously and is known as the Yang-Lee circle theorem [2], [13], , [15], [16], [17], [18], [19], [20].",
"Depending on the temperature one may distinguish different scenarios: In the low-temperature region of the Ising model ($T < T_c$ ), the set of zeros crosses the positive real $z$ -axis at $z = 1$ ($\\theta = 0$ ), which indicates the presence of a first-order phase transition as one traverses the $\\operatorname{Re}H = 0$ axis from positive to negative real $H$ (or vice versa).",
"On the other hand, in the high-temperature region ($T > T_c$ ) one observes a finite gap in the distribution $g(\\theta ) = 0$ for $|\\theta | < \\theta _g$ that closes as $T \\rightarrow T_c^{+}$ [3], [4].",
"Thus, for $T > T_c$ the free energy is analytic along the real $H$ axis.",
"However, at the edge of the gap $\\theta = \\pm \\theta _g$ , corresponding to imaginary values of the magnetic field $H = \\pm i |H_c(T)|$ , the distribution of zeros exhibits nonanalytic behavior, i.e., $g(\\theta ) \\simeq \\left( |\\theta | - \\theta _g \\right)^{\\sigma }$ , for $|\\theta | \\gtrsim \\theta _g$ , characterized by the exponent $\\sigma $ [5].",
"As pointed out by Fisher [21] this behavior can be identified with a thermodynamic singularity that yields a divergence in the isothermal susceptibility $\\chi = (\\partial M / \\partial H)_{T} \\sim |H-H_c(T)|^{\\sigma - 1}$ , where $M$ is the magnetization.",
"Thus, the Yang-Lee edge singularity at nonvanishing imaginary values of the field is similar to a conventional second order phase transition [21], [22].",
"In contrast to the well-known $\\phi ^4$ field theory that describes the critical point of the Ising model at $T = T_c$ and $H = 0$ , the field theory at the Yang-Lee edge point, the $\\phi ^3$ theory, admits no discrete reflection symmetry and is therefore characterized by only one independent (relevant) exponent.",
"In two dimensions the corresponding universality class has been identified with that of the simplest nonunitary conformal field theory (CFT), the minimal model $M_{2,5}$ , with central charge $c = -22/5$ [23].",
"This allowed to exploit conformal symmetry in two dimensions to calculate the scaling exponent $\\sigma (d=2) = -1/6$ , which has been confirmed with remarkable accuracy by series expansions [24], [25], as well as by comparing with experimental high-field magnetization data [7], [8].",
"Furthermore, using integral kernel techniques it is possible to establish the exact result $\\sigma (d=1) = -1/2$ [21], [22].",
"On the other hand, most of our knowledge in the region $2 < d < 6$ relies on appropriately resummed results from the $\\epsilon = 6 - d$ expansion [26], [27], [28], strong-coupling expansions [29], Monte Carlo methods [30], [31], and conformal bootstrap [32].",
"Note that in contrast to the Ising critical point (described by $\\phi ^4$ theory), the upper critical dimension of the Yang-Lee edge point (described by $\\phi ^3$ theory) is $d_c = 6$ and therefore, fluctuations are important even above dimension $d = 4$ .",
"Recently, there has been renewed interest in the Yang-Lee edge point for which the renormalization group (RG) $\\beta $ functions to four-loop order in the $\\epsilon $ expansion were determined in Ref.",
"[28] and the corresponding critical exponents (obtained from constrained Padé approximants) were compared to estimates from other methods.",
"In light of these developments, we examine the critical scaling properties of the Yang-Lee edge with the nonperturbative functional RG [33], [34] for dimensions $3 \\le d \\le 6$ .",
"In contrast to the $\\epsilon $ expansion, the functional RG does not rely on the expansion in a small parameter and is therefore ideally suited to investigate the critical behavior of the Yang-Lee edge away from $d_c = 6$ .",
"However, care must be taken to address possible systematic errors that arise from the truncation of the infinite hierarchy of flow equations.",
"We show that that these errors are under control and comment on the quality of different truncations.",
"In summary, we find that the obtained values for the critical exponents are in good agreement with previous results obtained in $d = 3$ dimensions using high-temperature series expansions [22], the three- and four-loop $\\epsilon $ expansion around $d = 6$ [26], [27] as well as other methods [29], [30], [31], [32].",
"We observe that derivative interactions have an important effect on the stability of the scaling solution and need to be taken into account properly in the framework of the nonperturbative functional RG.",
"The outline of this article is as follows: First, in Sec.",
", we give an overview of the nonperturbative functional RG and the truncations employed in this work.",
"In Sec.",
"we discuss the scaling properties of the critical equation of state and the mean-field theory at the Yang-Lee edge singularity.",
"In Sec.",
"we consider the general properties of RG flow trajectories and in particular their infrared (IR) behavior.",
"In Secs.",
"– we summarize our results for the critical exponents at the Yang-Lee edge singularity and analyze the expected systematic errors for the truncations employed in this work.",
"We close by comparing our estimates for the critical exponents to recent data from Refs.",
"[28] and [32] and conclude with an outlook on future work."
],
[
"Nonperturbative functional RG",
"In this work, we employ a RG scheme that relies on a truncation of a hierarchy of flow equations derived from an exact flow equation for the scale-dependent effective action $\\Gamma _k$ [33], [34], i.e., the generating functional of one-particle irreducible (1PI) diagrams (for reviews see, e.g., Refs.",
"[35], [36], [37], [38], [39]), where $k$ denotes the RG scale parameter.",
"The scale-dependent effective action is obtained from the functional Legendre transform $\\Gamma _k = \\sup _J \\left( \\int \\textrm {d}^dx \\hspace{1.0pt} J(x) \\phi (x) - W_k \\right) - \\Delta _k S ,$ of the scale-dependent generating functional of connected correlation functions $W_k = \\ln \\int [d\\varphi ] \\hspace{1.0pt} \\exp \\bigg \\lbrace - S - \\Delta _k S + \\int \\textrm {d}^dx \\, J(x) \\varphi (x)\\bigg \\rbrace ,$ with respect to the external source $J = J(x)$ ; $\\phi = \\delta W_k / \\delta J$ is the scalar field expectation value.",
"Here, we consider a classical action $S$ of a single-component scalar field $S = \\int \\textrm {d}^{d}x \\, \\left\\lbrace \\frac{1}{2} ( \\partial \\varphi )^2 + U_{\\Lambda }(\\varphi ) \\right\\rbrace ,$ and the classical potential $U_{\\Lambda }$ is specified in Sec.",
".",
"The additional term $\\Delta _k S$ in Eq.",
"(REF ) is a quadratic functional $\\Delta _k S = \\frac{1}{2} \\int \\textrm {d}^dx \\, \\textrm {d}^dy \\, \\varphi (x) R_k(x,y) \\varphi (y) ,$ and serves to regularize the theory in the IR; in particular, the regulator function $R_k(x,y) = R_k(-\\Box _x) \\delta ^{(d)}(x-y)$ , where $\\Box \\equiv \\partial _{\\mu }\\partial _{\\mu }$ , is chosen in such a way that it leads to a decoupling of IR modes.",
"We require that $\\lim _{k\\rightarrow 0} R_k = 0$ and $\\lim _{\\Lambda \\rightarrow \\infty } R_{k = \\Lambda } = \\infty $ , where $\\Lambda $ is a characteristic scale that regularizes the theory in the ultraviolet (UV) and can formally be sent to infinity.",
"In effect, this defines a one-parameter family of theories ($0 \\le k \\le \\Lambda $ ), which interpolates between the classical action, $S = \\lim _{k\\rightarrow \\Lambda } \\Gamma _k$ , and the full 1PI effective action, $\\Gamma = \\lim _{k\\rightarrow 0} \\Gamma _k$ .",
"Thus, the scale-dependent regulator function $R_k$ induces a functional RG flow $\\frac{\\partial }{\\partial s} \\Gamma _k = \\frac{1}{2} \\int \\!",
"\\frac{\\textrm {d}^dq}{(2\\pi )^d} \\frac{\\partial R_k(q)}{\\partial s} \\left[ \\Gamma _k^{(2)}(\\phi ; q) + R_k(q) \\right]^{-1} ,$ between these two limits, where $s = \\ln (k/\\Lambda )$ is a dimensionless scale parameter, and $\\delta ^{(d)}\\big (\\sum _{i = 1}^n p_i\\big ) \\Gamma _k^{(n)}(\\phi ; p_1, p_2, \\ldots , p_{n-1}) \\equiv (2\\pi )^{(n-1) d} {\\delta ^n \\Gamma _k[\\phi ]}/{\\delta \\phi (p_1) \\delta \\phi (p_2) \\cdots \\delta \\phi (p_n)}$ .",
"In principle, we may choose any (sufficiently smooth) regulator that satisfies the above limiting properties.",
"For details of our implementation and necessary requirements imposed on the regulator function see Secs.",
"– .",
"Table: Operators and canonical dimension of associated parameters and couplings that appear in the expansion of Γ k \\Gamma _k [cf.",
"Eq.",
"()].",
"Note that we drop the RG scale index kk, since the canonical dimensions are defined at the Gaussian fixed point of the RG β\\beta functions.Clearly, an exact solution for the full functional flow is not feasible in practice, so one has to rely on suitable approximations of Eq.",
"(REF ).",
"Here, we comment on the nature of our truncation and discuss its limitations.",
"We use a truncated expansion in derivatives for the scale-dependent effective action [40], [41] $&& \\hspace{-8.0pt} \\Gamma _k = \\int \\textrm {d}^dx \\hspace{2.0pt} \\bigg \\lbrace U_k(\\phi ) + \\frac{1}{2} Z_k(\\phi ) (\\partial \\phi )^{2} + \\frac{1}{2} W_{1,k}(\\phi ) ( \\Box \\hspace{1.0pt} \\phi )^{2} \\nonumber \\\\&& \\hspace{16.0pt} +\\: \\frac{1}{2} W_{2,k}(\\phi ) (\\partial \\phi )^{2} \\,\\Box \\hspace{1.0pt} \\phi + \\frac{1}{2} W_{3,k}(\\phi ) \\big [ ( \\partial \\phi )^{2} \\big ]^2 \\bigg \\rbrace ,$ where $U_k$ is the scale-dependent effective potential, and the scale-dependent functions $Z_k$ and $W_{a,k}$ , $a=1,2,3$ , parametrize the contributions to order $\\partial ^4$ (up to total derivative terms).",
"Furthermore, for each of these functions, we employ a finite series expansion in the fluctuation $\\delta \\phi _k = \\phi - \\bar{\\phi }_k$ around a field configuration $\\bar{\\phi }_k$ , which is assumed to be homogeneous in space [cf.",
"Sec.",
"].",
"In effect, this corresponds to an ansatz for $\\Gamma _k$ that includes only a finite set of independent operators, each of which is parametrized by a single parameter or coupling that is field independent, e.g., $Z_k(\\phi ) (\\partial \\phi )^2 = \\big ( \\bar{Z}_k^{(0)} + \\bar{Z}_k^{(1)} \\delta \\phi _k + \\ldots \\big ) (\\partial \\phi )^2$ , and $\\bar{Z}_k^{(n)}\\equiv Z_k^{(n)}(\\bar{\\phi }_k)$ , $n \\in \\mathbb {N}$ , and similar expansions apply to $U_k$ and $W_{a,k}$ .",
"The canonical dimensions of these parameters are displayed in Tab.",
"REF.",
"Clearly, above dimension $d = 2$ , $\\bar{Z}_k^{(n)}$ and $\\bar{W}_{a,k}^{(n)}$ are irrelevant as far as a counting of canonical dimensions goes, but this is not sufficient to conclude that this is also the case at a nontrivial (i.e., non-Gaussian) fixed point of the RG $\\beta $ functions.",
"Indeed, one of the objectives of this paper is to investigate their effect at the Yang-Lee edge point as well as on RG trajectories that approach this scaling solution in the IR.",
"We should point out that similar truncations of the scale-dependent effective action were considered also in Refs.",
"[42], [43], [44], [45] to establish the critical exponents at the Ising critical point.",
"Here, we study the scaling properties of Eq.",
"(REF ) in the presence of a nonvanishing external field, when the discrete reflection symmetry $\\phi \\leftrightarrow -\\phi $ of the Ising model is explicitly broken and the system is tuned to the Yang-Lee edge critical point.",
"The flow equations for $U_k$ , $Z_k$ , and $W_{a,k}$ , $a = 1,2,3$ , are derived from the exact functional flow equation for $\\Gamma _k$ [cf.",
"Eq.",
"(REF )] by applying functional derivatives and projecting them onto the appropriate momentum contributions, i.e., $&& \\hspace{-8.2pt} \\frac{\\partial }{\\partial s} U_k = \\frac{\\partial }{\\partial s} \\left.",
"\\Gamma _k[\\phi ] \\right|_{\\phi =\\textrm {const.",
"}}, \\\\&& \\hspace{-8.0pt} \\frac{\\partial }{\\partial s} Z_k = \\lim _{p\\rightarrow 0} \\frac{\\partial }{\\partial p^2} \\frac{\\partial }{\\partial s} \\Gamma _k^{(2)}(\\phi ; p) , \\\\&& \\hspace{-17.0pt} \\frac{\\partial }{\\partial s} W_{1,k} = \\lim _{p\\rightarrow 0} \\frac{\\partial }{\\partial (p^2)^2} \\frac{\\partial }{\\partial s} \\Gamma _k^{(2)}(\\phi ; p) , \\\\&& \\hspace{-17.0pt} \\frac{\\partial }{\\partial s} W_{2,k} = \\frac{1}{2} \\lim _{p_i\\rightarrow 0} \\frac{\\partial }{\\partial (p_1 \\hspace{-2.0pt}\\cdot \\hspace{-1.0pt} p_2)^2} \\frac{\\partial }{\\partial s} \\Gamma _k^{(3)}(\\phi ; p_1, p_2) , \\\\&& \\hspace{-16.5pt} \\frac{\\partial }{\\partial s} W_{3,k} = -\\frac{1}{4} \\lim _{p_i\\rightarrow 0} \\left[\\frac{\\partial }{\\partial (p_2 \\hspace{-2.0pt}\\cdot \\hspace{-1.0pt} p_3)} - \\frac{1}{2}\\frac{\\partial }{\\partial (p_1 \\hspace{-2.0pt}\\cdot \\hspace{-1.0pt} p_2)} \\right.",
"\\nonumber \\\\ && \\hspace{30.0pt} \\left.",
"-\\: \\frac{1}{2}\\frac{\\partial }{\\partial (p_1\\hspace{-2.0pt}\\cdot \\hspace{-1.0pt}p_3)} \\right] \\frac{\\partial }{\\partial p_1^2} \\frac{\\partial }{\\partial s} \\Gamma _k^{(4)}(\\phi ; p_1, p_2, p_3) , $ where $p\\cdot q \\equiv p_{\\mu }q_{\\mu }$ .",
"The corresponding RG flow equations for the field-independent parameters $\\bar{Z}_k^{(n)}$ and $\\bar{W}_{a,k}^{(n)}$ can be derived from Eqs.",
"(REF ) – () by suitable differentiation and successive projection onto the reference field configuration $\\bar{\\phi }_k$ that enters the series expansion.",
"We do not display them at this point but refer the reader to supplementary material available online [46].",
"The RG flow equations display the following chain of dependencies $U_k \\leftarrow \\lbrace Z_k , W_{1,k} \\rbrace \\leftarrow \\lbrace W_{2,k} , W_{3,k} \\rbrace \\leftarrow \\ldots ,$ where the ellipsis denotes higher order contributions that we have chosen to neglect in our ansatz, Eq.",
"(REF ).",
"That is, the RG flow equation for the scale-dependent effective potential $U_k$ depends on the quantities $Z_k$ and $W_{1,k}$ , but is independent of $W_{2,k}$ and $W_{3,k}$ etc.",
"We exploit this structure explicitly by truncating the hierarchy Eq.",
"(REF ) at the second level, i.e., we set $W_{2,k} = W_{3,k} = 0$ in Eq.",
"(REF ), while $U_k$ , $Z_k$ , and $W_k\\equiv W_{1,k}$ are expanded to some finite order in $\\delta \\phi _k$ .",
"Note that the order of the employed expansion might be different for each of these coefficients.",
"Similar approximations have led to reasonable estimates of the critical scaling exponents at the Ising critical point [44], [45] and we expect that this is also the case for the Yang-Lee edge critical point."
],
[
"Critical equation of state and mean-field scaling prediction",
"Here, we consider a classical potential of the following form $U_{\\Lambda } = \\frac{1}{2} t_{\\Lambda } \\varphi ^2 + \\frac{1}{4!}",
"\\lambda _{\\Lambda } \\varphi ^4 + h_{\\Lambda } \\varphi ,$ with a nonvanishing coupling to a symmetry-breaking field $h_{\\Lambda }$ , and $t_{\\Lambda } \\sim T - T_c$ , with $T_c$ the critical temperature at the Ising critical point.",
"Upon integration of the RG flow equations (REF ) – () down from the cutoff scale $\\Lambda $ to the IR, the parameters and couplings of the classical potential acquire a scale dependence.",
"In fact, the corresponding scale-dependent effective potential $U_k$ for $0\\le k < \\Lambda $ will typically include a large number of fluctuation-induced interactions.",
"The full effective potential is obtained only when the scale parameter $k$ is sent to zero and all modes have been integrated out, i.e., $U = \\lim _{k\\rightarrow 0} U_k$ .",
"In order to arrive at a critical point in the IR the relevant parameters of the classical action need to be tuned to their respective critical values, while all other parameters or couplings are kept constant.",
"That is, in the case of the Yang-Lee edge critical point, we fix $\\lambda _{\\Lambda }$ , $|h_{\\Lambda }| > 0$ , and tune $t_{\\Lambda }$ to its critical value $t_{\\Lambda ,c} = t_{\\Lambda ,c}(h_{\\Lambda }) > 0$ , for which $\\bar{U}^{(1)} \\equiv \\lim _{k\\rightarrow 0} \\bar{U}_k^{(1)} = 0$ and $\\bar{U}^{(2)} \\equiv \\lim _{k\\rightarrow 0} \\bar{U}_k^{(2)} = 0$ in the IR limit.",
"At the Yang-Lee edge critical point, the first and second derivative are evaluated at a nonvanishing, imaginary field expectation value $\\bar{\\phi }$ .",
"In the critical domain, the equation of state satisfies the scaling form $U^{\\prime }(\\phi ) = \\delta \\phi |\\delta \\phi |^{\\delta -1} f\\left(\\delta t_{\\Lambda } |\\delta \\phi |^{-1/\\beta }\\right) ,$ where $\\delta \\phi = \\phi - \\bar{\\phi }$ and $f = f(x)$ is a universal, dimensionless scaling function, which is uniquely defined up to normalization.",
"The critical exponents $\\beta $ and $\\delta $ characterize the asymptotic scaling behavior of the (residual) magnetization $\\delta \\phi $ for vanishing $U^{\\prime }(\\phi ) = \\delta h$ and $\\delta t_{\\Lambda } = t_{\\Lambda } - t_{\\Lambda ,c}$ , respectively.",
"Here, the parameter $\\delta h \\sim H - H_c$ , measures the deviation from the critical field strength $H_c = \\pm i |H_c(T)|$ , and $T > T_c$ for the range of values of $\\delta t_{\\Lambda }$ studied in this work.",
"Before we go on to consider the solution of the RG flow equations (REF ) – (), we discuss the mean-field scaling prediction.",
"Since there is no scale dependence in this case, we simply drop the $k$ (or $\\Lambda $ ) index on all parameters.",
"It is useful to express the potential in terms of an expansion in field differences $\\delta \\varphi = \\varphi - \\bar{\\varphi }$ around a reference field configuration $\\bar{\\varphi }$ , which is defined such that $U^{\\prime }(\\bar{\\varphi }) = 0$ .",
"According to the strategy outlined above, we fix $|h| > 0$ and inquire about possible critical points, by imposing in addition the condition that $U^{\\prime \\prime }(\\bar{\\varphi }) = 0$ .",
"We derive two independent scaling solutions, which we identify as $t_c = \\lambda /2 \\left( \\pm i 3 h/\\lambda \\right)^{2/3} .$ Assuming that $t_c > 0$ we see that the corresponding critical field value $h_c = \\pm i \\lambda /3 \\left( 2 t_c / \\lambda \\right)^{3/2}$ is imaginary in accordance with the Yang-Lee theorem [1], [2].",
"Near the critical point $U^{\\prime }(\\varphi )$ satisfies the scaling form (REF ) with $\\delta = 2$ and $\\beta = 1$ .",
"Other critical exponents that characterize the power-law singularities of various thermodynamic quantities can be determined via scaling relations [47].",
"That is, in the absence of fluctuations the anomalous dimension vanishes, $\\eta = 0$ , and we obtain the following scaling exponents: $\\alpha = -1$ , $\\gamma = 1$ , $\\nu = 1/2$ , and $\\nu _c = 1/4$ .",
"Note that the exponent $\\alpha $ is negative and therefore, at the mean-field level, the specific heat does not diverge at the Yang-Lee edge point."
],
[
"Solving the RG flow equations",
"To solve the RG equations we specify the classical action $S = \\int \\textrm {d}^{d}x \\, \\left\\lbrace \\frac{1}{2} (\\partial \\varphi )^2 + U_{\\Lambda }(\\varphi ) \\right\\rbrace $ , which is defined in terms of the short-distance potential $U_{\\Lambda }$ , and integrate the flow equations down to $s \\rightarrow -\\infty $ .",
"The classical potential is given in Eq.",
"(REF ) and the coefficients that parametrize the kinetic contribution to the action are $Z_{\\Lambda } = 1$ and $W_{\\Lambda } = 0$ .",
"We use a truncated series expansion for the scale-dependent effective potential $U_k$ as well as for the field-dependent renormalization factors $Z_k$ and $W_k$ ($0\\le k \\le \\Lambda $ ).",
"Such a strategy is often sufficient to extract the leading or subleading critical scaling behavior [49], [50], [44], [51], [45].",
"The employed expansion is organized around a nonvanishing, imaginary, and homogeneous field configuration $\\bar{\\phi }_k$ , which depends on the scale parameter $k$ , and is defined in the following way: 1) At the cutoff scale $\\Lambda $ , $\\bar{\\phi }_{k = \\Lambda } = \\bar{\\varphi }_{\\Lambda }$ is a solution to $U_{\\Lambda }^{\\prime \\prime }(\\bar{\\varphi }_{\\Lambda }) = \\tau $ , and 2) the scale derivative of $\\bar{U}_k^{(2)} \\equiv U_k^{\\prime \\prime }(\\bar{\\phi }_k)$ , evaluated at $\\bar{\\phi }_k = \\bar{\\phi } + \\bar{\\chi }_k$ , satisfies $\\frac{\\textrm {d}}{\\textrm {d}s} \\bar{U}_k^{(2)} &=& \\frac{\\partial }{\\partial s} \\bar{U}_k^{(2)} + \\bar{U}_k^{(3)} \\frac{\\textrm {d}\\bar{\\chi }_k}{\\textrm {d}s} = 0 .$ Of course, the imaginary field expectation value $\\bar{\\phi }$ is scale independent and therefore $\\textrm {d} \\bar{\\phi }_k / \\textrm {d}s = \\textrm {d}\\bar{\\chi }_k / \\textrm {d}s$ .",
"Note that $\\lim _{k\\rightarrow 0} \\bar{\\chi }_k = 0$ , i.e., $\\lim _{k\\rightarrow 0} \\bar{\\phi }_k = \\bar{\\phi }$ , only when $\\tau = 0$ and the system has been tuned to criticality.",
"Clearly, conditions 1) and 2) fix one parameter of the model $\\bar{U}_k^{(2)} = \\tau $ , at the expense of introducing another scale-dependent quantity, the field configuration $\\bar{\\chi }_k$ , for which we obtain $\\frac{\\textrm {d}\\bar{\\chi }_k}{\\textrm {d}s} = - \\big (\\bar{U}_k^{(3)}\\big )^{-1} \\frac{\\partial }{\\partial s} \\bar{U}_k^{(2)} .$ Note that the corresponding set of flow equations requires that $|\\bar{U}_k^{(3)}| > 0$ for all $0\\le k \\le \\Lambda $ .",
"This does not hold true in the vicinity of the Ising critical point and therefore, the chosen expansion point is not adequate to investigate the scaling properties for critical points on the $\\phi \\leftrightarrow -\\phi $ symmetry axis ($H = 0$ ).",
"Eq.",
"(REF ) fixes the second derivative of the scale-dependent effective potential at all scales and therefore the expansion of the scale-dependent effective potential reads $U_k = \\bar{U}_k^{(0)} + \\bar{U}_k^{(1)} \\delta \\phi _k + \\frac{1}{2} \\tau \\hspace{1.0pt} \\delta \\phi _k^2 + \\sum _{n = 3}^{n_{U}} \\frac{1}{n!}",
"\\bar{U}_k^{(n)} \\delta \\phi _k^n .$ Here, the sum runs up to some finite integer value $n_U$ , which defines our truncation for the scale-dependent effective potential with the prescribed expansion point.",
"The coefficients $\\bar{U}_k^{(n)}$ , $n\\in \\mathbb {N}$ , are related to the couplings and parameters of the classical potential at the short-distance cutoff $\\Lambda $ , i.e., $\\bar{U}_{\\Lambda }^{(0)} = \\bar{\\varphi }_{\\Lambda } \\left[h_{\\Lambda } + 1/12 \\hspace{2.0pt} (5 t_{\\Lambda } + \\tau ) \\bar{\\varphi }_{\\Lambda }\\right]$ , $\\bar{U}_{\\Lambda }^{(1)} = h_{\\Lambda } + (2 t_{\\Lambda } + \\tau ) / 3 \\hspace{2.0pt} \\bar{\\varphi }_{\\Lambda }$ , and $\\bar{U}_{\\Lambda }^{(3)} = \\lambda _{\\Lambda }/6 \\hspace{2.0pt}\\bar{\\varphi }_{\\Lambda }$ , $\\bar{U}_{\\Lambda }^{(4)} = \\lambda _{\\Lambda }$ , while $\\bar{U}_{\\Lambda }^{(n)} = 0$ , for $n > 4$ .",
"Similarly, the expansions for $Z_k$ and $W_k$ read $Z_k &=& \\sum _{n = 0}^{n_Z-1} \\frac{1}{n!}",
"\\bar{Z}_k^{(n)} \\delta \\phi _k^n , \\\\W_k &=& \\sum _{n = 0}^{n_W-1} \\frac{1}{n!}",
"\\bar{W}_k^{(n)}\\delta \\phi _k^n ,$ with $\\bar{Z}_{\\Lambda }^{(0)} = 1$ , $\\bar{Z}_{\\Lambda }^{(n)} = 0$ for $n > 0$ , and $\\bar{W}_{\\Lambda }^{(n)} = 0$ for $n\\in \\mathbb {N}$ .",
"We define $Z_k \\equiv 0$ if $n_Z = 0$ and $W_k \\equiv 0$ if $n_W = 0$ .",
"In the following, we denote these type of series truncations in short by the set of integers $(n_U, n_Z, n_W)$ .",
"$n_U$ is considered as a free parameter, while $n_Z$ and $n_W$ are chosen such that $\\max _{n_Z} \\dim \\bar{Z}_k^{(n_Z)} \\le \\dim \\bar{U}_k^{(n_U)}$ and $\\max _{n_W} \\dim \\bar{W}_k^{(n_W)} \\le \\dim \\bar{U}_k^{(n_U)}$ in $d = 6$ dimensions.",
"This choice defines what we consider to be consistent truncations (see Sec.",
").",
"Substituting Eqs.",
"(REF ) – (REF ) back into (REF ) – () we obtain a finite set of flow equations for the coefficients of the series expansion.",
"In this work, we consider expansions of order up to $(n_U, n_Z, n_W) = (7, 5, 0)$ and $(5, 3, 2)$ , which yields a coupled set of partial differential equations of up to 12 and 10 parameters, respectively.",
"The Yang-Lee scaling solution is identified by inspecting the behavior of the first and second derivatives of the effective potential, which should satisfy $\\bar{U}^{(1)} = \\bar{U}^{(2)} = 0$ , while $\\operatorname{Im}\\bar{U}^{(2n)} = \\operatorname{Re}\\bar{U}^{(2n+1)} = 0$ , for $n\\in \\mathbb {N}$ .",
"Note that all of these coefficients are defined at a reference field configuration $\\bar{\\phi }$ (where $\\lim _{k\\rightarrow 0} \\bar{\\chi }_k = 0$ ), which is imaginary, corresponding to the imaginary magnetic field $H_c = \\pm i |H_c(T)|$ , with $T > T_c$ .",
"We introduce the following short-hand notation for the renormalization factor $\\bar{Z}_k \\equiv Z_k^{(0)}(\\bar{\\phi }_{k})$ , which satisfies $\\bar{Z}_k \\sim (k/\\Lambda )^{-\\eta }$ at the critical point.",
"Starting from a set of initial values for the parameters and couplings in the classical action, which are tuned to their critical values, we may therefore define the anomalous dimension by the corresponding value in the IR: $\\eta = - \\lim _{k\\rightarrow 0} \\frac{\\partial }{\\partial s} \\ln \\bar{Z}_k .$ Note that the anomalous dimension at the Yang-Lee edge critical point is negative for all values of $1 \\le d < 6$ ."
],
[
"Critical scaling exponents and hyperscaling relations",
"The critical exponents at the Yang-Lee edge critical point are extracted by a stability analysis of the scaling solution with respect to perturbations with those operators included in our ansatz Eq.",
"(REF ).",
"That is, for any finite truncation of the scale-dependent effective action, we obtain a finite set critical exponents corresponding to the eigenvalues $\\lambda _n$ of the stability matrix, $\\gamma _{mn} = \\frac{\\partial \\beta _m\\big (\\lbrace \\bar{g}_{{}_{\\ast }, l} \\rbrace _{l\\in I}\\big )}{\\partial \\bar{g}_{n,k}} ,$ which is evaluated at the fixed point of the RG $\\beta $ functions, $\\beta _m \\equiv \\partial \\bar{g}_{m,k} / \\partial s$ , i.e., $\\beta _m\\big (\\lbrace \\bar{g}_{{}_{\\ast }, n}\\rbrace _{n\\in I}\\big ) = 0 .$ The $\\beta $ functions are derived for the dimensionless, renormalized parameters and couplings of the model, $\\bar{g}_{n,k}$ , $n\\in I = \\lbrace 1, 2, \\ldots , n_U + n_Z + n_W\\rbrace $ , which are given by $\\bar{g}_{1,k} = k^{-(d+2)/2} \\bar{Z}_k^{-1/2} \\bar{U}_k^{(1)}$ , $\\bar{g}_{2,k} = k^{(2-d)/2} \\bar{Z}_k^{1/2} \\bar{\\chi }_k$ , etc.",
"We order the eigenvalues $\\lambda _n$ , $n = 1,2, \\ldots $ , according to their values in $d = 6$ dimensions, where they are identical to the canonical dimension of the parameters and couplings associated with the operators that appear in $\\Gamma _k$ , e.g., $\\lambda _1(d = 6) = \\dim \\bar{U}^{(1)} \\ge \\lambda _2(d = 6) = \\dim \\bar{\\chi } \\ge \\ldots $ .",
"Of course, as the eigenvalues are analytically continued to dimensions below $d = 6$ , this ordering might change.",
"We observe that the largest eigenvalue $\\lambda _1 \\equiv 1/\\nu _c$ satisfies the following scaling relation $1/\\nu _c = ( d + 2 - \\eta ) / 2 ,$ and therefore, the critical exponent $\\nu _c$ is determined completely in terms of the anomalous dimension $\\eta $ .",
"The Yang-Lee edge critical point is known to exhibit another hyperscaling relation, which follows from the equation of motion of the $\\phi ^3$ theory [52] and can be written as $\\lambda _1 + \\lambda _2 = d ,$ with $\\lambda _2\\equiv 1/\\nu $ , from which we obtain $1/\\nu = ( d - 2 + \\eta ) / 2 .$ Furthermore, from scaling and hyperscaling relations, one can derive $\\sigma = \\frac{1}{\\delta } = \\frac{d - 2 + \\eta }{d + 2 - \\eta } ,$ and $\\beta = 1$ , independent of dimension [26].",
"Note, however, that for any finite truncation of $\\Gamma _k$ scaling relations between critical exponents need not necessarily be satisfied and therefore should be checked explicitly.",
"This applies to both Eq.",
"(REF ) and to Eq.",
"(REF ).",
"Taking Eq.",
"(REF ) for example, one may define the relative difference $\\Delta \\lambda _2 / [(d - 2 + \\eta )/2] = 2 \\lambda _2/ (d - 2 + \\eta ) - 1$ as an indicator for the quality of the employed truncation at the Yang-Lee edge fixed point.",
"We observe that the relative error in the scaling relation (REF ) increases with smaller dimensions.",
"For both the $(7,5,0)$ and $(5,3,2)$ truncations, we obtain a $15\\%$ error in $d = 5$ dimensions, a $60 - 70\\%$ error in $d = 4$ dimensions etc.",
"This is an indication that the considered series expansions are not fully converged yet.",
"Nevertheless, since we expect these scaling relations to hold for high enough orders, we employ Eq.",
"(REF ) in the following to determine the exponent $\\nu $ , keeping in mind that the corresponding estimates will be associated with an error that is likely to decrease only when higher-order truncations are considered.",
"In particular, the above numbers suggest that to reach a given precision, one will need to account for an increasing number of operators in $\\Gamma _k$ in lower dimensions.",
"Figure: Anomalous dimension η\\eta for different truncations of the scale-dependent effective action in d=3d = 3, 4, and 5 dimensions.Table: Numerical values for the anomalous dimension η\\eta and critical exponents σ\\sigma , ν c \\nu _c in d=3d = 3, 4, and 5 dimensions.",
"Here, we show our best estimates with errors to account for possible systematic effects (see Sec.",
").",
"These values were obtained with an exponential regulator (α=1\\alpha = 1) [cf.",
"Eq.",
"()] and the truncation of the type (7,5,0)(7,5,0).Table: Different estimates for the critical exponent σ\\sigma (as compiled in Ref. )",
"including results from the constrained three- and four-loop ϵ\\epsilon expansion , strong-coupling expansion , Monte Carlo methods , , and conformal bootstrap .",
"The values obtained from the functional RG, with an exponential regulator (α=1\\alpha = 1) and truncation of the type (7,5,0)(7,5,0), lie within error bars of Refs.",
", , , and are slightly larger the values provided by constrained Padé approximants of three- and four-loop ϵ\\epsilon expansion results , but are smaller than those obtained by conformal bootstrap methods .The scaling properties of the Yang-Lee edge are completely determined by the anomalous dimension $\\eta $ .",
"Therefore, we may use Eqs.",
"(REF ) and (REF ) to calculate the critical exponents $\\nu _c$ and $\\sigma $ .",
"Our results are summarized in Fig.",
"REF where we show the overall performance of different truncations in the range $3 \\le d \\le 6$ at the example of $\\sigma $ and $\\nu _c$ , contrasted against the one- and two-loop $\\epsilon $ expansion.",
"In Fig.",
"REF we show the values for $\\eta $ in $d = 3$ , 4, and 5 dimensions for all truncations employed in this work, and our best estimates for the critical exponents $\\eta $ , $\\nu _c$ , and $\\sigma $ are reported in Tab.",
"REF .",
"These values were obtained with the $(7,5,0)$ truncation for which, in contrast to the $(5,3,2)$ truncation, the values of $\\eta $ seem to be reasonably close to their asymptotic values that are reached in the infinite $n_U$ and $n_Z$ limit [cf.",
"Fig.",
"REF ].",
"That is, we observe that larger orders of the finite field expansion are necessary to reach the asymptotic scaling exponents and it seems that this order increases for dimensions well below the upper critical dimension $d_c = 6$ , which is consistent with our previous observation on the validity of scaling relations.",
"In Tab.",
"REF and REF we account for a systematic bias due to our choice of the IR regulator (see Sec.",
"for an in depth discussion of this issue).",
"We remark that the difference in the values of the anomalous dimension between different high-order truncations is typically larger than that obtained for the critical exponents $\\sigma $ and $\\nu _c$ , which is reflected in the errors for these quantities (cf.",
"Tab.",
"REF ).",
"This effect has also been observed with other methods and may be attributed to the scaling relations (REF ) and (REF ) that yield a smaller error for the exponents $\\nu _c$ and $\\sigma $ (see, e.g., Ref.",
"[28]).",
"Comparing our estimates for the critical exponent $\\sigma $ to a recent compilation of available data on the Yang-Lee edge critical scaling exponents provided in Ref.",
"[28], cf.",
"Tab.",
"REF , we find that our values lie within the error bounds provided by other methods, e.g., Refs.",
"[29], [30], [31].",
"They lie slightly above the values obtained from constrained Padé approximants of three- and four-loop $\\epsilon $ expansion results [28], but are in general smaller than those values obtained from a recent conformal bootstrap analysis [32].",
"Considering the fact, that our numerical implementation of the RG flow equations is not overly sophisticated (limiting the truncations that can be considered to a relatively small number of operators) it is quite remarkable that our present results are competitive with other data in the literature."
],
[
"Relevance of composite operators and quality of finite truncations",
"We observe that certain truncations of the scale-dependent effective action, of the type $(n_U, 0, 0)$ , $n_U > 3$ , are inadequate to investigate the Yang-Lee scaling behavior.",
"In fact, for these truncations, the Yang-Lee fixed point is unstable below $d \\approx 5.6$ .We remark that this observation depends on the choice of the IR regulator.",
"While the Lee-Yang edge fixed point is unstable for the smooth exponential regulator (REF ) ($\\alpha = 1$ ), this is not the case for the optimized Litim regulator [53], [54].",
"However, the latter is not immediately applicable at higher orders in the derivative expansion.",
"This is certainly surprising and in conflict with other available data [29], [30], [31], [32], [28].",
"However, this behavior can be understood by examining the effect of operator insertions at the level of the one-loop $\\epsilon = 6-d$ expansion, as considered in Refs.",
"[55], [56].",
"In particular, we consider the renormalization of quartic operators at the Yang-Lee fixed point.",
"This requires the simultaneous renormalization of all operators that carry the same canonical dimension as $\\delta \\phi _k^4$ , which mix under renormalization [57].",
"In $d = 6 -\\epsilon $ dimensions these operators can be listed as follows (up to total derivative contributions) $A_{1,k} &=& \\delta \\phi _k^4 /4!",
", \\\\A_{2,k} &=& k^{\\epsilon /2} \\delta \\phi _k \\left( \\partial \\delta \\phi _k \\right)^2 / 2 , \\\\A_{3,k} &=& k^{\\epsilon } \\left( \\Box \\hspace{1.0pt} \\delta \\phi _k \\right)^2 / 2 .",
"$ Note that they simply correspond to particular contributions in the finite series expansion of $U_k(\\phi )$ , $Z_k(\\phi ) (\\partial \\phi )^2$ , and $W_k(\\phi ) \\left(\\Box \\phi \\right)^2$ , respectively, around the homogeneous field expectation value $\\bar{\\phi }_k$ .",
"Different truncations of the scale-dependent effective action are distinguished by either including or neglecting some of these operators, (REF ) – ().",
"The $(n_U, 0, 0)$ -type truncations, for instance do not include operators $A_{2,k}$ and $A_{3,k}$ , while truncations of the type $(n_U, n_Z, 0)$ do not include $A_{3,k}$ .",
"Treating the operators (REF ) – () on an equal footing, both $A_{2,k}$ and $A_{3,k}$ turn out to be more relevant in $d < 6$ dimensions than the quartic interaction $A_{1,k}$ .",
"Indeed, from a one-loop calculation [55], [56], we obtain the following eigenvalues of the stability matrix: $\\lambda _4 = -2$ , $\\lambda _5 = -2 - \\epsilon /9$ , and $\\lambda _6 = -2 - 19 \\epsilon / 9$ .",
"Each of them corresponds to a different linear combination of operators (REF ) – ().",
"One can show that the dominant contribution to $\\lambda _4$ comes from $A_{3,k}$ , for $\\lambda _5$ it is the operator $A_{2,k}$ , and for $\\lambda _6$ it is $A_{1,k}$ that contributes the most.",
"Thus, one might conclude that any truncation that includes only the quartic interaction $A_{1,k}$ is ill-defined, as it neglects the more relevant contributions, namely $A_{2,k}$ and $A_{3,k}$ .",
"Interestingly, it is sufficient to consider truncations of the type $(n_U, n_Z, 0)$ to stabilize the Yang-Lee edge fixed point.",
"While $(n_U, 0, 0)$ -type truncations, $n_U > 3$ , fail to produce a Yang-Lee edge fixed point below $d\\approx 5.6$ , the $(n_U, n_Z, 0)$ truncations allow us to identify the corresponding scaling solution all the way down to $d = 3$ [cf.",
"Fig.",
"REF ].",
"In general, we expect that the scale-dependent effective action needs to respect the properties of the theory under simultaneous renormalization of operators with the same canonical dimension.",
"This is important to define consistent truncations that are adequate to describe the Yang-Lee edge critical point."
],
[
"Residual regulator dependence and principle of minimal sensitivity",
"To determine the critical scaling properties of a given model, we may in principle choose any regulator function $R_k = R_k(q)$ as long as it satisfies the appropriate limiting behavior $\\lim _{k\\rightarrow 0} R_k = 0$ and $\\lim _{\\Lambda \\rightarrow \\infty } R_{k = \\Lambda } = \\infty $ .",
"Indeed, if an exact solution to the functional flow equation for $\\Gamma _k$ were available, the calculated observables should not depend on the way we choose to regularize the theory in the IR and therefore must be independent of the regulator.",
"However, in practice, we are bound to consider truncations of the coupled infinite set of flow equations.",
"This yields a finite set of RG equations for which one observes a residual regulator dependence [58].",
"To investigate this effect, we define a one-parameter family of functions $R_{\\alpha , k} = \\alpha R_k ,$ with $\\alpha > 0$ , and consider the $\\alpha $ dependence of the critical exponents.",
"We employ the following set of exponential regulators $R_{\\alpha , k}^{\\textrm {exp}} = \\frac{\\alpha \\bar{Z}_k q^2}{\\exp (q^2 / k^2) - 1} ,$ for this analysis.Note that the regulator should be sufficiently smooth in momentum space if higher order approximations in the derivative expansion are considered (see, e.g., Ref.",
"[44]).",
"One may identify an optimal value of $\\alpha $ , which is determined by the principle of minimum sensitivity [44].",
"It states that the value of any given observable that is least sensitive to changes in $\\alpha $ can be considered the best estimate for that quantity.",
"Since by virtue of scaling relations all critical exponents at the Yang-Lee edge critical point can be expressed in terms of the anomalous dimension $\\eta $ , we apply this criterion to $\\eta = \\eta (\\alpha )$ , i.e., to find the optimal value, we require that $\\eta ^{\\prime }(\\alpha = \\alpha _{\\textrm {opt}}) = 0 .$ In Tab.",
"REF we compare the values of $\\eta (\\alpha )$ evaluated for $\\alpha = 1$ as well as $\\alpha = \\alpha _{\\textrm {opt}}$ in different dimensions and determine the relative error $\\Delta \\eta / \\eta (\\alpha _{\\textrm {opt}}) \\equiv \\left[\\eta (1) - \\eta (\\alpha _{\\textrm {opt}})\\right] / \\eta (\\alpha _{\\textrm {opt}})$ .",
"Largely independent of dimension, the anomalous dimension evaluated at $\\alpha = 1$ seems to be slightly overestimated with a relative error of approximately $3\\%$ .",
"From this comparison we conclude that $\\eta (\\alpha = 1)$ is typically already a good approximation to the optimal value $\\eta (\\alpha _{\\textrm {opt}})$ .",
"Table: Anomalous dimension η=η(α)\\eta =\\eta (\\alpha ) at the Yang-Lee edge critical point in dd dimensions, evaluated for the (deformed) exponential regulator R α,k exp (q)=αZ ¯ k q 2 exp(q 2 /k 2 )-1 -1 R_{\\alpha , k}^{\\textrm {exp}}(q) = \\alpha \\bar{Z}_k q^2 \\left[\\exp (q^2 / k^2) -1 \\right]^{-1} with α>0\\alpha > 0.",
"The optimal value of α\\alpha depends on the dimension, i.e., α opt =α opt (d)\\alpha _{\\textrm {opt}} = \\alpha _{\\textrm {opt}}(d) [cf.",
"Fig.",
"].",
"The shown values were obtained using a truncation of the scale-dependent effective action Γ k \\Gamma _k defined by the index set (4,2,0)(4,2,0).Figure: Rescaled anomalous dimension η(α)/|η(α opt )|\\eta (\\alpha ) / |\\eta (\\alpha _{\\textrm {opt}})| shown as a function of α\\alpha .",
"Different curves correspond to data obtained in d=3d = 3, 4, and 5 dimensions, respectively.",
"The optimal value α opt \\alpha _{\\textrm {opt}} for which the critical exponent is least sensitive to changes in the deformation parameter, i.e., η ' (α=α opt )=0\\eta ^{\\prime }(\\alpha = \\alpha _{\\textrm {opt}}) = 0, shifts to larger values as the dimension dd is lowered.",
"The displayed values were obtained for a truncation of the scale-dependent effective action Γ k \\Gamma _k of the type (4,2,0)(4,2,0).In principle, $\\alpha _{\\textrm {opt}}$ might depend on the dimension.",
"Indeed, as shown in Fig.",
"REF , the optimal value of $\\alpha $ shifts to larger values when the dimension $d$ is lowered and eventually stabilizes around $\\alpha \\approx 1.7$ .",
"Although the value of $\\alpha _{\\textrm {opt}}$ increases, the relative error in $\\eta $ remains roughly constant.",
"At this point, we remark that below $d = 4$ an ambiguity appears: $\\eta (\\alpha )$ develops a second extremum, a local maximum, for $\\alpha < 1$ [cf.",
"Fig.",
"REF ].",
"However, we do not consider this solution to be physical and define $\\alpha _{\\textrm {opt}}(d)$ as the analytically continued local minimum from $d = 6 - \\epsilon $ .",
"Since the search for fixed points of the RG $\\beta $ functions becomes quite demanding numerically for higher-order truncations, we use this information to limit our calculations to the case $\\alpha = 1$ and estimate the corresponding systematic error in $\\eta (\\alpha = 1)$ at the $3 - 5\\%$ level (within the considered one-parameter family of regulators).",
"This systematic effect in the estimation of the anomalous dimension has been accounted for and is indicated explicitly as a systematic error in the summary of our results in Tab.",
"REF and REF ."
],
[
"Conclusions",
"In this work we have examined the critical scaling properties of the Yang-Lee edge, or $\\phi ^3$ , theory in dimensions $3 \\le d \\le 6$ .",
"We find our results in good agreement with available data in the literature, which includes high-temperature series expansions, results from the $\\epsilon $ expansion, strong coupling expansion, and Monte Carlo methods.",
"While our results are consistent with the strong coupling expansion and Monte Carlo methods Refs.",
"[29], [30], [31], our estimates for the critical exponent $\\sigma $ are slightly larger than the values obtained from constrained Padé approximants for three- and four-loop $\\epsilon $ expansion results [28], and generally lie below those from a conformal bootstrap analysis [32].",
"We expect that truncations at higher orders in the derivative and field expansion will improve our estimates for the critical exponents.",
"However, more elaborate numerical treatment is necessary to study such truncations.",
"We have shown that the stability of nontrivial fixed point associated to the Yang-Lee edge singularity is sensitive to the insertion of operators that mix under renormalization.",
"This might seem surprising since a similar behavior is not observed in applications of the functional RG to establish the scaling behavior at the Ising critical point.",
"However, comparing our results with a stability analysis at the fixed point to one-loop order in the $\\epsilon = 6-d$ expansion provides a qualitative explanation for the observed lack of stability of the Yang-Lee edge fixed point for $(n_U,0,0)$ -type truncations ($n_U > 3$ ) of the scale-dependent effective action.",
"Finally, we remark on possible applications of this work.",
"Based on mean-field arguments, one expects another thermodynamic singularity in the low-temperature phase of the Ising model ($T < T_c$ ) with exactly the same critical exponents as those of the Yang-Lee edge point – the spinodal singularity.",
"The corresponding critical point appears on the metastable branch of the free energy and is usually associated with the classical limit of metastability.",
"However, its existence (beyond mean-field) as well as its scaling properties have been subject to some debate [59], [60], [61], [62].",
"It would be interesting to understand the relation between the Yang-Lee edge point and the spinodal singularity [63], [64].",
"We intend to address these issues in a future publication.",
"We thank B. Delamotte for valuable insights and for helpful advice regarding the numerical implementation of the RG flow equations.",
"This research is funded by the European Research Council under the European Union’s Seventh Framework Programme FP7/2007-2013, 339220.",
"This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Number DE-FG0201ER41195."
]
] | 1605.06039 |
[
[
"Effect of Temperature and Doping on Plasmon Excitations for an\n Encapsulated Double-Layer Graphene Heterostructure"
],
[
"Abstract We perform a comprehensive analysis of the spectrum of graphene plasmons which arise when a pair of sheets are confined between conducting materials.",
"The associated enhanced local fields may be employed in the manipulation of light on the nanoscale by adjusting the separation between the graphene layers, the energy band gap as well as the concentration of charge carriers in the conducting media surrounding the two-dimensional (2D) layers.",
"We present a theoretical formalism, based on the calculation of the surface response function, for determining the plasmon spectrum of an encapsulated pair of 2D layers and apply it to graphene.",
"We solve the coupled equations involving the continuity of the electric potential and discontinuity of the electric field at the interfaces separating the constituents of the hybrid structure.",
"We have compared the plasmon modes for encapsulated gapped and gapless graphene.",
"The associated nonlocal graphene plasmon spectrum coupled to the \"sandwich\" system show a linear acoustic plasmon mode as well as a low-frequency mode corresponding to in-phase oscillations of the adjacent 2D charge densities.",
"These calculations are relevant to the study of energy transfer via plasmon excitations when graphene is confined by a pair of thick conducting materials."
],
[
"Introduction",
"Many researchers have been devoting a great deal of effort to exploit the unique transport and optical properties of graphene.",
"In particular, an area of much interest to both experimentalists and theoreticians has been the study of plasmon excitations under various conditions of temperature and doping concentrations.",
"There have been many recent works focused on the study of these plasmon modes in graphene when it is free standing[1], lying on a substrate[2], [3] or encapsulated by two conducting materials.",
"[4] In this paper, we investigate the way in which the plasmon mode excitations[5] for a pair of graphene layers are affected by encapsulating conductors which are coupled non locally to the two-dimensional (2D) layers.",
"The unusual properties of free-standing graphene may be attributed to Bloch states in the corners of the hexagonal Brillouin zone of this 2D honeycomb crystal lattice.",
"For example, the Dirac fermions arising from this energy band structure lead to strongly enhanced and confined local fields through dipole-dipole coupling [6].",
"But, recently, novel properties have been predicted when graphene electrically interacts with a nearby metallic substrate separated by a thin insulator [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].",
"The graphene-insulator-metal plasmons have exhibited both a linear dispersion mode [9], [20] (a so-called acoustic plasmon) in the terahertz (THz) spectral regime and a plasmon mode which is 2D in nature (square root of the wave vector).",
"Such plasmon excitations are important for improving THz sensing, signal processing on electro-optic modulation[21] and communication technologies [22], [23], [24].",
"Reportedly, graphene encapsulated in hexagonal boron-nitride (hBN) displays an anomalous Hall effect at room temperature which may be interpreted as ballistic transport on a micrometer scale over a wide range of carrier concentration.",
"The encapsulation makes graphene virtually protected from its surroundings but at the same time allows the use of hBN as a top gate dielectric.",
"More generally, the properties of encapsulated graphene are currently being actively investigated due to recent advances in device fabrication techniques [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [25], [26].",
"One reason for the attention being paid to these heterostructures is the observed improvements in the electrical conductance of graphene interconnection when there is complete encapsulation by boron-nitride [19], [27], [28], [29], [30], [31].",
"Another is that when graphene interacts with a substrate such as SiC, hBN, graphite[32], [33], [34], an energy band gap opens up and the plasmon dispersion relation is modified.",
"The combination of a 2D layer interacting with a substrate presents theoreticians with a challenge to model the structure as well as to formulate the problem and eventually obtain a dispersion equation for the plasmon excitations.",
"There already is a formulation for graphene plasmons on a substrate [2], [35], [36] as well as when encapsulated by a pair of thick conductors [4].",
"These works were geared to help explain the reported relevant experimental data [37], [38], [39], [40], [41].",
"Our present investigation showed that the plasmons of an encapsulated double-layer graphene heterostructure are different in dispersive nature from the case when there are two graphene layers interacting with a single substrate.",
"In this paper, we present a formalism which yields the surface response function for an encapsulated pair of 2D layers.",
"We take into account the presence of a dielectric medium separating the 2D layer from the substrate.",
"Furthermore, the top encapsulating material could have finite thickness as shown schematically in Fig.",
"REF .",
"In general, this method is capable of obtaining the plasmon dispersion relation for an arbitrary number of encapsulated layers embedded within a background dielectric material which ensures that there is no electron tunneling in between the layers.",
"In addition, our method can be extended to the case when a protective coating is placed over the top substrate.",
"The surface response function is calculated in the vicinity of the surface and our central formula yields various already published results.",
"[2], [42], [43] This response function, thereby obtained is used to determine the plasmon modes which are exhibited as density plots.",
"Our method includes the effect of dielectric media non locally coupled which was not investigated by Badalyan and Peeters [44] who treated the effect of the surrounding dielectric media locally.",
"For further clarification, we compare the resulting calculated plasmon spectra for encapsulated gapless and gapped graphene in the presence and absence of the background dielectric.",
"Besides this, we investigated the effect of temperature on the plasmon dispersion.",
"A similar effect can be calculated for transition metal dichalcogenide monolayers, modeling them as we do for gapped graphene[45], [46], [47], [48].",
"On the other hand, the structure which we have examined here may play an important role in increasing the mobility of electrons in graphene and other 2D structure to improve the performance of nanoelectronic and spintronic devices.",
"The plasmon mode excitation of the structure could also be used as an illumination source for imaging a sample[49] or in optoelectronic devices as a sensor.",
"We now outline the rest of this paper as follows.",
"In Sec.",
", we present a detailed description of our method for calculating the surface response function for a pair of 2D layers sandwiched between two conducting substrates with arbitrary separation.",
"The calculated plasmon spectra for chosen energy band gap, carrier doping values and temperature effect are presented in Sec. .",
"We conclude our paper with a discussion of the highlights of our calculations in Sec.",
"."
],
[
" Theoretical Formalism",
"Let us consider a structure as shown in Fig.",
"REF where a 2D layer is placed on top of a substrate of thickness $\\ell _1$ with frequency-dependent dielectric function $\\epsilon _1(\\omega )$ .",
"Below this, there is a buffer layer between $z=\\ell _1$ and $z=\\ell _4$ with background dielectric constant $\\epsilon _b$ and a pair of 2D layers are embedded at $z=\\ell _2$ and at $z=\\ell _3$ .",
"This entire structure is now placed over a thick conducting substrate of dielectric function $\\epsilon _2(\\omega )$ which extends from $z=\\ell _4$ to infinity.",
"For this structure we determine the surface response function which is used to calculate the plasmon frequencies.",
"The electrostatic potential in the vicinity of the surface for the heterostructure is written as [50] $\\phi _<(z)=e^{-q_{\\parallel }z}-g(q_{\\parallel },\\omega )e^{q_{\\parallel }z}\\ , z \\lessapprox 0$ which introduces the surface response function $g(q_{\\parallel },\\omega )$ which is the object of our calculation.",
"To proceed further, we express the potential solutions of Poisson's equation in the various regions depicted in Fig.",
"REF as linear combinations of $e^{\\pm q_{\\parallel }z}$ by making use of the translational invariance parallel to the $xy$ -planar interfaces.",
"Referring to Fig.",
"REF , we write the potential functions in the different regions as follows: $\\phi _>(z) &=& a_1e^{q_{\\parallel }z}+b_1e^{-q_{\\parallel }z}\\ , \\ \\ 0\\le z\\le \\ell _1\\nonumber \\\\\\phi _{1<}(z)&=& t_1e^{q_{\\parallel }z}+r_1e^{-q_{\\parallel }z}\\ , \\ \\ \\ell _1\\le z\\le \\ell _2\\nonumber \\\\\\phi _{1>}(z) &=& t_2e^{q_{\\parallel }z}+r_2e^{-q_{\\parallel }z} \\ , \\ \\ \\ell _2\\le z\\le \\ell _3\\nonumber \\\\\\phi _{2<}(z) &=& f_1e^{q_{\\parallel }z}+k_1e^{-q_{\\parallel }z} \\ , \\ \\ \\ell _3\\le z\\le \\ell _4\\nonumber \\\\\\phi _{2>}(z) &=& k_2e^{-q_{\\parallel }z} \\ , \\ \\ z\\ge \\ell _4$ where $a_1$ , $b_1$ , $t_1$ , $r_1$ , $t_2$ , $r_2$ , $f_1$ , $k_1$ and $k_2$ are independent of $z$ .",
"These are readily determined by using the continuity of the potential and the discontinuity of the electric field across the interfaces.",
"We assume that the 2D layer at $z=0$ has induced surface charge density $\\sigma _1$ and that at $z=\\ell _2$ and $z=\\ell _3$ has charge density $\\sigma _2$ and $\\sigma _3$ , respectively.",
"Also, using linear response theory, we have $\\sigma _1=\\chi _1 \\phi _<(0)$ , $\\sigma _2=\\chi _2\\phi _{1<}(\\ell _2)$ and $\\sigma _3=\\chi _3\\phi _{2<}(\\ell _3)$ , where $\\chi _1$ , $\\chi _2$ and $\\chi _3$ are 2D susceptibilities.",
"After some algebra, we can obtain the surface response function for the structure shown in Fig.",
"REF with the cap 2D layer.",
"However, here we are concerned with a heterostructure without the protective layer at $z=0$ since this simplifies our calculation and which is reasonable to neglect if $\\ell _1$ is large.",
"For this, we set $\\sigma _1=0$ , $\\ell _1=d_1$ , $\\ell _2=d_1+d_2$ , $\\ell _3=d_1+d_2+d_3$ and $ \\ell _4=d_1+d_2+d_3+d_4$ .",
"In this notation, we obtain $g(q_\\parallel ,\\omega )$ for two encapsulated sheets.",
"Setting $d_1$ , $d_2=d/2$ , $d_3=d$ , $d_4=d/2$ , we have $g(q_{\\parallel },\\omega )=\\frac{{\\cal N}(q_{\\parallel },\\omega )}{{\\cal D}(q_{\\parallel },\\omega )} \\ .$ Here, ${\\cal N}(q_{\\parallel },\\omega )&&=2q_{\\parallel }^2\\epsilon _0^2\\epsilon _b^2\\left[\\sinh (q_{\\parallel }d_1)N_1+\\cosh (q_{\\parallel }d_1)\\epsilon _1N_2\\right]\\nonumber \\\\&&-2\\sinh (q_{\\parallel }d)N_3N_4\\chi _2\\chi _3+q_{\\parallel }\\epsilon _0\\epsilon _b\\left[\\sinh (q_{\\parallel }d)N_5\\epsilon _b(\\chi _2-\\chi _3)-\\sinh (2q_{\\parallel }d)N_6\\epsilon _b(\\chi _2+\\chi _3)\\right.\\nonumber \\\\&&+\\left.\\cosh (2q_{\\parallel }d)N_7(\\chi _2+\\chi _3)+\\cosh (q_{\\parallel }d)N_8(\\chi _2+\\chi _3)\\right]$ with $\\epsilon _0$ denoting the permittivity of free space, for convenience we suppress the frequency-dependence of some quantities, $N_1=\\sinh (2 q_{\\parallel }d) \\left(\\text{$\\epsilon $}_1^2 \\text{$\\epsilon $}_2-\\text{$\\epsilon $}_b^2\\right)+\\text{$\\epsilon $}_b \\left(\\text{$\\epsilon $}_1^2-\\text{$\\epsilon $}_2\\right) \\cosh (2 q_{\\parallel }d),$ $N_2=\\left(\\text{$\\epsilon $}_b^2-\\text{$\\epsilon $}_2\\right) \\sinh (2 q_{\\parallel }d)+(\\text{$\\epsilon $}_2-1) \\text{$\\epsilon $}_b \\cosh (2 q_{\\parallel }d),$ $N_3=\\text{$\\epsilon $}_2 \\sinh \\left(\\frac{q_{\\parallel }d}{2}\\right)+\\text{$\\epsilon $}_b \\cosh \\left(\\frac{q_{\\parallel }d}{2}\\right),$ $N_4=\\text{$\\epsilon $}_b \\cosh \\left(\\frac{q_{\\parallel }d}{2}\\right) \\left[\\sinh (q_{\\parallel }d_1)-\\text{$\\epsilon $}_1 \\cosh (q_{\\parallel }d_1)\\right]+\\text{$\\epsilon $}_1 \\sinh \\left(\\frac{q_{\\parallel }d}{2}\\right) \\left[\\cosh (q_{\\parallel }d_1)-\\text{$\\epsilon $}_1 \\sinh (q_{\\parallel }d_1)\\right],$ $N_5=\\left(\\text{$\\epsilon $}_1^2+\\text{$\\epsilon $}_2\\right) \\sinh (q_{\\parallel }d_1)-\\text{$\\epsilon $}_1 (\\text{$\\epsilon $}_2+1) \\cosh (q_{\\parallel }d_1),$ $N_6=\\left.\\left(\\text{$\\epsilon $}_1^2-\\text{$\\epsilon $}_2\\right) \\sinh (q_{\\parallel }d_1)+\\text{$\\epsilon $}_1 (\\text{$\\epsilon $}_2-1) \\cosh (q_{\\parallel }d_1)\\right),$ $N_7=\\sinh (q_{\\parallel }d_1) \\left(\\text{$\\epsilon $}_b^2-\\text{$\\epsilon $}_1^2 \\text{$\\epsilon $}_2\\right)+\\text{$\\epsilon $}_1 \\left(\\text{$\\epsilon $}_2-\\text{$\\epsilon $}_b^2\\right) \\cosh (q_{\\parallel }d_1),$ $N_8=\\sinh (q_{\\parallel }d_1) \\left(\\text{$\\epsilon $}_1^2 \\text{$\\epsilon $}_2+\\text{$\\epsilon $}_b^2\\right)-\\text{$\\epsilon $}_1 \\left(\\text{$\\epsilon $}_2+\\text{$\\epsilon $}_b^2\\right) \\cosh (q_{\\parallel }d_1),$ and ${\\cal D}(q_{\\parallel },\\omega )&&=2q_\\parallel ^2\\epsilon _0^2\\epsilon _b^2\\left[\\cosh (q_{\\parallel }d_1)\\epsilon _1D_1+\\sinh (q_{\\parallel }d_1)D_2\\right]\\nonumber \\\\&&+2\\sinh (q_{\\parallel }d)D_3D_4\\chi _2\\chi _3-q_{\\parallel }\\epsilon _0\\epsilon _b\\left[\\sinh (q_{\\parallel }d)D_5\\epsilon _b(\\chi _2-\\chi _3)+\\sinh (2q_{\\parallel }d)D_6\\epsilon _b(\\chi _2+\\chi _3)\\right.\\nonumber \\\\&&-\\left.\\cosh (q_{\\parallel }d)D_7(\\chi _2+\\chi _3)+\\cosh (2q_{\\parallel }d)D_8(\\chi _2+\\chi _3)\\right]$ with $D_1=\\left(\\text{$\\epsilon $}_2+\\text{$\\epsilon $}_b^2\\right) \\sinh (2q_{\\parallel }d)+(\\text{$\\epsilon $}_2+1) \\text{$\\epsilon $}_b \\cosh (2 q_{\\parallel }d),$ $D_2=\\sinh (2 q_{\\parallel }d) \\left(\\text{$\\epsilon $}_1^2 \\text{$\\epsilon $}_2+\\text{$\\epsilon $}_b^2\\right)+\\text{$\\epsilon $}_b \\left(\\text{$\\epsilon $}_1^2+\\text{$\\epsilon $}_2\\right) \\cosh (2 q_{\\parallel }d),$ $D_3=\\text{$\\epsilon $}_2 \\sinh \\left(\\frac{q_{\\parallel }d}{2}\\right)+\\text{$\\epsilon $}_b \\cosh \\left(\\frac{q_{\\parallel }d}{2}\\right),$ $D_4=\\text{$\\epsilon $}_b \\cosh \\left(\\frac{q_{\\parallel }d}{2}\\right) \\left[\\text{$\\epsilon $}_1 \\cosh (q_{\\parallel }d_1)+\\sinh (q_{\\parallel }d_1)\\right]+\\text{$\\epsilon $}_1 \\sinh \\left(\\frac{q_{\\parallel }d}{2}\\right) \\left[\\text{$\\epsilon $}_1 \\sinh (q_{\\parallel }d_1)+\\cosh (q_{\\parallel }d_1)\\right],$ $D_5=\\left(\\text{$\\epsilon $}_2-\\text{$\\epsilon $}_1^2\\right) \\sinh (q_{\\parallel }d_1)+\\text{$\\epsilon $}_1 (\\text{$\\epsilon $}_2-1) \\cosh (q_{\\parallel }d_1),$ $D_6=\\left(\\text{$\\epsilon $}_1^2+\\text{$\\epsilon $}_2\\right) \\sinh (q_{\\parallel }d_1)+\\text{$\\epsilon $}_1 (\\text{$\\epsilon $}_2+1) \\cosh (q_{\\parallel }d_1),$ $D_7=\\sinh (q_{\\parallel }d_1) \\left(\\text{$\\epsilon $}_1^2 \\text{$\\epsilon $}_2-\\text{$\\epsilon $}_b^2\\right)+\\text{$\\epsilon $}_1 \\left(\\text{$\\epsilon $}_2-\\text{$\\epsilon $}_b^2\\right) \\cosh (q_{\\parallel }d_1),$ $D_8=\\sinh (q_{\\parallel }d_1) \\left(\\text{$\\epsilon $}_1^2 \\text{$\\epsilon $}_2+\\text{$\\epsilon $}_b^2\\right)+\\text{$\\epsilon $}_1 \\left(\\text{$\\epsilon $}_2+\\text{$\\epsilon $}_b^2\\right) \\cosh (q_{\\parallel }d_1),$ where, $\\epsilon _1(\\omega )=\\epsilon _2(\\omega )=1- \\omega _p^2/\\omega ^2$ and $\\epsilon _b$ is constant.",
"It is of interest to note that Persson[42] calculated the surface response function for a 2D sheet lying on top of a substrate with dielectric function $\\epsilon _2(\\omega )$ with vacuum on the other side as $g_{\\text{P}}(q_{\\parallel },\\omega )=1-\\frac{2}{ 1+\\epsilon _2-\\frac{\\chi _3}{q_{\\parallel }\\epsilon _0}}$ which is readily recovered from our general result in Eq.",
"( REF ) by replacing $d_1=0$ , $d=0$ with $\\chi _2=0$ .",
"Similarly, Hwang and Das Sarma [43] obtained the plasmon dispersion equation for a pair of free standing 2D layers with a chosen separation between them.",
"When we make the substitution $\\epsilon _1(\\omega )=1$ , $\\epsilon _2(\\omega )=1$ and $\\epsilon _b=1$ in Eq.",
"(REF ), we obtain $g_{\\text{free-standing}}(q_{\\parallel },\\omega )=\\frac{e^{ -q_{\\parallel }\\left(3d+2 d_1\\right)}\\left[\\chi _2 \\chi _3 \\left(e^{2 q_{\\parallel }d }-1\\right)-2 q_{\\parallel } \\epsilon _0\\left(\\chi _2 e^{2 q_{\\parallel }d}+\\chi _3\\right)\\right]}{4 q_{\\parallel }^2 \\epsilon _0^2 \\left[\\big (1-\\frac{\\chi _2}{2 q_{\\parallel }\\epsilon _0}\\big )\\big (1-\\frac{\\chi _3}{2 q_{\\parallel } \\epsilon _0}\\big )-\\frac{\\chi _2}{2 q_{\\parallel } \\epsilon _0}\\frac{\\chi _3}{2 q_{\\parallel } \\epsilon _0}e^{-2 q_{\\parallel } d}\\right]}\\ .$ The zeros of the denominator in Eq.",
"( REF ) correspond to the plasmon poles and the resulting dispersion equation agrees with that in Ref.",
"[Dassarma].",
"The dispersion equation for plasmon excitations when a 2D layer is located at some distance from the surface of a thick conducting substrate, as considered by Gumbs, et al.",
"[2] can be successfully deduced from Eq.",
"(REF ) by substituting $d_1=0$ , $\\chi _3=0$ , $\\epsilon _b=1$ and $\\epsilon _2=\\epsilon $ .",
"The surface response function for this case is $g_{\\text{2D-substrate}}(q_{\\parallel },\\omega )=\\frac{\\left(\\epsilon -1\\right)\\left(2 q_{\\parallel } \\epsilon _0+\\chi _2\\right)-\\chi _2 \\left(\\epsilon +1\\right)e^{3 q_{\\parallel }d}}{2q_{\\parallel }\\epsilon _0(\\epsilon +1)e^{4 q_{\\parallel }d}\\left[1-\\frac{\\chi _2}{2 q_{\\parallel } \\epsilon _0}\\big (1+\\frac{1-\\epsilon }{1+\\epsilon }e^{-2 q_{\\parallel }\\frac{3d}{2}}\\big )\\right]}$ from which the dispersion equation is obtained by setting the factor in parenthesis in the denominator of this equation equal to zero and this agrees with the result in Ref. [GG].",
"We emphasize that in this special case, the distance from the conducting substrate to the 2D layer is $\\ 3d/2$ .",
"Next, we turn to the situation when there is a single 2D layer which is sandwiched between a conducting material of finite thickness on one side and by a semi-infinite conductor on the other side.",
"The surface response function of this structure can be deduced from Eq.",
"(REF ) by setting $\\chi _2=0$ and $d=0$ .",
"Then the surface response function obtained is as follows: $g_{\\text{single-layer}}(q_{\\parallel },\\omega )=1-\\frac{2\\left[\\big (\\epsilon _1+\\epsilon _2-\\frac{\\chi _3}{q_{\\parallel }\\epsilon _0}\\big )+\\big (\\epsilon _1-\\epsilon _2+\\frac{\\chi _3}{q_{\\parallel }\\epsilon _0}\\big )e^{-2q_{\\parallel }d_1}\\right]}{\\left[(1+\\epsilon _1)(\\epsilon _1+\\epsilon _2-\\frac{\\chi _3}{q_{\\parallel }\\epsilon _0})-(\\epsilon _1-1)(\\epsilon _1-\\epsilon _2+\\frac{\\chi _3}{q_{\\parallel }\\epsilon _0})e^{-2q_{\\parallel }d_1}\\right]}\\ .$ Equating the denominator of the second term to zero, we obtain the plasmon dispersion equation for one encapsulated layer with susceptibility $\\chi _3$ .",
"Furthermore, if one sets $\\chi _2$ and $\\chi _3$ equal to zero in Eqs.",
"(REF ) and (REF ), i.e., all the 2D layers in Fig.",
"REF are removed, we obtain an expression for the surface response function of a thick slab consisting of regions with different dielectric materials.",
"In particular, one may then also set $\\epsilon _2=1$ , $d=0$ , $d_1=L$ and $\\epsilon _1=\\epsilon $ to obtain the surface response function for a film of thickness $L$ given by $g_{{\\mbox{f}ilm}}(q_{\\parallel },\\omega )=2\\frac{g_\\infty (q_{\\parallel },\\omega )}{e^{q_\\parallel L}-g_\\infty ^2(q_{\\parallel },\\omega )e^{-q_\\parallel L}}\\sinh (q_\\parallel L)$ where $g_\\infty (q_{\\parallel },\\omega )=(\\epsilon -1)/(\\epsilon +1)$ The preceding calculations clearly confirm that our result in Eq.",
"(REF ) is very useful and can be employed for a variety of structures.",
"As we pointed out above, we may extend our calculations to include the influence of a protective top layer, as schematically demonstrated in Fig.",
"REF .",
"The plasmon frequency of an encapsulating substrate such as $\\text{Bi}_2\\text{Se}_3$[51] is calculated using $\\omega _p=\\sqrt{ n_{3D}e^2/(\\epsilon _b\\epsilon _0m^\\ast )}$ =270 meV for $n_{3D} \\approx 8.2 \\times 10^{18}\\text{cm}^{-3}$ and $m^\\ast =0.15 m_e$ representing the charge density of the conducting substrate and the effective mass of an electron given in terms of $m_e$ , the free electron rest mass, respectively.",
"Consequently, the background dielectric constant of a conductor can vary significantly from $\\epsilon _b=200-500$ for barium or strontium titanate, to many orders of magnitude greater for clean, copper-based metals and the plasmon frequency may vary over a wide range up to $10^{12}$ Hz, which corresponds to a few meV.",
"On the other hand, the zero temperature doping(Fermi energy) in graphene could be estimated as $E_F=\\hbar v_fk_F=\\hbar v_f \\sqrt{\\pi n_{2D}}=0.04$ eV [52] for two dimensional charge density, $n_{2D}=10^{15}\\text{m}^{-2}$ and Fermi wave vector, $k_F^{\\Delta }=\\sqrt{\\mu ^2-\\Delta ^2}/\\hbar v_F$ with $v_F$ , $\\mu $ and $\\Delta $ as Fermi velocity, chemical potential and half band gap respectively.",
"Thus, both quantities are of same order of magnitude, and using $\\mu $ as the unit for frequency is reasonable.",
"Now, in order to determine the plasmon modes for a pair of encapusulated 2D graphene layers numerically, the denominator, ${\\cal D}(q_{\\parallel },\\omega ,T)$ in Eq.",
"(REF ) is equated to zero with $\\chi _2=\\chi _3=-e^2\\Pi ^{(0)}_{2D}(q_{\\parallel },\\omega ,T)$ , where $\\Pi ^{(0)}_{2D}(q_{\\parallel },\\omega ,T)$ is the polarization function of graphene with $\\Pi ^{(0)}_{2D}(q_\\parallel ,\\omega ,T) = \\frac{1}{2 \\pi ^2}\\int d^2 {\\bf k} \\sum \\limits _{s, s^{\\prime } = \\pm 1}\\left\\lbrace 1+ s s^{\\prime }\\frac{ \\hbar ^2v_F^2({ \\bf k}+ { \\bf q})\\cdot { \\bf k}+\\Delta ^2}{ E_k\\,E_{|{\\bf k+q}|}} \\right\\rbrace \\frac{f_0(s E_{k}-\\mu ,T)-f_0(s^{\\prime } E_{|{\\bf k+q}|}-\\mu ,T)}{s E_k - s^{\\prime } E_{|{\\bf k+q}|}-\\hbar (\\omega + i 0^+)} \\ ,$ where, $f_0(s E_{k}-\\mu ,T)=(1+e^{(sE_k-\\mu )/k_BT})^{-1}$ is the Fermi-Dirac distribution function for subband energy, $s E_{k}=s\\sqrt{(\\hbar v_F k)^2+\\Delta ^2}$ with $s=\\pm 1$ and $T$ is temperature of the system.",
"To investigate the effect of encapsulation on gapless and gapped graphene, we employ the polarization function of Wunsch[1] and Pyatkovskiy[53] respectively whereas to see the effect of temperature on the plasmon modes, we make use of the results given in Ref.",
"[Ramezanali]."
],
[
"Numerical Results and Discussion",
"In Figs.",
"REF and REF , we present our numerical results for nonlocal plasmon excitations of a pair of gapless and gapped graphene layers encapsulated by dielectric materials above and below them.",
"On one side, the dielectric has finite thickness and on the other side, there is a semi-infinite material.",
"Figure REF portrays the plasmon modes for encapsulated double layer gapless graphene when the double layer system is in vacuum(left panel) and in background dielectric with $\\epsilon _b=2.4$ (right panel).",
"In total, four plasmon excitation modes are obtained, two of which originating from the origin are referred as the acoustic plasmon (AP), which is linear in the wave vector at long wavelengths and has the lowest frequency, and a mode lying above it which is 2D-like that we refer to as the optical plasmon (OP).",
"The other two plasmon modes originating from the bulk plasma frequency $\\omega _p$ and the surface plasmon frequency $\\omega _p/\\sqrt{2}$ are labeled as the upper hybrid plasmon (UHP) and the lower hybrid plasmon (LHP), respectively.",
"All plasmon modes survive in the longer wavelength region and get Landau damped as the mode enters the particle-hole mode region at shorter wavelengths.",
"Comparison of the results on the left-hand side of Fig.",
"REF with those on the right shows that due to embedding of the 2D layers in the dielcectric background the UHP mode flattens out and becomes less dispersive.",
"In fact, the AP mode is noticeably closer to the particle hole mode region and the range of wave vector for which this not Landau damped is reduced.",
"Besides this, the embedding of graphene layers in a dielectric background leads to the separation of the AP and OP mode in the shorter wavelength region where they were merged when there was no background dielectric screening.",
"Figure REF presents numerical results for encapsulated double-layer gapped graphene.",
"The results in the left panel of these plots are for a double-layer gapped graphene system lying in vacuum and the right panel figures are for cases when the layers are embedded in a background dielectric with constant $\\epsilon _b$ .",
"As in the case of gapless graphene, when gapped graphene layers are embedded in a dielectric, we observe four plasmon modes for which an AP and OP mode originate from the origin and the other two hybrid plasmon modes, the LHP and UHP stem from the surface and bulk plasma frequencies of the substrate.",
"A corresponding panel comparison of Fig.",
"REF with Fig.",
"REF clearly shows that due to the presence of the energy band gap, the particle-hole mode excitation region splits into two parts.",
"The reason is due to an increase in the energy required for an electron to transfer form the valence to the conduction band.",
"When the band gap is small of range $\\Delta =0.2\\hbar \\omega _p$ , the splitting is not large enough for the plasmon modes to enter.",
"However, as the band gap is increased, the splitting widens and can allow the plasmon modes originating from the bulk and surface plasma frequencies of the substrate not to undergo any Landau damping over a wide range of wave numbers.",
"As the band gap continues to be increased, the two hybrid modes merge into one for shorter wavelengths.",
"Beside this, we observe that when the 2D layers are embedded in a background dielectric material (right-hand side panels in the figures) the oscillator strengths of all modes become smaller and the AP mode lies closer to the particle-hole excitation region where it is Landau damped.",
"The AP and OP modes which are degenerate in the shorter wave length region now become non-coincident when the pair of graphene layers is embedded in the dielectric background.",
"Our numerical calculations for the effect of temperature on the heterostructure, displayed in Fig.",
"REF , encapsulating gapless graphene reveals that the behavior of the plasmon spectra may be affected non uniformly with respect to wave vector and frequency even at a chosen temperature.",
"At zero temperature, the Fermi energy is equal to $0.04 eV$ , the corresponding Fermi temperature is $T_F \\approx 450$ K and the Fermi wave vector is $k_F \\approx 10^7m^{-1}$ .",
"We may choose the plasma frequency $\\omega _p \\approx 10^{13} Hz$ so that we could take $T=0.5 T_F$ in our calculations.",
"At this temperature, the electron is thermally excited and the transition between the valence band and conduction band occurs continuously causing the boundary of the single-particle excitations to smear out which is a distinct difference from that seen in Figs.",
"REF and REF as an effect due to temperature.",
"This effect due to smearing resulted in the removal of sharp boundary of the particle hole excitation region in wave vector-frequency space for which there is none at absolute zero.",
"Due to this, the two hybrid modes originating from the vicinity of $\\omega _p$ seem to be affected the most.",
"The UHP mode dies off soon after it emerges with the bulk frequency in the long wavelength regime whereas the LHP mode survives over a wider range of wave vector and decays as $q_\\parallel $ is increased.",
"The AP and OP modes survive for longer wavelength and decay in the shorter wavelength region.",
"Also, another distinct behavior observed, when the double layer of graphene is embedded in a dielectric material is that the AP mode moves closer to the particle-hole region and becomes decayed at longer wavelengths.",
"In addition, the AP mode and OP mode become non-degenerate in the shorter wavelength regime."
],
[
" Concluding Remarks",
"We have presented a formalism for calculating the surface response function for a layered 2D structure which is sandwiched by conducting materials.",
"The 2D layers may be separated by dielectric materials with arbitrary thickness.",
"Our formulation uses a transfer matrix method for explicitly calculating the electrostatic potentials and electric fields along with linear response theory for the induced charge density on the 2D layers.",
"We could include the effect on the surface response function due to a 2D layer on the surface of the hybrid structure as illustrated in Fig.",
"REF .",
"However, the presence of this 2D layer was neglected in our calculations.",
"For comparison, we have presented numerical results when the encapsulated 2D layers are not embedded in a dielectric with that when these layers have a dielectric material on either side.",
"Our results are shown in Figs.",
"REF and REF for gapless and gapped graphene, respectively.",
"The effect due to temperature on these self-sustaining modes appears in our presented plots in Fig.",
"REF .",
"These plasmons may play an important role in fundamental studies involving strong light-matter interactions on the nanoscale.",
"Furthermore, the existence of a branch with acoustic dispersion could offer many-fold novel possibilities for the development of devices for detector, sensor and communication applications in the technologically important THz range, such as nanoscale waveguides or modulators.",
"We have demonstrated that we may tune the plasma frequency of a double layer graphene heterostructure by adjusting the doping concentrations.",
"Our model calculations show that these devices have potential for high-frequency operation and large-scale integration.",
"We only consider high doping concentrations so that the Fermi level is far away from the Dirac K point.",
"Otherwise, localization effects on the charge carriers cannot be neglected.",
"As a matter of fact, the broken electron-hole symmetry may be attributed to the mutual polarization of the closely spaced interacting layers and impurity scattering.",
"The encapsulated double-layer graphene plasmons we are predicting possess strong field confinement and very low damping.",
"This enables a new class of devices for tunable subwavelength heterostructures, strong light-matter interactions and nano-optoelectronic switches.",
"Although all of these prospects require low plasmon damping, our model calculations show that this may be achieved if the Fermi level is not too low so that impurity scattering may be neglected.",
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]
] | 1605.06070 |
[
[
"A new method to probe the thermal electron content of the Galaxy through\n spectral analysis of background sources"
],
[
"Abstract We present a new method for probing the thermal electron content of the Galaxy by spectral analysis of background point sources in the absorption-only limit to the radiative transfer equation.",
"In this limit, calculating the spectral index, $\\alpha$, of these sources using a natural logarithm results in an additive factor, which we denote $\\alpha_\\mathrm{EM}$, resulting from the absorption of radiation due to the Galactic thermal electron population.",
"We find that this effect is important at very low frequencies ($\\nu\\lesssim200$ MHz), and that the frequency spacing is critical.",
"We model this effect by calculating the emission measure across the sky.",
"A (smooth) thermal electron model for the Galaxy does not fit the observed emission measure distribution, but a simple, cloud-based model to represent the clumpy nature of the warm interstellar medium does.",
"This model statistically reproduces the Galactic emission measure distribution as obtained independently from $H_\\alpha$ data well.",
"We find that at the lowest frequencies ($\\sim10-50$ MHz), the observed spectral index for a large segment of the Galaxy below Galactic latitudes of $\\lesssim15^\\circ$ could be changed significantly (i.e., $\\alpha_\\mathrm{EM}\\gtrsim0.1$).",
"This method therefore provides a correction to low-frequency spectral index measurements of extragalactic sources, and provides a sensitive probe of the thermal electron distribution of the Galaxy using current and next-generation low-frequency radio telescopes.",
"We show that this effect should be robustly detectable individually in the strongest sources, and statistically in source samples at a level of $\\alpha_\\mathrm{EM}\\gtrsim0.18,0.06$, and 0.02 for source densities of 10, 100 and 1,000 sources per square degree."
],
[
"Introduction",
"A knowledge of the interactions between the phases that make up the interstellar medium (ISM) is fundamental in our attempt to understand the structure and evolution of our Galaxy.",
"The importance of thermal electrons, denoted $n_e$ , in the determination of the Galactic magnetic field is twofold.",
"Firstly, it is crucial for the determination of distances to pulsars through the dispersion measure, DM $\\equiv \\int n_eds$ in units of pc cm$^{-3}$ [15].",
"Secondly, it is required for an accurate determination of the strength and structure of the Galactic magnetic field, as $n_e$ enters into the determination of rotation measure (which is defined as RM $\\equiv 0.81\\int n_e\\mathbf {B}\\cdot d\\mathbf {s}$ rad m$^{-2}$ , with $n_e$ in cm$^{-3}$ , $\\mathbf {B}$ in micro-Gauss, and $d\\mathbf {s}$ in parsecs).",
"Thus the RM-grid method of exploring the magnetic field of our Galaxy (e.g., [4]), as well as the determination of small-scale structure through the RM synthesis technique ([3]) depend on a good knowledge of $n_e$ .",
"Although the sign of RM exclusively depends on the direction of the line-of-sight magnetic field, the electron density is needed for estimates of the magnitude of these magnetic fields.",
"In this paper, we explore the relationship between the spectral index of background point sources and the thermal electron content of the Galaxy.",
"Specifically, we show that using a natural logarithm in the calculation of the spectral index, assuming absorption-only, results in a additive factor to the spectral index that is proportional to the emission measure (EM).",
"This is equivalent to a multiplicative factor to flux.",
"We show that this parameter becomes significant at the lowest radio frequencies, such that so-called “in-band” spectral analysis at the lowest frequencies for the next-generation low-frequency radio telescopes such as the Low Frequency ARray (LOFAR) and the Murchison Widefield Array (MWA), might be severely affected by this phenomenon.",
"This method can thus also be used to obtain an estimate of the thermal electron content of the Galaxy.",
"Furthermore, we show that a simple smooth model for $n_e$ , as obtained from pulsar measurements (e.g, the modified Taylor-Cordes model; [24] after [25]), is inadequate to reproduce the distribution in EM as obtained from H$\\alpha $ observations [11].",
"We show that a simple “clumpy” model, based on that for pulsar DMs [17] can re-produce the observed EM distribution, as well as render the effect derived here an even more important factor in low-frequency radio observations.",
"Finally, we show that typical source densities as obtained with current and future low-frequency radio telescopes such as the LOFAR, MWA and eventually the Square Kilometre Array (SKA) imply that this effect should be statistically detected, even at high Galactic latitudes.",
"In addition, detection in individual sources should be possible for the strongest sources with currently planned or ongoing low-frequency surveys."
],
[
"Spectral Curvature at Low Frequencies",
"The solution to the equation of radiative transfer assuming absorption-only relates the intrinsic flux density, $S^{\\rm int}_\\nu $ , at frequency $\\nu $ to the observed flux density, $S^{\\rm obs}_\\nu $ , modified by an exponential term representing the optical depth, $\\tau _\\nu $ , of the photon (e.g., [21]): $S^{\\rm obs}_\\nu = S^{\\rm int}_\\nu e^{-\\tau _\\nu }.$ A value of $\\tau _\\nu \\gtrsim 1$ will result in the absorption of the intrinsic flux density, which may alter the observed flux density considerably.",
"Defining the spectral index using natural logarithms (instead of the canonical $\\log _{10}$ definition) leverages the information added by the optical depth term appearing in Equation REF in as simple a manner as possible: $\\alpha _{\\rm tot} & \\equiv & d\\ln (S^{\\rm obs}_{\\nu })/d\\ln (\\nu ) \\\\& \\approx & \\frac{\\ln ([S^{\\rm int}_{\\nu _1}e^{-\\tau _{\\nu _1}}]/[S^{\\rm int}_{\\nu _2}e^{-\\tau _{\\nu _2}}])}{\\ln (\\nu _1/\\nu _2)} \\nonumber \\\\& = & \\frac{\\ln (S^{\\rm int}_{\\nu _1}/S^{\\rm int}_{\\nu _2})}{\\ln (\\nu _1/\\nu _2)}+\\frac{(\\tau _{\\nu _2}-\\tau _{\\nu _1})}{\\ln (\\nu _1/\\nu _2)} \\\\& \\equiv &\\alpha _{\\rm int} + \\mbox{$\\alpha _{\\rm EM}$}\\nonumber ,$ where $\\nu _{1,2}$ refers to the frequencies at which the spectral index is calculated and $\\alpha _{\\rm int}$ is the spectral index intrinsic to the source.",
"We denote $\\mbox{$\\alpha _{\\rm EM}$}$ , which we defined above, as the “spectral thermal absorption parameter”, or STAP.",
"The optical depth is the integral over the line-of-sight $ds$ of the opacity $\\kappa _\\nu $ , which is a function of the square of the electron density (e.g.",
"[18]): $\\frac{\\kappa _\\nu }{\\textrm {cm}^{-1}}=1.78\\times 10^{-20} g^\\nu _{\\rm ff}\\left(\\frac{n_e}{\\textrm {cm}^{-3}}\\right)^2 \\left(\\frac{\\nu }{\\textrm {GHz}}\\right)^{-2}\\left(\\frac{T_e}{\\textrm {K}}\\right)^{-3/2},$ where $g_{\\rm ff}^\\nu $ is the Gaunt factor and is of order unity: $g^\\nu _{\\rm ff} = 10.6 + 1.9\\log \\left(\\frac{T_e}{\\textrm {K}}\\right) - 11.34\\log \\left(\\frac{\\nu }{\\textrm {GHz}}\\right).$ Hence, one can obtain a relation between the optical depth and emission measure EM $\\equiv \\int n_e^2ds$ pc cm$^{-6}$ ): $\\tau _\\nu = 0.055 \\overline{g_{\\rm ff}^\\nu } \\left( \\frac{\\nu }{\\textrm {GHz}}\\right)^{-2}\\left(\\frac{T_e}{\\textrm {K}}\\right)^{-3/2} \\left( \\frac{\\textrm {EM}}{\\textrm {pc cm}^{-6}} \\right),$ where $\\overline{g_{\\rm ff}^\\nu }$ is the average Gaunt factor.",
"Thus STAP is a function of EM.",
"Putting this all together, one obtains a simple relation for how the spectral index changes, $\\mbox{$\\alpha _{\\rm EM}$}$ , as a function of observing frequencies and thermal electron content along a line-of-sight: $\\mbox{$\\alpha _{\\rm EM}$}=5.5\\times 10^{-2} \\left(\\frac{\\textrm {EM}}{\\textrm {pc cm}^{-6}}\\right)\\left(\\frac{\\textrm {T}_e}{\\textrm {K}} \\right)^{-3/2} \\frac{1}{\\ln (\\nu _1 / \\nu _2)}$ $ \\times \\left[g_{\\rm ff}^{\\nu _2}\\left(\\frac{\\nu _2}{\\textrm {GHz}}\\right)^{-2} - g_{\\rm ff}^{\\nu _1} \\left(\\frac{\\nu _1}{\\textrm {GHz}}\\right)^{-2} \\right]$ where the factor of $5.5\\times 10^{-2}$ accounts for the unit conversion of EM to opacity $\\kappa _\\nu $ , and $\\nu _{1,2}$ are the frequencies in Hz at which the spectral index is calculated.",
"$T_e$ is the electron temperature and we assume a value of $T_e=10^4$ K throughout this paper.",
"The mean thermal electron density of the intergalactic medium is $\\sim 2.1\\times 10^{-7}$ cm$^{-3}$ [12], which makes its contribution negligible compared to that of the Galaxy, so that the absorption of background sources at low frequencies is due solely to the Galactic population of thermal electrons.",
"Figure: Plots of the total spectral index, α tot \\alpha _{\\rm tot}, illustrating the change from the intrinsic spectral index (α int \\alpha _{\\rm int}) that the STAP (i.e., α EM \\mbox{$\\alpha _{\\rm EM}$}) has as a function of frequency from 10 to 500 MHz for a number of different frequency widths.A canonical intrinsic spectral index of α int =-0.96\\alpha _{\\rm int}=-0.96 was used, and hence the STAP is any deviation away from a horizontal line.Plot (a) shows the STAP for separations of both 1 (thick) and 5 MHz (thin) lines as labelled, whilst plot (b) shows the same for separations in frequency of 10 and 50 MHz (thick and thin lines, respectively).The plots both show the same frequency and spectral index range, so as to illustrate the difference in the change between 1 and 5 MHz and 10 and 50 MHz steps.To study the effect of absorption, we first took a nominal source taken to have an (arbitrary) intrinsic spectral index of $\\alpha _{\\rm int}=-0.96$ .",
"We then calculated the spectral index for this source using Equation REF for a range of plausible EMs for the warm ionised medium (WIM); namely 0.1, 1.0, 10, 50 and 100 pc cm$^{-6}$ .",
"Figures REF (a) and (b) show the results for steps in frequency of 1 and 5 MHz (thick and thin lines of Figure REF (a), respectively), and 10 and 50 MHz (thick and thin lines of Figure REF (b), respectively).",
"These figures show a number of things.",
"Firstly, for reasonable (see below) EMs, such a change in spectral index may be detectable, and secondly, the spacing between spectral index calculations at low frequencies matters.",
"If a source is strong enough such that high signal-to-noise can be obtained in small (e.g., $\\sim 1$ MHz) channels, then this effect can be exploited to obtain information about the thermal electron content of the Galaxy even at relatively high latitudes (c.f.",
"Figure REF ).",
"We must note at this juncture that such effects are well known: at very-low-frequencies, the turnover of the Galactic synchrotron spectrum has been used to infer properties about the WIM as well [18].",
"Additionally, [13] explored absorption towards the envelopes of Hii regions because on large scales in our Galaxy, it was thought that the average density of thermal electrons along a particular line of sight is inadequate to produce significant amounts of absorption.",
"Using an average thermal electron density of $n_e\\approx 0.03$ cm$^{-3}$ at a frequencies of 150 and 330 MHz, this equates to optical depths of $\\tau _{150}\\sim 10^{-4}$ and $\\tau _{330}\\sim 10^{-3}$ (i.e., $\\tau _\\nu \\ll 1$ ), this would indeed seem to be the case.",
"However, given Figures REF (a) and (b), as well as the fact that no one has – to the best of our knowledge – investigated such effects in a systematic way, nor obtained the above simple-relation between the EM and spectral index, we tested a model for the distribution of thermal electrons in the Galaxy to test on a real catalogue in an attempt to explore this effect."
],
[
"The STAP, the modified-TC93 $n_e$ model and the Farnes, et al. “meta-catalogue”",
"We have obtained the modified Taylor-Cordes (modified-TC93; [24]) model for the thermal electron distribution of the Galaxy, and the “meta-catalogue” of [4].",
"This meta-catalogue is compiled of data from a number of radio catalogues, such as NRAO VLA Sky Survey (VLSS), AT20G, B3-VLA, GB6, NORTH6CM, Texas, and WENSS, in order to characterise the nature of polarised background galaxies for future experiments in the rotation measure grid (RM-grid; e.g., [6]).",
"We use the source coordinates in this catalogue to calculate the emission measure towards that particular location using the modified-TC93 model.",
"We integrate for a distance of 30 kpc in all directions for simplicity; doing this does not lead to significantly different results from using a more realistic integration length, such as stopping when two consecutive steps produce no change in the EM.",
"In Figure REF (a) we show the results of this calculation.",
"We chose to use the EM in these plots because it is frequency independent and shows that the EM range shown in Figures REF (a) and (b) are reasonable.",
"Calculating the STAP for the data in this catalogue would not be useful, since the majority of sources are measured at relatively high frequencies (between 1 and 2 GHz).",
"However, it is worth using such a catalogue so as to illustrate the coverage of a future catalogue from LOFAR, the MWA or SKA, which were not available at the time of writing.",
"Figure REF (a) shows, using the modified-TC93 model, that below latitudes of $10-15^{\\circ }$ , the average EM is somewhere between 1 and 100 pc cm$^{-6}$ .",
"Figure REF (a) and (b) implies that a change from the intrinsic spectral index of a source measured between 10 and 20 MHz can be up to 0.5 for a EM of 100 pc cm$^{-6}$ , but ostensibly undetectable for an EM of 1 pc cm$^{-6}$ (these are represented by the thick lines in Figure REF (b) for a frequency difference of 10 MHz).",
"At a more representative LOFAR low-band frequency of 50 MHz, but still using 10 MHz wide images for spectral index calculations, the spectral index change would be 0.005 and 0.05 for EMs of 1 and 100 pc cm$^{-6}$ , respectively.",
"However, as we show below (c.f.",
"Section REF ), a smooth model, such as that used here does not represent the known distribution of EMs across the sky; the WIM is known to be clumpy [5], [9].",
"Figure: Aitoff projection of EM for sources in the meta-catalogue obtained by (a) the smooth thermal electron distribution in the Galaxy as according to the modified-TC93 model, and (b) the smooth modified TC93 model in combination with our clumpy model using a filling factor of f=0.1f=0.1 and an (average) cloud–cloud length of L=0.43L=0.43 kpc according to Equation .",
"The colour-scale is in units of pc cm -6 ^{-6} and has had a (base–10) logarithm transfer function applied to it, so as to accentuate low-EM regions."
],
[
"The Clumpy WIM and STAP",
"In this section, we model the “clumpiness” of the WIM by extending the model of [17] developed to explain pulsar DM anomalies to EM."
],
[
"A simple model for a clumpy ISM",
"As in [17], we define a clump as a region where the number density of electrons is enhanced by a factor, $f$ , which is related to the number density, $N_c$ , and radius, $R_c$ , of the clump by: $f=N_c\\frac{4\\pi }{3}R_c^3.$ A source that lies behind a clump passes through it at distance $r_c\\le R_c$ from the centre of the clump will possess an emission measure of: $\\textrm {EM}(r_c) = \\frac{2n_e^2}{f^2}\\sqrt{R_c^2-r_c^2}.$ Thus, the average emission measure for a collection of lines-of-sight passing through an over-dense region is given by: $\\langle \\textrm {EM} \\rangle = \\int ^{R_c}_{0}\\frac{2n_e^2}{f^2}\\sqrt{R_c^2-r^2_c}\\frac{2\\pi r_cdr_c}{\\pi R_c^2}=\\frac{4n_e^2R_c}{3f^2}.$ Equation REF gives the average EM for a line-of-sight passing through one clump.",
"Thus to find the EM enhancement due to the clumpy WIM, we multiply this by the expected number of over-densities along a particular line-of-sight $N_{{\\rm cl}} = \\frac{d}{|\\sin (b)|L}, $ where $d$ is the height of the thermal electron content of the Galaxy, and $L$ is the distance between over-densities.",
"For this model, we take $d=1.75$ kpc [2], [7].",
"A better estimate for the electron density scale height may be about 1.4 kpc or even smaller [22].",
"If the scale height would be 1.4 kpc, this would decrease the estimate of EM$_{\\rm tot}$ to 20% since this factor appears only in the estimate of the number of clouds given in eq.",
"REF .",
"To obtain a value of mean free path, $L$ , we use DM$\\sin b$ , which represents the vertical contribution of electron content only, independent of direction (excepting directions that are very close to the Galactic plane).",
"The reason for this is that the length of the sight-line and number of clouds are both proportional to $1/\\sin (b)$ .",
"The clumpy component of the WIM is DM$_{\\rm cl} = \\textrm {DM}_{0, {\\rm cl}} / \\sin b$ and DM$_{\\rm cl} \\sin (b) = \\textrm {DM}_{0, {\\rm cl}}$ , where DM$_{0, {\\rm cl}}$ is the clumpy contribution of dispersion measure through a typical vertical size of the electron layer.",
"This property can be exploited by using a Poisson distribution for pulsars that possess a DM above a particular value: $P(k) = \\lambda ^k \\frac{e^{-\\lambda }}{k!",
"},$ where $P(k)$ is a probability to encounter $k$ clouds along a particular sight-line, and we draw the pulsars' DM value from the ATNF pulsar cataloguewww.atnf.csiro.au/people/pulsar/psrcat/ [16] with $DM\\sin (b)>24.5$ pc cm$^{-3}$, which is taken to infer the pulsar's height above the Galactic plane is higher than the scale height of thermal electronsThis is in torsion to the value of 16.5 pc cm$^{-3}$ given in [17], and comes from a re-evaluation of pulsar statistics from the ATNF database in the intervening 17 years..",
"The number of clouds along a particular line-of-sight is then $N_{\\rm cl}=\\lambda = D_p/L$ , where $D_p$ is the distance to pulsar and $L$ is the mean path-length between the clumps, was shown above to be the same for all pulsars.",
"If a sight-line does indeed encounter a clump, then, on average, the DM thus becomes $\\mathrm {DM} + \\langle \\textrm {dm}_1\\rangle $ .",
"If we let $F_\\mathrm {p}(\\lambda )$ be a function which generates a number distributed according to a Poisson distribution with the parameter $\\lambda $ , one can model pulsars with large DM$\\sin b$ as: $\\textrm {DM}\\sin (b) = 24.5\\sin (b) + [\\langle \\textrm {dm}_1\\rangle \\sin (b)]F_\\mathrm {p}(\\lambda ).$ It is possible to compare this value with the observed distribution by means of the Kolmogorov-Smirnov test.",
"In point of fact, it is not so important that in many cases the additional DM due to the clumpy WIM is not equal to $\\langle \\textrm {dm}_1\\rangle $ , because $F_\\mathrm {p}(\\lambda )$ is small (i.e., of order unity), and hence the difference between distributions according to the Kolmogorov-Smirnov test would not be large.",
"Additionally, if $F_\\mathrm {p}(\\lambda )\\gtrsim 1-2$ , then $\\langle \\textrm {dm}_1\\rangle $ is a good estimate of their mean.",
"Using the Kolmogorov-Smirnov test on 12 pulsars with the highest derived DM$\\sin (b)$ -value, we find a best estimate of $\\langle \\textrm {dm}_1\\rangle $ and $\\lambda $ .",
"Our method is not particularly sensitive to $\\lambda $ , and hence our analysis can only restrict the number of clouds to $\\lambda \\in [2, 4]$ .",
"It is, however, much more sensitive to $\\langle \\textrm {dm}_1\\rangle $ , which we restrict to $\\langle \\textrm {dm}_1\\rangle \\in [3.6, 3.8]$ pc cm$^{-3}$ ; almost identical to that from [17].",
"Our derived value of $L/R_c\\sim 20$ does, however, seem to be much larger than that found in [17].",
"We obtain $\\lambda = 2-4$ , which implies that $L \\equiv d/\\lambda = 0.87 -0.43$ kpc.",
"Figure: A plot of the same total spectral index (i.e., α tot =α int +α EM )\\alpha _{\\rm tot}=\\alpha _{\\rm int}+\\mbox{$\\alpha _{\\rm EM}$}) as a function of frequency for the same EMs as in Figure (thick lines for EMs as labelled), however, now we have added the clumpy component as modelled in Equation , assuming a line-of-sight of b=45 ∘ b=45^\\circ (thin lines according to the EM as labelled).Again, we assume a canonical spectral index of α int =-0.96\\alpha _{int}=-0.96, and calculate a spectral index at 10 MHz intervals (i.e., Figure (b), thick lines).This plot has the same range in frequency and spectral index as Figure (a) and (b) for easy comparison.The inset shows a zoom of the frequency region between 30 and 130 MHz."
],
[
"A New Model For the WIM",
"Taking into account the clumpy nature of the WIM, one can model STAP as a change in the emission measure due to the smooth and clumpy WIM components: $\\textrm {EM}_{{\\rm tot}} = \\textrm {EM}_{{\\rm sm}} + \\textrm {EM}_{{\\rm cl}} = \\textrm {EM}_{{\\rm sm}} + \\frac{4N_{{\\rm cl}}\\langle n_e^2 \\rangle R_c}{3f^2},$ where EM$_{{\\rm sm}}$ is the smooth component of the WIM and EM$_{{\\rm cl}}$ is the emission measure due to the clumpy component, and $D=d/\\sin (b)$ is the distance.",
"Figure REF (b) shows the results of the sum of the two components as in Equation REF .",
"Overall, the EM-sky qualitatively shows the same, stratified structure, but with an increased overall normalisation due to the clumps, with total EMs over significant parts of the sky in the range of 1 to 1000 pc cm$^{-6}$ .",
"Figure REF shows the change in total spectral index where, unlike Figure REF (a) and (b) which only calculate the smooth component of $\\mbox{$\\alpha _{\\rm EM}$}$ , the clumpy part of the WIM has been added, assuming a line-of-sight through $b=45^\\circ $ .",
"This figure shows that the spectral index when such a clumpy distribution is included is changed, comparative to that for a simple smooth distribution.",
"Quantifying this, at 50 MHz for a source towards which there exists an EM of 100 pc cm$^{-6}$ and calculating the spectral index in steps of 10 MHz, there is a difference in spectral index of 0.012 between the smooth-only model, and the smooth+clumpy model.",
"This is a measurable difference between the models, and the smooth+clumpy model gives a spectral index at this frequency of -0.916; a change of 0.05.",
"Even at frequencies of 110 MHz, which is the lower-end of the LOFAR high band (HBA), there is a difference in spectral index in this scenario of 0.01, suggesting that at least for strong sources, a careful spectral analysis may be able to mesure the thermal electron content of our Galaxy in those source directions."
],
[
"Model EM source distribution and application to ",
"It is instructive to compare our model to H$\\alpha $ data because it is a measure of EM that is independent to our method.",
"However, direct comparison between our EM map and an observed H$\\alpha $ map would only be possible at high latitudes.",
"At the intermediate and low latitudes, the H$\\alpha $ emission is so absorbed that the emission represents only the local galaxy (the nearest few kpc).",
"Also, due to the simplicity of our model, it does not contain the individual H II regions that dominate the observed H$\\alpha $ maps at low latitudes.",
"The H$\\alpha $ map displayed in Figure REF demonstrates this point: the map is dominated by local, individual H II regions.",
"However, the general magnitude of the EM distributions between the modeled and observed maps agrees qualitatively.",
"Therefore, we have analysed the EM-distribution (specifically, the EM$|\\sin (b)|$ -distribution) of our smooth, clumpy and total (smooth+clumpy) models, the results of which are shown in Figure REF .",
"Figure: All-sky observed Hα\\alpha map based on the WHAM and SHASSA surveys.Figure: Histogram plot of the EM|sinb||\\sin b| distribution for sources in the meta-catalogue obtained by excluding all sources with |b|<10 ∘ |b|<10^\\circ and obtained using f=0.1f=0.1, as well as L=0.43L=0.43 kpc.Relating Figure REF to that of Figure 3 of [11] demonstrates a number of things.",
"Firstly, it shows that the distribution of EM for the smooth model does not match that observed in the data, with a peak that is far too small; EM$|\\sin (b)|\\sim 0.1$ pc cm$^{-6}$ , where the data suggests it should be closer to EM$|\\sin (b)|\\sim 1.4$ pc cm$^{-6}$ .",
"This demonstrates that a model for the clumpy nature of the WIM is required for our model to accurately represent the EM-sky.",
"Secondly, the clumpy model that we derived in Section REF is a reasonable fit for the data presented in [11].",
"Figure REF shows that the value of the peak in EM for the total mode (smooth + clumpy) at EM$|\\sin (b)|\\sim 1.4$ pc cm$^{-6}$ almost exactly matches that found by [11] (see their Table 2).",
"Our distribution hints at a lognormal distribution, however, with a slight overdensity at high EM$|\\sin (b)|$ .",
"This suggests that our model is a good statistical approximation for the actual EM-sky."
],
[
"Consistency check of DMs towards Galactic Globular Clusters",
"As a consistency check that our method of determining the EM does not also significantly change the DM towards a large sample of pulsars with independently-derived DMs, we have checked the DM that our model predicts against a known pulsar DM population.",
"Since a precise knowledge of the distance is critical in determining the DM, we require test sources that have well-known distances as well.",
"Therefore, we have chosen pulsars observed in Globular clusters (GCs), since the distances towards the individual GC in question can be determined by independent means.",
"We have obtained the position, distance to (assuming an error of 0.5 kpc) and average DMs of the pulsars found in 28 GCs from the catalogue of Paulo FreireThe web-page and relevant references can be found at http://www.naic.edu/~pfreire/GCpsr.html to estimate the DM towards these pulsars.",
"Figure REF shows the results of this analysis – the observed DM versus our predicted DM – for the modified-TC93 only (plus signs) and modified-TC93 and our clumpy model (star-signs), with the dashed line showing a linear relationship between the observed and predicted DMs.",
"It should be noted that the datum near DM$_{model}\\sim 420$ pc cm$^{-3}$ is for the GC Terzan 5, which is located at $(l,b)=(3^\\circ .84,1^\\circ .69)$ at a distance of 10.5 kpc, which places it near the Galactic centre, and possibly explaining why it does not follow the DM$_{model}\\sim \\textrm {DM}_{obs}$ trend that the other data do.",
"This shows that the DMs towards these regions are not significantly altered by our analysis, and hence that we can be confident that our model does not lead to significantly altered DMs.",
"Figure: A comparison of the observed DMs (in pc cm -3 ^{-3}: pc/cc) from Globular clusters against that derived from our modified-TC93 model (plus-signs) and the modified-TC93 code plus the “clumpy” WIM model from Equation 2 of (star-signs)."
],
[
"Source counts and detecting the effect of the WIM",
"It was shown above that in our Galaxy the STAP will likely be observable for many parts of the Galaxy, particularly at relatively low latitudes of $|b|\\lesssim 15^\\circ $ .",
"Below, we estimate the required accuracy of low-frequency measurements to detect the STAP for a single source.",
"Also, we calculate the required source density at higher latitudes which would enable a statistical detection of STAP in an ensemble of sources."
],
[
"Detection of the STAP in a single source",
"Figure REF shows that very low-frequency observations are required for a detectable deviation of the measured spectral index from the intrinsic one.",
"In figure REF , we the total spectral index $\\alpha _{\\rm tot}$ as calculated according to eq.",
"() for two extragalactic sources.",
"The left plot shows data from 3C196 [14], [23], a strong source often used as calibrator source, which was chosen for the wealth of available observations.",
"The right plot depicts observations from 3C428 (obtained from the NED database), which is located at low Galactic latitude ($b = 1.31^{\\circ }$ ) so that the effect of the Galactic WIM would be higher.",
"The dashed lines show the intrinsic spectral index ($-0.88$ for 3C196 and $-1.33$ for 3C428 [14]) at the higher frequencies and the expected change in spectral index based on the EM obtained from our modified-TC93+clumpy model.",
"Figure: Total spectral index α tot \\alpha _{\\rm tot} as a function of frequency for 3C196 (top) and 3C428 (bottom).",
"The overlaid dashed line is the calculated α tot \\alpha _{\\rm tot} based on expected EM and intrinsic spectral index, as discussed in the text.However, the current low-frequency instruments LOFAR and MWA are planning or executing surveys at sufficiently low freqencies to etect this effect in principle in a single source.",
"As an example, we simulated detectability of a Galactic EM = 100 pc cm$^{-6}$ in the “Deep” survey (Tier 2) of the currently ongoing “Tier surveys” with LOFAR [20].",
"For the planned 30 MHz survey, assuming 1 MHz frequency channels over the bandwidth of 16 MHz, we fit equation () as a function of frequency channel.",
"A simple monte carlo analysis shows that this EM would be detectable at the $5\\sigma $ level in a source with flux $S_{30~\\mbox{MHz}} = 7$ Jansky."
],
[
"Statistical detection of the STAP in an ensemble of sources",
"We explore here how the sensitivity of next-generation radio telescopes such as LOFAR, MWA and especially the SKA may allow the statistical detection of this phenomenon at higher latitudes, where the low EM towards sources make STAP analysis currently difficult or impossible.",
"At latitudes higher than $b\\sim 30^\\circ $ , Figure REF (b) shows that an average of EM$\\sim 0.1-1$ pc cm$^{-6}$ , which Figure REF shows a change in spectral index at 30 MHz of $\\sim 0.01$ .",
"Given that [19] showed that the spectral index distribution of sources between 843 MHz and 1.4 GHz is Gaussian with a standard deviation of $\\sigma _\\alpha =0.56$ (after K-correction) for an average spectral index of $\\alpha _{\\rm meas}=-0.5$ , one can detect a change in the distribution that is statistically significant of: $\\Delta \\sigma \\gtrsim \\frac{\\sigma _{\\alpha }}{\\sqrt{N}},$ where $N$ is the number of sources per square degree.",
"For a source density of 10, 100 and 1000 sources per square degree, then, one can robustly detect a change in spectral index of $\\mbox{$\\alpha _{\\rm EM}$}\\sim 0.18,0.06$ , and 0.02.",
"This shows that, for source densities of 1000 sources per square degree, the thermal electron distribution at latitudes higher than $b\\sim 30^\\circ $ could be (statistically) explored.",
"Thus, sensitive, dedicated low-frequency surveys with telescopes such as LOFAR and MWA and eventually the SKA should be able to probe this effect and significantly enhance our knowledge of the thermal electron distribution of the Galaxy."
],
[
"Conclusions",
"In this paper, we have presented a new method for exploring the thermal electron density of our Galaxy by means of spectral analysis of background sources in the limit of a zero-source-function solution to the equation of radiative transfer.",
"We have shown the following: This limit results in an additive term to the intrinsic spectral index, if said index is calculated using natural logarithms.",
"This effect is particularly important at low frequencies, and especially-so at frequencies below $\\sim 100$ MHz, even for modest emission measures of between 1 and 100 pc cm$^{-6}$ .",
"Through the use of the meta-catalogue of [4] (used as a position-template for possible low-frequency point source catalogs), we find that these EM values are representative of large parts of the Galaxy.",
"A smooth distribution of thermal electrons does not represent the known EM distribution obtained from H$\\alpha $ data from the WHAM survey, which is an independent measure of the EM.",
"A simple model for the clumpy WIM, on the other hand, does indeed produce such a distribution, and increases the normalisation of EMs over the sky to 100-1000 pc cm$^{-6}$ , at least at relatively low Galactic latitudes (i.e., $|b|\\lesssim 15^\\circ $ ).",
"However, we also showed that because of this higher normalisation that the clumpy WIM gives, it should be possible to explore the Galactic distribution of thermal electrons at higher Galactic latitudes statistically with source counts of $\\sim 1000$ sources per square degree.",
"The latter two items confirm earlier work based on H$\\alpha $ observations [11] and pulsar dispersion measures [1], using lognormal distributions.",
"Given that an excellent knowledge of the thermal electron distribution is crucial for our knowledge of the magnetic field strength and structure in our Galaxy, which is itself an integral part of the Galactic ecology, this new method, which is independent of rotation measures but can be observed at the same frequencies, could make an important contribution to our knowledge of our magnetic Galaxy."
],
[
"Ackowlegements",
"DIJ would like to acknowledge enlightening conversations with Dominic Schnitzeler (who also kindly provided his modified-TC93 code), Bryan Gaensler, Jamie Farnes, Marco Iacobelli, Cameron van Eck, and Elmar Koerding.",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"MH acknowledges the support of research programme 639.042.915, which is partly financed by the Netherlands Organisation for Scientific Research (NWO).",
"The authors acknowledge use of the Southern H-Alpha Sky Survey Atlas (SHASSA), which is supported by the National Science Foundation."
]
] | 1605.06097 |
[
[
"Pattern Formation in Chemically Interacting Active Rotors with\n Self-Propulsion"
],
[
"Abstract We demonstrate that active rotations in chemically signalling particles, such as autochemotactic {\\it E. coli} close to walls, create a route for pattern formation based on a nonlinear yet deterministic instability mechanism.",
"For slow rotations, we find a transient persistence of the uniform state, followed by a sudden formation of clusters contingent on locking of the average propulsion direction by chemotaxis.",
"These clusters coarsen, which results in phase separation into a dense and a dilute region.",
"Faster rotations arrest phase separation leading to a global travelling wave of rotors with synchronized roto-translational motion.",
"Our results elucidate the physics resulting from the competition of two generic paradigms in active matter, chemotaxis and active rotations, and show that the latter provides a tool to design programmable self-assembly of active matter, for example to control coarsening."
],
[
"article170516 Pattern Formation in Chemically Interacting Active Rotors Electronic Supplementary Information to Pattern Formation in Chemically Interacting Active Rotors Benno Liebchen, Michael E. Cates, Davide Marenduzzo Here, we derive coarse grained equations of motion for the rotor density and orientation fields.",
"Since we are mainly interested in the competition between active rotations and chemotaxis - and in the corresponding physical mechanism allowing for structure formation - we use a variety of approximations to achieve a `minimal' rather than a rigorous coarse grained description of active rotors which focuses on this competition (and hence only covers a subset of possible phenomena).",
"This procedure leads to Eqs.",
"(-) considered in the main text.",
"Consider an ensemble of $N$ `signalling rotors', which self-propel with constant velocity $v_0$ along the directions ${\\bf p}_i=(\\cos \\theta _i,\\sin \\theta _i)$ .",
"In the uniform state, these directions rotate actively with natural frequencies $\\Omega _i$ , but generally also respond to chemical gradients, where $\\beta _R$ denotes the `chemotactic' coupling strength.",
"We further assume (steric) alignment interactions between different rotors which are sufficiently short ranged to allow us to replace their spatial dependence by a pseudopotential (a zero ranged `$\\delta $ '-interaction).",
"Denoting the rotational diffusion constant by $D_r$ , and by $\\xi (t)$ a Gaussian white noise with unit variance, we describe a single chemotactic rotor in 2D via the following Langevin equations $\\dot{\\bf r}_i &=& v_0 {\\bf p}_i \\\\\\dot{\\theta }_i &=& \\Omega _i + \\beta _R {\\bf p}_i \\times \\nabla c + G {\\sum \\limits _{j=1}^{N}} \\sin (\\theta _i - \\theta _j)\\delta ({\\bf r}_i -{\\bf r}_j) + \\sqrt{2D_r} \\xi _i(t) $ where we used the notation ${\\bf a}\\times {\\bf b}=a_1 b_2-a_2 b_1$ and $c({\\bf x},t)$ is the chemical field produced by the ensemble of `signalling' rotors with rate $k_0$ .",
"This field evolves as $ \\dot{c} = k_0 {\\sum \\limits _{i=1}^{N}}\\delta ({\\bf r}-{\\bf r}_i) - k_d c + D_c \\nabla ^2 c + \\epsilon (c-c_0)^3$ where $k_d$ is the decay rate.",
"Here, $D_c$ is the chemical diffusion constant and the term proportional to $\\epsilon $ prevents unlimited growth of $c$ in case of linear instability.",
"We now use Itôs Lemma and follow [1] to derive coarse grained equations of motion for the combined probability density $f_i({\\bf r},\\theta )=\\delta ({\\bf r}-{\\bf r}_i)\\delta (\\theta - \\theta _i)$ .",
"For rotors with identical frequencies $\\Omega _i \\rightarrow \\Omega $ we find for $f({\\bf r},\\theta )={\\sum \\limits _{i=1}^{N}}f_i({\\bf r},\\theta )$ $ \\dot{f} =-v_0 {\\bf p}\\cdot \\nabla f - G \\partial _\\theta \\int { {\\rm d} }{\\bf r}^{\\prime }{ {\\rm d} }{\\bf \\theta }^{\\prime }f({\\bf r}^{\\prime },\\theta ^{\\prime })F(\\theta -\\theta ^{\\prime },{\\bf r}-{\\bf r}^{\\prime }) f({\\bf r},\\theta )+D_r \\partial ^2_\\theta f-\\beta _R|\\nabla c|\\partial _\\theta [f \\sin (\\theta +\\delta )]-\\partial _\\theta \\Omega f -\\partial _\\theta \\sqrt{2D_r f}\\xi (t) $ Here $\\delta ={\\rm atan2}(\\partial _y c,-\\partial _x c)$ , where ${\\rm atan2}(y,x)$ is the (principle value of) the argument function ${\\rm arg}(x+Iy)$ and $-\\partial _\\theta \\sqrt{2D_r f}\\xi $ describes multiplicative noise, which we neglect in the following because we are interested in the mean field phenomenology.",
"From here, we follow [2] and expand $f$ in a Fourier series $f(r,\\theta )={\\sum \\limits _{k=-\\infty }^{\\infty }}f_k({\\bf r}){\\rm e}^{-Ik \\theta }$ with $f_k({\\bf r})=\\int f({\\bf r},\\theta ) {\\rm e}^{Ik \\theta }{ {\\rm d} }\\theta $ and identify $f_0 \\rightarrow \\rho ({\\bf r},t)$ and $\\left({\\rm Re}f_1,{\\rm Im} f_1\\right) \\rightarrow {\\bf w}$ .",
"This corresponds to a `continuum' approximation which is, due to the interaction term, only sensible in systems that are dense enough such that the interaction can be averaged over many neighbours.",
"Straightforward algebra leads to the following equation for the Fourier coefficients: $\\dot{f}_k =-{\\frac{v_0}{2}}\\left[(\\partial _x-I\\partial _y)f_{k+1} + (\\partial _x+I\\partial _y)f_{k-1}\\right]-k^2 D_r f_k +{\\frac{IG k}{2\\pi }} {\\sum \\limits _{m}^{}}f_{k-m} F_{-m}f_{m}+{\\frac{\\beta _R |\\nabla c| k}{2}} \\left(f_{k+1}{\\rm e}^{I\\delta } - f_{k-1} {\\rm e}^{-I\\delta }\\right) + Ik \\Omega f_k $ For $k=0$ we quickly find $\\dot{\\rho }= - v_0 \\nabla \\cdot {\\bf w}$ .",
"To achieve a closed equation for $f_1$ we neglect $f_k$ with $k\\ge 3$ and assume that $f_2$ is fast, i.e.",
"we set $\\dot{f}_2 \\rightarrow 0$ (compare [2], [4]).",
"This yields a closed equation of motion for $f_1$ and hence for ${\\bf w}$ .",
"This latter equation could in principle be solved numerically, but is rather involved and hardly allows us to analyse the interplay between chemotaxis and active rotations.",
"To highlight this interplay within a `minimal model' we apply a second layer of approximations and directly neglect all contributions which are both nonlinear and involve gradient terms (such terms would therefore contribute to the nonlinear saturation of unstable short wavelength modes)These neglected terms should be important at least deep in the nonlinear regime and could be the subject of further investigations..",
"This procedure leads to $\\dot{\\rho }&=& - v_0 \\nabla \\cdot {\\bf w} \\\\\\dot{\\bf w} &=& \\left({\\frac{G \\rho }{2}}-D_r\\right){\\bf w} + \\Omega {\\bf w}_\\perp - {\\frac{v_0}{2}}\\nabla \\rho + {\\frac{\\beta _R \\rho }{2}}\\nabla c + {\\frac{v_0^2 D_r}{b}}\\nabla ^2 {\\bf w} + \\left({\\frac{v_0^2 \\Omega }{2b}}\\right)\\nabla ^2 {\\bf w}_\\perp -\\left({\\frac{2G^2 D_r}{b}}\\right)|{\\bf w}^2|{\\bf w} - {\\frac{G^2 \\Omega }{b}}|{\\bf w}|^2 {\\bf w}_\\perp \\\\\\dot{c} &=& k_0 \\rho - k_d c + D_c \\nabla ^2 c + \\epsilon (c-c_0)^3 $ where ${\\bf w}_\\perp \\equiv (-w_y,w_x)$ and $b=16D_r^2 + 4\\Omega ^2$ .",
"Two solutions of these equations are: (i) the uniform unpolarized state $(\\rho ,{\\bf w},c)=(\\rho _0,{\\bf 0},\\rho _0 k_0/k_d)$ , and (ii) a polarized and coherently rotating state with uniform density $(\\rho ,w,\\phi ,c)=\\left(\\rho _0, w^\\ast ,\\Omega ^\\ast t, k_0 \\rho _0/k_d \\right)$ with $w^\\ast =\\sqrt{[G\\rho _0-2Dr](\\Omega ^2/(G^2D_r)+4D_r/G^2)}$ , $\\Omega ^\\ast =\\Omega (3/2-G\\rho _0/(4D_r))$ .",
"Here, $\\phi $ is defined via ${\\bf w}=w {\\bf p}$ with ${\\bf p}=(\\cos \\phi ,\\sin \\phi )^T$ representing the average (collective) self-propulsion direction.",
"If $G \\rho _0 < 2 D_r$ the unpolarized state is stable.",
"It becomes unstable in favour of the coherently rotating state when alignment interactions are strong enough to suppress dephasing by rotational noise ($G \\rho > 2 D_r$ ).",
"At the onset of polarization this state rotates with a frequency $\\Omega $ but slows down as more and more particles align.",
"In the main text, we focus on the regime of sufficiently strong alignment interactions and hence consider this coherently rotating state as the relevant uniform state whose dynamics we explore in presence of chemotaxis.",
"Also, if self-propulsion is not too strong, the term $(-v_0/2)\\nabla \\rho $ mainly reduces the chemotactic coupling in our simulations where we typically have $c\\sim \\rho $ and $\\beta _R\\rho _0 \\gg v_0$ .",
"Hence we also omit this term for simplicity as well as $\\nabla ^2$ terms in Eq.",
"(), which are not important at long wavelength or for almost uniform ${\\bf w}$ .",
"We now define the polarization ${\\bf P}$ which measures the degree of local alignment (per particle), as usual, via ${\\bf w}=\\rho {\\bf P}=\\rho |{\\bf P}| {\\bf p}$ .",
"Since we are mainly interested in the competition of the chemical alignment and active rotations, we do not describe spatial modulations of the `average' polarization but assume it as constant; $|{\\bf P}|=1$ for simplicity (smaller positive values of $|{\\bf P}|$ do not significantly alter our numerical results).",
"Assuming also $\\omega :=\\Omega -\\frac{G \\Omega }{b}{w}^2\\approx \\Omega -\\frac{G \\Omega }{b}{\\rho _0^2}$ in Eq.",
"() and defining $\\beta :={\\frac{\\beta _R}{2}}$ , after projecting Eq.",
"() onto $\\hat{\\bf p}_\\perp =(-\\sin \\phi ,\\cos \\phi ) $ our minimal description of chemotactic rotors reads $\\dot{\\rho }&=& -v_0 \\nabla \\cdot (\\rho {\\bf p}) + D_\\rho \\nabla ^2 \\rho + K \\nabla ^2 \\rho ^3 \\\\\\dot{\\phi }&=& \\omega + \\beta {\\bf p} \\times \\nabla c \\\\\\dot{c} &=& k_0 \\rho - k_d c + D_c \\nabla ^2 c + \\epsilon (c-c_0)^3 $ In Eq.",
"(REF ), we added a phenomenological term describing isotropic short ranged repulsions among colloids ($K \\nabla ^2 \\rho ^3$ ), whose main effect is to prevent strong gradients on too small scales in our simulations.",
"While we chose here a cubic term for convenience (retaining symmetry under $\\rho \\rightarrow -\\rho $ ), replacing this by a quadratic term, $\\nabla ^2 \\rho ^2$ leads, according to our simulations, to an almost identical phenomenology when modifying the coefficient $K$ appropriately.",
"Given the complexity of the system under consideration, our description in Eqs.",
"(REF -) is far from complete, but it highlights the competition between chemotactic alignment and active rotations, allowing us to focus on the physical mechanism underlying structure formation in signalling rotors.",
"Our approach can be straightforwardly extended to derive a more precise but rather complex coarse grained description of (chemotactic) active rotors.",
"Here, we generalize the above approach to rotors with non-identical frequencies $\\Omega _i$ , i.e.",
"we replace $\\Omega \\rightarrow \\Omega _i$ in Eq.",
"() and ask to which extend this alters Eqs.",
"(REF -).",
"Accordingly, this paragraph can be seen as an alternative to the paragraph `non-identical frequencies' in the main text.",
"The basic idea is to replace $f_i\\Omega _i$ by its mean plus a typical fluctuation $ \\Omega _i f_i \\longrightarrow \\bar{\\Omega }f + \\sqrt{\\Delta _\\Omega f} \\eta $ where $\\eta $ describes Gaussian random numbers with zero mean and unit variance $\\langle \\eta ({\\bf r},\\theta ,t)\\eta ({\\bf r}^{\\prime }\\theta ^{\\prime },t)\\rangle = \\delta ({\\bf r}-{\\bf r}^{\\prime })\\delta (\\theta -\\theta ^{\\prime })$ and $\\bar{\\Omega }$ ; $\\Delta _\\Omega $ are defined by $ \\langle {\\Omega _i \\Omega _j}\\rangle ={\\bar{\\Omega }}^2 + \\Delta _\\Omega \\delta _{ij} $ This replacement of course changes, in general, the dynamics of $f_i$ but not the statistical properties of a large rotor ensemble, since the distributions $\\lbrace \\Omega _i f_i\\rbrace $ and $\\lbrace \\bar{\\Omega }f + \\sqrt{\\Delta _\\Omega f} \\eta \\rbrace $ have identical statistical properties.",
"(This statement holds true at each point in time, independent of time correlations of $\\eta $ ).",
"Physically, we should understand Eq.",
"(REF ) as a local replacement and interpret the above averages as mesoscopic ones over all particles within the interaction range around a given point ${\\bf x}$ rather than a global average over the whole rotor ensemble.",
"Accordingly, we allow $\\bar{\\Omega }$ and $\\eta $ to fluctuate in space and time.",
"We now define a field $\\omega ({\\bf x},t)$ representing the deviation from the global (time-independent) average rotation frequency $\\bar{\\Omega }_0={\\sum \\limits _{i=1}^{N}}\\Omega _i/N$ .",
"In the following we do not worry about the specific form of $\\omega ({\\bf x},t)$ but assume it to be random, with a correlation time on the order of the `mixing time', i.e.",
"the time after which a given set of rotors in one `interaction domain' is replaced by another one.",
"In non-synchronized states, we expect that the mixing time is on the order of the time a particle needs to traverse a distance given by the range of the alignment interactions.",
"As this timescale is, for the assumed short ranged alignment interactions, short compared to all other relevant timescales in the system ($1/k_d, 1/k_0, 1/\\omega $ ) we allow it to tend to zero for simplicity.",
"Hence we assume $\\omega $ , and analogously also $\\eta $ , to represent spatiotemporal white noise.",
"Conversely, in the synchronized regime, rotors can move together for some time and hence we expect substantial time-correlations.",
"As these correlations should become relevant only after the onset of synchronization we do not need to care about them since our aim is to understand where synchronization sets in.",
"We now expand $\\sqrt{f}$ for modest deviations from isotropy as follows $\\sqrt{f}=\\sqrt{{\\sum \\limits _{k=-\\infty }^{\\infty }} f_k {\\rm exp}[-Ik \\theta ]} \\approx \\sqrt{f_0}/2+{\\frac{1}{2\\sqrt{f_0}}}{\\sum \\limits _{k=-\\infty }^{\\infty }} f_k {\\rm e}^{-Ik \\theta }.$ Using this approximation, the only modifications of our mean field results Eqs.",
"(REF -) due to non-identical rotor frequencies correspond to replacing $\\bar{\\Omega }&\\rightarrow & \\bar{\\Omega }_0 + \\omega ({{\\bf x},t}) \\\\D_r &\\rightarrow & D_r + \\sqrt{{\\frac{\\Delta _\\Omega }{4\\rho _0}}} $ in these equations.",
"Consequently, for non-identical rotors the unpolarized state becomes unstable for $G\\rho _0 > 2D_r + \\sqrt{{\\frac{\\Delta _\\Omega }{4\\rho _0}}}$ .",
"This is very similar to the result of the Kuramoto model (main text) which assumes a Lorentzian distribution for the frequencies, and justifies the assumption of locally coherent rotations underlying model (REF -).",
"The parameter space of Eqs.",
"(REF -) can be reduced to four dimensions (plus an effective density which is fixed by the initial state) by introducing the dimensionless quantities $\\tilde{x} = \\sqrt{k_d/D_\\rho }x$ , $\\tilde{t} = k_d t$ .",
"Defining $\\tilde{\\rho }= \\rho k_0 \\beta /(k_d v_0)$ , $\\tilde{c} = c \\beta /v_0$ and $\\tilde{\\bf p}=v_0 {\\bf p}/\\sqrt{D_\\rho k_d}$ we obtain (now omitting tildes) $\\dot{\\rho }&=& -\\nabla \\cdot (\\rho {\\bf p}) + \\nabla ^2 \\rho + \\kappa \\nabla ^2 \\rho ^3 \\\\\\dot{\\phi }&=& \\Omega + {\\bf p} \\times \\nabla c \\\\ \\dot{c} &=& \\rho - c + \\mathcal {D}_c \\nabla ^2 c + \\varepsilon (c_0 - c)^3 $ Here, $\\Omega =\\omega /k_d$ and $\\mathcal {D}_c=D_c/D_\\rho $ determine the linear behaviour together with the effective density $\\rho _0$ which is conserved in the course of the dynamics, while $\\kappa =K k_d^2 v_0^2/(D_\\rho k_0^2 \\beta ^2)$ and $\\varepsilon =\\epsilon v_0^2/(k_d \\beta ^2)$ control non-linear saturation effects (besides the $\\kappa $ term producing some contribution to the colloidal diffusion term).",
"As a key control parameter we identify $\\Delta :=\\Omega /\\rho _0 = v_0 \\omega /(\\beta k_0 \\rho _0)$ which measures (for given self-propulsion speed) the relative importance of active rotations and chemotaxis.",
"Promising candidates to physically realize our predictions are auto-chemotactic strains of E.coli which rotate naturally close to a wall or interface.",
"Here, one could measure the initial time lag followed by a delayed onset of clustering as an indication of the nonlinear locking instability.",
"In addition, one could also measure the suppression of rotations which arises exclusively at the interface between dense and dilute regions for comparatively low effective rotation frequencies (see below) and the sudden arrest of coarsening when for larger rotation frequencies.",
"The parameter values used in this work (Fig. )",
"correspond to typical experimental values; the most relevant dimensionless control parameters $\\Omega $ , $\\tilde{\\rho }$ and $\\mathcal {D}_c$ translate as follows.",
"(i) The choice $D_c/D_\\rho = 1$ approximately matches with measurements of $D_\\rho \\sim 2\\times 10^{-6}-1.5\\times 10^{-5} {\\rm cm}^2/{\\rm s}$ and $D_c \\sim 10^{-5} {\\rm cm}^2/{\\rm s}$ [6], [7], [8], [9].",
"(ii) Our parameter choice in Fig.",
"corresponds to $\\rho _0 \\beta k_0/k_d v_0 =50$ which can be matched for typical `chemotactic sensitivities' of $\\chi \\sim (1.5-75) \\times 10^{-5} {\\rm cm}^2/{\\rm s}$ [6], [7] with $\\chi \\sim \\beta v_0/(2D_r)$ when assuming $D_r \\sim 0.1-1/{\\rm s}$ and rates $k_d \\sim 0.09/{\\rm s}$ and $k_0\\sim 0.6/{\\rm s}$ typical for the chemoattractant of Dicty (cAMP [10], [11], rates for E.coli are unknown [9]) when using systems with $>5\\times 10^3$ bacteria per ${\\rm mm}^2$ or area fractions $>0.1$ .",
"(iii) The parameter $\\Omega =\\omega /k_d \\sim 2$ for typical swimming radii of $50\\mu {\\rm m}$ [12] and swimming speeds of $10\\mu {\\rm m}/{\\rm s}$ which is close to the value at which we observed the transition from a large cluster to a stripe pattern; see Figs.",
"i and l. Faster swimming leads to larger values of $\\Omega $ and could allow to observe the decrease of the wavelength of our travelling wave pattern.",
"Smaller values of $\\Omega $ corresponding to parameters in Figs.",
"d-f would require slower swimming velocities or faster decay (consumption) of the E.coli chemoattractant aspartate than for cAMP.",
"Alternatively, by working with magnetotactic bacteria which have a permanent dipole moment or active bimetallic colloids [13], slow rotations could be easily generated by exposing these particles to an external rotating magnetic field.",
"We now derive a reduced model for the rotor distribution $\\rho ({\\bf r},t)$ by assuming that ${\\bf p}$ is a fast variable, which is a good approximation for the parameter regime considered in the main text.",
"This allows us to adiabatically eliminate the orientation equation.",
"First, we demonstrate that in absence of active rotations and alignment interactions $\\Omega =G=0$ , Eqs.",
"(REF -) reduce to the well-known Keller-Segel model [14], [15] describing chemotaxis of signalling microorganisms, and are therefore consistent with previous studies of active colloids based on this model [16], [17], [18].",
"In particular, the Keller-Segel model for chemoattractive particles ($\\beta >0$ ) allows for cluster growth typically proceeding to phase separation.",
"Here, as we shall see, local alignment interactions ($G>0$ ) will support this instability in the sense that they allow for cluster growth already at very low particle densities.",
"Second, we will derive a corresponding reduced model in presence of active rotations, starting with Eqs.",
"(-) in the main text.",
"This model shows that active rotations suppress linear instability in its complete parameter space, even if strong alignment interactions are present.",
"Starting from Eq.",
"() and adiabatically eliminating $\\dot{\\bf w}\\rightarrow {\\bf 0}$ as well as neglecting gradient terms of order $\\nabla ^2$ , with $\\Omega =G=0$ we find ${\\bf w}=-v_0 \\nabla \\rho /(2D_r) + \\beta \\rho \\nabla c/{2D_r}$ .",
"Plugging this into Eq.",
"(REF ) leads together with Eq.",
"() to the following model which resembles the Keller-Segel model of chemotaxis (compare [14], [15], [16], [17], [18]) $\\dot{\\rho }&=& - {\\frac{v_0 \\beta }{2D_r}} \\nabla \\cdot (\\rho \\nabla c) + D \\nabla ^2 \\rho \\\\\\dot{c} &=& k_0 \\rho - k_d c + D_c \\nabla ^2 c + \\epsilon (c-c_0)^3$ Here $D=v_0^2/(2D_r)$ is the effective `active' diffusion constant.",
"Linearizing around the uniform solution $(\\rho ,c)=(\\rho _0,k_0 \\rho _0/k_d)$ and using $\\rho ^{\\prime }=\\rho - \\rho _0, c^{\\prime }=c-c_0$ leads to: $\\dot{\\rho }^{\\prime } &=& - B \\nabla ^2 c^{\\prime } + D\\nabla ^2 \\rho ^{\\prime } \\nonumber \\\\\\dot{c}^{\\prime } &=& k_0 \\rho ^{\\prime } - k_d c^{\\prime } + D_c \\nabla ^2 c^{\\prime } $ where $B={\\frac{v_0 \\beta \\rho _0}{2D_r}}$ .",
"We now test the stability of these equations with respect to plane wave perturbations.",
"Fourier transforming (REF ) and evaluating the respective linear stability problem we find a long wavelength instability for $B k_0>D k_d$ with instability band $q^2 < (Bk_0 - D k_d)/(D D_c)$ .",
"This instability is based on the positive feedback loop explained in our introduction (see main text) and hinges on aligning particles up the chemical gradient.",
"Accordingly, it is no surprise that local alignment interactions only help destabilize the uniform state and allow for cluster growth already at very low effective densities.",
"For a later comparison with the case of active rotations it is instructive to further reduce model (REF ).",
"Assuming that $c^{\\prime }$ is fast compared to the conserved colloidal field after adiabatic elimination $c^{\\prime }\\rightarrow 0$ and neglecting $\\mathcal {D}_c \\nabla ^2 c^{\\prime }$ Eq.",
"(REF ) simplifies to $ \\dot{\\rho }^{\\prime } = [D - (k_0/k_d)B]\\nabla ^2 \\rho ^{\\prime } $ This is a closed model for the particle density close to the uniform state where chemotaxis directly competes with the effective diffusion.",
"This equation reproduces the correct instability criterion, but leads to a short wavelength divergence, normally prevented by chemical diffusion.",
"To derive a reduced model for active rotors ($\\Omega >0$ ) we need to solve Eq.",
"() explicitly.",
"Rewriting Eq.",
"() leads to the Adler equation $\\dot{\\phi }= \\Omega + |\\nabla c|\\sin (\\phi +\\delta )$ , here with $\\delta ={\\rm atan2}(\\partial _y c, -\\partial _x c)$ .",
"We now solve this equation in the adiabatic limit of fast response of $\\phi $ to changes in the chemical field.",
"Since in presence of active rotations the collective propulsion direction ${\\bf p}$ as determined by $\\phi $ does not simply follow the chemical field but is determined by a competition between chemotaxis and active rotations, we may not simply set $\\dot{\\phi }\\rightarrow 0$ , but need to integrate Eq.",
"(), MT for a (quasi-)stationary chemical field to describe fast orientational response appropriately.",
"In this adiabatic approximation we find (compare also [19]) $ \\phi _{\\rm ad} = -{\\rm atan2}(\\partial _y c,-\\partial _x c) + 2\\arctan \\left[{\\frac{-|\\nabla c| + \\sqrt{\\Omega ^2 - |\\nabla c|^2} \\tan \\left(\\sqrt{\\Omega ^2 - |\\nabla c|^2}t/2 + \\theta _0 \\right)}{\\Omega }} \\right] $ Here $\\theta _0({\\bf x})$ is an integration constant that is fixed by the initial orientations of the rotors.",
"For $|\\nabla c|<\\Omega $ (REF ) describes an anharmonic and anisotropic periodic oscillation.",
"As the chemical gradient increases, the rotation frequency decreases as $\\sqrt{\\Omega ^2- |\\nabla c|^2}$ , i.e.",
"${\\bf p}$ rotates slower where chemical gradients are strong.",
"At $|\\nabla c|=\\Omega $ the colloids cease to rotate (on average), i.e.",
"we have a transition from rotations to locking.",
"Here, we have an equilibrium between chemotaxis and rotations, where rotations steer the director field away from the `optimal' swimming direction up the chemical gradient (which is approached for $|\\nabla c|\\rightarrow \\infty $ ), and towards a direction perpendicular to it.",
"From Eq.",
"(REF ) it is straightforward to derive the distance from the `optimal' angle as $ \\Delta :=\\phi _{\\rm ad}(\\Omega )-\\phi _{\\rm ad}(\\Omega \\rightarrow 0)=2\\arctan \\left[{\\frac{\\gamma }{1+\\sqrt{1-\\gamma ^2}}}\\right] $ where we used the notation $\\gamma :=\\Omega /|\\nabla c|$ .",
"For $\\Omega < 1/2$ we can approximate $\\Delta \\approx \\gamma $ and conversely, for $\\Omega $ close to $|\\nabla c|$ , we have $\\Delta \\approx \\pi /2 - \\sqrt{2(\\Omega /|\\nabla c|-1)}$ .",
"In other words, the advective particle flux up density (chemical) gradients ($\\propto \\cos \\Delta $ ) decreases slowly with increasing $\\Omega $ when $\\Omega $ is small, but decreases faster and faster for large $\\Omega $ .",
"We now use Eq.",
"(REF ) to formulate a reduced model for the density of rotors whose orientation responds quickly to changes in the chemical field.",
"For a quasi-instantaneous orientational dynamics Eqs.",
"(REF ,) lead to $\\dot{\\rho }&=& \\nabla ^2 \\rho + \\kappa \\nabla ^2 \\rho ^3- \\nabla \\cdot \\left(\\rho {\\bf p}_{\\rm ad}(t) \\right) \\nonumber \\\\\\dot{c} &=& \\rho - c + \\mathcal {D}_c \\nabla ^2 c $ where ${\\bf p}_{\\rm ad}=\\left(\\cos (\\phi _{\\rm ad}),\\sin (\\phi _{\\rm ad})\\right)^T$ and $\\phi _{\\rm ad}$ is given by Eq.",
"(REF ).",
"To understand the linear stability of the uniform solution $(\\rho ,c)=(\\rho _0,\\rho _0)$ of these equations with $\\theta ({\\bf x})={\\rm const}$ , we choose a coordinate system with x-axis parallel to ${\\bf p}(t=0)$ , i.e.",
"$\\theta =\\arctan \\left[(|\\nabla c|+\\Omega \\tan (\\delta /2))/\\sqrt{\\Omega ^2-|\\nabla c|^2}\\right]$ and expand ${\\bf p}_{\\rm ad}$ in $\\partial _x c,\\partial _y c$ : ${\\bf p}_{\\rm ad} = \\left(\\begin{matrix} \\cos \\Omega t\\\\ \\sin \\Omega t \\end{matrix}\\right)+ {\\frac{1-\\cos \\Omega t}{\\Omega }}\\left(\\begin{matrix} \\sin \\Omega t\\\\ -\\cos \\Omega t \\end{matrix}\\right)\\partial _x c+ {\\frac{\\sin \\Omega t}{\\Omega }}\\left(\\begin{matrix} -\\sin \\Omega t\\\\ \\cos \\Omega t \\end{matrix}\\right)\\partial _y c $ Hence, we obtain: $\\dot{\\rho }^{\\prime } &=& (1+3\\rho _0^2 \\kappa ) \\nabla ^2 \\rho ^{\\prime }- \\left(\\begin{matrix} \\cos \\Omega t\\\\ \\sin \\Omega t \\end{matrix}\\right)\\nabla \\rho ^{\\prime }+ {\\frac{\\sin ^2 \\Omega t + \\cos \\Omega t - \\cos ^2 \\Omega t}{\\Omega }}\\partial _x \\partial _y c - {\\frac{\\sin \\Omega t}{\\Omega }}\\left[1-\\cos \\Omega t\\right]\\partial ^2_x c - {\\frac{\\cos \\Omega t \\sin \\Omega t}{\\Omega }}\\partial ^2_y c\\nonumber \\\\\\dot{c}^{\\prime } &=& \\rho ^{\\prime } - c^{\\prime } + \\mathcal {D}_c\\nabla ^2 c^{\\prime } $ Remarkably, the active rotations enter the linearized equations as explicitly time-dependent `driving' terms.",
"To understand how this leads to linear stability of the uniform state, we first consider the instructive case of fast chemical dynamics $\\dot{c} \\rightarrow 0$ , before generalizing our approach in the next paragraph.",
"Neglecting also chemical diffusivity ($\\mathcal {D}_c \\rightarrow 0$ ) which only helps stabilizing the uniform state, we have $c^{\\prime } = \\rho ^{\\prime }$ .",
"Now using a plane wave Ansatz $\\rho ^{\\prime }=r(t){\\rm exp}[-I{\\bf q}\\cdot {\\bf x}]$ in Eqs.",
"(REF ), Floquet theory predicts $r(t) =r(0) \\exp [\\mu t] g(t)$ [20], where $g(t)$ is some $2\\pi /\\Omega $ -periodic function.",
"Here, $\\mu $ is called the Floquet exponent and serves as a linear stability parameter; if it is positive $r(t)$ oscillates with growing amplitude, but for $\\mu <0$ small density fluctuations around the uniform state decay.",
"According to the Liouville formula in Floquet theory [20], here $\\mu $ is given by the time-average of the right hand side of Eqs.",
"(REF ).",
"Thus, we have $\\mu = -(1+3\\rho _0^3 \\kappa ) {\\bf q}^2$ meaning that active rotations suppress the chemotactic instability, which occurs in their absence (Eq.",
"REF ).",
"More specifically, the time-dependent driving terms in Eq.",
"(REF ) effectively suppress the impact of self-advection on the dynamics: in the absence of rotations this leads to a spinodal instability (see Eq.",
"REF ).",
"This confirms the intuitive argument given in the introduction of the main text: close to the uniform state, chemical gradients are too weak to align the swimming directions of our rotors and so rotations dominate chemotaxis.",
"As we discuss in the main text, this picture breaks down away from the uniform state and can give rise to a nonlinear instability allowing for structure formation with chemically interacting rotors despite the fact that the uniform state is now linearly stable.",
"Finally, we generalize the linear stability analysis of the previous paragraph to cases where we may not assume that $c$ is a fast variable and have to account for corresponding delay effects.",
"We first plug the Ansatz $\\rho ^{\\prime }=r(t){\\rm exp}[-I{\\bf q}\\cdot {\\bf x}]$ , $c^{\\prime }=C(t){\\rm exp}[-I{\\bf q}\\cdot {\\bf x}]$ into Eqs.",
"(REF ), which leads to $\\left(\\begin{matrix} \\dot{r} \\\\ \\dot{C} \\end{matrix}\\right)=A \\left(\\begin{matrix} r \\\\ C \\end{matrix}\\right) $ where the $2\\pi /\\omega $ -periodic matrix $A=(a)_{ij}$ has the following components $a_{11} &=& -{\\bf q}^2 (1+3\\rho _0^2 \\kappa )-I[q_x \\cos \\Omega t+q_y \\sin \\Omega t]\\\\a_{12} &= & {\\frac{\\rho _0}{\\Omega }}\\left[q_x q_y(\\cos ^2\\Omega t -\\sin ^2\\Omega t-\\cos \\Omega t) + (q_y^2 - q_x^2)\\cos \\Omega t\\sin \\Omega t + q_x^2 \\sin \\Omega t\\right] \\\\a_{21} &= & 1 \\\\a_{22} &=& -\\mathcal {D}_c {\\bf q}^2$ Denoting the fundamental matrix solution to Eq.",
"(REF ) as $X$ , we have $\\dot{X}(t)=A(t)X(t)$ , where $X(t)=P(t){\\rm exp}[ t B]$ with $P(t+2\\pi /\\omega )=P(t)$ , according to the Floquet theorem [20].",
"The eigenvalues of $B$ are the Floquet exponents $\\mu _{1,2}$ which determine the linear stability of the present problem.",
"From here, we follow [21] to perform a Floquet-Magnus expansion, i.e.",
"we expand $P,B$ $P &=& {\\rm exp}[\\Lambda ]; \\quad \\Lambda = {\\sum \\limits _{k=1}^{\\infty }}\\Lambda _k; \\quad \\Lambda _k(t+T)=\\Lambda _k(t) \\\\B&=&{\\sum \\limits _{k=1}^{\\infty }} B_k$ and determine $B$ up to order $k=2$ .",
"To first order, we find $ \\mu _1 = -(1+3\\rho _0^2 \\kappa ){\\bf q}^2; \\quad \\mu _2 = -1 - \\mathcal {D}_c {\\bf q}^2 $ which suggests linear stability for all values of $\\Omega $ .",
"The general result at second order ($k=2$ ) is more involved and its physical meaning is not immediate; thus we perform an asymptotic expansion up to second order in $\\nu =1/\\Omega $ which yields $\\mu _{1,2} = -{\\frac{1}{2}}\\left[1+ (1+3\\rho _0^2 \\kappa + \\mathcal {D}_c) {\\bf q}^2 + \\left| 1+ (\\mathcal {D}_c - 1 - 3\\rho _0^2 \\kappa ){\\bf q}^2\\right|\\right] \\pm I{\\frac{2q_x {\\bf q}^2 \\rho _0}{\\Omega ^2 \\left| 1+ (\\mathcal {D}_c - 1 - 3\\rho _0^2 \\kappa ){\\bf q}^2\\right|}}$ The real parts of $\\mu _{1,2}$ (Lyapunov exponents) represent the average growth rate of small perturbations of the uniform state.",
"Thus, even if $c$ is not a fast variable, at least for large $\\Omega $ active rotations suppress linear stability.",
"Note that convergence of the Floquet-Magnus expansion can be shown here for large $\\Omega $ when choosing the 2-norm of $A$ to satisfy the convergence criterion in [21].",
"To generalize our approach further, for small $\\Omega $ we calculated the Floquet exponents $\\mu _{1,2}$ numerically for different portions of the parameter space.",
"Here, we found that $\\mu _{1,2}$ have negative real parts (linear stability of the uniform state) in most regions of the parameter regime, even for relatively slow driving.",
"However, interestingly, they can become positive if the effective $\\rho _0$ is large enough and $\\kappa ,\\mathcal {D}_c,\\Omega $ are sufficiently small.",
"This demonstrates that besides the nonlinear instability mechanism we discussed in the main text, also delay effects in the response of $c$ to changes in $\\rho $ could provide a route to structure formation in slowly rotating chemically interacting particles.",
"Although this linear route to structure formation applies only to a restricted subset of the parameter space, it could be an interesting topic for further investigations.",
"Video 1 shows the dynamics of signalling rotors at short and intermediate times for $\\Omega =0$ (this corresponds to Fig.",
"a-c, main text); $\\Omega =1.2$ (Fig.",
"g-i); $\\Omega =2.2$ (Fig.",
"j-l) and $\\Omega =7$ .",
"Initial states are identical in all four cases.",
"The two latter cases ($\\Omega =2.2$ and $\\Omega =7$ ) lead to travelling wave patterns on timescales beyond the ones shown in the video.",
"Other parameters are chosen as in Fig. .",
"Videos 2,3,4 show the rotor dynamics ($\\rho $ and ${\\bf p}$ ) on comparatively long timescales for $\\Omega =0.25$ (This case corresponds to Fig.",
"d-f, but for larger system size), $\\Omega =1.2$ (g-i) and $\\Omega =2.2$ (j-l), respectively.",
"Video 5 shows $c$ and ${\\bf p}$ for parameters as in Fig.",
"d-f and comparatively short timescales."
]
] | 1605.05982 |
[
[
"Macroscopic Quantum Measurements of noncommuting observables"
],
[
"Abstract Assuming a well-behaving quantum-to-classical transition, measuring large quantum systems should be highly informative with low measurement-induced disturbance, while the coupling between system and measurement apparatus is \"fairly simple\" and weak.",
"Here, we show that this is indeed possible within the formalism of quantum mechanics.",
"We discuss an example of estimating the collective magnetization of a spin ensemble by simultaneous measuring three orthogonal spin directions.",
"For the task of estimating the direction of a spin-coherent state, we find that the average guessing fidelity and the system disturbance are nonmonotonic functions of the coupling strength.",
"Strikingly, we discover an intermediate regime for the coupling strength where the guessing fidelity is quasi-optimal, while the measured state is almost not disturbed."
],
[
"Introduction",
"In everyday life we continuously perform measurements.",
"For instance, to locate our friends we perform some kind of position measurements; similarly to read this text.",
"Presumably all this can be described with the quantum formalism.",
"But, obviously, these measurements are not of the standard von Neumann projective kind.",
"They are highly noninvasive while still collecting a large amount of information in a global, single shot.",
"Additionally, from a physical point of view, we expect a fairly simple coupling between system and observer.",
"We call measurements that fulfill these requirements “macroscopic quantum measurements”.",
"Our goal is to see if and how macroscopic quantum measurement can be realized.",
"For concreteness, think of a magnet whose $N$ atoms are idealized by spin 1/2 particles and assume that all spins are aligned parallel to an unknown direction $\\vec{u} \\in \\mathbb {R}$ .",
"By a freely selectable measurement, we are asked to estimate $\\vec{u}$ .",
"We define the score function $\\cos ^{2} \\theta /2$ , where $\\theta $ is the angle between $\\vec{u}$ and the best guess $\\vec{w}$ .",
"This task can be seen as a simultaneous measurement of the magnetization in the three principal directions $x, y$ and $z$ .",
"The corresponding measurement operators are collective spin operators $S_x = \\frac{1}{2} \\sum _i \\sigma _x^{(i)}$ (and similarly for $y$ and $z$ ).",
"The maximal average guessing fidelity is $\\mathcal {F}_{\\mathrm {opt}} = (N+1)/(N+2) = 1 - 1/N + O(N^{-2})$ , which is achievable either with nontrivial, entangling projective measurements or with a continuous general measurement with elements $\\Omega (\\vec{r}) \\propto \\left| \\vec{r} \\right\\rangle \\!\\left\\langle \\vec{r}\\right| ^{\\otimes N}$ [1].",
"The same fidelity can be asymptotically reached by random measurements on individual spins [2].",
"Here, in order to realize the simultaneous measurement of $x$ , $y$ and $z$ , we build upon von Neumann's measurement model by coupling all three observables $S_x$ , $S_y$ and $S_z$ to three independent pointers.",
"Each pointer is then individually measured and the results are processed to guess $\\vec{w}$ [3].",
"(This is in the spirit of the Arthurs-Kelly model [4], where position and momentum are coupled to two individual pointers.)",
"In the strong coupling regime, this measurement corresponds to randomly choosing a direction $\\vec{n}$ and to strongly measure along this axis [3].",
"This measurement model leads to an average fidelity of 3/4 (i.e., far away from the optimal value) for large $N$ The average fidelity of 3/4 is exactly the same value as estimating a classical spin with a Stern-Gerlach-type experiment with randomly chosen measurement axis..",
"Figure: Average fidelity ℱ av \\mathcal {F}_{\\mathrm {av}} for guessing the unknown state as a function of the inverse coupling strength Δ\\Delta for N=1,⋯,4N = 1,\\dots ,4 (from bottom to top).",
"The dashed lines correspond to the optimal value ℱ opt \\mathcal {F}_{\\mathrm {opt}}.",
"For N≥2N\\ge 2, the global maximum is reached by some nonzero Δ\\Delta and is close to the optimal value.The coupling strength between system and pointers influences the guessing fidelity and the measurement-induced disturbance.",
"As shown by Poulin [6], the measurement of a single observable (e.g., only $S_x$ ) is more informative and more disturbing as the coupling strength increases for a natural choice of the initial pointer state and the final measurement.",
"For $N$ parallel spins measured in any direction, it turns out that the system is asymptotically undisturbed if the coupling is much less than $N^{-1/2}$ [6].",
"Poulin then suggested to perform many measurements in this weak coupling regime; potentially for several noncommuting observables.",
"He conjectured that this should eventually give maximal information about the initial state without disturbing it.",
"However, apart from practical questions, the limit of information gain and the measurement-induced disturbance of “infinitely many and infinitely weak” measurements is unclear so far.",
"Other contributions, like Ref.",
"[7], discuss the influence of the coupling strength, but do not compare to optimal strategies.",
"In this paper, we present an example of the surprising nonmonotonic relation between information gain, measurement-induced disturbance and the coupling strength when noncommuting observables are simultaneously measured.",
"By studying the example of estimating the direction of parallel spins, we find an intermediate regime for the coupling strength where the average guessing fidelity is numerically very close to the optimal value $\\mathcal {F}_{\\mathrm {opt}}$ ; both for small and large $N$ .",
"In the very same coupling regime, the measurement-induced disturbance is small, in particular as $N$ increases.",
"More specifically, the averaged Bloch vector of the post-measured states is almost identical with the initial state, meaning that repeating the protocol by independent observers gives almost the same estimates.",
"We claim that the quasi-optimal information gain and small measurement-induced disturbance is reminiscent of classical measurements.",
"In this sense, the presented model is an example of a quantum measurement that behaves classically for large system sizes."
],
[
"General measurement model and average fidelity",
"In the following, we denote a normalized spin state orientated in the direction of a general vector $\\vec{x} \\in \\mathbb {R}^3$ by $\\mathinner {|{\\vec{x}}\\rangle } \\in \\mathbb {C}^2$ .",
"Generally, we consider a scheme where the system state $\\mathinner {|{\\vec{u}}\\rangle }^{\\otimes N} $ is coupled to a measurement apparatus $| \\phi \\rangle $ –called pointer– through virtue of an interaction Hamiltonian $H_{\\mathrm {int}}$ .",
"Afterwards, the pointer is measured.",
"This procedure is often referred to as the von Neumann measurement model [8], [9], [10].",
"For simplicity, the coupling with strength $\\mu $ is uniformly switched on for a unit time and dominates meanwhile all other processes; thus $U = \\exp (-i \\mu H_{\\mathrm {int}})$ .",
"After this the joint state of system and pointer reads $U \\mathinner {|{\\phi }\\rangle } \\otimes \\mathinner {|{\\vec{u}}\\rangle }^{\\otimes N}$ .",
"The subsequent measurement of the pointer is modeled by a projective measurement with outcome $\\vec{r}$ for the eigenvectors $| \\psi _{\\vec{r}} \\rangle $ .",
"(The dimension of $\\vec{r}$ depends on the pointer space.)",
"This outcome is then classically processed.",
"Knowledge about the initial pointer state, the coupling and the final measurement allows one to calculate the optimal guess state $\\left| \\vec{w}_{\\vec{r}}\\right\\rangle $ .",
"So far, this is a fully general measurement procedure and is linked to the Kraus operator formalism via $E(\\vec{r}) = \\mathinner {\\langle {\\psi _{\\vec{r}}}|} U \\mathinner {|{\\phi }\\rangle }.$ To measure the accuracy of the measurement procedure we choose the average fidelity $\\left| \\left\\langle \\vec{w}_{\\vec{r}} \\right| \\left.",
"\\vec{u}\\right\\rangle \\right|^2$ .",
"The average score is then given by $\\mathcal {F}_{\\mathrm {av}} = \\int \\frac{d\\vec{u}}{4\\pi }~\\int d\\vec{r}~p(\\vec{r}|\\vec{u})~\\vert \\mathinner {\\langle {\\vec{w}_{\\vec{r}}|\\vec{u}}\\rangle }\\vert ^2,$ where $p(\\vec{r}|\\vec{u}) = \\Vert E(\\vec{r}) \\left| \\vec{u} \\right\\rangle ^{\\otimes N} \\Vert ^2$ is the probability to get outcome $\\vec{r}$ given that the initial state was $\\left| \\vec{u}\\right\\rangle ^{\\otimes N}$ .",
"A typical instance is a pointer with one spatial degree of freedom [6].",
"For example, the initial spatial wave function of the pointer is a Gaussian function with spread $\\Delta $ .",
"In order to measure an observable $A$ , one then defines $H_{\\mathrm {int}} = p\\otimes A$ , where $p$ represents the displacement operator in the pointer space, which is formally equivalent to the momentum operator.",
"Thus, the coupling induces a momentum kick on the pointer whose strength depends on the initial system state.",
"Finally, a position measurement of the pointer allows some inference about the system.",
"Information gain and invasiveness of this procedure manifest themselves in the relationship between the coupling strength $\\mu $ , the spread of the Gaussian, $\\Delta $ , and the spectrum of the eigenvalues of the operator $A$ .",
"By redefining $\\Delta $ , one can set $\\mu = 1$ without loss of generality.",
"Thus, the coupling strength of the measurement is proportional to $\\Delta ^{-1}$ .",
"A $\\Delta $ small compared to the spectral gap of $A$ corresponds to a strong coupling that resolves the individual eigenvalues.",
"The momentum kicks induced by two nearby eigenstates of $A$ are distinguishable.",
"On the other side, a $\\Delta $ large compared to the spectral gap of $A$ means that system states prepared in neighboring eigenstates cannot be well distinguished.",
"This is the regime of fuzzy measurements [10].",
"Note that the maximal information gain and the maximal disturbance are typically realized in the limit of a vanishing $\\Delta $ , that is, when the pointer model approaches the ideal projective measurement of $A$ .",
"Hence, even for large $N$ , the basic idea of small invasiveness and large information gain does not seem to be reachable with this simple construction."
],
[
"Specific modeling",
"To measure the direction of $| \\vec{u} \\rangle ^{\\otimes N}$ , a pointer with three spatial degrees of freedom seems to be more appropriate.",
"We choose $\\mathinner {|{\\phi }\\rangle } = \\frac{1}{(2 \\pi \\Delta ^2)^{3/4}} \\int {dx dy dz ~e^{-\\frac{(x^2+y^2+z^2)}{4 \\Delta ^2}} \\mathinner {|{x}\\rangle } \\mathinner {|{y}\\rangle } \\mathinner {|{z}\\rangle }},$ where $x$ , $y$ and $z$ denote the coordinates of the pointer.",
"The spatial distribution is a rotationally invariant Gaussian function with spread $\\Delta $ .",
"The direction of $| \\vec{u} \\rangle ^{\\otimes N} $ is determined by the three expectation values of the collective spin operators $S_x, S_y$ and $S_z$ .",
"Thus, a classically inspired interaction Hamiltonian reads $H_{\\mathrm {int}} = p_x \\otimes S_x + p_y \\otimes S_y + p_z \\otimes S_z = \\vec{p}\\cdot \\vec{S},$ which follows the pattern of the Arthurs-Kelly model.",
"The $p_k$ for $k = x,y,z$ represent the displacement operators in the $x$ , $y$ and $z$ directions, respectively.",
"Finally, a position measurement with outcome $\\vec{r} \\in \\mathbb {R}^3$ is performed on the pointer.",
"We simply guess $| \\vec{w}_{\\vec{r}} \\rangle = \\left| \\vec{r} \\right\\rangle $Note that this choice is not always optimal.",
"For some (nonoptimal) values of $\\Delta $ , the optimal choice is $| \\vec{w}_{\\vec{r}} \\protect \\rangle = | -\\vec{r} \\protect \\rangle $ .",
"For symmetry reasons the optimal guess has to lie on the axis of $\\vec{r}$ ..",
"The model is rotationally symmetric.",
"Without loss of generality, we hence consider a fixed input state $| \\vec{u} \\rangle = \\left| \\hat{e}_z \\right\\rangle \\equiv \\left| \\uparrow \\right\\rangle $ .",
"With $p(\\vec{r}) \\equiv p(\\vec{r}|\\hat{e}_z )$ , Eq.",
"(REF ) then simplifies to $\\mathcal {F}_{\\mathrm {av}} = 2\\pi \\int r^2 \\sin \\theta dr d\\theta p(\\vec{r}) \\cos ^2 \\theta /2,$ since $p(\\vec{r})$ does not depend on the azimuthal angle.",
"In the limit $\\Delta \\rightarrow 0$ , the measurement model is suboptimal and the average fidelity is $\\mathcal {F}_{\\mathrm {av}} = \\frac{1}{4} (3N+2)/(N+1)$ for even $N$ and $\\mathcal {F}_{\\mathrm {av}} = \\frac{1}{4} (3N+5)/(N+2)$ for odd $N$ [3].",
"These values are bounded by $\\frac{3}{4}$ and are hence far away from the optimal fidelity.",
"As we will show now, the three spatial dimensions of the pointer combined with noncommuting operators in Eq.",
"(REF ) involve a rich behavior for nonvanishing $\\Delta $ .",
"In contrast to the one-dimensional pointer, information gain and disturbance are not monotonically related to the coupling strength."
],
[
"Small number of spins",
"We start by calculating the average fidelity $\\mathcal {F}_{\\mathrm {av}}$ for one to four spins as a function of the spread $\\Delta $ of the Gaussian pointer.",
"In these cases the expression in Eq.",
"(REF ) is easy to work with due to the small number of spins.",
"The Kraus operators $E(\\vec{r})$ of the interaction, Eq.",
"(REF ), can be derived analytically.",
"The average fidelity is analytically calculated in the case of one single qubit, while for more spins the integration over the radial component of $\\vec{r}$ is performed numerically.",
"The results are shown in Fig.",
"REF .",
"For two or more spins the optimal $\\Delta $ is clearly distinct from zero and $\\mathcal {F}_{\\mathrm {opt}}$ (dashed lines in Fig.",
"REF ) can almost be reached.",
"Analyzing the graph shown in Fig.",
"REF in more detail reveals three distinct regions.",
"For very small values of $\\Delta $ (i.e., in the strong coupling regime) the limits predicted in [3] are recovered, yielding an average fidelity of $2/3$ for $N=1$ and 2 and $\\mathcal {F}_{av} = 7/10$ for $N=3$ and $N = 4$ , respectively.",
"On the other extreme, for $\\Delta > 1$ , we see that the average fidelity starts to decline rapidly.",
"This can be understood if one notices that after a certain value of $\\Delta $ the coupling is so weak that the procedure is essentially equivalent to randomly guessing, therefore yielding an average fidelity of $1/2$ .",
"The intermediate region is particularly intriguing because in the case of two and more spins the average fidelity is superior to what can be achieved when $\\Delta \\rightarrow 0$ .",
"This means that in this case a lesser coupling strength can achieve better results than a strong coupling."
],
[
"Large number of spins",
"For $N>4$ the full calculation becomes cumbersome without approximations.",
"To continue with larger $N$ , we calculate a lower bound on $\\mathcal {F}_{\\mathrm {av}}$ .",
"For this, we resolve the identity $\\mathbb {1} = \\left| \\vec{r} \\right\\rangle \\!\\left\\langle \\vec{r}\\right| ^{\\otimes N} + (\\mathbb {1}-\\left| \\vec{r} \\right\\rangle \\!\\left\\langle \\vec{r}\\right| ^{\\otimes N})$ and insert it in the expression for $p(\\vec{r})$ .",
"The reason for this choice is that $E(\\vec{r})$ is diagonal in the eigenbasis of $\\vec{r}\\cdot \\vec{S}$ and the Kraus operator leading to the optimal $\\mathcal {F}_{\\mathrm {av}}$ should have a large overlap with $\\left| \\vec{r} \\right\\rangle \\!\\left\\langle \\vec{r}\\right| ^{\\otimes N}$ M.-O.",
"Renou et al., in prep..",
"In the following, we bound $p(\\vec{r}) \\ge | \\left\\langle \\vec{r} \\right| ^{\\otimes N} E(\\vec{r}) \\left|\\uparrow \\right\\rangle |^{2N}$ .",
"For general $N$ , one has to find further simplifications for $p(\\vec{r})$ .",
"For an $\\mathcal {F}_{\\mathrm {av}}$ scaling as $1-O(1/N)$ (i.e., the optimal scaling), it is necessary that $\\left| \\left\\langle \\vec{r} \\right| \\left.",
"\\uparrow \\right\\rangle \\right|^{2} = 1-O(1/N)$ for almost all $\\vec{r}$ which have $O(1)$ support from $p(\\vec{r})$ .",
"Hence, the most important $\\left| \\vec{r} \\right\\rangle $ are $O(1/\\sqrt{N})$ -close to $\\left|\\uparrow \\right\\rangle $ .",
"Inspecting the matrix element $E_{\\vec{r}} = \\left\\langle \\vec{r} \\right| ^{\\otimes N} E(\\vec{r}) \\left| \\uparrow \\right\\rangle ^{\\otimes N}$ in the momentum basis of the pointer, $E_{\\vec{r}} \\propto \\int d \\vec{p} e^{i \\vec{r}\\cdot \\vec{p}} e^{- \\Delta ^2 p^2} \\left\\langle \\vec{r} \\right| ^{\\otimes N} e^{-i \\vec{p}\\cdot \\vec{S}} \\left| \\uparrow \\right\\rangle ^{\\otimes N},$ one notices that this implies $\\vec{p} = O(1/\\sqrt{N})$ , which in turn means that $\\Delta = O(\\sqrt{N})$ .",
"In this regime, it is advantageous to perform a Holstein-Primakoff transformation [13] to express $S_{+} = S_x + i S_y$ as $S_{+}=\\sqrt{1-a^{\\dagger }a/N} a$ with $[a^{\\dagger },a] = 1$ ; similarly for $S_{-}=S_{+}^{\\dagger }$ and $S_z = \\frac{1}{2} [S_{+},S_{-}] = \\frac{N}{2}- a^{\\dagger } a$.",
"The input state $\\left| \\uparrow \\right\\rangle ^{\\otimes N}$ is transformed to the vacuum $| 0 \\rangle $ , which is an eigenstate of $a$ with eigenvalue 0.",
"The state $| r \\rangle ^{\\otimes N}$ corresponds to a coherent state, since rotations on the Bloch sphere are approximated by planer displacements.",
"In summary, we have the mapping $\\left\\langle \\vec{r} \\right| ^{\\otimes N} e^{-i \\vec{p}\\cdot \\vec{S}} \\left|\\uparrow \\right\\rangle ^{\\otimes N} \\rightarrow \\left\\langle 0 \\right| \\exp [f(a,a^{\\dagger })] \\left| 0 \\right\\rangle $ with a lengthy function $f$ in $a$ and $a^{\\dagger }$ .",
"The next step is to find the optimal value of $\\Delta $ .",
"For this, we write $\\exp [f(a,a^{\\dagger })]$ in a Taylor series for $N\\rightarrow \\infty $ .",
"Keeping terms up to order $O(1)$ , one can analytically calculate $\\mathcal {F}_{\\mathrm {av}}$ and find a maximum for $\\Delta _{\\mathrm {opt}} = \\sqrt{N/8}$ .",
"Heuristic arguments indicate that the following results are asymptotically unchanged by adding $o(\\sqrt{N})$ terms to $\\Delta _{\\mathrm {opt}}$ .",
"In order to have more precise results for $\\mathcal {F}_{\\mathrm {av}}$ , we take into account terms up to order $O(1/N)$ in the power series of $\\exp [f(a,a^{\\dagger })]$ .",
"We are still able to analytically integrate over $\\vec{p}$ in Eq.",
"(REF ) and the radial part of $\\vec{r}$ in Eq.",
"(REF ).",
"The final integration over $\\theta $ is done numerically.",
"We find that $\\mathcal {F}_{\\mathrm {av}} \\gtrsim 1 - \\epsilon _N / N$ , where $\\epsilon _N$ seems to asymptotically converge to a value slightly above one (see Fig.",
"REF ).",
"This indicates that the approximate lower bound does asymptotically not coincide with $\\mathcal {F}_{\\mathrm {opt}} = 1-1/N + O(N^{-2})$ .",
"However, the relative difference between the two clearly goes to zero.",
"Moreover, taking into account more matrix elements in addition to Eq.",
"(REF ) could reduce the gap.",
"Figure: Scaling factor ϵ N ≲N(1-ℱ av )\\epsilon _N \\lesssim N(1-\\mathcal {F}_{\\mathrm {av}}) from the approximate lower bound on ℱ av \\mathcal {F}_{\\mathrm {av}} (upper, blue curve) compared to the optimal scaling factor N(1-ℱ opt )=(1+2/N) -1 N(1-\\mathcal {F}_{\\mathrm {opt}}) = (1+2/N)^{-1} (lower, orange curve).",
"For large NN, ϵ N \\epsilon _N seems to stay slightly above one, which indicates a nonoptimal lower bound.",
"For N<150N < 150, the approximation clearly does not hold."
],
[
"Measurement-induced disturbance",
"Interestingly, it turns out that the presented measurement scheme is hardly invasive.",
"To show this, we calculate the quantum fidelity between the pre- and post-measured state averaged over all measurement outcomes.",
"The post-measured state reads $\\rho _{\\mathrm {post}} = \\mathrm {Tr}_{\\text{pointer}} U \\left| \\phi \\right\\rangle \\!\\left\\langle \\phi \\right| \\otimes \\left| \\uparrow \\right\\rangle \\!\\left\\langle \\uparrow \\right| ^{\\otimes N} U^{\\dagger }$ .",
"The disturbance is then defined to be $D = 1 - \\left\\langle \\uparrow \\right| ^{\\otimes N} \\rho _{\\mathrm {post}} \\left| \\uparrow \\right\\rangle ^{\\otimes N}$ .",
"For calculating the trace of $\\rho _{\\mathrm {post}}$ it is most convenient to work in the momentum basis of the pointer space.",
"Then, simple manipulations give $D = 1- \\int d^3p \\left| \\phi (\\vec{p}) \\right|^2 \\left[ 1 - \\sin ^2(p/2) \\sin ^2 \\theta \\right]^N.$ For Gaussian pointers, this can in principle be calculated for any $N$ .",
"For small $N$ , we observe that the maximal $D$ is found for some $\\Delta >0$ , which does however not correspond to the $\\Delta $ leading to the maximal information (see Fig.",
"REF ).",
"For $N \\gg 1$ , a closed formula is only found for an approximated integrand.",
"Similar as before, we set $\\Delta = c \\sqrt{N/8}$ and do a Taylor series for $N \\rightarrow \\infty $ .",
"In lowest order, one finds the Lorentzian-like function $D = (1+c^2)^{-1}$ , which does not indicate any particular behavior at $c = 1$ , for which $\\mathcal {F}_{\\mathrm {av}}$ is maximal in the large $N$ limit.",
"Higher orders exhibit simple, but long expressions in $c$ , which simplify for $c = 1$ to $D = \\frac{1}{2} + \\frac{23}{1440 N^2} + O \\left( \\frac{1}{N^3} \\right).$ Hence, in the limit of large $N$ , the disturbance is close to the minimal disturbance $D_{\\min } = (N+1)/(2N +1)$ in the case of optimal guessing fidelity [14].",
"A nonunit value seems to indicate a rather severe change in the state.",
"We note, however, that the overlap between two states $| \\vec{u} \\rangle ^{\\otimes N}$ and $| \\vec{w} \\rangle ^{\\otimes N}$ generally decays exponentially in $N$ unless the angle between $\\vec{u}$ and $\\vec{w}$ is at most $O(1/\\sqrt{N})$ .",
"A constant overlap for large $N$ indicates an $O(1/\\sqrt{N})$ closeness between $\\left| \\uparrow \\right\\rangle ^{\\otimes N} $ and $\\rho _{\\mathrm {post}}$ on the Bloch sphere.",
"Figure: Disturbance measured by the quantum fidelity between the pre- and post-measured states as function of Δ\\Delta for N=1,2,3N = 1,2,3 (from bottom to top).",
"The solid lines correspond to the exact expression, Eq.",
"(), while the dashed curves are the lowest-order approximations for large NN, D≈1+8Δ 2 /N -1 D \\approx \\left( 1+ 8\\Delta ^2/N \\right)^{-1}.",
"The points on the curves indicate where Δ=Δ opt \\Delta = \\Delta _{\\mathrm {opt}}.This becomes even more evident when we consider the Bloch vector of the spin ensemble before and after measurement.",
"The expectation value $\\langle \\vec{S} \\rangle $ for both states can be calculated in a straightforward manner.",
"One has $\\langle S_x \\rangle = \\langle S_y \\rangle = 0$ for both $\\left|\\uparrow \\right\\rangle ^{\\otimes N}$ and $\\rho _{\\mathrm {post}}$ .",
"For the $z$ component, we find $\\langle S_z \\rangle _{\\left| \\uparrow \\right\\rangle ^{\\otimes N}} = N/2$ and $\\langle S_z \\rangle _{\\rho _{\\mathrm {post}}} = \\frac{1}{6}N\\left[ 1 + e^{-1/(8\\Delta ^2)}\\left( 2- 1/(2 \\Delta ^2) \\right) \\right] = N/2 - 1 + O(1/N)$ , where the last expression holds for $\\Delta = \\Delta _{\\mathrm {opt}}$ .",
"In words, the length of the post-measured Bloch vector is only minimally reduced.",
"A repetition of the measurement will give almost the same information about the Bloch vector than the first one.",
"This is in stark contrast to some other optimal measurement strategies.",
"For example, in Ref.",
"[2], the single-spin measurements in a random direction leaves behind a completely depolarized product state; therefore making it impossible for a different pointer to gain any information about the initial state in a subsequent measurement."
],
[
"Discussion and open questions",
"In summary, the simultaneous measurement of the noncommuting observables $S_x, S_y$ and $S_z$ can be made highly sensitive and hardly disturbing in a regime where the coupling to the pointers is relatively weak.",
"This holds true in particular for large spin ensembles.",
"Via an approximate Trotter expansion, our results also guide a quantitative analysis of proposals [6] for sequential measurements of noncommuting observables in the weak coupling regime.",
"Given the features of the model together with a fairly simple scheme, the discussed model is reminiscent of a classical measurement.",
"This observation allows us to speculate about a general class of “macroscopic quantum measurements” as measurements within the quantum formalism with high information gain, low disturbance and simple physical pointer models.",
"In other words, the present model could be a starting point to understand the emergence of classical behaviour of quantum measurements for large systems.",
"It might be interesting to study a connection to recent work on classifying different levels of simultaneous measurements [15].",
"Further research will be dedicated to other schemes of simultaneous measurements as a function of coupling strength, in particular when the observables have well defined macroscopic limits such as position and momentum.",
"It is likely that the very same measurement model can also be used to illustrate the classical appearance of macroscopic quantum states by changing the input state.",
"Suppose a superposition of macroscopically distinct states (i.e., a Schrödinger-cat state) [16] is perfectly prepared.",
"We expect that the coherence between these distinct states cannot be witnessed if the system is subject to a coupling as in Eq.",
"(REF ).",
"This reflects the basic idea of einselection [17], where the measurement apparatus takes the role of the environment.",
"In addition, it would be interesting to see whether the presented coupling could also serve as an illustrating example of quantum Darwinism [18].",
"Further open questions include a better characterization of $\\mathcal {F}_{\\mathrm {av}}$ for $N\\gg 1$ in terms of the precise scaling with $N$ as well as the robustness with respect to variations of $\\Delta _{\\mathrm {opt}}$ .",
"Also, more realistic initial states, that is, ones that better match the properties of real-world examples, are interesting to study.",
"A first attempt would include nonmaximally polarized and thermal states.",
"Acknowledgments.— We thank Pavel Sekatski and Serge Massar for stimulating discussions.",
"This work was supported by the National Swiss Science Foundation (SNSF), the Austrian Science Fund (FWF), grant number J3462, the COST Action No.",
"MP1006 and the European Research Council (ERC MEC)."
]
] | 1605.05956 |
[
[
"On the Sampling Strategy for Evaluation of Spectral-spatial Methods in\n Hyperspectral Image Classification"
],
[
"Abstract Spectral-spatial processing has been increasingly explored in remote sensing hyperspectral image classification.",
"While extensive studies have focused on developing methods to improve the classification accuracy, experimental setting and design for method evaluation have drawn little attention.",
"In the scope of supervised classification, we find that traditional experimental designs for spectral processing are often improperly used in the spectral-spatial processing context, leading to unfair or biased performance evaluation.",
"This is especially the case when training and testing samples are randomly drawn from the same image - a practice that has been commonly adopted in the experiments.",
"Under such setting, the dependence caused by overlap between the training and testing samples may be artificially enhanced by some spatial information processing methods such as spatial filtering and morphological operation.",
"Such interaction between training and testing sets has violated data independence assumption that is abided by supervised learning theory and performance evaluation mechanism.",
"Therefore, the widely adopted pixel-based random sampling strategy is not always suitable to evaluate spectral-spatial classification algorithms because it is difficult to determine whether the improvement of classification accuracy is caused by incorporating spatial information into classifier or by increasing the overlap between training and testing samples.",
"To partially solve this problem, we propose a novel controlled random sampling strategy for spectral-spatial methods.",
"It can greatly reduce the overlap between training and testing samples and provides more objective and accurate evaluation."
],
[
"Introduction",
"Spectral-spatial processing have attracted increasing attentions during the past several years.",
"Bringing spatial information into traditional single pixel based spectral analysis leads to better modelling of local structures in the image and facilitates more accurate land-cover and object classification.",
"While a large portion of the hyperspectral remote sensing community have focused their research on improving classification accuracy by developing a variety of spectral-spatial methods [1], [2], [3], [4], few attention has been paid to experimental settings.",
"Evaluation of hyperspectral image classification methods requires careful design of experiments such as appropriate benchmark data sets, sampling strategy to generate training and testing data, and appropriate and fair evaluation criteria [1], [5].",
"In the scope of supervised classification, we find that traditional experimental designs for spectral processing are often improperly used in the context of spectral-spatial processing, leading to unfair or biased performance evaluation.",
"This is particularly the case when training and testing samples are randomly drawn from the same image/scene which is a common setting in the hyperspectral classification research due to limited availability of benchmark data and high cost of ground truth data collection.",
"Fig.",
"REF shows a typical spectral-spatial hyperspectral image classification system built on a supervised learning scheme.",
"Training and testing samples are drawn from an image data set following a specific sampling strategy.",
"After image preprocessing which may involve spectral-spatial operations, feature extraction step fuses the spectral and spatial information to explore the most discriminative feature for different classes.",
"The extracted features are used to train a classifier that minimises the error on the training set.",
"In the testing step, the learned classifier is used to predict the classes of testing samples based on the extracted features.",
"The testing error is given by comparing the predicted labels with the ground truth, which can be used as a performance indicator for image preprocessing, feature extraction and classification methods.",
"Figure: Framework of a supervised hyperspectral image classification system that uses spectral-spatial features.In the experimental setting, the sampling strategy plays an important role in the classifier learning and evaluation.",
"Given a dataset including a hyperspectral image and its land-cover classes or other ground truth data, in most cases training and testing samples are not given in advance.",
"A sampling strategy has to be employed to create the training and testing sets [6], [7], [8].",
"Random sampling is a natural choice since it treats all labelled data equally and each sample would be selected with the same probability.",
"However, by this method some classes with small number of labeled samples may have much less selected samples than expectation.",
"Therefore, a more sophisticated sampling method, stratified random sampling, is often used [7].",
"To guarantee each class having sufficient samples, it firstly groups those labelled samples into subsets based on their class labels, and then random sampling is carried out within each subset.",
"In term of the number of training samples in each subset, it normally requires that proportion of each group should be the same as in the population.",
"Then the rest of samples are employed as testing samples in the testing step.",
"This method is very simple to implement, reproducible, and of statistical significance.",
"To the best of our knowledge, a number of hyperspectral classification methods adopted this option in the experimental setting [2], [9], [3], [10], [11].",
"In the following sections, we refer to the stratified random sampling as random sampling.",
"Before proceeding to the issue of random sampling, we have to re-affirm some basic principles for supervised learning.",
"Under statistical learning frame, a common assumption for inference purpose is that random variables are independent and identically distributed (i.i.d.).",
"The identical condition implies that training and testing samples are generated from the same data distribution.",
"The independent condition requires that the occurrence of each sample do not affect the probability of other samples.",
"i.i.d.",
"shall hold for data in different forms, for example, both raw spectral responses and extracted features.",
"Most supervised hyperspectral image classification approaches assume that data are i.i.d.. Pixels in the same class shall have similar spectral responses or spectral-spatial features so that a trained classifier can be generalised to predict the labels of unseen samples.",
"However, the independent assumption does not always hold if the training and testing samples are not carefully selected.",
"In general, arbitrary samples selected from a population by random sampling can be seen roughly independent from each other, or at least independent between the sets of training and testing samples.",
"However, for hyperspectral images, the random sampling is usually undertaken on the same image.",
"Consequently, those randomly selected training samples spread over the image and the testing samples will locate adjacent to them.",
"Then the independence assumption would become jeopardised due to the spatial correlation between training and testing samples.",
"This is not a problem for the traditional pixel based spectral analysis methods in which no spatial information is used.",
"However, when it comes to the spectral-spatial methods, the training and testing samples would inevitably interact with each other, and thus the dependence caused by overlap or partial overlap between the training and testing data could result in exaggerated classification accuracy.",
"To be more specific, the information from the testing set could be used in the training step by spatial operations, leading to a biased evaluation results.",
"The sampling problem was originally noticed by Friedl et al.",
"[5], who referred to overlap as auto-correlation.",
"Zhen et al.",
"[12] compared the influence of different sampling strategies to the classification accuracy.",
"However, none of these work has given theoretical analysis on the problems and provide an effective solution.",
"Therefore, it is necessary to revisit the sampling strategy and data dependence for supervised hyperspectral image classification, especially those based on spectral-spatial processing.",
"In-depth discussion on this issue can be made from both experiment and the computational learning theory points of view.",
"In this paper, we study the relationship between sampling strategies and the spectral-spatial processing in hyperspectral image classification, when the same image is used for training and testing.",
"We find that the experimental setting with random sampling makes data dependence on the whole image be increased by some spectral-spatial operations, and in turn increases the dependence between training and testing samples For the sake of conciseness and without confusion, we use “dependence between training and testing data\" and “data dependence\" interchangeably in the rest of the paper.. To address this problem, we propose an alternative controlled random sampling strategy to alleviate the side effect of traditional random sampling on the same hyperspectral image.",
"This leads to a fairer way to evaluate the effectiveness of spectral-spatial methods for hyperspectral classification.",
"In summary, the contribution of this paper are in three aspects: We point out that the traditional random sampling from the same image experimental setting is not suitable for supervised spectral-spatial classification algorithms.",
"This helps to re-examine the performance evaluation of various spectral-spatial classification methods.",
"We find that under the random sampling setting, spectral-spatial methods can enhance the data dependence and improve the classification accuracy.",
"We give a theoretical explanation to this phenomenon via computational learning theory.",
"We propose a novel controlled random sampling strategy which can greatly reduce the overlap between training and testing samples caused by spatial processing, such that more objective and accurate evaluation can be achieved.",
"The rest of this paper is organized as follows.",
"Section reviews the spectral-spatial processing that have been commonly used in hyperspectral image classification.",
"Section provides an in-depth analysis on the dependency between training and testing samples.",
"The spatial information embedded in the spectral-spatial processing under the experimental setting with random sampling is excavated and examined.",
"Section analyses the overlap between neighboring training and testing samples caused by spatial operations.",
"Such overlap increases the dependence between training and testing samples, which may lead to mistakenly using of the testing data in the training process.",
"Section discusses the relationship among spectral-spatial processing, data dependance and classification accuracy via computational learning theory.",
"A new sampling strategy is proposed in Section which reduces the influence of overlap between training and testing data.",
"To prove its advantage over random sampling, a series of experiments are developed and results are presented in Section .",
"At last the conclusions are drawn in Section ."
],
[
"Spectral-spatial Processing in Hyperspectral Image Classification",
"The advantage of using hyperspectral data in land cover classification is that spectral responses reflect the properties of components on the ground surface [7].",
"Therefore, raw spectral responses can be used directly as the discriminative features of different land covers.",
"At the same time, hyperspectral data also possesses the basic characteristic of the conventional images - the spatial information which corresponds to where a pixel locates in the image.",
"The spatial information can be represented in different forms, such as structural information including the size and shape of objects, textures which describe the granularity and patterns, and contextual information which can express the inter-pixel dependency [3].",
"This is also the foundation of development of most spectral-spatial methods for hyperspectral image classification.",
"In general, spectral-spatial information can contribute to hyperspectral image classification through three ways.",
"Firstly, in image preprocessing, it can be used for image denoising, morphology, and segmentation.",
"Image denoising enables the reduction of random noises introduced from sensor, photon effects, and calibration errors.",
"Several approaches have been exploited for this purpose, for example, smoothing filters, anisotropic diffusion, multi-linear algebra, wavelet shrinkage, and sparse coding methods [13].",
"In most cases, denoising can be done by applying a local filter with designed or learned kernel across the whole image.",
"In mathematical morphology, operations are performed to extract spatial structures of objects according to their spectral responses [14], [3].",
"Similar information is explored in image segmentation, which groups spatially neighboring pixels into clusters based on their spectral distribution [15], [9].",
"Secondly, common usage of joint spectral-spatial information lies in the feature extraction stage.",
"While traditional spectral features are extracted as responses at single pixel level in hyperspectral images, spectral-spatial feature extraction methods use spatial neighborhood to calculate features.",
"Typical examples include texture features such as 3D discrete wavelet [10], 3D Gabor wavelet [16], 3D scattering wavelet[17], and local binary patterns [18].",
"Morphological profiles, alternatively, use closing, opening, and geodesic operators to enhance spatial structures of objects [19], [20], [21].",
"Other spectral-spatial features include spectral saliency [22], spherical harmonics [23], and affine invariant descriptors [24].",
"Heterogeneous features can be further fused using feature selection or reduction approaches [25].",
"Thirdly, some image classification approaches rely on spatial relation between pixels for model building.",
"A direct way of doing so is calculating the similarity between a pixel and its surrounding pixels [26].",
"Markov random field, for example, treats hyperspectral image as dependent data and uses spectral information in the local neighborhood to help pixel class prediction [27], [9], [28].",
"Similar spatial structures are explored in conditional random fields [29], hypergraph modelling [30], and multi-scale analysis [11].",
"The spatial information can also be explored in constructing composite kernels in support vector machines [31].",
"While supervised learning approaches, such as K-nearest neighbors, linear discriminant analysis, Bayesian analysis, support vector machines, etc.",
"are widely used in these classification tasks [32], [33], some approaches adopt semi-supervised or active learning strategies [34], [35]."
],
[
"Spatial Information Embedded in Random Sampling",
"Random sampling makes the training and testing samples spread over the image, embedding plenty of underlying spatial information.",
"In this section, we point out that the embedded spatial information will mistakenly influence the classifier learning and evaluation.",
"We exploit this problem in a specific/extreme way, by which a hyperspectral classification task can even be done without spectral information.",
"In many benchmark hyperspectral datasets, pixels in the same class are not distributed randomly in the image.",
"On the contrary, they tend to exist in continuous regions and follow a certain spatial distribution, especially when objects in the same materials present in the scene.",
"Fig.",
"REF shows the false color composite and ground truth maps of five commonly used hyperspectral datasets, i.e., Botswana, Indian Pines (Indian), Kennedy Space Center (KSC) , Pavia University (PaviaU), and Salinas scene (Salinas) [36].",
"In these images, there are strong dependencies between the spatial locations of pixels and land cover classes.",
"This results in the potential using of the spatial structure and distribution of each single class.",
"In most cases, if random sampling is used for selecting training and testing samples in the same image, the class label of a testing sample can be easily inferred only by its spatial relation with the training samples.",
"This can be exemplified by Fig.",
"REF , in which 5%, 10% and 25% of training data are sampled from the Indian Pines and Pavia University datasets.",
"When it comes to 25% sampling rate, the spatial distribution of training samples (last column) is similar to the shape of the ground truth map (first column) in the spatial domain.",
"Figure: Random sampling strategy on Indian Pines and Pavia University datasets.",
"From left to right: the ground truth map, training set with 5% sampling rate, training set with 10% sampling rate, training set with 25% sampling rate.Table: Overall accuracy (OA), average accuracy (AA) and Kappa coefficient (κ\\kappa ) on five hyperspectral datasets when different feature/classifier combinations were used: spectral feature with SVM (Spe), spatial feature with SVM (Spa1) and spatial feature with KNN (Spa2) .To show the extent that the classification accuracy is impacted by spatial information, we performed experiments on five benchmark datasets in Fig.",
"REF .",
"In the experiment, a nonlinear support vector machine (SVM) was employed because the land cover classes are not linearly separable in the spatial domain.",
"The spatial coordinates were used as the spatial feature and no spectral information was included.",
"The parameters of the SVM were learned via five fold cross validation.",
"Three sampling rates were explored, i.e.",
"5%, 10%, and 25% to generate the training data from all labelled samples, while the rest of labelled data served as the testing samples.",
"In contrast to the spatial feature, the traditional spectral feature based methods was also implemented in which we followed the same setting as the spatial method.",
"Each test was repeated ten times in the experiment with random generation of training and testing samples.",
"The overall classification accuracies (OA), average accuracies (AA) and Kappa Coefficient ($\\kappa $ ) are shown in Table REF for different methods.",
"The comparison between accuracies using spectral feature with SVM (Spe) and spatial feature with SVM (Spa1) shows some surprising results.",
"Classification accuracy based on pure spatial feature has significantly outperformed the counterpart using pure spectral feature in all cases.",
"In terms of overall accuracy, the spatial method achieves more than $93.8\\%$ accuracy on all datasets when only $5\\%$ of training samples are used, while the spectral method has only around $75.5\\%-93.2\\%$ in accuracy.",
"When the sampling rate becomes $25\\%$ , the accuracy almost reaches $100\\%$ for the spatial feature which agrees with the perceptual intuition in Fig.",
"REF .",
"Essentially, these phenomena are caused by the random sampling strategy on the same image.",
"The results also show that higher sampling rate leads to increase of classification accuracy on all datasets.",
"Figure: Classification maps of the Indian Pines (including the unlabelled pixels) using only spectral or spatial features: (a) Spe, (b) Spa1 and (c) Spa2.In another point of view, the spatial classification can also be exploited in the local neighbourhood.",
"Since the training samples spread uniformly in the image, it would be easy to find a nearest training sample for any testing samples that belong to the same class.",
"An experiment was designed to test how the local information contributes the classification.",
"We employed the K-nearest neighbor (KNN) classifier and set the parameter K to 1.",
"The results are displayed in Table REF under the columns of Spa2.",
"It can be seen that the performance of Spa2 is comparable to the spatial method Spa1 on all datasets, which has significantly outperformed the spectral method on all datasets.",
"It should be noted that in the KNN classification, predicting the label of testing samples is only based on the nearest training pixels in their spatial neighbourhood.",
"This is similar to the mechanism of some spectral-spatial methods which also make use of the local spatial neighbourhood information but in a different way.",
"This experiment further proves that the training data provide too much information on the spatial domain for the classification task.",
"While classification based on spatial coordinates seems to perform better than the spectral information, it is infeasible in real applications in which unlabelled pixels are involved.",
"Those unlabelled pixels are prone to be classified into its nearby class, thus producing a thematic map dramatically different from the reality.",
"To exemplify this phenomenon, Fig.",
"REF shows the classification maps of the Indian Pines including the unlabelled pixels with 10% sampling rate.",
"Although Sp1 and Sp2 achieve higher classification accuracy than Spe, their classification maps are far away from the ground truth map.",
"Therefore this method is not acceptable in reality.",
"In summary, these two experiments show that random sampling from the same image makes an underestimated amount of spatial information be embedded in the training set and the testing set.",
"It is natural to raise the concern that they would interact with each other if spatial processing is applied to the image."
],
[
"Overlap between Training and Testing Data from the Same Image",
"The spectral-spatial methods make use of the spatial information in different forms and in different ways as introduced in Section .",
"When it comes to the random sampling strategy, a more severe problem may happen in the spectral-spatial analysis, especially for the feature extraction stage.",
"When only spectral responses are used, feature extraction is performed at single pixel, without exploring its spatial neighborhood.",
"Therefore, random sampling strategy provides a statistical solution for data splitting and there is no explicit overlap between training and testing samples.",
"However, the spectral-spatial methods usually exploit information from neighborhood pixels.",
"This is normally implemented by a sliding window with a specific size, for example, $3\\times 3$ , $5\\times 5$ and so on.",
"In each window, a kernel or filter is used to extract discriminative information.",
"Since the training and testing samples are drawn from the same image, their features are almost certain to overlap in the spatial domain due to the shared source of information.",
"Figure: Overlap between training and testing data on Indian Pines dataset under 5% sampling rate.Figure: The regions for feature extraction from a training sample (O) and a testing sample (+) overlap with each other, as represented in gray color.",
"The proportion of overlap is 2 3\\frac{2}{3} and 4 5\\frac{4}{5} for (a) 3×33 \\times 3 sliding window and (b) 5×55 \\times 5 sliding window, respectively.Fig.",
"REF shows the extent of overlap between training and testing data on the Indian Pines dataset.",
"In the figure, the white dots show the locations of training samples, and the surrounding white squares cover a $3\\times 3$ region used for spectral-spatial feature extraction.",
"The testing samples, however, may just lie in the the square and has its own surrounding regions.",
"This brings about a shared region between features extracted from the training and testing data such that they interact with each other and lose the mutual independence.",
"It is also evident that a larger filter leads to more overlap areas.",
"An example is shown in Fig.",
"REF in which a $3 \\times 3$ and $5 \\times 5$ window will result in $\\frac{2}{3}$ and $\\frac{4}{5}$ of overlap for adjacent training and testing samples, respectively.",
"Such overlap leads to using of the testing data for training purpose, and gives significant advantages to the spectral-spatial feature extraction approaches.",
"This violates the basic principle of supervised learning that training and testing data shall not interact with each other.",
"Depending on how feature is extracted, benefit of testing data may be explicit, for example when the spectral-spatial feature is extracted by concatenating the spectral responses of pixels in a neighborhood, or implicit, for example, by extracting texture features based on spatial frequency analysis such as discrete wavelet transform."
],
[
"Experiment with a Mean Filter Based Spectral-spatial Method",
"In order to estimate how the overlap impacts the accuracy of spectral-spatial method with random sampling strategy, an experiment was carried out on the Indian Pines dataset.",
"In this experiment, a linear SVM classifier was used to facilitate further comparison.",
"The features were constructed by applying a mean filter to calculate the mean of the spectral responses in a neighborhood of the hyperspectral images, which was mathematically formulated as follows: $f(x,y) = \\frac{1}{MN}\\sum _{i=x-\\frac{M}{2}}^{x+\\frac{M}{2}}\\sum _{j=y-\\frac{N}{2}}^{y+\\frac{N}{2}} S(i,j)$ where $M$ and $N$ are the width and height of neighborhood surrounding $(x,y)$ .",
"In the experiment, we set M and N both from 1 up to 27 with an interval of 2.",
"$S(i,j)$ represents the spectral response at location $(i,j)$ and $f(x,y)$ is the feature extracted on location $(x,y)$ which contains both spectral and spatial information.",
"This process can be considered as one of the simplest approaches to extract spectral-spatial features.",
"When the size of the neighborhood is $1\\times 1$ , this reduces to extracting spectral feature only.",
"Larger size of window results in more overlap.",
"The calculated rate of testing samples covered by the neighborhood of training samples is shown in Fig.",
"REF .",
"When 5% training data are sampled, 30.9% testing samples are covered by the $3\\times 3$ regions used to extract training features.",
"When random sampling rate increases to 25%, the extent of overlap becomes 86.4%.",
"The rise of sampling rate leads to rapid increase of overlap.",
"Furthermore, when the size of filter grows, the overlap rate also increases rapidly.",
"Eventually when the overlap rate reaches 100%, all testing samples are used in the training process.",
"Figure: Overlap of training and testing data on the Indian Pines with different size filters.Figure: Classification accuracies on the Indian Pines using a simple mean filter with different filter sizes.The experiment was repeated 10 times.",
"In each time, the indices of the training and testing pixels were randomly generated.",
"Features were generated using different settings of filter size and sampling rate.",
"Under each setting, the same training and testing samples were used for fair comparison.",
"The overall classification accuracies are shown in Fig.",
"REF .",
"Significant increase of the classification accuracy can be observed when spatial information is added to the spectral information.",
"When the size of neighborhood increases, more testing data contribute to the training step, therefore the classification accuracy increases.",
"It is also interesting to see that after the neighborhood increases to a specific size, the accuracy stops growing and tends to stable.",
"This is probably because that when the neighborhood becomes too large, unlabelled data or samples from other classes are involved in the feature extraction, which neutralizes the benefits of overlap."
],
[
"Non-overlap Measurement",
"Other than overlap, the increase of classification accuracy also owes to the better discriminative capability of spectral-spatial features.",
"With larger filter size, the feature includes more spatial information.",
"To demonstrate how the spatial neighborhood influences the effectiveness of spectral-spatial feature, we performed another experiment on those testing samples not overlapped with the training data.",
"Table: Classification accuracies on all testing samples and non-overlapped testing samples.Following the same setting as the previous experiment, we removed the testing samples that were covered by the training set and only test on the remaining samples.",
"Table REF shows the comparison of classification accuracy on all testing samples and non-overlap testing samples.",
"The results show that when testing on non-overlap testing samples, the accuracy is improved when the neighbourhood information is initially introduced by the $3 \\times 3$ mean filter.",
"However, when a larger size of filter is used, the accuracy of non-overlap testing samples does not increase and even decrease In Table REF , the null values are due to the absence of non-overlapped testing samples..",
"The decrease could be caused by the fact that the non-overlap testing samples are easily influenced by the samples from other classes in the neighborhood.",
"In contrast, the classification accuracy with overlapped testing samples has remarkable improvement when larger filter size is used.",
"Based on the above analysis, under the random sampling strategy, some filter-based spectral-spatial feature extraction methods would make the training and testing samples overlap and then interact with each other.",
"Subsequently, in the training process, information from testing samples are included to train the classifier, which in return is used to classify the testing samples in the testing step.",
"Although this kind of methods improves the classification results, they are not desired because they violate the basic assumption of supervise learning and their generalization is questionable.",
"So far we have only analysed a special case of spectral-spatial methods, it would be interesting to extend the analysis to a broader scope.",
"Next we try to discuss the data dependence and its impact on classification results by computational learning theory."
],
[
"Data Dependence and Classification Accuracy",
"Computational learning theory aims to analyse the computational complexity, feasibility of learning, and performance bound [37].",
"A widely known computational learning framework is the probably approximately correct (PAC) learning which estimates the sample complexity based on the required generalization error, probability of inference and complexity of a space of functions.",
"Another classic theory is the Vapnik-Chervonenkis theory (VC theory).",
"One of its functions is to bound the generalization ability of learning processes which is usually represented as the testing error $R(h)$ .",
"Before introducing the computational learning theory, some basic learning concepts shall be firstly introduced in the scope of i.i.d.",
"data.",
"In computational learning, instead of considering the classification accuracy, a more general term, generalization error bound, is usually derived to describe the ability of learning algorithm to predict the unseen data.",
"For a binary classification problem, given a hypothesis $h\\in H$ where $H$ are all hypotheses, a target hypothesis $c$ , and a sample set $S = ( x_1, x_2, ..., x_m)$ following a distribution $D$ , the empirical error (training error) $\\hat{R}(h)$ and the generalization error (testing error) $R(h)$ can be defined as: $\\hat{R}(h) = \\frac{1}{m}\\sum _{i=1}^{m} \\hspace{5.0pt}l( h(x_i), c(x_i))$ $R(h) = \\underset{x\\in D}{\\mathbb {E}} \\hspace{5.0pt}l(h(x), c(x))$ where $l$ is the error function and $\\mathbb {E}$ is the expectation.",
"Despite that the empirical error $\\hat{R}(h)$ can be calculated once the training data $S$ , its label $c(x_i)$ and the hypothesis $h$ are known, the generalization error can not be estimated directly.",
"In practice, simply decreasing $\\hat{R}(h)$ by building complex classification model may not always minimise $R(h)$ because it may lead to over-fitting.",
"In order to bound $R(h)$ , more factors have to be considered.",
"Based on PAC learning, the generalization bound can be calculated as: $R(h) \\le \\hat{R}(h) + \\frac{1}{m}(log|H|+log\\frac{1}{\\delta })$ which means that given training data of size $m$ and hypothesis complexity $|H|$ , the inequality of generalization holds with probability no less than $1-\\delta $ .",
"This definition conforms to our understanding of learning that more training data leads to better learning outcome.",
"Based on the inequality, the generalization bound can be tightened by increasing the training sample size $m$ or by decreasing the probability $1-\\delta $ which is equivalent to confidence of the inference.",
"The complexity of hypothesis is determined by the learning models.",
"When the hypothesis sets are infinite, the above bound is uninformative.",
"In order to impose generalization bound for infinite cases, the Redemacher complexity is introduced to measure the hypothesis complexity [38].",
"Specifically, it measures the variety of a set of functions by estimating the degree to which a hypothesis can fit random noise.",
"The Rademacher complexity based generalization bound on i.i.d.",
"data samples is defined as: $R(h) \\le \\hat{R}(h) + \\hat{\\mathfrak {R}}_s(H) + 3\\sqrt{\\frac{log\\frac{2}{\\delta }}{2m}}$ where $\\hat{\\mathfrak {R}}_s(H)$ is the empirical Rademacher complexity.",
"$1-\\delta $ is the probability or confidence and $m$ is the training sample size.",
"$\\hat{\\mathfrak {R}}_s(H)$ can be estimated by growth function or VC-dimension [37].",
"Even though these models provide generalization bounds for different learning algorithms, they are all based on the i.i.d.",
"assumption.",
"For non-i.i.d.",
"data, the generalization bound has not been fully studied due to the lack of statistical model for dependent data.",
"However, i.i.d.",
"does not always hold in practice.",
"In general, the samples in a hyperspectral image are not i.i.d., as the samples are spatially overlapping to each other in the image.",
"The data dependence will inevitably happen no matter how carefully the sampling strategy is designed.",
"In recent years, researchers begin to develop new learning theories on this topic.",
"Among all kinds of non-i.i.d.",
"data, some data types possess the property of asymptotic independence, which is weaker than independence but stronger than dependence, for instances, time series signal [39].",
"In order to define this kind of data, mixing condition is used to explicitly define the dependence of the future signal on the past signal based on decay.",
"A commonly used model in non-i.i.d.",
"scenario is the stationary $\\beta $ -mixing model [40].",
"Suppose events $A$ and $B$ are generated from a time sequence $\\alpha _{t\\in (-\\infty , +\\infty )}$ with an interval $k$ , the definition of $\\beta $ -mixing coefficient is $\\beta (k)= \\sup _{m}\\underset{B\\in \\alpha _{-\\infty }^{m}}{\\mathbb {E}}\\left[{\\sup _{A\\in \\alpha _{m+k}^{+\\infty }}\\left|Pr(A|B)- Pr(A)\\right|}\\right]$ This equation defines the dependence coefficient as the supremum of the difference between the conditional probability $Pr(A|B)$ and probability $Pr(A)$ when choosing arbitrary moment $m$ which separates event A and B.",
"The sequence $\\alpha $ is $\\beta $ -mixing if $\\beta (k)\\rightarrow 0$ when $k\\rightarrow +\\infty $ .",
"It implies that the dependence coefficient $\\beta (k)$ decreases with the increase of interval $k$ .",
"Several learning models have already been derived on stationary $\\beta $ -mixing data, such as VC-dimension bound[40], PAC learning [41] and Rademacher complexity [42].",
"In this work, the Rademacher complexity based generalization bound is employed since it associates the generalization bounds with $\\beta $ -mixing coefficient.",
"It uses a technique to transferring the original dependent data to independent blocks.",
"Let $2\\mu $ be the number of blocks and each block contains $k$ consecutive points, then the size of sample $m = 2\\mu k$ .",
"The original bound in Equation (REF ) is extended to $\\beta $ -mixing data as follows: $R(h) \\le \\hat{R}(h) + \\hat{\\mathfrak {R}}_s(H) + 3M\\sqrt{\\frac{log\\frac{2}{\\delta -4(\\mu -1)\\beta (k)}}{2\\mu }}$ where $M$ is the bound of a set of hypothesis $H$ .",
"Compared to the i.i.d.",
"case, this bound is not only related to the training error $\\hat{R}(h)$ , empirical Rademacher complexity $\\hat{\\mathfrak {R}}_s(H)$ , and probability $\\delta $ , but also relies on the $\\beta $ -mixing coefficient $\\beta (k)$ which implies the degree of dependence among data.",
"Considering the impact of $\\beta $ -mixing coefficient to the bound, this equation can be further simplified as: $R(h) \\le f(\\beta (k)) + C$ where $f(\\beta (k))$ is a monotonically decreasing function.",
"As a result, the generalization bound is tightened when the $\\beta (k)$ increases, i.e.",
"the dependence among data is enhanced.",
"Applying learning theory to hyperspectral image classification is challenging due to the complex statistical characteristic of hyperspectral images.",
"To our knowledge, similar work in is very rare.",
"In the following experiments, we show that hyperspectral images share the same properties of $\\beta $ -mixing data.",
"Spectral feature extracted at image pixels often have strong dependence to their surrounding regions [4].",
"However, it is still questionable whether such dependence decreases with the increasing distance between the central pixel and its neighbouring pixels.",
"In addition, since a hyperspectral image is a three-dimensional data, how the dependence is related to the spatial direction is still unknown.",
"To check how the dependence varies with the distance, we performed a simple statistical analysis on the Indian Pines dataset.",
"Here, the dependence between two pixels $X$ and $Y$ is approximated by the linear correlation coefficient of their spectral responses: $\\rho _{X, Y} =\\frac{cov(X,Y)}{\\sigma X \\sigma Y}$ where $cov$ and $\\sigma $ represent the covariance and standard deviation, respectively.",
"A random location was firstly selected on this image, then the correlation coefficient $\\rho $ was calculated between the pixel and its neighborhood pixels with different distances.",
"The result on a $9\\times 9$ patch is shown in Fig.",
"REF (a) in which the intensity implies the strength of the correlation.",
"In the center of the patch, the intensity is 1 due to self-correlation.",
"As expected, it does not show clear pattern at a single pixel.",
"However, after calculating the mean of patches centered at all locations in the image, the statistical result is shown in Fig.",
"REF (b).",
"It clearly shows that with the increasing interval, the correlation coefficient gradually drops in all directions.",
"This is consistent with the characteristic of $\\beta $ -mixing.",
"Figure: The correlation between a pixel and its 9×99\\times 9 neighbourhood on Indian Pines for (a) a random location; (b) the whole image (statistical result).Now we can safely assume that hyperspectral images are $\\beta $ -mixing, and explore the relationship between the generalization bound with data dependence.",
"Based on Equation (REF ), the bound is inversely related to $\\beta $ -mixing.",
"As a consequence, the classification accuracy can be increased by enhancing the dependence between training and testing data.",
"Recall that in the experiment with a mean filter based spectral-spatial method (Fig.",
"REF ), the accuracy increases with larger filters.",
"Similarly, the statistical results of the correlation coefficient on a $9\\times 9$ patch are calculated for the original image and the filtered images, which are represented by different colors.",
"For the sake of easy observation, we only draw the distance along X-axis in the positive direction.",
"The outcome is shown in Fig.",
"REF from which two trends can be observed.",
"Firstly, all curve drops continuously when the distance increases which means that the processed data agree with the properties of $\\beta $ -mixing.",
"Secondly, at the same distance, the larger the filter is, the stronger the dependence between the central pixel and its adjacent pixels become.",
"Therefore, the overlap enhances the data dependence which tighten the error bound of the final classification results.",
"Figure: The pixel correlation on Indian Pine processed by a mean filter with different sizes.The theories presented above have explained why mean filter improves the classification accuracy, and they can be extended to other spectral-spatial operations that increase the data dependence.",
"It should be noted that the above analysis is built on the assumption of random sampling for performance evaluation.",
"Under such experimental setting, the improvement of classification accuracy comes from not only incorporating spatial information into classifier but also enhancing the dependence between training and testing data.",
"The former is the main purpose of algorithm performance evaluation and the later should be avoided."
],
[
"A Controlled Random Sampling Strategy",
"Following the discussion in previous sections, since random sampling from the same image is not suitable for evaluation the spectral-spatial methods, it is necessary to develop a new sampling strategy to separate the training and testing sets without overlap.",
"It would be perfect if we could perform training and testing on two different images.",
"Unfortunately, this is still infeasible in most cases due to the limited availability of benchmark datasets and high cost of ground truth data collection.",
"Therefore, without changing much the current experimental setting, the goal is to significantly reduce the extent of data overlap and make the evaluation fair enough.",
"[t] Controlled Random Sampling Strategy Hyperspectral Image $I$ and sampling rate $s$ each class $c$ in $I$ Selects all unconnected partitions $P$ in the class $c$ each partition $p$ in $P$ Count the number of samples $n_p$ in the partition Calculate the number of training samples $n_t$ in the partition by $n_t = n_p \\times s $ Randomly select a seed point $q$ in the partition Applying the region-growing algorithm to extend $q$ to a region $r$ whose size is equal to $n_t$ Combine these regions $r$ to form training samples $R_c$ Combine the training samples $R_c$ and their corresponding class labels to get the whole training set $R$ Based on our analysis, the main problem of random sampling is that it makes the training and testing samples spatially adjacent to each other, leading to their overlap in the subsequent spatial operations.",
"On the other hand, as a classical method, it has advantages such as simplicity, reproducibility, and statistical significance.",
"As a result, the new sampling strategy should satisfies the following requirements.",
"Firstly, it shall avoid selecting samples homogeneously over the whole image so that the overlap between training and testing set can be minimised.",
"Secondly, those selected training samples should also be representative in the spectral domain, meaning that it shall adequately cover the spectral data variation in different classes.",
"There is a paradox between these two properties, as the spatial distribution and the spectral distribution are coupling with each other.",
"The first property tends to make the training samples clustered so that it generates less overlap between the training and testing data.",
"However, the second property prefers training samples being spatially distributed as random sampling does, and covering the spectral variation in different regions of the image.",
"Therefore, a good trade-off has to be achieved by the new sampling strategy.",
"Thirdly, because there is no prior knowledge, we do not know which samples are more important than the others.",
"Therefore the new method shall possess the property of randomness.",
"Figure: Controlled random sampling after Gaussian filterHere we propose a controlled random sampling strategy to achieve a compromise of the above considerations.",
"Similar to random sampling, a pre-defined proportion of samples in each class is to be randomly selected as the training samples and the rest data serve as the testing samples.",
"Those training samples shall be concentrated locally and dispersed globally.",
"We borrow the idea of region growing to create region-shape training samples [43].",
"The seed points are randomly selected from different partitions of classes to make the training samples disperse globally and randomly.",
"Then controlled random sampling proceeds with three steps.",
"Firstly, it selects the unconnected partitions for the each class and counts the samples in each partition.",
"This step is to find the spatial distribution of each class and make sure that the selected training samples in the next step cover the spectral variance at the most extent.",
"Secondly, for each partition, the training samples are generated by extending region from the seed pixel.",
"In terms of region growing, it expands in all directions and take account of 8-connected neighborhood pixels.",
"All the adjacent pixels of seed pixels are examined and if they are within the same class, they work as the new seed points.",
"This process is repeated until the amount of selected points reach a pre-defined number which is proportional to the number of pixels in the corresponding partition.",
"This guarantees that the total number of training samples meet the pre-defined proportion of the whole data population.",
"Thirdly, after the above steps are applied to all classes, those samples in the grown regions with their labels are chosen as the training samples and the rest of pixels work as the testing samples.",
"In case when there are more partitions than the required training samples, partitions are again randomly sampled.",
"A summary of this strategy is given in Algorithm .",
"Figure: Controlled random sampling strategy on the Indian Pines and Pavia University datasets.",
"From left to right: the ground truth map, training set with 5%, 10%, and 25% sampling rates, respectively.In Fig.",
"REF , we demonstrate different degrees of overlap between training and testing samples under random sampling and controlled random sampling strategies, after a Gaussian filter is applied.",
"In the left column of the figure, the training and testing data are represented by colored dots and white regions in each partition.",
"Applying the Gaussian filter creates the gray regions in the right column of the figure, representing the overlap between the training and testing data.",
"It can be noticed that all the training samples are impacted by the testing data under random sampling.",
"On the contrary, for controlled random sampling, only training samples at the edges of the training regions are influenced by the testing data.",
"This figure clearly shows that the overlap from controlled random sampling is significantly less than that from the traditional random sampling.",
"To further illustrate how the controlled random sampling works with real datasets, examples on Indian Pines and Pavia University are given in Fig.",
"REF with 5%, 10% and 25% sampling rates.",
"Compared to the random sampling strategy in Fig.",
"REF , it can be observed that the spatial structure of each class can no longer be inferred from the training data as random sampling does.",
"In the meantime, the training samples are still distributed across the whole image and a wide range of spectral variances are covered.",
"Though this approach can not completely eliminate overlap between the training and testing data, the influence of testing data in the training stage can be greatly reduced to limited pixels at the boundaries of each training region.",
"The experimental setting with the proposed sampling method can help us more accurately and objectively evaluate the performance of spectral-spatial methods."
],
[
"Experiments",
"To prove the usability and advantage of the proposed controlled random sampling against random sampling, we have developed a series of experiments to test these two strategies when they are used to evaluate spectral-spatial operations in different stages of image classification.",
"In the preprocessing step, we adopted a mean filter and a Gaussian filter as examples of smoothing and denoising operations.",
"Then, we performed experiments with raw spectral feature to examine the effectiveness of the proposed sampling method when evaluating the spectral responses without spatial processing.",
"Finally, two spectral-spatial feature extraction methods, i.e.",
"3D discrete wavelet and morphological profiles, were compared using two sampling methods.",
"In order to make the experiments more convincing, we adopted two widely used supervised classifiers, support vector machine (SVM) and random forest (RF) [44] to validate our results.",
"The SVM was implemented using the LIBSVM package [45], and the RF was implemented using the well-known Weka 3 data mining toolbox [46].",
"We present results on five benchmark datasets, i.e., Botswana, Indian Pines (Indian), Kennedy Space Center (KSC), Pavia University (PaviaU), and Salinas scene (Salinas)."
],
[
"Evaluation of Spectral-spatial Preprocessing Method",
"The spectral-spatial preprocessing step contributes to classification by improving the quality of hyperspectral images, reducing random noises, and enhancing specific features.",
"By varying the parameters of mean filter and Gaussian filter, their influence to the classification accuracy under two sampling strategies can be analysed.",
"We undertook experiments on both Indian Pines and Pavia University datasets with SVM and RF, respectively.",
"The results with mean filter are shown in Fig.",
"REF .",
"When traditional random sampling is used, the accuracy on the Indian Pines dataset increases with larger filter size when SVM and RF are adopted (Fig.",
"REF (a) and (b)).",
"For the Pavia University dataset (Fig.",
"REF (c) and (d)), the accuracies also increase with larger filter size but decrease when the size reaches a specific value, which is slightly different from the results on the Indian Pines image.",
"The reason may be that Pavia University has higher spatial resolution and interacts with filters in more complex way than the low spatial resolution Indian Pines data.",
"The results confirm that using a mean filter with relative large size can increase the classification accuracy, up to 92.4% on Indian Pines and 98.0% on Pavia University.",
"Essentially, it is mainly because larger filter leads to more overlap between the training and testing data.",
"In contrast, when adopting the controlled random sampling strategy, the classification accuracy firstly improves marginally, but then dramatically drops with larger size filters.",
"This is consistent with our expectation in evaluating the influence of spectral-spatial operations rather than the data dependence.",
"Therefore, the proposed sampling method successfully avoids the problem of random sampling.",
"Figure: PaviaU & RFWe then performed an experiment with Gaussian filter under the same setting to compare two sampling strategies.",
"Among different denoising and smoothing approaches, Gaussian filter is a basic but effective tool to reduce the random noise in hyperspectral images.",
"It works as a low-pass filter whose standard deviation controls the shape of filter and sets the threshold to remove the corresponding high frequency signal.",
"The larger the stand deviation is, the lower frequency the signal can be preserved and the image be more smoothed.",
"We applied a Gaussian filter on each band of hyperspectral images with a range of standard deviations.",
"The size of filters varies with the standard deviation so that the smoothing effect decays to nearly zero at the boundaries of filtering masks.",
"Then the smoothed image was fed into the classifier.",
"This experiment was repeated 10 times and the mean of overall accuracy was used as the evaluation criterion.",
"The standard deviation ranged from $2^{-1}$ to $2^3$ with an interval of 0.5 on the exponential term.",
"We plot the classification accuracy as a function of the standard deviation in Fig.",
"REF for random sampling and controlled sampling method, respectively.",
"From Fig.",
"REF (a)-(d), we can see that the accuracy continuously increases until a specific point when Gaussian filter with larger standard deviation is used with random sampling strategy.",
"This is consistent with the observation on the mean filter.",
"We can assume that the Gaussian filter influences the data dependence to varying extents under different standard deviations, such that the classification accuracy is impacted by the filter parameter.",
"This is also consistent with our earlier analysis that when data dependence is increased, the classification error bound will be tightened.",
"However, this is not desired when evaluating a preprocessing method for image classification as we would like to know what is the actual contribution from the operation itself.",
"Figure: PaviaU & RFCompared with the random sampling, the controlled random sampling presents a different trend between the accuracy and standard deviation.",
"The accuracy firstly improves marginally and then becomes stable or drops.",
"This indicates that smoothing with an appropriate Gaussian operator can remove noises, and thus contribute to the final image classification.",
"However, if the standard deviation of Gaussian is very large, too strong image smoothing does not help much for the discrimination of different classes since it may mix the training data with unlabelled data at boundaries of image regions, thus losing its adaptability.",
"Under the new sampling strategy, Gaussian filter is able to improve the classification but not very significantly and the training and testing data dependence caused by overlap is no longer the dominant factor to the classification.",
"Overall, these two experiments prove that the proposed sampling strategy is able to neutralize the improper benefit gained from enhancement of dependence between training and testing data."
],
[
"Raw Spectral Feature",
"We then performed an experiment to compare two sampling strategies when raw spectral features were used on the benchmark datasets.",
"The objective of this experiment is to examine the effectiveness and objectiveness of the proposed sampling method compared to random sampling.",
"As mentioned in Section , there is no issue with the experimental setting with random sampling when evaluating a pixel based spectral feature.",
"But we still do not know whether the proposed sampling method is qualified in such a task.",
"In the experiment, only the raw spectral features were used without any spatial processing.",
"Other settings were same as the previous experiment such as the classifiers, repetition of experimental runs, etc.",
"The overall accuracy and standard deviation under random sampling and the controlled random sampling strategies(*) are reported in Tables REF and REF for SVM and RF, respectively.",
"Following observations can be made from the results.",
"Firstly, higher sampling rate leads to increase of classification accuracy on all datasets.",
"This is the same and expected for both sampling methods.",
"Secondly, the standard deviation of the accuracy from the proposed sampling strategy is much higher than that of the random courter part.",
"This is due to the distinction of training data generated from the random seeds each time.",
"Lastly, there is a reduction on the classification accuracy when the proposed sampling strategy is used.",
"This is due to the fact that variations on the same class data in different regions are less sufficiently captured as some of them may not be included in the training samples when the proposed sampling strategy is used.",
"The difference of accuracies is more evident on Indian Pines, Pavia University and Salinas datasets as these scenes include large blocks of regions in the same class, which leads to more benefits from spectral variation covered by random sampling strategy.",
"For further illustrating this phenomenon, the classification maps on the Indian Pines and Pavia University under two sampling strategies are shown in Fig.",
"REF .",
"Compared to random sampling, those testing samples far away from the training regions are easily misclassified under controlled random sampling.",
"Despite the differences, this does not affect a fair evaluation of different algorithms with the proposed sampling strategy.",
"In this experiment, assuming that the goal is to evaluate SVM and RF, it can be concluded from the results that SVM is a preferred classifier since it generates higher classification accuracy.",
"Therefore, although the new sampling strategy has made the hyperspectral classification a more challenging problem and forces more rigorous evaluation to the feature extraction and classification approaches, it is still qualified in evaluating the algorithms in hyperspectral image classification."
],
[
"Spectral-spatial Features",
"Now we turn our attention to test the proposed sampling strategy with two typical spectral-spatial feature extraction methods, i.e., 3D discrete wavelet transform (3D-DWT) and morphological profile.",
"3D-DWT is a typical example of filter based methods.",
"The morphological profile is a widely adopted spatial feature extraction method, including a number of variations for hyperspectral image classification."
],
[
"3D discrete wavelet transform",
"The discrete wavelet transform is derived from the wavelet transform which is a mathematical tool for signal analysis.",
"Unlike Fourier transform, the advantage of wavelet transform is that the transformed signal provides time-frequency representation for the non-stationary signal, meaning that we can not only know whether a frequency component exists but also when it happens in a signal.",
"The definition of continuous wavelet transform is shown as following: $\\Psi _x^{\\psi }(\\tau ,s)= \\int x(t)\\cdot \\psi _{\\tau ,s}(t)dt$ where $\\psi _{\\tau ,s}$ is the basis functions (wavelet) with $s$ and $\\tau $ that control the scale and translation, respectively.",
"When it comes to discrete samples, DWT is implemented by a series of filters in the frequency domain.",
"Since hyperspectral images consist of three dimensions, 3D-DWT exploits the correlation along the wavelength axis, as well as along the spatial axes, so that both spatial and spectral structures of hyperspectral images can be more adequately mapped into the extracted features.",
"In the implementation, we followed the multiple scale setting as described in [10], however, without the feature selection step.",
"Firstly, the hyperspectral image was processed by a cascade of high pass filters and low pass filters.",
"In each level, the data was decomposed into high frequency part and low frequency part.",
"After three levels of decomposition, the original data was separated into 15 sub-cubes $C_1, C_2, ..., C_{15}$ based on the bandwidth, such that each of the sub-cubes contained different scales of information.",
"To further capture the spatial distribution of hyperspectral images, a mean filter was applied on the sub-cubes: $\\hat{C}_n(x,y,.)",
"= \\frac{1}{9}\\sum _{i=x-1}^{x+1}\\sum _{j=y-1}^{y+1}C(x,y,.",
")$ In order to keep the sub-cube and the original data cube at the same size, the filtered signals were not down-sampled as what the traditional DWT does.",
"Then these sub-cubes were concatenated into the wavelet features.",
"The multidimensional function was carried out along two spatial dimensions $x$ and $y$ , as well as the spectral dimension $\\lambda $ , respectively.",
"The final concatenation worked as the feature for the whole data cube and can be represented as: $f(x,y) = (\\hat{C}_{1}^{x},\\hat{C}_{2}^{x},..., \\hat{C}_{15}^{x}, \\hat{C}_{1}^{y},\\hat{C}_{2}^{y},..., \\hat{C}_{15}^{y}, \\hat{C}_{1}^{\\lambda },\\hat{C}_{2}^{\\lambda },..., \\hat{C}_{15}^{\\lambda })$ where $f(x,y)$ is the 3D-DWT feature at location (x,y).",
"The experimental results under random sampling strategy and controlled random sampling strategy(*) are shown in Table REF and Table REF for SVM and RF, respectively.",
"As expected, controlled random sampling strategy leads to lower accuracy compared to random sampling strategy on all datasets.",
"Interesting observation can be obtained by comparing these results with the results on the raw spectral feature in Tables REF and REF .",
"On one hand, 3D-DWT performs better than raw spectral feature under both sampling methods.",
"This indicates that the proposed method confirms that 3D-DWT is able to extract more discriminative information than raw spectral feature.",
"On the other hand, under experimental setting with random sampling, 3D-DWT significantly improves the accuracy on all datasets over raw spectral feature.",
"However, when testing it with the proposed controlled sampling strategy, the improvement can not reach the same level of significance, especially on Indian Pines, Pavia University, and Salinas datasets.",
"It means that 3D-DWT does not perform that significantly better than the raw spectral features as expected, when eliminating the advantage of introducing information from the testing data into the training stage."
],
[
"Morphological profile",
"To further analyse this issue, we undertook experiments on the mathematical morphology feature.",
"Morphological operations employ the structuring elements in the image, making it possible to enhance or alleviate structures based on the specific requirements from users.",
"The basic operators include erosion and dilation which expands and shrinks the structures, respectively.",
"Combining them results in the opening (erosion-dilation) and closing (dilation-erosion) operations.",
"These two processes can remove specific structures and noises without destroying the original primary structures in the image.",
"The results of processing are called morphological profiles.",
"Morphological profile based feature extraction method is able to explore the structures of objects based on the contrast and size of objects in the images, therefore, it has been widely studied for hyperspectral image classification [19], [21].",
"We followed a basic implementation of extended morphological profiles (EMP).",
"The details of this method and its variation can be found in a survey paper from Fauvel et al [3].",
"The spatial feature was extracted as follows $\\Omega ^{(n)}(I) = \\left[o^{(n)}(I),...,o^{(1)}(I), I, c^{(1)}(I),...,c^{(n)}(I)\\right]$ where $o^{(n)}(I)$ and $c^{(n)}(I)$ were the opening and closing operations with a disk-shape structural element of size $n$ , respectively.",
"As different sizes of structuring elements were used, the morphological profile $\\Omega ^{(n)}(I)$ was capable of integrating multi-scale information.",
"Before the feature extraction, a principle component analysis (PCA) step was applied to hyperspectral images to reduce the dimension of the data.",
"Then the morphological profiles were obtained on each of the $m$ primary components: $\\hat{\\Omega }^{(n)}_m(I) = \\left[\\Omega ^{(n)}_1(I), \\Omega ^{(n)}_2(I), ...,\\Omega ^{(n)}_m(I)\\right]$ In the last step, the morphological profiles were stacked with the spectral response to form the spectral-spatial feature.",
"The classification results with two sampling strategies are shown in Table REF and Table REF .",
"Similar to the results on 3D-DWT, although the morphological profile feature has achieved better performance than the raw spectral method when tested with random sampling strategy, the improvement is not as significant when controlled random sampling is used.",
"This is mainly because that spectral-spatial method does not take much advantage of the overlapped information between training and testing samples under the proposed method.",
"Directly comparing two completely different spectral-spatial methods may not make much sense since different features are more suitable to extract features on specific datasets or sensitive to specific classifiers.",
"Here we analyse the results from another point of view, which may explain the advantage of the proposed sampling over random sampling.",
"In Table REF , 3D-DWT achieves higher accuracy than EMP on both Indian Pines and Pavia University datasets when random sampling is adopted.",
"When adopting the new sampling strategy, 3D-DWT still performs slightly better than EMP on the Indian Pines, but EMP performs significantly better than 3D-DWT on the Pavia University.",
"This is consistent with the fact that the morphology method is capable of extracting more spatial structures than 3D-DWT on the dataset with high spatial resolution [19].",
"Under the proposed sampling method, the properties of the spectral-spatial method can be more accurately reflected and evaluated in the experiments.",
"This is impossible under random sampling because the classification result is strongly misled by the overlap between training and testing samples.",
"Overall, the proposed sampling strategy reveals more real discriminative ability of the spectral-spatial methods, which is the purpose of the evaluation."
],
[
"Conclusion",
"This paper presented a comprehensive study on the influence of the widely adopted sampling strategy for performance evaluation of the spectral-spatial methods in hyperspectral image classification.",
"We point out that random sampling has some problems because it has ignored the overlap and spatial dependency between training and testing samples when they are selected from the same image.",
"Based on the non-i.i.d.",
"characteristic of hyperspectral image data, we proved that the improvement of classification accuracy by some spectral-spatial methods are partly due to the enhancement of dependence between training and testing data, compared with sole spectral information based methods.",
"An alternative controlled random sampling strategy is proposed to alleviate these problems.",
"This new strategy provides a better way to evaluate the effectiveness of spectral-spatial operations and the corresponding classifiers.",
"Finally, it should be noted that the aim of this paper is not to criticize the spectral-spatial methods themselves or the exploration of spatial information.",
"The concern is only on the widely adopted evaluation approach, or more strictly speaking, on the experimental setting.",
"Under the experimental setting with random sampling, the performance evaluation may be not equally fair and unbiased for all spectral-spatial methods.",
"This is especially the case for the practice that training and testing are performed on the same image.",
"This problem is ultimately due to the lack of labelled hyperspectral data that are available for method evaluation.",
"Therefore, a more urgent task for the research community is to build more benchmark datasets to facilitate future spectral-spatial hyperspectral image analysis research.",
"Table: Classification accuracies (overall accuracy and standard deviation) using raw spectral feature, 3D discrete wavelet transform (3D-DWT) and extended morphological profile (EMP) with random sampling and controlled random sampling(*), when linear SVM is adopted.Table: Classification accuracies (overall accuracy and standard deviation) using raw spectral feature, 3D discrete wavelet transform (3D-DWT) and extended morphological profile (EMP) with random sampling and controlled random sampling(*), when RF is adopted.D.",
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[
[
"The multi-faceted Type II-L supernova 2014G from pre-maximum to nebular\n phase"
],
[
"Abstract We present multi-band ultraviolet, optical, and near-infrared photometry, along with visual-wavelength spectroscopy, of supernova (SN) 2014G in the nearby galaxy NGC 3448 (25 Mpc).",
"The early-phase spectra show strong emission lines of the high ionisation species He II/N IV/C IV during the first 2-3 d after explosion, traces of a metal-rich CSM probably due to pre-explosion mass loss events.",
"These disappear by day 9 and the spectral evolution then continues matching that of normal Type II SNe.",
"The post-maximum light curve declines at a rate typical of Type II-L class.",
"The extensive photometric coverage tracks the drop from the photospheric stage and constrains the radioactive tail, with a steeper decline rate than that expected from the $^{56}$Co decay if $\\gamma$-rays are fully trapped by the ejecta.",
"We report the appearance of an unusual feature on the blue-side of H$\\alpha$ after 100 d, which evolves to appear as a flat spectral feature linking H$\\alpha$ and the O I doublet.",
"This may be due to interaction of the ejecta with a strongly asymmetric, and possibly bipolar CSM.",
"Finally, we report two deep spectra at ~190 and 340 d after explosion, the latter being arguably one of the latest spectra for a Type II-L SN.",
"By modelling the spectral region around the Ca II, we find a supersolar Ni/Fe production.",
"The strength of the O I $\\lambda\\lambda$6300,6363 doublet, compared with synthetic nebular spectra, suggests a progenitor with a zero-age main-sequence mass between 15 and 19 M$_\\odot$."
],
[
"Introduction",
"The classification of supernovae (SNe) is mainly based on observational features.",
"The presence of hydrogen in the spectrum primarily splits SNe into Type I (hydrogen-poor) and Type II (hydrogen-rich).",
"Focusing on Type II SNe, the amount of hydrogen that the progenitor stars retained at the time of the explosion strongly affects the evolution of the SN and in particular the shape of the light curve (LC).",
"It can show a plateau (Type II-P SNe) or a linear decline after peak (Type II-L SNe).",
"Type II-P SNe are supposed to arise from progenitor stars in the mass range of $8-17$ M$_\\odot $ [101] that reach the evolution state at which the collapse of the iron core occurs.",
"It has also been suggested that the progenitors of Type II-L SNe could have larger masses at zero-age main-sequence (ZAMS) than Type II-P [32], [33], [1], [69].",
"In particular, Type II-L SNe are thought to originate from progenitor stars with less hydrogen ($1-2$ M$_\\odot $ ) and larger radii (few 1000 R$_\\odot $ ) with respect to those that give birth to Type II-P SNe [112], [9], [93], [5].",
"It has been proposed a physical continuity from Type II-L to Type IIb SNe, which show even less amount of hydrogen in the spectra at early phases, and then to Type I-b SNe, where there is no sign of hydrogen [85].",
"Whether Type II-P and II-L SNe are two clearly separated subclasses, and thus originated from two different types of progenitors, or a continuum exists between the two is still debated [91], [4], [2], [35], [36], [45], [47], [94], [97], [44], [120].",
"The lack of significant spectroscopic differences between the two classes leaves just the LC shape as the discriminating factor (or more quantitatively, the luminosity decline rate).",
"In the literature, there are several parameters proposed for a quantitative discrimination between the Type II-P and II-L SNe, based on different phase intervals for measuring the average decline rate in different optical bands, e.g., 3.5 mag 100 d$^{-1}$ in $B$ -band ($B_{100}$ ) by [91], 0.5 mag 50 d$^{-1}$ in $V$ -band ($V_{50}$ ) by [36], again 0.5 mag 50 d$^{-1}$ but in $R$ -band ($R_{50}$ ) by [74].",
"In addition to these sub-classes, Type IIn SNe are distinguished due to their evolution being significantly affected by the presence of circumstellar material (CSM) in the proximity of the star.",
"This CSM gets shocked by the ejecta, producing narrow emission lines superimposed on the SN spectrum.",
"These narrow lines are the traits that typically identify this subclass.",
"Thus the Type IIn SNe are classified as such purely based on their spectroscopic features.",
"The LCs of Type IIn SNe are usually quite heterogeneous, and depending on the masses, density and geometry of the CSM, they can fall linearly or they can stay bright even for many years [113].",
"The observed CSM can be generated by pre-SN mass loss events, typical of massive progenitors [101] or binary systems [16], or it can also be the result of almost static, photoionisation-confined shells created by the stellar wind of lower-mass red supergiants [77].",
"Thus the progenitors of Type IIn SNe could conceivably be spread over a wide range of masses and environments.",
"For a more thorough description on the SNe classes and subclasses, including the hydrogen-poor events, see [39] and [117].",
"There are only few Type II-L SNe which are nearby (less than about 30 Mpc), have been discovered shortly after the explosion, and were monitored until the nebular phase.",
"SN 2014G is a new entry that will enrich the statistics for the Type II-L SNe.",
"At coordinates $\\alpha \\,=\\,10^{\\rm {h}}54^{\\rm {m}}34^{\\rm {s}}.1$, $\\delta \\,=\\,+5417569$, it was discovered in the nearby galaxy NGC 3448 independently by two amateur astronomers, P. Wiggins and K. Itagaki, with the first detection on 2014 January 14.32 UT [84], and was initially reported in CBAT Transient Object Followup Reportshttp://www.cbat.eps.harvard.edu/unconf/tocp.html as PSN J10543413+5417569.",
"A useful constraint on the explosion epoch was given by the Master collaboration [75] which reported an unfiltered limit magnitude of 19.4 in a combination of 6 exposures of the host galaxy taken on January $10.85-10.88$ UT, during routine survey mode.",
"We then set the explosion epoch on January 12.6 UT (MJD 56669.6), with an uncertainty of $\\pm $ 1.7 d. The epoch of explosion will be taken as reference throughout the paper.",
"Our group took a classification spectrum less than a day after discovery [88].",
"This showed several narrow features superimposed on a blue continuum and a comparison made with GELATO [57] gave the best match with a Type IIn SN.",
"In the following days these narrow lines disappeared and the spectra started to resemble a more normal Type II SN.",
"The photometric evolution, on the other hand, showed a monotonic decline, which led to revise the classification to a Type II-L [28].",
"At the time of writing, another paper on SN 2014G was published [10].",
"They presented the photometry and polarimetry of the transient, but no spectroscopy.",
"Their work will be used as a useful comparison with our data and conclusions.",
"Although of minimum relevance, we point out that they set the explosion epoch on 2014 January 12.2 UT, so a minor 0.4 d discrepancy arises between our and their phase reference.",
"In the following, we will first briefly describe the host galaxy in Section .",
"In Section we describe the instrumentation and the reduction techniques used.",
"We will then present the photometric and the spectroscopic evolution in Sections and respectively.",
"Finally we will discuss the results in Section , focusing on the physical interpretation of the data for this transient."
],
[
"The host galaxy",
"The host galaxy Figure: RGB image of the SN 2014G field.",
"On the left, a reference image where the whole system Arp 205 (NGC 3448+UGC 6016) is clearly visible.",
"On the right, a post explosion image, where SN 2014G is marked in red.NGC 3448 is the host galaxy of SN 2014G and together with the dwarf companion UGC 6016 (visible in the left panel of Figure REF ) they make the system Arp 205 [6].",
"The morphology of NGC 3448 shows signatures of a tidal interaction with its companion [86].",
"It belongs to the amorphous class of galaxies [96], characterised by a smooth appearance and by their high star-formation rates.",
"The main structure is an edge-on disc, with two bright bulges and an optically-thick dust line crossing the central one (see right panel of Figure REF ).",
"[8] studied the optical rotation curve of NGC 3448 showing heliocentric velocities ranging from 1150 km s$^{-1}$ in the SW region up to 1400 km s$^{-1}$ in the NE region.",
"Two higher-velocity clumps departing from the “rigid” component were also identified and interpreted as foreground gas clouds of tidal origins and now infalling towards the galaxy.",
"Several knots and unresolved radio sources can also be seen which are probably associated with a complex of SN remnants [87].",
"Ultraviolet (UV) and radio analyses showed that NGC 3448 is indeed a starburst galaxy.",
"By a comparison with the nearby starburst galaxy M82, a SN rate of one event every $\\sim 10$ years was estimated.",
"More recent works produced drastically lower star formation rate of 1.4 M$_\\odot $ yr$^{-1}$ [71], which translates to just 0.01 SN per year.",
"SN 2014G is the first SN reported in NGC 3448.",
"It resides 44 west and 20 south of the centre [84] and then, correcting for the rotation curve in [8], it should have a heliocentric velocity of 1175 km s$^{-1}$ .",
"This value is in good agreement with the value we measured from the narrow lines visible in the early spectra.",
"However as the host is edge-on, and being the SN in the external part of the galaxy, its proper motion could lead to a miscalculation of the distance.",
"Thus, we prefer to use a redshift-independent measure [115] of the distance of NGC 3448 reported in the NASA Extragalactic Databasehttp://ned.ipac.caltech.edu (NED), which is $\\simeq 31.94\\pm 0.80$ [115], equivalent to a distance of $\\sim 24.5$ Mpc.",
"This value will be used through the paper.",
"The Galaxy reddening along the line of sight from the all-sky Galactic dust-extinction survey is E($B-V$ )$_{Gal}$ =0.01 mag [98].",
"A narrow Na I D absorption line is clearly visible in the first ten spectra and this can be used to estimate the reddening due to the host.",
"Averaging the equivalent width (EW) of the doublet, we measured EW$\\simeq 1.33$ Å.",
"Following the lower relation in [116] it yields E($B-V$ )$_{host}=0.20\\pm 0.11$ mag.",
"Thus a total reddening for SN 2014G E($B-V$ )$_{tot}=0.21\\pm 0.11$ mag was assumed.",
"B16 implemented the colour method [89] to estimate the local reddening and came up with a total reddening of E($B-V$ )$_{tot}=0.25$ mag, which is in good agreement with the one inferred here.",
"Table: Instrumental configurations used for the follow-up campaign of SN 2014G"
],
[
"Observations",
"Observations Figure: Photometric evolution of SN 2014G.",
"The optical bands in both Johnson/Cousins and SDSS filters are reported in top left and top right panels respectively.",
"Note that the SDSS filters are calibrated in AB mag, while Johnson/Cousins in Vega mag.",
"UV bands are shown in the bottom left panel and NIR in the bottom right.",
"The unfiltered observations of P. W. were calibrated as Cousins RR band.",
"Upper limits are indicated by an empty symbol with an arrow.",
"The lines connecting the points are simple interpolations with a spline.Our monitoring of SN 2014G lasted for $\\sim 1$ year, during which we gathered 407 photometric points distributed over 66 nights.",
"The wavelength coverage is from the UV to the near-infrared (NIR) domains, and in the optical both Johnson/Cousins and Sloan Digital Sky Surveyhttp://www.sdss.org (SDSS) filters were used.",
"In addition, 18 optical spectra were taken, covering the $3000-10000$ Å range (see Table REF ).",
"2 UV spectra taken by Swift + Ultraviolet/Optical Telescopehttp://swift.gsfc.nasa.gov/ [95] are also available but they were omitted from this analysis because of contamination by nearby stars, which would have required the acquisition of further data to disentangle.",
"Both the LC and the spectral evolution are well sampled, without significant temporal gaps.",
"All the CCD data have been corrected for overscan, bias and flatfields using standard procedures within IRAFhttp://iraf.noao.edu/.",
"Images from the Las Cumbres Observatory Global Telescope Networkhttp://lcogt.net/ [13] were automatically ingested and reduced using the lcogtsnpipe pipeline [120].",
"For the photometric measurements, the SNOoPYCappellaro, E. (2014).",
"SNOoPY: a package for SN photometry, http://sngroup.oapd.inaf.it/snoopy.html package has been used, which allowed, for each exposure, to extract the magnitude of the SN with the point-spread-function (PSF) fitting technique [111].",
"These magnitudes were then calibrated using the zero points and colour terms measured by reference to the magnitudes of field stars retrieved from the SDSS catalog (DR9).",
"For the UBVRI filters, we first converted the SDSS catalog magnitudes to Johnson/Cousins, following [18].",
"The magnitudes of the Johnson/Cousins bands are thus given in Vega system, while the SDSS ones are calibrated in AB mag.",
"The unfiltered images, considering the efficiency curve provided by the manufacturer of the camera, were calibrated to Cousins $R$ -band.",
"NIR data (JHK) were reduced with a modified version of the external NOTCam package for IRAF, including standard reduction steps of flat-field correction, sky background subtraction and stacking of the individual exposures for improved signal-to-noise ratio.",
"The JHK photometry was calibrated relatively to the magnitudes retrieved from the Two Micron All Sky Survey (2MASS) cataloghttp://www.ipac.caltech.edu/2mass/.",
"Some precautions had to be taken with the Swift data.",
"Swift is not equipped with a CCD but with a photon counter.",
"For this reason, in the analysis we performed aperture photometry using the specific tools and parameters within the HEASARCNASA's High Energy Astrophysics Science Archive Research Center.",
"software, and following [12].",
"For the optical spectra, the extractions were done using standard IRAF routines.",
"The spectra of comparison lamps and of standard stars acquired on the same night and with the same instrumental setting were used for the wavelength and flux calibrations, respectively.",
"A cross-check of the flux calibration with the photometry (if available from the same night) and the removal of the telluric bands with the standard star were also applied.",
"The complete list of all the telescopes and instrumentations used to gather the data is reported in Table REF , while all photometric measurements are reported in Appendix (Tables REF to REF )."
],
[
"Photometry",
"Photometry In Figure REF , the photometric evolution of SN 2014G in all bands is reported.",
"Swift UV observations (bottom left panel) started one day after discovery ($\\sim 3$ d after explosion) and lasted 23 d, during which the UV bands experienced a very steep and linear decline without showing any clear initial rise.",
"The first NIR epoch, instead, was taken over one month after explosion and thus did not cover the rise time.",
"In total, the NIR observations covered $\\sim 100$ d, but with only 5 epochs (bottom right panel).",
"Focusing on the optical bands, a more complex behaviour than in UV and NIR bands is evident.",
"The early dense coverage in UBVRIgri (with first epoch being around $2-3$ d after discovery) allowed us to accurately track the rise to maximum.",
"Fitting the curve with an order 3 polynomial, we inferred the peak epoch in different filters.",
"These peaks were reached between 5.9$\\pm $ 0.6 d in $U$ -band (M$^{max}_{U}=-19.2$ mag) and 15.6$\\pm $ 0.2 d in $i$ -band (M$^{max}_{i}=-18.1$ mag) after the explosion, in agreement with what found by B16.",
"In particular, the rise-time to maximum in $R$ -band is 14.4$\\pm $ 0.4 d (M$^{max}_{R}=-18.1$ mag), which, compared to the sample of [45], supports the scenario of Type II-L SNe having longer rise-time and higher peak magnitudes than Type II-P.",
"The LCs then declined linearly for $60-70$ d. Measuring the decline rates following the different prescriptions used in the literature, we found $B_{100}=4.8$ mag, $V_{50}=1.5$ mag and $R_{50}=1.1$ mag, thus confirming that SN 2014G comfortably matches the definitions used for Type II-L SNe (see slope limits reported in Section ).",
"A steeper drop of $1.5-2$ magnitudes in $\\sim 10$ d occurred around day 80.",
"This drop in particular is well sampled with very few comparable cases [119], [120], [127].",
"The LC then appears to settle on a radioactive tail.",
"The SN went behind the Sun after $\\sim 180$ d after explosion, and we were able to recover it at $\\sim 300$ d in four bands ($Vgri$ ).",
"Figure: Comparison of optical pseudo-bolometric LC of SN 2014G with those of SNe 1998S, 1999em, 2009kr, 2013by and 2013ej (see main text for references).",
"For SNe 2014G, 2013by and 2013ej both Johnson/Cousins UBVRI and SDSS ugri filters were used, while just the former were used for the rest of the SNe.",
"Note also that the U band was not available for SN 2009kr.",
"For comparison we included also the uvoir LC of SN 2014G, marked with green hollow circles.",
"The dashed magenta line marks the slope that the LC would follow assuming that all the energy of the 56 ^{56}Co decay was absorbed by the ejecta.Figure: B-VB-V and V-RV-R extinction-corrected colour evolution of SN 2014G compared to the other SNe considered in the text.",
"The legend is the same as in Figure .Figure: Optical spectral evolution of SN 2014G.",
"The spectra have been corrected for reddening and redshift, and shifted vertically for better display.",
"On the right of each spectrum, the epoch and the telescope used are reported.",
"The positions of major telluric absorption lines are marked with the ⨁\\bigoplus symbol.Figure: Top panel: Comparison between the normalised classification spectrum of SN 2014G and the WR star WN 49 .",
"Bottom panel: Comparison of early-phase spectra of SN 2014G with those of SNe 1998S and 2013cu .",
"The spectra of the two latter SNe were taken with a high resolution spectrograph, so a gaussian smoothing has been applied in order to match the resolution of the spectra of SN 2014G.",
"Moreover, the spectra have been scaled for better comparison.Figure: Spectral comparison of SN 2014G with SNe 1999em , 2009kr , 2013by and 2013ej at phase 34-3834-38 d. All the spectra are in rest frame and corrected for reddening.We then computed the uvoir bolometric LC of SN 2014G, starting from the extinction-corrected fluxes at each epoch, and using the trapezoidal rule, assuming zero flux at the integration boundaries.",
"All Johnson/Cousins UBVRI and SDSS ugri filters were used, along with the UV and NIR measurements.",
"We also computed a pseudo-bolometric LC, using optical bands only.",
"This was to allow meaningful comparisons with other SNe which do not have UV and NIR coverage.",
"For this purpose, we selected some representative SNe from the literature, i.e.",
"the Type II-L SNe 2009kr [32], 2013by [119] and 2013ej [58], [118], [127], the Type IIn SN 1998S [37] and the archetypal Type II-P SN 1999em [34].",
"The comparison is shown in Figure REF .",
"The shape of the LC of SN 2014G resembles those of SNe 2009kr and 2013ej.",
"With respect to both of these though, SN 2014G shows a more rapid evolution, with shorter rise time, shorter duration of the photospheric phase and slightly steeper decline.",
"The match with SN 2013by instead is striking, both in shape and luminosity, with the LCs of the two SNe matching almost perfectly.",
"The only small difference is in the radioactive tail, which is more luminous in SN 2014G, indicating a greater amount of $^{56}$ Ni synthesised.",
"We notice however that SN 2014G declines faster with respect to the $^{56}$ Co decay (dashed magenta line in Figure REF ).",
"One can argue that the missing flux could come from the NIR contribution, which is not optimally sampled by our data at these phases.",
"However [59] showed that at late phases the NIR contribution in Type II SNe is constant in time.",
"Thus underestimating the NIR contribution in SN 2014G would translate into a solid shift of the tail but not into a change of the slope.",
"We favour the idea that the steeper decline is due to a non-complete trapping of the $\\gamma $ -rays from radioactive decay (see Section REF ).",
"In Figure REF we show the colour evolution $B-V$ and $V-R$ of SN 2014G, together with those of the other Type II SNe considered so far, all corrected for reddening.",
"The behaviour of all SNe presented is quite similar, with a rapid increase of both colours, consistent with expectation from an expanding SN envelope.",
"Only SN 1998S differs from the others, likely due to the contribution of CSM-ejecta interaction that characterised this transient [72].",
"The excellent match of the colour evolution of SN 2014G with other Type II SNe supports the value of reddening adopted that was derived from the Na$\\,$I D (see Section ).",
"Figure: Comparison of nebular spectra of SN 2014G with those of SNe 1998S , 1999em and 2013ej .",
"All the spectra are in rest frame and corrected for reddening, and have been scaled for better comparison."
],
[
"Spectroscopy",
"Spectroscopy"
],
[
"Flash-ionised CSM lines",
"Figure REF shows the complete optical spectral evolution of SN 2014G from the classification spectrum to the nebular phase.",
"The first five spectra show a blue continuum typical of Type II SNe at early phases.",
"The first two, in particular, show emission lines that disappear after the first $\\sim 9-10$ d. These features have already been observed in a handful of early spectra of other CCSNe, such as SN 1998S [73], [19] or the Type IIb SN 2013cu [43].",
"The lines are emissions from highly ionised carbon and nitrogen, along with hydrogen and helium, and similar features are present also in the spectra of Wolf-Rayet (WR) winds.",
"Indeed because of this coincidence, a WR progenitor star was originally proposed for SN 2013cu by [43].",
"However, this interpretation was disputed by [49] who modelled the emission lines and identified their origin in a slow dense wind or CSM surrounding the precursor star when it exploded.",
"The wind velocity was estimated as $v_{\\rm wind} \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 100$~km\\,s$ -1$ together with a mass loss rate of $ M3103$M$$\\,yr$ -1$.",
"\\cite {Groh 2014} proposed that this ruled out a WR-star progenitor, and favoured a luminous blue variable (LBV) or yellow hypergiant (YHG) progenitor with a wind that was enhanced in nitrogen and depleted in carbon (see Section \\ref {subsec: progenitor}).",
"We find the same lines in SN 2014G as in SN 2013cu, indicating that the surrounding CSM may have similar density and composition.",
"A comparison among the early spectra of the above mentioned SNe is shown in Figure \\ref {fig: early_sp} in addition to the WN5 type nitrogen-rich WR star WN 49 \\cite {Hamann 1995}, to highlight the identification of these early-features.",
"The main features of the spectra arise from H and He$ ${\\scshape {II}} - the latter ion is seen as the line at $$4686.",
"In addition, both SNe 1998S and 2013cu show a prominent peak at $ 4630$~Å\\ which is probably a blend of N$ ${\\scshape {III}} and N$ ${\\scshape {V}}.Several lines ($$4057, $$5201 and $$7113) are likely to be attributable to N$ ${\\scshape {IV}}.",
"A strong C$ ${\\scshape {IV}} line at $$5803 is also evident in SN 2014G and this feature is missing in the other two SNe considered.",
"Peculiar, narrow emission lines are also present in a high resolution spectrum of the Type IIb SN 1993J at 3~d after explosion \\cite {Benetti 1994}.",
"Together with H$$\\ and He$ ${\\scshape {II}}, lines from [Fe$ ${\\scshape {X}}] and [Fe$ ${\\scshape {XIV}}] were identified, but these are not visible in SN 2014G.",
"In summary, we find the same narrow emission lines of high ionisation species in SN 2014G as in SNe 1998S and 2013cu.",
"They persist to at least 3~d after the estimated explosion date, and disappear by day 9 after explosion.$"
],
[
"Supernova lines",
"All the narrow emission lines disappear in the spectrum at 9.3 d and more typical Type II features start to appear since the spectrum at 16 d. After this phase, a broad H$\\alpha $ dominates the spectrum, with a P-Cygni profile characterised by an asymmetric emission and a shallow absorption.",
"A couple of other Balmer lines (H$\\beta $ and H$\\gamma $ ) are identified, along with several lines of metal ions, like Ca$\\,$II, Fe$\\,$II, Sc$\\,$II, Ba$\\,$II, and Ti$\\,$II.",
"A comparison with other SNe at these phases is shown in Figure REF and the match with the spectra of SNe 2009kr and 2013by is very good.",
"After $\\sim 80$ d the emission feature of [Ca$\\,$II] $\\lambda \\lambda $ 7291,7324 starts to become visible albeit weak at this epoch.",
"The appearance of these lines is approximately coincident with the sudden drop in the LC.",
"At a similar phase ($\\sim 100$ d) the H$\\alpha $ absorption disappears and the emission line feature changes in structure - the interpretation of this will be discussed in Section REF .",
"At this point, the spectra show a gradual transition to the nebular phase with the [O$\\,$I] $\\lambda \\lambda $ 6300,6363 doublet becoming prominent.",
"The forbidden Ca grows significantly in strength, with a red shoulder which can be attributed to [Ni$\\,$II] $\\lambda $ 7378 from stable $^{58}$ Ni [65], and a blue component of [Fe$\\,$II] $\\lambda $ 7155 (see Section REF).",
"In the very last spectrum a prominent Mg$\\,$I] $\\lambda $ 4571 is also present.",
"A comparison with other SNe in the nebular phases is shown in Figure REF .",
"At $\\sim 150-190$ d, distinct absorption components of H$\\alpha $ are still present in SNe 2013ej and 1999em, whereas SN 2014G does not show any absorption.",
"Instead at 103 d a narrow emission-like feature to the blue of the dominant H$\\alpha $ emission appears (see Section REF ).",
"SN 1998S also shows a similar, but much stronger, narrow emission at approximately the same wavelength as in SN 2014G but in SN 1998S the H$\\alpha $ line shows a complex blend with a triple peak structure.",
"We notice also that the appearance of [O$\\,$I] occurs much earlier in SN 2014G than the other SNe shown here.",
"Finally in the 342 d fully nebular spectrum H$\\alpha $ and the [O$\\,$I] doublet appears to be connected by a distict “bridge” that links the two features (see Section REF ).",
"Apart from this anomaly, the nebular spectrum of SN 2014G is remarkably similar to that of SN 2013ej.",
"Figure: Top panel: velocity evolution of the Balmer lines and some prominent metal lines of SN 2014G.",
"Middle panel: comparison of the Fe\\,II λ\\lambda 5169 velocity of SN 2014G with that of the other SNe considered so far.",
"Bottom panel: temperature evolution of SN 2014G and comparison with other SNe.",
"The values reported for SN 2009kr were measured from the spectra shown in Elias-Rosa et al.",
"(2010), while for the other SNe we used the values published in the literature."
],
[
"Line velocities and temperature evolution",
"We measured the velocity of the important lines in each spectra throughout the first 150 days or for as long as feasible.",
"The velocities were derived from the position of the minimum of their absorption features.",
"For H$\\alpha $ we measured also the FWHM of the emission component.",
"Both the minima of the absorptions and FWHM of the H$\\alpha $ emission were obtained from a fit with either a Gaussian, a Lorentzian or a low-order polynomial function, according to the best match with the shape of the feature.",
"The errors were estimated with a Monte Carlo technique, varying the flux of each pixel according to a normally distributed random value having variance equal to the noise of the continuum.",
"We did this procedure 100 times and then took the errors as the standard deviations of the fit parameters.",
"The evolution of H$\\alpha $ , H$\\beta $ , H$\\gamma $ , He$\\,$I+Na$\\,$I $\\lambda $ 5876, Fe$\\,$II $\\lambda $ 5018 and $\\lambda $ 5169, and Sc$\\,$II $\\lambda $ 5527 and $\\lambda $ 6245 are plotted in Figure REF .",
"We adopted the Fe$\\,$II $\\lambda $ 5169 as a probe of the photospheric velocity [53] and compared its evolution with that of the other SNe considered in this paper (Figure REF , middle panel).",
"At early phases the ejecta velocity of SN 2014G looks higher than all the other SNe considered.",
"Later on, however, it settles to a value of $\\sim 3300$ km s$^{-1}$ , in line with SNe 2009kr and 2013ej.",
"From the spectra we estimated also the temperature evolution, obtained fitting the continuum of each spectra with a black-body function.",
"The errors were calculated with the same Monte Carlo technique described above.",
"The complete evolution is reported in Figure REF , bottom panel, along with a comparison with the other SNe considered so far.",
"The ejecta in SN 2014G appear to be hotter in comparison to the other similar SNe, with a late-time temperature comparable with that of SN 1998S.",
"Moreover, we inferred the radius of the photosphere from the luminosity and the black-body temperature.",
"The evolution of the first 3 points is well described by a parabola, with the vertex coincident with our estimate of the explosion date, strengthen the assumption we made in Section ."
],
[
"Results and analysis",
"Discussion Figure: Hα\\alpha (right panel) and Hβ\\beta (left panel) profile history.",
"The evolution of several features are marked: the maximum of the Hα\\alpha emission (long-dashed blue line); the Hα\\alpha and Hβ\\beta minima (short-dashed green line); the [O I] doublet emission peaks (dot-dashed magenta line); the mysterious peak around 6400Å (dotted red line).",
"The rest frame zero velocity of both Hα\\alpha and Hβ\\beta are also marked by solid yellow lines."
],
[
"H$\\alpha $ evolution",
"Figure REF shows a zoom in of the H$\\alpha $ and H$\\beta $ profiles evolution (velocities are in the rest frame).",
"The blue side of H$\\alpha $ has been extended up to $-14000$ km s$^{-1}$ in order to include also the appearance of the [O$\\,$I] doublet $\\lambda \\lambda $ 6300,6364.",
"The peak of the H$\\alpha $ emission (blue dashed line) appears to be blue-shifted, by several thousands km s$^{-1}$ at early phases, to only 500 km s$^{-1}$ after $\\sim 80$ d. This is in contrast with the classical P-Cygni profile description, which predicts the emission to be at zero rest velocity.",
"This behaviour has been already reported in many SNe [15], [114].",
"This is thought to be a direct consequence of the steep density profile of the ejecta layers, which translates to more confined line emission and higher occultation of the receding part of the ejecta [3].",
"It is odd, however, that the emission never reaches the rest frame zero velocity, as one could expect at least at late phases.",
"The absorptions of both H$\\alpha $ and H$\\beta $ , on the other hand, evolve steadily starting from about 9000 km s$^{-1}$ to $6000-7000$ km s$^{-1}$ by day 100.",
"Observations of other Type II-L SNe showed that this class tend to have less prominent H$\\alpha $ P-Cygni absorptions then Type II-P[99], [50].",
"This could be simply a consequence of a progenitor with less hydrogen in the ejected envelope, naively interpreted as less absorbing material along the line of sight.",
"Models by [27] alternatively suggested that this could be the consequence of a particularly steep density profile of the ejecta.",
"[99] also proposed that weaker absorption can be the result of a smaller photosphere in comparison to the extent of the ejecta.",
"In this scenario the column of absorbing material is narrow and weighted to lower velocities.",
"The formation of a P-Cygni profile in SNe is not a trivial matter, and many factors can contribute simultaneously.",
"Many of these however point towards more diluted ejecta, supporting the idea of Type II-L SNe having progenitors with less-massive and more spatially extended hydrogen envelopes than Type II-P SNe.",
"At $\\sim 100$ d a transition occurs in the blue part of H$\\alpha $ : the absorption feature disappears and is filled with a narrow emission at $\\sim 6395$ Å.",
"This peak is also present in the next three spectra and appears to have an evolution in velocity, marked in Figure REF by the red dotted line.",
"No similar features are reported in the other SNe considered for comparison, with the possible exception of SN 1998S (see Figure REF ).",
"However, in the case of SN 1998S, there are two symmetric features around the H$\\alpha $ zero velocity emission, together with similar features around other Balmer lines, interpreted as the result of the interaction of the ejecta with a disk-like structure [73], [41].",
"In the case of SN 2014G only a blue peak is visible, and is limited only to the H$\\alpha $ line.",
"We could not find a plausible identification for the source of this emission.",
"Since its appearance is simultaneous with that of the [O$\\,$I] doublet, one might believe that these two features are related.",
"However, the velocity evolutions are quite distinct, with the [O$\\,$I] remaining nearly steady (magenta dot-dashed line in Figure REF ), while the mysterious peak moves redwards towards H$\\alpha $ .",
"At this phase, the same velocity evolution is visible in H$\\beta $ , which might suggest a hydrogen origin for the feature.",
"Assuming it is a high velocity feature of hydrogen, it evolves from $-7580$ km s$^{-1}$ in the spectrum at 103 d to $-6755$ km s$^{-1}$ at 187 d. High velocity hydrogen features have been identified before in other SNe spectra [60], although always in absorption and at much earlier phases.",
"The feature in SN 2014G is somewhat reminiscent of the so-called Bochum event [55]: the emergence of a blue and a red peak in the H$\\alpha $ profile of SN 1987A after 20 d. The origin of the two peaks is independent, thus the absence of the red one in SN 2014G is not an issue.",
"[56] suggested that the blue peak is indeed an “absorption deficit” rather than an emission.",
"This would be the result of a stratification of the hydrogen in the ejecta in 3 layers, the top and the bottom one being excited hydrogen, while the middle one would be constituted by ground state hydrogen with a low optical depth for H$\\alpha $ .",
"This peculiar geometry would give rise to an emission-like feature at the velocity corresponding to the middle layer.",
"However, even leaving aside the physical explanation for the complex structure to arise, this scenario does not comfortably explain our data, because a similar stratification would have created an anomalous feature also in H$\\beta $ , as in SN 1987A.",
"Therefore we suggest that a Bochum event-like explanation does not quantitatively match what we see in the spectral evolution.",
"Figure: Modelling of the 6200-68006200-6800 Å region of the SN 2014G spectrum at 342 d. We used gaussian profiles to reproduce Hα\\alpha and the [O\\,I] λλ\\lambda \\lambda 6300,6364 doublet and a boxy profile to reproduce the “bridge” in between.In the spectrum at 342 d the mysterious peak disappeared, leaving place to a wider feature connecting H$\\alpha $ and the [O$\\,$I] doublet.",
"The elapsed time between the +187 d and the last +342 d spectra is too long to assert, with confidence, any direct evolution of the narrow emission at $\\sim 6400$ Å to this “bridge”.",
"However the fact that we have two unusual features, at the same wavelength is peculiar and they may well be linked.",
"We modelled the line emissions using gaussian profiles, and subtracting their contribution to the data, we obtain a flat-top feature corresponding to the “bridge” and also a noticeable excess of flux in the red wing of H$_\\alpha $ .",
"Tentatively, we redid the fit with two boxy features symmetrically located with respect to the H$\\alpha $ peak, reproducing nicely the overall emission profile in this region, as can be seen in Figure REF .",
"The blue boxy profile is centred at $-5940$ km s$^{-1}$ and it is $\\sim 4000$ km s$^{-1}$ wide, while the red one is centred at 6270 km s$^{-1}$ and with the same width.",
"Note that we did not impose this condition, but this result came out from the fit.",
"Flat-top profiles are usually interpreted as the ejecta interacting with a spherical shell of CSM [61], [17], but it is quite unique to see two separate boxy profiles which are (roughly) symmetric with respect to the main emission.",
"This feature could be attributed to a strongly bipolar geometry, namely from a jet-like flow of the ejecta interacting with a spherical CSM [106].",
"Or, vice versa, from a spherically symmetric ejecta interacting with an asymmetric CSM.",
"The first scenario would create strong asymmetries in all the lines coming from the ejecta while here we see the boxy features only around H$\\alpha $ .",
"Thus we favour the second scenario, in which the most external part of spherical ejecta starts to interact with an highly bipolar CSM.",
"At this epoch (342 d) the interaction would thus be occurring only with the outer, hydrogen-rich part of the ejecta.",
"Although one would expect to see similar features around all Balmer lines, the lower optical depth of other lines than H$\\alpha $ could have prevented these features to be detectable.",
"Highly asymmetric CSM structures are well known to exist around massive evolved stars [11], [107], [109], [51].",
"The one surrounding the progenitor of SN 2014G would need to be composed of two polar lobes (like in the $\\eta $ -Car nebula), with a relatively narrow angle between their axis and the observer's point of view.",
"Given a maximum velocity of the ejecta of $\\sim 10000$ km s$^{-1}$ and the initial maximum velocity of the boxy feature of $\\sim 7600$ km s$^{-1}$ , taking the arccosine of the ratio of these two velocities we would infer an angle of $\\sim 40$ between the observing axis and the axis of the polar lobes.",
"In this scenario, the emission-like feature which appeared in the spectra at $\\sim 6400$ Å at $\\sim 100$ d could be interpreted as the beginning of the shock of the outer ejecta to the CSM.",
"The redwards shift then, could be due to the reverse-shock travelling inwards through the ejecta.",
"If this is the case, one may expect a red peak symmetrical to H$_\\alpha $ .",
"However, the flux of the red boxy profile that we use to fit the line profile at late phases is less than half the blue one.",
"So perhaps the red component at these phases is not bright enough to be detected, or there are radiative transfer effects across the CSM/ejecta interaction region that mask the receding material.",
"Overall the fact that there are two broad components at either side of the H$\\alpha $ emission peak, with approximately the same velocity, does suggest a bi-polar structure."
],
[
"$^{56}$ Ni mass",
"Once all hydrogen in the envelope has recombined, the LC of a Type II SN settles onto the so-called radioactive tail.",
"At this phase the energy source is the deposition of $\\gamma $ -rays and positrons originating from the decay chain $^{56}$ Ni$\\rightarrow ^{56}$ Co$\\rightarrow ^{56}$ Fe.",
"$^{56}$ Ni has an e-folding time of 8.8 d thus the first decay is dominant in the very first part of the LC.",
"However at these early phases the decay contribution is hidden by the hydrogen recombination power.",
"$^{56}$ Co, on the other hand, has a e-folding time of 111.4 d, thus the second decay of the chain is the one that shapes the LC when the recombination ends.",
"The radioactive decay rate translates directly into a well defined slope of the LC (0.98 mag 100 d$^{-1}$ ), while the amount of $^{56}$ Ni fixes the luminosity [14].",
"As reported in [63], the input energy during Co decay is $L_0(t)=9.92 \\times 10^{41} \\frac{M_{^{56}\\rm {Ni}}}{0.07 M_\\odot }\\left(\\rm {e}^{-t/111.4} - \\rm {e}^{-t/8.8}\\right)~\\rm {erg}~\\rm {s}^{-1}~~,$ where $M_{^{56}\\rm {Ni}}$ is the mass of $^{56}$ Ni ejected during the explosion.",
"In general in Type II SNe the $\\gamma $ -rays and positrons energy is fully trapped and thermalized, and the luminosity decline following the energy input, as shown with the dotted line in Figure REF .",
"SN 2014G however declines faster, suggesting that the $\\gamma $ -ray trapping is incomplete.",
"Figure: Zoom in of the radioactive tail of SN 2014G.",
"The green points correspond to the uvoir bolometric LC, while the red ones are the integrated flux from the spectra.",
"The red dashed line represents where the LC would be if it followed the 56 ^{56}Co decay in a full trapping regime.",
"The blue solid line is the fit to the data with Equation .",
"The luminosities extrapolated from the spectra have high uncertainties, thus were not included in the fit.The problem of a non-complete trapping of the $\\gamma $ -rays has been analysed by [21] for the case of stripped-envelope SNe.",
"They assume a simple model with spherical symmetry and homologous expansion, in which the $\\gamma $ -ray deposition is represented by a simple absorption process in radiative equilibrium.",
"Then a simple expression can describe the luminosity: $L(t)=L_0(t)\\times \\left(1-\\rm {e}^{-(\\tau _{tr}/t)^2}\\right)~~,$ where $L_0(t)$ comes from Equation REF and $\\tau _{tr}$ is the full-trapping characteristic time-scale defined as $\\tau _{tr}=\\left(D\\kappa _\\gamma \\frac{M_{ej}^2}{E_k}\\right)^\\frac{1}{2}~~,$ where $M_{ej}$ is the total ejecta mass, $E_k$ is the kinetic energy, $\\kappa _\\gamma $ is the $\\gamma $ -ray opacity and $D$ is a constant which depends on the density profile (i.e.",
"for a uniform density profile $D=9/40\\pi $).",
"The intent here is not to model the ejecta behaviour, rather to describe the radioactive contribution to the LC.",
"For this purpose, Equation REF is adequate and, as shown in Figure REF , a good fit can be obtained with the observed uvoir radioactive tail.",
"Given the high uncertainties of the luminosities extrapolated from the spectra, we decided to not include them in the fit.",
"A $^{56}$ Ni mass of $0.059\\pm 0.003$ M$_\\odot $ and $\\tau _{tr}=162\\pm 10$ d is inferred [62].",
"In Figure REF , the dashed line shows the expected luminosity decline with the amount of $^{56}$ Ni inferred and in case of complete trapping.",
"Assuming a typical explosion energy of 1 foe, a uniform density profile of the ejecta, and a fiducial $\\gamma $ -ray opacity $\\kappa _\\gamma =0.03$ cm$^2$ g$^{-1}$ [22], from Equation REF we can infer an indicative $M_{ej}\\sim 4.8$ M$_\\odot $ .",
"Taking instead the 2.1 foe of kinetic energy estimated by B16 from their LC modelling, we infer $M_{ej}\\sim 7.0$ M$_\\odot $ , in agreement with what they obtained from the modelling.",
"On the other hand, the $^{56}$ Ni mass found is slightly larger than the amount found by B16, who reported three different measurements: 0.045 M$_\\odot $ from the luminosity of radioactive tail, 0.055 M$_\\odot $ from a comparison with SN 1987A and 0.052 M$_\\odot $ from their LC modelling.",
"In fact, not considering the incomplete trapping of the $\\gamma $ -rays, lead to underestimate the nickel mass.",
"They considered a leakage of photons only in the LC modelling, but they might have underestimated the characteristic time-scale $\\tau _{tr}$ by considering the early tail only.",
"The $^{56}$ Ni inferred for SN 2014G is within typical values for Type II SNe [54], [83], [97], [120].",
"Nevertheless, the presence of incomplete trapping is unusual for a Type II SN.",
"It is clear from Equation REF that the factors which can cause the incomplete trapping are essentially three: a low mass of the ejecta, a high kinetic energies or peculiar density profiles.",
"[2] already found a significant number of Type II SNe with a $V$ -band declining faster than the $^{56}$ Co decay (with full trapping).",
"They attributed this behaviour to low-mass and highly-diluted ejecta, which would be unable to completely trap the $\\gamma $ -rays.",
"Since Type II-L SNe are supposed to arise from progenitors with rarefied envelopes [9], this could be a plausible scenario for SN 2014G.",
"In addition to this, from Figure REF , SN 2014G appears to have faster ejecta than the other SNe considered, possibly suggesting a particularly energetic event.Equation REF is appropriate if the Ni is in the centre of the ejecta.",
"However, if there is strong mixing, and a considerable amount of Ni is spread out in the outer layers of the ejecta, then the $\\gamma $ -rays of the decay would have higher escape probability, and would then heat the surrounding material less efficiently, decreasing the luminosity.",
"If this is the case, the ejecta mass could be higher than deduced from Equation REF .",
"One should note, however, that $^{56}$ Ni was strongly outmixed in SN 1987A, which still had full trapping for several hundred days."
],
[
"Ni/Fe production ratio",
"In the latest spectra, the [Ca$\\,$II] $\\lambda \\lambda $ 7291,7323 doublet showed a broad red shoulder.",
"[65] (hereafter referred to as J15) showed the case of SN 2012ec in which a prominent line was visible at this wavelength.",
"This line was identified as an emission feature from stable $^{58}$ Ni.",
"The spectral models predict a distinct [Ni$\\,$II] $\\lambda $ 7378 here, and in SN 2012ec the identification was made possible due to the relatively weak flux of the [Ca$\\,$II] doublet.",
"Moreover the [Ni$\\,$II] line at 1.939 $\\mu $ m was also identified in a NIR spectrum.",
"In the case of SN 2014G, however, the stronger [Ca$\\,$II] doublet and the higher blending makes the nickel identification less trivial.",
"The feature could also be simply the result of asymmetries in the ejecta, such as we argued for H$\\alpha $ .",
"However, in this case the excess is in the red, in contrast to H$\\alpha $ where the stronger asymmetric component was shifted bluewards.",
"Nevertheless, we see that the Mg$\\,$I] $\\lambda $ 4571 line has also an asymmetric red shoulder.",
"So we attempted to fit the [Ca$\\,$II] feature using a doublet composed of two lines with the same velocity profile as observed in the single Mg$\\,$I] $\\lambda $ 4571 line.",
"Alternatively, we fitted the whole feature with multiple gaussian profiles, including specifically lines at $\\lambda $ 7378 and $\\lambda $ 7412 representing [Ni$\\,$II].The first method did not give a satisfactory fit and we then conclude that the red shoulder of the [Ca] doublet is better fit with an additional feature of [Ni$\\,$II] (cfr.",
"Figure REF ).",
"This scenario is also physically supported by the fact that the material contributing to [Ca$\\,$II] emission is mostly situated in inner regions of the ejecta than the material emitting in Mg$\\,$I] [40], [82].",
"Therefore differences in the shape of the profiles between the two ions are to be expected.",
"J15 presented a new analytic method to determine the Ni/Fe ratio in nebular spectra of SNe in the $7100-7400$ Å region.",
"The physical regimes for which the method is valid was confirmed by inspecting the conditions in forward spectral simulation models.",
"Here, we apply the analytic method to SN 2014G, despite the quantitative measurement of [Ni$\\,$II] $\\lambda $ 7378 is not as easy as in the case of SN 2012ec, where the line is resolved.",
"In the above mentioned spectral region, there are 8 prominent emissions: [Ca$\\,$II] $\\lambda \\lambda $ 7291,7323, [Fe$\\,$II] $\\lambda $ 7155, $\\lambda $ 7172, $\\lambda $ 7388 and $\\lambda $ 7453, [Ni$\\,$II] $\\lambda $ 7378 and $\\lambda $ 7412 (see Figure REF ).",
"Following J15, we fixed the strength ratio between the lines of the same species: the Fe lines $\\lambda $ 7155 and $\\lambda $ 7453 come from the same atomic level, and thus their luminosity ratio is constant with L$_{7453}=0.31$ L$_{7155}$ .",
"Also the other two Fe lines come from the same level, thus we imposed L$_{7388}=0.74$ L$_{7172}$ .",
"We could also fix the ratio between the [Fe$\\,$II] $\\lambda $ 7155, $\\lambda $ 7172, despite coming from two different levels.",
"This ratio depends only weakly on the temperature, and following the J15 model we assumed L$_{7172}=0.24$ L$_{7155}$ .",
"J15 coupled each line to [Fe$\\,$II] $\\lambda $ 7155, but overall the line ratios among the iron lines were the same as ours.",
"On the other hand, the two Ni lines come from two different levels and their luminosity ratio depends non-negligibly on temperature.",
"However J15, based on their model with ${T}=3180$ K, fixed L$_{7412} = 0.31$ L$_{7378}$ .",
"We employed the same approach, but in appendix REF we investigated how to relax this constraint.",
"We used simple gaussians as fitting profiles, forcing all the lines to have the same velocity $\\Delta v$ (i.e.",
"the same FWHM) but allowing also a rigid shift $\\Delta \\lambda $ of the line centroids (however keeping fixed the relative position of each line).",
"In total we had 5 free parameters and our best fit with this set-up is shown in Figure REF .",
"The values inferred from this fit were L$_{7155} = 4.76\\times 10^{-16}$ erg s$^{-1}$, L$_{7291}=2.37\\times 10^{-15}$ erg s$^{-1}$, L$_{7378}=1.36\\times 10^{-15}$ erg s$^{-1}$, $\\Delta \\lambda =-6.7$ Å, $\\Delta v=2605$ km.",
"This gave the final result of L$_{7378}$ /L$_{7155}=2.9\\pm 0.2$.",
"Figure: GTC spectrum at 342 d between 7000 and 7600 Å and the Gaussian fit described in Section (red).",
"The [Fe\\,II] λ\\lambda 7155 and the [Ni\\,II] λ\\lambda 7378 and λ\\lambda 7412 lines are also marked (see legend).The iron and nickel content can be inferred by the ratio of the luminosity of the [Fe$\\,$II] $\\lambda $ 7155 and [Ni$\\,$II] $\\lambda $ 7378 lines following the relation $\\frac{L_{7378}}{L_{7155}}=4.9\\left(\\frac{n_{Ni}}{n_{Fe}}\\right)\\rm {e}^{0.28{eV}/kT}~,$ where $n_{Ni}$ and $n_{Fe}$ are the number densities of Ni$\\,$II and Fe$\\,$II, $k$ is the Boltzmann constant and $T$ the temperature (see J15 for the origin of the constants).",
"The temperature can be constrained from the luminosity of one Fe line and the total Fe mass, assuming local thermodynamic equilibrium (LTE).",
"The iron production is dominated by the $^{56}$ Ni$\\rightarrow ^{56}$ Co$\\rightarrow ^{56}$ Fe chain.",
"At these late phases, the Ni has all decayed into Co, and 95 per cent of it has already become Fe, thus the iron mass is constrained by the $^{56}$ Ni mass we measured in the previous Section (0.059 M$_\\odot $ ).",
"Then assuming that most of iron is in the form of Fe$\\,$II (J15 find it to be around 90 per cent at 370 d), we can write (see J15 for the derivation of the constants) $\\frac{L_{7155}}{M(^{56}{Ni})}=\\frac{8.67\\times 10^{43}}{15 + 0.006T}\\rm {e}^{-1.96{eV}/kT}~{erg~s^{-1}M}^{-1}_\\odot ~.$ This function is really steep, which translates in the temperature varying little for changes in the $L_{7155}/M(^{56}$ Ni) ratio (see Figure 6 in J15).",
"From this equation, we obtained $T=2701^{+44}_{-47}$ K. Then putting this value in Equation REF we infer $n_{Ni}/n_{Fe} = 0.18\\pm 0.02$ .",
"This is very similar to 0.19 found by J15 for SN 2012ec.",
"As they pointed out, these values are considerably higher than the 0.06 solar abundance ratio.",
"This is the fourth SN with a significantly supersolar Ni/Fe production [76], [78], [81].",
"Several others show solar or subsolar (J15), so there seems to be significant diversity.",
"Primordial Fe and Ni contamination could have contributed to the line flux we measured from the fit.",
"As deduced by J15, potentially this contamination could have had underestimated the Ni/Fe ratio in the iron zone, however less than $\\sim 1/3$ .",
"[66] demonstrated through nucleosynthesis simulations that a high Ni/Fe ratio, like the one found for SN 2014G, imply burning and ejection of the silicon-layer material in the progenitor, with neutron excess $\\eta \\sim 6\\times 10^{-3}$ .",
"Such a process is most easily achieved in lower mass progenitors (M$_{ZAMS}<13$ M$_\\odot $ ) exploding with a delay time of less than 1 second.",
"However, strongly asymmetric explosions may also achieve relatively high Ni/Fe ratios in more massive progenitors."
],
[
"The progenitor",
"The high ionisation lines of He$\\,$II/N$\\,$IV/C$\\,$IV seen in the early spectra of the Type IIb SN 2013cu addressed [43] to infer a WR progenitor for that transient.",
"However, [49] and [48] modelling the early spectra of SN 2013cu found that chemical composition, mass loss-rate and wind velocity are consistent with the properties of a LBV, a YHG or an extreme RSG.",
"Also, [100] studied the early spectra of SN 1998S and found a slow wind consistent with that of a RSG progenitor.",
"A similar interpretation was given by [108], and they conclude that the early-time WR-like spectrum has little to do with the spectral type of the progenitor before explosion.",
"In fact N-rich nebulae can also be found around evolved massive stars occupying the upper Hertzsprung-Russell diagram [104], [70], [103].",
"A sample of spectroscopy from young Type II SNe by [68] has recently been published showing that about 14 per cent of all Type II SNe which have spectra available within 10 days of explosion have these high ionisation features present.",
"They also suggest that the SNe with these features tend to have brighter peak magnitudes than average for Type II SNe and SN 2014G would fit well with that picture.",
"Although we present one object, in comparison to the sample of 12 of [68], our comprehensive dataset allows us to track this SN into its latest stages to probe the nucleosynthesis within the progenitor and estimate its mass.",
"From the modelling of the LC, B16 inferred a progenitor radius of 630 R$_\\odot $ .",
"Therefore the progenitor of SN 2014G was likely to be an extended supergiant star, with at least part of its H-rich envelope left at the time of explosion.",
"Current theories for the progenitors of Type II-L involve a more extended less-massive hydrogen envelope with respect to Type II-P SNe progenitors [9], and many aspects of SN 2014G favour this scenario.",
"First of all, the LC in the photospheric phase points towards a more diluted envelope, unable to sustain a flat luminosity plateau.",
"Secondly, the faster decline of the radioactive tail with respect to complete trapping of $\\gamma $ -rays from $^{56}$ Co decay suggests less-dense ejecta.",
"This is also consistent with the blue-shift of the H$\\alpha $ emission of SN 2014G, which shows a blue-shifted emission up to 5000 km s$^{-1}$ at 25 d and settles close to zero velocity already around 65 d. In fact [3] showed that the blue-shift should be higher in the early spectra for objects with more extended envelopes, and it should also evolve towards rest frame zero velocity much faster than for more compact objects.",
"We finally tried to constraint the mass of the progenitor.",
"[63], [64] showed how the spectral modelling of the nebular phase can actually be used to constrain the M$_{ZAMS}$ of the progenitors of Type II SNe [25], [26].",
"They find, in particular, that the flux of the [O$\\,$I] $\\lambda \\lambda $ 6300,6364, Na$\\,$I $\\lambda \\lambda $ 5890,5896 and Mg$\\,$I] $\\lambda $ 4571 lines show a strong correlation with the progenitor mass, with the oxygen doublet being the most important as a diagnostic of the core mass.",
"We took their synthetic spectra as a comparison for the nebular spectra of SN 2014G at 187 and 342 d, seeking the best match with the [O$\\,$I] lines flux.",
"We point out that, as it is possible to see from Figure REF , the “bridge” feature between the oxygen doublet and H$\\alpha $ does not significantly affect the oxygen flux, allowing a direct comparison between the data and the models.",
"In order to perform a correct comparison, the synthetic spectra had to be scaled to the same distance of SN 2014G and to the same amount of $^{56}$ Ni.",
"Moreover, having $\\tau _{tr}$ significantly higher than the one inferred for 2014G (470 and 530 d for the model of 15 and 19 M$_\\odot $ M$_{ZAMS}$ progenitor, respectively), we rescaled the models using Equation REF , to take account of the missing flux due to the higher leakage of $\\gamma $ -rays.",
"Finally we also corrected for the small difference in phases, reducing the cobalt contribution according to how much has decayed in the time between the epoch of the model and the epoch of the observed spectrum.",
"The comparison is reported in Figure REF .",
"The level of the quasi-continuum of the models matches that of the spectra of SN 2014G, suggesting that the opacity of the ejecta to $\\gamma $ -rays is in fact described sufficiently well by Equation REF .",
"From the comparison it is clear that the [O$\\,$I] $\\lambda \\lambda $ 6300,6364 doublet of SN 2014G sits between the model with a progenitor with M$_{ZAMS}$ of 15 and 19 M$_\\odot $ .",
"Therefore, we infer a progenitor with M$_{ZAMS}\\simeq 17\\pm 2$ M$_\\odot $ .",
"A particularly low hydrogen flux looks also evident from the comparison with the synthetic spectra.",
"This again favours the scenario of Type II-L SNe having progenitors with a reduced hydrogen envelope at the moment of explosion [93].",
"This is one of the first times that this method has been applied on a Type II-L SN [120], [127], and this is also arguably one of the most massive ZAMS progenitor for a Type IIP/L so far [102], favouring the scenario of Type II-L SNe arising from more massive ZAMS stars than Type II-P [32], [33].",
"Small differences between the observed spectra and the models are present, in particular for other tracers of M$_{ZAMS}$ , like Mg$\\,$I] $\\lambda $ 4571 and Na$\\,$I D. Modelling of the Mg and Na lines is, however, more complex than the modelling of [O$\\,$I] $\\lambda \\lambda $ 6300,6364, with increased sensitivity to density and ionisation conditions.",
"Both elements have only small fractions in the neutral state, and the exact value governs whether there is a cooling contribution or just a recombination contribution [67].",
"Moreover, sodium shows a somewhat erratic growth with M$_{ZAMS}$ [122].",
"A significant part of the Na$\\,$I line is caused by scattering in hydrogen-rich gas, and a reduced hydrogen-zone mass in SN 2014G may be responsible the weak line.",
"The oxygen on the other hand is mainly neutral, and thus its cooling emission less sensitive to ionisation ratios.",
"Thus, in lack of ad-hoc modelling, focusing only on the forbidden oxygen doublet is the preferred way to proceed.",
"As previously mentioned, we addressed the issue of the non-complete trapping observed in SN 2014G by down-scaling the models with Equation REF .",
"However, since the M$_{ZAMS}$ estimate is based on line comparison, one should focus on how the incomplete trapping is influencing the line formation, which is not trivial.",
"The mass loss in the [123] stellar evolution models used in J14 is computed with standard recipes at solar metallicity, and gives only minor mass loss for M$_{ZAMS} < 20$ M$_\\odot $ .",
"In the J14 models, the ejecta have a defined morphology, i.e.",
"a core where $^{56}$ Ni, Si/S, O and part of the He and H zones are macroscopically mixed, and an unmixed envelope.",
"The mixing structure is guided by the morphologies obtained in multi-dimensional simulations.",
"Thus the incomplete trapping of the $\\gamma $ -rays can be due to the following scenarios (see Section REF ): A less massive hydrogen envelope.",
"The envelope itself does not influence directly the line formation of core elements like oxygen.",
"If we then assume a core with unchanged properties, the core of SN 2014G would still be similar to the models, and the O$\\,$I luminosity should not be scaled with any trapping function.",
"The spectra would then be overall dim because of the lost of the hydrogen zone deposition.",
"But if the core structure has changed due to the hydrogen envelope loss (which is likely), weaker in-mixing of oxygen leads to dimmer oxygen lines.",
"Thus, the models are too bright and need to be scaled down.",
"High kinetic energies.",
"The ejecta are expanding faster due to higher energies, and the trapping is weaker.",
"Thus the models are too bright in all zones and a down-scaling is appropriate.",
"Higher $^{56}$ Ni mixing.",
"The whole ejecta are illuminated in a more dilute $\\gamma $ -field and $\\gamma $ -rays escape starts earlier.",
"Oxygen is heated less efficiently and thus a uniform down-scaling is appropriate.",
"We favour the idea that mixing played an important role, as it is also shown by the presence of carbon an nitrogen in the early spectra, and thus that a down-scaling of the models is necessary.",
"In Figure REF we showed, in grey, the 15 M$_\\odot $ ZAMS model scaled only to the $^{56}$ Ni mass of SN 2014G, but with the $\\gamma $ -ray deposition, including escape, computed in the models.",
"The difference in flux in all wavelength between the model and the observed spectrum is evident, again suggesting a down-scaling of the models in order to perform a satisfying comparison.",
"Everything considered, the comparison reported in Figure REF should be correct, and the estimate of a relatively massive ZAMS progenitor for SN 2014G consistent.",
"B16 modelled the LC of SN 2014G with a semi-analytical approach, inferring an ejecta mass of only 7 M$_\\odot $ , plus 2 M$_\\odot $ for the compact remnant.",
"Assuming a fiducial error of 1 M$_\\odot $ on their estimate, the $M_{\\rm ZAMS} \\simeq 17\\pm 2$ M$_\\odot $ estimated by us might indicate a mass loss between 5 and 11 M$_\\odot $ , from ZAMS to the explosion.",
"The “classical” empirical mass loss rate for RSGs given by [24] predicts that progenitors up to $M_{\\rm ZAMS} \\simeq 25$ M$_\\odot $ should retain enough mass to produce a Type II-P SN.",
"More recent works, however, showed how RSG stars could have experience much higher mass loss in their main sequence phase than previously thought, with several observations supporting this statement [105], [121], [23].",
"The mechanism at the origin of this enhanced mass loss is not clear, with pulsation-driven superwinds [124] or super-Eddington luminosities in the outer layers of the star [29] as two possible physical reasons.",
"In the former scenario, [124] find pulsations to occur from a M$_{ZAMS}$ of 17 M$_\\odot $ , and they also infer that a 20 M$_\\odot $ ZAMS progenitor would end its life as a yellow supergiant (YSG) of 6.1 M$_\\odot $ , 0.5 M$_\\odot $ of which is hydrogen.",
"[29] instead worked on models with enhanced mass loss due to the luminosity in the external layers exceeding the Eddington limit.",
"From their rotating models of ZAMS stars of 15 and 20 M$_\\odot $ they inferred a mass of $\\sim 11$ and 7 M$_\\odot $ respectively at the end of carbon burning phase.",
"According to these simulations then, the 17 M$_\\odot $ M$_{ZAMS}$ progenitor of SN 2014G inferred in this work could have reasonably led to a pre-explosion progenitor of only 9 M$_\\odot $ , as the one inferred by B16.",
"Without direct observations of the progenitor we cannot discriminate between a RSG or a YSG.",
"Several Type II SNe have observational evidence for YSG progenitors [79], [80], [31], [32], [42].",
"[46] showed how these YSG progenitors could actually be the end of the evolution of RSG stars with enhanced mass loss, which may qualitatively match our observations for SN 2014G.",
"We also have to take into consideration that a companion star in a close-binary system could also have been the culprit for the mass loss, through a Roche lobe overflow.",
"[125] showed how the mass loss in close binary for a ZAMS star of 17 M$_\\odot $ could have been much more extreme, leaving a star of only $\\sim 4$ M$_\\odot $ .",
"A fine tuning of distance$-$ mass ratio between the two stars could easily be able to reproduce the inferred mass loss [92], [20], [30].",
"In Section REF we suggest that an highly asymmetric CSM was the origin of the atypical H$\\alpha $ profile, which could be consistent with mass loss events in binary systems.",
"Both of these scenarios put strong constraints on current explosion theories, and the growing number of SNe with measured explosive burning products from nebular spectra gives hope for progress in understanding how CCSNe explode."
],
[
"Summary and conclusions",
"Summary and conclusions We presented uvoir photometry and optical spectroscopy for the Type II-L SN 2014G up to 342 d after explosion.",
"Our detailed dataset allowed us to investigate many different aspects of its evolution from very early after explosion to observations deep into the nebular phase.",
"The early spectra show narrow emission lines that disappear by day 9.",
"These high ionisation lines have been seen in other SNe of Types II-P/L and IIb and suggest that the CSM surrounding the star is photo-ionised by UV emission at shock breakout.",
"This study shows one example in which they exist in Type II-L SNe, and persist to at least 3 d after explosion and disappear by about 10 days.",
"One should be careful with classifications that are based on such early spectra, as a IIn classification would typically result.",
"The LC then evolved like a canonical Type II-L, with a linear decay of the LC lasting until $\\sim 80$ d after explosion.",
"The radioactive tail appeared to fall more rapidly than the $^{56}$ Co decay with full trapping.",
"We interpreted this as a leakage of $\\gamma $ -rays, possibly due to more diluted external layers of the ejecta and high levels of mixing.",
"Taking into account the missing flux, we were able to infer a $^{56}$ Ni mass of $0.059\\pm 0.003$ M$_\\odot $ .",
"We further presented extensive late-time spectral coverage of SN 2014G offering further insights into the progenitor and explosion.",
"H$\\alpha $ became optically thin already at 100 d, the spectra showed an early [O$\\,$I] doublet and an intriguing narrow emission between these two lines, which evolved later into a wider and flat-topped boxy profile.",
"We interpreted this feature as the interaction of the outer hydrogen rich ejecta with a strongly asymmetric CSM.",
"We computed a combined line profile of H$\\alpha $ in the 342 d spectrum with the sums of multiple individual velocity components.",
"We showed some evidence that there are two broad, boxy features roughly symmetrically distributed at either side of zero velocity that would explain the complex H$\\alpha $ region.",
"This symmetry led us to infer a strongly bipolar CSM geometry, with one of the lobes oriented toward us, at angle of $\\sim 40$ with the observer line of sight.",
"In the last spectrum the [Ca$\\,$II] $\\lambda \\lambda $ 7291,7323 doublet shows a distinct flux excess on the red side.",
"We found this flux excess consistent with the presence of emission lines, which we identified as [Ni$\\,$II].",
"Applying a semi-analytical line formation method, we were able to infer a Ni/Fe production ratio, obtaining a value of 0.18, 3 times higher than the solar ratio.",
"Finally we investigated and discussed the nature of the progenitor.",
"We compared the nebular spectra with the models of J14, focusing on the flux of the [O$\\,$I] $\\lambda 6300,6364$ doublet.",
"From this comparison, we inferred a progenitor with a ZAMS mass of 17 M$_\\odot $ .",
"From the LC modelling, B16 inferred a progenitor with an ejecta mass of 9 M$_\\odot $ , thus our result implies an extensive mass loss of the progenitor during its life.",
"Therefore, the progenitor of SN 2014G was likely to be a RSG or YSG which experienced extensive mass loss.",
"Traces of the mass loss are the photo-ionised metal-rich CSM inferable from the early spectra and the hints of interaction visible in the late ones.",
"Such extensive mass loss left the progenitor with a low-mass and highly diluted envelope.",
"Overall, SN 2014G is supporting the current theories of Type II-L SNe arising from massive hydrogen-depleted stars."
],
[
"Acknowledgments",
"GT, SB, EC, NE-R, AH, AP, LT, and MT are partially supported by the PRIN-INAF 2014 with the project Transient Universe: unveiling new types of stellar explosions with PESSTO.",
"NE-R acknowledges the support from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no.",
"267251 “Astronomy Fellowships in Italy” (AstroFIt).",
"The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement n$^{\\rm o}$ [291222] (PI : S. J. Smartt) and STFC grants ST/L000709/1.",
"AMG acknowledges financial support by the Spanish Ministerio Economía y Competitividad (MINECO) grant ESP2013-41268-R. Part of this material is based upon work supported by the National Science Foundation under Grant No.",
"1313484.",
"Part of this research was conducted by the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020.",
"This paper is based on observations collected with the 1.22m Galileo telescope of the Asiago Astrophysical Observatory, operated by the Department of Physics and Astronomy “G.",
"Galilei” of the Università of Padova; the 1.82-m Copernico Telescope and the Schmidt 67/92cm of INAF-Asiago Observatory; the Italian TNG operated on the island of La Palma by the Fundaciòn Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica); the NOT, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofìsica de Canarias; the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofìsica de Canarias, in the Island of La Palma; the Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council; and the Telescopi Joan Orò of the Montsec Astronomical Observatory, which is owned by the Generalitat de Catalunya and operated by the Institute for Space Studies of Catalunya (IEEC).",
"This paper is also based on observations made with the Swift and LCOGT Observatories: we thank their respective staffs for excellent assistance.",
"IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation."
],
[
"Temperature from the ratio of [Ni$\\,$",
"In Section REF we fixed the [Ni$\\,$II] $\\lambda $ 7378 and $\\lambda $ 7412 lines luminosity ratio following J15.",
"However, as mentioned before, these two lines arise from two different atomic levels and their luminosity ratio is non-negligibly temperature dependent.",
"This dependence, can be written as $\\frac{L_{7412}}{L_{7378}}=\\frac{n_8~A_{7412}~h\\nu _{7412}~\\beta _{7412}}{n_7~A_{7378}~h\\nu _{7378}~\\beta _{7378}}~~,$ where $n_7$ and $n_8$ are the number densities of level 7 and 8 (the levels from which the lines are from); $h$ is the Plank constant; $\\nu $ is the frequency of the line; $\\beta $ is the escape probability; $A_{7378}=0.23$ s$^{-1}$ and $A_{7412}=0.18$ s$^{-1}$ are atomic constants.",
"The number densities ratio can be written as $\\frac{n_8}{n_7}=\\frac{\\rm {e}^{-\\Delta E_8/kT}g_8}{\\rm {e}^{-\\Delta E_7/kT}g_7}~~,$ where $\\Delta E_7=13550$ cm$^{-1}$ and $\\Delta E_8=14995$ cm$^{-1}$ are the level 7 and 8 energy respectively; $g_7=8$ and $g_8=6$ are the statistical weights of the levels.",
"Assuming the lines to be optically thin (which they are according to the J15 model) then the escape probability are equal to 1 and thus we can write $\\frac{L_{7412}}{L_{7378}}=0.584\\rm {e}^{-0.18{eV}/kT}~~.$ Measuring the luminosity of the two Ni lines, then, we could have another estimate of the temperature.",
"So we untied the bond on the two Ni lines luminosity ratios fixed in the previous analysis and redid the fit (note that doing this we added a parameter to the fit).",
"We inferred a ratio of $L_{7412}/L_{7378}\\simeq 0.49$ (compare to the previous fixed 0.31) which gives the extremely high value of $T\\simeq 11600\\pm 8500$ K. This happens because Equation REF has the shape of an hyperbola with an horizontal asymptote at $L_{7412}/L_{7378}=0.584$ ; then when this ratio is above 0.4 (which happens roughly at $T=6000$ K) the temperature starts to raise exponentially.",
"Moreover for values $L_{7412}/L_{7378}>0.584$ the temperatures become negative.",
"SN 2012ec had a very weak [Ca$\\,$II] which made the [Ni$\\,$II] $\\lambda $ 7378 line the most prominent feature of this spectral region.",
"In our case the [Ca$\\,$II] doublet was much more intense, and the whole structure was still too blended in order to easily detach every single feature, despite the low number of free parameters in the fit.",
"This resulted in an unreliable measurement of the flux of the Ni lines and therefore in an unreliable estimate of the temperature $T$ with this method."
],
[
"Data",
"Data Here we report the complete dataset of our measurements.",
"Table: ugriz photometryTable: UBVRI photometry.Table: NO_CAPTIONTable: NIR photometry.Table: UV photometry.Table: Optical spectroscopy data."
]
] | 1605.06116 |
[
[
"An Algebraic Geometric Approach to Nivat's Conjecture"
],
[
"Abstract We study multidimensional configurations (infinite words) and subshifts of low pattern complexity using tools of algebraic geometry.",
"We express the configuration as a multivariate formal power series over integers and investigate the setup when there is a non-trivial annihilating polynomial: a non-zero polynomial whose formal product with the power series is zero.",
"Such annihilator exists, for example, if the number of distinct patterns of some finite shape $D$ in the configuration is at most the size $|D|$ of the shape.",
"This is our low pattern complexity assumption.",
"We prove that the configuration must be a sum of periodic configurations over integers, possibly with unbounded values.",
"As a specific application of the method we obtain an asymptotic version of the well-known Nivat's conjecture: we prove that any two-dimensional, non-periodic configuration can satisfy the low pattern complexity assumption with respect to only finitely many distinct rectangular shapes $D$."
],
[]
] | 1605.05929 |
[
[
"Exclusive $J/\\psi$ Production in Diffractive Process with AdS/QCD\n Holographic Wave Function in BLFQ"
],
[
"Abstract The AdS/QCD holographic wave function of basis light-front quantization (BLFQ) for vector meson $J/\\psi$ is applied in this manuscript.",
"The exclusive production of $J/\\psi$ in diffractive process is computed in dipole model with AdS/QCD holographic wave function.",
"We use IP-Sat and IIM model in the calculation of the differential cross section of the dipole scattering off the proton.",
"The prediction of AdS/QCD holographic wave function in BLFQ gives a good agreement to the experimental data."
],
[
"Introduction",
"Anti-de Sitter (AdS) and quantum chromodynamics (QCD) has been applied successfully in various fields [1], [2], [3], [4], [5].",
"In this manuscript, we use the AdS/QCD holographic wave function to calculate the $J/\\Psi $ production in photonproduction process.",
"The exclusive vector meson production at the HERA are a good probe for the structure of hadrons [6].",
"There are various approaches to compute the vector meson production in diffraction [7], [8], [9].",
"The dipole model was applied successfully to calculate the production of the vector mesons in diffractive process [10], [11].",
"According to the dipole picture, the virtual photon fluctuates into quark and antiquark pair which is called dipole in the diffractive process firstly.",
"Then, the dipole scatters off the proton by exchange gluons.",
"Finally, the dipole recombines a vector meson.",
"Thus, the amplitude of the vector meson production in the diffractive process contains three parts, they are the light-cone wave function of the photon, the cross section of the dipole scattering off the proton and wave function of the vector meson.",
"The light-cone wave function of the photon can be calculated in QED, and the differential cross section of the dipole scattering off the proton was firstly proposed and fitted by the experimental data several years ago [12], [13].",
"The wave function of the vector meson is very important in the photonproduction in the diffractive process.",
"It is a non-perturbative problem.",
"The wave function of vector meson can not be computed analytically.",
"Thus, it is modeled after the photon wave function.",
"There are various models for the wave function of the vector mesons, for example, NNPZ, DGKP, Boosted Gaussian and Gaus-LC [14], [22], [15], [16], [17], [18].",
"The wave function of vector meson has free parameters, which are determined by the decay constant and normalization condition.",
"But, they can not give the spectrum of the heavy quarkonium states.",
"The wave functions of heavy quarkonium was studied in non-relativistic potential model by Cornell group in 1970s [19], [20].",
"The Cornell potential model gives a good description to the spectrum of the charmonium states.",
"But it can not reproduce the decay width of the charmonium states.",
"On the other side, the wave function of heavy quarkonium were studied in decretized momentum basis [21] and basis of light-front quantization (BLFQ) [23], [26], [25], [24].",
"The AdS/QCD holographic wave function in BLFQ can reproduce the decay width of the charmonium states, and it is in the light-cone system.",
"Thus, we think it is also valid in the photonproduction in diffraction.",
"We should reproduce the $J/\\psi $ production in $\\gamma ^*p\\rightarrow J/\\psi p$ .",
"In this manuscript, we use the AdS/QCD holographic wave function in BLFQ and dipole model to compute the prediction of the cross section of the $J/\\psi $ in diffractive process, and compare the result with the experimental data.",
"The differential cross section of the dipole scattering off the proton is also included in the amplitude in the diffractive process.",
"In the literature, the IP-Sat and IIM models are successfully to describe the process of the differential cross section of the dipole scattering off the proton [28], [27], [18], [30], [31], [32], [29], [33].",
"The two models both have free parameters which are determined from the fit to HERA experimental data.",
"There are various parameter sets for IP-Sat model and IIM model.",
"In this manuscript, we apply the AdS/QCD holographic wave function in BLFQ of the $J/\\psi $ in the diffractive process.",
"Then, we reproduce the cross section of the $J/\\psi $ , and compare our prediction with the experimental data of HERA.",
"This paper is organized as follow.",
"The brief review of the dipole model and AdS/QCD holographic wave function in BLFQ will be presented in Sec. II.",
"The numerical results and some discussion will be presented in Sec.",
"III, and the conclusion will be presented in Sec. IV.",
"We begin with formulas of the differential cross section of the $\\gamma ^*p\\rightarrow Vp$ in the diffractive process.",
"The total cross section can be computed by integrating $t$ .",
"The differential cross section of vector meson in diffractive process can be computed as following: $\\frac{d\\sigma ^{\\gamma ^* p\\rightarrow Vp}}{dt}=\\frac{R_g^2(1+\\beta ^2)}{16\\pi }|\\mathcal {A}_{T,L}(x_p,Q^2,\\Delta )|^2,$ where $x_p=(Q^2+M_V^2)/(Q^2+W^2)$ in the diffractive process, and $t=-\\Delta ^2$ , $T,L $ denote the transverse and longitudinal amplitude.",
"The $1+\\beta ^2$ is the real part of the amplitude, $\\beta =\\tan (\\frac{\\pi }{2}\\lambda ),$ where $\\lambda $ is calculated as $\\lambda =\\frac{\\partial \\ln (\\mathrm {Im}\\mathcal {A}_{T,L}(x))}{\\partial \\ln 1/x}.$ The factor $R_g^2$ is the about the skewness effect [34], it gives $R_g=\\frac{2^{2\\lambda +3}}{\\sqrt{\\pi }}\\frac{\\Gamma (\\lambda +5/2)}{\\Gamma (\\lambda +4)}.$ The amplitude $\\mathcal {A}^{\\gamma ^*p\\rightarrow Vp}_{T,L}(x_p, Q^2, \\Delta )$ in Eq.",
"(REF ) is $\\mathcal {A}_{T,L}^{\\gamma ^*p\\rightarrow Vp}(x_p, Q^2,\\Delta )= i\\int d^2r\\int _0^1\\frac{dz}{4\\pi } \\int d^2b(\\psi _V^*\\psi _{\\gamma })_{T,L}(z,r,Q^2)e^{-i(b-(1-z)r)\\cdot \\Delta }\\frac{d\\sigma _{q\\bar{q}}}{d^2b},$ where $z$ is the fraction of the momentum carried by quark to the photon, $(\\psi ^*_V\\psi _\\gamma )_{T,L}(z,r,Q^2)$ is the overlap of the photon and the vector meson, and $\\frac{d\\sigma _{q\\bar{q}}}{d^2b}$ is the differential dipole cross section.",
"In the IP-Sat model, the differential dipole cross section is[18], [29] $\\frac{d\\sigma _{q\\bar{q}}}{d^2b}=2[1-\\exp (-\\frac{1}{2\\pi B_p}\\frac{\\pi ^2}{2N_c}r^2\\alpha _s(\\mu ^2)xg(x, \\mu ^2)T_p(b)],$ where $T_p(b)=\\exp (-b^2/2B_p)$ is the profile function, and $xg(x,\\mu ^2)$ is the gluon density, which is evolved from $\\mu _0^2$ to $\\mu ^2=\\mu _0^2+4/r^2$ by leading order DGLAP equation, the initial condition of the gluon density is $xg(x,\\mu _0^2)=A_gx^{-\\lambda _g}(1-x)^{5.6}.$ In the IIM model, the differential dipole cross section is written as [30], [32], [33] $\\frac{d\\sigma _{q\\bar{q}}}{d^2b}=2\\times {\\left\\lbrace \\begin{array}{ll}\\mathcal {N}_0(\\frac{rQs}{2})^{2(\\gamma _s+(1/\\kappa \\lambda Y)\\ln (2/rQs))},\\quad \\!",
"rQs\\le 2,\\\\1-\\exp \\big (-a\\ln ^2(b rQs)\\big ),\\quad \\quad rQs>2,\\end{array}\\right.",
"}$ where $Qs=(x/x_0)^{\\lambda /2}\\exp (-\\frac{b^2}{4\\gamma _sB_p})$ , $\\kappa =9.9$ , and $Y=\\ln (1/x)$ .",
"The parameters of the IP-Sat model and IIM model are determined from the fit to combined HERA data for the reduced cross section, the parameters we used are the same as Ref.",
"[29], [33]."
],
[
"AdS/QCD holographic wave function in BLFQ",
"The $\\psi ^*_V\\psi _\\gamma $ is the overlap of the photon and vector meson, we use the AdS/QCD wave function in the basis of light-front quantization.",
"The light vector meson is not considered in this manuscript, because this approach is not applicable for the light vector meson.",
"For more detail information about the AdS/QCD holographic wave function in BLFQ we refer the readers to the Ref. [26].",
"The wave function of the heavy quarkonium is eigenfunction of the eigenvalue equation $H_{eff}\\psi ^J_{m_j}=M^2_V\\psi ^J_{m_j}$ , where $J$ , $m_j$ are total spin and magnetic spin.",
"The Hamiltonian is $H_{eff}=q_\\perp ^2+\\kappa ^4\\zeta _\\perp ^4+\\frac{m_q^2}{z}+\\frac{m^2_{\\bar{q}}}{(1-z)}-\\frac{\\kappa ^4}{(m_q+m_{\\bar{q}})^2}\\partial _z(z(1-z)\\partial _z)+V_g,$ where $z$ is the fraction of the momentum carried by the quark $z=p_q^+/P^+ $ .",
"The variable $\\kappa $ is the strength of the potential , and $m_q$ is the effective quark mass, they are free parameters to fit from the spectrum of the charmonium states.",
"The relative transverse momentum is $k_\\perp =p_{q\\perp }-zP_\\perp $ , and $\\zeta _\\perp $ is the holographic coordinate $\\zeta _\\perp =\\sqrt{z(1-z)}r_\\perp ,$ with $r_\\perp $ the radius of the two quarks.",
"The conjugate of $\\zeta _\\perp $ is defined as $q_\\perp =k_\\perp /\\sqrt{z(1-z)}$ .",
"$V_g$ is the one gluon exchange potential between the quark and antiquark, the detail information of $V_g$ can be referred to the Ref. [26].",
"The AdS/QCD holographic wave function in BLFQ of charmonium states can be written as $\\psi ^J_{m_j}(z,\\mathbf {k}_\\perp ,s,\\bar{s})=\\sum _{m_j,m+s+\\bar{s}}\\widetilde{\\psi }_{m_j}^J(n,m,l,s,\\bar{s})\\phi _{nml}(\\mathbf {k}_\\perp /\\sqrt{z(1-z)},z).$ where $\\widetilde{\\psi }_{m_j}^J(n,m,l,s,\\bar{s})$ is the eigenfunction of the hamiltonian equation, and $s, \\bar{s}$ are the helicities of the quark and antiquark, $n$ is radial quantum number, $m$ is the angular momentum quantum number.",
"The $\\phi _{nml}(\\mathbf {k}_\\perp /\\sqrt{z(1-z)},z)$ is the product of 2D harmonic basis function $\\phi _{nml}(q_\\perp )$ and $\\chi _l(z)$ .",
"They are $\\phi _{nm}(q_\\perp )=\\frac{1}{\\kappa }\\sqrt{\\frac{4\\pi n!}{(n+|m|!",
")}}(q_\\perp /\\kappa )^{|m|}e^{-\\frac{1}{2}q_\\perp ^2/\\kappa ^2}L_n^{|m|}(q_\\perp ^2/\\kappa ^2)e^{im\\theta },$ where $L_n^m(x)$ is the Laguerre polynomial.",
"The $\\chi _l(z)$ is $\\chi _l(z)=\\sqrt{4\\pi (2l+2\\mu +1)}\\frac{\\sqrt{l!\\Gamma (l+2\\mu +1)}}{\\Gamma (l+\\mu +1)}z^{\\mu /2}(1-z)^{\\mu /2}P_l^{(\\mu ,\\mu )}(2z-1),$ where $\\mu =4m_q^2/\\kappa ^2$ , and the $P_l^{a,b}(x)$ is the Jaccobi polynomal.",
"The fourier transformation of 2D harmonic basis function is defined as $\\widetilde{\\phi }_{nnml}(r_\\perp )&=&\\int \\frac{d^2q_\\perp }{2\\pi ^2}e^{-iq_\\perp \\cdot r_\\perp }\\phi _{nml}(q_\\perp )\\\\&=&\\kappa \\sqrt{\\frac{ n!",
"}{\\pi (n+|m|!",
")}}(\\kappa r_\\perp )^{|m|}e^{-\\frac{1}{2}\\kappa ^2r_\\perp ^2}L_n^{|m|}(\\kappa ^2r_\\perp ^2)e^{im\\theta }(-1)^ni^{|m|}.$ Finally, the AdS/QCD holographic wave function in BLFQ in coordinate space can be written as $\\widetilde{\\psi }^V_{s,\\bar{s}}(z,r_\\perp )=\\sqrt{z(1-z)}\\sum _{n,m,l}\\widetilde{\\psi }^J_{m_j}(n,m,l,s,\\bar{s})\\widetilde{\\phi }_{nml}(\\sqrt{z(1-z)r_\\perp })\\chi _l(z).$ The light-cone wave function of the virtual photon can be computed from QED, the longitudinal virtual photon ($m=0$ ) is given by [22] $\\psi _{s,\\bar{s},m=0}(r,z,Q)=e_fe\\sqrt{N_c}\\delta _{s,-\\bar{s}}2Qz(1-z)\\frac{K_0(\\epsilon r)}{2\\pi }.$ The transverse virtual photon ($m=\\pm 1$ ) reads $\\psi _{s,\\bar{s},m=\\pm 1}(r,z,Q)=\\mp e_fe\\sqrt{2N_c}\\lbrace ie^{\\pm i\\theta }[z\\delta _{s,\\pm }\\delta _{\\bar{s},\\mp }-(1-z)\\delta _{s,\\mp }\\delta _{\\bar{s},{\\pm }}]\\partial _r \\mp m_q\\delta _{s,\\pm }\\delta _{\\bar{s},\\pm }\\rbrace \\frac{K_0(\\epsilon r)}{2\\pi },\\\\$ where $\\epsilon =\\sqrt{z(1-z)Q^2+m_q^2}$ , $N_c$ is the color number, and $K_0(x)$ and $K_1(x)$ are second kind Bessel functions.",
"If the eigenvalue equation is solved and the numerical expression of the $\\widetilde{\\psi }^J_{m_j}(n,m,l,s,\\bar{s})$ is providing, we can compute the overlap of the $\\psi ^*_V\\psi _\\gamma (r,z)_{T,L}$ .",
"Then, we can calculate the differential cross section of the diffractive process.",
"The Boosted Gaussian model is a successful ansatz [22], [18].",
"It is modeled after the structure of the photon, its transversely polarized vector meson function is $\\psi ^V_{s,\\bar{s}, m=\\pm 1}=\\mp \\sqrt{2N_c}\\frac{1}{z(1-z)}\\lbrace ie^{\\pm i\\theta }[z\\delta _{s,\\pm }\\delta _{\\bar{s},\\mp }-(1-z)\\delta _{s},\\mp \\delta _{\\bar{s},\\pm }]\\partial _r\\mp m_q\\delta _{s,\\pm }\\delta _{\\bar{s},\\pm }\\rbrace \\phi _T(z,r).$ The longitudinal polarized wave function is some different from the photon.",
"It is given by $\\psi ^V_{s,\\bar{s}, m=0}=\\sqrt{N_c}\\delta _{s,-\\bar{s}}\\Big [M_V+\\frac{m_q^2-\\nabla _r^2}{M_Vz(1-z)}\\Big ]\\phi _L(z,r),$ where $M_V$ is the mass of the vector meson, and $\\nabla _r^2=(1/r)\\partial _r+\\partial ^2_r$ .",
"The transversely and longitudinally scalar function of the Boosted Gaussian take the same form, it is written $\\phi ^V_{T,L}(z,r)=\\mathcal {N}_{T,L}z(1-z)\\exp \\Big (-\\frac{m_q^2\\mathcal {R}^2}{8z(1-z)}-\\frac{2z(1-z)r^2}{\\mathcal {R}^2}+\\frac{m_q^2\\mathcal {R}^2}{2}\\Big ),$ where $\\mathcal {N}$ and $\\mathcal {R}$ are free parameters to be determined from the normalization and decay width conditions.",
"We can see that there are many advantages for the AdS/QCD holographic wave function in BLFQ.",
"The Boosted Gaussian model is just a asantz, the AdS/QCD holographic wave function in BLFQ is from the first principle.",
"The Boosted Gaussian model can not reproduce the spectrum of the charmonium states, the AdS/QCD holographic wave function in BLFQ can reproduce the spectrum.",
"It is necessary to introduce other parameter for the excited states for Boosted Gaussian model.",
"But the parameters are same for the excited states in the AdS/QCD holographic wave function in BLFQ."
],
[
"Numerical result and discussion",
"We calculate the differential cross section and total cross section using IP-Sat and IIM model with the AdS/QCD holographic wave function in BLFQ with parameters with Nmax=8, $m_q=1.492$ GeV, and $\\kappa =0.963$ GeV, where the quark mass in the holographic wave function in BLFQ is different from the quark mass of dipole and the photon wave function.",
"The differential cross section are showed in Fig.",
"REF .",
"The total cross section are presented in Fig.",
"REF and Fig.",
"REF .",
"We integrate the $|t|$ up to 1 $\\mathrm {GeV}^2$ with $m_c=1.27$ GeV parameter set for IP-Sat and IIM model when we calculate the total cross section.",
"The differential cross section of the $J/\\psi $ in the diffractive process are presented in Fig.",
"REF , the differential cross section are computed in two wave functions and compared with the experimental data of H1.",
"The blue markers are H1 data.",
"The solid curves are computing using Boosted Gaussian wave function whose parameters can be found in Ref. [35].",
"The dashed curves are computing using AdS/QCD holographic wave function in BLFQ.",
"It is seen that the two wave functions both give a good agreement to the experimental data.",
"Figure: (Color online) The differential cross section of J/ψJ/\\psi in diffractive process using IP-Sat and IIM model as a function of |t||t|.",
"The solid curve is using the parameter set with m c =m_c=1.27 GeV with Boosted Gaussian model.",
"The dashed curves are using the parameter set with m c m_c=1.27 GeV with AdS/QCD holographic wave function in BLFQ.",
"The experimental data are taken from Refs.",
".Figure: (Color online) The total J/ψJ/\\psi cross section as a function of W compared with H1 and ZEUS experimental data , from IP-Sat and IIM model.",
"The solid curve is using the parameter set with m c =m_c=1.27 GeV with Boosted Gaussian model.",
"The dashed curves are using the parameter set with m c m_c=1.27 GeV with AdS/QCD holographic wave function in BLFQ.The total $J/\\psi $ cross section in diffractive process are showed in Fig.",
"REF and Fig.",
"REF .",
"IP-Sat and IIM model both are applied in the calculation.",
"The W dependence cross section at a fixed $Q^2$ are presented in Fig.",
"REF .",
"It is easy to find that the results of parameter set with $m_c=$ 1.27 GeV give a good description of the experimental data.",
"The cross section as a function of $Q^2$ at a fixed W are showed in Fig.",
"REF .",
"IP-Sat and IIM model are both applied in the calculation.",
"The two wave functions both give a good agreement to the experimental data of HI and ZEUS.",
"Figure: (Color online) (Color online) The total J/ψJ/\\psi cross section as a function of Q 2 +M V 2 Q^2+M_V^2 compared with H1 and ZEUS experimental data , from IP-Sat model and IIM model.",
"The solid curve is using the parameter set with m c =m_c=1.27 GeV with Boosted Gaussian model.",
"The dashed curves are using the parameter set with m c m_c=1.27 GeV with AdS/QCD holographic wave function in BLFQ.At the end of the day, we can see that the AdS/QCD holographic wave function in BLFQ gives a good description to the differential and total cross section of $J/\\psi $ in diffractive process.",
"The holographic wave function gives a little lower prediction than the Boosted Gaussian wave function with same dipole model parameters."
],
[
"conclusion",
"In this manuscript, we use AdS/QCD holographic wave function in calculation of $J/\\psi $ production in diffractive process.",
"We compute the prediction of differential and total cross section of $J/\\psi $ using IP-Sat and IIM model with AdS/QCD holographic wave function in BLFQ and compare the results with the experimental data.",
"There are two parameter sets in IP-Sat and IIM model, the values of mass of the charm quark are different in the two parameter sets.",
"The results show that the prediction using the parameter with $m_c=$ 1.27 GeV give a good description to the experimental data in the small $Q^2$ .",
"The two wave functions both give a good prediction in $\\gamma ^*p\\rightarrow J/\\psi p$ .",
"In the work, we only consider the production of ground state of charmonium.",
"The excited states of charmonium such as $\\psi (2s)$ , $\\psi (3s)$ are absent in the calculation.",
"The parameters of the excited states are the same as the ground state, it is not necessary to introduce new parameters for the excited states, the excited states of the heavy vector meson will be considered at the next step."
],
[
"Acknowledgements",
"We thank the authors of Ref.",
"[26] for providing the data file of the wave function, and thank communication with H. Mantysaari and A. H. Rezaenian.",
"This work is supported in part by the National 973 project in China (No: 2014CB845406)."
]
] | 1605.05859 |
[
[
"Studies of Spuriously Shifting Resonances in Time-dependent Density\n Functional Theory"
],
[
"Abstract Adiabatic approximations in time-dependent density functional theory (TDDFT) will in general yield unphysical time-dependent shifts in the resonance positions of a system driven far from its ground-state.",
"This spurious time-dependence is rationalized in [J. I. Fuks, K. Luo, E. D. Sandoval and N. T. Maitra, Phys.",
"Rev.",
"Lett.",
"{\\bf 114}, 183002 (2015)] in terms of the violation of an exact condition by the non-equilibrium exchange-correlation kernel of TDDFT.",
"Here we give details on the derivation and discuss reformulations of the exact condition that apply in special cases.",
"In its most general form, the condition states that when a system is left in an arbitrary state, in the absence of time-dependent external fields nor ionic motion, the TDDFT resonance position for a given transition is independent of the state.",
"Special cases include the invariance of TDDFT resonances computed with respect to any reference interacting stationary state of a fixed potential, and with respect to any choice of appropriate stationary Kohn-Sham reference state.",
"We then present several case studies, including one that utilizes the adiabatically-exact approximation, that illustrate the conditions and the impact of their violation on the accuracy of the ensuing dynamics.",
"In particular, charge-transfer across a long-range molecule is hampered, and we show how adjusting the frequency of a driving field to match the time-dependent shift in the charge-transfer resonance frequency, results in a larger charge transfer over time."
],
[
"Introduction",
"The art of making approximations in the ab initio quantum theory of many-body systems enables us to investigate realistic systems of various sizes, ranging from atoms to biological molecules with affordable computational resources.",
"Accurate but efficient approximations are crucial to reproduce the experimental results, improve our understanding of the mechanisms in play and make both qualitative and quantitative predictions.",
"Rapid progress in experimental spectroscopy has begun to unveil dynamics of the electronic degrees of freedom on the timescale of tens of attoseconds, which provides a touchstone for the assessment of ab initio electronic structure theories[1], [2], [3].",
"Based on Runge-Gross theorem, time-dependent density functional theory(TDDFT) is an in-principle exact and efficient theory[4], [5], [6], that, with functional approximations inherited from ground state density functional theory stands out prominently among all other methods.",
"The interacting one-body density $n({\\bf r},t)$ is obtained from the evolution of the Kohn-Sham (KS) system, a system of fictitious non-interacting particles.",
"The KS particles evolve under a one-body potential $v_{\\scriptscriptstyle \\rm S}({\\bf r}, t)$ , following $i \\partial _t \\varphi _k({\\bf r},t)= (-\\frac{\\nabla ^2}{2} +v_s({\\bf r},t))\\varphi _k({\\bf r},t)$ , such that the interacting density is reproduced from $n({\\bf r},t)=\\sum _k^{\\mathrm {occ}}|\\varphi _k({\\bf r},t)|^2$ .",
"The theory is in principle exact but the KS potential $v_s({\\bf r},t)$ contains an unknown contribution called exchange-correlation (xc) potential $v_{\\scriptscriptstyle \\rm XC}({\\bf r}, t)$ , which in practice needs to be approximated.",
"The latter is a functional of the density at all points in space and at all previous times as well as the initial interacting state $\\Psi _0,$ and initial KS state $\\Phi _0$ : $v_{\\scriptscriptstyle \\rm XC}[n;\\Psi _0, \\Phi _0](t)$ .",
"Most TDDFT calculations however are adiabatic, i.e.",
"the instantaneous density is plugged into a ground state functional, $v_{\\scriptscriptstyle \\rm XC}^{\\rm adia}[n;\\Psi _0, \\Phi _0](t) = v_{\\scriptscriptstyle \\rm XC}^{\\rm g.s.",
"}[n(t)]$ , neglecting all dependence on the history of the density and on the initial states $\\Psi _0$ and $\\Phi _0$ .",
"Adiabatic linear response TDDFT is extensively and successfully used to compute excitation energies of molecules and solids[5] , but TDDFT is not limited to linear response: the theorems state that the density response to any order in an external perturbation can be reproduced.",
"It is a promising candidate for modeling non-equilibrium electron dynamics since it can capture the correlation effects with a relatively cheap computational cost.",
"Although the performance of available functionals for non-equilibrium dynamics has been much less explored, promising results have been reported [7], [8], [9], [10], [11].",
"On the other hand, work on small systems where numerically-exact or high-level wavefunction methods are applicable, has shown that the approximate TDDFT functionals can yield significant errors in the simulated dynamics [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25].",
"There is an urgent need to better understand the origin of errors in TDDFT approximations in this realm, especially since many topical applications such as for example charge-transfer dynamics and excitonic coherences in light-harvesting systems [8], [26], [27] or attosecond control of electrons in real time [28], [1] to name a few, involve tracking the system as it evolves far from its ground state.",
"When simulating these experiments with TDDFT, one particular problem that has come to light in a number of recent studies, is that approximate TDDFT can present spurious time-dependence in the resonance positions in non-perturbative dynamics and discrepancies between resonant frequencies computed from different reference states [29], [20], [25], [23], [30], [24], [21], [31], [22].",
"Such an artifact can lead to inaccurate dynamics and can muddle the analysis of electron-ion interactions, coherent processes and quantum interferences, among others [8], [9], [32], [33], [34], [35].",
"The problem is not unique to TDDFT: it is inherent to any method that utilizes approximate potentials that depend on the time-evolving orbitals, such as time-dependent Hartree-Fock(TDHF) for instance [36].",
"Given that TDDFT is formally exact, there is hope that approximate functionals can be designed that minimize the spurious time-dependence of the resonances and thereby improve the accuracy of the predicted dynamics.",
"To this end, the exact condition derived in Ref.",
"[25] should be useful, and here we give details elaborating on the condition and its implications in various cases.",
"Any system can be perturbed continuously and left in a non-equilibrium state.",
"If we turn off the perturbation, it starts to evolve freely, the state being a superposition of the eigenstates of the static potential.",
"In this work, we define resonances, or response frequencies, of the system via the poles of the density-response of the field-free system; these appear as peaks in its absorption spectra.",
"From the principles of quantum mechanics, in the absence of ionic motion the resonances are independent of the instantaneous state of the system once any external field is turned off, and independent of the time the field is turned off.",
"The linear response of the system may show dynamics in the oscillator strengths, and new peaks may appear or disappear, but their positions remain constant.",
"An exact TDDFT simulation reproduces the physical resonances at all times, but approximate xc functionals in general display spuriously time-dependent resonances, also referred to as “peak-shifting”.",
"Ref.",
"[25] rationalized the spuriously time-dependent resonances in terms of the violation of an exact condition on the xc functional.",
"Here we elaborate on the derivation and provide a detailed analysis of examples that illustrate this effect.",
"The paper is organized as follows.",
"In Section we review the generalized non-equilibrium linear response formalism for TDDFT, introduced in Refs.",
"[25], [37], pointing out differences from the standard ground state linear response.",
"We discuss the difficulty of defining a pole structure for the non-equilibrium KS response function around an arbitrary state.",
"In Section we discuss in detail the exact condition presented in Ref.",
"[25] and reformulate it in several ways, including the special case of response around a stationary state.",
"In Section we show explicit time-dependent electronic spectra computed using approximate functionals for charge transfer dynamics.",
"In order to illustrate the non-adiabatic nature of this spurious “peak shifting\", in Section we analyze the electronic spectra for adiabatic dynamics using the exact ground state functional.",
"For this purpose we utilize a lattice model system, for which the exact ground state xc functional can be computed and then used to propagate the KS system.",
"For the same system, we present a proof of concept showing the impact of the violation of the exact condition on the simulated dynamics.",
"We conclude and outlook the future work in Section .",
"The non-equilibrium response function plays a crucial role in analyzing non-equilibrium dynamics, as has been recently highlighted in theoretical formulations of time-resolved photoabsorption spectroscopy [37], [38].",
"In fact the language of pump-probe dynamics is very useful for establishing the exact condition, although the relevance of the condition is not at all restricted to such a setup.",
"Consider then a pump pulse that drives the system out of its ground state.",
"At time $\\mathcal {T}^{\\prime }$ the pump pulse is turned off and is followed, after some delay $\\theta $ , by a weak probe pulse for duration $\\Delta \\theta $ , see Figure REF .",
"Figure: The cartoon shows the time series of the applied field in a typicalpump-probe experiment.",
"Note that v ext (0) v_{\\rm ext}^{(0)} is not shown since it is always on.",
"Attime 𝒯 ' \\mathcal {T}^{\\prime }, the pump is turned off and after a delay θ\\theta , a weak probe laseris turned on for duration of Δθ\\Delta \\theta .",
"In pump-probe spectroscopy, the experiment is repeated with differentdelays using the same probe.",
"At 𝒯+Δθ\\mathcal {T}+ \\Delta \\theta , the probe field has died off and only the staticnuclear field is present.",
"The times shaded in blue represents times whenonly the static potential v ext (0) v_{\\rm ext}^{(0)} is present.The response of the excited system is monitored by repeating the experiment for different delay $\\theta $ .",
"When the electronic response is measured, coupling of the electronic dynamics to ionic motion manifests itself as changes in the position of the spectral peaks with respect to $\\mathcal {T}^{\\prime }$ and $\\theta $ .",
"But when ions are clamped the peak positions should not move; they should be independent of both $\\theta $ and $\\mathcal {T}^{\\prime }$ .",
"For $t>\\mathcal {T}^{\\prime }$ and considering the ions clamped within the timescale of interest, the unperturbed Hamiltonian becomes static $\\hat{H}^{(0)} = \\hat{T} + \\hat{W} + \\hat{v}_{\\rm ext}^{(0)}$ , with $\\hat{H}^{(0)} \\Psi _n= E_n \\Psi _n$ .",
"Throughout the paper, the superscript ${(0)}$ indicates a quantity in the absence of time-dependent external fields.",
"Here $\\hat{T}$ and $\\hat{W}$ are the kinetic energy and the electron-electron interaction energy operators respectively.",
"The system is left in a superposition state which, at times $t$ greater than $\\mathcal {T}= \\mathcal {T}^{\\prime } +\\theta $ , in the absence of any probe pulse can be written, $\\Psi ^{(0)}(t\\ge \\mathcal {T}) = \\sum _n c_n(\\mathcal {T})\\Psi _n e^{-iE_n(t-\\mathcal {T})},$ with $c_n(\\mathcal {T})= c_n(\\mathcal {T}^{\\prime }) e^{-i E_n \\theta }$ and the $(0)$ superscript denotes a field-free evolution.",
"We denote the time-dependent density of this state as $n^{(0)}_\\mathcal {T}({\\bf r},t)$ , defined for times $t\\ge \\mathcal {T}$ , $n_\\mathcal {T}^{(0)}({\\bf r},t)= N \\sum _{\\sigma _1,...\\sigma _N} \\int \\mathop {}\\!\\mathrm {d}{\\bf r}_2...\\mathop {}\\!\\mathrm {d}{\\bf r}_N |\\Psi ^{(0)}({\\bf r}\\sigma _1,{\\bf r}_2\\sigma _2,...{\\bf r}_N\\sigma _N, t)|^2\\,,$ where the indices $\\sigma $ 's represent the spin.",
"The non-equilibrium response function, $\\tilde{\\chi }$ (we distinguish it from the ground state response function $\\chi $ by a tilde), describing the density response $\\delta n({\\bf r},t)$ to a perturbation $\\delta v_{\\rm ext}({\\bf r}^{\\prime },t^{\\prime })$ (probe) applied at time $t^{\\prime }<t$ reads, $\\tilde{\\chi }[n^{(0)}_\\mathcal {T}; \\Psi (\\mathcal {T})]({\\bf r},{\\bf r}^{\\prime },t,t^{\\prime }) = \\left.\\frac{\\delta n({\\bf r},t)}{\\delta v_{\\rm ext}({\\bf r}^{\\prime },t^{\\prime })}\\right|_{n^{(0)}_\\mathcal {T},\\Psi (\\mathcal {T})}\\,.$ In principle, $\\tilde{\\chi }$ depends on the unperturbed density $n^{(0)}(t)$ at times between $\\mathcal {T}$ and $t$ , and on the state at time $\\mathcal {T}$ , as follows from the Runge-Gross theorem [4].",
"Widely discussed in the literature is the ground state response function, which is a particular case of Eq.",
"(REF ) when the initial state is the ground state, $\\Psi (\\mathcal {T})=\\Psi _0$ , and the unperturbed density becomes the ground state density $n^{(0)}_\\mathcal {T}({\\bf r},0)=n_0({\\bf r})$ .",
"The ground state response function $\\chi [n_0]$ depends only on the time interval $\\tau =t-t^{\\prime }$ , $\\chi [n_0]({\\bf r}, {\\bf r}^{\\prime },t,t^{\\prime }) = \\left.\\frac{\\delta n({\\bf r},t)}{\\delta v_{\\rm ext}({\\bf r}^{\\prime },t^{\\prime })}\\right|_{n_0}\\equiv \\chi _0({\\bf r},{\\bf r}^{\\prime }, \\tau )\\;$ due to the time-translation invariance of the ground state.",
"Fourier transforming with respect to $\\tau $ yields the zero temperature Lehmann representation, see e.g.",
"Ref.",
"[39], $\\begin{split}\\chi _0({\\bf r}, {\\bf r}^{\\prime }, \\omega ) & = \\int \\!\\!",
"\\mathop {}\\!\\mathrm {d}\\tau \\chi _0({\\bf r}, {\\bf r}^{\\prime }, \\tau ) e^{-i \\omega \\tau } \\\\& = \\sum _{k} \\left( \\frac{f_{0k}({\\bf r})f_{k0}({\\bf r}^{\\prime })}{\\omega - \\omega _{k0}+i0^+} - \\frac{f_{0k}({\\bf r}^{\\prime })f_{k0}({\\bf r})}{\\omega + \\omega _{k0}+i0^+}\\right)\\,.\\end{split}$ where $f_{jl}({\\bf r}) = \\langle \\Psi _j \\vert \\hat{n}({\\bf r}) \\vert \\Psi _l \\rangle $ and $\\omega _{jl} = E_j -E_l$ .",
"$\\chi _0({\\bf r}, {\\bf r}^{\\prime }, \\omega )$ has poles at the frequencies corresponding to transitions from the ground state to other eigenstates $k$ of the system.",
"Unlike the ground state response function the generalized non-equilibrium response function $\\tilde{\\chi }[n^{(0)}_\\mathcal {T}; \\Psi (\\mathcal {T})]({\\bf r},{\\bf r}^{\\prime },t,t^{\\prime })$ defined in Eq.",
"(REF ) in principle depends both on $t$ and $t^{\\prime }$ independently.",
"Following derivations in standard linear response theory [39] but now generalized to an arbitrary initial state: $&& \\tilde{\\chi }[n^{(0)}_\\mathcal {T}; \\Psi (\\mathcal {T})]({\\bf r},{\\bf r}^{\\prime },t,t^{\\prime })\\nonumber \\\\& =& -i \\theta (t-t^{\\prime }) \\mathinner {\\langle {\\Psi (\\mathcal {T}) |[{\\hat{n}}({\\bf r},t), {\\hat{n}}({\\bf r}^{\\prime },t^{\\prime })] | \\Psi (\\mathcal {T})}\\rangle }.$ In Eq.",
"(REF ) the density operator is in the interaction picture: ${\\hat{n}}({\\bf r},t) = e^{i\\hat{H}^{(0)} t} {\\hat{n}}({\\bf r})e^{-i\\hat{H}^{(0)}t}$ .",
"Given that the response function is defined as the response of the density at time $t$ with respect to a perturbation at time $t^{\\prime }$ we define again $\\tau = t-t^{\\prime }$ , which we will Fourier transform with respect to, in order to obtain a spectral representation.",
"Then, inserting the expansion for the arbitrary state Eq.",
"(REF ) in Eq.",
"(REF ), we have $\\tilde{\\chi }_{t^{\\prime }}({\\bf r}, {\\bf r}^{\\prime }, \\tau ) = -i \\theta (\\tau ) \\sum _{m,n,k} P_{nm}(\\mathcal {T})e^{i \\omega _{nm} t^{\\prime }} \\left[ e^{i\\omega _{nk}\\tau } f_{nk}({\\bf r}) f_{km}({\\bf r}^{\\prime }) -e^{i\\omega _{km}\\tau } f_{nk}({\\bf r}^{\\prime }) f_{km}({\\bf r}) \\right]$ with $P_{jl}(\\mathcal {T}) = c_j^*(\\mathcal {T})c_l(\\mathcal {T})$ , where we choose to parameterize the response function via $t^{\\prime }$ .",
"Instead, one could parameterize it via $T =(t+t^{\\prime })/2$ , in which case we have, $\\tilde{\\chi }_{T}({\\bf r}, {\\bf r}^{\\prime }, \\tau )=-i\\theta (\\tau )\\sum _{n,m} P_{nm}(\\mathcal {T}) e^{i\\omega _{nm} T }\\sum _k\\left[e^{i \\frac{\\omega _{nk}+\\omega _{mk}}{2}\\tau } f_{nk}({\\bf r})f_{km}({\\bf r}^{\\prime })-e^{-i \\frac{\\omega _{nk}+\\omega _{mk}}{2}\\tau } f_{nk}({\\bf r}^{\\prime })f_{km}({\\bf r})\\right]\\,.$ Performing the Fourier transform yields, for the two choices of parametrization, $\\tilde{\\chi }_{t^{\\prime }}({\\bf r},{\\bf r}^{\\prime }, \\omega ) = \\chi ^{\\mathrm {diag}}({\\bf r},{\\bf r}^{\\prime },\\omega ) +\\left( \\sum _{n\\ne m} P_{nm}(\\mathcal {T}) e^{i\\omega _{nm} t^{\\prime }} \\sum _k\\left[\\frac{f_{nk}({\\bf r}) f_{km}({\\bf r}^{\\prime })}{\\omega -\\omega _{kn}+i0^+} - \\frac{f_{nk}({\\bf r}^{\\prime }) f_{km}({\\bf r}) }{\\omega + \\omega _{km} +i0^+}\\right]\\right)\\,,$ and $\\tilde{\\chi }_{T}({\\bf r},{\\bf r}^{\\prime },\\omega ) = \\chi ^{\\mathrm {diag}}({\\bf r},{\\bf r}^{\\prime },\\omega ) +\\left( \\sum _{n\\ne m} P_{nm}(\\mathcal {T})e^{i\\omega _{nm} T} \\sum _k\\left[\\frac{ f_{nk}({\\bf r})f_{km}({\\bf r}^{\\prime })}{\\omega - \\frac{\\omega _{kn} + \\omega _{km}}{2} +i0^+} -\\frac{ f_{nk}({\\bf r}^{\\prime })f_{km}({\\bf r})}{\\omega + \\frac{\\omega _{kn} + \\omega _{km}}{2} +i0^+}\\right]\\right) \\,,$ where the diagonal term corresponding to $m=n$ , an incoherent sum over populations of initially occupied states, has been isolated, $\\chi ^{\\mathrm {diag}}({\\bf r},{\\bf r}^{\\prime },\\omega ) = \\sum _n P_{nn}(\\mathcal {T}) \\sum _k \\left(\\frac{f_{nk}({\\bf r}) f_{kn}({\\bf r}^{\\prime })}{\\omega -\\omega _{kn} +i0^+} -\\frac{f_{nk}({\\bf r}^{\\prime }) f_{kn}({\\bf r}) }{\\omega + \\omega _{kn} +i0^+} \\right)$ Note that the second terms in the large curved parenthesis on the right of Eqs.",
"(REF ), (REF ), and (REF ), are simply complex conjugates of the first terms when evaluated at $-\\omega $ ; in the following we will replace such terms by the expression $+ c.c.",
"(\\omega \\rightarrow -\\omega )$ .",
"It may appear from the different pole structure of Eqs.",
"(REF ) and (REF ), that $\\tilde{\\chi }_{t^{\\prime }}$ and $\\tilde{\\chi }_{T}$ yield different density-responses, but in fact this is not the case.",
"Since they originate from the same $\\tilde{\\chi } ({\\bf r},{\\bf r}^{\\prime },t,t^{\\prime })$ they must yield the same density-response $\\delta n({\\bf r}, \\omega )$ , and we now show this explicitly; in particular the poles at half-sum frequencies in $\\tilde{\\chi }_{T}$ vanish once $\\delta n({\\bf r},\\omega )$ is computed.",
"The density-response in the frequency domain $\\delta n({\\bf r},\\omega ) = \\int \\!\\!\\mathop {}\\!\\mathrm {d}t\\; \\delta n({\\bf r},t)e^{i\\omega t}$ is computed via $\\delta n({\\bf r},t) = \\int \\!\\!",
"\\mathop {}\\!\\mathrm {d}{\\bf r}^{\\prime } \\mathop {}\\!\\mathrm {d}t^{\\prime } \\; \\tilde{\\chi }({\\bf r}, {\\bf r}^{\\prime }, t, t^{\\prime }) \\delta v({\\bf r}^{\\prime }, t^{\\prime })\\,.$ Taking the inverse Fourier transforms, $\\tilde{\\chi }_{t^{\\prime }}({\\bf r}, {\\bf r}^{\\prime }, \\tau ) = \\frac{1}{2\\pi } \\int \\!",
"\\mathop {}\\!\\mathrm {d}w_1\\, e^{-i\\omega _1 \\tau } \\tilde{\\chi }_{t^{\\prime }}({\\bf r}, {\\bf r}^{\\prime }, \\omega _1)$ and $\\delta v({\\bf r}^{\\prime },t^{\\prime }) =\\frac{1}{2\\pi } \\int \\!",
"\\mathop {}\\!\\mathrm {d}\\omega _2\\, e^{-i\\omega _2 t^{\\prime }} \\delta v({\\bf r}^{\\prime }, \\omega _2)$ , we have $\\delta n({\\bf r},\\omega ) = \\int \\!\\!\\mathop {}\\!\\mathrm {d}{\\bf r}^{\\prime }\\mathop {}\\!\\mathrm {d}t\\mathop {}\\!\\mathrm {d}t^{\\prime } \\frac{\\mathop {}\\!\\mathrm {d}\\omega _1}{2\\pi } \\frac{\\mathop {}\\!\\mathrm {d}\\omega _2}{2\\pi } e^{i \\omega t} e^{-i\\omega _1 \\tau }e^{-i\\omega _2 t^{\\prime }} \\tilde{\\chi }_{t^{\\prime }}({\\bf r}, {\\bf r}^{\\prime }, \\omega _1) \\delta v({\\bf r}^{\\prime }, \\omega _2)\\,.$ Inserting Eq.",
"(REF ) into Eq.",
"(REF ) yields two contributions: $\\delta n({\\bf r},\\omega ) = \\delta n^{\\mathrm {diag}}({\\bf r},\\omega ) +\\int \\!\\!",
"\\mathop {}\\!\\mathrm {d}{\\bf r}^{\\prime } \\left[\\sum _{n\\ne m} P_{nm}(\\mathcal {T}) \\sum _k\\frac{f_{nk}({\\bf r}) f_{km}({\\bf r}^{\\prime })}{\\omega -\\omega _{kn}+i0^+} + c.c.",
"(\\omega \\rightarrow -\\omega )\\right] \\delta v({\\bf r}^{\\prime }, \\omega + \\omega _{nm})\\,,$ where $\\delta n^{\\mathrm {diag}} ({\\bf r},\\omega ) = \\int \\!\\mathop {}\\!\\mathrm {d}{\\bf r}^{\\prime }\\, \\sum _n P_{nn}(\\mathcal {T}) \\sum _k\\left[\\frac{f_{nk}({\\bf r}) f_{kn}({\\bf r}^{\\prime })}{\\omega -\\omega _{kn} +i0^+} + c.c.",
"(\\omega \\rightarrow -\\omega )\\right] \\delta v({\\bf r}^{\\prime },\\omega )\\,.$ The first term in Eq.",
"(REF ) arises from the diagonal term, where the poles are at the true excitations: the energy differences between an occupied state $n$ and an unoccupied state $k$ .",
"The second term arises from the second term of Eq.",
"(REF ); the identities $\\int \\!",
"\\mathop {}\\!\\mathrm {d}t\\, e^{i\\omega t}e^{-i\\omega _1 t} = 2\\pi \\delta (\\omega -\\omega _1)$ and $\\int \\mathop {}\\!\\mathrm {d}t^{\\prime } e^{i\\omega _1t^{\\prime }} e^{-i\\omega _2 t^{\\prime }} e^{i\\omega _{nm} t^{\\prime }}= 2\\pi \\delta (\\omega _1-\\omega _2+\\omega _{nm})$ enabled us to readily perform the integrals.",
"If instead of $\\chi _{t^{\\prime }}$ , we insert $\\chi _{T}$ into Eq.",
"(REF ), we find exactly the same expression as Eq.",
"(REF ) after recognizing that the $\\mathop {}\\!\\mathrm {d}t$ and $\\mathop {}\\!\\mathrm {d}t^{\\prime }$ integrals in this case yield $\\delta (\\omega - \\omega _1 + \\omega _{nm}/2)$ and $\\delta (\\omega _2 - \\omega _1 - \\omega _{nm}/2)$ respectively.",
"The diagonal term has the usual structure showing resonant peaks where the perturbing potential has components at a frequency equal to an energy-difference between occupied and unoccupied states; the only difference with the ground-state response is that states other than the ground-state are occupied, as reflected in the $P_{nn}(\\mathcal {T})$ prefactor.",
"The off-diagonal term also has response peaks at the occupied–unoccupied energy differences: the peaks are at the zeroes of the denominator, at frequencies $\\omega _{kn}$ .",
"Considering that this is multiplied by the $(\\omega +\\omega _{nm})$ -frequency-component of the applied potential $\\delta v$ , then we see that this term is resonant when the applied potential resonates at $\\omega _{kn} +\\omega _{nm} = \\omega _{km}$ ."
],
[
"Generalized Non-Equilibrium Kohn-Sham Response",
"Generally the wavefunctions that are summed over in response function of an interacting system are inaccessible for a realistic system, so one turns to alternative methods to calculate it, such as TDDFT.",
"Unlike the interacting system, which, after the field is turned off evolves under the static potential $v_{\\rm ext}^{(0)}({\\bf r})$ , the KS system evolves in the potential $\\begin{split}v_{\\scriptscriptstyle \\rm S}[n_\\mathcal {T}^{(0)},\\Phi (\\mathcal {T})]({\\bf r},t) & = v_{\\rm ext}^{(0)}({\\bf r}) + v_{\\scriptscriptstyle \\rm HXC}[n_\\mathcal {T}^{(0)}; \\Psi (\\mathcal {T}),\\Phi (\\mathcal {T})]({\\bf r},t)\\\\ & \\equiv v_{\\scriptscriptstyle \\rm S}^{(0)}({\\bf r},t),\\end{split}$ which typically continues to evolve in time for $t \\ge \\mathcal {T}$ even in the absence of time-dependent external fields.",
"The time-dependence of the KS potential $v_{\\scriptscriptstyle \\rm S}^{(0)}[n_\\mathcal {T}^{(0)}, \\Phi (\\mathcal {T})]({\\bf r},t)$ is due to the potential itself being a functional of the non-stationary density $n_\\mathcal {T}^{(0)}({\\bf r}, t)$ and the KS initial state $\\Phi (\\mathcal {T})$ .",
"Similar as for the ground state response $\\chi _0$ , TDDFT can be used to find the interacting non-equilibrium response $\\tilde{\\chi }$ .",
"Since the physical and KS system must yield the same density-response, we can derive a Dyson-like equation linking the generalized non-equilibrium interacting response function $\\tilde{\\chi }$ Eq.",
"(REF ) with a generalized non-equilibrium KS response function $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}$ , $\\tilde{\\chi }^{-1}[n^{(0)}_{\\mathcal {T}}; \\Psi (\\mathcal {T})]=\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}^{-1}[n^{(0)}_{\\mathcal {T}}; \\Phi (\\mathcal {T})] - \\tilde{f}_{\\scriptscriptstyle \\rm HXC}[n^{(0)}_\\mathcal {T}; \\Psi (\\mathcal {T}), \\Phi (\\mathcal {T})]\\,$ via a generalized Hartree-xc kernel $\\tilde{f}_{\\scriptscriptstyle \\rm HXC}= 1/\\vert {\\bf r}- {\\bf r}^{\\prime }\\vert + \\tilde{f}_{\\scriptscriptstyle \\rm XC}$ , with, $\\tilde{f}_{\\scriptscriptstyle \\rm XC}[n^{(0)}_\\mathcal {T};\\Psi (\\mathcal {T}),\\Phi (\\mathcal {T})]({\\bf r},{\\bf r}^{\\prime },t,t^{\\prime }) =\\frac{\\delta v_{\\scriptscriptstyle \\rm XC}({\\bf r},t)}{\\delta n({\\bf r}^{\\prime },t^{\\prime })} \\Big \\vert _{n^{(0)}_\\mathcal {T},\\Psi (\\mathcal {T}),\\Phi (\\mathcal {T})}.$ Note that the dependence on the unperturbed density $n^{(0)}$ at times between $\\mathcal {T}$ and $t$ , and the interacting and KS initial states $\\Psi (\\mathcal {T})$ , $\\Phi (\\mathcal {T})$ in $\\tilde{\\chi }({\\bf r},{\\bf r}^{\\prime },t,t^{\\prime })[n^{(0)}_{\\mathcal {T}}; \\Psi (\\mathcal {T})]$ and $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}({\\bf r},{\\bf r}^{\\prime }, t,t^{\\prime })[n^{(0)}_{\\mathcal {T}}; \\Phi (\\mathcal {T})]$ follows from the Runge-Gross and van Leeuwen proofs [4], [40].",
"As a consequence of the time-dependence of $v_{\\scriptscriptstyle \\rm S}^{(0)}$ the bare KS eigenvalues and eigenvalue-differences also become time-dependent The time-dependent KS eigenvalues are referred as the eigenvalues when one solves a static KS equation with the instaneous xc potential..",
"The KS response function in Eq.",
"(REF ) is defined as $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}[n^{(0)}_\\mathcal {T},\\Phi (\\mathcal {T})]({\\bf r}, {\\bf r}^{\\prime }, t,t^{\\prime }) = \\left.\\frac{\\delta n({\\bf r},t)}{\\delta v_{\\scriptscriptstyle \\rm S}({\\bf r}^{\\prime },t^{\\prime })}\\right|_{n^{(0)}_\\mathcal {T},\\Phi (\\mathcal {T})}\\equiv \\tilde{\\chi }_{\\scriptscriptstyle \\rm S}({\\bf r}, {\\bf r}^{\\prime },t, t^{\\prime })\\,.$ Unlike $\\tilde{\\chi }$ , $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}$ in general does not have a simple spectral (Lehmann) representation.",
"The reason is the interaction picture for the KS system involves a time-dependent Hamiltonian, $H_{\\scriptscriptstyle \\rm S}^{(0)}(t)= -\\nabla ^2/2 +\\hat{v}_{\\scriptscriptstyle \\rm S}[n_\\mathcal {T}^{(0)},\\Phi (\\mathcal {T})](t)$ , therefore the non-equilibrium KS response [42], [39], $&& \\tilde{\\chi }_{\\scriptscriptstyle \\rm S}({\\bf r},{\\bf r}^{\\prime },t, t^{\\prime }) = -i \\theta (t-t^{\\prime }) \\mathinner {\\langle {\\Phi (\\mathcal {T}) |[{\\hat{n}}({\\bf r},t), {\\hat{n}}({\\bf r}^{\\prime },t^{\\prime })] | \\Phi (\\mathcal {T})}\\rangle }\\nonumber \\\\&&= -i \\theta (t-t^{\\prime }) \\sum _{l,k} \\varphi _{k}^{*}({\\bf r},t)\\varphi _{l}^{*}({\\bf r}^{\\prime },t^{\\prime }) \\varphi _{k}^{*}({\\bf r}^{\\prime },t^{\\prime })\\varphi _{l}({\\bf r},t) + c.c.\\,.$ involves time-ordered operators $\\hat{n}({\\bf r},t) = \\hat{T}e^{i\\int _0^t d\\tau \\hat{H}_{\\scriptscriptstyle \\rm S}^{(0)}(\\tau )}\\hat{n}({\\bf r})\\hat{T}e^{-i\\int _0^t d\\tau \\hat{H}_{\\scriptscriptstyle \\rm S}^{(0)}(\\tau )}$ , with $\\hat{T}$ being the time-ordering operator.",
"A simple interpretation of the Fourier transform of $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}({\\bf r},{\\bf r}^{\\prime },t, t^{\\prime })$ with respect to $\\tau $ , $\\tilde{\\chi }_{{\\scriptscriptstyle \\rm {S}},t^{\\prime }}({\\bf r},{\\bf r}^{\\prime },\\omega )$ or $\\tilde{\\chi }_{{\\scriptscriptstyle \\rm {S}},T}({\\bf r},{\\bf r}^{\\prime },\\omega )$ , as in the SEc.",
", in terms of eigenvalue differences of some static KS Hamiltonian is generally not possible.",
"Despite the fact that a pole structure for $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}$ may not be simple to define in general (see particular cases for which it can be done in Sec.",
"), when $\\tilde{\\chi }$ is constructed from $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}$ and $\\tilde{f}_{\\scriptscriptstyle \\rm HXC}$ via Eq.",
"(REF ), the interacting spectral representation is retrieved."
],
[
"Exact Conditions",
"As discussed in Sec.",
"REF the KS potential for non-stationary dynamics Eq.",
"(REF ) is generally time-dependent even after any external field is turned off.",
"This is the case both for the exact KS potential and also for adiabatic approximations, because the density $n^{(0)}_\\mathcal {T}$ is time-dependent.",
"The instantaneous eigenvalue-differences of the KS Hamiltonian are time-dependent because $v_{\\scriptscriptstyle \\rm HXC}[n_\\mathcal {T}^{(0)}; \\Psi (\\mathcal {T}),\\Phi (\\mathcal {T})]$ changes as the density evolves.",
"For ground state linear response, $f_{\\scriptscriptstyle \\rm XC}$ must shift the KS response frequencies to the physical ones and create missing multi-electron excitations.",
"For the present non-equilibrium response, the generalized $\\tilde{f}_{\\scriptscriptstyle \\rm XC}$ must additionally cancel spurious $\\mathcal {T}$ -dependence in Eq.",
"(REF ) to ensure $\\mathcal {T}$ -independent TDDFT resonances.",
"Cancellation of spurious $\\mathcal {T}$ -dependence is an exact condition on the xc functional [25].",
"We give the general form of this condition in (i) below, and in (ii) and (iii) we discuss implications for a few special cases.",
"(i) Condition 1: Invariance with respect to $\\mathcal {T}$ .",
"Consider the TDDFT prediction for the transition frequency $\\omega _i$ between two given interacting states.",
"This $\\omega _i$ is a pole of $\\left(\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}^{-1}[n^{(0)}_\\mathcal {T},\\Phi (\\mathcal {T})] - \\tilde{f}_{\\scriptscriptstyle \\rm HXC}[n^{(0)}_\\mathcal {T};\\Psi (\\mathcal {T}), \\Phi (\\mathcal {T})]\\right)^{-1}\\;.$ Then $\\omega _i$ should be invariant with respect to $\\mathcal {T}$ : $\\frac{\\mathop {}\\!\\mathrm {d}\\omega _i}{\\mathop {}\\!\\mathrm {d}\\mathcal {T}} = 0\\;.$ We shall call this Condition 1 in the following.",
"We note that as a system evolves under an external field, more states can become populated and so when the field is then turned off and the system's linear response probed, new frequencies may appear corresponding to transitions from states populated at the later time that were not populated at earlier times.",
"Likewise, some frequencies appearing in the response at earlier times may disappear.",
"The exact condition addresses those response frequencies present at both earlier and later times: the positions of these must be invariant.",
"(See also consistency condition discussed in ii).",
"Although expressed above in the context of pump-probe spectroscopy, the exact condition clearly applies to any non-equilibrium dynamics, where $\\omega _i$ are the field-free resonance positions of the system at some time $\\mathcal {T}$ .",
"It can be viewed in another way: let $\\Psi _\\mathcal {T}$ be any arbitrary interacting state, not necessarily a stationary state, of a system in a static potential $v_{\\rm ext}^{(0)}$ , and $n^{(0)}_\\mathcal {T}(t)$ be its field-free time-dependent density as it evolves in $v_{\\rm ext}^{(0)}$ .",
"The subscript $\\mathcal {T}$ is no longer a time label but instead labels a particular arbitrary state $\\Psi _{\\mathcal {T}}$ .",
"Then the response of this arbitrary state has poles at frequencies $\\omega _i$ , which for a given transition satisfy Eq.",
"(REF ): their positions are independent of the choice of this arbitrary state, i.e of $\\mathcal {T}$ .",
"(ii) Condition 2: Invariance with respect to any stationary interacting state of a given static potential Consider now the special case when the interacting system is in a stationary excited state, $\\Psi (\\mathcal {T})= \\Psi _k$ of the static potential that it lives in, $v_{\\rm ext}^{(0)}$ .",
"The density is then stationary, $n^{(0)}_\\mathcal {T}(t)=n_k$ .",
"The initial state chosen for the KS calculation must have the same density and its first time-derivative as the initial interacting state [40], [43], but need not itself be a stationary state of any potential, since it is possible for time-dependence of different KS orbitals to cancel each other out.",
"However, let us choose a KS initial state $\\Phi _l$ with density $n_k$ that is a stationary excited state of some static one-body potential $v_{\\scriptscriptstyle \\rm S}^{(0)}$ .",
"(Note that $v_{\\scriptscriptstyle \\rm S}^{(0)}$ is not the ground-state KS potential corresponding to the external potential $v_{\\rm ext}^{(0)}$ ; they do not have the same ground-state density, but rather the excited state $\\Phi _l$ of the former has the same density $n_k$ as the excited state $\\Psi _k$ of the latter).",
"Then, because the unperturbed KS system is static, we can find a Lehmann representation for the KS density-response function, and derive a matrix formulation of the Dyson equation Eq.",
"(REF ) in frequency-domain.",
"The poles of the frequency-domain Dyson equation are eigenvalues of the matrix equation [44].",
"Now let $\\lbrace \\Psi _1,...\\Psi _k...\\rbrace $ be a set of eigenstates of the same static potential $v_{\\rm ext}^{(0)}$ , and imagine finding an appropriate KS stationary state $\\Phi _{k}$ for each one, that is an excited state of some KS potential $v_{\\scriptscriptstyle \\rm S}^{(0)}[n_k;\\Phi _k]$ , reproducing the density, $n_k$ , of the $k$ th true excited state $\\Psi _k$ .",
"Note that, although we use here the same index $k$ to label the corresponding KS wavefunction instead of $l$ as in the previous paragraph, to avoid proliferation of subscripts, it need not be the $k$ th KS excitation of the potential in which it is an eigenstate, $v_{\\scriptscriptstyle \\rm S}[n_k; \\Phi _k]$ (see also Condition 3 (iii) shortly).",
"Then the exact condition is that the TDDFT response frequency predicted for a given transition between two states $\\Psi _{k}$ and $\\Psi _{k^{\\prime }}$ of a given potential $v_{\\rm ext}^{(0)}$ must be independent of which of these two states is chosen as the reference, i.e.",
"the TDDFT response frequency should be identical whether the response is calculated around $\\Psi _{k}$ or around $\\Psi _{k^{\\prime }}$ .",
"We call this Condition 2 in the following.",
"Other excitation frequencies of the system must be consistent.",
"For example, consider computing the response around (a) state $\\Psi _2$ and (b) state $\\Psi _3$ .",
"Then this condition expresses that $\\vert \\omega _{32}^{a}\\vert $ predicted in calculation (a) must equal $\\vert \\omega _{23}^{b}\\vert $ predicted in calculation (b), but also other frequencies have to be consistent, i.e.",
"we must have $\\omega _{k3}^{(b)} = \\omega _{k2}^{(a)} - \\omega _{32}^{(a)}$ .",
"Within a single-pole approximation (SPA) of this generalized Dyson equation (justified for a KS transition well-separated from all other transitions [45]), Condition 2 simplifies to the condition that, $\\omega ^{\\rm SPA} = \\omega _{{\\scriptscriptstyle \\rm {S}},q}^{k} + 2 \\operatorname{Re}\\int \\!\\!",
"\\mathop {}\\!\\mathrm {d}{\\bf r}\\int \\!\\!",
"\\mathop {}\\!\\mathrm {d}{\\bf r}^{\\prime }\\bar{\\Phi }_q^{k}({\\bf r}) \\tilde{f}^k_{\\scriptscriptstyle \\rm HXC}({\\bf r},{\\bf r}^{\\prime },\\omega _{{\\scriptscriptstyle \\rm {S}},q}^{k}) \\Phi _q^k({\\bf r}^{\\prime })$ is the same whether $\\Psi _{k}$ or $\\Psi _{k^{\\prime }}$ is chosen as the reference $k$ th interacting state.",
"The notation $\\omega _{{\\scriptscriptstyle \\rm {S}},q}^{k}$ means the $i\\rightarrow a$ KS excitation out of the potential $v_{\\scriptscriptstyle \\rm S}^{(0)}[n_k;\\Phi _k]$ and $q=(i,a)$ represents a double index.",
"We have defined the transition density $\\Phi _q^k({\\bf r}) = \\bar{\\varphi }_i^k({\\bf r}) \\varphi _a^k({\\bf r}),$ in which $\\bar{\\varphi }_i^k({\\bf r})$ denotes the complex conjugate of $\\varphi _i^k({\\bf r})$ .",
"Here $\\varphi _i^{k}, \\varphi _a^k$ are the initial occupied and unoccupied KS orbitals of potential $v_{\\scriptscriptstyle \\rm S}^{(0)}[n_k;\\Phi _k]$ , and $\\omega _{{\\scriptscriptstyle \\rm {S}},q}^{k}$ is their orbital energy difference.",
"The shorthand $\\tilde{f}_{\\scriptscriptstyle \\rm HXC}^{k}$ represents the generalized kernel $\\tilde{f}_{\\scriptscriptstyle \\rm HXC}[\\Psi _k,\\Phi _k]$ (the field-free density-dependence is redundant when beginning in a specified stationary state of the unperturbed potential, since that information is contained already in the initial states).",
"The expression Eq.",
"(REF ) is given for the spin-saturated case; for spin-polarized systems and non-degenerate KS poles, replace $\\tilde{f}_{\\scriptscriptstyle \\rm HXC}$ with $(1/\\vert {\\bf r}- {\\bf r}^{\\prime }\\vert +\\tilde{f}_{\\scriptscriptstyle \\rm XC}^{\\sigma ,\\sigma ^{\\prime }})$ , where $\\sigma $ and $\\sigma ^{\\prime }$ are spin indices.",
"Condition 2 and Eq.",
"(REF ) were given in Ref.",
"[25] but discussed within the adiabatic approximation, where, having a static density is enough to guarantee that the KS potential is static, and $\\Phi _k$ solves self-consistent field (SCF) equations for the static potential $v_{\\scriptscriptstyle \\rm S}^{(0)}[n_k; \\Phi _k]$ .",
"Like Condition 1, the degree to which Condition 2 is violated can be used to determine the accuracy of the dynamics, especially relevant when the dynamics involves just a few interacting excited states that get populated and depopulated in time.",
"Strictly speaking, to use Condition 2, one would need to know the exact density of the interacting excited states, but in practise one can often find appropriate KS excited states for a given functional approximation, whose densities are assumed to approximate the interacting ones [46].",
"This was done in Ref.",
"[25] for a few examples (see also section ), and we will also utilize this approximate Condition 2 in the next section.",
"(iii) Condition 3: Invariance with respect to the choice of a stationary non-interacting state $\\Phi _k$ .",
"Continuing with response of stationary states, not only must the TDDFT frequencies be independent of the interacting excited state of a fixed potential, but also they must be independent of the choice of the KS state.",
"Let $\\lbrace \\Phi _1,\\Phi _2...\\Phi _l...\\rbrace $ , each with density $n_k$ which is the density of a fixed interacting excited state $\\Psi _k$ of a given external potential $v_{\\rm ext}^{(0)}$ , be possible KS stationary states of different one-body potentials $\\lbrace v^{(0)}_{s,1},v^{(0)}_{s,2}...v^{(0)}_{s,l}\\rbrace $ .",
"Then Condition 3 states that, for a given transition, the frequency obtained via TDDFT response must be the same for any of these $\\Phi _l$ 's.",
"Again, like in (ii), one can express this condition directly in the frequency-domain in a generalized matrix formulation which within the single-pole approximation reduces to Eq.",
"(REF ) where here $k$ should be replaced by $l$ , labelling the particular KS state around which the TDDFT response is calculated."
],
[
"Spurious Time-dependent Spectra",
"We illustrate the significance of our exact conditions using two examples of charge-transfer dynamics.",
"Charge transfer(CT) is a crucial process in many topical applications today, including photovoltaic design where the system is initially photoexcited, and transport in nano-scale devices where the system begins in its ground state.",
"For each of these scenarios, we utilize model two-electron systems for which exact results are available to compare with, and for which we can thoroughly analyze how the spectral peaks shift as a function of time.",
"We will find both the Conditions 1 and Condition 2 are useful to understand the performance of the approximate functionals.",
"We also discuss a situation where Condition 3 is violated."
],
[
"Charge Transfer from a Photoexcited State",
"Our first example revisits an example of Ref.",
"[25], focusing here on the electronic spectra as the system evolves, obtained via the unrestricted exact-exchange approximation (EXX) and self-interaction-corrected local spin density approximation (SIC-LSD).",
"We consider two interacting electrons in a one-dimensional double well described by, $v_{\\rm ext}(x) = -\\frac{Z_L}{\\sqrt{(x+R)^2 + 1}} -\\frac{U_L}{\\cosh ^2(x+R)} - \\frac{U_R}{\\cosh ^2(x-R)}$ with parameters $Z_L = 2, U_L =2.9, U_R=1$ and $R = 3.5$ a.u.",
"Electrons interact via soft-Coulomb $w(x^{\\prime },x)=1/\\sqrt{(x^{\\prime }-x)^2+1}$ .",
"The spacing of the simulation box is $0.1$ a.u.",
"and we use zero boundary conditions at $\\pm 20.0$ a.u.",
"Atomic units are used throughout.",
"The real-time propagation is done using real-space code octopus [47], [48], [49] with a time step of $0.005$ a.u..",
"Figure: To the left the EXX potential v S ↑,i v_{\\scriptscriptstyle \\rm S}^{\\uparrow ,i} and the initial configuration for the photoexcitedKS state; to the right the final EXX potential v S ↑,f v_{\\scriptscriptstyle \\rm S}^{\\uparrow ,f} and thetarget CT state are plotted.",
"Here the position of the arrow denotes wheremost of the density sits.The model is specified in Section .As discussed in the text for this photoexcited CT dynamics the EXX potential the transferring ↑-\\uparrow -electron experiences(in the absence of time-dependent external fields) is nearly constant.The interacting system is prepared in an excited eigenstate of the unperturbed Hamiltonian and then evolved in the presence of a weak monochromatic laser resonant with the photoexcited-to-CT transition frequency $\\omega _{CT}$ .",
"The latter corresponds to the energy difference between the initial photoexcited state, denoted by $i$ , and the final target CT state, denoted $f$ , $\\omega _{CT}=E_f-E_i=0.289$ a.u..",
"The parameters are chosen such that Rabi oscillations between the photoexcited and the CT states are induced (although other states get lightly populated).",
"The exact interacting dynamics is compared with the results from the various TDDFT approximations.",
"For the latter, the initial state is obtained by promoting a KS particle from the ground state configuration to an unoccupied orbital (see the left-hand side of Fig.",
"REF ), such that the dominant configuration of the interacting initial state is mimicked.",
"This KS excited state is then relaxed via an SCF calculation to be the KS eigenstate of the initial KS potential $v_{\\scriptscriptstyle \\rm S}^{(0)}({\\bf r},0)$ , which prevents any dynamics before the applied field is turned on.",
"The KS system is propagated in the presence of a weak laser resonant with the TDDFT CT resonance of the given approximate functional computed via linear response around the initial KS photoexcited state.",
"(As is common practise, spin-polarized dynamics is run from the initial singly-excited KS determinant).",
"We note that if Condition 2 is violated, this value of the CT resonance is not the value determined from the usual linear response around the ground state, nor is it the value determined from linear response around the target CT state.",
"In fact, as pointed out in Ref.",
"[25], the deviation of the resonance predicted from linear response around the initial state, $\\omega _i$ , to that from target state, $\\omega _f$ , is strongly correlated with the performance of the functional approximation.",
"For example, for the two functionals we present here, for EXX $\\omega ^i=\\omega ^f =0.287$ a.u.",
"while for SIC-LSD $\\omega ^i =0.287$ a.u.",
"and $\\omega ^f = 0.237$ a.u..",
"Accordingly, we found in Ref.",
"[25] that EXX demonstrated near-perfect charge-transfer, while SIC-LSD began promisingly but ultimately failed to transfer the charge.",
"(Ref.",
"[25] also considered LSD, whose discrepancy between the frequencies determined from the initial and target state responses was even greater, and its dynamics was even worse).",
"For the same cases, we demonstrate here explicitly the violation of Condition 1, namely the unphysical time-dependence of the resonance positions, as a function of $\\mathcal {T}$ .",
"At various times $\\mathcal {T}$ during the evolution we turn off the monochromatic laser and perform a linear density-response calculation to obtain the spectra at these times.",
"The latter is done by applying a delta-kick [50] right after the laser is turned off, followed by a free evolution of some duration $T$ and then Fourier transforming the ensuing dipole difference between the kicked and un-kicked free propagations [22], [29], [30] $\\Delta d(t) = d^{\\mathrm {kicked}}(t)-d^{\\mathrm {un-kicked}}(t)\\,.$ We then plot the dipole spectrum $|\\Delta d(\\omega )|$ , denoted the “absolute spectrum\", $|\\Delta d(\\omega )| = \\vert \\int \\!\\mathop {}\\!\\mathrm {d}t\\, e^{-i \\omega t} \\Delta d(t)\\vert \\,,$ We evolved the field-free system for $T=5000~a.u,$ , resulting in a frequency-resolution of $2 \\pi / T \\approx 0.001$ a.u.. We plot the absolute spectrum $|\\Delta d(\\omega )|$ instead of the absorption spectrum $Im[d(w)]$ to simplify the analysis, since here we are only analyzing the position of the spectral peaks, not their oscillator strengths.",
"Fig.",
"REF shows the spectra for the exact, EXX, and SIC-LSD cases, each driven at their respective initial response frequencies $\\omega ^i$ , at different times $\\mathcal {T}=0, 400, 800, 1200$ a.u.",
"of the photoexcited CT dynamics shown in the inset.",
"As expected, the exact CT resonance peak only changes strength, but does not shift in position, remaining at $0.289$ a.u.",
"(see upper panel Fig.",
"REF ).",
"Note that the exact is obtained from solving the interacting Schrödinger equation and coincides with TDDFT with the exact functional, which satisfies all 3 Conditions.",
"We had seen in Ref.",
"[25] that EXX fulfills Condition 2 in this case, although the resonance position is off by about $0.002$ a.u.",
"from the exact.",
"Here we show explicitly in the middle panel of Fig.",
"REF that EXX for this dynamics also satisfies Condition 1; in fact even its peak strength changes in a similar way to the exact functional (see middle panel Fig.",
"REF ).",
"SIC-LSD is shown in the lower panel of Fig.",
"REF : peak-shifting signifying violation of Condition 1 is observed as a function of $\\mathcal {T}$ .",
"The resulting incomplete CT dynamics is evident in the inset.",
"Figure: Logarithm of absolute spectrum |Δd(ω)||\\Delta d(\\omega )|Eq.",
"() showing CT frequency for different laser durations 𝒯\\mathcal {T}.",
"Inset: dipole moments |d(t)||d(t)| for the photoexcited CT dynamics studied in Ref.",
".Upper panel: Exact.",
"Middle panel: EXX.",
"Lower panel: SIC-LSD.We note that the SIC-LSD peak drifts towards lower frequencies as the charge begins to transfer in time, retracing its path as the charge returns to the donor.",
"That the peak tracks the instantaneous dynamics is not unexpected, given the adiabatic nature of the approximation.",
"The direction of the peak shift (i.e.",
"towards lower frequencies) appears to be consistent with the fact that the linear response frequency computed at the target state, which gets partially occupied during the dynamics, is lower ($\\omega ^f=0.237$ a.u.)",
"than that computed at the initial state ($\\omega ^i=0.287$ a.u., see Table REF ).",
"One might ask what happens to the positions of the other resonances (transitions from the photoexcited state to other unoccupied states) during the dynamics.",
"In Fig.",
"REF a few peaks corresponding to excitations to higher delocalized states are analyzed as a function of $\\mathcal {T}$ .",
"For the exact dynamics the positions of the peaks do not change, but as the target CT state gets populated new peaks appear in the spectra, corresponding to transitions from the CT state.",
"The same is true for EXX which is shown in the middle panel of Fig.",
"REF .",
"This is consistent with the observation that EXX satisfies the exact condition.",
"As explained in Ref.",
"[25], this is because once the laser is turned off, the KS potential for the transfering electron within EXX is constant.",
"Figure: Logarithm of absolute spectrum |Δd(ω)||\\Delta d(\\omega )| Eq.",
"()for different laser durations 𝒯\\mathcal {T} correspondingto the photoexcited CT dynamics shown in insets of Fig.",
".Upper panel: Exact.",
"Middle panel: EXX.",
"Lower panel: SIC-LSD.Exact and EXX have constant positions for all shown resonances,while SIC-LSD has spuriously time-dependent resonances.",
"For SIC-LSD,ω 2 \\omega _{2} peak shifts up in energy as the system evolves while theω CT \\omega _{\\mathrm {CT}} shifts down,resulting in one unique broader peak for 𝒯=800\\mathcal {T}=800 a.u.",
"(see Fig.",
"and Table.",
").For SIC-LSD shown in lower panel of Fig.",
"REF not only the CT resonance but also the other peaks corresponding to transitions from the initially photoexcited state to other unoccupied states shift in time.",
"The direction of the shift during the transfer (from $\\mathcal {T}$ =0 to about $\\mathcal {T}$ = 800 au) is again towards the positions computed from the final target CT state (see Table REF last column), and then they shift back as the charge returns, as can be seen in Table REF .",
"In the table, the SIC-LSD response frequencies computed using a $\\delta $ -kick at different $\\mathcal {T}$ for a few transitions lying close to the CT peak are tabulated.",
"The last column shows the values of the resonances computed applying a strong kick in the target SCF CT state.",
"The $\\omega _2$ transition shifts up in energy and in the target configuration it lies higher in energy than the CT resonance, thus these two transitions cross as the system evolves becoming degenerate at $\\mathcal {T}=800$ a.u.",
"(see Fig.",
"REF ).",
"These resonant frequencies can also be computed from linear response around the SCF CT state, by shifting the transition frequencies from this state to the relevant unoccupied states by the value of the transition frequency between the CT and the locally excited state (see Table REF ).",
"That is, this table is a check on the \"consistency\" statement in Condition 2.",
"We have checked that the exact functional and EXX give identical numbers for the second and third columns, but as evident in Table REF , SIC-LSD shows some deviation from consistency.",
"It is interesting to note that the frequencies obtained from the shifted linear-response TDDFT calculations in column 3 of Table REF are close to, but not equal, to those obtained by the second-order response calculation in the last column in Table REF .",
"One can also check the consistency statement for excitations out of the CT state (3rd excited state), by shifting the linear-response TDDFT frequencies using the photo-excited state (4th state) as reference.",
"This is shown in Table REF .",
"It is evident that the consistency statement is more significantly violated in this case.",
"Consistency condition is likely to be violated whenever Condition 2 is significantly violated.",
"Table: SIC-LSD resonant frequencies (in atomic units) as computed in theinitial locally photoexcited SCF state (𝒯=0\\mathcal {T}=0) and after the laser has actedfor 𝒯=400,800\\mathcal {T}=400,800 and 1200 a.u..The values at each 𝒯\\mathcal {T} were obtained via linear response to a δ\\delta -kickperturbation of strength 0.0020.002 a.u.., for total propagation time 5000 a.u.",
"andresolution of 0.001250.00125 a.u., also shown in lower panel Fig.",
".ω 1 \\omega _1, ω 2 \\omega _2 and ω 4 \\omega _4 correspond to excitations from the initial photoexcited stateto higher, delocalized states and ω CT \\omega _{\\mathrm {CT}} is a de-excitation to the target CT state.The last column shows the SIC-LSD resonances as computed from the targeted SCF CT state,computed from second order response using a δ\\delta -kick perturbation of0.10.1 a.u.",
"strength and evolving for 2000 a.u., with a resolution of 0.0030.003a.u.",
"(see text).Table: Second column: SIC-LSD resonant frequencies (in atomic units) corresponding to transitions from theinitial locally photoexcited SCF state(which corresponds to the 4th excited state) to higher excited states, as computed from the initial photoexcited SCF state(same as shown in second column Table ).The last column shows the SIC-LSD resonances corresponding to the same transitions but computed via linear response from the targeted SCF CT state(the 3th excited state).",
"(See consistency condition in Section ).",
"The latter are obtainedas the difference between the transition frequencies from the SCF CT state to the same final states,after subtraction of the CT resonance as computed at the SCF CT state, namely ω CT =ω 34 =0.237\\omega _{\\mathrm {CT}}=\\omega _{34} = 0.237 a.u.Table: Second column: SIC-LSD resonant frequencies (in atomic units) corresponding to transitions from theSCF CT state (the 3th excited state) to higher excited states, as computed from the SCF CT state.",
"The last column shows the SIC-LSD resonances corresponding to thesame transitions but computed via linear response from the initial locally photoexcited SCF state (the 4th excited state).",
"(See consistency condition in Section ).The latter are obtained as the transition frequencies from the SCF locally excited state to the same final states, after subtraction of the CT resonance as computed at the SCF locally excited state,namely ω CT =ω 43 =-0.286\\omega _{\\mathrm {CT}}=\\omega _{43} = -0.286 a.u."
],
[
"Charge-transfer from a Ground State",
"The TDKS description of long-range CT beginning in a ground state is quite different than that beginning in a photoexcited state.",
"The reason is that when beginning in a ground state the natural choice for the KS initial state is a non-interacting ground state, which is a single-Slater determinant.",
"Such a choice places the transfering electron in a doubly occupied orbital, with the other electron occupying this orbital remaining in the donor.",
"The time-dependent orbital describing the transfering electron must at the same time describe an electron that remains in the donor.",
"The exact KS potential for a model molecule consisting of two closed-shell fragments in its ground state is depicted in Fig.",
"REF .",
"The doubly-occupied orbital initially localized on the donor becomes increasingly delocalized over both the donor and acceptor [17].",
"In the exact xc potential, a step feature develops over time.",
"Approximate functionals can not capture the step, and it is known [17], [19], [51], [52], [53] that they yield poor CT dynamics, failing to transfer the charge, even when the functional yields a good prediction for the CT excitation energy.",
"This is borne out in the dipole dynamics shown in the upper panel Fig.",
"3 in Ref.",
"[19], Fig.",
"4 in Ref.",
"[17] and upper panel Fig.",
"1 in Ref. [25].",
"Figure: SIC-LSD resonances for different laser durations𝒯=0,400,800,1200\\mathcal {T}=0,400,800,1200 a.u.. ω 1 \\omega _1, ω 2 \\omega _2 and ω 4 \\omega _4 shift up as chargetransfers, but ω CT \\omega _{CT} moves down.",
"It can be seen in table that the direction of the shift is towards the value of each resonance at the final target state.",
"For 𝒯=800\\mathcal {T}=800a.u.",
"ω 2 \\omega _{2} and ω CT \\omega _{CT} resonances become degenerate, resulting in the overlap of the two peaks (see lower panel Fig.",
").All the resonances return to their initial positions as the dipole moment d(t)d(t) shown in inset lower panel Fig.",
"returns to its initial value.In Ref.",
"[25], the failure of the approximate TDDFT dynamics was analyzed from the viewpoint of the violation of Condition 2 using a simple two-electron model and focusing on the EXX approximation.",
"The argument there also goes through for commonly used functionals.",
"Essentially, even when the CT excitation frequency, as determined by linear response from the ground state is accurate, the CT frequency as determined by linear response from the target CT state is poor – in fact it is vanishing, due to static correlation, with delocalized antibonding type of orbitals lying near the delocalized bonding-type of orbital.",
"In the target CT state, the bare KS frequency becomes very small, and so does the approximate $f_{\\scriptscriptstyle \\rm XC}$ -correction, resulting in large violations of Condition 2 by available functionals, $\\omega ^i>>\\omega ^f$ .",
"The dynamics is consequently very poor, yielding practically no CT, and very little dynamics occurs except at very short times.",
"Although Condition 2 is strongly violated, the lack of dynamics means that actually Condition 1 is satisfied; since there is very little change in the density, $v_{\\scriptscriptstyle \\rm XC}(t)$ remains about constant and so do the time-dependent spectra.",
"This example highlights the importance of considering both conditions together when understanding and predicting the performance of approximate functionals.",
"Figure: To the left the exact KS potential and initial KS ground state; to the right the exact final KSpotential and target CT state are plotted.",
"The model is specified in Section .As discussed in the text for the CT dynamics starting in the ground state the choice of a SSD forces the occupied time-dependent KS orbitalto describe both the electron that transfers to the acceptor, and the electron that remains in the donor.",
"The degeneracy between the bonding and antibonding orbitals in the target CT state is related to thebuilding up over time of a CT step in the exact xc potential .It is also interesting to analyse Condition 3 in this example.",
"Let's imagine running the same CT dynamics discussed above but backwards, i.e.",
"starting in the CT state and targeting the ground state.",
"We could choose as initial state a doubly occupied orbital, in which case, for a stretched molecule, the bare KS eigenvalue difference $\\omega _{\\scriptscriptstyle \\rm S}$ becomes very small due to the degeneracy between bonding and antibonding orbitals.",
"The EXX kernel together with the Hartree contribution, $f_{\\scriptscriptstyle \\rm HX}$ , equals half Hartree in this case, and the predicted resonance is very small and inaccurate, since the EXX kernel can not correct the vanishing $\\omega _{\\scriptscriptstyle \\rm S}$ .",
"On the other hand, we could choose as initial state a spin-broken configuration where $\\uparrow $ and $\\downarrow $ electrons occupy different orbitals as we did for the photoexcited CT dynamics studied in Section REF .",
"Such choice of KS initial state has a finite bare KS eigenvalue difference $\\omega _{\\scriptscriptstyle \\rm S}$ , and despite vanishing of the EXX kernel correction within SPA as discussed in Appendix , EXX gives a reasonably accurate prediction of the physical resonance ($0.287$ au).",
"This example illustrates how two distinct choices of KS initial state can lead to two different predictions of the TDDFT resonance, signifying violation of Condition 3."
],
[
"Adiabatically-Exact Propagation",
"To assess the impact of the adiabatic approximation itself on the fulfillment of the exact conditions, we now consider propagating self-consistently using the exact ground state xc functional.",
"This “adiabatically-exact” (AE) approximation [14], $v_{\\scriptscriptstyle \\rm XC}^{\\mathrm {AE}}[n;\\Psi _0,\\Phi _0](t) = v_{\\scriptscriptstyle \\rm XC}^{\\mathrm { exact-gs}}[n(t)]$ , is the best possible ground state approximation to the xc functional, when using explicit density functionals.",
"All discrepancies with the exact dynamics will be due to the adiabatic approximation.",
"Because this involves finding the potential whose ground state density coincides with the instantaneous one at each time-step, it is computationally quite laborious, so we simplify the model here and consider an asymmetric Hubbard dimer [51], [52], beginning the dynamics in the ground state.",
"The exact ground state functional can be found by Levy-Lieb contrained search within the small Hilbert space [54].",
"In Ref.",
"[51], [52] an asymmetric Hubbard dimer was used to study CT dynamics from the ground state (see Fig.",
"REF ), showing the same trends as CT dynamics in the LiCN molecule [19] and in the real-space one-dimensional model systems studied in Refs.",
"[17], [25].",
"We now consider the dynamics in the light of Condition 1 and Condition 2.",
"Further, we exploit these conditions to apply a modified field that enhances the amount of charge transfered.",
"The Hamiltonian of the 2-site Hubbard model reads (Fig.",
"REF ), $\\begin{split}\\hat{H}= & -T \\sum _\\sigma \\left( \\hat{c}_{L\\sigma }^\\dag \\hat{c}_{R\\sigma } +\\hat{c}_{R\\sigma }^\\dag \\hat{c}_{L\\sigma } \\right)+ U \\left( \\hat{n}_{L \\uparrow } \\hat{n}_{L \\downarrow } + \\hat{n}_{R\\uparrow } \\hat{n}_{R \\downarrow }\\right) \\\\& + \\frac{\\Delta v (t)}{2}(\\hat{n}_{L} -\\hat{n}_{R}),\\end{split}$ where $\\hat{c}_{L(R)\\sigma }^{\\dag }$ and $\\hat{c}_{L(R)\\sigma }$ are creation and annihilation operators for a spin-$\\sigma $ electron on the left(right) site $L(R)$ , respectively, and $\\hat{n}_{L(R)}=\\sum _{\\sigma =\\uparrow , \\downarrow }\\hat{c}_{L(R)\\sigma }^{\\dag } \\hat{c}_{L(R)\\sigma }$ are the site-occupancy operators.",
"The occupation difference $\\langle \\hat{n}_{L} -\\hat{n}_{R}\\rangle =\\Delta n$ represents the dipole in this model, $d=\\Delta n$ , and is the main variable [54]; the total number of fermions is fixed at $N=2$ .",
"A static potential difference, $\\Delta v^{0}= \\sum _\\sigma (v_{L\\sigma }^0 - v_{R \\sigma }^0)$ , renders the Hubbard dimer asymmetric.",
"The total external potential $\\Delta v (t)$ is given by $\\Delta v(t)= \\Delta v^{0}+ 2{\\mathcal {E}(t)}$ , where the last term represents a tunable electric field applied to induce CT between the sites.",
"An infinitely long-range molecule is represented by $T/U \\rightarrow 0$ .",
"We choose here the static potential difference as $\\Delta v^0=-1.5$ U, the hopping parameter $T=0.05$ U and the on-site interaction $U=1$ (see also Ref. [52]).",
"Figure: Two-site lattice model Eq.",
"() used to study a CT starting in the ground state.Initially, the two electrons occupy the site with the deeper on-site potential v L v_L, |Δn gs |≈2|\\Delta n_{gs}| \\approx 2.The target CT state consists of two open-shell fragments with approximately one electron on each, |Δn CT |≈0|\\Delta n_{\\rm CT}| \\approx 0.The KS Hamiltonian has the form of Eq.",
"(REF ) but with $U = 0$ and $\\Delta v(t)$ replaced by the KS potential difference, $\\Delta v_s[\\Delta n, \\Phi (t_0)](t) = v_{\\scriptscriptstyle \\rm HXC}[\\Delta n, \\Psi (t_0), \\Phi (t_0)](t) + \\Delta v(t).$ For a resonant applied field ${\\mathcal {E}(t)}= E_0 \\sin \\left(\\omega _{\\mathrm {CT}} t \\right)$ , with $\\omega _{CT}=(E_{\\mathrm {CT}}-E_{\\mathrm {gs}})=0.5177~a.u.$ , the interacting system achieves full population of the CT state after half a Rabi cycle (about 128 a.u.",
"), coinciding with a vanishing dipole moment (see inset in upper panel of Fig.",
"REF ).",
"AE dynamics is poor (inset in the middle panel) despite the AE resonance being very accurate, $\\omega ^{AE}_{gs}=0.5187~a.u.$ [51].",
"We follow an analogous procedure as described in Section to compute the linear response after different durations of the applied laser.",
"The time-step used is $0.01$ a.u.",
"and the total propagation time is $T=3000$ a.u., resulting in a frequency resolution of about $2\\pi /T \\approx 0.002$ a.u..",
"Figure: Logarithm of absolute spectrum |Δd(ω)||\\Delta d(\\omega )|Eq.",
"() showing CT frequency for different laser durations 𝒯\\mathcal {T}.",
"Inset: dipole moments |d(t)||d(t)| for the CT dynamics starting in the ground state studiedin Refs.",
", .",
"Upper panel: Exact.",
"Middle panel: AE.",
"Lower panel: EXX.Fig.",
"REF upper panel shows the exact dipole response $|\\Delta d(\\omega )|$ for different $\\mathcal {T}$ .",
"As expected the position of the peak is constant at $\\omega _{CT}$ for all $\\mathcal {T}$ .",
"In the middle panel of Fig.",
"REF the AE response for the corresponding $\\mathcal {T}$ is shown.",
"Peak-shifting as function of the laser duration $\\mathcal {T}$ is noticeable, with clear violation of Condition 1.",
"As time evolves, the density starts transferring to the other site and the KS potential Eq.",
"(REF ) follows the changes in the density.",
"The AE xc kernel is also time-dependent, but does not have the correct time-dependence to maintain fixed resonance positions.",
"The changes in the AE peak position follow the evolution of the density $\\Delta n(t)$ (compare peak migration in lower panel of Fig.",
"REF with evolution of AE dipole shown in the inset).",
"The AE peak shifts towards higher energies, this is consistent with our findings of Section REF , since here $\\omega ^f> \\omega ^i$ [52] and thus the peak shift is in the direction of the resonance computed at the final state.",
"In the lower panel of Fig.",
"REF the EXX spectra at different $\\mathcal {T}$ is shown.",
"The EXX peak shifts as the electron starts transferring and returns to its initial ground state position as the density $\\Delta n$ localizes back on the donor site around $\\mathcal {T}=60$ a.u.",
"(see inset lower panel of Fig.",
"REF ).",
"As the density transfers, the EXX peak shifts towards higher energies, although $ \\omega ^{EXX}_{CT}=\\omega ^f \\rightarrow 0$ .",
"Thus, in this case, the EXX peak does not shift towards $\\omega ^f$ .",
"Instead, it is consistent with the peak-shift directions observed in Ref.",
"[23], where the instantaneous spectra of small, closed-shell, laser-driven molecules beginning in their ground-state, was studied.",
"There, a single peak was observed in the TDDFT spectra that, as the system transitions onto a single-excited state migrates towards the value of the de-excitation energy from the doubly-excited state to the single-excited state.",
"Figure: Logarithm of absolute AE spectrum |Δd(ω)||\\Delta d(\\omega )| showing CT resonance for different laser durations 𝒯\\mathcal {T}.Inset: dipole moments |d(t)|=|Δn(t)||d(t)|=|\\Delta n(t)| for exact (black) and AE (red) for an applied laser ℰ(t)=1.8sin(0.618t){\\cal E}(t)=1.8 \\sin (0.618 t) starting in the ground state.In Fig.",
"REF we present the AE response $|\\Delta d(\\omega )|$ for non-resonant dynamics starting in the ground state.",
"The applied laser is detuned by $0.1$ a.u.",
"from $\\omega ^{AE}_{gs}$ (which for this problem as discussed before is very accurate) and is chosen 20 times stronger than the one applied in the resonant CT dynamics of Fig.",
"REF .",
"The exact dynamics is shown in the inset of Fig.",
"REF in black, along with the AE dynamics, shown in red in the inset.",
"As expected for the exact dynamics, the position of the CT peak is constant and only the intensity varies (not shown here).",
"Again AE presents spurious time-dependent CT resonance as a function of $\\mathcal {T}$ , signifying violation of Condition 1.",
"This example of non-resonant dynamics is presented to stress the fact that spurious peak shifting within TDDFT is not exclusive to resonant dynamics.",
"We next consider an example that violates Condition 1 but satisfies Condition 2, unlike the previous examples: resonant dynamics for a weakly-correlated ($T=U=1$ ), homogeneous ($\\Delta v^0=0$ ) Hubbard dimer [54].",
"Due to symmetry of the Hamiltonian, the ground-state and first-excited densities are identical, $\\Delta n_{gs}=\\Delta n^{*}$ .",
"Condition 2 is satisfied within AE if we start the dynamics in the ground state and target the first local excitation, namely $\\omega ^i_{AE}=\\omega ^f_{AE}$ , since the density of both stationary states is the same.",
"The AE dynamics however deviates from the exact as is shown in Ref.",
"[54] and this is due to violation of Condition 1.",
"Linear response calculations at different moments $\\mathcal {T}$ for resonant Rabi oscillations show significant peak-shifting, e.g.",
"$\\omega ^{AE}_{gs}=2.6~a.u.$ , $\\omega ^{AE}(\\mathcal {T}=15)=2.3~a.u.$ .",
"Figure: Exact CT dynamics starting in the ground state (black).",
"The AE dynamicsin the presence of the monochromatic laser is shown in red.",
"AE dynamicsin the presence of the chirped laser, which every 2020~a.u.",
"is adjusted to the instantaneous AE resonance ω AE (𝒯)\\omega ^{AE}(\\mathcal {T}) shown in middle panel Fig.",
", is shown in blue.",
"Inset: monochromatic laserℰ(t)(ω gs AE ,t)=0.09sin0.518t{\\mathcal {E}(t)(\\omega _{gs}^{AE},t)}= 0.09 \\sin \\left(0.518 t \\right) inred, chirped laser Eq.",
"() in blue.Finally we present a proof of principle directly demonstrating the effect of the spurious time-dependence of the electronic spectra on the TDDFT dynamics.",
"Peak-shifting as the system evolves means that the instantaneous TDDFT resonance is continuously detuned from the TDDFT resonance computed by perturbing the initial state.",
"But what if we would adjust for this spurious detuning by making the applied laser frequency-dependent, i.e.",
"by designing a chirped laser that adjusts its frequency according to changes in the evolving density in order to stay tuned with the instantaneous TDDFT resonance of the KS system?",
"In Fig.",
"REF , we show the results of propagating the AE system in the presence of a laser whose frequency is adjusted piecewise-in-time during the evolution to be approximately resonant with the KS system; that is every 20a.u., the frequency of the laser is changed such to be resonant with the instantaneous AE CT resonances shown in middle panel of Fig.",
"REF .",
"${\\cal E}[\\omega ^{AE}(\\mathcal {T})](t)=0.09 \\sin \\left(\\omega ^{AE}(\\mathcal {T})~t \\right),$ The chirped laser, Eq.",
"(REF ), is shown in blue in the inset of Fig.",
"REF , as a guide the monochromatic ${\\mathcal {E}(t)}= 0.09 \\sin \\left(0.518 t \\right)$ is plotted on top in red.",
"In Fig.",
"REF the exact dipole dynamics $\\Delta n(t)$ is in black, the AE dynamics under the monochromatic laser is in red and the AE dynamics under the chirped laser is in blue.",
"An improvement of the AE dynamics is observed for the laser that is approximately “optimally tuned” in the way above.",
"Notice that for the instantaneous resonances $\\omega ^{AE}(\\mathcal {T})$ in Eq.",
"(REF ) we have simply used the ones computed for the monochromatic driven AE dynamics of Fig.",
"REF .",
"Further improvements and eventually agreement with the exact dynamics is expected if the chirped laser is designed in a self-consistent way, i.e.",
"if $\\omega ^{AE}(\\mathcal {T})$ would be computed for the AE evolution in the presence of the chirped laser."
],
[
"Conclusions and Outlook",
"Recent observations of time-dependent spectra in TDDFT(or TDHF) have drawn much attention: Unphysical shifts in the position of their spectral peaks have been reported for electron dynamics far from equilibrium [21], [22], [29], [20], [25], [23], [30], [24].",
"Peak shifting is expected for coupled electron-ion dynamics or when computing resonances in the presence of time-dependent fields, but once all time-dependent fields are turned off electronic resonance positions are constant for pure electron dynamics.",
"The cancellation of spurious time-dependence to yield constant resonances has been rationalized as an exact condition for the xc functional of TDDFT in Ref. [25].",
"In this article we have elaborated on the theoretical aspects of the formulation and provided a detailed study of some representative examples that illustrate the relevance of the exact condition for the dynamics.",
"A generalized non-equilibrium response function was derived which applies to general non-equilibrium situations when the external fields are off [25], [37].",
"In contrast to the standard linear response formalism applied to systems around the ground state, the generalized non-equilibrium response function $\\tilde{\\chi }[n^{(0)}_\\mathcal {T}; \\Psi (\\mathcal {T})]$ deals with non-stationary densities $n^{(0)}_\\mathcal {T}$ in the absence of time-dependent fields and is not time-translationally invariant.",
"We showed that due to the lack of this symmetry the frequency-dependent response $\\tilde{\\chi }_{t^{\\prime }}({\\bf r}, {\\bf r}^{\\prime },\\omega )$ or $\\tilde{\\chi }_{T}({\\bf r}, {\\bf r}^{\\prime },\\omega )$ depends parametrically on a time-variable $t^{\\prime }$ or $T=(t+t^{\\prime })/2$ , respectively, and each exhibits a different pole structure.",
"The density response $\\delta n({\\bf r},\\omega )$ is independent of this choice as it should be.",
"The latter has poles at the physical resonances of the system, corresponding to transitions between eigenstates of the unperturbed Hamiltonian.",
"By virtue of the Runge-Gross theorem [4] the exact time-dependent KS system reproduces the non-equilibrium density response $\\delta n ({\\bf r},\\omega )$ exactly.",
"Therefore, a Dyson-like equation connects $\\tilde{\\chi }[n^{(0)}_\\mathcal {T}; \\Psi (\\mathcal {T})]$ with the non-equilibrium KS response $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}[n^{(0)}_\\mathcal {T}; \\Psi (\\mathcal {T}), \\Phi (\\mathcal {T})]$ via a generalized kernel $\\tilde{f}_{\\scriptscriptstyle \\rm HXC}[n^{(0)}_\\mathcal {T}; \\Psi (\\mathcal {T}), \\Phi (\\mathcal {T})]$ .",
"A simple Lehmann representation can be written down for $\\tilde{\\chi }$ but not for $\\tilde{\\chi }_{\\scriptscriptstyle \\rm S}$ because the KS potential is not static for non-stationary densities $n^{(0)}_\\mathcal {T}$ .",
"The exact condition was formulated in several different ways.",
"In the language of pump-probe experiments, the resonance positions must be independent of the moment $\\mathcal {T}^{\\prime }$ the pump is turned off and of the delay $\\theta $ between pump and probe.",
"More generally, in the absence of ionic motion, the TDDFT response frequencies (corresponding to transitions between two given interacting states) of a field-free system must be independent of the interacting state $\\Psi (\\mathcal {T})$ around which the response is calculated (Condition 1).",
"When the interacting system is in any excited stationary state, and the KS reference state is also chosen to be a stationary state of its potential, we can formulate the exact condition in a matrix form.",
"In such a case, the TDDFT resonance frequency for a specific transition between two stationary states must be independent of which of the two states the response is calculated around (Condition 2).",
"Further, a consistency condition relates the frequencies of transitions to other states from each reference state.",
"The TDDFT response frequencies must be invariant also with respect to the choice of KS state (Condition 3).",
"The exact conditions pose a very challenging task for the xc kernel, as we illustrated with several examples.",
"The first example was that of CT dynamics from a photo-excited state.",
"We used a model system of two electrons in an asymmetric double-well and started the KS propagation with one orbital promoted to a locally excited orbital; this electron was then driven to the other well by means of a weak resonant field.",
"The performance of the different functionals to accurately simulate this dynamics was related to the fulfillment of the exact condition in its different forms.",
"In Ref.",
"[25] it was observed that the degree of violation of Condition 2 was directly related to the performance of the approximate functional to reproduce the dynamics.",
"Here the time-dependent spectra for the same approximated functionals, including EXX and SIC-LSD are compared against the exact calculation at different moments in the evolution.",
"We find that in addition to Condition 2 also Condition 1 was violated within SIC-LSD, resulting in spurious peak shifting and consequently in poor dynamics.",
"The CT peak moved towards the SIC-LSD resonance computed around the final target state and this trend held also for the other resonances of the system.",
"SIC-LSD resonances were also shown to violate the consistency condition.",
"Unrestricted EXX was shown to fulfill Condition 2 for this particular case [25] and here we showed it also has fixed peak positions along the evolution (fulfillment of Condition 1), resulting in very accurate photoexcited CT dynamics.",
"In the next example, CT dynamics starting in the ground state, EXX trivially fulfilled Condition 1 but violated Condition 2, resulting in poor dynamics.",
"We also briefly discussed the violation of Condition 3 by EXX when we consider different initial KS states when running the dynamics backwards beginning in the CT state.",
"In order to assess the impact of the adiabatic approximation alone, independent of the effect of the choice of approximate ground-state functional, we used a two-site lattice model.",
"Given the small Hilbert space the exact ground state xc functional can be found [54] and was used to self-consistently propagate the KS system (AE propagation).",
"We tuned the parameters of the system to mimic long-range CT dynamics starting in the ground state [51], [52].",
"We observed that the AE CT dynamics violates both Condition 1 and Condition 2 and resulted in poor dynamics.",
"Violation of Condition 1 was observed for AE for both resonant and also off-resonant dynamics.",
"In the case of the weakly-correlated homogeneous Hubbard model studied in Ref.",
"[54] Condition 2 was satisfied but Condition 1 was violated within AE, resulting in inaccurate dynamics.",
"Perhaps the clearest impact of the influence of the exact condition on the dynamics was illustrated by the “optimally tuned\" laser that adjusted in a piece-wise manner to the instantaneous AE resonance (Figure REF and discussion).",
"This showed that the AE propagation in the presence of this chirped laser resulted in an improved charge transfer rate in the case of resonant CT dynamics.",
"We conclude that in order to be able to predict the performance of a given approximate functional all three Conditions need to be considered.",
"Further because the best possible ground state approximation for the density-functional fails to fulfill the exact conditions, resulting in poor dynamics, we stress the need to go beyond the adiabatic approximation.",
"We have shown the large impact that the violation of the exact conditions has on the ability of approximate functionals to reproduce the dynamics: the higher the degree of the violation, the poorer the simulated dynamics.",
"The examples presented here are drastic, but we have only analyzed small systems, perhaps a worst-case scenario for approximate TDDFT.",
"An important question for future work is the system-size scaling of the violation of the exact conditions.",
"When ionic motion is considered, the spectral peaks can be vibrationally broadened, perhaps relaxing the stringent exact conditions discussed here.",
"Development of an approximate functional or a propagation scheme that fulfills exactly or approximately the exact conditions is an important direction for future research.",
"Financial support from the National Science Foundation CHE-1152784 (for K.L.",
"), Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences under award DE-SC0015344(N.T.M., J.I.F.",
"), a grant of computer time from the Cuny High Performance Computing Center under NSF grants CNS-0855217 and CNS-0958379."
],
[
"$f_{\\scriptscriptstyle \\rm HXC}$ correction in EXX within the Single Pole Approximation",
"Within the single-pole approximation using EXX it can be shown that for a two-electron singlet state where the two electrons occupy different spatial orbitals, the $f_{\\scriptscriptstyle \\rm HXC}$ correction vanishes.",
"First, note the definition of the spin-resolved xc kernel [45], [44], $f_{\\scriptscriptstyle \\rm XC}^{\\sigma \\sigma ^{\\prime }}({\\bf r},{\\bf r}^{\\prime },t,t^{\\prime }) = \\frac{\\delta v_{\\scriptscriptstyle \\rm XC}^{\\sigma }({\\bf r},t)}{\\delta n_{\\sigma ^{\\prime }}({\\bf r}^{\\prime },t^{\\prime })}$ In adiabatic EXX, $v_{\\scriptscriptstyle \\rm XC}^{\\sigma }[n_\\uparrow ,n_\\downarrow ]({\\bf r}) = v_{\\scriptscriptstyle \\rm X}^\\sigma [n_\\uparrow ,n_\\downarrow ]({\\bf r}) =\\frac{\\delta E_{\\scriptscriptstyle \\rm X}[n_\\uparrow ,n_\\downarrow ]}{\\delta n_\\sigma ({\\bf r})}$ , where for two electrons $E_{\\scriptscriptstyle \\rm X}&=& -\\frac{1}{2}\\sum _{\\sigma =\\uparrow ,\\downarrow }\\sum _{ij}^{N_\\sigma } \\int \\!\\!\\int \\frac{\\phi _{i\\sigma }({\\bf r})\\phi _{i\\sigma }({\\bf r}^{\\prime }) \\phi _{j\\sigma }({\\bf r}^{\\prime }) \\phi _{j\\sigma }({\\bf r})}{|{\\bf r}- {\\bf r}^{\\prime }|} \\mathop {}\\!\\mathrm {d}{\\bf r}\\mathop {}\\!\\mathrm {d}{\\bf r}^{\\prime } \\nonumber \\\\ &=&-\\frac{1}{2}\\sum _{\\sigma =\\uparrow ,\\downarrow } \\int \\!\\!\\int \\frac{n_{\\sigma }({\\bf r}) n_{\\sigma }({\\bf r}^{\\prime })}{|{\\bf r}- {\\bf r}^{\\prime }|} \\mathop {}\\!\\mathrm {d}{\\bf r}\\mathop {}\\!\\mathrm {d}{\\bf r}^{\\prime }\\nonumber \\\\ &=&- E_{\\scriptscriptstyle \\rm H}[n_\\downarrow ] - E_{\\scriptscriptstyle \\rm H}[n_\\uparrow ]\\;.",
"$ This yields $v_{\\scriptscriptstyle \\rm X}^\\sigma [n_\\uparrow ,n_\\downarrow ] =-v_{\\scriptscriptstyle \\rm H}[n_\\sigma ]$ , and so using Eq.",
"REF , $f_{\\scriptscriptstyle \\rm XC}^{\\downarrow \\uparrow } =f_{\\scriptscriptstyle \\rm XC}^{\\uparrow \\downarrow } = 0 \\;\\; {\\rm while}\\;\\; f_{\\scriptscriptstyle \\rm XC}^{\\uparrow \\uparrow } = f_{\\scriptscriptstyle \\rm XC}^{\\downarrow \\downarrow } = -f_{\\scriptscriptstyle \\rm H}$ Thus for the parallel spin case the Hartree-exchange kernel $f_{\\scriptscriptstyle \\rm HX}^{\\sigma \\sigma }=0$ vanishes, however in the case of anti-parallel spin $f_{\\scriptscriptstyle \\rm HX}^{\\uparrow \\downarrow }= f_{\\scriptscriptstyle \\rm HX}^{\\downarrow \\uparrow }=f_{\\scriptscriptstyle \\rm H}$ .",
"Now for a non-degenerate Kohn-Sham pole, $\\omega _{\\scriptscriptstyle \\rm S}= \\epsilon _a - \\epsilon _i$ , the single-pole approximation for the TDDFT excitation energy depends only on the parallel-spin component of the kernel [45], [44]: $\\omega = \\omega _{\\scriptscriptstyle \\rm S}+ \\int \\Phi _q({\\bf r})\\left(f_{\\scriptscriptstyle \\rm H}+ f_{\\scriptscriptstyle \\rm XC}^{\\sigma \\sigma }\\right)\\Phi _q({\\bf r}^{\\prime }) d{\\bf r}d{\\bf r}^{\\prime }$ (where, as before, $\\Phi _q({\\bf r}) = \\phi _i({\\bf r})\\phi _a({\\bf r})$ ), clearly yielding a zero correction for the case of adiabatic EXX.",
"That is, within the single-pole approximation, the KS response frequencies are identical to the TDDFT excitations when the two electrons are occupying different orbitals (see Section REF ).",
"In contrast, when they occupy the same spatial orbital, as for a spin-saturated two-electron singlet (see section REF ), each Kohn-Sham pole is doubly-degenerate, and Eq.",
"REF is modified, see Ref.",
"[45], [44] for details; the kernel correction in EXX, $f_{\\scriptscriptstyle \\rm X}=-f_{\\scriptscriptstyle \\rm H}/2$ , does not vanish for this case."
]
] | 1605.05800 |
[
[
"Simulations of the Mg II k and Ca II 8542 lines from an Alfv\\'en\n Wave-heated flare chromosphere"
],
[
"Abstract We use radiation hydrodynamic simulations to examine two models of solar flare chromospheric heating: Alfv\\'en wave dissipation and electron beam collisional losses.",
"Both mechanisms are capable of strong chromospheric heating, and we show that the distinctive atmospheric evolution in the mid-to-upper chromosphere results in Mg II k-line emission that should be observably different between wave-heated and beam-heated simulations.",
"We also present Ca II 8542A profiles which are formed slightly deeper in the chromosphere.",
"The Mg II k-line profiles from our wave-heated simulation are quite different from those from a beam-heated model and are more consistent with IRIS observations.",
"The predicted differences between the Ca II 8542A in the two models are small.",
"We conclude that careful observational and theoretical study of lines formed in the mid-to-upper chromosphere holds genuine promise for distinguishing between competing models for chromospheric heating in flares."
],
[
"Introduction",
"In a solar flare, up to $10^{32-33}$ ergs is released when the coronal magnetic field reconfigures.",
"It is transported along the coronal magnetic field to be deposited in the dense chromosphere.",
"This results in plasma heating and broadband enhancement to the solar radiative output.",
"The sudden temperature increase also causes strong upflows (chromospheric `evaporation') and down flows (`condensations') observed via Doppler shifts in spectral lines.",
"Hard X-ray (HXR) footpoint sources indicating the presence of high-energy electrons are closely associated with enhancements to optical radiation, the locations of which are a subset of the UV and EUV ribbons that delineate the endpoints of just-reconnected field.",
"For an overview of flare observations, see e.g.",
"[15] and [43].",
"How energy is transported from the corona, where it is stored, to the chromosphere where it is dissipated is still a matter of debate.",
"The transport must be fast (sub-second in the flare impulsive phase), ducted along the coronal magnetic field and must involve the acceleration of electrons out of the thermal background.",
"It is possible that different flares have different combinations of transport by particle beams, conduction or MHD waves, and the challenge is to use observations and modelling to distinguish between these.",
"Coronal observations will tend to be ambiguous, as thermal conduction, strong flows, and optically-thin radiation from the hot, ionised plasma smear out and superpose, and hence obscure signatures of the energy transport.",
"The chromosphere, which produces the majority of flare radiation, has its own complications: it is optically thick in some lines, partially ionised, and highly structured both vertically and horizontally in temperature and density.",
"However, chromospheric line profiles in principle allow us to probe the conditions at different atmospheric depths, meaning there is the prospect of making progress in distinguishing different energy transport models on the basis of how they affect the chromosphere.",
"It is the purpose of this paper to model the profiles of some important diagnostic spectral lines generated in a chromosphere heated as a result of energy transport from the corona by electron beams and by Alfvén waves.",
"In the electron beam (EB) model [5], [24], electrons, accelerated to mildly relativistic speeds in or close to the coronal energy release site, stream along the field.",
"They deposit their energy via Coulomb collisions, primarily in the chromosphere which presents a collisional thick target.",
"This results in heating and radiation, including non-thermal hard X-ray (HXR) bremsstrahlung.",
"Using the collisional thick-target model (CTTM) the energy and number spectrum of the non-thermal coronal beam can be inferred from the HXR emission (e.g.",
"[22]) as has frequently been done using data from the RHESSI satellite [34].",
"The resulting electron spectra are frequently used as input to numerical codes for simulating the response of the chromosphere to flare energy input.",
"Flare energy transport by Alfvén waves (AWs) was first proposed by [13] to explain TMR heating and [14] restarted the discussion on Alfvénic perturbations as a potential flare energy transport mechanism.",
"Since a solar flare is, fundamentally, a large scale restructuring of the solar magnetic field, it is very reasonable to expect that AWs are produced.",
"[14] argued that a 5-10% Alfvénic perturbation to a 500G coronal field could supply adequate power to the flare chromosphere.",
"The dissipation of the wave, by various damping mechanisms, in the highly structured chromosphere would lead to heating and potentially also to in situ electron acceleration, but the details of electron acceleration have to be worked out.",
"The CTTM is attractive in that it neatly combines energy transport, HXR generation and footpoint heating, but it has difficulty explaining the low depths of some flare optical emission [40] and can also imply high coronal EB densities.",
"A high coronal EB density, given the inferred ambient densities suggest that the beam-return current relative speed is high enough that instabilities may follow.",
"Flare line and continuum radiation show that the temperature minimum region (TMR) is heated by up to a few hundred kelvin during flares [39], [38], [42].",
"Additionally, white light flare (WLF) images also show that the optical continuum enhancement originates low in the solar atmosphere; WL and HXR emission were found to be co-spatial and located at $<$ $0.5$ Mm above the photosphere in an event studied by [40].",
"To reach and heat this depth requires significantly higher power in high energy electrons than is observed [38], [44], [42].",
"For this reason radiative backwarming [37] is often used to explain WLF observations in the EB model.",
"The CTTM implies up to 10$^{36}~\\rm {electrons}~s^{-1}$ [23], [22] which, for a typical coronal density of $10^9~\\rm {cm}^{-3}$ is equivalent to accelerating all the electrons in a coronal volume of $(10,000~\\rm {km})^3$ each second.",
"A continual resupply of electrons for acceleration is needed, which could happen via the return current that is established by the ambient plasma.",
"As discussed by [14] and [28], observations imply coronal beam densities comparable to or greater than the ambient coronal density meaning a return current speed that is comparable to the beam speed.",
"In this case, the beam and its return current are likely to be unstable in the corona, and to dissipate a large fraction of their energy in turbulence and heating.",
"We note that in the scenario of a significantly greater ambient coronal density (on the order of $10^{11}$ cm$^{-3}$ ) then the replenishment of the beam does not pose as much of a problem.",
"Higher densities have been inferred in some flares [52], [18].",
"Atmospheric heating via damping of AWs generated by photospheric drivers and propagating upwards into the chromosphere has been proposed as a quiet Sun chromospheric heating mechanism, e.g.",
"by ion-neutral damping in the partially-ionised chromosphere.",
"[9], [27], [31].",
"Downwards-propagating AWs produced by flaring perturbations in the coronal magnetic field will also be damped in the chromospheric plasma, resulting in heating.",
"[13] suggested the resistive (Joule) dissipation of the currents associated with AWs as a means of heating the TMR, and calculated that $\\Delta T$ of 100-200 kelvin was possible for frequencies of order 1-10 Hz.",
"More recently, simulations by [47] of AWs traveling downwards through chromosphere with a realistic stratification of Alfvén speed and ionisation showed that for sufficiently high frequencies (around 1 Hz) a significant fraction of coronal AW energy can be transmitted to the deep chromosphere and damped by ion-neutral damping in the TMR (and electron resistivity lower down).",
"[14] and [41] propose that AWs are the dominant energy transport mechanism through the flaring corona, and discuss the viability of AWs in accelerating electrons to produce the observed HXR.",
"In the simulations of [47], heating in the upper chromosphere was also observed, and [45] investigated this further.",
"They updated the approach of [13] for describing the energy deposition by waves to use ambipolar resistivity instead of classical resistivity and implemented this as an energy input in the HYDRAD code [4].",
"The result was that AW damping in the mid-upper chromosphere produced strong heating and evaporation, and looked very similar to what is found in electron-beam driven simulations.",
"Heating was most efficient for perpendicular wave numbers $k_\\perp > 1\\times 10^{-4}$ cm$^{-1}$ and frequencies around 10 Hz.",
"We expect that beam-driven and wave-driven models of energy input will have different heating profiles, and different time evolution, which will form the basis of discriminating between models.",
"High spatial, spectral and temporal resolution data of chromospheric and transition region (TR) radiation in the near-UV (NUV) and far-UV (FUV) are now available from the Interface Region Imaging Spectrograph (IRIS; [10]) spacecraft.",
"For example, [26], [36] and [17] discuss the complex chromospheric Mg ii spectra observed during flares.",
"The Daniel K. Inouye Solar Telescope (DKIST) will also provide high resolution chromospheric observations in the optical and infrared (IR).",
"These resources provide the opportunity to probe models of energy transport in flares by comparing the synthetic spectra output by advanced models to observations.",
"In this paper we use the radiation hydrodynamics code RADYN (§ ) to describe the chromospheric temperature, density and ionisation profiles resulting from numerical experiments that simulate chromospheric heating by high fluxes of flare-generated AWs, and compare with those from a standard EB simulation (§ ).",
"We use an approximated form of AW heating developed by [13] and [45] as a heating term.",
"Finally, we synthesise the observational signatures that result from these experiments (the Mg ii h & k lines, and the Ca ii 8542Å line) (§ ) and present discussion and conclusions (§ , § ) The radiation hydrodynamics code RADYN is a well established code for investigating chromospheric dynamics.",
"Originally created by [7], [8], RADYN was used to study acoustic waves in the chromosphere, and was adapted by [1] to simulate the chromospheric response to flare energy deposition by an EB.",
"Later updates have included improved treatment of the EB, including a Fokker-Planck description, soft X-ray, extreme-UV (EUV) and UV radiation backwarming and photoionisation [2], [3].",
"We used the [3] version of RADYN for results presented in this paper, with our own modifications described in § REF .",
"RADYN solves the plane-parallel, coupled, non-linear equations of hydrodynamics, radiation transfer, charge conservation and atomic level populations on a 1D grid that extends from the sub-photosphere to the corona, representing one leg of a symmetric flux tube.",
"An adaptive grid [12] with 191 grid points resolves shocks and steep gradients.",
"Elements important for chromospheric energy balance are computed using non-Local Thermodynamic Equilibrium (nLTE) radiative transfer, with other atomic species included as background continuum opacity (assumed in LTE) using the Uppsala opacity package of [19].",
"A radiative loss function approximates the optically thin coronal radiation transfer by summing all transitions in the CHIANTI database [11], [30], apart from the transitions treated in detail (See [3] for a line list).",
"The atomic level populations are solved for a six-level-with-continuum hydrogen atom, a nine-level-with-continuum helium atom, a six-level-with-continuum Ca ii ion, and a four-level-with-continuum Mg ii ion.",
"Transitions (22 bound-bound transitions and 24 bound-free transitions) with up to 100 frequency points and 5 angular points are computed assuming complete redistribution (CRD), except the Lyman transitions which are truncated at 10 Doppler widths to approximate the effects of partial redistribution (PRD).",
"We return to this issue when discussion the Mg ii lines in § REF .",
"The product of the coronal/TR emission measure and emissivities (from CHIANTI) is integrated to find the XEUV spectrum, which is included as downward-directed incident radiation when solving the nLTE radiation transfer and ionisation equations.",
"Typically when simulating flares, the EB CTTM model has been used.",
"A non-thermal EB with a power law energy flux spectrum is introduced at the apex of the corona loop.",
"It deposits energy as it travels, heating the plasma with heating rate $Q_{\\rm BEAM}$ calculated from collisional losses."
],
[
"Alfvén Wave Dissipation and Heating",
"We follow [45] to include an additional heating rate term due to AWs, $Q_{AW}$ , in RADYN using the WKB approximation to obtain the period-averaged Poynting flux of the AW as a function of distance in a magnetic flux tube [13].",
"Collisions between ions, electrons and neutrals damp the Poynting flux, and the dissipation of Poynting flux gives the heating term for the plasma.",
"As noted by [45], this approximation is accurate if the parallel wavelength is less than or comparable to the gradient length scale of the Alfvén speed, which also means that reflections are assumed negligible.",
"As discussed in § , reflection at the corona-TR boundary has been shown to significant, so as in [45] we choose an initial Poynting flux giving a reasonable flux at the top of the chromosphere.",
"In the following, $i$ , $n$ , $e$ and $t$ subscripts refer to ions, neutrals, electrons and total.",
"The collisional frequencies are computed as follows.",
"The formula for $\\nu _{e,n}$ is quoted in [16] as $\\nu _{e,n} = 6.97\\times 10^{-14}~T^{0.1}~n_{H},$ where $T$ is temperature, and $n_H$ is the number density of neutral hydrogen.",
"[21] gives an expression for the electron-ion collision time, $\\tau _{e}$ $\\tau _e = \\frac{3}{4}\\left(\\frac{m_e}{2\\pi }\\right)^{1/2}\\left(\\frac{k_b~T}{n_e~\\lambda ~e^4}\\right)^{3/2}$ $\\nu _{e,i} = 1/\\tau _{e},$ where $m_e$ is the electron mass, $k_b$ is Boltzmann's constant, $n_e$ is electron number density, $e$ is electron charge and and $\\lambda $ is the Coulomb logarithm.",
"Finally, $\\nu _{n,i}$ is discussed in [47] (noting their typo in the first $T$ ) and [48] $\\nu _{n,i} = 2.65\\times 10^{-16}~T^{1/2}~(1-&0.083~{\\rm {log_{10}}}~T)^2~n_p \\\\& + 2.11\\times 10^{-15}~(n_e-n_p),$ where $n_p$ is the proton number density.",
"The parallel (to the field) and perpendicular resistivities of the plasma are defined as $\\eta _{||} = \\frac{m_e~(\\nu _{e,i} + \\nu _{e,n})}{n_e~e^2}$ $\\eta _{\\perp } & = \\eta _{||} + \\eta _C \\\\& = \\eta _{||} + \\frac{B^2~\\rho _n}{c^2~\\nu _{n,i}~\\rho _t^2~(1+\\xi ^2~\\theta ^2)},$ where $\\rho $ is mass density, $c$ the speed of light, $\\eta _C$ the Cowling resistivity, $\\xi $ the hydrogen ionisation fraction and $\\theta = \\omega /\\nu _{n,i}$ for $\\omega = 2\\pi f$ .",
"[13] derived an expression for the effective damping length of Alfvén waves, $L_D(z)$ , with height along the modelled flux tube.",
"We modify the [13] $L_D(z)$ to use ambipolar resistivity as proposed by [45] $L_D(z) & = \\left( \\frac{1}{L_{\\perp }(z)} + \\frac{1}{L_{||}(z)}\\right)^{-1} \\\\& = \\left( \\frac{\\eta _{||}~k_x^2~c^2}{4~\\pi ~v_A} + \\frac{\\eta _{\\perp }~w^2~c^2}{4~\\pi ~v_A^3}\\right)^{-1} \\\\& = \\frac{4~\\pi ~v_A^3}{c^2~(\\eta _{||~}k_x^2~v_A^2 + \\eta _{\\perp }~w^2)},$ where $v_A(z)$ is the Alfvén speed.",
"As in [45] we modify the Aflvén speed for the presence of neutrals, $v_{A}(z) = \\frac{B}{\\sqrt{4\\pi \\rho _{t}}}\\left(\\frac{1+\\xi \\theta ^{2}}{1+\\xi ^{2}\\theta ^{2}}\\right)^{1/2}.$ The period-averaged Poynting flux injected at the loop apex, $S_a$ , is then damped to give the flux as a function of height $S(z) = S_a~{\\rm {exp}}~\\left(- \\int _0^z \\frac{dz^\\prime }{L_D(z^\\prime )}\\right).$ The volumetric heating rate is then the change in Poynting flux with distance along the flux tube: $Q_{AW} = \\frac{dS}{dz}.$ A magnetic field strength $B(z)$ is imposed which depends on height as a function of pressure, $P(z)$ [53], with $B_{0}$ defined as the photospheric value (note, this is only used in calculating the wave damping, and is not updated in the hydrodynamic or radiation transfer solutions): $B(z) = B_{0}\\left(\\frac{P(z)}{P_{0}}\\right)^\\alpha .$ We choose $\\alpha = 0.139$ as in [47] and [45].",
"$B(z)$ is constant with time, and at each timestep is interpolated to the updated grid.",
"The perpendicular wavenumber varies as a function of $B(z)$ due to variations in the cross-section of the flux tube.",
"In this work we use the relation: $k_x(z) = k_{x,a}\\left(\\frac{B(z)}{B_a}\\right).$ where subscript $a$ denotes values at the loop apex.",
"This linear scaling is found when the magnetic field expands in one dimension, as in a magnetic arcade.",
"An alternative two-dimensional expansion leads to a square root dependence however comparison of simulations for both geometries [45] suggests that the conclusions of AW heating studies depend only weakly on the choice of geometrical scaling.",
"Flares are simulated with user inputs: $f$ , $k_{x,a}$ , $B_0$ & $S_{a}$ .",
"Currently $S_a$ can be varied as a function of time, and future work will allow $f$ , and $k_{x,a}$ to vary in time also."
],
[
"Simulations",
"We model a 10 Mm flux tube extending from below the photosphere ($z=0$ defined where $\\tau _{5000} = 1$ ) into the corona at temperature $T=1$ MK.",
"The pre-flare atmosphere is the QS.SL.LT model atmosphere discussed in [3].",
"This is the PF2 atmosphere used by [1] and [2], modified to include the XEUV backwarming of [3].",
"The PF2 atmosphere was originally created by adding a TR and corona to the [8] radiative equilibrium atmospheric model.",
"Non-radiative heating is applied to maintain the photospheric and coronal energy balance in grid cells with column mass greater than $7.6$ g cm$^{-2}$ (photosphere) and less than $1\\times 10^{-6}$ g cm$^{-2}$ (corona).",
"We use a fixed boundary condition in the sub-photosphere and a reflecting boundary condition at the top of the loop, to mimic the effect of disturbances from the other half of the flux tube.",
"Two simulations are compared here, one in which the flare energy transport mechanism is a non-thermal EB (F11) and one in which the energy transport is via AW dissipation (S11).",
"Both have the same injected energy flux of $10^{11}$ ergs cm$^{-2}$ s$^{-1}$ , which is constant for $t =10$ s, which is representative of the `dwell time' of a flare footpoint at a particular chromospheric position.",
"The additional EB simulation parameters are: $\\delta = 5$ and $E_c = 25$ keV.",
"The additional AW simulation parameters are: $f = 10$ Hz, $k_{x,a} = 4\\times 10^{-4}$ cm$^{-1}$ and $B_0 = 1000$ G. Figure REF shows the evolution of the atmosphere in each of the simulations, where F11 refers to the EB simulation and S11 the AW simulation.",
"The colour of the lines represents the time in the simulation, where we plot from $t = [0, 10]$ s in 0.5 s intervals.",
"The temperature, electron density, velocity, H ion fraction, He ii ion fraction and flare heating rate are shown.",
"In each case the lower panel shows the S11 atmosphere and the upper panel shows the F11 atmosphere.",
"We discuss features of the dynamics below.",
"We also show the energetics of the simulations at various times in Figure REF (F11) and Figure REF (S11), where positive quantities are heating and negative are cooling.",
"Figure: The evolution of the atmosphere for different energy transport mechanisms F11 (EB) and S11 (AW dissipation) where (a) & (b) show temperature, (c) & (d) show electron density, (e) & (f) show velocity with upflow negative, (g) & (h) show hydrogen ionisation fraction, (i) & (j) show He ii ionisation fraction, and (k) & (l) show flare heating rate per mass.",
"Note that the x-scale of the atmospheric velocity is larger than for the other panels, so as to show the large velocity achieved high in the loop.",
"Colour represents time with output plotted at 0.5 s intervals."
],
[
"EB Simulation (F11)",
"t$<$ 1 s: The temperature in the mid-upper chromosphere increases significantly over the background, to $T\\approx [6000-7000]$ K between $0.6-1$ Mm and $\\approx [40,000-85,000]$ K over $1.15-1.5$ Mm.",
"In the lower atmosphere, flare energy largely goes into ionisation of hydrogen, which becomes completely ionised at $>1$ Mm, and partially ionised between $0.5-1$ Mm.",
"Enhanced ionisation means a significantly increased electron density between $0.5-1.6$ Mm (and by more than three orders of magnitude at 1 Mm).",
"Helium ionisation also occurs at greater depths, with He ii quickly forming between $\\sim 1.05-1.6$ Mm.",
"In the upper atmosphere, just below the original TR position, T increases to $\\approx [85,000-90,000]$ K, so that the He ii fraction decreases again at $\\sim 1.5$ Mm as He iii starts to form.",
"A pressure wave starts at the TR ($1.6$ Mm) resulting from the sudden temperature increase to approximately $10^5$ K, producing an upward mass motion with a velocity of more than 50 km s$^{-1}$ , increasing with height.",
"Figure REF (a) & (b) show that beam energy input is mostly balanced by radiative losses.",
"t = 1-4 s: Energy input into the lower chromosphere at $0.6-1$ Mm largely results in increased hydrogen ionisation causing the temperature plateau to only very slowly increase in temperature.",
"The plateau extends to deeper layers, and electron density increases with further ionisation.",
"The transition from $T\\approx [7000-40,000]$ K, at $1-1.15$ Mm, steadily steepens.",
"At $1.15$ Mm the temperature increases by a few $\\times 10^4$ K to $T\\approx 60,000$ K, but radiative losses largely balance (and occasionally exceed) energy input between $1.15-1.4$ Mm meaning that the temperature changes little, and actually decreases at $\\sim $ 1.4 Mm.",
"Radiative losses decrease with time above this height and are no longer able to balance energy input, resulting in a temperature bubble in excess of $T=200,000$ K. Figure REF (c) illustrates the energetics at this time.",
"Within this bubble temperatures are high enough to almost completely ionise He to He iii.",
"Above $1.6$ Mm the temperature continues to increase but not smoothly.",
"Loop density is enhanced there by strong upflows ($v\\sim 150$ km s$^{-1}$ ), so the beam deposits more energy at greater height.",
"A strong conductive flux helps to increase temperature $> 2$ Mm.",
"t = 4-7.5 s: Conditions at heights $<~1.15$ Mm continue to evolve in a similar manner to previously.",
"The peak of the EB heating rate moves slightly higher, to $1.18$ Mm.",
"Losses are just unbalanced at this point allowing temperature to rise to $T=85,000$ K. Losses are able to balance, and at times exceed, energy input between $\\sim 1.2-1.35$ Mm resulting in a drop in temperature.",
"There is a corresponding drop in electron density as recombinations to He ii take place.",
"Note also at this time the amount of He iii in the mid-chromosphere around $1.18$ Mm increases due to high temperatures, so that a narrow region of almost fully ionised He begins to form.",
"Initially, the hot bubble at heights $> 1.4$ Mm is smoothed out as it is heated to $>400,000$ K, due to a conductive flux into the cooler material ahead of the bubble, which increases the temperature in those regions.",
"However, increased temperature at $\\sim 1.5$ Mm leads to an increased pressure which drives material away, making a narrow, under-dense region.",
"Radiative losses decrease as a result of decreased density allowing temperature to increase further.",
"Immediately ahead of this under-dense region is a locally over-dense region which, due to increased radiative losses, forms a local temperature minimum.",
"t$>$ 7.5 s: The final stage in the evolution is the formation of the large temperature bubble in the mid-upper chromosphere.",
"High chromospheric temperatures ionise a large proportion of the He ii to He iii at $\\sim ~1.2$ Mm.",
"Decreasing radiative losses from He ii can no longer balance the beam energy deposition, which produces an ever-increasing temperature at that location (in excess of $1.5$ MK), and further ionisation.",
"Figure REF (d) shows the decrease in radiative losses allowing temperature to quickly rise.",
"This high temperature bubble is very under-dense as material is pushed away by the strong pressure difference between the bubble and surrounding plasma, increasing the size of the high temperature region (Figure REF (e)).",
"Chromospheric condensations are much stronger at these times, reaching up to $v\\sim 45$ km s$^{-1}$ .",
"Since more mass is evaporated into the loop the heating rate again increases at greater heights, increasing the temperature of the corona and pushing the TR upwards.",
"The increase in density on either side of the bubble (material evacuated from the high pressure region) results in strong radiative losses that exceed energy input, creating very narrow, cool regions that permit recombination to He ii.",
"These regions propagate away from the shock.",
"Figure: Energy balance in the EB simulation.",
"Contributions to the energy balance are shown: total (black), viscous heating (red), work done by pressure (yellow), optically thick radiation computed in detail (green), optically thin radiation (blue), conductive flux (purple), the background heating function (black, dotted) and the flare heating rate (red, dashed).",
"Positive represents heating, and negative cooling.",
"Panel (a) shows that by t=0.25t = 0.25 s radiative losses effectively balance energy input in the lower atmosphere, but are unable to balance beam heating in the mid-upper chromosphere.",
"By t=1t = 1 s, however, strong optically thick losses balance beam energy.",
"At this time the pressure from enhanced temperature in the upper chromosphere results in upflows ∼1.7\\sim 1.7 Mm.",
"Panel (c) illustrates the decrease in temperature around ∼1.4\\sim 1.4 Mm at t=3.5t= 3.5 s and that conductive flux helps to increase the temperature in the upper atmosphere.",
"Panel (d) shows the onset of the high temperature bubble in the mid-chromosphere.",
"Radiative losses limit the temperature rise at this time, but as panel (e) shows, high temperatures increases ionisation in this region removing losses from He ii allowing the explosive temperature rise to >1>1 MK.Figure: Energy balance in the AW simulation.",
"Lines are as described in Figure .",
"Panel (a) shows the sudden energy input results in a large amount of unbalanced energy to heat and ionise the plasma.",
"Panel (b) shows that very quickly this energy input is well balanced by radiative losses in the lower atmosphere, and that the large temperature enhancement results in a strong pressure wave leading to high velocity upflows.",
"Panel (c) shows that losses exceed energy input at ∼1.4\\sim 1.4 Mm, leading to a decrease in temperature, and that a pressure wave pushes hot material upwards.",
"Panel (d) shows that a hot bubble begins to form, with the energy balanced immediately above the high temperature region at ∼1.5\\sim 1.5 Mm, and that a strong conductive flux is present in the upper atmosphere.",
"Panel (e) shows the complex dynamics of the temperature bubble.",
"The upward propagating pressure wave has made the bubble under-dense, decreasing the heating rate, but also reducing radiative losses so that temperature increases greatly here.",
"Immediately above the bubble is a local temperature minimum."
],
[
"Alfvén Wave Simulation (S11)",
"The lower panel of each atmospheric property in Figure REF shows the evolution of the atmosphere in response to AW dissipation.",
"Immediately clear from these figures is that the atmosphere evolves in a largely similar manner to the EB flare simulation, but also that the extreme temperature, low density bubble does not form in the mid chromosphere.",
"t $<$ 1 s: The location of the peak of the heating rate (Figure REF (l)) is within $\\sim 0.1$ Mm of the peak in the EB heating rate, but the peak is broader.",
"By $t=1$ s the temperature between $0.7-0.95$ Mm rises to $T\\approx [6000 - 7500]$ K. Mid-upper chromospheric temperatures show enhancements over pre-flare values, rising over a shallow gradient from $T\\sim 7500$ K at 0.95 Mm to $T\\sim 200,000$ K at 1.55 Mm.",
"The TR temperature is increased to $>1$ MK.",
"This initial enhancement to the mid-upper chromospheric temperature occurs rapidly, with radiative losses almost completely balancing energy input up to $\\sim 1.3$ Mm by $t=0.25$ s. Figure REF (a,b) illustrates the energy balance at these times showing that following the rapid ionisation the radiative losses increases sufficiently to mostly balance flare energy input.",
"Hydrogen is almost entirely ionised above 1 Mm, and ionisation continues gradually to greater depth (Figure REF (h)).",
"The elevated temperature in the mid-chromosphere results in ionisation of He, which is mostly ionised to He ii between $1.05-1.3$ Mm and to He iii above $1.4$ Mm, leading to an increase in the electron density between $0.9-1.6$ Mm.",
"The peak in $n_e$ occurs at the hydrogen ionisation boundary (between $0.9-1$ Mm), with $n_e\\sim 4\\times 10^{13}$ cm$^{-3}$ .",
"By $t=0.25-0.5$ s a strong pressure wave at $1.65$ Mm pushes chromospheric material into the corona with velocities in excess of $v\\sim 130$ km s$^{-1}$ , significantly higher than in the EB simulation.",
"Since the AW heating rate decreases sharply around 0.9 Mm, the ionisation of hydrogen below this occurs at later times in comparison to the EB simulation.",
"t = 1-5 s: Between $0.6-0.9$ Mm radiative losses almost completely balance the energy input, and so temperatures rise only modestly, to $T\\sim 8000$ K. Hydrogen ionisation also increases, creating a small region of locally high electron density $n_e = 1-1.5\\times 10^{13}$ cm$^{-3}$ at $\\sim 0.75$ Mm.",
"In the mid-chromosphere a temperature `pivot' point forms at $\\sim 1.15$ Mm, with temperature decreasing with time above this point, and increasing below.",
"As the temperature increases in the deep atmosphere, ionisation to He ii follows producing a small electron density increase at $1-1.05$ Mm.",
"The associated pressure changes results in upflows of a few $\\times 10$ km s$^{-1}$ .",
"The initial high velocity upflow reaches heights above $2.5$ Mm, with $v\\sim 200$ km s$^{-1}$ .",
"The temperature gradient between $1.15-1.45$ Mm flattens slightly from $T\\approx [35,000 - 90,000]$ K to $T\\approx [30,000 - 70,000]$ K. This occurs because hot plasma pushed up into the loop at a few $\\times 10$ km s$^{-1}$ , due to an increase in pressure above 1.15 Mm, leaves cooler material in its place, and because radiative losses above $1.25$ Mm begin to exceed the energy input which leads to cooling (see FigureREF (c)).",
"A narrow high temperature ($T\\sim 100,000$ K) bubble begins to form at $\\sim 1.4-1.5$ Mm.",
"The TR heats as a result of energy deposition and via a conductive flux from below.",
"Figure REF (d) shows the decrease in radiative losses that allow the formation of the high temperature at $1.4$ Mm, and the upwards propagating pressure wave.",
"t = 5-10 s: For the remainder of the simulation the dynamics evolves as before.",
"The atmosphere cools slightly between $1.25-1.4$ Mm, narrowing the high temperature bubble around $1.4-1.5$ Mm, making it more pronounced.",
"This results in a greater pressure difference that increases the flows (see Figure REF (f)), leading to an under-dense region (similar to the process that resulted in the high-altitude temperature bubble in the EB simulation).",
"The bubble cools slightly, but remains hot since, despite the heating rate in the bubble being reduced because of the lower density, the density change also significantly reduces radiative losses around 1.5 Mm.",
"Increased density ahead of the bubble leads to enhanced radiative losses and decreasing temperature, producing a local temperature minimum.",
"Figure REF (e) illustrates the energetics at this time."
],
[
"Ca ",
"The Ca ii 8542 Å line is part of the Ca ii subordinate infrared (IR) triplet, which are sensitive to the temperature at their formation height in the low chromosphere, and to magnetic structures, making them good tracers of solar and stellar activity [49], [35], [51].",
"Since this line is so sensitive to lower chromospheric temperature, and will be observed with high spatial and spectral resolution by DKIST, we investigate the differences in the spectral line profiles, and their formation properties, between our EB and AW simulations.",
"Figure REF (a) shows how the Ca ii 8542 Å line responds to flare energy input in the EB (F11) simulation.",
"Colour refers to simulation time (note that we plot $t = [0,~0.072,~0.25,~0.302,~0.5]$ s and then every 0.5 s thereafter).",
"The inset shows a zoom of the core, and the vertical dashed line indicates the rest wavelength.",
"The quiet Sun profile is in absorption but the line core immediately goes into emission in response to beam energy input.",
"Between $t=0.072$ and $t=0.25$ s the core intensity drops significantly, but the far wing intensity continues to rise (note that on Figure REF (a) we plot symbols for the $t=0.072$ s profile, since the colours at early times are very similar).",
"Over the next few seconds the core intensity increases again, reaching a peak of $\\sim 3.27\\times 10^{6}$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ at $t = 7.6$ s followed by a small decrease.",
"The core intensity changes little after $t\\sim 3$ s, but the wing intensity shows a strong enhancement.",
"The far wing intensity is initially $\\sim 1\\times 10^6$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ , rising to $\\sim 1.35\\times 10^6$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ .",
"The line was initially narrow and symmetric but over time a slight blueshift develops in the line core and the profile widens.",
"Figure: The Ca ii 8542 Å line, computed in (a) the EB simulation and (b) the AW simulation.",
"In both cases colour represents time, and the inset shows a closer view of the line core.",
"In panel (a) symbols are over-layed on the profile at t=0.072t=0.072 s to help it stand out against profiles from t<1t<1 s The equivalent is shown for the AW (S11) simulation in Figure REF (b).",
"We plot $t= 0, 0.064, 0.25, 0.5$ s and then every 0.5 s thereafter, and include an inset of the line core.",
"it is clear that the AW simulation produces profiles with a stronger asymmetry.",
"The line is slower to go into emission than in the EB simulation, and the intensity increases slowly over time.",
"Since the line is optically thick the peak intensity may not be the core of the line.",
"Instead we consider the line centroid (the centre of mass of the line).",
"The intensity of the line is lower than in the EB simulation, reaching a maximum line centroid intensity of $\\sim 3.16\\times 10^{6}$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ .",
"The peak intensity of the profile is located redward of the line centroid and peaks at $t\\sim 3.5$ with a value of $\\sim 3.22\\times 10^{6}$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ .The final intensity in the far wings is also lower than the EB simulation.",
"The line appears redshifted initially, but this decreases with time.",
"The final state is a small red asymmetry."
],
[
"Ca ",
"We can study the formation properties of the line in each simulation, by writing the formal solution of the radiative transfer equation for the emergent intensity as in [6]: $I_{\\nu } = \\frac{1}{\\mu }\\int _{z_0}^{z_1}S_{\\nu }~\\tau _{\\nu }e^{-\\tau _{\\nu }/\\mu }~\\frac{\\chi _{\\nu }}{\\tau _{\\nu }}~dz = \\frac{1}{\\mu }\\int _{z_0}^{z_1} C_{\\rm {I}}~dz,$ where $C_{\\rm {I}}$ is the contribution function to the emergent intensity and indicates how much emergent intensity originates from a certain height.",
"In Eq REF , ${\\mu }$ is the viewing angle, and the terms are a function of frequency $\\nu $ .",
"Since CRD is assumed, the source function, $S_{\\nu }$ (the ratio of emissivity to opacity), is independent of frequency across the line.",
"The term $\\tau _{\\nu }e^{-\\tau _{\\nu }/\\mu }$ describes the attenuation by the optical depth, $\\tau _{\\nu }$ .",
"The monochromatic opacity per unit volume, $\\chi _{\\nu }$ , is proportional to the density of emitting particles, so that the term ${\\chi _{\\nu }}/{\\tau _{\\nu }}$ is higher when there are large number of emitters at low optical depth (i.e photons are produced and can escape).",
"This is sensitive to mass motions, and shows velocity gradients.",
"Some example snapshots are shown in Figures REF & REF to illustrate the line formation.",
"In these figures the background images show the components of the contribution function, ${\\chi _{\\nu }}/{\\tau _{\\nu }}$ (top left), $S_{\\nu }$ (top right) & $\\tau _{\\nu }e^{-\\tau _{\\nu }/\\mu }$ (bottom left), and the contribution function itself, $C_{\\rm {I}}$ (bottom right).",
"These are inverse scale so that dark regions show large values of each term.",
"The $C_{\\rm {I}}$ images are normalised within each wavelength bin to better show the contribution to the wing intensity (which is much lower intensity than the line core).",
"These are shown as a function of wavelength and height, with wavelength expressed as a Doppler velocity, redshifts being positive.",
"The lines shown on each figure are the atmospheric velocity (blue, dashed), where positive is downflow, the $\\tau _{\\nu }=1$ curve (red, dashed), the line source function (green, dot-dashed), the Planck function (purple, dot-dashed), the Planck function at $t=0$ s (purple, dotted), and the emergent intensity (yellow, solid).",
"The source function, Planck function and emergent intensity are expressed in units of radiation temperature.",
"If the line is optically thick then the contribution to the emergent intensity originates from near the $\\tau _{\\nu }=1$ height.",
"Figure: Ca ii 8542Å line formation in the EB (F11) simulation at different times as indicated.",
"Each panel shows the image of the quantity labelled in the corner of the image.",
"Images are inverse scale.",
"The atmospheric velocity (blue, dashed), τ ν =1\\tau _{\\nu }=1 curve (red, dashed), line source function (green, dot-dashed), Planck function (orange, dot-dashed), Planck function at t=0t=0 s (orange, dotted), and emergent intensity (yellow, solid) are also plotted.",
"Positive velocity is redshift/downflow, and intensity is expressed in units of radiation temperature.",
"In the bottom right panels we have normalised the contribution function at each wavelength so that the image is not dominated by the line core, allowing details from the wings to be visible.Electron Beam model t = 0-0.25 s: see Figure REF (a).",
"The very high temperature where the beam energy is deposited causes a reduction in the population of the Ca ii 8542Å upper levels above 1 Mm.",
"The $\\tau _{\\nu } = 1$ layer forms below this at $\\sim 0.9$ Mm where there is an increase in Ca ii 8542Å upper level populations.",
"The contribution function peaks here with an additional small (optically thin) contribution to the line core, from between $0.9 - 1.1$ Mm.",
"The line source function is strongly coupled to the Planck function up to the core formation height (Figure REF (a), top left panel), which has increased due to enhanced temperature, and so the line core is in emission.",
"Moving away from the core, the wings are formed progressively deeper in the atmosphere.",
"Far wings are formed at only 0.2 Mm above the photosphere.",
"As time proceeds (not shown) the rapid increase in temperature at even greater depths starts to depopulate the upper levels further, driving down the formation height of the line core to $\\sim 0.75$ Mm.",
"The line source function at this altitude is still strongly coupled to the Planck function, which is smaller, so the emergent intensity decreases.",
"The line core still has an optically thin contribution from $0.75-1.1$ Mm.",
"t = 0.25-1.25 s: see Figure REF (b).",
"In the region $\\sim 0.8-1$ Mm the upper level of Ca ii begins to repopulate, and the $\\tau = 1$ height moves upward to $\\sim 0.85$ Mm where it remains for the rest of the simulation.",
"The line source function couples to the Planck function at ever increasing heights (up to 1 Mm), increasing the intensity of the optically thin component.",
"The source function at the core formation height increases significantly compared to only a small change in the wings, so the profiles look very narrow at these times.",
"A small downflow develops at $1.1$ Mm, causing an increase to the $\\chi _{\\nu }/\\tau _{\\nu }$ term redward of the line core.",
"This provides an additional small, optically thin contribution to the red wing, and marks the start of the red asymmetry in the line profile.",
"t = 1.5-6 s: see Figure REF (c).The upper level is repopulated across a wider range of heights so that emission from the near wings originates from higher layers, and intensity increases.",
"The $\\tau =1$ curve widens, increasing line width.",
"The condensation has increased in magnitude and moved lower in the atmosphere, making the optically thin contribution to the red wing more pronounced.",
"A small upflow at the line centre formation height ($\\sim $ 0.85Mm) blueshifts the core.",
"t = 6-8 s: The peak formation height is still $\\sim ~0.85$ Mm and since the source function is still strongly coupled to the Planck function, it changes only little.",
"Emergent intensity decreases slightly, largely due to the condensation which continues to move deeper in the atmosphere, adding more to the red wing and less to the line core as time progresses.",
"t = 8-10 s: As the under dense, high temperature bubble forms at $\\sim 1.2$ Mm it expands, starting a second (much larger) condensation.",
"There are not enough emitters at the height of the condensation to have much effect on the Ca ii 8542Å line, but the condensation does compress the atmosphere, resulting in a small increase to the optically thin emission in the red wing, increasing the asymmetry.",
"AW Model t = 0-0.25 s: (not shown) unlike in the EB simulation, at this time the line is still in absorption, as the source function is not strongly coupled to the Planck function, but decreases above $0.75$ Mm.",
"The electron density is much lower compared to the EB simulation since there is initially less H ionisation below 0.95 Mm at early times.",
"There is, however, a narrow region in the mid-chromosphere where there is a sufficient population of Ca ii to produce an optically thin component to the contribution function near 1.1 Mm.",
"t = 0.25-1.25 s: see Figure REF (a).",
"The temperature increase at greater depths depopulates the Ca ii 8542Å upper level between 0.65 - 0.9 Mm driving the formation height down to around 0.65 Mm, lower than in the EB simulation.",
"The lower temperature at this height compared to the formation height of the line in the EB simulation means a smaller line intensity.",
"The source function where the line forms is now more strongly coupled to the Planck function, so the line is in emission.",
"An upflow above 1 Mm shifts the absorption profile to the blue, meaning more blue-wing than red-wing absorption.",
"This creates a small but growing asymmetry.",
"In addition, the strength of the optically thin contribution increases, and is stronger on the red side of the profile due to a small condensation.",
"Together, these features have the effect of making the emergent profile appear redshifted.",
"Although the optically thin emission originates from a much narrower layer than in the EB simulation, the AW simulation has a somewhat higher temperature in this region, (the difference is $\\approx 2000$ K).",
"This raises the intensity of the optically thin component compared to in the EB simulation.",
"t = 1.25-6.5 s: see Figure REF (b,c).",
"Beginning around $t=2$ s the upper-level populations increase, raising the height of the $\\tau _{\\nu }=1$ layer.",
"Figure REF (b) shows the start of this process.",
"By $t=3.5$ s populations have increased enough that the location of the $\\tau _{\\nu }=1$ layer is raised and the line width increased.",
"The line core is slightly blueshifted, but the optically-thin redshifted component arising from the condensation means that the line is further broadened and peaks in the red.",
"By $t=6.5$ s the $\\tau _{\\nu } = 1$ height has risen to around 0.81 Mm, increasing the intensity of the whole line as it is formed in a region of higher temperature.",
"It takes longer for upper level populations in the AW simulation to reach a similar state to the EB simulation, meaning that the intensity increase is slower.",
"This occurs due to a difference in the length of time it takes for recombinations to increase the amount of Ca ii.",
"In both simulations the fraction of Ca iii to Ca ii increases significantly to large depth.",
"The ratio of $n_{\\rm {Ca~\\textsc {iii}}}/n_{\\rm {Ca~\\textsc {ii}}}$ is $\\sim 50\\%$ between 0.5-0.6 Mm, increasing quickly over a narrow height range so that calcium is almost all ionised to Ca iii above 0.6 Mm.",
"In the EB simulation, recombinations to Ca ii between $z\\sim 0.7-0.9$ Mm take place early ($t\\sim 1$ s) so that the Ca ii 8542Å upper level is subsequently populated at that height.",
"In contrast, the decreased electron density in the AW simulation relative to the EB simulation, at these heights, means that recombinations to Ca ii do not occur as quickly.",
"The electron density in the AW simulation begins to increase around $t\\sim 2.25$ s, which in turn allows recombinations, and for the Ca ii upper level to become populated over the next few seconds.",
"t = 6.5-10 s: There is little change over the remainder of the simulation, other than an intensity increase in the red wing as the density increases and the source function couples strongly to the Planck function.",
"Similarly to the EB simulation, the small condensation extends further into the wing versus the line core towards the end of the simulation, reducing the extent of the red peak, and broadening more of the red wing (Figure REF (d)).",
"Figure: Ca ii 8542Å line formation in the AW beam (S11) simulation at different times as indicated.",
"Lines are as described in Figure"
],
[
"Mg ",
"The Mg ii h & k resonance lines in the quiet Sun are formed over a wide range of chromospheric heights.",
"They usually appear as doubly peaked profiles with a central reversal.",
"The red and blue peaks of the k line are referred to as the k2r and k2v component, and the centrally reversed core as the k3 component.",
"[26] and [36] recently discussed these lines as observed in a solar flare.",
"They appeared as redshifted, single peaked profiles with a blue asymmetry at times of strongest redshift.",
"RADYN uses the assumption of CRD when computing line profiles but it has been shown by [33] that this is not valid for Mg ii.",
"Therefore we use the output RADYN atmospheres every 0.25 s as input to the RH radiative transfer code [50], which does use PRD, to synthesise the Mg ii spectra, using a 22-level Mg ii model atom.",
"RH can perform the PRD radiative transfer with either the fast approximation/Hybrid PRD scheme of [32] or with full angle-dependence.",
"Full angle-dependent PRD takes into account that in an atmosphere with strong velocity gradients the radiation field is non-isotropic, and requires computing the angle-dependent redistribution function (costly both in time and memory).",
"Instead it is possible to use the assumption of angle-averaged PRD and include transforms to/from the rest frame of particles (see [32] for details), to save on computational time whilst obtaining a good approximation to full angle-dependent PRD.",
"We performed some tests of full angle-dependent PRD compared to the Hybrid PRD finding that angle-dependent computations took significantly longer with little difference in the emergent profile.",
"We therefore used the Hybrid PRD.",
"Figure REF (a) shows the EB Mg ii k profiles, and Figure REF (b) shows the AW Mg ii k profiles.",
"In the EB simulation the line profiles have an obvious central reversal at all times.",
"This quickly becomes shallower, and the whole line becomes more intense.",
"From $t=0.25-0.5$ s k2r is stronger than k2v and the line reaches its maximum intensity.",
"Over the next few seconds of the simulation the line core appears redshifted and kr2 decreases so that the k2r and k2v are largely symmetric.",
"The blue wing develops an enhancement that moves steadily more blueward, making the line asymmetric.",
"Between $t=4-5.5$ s k2r is again stronger than k2v, and the redshift of the line core becomes smaller.",
"By $t=7$ s k2r and k2v have roughly equal intensity.",
"By the end of the simulation a strong enhancement to the red between 0.15 and 0.5 Å from line center has developed, and a weaker enhancement to the blue wing at $\\sim $ 0.75Å.",
"The line profiles in the AW simulation are very different from the EB profiles.",
"Before $t=0.25$ s the profiles are very similar, though the AW profiles are more intense (AW k2r intensity is $I_{k2r} = 21.0\\times 10^7$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ , and for EB $17.0\\times 10^7$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ ).",
"By $t = 1.25$ s, kr2 is very strong compared to k2v, and the line core is slightly blueshifted.",
"In relation to k2r, k2v continues to decrease.",
"The maximum intensity of k2r is at $t=2.25$ s, with $I_{k2r} = 22.5\\times 10^7$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ (compared to a maximum intensity in the EB simulation of $I = 19.0\\times 10^7$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ ).",
"At this point the central reversal is very shallow, the k2v is very weak, and k2r dominates.",
"By $t=5$ s the k2v decreases so much that it becomes difficult to discern its presence.",
"Instead the profile appears single peaked, with an extended blue wing or shoulder, and a wider blue wing between 0.1Å and 0.2Å blueward of line centre, than the red wing."
],
[
"Mg ",
"The computation of the Mg ii k line formation using RH differs from Ca II 8542Å using RADYN because using RH with PRD means that the source function is frequency-dependent and varies across the profile.",
"Figures REF & REF show the formation of the line in the EB and AW simulation respectively.",
"Four snapshots are shown, with the panels and lines as described in Section REF .",
"EB Model t = 0.25 s: The line core formation height drops from its pre-flare location just below the TR to $\\sim 1.1$ Mm, and k2r and k2v form slightly lower at $\\sim 1.05$ Mm.",
"This drop in formation height occurs because the Mg ii k upper level becomes depopulated above $\\sim 1.15$ Mm, due to heating, while populations around 1.1 Mm increase.",
"The central reversal occurs because at the k3 formation height, the source function has decoupled from the Planck function, and is decreasing with increasing height, whereas at the k2r and k2v peak formation height the source function is more strongly coupled to the Planck function, giving a higher intensity.",
"However, the high temperatures and density produces conditions such that the difference in formation height between k3 and k2r,v components is fairly small, and so the depth of the reversal feature is very much smaller than it is in the quiet Sun.",
"The line wings form deeper in the atmosphere, from $\\sim 0.7 - 1$ Mm in the wavelength range shown.",
"t = 0.25-1 s: see Figure REF (a).",
"A small downflow has formed immediately above the core formation height, which increases the number of emitters contributing to the red wing relative to the blue wing.",
"This widens the k2r peak and increases the intensity of k2r compared to k2v.",
"t = 1-3 s: see Figure REF (b).",
"The downflow develops and moves deeper, redshifting the core and the emission peaks.",
"The peak of the opacity is also in the red (see upper left panels in Figure REF ) so that red wing photons produced below the condensation are absorbed more than blue wing photons, steepening the extinction profile of the red wing.",
"Between $\\sim 1.1-1.2$ Mm an upflow moves some emitters upwards to locations at smaller optical depths, so that extra blue-wing photons are emitted from 0.10 - 0.15 Å bluewards of the rest wavelength.",
"t = 3-7 s: see Figure REF (c).",
"As the downflow moves deeper into the atmosphere it slows and the redshift of the core decreases.",
"The k2r peak is sometimes formed slightly higher (and is slightly more intense) than the k2v peak, but when the overall redshift becomes smaller this height difference also reduces.",
"As the upflow speed increases the contribution function for optically thin blue-wing emission is pushed further out, to around 0.15 - 0.20 Å from the core.",
"It originates from a height of $\\sim 1.3-1.4$ Mm t = 8-10 s: see Figure REF (d).",
"A hot bubble has formed at $\\sim 1.2$ Mm, creating a large condensation that travels downwards.",
"The flow associated with the bubble does not reach the core formation height, so has little effect on the k3 or k2 components, but it does result in peaks appearing in the red and blue wings.",
"The condensation creates a very narrow layer of enhanced electron and mass density.",
"The population of the Mg ii k upper level increases at the condensation height, resulting in a strong red-shifted source function.",
"The lower left panel of Figure REF (d) shows that the emitters in the condensation increase the attenuation of red wing photons, meaning that their contribution is almost exclusively from the condensation.",
"The steep velocity gradient results in a bump in the red wing at 0.15 - 0.45 Å from the rest wavelength.",
"Similarly, the upflow results in emission between $\\sim 0.60 - 0.80$ Å blueward of the rest wavelength.",
"Figure: The formation of the Mg ii k line in the EB simulation at four timesteps, as indicated on each panel.",
"The lines are as described in Figure .",
"Note the different scales used in (a).Figure: The formation of the Mg ii k line in the AW simulation at four timesteps, as indicated on each panel.",
"The lines are as described in Figure .",
"Note the different scales used in (b).AW Model t = 0.25 s: see Figure REF (a).",
"A shallow upflow is present in the chromosphere, from around the formation height of the k line at $\\sim 1.125$ Mm, to 1.55 Mm, resulting in a slightly blueshifted line core even at early times.",
"Above this a much stronger upflow carries chromospheric material into the TR.",
"The core is formed around $1.1-1.15$ Mm, and is centrally reversed, with a deeper reversal than the EB simulation as it is more decoupled from the Planck function.",
"The k2 components are symmetric about the core, and are formed at $\\sim 1.05$ Mm.",
"Since the AW atmosphere is hotter between 0.9-1.1 Mm, the k2 components are more intense than their EB counterparts.",
"The core forms higher than in the EB simulation since the latter is hotter above 1.1 Mm, depopulating the upper level of the k line more than in the AW simulation.",
"t = 0.25-2.0 s: see Figure REF (b).",
"The core becomes blueshifted as the $\\tau =1$ surface moves with the upflow.",
"The k2v and k3 formation height move closer.",
"The k2r peak is formed slightly lower, and is more coupled to the Planck function so more intense that the k2v peak.",
"The k2r peak appears wider than the k2v peak because the extinction on the blue side is higher - more photons are absorbed by the shifted opacity peak.",
"t = 2.0-5.0 s: see Figure REF (c).",
"The upflow steepens between 1.1 and 1.3 Mm, and results in blueshifted optically thin emission between $\\sim $ 0.10 - 0.15 Å blueward of the rest wavelength.",
"It is difficult to distinguish the core from the k2v peak.",
"The theoretical core is defined as the maximum of the $\\tau =1$ curve over the line, but by $t=5$ s this occurs at roughly the same formation height as the k2v peak.",
"This may be due to the upflow, which extends over a wide height range above the core formation height.",
"The opacity peak is shifted to the blue so that blue photons are preferentially absorbed relative to red, producing a red asymmetry.",
"t = 5.0-10.0 s: see Figure REF (d).",
"The line profile does not change very much, but features become more extreme.",
"The theoretical core position becomes more blueshifted, the optically thin component extends further into the blue wing, and the profile appears singly peaked with a blue `shoulder'.",
"The k2r peak is very intense, formed in a region of high temperature and electron density, where the source function couples to the Planck function.",
"Observationally k2r might be confused with the line core since the central reversal has all but vanished.",
"A small condensation near the k2r formation height broadens this feature."
],
[
"Summary",
"In the simulations presented, we find substantial differences in the atmospheric structure and flows in the AW simulations compared to the EB simulations.",
"These differences result primarily from the different heating profiles.",
"In the EB simulation, the heating is strongly concentrated at the column depth corresponding to the stopping depth of the electrons at the beam cut-off energy.",
"In the AW simulation the heating profile for the monochromatic wave is flatter.",
"Both simulations result in fast low-density upflows in the upper chromosphere.",
"For the energy flux used ($10^{11}\\rm {ergs~cm^{-2}~s^{-1}}$ ) the localised, strong heating in the EB simulation means that helium as well as hydrogen becomes very highly ionised, and the plasma has a reduced ability to radiate.",
"The resulting increase in temperature and pressure launches strong secondary, high-density, upflows and downflows in the mid-chromosphere.",
"A similar but much less pronounced process results in a secondary upflow in the AW simulation, but the secondary flows are much weaker in the AW simulation for the same energy flux.",
"Early in the EB simulation, the temperature in the lower chromosphere, from 0.6-0.9 Mm, is a few 100 K higher than in the AW simulation.",
"After a short time this reverses and the AW simulation is hotter.",
"The AW simulation is significantly hotter from $\\sim 0.9-1.1$ Mm, by $5,000-10,000$ K. In both models, H ionisation is complete above 1 Mm.",
"In the EB model, H ionisation is higher between 1 Mm and 0.7 Mm compared to the AW model, due to non-thermal ionisation from the electron beam.",
"After $\\sim 2$ s, ionisation at these depths in the AW simulation increases, and by the end of the simulation matches the EB simulation.",
"The electron density in the EB simulation is mostly higher than in the AW simulation.",
"These variations in the dynamic and thermodynamic properties of the atmosphere are reflected in the shape and variation of the lines in potentially distinguishable ways.",
"Ca ii 8542 Å goes into emission almost immediately after the heating starts in the EB simulation, and rises quickly to its peak emission.",
"In the AW simulation this takes a few seconds longer.",
"In the case of Mg ii k, the line peaks very quickly for both EB and AW heating.",
"For both Ca ii 8542 Å and Mg ii k, the line intensity increases with time as the locations of the peaks of the contribution functions tend to move down as deeper layers of the atmosphere heat.",
"Because the density in these deeper layers is higher, the emissivity is higher and the line source functions are more coupled to the Planck function.",
"The shape and intensity of the wings is determined by the deeper atmosphere.",
"Ca ii 8542 Å does not show a reversal in either the EB or AW simulation.",
"This is because the Ca ii 8542 Å source function and the Planck function are strongly coupled throughout the core formation region, so that the intensity increases with increasing temperature towards core formation heights.",
"In the EB simulation small upflows at the location of Ca ii 8542 Å core formation lead to a small blueshift.",
"A weak, optically thin redshifted component contributes in the red wing.",
"In the AW simulation the effect of redshifted, optically thin emission on the profile is more pronounced because the emission is stronger (higher temperature) and the downflow is larger.",
"In the EB simulation the core of Mg ii k has a central reversal.",
"The reversal occurs where the k3 source function has decoupled from the Planck function, leading to a drop in intensity compared to the k2v and k2r components which are formed where the source functions are more strongly coupled to the Planck function.",
"In the AW simulation, there is initially a Mg ii k central reversal, but over time the reversal becomes difficult to identify relative to the k2v peak.",
"The profile becomes quite asymmetric because the source function and the Planck function are strongly coupled in a region with small downflows, leading to a strong red peak, whereas the blue side of the line core is dominated by weaker, optically thin emission from upflowing plasma.",
"The line opacity is also primarily in the blue, increasing the asymmetry.",
"In the EB simulation there is very little optically thin Mg ii k emission near the line core, which remains quite symmetric.",
"However we do see hints of the effect of absorption of red photons by downflowing material and blue photons by upflowing material.",
"The far wings become enhanced by emission from the strong chromospheric condensation and upflow."
],
[
"Conclusions and Next Steps",
"We have presented the results of radiation hydrodynamics simulations of a solar flare in which the atmosphere is heated by a monochromatic Alfvén wave or by an electron beam with the same total energy flux and duration.",
"The electron beam calculation uses a well-established prescription for Coulomb heating and the treatment of Alfvén wave dissipation uses the WKB approximation which restricts it to waves of parallel wavelength less than the gradient scale length of the Alfvén speed.",
"Line profiles from two important chromospheric lines were synthesized for both simulations.",
"Our results show that Alfvén wave dissipation is effective in heating the chromosphere and producing the high velocity flows observed in solar flares; indeed the temperature around 1 Mm in the AW simulation is 5,000 - 10,000 K higher than in the EB simulation.",
"In most regards the dynamic response of the atmosphere is similar in the two cases, but the high-temperature shock in the mid-chromosphere that is a common feature of EB simulations is absent from the AW simulation.",
"This has a significant impact on the line profiles.",
"The Ca ii 8542Å profiles produced in both the AW and EB simulation are similar to some recent flare observations [46] but neither show the blue asymmetries observed by [29].",
"The AW simulation has a very small (0.05Å) redshift.",
"It takes slightly longer for the line intensity to increase in the AW simulation, but the delay we find here is likely to be an underestimate, as the transit time of the AW through the atmosphere is not captured in the simulation (the approximated form of AW energy input is a time average.)",
"The Mg ii k line profiles from the EB simulation do not provide a good match to observations, having a central reversal, and small amounts of broadening, redshift and asymmetry.",
"Observed lines are very broad, usually do not have a central reversal [26], [36] and have strong redshifts and asymmetries.",
"The Mg ii spectra from the AW simulation, after a few seconds of energy input, appear single peaked, redshifted and have an extended blue wing or shoulder, [26].",
"The `theoretical core' (peak of the $\\tau =1$ surface) is in fact formed in upflowing plasma, but more blue wing photons are absorbed than red wing photons, leading to the net red asymmetry [20].",
"This highlights the difficulty in interpreting observations of these optically thick lines.",
"We have used a radiation hydrodynamics treatment to demonstrate that Alfvén waves can in principle heat the solar chromosphere and lead to emission of important chromospheric flare lines.",
"This follows the results that Alfvén waves can heat the TMR and upper chromosphere [47], [45] including in flare simulations using a radiative loss function.",
"Dissipation of Alfvén waves therefore looks like a viable candidate for generating many of the observed flare chromospheric signatures.",
"We find that electron beam and Alfvén wave heating might be distinguishable via differences in the Mg ii line profiles and their evolution.",
"In our simulation of a single monochromatic wave, the more gradual heating profile of the Alfvén wave results in gentler flows and higher temperatures in the mid-chromosphere, producing single-peaked, asymmetric Mg ii k line profiles more similar to current observations.",
"However, we need to carry out a parameter study and also simulate the dissipation of a spectrum of waves.",
"Simulations containing both electron beam and Alfvén wave heating will also be instructive.",
"The chromospheric response to intense heating, even in a 1-D model, is complicated.",
"The shape of the emergent line profiles depends sensitively on the conditions - in particular the plasma flows - that arise at the line core formation heights.",
"From various published electron beam simulations [2], [25] strong upflows and downflows seem to be a common result of heating by a power-law beam with an energy flux in excess of $\\sim 10^{10}\\rm {ergs~cm^{-2}s^{-1}}$ because of the rapid ionisation of H and He in the chromosphere and the expansion that results.",
"It remains to be seen whether any such systematic behaviours arise from heating by Alfvén waves, which could aid in distinguishing the contribution of each.",
"Acknowledgments: The authors would like to thank Dr J. Reep for helpful discussions, and Dr J. Leenaarts for help with the RH code.",
"GSK would like to acknowledge the financial support of a PhD scholarship from the College of Science and Engineering, Univeristy of Glasgow.",
"LF acknowledges support from STFC consolidated grant ST/L000741/1.",
"The research leading these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement no.",
"606862 (F-CHROMA).",
"AJBR acknowledges support from STFC consolidated grant ST/K000993/1.",
"AJBR and LF acknowledge support from ISSI (Switzerland) for the International Team on `Magnetic Waves in Solar Flares: Beyond the “Standard\" Flare Model'.",
"JCA acknowledges funding support through the NASA Heliophysics Supporting Research and NASA Living With a Star programs."
]
] | 1605.05888 |
[
[
"Homogenized model of immiscible incompressible two-phase flow in double\n porosity media : A new proof"
],
[
"Abstract In this paper we give a new proof of the homogenization result for an immiscible incompressible two-phase flow in double porosity media obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikeli\\'c (1996) and in the paper of L. M. Yeh (2006) under some restrictive assumptions.",
"The microscopic model consists of the usual equations derived from the mass conservation laws for both fluids along with the standard Darcy-Muskat law relating the velocities to the pressure gradients and gravitational effects.",
"The problem is written in terms of the phase formulation, i.e.",
"where the phase pressures and the phase saturations are primary unknowns.",
"The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium is discontinuous.",
"The important difference with respect to the results of the cited papers is that the global pressure function as well as the saturation are also discontinuous.",
"This makes the initial mesoscopic problem more general and, physically, more realistic.",
"The convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique."
],
[
"Introduction",
"The modeling of displacement process involving two immiscible fluids in fractured porous media is important to many practical problems, including those in petroleum reservoir engineering, unsaturated zone hydrology, and soil science.",
"More recently, modeling multiphase flow has received an increasing attention in connection with the disposal of radioactive waste and sequestration of $CO_2$ .",
"Furthermore, fractured rock domains corresponding to the so-called Excavation Damaged Zone (EDZ) receives increasing attention in connection with the behaviour of geological isolation of radioactive waste after the drilling of the wells or shafts, see, e.g., [36].",
"In this paper we use the homogenization theory to derive a double porosity model describing the flow of incompressible fluids in fractured reservoirs.",
"The model corresponds physically to immiscible incompressible two-phase flow through fractured porous media.",
"Naturally fractured reservoirs can be modeled by two superimposed continua, a connected fracture system and a system of topologically disconnected matrix blocks.",
"The fracture system has low storage capacity but high conductivity, while the matrix block system has low conductivity and large storage capacity.",
"The majority of fluid transport will occur along flow paths through the fissure system.",
"When the system of fissures is so well developed that the matrix is broken into individual blocks or cells that are isolated from each other, there is consequently no flow directly from cell to cell, but only an exchange of fluid between each cell and the surrounding fissure system.",
"For more details on the physical formulation of such problems see, e.g., [15], [33], [39].",
"This paper continues the research published in [16] and [40], and the goal is to reformulate in a more systematic manner and in somewhat more general context the homogenization problem for an immiscible incompressible two-phase flow in double porosity media by weakening the standing assumptions.",
"Special attention is paid to developing a general approach to incorporating highly heterogeneous porous media with discontinuous capillary pressures.",
"During recent decades mathematical analysis and homogenization of multiphase flows in porous media have been the subject of investigation of many researchers owing to important applications in reservoir simulation.",
"There is an extensive literature on this subject.",
"We will not attempt a literature review here but will merely mention a few references.",
"A recent review of the mathematical homogenization methods developed for incompressible immiscible two-phase flow in porous media and compressible miscible flow in porous media can be viewed in [5], [29], [30].",
"Let us now turn to a brief review of the homogenization in double porosity media.",
"Here we restrict ourself to the mathematical homogenization method as described in [30] for flow and transport in porous media.",
"The interest for double porosity systems came at first from geophysics.",
"The notion of double porosity, or double permeability is borne from studies carried out on naturally fractured porous rocks, such as oil fields.",
"The double porosity model was first introduced in [14] and it is since used in a wide range of engineering specialities.",
"The first rigorous mathematical result on the subject was obtained in [13], where a linear parabolic equation with asymptotically degenerating coefficients describing a single-phase flow in fractured media was considered.",
"This result is then generalized in [17], [18], [32], [35] for non-periodic domains and various rates of contrast.",
"Linear double porosity models with thin fissures were considered in [11], [34].",
"A singular double porosity model was considered in [19].",
"Notice that the works [11], [18], [32], [34] are done in the framework of Khruslov's energy characteristic method which is close to the $\\Gamma $ -convergence method.",
"Let us also notice that the double porosity model was obtained in [30] (see Chapter 3) using the two-scale convergence method.",
"Non-linear double porosity models, elliptic and parabolic, including the homogenization in variable Sobolev spaces, were obtained in [7], [8], [23], [24], [26].",
"A study of discrete double-porosity models in the case of elastic energies has been recently done in [20].",
"Finally, in order to complete this brief review, we turn to the multiphase flow double porosity models.",
"These models were obtained e.g., in [16], [23], [40] (see also [30] and the references therein) and recently in [2], [6] for immiscible compressible two-phase flows.",
"A fully homogenized model for incompressible two-phase flow in double porosity media was obtained in [31].",
"This paper is concerned with a nonlinear degenerate system of diffusion-convection equations modeling the flow and transport of immiscible incompressible fluids through highly heterogeneous porous media, capillary and gravity effects being taken into account.",
"We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves.",
"The model to be presented herein is formulated in terms of the wetting phase saturation and the non-wetting phase pressure, and the feature of the global pressure as introduced in [12], [21] for incompressible immiscible flows is used to establish a priori estimates.",
"The governing equations are derived from the mass conservation laws of both fluids, along with constitutive relations relating the velocities to the pressures gradients and gravitational effects.",
"Traditionally, the standard Muskat-Darcy law provides this relationship.",
"Let us mention that the main difficulties related to the mathematical analysis of such equations are the coupling and the degeneracy of the diffusion term in the saturation equation.",
"Moreover the transmission conditions are nonlinear and the saturation is discontinuous at the interface separating the two media.",
"We start with a microscopic model defined on a domain with periodic microstructure.",
"We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves.",
"The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous.",
"Over the matrix domain, the permeability is scaled by $\\varepsilon ^\\theta $ , where $\\varepsilon $ is the size of a typical porous block and $\\theta >0$ is a parameter.",
"Our aim is to study the macroscopic behavior of solutions of this system of equations as $\\varepsilon $ tends to zero and give a rigorous mathematical derivation of upscaled models by means of the two-scale convergence method combined with the dilation technique.",
"Thus, we extend the results of [16], [40] to the case of highly heterogeneous porous media with discontinuous capillary pressures.",
"The rest of the paper is organized as follows.",
"In Section , we describe the physical model and formulate the corresponding mathematical problem.",
"We also provide the assumptions on the data and a weak formulation of the problem firstly in terms of phase pressures and secondly in terms of the global pressure and the saturation.",
"Section is then devoted to the derivation of the basic a priori estimates of the problem under consideration.",
"In Section we formulate the two-scale convergence results which will be used in the derivation of the homogenized system.",
"The key point here is the proof of the compactness result for the restriction-extension sequence of the wetting fluid saturation defined on the fracture set.",
"It is done by using the ideas from [40].",
"Section is devoted to the definition and the properties of the dilation operator and to the formulation of the convergence results for the dilated functions defined on the matrix part.",
"The key point of the section is the proof of the compactness result for the dilated saturations which is done by using the compactness result from [5].",
"The formulation of the main results of the paper is given in Section .",
"The resulting homogenized problem is a dual-porosity type model that contains a term representing memory effects which could be seen as source term or as a time delay for $\\theta = 2$ , and it is a single porosity model with effective coefficients for $0 < \\theta < 2$ or $\\theta > 2$ .",
"The proof of the convergence theorem in the critical case ($\\theta =2$ ) is done in subsection REF .",
"The key point here is subsection REF , where we prove the uniqueness of the solution to the local problem.",
"The proof is done by reducing the problem in the phase formulation to a boundary value problem for an imbibition equation and by using ideas from [38].",
"The proofs of the convergence theorems for non-critical cases ($\\theta >2$ or $0<\\theta <2$ ) are given in subsections REF , REF .",
"The effective model obtained in the case of moderate contrast ($0<\\theta <2$ , subsection REF ), up to our knowledge, is for the first time proposed and rigorously justified here."
],
[
"Formulation of the problem",
"The outline of this section is as follows.",
"First, in subsection REF we give a short description of the mathematical and physical model used in this study for immiscible incompressible two-phase flow in a periodic double porosity medium.",
"The notion of the global pressure is briefly recalled in subsection REF .",
"Finally, in subsection REF , we present the main assumptions on the data and we define the weak solution to our problem, first in terms of phase pressures and then an equivalent one in terms of the global pressure and saturation.",
"Figure: (a) The domain Ω\\Omega .",
"(b) The reference cell YY."
],
[
"Microscopic model",
"We consider a reservoir $\\Omega \\subset \\mathbb {R}^d$ ($d = 2, 3$ ) which is assumed to be a bounded, connected Lipschitz domain with a periodic microstructure.",
"More precisely, we will scale this structure by a parameter $\\varepsilon $ which represents the ratio of the cell size to the size of the whole region $\\Omega $ and we assume that $0 < \\varepsilon \\ll 1$ is a small parameter tending to zero.",
"Let $Y = (0, 1)^d$ be a basic cell of a fractured porous medium.",
"For the sake of simplicity and without loss of generality, we assume that $Y$ is made up of two homogeneous porous media $Y_\\mathsf {m}$ and $Y_\\mathsf {f}$ corresponding to the parts of the domain occupied by the matrix block and the fracture, respectively (see Fig.REF (b)).",
"Thus $Y = Y_\\mathsf {m}\\cup Y_\\mathsf {f}\\cup \\Gamma _{\\mathsf {f}\\mathsf {m}}$ , where $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ denotes the interface between the two media.",
"Let $\\Omega ^\\varepsilon _\\ell $ with $\\ell = \"\\mathsf {f}\"$ or $\"\\mathsf {m}\"$ denotes the open set corresponding to the porous medium with index $\\ell $ .",
"Then $\\Omega = \\Omega ^\\varepsilon _\\mathsf {m}\\cup \\Gamma ^\\varepsilon _{\\mathsf {f}\\mathsf {m}} \\cup \\Omega ^\\varepsilon _\\mathsf {f}$ , where $\\Gamma ^\\varepsilon _{\\mathsf {f}\\mathsf {m}} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\partial \\Omega ^\\varepsilon _\\mathsf {f}\\cap \\partial \\Omega ^\\varepsilon _\\mathsf {m}\\cap \\Omega $ and the subscripts $\"\\mathsf {m}\"$ , $\"\\mathsf {f}\"$ refer to the matrix and fracture, respectively (see Fig.REF (a)).",
"For the sake of simplicity, we assume that $\\Omega ^\\varepsilon _\\mathsf {m}\\cap \\partial \\Omega =\\emptyset $ .",
"We also introduce the notation: $\\Omega _T \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\Omega \\times (0,T), \\quad \\Omega ^\\varepsilon _{\\ell ,T} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\Omega ^\\varepsilon _\\ell \\times (0,T), \\quad \\Sigma ^\\varepsilon _T \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\Gamma ^\\varepsilon _{\\mathsf {f}\\mathsf {m}} \\times (0,T), \\quad {\\rm where}\\,\\, T > 0 \\,\\, {\\rm is \\,\\, fixed.",
"}$ Before describing the equations of the model, we give some notation: $\\Phi ^\\varepsilon (x) = \\Phi (x, \\frac{x}{\\varepsilon })$ is the porosity of the reservoir $\\Omega $ ; $K^\\varepsilon (x) = K(x, \\frac{x}{\\varepsilon })$ is the absolute permeability tensor of $\\Omega $ ; $\\varrho _w$ , $\\varrho _n$ are the densities of wetting and nonwetting fluids, respectively; $S^\\varepsilon _{\\ell , w} = S^\\varepsilon _{\\ell , w}(x, t)$ , $S^\\varepsilon _{\\ell , n}= S^\\varepsilon _{\\ell , n}(x, t)$ are the saturations of wetting and nonwetting fluids in $\\Omega ^\\varepsilon _\\ell $ , respectively; $k_{r,w}^{(\\ell )} = k_{r,w}^{(\\ell )}(S^\\varepsilon _{\\ell , w})$ , $k_{r,n}^{(\\ell )} = k_{r,n}^{(\\ell )}(S^\\varepsilon _{\\ell , n})$ are the relative permeabilities of wetting and nonwetting fluids in $\\Omega ^\\varepsilon _\\ell $ , respectively; $p^\\varepsilon _{\\ell , w} =p^\\varepsilon _{\\ell , w}(x,t)$ , $p^\\varepsilon _{\\ell , n} = p^\\varepsilon _{\\ell , n}(x,t)$ are the phase pressures of wetting and nonwetting fluids in $\\Omega ^\\varepsilon _\\ell $ , respectively.",
"Here $\\ell = \\mathsf {f}, \\mathsf {m}$ .",
"The conservation of mass in each phase can be written as (see, e.g., [21], [22], [28]): $\\left\\lbrace \\begin{array}[c]{ll}\\displaystyle \\Phi ^\\varepsilon (x) \\frac{\\partial }{\\partial t}\\left[S^\\varepsilon _{\\ell , w}\\,\\varrho _w(p^\\varepsilon _{\\ell , w})\\right] +{\\rm div} \\big \\lbrace \\varrho _w(p^\\varepsilon _{\\ell , w}) \\, \\vec{q}^{\\,\\varepsilon }_{\\ell , w} \\big \\rbrace = F^\\varepsilon _{\\ell ,w}(x,t)\\quad {\\rm in}\\,\\, \\Omega ^\\varepsilon _{\\ell ,T}; \\\\[3mm]\\displaystyle \\Phi (x) \\frac{\\partial }{\\partial t} \\left[S^\\varepsilon _{\\ell , n}\\, \\varrho _n(p^\\varepsilon _{\\ell , n})\\right] +{\\rm div}\\big \\lbrace \\varrho _n(p^\\varepsilon _{\\ell , n}) \\, \\vec{q}^{\\,\\varepsilon }_{\\ell , n} \\big \\rbrace = F^\\varepsilon _{\\ell ,n}(x,t)\\quad {\\rm in}\\,\\, \\Omega ^\\varepsilon _{\\ell ,T}, \\\\[3mm]\\end{array}\\right.$ where the velocities of the wetting and nonwetting fluids $\\vec{q}^{\\,\\varepsilon }_{\\ell , w}$ , $\\vec{q}^{\\,\\varepsilon }_{\\ell , n}$ are defined by Darcy-Muskat's law: $\\vec{q}^{\\,\\varepsilon }_{\\ell , w} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}-K^\\varepsilon (x) \\lambda _{\\ell , w}(S^\\varepsilon _{\\ell , w})\\left[\\nabla p^\\varepsilon _{\\ell , w} - \\varrho _w(p^\\varepsilon _{\\ell , w})\\, \\vec{g}\\right],\\quad \\!",
"{\\rm with}\\,\\, \\lambda _{\\ell , w}(S^\\varepsilon _{\\ell , w}) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\frac{k_{r,w}^{(\\ell )}}{\\mu _{w}}(S^\\varepsilon _{\\ell , w});$ $\\vec{q}^{\\,\\varepsilon }_{\\ell , n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}- K^\\varepsilon (x) \\lambda _{\\ell , n}(S^\\varepsilon _{\\ell , n})\\left[\\nabla p^\\varepsilon _{\\ell , n} - \\varrho _n(p^\\varepsilon _{\\ell , n})\\, \\vec{g}\\right], \\quad {\\rm with}\\,\\, \\lambda _{\\ell , n}(S_{\\ell , n}) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\frac{k_{r,n}^{(\\ell )}}{\\mu _{n}}(S^\\varepsilon _{\\ell , n}).$ Here $\\vec{g}$ , $\\mu _w, \\mu _n$ are the gravity vector and the viscosities of the wetting and nonwetting fluids, respectively.",
"The source terms $F^\\varepsilon _{\\ell ,w}, F^\\varepsilon _{\\ell ,n}$ are given by: $F^\\varepsilon _{\\ell ,w} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varrho _w(p^\\varepsilon _{\\ell , w}) S^I_{\\ell , w} f_I(x,t) -\\varrho _w(p^\\varepsilon _{\\ell , w}) S^\\varepsilon _{\\ell , w} f_P(x,t);$ $F^\\varepsilon _{\\ell ,n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varrho _n(p^\\varepsilon _{\\ell , n}) S^I_{\\ell , n} f_I(x,t) -\\varrho _n(p^\\varepsilon _{\\ell , n}) S^\\varepsilon _{\\ell , n} f_P(x,t),$ where $f_I, f_P \\geqslant 0$ are injection and productions terms and $S^I_{\\ell , w}, S^I_{\\ell , n}$ are known injection saturations.",
"From now on we deal with two incompressible fluids, that is the densities of the wetting and nonwetting fluids are constants, which for the sake of simplicity and brevity, will be taken equal to one, i.e.",
"$\\varrho _w(p^\\varepsilon _{\\ell ,w}) = \\varrho _n(p^\\varepsilon _{\\ell ,n}) = 1$ .",
"The model is completed as follows.",
"By the definition of saturations, one has $S^\\varepsilon _{\\ell , w} + S^\\varepsilon _{\\ell , n} = 1$ with $S^\\varepsilon _{\\ell , w}, S^\\varepsilon _{\\ell , n} \\geqslant 0$ .",
"We set $S^\\varepsilon _\\ell \\mathop {=}\\limits ^{\\hbox{\\tiny def}}S^\\varepsilon _{\\ell , w}$ .",
"Then the curvature of the contact surface between the two fluids links the difference in the pressures of the two phases to the saturation by the capillary pressure law: $P_{\\ell ,c}(S^\\varepsilon _\\ell ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}p^\\varepsilon _{\\ell , n} - p^\\varepsilon _{\\ell , w} \\quad {\\rm with} \\,\\,P^\\prime _{\\ell ,c}(s) < 0\\,\\, {\\rm for\\,\\, all}\\,\\,s \\in (0, 1)\\,\\, {\\rm and} \\,\\, P_{\\ell ,c}(1) = 0,$ where $P^\\prime _{\\ell ,c}(s)$ denotes the derivative of the function $P_{\\ell ,c}(s)$ .",
"Now due to the assumptions on the densities of the liquids, we rewrite the system (REF ) as follows: $\\left\\lbrace \\begin{array}[c]{ll}0 \\leqslant {S}^\\varepsilon \\leqslant 1 \\quad {\\rm in}\\,\\, \\Omega _T;\\\\[3mm]\\displaystyle \\Phi ^\\varepsilon (x) \\frac{\\partial {S}^\\varepsilon }{\\partial t} - {\\rm div}\\, \\left\\lbrace K^\\varepsilon (x)\\lambda _{w}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\left(\\nabla {\\mathsf {p}}^\\varepsilon _{w} - \\vec{g}\\right)\\right\\rbrace = F^\\varepsilon _{w} \\quad {\\rm in}\\,\\, \\Omega _{T}; \\\\[4mm]\\displaystyle -\\Phi ^\\varepsilon (x) \\frac{\\partial {S}^\\varepsilon }{\\partial t} -{\\rm div}\\, \\left\\lbrace K^\\varepsilon (x) \\lambda _{n} \\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\left(\\nabla {\\mathsf {p}}^\\varepsilon _{n} - \\vec{g} \\right)\\right\\rbrace = F^\\varepsilon _{n} \\quad {\\rm in}\\,\\, \\Omega _{T}; \\\\[4mm]P_{c}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right) = {\\mathsf {p}}^\\varepsilon _{n} - {\\mathsf {p}}^\\varepsilon _{w}\\quad {\\rm in}\\,\\, \\Omega _{T},\\end{array}\\right.$ where $\\lambda _{\\ell ,n}(S^\\varepsilon _\\ell ) := \\lambda _{\\ell ,n}(1-S^\\varepsilon _\\ell )$ and each function $u^\\varepsilon := {S}^\\varepsilon , {\\mathsf {p}}^\\varepsilon _{w}, {\\mathsf {p}}^\\varepsilon _{n}, F^\\varepsilon _{w}, F^\\varepsilon _{n}$ is defined as: $u^\\varepsilon \\mathop {=}\\limits ^{\\hbox{\\tiny def}}u^\\varepsilon _{\\mathsf {f}}(x, t)\\, {\\bf 1}^\\varepsilon _\\mathsf {f}(x) + u^\\varepsilon _{\\mathsf {m}}(x, t)\\, {\\bf 1}^\\varepsilon _\\mathsf {m}(x).$ Here ${\\bf 1}^\\varepsilon _\\ell = {\\bf 1}_\\ell (\\frac{x}{\\varepsilon })$ is the characteristic function of the subdomain $\\Omega ^\\varepsilon _\\ell $ for $\\ell = \\mathsf {f}, \\mathsf {m}$ .",
"The exact form of the porosity function and the absolute permeability tensor corresponding to the double porosity model will be specified in conditions (A.1), (A.2) in subsection REF below.",
"Model (REF ) have to be completed with appropriate interface, boundary and initial conditions.",
"Interface conditions.",
"The continuity at the interface $\\Gamma ^\\varepsilon _{\\mathsf {f}\\mathsf {m}}$ of the phase fluxes and the phase pressures, gives the following transmission conditions: $\\left\\lbrace \\begin{array}[c]{ll}\\vec{q}^{\\,\\varepsilon }_{\\mathsf {f},w} \\cdot \\vec{\\nu }= \\vec{q}^{\\,\\varepsilon }_{\\mathsf {m},w}\\cdot \\vec{\\nu }\\quad {\\rm and} \\quad \\vec{q}^{\\,\\varepsilon }_{\\mathsf {f},n} \\cdot \\vec{\\nu }= \\vec{q}^{\\,\\varepsilon }_{\\mathsf {m},n}\\cdot \\vec{\\nu }\\quad {\\rm on}\\,\\, \\Sigma ^\\varepsilon _T; \\\\[2mm]p^\\varepsilon _{\\mathsf {f},w} = p^\\varepsilon _{\\mathsf {m},w} \\quad {\\rm and} \\quad p^\\varepsilon _{\\mathsf {f},n} = p^\\varepsilon _{\\mathsf {m},n} \\quad {\\rm on}\\,\\, \\Sigma ^\\varepsilon _T, \\\\\\end{array}\\right.$ where $\\Sigma ^\\varepsilon _T$ is defined in (REF ), $\\vec{\\nu }$ is the unit outer normal on $\\Gamma ^\\varepsilon _{\\mathsf {f}\\mathsf {m}}$ , and the fluxes $\\vec{q}^{\\,\\varepsilon }_{\\ell ,w}, \\vec{q}^{\\,\\varepsilon }_{\\ell ,n}$ , under the assumption on the densities of the liquids, are equal to the velocities (REF ), (REF ).",
"Remark 1 It is important to notice that in contrast to the functions ${\\mathsf {p}}^\\varepsilon _n, {\\mathsf {p}}^\\varepsilon _w$ , the saturation ${S}^\\varepsilon $ may have a jump at the interface $\\Gamma ^\\varepsilon _{\\mathsf {f}\\mathsf {m}}$ .",
"Namely, it is easy to see from the transmission conditions (REF ) for the phase pressures that $P_{\\mathsf {f},c}(S^\\varepsilon _1) = P_{\\mathsf {m},c}(S^\\varepsilon _2)$ on $\\Sigma ^\\varepsilon _T$ which gives a discontinuity of the saturation at the interface.",
"Now we specify the boundary and initial conditions.",
"We suppose that the boundary $\\partial \\Omega $ consists of two parts $\\Gamma _{1}$ and $\\Gamma _{2}$ such that $\\Gamma _{1} \\cap \\Gamma _{2} = \\emptyset $ , $\\partial \\Omega =\\overline{\\Gamma }_{1} \\cup \\overline{\\Gamma }_{2}$ .",
"Boundary conditions: $\\left\\lbrace \\begin{array}[c]{ll}{\\mathsf {p}}^\\varepsilon _{w}(x, t) = {\\mathsf {p}}^\\varepsilon _{n}(x, t) = 0 \\quad {\\rm on} \\,\\,\\Gamma _{1} \\times (0,T); \\\\[2mm]\\vec{q}^{\\,\\varepsilon }_{\\mathsf {f},w} \\cdot \\vec{\\nu }= \\vec{q}^{\\,\\varepsilon }_{\\mathsf {f},n} \\cdot \\vec{\\nu }= 0\\quad {\\rm on} \\,\\, \\Gamma _{2} \\times (0,T).\\\\\\end{array}\\right.$ Initial conditions: ${\\mathsf {p}}^\\varepsilon _{w}(x, 0) = {\\mathsf {p}}_{w}^{\\bf 0}(x) \\quad {\\rm and}\\quad {\\mathsf {p}}^\\varepsilon _{n}(x, 0) = {\\mathsf {p}}_{n}^{\\bf 0}(x)\\quad {\\rm in} \\,\\, \\Omega .$"
],
[
"A fractional flow formulation",
"In the sequel, we will use a formulation obtained after transformation using the concept of the global pressure introduced in [12], [21].",
"For each subdomain $\\Omega ^\\varepsilon _\\ell $ , the global pressure, ${\\mathsf {P}}^\\varepsilon _\\ell $ , is defined by: $p^\\varepsilon _{\\ell ,w} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {P}}^\\varepsilon _\\ell + {\\mathsf {G}}_{\\ell ,w}(S^\\varepsilon _\\ell )\\quad {\\rm and} \\quad p^\\varepsilon _{\\ell ,n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {P}}^\\varepsilon _\\ell + {\\mathsf {G}}_{\\ell ,n}(S^\\varepsilon _\\ell ),$ where the functions ${\\mathsf {G}}_{\\ell ,w}(s)$ , ${\\mathsf {G}}_{\\ell ,n}(s)$ are given by: ${\\mathsf {G}}_{\\ell ,n}(S^\\varepsilon _\\ell ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {G}}_{\\ell ,n}(0) + \\int \\limits _0^{S^\\varepsilon _\\ell } \\frac{\\lambda _{\\ell ,w}(s)}{\\lambda _{\\ell }(s)} \\,P^\\prime _{\\ell ,c}(s)\\, ds\\quad {\\rm and} \\quad {\\mathsf {G}}_{\\ell , w}(S^\\varepsilon _\\ell ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {G}}_{\\ell ,n}(S^\\varepsilon _\\ell )- P_{\\ell ,c}\\left(S^\\varepsilon _\\ell \\right),$ where $\\lambda _{\\ell }(s) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\lambda _{\\ell ,w}(s) + \\lambda _{\\ell ,n}(s)$ and ${\\mathsf {G}}_{\\ell ,n}(0)$ is a constant chosen to ensure $p^\\varepsilon _{\\ell ,w} \\leqslant {\\mathsf {P}}^\\varepsilon _\\ell \\leqslant p^\\varepsilon _{\\ell ,n}$ .",
"Notice that from (REF ) we get: $\\lambda _{\\ell ,w}(S^\\varepsilon _\\ell ) \\nabla {\\mathsf {G}}_{\\ell ,w}(S^\\varepsilon _\\ell ) = \\nabla \\beta _\\ell (S^\\varepsilon _\\ell )\\quad {\\rm and} \\quad \\lambda _{\\ell ,n}(S^\\varepsilon _\\ell ) \\nabla {\\mathsf {G}}_{\\ell ,n}(S^\\varepsilon _\\ell ) = - \\nabla \\beta _\\ell (S^\\varepsilon _\\ell ),$ where $\\beta _\\ell (s) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\int \\limits _0^s \\alpha _\\ell (\\xi )\\, d\\xi \\quad {\\rm with} \\,\\,\\,\\alpha _\\ell (s) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\frac{\\lambda _{\\ell ,n}(s)\\, \\lambda _{\\ell ,w}(s)}{\\lambda _\\ell (s)} \\left| P^\\prime _{\\ell ,c}(s) \\right|.$ Furthermore, we have the following important relation: $\\lambda _{\\ell ,n} (S^\\varepsilon _\\ell ) |\\nabla p^\\varepsilon _{\\ell ,n}|^2 +\\lambda _{\\ell ,w} (S^\\varepsilon _\\ell ) |\\nabla p^\\varepsilon _{\\ell ,w}|^2=\\lambda _{\\ell } (S^\\varepsilon _\\ell ) |\\nabla {\\mathsf {P}}^\\varepsilon _\\ell |^2 +\\left|\\nabla \\mathfrak {b}_\\ell (S^\\varepsilon _\\ell ) \\right|^2,$ where $\\mathfrak {b}_\\ell (s) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\int \\limits _0^s \\mathfrak {a}_\\ell (\\xi )\\, d\\xi \\quad {\\rm with} \\,\\, \\mathfrak {a}_\\ell (s) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\sqrt{\\frac{\\lambda _{\\ell ,n}(s)\\, \\lambda _{\\ell ,w}(s)}{\\lambda _\\ell (s)}}\\, \\left| P^\\prime _{\\ell ,c}(s) \\right|.$ Now if we use the global pressure and the saturation as new unknown functions then (REF ) reads: $\\left\\lbrace \\begin{array}[c]{ll}0 \\leqslant S^\\varepsilon _\\ell \\leqslant 1 \\quad {\\rm in}\\,\\, \\Omega ^\\varepsilon _{\\ell ,T};\\\\[3mm]\\displaystyle \\Phi ^\\varepsilon (x) \\frac{\\partial S^\\varepsilon _\\ell }{\\partial t} -{\\rm div}\\, \\bigg \\lbrace K^\\varepsilon (x) \\left[\\lambda _{\\ell ,w} (S^\\varepsilon _\\ell ) \\nabla {\\mathsf {P}}^\\varepsilon _\\ell +\\nabla \\beta _\\ell (S^\\varepsilon _\\ell ) - \\lambda _{\\ell ,w} (S^\\varepsilon _\\ell ) \\vec{g} \\right] \\bigg \\rbrace = F^\\varepsilon _{\\ell ,w} \\quad {\\rm in}\\,\\, \\Omega ^\\varepsilon _{\\ell ,T}; \\\\[4mm]\\displaystyle -\\Phi ^\\varepsilon (x) \\frac{\\partial S^\\varepsilon _\\ell }{\\partial t}-{\\rm div}\\, \\bigg \\lbrace K^\\varepsilon (x) \\left[ \\lambda _{\\ell ,n} (S^\\varepsilon _\\ell )\\nabla {\\mathsf {P}}^\\varepsilon _\\ell - \\nabla \\beta _\\ell (S^\\varepsilon _\\ell )-\\lambda _{\\ell ,n}(S^\\varepsilon _\\ell ) \\vec{g} \\right] \\bigg \\rbrace =F^\\varepsilon _{\\ell ,n}\\quad {\\rm in}\\,\\, \\Omega ^\\varepsilon _{\\ell ,T}.\\\\\\end{array}\\right.$ The system (REF ) is completed by the following boundary, interface and initial conditions.",
"Boundary conditions: $\\left\\lbrace \\begin{array}[c]{ll}{S}^\\varepsilon = 1 \\,\\,\\,{\\rm and}\\,\\,\\,{\\mathsf {P}}^\\varepsilon = {\\mathsf {P}}_{\\Gamma _1} \\quad {\\rm on} \\,\\,\\Gamma _{1} \\times (0,T); \\\\[2mm]\\vec{q}^{\\,\\varepsilon }_{\\mathsf {f},w} \\cdot \\vec{\\nu }= \\vec{q}^{\\,\\varepsilon }_{\\mathsf {f},n} \\cdot \\vec{\\nu }= 0\\quad {\\rm on} \\,\\, \\Gamma _{2} \\times (0,T),\\\\\\end{array}\\right.$ where ${\\mathsf {P}}_{\\Gamma _1}$ is a given constant and $\\vec{q}^{\\,\\varepsilon }_{\\ell , w}, \\vec{q}^{\\,\\varepsilon }_{\\ell , n}$ are defined by $\\vec{q}^{\\,\\varepsilon }_{\\ell , w} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}-K^\\varepsilon (x) \\left[\\lambda _{\\ell ,w} (S^\\varepsilon _\\ell ) \\nabla {\\mathsf {P}}^\\varepsilon _\\ell +\\nabla \\beta _\\ell (S^\\varepsilon _\\ell ) - \\lambda _{\\ell ,w} (S^\\varepsilon _\\ell ) \\vec{g} \\right];$ $\\vec{q}^{\\,\\varepsilon }_{\\ell , n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}- K^\\varepsilon (x) \\left[ \\lambda _{\\ell ,n} (S^\\varepsilon _\\ell )\\nabla {\\mathsf {P}}^\\varepsilon _\\ell - \\nabla \\beta _\\ell (S^\\varepsilon _\\ell )- \\lambda _{\\ell ,n}(S^\\varepsilon _\\ell ) \\vec{g} \\right].$ Interface conditions: $\\left\\lbrace \\begin{array}[c]{ll}\\vec{q}^{\\,\\,\\varepsilon }_{\\mathsf {f},w} \\cdot \\vec{\\nu }=\\vec{q}^{\\,\\,\\varepsilon }_{\\mathsf {m},w}\\cdot \\vec{\\nu }\\,\\,\\, {\\rm and} \\,\\,\\,\\vec{q}^{\\,\\,\\varepsilon }_{\\mathsf {f},n} \\cdot \\vec{\\nu }=\\vec{q}^{\\,\\,\\varepsilon }_{\\mathsf {m},n}\\cdot \\vec{\\nu }\\quad {\\rm on}\\,\\, \\Sigma ^\\varepsilon _T; \\\\[3mm]{\\mathsf {P}}^\\varepsilon _\\mathsf {f}+ {\\mathsf {G}}_{\\mathsf {f},j}(S^\\varepsilon _\\mathsf {f}) ={\\mathsf {P}}^\\varepsilon _\\mathsf {m}+ {\\mathsf {G}}_{\\mathsf {m},j}(S^\\varepsilon _\\mathsf {m})\\quad {\\rm on}\\,\\, \\Sigma ^\\varepsilon _T \\quad (j = w, n);\\\\[3mm]P_{\\mathsf {f},c}\\left(S^\\varepsilon _\\mathsf {f}\\right) = P_{\\mathsf {m},c}\\left(S^\\varepsilon _\\mathsf {m}\\right)\\quad {\\rm on}\\,\\, \\Sigma ^\\varepsilon _T.\\end{array}\\right.$ Note that the global pressure function might be discontinuous at the interface.",
"This makes the compactness result in Section non-trivial.",
"Initial conditions: $S^\\varepsilon _\\ell (x, 0) = S^{\\bf 0}_\\ell (x) \\,\\, {\\rm and} \\,\\,{\\mathsf {P}}^\\varepsilon _\\ell (x, 0) = {\\mathsf {P}}^{\\bf 0}_\\ell (x) \\quad {\\rm in} \\,\\, \\Omega .$"
],
[
"Weak formulations of the problem",
"Let us begin this subsection by stating the following assumptions.",
"(A.1) The porosity $\\Phi ^\\varepsilon $ is given by $\\Phi ^\\varepsilon (x) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\Phi ^\\varepsilon _{\\mathsf {f}}(x)\\, {\\bf 1}^\\varepsilon _\\mathsf {f}(x) + \\Phi ^\\varepsilon _{\\mathsf {m}}(x)\\, {\\bf 1}^\\varepsilon _\\mathsf {m}(x) =\\Phi ^\\varepsilon _{\\mathsf {f}}(x)\\, {\\bf 1}^\\varepsilon _\\mathsf {f}(x) + \\Phi _{\\mathsf {m}}\\left(\\frac{x}{\\varepsilon }\\right)\\, {\\bf 1}^\\varepsilon _\\mathsf {m}(x),$ where $\\Phi ^\\varepsilon _{\\mathsf {f}} \\in L^{\\infty }(\\Omega )$ and there are positive constants $0 < \\phi ^{\\ell }_- < \\phi ^{\\ell }_+ < 1$ , $\\ell = \\mathsf {f}, \\mathsf {m}$ , that do not depend on $\\varepsilon $ and such that $0 < \\phi ^{\\mathsf {f}}_- \\leqslant \\Phi ^\\varepsilon _{\\mathsf {f}}(x) \\leqslant \\phi ^{\\mathsf {f}}_+ < 1$ a.e.",
"in $\\Omega $ .",
"Moreover, $\\Phi ^\\varepsilon _{\\mathsf {f}} \\longrightarrow \\Phi ^{\\rm H}_{\\mathsf {f}}$ strongly in $L^2(\\Omega )$ .",
"$\\Phi _\\mathsf {m}= \\Phi _\\mathsf {m}(y)$ is $Y$ -periodic, $\\Phi _{\\mathsf {m}} \\in L^{\\infty }(Y)$ and such that $0 < \\phi ^{\\mathsf {m}}_- \\leqslant \\Phi _\\mathsf {m}(y) \\leqslant \\phi ^{\\mathsf {m}}_+ < 1$ a.e.",
"in $Y$ .",
"(A.2) The permeability $K^\\varepsilon (x) = K^\\varepsilon (x, \\frac{x}{\\varepsilon })$ is defined as $K^\\varepsilon (x, y) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}K(x, y)\\, {\\bf 1}^\\varepsilon _\\mathsf {f}(x) + \\varkappa (\\varepsilon )\\, K(x, y)\\, {\\bf 1}^\\varepsilon _\\mathsf {m}(x),$ where $\\varkappa (\\varepsilon ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varepsilon ^\\theta $ with $\\theta > 0$ and $K \\in (L^{\\infty }(\\Omega \\times Y))^{d\\times d}$ .",
"Moreover, there exist constants $k_{\\rm min}, k^{\\rm max}$ such that $0 < k_{\\rm min} < k^{\\rm max}$ and $k_{\\rm min} |\\xi |^2 \\le (K(x, y)\\,\\xi , \\xi ) \\le k^{\\rm max} |\\xi |^2 \\,\\, {\\rm for \\, all \\,\\xi \\in \\mathbb {R}^d, \\,\\, a.e.",
"\\, in}\\,\\, \\Omega \\times Y.$ (A.3) The capillary pressure function $P_{\\ell ,c}(s) \\in C^1([0, 1]; \\mathbb {R}^+)$ , $\\ell = \\mathsf {f}, \\mathsf {m}$ .",
"Moreover, $P_{\\ell ,c}^\\prime (s) < 0$ in $[0, 1]$ , $P_{\\ell ,c}(1) = 0$ and $P_{\\mathsf {f},c}(0) = P_{\\mathsf {m},c}(0)$ .",
"(A.4) The functions $\\lambda _{\\ell ,w}, \\lambda _{\\ell ,n}$ belong to the space $C([0, 1]; \\mathbb {R}^+)$ and satisfy the following properties: (i) $0 \\leqslant \\lambda _{\\ell ,w}, \\lambda _{\\ell ,n} \\leqslant 1$ in $[0, 1]$ ; (ii) $\\lambda _{\\ell ,w}(0) = 0$ and $\\lambda _{\\ell ,n}(1) = 0$ ; (iii) there is a positive constant $L_0$ such that $\\lambda _{\\ell }(s) = \\lambda _{\\ell ,w}(s) + \\lambda _{\\ell ,n}(s)\\geqslant L_0 > 0$ in $[0, 1]$ .",
"(A.5) The functions $\\alpha _\\ell \\in C([0, 1]; \\mathbb {R}^+)$ .",
"Moreover, $\\alpha _\\ell (0) = \\alpha _\\ell (1) = 0$ and $\\alpha _\\ell > 0$ in $(0, 1)$ .",
"(A.6) The functions $\\beta ^{-1}_\\ell $ , inverse of $\\beta _\\ell $ defined in (REF ) are Hölder functions of order $\\gamma \\in (0, 1)$ in $[0, \\beta _\\ell (1)]$ .",
"Namely, there exists a positive constant $C_\\beta $ such that for all $s_1, s_2 \\in [0, \\beta (1)]$ , we have: $\\left|\\beta ^{-1}_\\ell (s_1) - \\beta ^{-1}_\\ell (s_2) \\right| \\leqslant C_\\beta \\, |s_1 - s_2|^\\gamma .$ (A.7) The initial data for the pressures are such that ${\\mathsf {p}}^{\\bf 0}_{n},{\\mathsf {p}}^{\\bf 0}_{w} \\in L^2(\\Omega )$ .",
"(A.8) The initial data for the saturation $S^{\\bf 0}$ is given by $P_{\\ell ,c}(S^{\\bf 0}_\\ell ) ={\\mathsf {p}}^{\\bf 0}_{\\ell , n} - {\\mathsf {p}}^{\\bf 0}_{\\ell , w}$ and is such that ${S}^{\\bf 0} \\in L^\\infty (\\Omega )$ and $0 \\leqslant {S}^{\\bf 0} \\leqslant 1$ a.e.in $\\Omega $ .",
"(A.9) The source terms $F^\\varepsilon _{w}, F^\\varepsilon _{n}$ are equal to zero on the matrix part, i.e.",
"$F^\\varepsilon _{w} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\bf 1}^\\varepsilon _\\mathsf {f}(x)\\, \\big [S^I_{\\mathsf {f}, w} f_I(x,t) - S^\\varepsilon _{\\mathsf {f}} f_P(x,t)\\big ]$ and $F^\\varepsilon _{n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\bf 1}^\\varepsilon _\\mathsf {f}(x)\\, \\big [S^I_{\\mathsf {f}, n} f_I(x,t) - (1 - S^\\varepsilon _{\\mathsf {f}}) f_P(x,t)\\big ],$ where $f_I, f_P \\in L^2(\\Omega _T)$ and $0 \\leqslant S^I_{\\mathsf {f}, w}, S^I_{\\mathsf {f}, n} \\leqslant 1$ .",
"The assumptions (A.1)–(A.9) are classical and physically meaningful for existence results and homogenization problems of two-phase flow in porous media.",
"They are similar to the assumptions made in [12], [21] that dealt with the existence of a weak solution of the studied problem.",
"We next introduce the following Sobolev space: $H^1_{\\Gamma _{1}}(\\Omega ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace u \\in H^1(\\Omega ) \\,:\\,u = 0 \\,\\, {\\rm on}\\,\\, \\Gamma _{1} \\right\\rbrace .$ The space $H^1_{\\Gamma _{1}}(\\Omega )$ is a Hilbert space.",
"The norm in this space is given by $\\Vert u \\Vert _{H^1_{\\Gamma _{1}}(\\Omega )} = \\Vert \\nabla u \\Vert _{(L^2(\\Omega ))^d}$ .",
"Definition 2.1 (Weak solution in terms of phase pressures) We say that the functions $\\langle {\\mathsf {p}}^\\varepsilon _{w}, {\\mathsf {p}}^\\varepsilon _{n} , {S}^\\varepsilon \\rangle $ is a weak solution of problem (REF ) if (i) $0 \\leqslant {S}^\\varepsilon \\leqslant 1$ a.e.",
"in $\\Omega _T$ and $P_{\\ell ,c}(S^\\varepsilon _\\ell ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {p}}^\\varepsilon _{\\ell , n} - {\\mathsf {p}}^\\varepsilon _{\\ell , w}$ for $\\ell \\in \\lbrace \\mathsf {f},\\mathsf {m}\\rbrace $ .",
"(ii) The functions ${\\mathsf {p}}^\\varepsilon _{w}, {\\mathsf {p}}^\\varepsilon _{n}$ are such that ${\\mathsf {p}}^\\varepsilon _{w}\\,, {\\mathsf {p}}^\\varepsilon _{n}\\,,\\sqrt{\\lambda _w(x, {S}^\\varepsilon )}\\, \\nabla {\\mathsf {p}}^\\varepsilon _w\\,,\\sqrt{\\lambda _n(x, {S}^\\varepsilon )}\\, \\nabla {\\mathsf {p}}^\\varepsilon _n \\in L^2(\\Omega _T).$ (iii) The boundary conditions (REF ) and the initial conditions (REF ) are satisfied.",
"(iv) For any $\\varphi _w, \\varphi _n \\in C^1([0, T]; H^1_{\\Gamma _{1}}(\\Omega ))$ satisfying $\\varphi _w(T) = \\varphi _n(T) = 0$ , we have: $-\\int \\limits _{\\Omega _{T}} \\Phi ^\\varepsilon (x) {S}^\\varepsilon \\frac{\\partial \\varphi _w}{\\partial t}\\, dx dt - \\int \\limits _{\\Omega } \\Phi ^\\varepsilon {S}^{\\bf 0} \\varphi _w^{\\bf 0}\\, dx+\\int \\limits _{\\Omega _{T}} K^\\varepsilon (x) \\lambda _{w}\\left(\\frac{x}{\\varepsilon }, { S}^\\varepsilon \\right)\\left(\\nabla {\\mathsf {p}}^\\varepsilon _w - \\vec{g} \\right) \\cdot \\nabla \\varphi _w\\, dx dt= \\int \\limits _{\\Omega _{T}} F^\\varepsilon _{w}\\, \\varphi _w\\, dx dt;$ $\\int \\limits _{\\Omega _{T}} \\Phi ^\\varepsilon (x) {S}^\\varepsilon \\frac{\\partial \\varphi _n}{\\partial t}\\, dx dt + \\int \\limits _{\\Omega } \\Phi ^\\varepsilon {S}^{\\bf 0} \\varphi _n^{\\bf 0}\\, dx+\\int \\limits _{\\Omega _{T}} K^\\varepsilon (x) \\lambda _{n}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\left(\\nabla {\\mathsf {p}}^\\varepsilon _n - \\vec{g} \\right) \\cdot \\nabla \\varphi _n\\, dx dt= \\int \\limits _{\\Omega _{T}} F^\\varepsilon _{n}\\,\\varphi _n\\, dx dt,$ where $\\varphi _w^{\\bf 0} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varphi _w(0, x)$ , $\\varphi _n^{\\bf 0} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varphi _n(0, x)$ , and the function ${S}^{\\bf 0} = {S}^{\\bf 0}(x)$ is defined by the initial condition (REF ) and the capillary pressure relation (REF ).",
"Let us also give an equivalent definition of a weak solution in terms of the global pressure and the saturation.",
"Definition 2.2 (Weak solution in terms of global pressure and saturation) We say that the pair of functions $\\langle {S}^\\varepsilon , {\\mathsf {P}}^\\varepsilon \\rangle $ is a weak solution of problem (REF ) if (i) $0 \\leqslant {S}^\\varepsilon \\leqslant 1$ a.e.",
"in $\\Omega _T$ .",
"(ii) The global pressure function ${\\mathsf {P}}^\\varepsilon _\\ell \\in L^2(0, T; H^1(\\Omega ^\\varepsilon _\\ell ))$ and, for any $\\varepsilon > 0$ , the saturation function $S^\\varepsilon _\\ell $ is such that $\\beta _\\ell (S^\\varepsilon _\\ell ) \\in L^2(0, T; H^1(\\Omega ^\\varepsilon _\\ell ))$ .",
"(iii) The boundary conditions (REF ) and the initial conditions (REF ) are satisfied.",
"(iv) For any $\\varphi _w, \\varphi _n \\in C^1([0, T]; H^1_{\\Gamma _{1}}(\\Omega ))$ satisfying $\\varphi _w(T) = \\varphi _n(T) = 0$ , we have: $-\\int \\limits _{\\Omega _{T}} \\Phi ^\\varepsilon (x) {S}^\\varepsilon \\frac{\\partial \\varphi _w}{\\partial t}\\, dx dt - \\int \\limits _{\\Omega } \\Phi ^\\varepsilon (x) {S}^{\\bf 0} \\varphi _w^{\\bf 0}\\, dx +\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} K^\\varepsilon (x)\\bigg \\lbrace \\lambda _{\\,\\mathsf {f},w} (S^\\varepsilon _\\mathsf {f})\\left(\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}- \\vec{g} \\right) + \\nabla \\beta _\\mathsf {f}(S^\\varepsilon _\\mathsf {f})\\bigg \\rbrace \\cdot \\nabla \\varphi _w\\, dx dt +$ $+ \\varkappa (\\varepsilon )\\,\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} K^\\varepsilon (x)\\bigg \\lbrace \\lambda _{\\,\\mathsf {m},w} (S^\\varepsilon _\\mathsf {m})\\left(\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {m}- \\vec{g} \\right) + \\nabla \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m})\\bigg \\rbrace \\cdot \\nabla \\varphi _w\\, dx dt= \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} F^\\varepsilon _{w}\\, \\varphi _w\\, dx dt;$ $\\int \\limits _{\\Omega _{T}} \\Phi ^\\varepsilon (x) {S}^\\varepsilon \\frac{\\partial \\varphi _n}{\\partial t}\\, dx dt + \\int \\limits _{\\Omega } \\Phi ^\\varepsilon (x) {S}^{\\bf 0} \\varphi _n^{\\bf 0}\\, dx +\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} K^\\varepsilon (x)\\bigg \\lbrace \\lambda _{\\,\\mathsf {f},n} (S^\\varepsilon _\\mathsf {f})\\left(\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}- \\vec{g} \\right) - \\nabla \\beta _\\mathsf {f}(S^\\varepsilon _\\mathsf {f})\\bigg \\rbrace \\cdot \\nabla \\varphi _n\\, dx dt +$ $+ \\varkappa (\\varepsilon )\\,\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} K^\\varepsilon (x)\\bigg \\lbrace \\lambda _{\\,\\mathsf {m},n} (S^\\varepsilon _\\mathsf {m})\\left(\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {m}- \\vec{g} \\right) - \\nabla \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m})\\bigg \\rbrace \\cdot \\nabla \\varphi _n\\, dx dt = \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} F^\\varepsilon _{n}\\, \\varphi _n\\, dx dt.$ Existence theorem for the weak solutions defined in Definition REF and Definition REF is given in [10] in more general case of compressible fluids.",
"Notational convention.",
"In what follows $C, C_1,..$ denote generic constants that do not depend on $\\varepsilon $ ."
],
[
"A priori uniform estimates",
"The uniform estimates for the initial system (REF ) or the equivalent one (REF ) are given by the following lemma: Lemma 3.1 Let $\\langle {\\mathsf {p}}^\\varepsilon _w, {\\mathsf {p}}^\\varepsilon _n , S^\\varepsilon \\rangle $ be a solution to problem (REF ).",
"Then under assumptions (A.1)-(A.9) the following uniform in $\\varepsilon $ estimates hold true: $\\big \\Vert \\sqrt{\\lambda _{\\mathsf {f},w} \\left(S^\\varepsilon _\\mathsf {f}\\right)}\\, \\nabla p^\\varepsilon _{\\mathsf {f},w}\\big \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}+\\big \\Vert \\sqrt{\\lambda _{\\mathsf {f},n} \\left(S^\\varepsilon _\\mathsf {f}\\right)}\\, \\nabla p^\\varepsilon _{\\mathsf {f},n} \\big \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}+$ $+\\varkappa ^{\\frac{1}{2}}(\\varepsilon )\\,\\big \\Vert \\sqrt{\\lambda _{\\mathsf {m},w} \\left(S^\\varepsilon _\\mathsf {m}\\right)}\\, \\nabla p^\\varepsilon _{\\mathsf {m},w}\\big \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})}+\\varkappa ^{\\frac{1}{2}}(\\varepsilon )\\,\\big \\Vert \\sqrt{\\lambda _{\\mathsf {m},n} \\left(S^\\varepsilon _\\mathsf {m}\\right)}\\, \\nabla p^\\varepsilon _{\\mathsf {m},n} \\big \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})} \\leqslant C;$ $\\left\\Vert \\nabla \\beta _\\mathsf {f}(S^\\varepsilon _\\mathsf {f})\\right\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}+\\left\\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\right\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}+\\varkappa ^{\\frac{1}{2}}(\\varepsilon )\\,\\left\\Vert \\nabla \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m})\\right\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})}+\\varkappa ^{\\frac{1}{2}}(\\varepsilon )\\,\\left\\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\right\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})}\\leqslant C,$ where $\\varkappa (\\varepsilon ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varepsilon ^\\theta $ with $\\theta > 0$ .",
"Proof of Lemma REF .",
"Notice that the uniform boundedness results (REF ), (REF ) were already proved by many authors (see, e.g., [40] and the references therein) in the case when the source terms in (REF ) were assumed to be zero.",
"We also refer here to [5] and the references therein, where the uniform boundedness results were obtained in the case of compressible two-phase flows in porous media.",
"Here, for reader's convenience, we recall the proof of the bounds (REF ), (REF ) focusing on the terms involving the source functions $F^\\varepsilon _{w}, F^\\varepsilon _{n}$ .",
"We start our analysis by obtaining the uniform bound (REF ).",
"To this end we multiply the first equation in (REF ) by ${\\mathsf {p}}^\\varepsilon _{w}$ , the second equation in (REF ) by ${\\mathsf {p}}^\\varepsilon _{n}$ and then integrate over the domain $\\Omega $ .",
"Taking into account the boundary conditions (REF ) after integration by parts, we get the following energy equality: $- \\frac{d}{d t}\\int \\limits _\\Omega \\Phi ^\\varepsilon (x)\\,\\digamma ({S}^\\varepsilon ) \\, dx +\\int \\limits _\\Omega \\left\\lbrace K^\\varepsilon (x) \\lambda _{w}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\left(\\nabla {\\mathsf {p}}^\\varepsilon _{w} - \\vec{g}\\right)\\right\\rbrace \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{w} \\, dx +$ $+\\int \\limits _\\Omega \\left\\lbrace K^\\varepsilon (x) \\lambda _{n}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\left(\\nabla {\\mathsf {p}}^\\varepsilon _{n} - \\vec{g}\\right)\\right\\rbrace \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{n} \\, dx=\\int \\limits _\\Omega \\left[F^\\varepsilon _{w}(x, t)\\,{\\mathsf {p}}^\\varepsilon _{w} + F^\\varepsilon _{n}(x, t)\\,{\\mathsf {p}}^\\varepsilon _{n}\\right],$ where $\\digamma ({S}^\\varepsilon ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\digamma _\\mathsf {f}(S^\\varepsilon _\\mathsf {f})\\, {\\bf 1}^\\varepsilon _\\mathsf {f}(x) +\\digamma _\\mathsf {m}(S^\\varepsilon _\\mathsf {m})\\, {\\bf 1}^\\varepsilon _\\mathsf {m}(x)\\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\bf 1}^\\varepsilon _\\mathsf {f}(x)\\, \\int \\limits _1^{S^\\varepsilon _\\mathsf {f}} P_{\\mathsf {f},c}(u)\\, du +{\\bf 1}^\\varepsilon _\\mathsf {m}(x)\\, \\int \\limits _1^{S^\\varepsilon _\\mathsf {m}} P_{\\mathsf {m},c}(u)\\, du.$ The equality (REF ) is the desired energy equality which will be used below to obtain the necessary bounds that are uniform in $\\varepsilon $ .",
"To this end we integrate (REF ) over the interval $(0, T)$ to get: $- \\int \\limits _\\Omega \\Phi ^\\varepsilon (x)\\,\\digamma ({S}^\\varepsilon ) \\, dx +\\int \\limits _{\\Omega _T}\\left\\lbrace K^\\varepsilon (x) \\lambda _{w}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\left(\\nabla {\\mathsf {p}}^\\varepsilon _{w} - \\vec{g}\\right)\\right\\rbrace \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{w} \\, dx dt +$ $+\\int \\limits _{\\Omega _T}\\left\\lbrace K^\\varepsilon (x) \\lambda _{n}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\left(\\nabla {\\mathsf {p}}^\\varepsilon _{n} - \\vec{g}\\right)\\right\\rbrace \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{n} \\, dx dt = \\mathbb {J}^\\varepsilon _{w,n} -\\int \\limits _\\Omega \\Phi ^\\varepsilon (x)\\,\\digamma \\left({S}^\\varepsilon (x, 0)\\right) \\, dx,$ where $\\mathbb {J}^\\varepsilon _{w,n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\int \\limits _{\\Omega _T} \\left[F^\\varepsilon _{w}(x, t)\\,{\\mathsf {p}}^\\varepsilon _{w} +F^\\varepsilon _{n}(x, t)\\,{\\mathsf {p}}^\\varepsilon _{n}\\right]\\, dx dt.$ First, we notice that due to the positiveness of the porosity function $\\Phi ^\\varepsilon $ and the definition of the function $\\digamma \\left({S}^\\varepsilon \\right)$ we have that the first term on the left-hand side of (REF ) is bounded from below by a constant which does not depend on $\\varepsilon $ .",
"It is also easy to see from conditions (A.1), (A.3) that the second term on the right-hand side of (REF ) is uniformly bounded in $\\varepsilon $ .",
"Then from (REF ) we get the following inequality: $\\int \\limits _{\\Omega _T}K^\\varepsilon (x) \\lambda _{w}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\nabla {\\mathsf {p}}^\\varepsilon _{w} \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{w} \\, dx dt +\\int \\limits _{\\Omega _T}K^\\varepsilon (x) \\lambda _{n}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\nabla {\\mathsf {p}}^\\varepsilon _{n} \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{n} \\, dx dt \\leqslant $ $\\leqslant C+\\int \\limits _{\\Omega _T}K^\\varepsilon (x) \\lambda _{w}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right) \\vec{g} \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{w} \\, dx dt + \\int \\limits _{\\Omega _T}K^\\varepsilon (x) \\lambda _{n}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right) \\vec{g} \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{n} \\, dx dt + \\mathbb {J}^\\varepsilon _{w,n}.$ With the help of Young's inequality the second and the third terms in the right-hand side of (REF ) can be absorbed by the first and second term in the left-hand side of (REF ).",
"Namely, we get: $\\int \\limits _{\\Omega _T}K^\\varepsilon (x) \\lambda _{w}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\nabla {\\mathsf {p}}^\\varepsilon _{w} \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{w} \\, dx dt +\\int \\limits _{\\Omega _T} K^\\varepsilon (x) \\lambda _{n}\\left(\\frac{x}{\\varepsilon }, {S}^\\varepsilon \\right)\\nabla {\\mathsf {p}}^\\varepsilon _{n} \\cdot \\nabla {\\mathsf {p}}^\\varepsilon _{n} \\, dx dt\\leqslant C \\big [1 + \\mathbb {J}^\\varepsilon _{w,n}\\big ].$ Now it remains to estimate $\\mathbb {J}^\\varepsilon _{w,n}$ .",
"Due to condition (A.9), it can be written as: $\\mathbb {J}^\\varepsilon _{w,n} =\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\big [S^I_{\\mathsf {f}, w} f_I(x,t) - S^\\varepsilon _{\\mathsf {f}} f_P(x,t)\\big ]\\,p^\\varepsilon _{\\mathsf {f},w}\\, dx dt+\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}}\\big [S^I_{\\mathsf {f}, n} f_I(x,t) - (1 - S^\\varepsilon _{\\mathsf {f}}) f_P(x,t)\\big ] \\,p^\\varepsilon _{\\mathsf {f},n}\\, dx dt \\mathop {=}\\limits ^{\\hbox{\\tiny def}}$ $\\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\mathbb {J}^\\varepsilon _{w} + \\mathbb {J}^\\varepsilon _{n}.$ Consider, first, the term $\\mathbb {J}^\\varepsilon _{w}$ .",
"From the boundedness of the saturation functions, Cauchy's inequality and condition (A.9), we get: $\\big |\\mathbb {J}^\\varepsilon _{w} \\big |\\leqslant \\left[\\Vert f_I \\Vert _{L^2(\\Omega _T)}+\\Vert f_P \\Vert _{L^2(\\Omega _T)}\\right] \\Vert p^\\varepsilon _{\\mathsf {f},w} \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}\\leqslant C_1\\, \\Vert p^\\varepsilon _{\\mathsf {f},w} \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}.$ In a similar way, $\\big |\\mathbb {J}^\\varepsilon _{n} \\big | \\leqslant C_2\\, \\Vert p^\\varepsilon _{\\mathsf {f},n} \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}.$ Now using condition (A.2), (REF ), (REF ), and (REF ), from the inequality (REF ), we get: $\\mathbb {L}^\\varepsilon \\mathop {=}\\limits ^{\\hbox{\\tiny def}}k_{\\rm min}\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\lambda _{\\mathsf {f},w}(S^\\varepsilon _\\mathsf {f}) \\big |\\nabla p^\\varepsilon _{\\mathsf {f},w} \\big |^2\\, dx dt +k_{\\rm min}\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\lambda _{\\mathsf {f},n}(S^\\varepsilon _\\mathsf {f}) \\big |\\nabla p^\\varepsilon _{\\mathsf {f},n} \\big |^2\\, dx dt +$ $+\\varkappa (\\varepsilon )\\, k_{\\rm min}\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} \\lambda _{\\mathsf {m},w}(S^\\varepsilon _\\mathsf {m}) \\big |\\nabla p^\\varepsilon _{\\mathsf {m},w} \\big |^2\\, dx dt +\\varkappa (\\varepsilon )\\, k_{\\rm min}\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} \\lambda _{\\mathsf {m},n}(S^\\varepsilon _\\mathsf {m}) \\big |\\nabla p^\\varepsilon _{\\mathsf {m},n} \\big |^2\\, dx dt\\leqslant $ $\\leqslant C_3 \\left[1 + \\Vert p^\\varepsilon _{\\mathsf {f},w} \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} +\\Vert p^\\varepsilon _{\\mathsf {f},n} \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}\\right].$ Consider the right-hand side of (REF ).",
"From (REF ) we have: $\\Vert p^\\varepsilon _{\\mathsf {f},w} \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} + \\Vert p^\\varepsilon _{\\mathsf {f},n} \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}\\leqslant $ $\\leqslant \\left[\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} +\\Vert {\\mathsf {G}}_{\\mathsf {f},w}(S^\\varepsilon _\\mathsf {f}) \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} +\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} +\\Vert {\\mathsf {G}}_{\\mathsf {f},n}(S^\\varepsilon _\\mathsf {f}) \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}\\right].$ Then, taking into account that the functions ${\\mathsf {G}}_{\\mathsf {f},w}(S^\\varepsilon _\\mathsf {f}), {\\mathsf {G}}_{\\mathsf {f},n}(S^\\varepsilon _\\mathsf {f})$ are uniformly bounded in $\\varepsilon $ , the inequality (REF ) takes the form: $\\mathbb {L}^\\varepsilon \\leqslant C_4 \\left[1 + \\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} \\right].$ Taking into account the boundary condition ${\\mathsf {P}}^\\varepsilon = {\\mathsf {P}}_{\\Gamma _1} = {\\rm Const}$ on $\\Gamma _{1} \\times (0,T)$ and applying Friedrich's inequality we obtain that $\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} \\leqslant C_5\\, \\left[1 +\\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} \\right].$ Finally, in view of (REF ), the inequality (REF ) takes the form: $\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\lambda _{\\mathsf {f},w}(S^\\varepsilon _\\mathsf {f}) \\big |\\nabla p^\\varepsilon _{\\mathsf {f},w} \\big |^2\\, dx dt +\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\lambda _{\\mathsf {f},n}(S^\\varepsilon _\\mathsf {f}) \\big |\\nabla p^\\varepsilon _{\\mathsf {f},n} \\big |^2\\, dx dt +\\varkappa (\\varepsilon )\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} \\lambda _{\\mathsf {m},w}(S^\\varepsilon _\\mathsf {m}) \\big |\\nabla p^\\varepsilon _{\\mathsf {m},w} \\big |^2\\, dx dt +$ $+\\varkappa (\\varepsilon )\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} \\lambda _{\\mathsf {m},n}(S^\\varepsilon _\\mathsf {m}) \\big |\\nabla p^\\varepsilon _{\\mathsf {m},n} \\big |^2\\, dx dt\\leqslant C_6 \\left[1 + \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} \\right].$ In order to complete the derivation of the uniform estimate, we make use of the equality (REF ).",
"We estimate the norm of $\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}$ using the Cauchy inequality as follows: $C_6\\, \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} \\leqslant C_6\\, \\frac{\\eta }{2}\\, \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert ^2_{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} +C_6\\, \\frac{1}{2\\eta },$ where $\\eta > 0$ is an arbitrary number.",
"Moreover, it follows from (REF ) that $\\lambda _{\\mathsf {f}} (S^\\varepsilon _\\mathsf {f}) |\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}|^2 \\leqslant \\lambda _{\\mathsf {f},n} (S^\\varepsilon _\\mathsf {f}) |\\nabla p^\\varepsilon _{\\mathsf {f},n}|^2 +\\lambda _{\\mathsf {f},w} (S^\\varepsilon _\\mathsf {f}) |\\nabla p^\\varepsilon _{\\mathsf {f},w}|^2.$ Now (REF ) allows us to rewrite (REF ) in the form: $\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\lambda _{\\mathsf {f},w}(S^\\varepsilon _\\mathsf {f}) \\big |\\nabla p^\\varepsilon _{\\mathsf {f},w} \\big |^2\\, dx dt +\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\lambda _{\\mathsf {f},n}(S^\\varepsilon _\\mathsf {f}) \\big |\\nabla p^\\varepsilon _{\\mathsf {f},n} \\big |^2\\, dx dt +\\varkappa (\\varepsilon )\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} \\lambda _{\\mathsf {m},w}(S^\\varepsilon _\\mathsf {m}) \\big |\\nabla p^\\varepsilon _{\\mathsf {m},w} \\big |^2\\, dx dt +$ $+\\varkappa (\\varepsilon )\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} \\lambda _{\\mathsf {m},n}(S^\\varepsilon _\\mathsf {m}) \\big |\\nabla p^\\varepsilon _{\\mathsf {m},n} \\big |^2\\, dx dt\\leqslant C_6 + C_6\\, \\frac{\\eta }{2}\\, \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert ^2_{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} +C_6\\, \\frac{1}{2\\eta }.$ Let us estimate the second term on the right-hand side of (REF ).",
"From condition (A.4) and (REF ), we have: $\\begin{split}C_6\\, \\frac{\\eta }{2}\\, \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert ^2_{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}&\\leqslant \\frac{C_6\\eta }{2 L_0}\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\lambda _{\\mathsf {f}} (S^\\varepsilon _\\mathsf {f})\\,\\big |\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\big |^2\\, dx dt\\\\&\\leqslant \\frac{C_6\\eta }{2 L_0}\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}}\\left[\\lambda _{\\mathsf {f},w}(S^\\varepsilon _\\mathsf {f}) \\big |\\nabla p^\\varepsilon _{\\mathsf {f},w} \\big |^2 +\\lambda _{\\mathsf {f},n}(S^\\varepsilon _\\mathsf {f}) \\big |\\nabla p^\\varepsilon _{\\mathsf {f},n} \\big |^2 \\right]\\, dx dt.\\end{split}$ We set $\\eta = \\frac{L_0}{C_6}$ and, finally, obtain from (REF ) the desired inequality (REF ).",
"Now we turn to the uniform bound (REF ).",
"It immediately follows from (REF ) equality (REF ) and the following inequality: $\\left|\\nabla \\beta _\\ell (S^\\varepsilon _\\ell ) \\right| \\leqslant C\\,\\left|\\nabla \\mathfrak {b}_\\ell (S^\\varepsilon _\\ell ) \\right|$ .",
"This completes the proof of Lemma REF .",
"Lemma 3.2 Let $\\langle {\\mathsf {p}}^\\varepsilon _w, {\\mathsf {p}}^\\varepsilon _n, S^\\varepsilon \\rangle $ be a solution to problem (REF ) and $\\varkappa (\\varepsilon ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varepsilon ^\\theta $ with $\\theta \\leqslant 2$ .",
"Then under assumptions (A.1)-(A.9) the following uniform in $\\varepsilon $ estimate holds true: $\\left\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\right\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})} \\leqslant C.$ Proof of Lemma REF .",
"In contrast to the papers [16], [40], where the standing assumptions allow to prove the continuity of the global pressure on the interface $\\Sigma ^\\varepsilon _T$ , in our case the global pressure is discontinuous on $\\Sigma ^\\varepsilon _T$ .",
"So the method which allowed to prove (REF ) by use of the extension operator from the subdomain $\\Omega ^\\varepsilon _\\mathsf {f}$ to the whole $\\Omega $ cannot be applied here.",
"To avoid this difficulty we make use of the ideas from [27] (see also [9]).",
"Since ${\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\in L^2(0, T; H^1(\\Omega ^\\varepsilon _\\mathsf {m}))$ and ${\\mathsf {P}}^\\varepsilon _\\mathsf {f}-{\\mathsf {P}}_{\\Gamma _1} \\in L^2(0, T; H^1_{\\Gamma _{1}}(\\Omega ^\\varepsilon _\\mathsf {f}))$ , then we have: $\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega ^\\varepsilon _{m,T})} \\leqslant C\\, \\left[\\varepsilon \\, \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})}+ \\sqrt{\\varepsilon }\\, \\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Sigma ^\\varepsilon _T)} \\right].$ Then due to the definition of the global pressure ${\\mathsf {P}}^\\varepsilon _\\mathsf {m}$ , (REF ), and the interface condition (REF ) written in terms of the global pressure, one obtains the following estimate: $\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Sigma ^\\varepsilon _T)} \\leqslant \\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {m}+{\\mathsf {G}}_{\\mathsf {m}, w}(S^\\varepsilon _\\mathsf {m}) \\Vert _{L^2(\\Sigma ^\\varepsilon _T)}+\\Vert {\\mathsf {G}}_{\\mathsf {m}, w}(S^\\varepsilon _\\mathsf {m}) \\Vert _{L^2(\\Sigma ^\\varepsilon _T)} =\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {f}+ {\\mathsf {G}}_{\\mathsf {f}, w}(S^\\varepsilon _\\mathsf {f}) \\Vert _{L^2(\\Sigma ^\\varepsilon _T)}+$ $+ \\Vert {\\mathsf {G}}_{\\mathsf {m}, w}(S^\\varepsilon _\\mathsf {m}) \\Vert _{L^2(\\Sigma ^\\varepsilon _T)}\\leqslant \\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Sigma ^\\varepsilon _T)}+\\Vert {\\mathsf {G}}_{\\mathsf {f}, w}(S^\\varepsilon _\\mathsf {f}) \\Vert _{L^2(\\Sigma ^\\varepsilon _T)}+\\Vert {\\mathsf {G}}_{\\mathsf {m}, w}(S^\\varepsilon _\\mathsf {m}) \\Vert _{L^2(\\Sigma ^\\varepsilon _T)}.$ Now, taking into account the boundedness of ${\\mathsf {G}}_{\\ell , w}(S^\\varepsilon _\\ell )$ , the geometry of $\\Omega ^\\varepsilon _{\\mathsf {m},T}$ , (REF ), and the estimate: $\\sqrt{\\varepsilon }\\, \\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Sigma ^\\varepsilon _T)} \\leqslant C\\, \\left[\\varepsilon \\, \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})}+ \\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} \\right]$ we obtain $\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})} \\leqslant C \\left( \\varepsilon \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})} + 1\\right)= C \\left( \\frac{\\varepsilon }{\\varkappa ^{\\frac{1}{2}}(\\varepsilon )} \\, \\varkappa ^{\\frac{1}{2}}(\\varepsilon ) \\Vert \\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})} + 1\\right).$ By using (REF ), from (REF ) we get $\\Vert {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})} \\leqslant C \\left( {\\varepsilon }{\\varkappa ^{-\\frac{1}{2}}(\\varepsilon )} + 1\\right),$ which means that for $\\varkappa (\\varepsilon ) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varepsilon ^\\theta $ with $\\theta \\leqslant 2$ the desired inequality (REF ) is obtained.",
"Lemma REF is proved.",
"Let us pass to the uniform bounds for the time derivatives of ${S}^\\varepsilon $ .",
"In a standard way (see, e.g., [6]) we get: Lemma 3.3 Let $\\langle {\\mathsf {p}}^\\varepsilon _w, {\\mathsf {p}}^\\varepsilon _n, S^\\varepsilon \\rangle $ be a solution to problem (REF ).",
"Then under assumptions (A.1)-(A.9) the following uniform in $\\varepsilon $ estimate holds true: $\\left\\lbrace \\partial _t(\\Phi ^\\varepsilon _\\ell S^\\varepsilon _\\ell ) \\right\\rbrace _{\\varepsilon >0} \\quad {\\rm is \\,\\, uniformly \\,\\,bounded \\,\\, in} \\,\\, L^2(0,T;H^{-1}(\\Omega ^\\varepsilon _\\ell )),$ where the functions $\\Phi ^\\varepsilon _\\mathsf {f}, \\Phi ^\\varepsilon _\\mathsf {m}$ are defined in condition (A.1)."
],
[
"Compactness and convergence results",
"The outline of this section is as follows.",
"First, in subection REF we extend the function $S^\\varepsilon _\\mathsf {f}$ from the subdomain $\\Omega ^\\varepsilon _\\mathsf {f}$ to the whole $\\Omega $ and obtain uniform estimates for the extended function $\\widetilde{S}^\\varepsilon _\\mathsf {f}$ .",
"Then in subsection REF , using the uniform estimates for the function $\\widetilde{\\mathsf {P}}^\\varepsilon _\\mathsf {f}$ and the corresponding bounds for $\\widetilde{S}^\\varepsilon _\\mathsf {f}$ , we prove the compactness result for the family $\\lbrace \\widetilde{S}^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$ .",
"Finally, in subsection REF we formulate the two-scale convergence which will be used in the derivation of the homogenized system."
],
[
"Extensions of the functions ${\\mathsf {P}}^\\varepsilon _\\mathsf {f}$ , {{formula:d6df24fd-06ce-498c-ae90-1c730a22ef99}}",
"The goal of this subsection is to extend the functions ${\\mathsf {P}}^\\varepsilon _\\mathsf {f}$ , $S^\\varepsilon _\\mathsf {f}$ defined in the subdomain $\\Omega ^\\varepsilon _\\mathsf {f}$ to the whole $\\Omega $ and derive the uniform in $\\varepsilon $ estimates for the extended functions.",
"Extension of the function ${\\mathsf {P}}^\\varepsilon _\\mathsf {f}$ .",
"First, we introduce the extension operator from the subdomain $\\Omega ^\\varepsilon _{\\mathsf {f}}$ to the whole $\\Omega $ .",
"Taking into account the results of [1] we conclude that there exists a linear continuous extension operator $\\Pi ^\\varepsilon : H^1(\\Omega ^\\varepsilon _{\\mathsf {f}}) \\longrightarrow H^1(\\Omega )$ such that: (i) $\\Pi ^\\varepsilon u = u$ in $\\Omega ^\\varepsilon _{\\mathsf {f}}$ and (ii) for any $u \\in H^1(\\Omega ^\\varepsilon _{\\mathsf {f}})$ , $\\Vert \\Pi ^\\varepsilon u \\Vert _{L^2(\\Omega )} \\leqslant C\\, \\Vert u \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f}})}\\quad {\\rm and} \\quad \\Vert \\nabla (\\Pi ^\\varepsilon u) \\Vert _{L^2(\\Omega )} \\leqslant C\\, \\Vert \\nabla u \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f}})},$ where $C$ is a constant that does not depend on $u$ and $\\varepsilon $ .",
"Now it follows from (REF ) and the Dirichlet boundary condition on $\\Gamma _{1}$ , that $\\left\\Vert \\nabla (\\Pi ^\\varepsilon P^\\varepsilon _\\mathsf {f}) \\right\\Vert _{L^2(\\Omega _{T})} +\\left\\Vert \\Pi ^\\varepsilon P^\\varepsilon _\\mathsf {f}\\right\\Vert _{L^2(\\Omega _{T})} \\leqslant C.$ Notational convention.",
"In what follows the extension of any function $f$ will be denoted by $\\widetilde{f}$ instead of $\\Pi ^\\varepsilon f$ .",
"Extension of the function $S^\\varepsilon _\\mathsf {f}$ .",
"In order to extend $S^\\varepsilon _\\mathsf {f}$ , following the ideas of [16], we make use of the function $\\beta _\\mathsf {f}$ defined in (REF ).",
"It is evident that $\\beta _\\mathsf {f}$ is a monotone function of $s$ .",
"Let us introduce the function: $\\beta ^\\varepsilon _\\mathsf {f}(x, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\beta _\\mathsf {f}(S^\\varepsilon _\\mathsf {f}) = \\int \\limits _0^{S^\\varepsilon _\\mathsf {f}} \\alpha _\\mathsf {f}(u)\\, du.$ Then it follows from condition (A.5) that $0 \\leqslant \\beta ^\\varepsilon _\\mathsf {f}\\leqslant \\max _{s\\in [0, 1]}\\alpha _\\mathsf {f}(s) \\quad {\\rm a.e.\\,\\, in} \\,\\, \\Omega ^\\varepsilon _{\\mathsf {f},T}.$ It is also clear from (REF ) that $\\Vert \\nabla \\beta ^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {f},T})} \\leqslant C.$ Hence, $0 \\leqslant \\widetilde{\\beta }^\\varepsilon _\\mathsf {f}\\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\Pi ^\\varepsilon \\beta ^\\varepsilon _\\mathsf {f}\\leqslant \\max _{s\\in [0, 1]}\\alpha _{\\mathsf {f}}(s) \\,\\, {\\rm a.e.\\,\\, in} \\,\\, \\Omega _T\\quad {\\rm and} \\quad \\Vert \\nabla \\widetilde{\\beta }^\\varepsilon _\\mathsf {f}\\Vert _{L^2(\\Omega _{T})} \\leqslant C.$ Now we can extend $S^\\varepsilon _\\mathsf {f}$ from $\\Omega ^\\varepsilon _\\mathsf {f}$ to the whole $\\Omega $ .",
"We denote this extension by $\\widetilde{S}^\\varepsilon _\\mathsf {f}$ and define it as follows: $\\widetilde{S}^\\varepsilon _\\mathsf {f}\\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\, (\\beta _\\mathsf {f})^{-1}(\\widetilde{\\beta }^\\varepsilon _\\mathsf {f}).$ This implies that $\\int \\limits _{\\Omega _T} \\big |\\nabla \\beta _\\mathsf {f}\\big (\\,\\widetilde{S}^\\varepsilon _\\mathsf {f}\\,\\big )\\big |^2 \\,\\,dx\\, dt =\\int \\limits _{\\Omega _T}\\big |\\nabla \\widetilde{\\beta }^\\varepsilon _\\mathsf {f}\\big |^2 \\,\\,dx\\, dt \\leqslant C\\quad {\\rm and} \\quad 0 \\leqslant \\widetilde{S}^\\varepsilon _\\mathsf {f}\\leqslant 1 \\,\\, {\\rm a.e.\\,\\, in}\\,\\, \\Omega _T.$"
],
[
"Compactness results for the sequence $\\lbrace \\widetilde{S}^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$",
"In this subsection we establish the compactness and corresponding convergence results for the sequence $\\lbrace \\widetilde{S}^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$ constructed in the previous section.",
"Proposition 4.1 Under our standing assumptions there is a function $S$ such that $0 \\leqslant S \\leqslant 1$ in $\\Omega _T$ and (up to a subsequence) $\\widetilde{S}^\\varepsilon _\\mathsf {f}\\longrightarrow S \\,\\, {\\rm strongly\\,\\, in}\\,\\, L^q(\\Omega _T)\\,\\,{\\rm for\\, all} \\,\\, 1 \\leqslant q < +\\infty .$ Proof of Proposition REF .",
"In the proof of Proposition REF we follow the lines of [16], [40].",
"Namely, first, we establish the modulus of continuity in time for $\\widetilde{\\beta }^\\varepsilon _\\mathsf {f}$ and then apply the compactness result from [37].",
"The derivation of the modulus of continuity in time is based on the lemma obtained earlier in [40], (see also [6]).",
"Lemma 4.1 For $h$ sufficiently small, we have: $\\int \\limits _h^T \\int \\limits _{\\Omega ^\\varepsilon _\\mathsf {f}} \\big [S^\\varepsilon _\\mathsf {f}(t) - S^\\varepsilon _\\mathsf {f}(t - h)\\big ]\\,\\big [\\beta ^\\varepsilon _\\mathsf {f}(t) - \\beta ^\\varepsilon _\\mathsf {f}(t - h) \\big ]\\, dx\\, dt \\leqslant C\\, h\\quad {\\rm with} \\,\\, \\beta ^\\varepsilon _\\mathsf {f}\\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\beta _\\mathsf {f}(S^\\varepsilon _\\mathsf {f}),$ where $C$ is a constant that does not depend on $\\varepsilon , h$ .",
"Corollary 4.2 For $h$ sufficiently small, we have: $\\int \\limits _{\\Omega ^h_T}\\big |\\widetilde{\\beta }^\\varepsilon _\\mathsf {f}(t) - \\widetilde{\\beta }^\\varepsilon _\\mathsf {f}(t - h) \\big |^2\\, dx\\, dt\\leqslant C h\\quad {\\rm with} \\,\\, \\Omega ^h_T \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\Omega \\times (h, T).$ Proof of Corollary REF .",
"First, let us show that the bound (REF ) implies: $\\int \\limits _h^T \\int \\limits _{\\Omega ^\\varepsilon _\\mathsf {f}}\\big |\\beta ^\\varepsilon _\\mathsf {f}(t) - \\beta ^\\varepsilon _\\mathsf {f}(t - h) \\big |^2\\, dx\\, dt \\leqslant C\\, h.$ In fact, it is clear that due to the definition of the function $\\beta _\\mathsf {f}$ and condition (A.6) we have: $\\left|\\beta _\\mathsf {f}(S^\\varepsilon _\\mathsf {f}(t)) - \\beta _\\mathsf {f}(S^\\varepsilon _\\mathsf {f}(t-h)) \\right| =\\left|\\int \\limits ^{S^\\varepsilon _\\mathsf {f}(t)}_{S^\\varepsilon _\\mathsf {f}(t-h)} \\alpha _\\mathsf {f}(\\xi )\\,d\\xi \\right|\\leqslant \\max _{s\\in [0,1]} \\alpha _\\mathsf {f}(s)\\, |S^\\varepsilon _\\mathsf {f}(t) - S^\\varepsilon _\\mathsf {f}(t - h)|.$ Then from (REF ) we get: $\\int \\limits _h^T \\int \\limits _{\\Omega ^\\varepsilon _\\mathsf {f}}\\big |\\beta ^\\varepsilon _\\mathsf {f}(t) - \\beta ^\\varepsilon _\\mathsf {f}(t - h) \\big |^2\\, dx\\, dt \\leqslant C\\,\\int \\limits _h^T \\int \\limits _{\\Omega ^\\varepsilon _\\mathsf {f}} \\big [S^\\varepsilon _\\mathsf {f}(t) - S^\\varepsilon _\\mathsf {f}(t - h)\\big ]\\,\\big [\\beta ^\\varepsilon _\\mathsf {f}(t) - \\beta ^\\varepsilon _\\mathsf {f}(t - h) \\big ]\\, dx\\, dt \\leqslant C\\, h$ and the desired bound (REF ) is obtained.",
"Now using the property (REF ) of the extension operator, from (REF ) we get (REF ).",
"This completes the proof of Corollary REF .",
"Now we are in position to complete the proof of Proposition REF .",
"First, we observe that the sequence $\\lbrace \\widetilde{\\beta }^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$ is uniformly bounded in the space $L^2(0, T; H^1_{\\Gamma _1}(\\Omega ))$ and this sequence satisfies (REF ).",
"Then it follows from [37] that $\\lbrace \\widetilde{\\beta }^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$ is a compact set in the space $L^2(\\Omega _T)$ and we have that $\\widetilde{\\beta }^\\varepsilon _\\mathsf {f}\\rightarrow \\beta ^\\star $ strongly in $L^2(\\Omega _T)$ and due to the uniform boundedness of the function $\\widetilde{\\beta }^\\varepsilon _\\mathsf {f}$ in the space $L^\\infty (\\Omega _T)$ , $\\widetilde{\\beta }^\\varepsilon _\\mathsf {f}\\rightarrow \\beta ^\\star \\,\\, {\\rm strongly\\,\\, in}\\,\\, L^q(\\Omega _T)\\,\\,{\\rm for\\, all} \\,\\, 1 \\leqslant q < +\\infty .$ Now we recall that the extended saturation function $\\widetilde{S}^\\varepsilon _\\mathsf {f}$ is defined by $\\widetilde{S}^\\varepsilon _\\mathsf {f}\\mathop {=}\\limits ^{\\hbox{\\tiny def}}(\\beta _\\mathsf {f})^{-1}(\\widetilde{\\beta }^\\varepsilon _\\mathsf {f})$ .",
"We set $S \\mathop {=}\\limits ^{\\hbox{\\tiny def}}(\\beta _\\mathsf {f})^{-1}(\\beta ^\\star ).$ Then from condition (A.6) we have: $\\Vert \\widetilde{S}^\\varepsilon _\\mathsf {f}- S \\Vert _{L^q(\\Omega _T)} =\\Vert (\\beta _\\mathsf {f})^{-1}(\\widetilde{\\beta }^\\varepsilon _\\mathsf {f}) - (\\beta _\\mathsf {f})^{-1}(\\beta ^\\star ) \\Vert _{L^q(\\Omega _T)}\\leqslant C_\\beta \\, \\Vert \\widetilde{\\beta }^\\varepsilon _\\mathsf {f}- \\beta ^\\star \\Vert ^{\\gamma }_{L^{q\\gamma }(\\Omega _T)}.$ This inequality along with (REF ) implies (REF ) and Proposition REF is proved."
],
[
"Two-scale convergence results",
"In this subsection, taking into account the compactness results from the previous section, we formulate the convergence results for the sequences $\\lbrace \\widetilde{P}^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$ , $\\lbrace \\widetilde{S}^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$ .",
"In this paper the homogenization process for the problem is rigorously obtained by using the two-scale approach, see, e.g., [3].",
"For the reader's convenience, let us recall the definition of the two-scale convergence.",
"Definition 4.3 A sequence of functions $\\lbrace v^\\varepsilon \\rbrace _{\\varepsilon >0} \\subset L^2(\\Omega _T)$ two-scale converges to $v \\in L^2(\\Omega _T\\times Y)$ if $\\Vert v^\\varepsilon \\Vert _{L^2(\\Omega _T)} \\leqslant C$ , and for any test function $\\varphi \\in C^\\infty (\\overline{\\Omega _T}; C_\\#(Y))$ the following relation holds: $\\lim _{\\varepsilon \\rightarrow 0} \\int \\limits _{\\Omega _T} v^\\varepsilon (x, t)\\, \\varphi \\left(x, \\frac{x}{\\varepsilon }, t\\right)\\, dx\\, dt =\\int \\limits _{\\Omega _T \\times Y} v(x, y, t)\\,\\varphi (x, y, t) \\, dy\\, dx\\, dt.$ This convergence is denoted by $v^\\varepsilon (x, t) \\stackrel{2s}{\\rightharpoonup }v(x, y, t)$ .",
"Following [4] we also introduce the two-scale convergence on periodic surfaces: Definition 4.4 A sequence of functions $\\lbrace v^\\varepsilon \\rbrace _{\\varepsilon >0} \\subset L^2(\\Sigma ^{\\varepsilon }_T)$ two-scale converges to $v \\in L^2(\\Omega _T; L^2(\\Gamma _{\\mathsf {f}\\mathsf {m}}))$ on $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ if for any test function $\\varphi \\in C^\\infty (\\overline{\\Omega _T}; C_\\#(Y))$ the following relation holds: $\\lim _{\\varepsilon \\rightarrow 0} \\varepsilon \\int \\limits _{\\Sigma ^{\\varepsilon }_T} v^\\varepsilon (x, t)\\,\\varphi \\left(x, \\frac{x}{\\varepsilon }, t\\right)\\, dH^{d-1}(x)\\, dt =\\int \\limits _{\\Omega _T}\\int \\limits _{\\Gamma _{\\mathsf {f}\\mathsf {m}}} v(x, y, t)\\,\\varphi (x, y, t) \\, dH^{d-1}(y)\\, dx\\, dt,$ where, as before $\\Sigma ^{\\varepsilon }_T \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\Gamma _{\\mathsf {f}\\mathsf {m}}^{\\varepsilon }\\times (0,T)$ , and $dH^{d-1}$ is the $(d-1)$ -dimensional Hausdorff measure.",
"This convergence is denoted by $v^\\varepsilon (x, t) \\stackrel{2s-\\Gamma _{\\mathsf {m}\\mathsf {f}}}{\\rightharpoonup }v(x, y, t)$ .",
"Now we summarize the convergence results for the sequences $\\lbrace \\widetilde{P}^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$ and $\\lbrace \\widetilde{S}^\\varepsilon _\\mathsf {f}\\rbrace _{\\varepsilon >0}$ .",
"We have: Lemma 4.2 For any rate of contrast there exist a function $S$ such that $0 \\leqslant S \\leqslant 1$ a.e.",
"in $\\Omega _T$ , $\\beta _\\mathsf {f}(S)-\\beta _\\mathsf {f}(1) \\in L^2(0, T; H^1_{\\Gamma _1}(\\Omega ))$ , and functions ${\\mathsf {P}}-{\\mathsf {P}}_{\\Gamma _1} \\in L^2(0, T; H^1_{\\Gamma _1}(\\Omega ))$ , ${\\mathsf {w}}_p, {\\mathsf {w}}_s \\in L^2(\\Omega _T; H^1_{per}(Y))$ such that up to a subsequence: $\\widetilde{S}^\\varepsilon _\\mathsf {f}(x, t) \\longrightarrow S(x, t) \\,\\,{\\rm strongly\\,\\, in}\\,\\, L^q(\\Omega _T)\\,\\, \\forall \\ 1 \\leqslant q < +\\infty ;$ $\\widetilde{\\mathsf {P}}^\\varepsilon _\\mathsf {f}(x, t) \\rightharpoonup {\\mathsf {P}}(x, t) \\,\\,{\\rm weakly\\,\\, in}\\,\\, L^2(0, T; H^1(\\Omega ));$ $\\nabla \\widetilde{\\mathsf {P}}^\\varepsilon _\\mathsf {f}(x, t)\\stackrel{2s}{\\rightharpoonup }\\nabla {\\mathsf {P}}(x, t) + \\nabla _y {\\mathsf {w}}_p(x, t, y);$ $\\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}) \\longrightarrow \\beta _\\mathsf {f}(S) \\,\\,{\\rm strongly\\,\\, in}\\,\\, L^q(\\Omega _T)\\,\\, \\forall \\ 1 \\leqslant q < +\\infty ;$ $\\nabla \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}) (x, t) \\stackrel{2s}{\\rightharpoonup }\\nabla \\beta _\\mathsf {f}(S)(x, t) + \\nabla _y {\\mathsf {w}}_s(x, t, y);$ $\\widetilde{\\mathsf {P}}^\\varepsilon _\\mathsf {f}(x, t) \\stackrel{2s-\\Gamma _{\\mathsf {m}\\mathsf {f}}}{\\rightharpoonup }{\\mathsf {P}}(x, t);$ $\\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}(x, t)) \\stackrel{2s-\\Gamma _{\\mathsf {m}\\mathsf {f}}}{\\rightharpoonup }\\beta _\\mathsf {f}(S(x, t)) .$ The Proof of Lemma REF is based on the a priori estimates for the functions $\\beta _\\mathsf {f}(S^\\varepsilon _\\mathsf {f})$ and ${\\mathsf {P}}^\\varepsilon _\\mathsf {f}$ obtained in Section , the extension results from Subsection REF , and Proposition REF .",
"The two-scale convergence results (REF ) and (REF ) are obtained by arguments similar to those in [3].",
"The two-scale convergence (REF ) and (REF ) can be proved by applying Proposition 2.6 in [4].",
"Lemma REF is proved.",
"Note also that the notion of strong two-scale convergence on periodic surfaces can be introduced in analogy with the ordinary strong two-scale convergence.",
"Definition 4.5 A sequence $\\lbrace v^\\varepsilon \\rbrace _{\\varepsilon >0} \\subset L^2(\\Sigma ^{\\varepsilon }_T)$ converges the two-scale strongly to $v \\in L^2(\\Omega _T; L^2(\\Gamma _{\\mathsf {f}\\mathsf {m}}))$ on $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ if $\\lim _{\\varepsilon \\rightarrow 0} \\varepsilon \\int \\limits _{\\Sigma ^{\\varepsilon }_T} | v^\\varepsilon (x, t)- v\\left(x, \\frac{x}{\\varepsilon }, t\\right)|^2\\,dH^{d-1}(x)\\, dt = 0.$ It is easy to verify that the strong two-scale convergence on periodic surfaces implies the two-scale convergence on periodic surfaces with the same limit.",
"Using the strong convergence (REF ) and the boundedness of $\\nabla \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f})$ given in Lemma REF we get: $\\varepsilon \\Vert \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}) - \\beta _\\mathsf {f}(S)\\Vert _{L^2(\\Sigma ^{\\varepsilon }_T)}^2 \\leqslant C \\left[\\varepsilon ^2 \\Vert \\nabla \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}) -\\nabla \\beta _\\mathsf {f}(S)\\Vert _{L^2(\\Omega _{\\mathsf {f},T}^\\varepsilon )}^2 +\\Vert \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}) - \\beta _\\mathsf {f}(S) \\Vert _{L^2(\\Omega _{\\mathsf {f},T}^\\varepsilon )}^2\\right],$ which tends to zero on a given subsequence as $\\varepsilon \\rightarrow 0$ .",
"Therefore, we conclude that the sequence $\\lbrace \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f})\\rbrace _{\\varepsilon >0}$ converges strongly two-scale on the surface $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ to $\\beta _\\mathsf {f}(S)$ .",
"Furthermore, we have: Lemma 4.3 Let $\\lbrace \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f})\\rbrace $ be a subsequence from Lemma REF .",
"Then for any Lipschitz function ${\\cal M}\\colon [0, \\beta _\\mathsf {f}(1)]\\rightarrow \\mathbb {R}$ the sequence $\\lbrace {\\cal M}(\\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}))\\rbrace _{\\varepsilon >0}$ converges strongly two-scale on the surface $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ to ${\\cal M}(\\beta _\\mathsf {f}(S))$ .",
"Lemma REF follows immediately from the estimate $\\Vert {\\cal M}(\\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f})) - {\\cal M}(\\beta _\\mathsf {f}(S))\\Vert _{L^2(\\Sigma ^{\\varepsilon }_T)}^2 \\le L_{\\cal M}^2 \\Vert \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}) - \\beta _\\mathsf {f}(S)\\Vert _{L^2(\\Sigma ^{\\varepsilon }_T)}^2,$ where $L_{\\cal M}$ is the Lipschitz constant which does not depend on $\\varepsilon $ ."
],
[
"Dilation operator and convergence results",
"It is known that due to the nonlinearities and the strong coupling of the problem, the two-scale convergence does not provide an explicit form for the source terms appearing in the homogenized model, see for instance [16], [23], [40].",
"To overcome this difficulty the authors make use of the dilation operator.",
"Here we refer to [13], [16], [23], [40] for the definition and main properties of the dilation operator.",
"Let us also notice that the notion of the dilation operator is closely related to the notion of the unfolding operator.",
"We refer here, e.g., to [25] for the definition and the properties of this operator.",
"The outline of this section is as follows.",
"First, in subsection REF we introduce the definition of the dilation operator and describe its main properties.",
"Then in subsection REF we obtain the equations for the dilated saturation and the global pressure functions and the corresponding uniform estimates.",
"Finally, in subsection REF we consider the convergence results for the dilated functions."
],
[
"Definition and preliminary results",
"Definition 5.1 For a given $\\varepsilon > 0$ , we define a dilation operator $\\mathfrak {D}^\\varepsilon $ mapping measurable functions defined in $\\Omega ^\\varepsilon _{\\mathsf {m},T}$ to measurable functions defined in $\\Omega _T \\times Y_\\mathsf {m}$ by $\\left(\\mathfrak {D}^\\varepsilon \\varphi \\right)(x, y, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace \\begin{array}[c]{ll}\\varphi \\left( c^\\varepsilon (x) + \\varepsilon \\, y, t\\right),\\quad {\\rm if}\\,\\, c^\\varepsilon (x) + \\varepsilon \\, y \\in \\Omega ^\\varepsilon _\\mathsf {m}; \\\\[2mm]0,\\quad {\\rm elsewhere}, \\\\\\end{array}\\right.$ where $c^\\varepsilon (x) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varepsilon \\, k$ if $x \\in \\varepsilon \\, (Y + k)$ with $k \\in \\mathbb {Z}^d$ denotes the lattice translation point of the $\\varepsilon $ -cell domain containing $x$ .",
"The basic properties of the dilation operator are given by the following lemma (see [13], [40]).",
"Lemma 5.1 Let $\\varphi , \\psi \\in L^2(0, T; H^1(\\Omega ^\\varepsilon _\\mathsf {m}))$ .",
"Then we have: $\\nabla _y \\mathfrak {D}^\\varepsilon \\varphi = \\varepsilon \\, \\mathfrak {D}^\\varepsilon (\\nabla _x \\varphi )\\quad {\\rm a.e.\\,\\,in}\\,\\,\\Omega _T \\times Y_\\mathsf {m};$ $\\Vert \\mathfrak {D}^\\varepsilon \\varphi \\Vert _{L^2(\\Omega _T \\times Y_\\mathsf {m})} =\\Vert \\varphi \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})}; \\,\\,\\Vert \\nabla _y \\mathfrak {D}^\\varepsilon \\varphi \\Vert _{L^2(\\Omega _T \\times Y_\\mathsf {m})}= \\varepsilon \\,\\Vert \\mathfrak {D}^\\varepsilon \\nabla _x\\, \\varphi \\Vert _{L^2(\\Omega _T \\times Y_\\mathsf {m})}= \\varepsilon \\,\\Vert \\nabla _x\\, \\varphi \\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})};$ $\\left(\\mathfrak {D}^\\varepsilon \\varphi , \\mathfrak {D}^\\varepsilon \\psi \\right)_{L^2(\\Omega _T \\times Y_\\mathsf {m})} =\\left(\\varphi , \\psi \\right)_{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})}.$ The following lemma gives the link between the two-scale and the weak convergence (see, e.g., [16]).",
"Lemma 5.2 Let $\\lbrace \\varphi ^\\varepsilon \\rbrace _{\\varepsilon >0}$ be a uniformly bounded sequence in $L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})$ satisfying: (i) $\\mathfrak {D}^\\varepsilon \\varphi ^\\varepsilon \\rightharpoonup \\varphi ^0$ weakly in $L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))$ ; (ii) ${\\bf 1}^\\varepsilon _\\mathsf {m}(x)\\varphi ^\\varepsilon \\stackrel{2s}{\\rightharpoonup }\\varphi ^* \\in L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))$ .",
"Then $\\varphi ^0 = \\varphi ^*$ a.e.",
"in $\\Omega _T \\times Y_\\mathsf {m}$ .",
"Finally, we also have the following result (see, e.g., [23], [40]).",
"Lemma 5.3 If $\\varphi ^\\varepsilon \\in L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})$ and ${\\bf 1}^\\varepsilon _\\mathsf {m}(x) \\varphi ^\\varepsilon \\stackrel{2s}{\\rightarrow }\\varphi \\in L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))$ then $\\mathfrak {D}^\\varepsilon \\varphi ^\\varepsilon $ converges to $\\varphi $ strongly in $L^2(\\Omega _T \\times Y_\\mathsf {m})$ .",
"Here $\\stackrel{2s}{\\rightarrow }$ denotes the strong two-scale convergence.",
"If $\\varphi \\in L^2(\\Omega _T)$ is considered as an element of $L^2(\\Omega _T \\times Y_\\mathsf {m})$ constant in $y$ , then $\\mathfrak {D}^\\varepsilon \\varphi $ converges strongly to $\\varphi $ in $L^2(\\Omega _T \\times Y_\\mathsf {m})$ .",
"The dilation operator shows the same properties with respect to the two-scale convergence on periodic surfaces.",
"For a given function $v\\in L^2(\\Sigma ^{\\varepsilon }_T)$ and from definition of the dilation operator we have $\\mathfrak {D}^\\varepsilon (v)\\in L^2(\\Omega _T; L^2(\\Gamma _{\\mathsf {f}\\mathsf {m}}))$ and $\\sqrt{\\varepsilon }\\Vert v\\Vert _{L^2(\\Sigma ^{\\varepsilon }_T)} = \\Vert \\mathfrak {D}^\\varepsilon (v)\\Vert _{L^2(\\Omega _T; L^2(\\Gamma _{\\mathsf {f}\\mathsf {m}}))}.$ We have also the following lemma: Lemma 5.4 If $\\lbrace v^\\varepsilon \\rbrace _{\\varepsilon >0} \\subset L^2(\\Sigma ^{\\varepsilon }_T)$ is a sequence that converges to $v \\in L^2(\\Omega _T; L^2(\\Gamma _{\\mathsf {f}\\mathsf {m}}))$ in the two-scale sense on $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ , then the sequence $\\lbrace \\mathfrak {D}^\\varepsilon (v^\\varepsilon )\\rbrace _{\\varepsilon >0}$ converges weakly to the same limit, that is $\\mathfrak {D}^\\varepsilon (v^\\varepsilon ) \\rightharpoonup v$ in $L^2(\\Omega _T; L^2(\\Gamma _{\\mathsf {f}\\mathsf {m}}))$ .",
"If $\\lbrace v^\\varepsilon \\rbrace _{\\varepsilon >0} \\subset L^2(\\Sigma ^{\\varepsilon }_T)$ converges strongly to $v \\in L^2(\\Omega _T; L^2(\\Gamma _{\\mathsf {f}\\mathsf {m}}))$ in the two-scale sense on $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ , then the sequence $\\lbrace \\mathfrak {D}^\\varepsilon (v^\\varepsilon )\\rbrace _{\\varepsilon >0}$ converges strongly to the same limit in $L^2(\\Omega _T; L^2(\\Gamma _{\\mathsf {f}\\mathsf {m}}))$ .",
"Due to Lemma REF , one can apply Lemma REF to the sequence $\\lbrace {\\cal M}(\\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}))\\rbrace _{\\varepsilon >0}$ and find a subsequence, such that $\\int \\limits _{\\Omega _T} \\int \\limits _{\\Gamma _{\\mathsf {f}\\mathsf {m}}} \\left|{\\cal M}(\\beta _\\mathsf {f}(\\mathfrak {D}^\\varepsilon (\\widetilde{S}^\\varepsilon _\\mathsf {f}))) - {\\cal M}(\\beta _\\mathsf {f}(S))\\right|^2 \\, dH^{d-1}(y)\\, dx\\, dt \\rightarrow 0$ when $\\varepsilon \\rightarrow 0$ , for any Lipschitz function ${\\cal M}$ .",
"As a consequence we have.",
"Corollary 5.2 Let ${\\cal M}\\colon [0, \\beta _\\mathsf {f}(1)]\\rightarrow \\mathbb {R}$ be a Lipschitz function.",
"Then there is a subsequence $\\varepsilon = \\varepsilon _k$ of the sequence $\\lbrace {\\cal M}(\\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}))\\rbrace _{\\varepsilon >0}$ , still denoted by $\\varepsilon $ , such that for a.e.",
"$x\\in \\Omega $ $\\int \\limits _0^T \\int \\limits _{\\Gamma _{\\mathsf {f}\\mathsf {m}}} \\left|{\\cal M}(\\beta _\\mathsf {f}(\\mathfrak {D}^\\varepsilon (\\widetilde{S}^\\varepsilon _\\mathsf {f}(x,y,t)))) - {\\cal M}(\\beta _\\mathsf {f}(S(x,y,t)))\\right|^2 \\, dH^{d-1}(y) dt \\rightarrow 0 \\quad \\text{as }\\; \\varepsilon \\rightarrow 0.$"
],
[
"The dilated functions $\\mathfrak {D}^\\varepsilon S^\\varepsilon _\\mathsf {m}, \\mathfrak {D}^\\varepsilon P^\\varepsilon _\\mathsf {m}$ \nand their properties",
"In this section we derive the equations for the dilated functions $\\mathfrak {D}^\\varepsilon S^\\varepsilon _\\mathsf {m},\\mathfrak {D}^\\varepsilon P^\\varepsilon _\\mathsf {m}$ and obtain the corresponding uniform estimates.",
"In what follows we also make use of the notation: $\\mathfrak {D}^\\varepsilon S^\\varepsilon _\\mathsf {m}\\mathop {=}\\limits ^{\\hbox{\\tiny def}}s^\\varepsilon _\\mathsf {m}\\quad {\\rm and} \\quad \\mathfrak {D}^\\varepsilon {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\mathop {=}\\limits ^{\\hbox{\\tiny def}}p^\\varepsilon _\\mathsf {m}.$ The equations for the dilated functions $s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}$ are given by the following lemma.",
"Lemma 5.5 For $x \\in \\Omega $ , the functions $s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}$ satisfy the following system of equations: $\\Phi _\\mathsf {m}(y) \\frac{\\partial s^\\varepsilon _\\mathsf {m}}{\\partial t} - \\frac{\\varkappa (\\varepsilon )}{\\varepsilon ^2}\\,{\\rm div}_y\\, \\bigg \\lbrace K(x, y) \\left[\\lambda _{\\mathsf {m},w} (s^\\varepsilon _\\mathsf {m}) \\nabla _y p^\\varepsilon _\\mathsf {m}+\\nabla _y \\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) - \\varepsilon \\,\\lambda _{\\mathsf {m},w} (s^\\varepsilon _\\mathsf {m}) \\vec{g}\\, \\right] \\bigg \\rbrace = 0;$ $- \\Phi _\\mathsf {m}(y) \\frac{\\partial s^\\varepsilon _\\mathsf {m}}{\\partial t}- \\frac{\\varkappa (\\varepsilon )}{\\varepsilon ^2}\\, {\\rm div}_y\\, \\bigg \\lbrace K(x, y)\\, \\left[ \\lambda _{\\mathsf {m},n} (s^\\varepsilon _\\mathsf {m})\\nabla _y p^\\varepsilon _\\mathsf {m}- \\nabla _y \\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m})- \\varepsilon \\,\\lambda _{\\mathsf {m},n}(s^\\varepsilon _\\mathsf {m})\\,\\vec{g}\\,\\right] \\bigg \\rbrace = 0,$ in the space $L^2(0, T; H^{-1}(Y_\\mathsf {m}))$ .",
"The Proof of Lemma REF is given in [16], [40].",
"The system of equations (REF )-(REF ) is provided with the following boundary conditions: $\\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) = {\\cal M}(\\beta _\\mathsf {f}(\\mathfrak {D}^\\varepsilon \\tilde{S^\\varepsilon _\\mathsf {f}}))\\quad {\\rm on}\\; \\Gamma _{\\mathsf {f}\\mathsf {m}}$ for $(x, t) \\in \\Omega ^\\varepsilon _\\mathsf {m}\\times (0,T)$ , where ${\\cal M} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\beta _{\\mathsf {m}} \\circ ( P_{\\mathsf {m},c})^{-1}\\circ P_{\\mathsf {f},c} \\circ (\\beta _{\\mathsf {f}})^{-1}.$ Note that under our hypothesis function $ {\\cal M}$ is Lipschitz continuous.",
"We also have ${\\mathsf {p}}^\\varepsilon _\\mathsf {m}+ {\\mathsf {G}}_{\\mathsf {m}, w}(s^\\varepsilon _\\mathsf {m}) =\\mathfrak {D}^\\varepsilon {\\mathsf {P}}^\\varepsilon _\\mathsf {f}+ {\\mathsf {G}}_{\\mathsf {f}, w}(\\mathfrak {D}^\\varepsilon \\tilde{S^\\varepsilon _\\mathsf {f}})\\,\\,{\\rm and} \\,\\,{\\mathsf {p}}^\\varepsilon _\\mathsf {m}+ {\\mathsf {G}}_{\\mathsf {m},n}(s^\\varepsilon _\\mathsf {m})= \\mathfrak {D}^\\varepsilon {\\mathsf {P}}^\\varepsilon _\\mathsf {f}+ {\\mathsf {G}}_{\\mathsf {f},n}(\\mathfrak {D}^\\varepsilon \\tilde{S^\\varepsilon _\\mathsf {f}})$ on $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ for $(x, t) \\in \\Omega ^\\varepsilon _\\mathsf {m}\\times (0,T)$ .",
"The initial conditions are $s^\\varepsilon _\\mathsf {m}(x, y, 0) = (\\mathfrak {D}^\\varepsilon S^{\\bf 0}_\\mathsf {m})(x, y) \\quad {\\rm and} \\quad p^\\varepsilon _\\mathsf {m}(x, y, 0) = (\\mathfrak {D}^\\varepsilon {\\mathsf {P}}^{\\bf 0}_\\mathsf {m})(x, y) \\quad {\\rm in} \\,\\,\\Omega ^\\varepsilon _\\mathsf {m}\\times Y_\\mathsf {m},$ where $S^{\\bf 0}_\\mathsf {m}, {\\mathsf {P}}^{\\bf 0}_\\mathsf {m}$ are the restrictions to the domain $\\Omega ^\\varepsilon _\\mathsf {m}$ of the functions $S^{\\bf 0}, {\\mathsf {P}}^{\\bf 0}$ defined in (REF ) and the dilations of the functions defined on the fracture system can be defined in a way similar to one already used for the functions defined on the matrix part.",
"Now we establish a priori estimates for the functions $s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}$ .",
"They are given by the following lemma.",
"Lemma 5.6 Let $\\langle s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}\\rangle $ be a solution to problem (REF )-(REF ).",
"Then: (i) For any rate of contrast ($\\theta > 0$ ), $0 \\leqslant s^\\varepsilon _\\mathsf {m}\\leqslant 1 \\quad {\\rm a.e.\\,\\, in\\,\\,} \\Omega _T \\times Y_\\mathsf {m};$ $\\Vert \\partial _t(\\Phi _\\mathsf {m}\\, s^\\varepsilon _\\mathsf {m}) \\Vert _{L^2(\\Omega _T; H^{-1}_{per}(Y_\\mathsf {m}))}\\leqslant C.$ (ii) For the high contrast in the critical case ($\\theta = 2$ ), $\\Vert \\nabla _y \\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) \\Vert _{L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))} \\leqslant C;$ $\\Vert p^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega _T; H^1_{per}(Y_\\mathsf {m}))} \\leqslant C.$ (iii) For the moderate contrast ($0 < \\theta < 2$ ), $\\varepsilon ^{\\frac{\\theta }{2}-1}\\,\\Vert \\nabla _y \\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) \\Vert _{L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))}+\\Vert \\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) \\Vert _{L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))}\\leqslant C;$ $\\varepsilon ^{\\frac{\\theta }{2}-1}\\,\\Vert \\nabla _y p^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))}+\\Vert p^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))}\\leqslant C.$ Proof of Lemma REF .",
"Statement (REF ) is evident.",
"The bound (REF ) with $\\Phi _\\mathsf {m}= \\Phi _\\mathsf {m}(y)$ follow from Lemma REF and Lemma REF .",
"The uniform estimates for $p^\\varepsilon _\\mathsf {m}$ in (REF ) and (REF ) follow from the uniform bound (REF ) and Lemma REF .",
"The uniform estimates for the gradients of the functions $\\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m})$ and $p^\\varepsilon _\\mathsf {m}$ easy follow from the uniform bounds (REF ) and Lemma REF .",
"Lemma REF is proved.",
"Remark 2 Notice that in what follows we do not need the uniform estimates for the dilated functions in the case of the very high contrast."
],
[
"Convergence results for the dilated functions",
"In this subsection we establish convergence results which will be used below to obtain the homogenized system.",
"From Lemmas REF , REF we get the following convergence results.",
"Lemma 5.7 Let $\\langle s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}\\rangle $ be a solution to problem (REF )-(REF ), (REF )-(REF ).",
"Then (up to a subsequence), (i) For the high contrast in the critical case ($\\theta = 2$ ), ${\\bf 1}^\\varepsilon _\\mathsf {m}(x) S^\\varepsilon _\\mathsf {m}\\stackrel{2s}{\\rightharpoonup }s \\in L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))\\quad {\\rm and} \\quad s^\\varepsilon _\\mathsf {m}\\rightharpoonup s \\,\\, {\\rm weakly\\,\\, in}\\,\\, L^2(\\Omega _T \\times Y_\\mathsf {m});$ ${\\bf 1}^\\varepsilon _\\mathsf {m}(x)\\, {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\stackrel{2s}{\\rightharpoonup }p \\in L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))\\quad {\\rm and} \\quad p^\\varepsilon _\\mathsf {m}\\rightharpoonup p \\,\\, {\\rm weakly\\,\\, in}\\,\\, L^2(\\Omega _T; H^1(Y_\\mathsf {m}));$ ${\\bf 1}^\\varepsilon _\\mathsf {m}(x)\\, \\nabla _x {\\mathsf {P}}^\\varepsilon _\\mathsf {m}\\stackrel{2s}{\\rightharpoonup }\\nabla _y p \\in L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}));$ ${\\bf 1}^\\varepsilon _\\mathsf {m}(x) \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m}) \\stackrel{2s}{\\rightharpoonup }\\beta ^*\\quad {\\rm and} \\quad \\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) \\rightharpoonup \\beta ^* \\,\\, {\\rm weakly\\,\\, in}\\,\\, L^2(\\Omega _T; H^1(Y_\\mathsf {m}));$ ${\\bf 1}^\\varepsilon _\\mathsf {m}(x)\\, \\nabla _x \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m}) \\stackrel{2s}{\\rightharpoonup }\\nabla _y \\beta ^*.$ (ii) For the very high contrast ($\\theta > 2$ ), ${\\bf 1}^\\varepsilon _\\mathsf {m}(x) S^\\varepsilon _\\mathsf {m}\\stackrel{2s}{\\rightharpoonup }s \\in L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m})).$ (iii) For the moderate contrast ($0 < \\theta < 2$ ), ${\\bf 1}^\\varepsilon _\\mathsf {m}(x) S^\\varepsilon _\\mathsf {m}\\stackrel{2s}{\\rightharpoonup }s \\in L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))\\quad {\\rm and} \\quad s^\\varepsilon _\\mathsf {m}\\rightharpoonup s \\,\\, {\\rm weakly\\,\\, in}\\,\\, L^2(\\Omega _T \\times Y_\\mathsf {m});$ ${\\bf 1}^\\varepsilon _\\mathsf {m}(x) \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m}) \\stackrel{2s}{\\rightharpoonup }\\beta ^*_1\\quad {\\rm and} \\quad {\\bf 1}^\\varepsilon _\\mathsf {m}(x)\\, \\varepsilon ^{\\theta } \\nabla \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m}) \\stackrel{2s}{\\rightharpoonup }\\beta _1;$ $\\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) \\rightharpoonup \\beta ^*_1 \\,\\, {\\rm weakly\\,\\, in}\\,\\, L^2(\\Omega _T; H^1(Y_\\mathsf {m})).$ It is important to notice that the convergence results of Lemma REF are not sufficient for derivation of the equations for the limit functions $\\langle s, p \\rangle $ which involve only these functions and not the undefined limits appearing in (REF ), (REF ), (REF ) and (REF ).",
"In order to overcome this difficulty, we introduce the restrictions of the functions $s^\\varepsilon _\\mathsf {m}$ , $p^\\varepsilon _\\mathsf {m}$ which are defined below.",
"For these functions we obtain more estimates which allow us to obtain the desired equations.",
"For this, we make use of the density arguments.",
"Namely, following [23] (see also [6]), we fix $x \\in \\Omega $ and define the restrictions of $s^\\varepsilon _\\mathsf {m}$ , $p^\\varepsilon _\\mathsf {m}$ to the $\\varepsilon $ -cell containing the point $x$ .",
"These functions are defined in the domain $Y_\\mathsf {m}\\times (0, T)$ and are constants in the slow variable $x$ .",
"In order to obtain the uniform estimates for the restricted functions (they are similar to the corresponding estimates for ${\\mathsf {P}}^\\varepsilon _\\mathsf {f}$ , $S^\\varepsilon _\\mathsf {f}$ from Section ) we make use of the estimates (REF )-(REF ).",
"The scheme is as follows.",
"First, for any natural ${\\bf n}$ , we introduce the set of points $x \\in \\Omega $ such that the corresponding norms for the restricted functions are not uniformly bounded in $\\varepsilon $ .",
"It turns out that the measure of this set is asymptotically small as ${\\bf n} \\rightarrow +\\infty $ (see Propositions REF , REF below).",
"Then taking into account this fact and using the estimates (REF )-(REF ), we, finally, obtain the desired uniform estimates for the restricted functions (see Lemma REF below).",
"Let us first denote a periodicity cell $ \\varepsilon \\big (Y + k\\big )$ which contains point $x_0$ by $K^\\varepsilon _{x_0}$ .",
"For given $x_0$ and $\\varepsilon $ the index $k\\in \\mathbb {Z}^d$ which defines the cell $K^\\varepsilon _{x_0}$ can be uniquely defined and therefore we have a well defined function $k(x_0, \\varepsilon ) \\in \\mathbb {Z}^d$ such that $K^\\varepsilon _{x_0} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varepsilon \\big (Y + k(x_0, \\varepsilon )\\big )$ .",
"Due to the definition of the dilation operator dilated functions are constant in $x$ on $K^\\varepsilon _{x_0}$ .",
"The restricted functions are given by: $s^\\varepsilon _{\\mathsf {m},x_0}(y, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace \\begin{array}[c]{ll}s^\\varepsilon _\\mathsf {m},\\quad {\\rm for}\\,\\, x \\in K^\\varepsilon _{x_0}; \\\\[2mm]0, \\quad {\\rm if \\,\\, not}; \\\\\\end{array}\\right.\\quad p^\\varepsilon _{\\mathsf {m},x_0}(y, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace \\begin{array}[c]{ll}p^\\varepsilon _\\mathsf {m},\\quad {\\rm for}\\,\\, x \\in K^\\varepsilon _{x_0}; \\\\[2mm]0, \\quad {\\rm if \\,\\, not}.",
"\\\\\\end{array}\\right.$ For any $\\varepsilon > 0$ , the pair $\\langle s^\\varepsilon _{\\mathsf {m},x_0}, p^\\varepsilon _{\\mathsf {m},x_0} \\rangle $ is a solution to problem (REF )-(REF ), (REF )-(REF ) in $Y_\\mathsf {m}\\times (0,T)$ .",
"Now we estimate the measure of the set of points $x \\in \\Omega $ such that the corresponding norms for the restricted functions are not uniformly bounded in $\\varepsilon $ .",
"The following result holds true.",
"Proposition 5.3 Let $f^\\varepsilon _\\mathsf {m}= f^\\varepsilon _\\mathsf {m}(x, y, t)$ be a dilated function such that $\\Vert f^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))} \\leqslant C$ and let $A_{\\bf n}$ be a set of points defined by $A_{\\bf n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace x \\in \\Omega \\,:\\, \\mathop {\\underline{\\rm lim}}_{\\varepsilon \\rightarrow 0}\\Vert \\widehat{f}^\\varepsilon _{\\mathsf {m},k(x,\\varepsilon )}\\Vert _{L^2(0,T; L^2_{per}(Y_\\mathsf {m}))} \\geqslant {\\bf n} \\right\\rbrace ,$ where for fixed $k\\in \\mathbb {Z}^d$ $\\widehat{f}^\\varepsilon _{\\mathsf {m},k}(y, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace \\begin{array}[c]{ll}f^\\varepsilon _\\mathsf {m}(\\varepsilon k, y, t),\\quad {\\rm if}\\,\\, k \\,\\, {\\rm is \\,\\, such\\,\\, that}\\,\\,\\,\\varepsilon (Y_\\mathsf {m}+ k) \\cap \\Omega \\ne \\emptyset ; \\\\[2mm]0, \\quad {\\rm if \\,\\, not}.",
"\\\\\\end{array}\\right.$ Then $\\sqrt{|A_{\\bf n}|} \\leqslant {C}/{\\bf n}$ .",
"Proof of Proposition REF .",
"Let $f^\\varepsilon _\\mathsf {m}= f^\\varepsilon _\\mathsf {m}(x, y, t)$ be a function that satisfies (REF ).",
"Then we can write $\\Vert f^\\varepsilon _\\mathsf {m}\\Vert ^2_{L^2(\\Omega _T; L^2_{per}(Y_\\mathsf {m}))} =\\sum _{k=1}^{N_\\varepsilon } \\big |\\varepsilon Y_\\mathsf {m}\\big |\\, \\Vert \\widehat{f}^\\varepsilon _{\\mathsf {m},k}\\Vert ^2_{L^2(0,T; L^2_{per}(Y_\\mathsf {m}))},$ where, due to (REF ), we have that $\\sum _{k=1}^{N_\\varepsilon } \\big |\\varepsilon Y_\\mathsf {m}\\big |\\, \\Vert \\widehat{f}^\\varepsilon _{\\mathsf {m},k}\\Vert ^2_{L^2(0,T; L^2_{per}(Y_\\mathsf {m}))} \\leqslant C^2.$ Now, for any ${\\bf n} \\in \\mathbb {N}$ and $\\varepsilon >0$ , let us introduce the set of \"bad points\" $A^\\varepsilon _{\\bf n}$ defined by: $A^\\varepsilon _{\\bf n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace x \\in \\Omega \\,:\\, \\Vert \\widehat{f}^\\varepsilon _{\\mathsf {m},k(x,\\varepsilon )}\\Vert _{L^2(0,T; L^2_{per}(Y_\\mathsf {m}))} > {\\bf n} \\right\\rbrace .$ Let us estimate the measure of the set $A^\\varepsilon _{\\bf n}$ .",
"It follows from (REF ) and (REF ) that $C^2 \\geqslant \\sum _{k=1}^{N_\\varepsilon } \\big |\\varepsilon Y_\\mathsf {m}\\big |\\, \\Vert \\widehat{f}^\\varepsilon _{\\mathsf {m},k(x,\\varepsilon )}\\Vert ^2_{L^2(0,T; L^2_{per}(Y_\\mathsf {m}))} \\geqslant \\sum _{x \\in A^\\varepsilon _{\\bf n}} \\big |\\varepsilon Y_\\mathsf {m}\\big |\\, {\\bf n}^2 = {\\bf n}^2 \\, | A^\\varepsilon _{\\bf n}|.$ Therefore, $| A^\\varepsilon _{\\bf n}|\\leqslant C^2/{\\bf n}^2$ .",
"By definition of limit inferior, for any $\\eta >0$ we have $A_{\\bf n}\\subseteq \\mathop {\\underline{\\rm lim}}_{\\varepsilon \\rightarrow 0} A^\\varepsilon _{{\\bf n}-\\eta }$ , (where $\\varepsilon $ denotes a sequence of real numbers).",
"Due to the continuity of the measure we get $|A_{\\bf n}|\\leqslant \\mathop {\\underline{\\rm lim}}_{\\varepsilon \\rightarrow 0} |A^\\varepsilon _{{\\bf n}-\\eta }|\\leqslant C^2/({\\bf n}-\\eta )^2$ .",
"Proposition REF is proved.",
"We note that previously defined restricted functions are linked to ones appearing in Proposition REF by the following relation: $f^\\varepsilon _{\\mathsf {m},x_0}(y,t) = \\widehat{f}^\\varepsilon _{\\mathsf {m},k(x_0,\\varepsilon )}(y,t).$ In a similar way, taking into account the uniform estimate (REF ), we prove the following proposition.",
"Proposition 5.4 Let $f^\\varepsilon _\\mathsf {m}= f^\\varepsilon _\\mathsf {m}(x, y, t)$ be a dilated function such that $\\Vert f^\\varepsilon _\\mathsf {m}\\Vert _{L^2(\\Omega _T; H^{-1}_{per}(Y_\\mathsf {m}))} \\leqslant C$ and let $B_{\\bf n}$ be a set of points defined by $B_{\\bf n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace x \\in \\Omega \\,:\\, \\mathop {\\underline{\\rm lim}}_{\\varepsilon \\rightarrow 0}\\Vert \\widehat{f}^\\varepsilon _{\\mathsf {m},k(x,\\varepsilon )}\\Vert _{L^2(0,T; H^{-1}_{per}(Y_\\mathsf {m}))} \\geqslant {\\bf n} \\right\\rbrace ,$ where the function $\\widehat{f}^\\varepsilon _{\\mathsf {m},k}$ is defined in (REF ).",
"Then $\\sqrt{|B_{\\bf n}|} \\leqslant C/{\\bf n}$ .",
"Now let us introduce ${A}_{\\bf n}$ , the set of \"bad points\" for the functions appearing in (REF )-(REF ).",
"We set: ${A}_{1,{\\bf n}} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace x \\in \\Omega \\,:\\,\\mathop {\\underline{\\rm lim}}_{\\varepsilon \\rightarrow 0} \\varepsilon ^{\\theta /2 -1}\\, \\Vert \\nabla _y \\beta _\\mathsf {m}(s^\\varepsilon _{\\mathsf {m},x})\\Vert _{L^2(0,T; L^2_{per}(Y_\\mathsf {m}))} \\geqslant {\\bf n} \\right\\rbrace ;$ ${A}_{2,{\\bf n}} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace x \\in \\Omega \\,:\\,\\mathop {\\underline{\\rm lim}}_{\\varepsilon \\rightarrow 0}\\Vert p^\\varepsilon _{\\mathsf {m},x}\\Vert _{L^2(0,T; L^2_{per}(Y_\\mathsf {m}))} \\geqslant {\\bf n} \\right\\rbrace ;$ ${A}_{3,{\\bf n}} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace x \\in \\Omega \\,:\\,\\mathop {\\underline{\\rm lim}}_{\\varepsilon \\rightarrow 0} \\varepsilon ^{\\theta /2 -1}\\, \\Vert \\nabla _y p^\\varepsilon _{\\mathsf {m},x}\\Vert _{L^2(0,T; L^2_{per}(Y_\\mathsf {m}))} \\geqslant {\\bf n} \\right\\rbrace ;$ ${A}_{4,{\\bf n}} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace x \\in \\Omega \\,:\\,\\mathop {\\underline{\\rm lim}}_{\\varepsilon \\rightarrow 0}\\Vert \\partial _t(\\Phi _\\mathsf {m}\\, s^\\varepsilon _{\\mathsf {m},x})\\Vert _{L^2(0, T; H^{-1}_{per}(Y_\\mathsf {m}))} \\geqslant {\\bf n} \\right\\rbrace .$ Here $s^\\varepsilon _{\\mathsf {m},x}, p^\\varepsilon _{\\mathsf {m},x}$ are defined in (REF ).",
"Then ${A}_{\\bf n} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\bigcup _{\\ell =1}^4 {A}_{\\ell ,{\\bf n}}$ and, due to Propositions REF , REF , the measure of this set satisfies the estimate $\\sqrt{|{A}_{\\bf n}|} \\leqslant {C}/{\\bf n}$ .",
"The following result holds.",
"Lemma 5.8 Let $s^\\varepsilon _{\\mathsf {m},x_0}, p^\\varepsilon _{\\mathsf {m},x_0}$ be the functions defined in (REF ) and $0 <\\theta \\leqslant 2$ .",
"Then for any $x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ , there is a subsequence $\\varepsilon = \\varepsilon _k$ still denoted by $\\varepsilon $ such that: $0 \\leqslant s^\\varepsilon _{\\mathsf {m},x_0} \\leqslant 1 \\quad {\\rm a.e.\\,\\, in\\,\\,} Y_\\mathsf {m}\\times (0,T);$ $\\Vert \\nabla _y \\beta _\\mathsf {m}(s^\\varepsilon _{\\mathsf {m},x_0}) \\Vert _{L^2(0, T; L^2_{per}(Y_\\mathsf {m}))} \\leqslant C \\varepsilon ^{1-\\theta /2};$ $\\Vert p^\\varepsilon _{\\mathsf {m},x_0} \\Vert _{L^2(0, T; L^2_{per}(Y_\\mathsf {m}))} \\leqslant C;\\quad \\Vert \\nabla p^\\varepsilon _{\\mathsf {m},x_0} \\Vert _{L^2(0, T; L^2_{per}(Y_\\mathsf {m}))} \\leqslant C\\varepsilon ^{1-\\theta /2};$ $\\Vert \\partial _t(\\Phi _\\mathsf {m}\\, s^\\varepsilon _{\\mathsf {m},x_0}) \\Vert _{L^2(0, T; H^{-1}_{per}(Y_\\mathsf {m}))}\\leqslant C,$ where $C = C({\\bf n})$ is constant that does not depend on $x_0$ and $\\varepsilon $ , and ${\\bf n}$ is an arbitrary natural number.",
"Proof of Lemma REF .",
"First, we notice that the estimate (REF ) follows immediately from (REF ).",
"Let us prove, for example, (REF ).",
"Taking into account that $x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ , from the definition of the set ${A}_{1,{\\bf n}}$ , we obtain immediately the existence of a subsequence on which (REF ) holds with constant $C$ depending only on ${\\bf n}$ .",
"The estimates (REF )-(REF ) are obtained in a similar way.",
"Lemma REF is proved.",
"Using these estimates and applying Lemma 4.2 from [5], we obtain the following compactness result.",
"Proposition 5.5 Assume $0 < \\theta \\leqslant 2$ .",
"For any $x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ , on a subsequence extracted in Lemma REF , the family $\\lbrace s^\\varepsilon _{\\mathsf {m},x_0}\\rbrace _{\\varepsilon >0}$ is a compact set in the space $L^q(Y_\\mathsf {m}\\times (0,T))$ for all $q \\in [1, \\infty )$ .",
"In the case $\\theta < 2$ every limit point of the sequence $\\lbrace s^\\varepsilon _{\\mathsf {m},x_0}\\rbrace _{\\varepsilon >0}$ is independent of the fast variable $y$ ."
],
[
"Homogenization results",
"In this section we formulate and prove the main results of the paper corresponding to the homogenized models for various rates of contrast.",
"First, we introduce the notation.",
"- $S$ , $P_w$ , $P_n$ denote the homogenized wetting liquid saturation, wetting liquid pressure, and nonwetting liquid pressure, respectively.",
"- $\\Phi ^\\star = \\Phi ^\\star (x)$ denotes the effective porosity and is given by: $\\Phi ^\\star (x) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\Phi ^{\\rm H}_{\\mathsf {f}}(x)\\, \\frac{|Y_\\mathsf {f}|}{|Y_\\mathsf {m}|},$ where $\\Phi ^{\\rm H}_{\\mathsf {f}}$ is defined in condition (A.1) and $|Y_\\ell |$ is the measure of the set $Y_\\ell $ ($\\ell = \\mathsf {f}, \\mathsf {m}$ ).",
"- $F^\\star _w, F^\\star _n$ denote the effective source terms and are given by: $F^\\star _w(x, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}F^{\\rm H}_{w}(x, t)\\, \\frac{|Y_\\mathsf {f}|}{|Y_\\mathsf {m}|} \\quad {\\rm and} \\quad F^\\star _n(x, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}F^{\\rm H}_{n}(x, t)\\, \\frac{|Y_\\mathsf {f}|}{|Y_\\mathsf {m}|},$ where $F^{\\rm H}_w(x, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}S^I_{\\mathsf {f}, w}\\, f_I(x,t) - S\\, f_P(x,t) \\quad {\\rm and}\\quad F^{\\rm H}_n(x, t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}S^I_{\\mathsf {f}, n}\\, f_I(x,t) - (1 - S)\\, f_P(x,t)$ and where the functions $S^I_{\\mathsf {f}, w}, S^I_{\\mathsf {f}, n}, f_I, f_P$ are defined in (REF ), (REF ), respectively (see also (A.9)).",
"- $\\mathbb {K}^\\star = \\mathbb {K}^\\star (x)$ is the homogenized tensor with the entries $\\mathbb {K}^\\star _{ij}$ defined by: $\\mathbb {K}^{\\star }_{ij}(x) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\frac{1}{|Y_\\mathsf {m}|}\\, \\int \\limits _{Y_\\mathsf {f}}\\, K(x, y)\\,\\left[\\nabla _y \\xi _i + \\vec{e}_i \\right]\\cdot \\left[\\nabla _y \\xi _j + \\vec{e}_j \\right]\\, dy,$ where $\\xi _j = \\xi _j(x, y)$ ($j = 1,\\ldots ,d$ ) is a $Y$ -periodic solution to the auxiliary cell problem: $\\left\\lbrace \\begin{array}[c]{ll}- {\\rm div}_y\\, \\big \\lbrace K(x, y) \\nabla _y \\xi _j\\big \\rbrace = 0 \\quad {\\rm in} \\,\\, Y_{\\mathsf {f}}; \\\\[2mm]\\nabla _y \\xi _j \\cdot \\vec{\\nu }_y = - \\vec{e}_j \\cdot \\vec{\\nu }_y\\quad {\\rm on} \\,\\, \\Gamma _{\\mathsf {f}\\mathsf {m}};\\\\[2mm]y \\mapsto \\xi _j(y)\\quad Y-{\\rm periodic}.",
"\\\\\\end{array}\\right.$"
],
[
"High contrast media: critical case, ${\\theta }{\\bf =2}$",
"We study the asymptotic behavior of the solution to problem (REF ), (REF )-(REF ) in the case $\\varkappa (\\varepsilon ) = \\varepsilon ^2$ as $\\varepsilon \\rightarrow 0$ .",
"In particular, we are going to show that the effective model, expressed in terms of the homogenized phase pressures, reads: $\\left\\lbrace \\begin{array}[c]{ll}0 \\leqslant S \\leqslant 1 \\quad {\\rm in} \\,\\, \\Omega _T; \\\\[2mm]\\displaystyle \\Phi ^\\star (x)\\, \\frac{\\partial S}{\\partial t}- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},w}(S) \\big (\\nabla P_w - \\vec{g} \\big ) \\bigg \\rbrace ={Q}_w + F^\\star _w \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]\\displaystyle - \\Phi ^\\star (x)\\, \\frac{\\partial S}{\\partial t}- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},n}(S)\\big (\\nabla P_n - \\vec{g} \\big ) \\bigg \\rbrace ={Q}_n + F^\\star _n \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]P_{\\mathsf {f},c}(S) = P_n - P_w \\quad {\\rm in} \\,\\, \\Omega _T.\\end{array}\\right.$ For almost every point $x \\in \\Omega $ a matrix block $Y_\\mathsf {m}\\subset \\mathbb {R}^d$ is suspended topologically.",
"The system for flow in a matrix block is given by the so-called imbibition equation: $\\left\\lbrace \\begin{array}[c]{ll}\\displaystyle \\Phi _m(y) \\frac{\\partial s}{\\partial t}\\, - {\\rm div}_y \\big \\lbrace K(x, y) \\nabla _y \\beta _m(s)\\big \\rbrace = 0\\quad {\\rm in}\\,\\, Y_\\mathsf {m}\\times \\Omega _T; \\\\[3mm]s(x, y, t) = {P}(S(x,t)) \\quad {\\rm on}\\,\\, \\Gamma _{\\mathsf {f}\\mathsf {m}} \\times \\Omega _T; \\\\[2mm]s(x, y, 0) = S_m^{\\,\\bf 0}(x) \\quad {\\rm in}\\,\\, Y_\\mathsf {m}\\times \\Omega .\\\\\\end{array}\\right.$ Here $s$ denotes the wetting liquid saturation in the block $Y_\\mathsf {m}$ and the function ${P}(S)$ is defined by ${P}(S)\\, \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\, (P_{c,m}^{-1} \\circ P_{c,f})(S).$ For any $x \\in \\Omega $ and $t > 0$ , the matrix-fracture sources are given by: ${Q}_w \\mathop {=}\\limits ^{\\hbox{\\tiny def}}- \\frac{1}{|Y_\\mathsf {m}|}\\, \\int \\limits _{Y_\\mathsf {m}} \\Phi _\\mathsf {m}(y)\\frac{\\partial s}{\\partial t}(x, y, t) \\,dy = - {Q}_n.$ The boundary conditions for the effective system (REF ) are given by: $\\left\\lbrace \\begin{array}[c]{ll}P_{w} = P_{n} = 0 \\quad {\\rm on} \\,\\, \\Gamma _{1} \\times (0,T); \\\\[3mm]\\mathbb {K}^\\star \\,\\lambda _{n}(S) \\left(\\nabla P_w - \\vec{g} \\right) \\cdot \\vec{\\nu }=\\mathbb {K}^\\star \\, \\lambda _{w}(S) (\\nabla P_n - \\vec{g})\\cdot \\vec{\\nu }= 0 \\quad {\\rm on} \\,\\,\\Gamma _{2} \\times (0,T).\\\\\\end{array}\\right.$ Finally, the initial conditions read: $P_{w}(x, 0) = {\\mathsf {p}}_{w}^{\\bf 0}(x) \\quad {\\rm and}\\quad P_{n}(x, 0) = {\\mathsf {p}}_{n}^{\\bf 0}(x)\\quad {\\rm in} \\,\\, \\Omega .$ The first main result of the paper is given by the following theorem.",
"Theorem 6.1 Let $\\varkappa (\\varepsilon ) = \\varepsilon ^2$ and let assumptions (A.1)-(A.9) be fulfilled.",
"Then the solution of the initial problem (REF ), (REF )-(REF ) converges (up to a subsequence) in the two-scale sense to a weak solution of the homogenized problem (REF ), (REF ), (REF )-(REF ).",
"Proof of Theorem REF .",
"It is done in several steps.",
"We start our analysis by considering the system (REF ).",
"The main difficulty with the initial unknown functions ${\\mathsf {p}}^\\varepsilon _{w}, {\\mathsf {p}}^\\varepsilon _{n}$ in this system is that they do not possess the uniform $H^1$ -estimates (see Lemma REF ).",
"It is important to notice that in the case of two-phase incompressible flow it is possible to find appropriate but rather strong conditions which allow us to deal directly with the phase pressures in a space wider than $H^1$ (see [40]).",
"To overcome the difficulties appearing due to the absence of the uniform $H^1$ -estimates, the authors usually pass to the equivalent formulation of the problem in terms of the global pressure and saturation.",
"In our case it is done in subsection REF and the corresponding weak formulation of the problem is then given in subsection REF .",
"Then using the convergence and compactness results from subsection we pass to the limit in equations (REF ), (REF ).",
"This is done in subsections REF and REF .",
"In order, to pass to the homogenized phase pressures we make use of the change of the unknown functions.",
"Namely, we set, by the definition of the global pressure: $P_w \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {P}} + {\\mathsf {G}}_{\\mathsf {f},w}(S)$ and $P_n \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {P}} + {\\mathsf {G}}_{\\mathsf {f},n}(S)$ .",
"Then we rewrite the limit system obtained in terms of the global pressure and saturation in terms of the homogenized phase pressures.",
"The passage to the limit in the matrix blocks makes use of the dilation operator (see Section above).",
"Then we pass to the equivalent problem for the imbibition equation and, finally, obtain system (REF )."
],
[
"Passage to the limit in equation (",
"We set: $\\varphi _w\\left(x, \\frac{x}{\\varepsilon }, t \\right) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varphi (x, t) + \\varepsilon \\, \\zeta \\left(x, \\frac{x}{\\varepsilon }, t \\right)=\\varphi (x, t) + \\varepsilon \\, \\zeta _1(x, t)\\, \\zeta _2\\left( \\frac{x}{\\varepsilon }\\right)\\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varphi (x, t) + \\varepsilon \\, \\zeta ^\\varepsilon (x, t),$ where $\\varphi \\in {D}(\\Omega _T), \\zeta _1 \\in {D}(\\Omega _T),\\zeta _2 \\in C^\\infty _{per}(Y)$ , and plug the function $\\varphi _w$ in (REF ).",
"This yields: $-\\int \\limits _{\\Omega _{T}} {\\bf 1}^\\varepsilon _\\mathsf {f}(x)\\, \\Phi ^\\varepsilon _\\mathsf {f}(x)\\, \\widetilde{S}^\\varepsilon _\\mathsf {f}\\left[ \\frac{\\partial \\varphi }{\\partial t} +\\varepsilon \\frac{\\partial \\zeta ^\\varepsilon }{\\partial t} \\right]\\,\\, dx\\, dt+$ $+\\int \\limits _{\\Omega _{T}} {\\bf 1}^\\varepsilon _\\mathsf {f}(x)\\, K\\left(x, \\frac{x}{\\varepsilon }\\right)\\bigg \\lbrace \\lambda _{\\mathsf {f},w} (\\widetilde{S}^\\varepsilon _\\mathsf {f})\\left(\\nabla \\widetilde{\\mathsf {P}}^\\varepsilon _\\mathsf {f}- \\vec{g}\\right)+ \\nabla \\beta _\\mathsf {f}(\\widetilde{S}^\\varepsilon _\\mathsf {f}) \\bigg \\rbrace \\cdot \\left[\\nabla \\varphi +\\varepsilon \\nabla _x \\zeta ^\\varepsilon + \\nabla _y \\zeta ^\\varepsilon \\right]\\,\\, dx\\, dt-$ $-\\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} \\Phi _\\mathsf {m}\\left(\\frac{x}{\\varepsilon }\\right) S^\\varepsilon _\\mathsf {m}\\left[ \\frac{\\partial \\varphi }{\\partial t} +\\varepsilon \\frac{\\partial \\zeta ^\\varepsilon }{\\partial t} \\right]\\,\\, dx\\, dt+$ $+\\varepsilon ^2\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} K\\left(x, \\frac{x}{\\varepsilon }\\right) \\bigg \\lbrace \\lambda _{\\mathsf {m},w} (S^\\varepsilon _\\mathsf {m})\\left(\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {m}- \\vec{g}\\right)+ \\nabla \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m}) \\bigg \\rbrace \\cdot \\left[\\nabla \\varphi +\\varepsilon \\nabla _x \\zeta ^\\varepsilon + \\nabla _y \\zeta ^\\varepsilon \\right]\\,\\, dx\\, dt =$ $= \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {f},T}} \\big (S^I_{\\mathsf {f}, w} f_I(x,t) - S^\\varepsilon _{\\mathsf {f}} f_P(x,t)\\big )\\,\\left[\\varphi + \\varepsilon \\, \\zeta ^\\varepsilon \\right]\\, dx dt.$ Taking into account Lemma REF and the convergence results of Lemma REF and Lemma REF , we pass to the limit in (REF ) as $\\varepsilon \\rightarrow 0$ and obtain the following homogenized equation: $-\\, |Y_\\mathsf {f}|\\, \\int \\limits _{\\Omega _T} \\Phi ^{\\rm H}_\\mathsf {f}(x) S(x, t) \\frac{\\partial \\varphi }{\\partial t} \\,dx\\,dt +$ $+\\int \\limits _{\\Omega _T \\times Y_\\mathsf {f}} K(x, y) \\bigg \\lbrace \\lambda _{\\,\\mathsf {f},w}(S) \\left[\\nabla {\\mathsf {P}}+ \\nabla _y {\\mathsf {w}}_p - \\vec{g}\\right] + \\nabla \\beta _\\mathsf {f}(S) + \\nabla _y {\\mathsf {w}}_s \\bigg \\rbrace \\cdot \\left[\\nabla \\varphi + \\zeta _1 \\nabla _y \\zeta _2 \\right] \\,dy\\, dx\\, dt=$ $=\\int \\limits _{\\Omega _T \\times Y_\\mathsf {m}} \\Phi _\\mathsf {m}(y)\\, s(x, y, t) \\, \\frac{\\partial \\varphi }{\\partial t}\\,\\,dy\\, dx\\, dt + |Y_\\mathsf {f}|\\, \\int \\limits _{\\Omega _T} F^{\\rm H}_w\\, \\varphi \\, dx\\, dt,$ where $F^{\\rm H}_w$ is given by (REF )."
],
[
"Passage to the limit in equation (",
"Equation (REF ) is treated in the same way as equation (REF ).",
"Taking the test function of the form (REF ) and using the same arguments we can pass to a limit $\\varepsilon \\rightarrow 0$ and obtain the following homogenized equation: $|Y_\\mathsf {f}|\\, \\int \\limits _{\\Omega _T} \\Phi ^{\\rm H}_\\mathsf {f}(x) S(x, t)\\frac{\\partial \\varphi }{\\partial t} \\,dx\\,dt +$ $+\\int \\limits _{\\Omega _T \\times Y_\\mathsf {f}} K(x, y) \\bigg \\lbrace \\lambda _{\\mathsf {f},n}(S) \\left[\\nabla {\\mathsf {P}}+ \\nabla _y {\\mathsf {w}}_p - \\vec{g}\\right] - \\nabla \\beta _\\mathsf {f}(S) - \\nabla _y {\\mathsf {w}}_s \\bigg \\rbrace \\cdot \\left[\\nabla \\varphi + \\zeta _1 \\nabla _y \\zeta _2 \\right] \\,dy\\, dx\\, dt=$ $= -\\int \\limits _{\\Omega _T \\times Y_\\mathsf {m}} \\Phi _\\mathsf {m}(y)\\, s(x, y, t) \\, \\frac{\\partial \\varphi }{\\partial t}\\,\\,dy\\, dx\\, dt + |Y_\\mathsf {f}|\\, \\int \\limits _{\\Omega _T} F^{\\rm H}_n\\, \\varphi \\, dx\\, dt.$"
],
[
"Identification of the corrector functions ${\\mathsf {w}}_p$ , {{formula:64ccfb57-f58a-418d-b7ec-d4beba0f6183}} and homogenized equations",
"In this section we identify the corrector functions ${\\mathsf {w}}_p$ , ${\\mathsf {w}}_s$ appearing in the equations (REF ), (REF ) and obtain the desired homogenized system (REF ).",
"Consider the equations (REF ), (REF ).",
"Setting $\\varphi \\equiv 0$ , we get: $\\int \\limits _{Y_{\\mathsf {f}}} K(x, y)\\,\\bigg \\lbrace \\lambda _{\\,\\mathsf {f},w}(S)\\big [\\nabla P + \\nabla _{y} {\\mathsf {w}}_p - \\vec{g} \\big ]+ \\big [\\nabla \\beta _\\mathsf {f}+ \\nabla _{y} {\\mathsf {w}}_s\\big ] \\bigg \\rbrace \\cdot \\nabla _y \\zeta _2(y)\\,dy = 0$ and $\\int \\limits _{Y_{\\mathsf {f}}} K(x, y)\\,\\bigg \\lbrace \\lambda _{\\,\\mathsf {f},n}(S)\\big [\\nabla P + \\nabla _{y} {\\mathsf {w}}_p - \\vec{g} \\big ]- \\big [\\nabla \\beta _\\mathsf {f}+ \\nabla _{y} {\\mathsf {w}}_s\\big ] \\bigg \\rbrace \\cdot \\nabla _y \\zeta _2(y)\\,dy = 0.$ Now adding (REF ) and (REF ) and taking into account condition (A.4) and the fact that the saturation $S$ does not depend on the fast variable $y$ , we obtain: $\\int \\limits _{Y_{\\mathsf {f}}} K(x, y)\\,\\bigg \\lbrace \\nabla P + \\nabla _{y} {\\mathsf {w}}_p -\\vec{g} \\bigg \\rbrace \\cdot \\nabla _y \\zeta _2(y)\\,dy = 0.$ Then we proceed in a standard way (see, e.g., [30]).",
"Let $\\xi _j = \\xi _j(x, y)$ ($j = 1,..,d$ ) be the $Y$ -periodic solution of the auxiliary cell problem (REF ).",
"Then the function ${\\mathsf {w}}_p$ can be represented as: ${\\mathsf {w}}_p(x, y, t) = \\sum ^d_{j=1} \\xi _j(x, y) \\left[\\frac{\\partial \\, P}{\\partial x_j}(x, t) - g_j\\right].$ Now we turn to the identification of the function ${\\mathsf {w}}_s$ .",
"From (REF ) and (REF ), we get: $\\int \\limits _{Y_{\\mathsf {f}}} K(x, y)\\, \\bigg \\lbrace \\nabla \\beta _{\\mathsf {f}} + \\nabla _{y} {\\mathsf {w}}_s \\bigg \\rbrace \\cdot \\nabla _y \\zeta _2(y)\\,dy = 0.$ Then as in the previous case, we obtain that ${\\mathsf {w}}_s(x, y, t) = \\sum ^d_{j=1} \\xi _j(x, y) \\frac{\\partial \\, \\beta _\\mathsf {f}(S)}{\\partial x_j}(x, t).$"
],
[
"Effective equations in terms of the global pressure and saturation",
"We start by obtaining the corresponding homogenized equation for the wetting phase.",
"Choosing $\\zeta _2 = 0$ in (REF ), we get: $\\Phi ^\\star (x) \\, \\dfrac{\\partial S}{\\partial t} -{\\rm div}_x \\bigg \\lbrace \\mathbb {K}^\\star (x)\\,\\big [\\lambda _{\\,\\mathsf {f},w}(S)\\,\\nabla P + \\nabla \\beta _{\\mathsf {f}}(S) - \\lambda _{\\,\\mathsf {f},w}(S)\\, \\vec{g}\\big ]\\bigg \\rbrace =$ $=- \\frac{1}{|Y_{m}|}\\, \\int \\limits _{Y_{\\mathsf {m}}} \\Phi _{m}(y)\\dfrac{\\partial s}{\\partial t}(x, y, t)\\,dy + F^\\star _w(x, t),$ where the effective porosity $\\Phi ^\\star $ , the effective source term $F^\\star _w$ , and the homogenized permeability tensor $\\mathbb {K}^\\star $ are defined in (REF ), (REF ) and (REF ), respectively.",
"In a similar way, choosing $\\zeta _2 = 0$ in equation (REF ), we derive the second homogenized equation: $- \\Phi ^\\star (x) \\, \\dfrac{\\partial S}{\\partial t} -{\\rm div}_x \\bigg \\lbrace \\mathbb {K}^\\star (x)\\,\\big [\\lambda _{\\,\\mathsf {f},n}(S)\\,\\nabla P + \\nabla \\beta _{\\mathsf {f}}(S) - \\lambda _{\\,\\mathsf {f},n}(S)\\, \\vec{g}\\big ]\\bigg \\rbrace =$ $=\\frac{1}{|Y_{m}|}\\, \\int \\limits _{Y_{\\mathsf {m}}} \\Phi _{m}(y)\\dfrac{\\partial s}{\\partial t}(x, y, t)\\,dy + F^\\star _n(x, t),$ where $F^\\star _n$ denotes the effective source term defined in (REF )."
],
[
"Effective equations in terms of the phase pressures",
"Let us introduce now the functions that is naturally to call the homogenized phase pressures.",
"Namely, we set, by the definition: $P_{w} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {P}} + {\\mathsf {G}}_{\\mathsf {f},w}(S) \\quad {\\rm and}\\quad P_n \\mathop {=}\\limits ^{\\hbox{\\tiny def}}{\\mathsf {P}} + {\\mathsf {G}}_{\\mathsf {f},n}(S),$ where the functions ${\\mathsf {G}}_{\\mathsf {f},w}, {\\mathsf {G}}_{\\mathsf {f},n}$ are defined in Section REF .",
"Then it easy to see that the homogenized equations can be rewritten as follows: $\\left\\lbrace \\begin{array}[c]{ll}\\displaystyle \\Phi ^\\star (x)\\, \\frac{\\partial S}{\\partial t}- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},w}(S) \\big (\\nabla P_w - \\vec{g} \\big ) \\bigg \\rbrace ={Q}_w + F^\\star _w \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]\\displaystyle - \\Phi ^\\star (x)\\, \\frac{\\partial S}{\\partial t}- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},n}(S)\\big (\\nabla P_n - \\vec{g} \\big ) \\bigg \\rbrace ={Q}_n + F^\\star _n \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]P_c(S) = P_n - P_w \\quad {\\rm in} \\,\\, \\Omega _T.\\end{array}\\right.$"
],
[
"Flow equations in the matrix block",
"In this section, following the ideas of the papers [6], [16], [23], we obtain the system (REF ) describing the behavior of the function $s$ which is involved in the definition of the matrix-fracture source term.",
"Briefly, we pass to the limit in the equations for the dilated functions for fixed $k$ and then by density arguments the limit equations will be obtained.",
"We recall that the equations for the dilated functions are already obtained in Lemma REF from subsection REF .",
"Namely, for almost all $x \\in \\Omega $ , the functions $s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}$ satisfy the following variational problem: for all $\\phi _n,\\phi _w \\in L^2(0,T;H^1_0(Y_{\\mathsf {m}}))\\cap H^1(0,T; L^2(Y_{\\mathsf {m}}))$ , $\\phi _n(T)=\\phi _w(T) =0$ , $- \\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\Phi _\\mathsf {m}(y) s^\\varepsilon _\\mathsf {m}\\frac{\\partial \\phi _w}{\\partial t}\\, dy- \\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\Phi _\\mathsf {m}(y) (\\mathfrak {D}^\\varepsilon S^{\\bf 0}_\\mathsf {m})\\frac{\\partial \\phi _w}{\\partial t}(0)\\, dy$ $+ \\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\bigg \\lbrace K(x, y) \\left[\\lambda _{\\mathsf {m},w} (s^\\varepsilon _\\mathsf {m}) \\nabla _y p^\\varepsilon _\\mathsf {m}+\\nabla _y \\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) - \\varepsilon \\,\\lambda _{\\mathsf {m},w} (s^\\varepsilon _\\mathsf {m}) \\vec{g}\\, \\right] \\bigg \\rbrace \\cdot \\nabla _y \\phi _w \\, dy = 0;$ $\\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\Phi _\\mathsf {m}(y) s^\\varepsilon _\\mathsf {m}\\frac{\\partial \\phi _n }{\\partial t}\\, dy+ \\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\Phi _\\mathsf {m}(y) (\\mathfrak {D}^\\varepsilon S^{\\bf 0}_\\mathsf {m})\\frac{\\partial \\phi _n }{\\partial t}\\, dy$ $+\\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\bigg \\lbrace K(x, y) \\left[ \\lambda _{\\mathsf {m},n} (s^\\varepsilon _\\mathsf {m}) \\nabla _y p^\\varepsilon _\\mathsf {m}-\\nabla _y \\beta _\\mathsf {m}(s^\\varepsilon _\\mathsf {m}) - \\varepsilon \\, \\lambda _{\\mathsf {m},n}(s^\\varepsilon _\\mathsf {m})\\,\\vec{g}\\,\\right] \\bigg \\rbrace \\cdot \\nabla _y \\phi _n \\, dy= 0$ with the boundary conditions (REF ).",
"The uniform estimates for the functions $s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}$ imply the convergence results of $\\langle s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}\\rangle $ to $\\langle s, p \\rangle $ in a weak sense (see Lemma REF ).",
"Thus, the limit behavior of the dilated functions $s^\\varepsilon _\\mathsf {m}$ , $p^\\varepsilon _\\mathsf {m}$ is determined.",
"However, the convergence results of Lemma REF are not sufficient for derivation of the equations for the limit functions $\\langle s, p \\rangle $ .",
"To overcome this difficulty, in Section REF we pass to the restrictions of the functions $s^\\varepsilon _\\mathsf {m}, p^\\varepsilon _\\mathsf {m}$ to $K^\\varepsilon _{x_0}$ defined in (REF ).",
"Evidently, they are constants in the slow variable $x$ .",
"Introducing the set ${A}_{\\bf n}$ of \"bad points\" (REF ), by Lemma REF we have the uniform estimates (REF )-(REF ) for the functions $s^\\varepsilon _{\\mathsf {m},x_0}, p^\\varepsilon _{\\mathsf {m},x_0}$ .",
"For any $\\varepsilon > 0$ , the pair of functions $\\langle s^\\varepsilon _{\\mathsf {m},x_0}, p^\\varepsilon _{\\mathsf {m},x_0} \\rangle $ is a solution to problem (REF ), (REF ) in $Y_\\mathsf {m}\\times (0,T)$ .",
"Moreover, the compactness result, i.e., Proposition REF is established for the family $\\lbrace s^\\varepsilon _{\\mathsf {m},x_0}\\rbrace _{\\varepsilon >0}$ .",
"Having established these results, we are in position to complete the proof of Theorem REF .",
"The uniform estimates for the functions $s^\\varepsilon _{\\mathsf {m},x_0}, p^\\varepsilon _{\\mathsf {m},x_0}$ from Lemma REF and the compactness result formulated in Proposition REF allow us to obtain the following convergence results.",
"Lemma 6.1 Let $x_0\\in \\Omega \\setminus {A}_{\\bf n}$ .",
"There exist functions $s_{x_0}, p_{x_0}$ , and $\\beta _\\mathsf {m}(s_{x_0})$ such that up to a subsequence: $s^\\varepsilon _{\\mathsf {m},x_0} \\rightarrow s_{x_0} \\,\\, {\\rm strongly\\,\\, in}\\,\\, L^q(Y_\\mathsf {m}\\times (0,T))\\,\\,\\forall \\ 1 \\leqslant q < +\\infty ;$ $p^\\varepsilon _{\\mathsf {m},x_0} \\rightharpoonup p_{x_0} \\,\\, {\\rm weakly\\,\\, in}\\,\\, L^2(0, T; H^1_{per}(Y_\\mathsf {m}));$ $\\beta _\\mathsf {m}(s^\\varepsilon _{\\mathsf {m},x_0}) \\rightharpoonup \\beta _\\mathsf {m}(s_{x_0}) \\,\\, {\\rm weakly\\,\\, in}\\,\\,L^2(0, T; H^1_{per}(Y_\\mathsf {m}));$ $\\beta _\\mathsf {m}(s^\\varepsilon _{\\mathsf {m},x_0}) \\rightarrow \\beta _\\mathsf {m}(s_{x_0}) \\,\\, {\\rm strongly\\,\\, in}\\,\\,L^q(Y_\\mathsf {m}\\times (0,T))\\,\\, \\forall \\ 1 \\leqslant q < +\\infty ;$ $\\beta _\\mathsf {m}(s^\\varepsilon _{\\mathsf {m},x_0})\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}} \\rightarrow \\beta _\\mathsf {m}(s_{x_0})\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}}\\,\\, {\\rm weakly\\,\\, in}\\,\\, L^2(0, T; L^2(\\Gamma _{\\mathsf {m}\\mathsf {f}}));$ $p^\\varepsilon _{\\mathsf {m},x_0}\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}} \\rightarrow p_{x_0}\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}} \\,\\,{\\rm weakly\\,\\, in}\\,\\, L^2(0, T; L^2(\\Gamma _{\\mathsf {m}\\mathsf {f}})).$ As in subsections REF , REF we can easily pass to the limit in (REF ) and (REF ).",
"We get the following system of equations: $- \\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\Phi _\\mathsf {m}(y) &s_{x_0} \\frac{\\partial \\phi _w}{\\partial t}\\, dy- \\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\Phi _\\mathsf {m}(y) S^{\\bf 0}_{x_0}\\frac{\\partial \\phi _w}{\\partial t}(0)\\, dy \\\\&+ \\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\bigg \\lbrace K(x, y) \\big [\\lambda _{\\mathsf {m},w} (s_{x_0}) \\nabla _y p_{x_0} +\\nabla _y \\beta _\\mathsf {m}(s_{x_0})\\big ] \\bigg \\rbrace \\cdot \\nabla _y \\phi _w \\, dy= 0; \\nonumber \\\\\\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\Phi _\\mathsf {m}(y) & s_{x_0} \\frac{\\partial \\phi _n }{\\partial t}\\, dy+ \\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\Phi _\\mathsf {m}(y) S^{\\bf 0}_{x_0}\\frac{\\partial \\phi _n }{\\partial t}\\, dy \\\\&+\\int \\limits _0^T\\int \\limits _{Y_{\\mathsf {m}}} \\bigg \\lbrace K(x, y)\\big [ \\lambda _{\\mathsf {m},n} (s_{x_0}) \\nabla _y p_{x_0} -\\nabla _y \\beta _\\mathsf {m}(s_{x_0}) \\big ] \\bigg \\rbrace \\cdot \\nabla _y \\phi _n \\, dy= 0,\\nonumber $ where we have used the fact that $\\mathfrak {D}^\\varepsilon S^{\\bf 0}_{x_0}\\rightarrow S^{\\bf 0}_{x_0}$ strongly in $L^2(Y_m)$ for almost all $x_0\\in \\Omega $ .",
"Now we turn to the boundary condition for $s_{x_0}$ on $\\Gamma _{\\mathsf {m}\\mathsf {f}}$ .",
"From Corollary REF we know that for a.e.",
"$x_0$ , ${\\cal M}(\\beta _\\mathsf {f}(\\mathfrak {D}^\\varepsilon (\\widetilde{S}^\\varepsilon _\\mathsf {f}(x_0,\\cdot ,\\cdot )))) \\rightarrow {\\cal M}(\\beta _\\mathsf {f}(S(x_0,\\cdot ,\\cdot )))\\quad \\text{strongly in }\\; L^2(0,T; L^2(\\Gamma _{\\mathsf {m}\\mathsf {f}})),$ where ${\\cal M}$ is the function given in (REF ).",
"Therefore, for a.e.",
"$x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ , from (REF ) and (REF ) we have: $\\beta _\\mathsf {m}(s_{x_0})\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}} ={\\cal M}(\\beta _\\mathsf {f}(S(x_0,\\cdot ,\\cdot )))\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}},$ or, equivalently $s_{x_0} = {P}(S(x_0,\\cdot )) \\quad \\text{on }\\; \\Gamma _{\\mathsf {m}\\mathsf {f}}\\times (0,T).$ Note also that it follows from (REF ) that the convergence in (REF ) is strong in $L^2(0, T; L^2(\\Gamma _{\\mathsf {m}\\mathsf {f}}))$ .",
"This, together with convergence (REF ) and Lipschitz continuity of the functions ${\\mathsf {G}}_{\\ell ,g} $ , ${\\mathsf {G}}_{\\ell ,w} $ , enables us to pass to the limit in the boundary condition for dilated global pressure (REF ) using the two-scale convergence on $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ , and get $p_{x_0} + {\\mathsf {G}}_{\\mathsf {m}, w}(s_{x_0})={\\mathsf {P}}(x_0, \\cdot )+ {\\mathsf {G}}_{\\mathsf {f}, w}(S(x_0,\\cdot ))\\quad \\text{on }\\; \\Gamma _{\\mathsf {m}\\mathsf {f}}\\times (0,T).$ In the same way we also get $p_{x_0} + {\\mathsf {G}}_{\\mathsf {m}, g}(s_{x_0})={\\mathsf {P}}(x_0, \\cdot )+ {\\mathsf {G}}_{\\mathsf {f}, g}(S(x_0,\\cdot ))\\quad \\text{on }\\; \\Gamma _{\\mathsf {m}\\mathsf {f}}\\times (0,T).$ Thus the system which is satisfied by the limit $\\langle s_{x_0}, p_{x_0}\\rangle $ is obtained for any $x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ .",
"Now it remains to make the link between the functions $s_{x_0}, p_{x_0}$ and the limits $s, p$ of the sequences $\\lbrace s^\\varepsilon _\\mathsf {m}\\rbrace _{\\varepsilon >0}$ , $\\lbrace p^\\varepsilon _\\mathsf {m}\\rbrace _{\\varepsilon >0}$ .",
"First, we observe that the convergent subsequence in Lemma REF depends on point $x_0 \\notin {A}_{\\bf n}$ .",
"To avoid this difficulty we will prove (see subsection REF below) that the problem (REF ), () with the corresponding boundary conditions (REF ), (REF ), and (REF ) has a unique weak solution.",
"Then the convergence results from Lemma REF hold for the whole sequences, as $\\varepsilon \\rightarrow 0$ .",
"Since the functions $s_{x_0} = s(x_{x_0}, y, t), p_{x_0} = p(x_{x_0}, y, t)$ satisfy (REF )-(REF ) for almost all $x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ , we conclude that $s$ and $p$ are weak solution of the following system of equations: $\\left\\lbrace \\begin{array}[c]{ll}0 \\leqslant s \\leqslant 1 \\quad {\\rm in} \\,\\, Y_\\mathsf {m}\\times \\Omega _T; \\\\[2mm]\\displaystyle \\Phi _\\mathsf {m}(y) \\frac{\\partial s}{\\partial t} -{\\rm div}_y\\, \\bigg \\lbrace K(x, y) \\left[\\lambda _{\\mathsf {m},w} (s) \\nabla _y p + \\nabla _y \\beta _\\mathsf {m}(s) \\right] \\bigg \\rbrace = 0 \\quad {\\rm in} \\,\\, Y_\\mathsf {m}\\times \\Omega _T; \\\\[5mm]\\displaystyle - \\Phi _\\mathsf {m}(y) \\frac{\\partial s}{\\partial t}-{\\rm div}_y\\, \\bigg \\lbrace K(x, y) \\left[ \\lambda _{\\mathsf {m},n} (s)\\nabla _y p - \\nabla _y \\beta _\\mathsf {m}(s) \\right] \\bigg \\rbrace = 0\\quad {\\rm in} \\,\\, Y_\\mathsf {m}\\times \\Omega _T.\\\\[2mm]\\end{array}\\right.$ The system is completed by the corresponding boundary and initial conditions: $\\left\\lbrace \\begin{array}[c]{ll}{\\mathsf {P}} + {\\mathsf {G}}_{\\mathsf {f},w}(S) =p + {\\mathsf {G}}_{\\mathsf {m},w}(s) \\quad {\\rm on}\\,\\, \\Gamma _{\\mathsf {f}\\mathsf {m}} \\times \\Omega _T;\\\\[4mm]{\\mathsf {P}} + {\\mathsf {G}}_{\\mathsf {f},n}(S) =p + {\\mathsf {G}}_{\\mathsf {m},n}(s) \\quad {\\rm on}\\,\\, \\Gamma _{\\mathsf {f}\\mathsf {m}} \\times \\Omega _T; \\\\[4mm]s(x, y, t) = {P}(S(x,t)) \\quad {\\rm on}\\,\\, \\Gamma _{\\mathsf {f}\\mathsf {m}} \\times \\Omega _T ,\\\\[2mm]s(x, y, 0) = S^{\\,\\bf 0}(x) \\quad {\\rm in}\\,\\, Y_\\mathsf {m}\\times \\Omega .\\end{array}\\right.$ Thus, we have identified $s$ and $p$ for $x\\in \\Omega \\setminus {A}_{\\bf n}$ .",
"Since by Propositions REF , REF , the measure of the set ${A}_{\\bf n}$ goes to zero as ${\\bf n}\\rightarrow \\infty $ we conclude that our conclusion holds a.e.",
"in $\\Omega $ .",
"The proof of the uniqueness of the solution to problem (REF ) will be done as follows.",
"First, we reduce the system (REF ) to a boundary value problem for the so-called imbibition equation and then make use of the uniqueness result from [38].",
"Equation (REF )$_1$ is the well known generalized porous medium equation (see, e.g., [38]).",
"Lemma 6.2 Let $s = s(x, y, t)$ be the solution of the cell problem (REF )-(REF ).",
"Then $s$ satisfies the boundary value problem (REF ).",
"Proof of Lemma REF .",
"First we observe that it follows from the boundary conditions (REF ) that the function $s$ does not depend on $y$ on $\\Gamma _{\\mathsf {f}\\mathsf {m}} \\times \\Omega _T$ .",
"Then the global pressure $p$ does not depend on $y$ on $\\Gamma _{\\mathsf {f}\\mathsf {m}} \\times \\Omega _T$ .",
"Namely, we can write that $p(x, y, t) = p_{\\Gamma }(x, t) \\quad {\\rm on} \\,\\, \\Gamma _{\\mathsf {f}\\mathsf {m}} \\times \\Omega _T.$ By summing the two equations in (REF ) we get: $- {\\rm div}\\, \\big \\lbrace K(x, y)\\, \\lambda _{\\mathsf {m}}(s) \\nabla _y p \\big \\rbrace = 0 \\quad {\\rm in}\\,\\,Y_\\mathsf {m}\\times \\Omega .$ Then multiplying the equation (REF ) by $(p - p_{\\Gamma })$ and integrating over $Y_\\mathsf {m}\\times \\Omega _T$ , using (REF ) and conditions (A.2), (A.4) we obtain: $0 = \\int \\limits _{Y_\\mathsf {m}\\times \\Omega _T} K(x, y)\\,\\lambda _{\\mathsf {m}}(s) \\nabla _y p \\cdot \\nabla _y p \\,\\, dx\\,dy\\,dt \\geqslant k_{\\rm min}\\,\\, L_0\\, \\int \\limits _{Y_\\mathsf {m}\\times \\Omega _T} |\\nabla _y p|^2 \\,\\, dx\\,dy\\,dt,$ which gives $\\nabla _y p = 0 \\quad {\\rm a.e.\\,\\, in} \\,\\, Y_\\mathsf {m}\\times \\Omega _T$ .",
"This result allows us to reduce the two equations in the problem (REF ) to only one, as announced in (REF ).",
"This completes the proof of Lemma REF .",
"Now we turn to the proof of the uniqueness of the solution to (REF ).",
"This proof is given in Theorem 5.3 from [38].",
"For reader's convenience we discuss it briefly in the following lemma.",
"Lemma 6.3 Under our standing assumptions, there is a unique weak solution to problem (REF ).",
"Proof of Lemma REF .",
"First, we introduce the weak formulation of problem (REF ).",
"Omitting, for the sake of simplicity, the dependence on the slow variable $x$ , we have: for any function $\\eta \\in C^1(\\overline{Y}_{\\mathsf {m},T})$ , where $Y_{\\mathsf {m},T} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}Y_\\mathsf {m}\\times (0,T)$ , vanishing on $\\Gamma _{\\mathsf {f}\\mathsf {m}}$ and such that $\\eta (x, T) = 0$, $\\int \\limits _{Y_{\\mathsf {m},T}} \\left\\lbrace K(y)\\, \\nabla _y \\beta _\\mathsf {m}(s) \\cdot \\nabla _y \\eta -s\\, \\frac{\\partial \\eta }{\\partial t} \\right\\rbrace \\, dy\\, dt =\\int \\limits _{Y_\\mathsf {m}} S_m^{\\bf 0}(x)\\, \\eta (y, 0)\\, dy,$ .",
"Suppose now that we have two solutions $s_1$ and $s_2$ satisfying (REF ).",
"Then denoting $W_i \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\beta _\\mathsf {m}(s_i)$ , from (REF ), we have: $\\int \\limits _{Y_{\\mathsf {m},T}} \\left\\lbrace K(y)\\, \\nabla _y (W_1 - W_2) \\cdot \\nabla _y \\eta -(s_1 - s_2)\\, \\frac{\\partial \\eta }{\\partial t} \\right\\rbrace \\, dy\\, dt = 0$ for all $\\eta $ .",
"Then we use as a special test function $\\eta = \\widehat{\\eta }$ , see e.g.",
"[38]: $\\widehat{\\eta }\\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\left\\lbrace \\begin{array}[c]{ll}\\displaystyle \\int _t^T \\big [W_1(x, \\varsigma ) - W_2(x, \\varsigma )\\big ]\\, d\\varsigma \\quad {\\rm if}\\,\\, 0 < t < T;\\\\[5mm]0, \\quad {\\rm if}\\,\\, t \\geqslant T.\\end{array}\\right.$ Then, plugging (REF ) in (REF ), we get: $\\int \\limits _{Y_{\\mathsf {m},T}} (s_1 - s_2)\\, (W_1 - W_2) \\, dy\\, dt +\\int \\limits _{Y_{\\mathsf {m},T}} K(y)\\, \\nabla _y (W_1 - W_2) \\cdot \\left\\lbrace \\int \\limits _t^T \\nabla _y (W_1 - W_2)\\, d\\varsigma \\right\\rbrace \\, dy\\, dt = 0.$ Integration of the last term leads to the following relation: $\\int \\limits _{Y_{\\mathsf {m},T}} (s_1 - s_2)\\, \\big (\\beta _\\mathsf {m}(s_1) - \\beta _\\mathsf {m}(s_2)\\big ) \\, dy\\, dt +\\frac{1}{2}\\, \\int \\limits _{Y_\\mathsf {m}} K(y)\\,\\left[\\int \\limits _0^T \\nabla _y \\big (\\beta _\\mathsf {m}(s_1) - \\beta _\\mathsf {m}(s_2)\\big )\\, d\\varsigma \\right]^2\\, dy = 0.$ Due to the monotonicity of the function $\\beta _\\mathsf {m}$ , the first term in (REF ) is non-negative.",
"Therefore we can conclude that $s_1 = s_2$ a.e.",
"in $Y_{\\mathsf {m},T}$ .",
"Lemma REF is proved.",
"This completes the proof of Theorem REF ."
],
[
"Very high contrast media: ${\\theta }{\\bf >2}$",
"We study the asymptotic behavior of the solution to problem (REF ), (REF )-(REF ) as $\\varepsilon \\rightarrow 0$ in the case $\\varkappa (\\varepsilon ) = \\varepsilon ^\\theta $ with $\\theta > 2$ .",
"In particular, we are going to show that the effective model reads: $\\left\\lbrace \\begin{array}[c]{ll}0 \\leqslant S \\leqslant 1 \\quad {\\rm in} \\,\\, \\Omega _T; \\\\[2mm]\\displaystyle \\Phi ^\\star (x)\\, \\frac{\\partial S}{\\partial t}- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},w}(S) \\big (\\nabla P_w - \\vec{g} \\big ) \\bigg \\rbrace =F^\\star _w \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]\\displaystyle - \\Phi ^\\star (x)\\, \\frac{\\partial S}{\\partial t}- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},n}(S)\\big (\\nabla P_n - \\vec{g} \\big ) \\bigg \\rbrace =F^\\star _n \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]P_{\\mathsf {f},c}(S) = P_n - P_w \\quad {\\rm in} \\,\\, \\Omega _T,\\end{array}\\right.$ where the effective porosity $\\Phi ^\\star $ , the effective source terms $F^\\star _w, F^\\star _n$ , and the homogenized permeability tensor $\\mathbb {K}^\\star $ in (REF ) are defined in (REF ), (REF ) and (REF ), respectively.",
"The boundary and the initial conditions for the system (REF ) are given by (REF ), (REF ).",
"We see that in this case the matrix blocks have a vanishing, as $\\varepsilon \\rightarrow 0$ , influence on the effective flow.",
"This means that in the case of very high contrast, the medium behaves as a perforated one.",
"The second main result of the paper is as follows.",
"Theorem 6.2 Let $\\varkappa (\\varepsilon ) = \\varepsilon ^\\theta $ with $\\theta > 2$ and let assumptions (A.1)-(A.9) be fulfilled.",
"Then the solution of the initial problem (REF ), (REF )-(REF ) converges (up to a subsequence) in the two-scale sense to a weak solution of the homogenized problem (REF ), (REF ), (REF ).",
"Proof of Theorem REF .",
"Let $\\theta >2$ .",
"In the proof of Theorem REF we follow the lines of the proof of Theorem REF .",
"Namely, arguing as in Sections REF , REF , REF , we obtain the homogenized equations (REF ), (REF ).",
"Now we want to show that in the case of the very high contrast, the model behaves as in the perforated media, i.e., the matrix blocks are totally impermeable and the additional matrix-source term equals zero.",
"As in the paper [40], we prove the following result.",
"Lemma 6.4 The following equation holds true: $\\Phi _\\mathsf {m}(y)\\, \\frac{\\partial s}{\\partial t}(x, y, t) = 0 \\quad {\\rm in} \\,\\,\\, Y_\\mathsf {m}\\times \\Omega _T.$ Proof of Lemma REF .",
"Let us define the function: ${F}^{\\,\\varepsilon }(x,t) \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\varepsilon ^{\\frac{\\theta }{2}}\\,K^\\varepsilon (x)\\, \\bigg \\lbrace \\lambda _{\\mathsf {m},w} (S^\\varepsilon _\\mathsf {m})\\left(\\nabla {\\mathsf {P}}^\\varepsilon _\\mathsf {m}- \\vec{g}\\right)+ \\nabla \\beta _\\mathsf {m}(S^\\varepsilon _\\mathsf {m}) \\bigg \\rbrace .$ By using the estimate (REF ) and the assumptions (A.2), (A.4), we get the uniform bound: $\\left\\Vert {F}^{\\,\\varepsilon } \\right\\Vert _{L^2(\\Omega ^\\varepsilon _{\\mathsf {m},T})} \\leqslant C.$ Let define a function: $\\varphi _w\\left(x, \\frac{x}{\\varepsilon }, t \\right) \\in {D}(\\Omega _T; C^\\infty _{per}(Y))\\quad {\\rm such\\,\\, that\\,\\,} \\varphi _w = 0 \\,\\,\\, {\\rm for}\\,\\,\\, y\\in Y_\\mathsf {f}.$ Plugging $\\varphi _w$ in (REF ), and taking into account condition (A.9), we get: $-\\int \\limits _{\\Omega _{T}} {\\bf 1}^\\varepsilon _\\mathsf {m}(x)\\, \\Phi ^\\varepsilon _\\mathsf {m}(x)\\, S^\\varepsilon _\\mathsf {m}\\,\\frac{\\partial \\varphi _w}{\\partial t} \\, dx\\, dt+\\varepsilon ^{\\frac{\\theta }{2}}\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} {F}^{\\,\\varepsilon } \\, \\nabla _x \\varphi _w \\, dx\\, dt+\\varepsilon ^{\\frac{\\theta }{2}-1}\\, \\int \\limits _{\\Omega ^\\varepsilon _{\\mathsf {m},T}} {F}^{\\,\\varepsilon } \\, \\nabla _y \\varphi _w \\, dx\\, dt = 0,$ We pass to the two-scale limit in (REF ) using (REF ).",
"We obtain: $\\int \\limits _{\\Omega _{T}\\times Y_\\mathsf {m}} \\Phi _\\mathsf {m}(y)\\, s(x, y, t)\\,\\frac{\\partial \\varphi _w}{\\partial t} \\, dx\\, dt\\, dy = 0.$ This completes the proof of Lemma REF .",
"Finally, from the equations (REF ), (REF ) in view of Lemma REF , arguing as in subsection REF , we arrive to the desired system (REF ).",
"This completes the proof of Theorem REF ."
],
[
"Moderate contrast media: ${\\bf 0<}{\\theta }{\\bf <2}$",
"We study the asymptotic behavior of the solution to problem (REF ) as $\\varepsilon \\rightarrow 0$ in the case $\\varkappa (\\varepsilon ) = \\varepsilon ^\\theta $ with $0 < \\theta < 2$ .",
"In particular, we are going to show that the effective model reads: $\\left\\lbrace \\begin{array}[c]{ll}0 \\leqslant S \\leqslant 1 \\quad {\\rm in} \\,\\, \\Omega _T; \\\\[2mm]\\displaystyle \\frac{\\partial }{\\partial t}\\left[\\Phi ^\\star (x)\\, S + \\widehat{\\Phi }_\\mathsf {m}\\, {P}(S) \\right]- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},w}(S) \\big (\\nabla P_w - \\vec{g} \\big ) \\bigg \\rbrace = F^\\star _w \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]\\displaystyle - \\frac{\\partial }{\\partial t}\\left[\\Phi ^\\star (x)\\, S + \\widehat{\\Phi }_\\mathsf {m}\\, {P}(S) \\right]- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},n}(S)\\big (\\nabla P_n - \\vec{g} \\big ) \\bigg \\rbrace = F^\\star _n \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]P_{\\mathsf {f},c}(S) = P_n - P_w \\quad {\\rm in} \\,\\, \\Omega _T,\\end{array}\\right.$ where the effective porosity $\\Phi ^\\star $ , the effective source terms $F^\\star _w, F^\\star _n$ , and the homogenized permeability tensor $\\mathbb {K}^\\star $ in (REF ) are defined in (REF ), (REF ) and (REF ), respectively.",
"The boundary conditions and the initial conditions for the system (REF ) are given by (REF ), (REF ).",
"In this case we observe a complete decoupling between microscale and macroscale, which is not the case for the critical scaling $\\theta = 2$ .",
"The third main result of the paper is as follows.",
"Theorem 6.3 Let $\\varkappa (\\varepsilon ) = \\varepsilon ^\\theta $ with $0 < \\theta < 2$ and let assumptions (A.1)-(A.9) be fulfilled.",
"Then the solution of the initial problem (REF ), (REF )-(REF ) converges (up to a subsequence) in the two-scale sense to a weak solution of the homogenized problem (REF ), (REF ), (REF ).",
"Let $0 < \\theta < 2$ .",
"In the proof of Theorem REF we follow the lines of the proof of Theorem REF .",
"Namely, arguing as in Sections REF , REF , REF , we obtain the homogenized equations (REF ), (REF ).",
"Namely, in the case of the moderate contrast we have: $\\left\\lbrace \\begin{array}[c]{ll}0 \\leqslant S \\leqslant 1 \\quad {\\rm in} \\,\\, \\Omega _T; \\\\[2mm]\\displaystyle \\Phi ^\\star (x)\\, \\frac{\\partial S}{\\partial t}- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},w}(S) \\big (\\nabla P_w - \\vec{g} \\big ) \\bigg \\rbrace =\\widehat{{Q}}_w + F^\\star _w \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]\\displaystyle - \\Phi ^\\star (x)\\, \\frac{\\partial S}{\\partial t}- {\\rm div}_x\\, \\bigg \\lbrace \\mathbb {K}^\\star (x)\\, \\lambda _{\\,\\mathsf {f},n}(S)\\big (\\nabla P_n - \\vec{g} \\big ) \\bigg \\rbrace =\\widehat{{Q}}_n + F^\\star _n \\quad {\\rm in} \\,\\, \\Omega _T;\\\\[5mm]P_{\\mathsf {f},c}(S) = P_n - P_w \\quad {\\rm in} \\,\\, \\Omega _T,\\end{array}\\right.$ where the effective porosity $\\Phi ^\\star $ , the effective source terms $F^\\star _w, F^\\star _n$ , and the homogenized permeability tensor $\\mathbb {K}^\\star $ in (REF ) are defined in (REF ), (REF ) and (REF ), respectively.",
"For any $x \\in \\Omega $ and $t > 0$ , the matrix-fracture source terms $\\widehat{{Q}}_w$ , $\\widehat{{Q}}_n$ in (REF ) have the form: $\\widehat{{Q}}_w \\mathop {=}\\limits ^{\\hbox{\\tiny def}}- \\widehat{\\Phi }_\\mathsf {m}\\,\\frac{\\partial s}{\\partial t}(x, t)= - \\widehat{{Q}}_n \\quad {\\rm with} \\,\\,\\,\\widehat{\\Phi }_\\mathsf {m}\\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\frac{1}{|Y_\\mathsf {m}|}\\, \\int \\limits _{Y_\\mathsf {m}} \\Phi _\\mathsf {m}(y)\\,dy.$ In order to complete the proof of Theorem REF , we have to identify the saturation function $s$ appearing on the right-hand side of equations in (REF ).",
"The following result holds true: Lemma 6.5 Let $s$ be the weak limit of $\\lbrace \\mathfrak {D}^\\varepsilon S_{\\mathsf {m}}^\\varepsilon \\rbrace _{\\varepsilon >0}$ and $S$ is the saturation function defined in (REF ).",
"Then $s = {P}(S) \\quad {\\rm a.e.\\,\\, in}\\,\\, \\Omega _T\\quad {\\rm with} \\,\\, {P}(S) = (P_{c,m}^{-1} \\circ P_{c,f})(S).$ Proof of Lemma REF.",
"Applying Lemma REF and Proposition REF we conclude that, for any $x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ , $\\beta _\\mathsf {m}(s^\\varepsilon _{\\mathsf {m},x_0}) &\\rightarrow \\beta _\\mathsf {m}(s_{x_0})\\quad \\text{weakly in }\\;L^2(0,T; H^1(Y_{\\mathsf {m}})),\\\\s^\\varepsilon _{\\mathsf {m},x_0} &\\rightarrow s_{x_0} \\quad \\text{a.e.",
"in }\\; Y_{\\mathsf {m}}\\times (0,T),$ and the limit $s_{x_0}$ does not depend of the fast variable $y$ .",
"Due to continuity of the trace operator we also have: $\\beta _\\mathsf {m}(s^\\varepsilon _{\\mathsf {m},x_0})\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}} \\rightarrow \\beta _\\mathsf {m}(s_{x_0})\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}}\\quad \\text{weakly in }\\; L^2(0,T; L^2(\\Gamma _{\\mathsf {m}\\mathsf {f}})).$ On the other hand we know that, for a.e.",
"$x_0\\in \\Omega $ , ${\\cal M}(\\beta _\\mathsf {f}(\\mathfrak {D}^\\varepsilon (\\widetilde{S}^\\varepsilon _\\mathsf {f}(x_0,\\cdot ,\\cdot ))))\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}} = \\beta _{\\mathsf {m}}(s^\\varepsilon _{\\mathsf {m},x_0})\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}} \\quad {\\rm with} \\,\\,\\,{\\cal M} \\mathop {=}\\limits ^{\\hbox{\\tiny def}}\\beta _{\\mathsf {m}} \\circ ( P_{\\mathsf {m},c})^{-1}\\circ P_{\\mathsf {f},c} \\circ (\\beta _{\\mathsf {f}})^{-1}$ a.e.",
"on $\\Gamma _{\\mathsf {m}\\mathsf {f}}\\times (0,T)$ .",
"For a.e.",
"$x_0\\in \\Omega $ , from Corollary REF we have that ${\\cal M}(\\beta _\\mathsf {f}(\\mathfrak {D}^\\varepsilon (\\widetilde{S}^\\varepsilon _\\mathsf {f}(x_0,\\cdot ,\\cdot )))) \\rightarrow {\\cal M}(\\beta _\\mathsf {f}(S(x_0,\\cdot ,\\cdot )))\\quad \\text{strongly in }\\;L^2(0,T; L^2(\\Gamma _{\\mathsf {m}\\mathsf {f}}))$ and therefore, for a.e.",
"$x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ , $\\beta _\\mathsf {m}(s_{x_0})\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}} ={\\cal M}(\\beta _\\mathsf {f}(S(x_0,\\cdot ,\\cdot )))\\big |_{\\Gamma _{\\mathsf {m}\\mathsf {f}}}.$ Since these functions are independent of $y$ we have that $\\beta _\\mathsf {m}(s_{x_0}) ={\\cal M}(\\beta _\\mathsf {f}(S(x_0,\\cdot ))$ in $L^2(0,T)$ , or, equivalently, $s_{x_0} = {P}(S(x_0,\\cdot ))$ .",
"Now, for a chosen $x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ , we can find a subsequence such that $s^\\varepsilon _{\\mathsf {m},x_0} &\\rightarrow {P}(S(x_0,\\cdot )) \\quad \\text{a.e.",
"in }\\;Y_{\\mathsf {m}}\\times (0,T).$ Since the limit is uniquely defined by the limit $S$ of the sequence $\\mathfrak {D}^\\varepsilon (\\widetilde{S}^\\varepsilon _\\mathsf {f})$ we conclude that the whole sequence converge to the same limit (that is the whole subsequence for which $\\mathfrak {D}^\\varepsilon (\\widetilde{S}^\\varepsilon _\\mathsf {f})$ converges).",
"Now we can repeat our procedure for almost any $x_0 \\in \\Omega \\setminus {A}_{\\bf n}$ and conclude that $s = {P}(S)$ a.e.",
"in $(\\Omega \\setminus {A}_{\\bf n})\\times (0,T)$ .",
"Thanks to Propositions REF , REF , the measure of the set ${A}_{\\bf n}$ goes to zero as ${\\bf n}\\rightarrow \\infty $ and the desired equality (REF ) is proved.",
"Now we complete easily the proof of Theorem REF .",
"Taking into account (REF ) we can rewrite (REF ) and thereby obtain (REF ).",
"Theorem REF is proved."
],
[
"Acknowledgments",
"The work of M. Jurak and A. Vrbaški was funded by Croatian science foundation project no 3955.",
"The work of L. Pankratov was funded by the RScF research project N 15-11-00015.",
"This work was partially supported by ISIFoR (http://www.carnot-isifor.eu/) France.",
"The supports are gratefully acknowledged.",
"Most of the work on this paper was done when M. Jurak and L. Pankratov were visiting the Applied Mathematics Laboratory of the University of Pau & CNRS."
]
] | 1605.05984 |
[
[
"Topological dynamics of Zadeh's extension on the space of upper\n semi-continuous fuzzy sets"
],
[
"Abstract In this paper, some characterizations about transitivity, mildly mixing property, $\\mathbf{a}$-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh's extensions restricted on some invariant closed subsets of the space of all upper semi-continuous fuzzy sets with the level-wise metric are obtained.",
"In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and $\\mathbf{a}$-transitive, equicontinuous, uniformly rigid) if and only if the Zadeh's extension is transitive (resp., mildly mixing, $\\mathbf{a}$-transitive, equicontinuous, uniformly rigid)."
],
[
"Introduction",
"A dynamical system is a pair $(X, T)$ , where $X$ is a nontrivial compact metric space with a metric $d$ and $T: X\\longrightarrow X$ is a continuous surjection.",
"A nonempty invariant closed subset $Y\\subset X$ (i.e., $T(Y) \\subset Y$ ) defines naturally a subsystem $(Y,T|_{Y})$ of $(X, T)$ .",
"Throughout this paper, let $\\mathbb {N}=\\lbrace 1, 2, 3, \\ldots \\rbrace $ , $\\mathbb {Z}^{+}=\\lbrace 0, 1, 2, \\ldots \\rbrace $ and $I=[0, 1]$ .",
"For any $n\\in \\mathbb {N}$ , write $(X^{n}, T^{(n)})$ as the $n$ -fold product system $(\\underbrace{X\\times \\dots \\times X}\\limits _{n}, \\underbrace{T\\times \\dots \\times T}\\limits _{n})$ .",
"Sharkovsky's amazing discovery [28], as well as Li and Yorke's famous work which introduced the concept of `chaos' known as Li-Yorke chaos today [22], have provoked the recent rapid advancement of research on discrete chaos theory.",
"The essence of Li-Yorke chaos is the existence of uncountable scrambled sets.",
"Another well-known definition of chaos was given by Devaney [6], according to which a continuous map $T$ is said to be chaotic in the sense of Devaney if it satisfies the following three properties: $T$ is (topologically) transitive, i.e., for every pair of nonempty open sets $U, V$ of $X$ , there exists $n\\in \\mathbb {Z}^{+}$ such that $T^{n}(U)\\cap V\\ne \\emptyset $ ; The set of periodic points of $T$ is dense in $X$ ; $T$ has sensitive dependence on initial conditions (briefly, is sensitive), i.e., there exists $\\varepsilon >0$ such that for any $x\\in X$ and any neighborhood $U$ of $x$ , there exist $y\\in U$ and $n\\in \\mathbb {Z}^{+}$ satisfying $d(T^{n}(x), T^{n}(y))>\\varepsilon $ .",
"Banks et al.",
"[2] proved that every transitive map whose periodic points are dense in $X$ is sensitive, which implies that the above condition (REF ) is redundant, while Huang and Ye [12] showed that every transitive map containing a periodic point is chaotic in the sense of Li-Yorke.",
"Given a dynamical system $(X, T)$ , one can naturally obtain three associated systems induced by $(X, T)$ .",
"The first one is $({K}(X), T_{K})$ on the hyperspace ${K}(X)$ consisting of all nonempty closed subsets of $X$ with the Hausdorff metric.",
"The second one is $(M(X), T_M)$ on the space $M(X)$ consisting of all Borel probability measures with the Prohorov metric.",
"And the last one is its Zadeh's extension $(\\mathbb {F}_{0}(X), T_{F})$ (more generally $g$ -fuzzification $(\\mathbb {F}_{0}(X), T_{F}^{g})$ ) on the space $\\mathbb {F}_{0}(X)$ consisting of all nonempty upper semi-continuous fuzzy sets with the level-wise metric induced by the extended Hausdorff metric.",
"A systematic study on the connections between dynamical properties of $(X, T)$ and its two induced systems $(K(X), T_{K})$ and $(M(X), T_{M})$ was initiated by Bauer and Sigmund in [5], and later has been widely developed by several authors.",
"For more results on this topic, one is referred to [3], [13], [8], [19], [20], [23], [24], [26], [29], [27], [30], [31] and references therein.",
"Román-Flores and Chalco-Cano [25] studied some chaotic properties (for example, transitivity, sensitive dependence, periodic density) for the Zadeh's extension of a dynamical system.",
"Then, Kupka [15] investigated the relations between Devaney chaos in the original system and in the Zadeh's extension and proved that Zadeh's extension is periodically dense in $\\mathbb {F}(X)$ (resp.",
"$\\mathbb {F}^{\\ge \\lambda }(X)$ for any $\\lambda \\in (0, 1]$ ) if and only if so is $(K(X), T_{K})$ (see Lemma REF ).",
"Recently, Kupka [16] introduced the notion of $g$ -fuzzification to generalize Zadeh's extension and obtained some basic properties of $g$ -fuzzification.",
"In [17], he continued in studying chaotic properties (for example, Li-Yorke chaos, distributional chaos, $\\omega $ -chaos, transitivity, total transitivity, exactness, sensitive dependence, weakly mixing, mildly mixing, topologically mixing) of $g$ -fuzzification and showed that if the $g$ -fuzzification $(\\mathbb {F}^{=1}(X), T^{g}_{F})$ has the property $P$ , then $(X, T)$ also has the property $P$ , where $P$ denotes the following properties: exactness, sensitive dependence, weakly mixing, mildly mixing, or topologically mixing.",
"Meanwhile, he posed the following question: [17] Does the $P$ -property of $(X, T)$ imply the $P$ -property of $(\\mathbb {F}^{=1}(X), T^{g}_{F})$ ?",
"We [33] obtained a sufficient condition on $g\\in D_{m}(I)$$D_{m}(I)$ is the set of all nondecreasing right-continuous functions $g: I\\longrightarrow I$ with $g(0)=0$ and $g(1)=1$ .",
"to ensure that for every dynamical system $(X, T)$ , its $g$ -fuzzification $(\\mathbb {F}^{=1}(X), T_{F}^{g})$ is not transitive (thus, not weakly mixing) and constructed a sensitive dynamical system whose $g$ -fuzzification is not sensitive for any $g\\in D_{m}(I)$ ; giving a negative answer to Question .",
"In this paper, we further investigate the relationships between some dynamical properties (for example, transitivity, weakly mixing, mildly mixing, equicontinuity, uniform rigidity) of $(K(X), T_{K})$ and $(\\mathbb {F}_{0}(X), T_{F})$ through further developing the results in [15].",
"In this study, we prove that dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and $\\mathbf {a}$ -transitive, equicontinuous, uniformly rigid) if and only if the Zadeh's extension is transitive (resp., mildly mixing, $\\mathbf {a}$ -transitive, equicontinuous, uniformly rigid).",
"This paper is organized as follows: in Section , some basic definitions and notations are introduced.",
"In Section and , the transitivity, the weakly mixing property and the mildly mixing property of Zadeh's extension are studied.",
"Then, in Sections and , some results on the equicontinuity and the uniform rigidity are obtained.",
"Let $\\mathcal {P}$ be the collection of all subsets of $\\mathbb {Z}^{+}$ .",
"A collection ${F}\\subset \\mathcal {P}$ is called a Fürstenberg family (briefly, a family) if it is hereditary upwards, i.e., $F_{1}\\subset F_{2}$ and $F_{1}\\in {F}$ imply $F_{2}\\in {F}$ .",
"A family ${F}$ is proper if it is a proper subset of $\\mathcal {P}$ , i.e., neither empty nor the whole $\\mathcal {P}$ .",
"It is easy to see that ${F}$ is proper if and only if $\\mathbb {Z}^{+}\\in {F}$ and $\\emptyset \\notin {F}$ .",
"Let ${F}_{inf}$ be a family of all infinite subsets of $\\mathbb {Z}^+$ .",
"Given a family ${F}$ , define its dual family as $\\kappa {F}=\\left\\lbrace F\\in \\mathcal {P}:F\\cap F^{\\prime }\\ne \\emptyset \\text{ for all } F^{\\prime }\\in {F}\\right\\rbrace =\\left\\lbrace F\\in \\mathcal {P}:\\mathbb {Z}^+\\setminus F\\notin {F}\\right\\rbrace .$ Clearly, $\\kappa {F}_{inf}$ is the family of all cofinite subsets.",
"It is easy to check that $\\kappa {F}$ is a family, and is proper if and only if ${F}$ is so.",
"Given two families ${F}_1$ and ${F}_2$ , define ${F}_{1}\\cdot {F}_{2}=\\left\\lbrace F_{1}\\cap F_{2}:F_{1}\\in {F}_{1}, F_{2}\\in {F}_{2}\\right\\rbrace $ .",
"A family ${F}$ is full if ${F}\\cdot \\kappa {F}\\subset {F}_{inf}$ .",
"A family ${F}$ is a filter if it is proper and satisfies ${F}\\cdot {F}\\subset {F}$ ; and it is a filterdual if its dual family $\\kappa {F}$ is a filter.",
"For a family ${F}$ , let ${F}-{F}=\\lbrace F-F: F\\in {F}\\rbrace $ , where $F-F=\\lbrace i-j: i, j\\in F\\rbrace \\cap \\mathbb {Z^+}$ .",
"A subset $S$ of $\\mathbb {Z}^+$ is syndetic if it has a bounded gap, i.e., if there is $N \\in \\mathbb {N}$ such that $\\lbrace i, i+1, \\ldots ,i+N\\rbrace \\cap S \\ne \\emptyset $ for every $i \\in \\mathbb {Z}^+$ ; $S$ is thick if it contains arbitrarily long runs of positive integers, i.e., for every $n \\in \\mathbb {N}$ there exists some $a_n\\in \\mathbb {Z}^+$ such that $\\lbrace a_n, a_n+1, \\ldots , a_n+n\\rbrace \\subset S$ .",
"The set of all thick subsets of $\\mathbb {Z}^{+}$ and all syndetic subsets of $\\mathbb {Z}^{+}$ are denoted by ${F}_{t}$ and ${F}_{s}$ , respectively.",
"Clearly, they are both families.",
"Let $\\lbrace p_{i}\\rbrace _{i=1}^{\\infty }$ be an infinite sequence in $\\mathbb {N}$ and $FS(\\lbrace p_{i}\\rbrace _{i=1}^{\\infty })=\\left\\lbrace p_{i_1}+p_{i_2}+\\cdots +p_{i_n}: 1\\le i_1<i_2<\\cdots <i_n,\\ n\\in \\mathbb {N}\\right\\rbrace .$ A subset $A\\subset \\mathbb {Z}^+$ is an IP-set if it equals to some $FS(\\lbrace p_{i}\\rbrace _{i=1}^{\\infty })$ .",
"Denote the family generated by all IP-sets by ${F}_{IP}$ .",
"It follows from Hindman's Theorem [10] that ${F}_{IP}$ is a filterdual.",
"For a family ${F}$ , a dynamical system $(X, T)$ is called ${F}$ -transitive if $N(U, V)\\in {F}$ for every pair of nonempty open subsets $U, V\\subset X$ ; and it is ${F}$ -mixing if $(X \\times X, T\\times T)$ is ${F}$ -transitive.",
"Lemma 1 [1] Let $(X, T)$ be a dynamical system and ${F}$ be a full family.",
"Then, the following statements are equivalent: $(X, T)$ is ${F}$ -mixing; $\\forall n\\in \\mathbb {N}$ , $T^{(n)}$ is ${F}$ -mixing; $(X, T)$ is weakly mixing and ${F}$ -transitive."
],
[
"Topological dynamics",
"For $U, V\\subset X$ , define the return time set from $U$ to $V$ as $ N(U, V)=\\lbrace n\\in \\mathbb {Z}^{+}:T^{n}(U)\\cap V\\ne \\emptyset \\rbrace $ .",
"In particular, $N(x, V)=\\left\\lbrace n\\in \\mathbb {Z}^{+}:T^{n}(x)\\in V\\right\\rbrace $ for $x\\in X$ .",
"A dynamical system $(X, T)$ is (topologically) weakly mixing if $(X \\times X, T\\times T)$ is transitive; topologically mixing if for every pair of nonempty open subsets $U, V$ of $X$ , $N(U, V)$ is cofinite, i.e., there exists $m\\in \\mathbb {N}$ such that $[m, +\\infty )\\subset N(U, V)$ .",
"It is well known that $(X, T)$ is transitive (resp., weakly mixing) if and only if it is ${F}_{inf}$ -transitive (resp., ${F}_{t}$ -transitive) (see [7]).",
"A point $x \\in X$ is a transitive point of $T$ if its orbit $\\mathrm {orb}(x, T):=\\lbrace x, T(x), T^{2}(x), \\ldots \\rbrace $ is dense in $X$ .",
"The set of all transitive points of $T$ is denoted by $\\mathrm {Tran}(T)$ .",
"It is well known that if $(X, T)$ is transitive, then $\\mathrm {Tran}(T)$ is a dense $G_{\\delta }$ -set.",
"The $\\omega $ -limit set of $x$ is the set of limit points of its orbit sequence $\\omega (x, T)=\\bigcap _{m=0}^{+\\infty }\\overline{\\lbrace T^{n}(x): n\\ge m\\rbrace }$ .",
"A point $x\\in X$ is a recurrent point of $T$ if $x\\in \\omega (x, T)$ , i.e., there exists $m_{i}\\longrightarrow +\\infty $ such that $T^{m_{i}}(x)\\longrightarrow x$ .",
"A well known result of Birkhoff states that every dynamical system admits a recurrent point.",
"Lemma 2 [7] Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $(X, T)$ is weakly mixing; For any pair of nonempty open subsets $U, V$ of $X$ , $N(U, U)\\cap N(U, V)\\ne \\emptyset $ ; For any pair of nonempty open subsets $U, V$ of $X$ , $N(U, V)\\cap N(V, V)\\ne \\emptyset $ ; $(X, T)$ is ${F}_{t}$ -transitive.",
"Recently, Moothathu [21] introduced the notion of multi-transitivity.",
"A dynamical system $(X, T)$ is called multi-transitive if for any $n\\in \\mathbb {N}$ , the product system $(X^n, T\\times T^2\\times \\cdots \\times T^n)$ is transitive.",
"He also proved that a minimal system is multi-transitive if and only if it is weakly mixing and asked whether there are implications between the multi-transitivity and the weak mixing property for general (not necessarily minimal) systems.",
"Then, Kwietniak and Oprocha [14] showed that in general there is no connection between the multi-transitivity and the weakly mixing property by constructing examples of weakly mixing but non-multi-transitive and multi-transitive but non-weakly mixing systems.",
"To generalize the concept of multi-transitivity, Chen et al.",
"[4] introduced the notion of multi-transitivity with respect to a vector.",
"Let $\\mathbf {a}=(a_{1}, a_{2}, \\cdots , a_n)$ be a vector in $\\mathbb {N}^n$ .",
"A dynamical system $(X, T)$ is multi-transitive with respect to the vector $\\mathbf {a}$ (briefly, $\\mathbf {a}$ -transitive) if the product system $(X^{n}, T^{(\\mathbf {a})})$ is transitive, where $T^{(\\mathbf {a})}=T^{a_{1}}\\times T^{a_{2}}\\times \\cdots \\times T^{a_{r}}$ .",
"Lemma 3 [33] Let $(X, T)$ be a dynamical system and $\\mathbf {a}=(a_{1}, \\ldots , a_{n})\\in \\mathbb {N}^{n}$ .",
"Then, the following statements are equivalent: $(X, T)$ is weakly mixing and $\\mathbf {a}$ -transitive; For any $m\\in \\mathbb {N}$ , $(X^{m}, T^{b_{1}}\\times T^{b_{2}}\\times \\cdots \\times T^{b_{m}})$ is transitive, where $b_{i}\\in \\lbrace a_{i}: 1\\le i\\le n\\rbrace $ , $i=1, 2, \\ldots , m$ .",
"A dynamical system $(X, T)$ is weakly disjoint with another dynamical system $(Y, S)$ if their product system $(X\\times Y, T\\times S)$ is transitive.",
"A dynamical system is mildly mixing if it is weakly disjoint with every transitive system.",
"Huang and Ye [11] proved that a dynamical system is mildly mixing if and only if it is $\\kappa ({F}_{IP}-{F}_{IP})$ -transitive.",
"A dynamical system $(X, T)$ is equicontinuous if for any $\\varepsilon >0$ , there exists $\\delta >0$ such that for any $x, y\\in X$ with $d(x, y)< \\delta $ and any $n\\in \\mathbb {Z}^{+}$ , $d(T^{n}(x), T^{n}(y))<\\varepsilon $ ."
],
[
"Hyperspace $K(X)$",
"Let $K(X)$ be the hyperspace on $X$ , i.e., the space of all nonempty closed subsets of $X$ with the Hausdorff metric $d_{H}$ defined by $d_{H}(A, B)=\\max \\left\\lbrace \\max _{x\\in A}\\min _{y\\in B}d(x, y), \\max _{y\\in B}\\min _{x\\in A}d(x, y)\\right\\rbrace ,\\quad \\forall A, B\\in K(X).$ It is known that $(K(X), d_{H})$ is also a compact metric space (see [13]).",
"The system $(X, T)$ induces naturally a set-valued dynamical system $(K(X), T_{K})$ , where $T_{K}: K(X)\\longrightarrow K(X)$ is defined as $T_{K}(A)=T(A)$ for any $A\\in K(X)$ .",
"For any finite collection $A_{1}, \\ldots , A_{n}$ of nonempty subsets of $X$ , take $\\langle A_{1}, \\ldots , A_{n}\\rangle =\\left\\lbrace A\\in K(X): A\\subset \\bigcup _{i=1}^{n}A_{i},A\\cap A_{i}\\ne \\emptyset \\text{ for all } i=1, \\ldots , n\\right\\rbrace .$ It follows from [13] that the topology on $K(X)$ given by the metric $d_{H}$ is same as the Vietoris or finite topology, which is generated by a basis consisting of all sets of the following form, $\\langle U_{1}, \\ldots , U_{n}\\rangle ,\\text{ where } U_{1}, \\ldots , U_{n} \\text{ are an arbitrary finite collection of nonempty open subsets of} X.$ Under this topology $\\mathcal {F}(X)$ , the set of all finite subsets of $X$ , is dense in $K(X)$ ."
],
[
"Zadeh's extension",
"Let $I= [0, 1]$ .",
"A fuzzy set $A$ in space $X$ is a function $A: X\\longrightarrow I$ .",
"Given a fuzzy set $A$ , its $\\alpha $ -cuts (or $\\alpha $ -level sets) $[A]_{\\alpha }$ and support $\\mathrm {supp}(A)$ are defined respectively by $[A]_{\\alpha }=\\lbrace x\\in X: A(x)\\ge \\alpha \\rbrace , \\quad \\forall \\alpha \\in I,$ and $\\mathrm {supp}(A)=\\overline{\\left\\lbrace x\\in X: A(x)>0\\right\\rbrace }.$ Let $\\mathbb {F}(X)$ denote the set of all upper semicontinuous fuzzy sets defined on $X$ and set $\\mathbb {F}^{\\ge \\lambda }(X)=\\left\\lbrace A\\in \\mathbb {F}(X): A(x)\\ge \\lambda \\text{ for some }x\\in X\\right\\rbrace ,$ $\\mathbb {F}^{=\\lambda }(X)=\\left\\lbrace A\\in \\mathbb {F}(X): \\max \\lbrace A(x): x\\in X\\rbrace =\\lambda \\right\\rbrace .$ Especially, let $\\mathbb {F}^{=1}(X)$ denote the system of all normal fuzzy sets on $X$ .",
"Define $\\emptyset _{X}$ as the empty fuzzy set ($\\emptyset _{X}\\equiv 0$ ) in $X$ , and $\\mathbb {F}_{0}(X)$ as the set of all nonempty upper semicontinuous fuzzy sets.",
"Since the Hausdorff metric $d_{H}$ is measured only between two nonempty closed subsets in $X$ , one can consider the following extension of the Hausdorff metric: $d_{H}(\\emptyset , \\emptyset )=0 \\text{ and } d_{H}(\\emptyset , A)=d_{H}(A, \\emptyset )=\\mathrm {diam}(X), \\quad \\forall A\\in {K}(X).$ Under this Hausdorff metric, one can define a levelwise metric $d_{\\infty }$ on $\\mathbb {F}(X)$ by $d_{\\infty }(A, B)=\\sup \\left\\lbrace d_{H}([A]_{\\alpha }, [B]_{\\alpha }): \\alpha \\in (0, 1]\\right\\rbrace , \\quad \\forall A, B\\in \\mathbb {F}(X).$ It is well known that the spaces $(\\mathbb {F}(X), d_{\\infty })$ and $(\\mathbb {F}^1(X), d_{\\infty })$ are complete, but not compact and not separable (see [16] and references therein).",
"A fuzzy set $A\\in \\mathbb {F}(X)$ is piecewise constant if there exists a finite number of sets $D_{i}\\subset X$ such that $\\bigcup \\overline{D_{i}}=X$ and $A|_{\\mathrm {int}\\overline{D_{i}}}$ is constant.",
"In this case, a piecewise constant $A$ can be represented by a strictly decreasing sequence of closed subsets $\\lbrace A_{1}, A_{2}, \\ldots , A_{k}\\rbrace \\subset K(X)$ and a strictly increasing sequence of reals $\\lbrace \\alpha _{1}, \\alpha _{2}, \\ldots , \\alpha _{k}=\\max \\lbrace A(x): x\\in X\\rbrace \\rbrace \\subset (0, 1]$ if $[A]_{\\alpha }=A_{i+1}, \\text{ whenever } \\alpha \\in (\\alpha _{i}, \\alpha _{i+1}].$ Fix any two piecewise constants $A, B\\in \\mathbb {F}(X)$ which are represented by strictly decreasing sequences of closed subsets $\\lbrace A_{1}, A_{2}, \\ldots , A_{k}\\rbrace $ , $\\lbrace B_{1}, B_{2}, \\ldots , B_{s}\\rbrace \\subset K(X)$ and strictly increasing sequences of reals $\\lbrace \\alpha _{1}, \\alpha _{2}, \\ldots , \\alpha _{k}\\rbrace $ , $\\lbrace \\beta _{1}, \\beta _{2}, \\ldots , \\beta _{s}\\rbrace \\subset (0, 1]$ with $[A]_{\\alpha }=A_{i+1}, \\quad \\forall \\alpha \\in (\\alpha _{i}, \\alpha _{i+1}] \\text{ and }[B]_{\\alpha }=B_{i+1}, \\quad \\forall \\beta \\in (\\beta _{i}, \\beta _{i+1}],$ respectively.",
"Arrange all reals $\\alpha _{1}, \\alpha _{2}, \\ldots , \\alpha _{k}, \\beta _{1}, \\beta _{2}, \\ldots , \\beta _{s}$ by the natural order `$<$ ' and denote them by $\\gamma _{1}, \\gamma _{2}, \\ldots , \\gamma _{n}$ $(n\\le k+s)$ .",
"Then, it can be verified that for any $1\\le t<n$ , there exist $1\\le i\\le k$ and $1\\le j\\le s$ such that for any $\\gamma \\in (\\gamma _{t}, \\gamma _{t+1}]$ , $[A]_{\\gamma }=A_{i} \\text{ and } [B]_{\\gamma }=B_{j}.$ This implies that there exist (not necessarily strictly) decreasing sequences of closed subsets $\\lbrace C_{1}, C_{2}, \\ldots , C_{n}\\rbrace $ , $\\lbrace D_{1}, D_{2}, \\ldots , D_{n}\\rbrace \\subset K(X)$ and a strictly increasing sequence of reals $\\gamma _{1}, \\gamma _{2}, \\ldots , \\gamma _{n}\\subset (0, 1]$ such that $[A]_{\\gamma }=C_{i+1} \\text{ and } B_{\\gamma }=D_{i+1}, \\text{ whenever }\\gamma \\in (\\gamma _{i}, \\gamma _{i+1}].$ To generalize the concept of Zadeh's extension, Kupka [16] introduced the notion of $g$ -fuzzification.",
"Zadeh's extension (also called usual fuzzification) of a dynamical system $(X, T)$ is a map $T_{F}: \\mathbb {F}(X)\\longrightarrow \\mathbb {F}(X)$ defined by $T_{F}(A)(x)=\\sup \\left\\lbrace A(y): y\\in T^{-1}(x)\\right\\rbrace ,\\quad \\forall A\\in \\mathbb {F}(X), \\forall x\\in X.$ Lemma 4 [15] Let $(X, T)$ be a dynamical system and $\\lambda \\in (0, 1]$ .",
"Then, the set of piecewise constants is dense in $\\mathbb {F}(X)$ , $\\mathbb {F}^{\\ge \\lambda }(X)$ and $\\mathbb {F}^{=\\lambda }(X)$ .",
"$(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is periodically dense in $\\mathbb {F}^{=\\lambda }(X)$ if and only if $(K(X), T_{K})$ is periodically dense in $K(X)$ .",
"For any $A, B\\in K(X)$ , any $n\\in \\mathbb {N}$ and any $\\lambda \\in (0, 1]$ , $d_{H}(T_{K}^{n}(A), T_{K}^{n}(B))=d_{\\infty }(T_{F}^{n}(\\lambda \\cdot \\chi _{A}),T_{F}^{n}(\\lambda \\cdot \\chi _{B})).$ This shows that for any fixed $\\lambda \\in (0, 1]$ , the subsystem $(\\mathbb {F}_{\\lambda \\cdot \\chi }:=\\lbrace \\lambda \\cdot \\chi _{A}\\in \\mathbb {F}_{0}(X):A\\in K(X)\\rbrace , T_{F}|_{\\mathbb {F}_{\\lambda \\cdot \\chi }})$ is topologically conjugated to $(K(X), T_{K})$ .",
"Let $D_{m}(I)$ be the set of all nondecreasing right-continuous functions $g: I\\longrightarrow I$ with $g(0)=0$ and $g(1)=1$ .",
"For a dynamical system $(X, T)$ and for any $g\\in D_{m}(I)$ , define a map $T_{F}^{g}: \\mathbb {F}(X)\\longrightarrow \\mathbb {F}(X)$ by $T_{F}^{g}(A)(x)=\\sup \\left\\lbrace g(A(y)): y\\in T^{-1}(x)\\right\\rbrace ,\\quad \\forall A\\in \\mathbb {F}(X), \\forall x\\in X,$ which is called the $g$ -fuzzification of the dynamical system $(X, T)$ .",
"Clearly, $T_{F}=T_{F}^{\\mathrm {id}_{I}}$ , where $\\mathrm {id}_{I}$ is the identity map defined on $I$ .",
"Also, define the $\\alpha $ -cut $[A]_{\\alpha }^{g}$ of a fuzzy set $A\\in \\mathbb {F}(X)$ with respect to $g\\in D_{m}(I)$ by $[A]_{\\alpha }^{g}=\\left\\lbrace x\\in \\mathrm {supp}(A): g(A(x))\\ge \\alpha \\right\\rbrace .$ For any $g\\in D_{m}(I)$ , the right-continuity of $g$ implies that $\\min g^{-1}([x, 1])$ exists for any $x\\in [0, 1]$ .",
"Since $g$ is nondecreasing, $\\min g^{-1}([x, 1])>0$ holds for any $x\\in (0, 1]$ .",
"Define $\\xi _{g}: [0, 1]\\longrightarrow [0,1]$ by $\\xi _{g}(x)=\\min g^{-1}([x, 1])$ for any $x\\in [0, 1]$ .",
"Clearly, $\\xi _{g}$ is nondecreasing.",
"Recently, we [32] proved the following result: Lemma 5 [32] Let $(X, T)$ be a dynamical system, $g\\in D_{m}(I)$ and $T_{F}^{g}$ be the $g$ -fuzzification of $T$ .",
"Then, for any $n\\in \\mathbb {N}$ , any $A\\in \\mathbb {F}(X)$ and any $\\alpha \\in (0, 1]$ , $\\left[(T_{F}^{g})^{n}(A)\\right]_{\\alpha }=T^{n}([A]_{\\xi _{g}^{n}(\\alpha )})$ .",
"In particular, $\\left[T_{F}^{n}(A)\\right]_{\\alpha }=T^{n}([A]_{\\alpha })$ ."
],
[
"Transitivity of $(\\mathbb {F}^{=\\lambda }(X), T_{F})$",
"Banks [3] proved the following result on the transitivity of $(K(X), T_{K})$ .",
"Lemma 6 [3] Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $(X, T)$ is weakly mixing; $(K(X), T_{K})$ is weakly mixing; $(K(X, T_{K})$ is transitive.",
"Inspired by Lemma REF , this section is devoted to studying the transitivity of Zadeh's extension $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ .",
"In particular, it is proved that both the transitivity and the weakly mixing property of $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ are equivalent to the weakly mixing property of $(X, T)$ (see Theorem REF ).",
"Theorem 1 Let $(X, T)$ be a dynamical system and $C$ be an invariant closed subset of $\\mathbb {F}_{0}(X)$ .",
"If $(C, T_{F}|_{C})$ is transitive, then for any $A, B\\in C$ , $\\max \\lbrace A(x): x\\in X\\rbrace =\\max \\lbrace B(x): x\\in X\\rbrace $ , i.e., there exists $\\lambda \\in [0, 1]$ such that $C\\subset \\mathbb {F}^{=\\lambda }(X)$ .",
"Proof.",
"Suppose that there exist $A, B\\in C$ such that $\\xi :=\\max \\lbrace A(x): x\\in X\\rbrace <\\max \\lbrace B(x): x\\in X\\rbrace :=\\eta .$ Choose $a=\\min \\left\\lbrace \\frac{\\eta -\\xi }{4}, \\frac{\\mathrm {diam}(X)}{4}\\right\\rbrace >0$ and set $U=B_{d_{\\infty }}(A, a)\\cap C$ and $V=B_{d_{\\infty }}(B, a)\\cap C$ .",
"Clearly, $U$ and $V$ are nonempty open subsets of $C$ .",
"For any $D\\in U$ , $a>d_{\\infty }(A,D)\\ge d_{H}([A]_{a}, [D]_{a})=d_{H}(\\emptyset , [D]_{a}),$ implying that $[D]_{a}=\\emptyset $ .",
"Then, for any $n\\in \\mathbb {Z}^{+}$ , $d_{H}([T_{F}^{n}(D)]_{a}, [B]_{a})&=&d_{H}(T_{K}^{n}([D]_{a}), [B]_{a})\\\\&=&d_{H}(\\emptyset , [B]_{a}) \\quad (\\text{as } [B]_{a}\\ne \\emptyset )\\\\&=& \\mathrm {diam}(X).$ This implies that $T_{F}^{n}(U)\\cap V=\\emptyset $ , i.e., $(C, T_{F}|_{C})$ is not transitive, which is a contradiction.",
"$\\blacksquare $ Lemma 7 Let $(X, T)$ be a dynamical system and $\\lambda \\in (0, 1]$ .",
"If $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is transitive, then $(K(X), T_{K})$ is weakly mixing.",
"Proof.",
"Applying Lemma REF , it suffices to prove that $T_{K}$ is transitive.",
"For any pair of nonempty open subsets $U, V$ of $K(X)$ , there exist $A\\in U$ , $B\\in V$ and $0<\\delta <\\frac{\\mathrm {diam}(X)}{2}$ such that $B_{d_{H}}(A, \\delta ):=\\lbrace C\\in K(X): d_{H}(C, A)<\\delta \\rbrace \\subset U$ and $B_{d_{H}}(B, \\delta )\\subset V$ .",
"Noting that $U_{1}:=B_{d_{\\infty }}(\\lambda \\cdot \\chi _{A}, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)$ and $V_{1}:=B_{d_{\\infty }}(\\lambda \\cdot \\chi _{B}, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)$ are nonempty open subsets of $\\mathbb {F}^{=\\lambda }(X)$ , since $T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}$ is transitive, there exists $n\\in \\mathbb {Z}^{+}$ such that $T_{F}^{n}(U_{1})\\cap V_{1}\\ne \\emptyset .$ Then, there exists a point $E\\in U_1$ such that $T_{F}^{n}(E)\\in V_{1}$ .",
"This implies that $d_{H}\\left(\\left[T_{F}^{n}(E)\\right]_{\\lambda }, [\\lambda \\cdot \\chi _B]_{\\lambda }\\right)=d_{H}(T_{K}^{n}([E]_{\\lambda }), B)<\\delta .$ Since $E\\in B_{d_{\\infty }}(\\lambda \\cdot \\chi _A, \\delta )$ , it can be verified that $d_{H}(A, [E]_{\\lambda })<\\delta .$ Clearly, $[E]_{\\lambda }\\ne \\emptyset .$ Then, $[E]_{\\lambda }\\in B_{d_{H}}(A, \\delta )\\subset U.$ Combining this with (REF ), it follows that $T_{K}^{n}([F]_{\\lambda })\\in T_{K}^{n}(U)\\cap V\\ne \\emptyset .$ $\\blacksquare $ Lemma 8 Let $(X, T)$ be a dynamical system and $\\lambda \\in (0, 1]$ .",
"If $(K(X), T_{K})$ is weakly mixing, then $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is weakly mixing.",
"Proof.",
"Given any pair of nonempty open subsets $U, V$ of $\\mathbb {F}^{=\\lambda }(X)$ , applying Lemma REF implies that there exist piecewise constants $P\\in U$ , $Q\\in V$ and $\\delta >0$ such that $B_{d_{H}}(P, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\subset U$ and $B_{d_{H}}(Q, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\subset V$ .",
"Since $P$ and $Q$ are piecewise constants and $P, Q\\in \\mathbb {F}^{=\\lambda }(X)$ , it follows from Remark REF that there exist strictly increasing sequence of reals $\\lbrace \\alpha _{1}, \\ldots , \\alpha _{k}=\\lambda \\rbrace \\subset [0, 1]$ and decreasing sequences of closed subsets $\\lbrace C_{1}, \\ldots , C_{k}\\rbrace \\subset K(X)$ , $\\lbrace D_{1}, \\ldots , D_{k}\\rbrace \\subset K(X)$ such that $[P]_{\\alpha }=C_{i+1}, \\ [Q]_{\\alpha }=D_{i+1}, \\text{ whenever } \\alpha \\in (\\alpha _{i}, \\alpha _{i+1}].$ Noting that $B_{d_{H}}(C_{1}, \\frac{\\delta }{2}), \\ldots , B_{d_{H}}(C_{k}, \\frac{\\delta }{2})$ and $B_{d_{H}}(C_{1}, \\frac{\\delta }{2}), \\ldots , B_{d_{H}}(C_{k}, \\frac{\\delta }{2})$ are nonempty open subsets of $K(X)$ , since $T_{K}$ is weakly mixing, it follows that there exists $n\\in \\mathbb {Z}^{+}$ such that for any $1\\le i\\le k$ , $T_{K}^{n}\\left(B_{d_{H}}\\left(C_{i}, \\frac{\\delta }{2}\\right)\\right)\\cap B_{d_{H}}\\left(C_{i}, \\frac{\\delta }{2}\\right) \\ne \\emptyset ,$ and $T_{K}^{n}\\left(B_{d_{H}}\\left(C_{i}, \\frac{\\delta }{2}\\right)\\right)\\cap B_{d_{H}}\\left(D_{i}, \\frac{\\delta }{2}\\right) \\ne \\emptyset ,$ implying that there exist $G_{i}, E_{i}\\in B_{d_{H}}(C_{i}, \\frac{\\delta }{2})$ such that $T_K^{n}(G_{i})\\in B_{d_{H}}\\left(C_{i}, \\frac{\\delta }{2}\\right) \\text{ and }T_{K}^{n}(E_{i})\\in B_{d_{H}}\\left(D_{i}, \\frac{\\delta }{2}\\right).$ Define respectively two fuzzy sets $G: X\\longrightarrow I$ and $E: X\\longrightarrow I$ by $[G]_{\\alpha }=\\bigcup _{j=i+1}^{k}G_{j} \\text{ and }[E]_{\\alpha }=\\bigcup _{j=i+1}^{k}E_{j}, \\quad \\forall \\alpha \\in (\\alpha _{i}, \\alpha _{i+1}].$ Clearly, $G, E\\in \\mathbb {F}^{=\\lambda }(X)$ .",
"Since $G_{i}, E_{i}\\in B_{d_{H}}(C_{i}, \\frac{\\delta }{2})$ , it can be verified that for any $\\alpha \\in (\\alpha _{i}, \\alpha _{i+1}]$ , $d_{H}([G]_{\\alpha }, [P]_{\\alpha })=d_{H}\\left(\\bigcup _{j=i+1}^{k}G_{j}, C_{i+1}\\right)<\\frac{\\delta }{2},$ and $d_{H}([E]_{\\alpha }, [P]_{\\alpha })<d_{H}\\left(\\bigcup _{j=i+1}^{k}E_{j}, C_{i+1}\\right)<\\frac{\\delta }{2}.$ This implies that $G, E\\in B_{d_{\\infty }}(P, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\subset U.$ Meanwhile, since $T_{K}^{n}(G_{i})\\in B_{d_{H}}(C_{i}, \\frac{\\delta }{2})$ and $T_{K}^{n}(E_{i})\\in B_{d_{H}}(D_{i}, \\frac{\\delta }{2})$ , it can be verified that for any $\\alpha \\in (\\alpha _{i}, \\alpha _{i+1}]$ , $d_{H}([T_{F}^{n}(G)]_{\\alpha }, [P]_{\\alpha })=d_{H}(T_{K}^{n}([G]_{\\alpha }), [P]_{\\alpha })=d_{H}\\left(\\bigcup _{j=i+1}^{k}T_{K}^{n}(G_{j}), C_{i+1}\\right)<\\frac{\\delta }{2},$ and $d_{H}([T_{F}^{n}(E)]_{\\alpha }, [Q]_{\\alpha })=d_{H}(T_{K}^{n}([E]_{\\alpha }), [Q]_{\\alpha })=d_{H}\\left(\\bigcup _{j=i+1}^{k}T_{K}^{n}(E_{j}), D_{i+1}\\right)<\\frac{\\delta }{2}.$ Then, $\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(G)\\in B_{d_{H}}(P, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\subset U$ and $\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(E)\\in B_{d_{H}}(Q, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\subset V.$ Therefore, $\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(G)\\in T_{F}^{n}(U)\\cap U\\ne \\emptyset \\text{ and }\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(E)\\in T_{F}^{n}(U)\\cap V\\ne \\emptyset .$ This, together with Lemma REF , implies that $(\\mathbb {F}^{=\\lambda }(X), T_{F})$ is weakly mixing.",
"$\\blacksquare $ Theorem 2 Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $(X, T)$ is weakly mixing; $(K(X), T_{K})$ is transitive; $(K(X), T_{K})$ is weakly mixing; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is transitive; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is weakly mixing.",
"Proof.",
"Applying Lemma REF implies that (REF )$\\Longleftrightarrow $ (REF )$\\Longleftrightarrow $ (REF ).",
"It follows from Lemma REF and Lemma REF that (REF )$\\Longrightarrow $ (REF ) $\\Longrightarrow $ (REF )$\\Longrightarrow $ (REF ).",
"$\\blacksquare $ Corollary 1 Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $({K}(X), T_{K})$ is Devaney chaotic; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is Devaney chaotic; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is Devaney chaotic.",
"Proof.",
"It follows immediately from Theorem REF and Lemma REF .",
"$\\blacksquare $ In [18], Lan et al.",
"proved that $(X, T)$ is weakly mixing if and only if $(\\mathbb {F}^{=1}(X), T_{F}|_{\\mathbb {F}^{=1}(X)})$ is weakly mixing.",
"However, their proof is not correct.",
"Because the proof of [18] is based on [18] which claimed that for any nonempty open subset $U\\subset \\mathbb {F}^{=1}(X)$ , $r(U):=\\left\\lbrace A\\in K(X):\\exists u\\in U \\text{ such that } A\\subset [u]_{0}\\right\\rbrace $ is also a nonempty open subset of $X$ .",
"It is clear that $r(U)$ is not a open subset of $X$ , because $K(X)\\supset r(U)\\nsubseteq X$ .",
"Meanwhile, noting that for any $u\\in U$ , $[u]_{0}=X$ , it is easy to see that $r(U)=K(X)$ .",
"Kupka [15] obtained that the Devaney's chaoticity of $(X, T)$ does not imply the same of $(\\mathbb {F}^{=1}(X), T_{F}|_{\\mathbb {F}^{=1}(X)})$ .",
"According to [19], there exists a dynamical system $(X, T)$ such that $(K(X), T_{K})$ is Devaney chaotic, while $(X, T)$ is not Devaney chaotic, showing that the answer to [23] is negative.",
"This, together with Corollary REF , shows that the Devaney's chaoticity of $(\\mathbb {F}^{=1}(X), T_{F}|_{\\mathbb {F}^{=1}(X)})$ does not imply the Devaney's chaoticity of $(X, T)$ .",
"Applying Theorem REF and Corollary REF yields that [15] holds trivially and that the converses of [15] are true.",
"About the weakly mixing property of dynamical systems, Liao et al.",
"[23] provided the following question: [23] Which systems, besides $T_{K}$ , have the equivalence between the transitivity and the weakly mixing property?",
"As a partial answer to Question , applying Theorem REF , we know that the Zadeh's extension restricted on the space of normal fuzzy sets has the equivalence between the transitivity and the weakly mixing property.",
"Lemma 9 Let $(X, T)$ be a dynamical system and ${F}$ be a full family.",
"Then, the following statements are equivalent: $(X, T)$ is ${F}$ -mixing; $(K(X), T_{K})$ is ${F}$ -transitive; $(K(X), T_{K})$ is ${F}$ -mixing.",
"Slightly modifying the proofs of Lemma REF and Lemma REF , applying Lemma REF and Lemma REF , it is not difficult to prove the following.",
"Theorem 3 Let $(X, T)$ be a dynamical system and $\\lambda \\in (0, 1]$ .",
"Then, $(X, T)$ is mixing if and only if $(K(X), T_{K})$ is mixing if and only if $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is mixing.",
"Corollary 2 Let $(X, T)$ be a dynamical system and ${F}$ be a full family.",
"Then, the following statements are equivalent: $(X, T)$ is ${F}$ -mixing; $(K(X), T_{K})$ is ${F}$ -transitive; $(K(X), T_{K})$ is ${F}$ -mixing; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is ${F}$ -transitive; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is ${F}$ -mixing; $\\exists \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is ${F}$ -transitive; $\\exists \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is ${F}$ -mixing."
],
[
"Mildly mixing property and $\\mathbf {a}$ -transitivity of {{formula:285fcf25-77f9-48e6-819c-f77464f999d1}}",
"Bauer and Sigmund [5] proved the equivalence of the mildly mixing property between $(X, T)$ and $(K(X), T_{K})$ .",
"Lemma 10 [5] A dynamical system $(X, T)$ is mildly mixing if and only if $(K(X), T_{K})$ is mildly mixing.",
"Similarly to the proof of [17], it can be verified that the following result holds.",
"Lemma 11 Let $(X, T)$ be a dynamical system and $\\lambda \\in (0, 1]$ .",
"If $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is mildly mixing, then $(X, T)$ is mildly mixing.",
"Lemma 12 If $(K(X), T_{K})$ is mildly mixing, then for any $\\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is mildly mixing.",
"Proof.",
"It suffices to check that for any transitive system $(Y, S)$ , $(\\mathbb {F}^{=\\lambda }(X)\\times Y, T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\times S)$ is transitive.",
"For any pair of nonempty open subsets $W, V$ of $\\mathbb {F}^{=\\lambda }(X)\\times Y$ , it follows from Lemma REF and Remark REF that there exist piecewise constants $A, B\\in \\mathbb {F}^{=\\lambda }(X)$ which are represented by decreasing sequences of closed subsets $\\lbrace A_{1}, A_{2}, \\ldots , A_{k}\\rbrace $ , $\\lbrace B_{1}, B_{2}, \\ldots , B_{k}\\rbrace \\subset K(X)$ and a strictly increasing sequence of reals $\\lbrace \\alpha _{1}, \\alpha _{2}, \\ldots , \\alpha _{k}=\\lambda \\rbrace \\subset (0, 1]$ such that $[A]_{\\alpha }=A_{i+1}, \\ [B]_{\\alpha }=B_{i+1}, \\text{ whenever } \\alpha \\in (\\alpha _{i}, \\alpha _{i+1}],$ $y_{1}, y_{2}\\in Y$ and $\\delta >0$ such that $\\left[B_{d_{\\infty }}(A, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\right]\\cap B(y_{1}, \\delta )\\subset W,$ and $\\left[B_{d_{\\infty }}(B, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\right]\\cap B(y_{2}, \\delta )\\subset V.$ Clearly, $(\\underbrace{K(X)\\times \\cdots \\times K(X)}\\limits _{k}\\times Y,\\underbrace{T_{K}\\times \\cdots \\times T_{K}}\\limits _{k}\\times S)$ is transitive, as $T_{K}$ is mildly mixing.",
"This implies that there exists $n\\in \\mathbb {Z}^+$ such that for any $1\\le i\\le k$ , $T_{K}^{n}(B_{d_{H}}(A_{i}, \\delta ))\\cap B_{d_{H}}(B_{i}, \\delta )\\ne \\emptyset ,$ and $S^{n}(B(y_1, \\delta ))\\cap B(y_2, \\delta )\\ne \\emptyset .$ Then there exist $C_{i}\\in B_{d_{H}}(A_{i}, \\delta )$ and $y\\in B(y_{1}, \\delta )$ such that $d_{H}(T_{K}^{n}(C_{i}), B_{i})<\\delta \\text{ and } d(S^{n}(y), y_{2})<\\delta .$ Take a piecewise constant $C\\in \\mathbb {F}^{=\\lambda }(X)$ as $[C]_{\\alpha }=\\bigcup _{j=i+1}^{k}C_{j}, \\text{ whenever } \\alpha \\in (\\alpha _{i}, \\alpha _{i+1}].$ It can be verified that the following statements hold: $d_{H}\\left(\\bigcup _{j=i+1}^{k}C_{j}, A_{i+1}\\right)=d_{H}\\left(\\bigcup _{j=i+1}^{k}C_{j}, \\bigcup _{j=i+1}^{k}A_{j}\\right)<\\delta $ ; $d_{H}\\left(T_{K}^{n}(\\bigcup _{j=i+1}^{k}C_{j}), B_{i+1}\\right)=d_{H}\\left(\\bigcup _{j=i+1}^{k}T_{K}^{n}(C_{j}),\\bigcup _{j=i+1}^{k}B_{j}\\right)<\\delta $ ; $d(S^{n}(y), y_{2})<\\delta $ .",
"Therefore, $\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(C)\\in \\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}\\left(B_{d_{\\infty }}(A, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\right)\\cap \\left[B_{d_{\\infty }}(B, \\delta )\\cap \\mathbb {F}^{=\\lambda }(X)\\right]\\ne \\emptyset ,$ and $S^{n}(y)\\in S^{n}(B(y_{1}, \\delta ))\\cap B(y_{2}, \\delta )\\ne \\emptyset ,$ implying that $\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\times S\\right)^{n}(W)\\cap V\\ne \\emptyset .$ Hence, $T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\times S$ is transitive.",
"$\\blacksquare $ Summing up Lemma REF , Lemma REF and Lemma REF , one has Theorem 4 Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $(X, T)$ is mildly mixing; $(K(X), T_{K})$ is mildly mixing; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is mildly mixing; $\\exists \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is mildly mixing.",
"Applying Lemma REF and Lemma REF , similarly to the proof of Lemma REF , it can be verified that the following holds.",
"Theorem 5 Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $(X, T)$ weakly mixing and $\\mathbf {a}$ -transitive; $(K(X), T_{K})$ is $\\mathbf {a}$ -transitive; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is $\\mathbf {a}$ -transitive; $\\exists \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is $\\mathbf {a}$ -transitive."
],
[
"Equicontinuity of $(\\mathbb {F}_{0}(X), T_{F})$",
"Based on Bauer and Sigmund's result which states that $(X, T)$ is equicontinuous if and only if $(K(X), T_{K})$ is equicontinuous, this section proves the equivalence of the equicontinuity between $(X, T)$ and $(\\mathbb {F}_{0}(X), T_{F})$ .",
"Lemma 13 [5] A dynamical system $(X, T)$ is equicontinuous if and only if $(K(X), T_{K})$ is equicontinuous.",
"Proposition 1 Let $(X, d)$ be a complete metric space, $T: X\\longrightarrow X$ be a continuous map and $A\\subset X$ be a dense subset of $X$ .",
"If $T|_{A}: A\\longrightarrow X$ is equicontinuous, then $T: X\\longrightarrow X$ is equicontinuous.",
"Proof.",
"For any fixed $\\varepsilon >0$ , there exists $\\delta >0$ such that for any $x, y\\in A$ with $d(x, y)<\\delta $ and any $n\\in \\mathbb {Z}^+$ , $d(T^{n}(x), T^{n}(y))<\\frac{\\varepsilon }{2}.$ For any $x^{\\prime }, y^{\\prime }\\in X$ with $d(x^{\\prime }, y^{\\prime })<\\frac{\\delta }{2}$ and any $n\\in \\mathbb {Z}^{+}$ , as $T^{n}$ is continuous at $x^{\\prime }$ and $y^{\\prime }$ , there exists $0<\\delta ^{\\prime }<\\frac{\\delta }{4}$ such that for any $z_{1}\\in B(x^{\\prime }, \\delta ^{\\prime })$ and any $z_{2}\\in B(y^{\\prime }, \\delta ^{\\prime })$ , $d(T^{n}(z_{1}), T^{n}(x^{\\prime }))<\\frac{\\varepsilon }{4} \\text{ and } d(T^{n}(z_{2}), T^{n}(y^{\\prime }))<\\frac{\\varepsilon }{4}.$ Choose $p\\in B(x^{\\prime }, \\delta ^{\\prime })\\cap A$ and $q\\in B(y^{\\prime }, \\delta ^{\\prime })\\cap A$ .",
"Clearly, $d(p, q)\\le d(p, x^{\\prime })+ d(x^{\\prime }, y^{\\prime })+d(y^{\\prime }, q)<\\delta ^{\\prime }+\\frac{\\delta }{2}+\\delta ^{\\prime }<\\delta .$ This, together with (REF ), implies that $d(T^{n}(p), T^{n}(q))<\\frac{\\varepsilon }{2}.$ Then, $d(T^{n}(x^{\\prime }), T^{n}(y^{\\prime }))&\\le & d(T^{n}(x^{\\prime }), T^{n}(p))+d(T^{n}(p), T^{n}(q))+d(T^{n}(q), T^{n}(y^{\\prime }))\\\\&<& \\frac{\\varepsilon }{4}+\\frac{\\varepsilon }{2}+\\frac{\\varepsilon }{4}=\\varepsilon .$ Hence, $T$ is equicontinuous.",
"$\\blacksquare $ Lemma 14 Let $(X, T)$ be a dynamical system.",
"If $(K(X), T_{K})$ is equicontinuous, then $(\\mathbb {F}_{0}(X), T_{F})$ is equicontinuous.",
"Proof.",
"For any fixed $\\varepsilon >0$ , as $T_{K}$ is equicontinuous, there exists $0<\\delta <\\frac{\\mathrm {diam}(X)}{2}$ such that for any $E, F\\in K(X)$ with $d_{H}(E, F)<\\delta $ and any $n\\in \\mathbb {Z}^+$ , $d_{H}(T_{K}^{n}(E), T_{K}^{n}(F))<\\frac{\\varepsilon }{2}$ .",
"For any $A, B\\in \\mathbb {F}_{0}(X)$ with $d_{\\infty }(A, B)<\\delta $ , it is clear that $\\beta :=\\max \\lbrace A(x): x\\in X\\rbrace =\\max \\lbrace B(x): x\\in X\\rbrace $ , as $\\delta <\\frac{\\mathrm {diam}(X)}{2}$ , implying that for any $\\alpha \\in (0, \\beta ]$ , $[A]_{\\alpha } \\ne \\emptyset \\ne [B]_{\\alpha }$ and $d_{H}([A]_{\\alpha }, [B]_{\\alpha })<\\delta $ .",
"Then, for any $n\\in \\mathbb {Z}^+$ , $d_{\\infty }(T_{F}^{n}(A), T_{F}^{n}(B))&=&\\sup \\left\\lbrace d_{H}(T_{K}^{n}([A]_{\\alpha }), T_{K}^{n}([B]_{\\alpha })):\\alpha \\in (0, 1]\\right\\rbrace \\\\&=&\\sup \\left\\lbrace d_{H}(T_{K}^{n}([A]_{\\alpha }), T_{K}^{n}([B]_{\\alpha })):\\alpha \\in (0, \\beta ]\\right\\rbrace \\le \\frac{\\varepsilon }{2}<\\varepsilon .$ So, $(\\mathbb {F}_{0}(X), T_{F})$ is equicontinuous.",
"$\\blacksquare $ Lemma 15 If $(\\mathbb {F}_{0}(X), T_{F})$ is equicontinuous, then $(K(X), T_{K})$ is equicontinuous.",
"Proof.",
"The result yields by (REF ).",
"$\\blacksquare $ Theorem 6 The following statements are equivalent: $(X, T)$ is equicontinuous; $(K(X), T_{K})$ is equicontinuous; $(\\mathbb {F}_{0}(X), T_{F})$ is equicontinuous.",
"Proof.",
"Applying Lemma REF , Lemma REF and Lemma REF , this holds trivially.",
"$\\blacksquare $"
],
[
"Uniform rigidity and proximality of $(\\mathbb {F}_{0}(X), T_{F})$",
"Let $n\\in \\mathbb {N}$ , according to Glasner and Maon [9], a dynamical $(X, T)$ is $n$ -rigid if every $n$ -tuple $(x_{1}, x_{2}, \\ldots , x_{n})\\in X^{n}$ is a recurrent point of $T^{(n)}$ ; weakly rigid if $(X, T)$ is $n$ -rigid for any $n\\in \\mathbb {N}$ ; rigid if there exists $m_i\\longrightarrow +\\infty $ such that $T^{m_i} \\longrightarrow \\mathrm {id}_{X}$ pointwise, where $\\mathrm {id}_{X}$ is the identity map on $X$ ; uniformly rigid if there exists $m_{i}\\longrightarrow +\\infty $ such that $T^{m_{i}}\\longrightarrow \\mathrm {id}_{X}$ uniformly on $X$ .",
"It can be verified that a dynamical system $(X, T)$ is uniformly rigid if and only if for any $\\varepsilon >0$ , there exists $n\\in \\mathbb {N}$ such that for any $x\\in X$ , $d(T^{n}(x), x)<\\varepsilon $ .",
"It is known that every transitive map containing an equicontinuous pointA point $x\\in X$ is equicontinuous if for any $\\varepsilon >0$ , there exists $\\delta >0$ such that for any $y\\in X$ with $d(x, y)<\\delta $ and any $n\\in \\mathbb {Z}^+$ , $d(T^{n}(x), T^{n}(y))<\\varepsilon $ .",
"is uniformly rigid.",
"Recently, we [33] proved that a dynamical system $(X, T)$ is uniformly rigid if and only if $(M(X), T_{M})$ is uniformly rigid.",
"The following result is obtained by Li et al.",
"[20].",
"Lemma 16 [20] Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $(X, T)$ is uniformly rigid; $(K(X), T_{K})$ is uniformly rigid; $(K(X), T_{K})$ is rigid; $(K(X), T_{K})$ is weakly rigid.",
"Proposition 2 Let $(X, d)$ be a complete metric space, $T: X\\longrightarrow X$ be a continuous map and $A\\subset X$ be an invariant dense subset of $X$ .",
"If $T|_{A}: A\\longrightarrow A$ is uniformly rigid, then $T: X\\longrightarrow X$ is uniformly rigid.",
"Proof.",
"For any $\\varepsilon >0$ , as $T|_{A}$ is uniformly rigid, then there exists $n\\in \\mathbb {N}$ such that for any $x\\in A$ , $d(T^{n}(x), x)<\\frac{\\varepsilon }{2}$ .",
"For any $y\\in X$ , as $T^{n}$ is continuous and $\\overline{A}=X$ , there exists $x^{\\prime }\\in A$ such that $d(x^{\\prime }, y)<\\frac{\\varepsilon }{4}$ and $d(T^{n}(y), T^{n}(x^{\\prime }))<\\frac{\\varepsilon }{4}$ .",
"This implies that $d(T^{n}(y), y)\\le d(T^{n}(y), T^{n}(x^{\\prime }))+ d(T^{n}(x^{\\prime }), x^{\\prime })+d(x^{\\prime }, y)<\\frac{\\varepsilon }{4}+\\frac{\\varepsilon }{2}+\\frac{\\varepsilon }{4}=\\varepsilon .$ Thus, $(X, T)$ is uniformly rigid.",
"$\\blacksquare $ Lemma 17 If $(K(X), T_{K})$ is uniformly rigid, then $(\\mathbb {F}_{0}(X), T_{F})$ is uniformly rigid.",
"Proof.",
"For any $\\varepsilon >0$ , as $T_{K}$ is uniformly rigid, there exists $n\\in \\mathbb {N}$ such that for any $E\\in K(X)$ , $d_{H}(T_{K}^{n}(E), E)<\\frac{\\varepsilon }{2}$ .",
"For any $A\\in \\mathbb {F}_{0}(X)$ , it can be verified that $d_{\\infty }(T_{F}^{n}(A), A)=\\sup \\left\\lbrace d_{H}(T_{K}^{n}([A]_{\\alpha }), [A]_{\\alpha }): \\alpha \\in (0, 1]\\right\\rbrace \\le \\frac{\\varepsilon }{2}<\\varepsilon .$ Therefore, $T_{F}$ is uniformly rigid.",
"$\\blacksquare $ Theorem 7 Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $(X, T)$ is uniformly rigid; $(K(X), T_{K})$ is uniformly rigid; $(\\mathbb {F}_{0}(X), T_{F})$ is uniformly rigid; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is uniformly rigid; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{\\ge \\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is uniformly rigid; $\\exists \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is uniformly rigid; $\\exists \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{\\ge \\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is uniformly rigid.",
"Proof.",
"This follows by Lemma REF , Lemma REF and (REF ).",
"$\\blacksquare $ Recall that a dynamical system $(X, T)$ is proximal if for any $x, y\\in X$ , $\\liminf _{n\\rightarrow \\infty }d(T^{n}(x), T^{n}(y))=0$ .",
"Theorem 8 Let $(X, T)$ be a dynamical system.",
"Then, the following statements are equivalent: $(K(X), T_{K})$ is proximal; $\\lim _{n\\rightarrow \\infty }\\mathrm {diam}(T^{n}(X))=0$ ; $\\forall \\lambda \\in (0, 1]$ , $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is proximal.",
"Proof.",
"(3) $\\Longrightarrow $ (1) follows by Remark REF .",
"(1) $\\Longrightarrow $ (2).",
"Fix any $x\\in X$ .",
"As $T_{K}$ is proximal, $\\liminf _{n\\rightarrow \\infty }d_{H}\\left(T_{K}^{n}(X), T_{K}^{n}(\\lbrace x\\rbrace )\\right)=0$ .",
"This, together with the fact that $\\lbrace T^n (X)\\rbrace $ is a decreasing sequence, implies that $\\lim _{n\\rightarrow \\infty }\\mathrm {diam}(T^{n}(X))=0$ .",
"(2) $\\Longrightarrow $ (3).",
"For nay $A, B\\in \\mathbb {F}^{=\\lambda }(X)$ and any $n\\in \\mathbb {Z}^{+}$ , one has $d_{\\infty }\\left(\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(A),\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(B)\\right)&=&\\sup \\left\\lbrace d_{H}(T^{n}([A]_{\\alpha }), T^{n}([B]_{\\alpha })): \\alpha \\in (0, 1]\\right\\rbrace \\\\&=&\\sup \\left\\lbrace d_{H}(T^{n}([A]_{\\alpha }), T^{n}([B]_{\\alpha })): \\alpha \\in (0, \\lambda ]\\right\\rbrace \\le \\mathrm {diam}(T^{n}(X)).$ This implies that $\\liminf _{n\\rightarrow \\infty }d_{\\infty }\\left(\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(A),\\left(T_{F}|_{\\mathbb {F}^{=\\lambda }(X)}\\right)^{n}(B)\\right)=0$ .",
"$\\blacksquare $ Because there exists proximal system which is topologically mixing, this shows that the proximality of $(\\mathbb {F}^{=\\lambda }(X), T_{F}|_{\\mathbb {F}^{=\\lambda }(X)})$ is strictly stronger that the proximality of $(X, T)$ .",
"Clearly, for any $A, B\\in \\mathbb {F}_{0}(X)$ with $\\xi :=\\max \\lbrace A(x): x\\in X\\rbrace <\\max \\lbrace B(x): x\\in X\\rbrace :=\\eta $ and any $n\\in \\mathbb {Z}^{+}$ , $d_{\\infty }(T_{F}^{n}(A), T_{F}^{n}(B))\\ge d_{H}(T^{n}([A]_{\\frac{\\xi +\\eta }{2}}),T^{n}([B]_{\\frac{\\xi +\\eta }{2}}))=d_{H}(\\emptyset , T^{n}([B]_{\\frac{\\xi +\\eta }{2}}))=\\mathrm {diam}(X).$ This implies that $(\\mathbb {F}_{0}(X), T_{F})$ is not proximal."
],
[
"Acknowledgments",
"We thank the referees for their careful reading and valuable suggestions which helped us to improve the quality of this paper."
]
] | 1605.06010 |
[
[
"A Hierarchical Latent Variable Encoder-Decoder Model for Generating\n Dialogues"
],
[
"Abstract Sequential data often possesses a hierarchical structure with complex dependencies between subsequences, such as found between the utterances in a dialogue.",
"In an effort to model this kind of generative process, we propose a neural network-based generative architecture, with latent stochastic variables that span a variable number of time steps.",
"We apply the proposed model to the task of dialogue response generation and compare it with recent neural network architectures.",
"We evaluate the model performance through automatic evaluation metrics and by carrying out a human evaluation.",
"The experiments demonstrate that our model improves upon recently proposed models and that the latent variables facilitate the generation of long outputs and maintain the context."
],
[
"Introduction",
"Deep recurrent neural networks (RNNs) have recently demonstrated impressive results on a number of difficult machine learning problems involving the generation of sequential structured outputs [9], including language modelling [10], [18] machine translation [28], [5], dialogue [27], [24] and speech recognition [9].",
"While these advances are impressive, the underlying RNNs tend to have a fairly simple structure, in the sense that the only variability or stochasticity in the model occurs when an output is sampled.",
"This is often an inappropriate place to inject variability [2], [6], [1].",
"This is especially true for sequential data such as speech and natural language that possess a hierarchical generation process with complex intra-sequence dependencies.",
"For instance, natural language dialogue involves at least two levels of structure; within a single utterance the structure is dominated by local statistics of the language, while across utterances there is a distinct source of uncertainty (or variance) characterized by aspects such as conversation topic, speaker goals and speaker style.",
"In this paper we introduce a novel hierarchical stochastic latent variable neural network architecture to explicitly model generative processes that possess multiple levels of variability.",
"We evaluate the proposed model on the task of dialogue response generation and compare it with recent neural network architectures.",
"We evaluate the model qualitatively through manual inspection, and quantitatively using a human evaluation on Amazon Mechanical Turk and using automatic evaluation metrics.",
"The results demonstrate that the model improves upon recently proposed models.",
"In particular, the results highlight that the latent variables help to both facilitate the generation of long utterances with more information content, and to maintain the dialogue context."
],
[
"Recurrent Neural Network Language Model",
"A recurrent neural network (RNN), with parameters $\\theta $ , models a variable-length sequence of tokens $(w_1, \\dots , w_M)$ by decomposing the probability distribution over outputs: $P_{\\theta }(w_1, \\dots , w_M) = \\prod _{m=2}^M P_{\\theta }(w_m \\mid w_1, \\dots , w_{m-1})P(w_1).$ The model processes each observation recursively.",
"At each time step, the model observes an element and updates its internal hidden state, $h_m = f(h_{m-1}, w_m)$ , where $f$ is a parametrized non-linear function, such as the hyperbolic tangent, the LSTM gating unit [12] or the GRU gating unit [5].We concatenate the LSTM cell and cell input hidden states into a single state $h_m$ for notational simplicity.",
"The hidden state acts as a sufficient statistic, which summarizes the past sequence and parametrizes the output distribution of the model: $P_{\\theta }(w_{m+1} \\mid w_1, \\dots , w_m) = P_{\\theta }(w_{m+1} \\mid h_m)$ .",
"We assume the outputs lie within a discrete vocabulary $V$ .",
"Under this assumption, the RNN Language Model (RNNLM) [18], the simplest possible generative RNN for discrete sequences, parametrizes the output distribution using the softmax function applied to an affine transformation of the hidden state $h_m$ .",
"The model parameters are learned by maximizing the training log-likelihood using gradient descent."
],
[
"Hierarchical Recurrent Encoder-Decoder",
"The hierarchical recurrent encoder-decoder model (HRED) [26], [24] is an extension of the RNNLM.",
"It extends the encoder-decoder architecture [5] to the natural dialogue setting.",
"The HRED assumes that each output sequence can be modelled in a two-level hierarchy: sequences of sub-sequences, and sub-sequences of tokens.",
"For example, a dialogue may be modelled as a sequence of utterances (sub-sequences), with each utterance modelled as a sequence of words.",
"Similarly, a natural-language document may be modelled as a sequence of sentences (sub-sequences), with each sentence modelled as a sequence of words.",
"The HRED model consists of three RNN modules: an encoder RNN, a context RNN and a decoder RNN.",
"Each sub-sequence of tokens is deterministically encoded into a real-valued vector by the encoder RNN.",
"This is given as input to the context RNN, which updates its internal hidden state to reflect all information up to that point in time.",
"The context RNN deterministically outputs a real-valued vector, which the decoder RNN conditions on to generate the next sub-sequence of tokens.",
"For additional details see [26], [24]."
],
[
"A Deficient Generation Process",
"In the recent literature, it has been observed that the RNNLM and HRED, and similar models based on RNN architectures, have critical problems generating meaningful dialogue utterances [24], [15].",
"We believe that the root cause of these problems arise from the parametrization of the output distribution in the RNNLM and HRED, which imposes a strong constraint on the generation process: the only source of variation is modelled through the conditional output distribution.",
"This is detrimental from two perspectives: from a probabilistic perspective, with stochastic variations injected only at the low level, the model is encouraged to capture local structure in the sequence, rather than global or long-term structure.",
"This is because random variations injected at the lower level are strongly constrained to be in line with the immediate previous observations, but only weakly constrained to be in line with older observations or with future observations.",
"One can think of random variations as injected via i.i.d.",
"noise variables, added to deterministic components, for example.",
"If this noise is injected at a higher level of representation, spanning longer parts of the sequence, its effects could correspond to longer-term dependencies.",
"Second, from a computational learning perspective, the state $h_m$ of the RNNLM (or decoder RNN of HRED) has to summarize all the past information up to time step $m$ in order to (a) generate a probable next token (short term goal) and simultaneously (b) to occupy a position in embedding space which sustains a realistic output trajectory, in order to generate probable future tokens (long term goal).",
"Due to the vanishing gradient effect, shorter-term goals will have more influence: finding a compromise between these two disparate forces will likely lead the training procedure to model parameters that focus too much on predicting only the next output token.",
"In particular for high-entropy sequences, the models are very likely to favour short-term predictions as opposed to long-term predictions, because it is easier to only learn $h_m$ for predicting the next token compared to sustaining a long-term trajectory, which at every time step is perturbed by a highly noisy source (the observed token)."
],
[
"Latent Variable Hierarchical Recurrent Encoder-Decoder (VHRED)",
"Motived by the previous discussion, we now introduce the latent variable hierarchical recurrent encoder-decoder (VHRED) model.",
"This model augments the HRED model with a latent variable at the decoder, which is trained by maximizing a variational lower-bound on the log-likelihood.",
"This allows it to model hierarchically-structured sequences in a two-step generation process—first sampling the latent variable, and then generating the output sequence—while maintaining long-term context.",
"Let $\\mathbf {w}_1, \\dots , \\mathbf {w}_N$ be a sequence consisting of $N$ sub-sequences, where $\\mathbf {w}_n = (w_{n,1}, \\dots , w_{n,M_n})$ is the $n$ 'th sub-sequence and $w_{n,m} \\in V$ is the $m$ 'th discrete token in that sequence.",
"The VHRED model uses a stochastic latent variable $\\mathbf {z}_n \\in \\mathbb {R}^{d_z}$ for each sub-sequence $n=1,\\dots ,N$ conditioned on all previous observed tokens.",
"Given $\\mathbf {z}_n$ , the model next generates the $n$ 'th sub-sequence tokens $\\mathbf {w}_n = (w_{n,1}, \\dots , w_{n,M_n})$ : $P_{\\theta }(\\mathbf {z}_n \\mid \\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}) & = \\mathcal {N}(\\mu _{\\text{prior}}(\\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}), \\Sigma _{\\text{prior}}(\\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1})), \\\\P_{\\theta }(\\mathbf {w}_n \\mid \\mathbf {z}_n, \\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}) & = \\prod _{m=1}^{M_n} P_{\\theta }(w_{n,m} \\mid \\mathbf {z}_n, \\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}, w_{n,1}, \\dots , w_{n,m-1}),$ where $\\mathcal {N}(\\mu , \\Sigma )$ is the multivariate normal distribution with mean $\\mu \\in \\mathbb {R}^{d_z}$ and covariance matrix $\\Sigma \\in \\mathbb {R}^{{d_z} \\times {d_z}}$ , which is constrained to be a diagonal matrix.",
"The VHRED model (figure REF ) contains the same three components as the HRED model.",
"The encoder RNN deterministically encodes a single sub-sequence into a fixed-size real-valued vector.",
"The context RNN deterministically takes as input the output of the encoder RNN, and encodes all previous sub-sequences into a fixed-size real-valued vector.",
"This vector is fed into a two-layer feed-forward neural network with hyperbolic tangent gating function.",
"A matrix multiplication is applied to the output of the feed-forward network, which defines the multivariate normal mean $\\mu _{\\text{prior}}$ .",
"Similarly, for the diagonal covariance matrix $\\Sigma _{\\text{prior}}$ a different matrix multiplication is applied to the net's output followed by softplus function, to ensure positiveness [6].",
"The model's latent variables are inferred by maximizing the variational lower-bound, which factorizes into independent terms for each sub-sequence: $\\log P_{\\theta }(\\mathbf {w}_1, \\dots , \\mathbf {w}_N) \\ge & \\sum _{n=1}^N - \\text{KL} \\left[ Q_{\\psi }(\\mathbf {z}_n \\mid \\mathbf {w}_1, \\dots , \\mathbf {w}_n) || P_{\\theta }(\\mathbf {z}_n \\mid \\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}) \\right] \\nonumber \\\\& + \\mathbb {E}_{Q_{\\psi }(\\mathbf {z}_n \\mid \\mathbf {w}_1, \\dots , \\mathbf {w}_n)} \\left[ \\log P_{\\theta }(\\mathbf {w}_n \\mid \\mathbf {z}_n, \\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}) \\right], $ where $\\text{KL}[Q || P]$ is the Kullback-Leibler (KL) divergence between distributions $Q$ and $P$ .",
"The distribution $Q_{\\psi }(\\mathbf {z} \\mid w_1, \\dots , w_M)$ is the approximate posterior distribution (also known as the encoder model or recognition model), which aims to approximate the intractable true posterior distribution: $Q_{\\psi }(\\mathbf {z}_n \\mid \\mathbf {w}_1, \\dots , \\mathbf {w}_N) = Q_{\\psi }(\\mathbf {z}_n \\mid \\mathbf {w}_1, \\dots , \\mathbf {w}_n) & = \\mathcal {N}(\\mu _{\\text{posterior}}(\\mathbf {w}_1, \\dots , \\mathbf {w}_n), \\Sigma _{\\text{posterior}}(\\mathbf {w}_1, \\dots , \\mathbf {w}_n)) \\nonumber \\\\& \\approx P_{\\psi }(\\mathbf {z}_n \\mid \\mathbf {w}_1, \\dots , \\mathbf {w}_N),$ where $\\mu _{\\text{posterior}}$ defines the approximate posterior mean and $\\Sigma _{\\text{posterior}}$ defines the approximate posterior covariance matrix (assumed diagonal) as a function of the previous sub-sequences $\\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}$ and the current sub-sequence $\\mathbf {w}_{n}$ .",
"The posterior mean $\\mu _{\\text{posterior}}$ and covariance $\\Sigma _{\\text{posterior}}$ are determined in the same way as the prior, via a matrix multiplication with the output of the feed-forward network, and with a softplus function applied for the covariance.",
"Figure: Computational graph for VHRED model.",
"Rounded boxes represent (deterministic) real-valued vectors.",
"Variables 𝐳\\mathbf {z} represent latent stochastic variables.At test time, conditioned on the previous observed sub-sequences $(\\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1})$ , a sample $\\mathbf {z}_n$ is drawn from the prior $\\mathcal {N}(\\mu _{\\text{prior}}(\\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}), \\Sigma _{\\text{prior}}(\\mathbf {w}_1, \\dots , \\mathbf {w}_{n-1}))$ for each sub-sequence.",
"This sample is concatenated with the output of the context RNN and given as input to the decoder RNN as in the HRED model, which then generates the sub-sequence token-by-token.",
"At training time, for $n=1,\\dots ,N$ , a sample $\\mathbf {z}_n$ is drawn from the approximate posterior $\\mathcal {N}(\\mu _{\\text{posterior}}(\\mathbf {w}_1, \\dots , \\mathbf {w}_{n}), \\Sigma _{\\text{posterior}}(\\mathbf {w}_1, \\dots , \\mathbf {w}_{n}))$ and used to estimate the gradient of the variational lower-bound given by Eq.",
"(REF ).",
"The approximate posterior is parametrized by its own one-layer feed-forward neural network, which takes as input the output of the context RNN at the current time step, as well as the output of the encoder RNN for the next sub-sequence.",
"The VHRED model greatly helps to reduce the problems with the generation process used by the RNNLM and HRED model outlined above.",
"The variation of the output sequence is now modelled in two ways: at the sequence-level with the conditional prior distribution over $\\mathbf {z}$ , and at the sub-sequence-level (token-level) with the conditional distribution over tokens $w_1,\\dots , w_M$ .",
"The variable $\\mathbf {z}$ helps model long-term output trajectories, by representing high-level information about the sequence, which in turn allows the variable $h_m$ to primarily focus on summarizing the information up to token $M$ .",
"Intuitively, the randomness injected by the variable $\\mathbf {z}$ corresponds to higher-level decisions, like topic or sentiment of the sentence."
],
[
"Experimental Evaluation",
"We consider the problem of conditional natural language response generation for dialogue.",
"This is an interesting problem with applications in areas such as customer service, technical support, language learning and entertainment [29].",
"It is also a task domain that requires learning to generate sequences with complex structures while taking into account long-term context [17], [27].",
"We consider two tasks.",
"For each task, the model is given a dialogue context, consisting of one or more utterances, and the goal of the model is to generate an appropriate next response to the dialogue.",
"We first perform experiments on a Twitter Dialogue Corpus [22].",
"The task is to generate utterances to append to existing Twitter conversations.",
"The dataset is extracted using a procedure similar to Ritter et al.",
"[22], and is split into training, validation and test sets, containing respectively $749,060$ , $93,633$ and $10,000$ dialogues.",
"Each dialogue contains $6.27$ utterances and $94.16$ tokens on average.Due to Twitter's terms of service we are not allowed to redistribute Twitter content.",
"Therefore, only the tweet IDs can be made public.",
"These are available at: www.iulianserban.com/Files/TweetIDs.zip.",
"The dialogues are fairly long compared to recent large-scale language modelling corpora, such as the 1 Billion Word Language Model Benchmark [4], which focus on modelling single sentences.",
"We also experiment on the Ubuntu Dialogue Corpus [17], which contains about $500,000$ dialogues extracted from the #Ubuntu Internet Relayed Chat channel.",
"Users enter the chat channel with a Ubuntu-related technical problem, and other users try to help them.",
"For further details see Appendix REF .The pre-processed Ubuntu Dialogue Corpus used is available at www.iulianserban.com/Files/UbuntuDialogueCorpus.zip.",
"We chose these corpora because they are large, and have different purposes—Ubuntu dialogues are typically goal driven, where as Twitter dialogues typically contain social interaction (\"chit-chat\")."
],
[
"Training and Evaluation Procedures",
"We optimize all models using Adam [13].",
"We choose our hyperparameters and early stop with patience using the variational lower-bound [9].",
"At test time, we use beam search with 5 beams for outputting responses with the RNN decoders [10].",
"For the VHRED models, we sample the latent variable $\\mathbf {z}_n$ , and condition on it when executing beam search with the RNN decoder.",
"For Ubuntu we use word embedding dimensionality of size 300, and for Twitter we use word embedding dimensionality of size 400.",
"All models were trained with a learning rate of $0.0001$ or $0.0002$ and with mini-batches containing 40 or 80 training examples.",
"We use a variant of truncated back-propagation and we apply gradient clipping.",
"Further details are given in Appendix REF .",
"Baselines On both Twitter and Ubuntu we compare to an LSTM model of 2000 hidden units.",
"On Ubuntu, the HRED model has 500, 1000 and 500 hidden units for the encoder, context and decoder RNNs respectively.",
"The encoder RNN is a standard GRU RNN.",
"On Twitter, the HRED model encoder RNN is a bidirectional GRU RNN encoder, where the forward and backward RNNs each have 1000 hidden units, and context RNN and decoder RNN have each 1000 hidden units.",
"For reference, we also include a non-neural network baseline, specifically the TF-IDF retrieval-based model proposed in [17].",
"VHRED The encoder and context RNNs for the VHRED model are parametrized in the same way as the corresponding HRED models.",
"The only difference in the parametrization of the decoder RNN is that the context RNN output vector is now concatenated with the generated stochastic latent variable.",
"Furthermore, we initialize the feed-forward networks of the prior and posterior distributions with values drawn from a zero-mean normal distribution with variance $0.01$ and with biases equal to zero.",
"We also multiply the diagonal covariance matrices of the prior and posterior distributions with $0.1$ to make training more stable, because a high variance makes the gradients w.r.t.",
"the reconstruction cost unreliable, which is fatal at the beginning of the training process.",
"The VHRED's encoder and context RNNs are initialized to the parameters of the corresponding converged HRED models.",
"We also use two heuristics proposed by Bowman et al.",
"[3]: we drop words in the decoder with a fixed drop rate of $25\\%$ and multiply the KL terms in eq.",
"(REF ) by a scalar, which starts at zero and linearly increases to 1 over the first $60,000$ and $75,000$ training batches on Twitter and Ubuntu respectively.",
"Applying these heuristics helped substantially to stabilize the training process and make the model use the stochastic latent variables.",
"We experimented with the batch normalization training procedure for the feed-forward neural networks.",
"but found that this made training very unstable without any substantial gains in performance w.r.t.",
"the variational bound.",
"Evaluation Accurate evaluation of dialogue system responses is a difficult problem [8], [20].",
"Inspired by metrics for machine translation and information retrieval, researchers have begun adopting word-overlap metrics, however Liu et al.",
"[16] show that such metrics have little correlation with human evaluations of response quality.",
"We therefore carry out a human evaluation to compare responses from the different models.",
"We also compute several statistics and automatic metrics on model responses to characterize differences between the model-generated responses.",
"We carry out the human study for the Twitter Dialogue Corpus on Amazon Mechanical Turk (AMT).",
"We do not conduct AMT experiments on Ubuntu as evaluating these responses usually requires technical expertise, which is not prevalent among AMT users.",
"We set up the evaluation study as a series of pairwise comparison experiments.Source code for the AMT experiments will be released upon publication.",
"We show human evaluators a dialogue context along with two potential responses, one generated from each model (conditioned on dialogue context).",
"We ask participants to choose the response most appropriate to the dialogue context.",
"If the evaluators are indifferent to either of the two responses, or if they cannot understand the dialogue context, they can choose neither response.",
"For each pair of models we conduct two experiments: one where the example contexts contain at least 80 unique tokens (long context), and one where they contain at least 20 (not necessarily unique) tokens (short context).",
"This helps compare how well each model can integrate the dialogue context into its response, since it has previously been hypothesized that for long contexts hierarchical RNNs models fare better [24], [26].",
"Screenshots and further details of the experiments are in Appendix REF ."
],
[
"Results of Human Evaluation",
"The results (table REF ) show that VHRED is clearly preferred in the majority of the experiments.",
"In particular, VHRED is strongly preferred over the HRED and TF-IDF baseline models for both short and long context settings.",
"VHRED is also preferred over the LSTM baseline model for long contexts; however, the LSTM is preferred over VHRED for short contexts.",
"We believe this is because the LSTM baseline tends to output much more generic responses (see table REF ); since it doesn't model the hierarchical input structure, the LSTM model has a shorter memory span, and thus must output a response based primarily on the end of the last utterance.",
"Such `safe' responses are reasonable for a wider range of contexts, meaning that human evaluators are more likely to rate them as appropriate.",
"However, we argue that a model that only outputs generic responses is undesirable for dialogue, as this leads to uninteresting and less engaging conversations.",
"Conversely, the VHRED model is explicitly designed for long contexts, and to output a diverse set of responses due to the sampling of the latent variable.",
"Thus, the VHRED model generates longer sentences with more semantic content than the LSTM model (see tables REF -REF ).",
"This can be `riskier' as longer utterances are more likely to contain small mistakes, which can lead to lower human preference for a single utterance.",
"However, we believe that response diversity is crucial to maintaining interesting conversations — in the dialogue literature, generic responses are used primarily as `back-off' strategies in case the agent has no interesting response that is relevant to the context [25].",
"The above hypotheses are confirmed upon qualitative assessment of the generated responses (table REF ).",
"VHRED generates longer and more meaningful responses compared to the LSTM model, which generates mostly generic responses.",
"Additionally, we observed that the VHRED model has learned to better model smilies, slang (see first example in table REF ) and can even continue conversations in different languages (see fifth example).There is a notable amount of Spanish and Dutch conversations in the corpus.",
"Such aspects are not measured by the human study.",
"Further, VHRED appears to be better at generating stories or imaginative actions compared to the generative baseline models (see third example).",
"The last example in table REF is a case where the VHRED generated response is more interesting, yet may be less preferred by humans as it is slightly incompatible with the context, compared to the generic LSTM response.",
"In the next section, we back these examples quantitatively, showing that the VHRED model learns to generate longer responses with more information content that share semantic similarity to the context and ground-truth response."
],
[
"Results of Metric-based Evaluation",
"To show the VHRED responses are more on-topic and share semantic similarity to the ground-truth response, we consider three textual similarity metrics based on word embeddings.",
"The Embedding Average (Average) metric projects the model response and ground truth response into two separate real-valued vectors by taking the mean over the word embeddings in each response, and then computes the cosine similarity between them [19].",
"This metric is widely used for measuring textual similarity.",
"The Embedding Extrema (Extrema) metric similarly embeds the responses by taking the extremum (maximum of the absolute value) of each dimension, and afterwards computes the cosine similarity between them.The Embedding Greedy (Greedy) metric is more fine-grained; it uses cosine similarity between word embeddings to find the closest word in the human-generated response for each word in the model response.",
"Given the (non-exclusive) alignment between words in the two responses, the mean over the cosine similarities is computed for each pair of questions [23].",
"Since this metric takes into account the alignment between words, it should be more accurate for long responses.",
"While these metrics do not strongly correlate with human judgements of generated responses, we interpret them as measuring topic similarity: if the model generated response has similar semantic content to the ground truth human response, then the metrics will yield a high score.",
"To ease reproducibility, we use the publicly available Word2Vec word embeddings trained on the Google News Corpus.https://code.google.com/archive/p/word2vec/ We compute these metrics in two settings: one where the models generate a single response (1-turn), and one where they generate the next three consecutive utterances (3-turns) (table REF ).",
"Overall, VHRED seems to better capture the ground truth response topic than either the LSTM or HRED models.",
"The fact that VHRED does better in particular in the setting where the model generates three consecutive utterances strongly suggests that hidden states in both the decoder and context RNNs of the VHRED models are better able to follow trajectories which remain on-topic w.r.t the dialogue context.",
"This supports our computational hypothesis that the stochastic latent variable helps modulate the training procedure to achieve a better trade-off between short-term and long-term generation.",
"We also observed the same trend when computing the similarity metrics between the model generated responses and the corresponding context, which further reinforces this hypothesis.",
"Table: Response information content on 1-turn generation as measured by average utterance length |U||U|, word entropy H w =-∑ w∈U p(w)logp(w)H_w = -\\sum _{w \\in U} p(w) \\log p(w) and utterance entropy H U H_U with respect to the maximum-likelihood unigram distribution of the training corpus pp.To show that the VHRED responses contain more information content than other model responses, we compute the average response length and average entropy (in bits) w.r.t.",
"the maximum likelihood unigram model over the generated responses (table REF ).",
"The unigram entropy is computed on the preprocessed tokenized datasets.",
"VHRED produces responses with higher entropy per word on both Ubuntu and Twitter compared to the HRED and LSTM models.",
"VHRED also produces longer responses overall on Twitter, which translates into responses containing on average $~6$ bits of information more than the HRED model.",
"Since the actual dialogue responses contain even more information per word than any of the generative models, it reasonable to assume that a higher entropy is desirable.",
"Thus, VHRED compares favourably to recently proposed models in the literature, which often output extremely low-entropy (generic) responses such as OK and I don't know [24], [15].",
"Finally, the fact that VHRED produces responses with higher entropy suggests that its responses are on average more diverse than the responses produced by the HRED and LSTM models.",
"This implies that the trajectories of the hidden states of the VHRED model traverse a larger area of the space compared to the hidden states of the HRED and LSTM baselines, which further supports our hypothesis that the stochastic latent variable helps the VHRED model achieve a better trade-off between short-term and long-term generation."
],
[
"Related Work",
"The use of a stochastic latent variable learned by maximizing a variational lower bound is inspired by the variational autoencoder (VAE) [14], [21].",
"Such models have been used predominantly for generating images in the continuous domain [11].",
"However, there has also been recent work applying these architectures for generating sequences, such as the Variational Recurrent Neural Networks (VRNN) [6], which was applied for speech and handwriting synthesis, and Stochastic Recurrent Networks (STORN) [1], which was applied for music generation and motion capture modeling.",
"Both the VRNN and STORN incorporate stochastic latent variables into RNN architectures, but unlike the VHRED they sample a separate latent variable at each time step of the decoder.",
"This does not exploit the hierarchical structure in the data, and thus does not model higher-level variability.",
"Similar to our work is the Variational Recurrent Autoencoder [7] and the Variational Autoencoder Language Model [3], which apply encoder-decoder architectures to generative music modeling and language modeling respectively.",
"The VHRED model is different from these in the following ways.",
"The VHRED latent variable is conditioned on all previous sub-sequences (sentences).",
"This enables the model to generate multiple sub-sequences (sentences), but it also makes the latent variables co-dependent through the observed tokens.",
"The VHRED model builds on the hierarchical architecture of the HRED model, which makes the model applicable to generation conditioned on long contexts.",
"It has a direct deterministic connection between the context and decoder RNN, which allows the model to transfer deterministic pieces of information between its components.Our initial experiments confirmed that the deterministic connection between the context RNN to the decoder RNN was indeed beneficial in terms of lowering the variational bound.",
"Crucially, VHRED also demonstrates improved results beyond the autoencoder framework, where the objective is not input reconstruction but the conditional generation of the next utterance in a dialogue."
],
[
"Discussion",
"We have introduced a novel latent variable neural network architecture, called VHRED.",
"The model uses a hierarchical generation process in order to exploit the structure in sequences and is trained using a variational lower bound on the log-likelihood.",
"We have applied the proposed model on the difficult task of dialogue response generation, and have demonstrated that it is an improvement over previous models in several ways, including quality of responses as measured in a human study.",
"The empirical results highlight the advantages of the hierarchical generation process for modelling high-entropy sequences.",
"Finally, it is worth noting that the proposed model is very general.",
"It can in principle be applied to any sequential generation task that exhibits a hierarchical structure, such as document-level machine translation, web query prediction, multi-sentence document summarization, multi-sentence image caption generation, and others.",
"3pt Appendix Dataset Details Our Twitter Dialogue Corpus was extracted in 2011.",
"We perform a minimal preprocessing on the dataset to remove irregular punctuation marks and tokenize it using the Moses tokenizer: https://github.com/moses-smt/mosesdecoder/blob/master/scripts/tokenizer/tokenizer.perl.",
"We use the Ubuntu Dialogue Corpus v2.0 extracted in Jamuary 2016 from: http://cs.mcgill.ca/~jpineau/datasets/ubuntu-corpus-1.0/.",
"The preprocessed version of the dataset will be made available to the public.",
"Model Details The model implementations will be released to the public upon acceptance of the paper.",
"Training and Generation We validate each model on the entire validation set every 5000 training batches.",
"As mentioned in the main text, at test time we use beam search with 5 beams for outputting responses with the RNN decoders [10].",
"We define the beam cost as the log-likelihood of the tokens in the beam divided by the number of tokens it contains.",
"This is a well-known modification, which is often applied in machine translation models.",
"In principle, we could sample from the RNN decoders of all the models, but is well known that such sampling produces poor results in comparison to the beam search procedure.",
"It also introduces additional variance into the evaluation procedure, which will make the human study very expensive or even impossible within a limited budget.",
"Baseline Models On Ubuntu, the gating function between the context RNN and decoder RNN is a one-layer feed-forward neural network with hyperbolic tangent activation function.",
"On Twitter, the HRED decoder RNN computes a 1000 dimensional real-valued vector for each hidden time step, which is multiplied with the output context RNN.",
"The result is feed through a one-layer feed-forward neural network with hyperbolic tangent activation function, which the decoder RNN then takes as input.",
"Furthermore, the encoder RNN initial state for each utterance is initialized to the last hidden state of the encoder RNN from the previous utterance.",
"We found that this worked slightly better for the VHRED model, but a more careful choice of hyperparameters is likely to make this additional step unnecessary for both the HRED and VHRED models.",
"Latent Variable Parametrization We here describe the formal definition of the latent variable prior and approximate posterior distributions.",
"Let $w_1,\\dots ,w_T$ be discrete tokens in vocabulary $V$ , which correspond to one sequence (e.g.",
"one dialogue).",
"Let $h_{t,con} \\in \\mathbb {R}^{d_{h,con}}$ be the hidden state of the HRED context encoder at time $t$ .",
"Then the prior mean and covariance matrix are given as: $\\bar{h}_{t,con} &= \\text{tanh}(H_{l_2,prior} \\text{tanh}(H_{l_1,prior}h_{t,con} + b_{l_1,prior}) + b_{l_2,prior} ), \\\\\\mu _{t,\\text{prior}} &= H_{\\mu ,prior} \\bar{h}_{t,con} + b_{\\mu ,prior}, \\\\\\Sigma _{t,\\text{prior}} &= \\text{diag} ( \\log (1 + \\exp ( H_{\\Sigma ,prior} \\bar{h}_{t,con} + b_{\\Sigma ,prior}))),$ where the parameters are $H_{l_1,prior} \\in \\mathbb {R}^{d_z \\times d_{h,con}}$ , $H_{\\Sigma ,prior}, H_{\\mu ,prior}, H_{l_2,prior} \\in \\mathbb {R}^{d_z \\times d_z}$ and $b_{l_1,prior}, b_{l_2,prior}, b_{\\mu ,prior}, b_{\\Sigma ,prior} \\in \\mathbb {R}^{d_z}$ , and where $\\text{diag}(\\mathbf {x})$ is a function mapping a vector $\\mathbf {x}$ to a matrix with diagonal elements $\\mathbf {x}$ and all off-diagonal elements equal to zero.",
"At generation time, the latent variable is sampled at the end of each utterance: $\\mathbf {z}_t \\sim \\mathcal {N}(\\mu _{t.\\text{prior}}, \\Sigma _{t,\\text{prior}})$ .",
"The equations for the approximate posterior are similar.",
"Let $h_{t,p} \\in \\mathbb {R}^{d_{h,con}+d_{h,enc}}$ be the concatenation of $h_{t,con}$ and the hidden state of the encoder RNN at the end of the next sub-sequence, which we assume has dimensionality $d_{h,enc}$ .",
"The approximate posterior mean and covariance matrix are given as: $\\bar{h}_{t,p} &= \\text{tanh}(H_{l_2,posterior} \\text{tanh}(H_{l_1,posterior}h_{t,p} + b_{l_1,posterior})) + b_{l_2,posterior}) ), \\\\\\mu _{t,\\text{posterior}} &= H_{\\mu ,posterior} \\bar{h}_{t,p} + b_{\\mu ,posterior}, \\\\\\Sigma _{t,\\text{posterior}} &= \\text{diag} ( \\log (1 + \\exp ( H_{\\Sigma ,posterior} \\bar{h}_{t,p} + b_{\\Sigma ,posterior}) )),$ where $H_{l_1,posterior} \\in \\mathbb {R}^{d_z \\times (d_{h,con} + d_{h,enc})}$ , $H_{\\Sigma ,posterior}, H_{\\mu ,posterior}, H_{l_2,posterior} \\in \\mathbb {R}^{d_z \\times d_z}$ and $b_{l_1,posterior}, b_{l_2,posterior}, b_{\\mu ,posterior}, b_{\\Sigma ,posterior} \\in \\mathbb {R}^{d_z}$ are its parameters.",
"The derivative of the variational bound is computed by sampling one latent variable at the end of each utterance: $\\mathbf {z}_t \\sim \\mathcal {N}(\\mu _{t.\\text{posterior}}, \\Sigma _{t,\\text{posterior}})$ .",
"Model Examples Model responses for Ubuntu are shown in Table REF .",
"All model responses are available for download at www.iulianserban.com/Files/UbuntuDialogueCorpus.zip and http://www.iulianserban.com/Files/TwitterDialogueCorpus.zip.",
"Table: Ubuntu model examples.",
"The →\\rightarrow token indicates a change of turn.",
"Human Study on Amazon Mechanical Turk Setup We choose to use crowdsourcing platforms such as AMT rather than carrying out in-lab experiments, even though in-lab experiments usually exhibit less noise and result in higher agreement between human annotators.",
"We do this because AMT experiments involve a larger and more heterogeneous pool of annotators, which implies less cultural and geographic biases, and because such experiments are easier to replicate, which we believe is important for benchmarking future research on these tasks.",
"Allowing the AMT human evaluators to not assign preference for either response is important, since there are several reasons why humans may not understand the dialogue context, which include topics they are not familiar with, slang language and non-English language.",
"We refer to such evaluations as `indeterminable'.",
"The evaluation setup resembles the classical Turing Test where human judges have to distinguish between human-human conversations and human-computer conversations.",
"However, unlike the original Turing Test, we only ask human evaluators to consider the next utterance in a given conversation and we do not inform them that any responses were generated by a computer.",
"Apart from minimum context and response lengths we impose no restrictions on the generated responses.",
"Selection Process At the beginning of each experiment, we briefly instruct the human evaluator on the task and show them a simple example of a dialogue context and two potential responses.",
"To avoid presentation bias, we shuffle the order of the examples and the order of the potential responses for each example.",
"During each experiment, we also show four trivial `attention check' examples that any human evaluator who has understood the task should be able to answer correctly.",
"We discard responses from human evaluators who fail more than one of these checks.",
"We select the examples shown to human evaluators at random from the test set.",
"We filter out all non-English conversations and conversations containing offensive content.",
"This is done by automatically filtering out all conversations with non-ascii characters and conversations with profanities, curse words and otherwise offensive content.",
"This filtering is not perfect, so we manually skim through many conversations and filter out conversations with non-English languages and offensive content.",
"On average, we remove about $1/80$ conversations manually.",
"To ensure that the evaluation process is focused on evaluating conditional dialogue response generation (as opposed to unconditional single sentence generation), we constrain the experiment by filtering out examples with fewer than 3 turns in the context.",
"We also filter out examples where either of the two presented responses contain less than 5 tokens.",
"We remove the special token placeholders and apply regex expressions to detokenize the text.",
"Execution We run the experiments in batches.",
"For each pairs of models, we carry out $3-5$ human intelligence tests (HITs) on AMT.",
"Each HIT contains $70-90$ examples (dialogue context and two model responses) and is evaluated by $3-4$ unique humans.",
"In total we collect 5363 preferences in 69 HITs.",
"The following are screenshots from one actual Amazon Mechanical Turk (AMT) experiment.",
"These screenshots show the introduction (debriefing) of the experiment, an example dialogue and one dialogue context with two candidate responses, which human evaluators were asked to choose between.",
"The experiment was carried out using psiturk, which can be downloaded from www.psiturk.org.",
"The source code will be released upon publication.",
"Figure: Screenshot of the introduction (debriefing) of the experiment.Figure: Screenshot of the introductory dialogue example.Figure: Screenshot of one dialogue context with two candidate responses, which human evaluators were asked to choose between."
],
[
"Dataset Details",
"Our Twitter Dialogue Corpus was extracted in 2011.",
"We perform a minimal preprocessing on the dataset to remove irregular punctuation marks and tokenize it using the Moses tokenizer: https://github.com/moses-smt/mosesdecoder/blob/master/scripts/tokenizer/tokenizer.perl.",
"We use the Ubuntu Dialogue Corpus v2.0 extracted in Jamuary 2016 from: http://cs.mcgill.ca/~jpineau/datasets/ubuntu-corpus-1.0/.",
"The preprocessed version of the dataset will be made available to the public."
],
[
"Model Details",
"The model implementations will be released to the public upon acceptance of the paper."
],
[
"Training and Generation",
"We validate each model on the entire validation set every 5000 training batches.",
"As mentioned in the main text, at test time we use beam search with 5 beams for outputting responses with the RNN decoders [10].",
"We define the beam cost as the log-likelihood of the tokens in the beam divided by the number of tokens it contains.",
"This is a well-known modification, which is often applied in machine translation models.",
"In principle, we could sample from the RNN decoders of all the models, but is well known that such sampling produces poor results in comparison to the beam search procedure.",
"It also introduces additional variance into the evaluation procedure, which will make the human study very expensive or even impossible within a limited budget.",
"On Ubuntu, the gating function between the context RNN and decoder RNN is a one-layer feed-forward neural network with hyperbolic tangent activation function.",
"On Twitter, the HRED decoder RNN computes a 1000 dimensional real-valued vector for each hidden time step, which is multiplied with the output context RNN.",
"The result is feed through a one-layer feed-forward neural network with hyperbolic tangent activation function, which the decoder RNN then takes as input.",
"Furthermore, the encoder RNN initial state for each utterance is initialized to the last hidden state of the encoder RNN from the previous utterance.",
"We found that this worked slightly better for the VHRED model, but a more careful choice of hyperparameters is likely to make this additional step unnecessary for both the HRED and VHRED models.",
"We here describe the formal definition of the latent variable prior and approximate posterior distributions.",
"Let $w_1,\\dots ,w_T$ be discrete tokens in vocabulary $V$ , which correspond to one sequence (e.g.",
"one dialogue).",
"Let $h_{t,con} \\in \\mathbb {R}^{d_{h,con}}$ be the hidden state of the HRED context encoder at time $t$ .",
"Then the prior mean and covariance matrix are given as: $\\bar{h}_{t,con} &= \\text{tanh}(H_{l_2,prior} \\text{tanh}(H_{l_1,prior}h_{t,con} + b_{l_1,prior}) + b_{l_2,prior} ), \\\\\\mu _{t,\\text{prior}} &= H_{\\mu ,prior} \\bar{h}_{t,con} + b_{\\mu ,prior}, \\\\\\Sigma _{t,\\text{prior}} &= \\text{diag} ( \\log (1 + \\exp ( H_{\\Sigma ,prior} \\bar{h}_{t,con} + b_{\\Sigma ,prior}))),$ where the parameters are $H_{l_1,prior} \\in \\mathbb {R}^{d_z \\times d_{h,con}}$ , $H_{\\Sigma ,prior}, H_{\\mu ,prior}, H_{l_2,prior} \\in \\mathbb {R}^{d_z \\times d_z}$ and $b_{l_1,prior}, b_{l_2,prior}, b_{\\mu ,prior}, b_{\\Sigma ,prior} \\in \\mathbb {R}^{d_z}$ , and where $\\text{diag}(\\mathbf {x})$ is a function mapping a vector $\\mathbf {x}$ to a matrix with diagonal elements $\\mathbf {x}$ and all off-diagonal elements equal to zero.",
"At generation time, the latent variable is sampled at the end of each utterance: $\\mathbf {z}_t \\sim \\mathcal {N}(\\mu _{t.\\text{prior}}, \\Sigma _{t,\\text{prior}})$ .",
"The equations for the approximate posterior are similar.",
"Let $h_{t,p} \\in \\mathbb {R}^{d_{h,con}+d_{h,enc}}$ be the concatenation of $h_{t,con}$ and the hidden state of the encoder RNN at the end of the next sub-sequence, which we assume has dimensionality $d_{h,enc}$ .",
"The approximate posterior mean and covariance matrix are given as: $\\bar{h}_{t,p} &= \\text{tanh}(H_{l_2,posterior} \\text{tanh}(H_{l_1,posterior}h_{t,p} + b_{l_1,posterior})) + b_{l_2,posterior}) ), \\\\\\mu _{t,\\text{posterior}} &= H_{\\mu ,posterior} \\bar{h}_{t,p} + b_{\\mu ,posterior}, \\\\\\Sigma _{t,\\text{posterior}} &= \\text{diag} ( \\log (1 + \\exp ( H_{\\Sigma ,posterior} \\bar{h}_{t,p} + b_{\\Sigma ,posterior}) )),$ where $H_{l_1,posterior} \\in \\mathbb {R}^{d_z \\times (d_{h,con} + d_{h,enc})}$ , $H_{\\Sigma ,posterior}, H_{\\mu ,posterior}, H_{l_2,posterior} \\in \\mathbb {R}^{d_z \\times d_z}$ and $b_{l_1,posterior}, b_{l_2,posterior}, b_{\\mu ,posterior}, b_{\\Sigma ,posterior} \\in \\mathbb {R}^{d_z}$ are its parameters.",
"The derivative of the variational bound is computed by sampling one latent variable at the end of each utterance: $\\mathbf {z}_t \\sim \\mathcal {N}(\\mu _{t.\\text{posterior}}, \\Sigma _{t,\\text{posterior}})$ ."
],
[
"Model Examples",
"Model responses for Ubuntu are shown in Table REF .",
"All model responses are available for download at www.iulianserban.com/Files/UbuntuDialogueCorpus.zip and http://www.iulianserban.com/Files/TwitterDialogueCorpus.zip.",
"Table: Ubuntu model examples.",
"The →\\rightarrow token indicates a change of turn."
],
[
"Setup",
"We choose to use crowdsourcing platforms such as AMT rather than carrying out in-lab experiments, even though in-lab experiments usually exhibit less noise and result in higher agreement between human annotators.",
"We do this because AMT experiments involve a larger and more heterogeneous pool of annotators, which implies less cultural and geographic biases, and because such experiments are easier to replicate, which we believe is important for benchmarking future research on these tasks.",
"Allowing the AMT human evaluators to not assign preference for either response is important, since there are several reasons why humans may not understand the dialogue context, which include topics they are not familiar with, slang language and non-English language.",
"We refer to such evaluations as `indeterminable'.",
"The evaluation setup resembles the classical Turing Test where human judges have to distinguish between human-human conversations and human-computer conversations.",
"However, unlike the original Turing Test, we only ask human evaluators to consider the next utterance in a given conversation and we do not inform them that any responses were generated by a computer.",
"Apart from minimum context and response lengths we impose no restrictions on the generated responses.",
"At the beginning of each experiment, we briefly instruct the human evaluator on the task and show them a simple example of a dialogue context and two potential responses.",
"To avoid presentation bias, we shuffle the order of the examples and the order of the potential responses for each example.",
"During each experiment, we also show four trivial `attention check' examples that any human evaluator who has understood the task should be able to answer correctly.",
"We discard responses from human evaluators who fail more than one of these checks.",
"We select the examples shown to human evaluators at random from the test set.",
"We filter out all non-English conversations and conversations containing offensive content.",
"This is done by automatically filtering out all conversations with non-ascii characters and conversations with profanities, curse words and otherwise offensive content.",
"This filtering is not perfect, so we manually skim through many conversations and filter out conversations with non-English languages and offensive content.",
"On average, we remove about $1/80$ conversations manually.",
"To ensure that the evaluation process is focused on evaluating conditional dialogue response generation (as opposed to unconditional single sentence generation), we constrain the experiment by filtering out examples with fewer than 3 turns in the context.",
"We also filter out examples where either of the two presented responses contain less than 5 tokens.",
"We remove the special token placeholders and apply regex expressions to detokenize the text.",
"We run the experiments in batches.",
"For each pairs of models, we carry out $3-5$ human intelligence tests (HITs) on AMT.",
"Each HIT contains $70-90$ examples (dialogue context and two model responses) and is evaluated by $3-4$ unique humans.",
"In total we collect 5363 preferences in 69 HITs.",
"The following are screenshots from one actual Amazon Mechanical Turk (AMT) experiment.",
"These screenshots show the introduction (debriefing) of the experiment, an example dialogue and one dialogue context with two candidate responses, which human evaluators were asked to choose between.",
"The experiment was carried out using psiturk, which can be downloaded from www.psiturk.org.",
"The source code will be released upon publication.",
"Figure: Screenshot of the introduction (debriefing) of the experiment.Figure: Screenshot of the introductory dialogue example.Figure: Screenshot of one dialogue context with two candidate responses, which human evaluators were asked to choose between."
]
] | 1605.06069 |
[
[
"Controlling the Dark Exciton Spin Eigenstates by External Magnetic Field"
],
[
"Abstract We study the dark exciton's behavior as a coherent physical two-level spin system (qubit) using an external magnetic field in the Faraday configuration.",
"Our studies are based on polarization-sensitive intensity autocorrelation measurements of the optical transition resulting from the recombination of a spin-blockaded biexciton state, which heralds the dark exciton and its spin state.",
"We demonstrate control over the dark exciton eigenstates without degrading its decoherence time.",
"Our observations agree well with computational predictions based on a master equation model."
],
[
"3 Controlling the Dark Exciton Spin Eigenstates by External Magnetic Field L. Gantz Faculty of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000 Israel The Physics Department and the Solid State Institute, Technion-Israel Institute of Technology, Haifa 32000 Israel E. R. Schmidgall The Physics Department and the Solid State Institute, Technion-Israel Institute of Technology, Haifa 32000 Israel Department of Physics, University of Washington, Seattle WA 98195 United States I. Schwartz The Physics Department and the Solid State Institute, Technion-Israel Institute of Technology, Haifa 32000 Israel Y.",
"Don The Physics Department and the Solid State Institute, Technion-Israel Institute of Technology, Haifa 32000 Israel E. Waks The Joint Quantum Institute and the Institute for Research in Electronics and Applied Physics, University of Maryland G. Bahir Faculty of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000 Israel The Solid State Institute, Technion-Israel Institute of Technology, Haifa 32000 Israel D. Gershoni The Physics Department and the Solid State Institute, Technion-Israel Institute of Technology, Haifa 32000 Israel The Solid State Institute, Technion-Israel Institute of Technology, Haifa 32000 Israel We study the dark exciton’s behavior as a coherent physical two-level spin system (qubit) using an external magnetic field in the Faraday configuration.",
"Our studies are based on polarization-sensitive intensity autocorrelation measurements of the optical transition resulting from the recombination of a spin-blockaded biexciton state, which heralds the dark exciton and its spin state.",
"We demonstrate control over the dark exciton eigenstates without degrading its decoherence time.",
"Our observations agree well with computational predictions based on a master equation model.",
"Reliable quantum two-level systems (TLS) are the building blocks for quantum information processing (QIP).",
"Solid state quantum bits (qubits) that can also be well-controlled are required for QIP to become a viable technology.",
"An important prerequisite of a solid state qubit is that it has a long coherence time, in which its quantum state is not randomized by spurious interactions with its environment [1], [2].",
"Semiconductor quantum dots (QDs) confine charge carriers into a three dimensional nanometer scale region, thus acting in many ways as isolated 'artificial atoms,’ whose properties can be engineered.",
"They are also compatible with modern microelectronics, making them particularly attractive as solid state qubits.",
"Many efforts have been devoted to prepare, control, and measure the quantum states of charge carriers in QDs[3], [4], [5], [6], [7].",
"One of the more studied TLS in QDs is their fundamental optical excitation, which results in a QD confined electron-hole (e-h) pair.",
"Since light interacts very weakly with the electronic spin, the photogenerated e-h pair has antiparallel spin projections on the incident light direction[8].",
"Such an e-h pair is called a bright exciton (BE).",
"The coherent properties of the BE have been extensively studied [9], [10], [11].",
"The main advantages of the BE qubit are its accessibility to coherent control by light and its neutrality, which results in insensitivity to vicinal electrostatic fluctuations.",
"The main disadvantage is in its short radiative lifetime ($\\sim $ 1 ns).",
"In contrast, dark excitons (DEs) - formed by e-h pairs with parallel spin projections, are almost optically inactive.",
"[12], [13] Due to small BE-DE mixing, induced by the QD deviation from symmetry, DEs may still have some residual optical activity.",
"[14], [15] However, their radiative lifetimes are orders of magnitude longer than that of the BEs [16].",
"DEs, like BEs, are neutral and therefore have a long spin coherence time [16].",
"Recently, it was demonstrated that the DE can be optically initiated in a coherent state by an ultrashort resonant optical pulse [17], and that its quantum state can be coherently controlled and reset using short optical pulses [16], [18], making it thus an attractive matter spin qubit.",
"In this work, we present further experimental study of the DE as a coherent TLS under an external magnetic field and demonstrate full control over its eigenstates.",
"Even at zero magnetic field, due to the short range e-h exchange interaction [19], the DE spin states are not degenerate.",
"The spin eigenstates are the symmetric ${\\left|{S_2}\\right\\rangle }$ and anti-symmetric ${\\left|{A_2}\\right\\rangle }$ coherent superpositions of the DE spin up (${\\left|{+2}\\right\\rangle }$ ) and spin down (${\\left|{-2}\\right\\rangle }$ ) states [20].",
"At non-vanishing external magnetic fields, however, when the Zeeman splitting is larger than the exchange interaction, the eigenstates become the ${\\left|{+2}\\right\\rangle }$ and ${\\left|{-2}\\right\\rangle }$ spin states.",
"Our experimental data is corroborated by a theoretical model which produces excellent agreement with measured photoluminescence (PL) intensity correlations under various magnetic fields and optical excitation intensities.",
"This agreement shows that the externally applied field controls the DE as a qubit, without reducing its inherently long coherence time.",
"At zero magnetic field, due to the short range e-h exchange interaction, the DE eigenstates are the symmetric $\\left|{S_2}\\right\\rangle =1/\\sqrt{2}\\left[{\\left|{+2}\\right\\rangle +\\left|{-2}\\right\\rangle }\\right]$ and anti-symmetric $\\left|{A_2}\\right\\rangle =1/\\sqrt{2}\\left[{\\left|{+2}\\right\\rangle -\\left|{-2}\\right\\rangle }\\right]$ coherent superposition of the spin up (${\\left|{+2}\\right\\rangle }$ ) and spin down (${\\left|{-2}\\right\\rangle }$ ) states, where the anti-symmetric state is lower in energy [16].",
"These states are schematically described in Figure.REF The DE can be optically excited, thereby generating a spin blockaded biexciton $XX_{{T_{3}}}^{0}$ [21].",
"This biexciton is comprised of two electrons in a singlet configuration at their ground level (total spin projection zero), and two holes with parallel spins forming a triplet (total spin projection ±3), occupying the ground and second hole levels [21].",
"Likewise, as first demonstrated here, the lower and higher eigenstates of the $XX_{{T_{3}}}^{0}$ qubit are also the anti-symmetric $\\left|{A_3}\\right\\rangle =1/\\sqrt{2}\\left[{\\left|{+3}\\right\\rangle -\\left|{-3}\\right\\rangle }\\right]$ and symmetric $\\left|{S_3}\\right\\rangle =1/\\sqrt{2}\\left[{\\left|{+3}\\right\\rangle +\\left|{-3}\\right\\rangle }\\right]$ coherent superpositions of the spin up ($\\left|{+3}\\right\\rangle $ ) and spin down ($\\left|{-3}\\right\\rangle $ ), respectively.",
"The DE and $XX_{{T_{3}}}^{0}$ form an optical $\\Pi $ -system since optical transitions are allowed between the ${\\left|{+2}\\right\\rangle }$ (${\\left|{-2}\\right\\rangle }$ ) DE state to and from the $\\left|{+3}\\right\\rangle $ ($\\left|{-3}\\right\\rangle $ ) biexciton state by right (left) handed circularly polarized light only.",
"At zero magnetic field, the DE and $XX_{{T_{3}}}^{0}$ eigenstates are therefore optically connected by linear cross polarized optical transitions denoted as horizontal (H) and vertical (V), where the H direction is chosen such that it coincides with the polarization of the ground state BE optical transition [21].",
"The system is schematically described in Figure.REF (a).",
"The time independent Hamiltonian of the DE and the $XX_{{T_{\\pm 3}}}^{0}$ in the presence of a magnetic field in the Faraday configuration as expressed in the basis $\\left\\lbrace {\\left|{+2}\\right\\rangle ,\\left|{-2}\\right\\rangle ,\\left|{+3}\\right\\rangle ,\\left|{-3}\\right\\rangle }\\right\\rbrace $ is given by: $\\hat{H}=\\frac{1}{2}\\left({\\begin{array}{cccc}{-{\\mu _{B}}\\left({{g_{e}}-{g_{h}}}\\right)B} & {\\hbar \\omega _{2}} & {} & {}\\\\{\\hbar \\omega _{2}} & {{\\mu _{B}}\\left({{g_{e}}-{g_{h}}}\\right)B} & {} & {}\\\\{} & {} & {2(\\Delta +{\\mu _{B}}{g_{2h}}B)} & {\\hbar \\omega _{3}}\\\\{} & {} & {\\hbar \\omega _{3}} & {2(\\Delta -{\\mu _{B}}{g_{2h}}B)}\\end{array}}\\right)$ This Hamiltonian represents two decoupled Hamiltonians, one for the DE and one for the $XX_{{T_{\\pm 3}}}^{0}$ , where ${\\mu _{B}}={{e\\hbar }\\mathord {\\left\\bad.",
"{\\vphantom{{e\\hbar }{2{m_{e}}c}}}\\right.\\hspace{0.0pt}}{2{m_{e}}c}}$ is the Bohr magnetron, B the magnitude of the magnetic field (normal to the sample surface), $g_{e}$ and $g_{h}$ are the electron and hole gyromagnetic ratios in the direction of the magnetic field, and $g_{2h}$ is the gyromagnetic ratio of the two heavy holes in triplet configuration.",
"The sign convention for the gyromagnetic factors is such that positive factors mean that electron (heavy hole) with spin parallel (antiparallel) to the magnetic field direction is lower in energy than that with spin antiparallel (parallel) [20].",
"We note that the triplet gyromagnetic ratio is not a simple sum of the gyromagnetic ratios of the individual holes [22].",
"The energy difference between the DE and the $XX_{{T_{\\pm 3}}}^{0}$ is $\\Delta $ , and $\\hbar {\\omega _{2}}$ and $\\hbar {\\omega _{3}}$ are the energy differences between the DE and $XX_{{T_{\\pm 3}}}^{0}$ eigenstates, respectively.",
"All energies are defined at zero magnetic field.",
"From this Hamiltonian, one calculates the energies and eigenstates of the system.",
"Figure.",
"REF a schematically describes the DE energy level structure, their magnetic field dependence, and the optical transitions between their eigenstates.",
"The externally applied magnetic field modifies the eigenstates of both qubits: [20] $\\begin{array}{l}{\\left|+\\right\\rangle _{i}}={N^{i}}_{+}\\left[{\\left|{+i}\\right\\rangle +\\left({\\frac{{\\beta _{i}}}{{\\omega _{i}}}+\\sqrt{1+\\frac{{\\beta _{i}^{2}}}{{\\omega _{i}^{2}}}}}\\right)\\left|{-i}\\right\\rangle }\\right]\\\\{\\left|-\\right\\rangle _{i}}={N^{i}}_{-}\\left[{\\left|{+i}\\right\\rangle +\\left({\\frac{{\\beta _{i}}}{{\\omega _{i}}}-\\sqrt{1+\\frac{{\\beta _{i}^{2}}}{{\\omega _{i}^{2}}}}}\\right)\\left|{-i}\\right\\rangle }\\right]\\end{array}$ where $i=2,3$ $N_{\\pm }^{i}$ are normalization factors and ${\\beta _{2}}={\\mu _{B}}\\left({{g_{e}}-{g_{h}}}\\right)B$ and ${\\beta _{3}}={\\mu _{B}}{g_{2h}}B$ are the magnetic energies.",
"The energy difference between the two eigenstates is given by their Zeeman splitting: ${\\Delta _{i}}(B)=\\sqrt{\\beta _{i}^{2}+(\\hbar \\omega _{i})^{2}}$ .",
"If one defines $\\tan \\theta _{B}^{i}=\\left({\\frac{{\\beta _{i}}}{{\\hbar \\omega _{i}}}}\\right)$ , EQ REF can be expressed more conveniently as: $\\begin{array}{l}\\begin{array}{l}{\\left|+\\right\\rangle _{i}}=\\cos \\left({\\frac{\\pi }{4}+\\frac{{\\theta _{B}^{i}}}{2}}\\right)\\left|{+i}\\right\\rangle +\\sin \\left({\\frac{\\pi }{4}+\\frac{{\\theta _{B}^{i}}}{2}}\\right)\\left|{-i}\\right\\rangle \\\\{\\left|-\\right\\rangle _{i}}=\\cos \\left({\\frac{\\pi }{4}-\\frac{{\\theta _{B}^{i}}}{2}}\\right)\\left|{+i}\\right\\rangle -\\sin \\left({\\frac{\\pi }{4}-\\frac{{\\theta _{B}^{i}}}{2}}\\right)\\left|{-i}\\right\\rangle \\end{array}\\end{array}$ Figure.",
"REF b presents an intuitive geometrical interpretation for the angle $\\theta _{B}$ and the DE Bloch sphere.",
"Since in the Faraday configuration the magnetic field direction is aligned with the direction of the $\\left|{+2}\\right\\rangle $ spin state, it follows that $\\pi /2-\\theta _{B}$ is the angle between the Bloch sphere eigenstate axis and the direction of the magnetic field.",
"Thus, as the magnitude of the external field (B) increases $\\theta _{B}^{i}$ approaches ${\\pi \\mathord {\\left\\bad.",
"{\\vphantom{\\pi 2}}\\right.\\hspace{0.0pt}}2}$ and the eigenstates gradually change their nature.",
"Once the Zeeman energies significantly exceed the exchange energies, the eigenstates become the $\\left|{\\pm 2}\\right\\rangle $ and $\\left|{\\pm 3}\\right\\rangle $ spin states for the DE and the $XX_{{T_{\\pm 3}}}^{0}$ , respectively.",
"Figure: (a) Schematic description of the energy levels, andspin wavefunctions of the DE and the XX T 3 0 XX_{{T_{3}}}^{0} –biexciton as function of an externally applied magnetic field in Faradayconfiguration.",
"↑\\uparrow (⇓\\Downarrow ) represents spin up (down)electron (hole).",
"The blue and purple solid (dashed) lines representthe energies of the low and high energy eigenstates of the DE (biexciton)respectively.",
"The spin eigenstates are written to the right and leftsides of the figure for zero and high field, respectively.",
"Verticalarrows connecting the DE and biexciton eigenstates mark allowed polarized optical transitions between the eigenstatesat zero and high field.",
"(b) Schematic representation of the changes thatthe external field induces on the Bloch sphere of the DE qubit.",
"Shownare three cases: (i) zero field(ii) cross section of the sphere at arbitrarymagnetic field,(iii) strong magnetic field.The eigenstates A 2 \\left|{A_2}\\right\\rangle , S 2 \\left|{S_2}\\right\\rangle , at zero field and ± 2 \\left|\\pm \\right\\rangle _2 at finite field,and the angle θ B {\\theta _{B}} are defined in the text and in EQ.",
".The eignestates are always at the poles of the sphere, north pole being the lower energy one.The pink dot represents the +2\\left|{+2}\\right\\rangle state, heralded by detecting Rpolarized biexciton photon.",
"The blue circle represents the counter clockwise temporal evolution of the DE state followingits heralding.In self assembled InGaAs QDs, the out of plane g-factors of the electron and the heavy hole are known to be both positive [20], [23] with that of the electron larger than that of the hole.",
"As a result the lower energy eigenstate contains an increasing contribution from the $\\left|{+2}\\right\\rangle $ spin state while the higher energy contains an increasing contribution from the $\\left|{-2}\\right\\rangle $ state, as the magnetic field increases.",
"The behavior of the $XX_{{T_{3}}}^{0}$ is similar, because as we show below, the Zeeman splitting of the optical transition from this state to the DE is opposite in sign to the Zeeman splitting of the BE transitions.",
"As can be seen in Figure REF , at the limit of high magnetic field, the DE - $XX_{{T_{3}}}^{0}$ system forms two separate TLSs, in which the DE spin up ($\\left|{+2}\\right\\rangle $ ) and spin down ($\\left|{-2}\\right\\rangle $ ) eigenstates are optically connected to the spin up ($\\left|{+3}\\right\\rangle $ ) and spin down ($\\left|{-3}\\right\\rangle $ ) eigenstates by a right (R) or left (L) hand circularly polarized transition, respectively.",
"The externally applied field thus changes the polarization of the optical transitions between the DE and $XX_{{T_{3}}}^{0}$ eigenstates from linearly cross polarized transitions into elliptically –cross- polarized ones as the field increases and eventually the optical transitions become cross-circularly polarized.",
"The state of a TLS (or a qubit) is conventionally described as a point on the surface of a unit sphere (Bloch sphere).",
"The north pole of the Bloch sphere describes the lower energy eigenstate and the sphere's south-pole describes the higher energy eigenstate.",
"The surface of the sphere describes all possible coherent superpositions of the TLS eigenstates.",
"Each superposition is therefore uniquely defined by a polar angle ($\\varphi $ ) and an azimuthal angle ($\\theta $ ): $\\left|{\\psi }\\right\\rangle =\\cos \\left({\\frac{\\theta }{2}}\\right)\\left|d\\right\\rangle +{e^{-i{\\varphi ^{}}}}\\sin \\left({\\frac{\\theta }{2}}\\right)\\left|u\\right\\rangle $ where $\\left|d\\right\\rangle $ ($\\left|u\\right\\rangle $ ) is the lower (higher) energy eigenstate at the north (south) pole of the Bloch sphere.",
"When a coherent superposition of a given TLS is formed, the relative phase between the two eigenstates evolves in time, due to the energy difference between the two eigenstates $\\Delta $ [8].",
"This evolution can be described as a counter-clockwise precession around an axis connecting the Bloch sphere's poles at a rate $\\Delta /\\hbar $ .",
"The evolution is therefore described such that $\\varphi \\left(t\\right)=\\varphi \\left({t=0}\\right)-{\\frac{\\Delta }{\\hbar }}t$ , while $\\theta \\left(t\\right)=\\theta \\left({t=0}\\right)$ remains unchanged as shown in Figure .1b (i), for the case in which a detection of an R circularly polarized $XX^0_{T_3}$ - biexciton photon initiated the DE in the $\\left|{+2}\\right\\rangle $ coherent state.",
"In this case $\\theta \\left({t=0}\\right)=\\pi /2$ , $\\varphi \\left({t=0}\\right)=0$ and $\\Delta /\\hbar =\\omega _2$ .",
"The externally applied field, induces changes on the DE and $XX_{{T_{3}}}^{0}$ eigenstates as described by EQ.",
"REF .",
"These changes can be described as geometrical rotations of their Bloch spheres in space, such that the new direction of the sphere's axis is given by: $\\frac{1}{{\\Delta _{i}}}\\left({{\\beta _{i}},0,{\\hbar \\omega _{i}}}\\right)$ where ${\\beta _{i}}$ , ${\\omega _{i}}$ and ${\\Delta _{i}}$ are defined above.",
"Thus, there is an angle ${\\theta _B}^i=\\tan ^{-1}(\\beta _{i}/\\hbar \\omega _{i})$ between the sphere's axis in the presence of the external field and the axis in the absence of the field, as described in Figure.",
"1b (ii).",
"Relative to the new axis the qubit spin state evolves in time like $\\left|{\\psi \\left(t\\right)}\\right\\rangle =\\cos \\left({\\frac{\\theta }{2}}\\right)\\left|-\\right\\rangle _{i} +{e^{-i{\\varphi ^{}}\\left(t\\right)}}\\sin \\left({\\frac{\\theta }{2}}\\right)\\left|+\\right\\rangle _{i} $ Here, detection of an R polarized $XX^{0}_{T_3}$ - biexciton photon, which initiates the DE in the $\\left|{+2}\\right\\rangle $ coherent state, defines that $\\theta \\left({t=0}\\right)=\\pi /2-\\theta _ B$ , $\\varphi \\left({t=0}\\right)=0$ and $\\Delta /\\hbar =\\Delta _2/\\hbar $ .",
"This situation is schematically described in Figure .1b (ii) and (iii).",
"For probing the DE precession and its dependence on the externally applied magnetic field we used continuous wave (CW) resonant optical excitation of the DE to the $XX_{{T_{3}}}^{0}$ -biexciton.",
"In the presence of such a CW resonance light field the two TLSs are coupled and the time independent Hamiltonian is given by $\\hat{H}=\\left({\\begin{array}{cccc}{-{\\Omega _{B}}\\left({{g_{e}}-{g_{h}}}\\right)} & {{\\hbar \\omega _{2}}\\mathord {\\left\\bad.",
"{\\vphantom{{\\omega _{2}}2}}\\right.\\hspace{0.0pt}}2} & {\\hbar \\Omega _{R}} & {}\\\\{{\\hbar \\omega _{2}}\\mathord {\\left\\bad.",
"{\\vphantom{{\\omega _{2}}2}}\\right.\\hspace{0.0pt}}2} & {{\\Omega _{B}}\\left({{g_{e}}-{g_{h}}}\\right)} & {} & {\\hbar \\Omega _{L}}\\\\{\\hbar \\Omega _{R}} & {} & {\\delta +{\\Omega _{B}}{g_{2h}}} & {{\\hbar \\omega _{3}}\\mathord {\\left\\bad.",
"{\\vphantom{{\\omega _{3}}2}}\\right.\\hspace{0.0pt}}2}\\\\{} & {\\hbar \\Omega _{L}} & {{\\hbar \\omega _{3}}\\mathord {\\left\\bad.",
"{\\vphantom{{\\omega _{3}}2}}\\right.\\hspace{0.0pt}}2} & {\\delta -{\\Omega _{B}}{g_{2h}}}\\end{array}}\\right)$ where ${{{\\Omega _{B}}={\\mu _{B}}B}\\mathord {\\left\\bad.",
"{\\vphantom{{{\\Omega _{B}}={\\mu _{B}}B}2}}\\right.\\hspace{0.0pt}}2}$ , ${\\Omega _{R(L)}}$ is the Rabi frequency for right- R (left – L) hand circularly polarized light.",
"The detuning of the exciting laser energy from the resonant transition between the DE and $XX^0_{T_3}$ - biexciton is assumed to be zero in our experiments.",
"EQ.",
"REF shows that the optical coupling depends on the light polarization and the spin state of the DE.",
"A circularly polarized R (L) photon, is absorbed in proportion to the magnitude of the DE spin state projection on the $\\left|{+2}\\right\\rangle $ ($\\left|{-2}\\right\\rangle $ ) state.",
"The $XX_{{T_{\\pm 3}}}^{0}$ -biexciton then starts to precess while it radiatively recombines into an excited DE state.",
"Detection of a right (left) hand circularly polarized photon heralds the system in a well-defined DE state given by $\\left|{+2}\\right\\rangle $ ($\\left|{-2}\\right\\rangle $ ).",
"The DE then precesses until a second photon is absorbed, and the process repeats itself.",
"Therefore, time resolved intensity autocorrelation measurements of the $XX_{{T_{\\pm 3}}}^{0}$ spectral line in the circularly polarized basis, provide a straightforward experimental way for probing the dynamics of the system [16].",
"In the absence of an external field and at low resonant excitation intensities, such measurements show a temporally oscillating signal at the frequency $\\omega _2$ .",
"The visibility of the oscillations in the degree of circular polarization can be used as a measure for $\\theta \\left({t=0}\\right)$ (where $t=0$ is the time of detecting the first polarized photon), and the phase of the signal as a measure for $\\varphi \\left({t=0}\\right)$ [17].",
"Our experiments used QDs grown by molecular beam epitaxy (MBE) on a [001]-oriented GaAs substrate.",
"One layer of self-assembled InGaAs QDs was deposited in the center of a one-wavelength microcavity sandwiched between an upper and lower set of AlAs/GaAs quarter-wavelength layer Bragg mirrors.",
"The sample was placed inside a tube, immersed in liquid helium, maintaining a temperature of $4.2$ K. Conducting coil outside the tube was used for generating an external magnetic field along the tube axis, permitting this way optical studies in Faraday configuration.",
"A x60, 0.85 numerical aperture microscope objective was used to focus the excitation lasers on the sample surface and to collect the emitted light.",
"We used low intensity high above bandgap energy 445nm diode CW laser light to photogenerate a steady state population of DEs in the QD in a statistical manner [24].",
"In addition, by using a grating stabilized tunable CW diode laser, we resonantly excited the DE population in one of the QDs into a $XX_{{T_{3}}}^{0}$ population [16].",
"Figure: (a) Rectilinear polarization sensitive PL spectra ofthe QD at zero magnetic field.",
"Solid black (red) line represents horizontal- H (vertical - V) polarization.",
"(b) The degree of rectilinear (black)and circular (orange) polarizations as a function of the emitted photonenergy.",
"(c) Circular polarization sensitive PL spectra at B=0.2B=0.2T.Red (black) line represents right-R (left- L) hand circular polarization.",
"(d) The degree of rectilinear (black) and circular (orange) polarizationsas a function of the emitted photon energy at B=0.2B=0.2T.",
"Note thatthe Zeeman splitting of the XX T 3 0 XX_{{T_{3}}}^{0} line is opposite insign to that of the negative, neutral and positive excitons.Figure REF shows polarization sensitive PL spectra of the single QD under study.",
"The PL was excited using 445nm non-resonant cw laser light.",
"Figure REF a (REF c) presents the measured spectra in the two linear (circular) polarizations, in the absence (presence of $B=0.2$ T) external magnetic field.",
"Figure REF b (REF d) presents the obtained degrees of linear (circular) polarizations as a function of the emitted photon energy in the absence (presence of $B=0.2$ T) external magnetic field.",
"In Figure.",
"2a the solid black (red) line represents horizontal - H (vertical - V) polarization and in Figure REF c black (red) line represents left- L (right-R) hand circular polarization.",
"Black (orange) lines in Figure REF b and REF d represent the degree of linear (circular) polarization.",
"The various exciton and biexciton lines are identified in Figure REF a.",
"Even in the absence of a magnetic field, one can clearly observe in Figure REF a and REF b that the BE spectral line is split into two cross linearly polarized components.",
"This split, measured to be $27\\pm 3 \\mu $ eV is common to self assembled QDs.",
"It results from the anisotropic e-h exchange interaction, mainly due to the QD deviation from cylindrical symmetry.",
"[20], [25] The DE degeneracy is also removed mainly due to the short range e-h exchange interaction [20], [25].",
"However, since the splits $\\omega _{2}$ and $\\omega _{3}$ are smaller than the radiative linewidth, the linearly polarized components of the $XX_{{T_{3}}}^{0}$ biexciton line cannot be spectrally resolved.",
"Therefore, only one, unpolarized spectral line is observed.",
"An upper bound for $\\omega _{3}$ < $0.2 ns^{-1} $ corresponding to split of less than $0.82\\mu $ eV is deduced directly from the degree of circular polarization memory of the $XX_{{T_{3}}}^{0}$ biexciton line at zero magnetic field [16].",
"At a sufficiently large magnetic field the line splits into two components.",
"The lower energy transition is R-circularly polarized and the upper energy one is L-circularly polarized.",
"At a magnitude of $0.2$ T, the splitting amounts to $13.6\\pm 3$$\\mu $ eV and it exceeds the measured linewidth of $11.4\\pm 3$$\\mu $ eV in the absence of external field.",
"We note that the measured Zeeman splitting of the $XX_{{T_{\\pm 3}}}^{0}$ line is opposite in sign to those of the ${X^{+1}}$ , the ${X^{-1}}$ , and the ${X^{0}}$ excitonic lines.",
"It follows from simple considerations that the expected Zeeman splitting of the charged and neutral excitonic spectral lines is proportional to the sum of the hole and electron g-factors $\\left({{g_{h}}+{g_{e}}}\\right)$ .",
"Therefore the R polarized part of these spectral lines is expected to be higher in energy than the L polarized part.",
"This is indeed what we experimentally observe.",
"Since the $XX_{{T_{\\pm 3}}}^{0}$ line splits in proportion to ${g_{2h}}+{g_{e}}-g_{h}^{*}$ our experimental observations indicate that the sign of ${g_{2h}}-g_{h}^{*}$ is negative, and its magnitude in this particular QD is larger than that of the electron g-factor.",
"These observations are in agreement with the energy level diagram of Figure.",
"1a.",
"The dependences of the Zeeman splitting of the various spectral lines on the g-factors are summarized in Table REF .",
"Table: The measured Zeeman splitting of various spectral lines.",
"The DE splitting was measured from a similar dot from the same sample.In order to probe the precession of the DE, we excite the sample with low intensity 445 nm CW laser light.",
"This non-resonant excitation photogenerates the QD confined BE and DE in a statistical manner.",
"The BE recombines radiatively within about 1ns, while the DE remains in the QD until it decays radiatively or an additional charge carrier enters the QD, whichever comes first.",
"The rate by which additional carriers enter depends linearly on the power of the (blue) laser light ($P_{b}$ ).",
"One can tune $P_{b}$ such that the average time between consecutive arrivals of carriers to the QD is comparable to the radiative lifetime of the DE.",
"[16] An additional circularly polarized CW laser light, resonantly tuned to the DE-$XX_{{T_{\\pm 3}}}^{0}$ transition is then used for probing the DE precession [26].",
"Figure: (a) Intensity autocorrelation measurements g 2 τ{g^{\\left(2\\right)}}\\left(\\tau \\right)of the emission line XX T 3 0 XX_{{T_{3}}}^{0}without magnetic field.",
"Blue(red) line represents low- (high-) intensity non-resonant excitationwith blue light P b =0.1μP_{b}=0.1\\mu W (P b =0.55P_{b}=0.55µW) and lowintensity resonant excitation with red light (P r =3.5P_{r}=3.5µW).Inset shows the Fourier transform of the low intensity measurement(blue filled line) and the fitted model calculations (solid blackline), revealing DE precession frequency of 417MHz ± 3MHzcorresponding to a 2.392.39ns precession period.",
"(b)-(c) present thecolor matched measurements in (a) for a limited temporal window (markedby dashed vertical lines in (a)).",
"The measured data points (dots)are overlaid by our model simulations convoluted by the temporal responseof the detectors (solid lines).In Figure REF (a-c), we present measured and fitted intensity autocorrelation functions at zero applied magnetic field.",
"As defined, the functions are normalized to unity at $t\\rightarrow \\infty $ .",
"Two measurements in two vastly different powers of the blue laser are presented together in Figure REF (a) to clearly demonstrate the reduction of the DE lifetime resulting from the increase in the non-resonant excitation power.",
"The DE lifetime decreases significantly as the non-resonant blue light ($P_{b}$ ) power increases as a result of the increase in the flux of carriers accumulating in the QD.",
"All other experimental conditions, in particular the intensity of the resonant laser ($P_{r}$ ), were kept the same.",
"The measured data points (dots) are overlaid by our model best fits (continuous lines in Figure REF (b,c)) using the parameters listed in Table REF convoluted with the system temporal response function [12].",
"The temporal oscillations in the correlation function resulting from the precession of the DE [16], are clearly observed as well.",
"The inset to Figure REF (a) shows the Fourier transform of the measured and calculated correlation functions under weak blue excitation.",
"From these measurements we calculate a precession frequency of $417\\pm 3$ MHz, which corresponds to a precession period of $2.39\\pm 0.03$ ns and a natural splitting of $1.7\\pm 0.02$ meV between the two DE eigenstates.",
"This splitting, in the absence of magnetic field is due to the short-range e-h exchange interaction [20], [14].",
"The measured full width at half maximum of the DE frequency at this intensity is 25 MHz and it increases with the excitation power of both the blue and red laser.",
"This power induced broadening is a consequence of the polarization oscillation decay, induced by the resonant CW excitation.",
"Much longer polarization decay times are measured under pulsed excitation [16].",
"Figure: Schematic description of the levels used in our model.The DE – XX T 3 0 XX_{{T_{3}}}^{0} biexciton form a Π\\Pi -system,with circularly polarized selection rules for optical transitions.",
"Upwards red arrows represent resonant excitation and curly downwards arrows representspontaneous emission.",
"Thecharge level represents all states which do not participate in the optical transitions, in particular singly positive or negative charge.The non-resonant optical charging and discharging rates, marked by upward and downwards vertical blue arrowsare proportional to the non resonant excitation rate G b G_b.",
"They were deduced from a set of power dependent measurementsat zero magnetic field.",
"Thevarious rates are defined in Table .To model our measurements, we added to the Hamiltonian presented in Eq REF a vacuum state and a charge state as shown in Figure REF , which schematically describes the various states of the system and the transition rates between these states.",
"For the sake of simplicity, we only included one additional auxiliary charged state in our model.",
"This charge level represents all states which do not participate in the optical transitions, such as a singly positive or negative charged QD.",
"With this degree of simplicity, however, we had to estimate the various proportionality constants to $G_{b}$ by which the charged state is connected to other states by the blue laser excitation (see Figure REF ).",
"These non-resonant optical charging and discharging rates, marked by upward and downwards vertical blue arrows are proportional to the non resonant excitation rate $G_b$ .",
"They were deduced from a set of power dependent measurements at zero magnetic field.",
"The various rates used in our model are defined in Table REF .",
"We then solved the system's master equation, which includes a Lindblad dissipation part in addition to the Hamiltonian $\\frac{d}{{dt}}\\rho \\left(t\\right)=-\\frac{i}{\\hbar }\\left[{H,\\rho \\left(t\\right)}\\right]+\\sum \\limits _{k}{\\left({{L_{k}}\\rho \\left(t\\right)L_{k}^{^{\\dag }}-\\frac{1}{2}\\rho \\left(t\\right)L_{k}^{^{\\dag }}{L_{k}}-\\frac{1}{2}L_{k}^{^{\\dag }}{L_{k}}\\rho \\left(t\\right)}\\right)}$ where $L_{k}$ represents the various non-Hermitian dissipation rates.",
"The various parameters used as input to the model are listed and referenced in Table 2.",
"$G_{b}$ in Figure REF represents the rate by which electrons and holes are equally added non coherently to the QD by the non-resonant blue laser excitation and it is therefore proportional to the power of blue laser ($P_{b}$ ).",
"Since the DE radiative lifetime is very long, $G_{b}$ essentially defines the DE lifetime, and the probability to find a DE in the QD.",
"Therefore $G_{b}$ can be deduced directly from the decay of the autocorrelation measurements to its steady state (see Figure REF ).",
"Likewise $\\Omega _{R(L)}$ was set proportional to the square root of the R (L) circularly polarized red laser power $P_{r}$ , as deduced from the power needed to saturate the PL under excitation with this source.",
"At saturation the co (cross)-circular intensity autocorrelation signal exhibits no oscillations, while the lower the power is, the more oscillations are observed.",
"This feature facilitated quite sensitive fitting of $\\Omega _{R(L)}$ so that the observed and calculated number of oscillations match.",
"We use the quantum regression theorem [27] to solve the master equation and thus to describe the temporal evolution of the system.",
"From the numerical solution, we calculated the polarization sensitive intensity autocorrelation measurement of the $XX_{T_{\\pm 3}}^{0}$ line, where detection of the first photon sets the initial system conditions, and the time by which the second photon is detected defines the time by which the system evolution is calculated [28].",
"The calculations were repeated for various blue light and resonance excitation intensities, and as a function of the magnitude of the externally applied magnetic field.",
"Figure: Measured (symbols) and calculated (lines) circularlyco-polarized intensity autocorrelation functions (g 2 τ{g^{\\left(2\\right)}}\\left(\\tau \\right))of the emission from the XX T 3 0 XX_{{T_{3}}}^{0} under weak non-resonant and resonant excitation condition for various externallyapplied magnetic fields in Faraday configuration.",
"The solid lines present the best fitted calculations convolutedwith the temporal response of the detectors.",
"The curves are verticallyshifted for clarity and the zero for each measurement is marked bya color-matched horizontal line.Figure REF shows co-circular polarization sensitive intensity autocorrelation measurements of the emission from the $XX_{T_{\\pm 3}}^{0}$ biexciton line under weak non-resonant ($P_{b}$ ) and resonant ($P_{r}$ ) excitation powers, for various externally applied magnetic fields.",
"Here as well, the measured data points (dots) are overlaid by our model simulations (dashed lines), convoluted by the detectors temporal response.",
"Figure: Measured (symbols) and calculated (lines) circularlyco-polarized intensity autocorrelation functions (g 2 τ{g^{\\left(2\\right)}}\\left(\\tau \\right))of the emission from the XX T 3 0 XX_{{T_{3}}}^{0} line under various externallyapplied magnetic fields in Faraday configuration, under quasi-resonantexcitation.",
"The solid lines present the best fitted calculations convolutedwith the temporal response of the detectors.",
"The curves are verticallyshifted for clarity and the zero for each measurement is marked bya color-matched horizontal line.Table: Physical values used in model calculations.",
"g-factors for the electron and hole were taken from ref [23] and slightly modified to best fit our measurements (see table ).The specific DE-biexciton resonance that we discussed so far is such that an electron is added to the first level and a heavy hole is added to the second level thereby directly exciting the $XX_{T_{\\pm 3}}^{0}$ biexciton.",
"The use of this resonance is not very convenient for two reasons: a) The oscillator strength of the resonance is relatively weak due to the different parities of the electron and heavy-hole envelope functions.",
"b) The width of this resonance is relatively narrow, since it is set by the radiative recombination lifetime of the state (700 ps).",
"As a result, excitation to this resonant is very sensitive to the detuning from resonance, which becomes highly sensitive to the externally applied magnetic field.",
"We therefore repeated the measurements using a DE-biexciton resonance in which, as before, the electron is added to the first level but the heavy hole is added to the fourth level.",
"This excited biexciton state has significantly larger oscillator strength, since the electron and hole envelope wavefunctions are of same parity.",
"Moreover, this excited biexciton state relaxes non-radiatively, by a spin conserving process in which a phonon is emitted, to the $XX_{{T_{3}}}^{0}$ ground state.",
"The process occurs within 70ps (see Supplementary Information of Ref 12).",
"As a result, the width of the resonance is significantly broader than that of the $XX_{{T_{3}}}^{0}$ , and consequently its excitation is less sensitive to detuning and to variations in the externally applied field.",
"These advantages, make the experiments less demanding, while hardly affecting our conclusions regarding the influence of the externally applied field on the DE as a qubit.",
"Figure REF shows, co-circular polarization sensitive intensity autocorrelation measurements of the $XX_{T_{\\pm 3}}^{0}$ emission line for various magnetic field intensities, under fixed weak non-resonant ($P_{b}$ ) and quasi-resonant ($P_{r}$ ) excitation powers.",
"The measured data points (dots) are overlaid by our model simulations (dashed lines).",
"For these simulations the excited biexciton levels were added to the model, together with their non-radiative, spin preserving relaxation channels.",
"The observed reduction in the visibility of the oscillations as the magnetic field increases is observed in both resonant and quasi-resonant excitations.",
"The source of this reduction is explained in Figure REF (b) as resulting from the field induced changes in the DE qubit eigenstates.",
"For example, at a field of $B=0.2$ T the DE splitting was calculated in Figure REF to be 4µeV, which is larger than the measured zero magnetic field splitting of $1.7$ µeV.",
"Hence, as expected, no oscillations are observed, and the system can be described as two separated TLSs.",
"In Figure REF (a), we present as an example, the measured (points) and best fitted model calculations (convoluted with the detector response, solid lines) polarization sensitive intensity auto correlation functions of the $XX_{T_{\\pm 3}}^{0}$ line at $B=8$ mT for the quasi-resonant excitation case.",
"The blue (red) color represents co- (cross-) circular polarizations of the first and second detected photon.",
"From the two autocorrelation functions $g_{\\parallel }^{\\left(2\\right)}\\left(\\tau \\right)$ and $g_{\\bot }^{\\left(2\\right)}\\left(\\tau \\right)$ , the temporal response of the degree of circular polarization (DCP) $D\\left(\\tau \\right)$ can be readily obtained: $D\\left(\\tau \\right)=\\frac{{g_{\\parallel }^{\\left(2\\right)}\\left(\\tau \\right)-g_{\\bot }^{\\left(2\\right)}\\left(\\tau \\right)}}{{g_{\\parallel }^{\\left(2\\right)}\\left(\\tau \\right)+g_{\\bot }^{\\left(2\\right)}\\left(\\tau \\right)}}$ In Figure REF (b), we present $D\\left(\\tau \\right)$ , obtained from Figure REF (a), where data points present the measured value and the orange dashed line represents the DCP obtained from the best fitted numerical model without convolving the detector response function.",
"The DCP can be also obtained analytically, using the following considerations: Recalling that the DCP is given by: $D\\left(\\tau \\right)=\\frac{{{{\\left|{\\left\\langle {+2}\\right.\\left|{\\psi \\left(\\tau \\right)}\\right\\rangle }\\right|}^{2}}-{{\\left|{\\left\\langle {-2}\\right.\\left|{\\psi \\left(\\tau \\right)}\\right\\rangle }\\right|}^{2}}}}{{{{\\left|{\\left\\langle {+2}\\right.\\left|{\\psi \\left(\\tau \\right)}\\right\\rangle }\\right|}^{2}}+{{\\left|{\\left\\langle {-2}\\right.\\left|{\\psi \\left(\\tau \\right)}\\right\\rangle }\\right|}^{2}}}}$ and substituting $\\left|{\\psi \\left(\\tau \\right)}\\right\\rangle $ using Eq REF and Eq REF one obtains: $D\\left(\\tau \\right)=\\left[{{{\\cos }^{2}}\\left({\\frac{{{\\Delta _{2}}\\tau }}{{2\\hbar }}}\\right)-{{\\sin }^{2}}\\left({\\frac{{{\\Delta _{2}}\\tau }}{{2\\hbar }}}\\right)\\cos \\left({2{\\theta _{B}}}\\right)}\\right]$ Eq REF describes the temporal evolution of the DCP assuming that the first detected photon is R polarized ($\\varphi (\\tau =0)=0$ in Eq REF ), the radiative decay is instantaneous, and the coherence of the DE is infinitely long.",
"The fact that the biexciton precesses and has a finite radiative lifetime ($\\tau _R=700 ps$ ) adds a prefactor $A_v=0.84$ to Eq REF .",
"This prefactor was deduced directly from the polarization memory measurements.",
"Assuming, in addition, that the DCP decays exponentially with a characteristic time $T_D$ , due to the optical re-excitation and the decoherence of the DE, transforms Eq REF into: $D\\left( \\tau \\right) = {A_V}\\left[ {{{\\cos }^2}\\left( {\\frac{{{\\Delta _2}\\tau }}{{2\\hbar }}} \\right) - {{\\sin }^2}\\left( {\\frac{{{\\Delta _2}\\tau }}{{2\\hbar }}} \\right)\\cos \\left( {2{\\theta _B}} \\right)} \\right] \\cdot {e^{ - {\\tau \\mathord {\\left\\bad.",
"{\\vphantom{\\tau {{T_D}}}} \\right.\\hspace{0.0pt}} {{T_D}}}}}$ From Eq REF the visibility of the DCP oscillations and its dependence on the magnetic field can be straightforwardly calculated for the case $\\tau <<T_D$ : $V(\\theta _B)=[D(\\theta _B)_{max} - D(\\theta _B)_{min}]/2 =A_v^{^{\\prime }}(1+\\cos 2\\theta _B)/2 =A_v^{^{\\prime }}\\cos ^2\\theta _B$ where $ D(\\theta _B)_{max}$ and $ D(\\theta _B)_{min}$ are obtained from Eq REF for $\\frac{{{\\Delta _{2}}\\tau }}{{2\\hbar }}=0$ and $\\frac{{{\\Delta _{2}}\\tau }}{{2\\hbar }}=\\pi $ , respectively, and $A_v^{^{\\prime }}<A_v$ , includes corrections due to the exponential decay of the DCP.",
"The best fitted numerical model to the data of each of the measurements, presented in Figure REF , represents the measured evolution of the DE after the quantified finite temporal response of the experimental setup was considered.",
"Therefore, to the best numerical model fits, those without the convoluted spectral response of the system, we fitted the analytical expression of Eq REF , as shown in Figure REF (b) by the dashed black line.",
"The observed decay of the DCP ($T_{D}$ ) has two main contributions.",
"The first one results from the actual decoherence of the DE spin qubit due to its interaction with the nuclei spins $T_{2}$ .",
"The second one results from the spontaneous nature of the $XX_{T_{\\pm 3}}^{0}$ radiative recombination and its re-excitation using CW light field.",
"In order to estimate $T_{2}$ , the second contribution should be reduced to minimum.",
"Using weak pulsed excitation rather than CW, we previously showed that the coherence time of the DE has a lower bound of about 100 ns[16].",
"The obtained visibilities and DCP decay times are summarized in the upper and lower insets to Figure REF (b), respectively.",
"As expected, the increase in the magnetic field does not affect the coherence of the DE as clearly seen in the lower inset to Figure REF (b).",
"Clearly, the decay of the DCP ($T_{D}$ ) is almost field independent.",
"Moreover, since the obtained $T_{D}$ of about 8ns is about an order of magnitude shorter than that measured under pulsed excitation in Ref [16], one can safely deduce that the dominant mechanism, which defines the DCP oscillations decay time $T_{D}$ in our measurements is the resonantly exciting laser field.",
"Figure: (a) Measured (points) and fitted (solid lines) polarizationsensitive correlation functions of the co- (blue) and cross- (red)circular polarization intensity autocorrelation at B=8B=8mT.",
"Solidlines are the results of our numerical model best fitted calculationsconvoluted with the temporal response of the detectors.",
"(b) Measured(blue dots - obtained from (a)), and calculated (orange line - obtainedfrom the calculations without the convolution with the detector response),time resolved DCP.",
"The dashed black line representsthe best fitted analytical expression (Eq.",
")to the numerical model.",
"The upper inset shows the visibility of the polarizationoscillations as a function of the magnetic field.The dashed black line describes the expected dependence as deduced from the numerical calculations.The solid orange line describes the analytical expression following Eq .",
"The lower inset shows thepolarization decay time (T D T_{D}) of the DCP as a function of the magnetic field.In contrast, the upper inset to Figure REF (b) shows that the visibility of the DCP oscillations depends on the externally applied field.",
"This dependence is readily understood from Figure REF (b.ii) and Eqs.",
"REF and REF , as resulting from the magnetic field induced changes of the eigenstates of the DE qubit.",
"Symbols in the upper inset represent the measured visibility (derived from the first valley and second peak of the modeled DCP).",
"The expected dependence from the numerical model is represented by a dash black line and that expected from Eq.",
"REF is represented by a solid orange line.",
"The slight difference between the full numerical model and the simple analytical one is the absence of the effect of other levels (such as the DE biexciton) in the analytical model.",
"In summary, we present an experimental and theoretical study of the quantum dot confined dark exciton as a coherent two level system subject to an externally applied magnetic field.",
"Experimentally, we used polarization sensitive intensity autocorrelation measurements of the optical transition which connect a $XX_{{T_{3}}}^{0}$ biexciton state with the dark exciton state.",
"Detection of a circularly polarized photon from this transition heralds the dark exciton and its spin state.",
"By applying an external magnetic field in the Faraday configuration, we measured the Zeeman splitting of various lines and accounted for our measurements by determining the g-factors of the electron, the hole and that of two holes in a triplet spin state.",
"We then used the external field as a tuning knob for varying the dark exciton eigenstates.",
"We showed that this external control knob does not affect the long coherence time of the dark exciton.",
"Theoretically, we were able to describe all our measurements using a Lindblad type master equation model with a minimal number of free fitting parameters.",
"Ultimately, our work provides a better understanding of the fundamentals of quantum dot excitations and may enable their use in future technologies.",
"The support of the U.S. Israel Binational Science Foundation (BSF), the Israeli Science Foundation (ISF), The Lady Davis Fellowship Trust and the Israeli Nanotechnology Focal Technology Area on “Nanophotonics for Detection” are gratefully acknowledged."
]
] | 1605.05748 |
[
[
"Interests Diffusion on a Semantic Multiplex"
],
[
"Abstract Exploiting the information about members of a Social Network (SN) represents one of the most attractive and dwelling subjects for both academic and applied scientists.",
"The community of Complexity Science and especially those researchers working on multiplex social systems are devoting increasing efforts to outline general laws, models, and theories, to the purpose of predicting emergent phenomena in SN's (e.g.",
"success of a product).",
"On the other side the semantic web community aims at engineering a new generation of advanced services tailored to specific people needs.",
"This implies defining constructs, models and methods for handling the semantic layer of SNs.",
"We combined models and techniques from both the former fields to provide a hybrid approach to understand a basic (yet complex) phenomenon: the propagation of individual interests along the social networks.",
"Since information may move along different social networks, one should take into account a multiplex structure.",
"Therefore we introduced the notion of \"Semantic Multiplex\".",
"In this paper we analyse two different semantic social networks represented by authors publishing in the Computer Science and those in the American Physical Society Journals.",
"The comparison allows to outline common and specific features"
],
[
"Introduction",
"Understanding how and to what extent people influence (and are influenced by) the different social networks they belong to, it is a very important issue.",
"Profiling people may allow improvement in public services, while representing a powerful means to enhance marketing capabilities by vendors.",
"In the present paper, interest represents a general concept that may refer to very detailed entities such as real product on the market or to abstract entities such as music or literature.",
"Some authors refer to \"memes\" as basic ideas propagating on social networks, however the definition of interest employed here is more general.",
"In this paper we combine semantic and complexity techniques to provide insights on the interest diffusion on social networks.",
"Our framework consists of the following main components: a method to gather information about the members; methods to perform semantic analysis of the domain of interest; a procedure to infer members' interests; and a model for interests propagation in the network.",
"We studied interest diffusion on single domain Semantic Social Networks in some previous works [5], [4].",
"This work represents a first attempt to provide insights on a Semantic Multiplex.",
"We are introducing the former term to refer to a representation of a social network with multiple relationship channels (i.e.",
"a multiplex) linked to a semantic representation of a domain of interest.",
"We studied the special case of the American Physical Society and Computer Science scientific communities, that, in fact, form a \"Semantic Multiplex\".",
"The links in the scientific social network were limited to co-authorship; members were attributed links to the domain of interest according to their publications; finally links between the concepts were attributed by the semantic analysis of publications.",
"It is worth stressing that both the domain of interest and the social network do exhibit a typical multilayer structure as each member may belong to different social networks; conversely, each concept may belong to several semantic layers.",
"The interests represent links between members and concepts.",
"Therefore a Semantic Multiplex is a bipartite graph were each of the two halves are multiplexes [2].",
"Fig.",
"REF provides a pictorial representation of our conceptualisation instanced for the test case: members may belong to one of two networks of authors and they exhibit some links to the semantic space representing the (stratified) domain of interest.",
"The two sets, authors and interests, form a bipartite network; while the social and the semantic networks do exhibit internal links representing friendship and closeness of concepts, respectively.",
"This means that three different types of links are accounted.",
"Alternatively one may image a \"Semantic Multiplex\" as a stacking of different \"Semantic Social Networks\" as introduced in [5].",
"Along the line with our previous research, we assumed diffusion to be the basic mechanism driving the interest spreading in the social network.",
"We further assumed that members have a limited tendency to change their interests $x_i$ , a capability to influence their neighbours $x_{ji}$ (i.e.",
"authority), and a tendency $x_{si}$ to be influenced by ongoing general trends $L_{s}$ : $L_i (c_k,t+\\Delta t) = \\left[1-x_i-x_{is}\\right] \\cdot L_i(c_k,t)+ \\frac{1}{|N_i|}\\cdot \\sum _{j \\in N_i} x_{ij} \\cdot L_j(c_k,t)+x_{is} \\cdot L_s(c_k,t)$ where $L_i=L_i(c_k)$ represents the level of interest of member $i-th$ in a topic $c_k$ at time $t$ and the $x_{ij}$ , $x_{ij}$ represent the average opinion change due to neighbours and general trend, respectively.",
"When time increment is brought to zero, one is left a set diffusion-like equations.",
"$\\frac{d}{dt} L_i (c_k,t) = -\\left[a_i+a_{is}\\right] \\cdot L_i(c_k,t)+ \\frac{1}{|N_i|}\\cdot \\sum _{j \\in N_i} a_{ij} \\cdot L_j(c_k,t)+a_{is} \\cdot L_s(c_k,t).$ The $a_{ij}$ and $a_{sj}$ are the rate of interest change in unit time and $a_i=\\sum _j a_{ij}$ .",
"The total attitude of a member to change her or his interests in unit time is represented by $a_i+a_{si}$ i.e.",
"the total susceptibility.",
"In the case of scientific SN's, one can estimate $L_i(c_k)$ 's by the fraction of publications produced in topic $c_k$ .",
"This allows to apply the general model to the semantic social network represented by the authors and the subjects of their publications.",
"Authors form a social network by co-authorship, while the topics of the publications can be represented as a semantic network which can be organised as a taxonomy.",
"The interest of an author in a subject is expressed by a publication in that field.",
"Applying the above defined paradigm to real set of data, provides an analytic tool to measure individual features, such as members' susceptibilities and authorities.",
"Authority is quantity that measure the capability to influence other members that we quantify by $A_i=\\sum _j a_{ji} $ .",
"It is worth stressing that $A_i $ and $a_i$ are different because the probability of being influenced (i.e.",
"susceptibility) is different from the probability to influence other people: $a_{ji} \\ne a_{ij}$ .",
"The social network endowed by the $a_{ji}$ weights, may be seen a weighted directed network."
],
[
"The Computer Science and American Physical Society Research Communities",
"Although the approach applies to any type of social network, here we test it against two different scientific research communities: the Computer Science (CS) and the American Physical Society (APS) ones.",
"More precisely, we employ the DBLP (Digital Bibliography and Library Project) and the APS databases as exclusive certified sources.",
"This means that a semantic representation of a domain of interest and co-authorships networks for the two selected cases were extracted, respectively, from the DBLPDBLP: Digital Bibliography & Library Project.",
"http://www.informatik.uni-trier.de/ ley/db/ computer science bibliography and the APS (American Physical Society) datasetAPS datasets can be retrieved at http://journals.aps.org/datasets.",
"Both the sources above provide (real-time) rich datasets covering a long time period on both the social network and authors interests.",
"Such amount of data guarantees that, if existing, the interest diffusion phenomenon can be observed and the individual features estimated.",
"Despite some limitations related to the simplicity of the diffusion theory and to some intrinsic limits in the semantic analysis, the basic mechanism for interest contagion is a diffusion process [5].",
"Several scientific papers using the DBLP dataset demonstrate its validity as a standalone case study ([7], [27], [11], [13], and [1]).",
"However most of them address structural and topological properties of the social network and only partially the social network dynamics.",
"The second standalone case study concerns basically the research social network in the field of physics.",
"The analysis of the APS publications is a also representative application of the general interest diffusion phenomenon as it covers a large time period (even larger than that covered by DBLP).",
"Furthermore, as the DBLP case study, the scientific community deems the APS dataset to be relevant for studies in the field of social and complex networks as demonstrated by the existence of several outstanding papers using it (e.g.",
"[2], [6]).",
"In particular, the following APS journals were considered for this analysis: Physical Review (PR), Physical Review A (PRA), Physical Review B (PRB), Physical Review C (PRC), Physical Review D (PRD), Physical Review E (PRE), PRI, Physical Review Letters (PRL), Physical Review Special Topics - Accelerators and Beams (PRST-AB), Physical Review X (PRX), and Reviews of Modern Physics (RMP).",
"Here the goal of the experimental evaluation is to test the theory (that is eq.",
"REF ) against the two real cases and comparing the resulting characteristics of the two communities.",
"In order to perform the analysis, we need to acquire the information about the topics defining the scope of the computer science domain and the evolution dynamics of both the social relationships and the interests of the authors.",
"In principle, the above information could be extracted from different sources, however the DBLP dataset and APS provide both the information through a single or a set of XML documents [35].",
"The analysis of each community followed our methodology for experimental assessment of social networks and, specifically, consisted of the following steps: papers selection, according to their type (e.g., journal, conference, book chapter) and year; interests and topics identification; papers indexing; identification of social network topology and its temporal evolution; semantic profiling of the members; analysis of the trends; assessment of the validity of the interests propagation theory and estimation of individual features."
],
[
"Papers Selection",
"The DPLP and APS databases are evolving entities.",
"Results presented in this work refer to the datasets as published by November 2013.",
"For DBLP the observation period was limited to the years from 1950 to 2012.",
"In such a temporal range, the number of considered papers is 2246098 and the authors 1337195.",
"In order to study the evolution of authors' interests it is necessary to observe some change in their semantic profile during time; therefore only authors that published papers in, at least, two different years can be analysed.",
"Those authors were named \"treatable\".",
"It is worth noting that only 519886 authors out of 1337195 are treatable as far as 2012.",
"It is reasonable to image that this is mainly due to students that just publish one work and then leave the world of research for other activities.",
"It should be noted that not-treatable authors are intrinsically untreatable, i.e.",
"they are so independently from the specific capability of a suitable set of topics to index papers.",
"Similarly to DBLP, the APS (American Physical Society) dataset provides a list of scientific papers in journals of physics.",
"The results presented here refer to the dataset provided by the American Physical Society including papers up through 2013.",
"The observation period was limited to the years from 1955 to 2005.",
"In this temporal range, the number of considered papers is 357553.",
"The member identification in the DBLP suffers from ambiguities.",
"There are authors' names under which papers by different members are gathered (polysemy) and, viceversa, there are authors that sign different papers with slightly different names (synonymy).",
"This issue is particularly relevant for Asian names.",
"The problem is currently approached by different authors with promising results [8], [30], [17], [24], [21].",
"A suitable wider set of sources will allow to detect a more complete set of topics and, hence, to define more accurate profiles.",
"This problem is mostly solved for the APS dataset since it provides information on affiliations that can be used to disambiguate authors, improve the quality of the analysis and, hence, of the overall assessment of the theory.",
"The members of the social network identified with a disambiguation engine (see [4]) are 257391.",
"Only 133058 members out of 257391 are treatable in the considered time period."
],
[
"Identification of Interests and Topics",
"In principle the text of all communications form the potential corpus supporting the semantic network.",
"However we limited the analysis to the titles.",
"There are other possible choices such as including the abstract or the introduction.",
"Nevertheless it is worth noting that a lot of information contained in the papers (and, specifically, in those sections) do not refer to their specific contents, but to the general state of the art in the field and, hence, the semantic analysis of the full text (or the abstract and the introduction) could include spurious terms not specific to the subject.",
"Finally the analysis of millions of full papers is extremely time consuming and it may be not sustainable.",
"For all the reasons above, limiting the analysis to the titles seems the most appropriate and handy choice.",
"A TermExtractor web application [25] and a software platform developed for natural language processing [4] were employed to extract preliminary multi-lexemes from the corpus of titles.",
"A lexeme is basically a set of words sharing a common root, or, more precisely, the class of equivalence of a word and all its linguistic inflected forms.",
"This tool is based on natural language processing techniques [19] and allows extracting the shared terminology of a community from the available documents in a given domain.",
"Then the multi-lexemes were validated by human processing devoted to remove general-purpose ones that are not specific of the computer science (or APS) domain and to merge synonyms.",
"The list of candidate topics was also double-checked to assure lack of possible polysemic multi-lexemes.",
"The resulting list of lexemes is the best approximation of the set of basic topics $\\lbrace c_k\\rbrace $ we were able to build up.",
"Recently, based on Latent Dirichlet Allocation, new methods have been employed for automated topic extraction [15].",
"These novel techniques will possibly improve the quality of the set.",
"The selection of the basic set of topics $C=\\lbrace c_1, c_2, \\ldots , c_N\\rbrace $ plays a crucial role to \"tame\" the domain of interest.",
"It is worth stating that, in order the diffusion theory to work, the $c^{\\prime }s$ must form a \"basis\" for the algebra of interests.",
"The most relevant relations among concepts (and, hence, among interests) are generalization (specialization) and similarity.",
"From the algebraic point of view these represent inclusion relationships.",
"The two constraints here imposed on the set $C$ are \"completeness\" and \"independence\" respectively.",
"A set of concepts will be named \"complete\" when each concept can be seen as the union of a subset of basic concepts: $\\forall c \\exists \\lbrace i_1, i_2, \\ldots , i_m\\rbrace : c=\\cup _{k=1}^m c_{i_k}.$ this means that each interest is the combination of a set of the basic interests.",
"On the other side a set of concepts will be named \"independent\" when each pair is disjoint, that is, it does not exist a concept representing a common specialisation of both: $\\forall c_i, c_j : \\ c_{i} \\cap c_j =\\emptyset .$ When eq.",
"(REF ) holds, the decomposition of eq.",
"(REF ) is unique.",
"The basic topics are identified with a subset of the multi-lexemes.",
"The quality of the results strongly depends on the capability of the selected set of multi-lexemes to fullfil the required constraints.",
"Once the set of topics is assessed, it is possible to attribute a subset of them to each scientific product.",
"Conversely, each topic can be given a frequency as the number of papers referring to it ($\\nu (c_k,t)$ ).",
"7632 topics were identified for DBLP and 30967 topics for APS."
],
[
"Identification of Social Networks Topology and Semantic Profiling",
"For each year, the nodes of the social network are given by the authors that have written papers by that year and the edges are given by the co-authorships.",
"According to this assumption, the social network grows with the time.",
"At a given a year, a link is attributed to all authors that have published a product together by that time.",
"Links are not removed and, therefore, the social network shows increasing complexity.",
"It is worth noting that in real life authors stop to publish (e.g., for retirement or job change) or give up their collaboration.",
"However such phenomena was not taken into account since the work was limited to the information available in the DBLP or APS datasets.",
"Moreover, all members are treated on the same ground regardless of their notoriety or scientific production.",
"The former approximations may affect the quality of these results.",
"Figure REF shows snapshots of the structure of the DBLP and APS social networks in 1980.",
"In accordance with what observed in [14], both the networks can be partitioned in three regions: single authors who do not participate in the network; isolated communities which display star structures; and a huge component characterised by a well-connected core region.",
"Figure: A graph representation of Computer Science (left) and APS (right) networks: authors are red dots while links are blue.Three regions are clearly identified: a huge core, disconnected small communities and a crown of isolated authors.For each year in the observation period, a semantic profile can be given to each author by means of the relative frequencies of \"expressions of interest\" that in this contest coincide with publications.",
"Treatable authors are named \"semantically treatable\" once they acquire a not-trivial semantic profile.",
"Their number is affected by the present capability to identify interests related to the domain and to index all papers.",
"In principle, it is possible to estimate both trends and neighbours susceptibilities of semantically treatable authors.",
"However, the higher is the number of publications of an author, the more precise is the estimation of susceptibility.",
"By sewing together all the semantic profiles of the different members one achieves the total interest graph.",
"To reconstruct the time evolution of the social network, for each year, a link to all pairs of members sharing a paper published by that year was attributed.",
"Weights were not attributed to links depending neither on the age of the shared publications nor on their number or scientific relevance.",
"The former are two strong hypotheses that may be relaxed in future (ongoing) work.",
"In fact, human contacts that took place in a remote past may have ceased; moreover, a successful publication may stimulate further scientific common activity more than a coarse one."
],
[
"Numerical Results",
"We performed a maximum likelihood fit to extract information on members susceptibility to both trend and neighbours [5], [4].",
"For both the networks, the hypothesis that neighbours influence the future propagation is necessary as it exhibits the maximum likelihood even upon normalising $\\chi ^2$ with the number of degrees of freedom.",
"The analytical $\\chi ^2$ minimization provides best fit quantities for the majority of members, however in some cases those solutions are not feasible (i.e.",
"the [0,1] range constraint is violated).",
"In those cases the most likely values are achieved at boundaries.",
"Negative and super-unitary values may represent real social characteristics of members (not accounted in the present model) or may result from the incompleteness of the semantic analysis.",
"As it is widely discussed in [5], probably both interpretations contribute.",
"Figure: Three dimensional histogram of the frequencies of the fitted x ^ i \\hat{x}_i and x ^ is \\hat{x}_{is} for the computer science case study (left) and for the APS (right) dataset.The authority coefficients (authorities, shortly) of DBLP authors spans the [0,52] range; their mean value is $\\bar{a} = 0.44$ and its standard deviation is about twice that value (0.89).",
"While there certainly exist real authors with some hundred collaborators, some of the observed ones may be fictitious.",
"It is known that there exist different authors with the same name (given name and family name); those people are very often treated as a single author in several datasets.",
"This problem is known as \"ambiguity\" of the papers indexing; it results in gathering different authors into a single member of our social network.",
"Due to the many to one transliteration of the original names in latin characters, Asian members are mostly prone to such effect.",
"The problem is known to affect DBLP data analysis [12].",
"Currently, DBLP administrators only use the surname and given name for the disambiguation, while providing room for explicit requests from the authors.",
"Evidently this is not enough to prevent disambiguation.",
"In order to check this phenomenon, a list of frequent Chinese full names by combining 50 very common Chinese given names [31] with 100 frequent Chinese family names [32] was built.",
"Several high values of authority correspond to entries associated with the constructed set of frequent Chinese full names.",
"Table REF presents some individual features of some famous authors in computer science.",
"As expected they all exhibit high levels of authority.",
"Table REF reports the neighbours' and trends' susceptibilities and the authority of famous scientists winning the Nobel Prize in Physics.",
"This table shows a relationship between the number of coauthors and the authority even for the most famous scientists.",
"This suggests to consider a different index of authority obtained by normalizing the actual authority parameter against the number of coauthors.",
"This will be further investigated in the future.",
"Table: Famous authors in Computer ScienceTable: Some authors winning the Nobel Prize in Physics.",
"The table shows, respectively, the name, the year of the Nobel Prize, the neighbours' and trends' susceptibilities, the authority, the number of papers considered for this analysis and the number of considered coauthors.Then the relationship between the success of an author with its authority was investigated.",
"The number of published papers (including proceedings) was employed as an index of success.",
"A more appropriate index should be the total number of citations [22], [29] or the h-index which were not available.",
"As shown in Figure REF , the higher is the success index, the higher is the authority.",
"We are not able to provide evidence of significant dependence of trende susceptibility upon success.",
"This means that there are successful authors of different types: some of them follow trends; some propose new topics and some continue working mostly on the same topics.",
"Figure: Scatter plots of the authority of different semantically treatable authors versus their success index for the two test cases: computer science (left) and APS (right).Beside the limits of the theory there are other factors that need further treatment and possibly hindered the analysis of the DBLP: the semantic analysis and the disambiguation.",
"Currently, interests are extracted from the titles of the papers by means of natural language techniques.",
"Even assuming that there exists a basic set of disjoint topics covering the domain of interest, the semantic analysis may only lead to an approximation of it.",
"Small sets of basic topics are not capable to index all papers, while larger ones tend to contain synonyms, similar multi-lexemes and, above all, concepts not representing interests (e.g.",
"the fake ones resulting from the words: report, surveys etc).",
"As already discussed, in this experimentation, an effect of the lack of coverage of the domain is the presence of null values of $x_i$ and $x_{is}$ .",
"This issue affects the quality and the completeness [3] of the identified topics and has an impact on semantic profile estimation.",
"Figure REF presents the scatter plot of the number of coauthors versus the authority for each author in the DBLP and APS datasets.",
"In both cases, authors with high authority have a huge number of coauthors.",
"However, while the disambiguation plays a relevant role for the DBLP case, it is reasonable that in APS dataset, due to availability of the affiliation, authors with large neighbours are real.",
"This is probably due to the existence of several papers (especially in high energy physics) describing experiments involving hundreds of scientists.",
"Figure: Scatter plot of co-authorship vs authority for both datasets (DBLP left, APS right).",
"Red crosses correspond to authors with very common asian names who where impossible to disambiguate."
],
[
"Comparing the two scientific communities",
"Despite some slight overlap, the computer science and the APS community are definitely distinct.",
"Generally speaking the authors of both communities exhibit the same average susceptibility to trends, however the APS authors are twice more susceptible to co-authors influence.",
"This means that the new ideas do circulate more effectively in the APS community.",
"In other words physicists tend to share the new interests more than computer scientists.",
"Possibly due to the large teams involved in very expensive facilities such as those at CERN in Geneve, ERFS in Lausanne, or neutron facility ion Rutherford, real large collaborations are observed in the APS community, while some much smaller group exists in computer science.",
"In both cases authority is related to success in the average even though wide fluctuations are observed.",
"On the other side the trend susceptibility does not seem to be related to the success in any of the two communities.",
"In fact we observe successful authors that have been working for long time on a specific topic and others that have changed their field of interest continuously.",
"Table REF reports some average quantities for the set of authors belonging to both communities.",
"As can be seen, the tendency of APS community is confirmed also for this special sample, as well as the basic susceptibility to general trends.",
"Again even the same authors do exhibit a larger average authority, thus indicating that it is the habit of publish more in the community and not a specific human characteristic responsible for the effect.",
"The correlations between the susceptibilities and the authorities as measured in the different disciplines are all slightly positive.",
"This means that there exists a very soft effect due to human character, but the leading effect is the position occupied in the community to determine the authority of susceptibility.",
"This result is also evident from Figure REF which shows the measured characteristics of the different authors as measured in the APS community versus the same quantity measured in the computer science community.",
"If the measured features were related to some human intrinsic characteristics, one expects the point to concentrate on the diagonal.",
"On the contrary they are concentrated on the axes.",
"Moreover the few points falling on the diagonal tend to be concentrated at low values, especially for the authorities.",
"This confirms the general idea that authority is strongly dependent on the field and normally the highest in a field the lowest in an other.",
"Table: Comparing Member Characteristics of Computation (DBLP) and APS communities.",
"Data in parentheses refer to the equivalent value calculated on the whole DBLP or APS sample.Figure: Scatterplot of neighbours' (left) and trends' (right) susceptibilities as measured for one author in the computer science and physics datasets.To improve legibility color coding is also used: blue corresponds to authors with high susceptibility in Computer Science, while red indicates high susceptibility on APS activities.Figure: Authorities measured in the two different dataset for the authors writing in both the computer science and physics disciplines.",
"The colours of the different points areset in a way that blue corresponds to authors with high authority in computer science, while red indicates authority in APS topics.",
"The inset focuses on the most populate region."
],
[
"Discussion",
"This work represents a first application of a model for diffusion of interests to a real social network observed from two different layers.",
"A social network endowed with a semantic domain of interest and its relation with members is named \"Semantic Social Networks\", since here we are dealing with several interacting Semantic Social Networks we introduced the concept of \"Semantic Multiplex\" (abridged form for Semantic Social Multiplex) which represents a multiplex (a social network with different relation layers) endowed with a semantic representation of a domain of interest.",
"As a propagation model, we assumed that interests spread according to a diffusion mechanism, while being continuously created.",
"The human behaviour is described by means of two basic behavioural characteristics (the susceptibility and the authority) that quantify the tendency to be influenced by \"friends\" and the environment.",
"By that means also the capability to influence others is inferred.",
"The model was tested against two interdependent scientific co-authorship networks: the APS and computer science communities.",
"This set provides a prototype of a \"Semantic Multiplex\".",
"Despite some intrinsic limitations of the model, authors' characteristics were extracted and compared.",
"One of the most interesting features is that authority and susceptibility are not absolute psychological characteristics of authors, but depend on the contest they work.",
"The majority of authors, even when they are leaders in one field (and hence strongly influence that community), do not exhibit the same authority in the other.",
"In other words one can be source of new ideas in a sector while being strongly receptive in an other.",
"Generally speaking the APS community is more coesive and authors tend to share collaborators' interests more than those working in computer science.",
"It is our opinion that general results are robust against the model approximation, however it worth stating some of the most relevant: The aging of the links [34] [28].",
"In this model, once a link is established it is supposed to hold forever; whereas, in reality, links can also vanish for several reasons related to competitiveness, displacements, personal frictions etc.",
"In general, here the strengths of links and their evolution are not taken into account.",
"The multiplex problem [18], [9], [10].",
"This paper deals with two interacting social layer only (based on co-authorship), however, people have different types of relationships (such as for instance working in the same company; participating same meetings etc) and, hence, other layers should be accounted for the interest diffusion.",
"The semantic profiling [16], [26].",
"Generally speaking, one is not allowed to assume that there exist a set of disjoint topics covering all possible interests.",
"Concepts are normally overlapping and, possibly, one interest may induce another one in a \"close\" topic.",
"In the present model, in order to provide members with a semantic profile, the existence of a basic set of disjoint topics is strictly required.",
"Further developing will enter the semantic structure of the domain of interest and will lead to more complex modelling.",
"Psychological types.",
"The present model only allows people to be influenced by friends (or by the environment) or to be independent on them.",
"However, there are several reasons for which a person may deliberately decide to do things, not just disregarding friends' positions, but in contrast with them.",
"This may take place for competition or just for spirit of independence.",
"This type of behaviour is usually referred to as \"anti conformity\" and it has been studied elsewhere [20], [33], [23].",
"To conclude, one may state that the preliminary analysis performed on the APS and DBLP datasets demonstrates that diffusion of interest on social networks is a reality and historical data can be analysed to provide information on members' profiles and their human relationships.",
"The Computer Science and the APS communities form a Semantic Multiplex.",
"The analysis on their common authors provides insights on the similarity and differences of the two scientific communities."
],
[
"Acknowledgements",
"We acknowledge interesting discussions with Antonio Scala, Walter Quattrociocchi and Alberto Tofani and the American Physical Society for providing us the dataset."
]
] | 1605.05870 |
[
[
"Vector perturbations of galaxy number counts"
],
[
"Abstract We derive the contribution to relativistic galaxy number count fluctuations from vector and tensor perturbations within linear perturbation theory.",
"Our result is consistent with the the relativistic corrections to number counts due to scalar perturbation, where the Bardeen potentials are replaced with line-of-sight projection of vector and tensor quantities.",
"Since vector and tensor perturbations do not lead to density fluctuations the standard density term in the number counts is absent.",
"We apply our results to vector perturbations which are induced from scalar perturbations at second order and give numerical estimates of their contributions to the power spectrum of relativistic galaxy number counts."
],
[
"Introduction",
"Within the last decade, cosmology has become a precision science, especially thanks to the very accurate measurements of the temperature fluctuations and the polarisation of the cosmic microwave background with the Planck satellite [1].",
"These measurements have allowed us to determine cosmological parameters with a precision of 1% and better.",
"Now we plan to continue this success story with very precise and deep large scale observations of the galaxies distribution.",
"Several observational projects are presently under way or planned [2], [3], [4], [5], [6], [7].",
"In order to profit maximally from these future data, we have to understand very precisely what we are measuring.",
"With perturbation theory and N-body simulations we compute the spatial matter density distribution in the Universe, while we observe galaxies in different directions on the sky and at different redshifts.",
"The relation between the matter density and galaxies is the so called biasing problem.",
"On large scales we expect biasing to be linear and in the simplest cases not scale dependent.",
"Another problem is the fact that we observe redshifts and directions while the matter density fluctuations are calculated in real (physical) space.",
"In order to convert angles and redshifts into physical distances we have to assume cosmological parameters.",
"On the other hand, we would like to use the observed galaxy distribution to infer cosmological parameters.",
"Therefore we have to calculate the density fluctuations in angular and redshift space to compare it directly with observations.",
"This leads to several additional terms in the observed galaxy number counts due to the fact that also directions and redshifts are perturbed in the presence of fluctuations.",
"In the last couple of years, the truly observable density fluctuations have been determined in angle and redshift space [8], [9], [10].",
"In addition to the usual galaxy fluctuations there are contributions from redshift space distortions (RSD), lensing, Shapiro time delay, an integrated Sachs-Wolfe (ISW) term and several other contributions from the gravitational potential which are due to the perturbations of the observed direction and redshift.",
"This approach has been put into context in [11] and with the general \"cosmic rulers\" and \"cosmic clocks\" formalism in the nice review [12].",
"Galaxy number counts have recently also been calculated to second order [13], [14], [15] and the bispectrum has been determined [16].",
"In this work we determine the galaxy number counts from vector and tensor perturbations (see also [17]).",
"This is relevant for different reasons.",
"First of all, the non-linearities of structure formation induce vector and tensor fluctuations as first discussed in [18] and then further in [19], [20], [21].",
"The first estimate of the vector power spectrum was carried out in [22] and it has been shown recently [23] that the induced frame dragging which is a vector perturbations can become quite substantial, of the order of 1%.",
"For discussion on small scales non linear effects see [24], [25].",
"Furthermore, if cosmology is not standard $\\Lambda $ CDM, e.g.",
"if there is a contribution from cosmic strings, the presence of vector perturbations may be a very interesting diagnostic.",
"The remainder of this paper is organized as follows.",
"In the next section we derive the expression for perturbations of number counts from vector perturbations.",
"We also repeat the expression for tensor perturbations for completeness.",
"In section we apply our result to second order vector perturbations.",
"This gives a good indication of the order of magnitude of vector perturbations induced at second order in the number counts.",
"In Section we summarize our findings and conclude.",
"Notation: We work with a flat Friedmann-Lemaître (FL) background using conformal time denoted by $t$ , such that $ \\text{d}s^2 =a^2(t)\\left( -\\text{d}t^2+\\delta _{ij} \\text{d}x^i \\text{d}x^j\\right) \\,.$ Spatial vectors are indicated by boldface symbols and by latin indices, while the 4 spacetime indices are greek.",
"A photon geodesic in this background which arrives at position ${\\mathbf {x}}_0$ at time $t_0$ and which has been emitted at affine parameter $\\lambda =0$ at time $t_s$ , moving in direction ${\\mathbf {n}}$ is then given by $(x^\\mu (\\lambda )) =(t_s+\\lambda , {\\mathbf {x}}_0+(\\lambda +t_s-t_0){\\mathbf {n}})$ .",
"Here $\\lambda = t-t_s = r_s-r$ , where $r$ denotes the comoving distance $r=|{\\mathbf {x}}(\\lambda ) -{\\mathbf {x}}_0|$ , hence $dr=-d\\lambda $ .",
"We can of course choose ${\\mathbf {x}}_0=0$ .",
"We denote the derivative w.r.t.",
"comoving time $t$ by an overdot such that the Hubble parameter, $H$ , is given by $H=\\dot{a}/a^2$ and the conformal Hubble parameter is ${\\cal H}= \\dot{a}/a=a H$ ."
],
[
"Vector & tensor contribution to galaxy number counts",
"We consider the number of galaxies in direction $-{\\mathbf {n}}$ at redshift $z$ , called $N({\\mathbf {n}},z)d\\Omega _{{\\mathbf {n}}}dz$ .",
"The average over angles gives their redshift distribution, $\\langle N\\rangle (z)dz$ .",
"The galaxy density perturbation at fixed redshift in direction ${\\mathbf {n}}$ is given by $\\begin{aligned}\\delta _z({\\mathbf {n}},z) =& \\frac{\\rho _g({\\mathbf {n}},z)-\\langle \\rho _g\\rangle (z)}{\\langle \\rho _g\\rangle (z)}=\\frac{\\frac{N({\\mathbf {n}},z)}{V({\\mathbf {n}},z)}-\\frac{\\langle N\\rangle (z)}{V(z)}}{\\frac{\\langle N\\rangle (z)}{V(z)}}\\\\=& \\frac{N({\\mathbf {n}},z)-\\langle N\\rangle (z)}{\\langle N\\rangle (z)}-\\frac{\\delta V({\\mathbf {n}},z)}{V(z)}~,\\end{aligned}$ where $V({\\mathbf {n}},z)$ is the physical survey volume density per redshift bin, per solid angle and $\\rho _g$ denotes the galaxy density.",
"The volume is also a perturbed quantity since the solid angle of observation as well as the redshift bin are distorted between the source and the observer.",
"Hence $V({\\mathbf {n}},z)=V(z)+\\delta V({\\mathbf {n}},z)$ .",
"The observed perturbation of the galaxy number density is $\\frac{N({\\mathbf {n}},z)-\\langle N\\rangle (z)}{\\langle N\\rangle (z)}=\\delta _z({\\mathbf {n}},z)+\\frac{\\delta V({\\mathbf {n}},z)}{V(z)} \\equiv \\Delta ({\\mathbf {n}},z)\\,.$ The redshift density perturbation $\\delta _z({\\mathbf {n}},z)$ , the volume perturbation $\\delta V({\\mathbf {n}},z)/V(z)$ and hence the galaxy number counts $\\Delta ({\\mathbf {n}},z)$ are gauge invariant quantities [9].",
"Vector perturbations do not lead to density fluctuations, their contributions to the number count fluctuation comes from two terms: the redshift perturbation $\\delta z$ which contributes to $\\delta _z({\\mathbf {n}},z)$ and the volume perturbation $\\delta V$ .",
"We start by relating the redshift density perturbation $\\delta _z({\\mathbf {n}},z)$ to the metric and energy-momentum tensor perturbations.",
"Expanding in Taylor series $\\langle \\rho _g\\rangle (z)\\equiv \\bar{\\rho }_g(z) =\\bar{\\rho }_g(\\bar{z})+\\partial _{\\bar{z}} \\bar{\\rho }_g \\,\\delta z({\\mathbf {n}},z)$ , where $z=\\bar{z} + \\delta z$ , we obtain [9]: $\\delta _z({\\mathbf {n}},z)= \\delta _g(r(z){\\mathbf {n}},t(z))-\\frac{\\text{d}\\bar{\\rho }_g(\\bar{z})}{\\text{d}\\bar{z}} \\frac{\\delta z ({\\mathbf {n}},z)}{\\bar{\\rho }(\\bar{z})}= -\\frac{3}{1+\\bar{z}} \\delta z({\\mathbf {n}},z) \\,,$ where $r(z) = t_0-t(z)$ and $t(z)$ is the conformal time at redshift $z$ .",
"For the second equal sign we have set the density fluctuation $\\delta _g$ in real space to zero (vector perturbations) and, since $a^3 \\bar{\\rho }_g = \\text{const.",
"}$ , $\\partial _{\\bar{z}} \\bar{\\rho }_g=3 \\bar{\\rho }_g / (1+\\bar{z})$ .",
"Next we compute the redshift fluctuation in a perturbed FL universe with vector and tensor perturbations only.",
"We choose the metric as $ \\text{d}s^2 =a^2(t)\\left[ -\\text{d}t^2 -2S_i \\,\\text{d}t \\text{d}x^i + (\\delta _{ij}+2H_{ij}) \\text{d}x^i \\text{d}x^j\\right] \\,.$ where $S_i$ is a transverse vector and $H_{ij}$ is a transverse-traceless tensor, i.e., $\\partial ^i S_i=0$ , $\\partial ^i H_{ij}=0$ and $H^i_i=0$ .Spatial indices of perturbed quantities are raised and lowered with $\\delta ^{ij}$ .",
"In the perturbed universe a photon emitted by a galaxy, the source $s$ , arrives at the observer $o$ with redshift $1+z= \\frac{(n^\\alpha u_\\alpha )_s}{(n^\\alpha u_\\alpha )_o}.$ Here we have introduced the perturbed photon momentum $n= a^{-2}(1+ \\delta n^0,{\\mathbf {n}}+ \\delta {\\mathbf {n}})$ , where ${\\mathbf {n}}$ is the unperturbed radial direction.With this convention the direction of observation is $-{\\mathbf {n}}$ .",
"The observer 4-velocity is $u=a^{-1}(1,{\\mathbf {v}})$ and one should keep in mind that the peculiar velocity ${\\mathbf {v}}$ is of the same order as the metric fluctuations.",
"A brief first order calculation, ignoring unobservable contributions at the observer position, yields $(1+z) \\simeq (1+\\bar{z})\\left( 1+ \\delta n^0_s-\\delta n^0_o +(S_i v^i)_s -(v_i n^i)_s \\right)=(1+\\bar{z})\\left( 1+ \\delta z\\right)\\,.$ Solving the geodesic equation $\\frac{\\text{d}}{\\text{d}\\lambda } \\delta n^0=-\\Gamma ^0_{\\alpha \\beta }n^\\alpha n ^\\beta $ we obtain $\\delta n^0_o-\\delta n^0_s= (S_i v^i)_s +\\int _0^{r_s} \\text{d}r \\, \\dot{S}_i n^i - \\int _0^{r_s} \\text{d}r \\, \\dot{H}_{ij} n^i n^j \\,.$ Inserting this result in eqs.",
"(REF ) and (REF ) we find $\\delta _z({\\mathbf {n}},z)= 3\\left( v_i n^i + \\int _0^{r_s} \\text{d}r \\, \\dot{S}_i n^i -\\int _0^{r_s} \\text{d}r \\, \\dot{H}_{ij} n^i n^j \\right) = -\\frac{3 \\,\\delta z}{1+ \\bar{z}} \\,.$ To compute the volume perturbation $\\delta V({\\mathbf {n}},z)/V(z)$ , let us express the spatial volume element in terms of 'observable' quantities such as the angles at the observer position and the perturbed redshift.",
"An observer moving with 4-velocity $u^\\mu $ sees a spatial volume element $\\text{d}V &= \\sqrt{-g} \\, \\epsilon _{\\mu \\nu \\alpha \\beta }u^\\mu \\text{d}x^\\nu \\text{d}x^\\alpha \\text{d}x^\\beta = \\sqrt{-g} \\, \\epsilon _{\\mu \\nu \\alpha \\beta }u^\\mu \\frac{\\partial x^\\nu }{\\partial z} \\frac{\\partial x^\\alpha }{\\partial \\theta _s} \\frac{\\partial x^\\beta }{\\partial \\phi _s} \\left|\\mathbf {J} \\right| \\text{d}z \\text{d}\\theta \\text{d}\\phi \\\\&\\equiv v(z,\\theta ,\\phi ) \\text{d}z \\text{d}\\theta \\text{d}\\phi \\nonumber \\,,$ where we have introduced the volume density $v$ such that $\\delta V({\\mathbf {n}},z)/V(z)=\\delta v({\\mathbf {n}},z)/v(z)$ and $|\\mathbf {J}|$ is the determinant of the Jacobian matrix, $\\mathbf {J}$ , of the transformation from the angles at the source $(\\theta _s,\\phi _s)$ to the the angles at the observer $(\\theta ,\\phi )$ .",
"Given the unperturbed radial trajectory $(\\theta ,\\phi )=(\\theta _s,\\phi _s)$ we can write, at first order, $\\theta _s=\\theta + \\delta \\theta $ and $\\phi _s= \\phi + \\delta \\phi $ , so that $\\left|\\mathbf {J} \\right|= \\left(1+ \\partial _\\theta \\delta \\theta + \\partial _\\phi \\delta \\phi \\right)$ .",
"In the absence of scalar perturbations and given our gauge choice for vector perturbations the expression for the metric determinant is simply $\\sqrt{-g}=a^4 r^2 \\sin {\\theta _s} = a^4 \\bar{r}^2 \\sin {\\theta } \\left( 1+\\cot {\\theta } \\delta \\theta +\\frac{2}{\\bar{r}} \\delta r \\right) $ , where we consider the fact that $r= \\bar{r} + \\delta r$ and we evaluate everything in terms of the observed redshift and angles at the observer.",
"With this we can express $v$ as $v= a^3 \\bar{r}^2 \\sin {\\theta } \\left( 1+\\cot {\\theta } \\delta \\theta +\\frac{2}{\\bar{r}} \\delta r \\right) \\left( \\frac{\\text{d}r}{\\text{d}z}+ \\frac{a}{{\\cal H}} v_r \\right) \\left( 1+ \\frac{\\partial \\delta \\theta }{\\partial \\theta } + \\frac{\\partial \\delta \\phi }{\\partial \\phi } \\right) \\,.$ Since at lowest order, on a photon geodesic, $\\text{d}t = \\text{d}\\lambda $ , the derivative of comoving distance $r$ w.r.t.",
"redshift, to first order, is given by $\\frac{\\text{d}r}{\\text{d}z}=\\frac{\\text{d}\\bar{r}}{\\text{d}\\bar{z}}+\\frac{\\text{d}\\delta r}{\\text{d}\\bar{z}}-\\frac{\\text{d}\\delta z}{\\text{d}\\bar{z}}\\frac{\\text{d}\\bar{r}}{\\text{d}\\bar{z}}=\\frac{a}{{\\cal H}} \\left( 1-\\frac{\\text{d}\\delta r}{\\text{d}\\lambda }+\\frac{a}{{\\cal H}} \\frac{\\text{d}\\delta z}{\\text{d}\\lambda } \\right)\\,.$ Inserting this and $a=(1+\\bar{z})^{-1}$ in the volume element $v$ we obtain $v=\\frac{\\bar{r}^2 \\sin {\\theta }}{(1+\\bar{z})^4{\\cal H}} \\left(1+ \\frac{\\partial \\delta \\theta }{\\partial \\theta } +\\cot {\\theta } \\, \\delta \\theta + \\frac{\\partial \\delta \\phi }{\\partial \\phi } -\\frac{\\text{d}\\delta r}{\\text{d}\\lambda } +\\frac{2}{\\bar{r}} \\delta r +\\frac{1}{(1+\\bar{z}){\\cal H}}\\frac{\\text{d}\\delta z}{\\text{d}\\lambda } - v_in^i\\right) \\,.$ We are interested in the fluctuation of the volume density $\\delta v = v(z)-\\bar{v}(z)$ .",
"The unperturbed volume element is simply $\\bar{v} (\\bar{z})=\\frac{a^4}{{\\cal H}}\\bar{r}^2 \\sin {\\theta }$ but we need to evaluate it at the observed (perturbed) redshift.",
"We use $\\bar{v} (\\bar{z}) = \\bar{v}( z)- \\frac{\\text{d}\\bar{v}}{\\text{d}\\bar{z}} \\delta z \\,,$ and $\\frac{\\text{d}\\bar{v}}{\\text{d}\\bar{z}}= \\bar{v} (\\bar{z}) \\left(4-\\frac{2}{\\bar{r} {\\cal H}}-\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right)\\frac{1}{1+\\bar{z}} \\,.$ Combining eq.",
"(REF ) with eqs.",
"(REF –REF ) we find $\\frac{\\delta v}{v}=(\\cot {\\theta }+\\partial _\\theta ) \\delta \\theta + \\partial _\\phi \\delta \\phi -\\frac{\\text{d}\\delta r}{\\text{d}\\lambda } +\\frac{2}{\\bar{r}} \\delta r +\\frac{1}{(1+\\bar{z}){\\cal H}} \\frac{\\text{d}\\delta z}{\\text{d}\\lambda }- v_in^i+ \\left(4-\\frac{2}{\\bar{r} {\\cal H}}-\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right)\\frac{\\delta z}{1+\\bar{z}} \\,.$ Considering this equation, we are still missing the geodesic displacements $\\delta x^j(\\lambda )$ in order to express the volume fluctuation in terms of the metric potentials and the peculiar velocities.",
"To find them we write $\\frac{\\text{d}x^\\alpha }{\\text{d}t}= \\frac{\\text{d}x^\\alpha }{\\text{d}\\lambda }\\frac{\\text{d}\\lambda }{\\text{d}t}=\\frac{n^\\alpha }{1+\\delta n^0} \\,,$ and we use the photon geodesic equation to find the $\\delta n^i$ .",
"Together with eq.",
"(REF ) we can express the integrals of (REF ) in terms of metric perturbations to find $\\delta r= \\int _0^{r_s} \\text{d}r \\, S_i n^i - \\int _0^{r_s} \\text{d}r \\, H_{ij} n^i n^j \\,,$ $(\\cot {\\theta }+\\partial _\\theta ) \\delta \\theta + \\partial _\\phi \\delta \\phi = - \\int _0^{r_s} \\text{d}r \\, \\frac{1}{r} \\left( \\nabla _\\Omega \\cdot S_\\Omega - \\nabla _\\Omega \\cdot (H_{ij}n^j)_\\Omega \\right)- \\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} \\nabla ^2_\\Omega \\left( S_in^i - H_{ij} n^i n^j \\right) \\,,$ where the subscript $\\Omega $ denotes the angular part of a vector $\\vec{A}_\\Omega =A_i \\hat{e}^i_\\theta + A_i \\hat{e}^i_\\phi $ and we denote the angular divergence and the angular Laplacian respectively by $\\nabla _\\Omega \\cdot \\vec{A}_\\Omega = (\\cot {\\theta }+\\partial _\\theta ) A_\\theta + \\partial _\\phi A_\\phi \\,,$ and $\\nabla ^2_\\Omega = \\left( \\cot {\\theta }\\partial _\\theta +\\partial ^2_\\theta +\\frac{1}{\\sin ^2{\\theta }} \\partial ^2_\\phi \\right) \\,.$ Combining eq.",
"(REF ) with eqs.",
"(REF ) and (REF ), and using the redshift perturbation given in eq.",
"(REF ) which yields $\\frac{1}{(1+\\bar{z}){\\cal H}} \\frac{\\text{d}\\delta z}{\\text{d}\\lambda } = v_i n^i -\\frac{1}{{\\cal H}} \\frac{\\text{d}}{\\text{d}\\lambda } (v_in^i)+\\frac{1}{{\\cal H}} \\left(\\dot{S}_i n^i -\\dot{H}_{ij} n^i n^j \\right) + \\int _0^{r_s} \\text{d}r \\, \\left(\\dot{S}_i n^i - \\dot{H}_{ij} n^i n^j \\right) \\,,$ we find $\\begin{aligned}\\frac{\\delta v}{v}= & \\left(S_in^i -H_{ij}n^in^j \\right) - \\int _0^{r_s} \\text{d}r \\, \\frac{1}{r} \\left( \\nabla _\\Omega \\cdot S_\\Omega - \\nabla _\\Omega \\cdot (H_{ij}n^j)_\\Omega \\right) -\\frac{1}{{\\cal H}} \\frac{\\text{d}}{\\text{d}\\lambda } (v_in^i)- \\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} \\nabla ^2_\\Omega \\left( S_in^i -H_{ij} n^i n^j \\right) \\\\&+\\frac{1}{{\\cal H}} \\left(\\dot{S}_i n^i -\\dot{H}_{ij} n^i n^j \\right) + \\int _0^{r_s} \\text{d}r \\, \\left(\\dot{S}_i n^i - \\dot{H}_{ij} n^i n^j \\right) +\\frac{2}{r_s} \\int _0^{r_s} \\text{d}r \\, \\left( S_i n^i - H_{ij} n^i n^j \\right)\\\\&-\\left(4-\\frac{2}{\\bar{r} {\\cal H}}-\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) \\left(v_i n^i + \\int _0^{r_s} \\text{d}r \\, \\left(\\dot{S}_i n^i - \\dot{H}_{ij} n^i n^j \\right)\\right) \\,.\\end{aligned}$ Adding the results given in eqs.",
"(REF ) and (REF ) we finally obtain the galaxy number count fluctuations for vector and tensor modes in a perturbed FL universe: $\\Delta ({\\mathbf {n}},z) = & \\left(S_in^i -H_{ij}n^in^j -v_in^i\\right)+ \\frac{1}{{\\cal H}} \\left(\\dot{S}_i n^i -\\dot{H}_{ij} n^i n^j - \\dot{v}_i n^i +\\partial _r (v_i n^i) \\right) \\nonumber \\\\& - \\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} \\nabla ^2_\\Omega \\left( S_in^i -H_{ij} n^i n^j \\right) -2\\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} \\left(S_in^i -H_{ij}n^in^j \\right) \\\\&- \\int _0^{r_s} \\text{d}r \\, \\partial _r\\left( S_i n^i - H_{ij} n^i n^j \\right) +\\left(\\frac{2}{\\bar{r} {\\cal H}}+\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) \\left( v_i n^i + \\int _0^{r_s} \\text{d}r \\, \\left(\\dot{S}_i n^i - \\dot{H}_{ij} n^i n^j \\right)\\right) \\nonumber \\,.$ For the last equation we have used the fact that, with our normalization of the affine parameter $\\text{d}t = \\text{d}\\lambda $ , the chain rule reads $\\frac{\\text{d}A}{\\text{d}\\lambda }= \\dot{A} + {\\mathbf {n}}\\cdot \\nabla A = \\dot{A} - \\partial _r A$ .",
"We have also exploited the transversality conditions, $\\partial ^i S_i=0$ and $\\partial ^i H_{ij}=0$ which imply $\\frac{1}{r} \\nabla _\\Omega \\cdot S_\\Omega = \\left(\\frac{2}{r}+\\partial _r \\right) S_i n^i$ and equivalently for $H_{ij}$ .",
"Furthermore we assume that galaxies move along geodesic and use their geodesic equation, $\\dot{{\\mathbf {v}}}\\cdot {\\mathbf {n}}- \\dot{{\\mathbf {S}}}\\cdot {\\mathbf {n}}+{\\cal H}( {\\mathbf {v}}\\cdot {\\mathbf {n}}- {\\mathbf {S}}\\cdot {\\mathbf {n}})=0$ , to rewrite eq.",
"(REF ) as $\\Delta ({\\mathbf {n}},z) = & -H_{ij}n^in^j + \\frac{1}{{\\cal H}} \\left(-\\dot{H}_{ij} n^i n^j +\\partial _r (v_i n^i) \\right) - \\int _0^{r_s} \\text{d}r \\, \\partial _r\\left( S_i n^i - H_{ij} n^i n^j \\right) \\nonumber \\\\& - \\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} \\nabla ^2_\\Omega \\left( S_in^i -H_{ij} n^i n^j \\right) -2\\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} \\left(S_in^i -H_{ij}n^in^j \\right) \\\\& +\\left(\\frac{2}{\\bar{r} {\\cal H}}+\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) \\left( v_i n^i + \\int _0^{r_s} \\text{d}r \\, \\left(\\dot{S}_i n^i - \\dot{H}_{ij} n^i n^j \\right)\\right) \\nonumber \\,.$ Equation (REF ) is the main result of this section.",
"Let us comment on it before we move on to the study of a numerical application.",
"We first notice that since vector and tensor perturbations do not produce density fluctuation we have no density term in the number counts which is the biggest contribution in the case of scalar perturbation.",
"In the first line we have two terms coming from the tensor metric potential, the redshift-space distortion term and the last term that accounts for the volume distortion along the line of sight: $&&\\Delta ^{\\text{P1}} ({\\mathbf {n}},z) = -H_{ij}n^in^j \\\\&& \\Delta ^{\\text{P2}} ({\\mathbf {n}},z)= -\\frac{1}{{\\cal H}} \\dot{H}_{ij} n^i n^j \\\\&&\\Delta ^{\\text{RSD}} ({\\mathbf {n}},z) = \\frac{1}{{\\cal H}} \\partial _r (v_i n^i) \\\\&&\\Delta ^{\\text{Vr}} ({\\mathbf {n}},z)= - \\int _0^{r_s} \\text{d}r \\, \\partial _r\\left( S_i n^i - H_{ij} n^i n^j \\right) \\,.$ The second line of eq.",
"(REF ) contains the lensing term which accounts for angular distortion of the volume and the third line represents a Doppler term and an Integrated Sachs-Wolfe term: $&&\\Delta ^{\\text{Len}} ({\\mathbf {n}},z) = - \\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} (2+ \\nabla ^2_\\Omega ) \\left( S_in^i -H_{ij} n^i n^j \\right) \\\\&& \\Delta ^{\\text{Dop}} ({\\mathbf {n}},z)= \\left(\\frac{2}{\\bar{r} {\\cal H}}+\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) v_i n^i \\\\&& \\Delta ^{\\text{ISW}} ({\\mathbf {n}},z)= \\left(\\frac{2}{\\bar{r} {\\cal H}}+\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) \\int _0^{r_s} \\text{d}r \\, \\left(\\dot{S}_i n^i - \\dot{H}_{ij} n^i n^j \\right) \\,.",
"$ In the number counts all the terms that are not integrated are evaluated at the unperturbed source position (in direction $-{\\mathbf {n}}$ at the observed redshift $z=z_s$ ) while the terms inside integrals are evaluated along the unperturbed line of sight (Born approximation) at conformal distance $r$ and conformal time $t_0-r$ ."
],
[
"Application to second order perturbation theory",
"We now apply our main formula (REF ) to vector perturbations present in a standard $\\Lambda $ CDM universe.",
"At first order the situation is not promising since standard inflationary scenarios do not produce vector perturbations and even if they would, vector perturbations decay without the presence of a non-standard source term, e.g.",
"cosmic strings.",
"However, at second order, non linearities in the scalar sector source vector modes and here we target these scalar-induced vector modes as a test of eq.",
"(REF ).",
"We use the following perturbation scheme for the metric potentials ${\\left\\lbrace \\begin{array}{ll}g_{00}= -a^2 \\left( 1+2 \\sum \\frac{1}{n!}",
"\\psi ^{(n)} \\right) \\\\g_{0i}= -a^2 \\sum \\frac{1}{n!}",
"S_i^{(n)} \\\\g_{ij}= a^2 \\left( \\left( 1-2 \\sum \\frac{1}{n!}",
"\\phi ^{(n)} \\right) \\delta _{ij}+\\sum \\frac{1}{n!}",
"H_{ij}^{(n)} \\right) \\,,\\\\\\end{array}\\right.",
"}$ for the energy-momentum tensor $\\rho = \\bar{\\rho }+ \\sum \\frac{1}{n!}",
"\\delta ^{(n)} \\rho $ , $p= \\bar{p} + \\sum \\frac{1}{n!}",
"\\delta ^{(n)} p$ and for the 4-velocity $u^\\mu = a^{-1} \\left(1+\\delta u^0, \\sum \\frac{1}{n!",
"}{\\mathbf {v}}^{(n)} \\right)$ .",
"Here we have used Newtonian gauge for the scalar perturbations which (locally) is well defined at every order.",
"At first order, $\\psi ^{(1)}$ and $\\phi ^{(1)}$ are the usual Bardeen potentials.",
"We neglect second order scalar and tensor fluctuations as well as first order vectors and tensors.",
"The metric, up to second order, is then written $ \\text{d}s^2 =a^2(t)\\left( -(1+2\\psi ^{(1)})\\text{d}t^2 -S_i^{(2)} \\,\\text{d}t \\text{d}x^i + (1-2 \\phi ^{(1)})\\delta _{ij} \\text{d}x^i \\text{d}x^j\\right) \\,.$ Within $\\Lambda $ CDM we can identify the two Bardeen potentials, $\\psi ^{(1)}=\\phi ^{(1)}=\\psi $ .",
"It is worth pointing out that our metric vector potential and the second order peculiar velocity are pure vector quantities: ${\\mathbf {S}}= {\\mathbf {S}}^V$ and ${\\mathbf {v}}_{(2)}={\\mathbf {v}}_{(2)}^V$ , where with $V$ we denote the transverse part of a vector that we can extract in Fourier space with the projection operator $P_{ij}$ which acts as $A^V_i= P_{ij} A^j= \\left(\\delta _{ij}-\\frac{k_ik_j}{k^2}\\right) A^j\\,.$ Following [22] we also define ${\\mathbf {\\Omega }}_{(2)} = {\\mathbf {v}}_{(2)}-{\\mathbf {S}}= {\\mathbf {\\Omega }}_{(2)}^V$ .",
"The covariant 4-velocity of the fluid is obtained via the normalization condition $g_{\\mu \\nu }u^\\mu u^\\nu =-1$ $u_\\mu = a \\left(-1-\\psi +\\frac{1}{2} \\psi ^2 -\\frac{1}{2} {\\mathbf {v}}^{(1)} \\cdot {\\mathbf {v}}^{(1)}\\,,\\, {\\mathbf {v}}^{(1)} -2 \\psi \\, {\\mathbf {v}}^{(1)} + \\frac{1}{2} {\\mathbf {\\Omega }}^{(2)} \\right) \\,.$ With this, modeling matter as a perfect fluid, we can construct the energy momentum tensor $T_{\\mu \\nu }= (\\rho +p)u_\\mu u_\\nu +p\\, g_{\\mu \\nu }$ .",
"At first order, the Einstein constraint equations reduce to $& 4 \\pi G a^2 \\delta \\rho = \\nabla ^2 \\psi - 3 {\\cal H}({\\cal H}\\psi + \\dot{\\psi }) \\,,\\\\& 4 \\pi G (1+ \\omega ) \\,\\bar{\\rho }\\,a^2 v_j^{(1)}=\\partial _j ({\\cal H}\\psi +\\dot{\\psi }) \\,,$ where $\\omega =p/\\rho $ .",
"At second order we use $T_j^{0 (2)}=\\frac{1}{2}\\left(\\bar{\\rho }\\, \\Omega _j +2v_j^{(1)}(\\delta \\rho -3\\bar{\\rho }\\psi ) \\right)$ and the $0i$ Einstein equation is $\\Omega _i&=\\frac{1}{6(1+\\omega ){\\cal H}^2} \\left(- \\nabla ^2 S_i +\\frac{16 \\nabla ^2 \\psi }{3{\\cal H}^2} \\partial _i({\\cal H}\\psi +\\dot{\\psi })-8 {\\cal H}\\psi \\partial _i \\psi -\\frac{16}{{\\cal H}}\\dot{\\psi }\\partial _i \\dot{\\psi }-8(3 \\dot{\\psi }\\partial _i \\psi +5 \\psi \\partial _i \\dot{\\psi }) \\right)^V\\\\&=\\frac{1}{6(1+\\omega ){\\cal H}^2} \\left(- \\nabla ^2 S_i +\\frac{16 \\nabla ^2 \\psi }{3{\\cal H}^2} \\partial _i({\\cal H}\\psi +\\dot{\\psi }) -8(3 \\dot{\\psi }\\partial _i \\psi +5 \\psi \\partial _i \\dot{\\psi }) \\right)^V \\,,$ where in the second line we ignored the pure gradient terms: $\\psi \\partial _i \\psi \\propto \\partial _i \\psi ^2$ and $\\dot{\\psi }\\partial _i \\dot{\\psi }\\propto \\partial _i \\dot{\\psi }^2$ which have vanishing vector projections.",
"Since both the left hand side and the right hand side are pure vector terms, they are fixed by their curl.",
"We can than write $\\partial _{[i}\\Omega _{j]}=\\partial _{[i}(\\cdots )_{j]}$ , where $_{[i}(\\cdots )_{j]}$ denotes anti-symmetrization, as $6(1+\\omega ){\\cal H}^2 \\partial _{[i}\\Omega _{j]}=\\partial _{[i} \\left( - \\nabla ^2 S_{j]}+8 \\left( 2 \\dot{\\psi }\\partial _{j]}\\psi +\\frac{2}{3 {\\cal H}^2} \\nabla ^2 \\psi \\partial _{j]} ({\\cal H}\\psi + \\dot{\\psi })\\right) \\right) \\,,$ and conclude that $6(1+\\omega ){\\cal H}^2 \\Omega _{j}= - \\nabla ^2 S_{j}+8 \\left( 2 \\dot{\\psi }\\partial _{j}\\psi +\\frac{2}{3 {\\cal H}^2} \\nabla ^2 \\psi \\partial _{j} ({\\cal H}\\psi + \\dot{\\psi })\\right)^V \\,,$ in agreement with eq.",
"(18) of [22].",
"Figure: The dimensionless power spectra of the Bardeen potential 𝒫 ψ {\\cal P}_\\psi (dashed) and of the scalar induced vectors 𝒫 S {\\cal P}_S (solid), for different redshifts: z=0z=0 (black), z=1z=1 (green) and z=3z=3 (orange).The vorticity in the fluid is defined as $\\omega _{\\mu \\nu }= F_\\mu ^\\lambda F_\\nu ^\\sigma (u_{\\lambda ;\\sigma }-u_{\\sigma ;\\lambda })$ , with $F_{\\mu \\nu }= g_{\\mu \\nu }+u_\\mu u_\\nu $ [26].",
"In [22] it is shown that in a perfect fluid there is no generation of vorticity at any order.",
"This allows us to set $0=\\omega _{ij}= \\partial _{[i}\\Omega _{j]}+6 \\, v^{(1)}_{[i}\\partial _{j]}\\psi +2 \\, v^{(1)}_{[i}\\dot{v}^{(1)}_{j]} \\,.$ Inserting eqs.",
"(REF ) and (REF ) in this expression we obtain $\\nabla ^2 S_{i}=\\frac{16}{3 {\\cal H}^2 \\Omega _m (1+\\omega _m)}\\left( \\nabla ^2 \\psi \\, \\partial _{i} ({\\cal H}\\psi + \\dot{\\psi }) \\right)^V \\,,$ where $\\omega _m=p_m/\\rho _m$ and we shall set it to 0 in the following.",
"Using the fact that for pressureless matter $\\psi ({\\mathbf {x}},t)= g(t)\\psi ({\\mathbf {x}},t_0)$ , we find that $\\dot{\\psi }\\partial _i\\psi = (\\dot{g}/g)\\partial _i(\\psi ^2/2)$ so that $ (\\dot{\\psi }\\partial _{i}\\psi )^V=0$ .",
"Inserting this and (REF ) in eq.",
"(REF ) yields ${\\mathbf {\\Omega }}=0$ and ${\\mathbf {v}}_{(2)}={\\mathbf {S}}$ .",
"From eq.",
"(REF ) we can conclude that the scalar-induced vector power spectrum $P_S(k,z)$ is a convolution of the scalar power spectrum $P_\\psi (k,z)$ .",
"We can furthermore factorize the gravitational potential as $\\psi (k,z)=\\psi ^{(\\text{in})}(k) T(k) g(z)$ , where $T(k)$ is the transfer function, a good approximation to it can be found in [27], and $g(z)$ is the growth factor which, in a $\\Lambda $ CDM cosmology can be approximated as $g(z)=\\frac{5}{2} g_{\\infty } \\Omega _m(z) \\left(\\Omega _m^{4/7}(z) -\\Omega _\\Lambda +\\left( 1+\\frac{1}{2} \\Omega _m(z)\\right) \\left( 1+\\frac{1}{70}\\Omega _\\Lambda ) \\right) \\right)^{-1} \\,.$ The prefactor $g_{\\infty }$ is chosen such that $g(0)=1$ .",
"With this the dimensionless power spectrum of the Bardeen potential is given by, ${\\cal P}_\\psi (k,z)=k^3/(2\\pi ^2) P_\\psi (k,z)= {\\cal P}(k)T^2(k) g^2(z)$ , where we define the primordial power spectrum ${\\cal P}(k)$ by ${\\cal P}(k)= A_s \\left( \\frac{k}{k_*} \\right)^{n_s-1} \\,,$ where $k_*$ is an (arbitrary) pivot scale.",
"In Fourier space eq.",
"(REF ) becomes $S_i({\\mathbf {k}}) = -\\frac{i k^{-2}}{(2\\pi )^3} \\frac{16}{3 {\\cal H}^2 \\Omega _m}\\int \\text{d}^3 {\\mathbf {q}}\\, q^2P_{ij}({\\mathbf {k}}) (q^j -k^j) \\psi ({\\mathbf {q}}) \\left({\\cal H}\\psi ({\\mathbf {k}}-{\\mathbf {q}})+ \\dot{\\psi }({\\mathbf {k}}-{\\mathbf {q}}) \\right) \\,.$ Defining $\\left\\langle {S_i({\\mathbf {k}}) \\, S_j^*({\\mathbf {k}}^{\\prime })}\\right\\rangle = (2\\pi )^3 \\frac{P_{ij}}{2}P_S(k) \\delta ({\\mathbf {k}}-{\\mathbf {k}}^{\\prime })$ , the power spectrum of vector perturbations, we find $P_S(k,z) =& \\frac{4}{(2\\pi )^3} \\frac{64 k^{-4}}{9 {\\cal H}^2 \\Omega _m^2} g(z)^2 (g(z)-(1+z)g^{\\prime }(z))^2 \\times \\\\& \\int \\text{d}^3 {\\mathbf {q}}\\, q^2( 2 k_i q^i - k^2) \\left( q^2 - \\frac{(k_i q^i)^2}{k^2}\\right) T^2(q)P_\\psi ^{(\\text{in})}(q) T^2(|{\\mathbf {k}}-{\\mathbf {q}}|)P_\\psi ^{(\\text{in})} (|{\\mathbf {k}}-{\\mathbf {q}}|) \\,,$ which can be simplified to [22] ${\\cal P}_S(k,z) = 4 \\frac{8 A_s^2}{9 {\\cal H}^2 \\Omega _m^2} g(z)^2 (g(z)-(1+z)g^{\\prime }(z))^2 \\, k^2 \\, \\Pi (k) \\,,\\qquad \\mbox{where}$ $\\Pi (k) = \\int _0^\\infty \\!\\!",
"\\!\\!\\!\\text{d}x \\!\\!\\int _{|x-1|}^{x+1}\\!\\!",
"\\!\\!\\!\\!",
"\\!\\!\\!\\text{d}y \\,\\frac{(y^2-x^2)((x+y)^2-1)((y-x)^2-1)}{y^2} \\left( \\frac{kx}{k_*} \\right)^{n_s-1}\\!\\!",
"\\left( \\frac{ky}{k_*} \\right)^{n_s-1}\\!\\!\\!\\!\\!",
"T^2( k x) T^2 (k y) \\,.$ Table: The color coding used in the plots for the auto correlation angular power spectra C ℓ (z s ,z s ' )C_\\ell (z_s,z^{\\prime }_s) of the different contribution to Δ vec (𝐧,z)\\Delta ^{\\text{vec}} ({\\mathbf {n}},z) in eq.",
"().Figure: The angular power spectrum for the different terms at redshifts z s =z s ' =0.1z_s=z_s^{\\prime }=0.1 (left) and z s =z s ' =1z_s=z_s^{\\prime }=1 (right).",
"We use the following color coding: redshift space distortion (green), lensing term (magenta), radial volume distortion term (orange), Doppler term (blue) and ISW effect term (red).We now go back to the galaxy number counts.",
"With the results of this section we can rewrite the vector contributions to eq.",
"(REF ) for a vorticity-free fluid as $\\begin{aligned}\\Delta ^{\\text{vec}} ({\\mathbf {n}},z) &= \\frac{1}{2{\\cal H}} \\partial _r ({\\mathbf {S}}\\cdot {\\mathbf {n}}) - \\frac{1}{2} \\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} \\nabla ^2_\\Omega \\left({\\mathbf {S}}\\cdot {\\mathbf {n}}\\right) -\\int _0^{r_s} \\text{d}r \\, \\frac{r_s-r}{r_s r} \\left({\\mathbf {S}}\\cdot {\\mathbf {n}}\\right) \\\\& -\\frac{1}{2} \\int _0^{r_s} \\text{d}r \\, \\partial _r\\left( {\\mathbf {S}}\\cdot {\\mathbf {n}}\\right) + \\frac{1}{2} \\left(\\frac{2}{r_s {\\cal H}}+\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) \\left( {\\mathbf {S}}\\cdot {\\mathbf {n}}+ \\int _0^{r_s} \\text{d}r \\, (\\dot{{\\mathbf {S}}}\\cdot {\\mathbf {n}})\\right) \\,.",
"\\end{aligned}$ The first term is the vector-redshift space distortion, the second and third terms are the lensing contributions.",
"In the second line the first term is the radial distortion of the volume and the last two terms come from the redshift perturbation of the volume: a Doppler term and the vector-type integrated Sachs-Wolfe (ISW) term (see table REF ).",
"Since, at fixed redshift, (REF ) is a function on the sphere we expand it in spherical harmonics with redshift dependent amplitudes $\\Delta ^{\\text{vec}} ({\\mathbf {n}},z) = \\sum _{\\ell m} \\, \\delta _{\\ell m} (z) Y_{\\ell m} ({\\mathbf {n}}) \\,,$ and we denote the angular power spectrum of vector galaxy number counts by $C_\\ell (z,z^{\\prime }) = \\left\\langle {\\delta _{\\ell m}(z) \\delta _{\\ell ^{\\prime } m^{\\prime }}^*(z^{\\prime })}\\right\\rangle \\,.$ The computation of the angular correlators is straighforward given that, with our Fourier convention, $\\partial _r ({\\mathbf {S}}\\cdot {\\mathbf {n}})= -i \\int \\frac{\\text{d}^3 {\\mathbf {k}}}{(2 \\pi )^3} n^i k_i \\, n^j S_j(k) e^{i {\\mathbf {k}}\\cdot {\\mathbf {n}}r} \\,.$ It is useful to factorize the scalar-induced vector power spectrum of eq.",
"(REF ) as ${\\cal P}_S(k,z,z^{\\prime })= g_S(z) g_S(z^{\\prime }) k^2 \\Pi (k)$ with $g_S(z)= \\frac{4\\sqrt{2} A_s}{3 {\\cal H}(z) \\Omega _m(z)} g(z) (g(z)-(1+z)g^{\\prime }(z))\\,.$ Figure: Different terms for the transversal power spectrum C ℓ (z s ,z s )C_\\ell (z_s,z_s) at fixed multipoles ℓ=5\\ell =5 (left) and ℓ=20\\ell =20 (right) as a function of redshift.",
"Color coding as in figure .We present the angular power spectra for the auto-correlations of the different effects defined in eqs.",
"(REF )–(REF ).",
"The expressions for the cross-correlations are given in Appendix .",
"We denote the comoving distance to the source redshift $z_s$ by $r_s$ , ${\\cal H}$ is the Hubble parameter at $z_s$ and ${\\cal H}^{\\prime }$ is the Hubble parameter at $z^{\\prime }_s$ .",
"$&C_\\ell ^{\\text{RSD}}(z_s,z^{\\prime }_s)&= \\frac{\\pi }{2} \\frac{\\ell (\\ell +1)}{r_s^2 r_s^{\\prime 2} {\\cal H}{\\cal H}^{\\prime }} \\int \\frac{\\text{d}k}{k^3} \\Bigl [ \\Bigl ((\\ell -1) j_\\ell (k r_s) - k r_s j_{\\ell +1}(k r_s) \\Bigr ) \\Bigl ((\\ell -1) j_\\ell (k r_s^{\\prime }) - k r_s^{\\prime } j_{\\ell +1}(k r_s^{\\prime }) \\Bigr ){\\cal P}_S(k,z_s,z_s^{\\prime }) \\Bigr ]\\\\&C_\\ell ^{\\text{Len}}(z_s,z^{\\prime }_s)&= \\frac{\\pi }{2}\\ell (\\ell +1) (\\ell ^2+\\ell -2)^2 \\int _0^{r_s}\\text{d}r \\int _0^{r_s^{\\prime }}\\text{d}r^{\\prime } \\,W_L (r) W_L (r^{\\prime }) \\int \\frac{\\text{d}k}{k^3} \\frac{j_\\ell (k r) }{r} \\frac{j_\\ell (k r^{\\prime }) }{r^{\\prime }} {\\cal P}_S(k,z,z^{\\prime })\\\\&C_\\ell ^{\\text{Vr}}(z_s,z^{\\prime }_s)&= \\frac{\\pi }{2} \\ell (\\ell +1) \\int _0^{r_s}\\!\\!\\!\\text{d}r \\int _0^{r_s^{\\prime }}\\!\\!\\!\\text{d}r^{\\prime } \\!\\!\\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\left(\\frac{(\\ell -1) j_\\ell (k r) - k r j_{\\ell +1}(k r)}{r^2} \\right) \\!",
"\\left(\\frac{(\\ell -1) j_\\ell (k r^{\\prime }) - k r^{\\prime } j_{\\ell +1}(k r^{\\prime })}{r^{\\prime 2}} \\right) {\\cal P}_S(k,z,z^{\\prime }) \\Biggr ] \\\\&C_\\ell ^{\\text{Dop}}(z_s,z^{\\prime }_s)&= \\frac{\\pi }{2} \\ell (\\ell +1) \\left(\\frac{2}{r_s {\\cal H}}+\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int \\frac{\\text{d}k}{k^3} \\frac{j_\\ell (k r_s) }{r_s} \\frac{j_\\ell (k r_s^{\\prime }) }{r_s^{\\prime }} {\\cal P}_S(k,z_s,z_s^{\\prime }) \\\\&C_\\ell ^{\\text{ISW}}(z_s,z_s^{\\prime })&= \\frac{\\pi }{2} \\ell (\\ell +1) \\left(\\frac{2}{r_s {\\cal H}}+\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int _0^{r_s}\\text{d}r \\int _0^{r_s^{\\prime }}\\text{d}r^{\\prime } \\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\frac{j_\\ell (k r) }{r} \\frac{j_\\ell (k r^{\\prime }) }{r^{\\prime }} {\\cal P}_{\\dot{S}}(k,z,z^{\\prime }) \\Biggr ] \\,,$ where $W_L(r) = \\frac{r_s-r}{r \\,r_s}$ and ${\\cal P}_{\\dot{S}}(k,z,z^{\\prime })= \\dot{g}_S(z) \\dot{g}_S(z^{\\prime }) k^2 \\Pi (k)$ .",
"As vector perturbations do not affect the density of galaxies, all the contributions relate to gravitational effects on the propagation of light.",
"We calculate these contributions numerically for a flat $\\Lambda $ CDM model with Planck [1] cosmological parameters.",
"More precisely, we choose $\\Omega _bh^2=0.022$ , $\\Omega _m h^2=0.12$ , $n_s=0.96$ , $A_s=2.21 \\times 10^{-9}$ at the pivot scale $k_*=0.05 \\, \\, \\text{Mpc}^{-1}$ .",
"The Hubble constant at present time is $H_0 = h \\times 100$ km/s/Mpc with $h=0.67$ .",
"If we correlate perturbations at fixed redshift $C_\\ell (z_s,z_s)$ we obtain the transversal power spectrum but we can also correlate perturbations at different redshifts to obtain the radial power spectrum $C_\\ell (z_s,z^{\\prime }_s)$ .",
"In figures REF –REF we plot the transversal and radial angular power spectra for the different terms.",
"Comparing them with the effects induced by scalar perturbations we see that the amplitude of the corresponding vector terms is suppressed by 2 orders of magnitudes in the case of the relativistic terms and up to 4–5 orders of magnitudes in the case of RSD, see figure REF .",
"The standard density term is however absent and this means that, in total, the vector number counts amplitude can be suppressed up to 6 orders of magnitudes at low redshifts.",
"The RSD is the dominant contribution only at low redshift while the lensing term starts to dominate for $z_s\\gtrsim 0.2$ .",
"Like for scalar perturbations, the radial power spectra terms are largely dominated by the integrated terms, especially the lensing term.",
"Therefore, in radial spectra with $z_s\\ne z_s^{\\prime }$ the vector contribution is less suppressed.",
"Note also that all the results presented here have been obtained with a $\\delta $ -function window.",
"Admitting a wider window function in redshift would significantly reduce the density term and the redshift space distortion without affecting integrated terms like lensing.",
"[28], [29].",
"Figure: Most relevant terms for the radial power spectrum C ℓ (z s ,z S ' )C_\\ell (z_s,z_{S^{\\prime }}) at z s =0.5z_s=0.5 (left) and z s =0.1z_s=0.1 (right), both for fixed multipole ℓ=20\\ell =20.",
"The lensing term (magenta), the volume contribution (orange), the ISW effect (red).",
"For the cross spectra: the correlation between lensing and ISW effect (black, dashed), the RSD-lensing correlation (cyan, dashed) and the lensing-volume distortion (dashed, dark green).",
"Cross spectra are dashed and negative contributions are dot-dashed.Figure: Most relevant terms for the radial power spectrum C ℓ (z s ,z s ' )C_\\ell (z_s,z_{s^{\\prime }}) as a function of multipoles for z s =0.5z_s=0.5, z s ' =1z^{\\prime }_s=1 (left) and z s =1z_s=1, z s ' =1.5z^{\\prime }_s=1.5 (right).",
"Cross spectra are dashed.",
"Color coding as in figure .Figure: The dominant fractional contributions ΔC ℓ =(C ℓ -C ℓ tot )/C ℓ tot \\Delta C_\\ell = (C_\\ell -C_\\ell ^{\\text{tot}})/C_\\ell ^{\\text{tot}} to the total effect of vector perturbations due to the most relevant terms at z s =z s ' =0.1z_s=z_s^{\\prime }=0.1 (left) and z s =z s ' =1z_s=z_s^{\\prime }=1 (right).",
"Color coding as in figures – and we also plot the RSD-doppler correlation (dashed, gray).Figure: Comparison of the different terms in the case of scalar perturbation and in the case of scalar-induced vectors.",
"If we refer to the CLASSgal terminolgy we have the lensing term (pink), the RSD term (green) and the GR terms (light blue).",
"We also plot the total C ℓ C_\\ell (black)."
],
[
"Conclusions",
"We have computed the galaxy number counts for vector and tensor perturbations in linear perturbation theory.",
"We have obtained a general expression which can be applied for all situations where linear cosmological perturbation theory is valid for vector and tensor perturbations.",
"We have employed it to compute the contribution the galaxy number counts from vector perturbations which are induced from the usual scalar perturbations at second order in perturbation theory.",
"While these terms are certainly present in the standard $\\Lambda $ CDM cosmology, they are very small.",
"Since within the perfect fluid approximation no vorticity is generated, the only 'standard term', the redshift space distortion is also very small.",
"For intermediate to large redshifts, $z \\gtrsim 0.2$ , the lensing term dominates the result for both radial and transversal correlations.",
"It is however 4 to 5 orders of magnitude smaller than the corresponding signal due to scalar perturbations.",
"This means that only if the amplitude of the scalar lensing contribution can be measured to an accuracy of better than 1%, it might be feasible to see this vector contribution.",
"This seems to be difficult, but the scalar lensing contribution by far dominates the radial correlation function and will probably be measured with good accuracy in the future.",
"Furthermore, it has been found in simulations [23] that higher order non-linear contributions tend to enhance vector perturbations.",
"However, this effect is strong only on small scales which are relevant in angular power spectra only at high multipoles [30].",
"Interestingly, when going to higher redshifts, up to redshift $z=3$ , the total vector to scalar ratio is increasing, see figure REF , even though the second order vectors are smaller at higher redshift.",
"This is due to the fact that at higher redshift the lensing term increaes while the density and redshift space distortions decrease [9].",
"Therefore the lensing term becomes more relevant and for this term vector perturbations are least suppressed.",
"Nevertheless, it seems not very promising to detect vector perturbations in the number counts with presently planned observations, if they are not larger than what is expected within $\\Lambda $ CDM.",
"This probably stems from the fact that number counts are an inherently scalar quantity which is expected to be dominated by scalar perturbations.",
"It has recently been suggested [31] that intrinsically spin-2 quantities like the alignment of the ellipticity of galaxies might be more promising.",
"Another intriguing possibility might be measuring the alignment or the correlation of the spins of distant galaxies.",
"This work is financially supported by the Swiss National Science Foundation."
],
[
"Cross-correlations",
"For completeness we present here the results of eq.",
"(REF ) also for the cross-correlations between the different terms of eqs.",
"(REF –REF ).",
"$& C_\\ell ^{\\text{RSD-Len}}(z_s,z_s^{\\prime })&=- \\frac{\\pi }{2} \\frac{\\ell (\\ell +1)}{r_s^2 {\\cal H}}(\\ell ^2+\\ell -2) \\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\Bigl ((\\ell -1) j_\\ell (k r_s) - k r_s j_{\\ell +1}(k r_s) \\Bigr ) \\int _0^{r_s^{\\prime }}\\text{d}r^{\\prime } \\,W_L (r^{\\prime }) \\frac{j_\\ell (k r^{\\prime }) }{r^{\\prime }} {\\cal P}_S(k,z_s,z^{\\prime }) \\Biggr ]\\\\&C_\\ell ^{\\text{RSD-Vr}}(z_s,z^{\\prime }_s)&=- \\frac{\\pi }{2} \\frac{\\ell (\\ell +1)}{r_s^2 {\\cal H}}\\!",
"\\!",
"\\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\Bigl ((\\ell -1) j_\\ell (k r_s) - k r_s j_{\\ell +1}(k r_s) \\Bigr ) \\!\\!",
"\\int _0^{r_s^{\\prime }}\\!\\!\\!\\text{d}r^{\\prime } \\!\\left(\\frac{(\\ell -1) j_\\ell (k r^{\\prime }) - k r^{\\prime } j_{\\ell +1}(k r^{\\prime })}{r^{\\prime 2}} \\right) \\!",
"{\\cal P}_S(k,z_s,z^{\\prime }) \\Biggr ] \\\\&C_\\ell ^{\\text{RSD-Dop}}(z_s,z_s^{\\prime })&=- \\frac{\\pi }{2} \\frac{\\ell (\\ell +1)}{r_s^2 {\\cal H}} \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\Bigl ((\\ell -1) j_\\ell (k r_s) - k r_s j_{\\ell +1}(k r_s) \\Bigr ) \\frac{j_\\ell (k r_s^{\\prime }) }{r_s^{\\prime }} {\\cal P}_S(k,z_s,z_s^{\\prime }) \\Biggr ] \\\\&C_\\ell ^{\\text{RSD-ISW}}(z_s,z_s^{\\prime })&=- \\frac{\\pi }{2} \\frac{\\ell (\\ell +1)}{r_s^2 {\\cal H}} \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\Bigl ((\\ell -1) j_\\ell (k r_s) - k r_s j_{l+1}(k r_s) \\Bigr ) \\int _0^{r_s^{\\prime }}\\text{d}r^{\\prime } \\frac{j_\\ell (k r^{\\prime }) }{r^{\\prime }} {\\cal P}_{S \\dot{S}}(k,z_s,z^{\\prime }) \\Biggr ] \\\\&C_\\ell ^{\\text{Len-Vr}}(z_s,z_s^{\\prime })&= \\frac{\\pi }{2}\\ell (\\ell +1) (\\ell ^2+\\ell -2) \\int _0^{r_s}\\!\\!\\!\\text{d}r \\int _0^{r_s^{\\prime }}\\!\\!\\!\\text{d}r^{\\prime } \\,W_L (r) \\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\frac{j_\\ell (k r) }{r} \\left(\\frac{(\\ell -1) j_\\ell (k r^{\\prime }) - k r^{\\prime } j_{\\ell +1}(k r^{\\prime })}{r^{\\prime 2}} \\right) {\\cal P}_S(k,z,z^{\\prime }) \\Biggr ] \\\\&C_\\ell ^{\\text{Len-Dop}}(z_s,z_s^{\\prime })&= \\frac{\\pi }{2}\\ell (\\ell +1) (\\ell ^2+\\ell -2) \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int \\frac{\\text{d}k}{k^3} \\frac{j_\\ell (k r_s^{\\prime }) }{r_s^{\\prime }} \\int _0^{r_s}\\text{d}r \\,\\Bigl [ W_L (r) \\frac{j_\\ell (k r) }{r} {\\cal P}_{S }(k,z,z_s^{\\prime }) \\Biggr ] \\\\&C_\\ell ^{\\text{Len-ISW}}(z_s,z_s^{\\prime })&= \\frac{\\pi }{2}\\ell (\\ell +1) (\\ell ^2+\\ell -2) \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int _0^{r_s}\\text{d}r \\int _0^{r_s^{\\prime }}\\text{d}r^{\\prime } \\,\\Biggl [ W_L (r) \\int \\frac{\\text{d}k}{k^3} \\frac{j_\\ell (k r)}{r} \\frac{j_\\ell (k r^{\\prime }) }{r^{\\prime }} {\\cal P}_{S \\dot{S}}(k,z,z^{\\prime }) \\Biggr ]\\\\&C_\\ell ^{\\text{Vr-Dop}}(z_s,z_s^{\\prime })&= \\frac{\\pi }{2}\\ell (\\ell +1) \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\frac{j_\\ell (k r_s^{\\prime }) }{r_s^{\\prime }} \\int _0^{r_s}\\text{d}r \\left(\\frac{(\\ell -1) j_\\ell (k r) - k r j_{\\ell +1}(k r)}{r^2} \\right) {\\cal P}_S(k,z,z_s^{\\prime }) \\Biggr ] \\\\&C_\\ell ^{\\text{Vr-ISW}}(z_s,z_s^{\\prime })&= \\frac{\\pi }{2}\\ell (\\ell +1) \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int _0^{r_s}\\text{d}r \\int _0^{r_s^{\\prime }}\\text{d}r^{\\prime } \\int \\frac{\\text{d}k}{k^3} \\Biggl [ \\left(\\frac{(\\ell -1) j_\\ell (k r) - k r j_{\\ell +1}(k r)}{r^2} \\right) \\frac{j_\\ell (k r^{\\prime }) }{r^{\\prime }} {\\cal P}_{S \\dot{S}}(k,z,z^{\\prime }) \\Biggr ]\\\\&C_\\ell ^{\\text{Dop-ISW}}(z_s,z_s^{\\prime })&= \\frac{\\pi }{2}\\ell (\\ell +1) \\left(\\frac{2}{r_s {\\cal H}}+\\frac{\\dot{{\\cal H}}}{{\\cal H}^2} \\right) \\left(\\frac{2}{r_s^{\\prime } {\\cal H}^{\\prime }}+\\frac{\\dot{{\\cal H}}^{\\prime }}{{\\cal H}^{\\prime 2}} \\right) \\int \\frac{\\text{d}k}{k^3} \\frac{j_\\ell (k r_s) }{r_s} \\int _0^{r_s^{\\prime }}\\text{d}r^{\\prime } \\Bigl [ \\frac{j_\\ell (k r^{\\prime }) }{r^{\\prime }} {\\cal P}_{S \\dot{S}}(k,z_s,z^{\\prime }) \\Bigr ] \\,,$ where ${\\cal P}_{S \\dot{S}}(k,z,z^{\\prime }) = g_S(z) \\dot{g}_S(z^{\\prime }) k^2 \\Pi (k)$ ."
]
] | 1605.05974 |
[
[
"Simulation study of a new InGaN p-layer free Schottky based solar cell"
],
[
"Abstract On the road towards next generation high efficiency solar cells, the ternary Indium Gallium Nitride (InGaN) alloy is a good passenger since it allows to cover the whole solar spectrum through the change in its Indium composition.",
"The choice of the main structure of the InGaN solar cell is however crucial.",
"Obtaining a high efficiency requires to improve the light absorption and the photogenerated carriers collection that depend on the layers parameters, including the Indium composition, p-and n-doping, device geometry.. .",
"Unfortunately, one of the main drawbacks of InGaN is linked to its p-type doping, which is very difficult to realize since it involves complex technological processes that are difficult to master and that highly impact the layer quality.",
"In this paper, the InGaN p-n junction (PN) and p-in junction (PIN) based solar cells are numerically studied using the most realistic models, and optimized through mathematically rigorous multivariate optimization approaches.",
"This analysis evidences optimal efficiencies of 17.8% and 19.0% for the PN and PIN structures.",
"It also leads to propose, analyze and optimize player free InGaN Schottky-Based Solar Cells (SBSC): the Schottky structure and a new MIN structure for which the optimal efficiencies are shown to be a little higher than for the conventional structures: respectively 18.2% and 19.8%.",
"The tolerance that is allowed on each parameter for each of the proposed cells has been studied.",
"The new MIN structure is shown to exhibit the widest tolerances on the layers thicknesses and dopings.",
"In addition to its being player free, this is another advantage of the MIN structure since it implies its better reliability.",
"Therefore, these new InGaN SBSC are shown to be alternatives to the conventional structures that allow removing the p-type doping of InGaN while giving photovoltaic (PV) performances at least comparable to the standard multilayers PN or PIN structures."
],
[
"Introduction",
"The Indium Gallium Nitride (InGaN) ternary alloy has attracted attention as a potentially ideal candidate for high efficiency solar cells.",
"Indeed, its bandgap can cover the whole solar spectrum, solely by changing its Indium composition [1], [2].",
"The InGaN alloy also counts among its advantages a high absorption coefficient [3], [4] as well as a good radiation tolerance [5], allowing its operation in extreme conditions.",
"However, one of its main drawbacks is the difficulty of its p-doping, owing mainly to the high residual donors' concentration, the lack of ad.",
"hoc.",
"acceptors [6] and the complex technological processes that are difficult to master and that highly impact the layer quality [7], [8].",
"The other drawbacks concern the difficulty to realize ohmic contacts [1], the poor InGaN material quality and the difficulty to grow InGaN with Indium content high enough to allow the optimal covering of the whole solar spectrum [9], [10].",
"For these reasons the InGaN based solar cell is still in early development stages and the reported PV efficiency is still very low to be competitive with other well established thin films technologies [11].",
"That is the reason why we present a comprehensive comparative study of $PN$ , $PIN$ and p-layer free Schottky Based Solar Cells ($SBSC$ ) structures using realistic physical models and rigorous mathematical optimization approaches and propose a new efficient p-layer free solar cell design with performances higher and tolerances wider than the previously studied Schottky structure [12].",
"The following section describes the physical modeling and simulation methodology for the InGaN solar cell structures and discusses their main physical models and material parameters.",
"Section presents the optimal results for the $PN$ and $PIN$ structures and discusses the impact of the p-layer parameters.",
"Section propose the replacement of the p-layer by a Schottky contact and discusses the performances of the resulting Schottky based solar cells, evidencing, in particular, better fabrication tolerances for the new $MIN$ structure.",
"Section presents the results obtained using actual recently published InGaN experimental composition[13], before section concludes.",
"The physical modeling used throughout this paper to carry out the device simulations and optimizations presented in the next sections has been conceived with the less possible approximations and based, whenever possible, on actual measurements.",
"It is summarized in this section.",
"The mobilities for electrons and holes, needed for the drift-diffusion model, were calculated using the Caughey-Thomas expressions [14]: $\\mu _m=\\mu _{1m}\\left(\\frac{T}{300}\\right)^{\\alpha _m}+\\frac{\\mu _{2m}\\left(\\frac{T}{300}\\right)^{\\beta _m}- \\mu _{1m}\\left(\\frac{T}{300}\\right)^{\\alpha _m}}{1+\\left(\\frac{N}{N_m^{\\mathrm {crit}}\\left(\\frac{T}{300}\\right)^{\\gamma _m}}\\right)^{\\delta _m}},$ where $m$ is either $n$ or $p$ , $\\mu _n$ being the electrons mobility and $\\mu _p$ that of holes.",
"$T$ is the absolute temperature.",
"$N$ is the doping concentration.",
"$N^{\\mathrm {crit}}$ and the $n$ or $p$ subscripted $\\alpha $ , $\\beta $ , $\\delta $ and $\\gamma $ are the model parameters which depend on the Indium composition [15].",
"In addition to the mobility model, were taken into account the bandgap narrowing effect [16], as well as the Shockley–Read–Hall (SRH) [17] and direct and Auger recombination models using the Fermi statistics [18]."
],
[
"Light absorption modeling",
"Modeling InGaN based solar cells also implies the need for a precise model of light absorption in the whole solar spectrum and for all $x$ Indium composition.",
"We used a phenomenological model for InGaN that was proposed previously [15] as $ {\\alpha }^{(\\mathrm { cm^{-1}})} = {10^5}^{(\\mathrm { cm^{-1}})} \\sqrt{C\\left(E_{ph}-E_g\\right)+D\\left(E_{ph}-E_g\\right)^2},$ where $E_{ph}$ is the incoming photon energy, $E_g$ is the material bandgap at a given Indium composition, $C$ and $D$ are empirical parameters depending on the Indium composition.",
"For the refraction index we used the Adachi model [19], defined for InGaN and for a given photon energy as $n\\left(E_{ph}\\right)=\\sqrt{\\frac{A}{\\left(\\frac{E_{ph}}{E_g}\\right)^2}\\left[2-\\sqrt{1+\\frac{E_{ph}}{E_g}}-\\sqrt{1-\\frac{E_{ph}}{E_g}}\\right]+B },$ where $A$ and $B$ are also empirical parameters depending on the Indium composition."
],
[
"Material parameters",
"The material dependent parameters have been determined for GaN and InN binaries, either from experimental work or ab initio calculations [20], [15].",
"A review of their values is given in Table REF .",
"In the following, the values for the material parameters of InGaN, for any Indium composition $x\\in \\left[0,1\\right]$ , were linearly interpolated in between the GaN and InN binaries, except for the bandgap $E_g$ and the electronic affinity $\\chi $ where we used the modified Vegard Law with a bowing factor $b=1.43 \\mathrm {eV}$ for the bandgap and $b=0.8 \\mathrm {eV}$ for the affinity [22], [15] respectively.",
"Table: Values for CC and DD in equation () as found by Brown et.",
"al.",
"in .For the recombination models, we chose a relatively low carrier lifetime value of 1ns, much lower than the value of 40ns reported for GaN [23], in order to get as realistic results as can be.",
"For the light absorption model, the values of $C$ and $D$ in equation (REF ) are taken from the experimental measurement reported in [15] and summarized in Table REF .",
"We approximated their dependency on the Indium composition ${x}$ by a polynomial fit, of the $\\mathrm {4}^{th}$ degree for the former, and quadratic for the latter: $C &=& 3.525 - 18.29{x}+ 40.22{x}^2 - 37.52{x}^3 + 12.77{x}^4,\\\\D &=& -0.6651 + 3.616{x}- 2.460{x}^2.$ The $A$ and $B$ parameters in the refraction index equation (REF ) are experimentally measured [20], [15] for GaN ($A^{\\mathrm {GaN}}=9.31$ and $B^{\\mathrm {GaN}}=3.03$ ) and InN ($A^{\\mathrm {InN}}=13.55$ and $B^{\\mathrm {InN}}=2.05$ ) and linearly interpolated for InGaN.",
"Finally, we have chosen to shine on the cell the ASTM-G75-03 solar spectrum taken from the National Renewable Energy Laboratory databasehttp://rredc.nrel.gov/solar/spectra/am1.5/astmg173/astmg173.html."
],
[
"Simulation Methodology",
"The devices are simulated in the framework of a drift-diffusion model using the Atlas$^{\\text{®}}$ device simulation software from the Silvaco$^{\\text{®}}$ suite, in which we implemented our physical models.",
"Atlas$^{\\text{®}}$ solves, in two dimensions, the drift-diffusion nonlinear partial differential problem using the Newton coupled and the Gummel decoupled methods [24].",
"The solar cell analyzed characteristics were the spectral response, the I-V characteristics, the inner electric field and potential distributions as well as the recombination rate variations.",
"We used mathematically rigorous multivariate optimization methods to find the optimum efficiency with respect to a given set of parameters (as later shown in tables REF and REF ).",
"This methodology is far more rigorous than the usual single parametric analysis, where one parameter is varying while the other parameters are kept constant.",
"It yields for instance the absolute optimum efficiency as a function of the physical parameters.",
"We have used three mathematical optimization methods that give very similar results within a comparable amount of computing time (typically a few hours per simulation with a highly optimized code): the truncated Newton algorithm (TNC) [25], the Sequential Least SQuares Programming (SLSQP) method [26] and the L-BFGS-B quasi-Newton method [25].",
"The optimization work has been done with a Python package we developed in the SAGE [27] interface to the SciPy [28], [29] optimizers, using the Atlas$^{\\text{®}}$ simulator as the backend engine."
],
[
"Optimal Results",
"The $PN$ and $PIN$ solar cells are schematically shown in Figure REF .",
"These devices could be realized in practice using the developed growth techniques of InGaN on GaN/sapphire substrates and the device realization techniques [30].",
"The first device, shown on figure REF , is an InGaN $PN$ structure.",
"Its optimization, and eventually its practical realization with a competitive efficiency, is the sine qua none condition to actually manufacturing the high efficiency multijunction next-generation solar cells [31], [32], [33].",
"The physical parameters for which the optimum was sought are shown on table REF : the relevant five parameters for the $PN$ structure are the thickness and dopings of the two layers, along with their common Indium composition.",
"The second design is based on a $PIN$ structure where the \"intrinsic\" layer consists in an n-doped layer with a relatively low doping concentration.",
"The standard intrinsic layer (i-layer) has been replaced by a slightly doped n-layer for two reasons: on the one hand, the elaborated InGaN usually exhibits residual n-doping [34], [35] and, on the other hand as we will demonstrate later in this section, the optimal efficiency for a $PIN$ solar cell is obtained for an intermediate n-doped layer and not for the quasi-intrinsic layer.",
"The resulting structure is shown on figure REF .",
"The seven optimization parameters are shown in table REF , with the thickness and dopings of the three layers along with their common Indium composition.",
"The minimum value of the quasi-intrinsic layer doping has been set lower than the usually reported residual doping value in InGaN [34].",
"To optimize these devices, we used the mathematical optimization methods presented in section REF .",
"These methods are constrained and therefore need a parameter range, and, as for the non-constrained methods, a starting point.",
"We defined the parameter range to ensure the physical meaning and the technological feasibility of each parameter.",
"The range chosen for each parameter is shown on the second line of table REF .",
"We then chose to run the optimization with several randomly chosen starting points to get an insight into the precision of our computation and ensure that the found optimum is absolute.",
"For the $PN$ structure, we found a maximum cell efficiency of $17.8\\%$ and optimal values for the physical parameters.",
"The optimal thickness of the $P$ layer is found to be $L_p=0.01 \\mathrm {\\mu m}$ .",
"That of the $N$ layer thickness is $L_n=1.00 \\mathrm {\\mu m}$ .",
"The optimal doping of the $P$ layer is $N_a=1.0\\times 10^{19} \\mathrm {cm^{-3}}$ .",
"That of the $N$ layer is $N_d=3.9\\times 10^{16} \\mathrm {cm^{-3}}$ .",
"The optimal Indium composition is ${x}=0.56$ .",
"The corresponding open-circuit voltage ($V_{OC}$ ) is $0.855 \\mathrm {V}$ with a short-circuit current ($J_{SC}$ ) of $26.75 \\mathrm {mA/cm^{2}}$ and a fill factor ($FF$ ) of $77.85\\%$ .",
"For the $PIN$ structure, we found a maximum cell efficiency of $19.0\\%$ for the optimal values of the following parameters: a $P$ layer thickness of $L_p=0.01 \\mathrm {\\mu m}$ , an $I$ layer thickness of $L_i=0.54 \\mathrm {\\mu m}$ , a $N$ layer thickness of $L_n=0.50 \\mathrm {\\mu m}$ , a $P$ layer doping of $N_a=1.0\\times 10^{19} \\mathrm {cm^{-3}}$ , an $I$ layer doping of $N_i=5.8\\times 10^{16} \\mathrm {cm^{-3}}$ , a $N$ layer doping of $N_d=5.0\\times 10^{17} \\mathrm {cm^{-3}}$ and an Indium composition of ${x}=0.59$ .",
"The corresponding PV parameters are $V_{OC}=0.875 \\mathrm {V}$ , $J_{SC}=\\mathrm {27.36 mA/cm^{2}}$ and $FF=79.39\\%$ .",
"All these parameters with their tolerance range, as defined below, are reported in table REF .",
"In practice, it is indeed necessary for an optimal parameter to have a wide tolerance range in which it can vary without lowering the cell efficiency too much.",
"We have performed the tolerance analysis on each parameter, while keeping all the others at their optimal value.",
"We have thus defined a tolerance range, which is the range of values of a given parameter for which the efficiency $\\eta $ remains above $90\\%$ of its maximum value.",
"The tolerance range is shown on table REF , just below the optimal value.",
"For instance, for the $PN$ structure, the efficiency value remains between $16.0\\%$ and $17.8\\%$ for a p-layer doping $N_a$ varying between $4.4\\times 10^{16} \\mathrm {cm^{-3}}$ and $1.0\\times 10^{19} \\mathrm {cm^{-3}}$ , the other parameters remaining at their optimal values.",
"Table REF shows that, on the one hand, the $PIN$ solar cell has an efficiency slightly higher than that of the $PN$ solar cell and, on the other hand, the tolerance ranges for layers thicknesses in the $PIN$ structure are wider than in the $PN$ structure.",
"This latter property is a considerable advantage of the $PIN$ structure in the practical cell realization.",
"For instance, the $PIN$ structure has a tolerance range of $[0.10 - 1.00] \\mathrm {\\mu m}$ for the n-layer thickness, almost twice wider than that of the $PN$ structure.",
"The wider tolerance range for the n-doping in the $PIN$ structure also allows increasing the n-layer doping value without noticeably impacting the efficiency, for designing low resistance ohmic contacts [36]."
],
[
"Impact of the p-layer parameters",
"Figures REF and REF show the efficiency as a function of the p-layer doping for various p-layer thicknesses (p-thicknesses) for $PN$ and $PIN$ structures respectively.",
"These results show that the efficiency is optimal for a given p-doping which increases when the p-thickness decreases.",
"Figures REF and REF display the optimal p-doping variation with the p-thicknesses for the $PN$ and $PIN$ structures respectively.",
"For the $PN$ structure, the corresponding efficiency varies from $17.8\\%$ down to $15.6\\%$ with the thickness of the p-layer varying from $0.01\\mu m $ to $0.10\\mu m $ .",
"The corresponding efficiency for the $PIN$ structure, varies from $19.8\\%$ down to $16.5\\%$ with the thickness of the p-layer varying from $0.01\\mu m $ to $0.10\\mu m $ .",
"Figures REF and REF display the I-V characteristics for some p-thicknesses values, for the $PN$ and $PIN$ structures respectively.",
"The I-V curves for the $PN$ structure were obtained for the thicknesses of the p-layer of $0.01\\mathrm {\\mu m}$ , $0.04\\mathrm {\\mu m}$ and $0.10\\mathrm {\\mu m}$ corresponding to the optimal p-doping values of $1.0\\times 10^{19}\\mathrm {cm^{-3}}$ , $1.3\\times 10^{17}\\mathrm {cm^{-3}}$ and $3.0\\times 10^{16}\\mathrm {cm^{-3}}$ respectively.",
"For the $PIN$ structure, the I-V curves were obtained for the same thicknesses corresponding to the optimal p-doping values of $1.0\\times 10^{19}\\mathrm {cm^{-3}}$ , $1.8\\times 10^{17}\\mathrm {cm^{-3}}$ and $2.9\\times 10^{16}\\mathrm {cm^{-3}}$ respectively.",
"For both the $PN$ and the $PIN$ structures, the short-circuit current $J_{SC}$ remains almost constant while the open-circuit voltage $V_{OC}$ increases when decreasing the p-thickness and increasing the p-doping along the optimal curve of figures REF and REF .",
"For a given thickness, say $0.10 \\mu m$ , the maximum electric field value obviously increases with the p-doping while the space charge region (SCR) width decreases, as well as the recombination rate.",
"This is mainly due to the SRH recombination mechanism.",
"These two variations lead to an increase of the $V_{OC}$ and, in the same time, to a decrease of $J_{SC}$ .",
"These two opposing trends of $J_{SC}$ and $V_{OC}$ lead to maximum efficiency points depending on the p-doping and thickness of the p-layer, as shown in Figures REF and REF .",
"Figures REF and REF summarize this variation showing that even if the absolute optimal doping is high, the efficiency remains relatively high for wide doping and thickness ranges.",
"For instance, for the $PIN$ structure, a p-thickness of $0.04 \\mu m$ and a p-doping of $1.8\\times 10^{17} \\mathrm {cm^{-3}}$ lead to an efficiency of $17.5\\%$ .",
"It remains relatively close to the optimal one obtained for a thickness of $0.01 \\mu m$ and a doping of $1.0\\times 10^{19} \\mathrm {cm^{-3}}$ .",
"This point concerning the tolerance range, as previously underlined for $PN$ and $PIN$ solar cells, is of great importance for the practical solar cell realization and it will be discussed for $SBSC$ in the following section.",
"Figures REF and REF show that the efficiency variation with the p-layer thickness and doping is mainly due to variations in $V_{OC}$ .",
"Indeed, the short-circuit current $J_{SC}$ remains almost constant owing to increasing p-doping associated to decreasing thickness, whereas $V_{OC}$ increases owing to increasing p-doping only.",
"All these results show that the optimal performances of both the $PN$ and $PIN$ structures are obtained for a relatively thin p-layer (10 $nm$ ) with a relatively high p-doping value of $1.0\\times 10^{19}\\mathrm {cm^{-3}}$ .",
"Considering on the one hand that the optimal thickness is much lower than the mean penetration depth and diffusion length in InGaN and on the other hand that the optimal doping is relatively high, we propose an alternative that allows the removal of the p-layer.",
"These alternatives are Schottky Based Solar Cells (SBSC), which correspond on the one hand to a Schottky junction and, on the other hand to a new structure."
],
[
"Schottky Based Solar Cells",
"As demonstrated in the previous section, the optimal $PN$ and $PIN$ solar cell efficiencies were obtained for p-layer thicknesses much lower than the light penetration depth and for a relatively high p-doping.",
"The $Schottky$ solar cell, obtained by replacing the p-layer in the $PN$ structure by a relatively high workfunction metal, was previously demonstrated as a reliable alternative to the InGaN $PN$ solar cell [12].",
"Similarly, replacing the p-layer in the $PIN$ solar cell by a rectifying metal/InGaN contact leads to the new $MIN$ (Metal-IN) structure.",
"Figure REF schematically displays these $Schottky$ and $MIN$ structures.",
"The $Schottky$ and $MIN$ solar cells were optimized with respect to their most important parameters: $L_i$ and $L_n$ , the thicknesses of the $I$ and $N$ layers respectively and where applicable, $N_i$ and $N_d$ , the doping levels of the $I$ and $N$ layers respectively and where applicable, the Indium composition ${x}$ and the metal workfunction $W_f$ .",
"The optimization was conducted in the same way as in the previous section.",
"As was also done in the previous section, the optimum efficiency is reported in table REF , along with the associated photovoltaic parameters as well as the corresponding parameters and their tolerance range, with the same definition as in the previous section.",
"For the Schottky structure, we found a maximum cell efficiency of $18.2\\%$ for the following optimal parameter values: $L_n=0.86 \\mathrm {\\mu m}$ , $N_d=6.5\\times 10^{16} \\mathrm {cm^{-3}}$ , ${x}=0.56$ and $W_f= 6.30 \\mathrm {eV}$ .",
"The corresponding open-circuit voltage is $V_{OC}=0.863 \\mathrm {V}$ with a short-circuit current of $J_{SC}= \\mathrm {26.80 mA/cm^{2}}$ and a fill factor of $FF=78.82\\%$ .",
"For the $MIN$ structure, the maximum cell efficiency is $19.8\\%$ for the following parameters values: $L_i=0.61 \\mathrm {\\mu m}$ , $L_n=0.83 \\mathrm {\\mu m}$ , $N_i=6.1\\times 10^{16} \\mathrm {cm^{-3}}$ , $N_d=3.6\\times 10^{17} \\mathrm {cm^{-3}}$ , ${x}=0.60$ and $W_f=6.30 \\mathrm {eV}$ .",
"The corresponding open-circuit voltage is $V_{OC}=0.835 \\mathrm {V}$ with a short-circuit current of $J_{SC}=\\mathrm {30.29 mA/cm^{2}}$ and a fill factor of $FF=78.39\\%$ .",
"Figure REF shows the current-voltage characteristics of the $Schottky$ and $MIN$ solar cells.",
"We observe that the $Schottky$ structure has a lower $J_{SC}$ and a higher $V_{OC}$ compared to the $MIN$ structure.",
"This is due to the different optimal Indium composition: the $Schottky$ structure has an optimal Indium composition of $56\\%$ , that is lower than the optimal value for the $MIN$ structure ($60\\%$ ).",
"$V_{OC}$ increases as the Indium concentration decreases, owing to the widening of the bandgap.",
"Simultaneously, $J_{SC}$ decreases as the direct consequence of a lower solar light absorption.",
"Figure REF shows the variation of the PV efficiency as a function of the i-layer thickness (i-thickness), whereas figure REF shows it as a function of i-doping, for different i-thicknesses.",
"The optimal i-thickness value, as shown in figure REF , is about $0.60 \\mathrm {\\mu m}$ as a consequence of the trade-off between the solar light absorption, increasing with the thickness, and the diffusion length that need to remain relatively higher than the layer thickness.",
"The same figure REF shows that the optimal i-doping value is $6.1\\times 10^{16}cm^{-3}$ , corresponding to the optimal Space Charge Region (SCR) in the device.",
"In addition to its main advantage of being p-layer free, the $MIN$ structure has another decisive advantage over the $PN$ and even the Schottky structures: the wider tolerance ranges of its optimal parameters, as Table REF shows.",
"This is due to the additional degree of freedom obtained with the i-layer.",
"Indeed, for the Schottky structure, the tolerance range of the n-layer thickness is $[0.53 - 1.00] \\mu m$ , while for the $MIN$ structure, it is $[0.10 - 1.00] \\mu m$ .",
"This gives the $MIN$ structure a wider n-layer manufacturing tolerance than the Schottky structure.",
"This tolerance range is important when actual device fabrication is considered.",
"For the n-doping, the $Schottky$ structure has a tolerance range of $[1.0\\times 10^{16}-3.0\\times 10^{17}] cm^{-3}$ , while, for the $MIN$ structure, the tolerance range is $[1.8\\times 10^{16}-1.0\\times 10^{19}] cm^{-3}$ .",
"This allows to design heavily dopped n-layer to elaborate low resistance ohmic contact on InGaN, one of the major challenges in the III-Nitride solar cell processing [1], and without noticeably impacting the PV performances."
],
[
"MIN structure with actual experimental InGaN composition, thickness and metal workfunction",
"The above presented optimisation work lead to an optimal InGaN composition of ${x}=0.60$ which is not yet experimentally achieved with sufficient material quality, although some very recent papers suggest that these compositions are in the process of being accessible [13], [37], [38], [39].",
"In this section, we propose to use one actual recent Indium composition obtained by Fabien et al.",
"[13], that is ${x}=0.22$ for large-area solar cells, and to evaluate the maximum efficiency that it allows.",
"Furthermore, a thickness constraint is linked to a composition constraint.",
"We therefore limited the reachable thickness to $0.4\\mathrm {\\mu m}$ .",
"Table: The dominating deep-level defect parameters in InGaN as experimentally measured and reported in , for the x=0.09x = 0.09 Indium composition, in for x=0.13x = 0.13 and in for x=0.20x = 0.20.",
"The defect energy is measured relatively to the conduction band edge.Even though, the actually grown layers can have a high density of defects[9].",
"To take it into account, we introduced, on the one hand, valence and conduction band Urbach tails in the simulation, with an energy of $0.125 \\mathrm {eV}$ as experimentally obtained in [44], and, on the other hand, a Gaussian distribution of defects in the bandgap.",
"We used defects that were experimentally studied in the literature using the well known Deep Level (Transient & Optical) Spectroscopy (DLTS and DLOS), the Steady-State PhotoCapacitance (SSPC) and the Lighted Capacitance-Voltage (LCV) techniques [40], [41], [42], [43] as summarized in table REF .",
"The capture cross section that we chose to include in the simulation is the highest experimental value reported in [43].",
"Table: Optimum efficiency η\\eta obtained for the MINMIN solar cell with a recently published actual experimental x=0.22x=0.22 Indium composition and layer thicknesses limited to 0.4μm0.4\\mathrm {\\mu m}.",
"The associated open-circuit voltage V OC V_{OC}, short-circuit current J SC J_{SC} and Fill Factor FFFF, along with the corresponding physical and material parameters, are equally shown.",
"For each parameter, a range and a tolerance range are given.",
"The range, within which the optimum value of a given parameter is sought, is on the second line of the table.",
"The tolerance range is given just below each parameter optimal value.",
"It corresponds to the set of values of that parameter for which the efficiency η\\eta remains above 90%90\\% of its maximum, the other parameters being kept at their optimum values.The optimization process was then run within these constraints and yield the optimal parameters summarized in table REF .",
"As could be expected the layer thicknesses as well as the Indium concentration were found at their maximum authorized value, $0.4\\mathrm {\\mu m}$ and $x=0.22$ respectively, yielding a 7.25% maximum efficiency.",
"However, the computed tolerances deserve attention, since they are higher than one fourth, or even one half, of the optimal values, as far as the thicknesses are concerned.",
"As this was carried out without defects included, we then evaluated the MIN cell efficiency while varying the total density of states from $1.0\\times 10^{13} \\mathrm {cm^{-3}}$ to $1.0\\times 10^{17} \\mathrm {cm^{-3}}$ .",
"This latter density is even higher than the dominating defects concentration reported in [40], [41], [42], [43].",
"Figure: The photovoltaic efficiency of the InGaN MIN solar cell with the actual experimental Indium composition, with defect density for two Gaussian distributions.",
"The cell parameters are fixed to their optimal values shown in table .Figure REF shows the MIN solar cell photovoltaic efficiency, with the actual experimental Indium composition, with respect to the defect concentration for two decay energy $\\delta $ values of $0.05 \\mathrm {eV}$ and $0.10 \\mathrm {eV}$ .",
"The efficiency remains close to its maximum value as long as the defect concentration is smaller than the i-layer doping concentration ($6.1\\times 10^{16}\\mathrm {cm}^{-3}$ ).",
"When the defect concentration becomes comparable to the optimal i-layer doping concentration, the solar cell efficiency decreases within a concentration range that depends on the distribution decay energy.",
"This result means that the defects concentration must be kept lower but not necessarily much lower than the doping concentration.",
"The demonstrated wide tolerance of the MIN structure can allow keeping as low as possible the negative impact of the defects on the overall solar cell efficiency, by adjusting accordingly the InGaN doping.",
"A compromise can therefore be found to limit the effect of the defects density that is relatively high in the presently elaborated InGaN layers.",
"Table: Optimum efficiency η\\eta obtained for a MINMIN solar cell with various usual metal work functions W f W_f, lower than the optimal 6.30 eV 6.30\\mathrm {eV} yield by the optimization process; and for the optimal x=0.60{x}=0.60 Indium composition alongside the x=0.22{x}=0.22 composition published in .Finally, one may spot that the optimal $6.30\\mathrm {eV}$ work function obtained in table tabexpoptimum seems to be relatively high when compared to the most reported values in the literature for Platinum (Pt), which is the ideal candidate for the practical realization of the MIN solar cell.",
"However, a closer look at the reported values reveals a large dispersion in the Platinum work function measurements, from 5.65eV in [45] to 6.35eV in the historical works by Lee Alvin Dubridge from Caltech (see e.g.",
"[46]), through 5.93eV [47] and 6.10eV [48].",
"To take these discrepancies into account, as well the possibility to use lower work function metals for the practical realization of the MIN solar cell, we have evaluated the foreseen efficiency for a set of possible work functions for both the optimal $x=0.60$ Indium composition and the $x=0.22$ composition reported in [13].",
"The results are summarized in table REF ."
],
[
"Conclusion",
"We investigated the photovoltaic performances of InGaN based $PN$ , $PIN$ and $SBSC$ structures, using rigorous multivariate numerical optimization methods to simultaneously optimize the main physical and geometrical parameters of the solar cell structures.",
"We have found optimal photovoltaic efficiencies of $17.8\\%$ and $19.0\\%$ for the $PN$ and $PIN$ structures respectively.",
"The optimization results led us to propose a new p-layer free $SBSC$ structure called $MIN$ , the optimal efficiency of which is a higher $19.8\\%$ for an Indium composition of a yet-to-reach ${x}=0.60$ , and as high as $7.25\\%$ for a recent experimental Indium composition of ${x}=0.22$ for a $0.4\\mathrm {\\mu m}$ thin layer that is not free of cristalline defects, the density of which we took into account.",
"In addition, the $MIN$ structures has been shown to allow wider tolerance ranges on its physical and geometrical parameters, which allows to enhance its practical feasibility and reliability.",
"The wider tolerance ranges of the new $MIN$ structure allow, for example, when compared to the previously studied Schottky structure, the easier realization of low resistance ohmic contacts, solely by raising the n-doping, as it was shown not to impair the efficiency."
]
] | 1605.05936 |
[
[
"Solving S-unit, Mordell, Thue, Thue-Mahler and generalized\n Ramanujan-Nagell equations via Shimura-Taniyama conjecture"
],
[
"Abstract In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations.",
"As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes.",
"To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods.",
"In addition we used the resulting data to motivate various conjectures and questions, including Baker's explicit abc-conjecture and a new conjecture on S-integral points of any hyperbolic genus one curve over Q.",
"In the second part we establish new results for certain old Diophantine problems (e.g.",
"the difference of squares and cubes) related to Mordell equations, and we prove explicit height bounds for cubic Thue, cubic Thue--Mahler and generalized Ramanujan--Nagell equations.",
"As a byproduct, we obtain here an alternative proof of classical theorems of Baker, Coates and Vinogradov-Sprindzuk.",
"In fact we get refined versions of their theorems, which improve the actual best results in many fundamental cases.",
"Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Parshin, Szpiro) combined with the Shimura-Taniyama conjecture, and they all do not use lower bounds for linear forms in (elliptic) logarithms.",
"In the third part we solve the problem of constructing an efficient sieve for the S-integral points of bounded height on any elliptic curve E over Q with given Mordell-Weil basis of E(Q)."
],
[
"Introduction",
"In this paper we combine the method of Faltings [60] (Arakelov, Paršin, Szpiro) with the Shimura–Taniyama conjecture [164], [158], [12] in order to study various classical Diophantine problems, including $S$ -unit equations, Mordell equations, cubic Thue equations, cubic Thue–Mahler equations and generalized Ramanujan–Nagell equations.",
"We now begin to discuss the Diophantine equations.",
"Let $S$ be a finite set of rational prime numbers.",
"Write $N_S=1$ if $S$ is empty and $N_S=\\prod _{p\\in S} p$ otherwise.",
"We denote by $\\mathcal {O}^\\times $ the units of $\\mathcal {O} ={\\mathbb {Z}}[1/N_S]$ and we consider the $S$ -unit equation $x+y=1, \\ \\ \\ (x,y)\\in \\mathcal {O}^\\times \\times \\mathcal {O}^\\times .$ Many important Diophantine conjectures can be reduced to the study of $S$ -unit equations.",
"For example, the $abc$ -conjecture of Masser–Oesterlé is equivalent to a certain height bound for the solutions of (REF ).",
"On using Diophantine approximations in the style of Thue and Siegel, Mahler [108] showed that (REF ) has only finitely many solutions.",
"Furthermore there already exists a practical method of de Weger [162] which solves $S$ -unit equations by using the theory of logarithmic forms [33], see Section REF .",
"For a detailed discussion of (general) $S$ -unit equations we refer to the recent book of Evertse–Győry [52].",
"Next we take a nonzero $a\\in \\mathcal {O}$ and we consider the Mordell equation $y^2=x^3+a, \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}.$ This Diophantine equation is a priori more difficult than (REF ).",
"Further, if $\\mathcal {O}={\\mathbb {Z}}$ then resolving (REF ) is equivalent to solving the classical problem, going back at least to Bachet (1621), of finding all perfect squares and perfect cubes with given difference.",
"In the case $\\mathcal {O}={\\mathbb {Z}}$ , Mordell [118], [119] showed finiteness of (REF ) via Diophantine approximation, and Baker–Davenport [4], [13] and Masser, Zagier [112], [167] introduced practical approaches solving (REF ) via the theory of logarithmic forms (see Section ).",
"Furthermore we shall see that a special class of Mordell equations (REF ) covers in particular any generalized Ramanujan–Nagell equation discussed in (REF ) below.",
"Finally, we let $m\\in \\mathcal {O}$ be nonzero and we suppose that $f\\in \\mathcal {O}[x,y]$ is a homogeneous polynomial of degree three with nonzero discriminant.",
"Consider the cubic Thue equation $f(x,y)=m, \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}.$ Thue (1909) proved that (REF ) has only finitely many solutions in the case $\\mathcal {O}={\\mathbb {Z}}$ .",
"In general, equation (REF ) is essentially equivalent to the cubic Thue–Mahler equation recalled in (REF ) below.",
"Baker–Davenport [4], [13] and Tzanakis–de Weger [152], [154] obtained practical approaches solving in particular cubic Thue and Thue–Mahler equations via the theory of logarithmic forms [33], see also the discussions in Section ."
],
[
"Algorithms",
"We construct two types of algorithms which we use to solve $S$ -unit equations (REF ), Mordell equations (REF ), cubic Thue equations (REF ), cubic Thue–Mahler equations (REF ) and generalized Ramanujan–Nagell equations (REF ).",
"Both types do not use the theory of logarithmic forms.",
"Before we discuss our algorithms in more detail, we describe the general strategy."
],
[
"General strategy",
"As in [89] we use the method of Faltings (Arakelov, Paršin, Szpiro) which in our situation is applied as follows: Let $Y(\\mathcal {O})$ be the set of solutions of any of the above equations.",
"Then there is an effective map $\\phi $ (Paršin construction) from $Y(\\mathcal {O})$ to the set $M(T)$ of isomorphism classes of elliptic curves over a controlled open $T\\subset \\textnormal {Spec}({\\mathbb {Z}})$ , $\\phi : Y(\\mathcal {O})\\rightarrow M(T).$ Here effective means that one can compute $\\phi ^{-1}(E)$ for each $E$ in $M(T)$ .",
"To determine $Y(\\mathcal {O})$ , it thus suffices to compute $M(T)$ (effective Shafarevich theorem).",
"For this purpose we use two types of algorithms: The first type applies Cremona's algorithm [39] involving modular symbols, and the second type combines our optimized height bounds (see Section REF ) with efficient sieves.",
"Both types of algorithms crucially rely on a geometric version of the Shimura–Taniyama conjecture [12] using inter alia the Tate conjecture [60], and on isogeny estimates based on the method of Mazur [114], [93] or Faltings [60], [127].",
"In fact the strategy of combining modularity with Faltings' method gives effective finiteness results for considerably more general Diophantine problems, see [89], [92].",
"However in the present paper we focus on optimizing the strategy for the fundamental Diophantine equations appearing in the title, and in the future we plan to work out algorithms for other Diophantine problems of interest."
],
[
"Algorithms via modular symbols",
"We next discuss in more detail our first type of algorithms.",
"They crucially rely on Cremona's algorithm [39] using modular symbols in order to compute all elliptic curves over ${\\mathbb {Q}}$ of given conductor.",
"This allows to determine $M(T)$ , since the curves in $M(T)$ have bounded conductor.",
"Then we compute $Y(\\mathcal {O})$ by enumerating $\\phi ^{-1}(E)$ for each $E$ in $\\phi (Y(\\mathcal {O}))$ .",
"Here we exploit that the maps $\\phi $ are effective by classical constructions going back at least to Cayley, Mordell and Frey–Hellegouarch.",
"To illustrate the utility of our first type of algorithms we computed several examples.",
"For instance we solved the $S$ -unit equation (REF ) for all sets $S$ with $N_S\\le 20000$ , and we solved several Mordell equations (REF ).",
"In fact, as already pointed out in [89], our first type of algorithms can in principle solve any Diophantine equation inducing integral points on a moduli scheme of elliptic curves with an effective Paršin construction $\\phi $ , see (REF ) and Section REF .",
"This class of equations contains in particular all equations considered in this paper.",
"Here we mention that the possibility of solving certain cubic Thue–Mahler equations via modular symbols was already discussed in Bennett–Dahmen [14], see also the recent works of Kim [95] and Bennett–Billerey [8].",
"Our first type of algorithms is very fast for “small\" parameters, since in this case we can use Cremona's database listing all elliptic curves over ${\\mathbb {Q}}$ of conductor at most 350000 (as of August 2014).",
"In particular these algorithms directly benefit from the ongoing extension of such databases.",
"However the approach via modular symbols can usually not compete with the actual most efficient methods solving our equations of interest.",
"Thus we worked out a second type of algorithms."
],
[
"Algorithms via height bounds",
"We now give a more detailed description of our second type of algorithms.",
"They rely on an effective Shafarevich theorem in the form of explicit bounds for the Faltings height $h_F$ on $M(T)$ .",
"The space $M(T)$ can be very complicated and it is usually a difficult task to compute $M(T)$ , see the discussions surrounding (REF ).",
"Hence instead of first computing $M(T)$ and then $Y(\\mathcal {O})=\\phi ^{-1}(M(T))$ , we often directly work with $Y(\\mathcal {O})$ by using that the height $\\phi ^*h_F$ is bounded on $Y(\\mathcal {O})$ .",
"This has the advantage that we can exploit extra structures on $Y(\\mathcal {O})$ in order to construct efficient sieves for solutions of bounded height."
],
[
"$S$ -unit equation.",
"To solve $S$ -unit equations (REF ) it is natural to consider the set $\\Sigma (S)$ of solutions of (REF ) modulo symmetry.",
"Here solutions $(x,y)$ and $(x^{\\prime },y^{\\prime })$ of (REF ) are called symmetric if $x^{\\prime }$ or $y^{\\prime }$ lies in $\\lbrace x,\\frac{1}{x},\\frac{1}{1-x}\\rbrace $ .",
"One can directly write down all solutions which are symmetric to a given solution and thus it suffices to determine $\\Sigma (S)$ in order to solve (REF ).",
"In fact the number of solutions of (REF ) is either zero or $6\\vert {}\\Sigma (S){}\\vert -3$ where $\\vert {}\\Sigma {}\\vert $ denotes the cardinality of a set $\\Sigma $ .",
"For any $n\\in {\\mathbb {Z}}_{\\ge 1}$ we denote by $S(n)$ the set of the $n$ smallest rational primes.",
"Our Algorithm REF allows to efficiently solve (REF ), even for sets $S$ with relatively large $\\vert {}S{}\\vert $ .",
"To demonstrate this we solved large classes of $S$ -unit equations (REF ) by using Algorithm REF .",
"In particular, we obtained the following result.",
"Theorem A.",
"Suppose that $n\\in \\lbrace 1,2,\\ldots ,16\\rbrace $ .",
"Then the cardinality $\\#$ of the set $\\Sigma (S(n))$ is given in the following table.",
"Table: NO_CAPTIONWe mention that among all sets $S$ of cardinality $n$ the set $S(n)$ is usually the most difficult case for solving (REF ).",
"In [168], Zagier explained that the cardinality of $\\Sigma (S)$ plays an important role in certain questions on polylogarithms.",
"In particular he states a table attributed to Gross–Vojta, which for each $n\\in \\lbrace 1,\\cdots ,8\\rbrace $ lists a lower bound for the cardinality of $\\Sigma (S(n))$ .",
"Theorem A proves that the entries of this table are not only lower bounds, but in fact the correct values.",
"Further we point out that the cases $n\\in \\lbrace 1,\\cdots ,6\\rbrace $ in Theorem A are not new.",
"They were previously known by the work of de Weger [161].",
"Our Algorithm REF substantially improves de Weger's method in [161] in the following sense: Instead of using inequalities based on the theory of logarithmic forms as done by de Weger, we apply our optimized height bounds (see Section REF ).",
"These optimized bounds are strong enough such that we can omit de Weger's reduction process.",
"Then to enumerate all solutions of (REF ) of bounded height, we use de Weger's sieve which is efficient as long as $\\vert {}S{}\\vert $ is small (e.g.",
"$\\vert {}S{}\\vert <6$ ).",
"To deal efficiently with sets $S$ of larger cardinality, we were forced to introduce new ideas: In Section REF we take into account certain geometric considerations to construct a refined sieve, and in Section REF we develop a refined enumeration algorithm for solutions of (REF ) with very small height.",
"Our new ideas are crucial to efficiently solve (REF ) for sets $S$ with $\\vert {}S{}\\vert \\ge 6$ .",
"Furthermore we prove that our refinements are substantial in the sense that they considerably improve the running time in theory and in practice, see (REF ) and Section REF .",
"In general we conducted some effort to optimize Algorithm REF .",
"We refer to Section REF where we explain and motivate our optimizations.",
"Also we developed a method which (automatically) chooses parameters that are close to optimal in the generic case.",
"This was necessary to obtain our database $\\mathcal {D}_1$ listing the solutions of the $S$ -unit equation (REF ) for many distinct sets $S$ , including all sets $S$ with $N_S\\le 10^7$ and all sets $S\\subseteq S(16)$ .",
"Theorem B.",
"For each finite set of rational primes $S$ considered in $\\mathcal {D}_1$ , the database $\\mathcal {D}_1$ contains all solutions of the $S$ -unit equation (REF ).",
"Another useful feature of Algorithm REF is that it allows to prove properties of $abc$ -triples with bounded radical.",
"For example, on using our algorithm we verified Baker's explicit $abc$ -conjecture [7] for all $abc$ -triples with radical at most $10^7$ or with radical composed of primes in $S(16)$ .",
"Furthermore we used our database $\\mathcal {D}_1$ to motivate several new questions.",
"In particular, in view of the construction of the refined sieve, we make the following conjecture describing a property of integral points of $\\mathbb {P}_{\\mathbb {Z}}^1-\\lbrace 0,1,\\infty \\rbrace $ which is rather unexpected from a general Diophantine geometry perspective.",
"Conjecture 1.",
"There exists $c\\in {\\mathbb {Z}}$ with the following property: If $n\\in {\\mathbb {Z}}_{\\ge 1}$ then any finite set of rational primes $S$ with $\\vert {}S{}\\vert \\le n$ satisfies $\\vert {}\\Sigma (S){}\\vert \\le \\vert {}\\Sigma (S(n)){}\\vert +c$ .",
"Theorem B shows that Conjecture 1 holds with $c=0$ for all sets $S$ in $\\mathcal {D}_1$ and this motivates to ask whether any set of rational primes $S$ with $\\vert {}S{}\\vert \\le n$ satisfies $\\vert {}\\Sigma (S){}\\vert \\le \\vert {}\\Sigma (S(n)){}\\vert $ ?",
"We remark that additional applications of Algorithm REF are given in Section REF .",
"To conclude the discussion we point out that the geometric main idea (described in Section REF ) underlying our refined sieve is applicable in many other situations where sieves of de Weger type are applied.",
"Here one can mention for example the practical resolution of $S$ -unit and Thue–Mahler equations over number fields.",
"We leave this for the future."
],
[
"Mordell equation.",
"To solve the Mordell equation (REF ) via height bounds, we constructed Algorithm REF .",
"Generically, this algorithm allows to deal efficiently with huge parameters.",
"To illustrate this feature we used Algorithm REF to create our database $\\mathcal {D}_2$ listing the solutions of (REF ) for large classes of pairs $(a,S)$ with $a\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ , including the classes $\\lbrace \\vert {}a{}\\vert \\le 10,S\\subseteq S(10^5)\\rbrace $ , $\\lbrace \\vert {}a{}\\vert \\le 100,S\\subseteq S(10^3)\\rbrace $ and $\\lbrace \\vert {}a{}\\vert \\le 10^4,S\\subseteq S(300)\\rbrace $ .",
"Theorem C. For each pair $(a,S)$ considered in $\\mathcal {D}_2$ , the database $\\mathcal {D}_2$ contains all solutions of the Mordell equation (REF ) defined by $(a,S)$ .",
"Here we point out that Gebel–Pethő–Zimmer [71] already established the important special case $\\lbrace \\vert {}a{}\\vert \\le 10^4, S=\\emptyset \\rbrace $ by using their algorithm [68] based on the elliptic logarithm approach introduced by Masser and Zagier.",
"Algorithm REF substantially improves the latter approach for (REF ) in the following sense: Instead of using inequalities based on the theory of logarithmic forms, we apply our optimized height bounds (see Section REF ).",
"Our bounds are considerably stronger in practice, which leads to significant running time improvements as illustrated in Section REF .",
"Then to enumerate all solutions of (REF ) with bounded height, we use the elliptic logarithm sieve constructed in Section .",
"Here our construction combines a geometric interpretation of the known elliptic logarithm reduction with conceptually new ideas described in Section REF .",
"The elliptic logarithm sieve is very efficient and it considerably improves in all aspects (see Section REF ) the known methods enumerating solutions of (REF ).",
"However, our sieve requires an explicit Mordell–Weil basis of the group $E_a({\\mathbb {Q}})$ associated to the elliptic curve $E_a$ defined by (REF ).",
"While it is usually possible to determine such a basis in practice, there is so far no general effective method.",
"In fact the dependence on a Mordell–Weil basis is a disadvantage of Algorithm REF compared to the classical approach of Baker–Davenport which can be applied in the important case $S=\\emptyset $ .",
"Their approach is very efficient in solving (REF ) for varying $a\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ with $\\vert {}a{}\\vert $ at most some given bound, see Bennett–Ghadermarzi [18].",
"On the other hand, an advantage of Algorithm REF over the known algorithms is that it can efficiently solve (REF ) for large sets $S$ .",
"This feature allows to study the Diophantine problem for hyperbolic curves described in the next paragraph.",
"Other important features of Algorithm REF are the following: It can efficiently solve (REF ) for parameters $a$ with huge height and its underlying correctness proofs are complete (even when $2\\in S$ or when $S$ contains bad reduction primes of $E_a$ ).",
"For example these two features are crucial to efficiently determine all elliptic curves over $\\textnormal {Spec}({\\mathbb {Z}})-S$ by solving certain equations (REF ), see below."
],
[
"Points of hyperbolic curves.",
"Suppose that $T$ and $B$ are nonempty open subschemes of $\\textnormal {Spec}({\\mathbb {Z}})$ , and assume that $T\\subseteq B$ .",
"Let $Y\\rightarrow B$ be an arbitrary hyperbolic curve of genus $g$ , see for example [117] for the definition.",
"We denote by $Y(T)$ the set of $T$ -points of $Y$ and we now consider the following Diophantine problem.",
"Problem.",
"Describe the set $Y(T)$ in terms of $T$ , with $T\\subseteq B$ varying.",
"If $g\\ge 2$ then a result of Faltings [60] implies that the cardinality of $Y(T)$ is uniformly bounded in terms of $T$ , which in some sense solves the problem for $g\\ge 2$ .",
"Over the last decades the case $g=0$ was successfully studied by many authors, including Bombieri, Erdös, Evertse, Győry, Moree, Silverman, Stewart, Tijdeman [59], [57], [54], [24], [55] and more recently Harper, Konyagin, Lagarias, Soundararajan [101], [106], [76].",
"However the situation completely changes for $g=1$ .",
"In this case, the problem is essentially not investigated in the literature and is widely open.",
"On using Algorithm REF we study the problem for the families of hyperbolic genus one curves defined by Mordell equations (REF ) and cubic Thue equations (REF ).",
"In particular, motivated by Theorem C and the construction of the elliptic logarithm sieve, we propose the following conjecture.",
"Conjecture 2.",
"There are constants $c_a$ and $c_r$ , depending only on $a$ and $r$ respectively, such that any finite nonempty set of rational primes $S$ satisfies $\\vert {}Y_a(\\mathcal {O}){}\\vert \\le c_a \\vert {}S{}\\vert ^{c_r}$ .",
"Here $Y_a(\\mathcal {O})$ denotes the set of solutions of the Mordell equation (REF ) defined by $(a,S)$ and $r$ is the rank of the free part of the finitely generated abelian group $E_a({\\mathbb {Q}})$ .",
"The conjectured bound is polynomial in terms of $\\vert {}S{}\\vert $ , while as far as we know all conjectures and results in the literature provide exponential bounds such as in Evertse–Silverman [56].",
"We construct an infinite family of sets $S[b]$ which shows that the exponent $c_r$ has to be at least $\\tfrac{r}{r+2}$ .",
"Furthermore Theorem C strongly indicates that $c_r=\\tfrac{r}{r+2}$ would be still far from optimal for many sets $S$ of interest, including the sets $S(n)$ .",
"On taking into account Theorem C, we ask whether one can replace in Conjecture 2 the quantity $\\vert {}S{}\\vert $ by the logarithm of the largest prime in $S$ ?",
"We motivate this question by constructing a probabilistic model.",
"Together with a classical Diophantine approximation result of Siegel (1929) and known estimates for the de Bruijn function, this model predicts a bound for $\\vert {}Y_a(\\mathcal {O}){}\\vert $ in terms of $S$ which would be optimal in view of the family $S[b]$ ."
],
[
"Effective Shafarevich theorem.",
"Now we take $T=\\textnormal {Spec}({\\mathbb {Z}})-S$ and we identify $M(T)$ with the set $M(S)$ of ${\\mathbb {Q}}$ -isomorphism classes of elliptic curves over ${\\mathbb {Q}}$ with good reduction outside $S$ .",
"In the 1960s Shafarevich showed that $M(S)$ is finite: He reduced the problem to Mordell equations (REF ) and then he applied Diophantine approximations.",
"Coates [37] made Shafarevich's proof effective by using the theory of logarithmic forms.",
"In fact there already exist several practical methods which allow to determine the space $M(S)$ .",
"We refer to Section REF for an overview.",
"On combining Shafarevich's reduction with our Algorithm REF for Mordell equations (REF ), we obtain Algorithm REF which allows to compute $M(S)$ .",
"To illustrate the practicality of our approach, we determined the space $M(S)$ for each set $S\\in \\mathcal {S}$ .",
"Here $\\mathcal {S}$ is a family of sets which contains in particular the set $S(5)$ and all sets $S$ with $N_S\\le 10^3$ .",
"Motivated by our data, we conjecture that one can replace in Conjecture 1 the moduli scheme $\\mathbb {P}^1_{{\\mathbb {Z}}[1/2]}-\\lbrace 0,1,\\infty \\rbrace $ of Legendre elliptic curves by the moduli stack $\\mathcal {M}_{1,1}$ of elliptic curves.",
"In other words for any $n\\in {\\mathbb {Z}}_{\\ge 1}$ our conjecture says that among all sets $S$ with $\\vert {}S{}\\vert \\le n$ the cardinality of $M(S)$ is maximal (up to an absolute constant) when $S=S(n)$ .",
"For many sets $S\\in \\mathcal {S}$ it seems that computing the space $M(S)$ is out of reach for the known methods, see the discussions in Section REF .",
"In particular, our approach is significantly more efficient than the method of Cremona–Lingham [34].",
"They use a different reduction to Mordell equations (REF ) which involves $j$ -invariants, and then they solve (REF ) via the algorithm of Pethő–Zimmer–Gebel–Herrmann [126] based on the theory of logarithmic forms.",
"The input of Algorithm REF requires a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for $2\\cdot 6^{\\vert {}S{}\\vert }$ distinct integers $a$ .",
"Thus our approach is not practical for large $\\vert {}S{}\\vert $ .",
"Next, we mention that the problem of explicitly describing the space $M(T)=M(S)$ is of interest for many reasons.",
"For instance, in [89] the moduli formalism was used to reduce many Diophantine problems to the study of $M(T)$ .",
"On combining this strategy with our database listing the set $M(T)=M(S)$ for $S\\in \\mathcal {S}$ , we can directly solve any Diophantine problem inducing $T$ -points on moduli schemes $Y$ of elliptic curves with effective Paršin construction $\\phi :Y(T)\\rightarrow M(T)$ ; see Section REF for details and explicit examples.",
"Here it suffices to know the image of $\\phi $ in $M(T)$ , which is often much smaller than the whole space $M(T)$ .",
"Taking this into account, we simplified and optimized the strategy for several classical Diophantine problems.",
"In particular, we worked out the cases of cubic Thue equations (REF ), cubic Thue–Mahler equations (REF ) and generalized Ramanujan–Nagell equations (REF ).",
"This led to the following algorithms and results."
],
[
"Thue equation.",
"We constructed Algorithm REF which allows to solve the cubic Thue equation (REF ).",
"Our approach is efficient in the generic case and it can deal with large sets $S$ .",
"To illustrate this we used Algorithm REF in order to compile the database $\\mathcal {D}_3$ containing the solutions of (REF ) for large classes of parameter triples $(f,S,m)$ , where $m\\in {\\mathbb {Z}}$ is nonzero and $f\\in {\\mathbb {Z}}[x,y]$ is homogeneous of degree three with nonzero discriminant $\\Delta $ (see Section ).",
"In particular our database $\\mathcal {D}_3$ covers all $(f,S,m)$ such that $m=1$ and such that $(\\Delta ,S)$ lies in $\\lbrace \\vert {}\\Delta {}\\vert \\le 10^4,S\\subseteq S(100)\\rbrace $ , $\\lbrace \\vert {}\\Delta {}\\vert \\le 100,S\\subseteq S(10^3)\\rbrace $ or $\\lbrace \\vert {}\\Delta {}\\vert \\le 20,S\\subseteq S(10^5)\\rbrace $ ; see Section REF for more information and additional examples.",
"Theorem D. For each triple $(f,S,m)$ considered in $\\mathcal {D}_3$ , the database $\\mathcal {D}_3$ contains all solutions of the cubic Thue equation (REF ) defined by $(f,S,m)$ .",
"This gives in particular a new proof of several results in the literature (see Section ) which determined the solutions of specific cubic Thue equations (REF ).",
"Furthermore Theorem D motivates new conjectures and questions on the number of solutions of (REF ), see Section REF .",
"We next describe the main ingredients of Algorithm REF .",
"As in [89] we reduce the problem to Mordell equations: This reduction uses classical invariant theory which provides an explicit morphism $\\varphi :X\\rightarrow Y$ over $T=\\textnormal {Spec}(\\mathcal {O})$ , where $X$ and $Y$ are the closed subschemes of $\\mathbb {A}^2_T$ given by the Thue equation (REF ) and by the Mordell equation (REF ) with $a=432\\Delta m^2$ respectively.",
"Then we compute $Y(T)$ using our Algorithm REF for Mordell equations and we apply triangular decomposition in order to finally determine $X(T)=\\varphi ^{-1}(Y(T))$ .",
"Here we recall that Algorithm REF requires a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ .",
"Although it turned out that it is usually possible to determine such a basis in practice, the dependence on a Mordell–Weil basis is a disadvantage of our approach compared to the known methods discussed in Section .",
"On the other hand, an advantage of our approach is that it can solve (REF ) for huge sets $S$ .",
"Here it seems that already sets $S$ with $\\vert {}S{}\\vert \\ge 10$ are out of reach for the known methods solving (REF )."
],
[
"Thue–Mahler equation.",
"Let $f\\in \\mathcal {O}[x,y]$ be a homogeneous polynomial of degree three with nonzero discriminant $\\Delta $ , and let $m\\in \\mathcal {O}$ be nonzero.",
"We constructed Algorithm REF which allows in particular to solve the classical cubic Thue–Mahler equation $f(x,y)=mz,$ where $x,y,z\\in {\\mathbb {Z}}$ with $z\\in \\mathcal {O}^\\times $ and $\\gcd (x,y)=1$ .",
"To demonstrate the practicality of our approach, we used Algorithm REF in order to create the database $\\mathcal {D}_4$ listing the solutions of (REF ) for many triples $(f,S,m)$ with $m=1$ and $f\\in {\\mathbb {Z}}[x,y]$ as above.",
"In particular $\\mathcal {D}_4$ covers all such triples with $(\\Delta ,S)$ in $\\lbrace \\vert {}\\Delta {}\\vert \\le 3000,S\\subseteq S(2)\\rbrace $ , $\\lbrace \\vert {}\\Delta {}\\vert \\le 10^3,S\\subseteq S(3)\\rbrace $ , $\\lbrace \\vert {}\\Delta {}\\vert \\le 100,S\\subseteq S(4)\\rbrace $ or $\\lbrace \\vert {}\\Delta {}\\vert \\le 16,S\\subseteq S(5)\\rbrace $ ; see Section REF for more information.",
"Theorem E. For each triple $(f,S,m)$ considered in $\\mathcal {D}_4$ , the database $\\mathcal {D}_4$ contains all solutions of the cubic Thue–Mahler equation (REF ) defined by $(f,S,m)$ .",
"We mention that $\\mathcal {D}_4$ contains in addition the solutions of (REF ) for various other $(f,S,m)$ of interest, including cases with $S=S(6)$ .",
"In fact Theorem E gives in particular a new proof of several results in the literature (see Section ) which solved specific equations (REF ).",
"We next describe the main ingredients of Algorithm REF .",
"On using an elementary standard reduction, we reduce (REF ) to $3^{\\vert {}S{}\\vert }$ distinct cubic Thue equations (REF ) and these equations are then solved via Algorithm REF .",
"Here the applications of Algorithm REF require $3^{\\vert {}S{}\\vert }$ distinct Mordell–Weil bases.",
"Hence our approach is not practical when $\\vert {}S{}\\vert $ is large.",
"However for small $\\vert {}S{}\\vert $ it turned out that it is usually possible to determine the required Mordell–Weil bases and then our approach is indeed efficient as illustrated in Section REF ."
],
[
"Generalized Ramanujan–Nagell equations.",
"Let now $b$ and $c$ be arbitrary nonzero elements of $\\mathcal {O}$ .",
"On using our approach for Mordell equations (REF ), we obtained Algorithm REF which allows to solve the generalized Ramanujan–Nagell equation $x^2+b=cy, \\ \\ \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}^\\times .$ There is a vast literature devoted to the study of (special cases of) this Diophantine problem.",
"See for example the results, discussions and references in Bugeaud–Shorey [28], Bennett–Skinner [29] and Saradha–Srinivasan [139].",
"To illustrate the practicality of our approach, we used Algorithm REF in order to create the database $\\mathcal {D}_5$ listing the solutions of (REF ) for many triples $(b,c,S)$ with $c=1$ and $b\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ .",
"In particular our database $\\mathcal {D}_5$ covers all such triples with $(b,S)$ contained in $\\lbrace \\vert {}b{}\\vert \\le 12,S\\subseteq S(5)\\rbrace $ , $\\lbrace \\vert {}b{}\\vert \\le 35,S\\subseteq S(4)\\rbrace $ , $\\lbrace \\vert {}b{}\\vert \\le 250,S\\subseteq S(3)\\rbrace $ or $\\lbrace \\vert {}b{}\\vert \\le 10^3,S\\subseteq S(2)\\rbrace $ .",
"Theorem F. For each triple $(b,c,S)$ considered in $\\mathcal {D}_5$ , the database $\\mathcal {D}_5$ contains all solutions of the generalized Ramanujan–Nagell equation (REF ) defined by $(b,c,S)$ .",
"This theorem gives in particular a new proof of many results in the literature (see Section REF ) which solved special cases of (REF ).",
"If $b\\in {\\mathbb {Z}}$ is nonzero and $c=1$ , then Pethő–de Weger [124] obtained a practical approach to find all solutions $(x,y)$ of (REF ) with $x,y\\in {\\mathbb {Z}}_{\\ge 0}$ .",
"Their method involves binary recurrence sequences and the theory of logarithmic forms.",
"Our approach is completely different: On using an elementary construction, we reduce (REF ) to certain Mordell equations (REF ) which we then solve via Algorithm REF .",
"Here the involved Mordell curves usually have huge height.",
"This is no problem for Algorithm REF and it turned out that the bottleneck of our approach is finding the $3^{\\vert {}S{}\\vert }$ distinct Mordell–Weil bases required for the applications of Algorithm REF .",
"In light of this, we worked out a refinement of Algorithm REF in the following special case of (REF ).",
"For arbitrary nonzero $b,c,d$ in ${\\mathbb {Z}}$ with $d\\ge 2$ , consider the classical Diophantine problem $x^2+b=cd^n, \\ \\ \\ \\ \\ (x,n)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}.$ Now, the crucial advantage of our refinement (see Algorithm REF ) is that it only requires three distinct Mordell–Weil bases in order to find all solutions of (REF ).",
"On using Algorithm REF , we solved (REF ) for all triples $(7,1,d)$ with $d\\le 888$ ; we note that here the case $d=2$ corresponds to the classical Ramanujan–Nagell equation.",
"Furthermore, in Section REF we worked out additional applications of Algorithms REF and REF .",
"For example, we apply our approach to the problem of finding all coprime $S$ -units $x,y\\in {\\mathbb {Z}}$ with $x+y$ a square or a cube.",
"Here we solve several new cases of this problem.",
"Also, we show that our approach is a useful tool to study conjectures of Terai on Pythagorean triples."
],
[
"Diophantine problems related to Mordell equations",
"We next discuss certain old Diophantine problems which are related to Mordell equations (REF ).",
"After presenting new results for primitive solutions of (REF ), we state a corollary on the greatest prime divisor of the difference of coprime squares and cubes.",
"We also give new height bounds for the solutions of cubic Thue equations (REF ), cubic Thue–Mahler equations (REF ) and generalized Ramanujan–Nagell equations (REF ).",
"As a byproduct, we obtain in this section alternative proofs of classical theorems of Baker [4], Coates [35], [36], [37] and Vinogradov–Sprindžuk [160]."
],
[
"Primitive solutions of Mordell equations",
"Following Bombieri–Gubler [17], we say that $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ is primitive if $\\pm 1$ are the only $n\\in {\\mathbb {Z}}$ with $n^{6}$ dividing $\\gcd (x^3,y^2)$ .",
"In particular $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ is primitive if $x,y$ are coprime.",
"To measure the number $a\\in \\mathcal {O}$ and the finite set $S$ , we take $a_S=1728N_S^2\\prod p^{\\min (2,{\\textnormal {ord}}_p(a))}$ with the product extended over all rational primes $p\\notin S$ .",
"Let $h$ be the usual logarithmic Weil height [17], with $h(n)=\\log \\vert {}n{}\\vert $ for $n\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ .",
"Building on the arguments of [89], we establish the following result (take $\\mu =0$ in Theorem REF ).",
"Theorem G. Let $a\\in {\\mathbb {Z}}$ be nonzero.",
"Assume that $y^2=x^3+a$ has a solution in ${\\mathbb {Z}}\\times {\\mathbb {Z}}$ which is primitive.",
"Then any $(x,y)\\in \\mathcal {O}\\times \\mathcal {O}$ with $y^2=x^3+a$ satisfies $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le a_S\\log a_S.$ We now discuss several aspects of this result.",
"A useful feature of Theorem G is that it does not involve $\\vert {}a{}\\vert $ .",
"To illustrate this we take $n\\in {\\mathbb {Z}}_{\\ge 1}$ , we let $\\mathcal {F}_n$ be the infinite family of integers $a$ with radical $\\textnormal {rad}(a)$ at most $n$ , and we put $a_{*}=a_\\emptyset $ .",
"Then it holds $a_*\\le 1728\\textnormal {rad}(a)^2$ and Theorem REF with $\\mu =0$ directly implies the following corollary.",
"Corollary H. For any integer $n\\ge 1$ , the set of primitive $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ with $y^2-x^3\\in \\mathcal {F}_n$ is finite and can in principle be determined.",
"Furthermore if $a\\in {\\mathbb {Z}}$ satisfies $\\log \\vert {}a{}\\vert \\ge a_*\\log a_*$ , then there are no primitive $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ with $y^2-x^3=a$ .",
"It holds that $(3m^{3n})^2=(2m^{2n})^3+m^{6n}$ for all $m,n\\in {\\mathbb {Z}}$ .",
"Hence we see that one can not remove the assumption in Theorem G. However, one can weaken the assumption by considering a certain class of (almost primitive) solutions of (REF ) which fits into Szpiro's small points philosophy [148]; see Theorem REF and the discussions given there.",
"We also deduce Corollaries REF and REF on the difference of perfect squares and perfect cubes.",
"On taking for example $\\varepsilon =\\frac{1}{10}$ in Corollary REF , one obtains the following result.",
"Corollary I.",
"Suppose that $x,y\\in {\\mathbb {Z}}$ are coprime, and write $X=\\max (\\vert {}x{}\\vert ,\\vert {}y{}\\vert )$ .",
"Then the greatest rational prime divisor $p$ of $y^2-x^3$ exceeds $(1-\\frac{1}{10})\\log \\log X-20.$ This improves the old theorem of Coates [37] in which he established the lower bound $10^{-3}(\\log \\log X)^{1/4}$ .",
"Similarly our Corollary REF refines [37]."
],
[
"Comparison with literature.",
"We point out that Coates' method, which uses early estimates for logarithmic forms, is completely different to the method applied in this paper.",
"In fact it is possible to prove a weaker version of our Theorem REF and to improve Coates' results [37] by using more recent estimates for logarithmic forms.",
"However, it turns out that without introducing new ideas the actual best lower bounds for linear forms in logarithms (see Baker–Wüstholz [33] for an overview) do not give inequalities as strong as those provided by Theorem REF and Corollaries REF and REF .",
"For example, let us consider our asymptotic version of Corollary I established in Corollary REF .",
"For any $\\varepsilon >0$ this version gives that the prime $p$ in Corollary I exceeds $\\alpha \\log \\log X+\\beta , \\ \\ \\ \\ \\ \\alpha =1-\\varepsilon ,$ where $\\beta $ denotes an effective constant depending only on $\\varepsilon $ .",
"This improves the actual best factor $\\alpha =\\tfrac{1}{84}-\\varepsilon $ contained in the general result of Bugeaud [32], which was proven by using inter alia a direct and ingenious reduction to lower bounds for logarithmic forms.",
"Here it seems possible that one can slightly improve the factor $\\alpha =\\tfrac{1}{84}-\\varepsilon $ by updating Bugeaud's approach with the actual best lower bounds for logarithmic forms.",
"However, the presence of the usual quantity $h(\\alpha _1)\\cdot \\cdots \\cdot h(\\alpha _n)$ in these lower bounds shows that this approach will always produce a factor $\\alpha $ which is smaller than $\\tfrac{1}{36}-\\varepsilon $ .",
"In view of this, our result (REF ) seems to be out of reach for the present state of the art in the theory of logarithmic forms.",
"See also the related discussions given at the end of Section ."
],
[
"Idea of proof.",
"To prove our results for primitive solutions of (REF ), we go into the proof of [89] which combines the Shimura–Taniyama conjecture with the method of Faltings (Arakelov, Paršin, Szpiro) as outlined in Section REF ; see also Section REF .",
"Then we exploit that primitive solutions of (REF ) induce, via the Paršin construction $\\phi $ , elliptic curves with useful extra properties.",
"For instance, to obtain the factor $\\alpha =1-\\varepsilon $ in (REF ), we use that the Paršin construction $\\phi $ maps coprime solutions to elliptic curves which have semistable reduction over ${\\mathbb {Z}}[1/6]$ .",
"The corollaries are then direct consequences of our results for primitive/coprime solutions and of the prime number theorem."
],
[
"Height bounds for cubic Thue and Thue–Mahler equations",
"Baker [4] applied his theory of logarithmic forms in order to establish in particular an effective finiteness result for any cubic Thue equation (REF ) in the case when $\\mathcal {O}={\\mathbb {Z}}$ .",
"In the general case (REF ) is essentially equivalent to the cubic Thue–Mahler equation (REF ).",
"Mahler [108] showed via Diophantine approximations that (REF ) has only finitely many solutions.",
"Furthermore Coates [35], [36] and Vinogradov–Sprindžuk [160] independently proved effective finiteness via the theory of logarithmic forms.",
"We refer to Baker–Wüstholz [33] and Evertse–Győry [52], [53] for an overview on generalizations and improvements of finiteness results for Thue and Thue–Mahler equations."
],
[
"Height bounds.",
"A new effective finiteness proof for any cubic Thue equation (REF ) was obtained in [89].",
"On working out explicitly the arguments of [89], we get explicit height bounds for the solutions of cubic Thue and Thue–Mahler equations.",
"To state our results we denote by $h(f-m)$ the maximum of the logarithmic Weil heights of the coefficients of the polynomial $f-m\\in \\mathcal {O}[x,y]$ .",
"We put $a=432\\Delta m^2$ with $\\Delta $ the discriminant of $f$ .",
"The next corollary may be viewed as a refinement of [89] in the case of moduli schemes (see Section REF ) corresponding to (REF ) or (REF ).",
"Corollary J.",
"The following statements hold.",
"(i) Define the number $n$ by putting $n=2$ if $f,m\\in {\\mathbb {Z}}[x,y]$ and $n=10$ otherwise.",
"Then any solution $(x,y)$ of the cubic Thue equation (REF ) satisfies $\\max \\bigl (h(x),h(y)\\bigl )\\le a_S\\log a_S+43nh(f-m).$ (ii) If $(x,y,z)$ is a solution of the cubic Thue–Mahler equation (REF ) then $\\max \\bigl (h(x),h(y),\\tfrac{1}{3}h(z)\\bigl )\\le 2a_S\\log a_S+86nh(f-m).$ In Corollary REF we shall establish a more precise version of Corollary J which provides sharper but more complicated bounds.",
"Furthermore we shall show in Corollary REF that statement (ii) holds more generally for any primitive solution $(x,y,z)$ of the general cubic Thue–Mahler equation (REF ); the definition of such solutions is given in Definition REF ."
],
[
"Comparison with literature.",
"We now compare Corollary J with corresponding results in the literature.",
"On using the theory of logarithmic forms, Bugeaud–Győry [16], Bugeaud [31], Győry–Yu [72] and Juricevic [84] obtained the actual best height boundsWe point out that these results hold for Diophantine equations which are considerably more general than (REF ) and (REF ), and some of these results deal moreover with arbitrary number fields.",
"for the solutions of (REF ) and (REF ).",
"We do not state these rather complicated bounds, but we mention that each of them has certain advantages and disadvantages.",
"To compare these results with Corollary J, we may and do assume that $f\\in {\\mathbb {Z}}[x,y]$ and $m\\in {\\mathbb {Z}}$ .",
"Then it follows that $a_S\\le 2^83^5\\Delta _2\\bigl (\\textnormal {rad}(m)N_S\\bigl )^2$ where $\\Delta _2=\\min \\bigl (\\textnormal {rad}(\\Delta )^2,\\vert {}\\Delta {}\\vert \\bigl )$ , and standard height arguments lead to $\\vert {}\\Delta {}\\vert \\le 3^5H^4$ for $H=\\max _i \\vert {}a_i{}\\vert $ the maximum of the absolute values of the coefficients $a_i$ of $f$ .",
"Therefore Corollary J gives estimates which are asymptotically of the form $H^4\\log H$ , improving the actual best bounds $(H\\log H)^4$ in terms of $H$ .",
"In particular in the classical case, when $m$ is fixed (usually $m=1$ ) and $\\mathcal {O}={\\mathbb {Z}}$ , our Corollary J improves the actual best results in all aspects.",
"Furthermore Corollary J improves the known estimates in terms of $S$ for infinitely many sets $S$ , including all sets $S$ with $\\vert {}S{}\\vert \\le 3$ .",
"On the other hand, our results are worse in terms of $m$ and the bound [72] is significantly better in terms of $S$ for infinitely many sets $S$ including all sets $S=S(n)$ with $n$ large.",
"Finally we mention that our estimates (see also Corollary REF ) involve small absolute constants and hence they considerably improve the actual best height bounds for all parameters which are not that large.",
"This might be of interest for the practical resolution of (REF ) and (REF )."
],
[
"Idea of proof.",
"Following [89], we deduce our height bounds for cubic Thue equations (REF ) from a result for Mordell equations (Theorem REF ) discussed above.",
"This deduction uses classical invariant theory which provides an explicit morphism $\\varphi :X\\rightarrow Y$ over $T=\\textnormal {Spec}(\\mathcal {O})$ , where $X$ and $Y$ are the closed subschemes of $\\mathbb {A}^2_T$ given by the Thue equation (REF ) and by the Mordell equation (REF ) with $a=432\\Delta m^2$ respectively.",
"Then in Proposition REF we control the Weil height of any $P\\in X(T)$ in terms of the Weil height of $\\varphi (P)\\in Y(T)$ .",
"To prove Proposition REF we apply inter alia an effective arithmetic Nullstellensatz over the hypersurface in $\\mathbb {A}_T^3$ given by $f-mz^3$ .",
"In fact we use here the Nullstellensatz of D'Andrea–Krick–Sombra [48] which leads to small constants.",
"Finally, we deduce our height bounds for cubic Thue–Mahler equations (REF ) by invoking an elementary standard construction which reduces (REF ) to Thue equations (REF ).",
"Alternatively, one can obtain explicit height bounds for the solutions of (REF ) and (REF ) by directly applying [89] with suitable moduli problems; see Sections REF and REF ."
],
[
"Height bounds for generalized Ramanujan–Nagell equations",
"An elementary construction reduces the generalized Ramanujan–Nagell equation (REF ), and the more classical special case (REF ), to Mordell equations (REF ).",
"In light of this, results of Mordell (1922) and Mahler (1933) give finiteness for (REF ) and (REF ) respectively.",
"Moreover effective finiteness follows from Baker [5] in the case (REF ) and from Coates [37] in the case (REF ).",
"Our height bounds for Mordell equations lead to the following result which may be viewed as a refinement of [89] in the case of moduli schemes (see Section REF ) corresponding to generalized Ramanujan–Nagell equations (REF ).",
"Corollary K. If $(x,y)$ satisfies the generalized Ramanujan–Nagell equation (REF ), then $\\max \\bigl (2h(x),h(y)\\bigl )\\le 2a_S+h(a)+3h(c), \\ \\ \\ a=bc^2.$ In Corollary REF we shall give a more precise version of this height bound.",
"Furthermore we shall deduce Corollary REF which provides explicit height bounds for “coprime\" $u,v\\in \\mathcal {O}^\\times $ with $u+v$ a square or cube in ${\\mathbb {Q}}$ .",
"To discuss an application, we take arbitrary coprime $m,n\\in {\\mathbb {Z}}$ and we consider the following simple condition in terms of $r=\\textnormal {rad}(mn)$ : $(*)$ The natural logarithm of $\\vert {}m{}\\vert $ or $\\vert {}n{}\\vert $ exceeds $(90r)^2\\log (9r)$ .",
"Now, the height bounds in Corollary REF imply our next result which shows that condition $(*)$ is in fact sufficient to rule out that $m+n$ is a perfect square or cube.",
"Corollary L. Suppose that $m$ and $n$ are arbitrary coprime rational integers.",
"If condition $(*)$ holds, then $m+n$ is not a perfect square or cube.",
"One can obtain versions of our corollaries by using height bounds for the solutions of Mordell or Thue–Mahler equations which are based on the actual state of the art in the theory of logarithmic forms.",
"For a comparison of the resulting bounds with our estimates, we refer to the analogous discussions given above and in Section REF .",
"We further mention that the strong $abc$ -conjecture of Masser–Oesterlé in Remark REF directly implies versions of our corollaries which are asymptotically considerably better; notice that these implications are (as usual) not compatible with any exponential version of the $abc$ -conjecture."
],
[
"Proof of the height bounds",
"There is a long tradition of proving effective height bounds for Thue equations, Mordell equations, Thue–Mahler equations and $S$ -unit equations.",
"In fact during the last few decades, one conducted quite some effort to refine the initial effective bounds of Baker [4], [5], Coates [35], [36], [37] and Győry [73].",
"See for example [33], [72] for an overview on these refinements which allExcept Bombieri's refinement (see Bombieri–Cohen [11]) of Thue's method using Diophantine approximations.",
"This method is relatively new and it is essentially self-contained.",
"So far it leads to height bounds which are (slightly) worse compared to those coming from the theory of logarithmic forms.",
"ultimately rely on the theory of logarithmic forms.",
"We now discuss the strategy underlying the proofs of our height bounds for the Diophantine problems considered in the present paper.",
"The symbol (ST) refers to the geometric version of the Shimura–Taniyama conjecture [12] which relies inter alia on the Tate conjecture [60]."
],
[
"$S$ -unit equation.",
"Let $(x,y)$ be a solution of the $S$ -unit equation (REF ).",
"In the 1990s Frey [66] (see also Murty [121]) remarked that (ST) provides an alternative approach to bound $h(x)$ , and in 2011 it was independently shown by Murty–Pasten and by the first mentioned author that Frey's ideas together with (ST) lead to effective bounds for $h(x)$ ; see [120] and [89].",
"Here [120] works with the coprime Hecke algebra which leads to $h(x)\\ll N_S\\log N_S$ , while [89] uses the full Hecke algebra which in general leads to the (slightly) weaker bound $h(x)\\ll N_S(\\log N_S)(\\log \\log N_S)$ .",
"Since there are also many situations in which the full Hecke algebra provides the best bounds, we use in this paper the coprime and the full Hecke algebra approach.",
"To work out the optimized height bounds for Algorithm REF , we follow closely the arguments of [120] and [89] and we conduct some effort to refine the involved estimates.",
"Asymptotically, the actual best result $h(x)\\ll N_S^{1/3}(\\log N_S)^3$ is due to Stewart–Yu [146].",
"However, for all sets $S$ with $N_S\\le 2^{100}$ and for many other sets $S$ of practical interest, our optimized height bounds are considerably stronger than those coming from the theory of logarithmic forms; see Section REF .",
"We note that this alternative approach for $S$ -unit equations (REF ) is in fact (see [89]) a special case of the method of Faltings (Arakelov, Paršin, Szpiro) combined with (ST) as described in Section REF above."
],
[
"Other Diophantine problems.",
"Many classical Diophantine problems can be reduced to $S$ -unit equations.",
"However in most cases the known (unconditional) reductions involve number fields larger than ${\\mathbb {Q}}$ .",
"In particular one can not combine these reductions with (unconditional) results for (REF ) in order to deal with the Diophantine problems considered in the present paper.",
"Instead we use that all these problems induce integral points on moduli schemes of elliptic curves.",
"Given this observation, we can apply the strategy of [89] which provides explicit height bounds for integral points on moduli schemes of elliptic curves.",
"This strategy consists of combining (ST) with Faltings' method in a way which is similar as described in Section REF above.",
"Here important ingredients of the proof are the height-conductor inequality [89] (proven independently in [120], see Section REF ) and the moduli formalism.",
"Besides providing a useful geometrical interpretation of various classical Diophantine problems, the moduli formalism allows to find new explicit applications of the method.",
"Indeed we discovered many results of the present paper by searching for moduli schemes with “interesting\" defining equations: Given a priori the information that the equation defines a moduli scheme with effective Paršin construction $\\phi $ , one can explicitly work out the strategy of [89] to get effective finiteness results; see [89] and Section REF .",
"Furthermore, a posteriori one can often obtain here simpler (but less conceptual) proofs by removing the moduli formalism.",
"In light of the practical purpose of the present article, we conducted some effort to simplify our proofs as much as possible in the case of Mordell, cubic Thue, cubic Thue–Mahler and generalized Ramanujan–Nagell equations.",
"For instance on working out the method for the moduli schemes defined by Mordell equations (REF ), one obtains the actual best height bounds [89] for the solutions of (REF ).",
"To prove the optimized height bounds for Algorithm REF , we follow and simplify the arguments of [89].",
"Here we try to optimally estimate the involved quantities.",
"We note that a priori the arguments of [89] prove all our explicit simplified height bounds, but of the form $X(\\log X)(\\log \\log X)$ .",
"To remove here in addition the factor $\\log \\log X$ , we go into the proof of [89] and we now use an idea of Murty–Pasten [120] involving the coprime Hecke algebra.",
"Their idea was not known to the author of [89].",
"However, for each considered Diophantine problem, there are also many situations in which the full Hecke algebra approach of [89] provides the best bounds.",
"Hence we work out all our height bounds using the coprime and the full Hecke algebra approach."
],
[
"Plan of the paper.",
"In Section we briefly discuss some tools which are crucial for our results and algorithms.",
"In particular, we recall a geometric version of the Shimura–Taniyama conjecture which relies inter alia on the Tate conjecture.",
"In Section we present two algorithms for $S$ -unit equations.",
"The first approach via modular symbols is worked out in Section REF .",
"Then in Section REF we conduct some effort to construct the second algorithm which uses height bounds.",
"Here, after discussing in Section REF a slight variation of de Weger's method, we construct our refined sieve in Section REF and we develop a refined enumeration in Section REF .",
"In Section REF we present various applications of our algorithm via height bounds.",
"In particular, we discuss our database $\\mathcal {D}_1$ containing the solutions of large classes of $S$ -unit equations and we use our data to motivate several Diophantine conjectures related to $S$ -unit equations.",
"Section contains two algorithms which allow to solve the Mordell equation.",
"The first approach via modular symbols is worked out in Section REF .",
"Then in Section REF we construct the second algorithm via height bounds.",
"Here, after discussing the main ingredients of our algorithm, including our initial height bounds in Section REF and the elliptic logarithm sieve, we put everything together in Section REF .",
"We also present various applications of our algorithm via height bounds.",
"In Section REF we apply this algorithm to study the problem of finding all elliptic curves over ${\\mathbb {Q}}$ with good reduction outside $S$ , and in Section REF we solve certain classes of Diophantine equations by combining our algorithm with the moduli formalism.",
"Then in Section REF we discuss our database $\\mathcal {D}_2$ containing the solutions of large classes of Mordell equations and we motivate new conjectures/questions.",
"In Section REF we compare our algorithms with other methods.",
"In Section we present algorithms for cubic Thue and Thue–Mahler equations.",
"Our algorithms use a construction from classical invariant theory which allows to reduce the Diophantine problems to Mordell equations.",
"After working out some useful properties of this construction in Section REF , we discuss our algorithms via modular symbols in Section REF .",
"Then in Section REF we explain our algorithms via height bounds and we give various applications.",
"In particular, we discuss in Section REF parts of our databases $\\mathcal {D}_3$ and $\\mathcal {D}_4$ containing the solutions of large classes of cubic Thue and Thue–Mahler equations respectively.",
"In Section REF we compare our algorithms with the known methods.",
"Section contains two algorithms for the generalized Ramanujan–Nagell equation.",
"After presenting our approach via modular symbols in Section REF , we explain the algorithm via height bounds in Section REF and we discuss some applications including our database $\\mathcal {D}_5$ .",
"Then we compare our algorithms with the known methods in Section REF .",
"In Sections , and we consider certain classical Diophantine problems related to Mordell equations.",
"In particular, in Section we study properties of (almost) primitive solutions of Mordell equations and we deduce explicit lower bounds for the largest prime divisor of the difference of coprime squares and cubes.",
"In Sections and we give new explicit height bounds for the solutions of cubic Thue and Thue–Mahler equations and of generalized Ramanujan–Nagell equations.",
"Then in Section we prove our results for (almost) primitive solutions and we work out the optimized height bounds for the solutions of Mordell and $S$ -unit equations which are used in our algorithms.",
"In Section we construct the elliptic logarithm sieve.",
"It allows to efficiently find all integral points of bounded height on any elliptic curve $E$ over ${\\mathbb {Q}}$ with given Mordell–Weil basis of $E({\\mathbb {Q}})$ .",
"We refer to the introduction of Section for an overview of the main ideas of our construction.",
"The elliptic logarithm sieve is of independent interest and thus we made the presentation of Section independent of the rest of this paper."
],
[
"Notation.",
"We shall use throughout the following (standard) notations and conventions.",
"By $\\log $ we mean the principal value of the natural logarithm.",
"We define the product taken over the empty set as 1.",
"For any set $M$ , we denote by $\\vert M\\vert $ the (possibly infinite) number of distinct elements of $M$ .",
"Let $f_1$ and $f_2$ be real valued functions on $M$ .",
"We write $f_1=O(f_2)$ if there is a constant $c$ such that $f_1\\le cf_2$ .",
"Further $f_1=O_\\varepsilon (f_2^\\varepsilon )$ means that for any real number $\\varepsilon >0$ there is a constant $c(\\varepsilon )$ depending only on $\\varepsilon $ such that $f_1\\le c(\\varepsilon ) f_2^\\varepsilon $ .",
"For any $n\\in {\\mathbb {Z}}_{\\ge 1}$ , we say that $\\mathcal {E}\\subset \\mathbb {R}^n$ is an ellipsoid centered at the origin if $\\mathcal {E}=\\lbrace x\\in \\mathbb {R}^n\\,;\\,q(x)\\le c\\rbrace $ for some positive definite quadratic form $q:{\\mathbb {R}}^n\\rightarrow {\\mathbb {R}}$ and some positive real number $c$ .",
"We denote by $\\vert {}z{}\\vert $ the usual complex absolute value of $z\\in \\mathbb {C}$ .",
"If $m,n\\in {\\mathbb {Z}}$ then the symbol $m\\mid n$ (resp.",
"$m\\nmid n)$ means that $m$ divides $n$ (resp.",
"$m$ does not divide $n$ ).",
"Further $\\gcd (a_1,\\cdots ,a_n)$ denotes the greatest common divisor of $a_1,\\cdots ,a_n\\in {\\mathbb {Z}}$ .",
"The radical of $n\\in {\\mathbb {Z}}$ is given by $\\textnormal {rad}(n)=\\prod p$ with the product taken over all rational primes $p$ dividing $n$ .",
"If $\\alpha \\in {\\mathbb {Q}}$ is nonzero and if $p$ is a rational prime, then we write ${\\textnormal {ord}}_p(\\alpha )\\in {\\mathbb {Z}}$ for the order of $p$ in $\\alpha $ .",
"We denote by $h(\\alpha )$ the usual absolute logarithmic Weil height of $\\alpha \\in {\\mathbb {Q}}$ , with $h(0)=0$ and $h(\\alpha )=\\log \\max (\\vert {}m{}\\vert ,\\vert {}n{}\\vert )$ if $\\alpha =m/n$ for coprime $m,n\\in {\\mathbb {Z}}$ .",
"Finally for any real number $x\\in \\mathbb {R}$ , we write $\\left\\lfloor {x}\\right\\rfloor =\\max (n\\in {\\mathbb {Z}}\\,;\\,n\\le x)$ and $\\left\\lceil {x}\\right\\rceil =\\min (n\\in {\\mathbb {Z}}\\,;\\,n\\ge x)$ ."
],
[
"Computer, software and algorithms.",
"Unless mentioned otherwise, we used a standard personal working computer at the MPI Bonn for our computations.",
"Our algorithms are all implemented in Sage and we shall use functions of the computer algebra systems Pari [123], Sage [130] and Magma [107].",
"In what follows, we shall sometimes refer by (PSM) to these computer packages in order to simplify the notation.",
"For each of our algorithms, we conducted some effort to motivate our constructions (in theory and in practice), to explain our choice of parameters, to discuss important complexity aspects, to give detailed correctness proofs and to circumvent potential numerical issues.",
"We shall also list the running times of our algorithms for many examples.",
"The listed times are always upper bounds.",
"In fact some of them were obtained by using older versions of our algorithms, and in many cases the running times would now be significantly better when using the most recent versions (as of February 2016) of our algorithms."
],
[
"Acknowledgements.",
"The research presented in this paper was initiated when we were members at the IAS Princeton (2011/12), it was continued at the IHÉS (2012/13) and it was completed at the MPIM Bonn (2013-15).",
"We are grateful to these institutions for providing excellent working conditions.",
"The authors were supported by the NSF grant No.",
"DMS-0635607 (2011/12) and by EPDI fellowships (2012-14).",
"We would like to thank the MPI, in particular Gerd Faltings and Pieter Moree, for support in (2013-15).",
"Further, we would like to thank Richard Taylor for motivating and very useful initial discussions.",
"We are grateful to Yuri Bilu, Enrico Bombieri, Sander Dahmen, Jan-Hendrik Evertse, Kálmán Győry, Pieter Moree, Hector Pasten and Don Zagier for encouraging discussions and/or for informing us about useful literature.",
"Also, we learned a lot from Yuri Bilu, John Cremona, Stephen Donnelly, Nuno Freitas, Steffen Müller, Martin Raum and Samir Siksek.",
"We would like to thank all of them for explaining various aspects of computational number theory and/or for answering questions."
],
[
"Data and earlier versions.",
"The present version of the paper (Feb. 2016) extends in particular all results and algorithms presented in our earlier versions of the paper (April 2013 and Nov. 2014).",
"However we removed certain applications and discussions of our algorithms via modular symbols, since they are meanwhile obsolete in view of more recent results.",
"Our data is uploaded on: https://www.math.u-bordeaux.fr/~bmatschke/data/."
],
[
"Shimura–Taniyama conjecture",
"A crucial ingredient for all of our results and algorithms is a (geometric) version of the Shimura–Taniyama conjecture which relies inter alia on the Tate conjecture.",
"In the case of our first type of algorithms, another important ingredient is an algorithm of Cremona using modular symbols.",
"In this section we first introduce some notation and then we briefly discuss these ingredients in order to emphasize that they do not depend on results proven by (classical) transcendence or Diophantine approximation techniques.",
"Let $N\\ge 1$ be an integer.",
"Consider the classical congruence subgroup $\\Gamma _0(N)\\subset \\textnormal {SL}_2({\\mathbb {Z}})$ , let $X_0(N)=X(\\Gamma _0(N))_{\\mathbb {Q}}$ be the smooth, projective and geometrically connected model over ${\\mathbb {Q}}$ of the modular curve associated to $\\Gamma _0(N)$ and denote by $S_2(\\Gamma _0(N))$ the complex vector space of cuspforms of weight 2 with respect to $\\Gamma _0(N)$ .",
"See for example [49] for the definitions.",
"We denote by $J_0(N)=\\textnormal {Pic}^0(X_0(N))$ the Jacobian variety of $X_0(N)$ .",
"Let $\\mathbb {T}_{\\mathbb {Z}}$ be the subring of the endomorphism ring of $J_0(N)$ , which is generated over ${\\mathbb {Z}}$ by the usual Hecke operators $T_n$ for all $n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"For any $f\\in S_2(\\Gamma _0(N))$ , we denote by $a_n(f)$ the $n$ -th Fourier coefficient of $f$ and we say that $f$ is rational if $a_n(f)\\in {\\mathbb {Q}}$ for all $n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"Further, we say that $f\\in S_2(\\Gamma _0(N))$ is a newform (of level $N$ ) if $a_1(f)=1$ , if $f$ lies in the new part of $S_2(\\Gamma _0(N))$ and if $f$ is an eigenform for all Hecke operators on $S_2(\\Gamma _0(N))$ .",
"We now suppose that $f\\in S_2(\\Gamma _0(N))$ is a rational newform.",
"Let $I_f$ be the kernel of the ring homomorphism $\\mathbb {T}_{\\mathbb {Z}}\\rightarrow {\\mathbb {Z}}[\\lbrace a_n(f)\\rbrace ]$ which is induced by $T_n\\mapsto a_n(f)$ .",
"It turns out that the image $I_fJ_0(N)$ of $J_0(N)$ under $I_f$ is connected, and the corresponding quotient $E_f=J_0(N)/I_fJ_0(N)$ is an elliptic curve over ${\\mathbb {Q}}$ since $f$ is rational.",
"We next define the modular degree and congruence number of $f$ .",
"On composing the usual embedding $X_0(N)\\hookrightarrow J_0(N)$ , which sends the cusp $\\infty $ of $X_0(N)$ to the zero element of $J_0(N)$ , with the natural projection $J_0(N)\\rightarrow E_f$ , we obtain a finite morphism $\\varphi :X_0(N)\\rightarrow E_f$ of curves over ${\\mathbb {Q}}$ .",
"The modular degree $m_f$ of $f$ is defined as the degree of $\\varphi $ : $m_f=\\textnormal {deg}(\\varphi ).$ The congruence number $r_f$ of $f$ is defined as the largest integer such that there exists a cusp form $f_c\\in S_2(\\Gamma _0(N))$ with all Fourier coefficients in ${\\mathbb {Z}}$ and $(f,f_c)=0 \\textnormal { and } a_n(f)\\equiv a_n(f_c)\\textnormal { mod } (r_f) \\textnormal { for all } n\\in {\\mathbb {Z}}_{\\ge 1}.$ Here $(\\ ,\\,)$ denotes the usual Petersson inner product.",
"Let $\\mathcal {E}(N)$ be the set of all elliptic curves $E$ over ${\\mathbb {Q}}$ which are of the form $E=E_f$ for some rational newform $f\\in S_2(\\Gamma _0(N))$ .",
"We say that an elliptic curve $E$ over ${\\mathbb {Q}}$ is modular if there exists a positive integer $N$ such that the curve $E$ is ${\\mathbb {Q}}$ -isogenous to some elliptic curve in $\\mathcal {E}(N)$ .",
"For any given $N\\in {\\mathbb {Z}}_{\\ge 1}$ , Cremona's algorithm [39] computes in particular the coefficients of minimal Weierstrass equations of all modular elliptic curves over ${\\mathbb {Q}}$ .",
"A short description of this algorithm may be as follows: One considers $X_0(N)$ , computes its first homology using $M$ -symbols (after Manin [110]), computes the action of sufficiently many Hecke operators on it, and determines the one-dimensional eigenspaces with rational eigenvalues.",
"By induction on the divisors of $N$ , this yields the rational newforms in $S_2(\\Gamma _0(N))$ , and their period lattices allow then to compute the set $\\mathcal {E}(N)$ .",
"Finally on using a theorem of Mazur [114], one computes all elliptic curves over ${\\mathbb {Q}}$ which are ${\\mathbb {Q}}$ -isogenous to some curve in $\\mathcal {E}(N)$ and one determines their minimal Weierstrass equations.",
"Building on the key breakthroughs by Wiles [164] and by Taylor–Wiles [158], the Shimura–Taniyama conjecture was finally established by Breuil–Conrad–Diamond–Taylor [12].",
"This conjecture implies (its geometric version saying) that any elliptic curve over ${\\mathbb {Q}}$ of conductor $N$ is ${\\mathbb {Q}}$ -isogenous to some curve in $\\mathcal {E}(N)$ .",
"This implication uses the Tate conjecture [60].",
"We point out that Faltings' proof of the Tate's conjecture does not use transcendence theory or classical Diophantine approximations."
],
[
"Algorithms for $S$ -unit equations",
"Let $S$ be a finite set of rational primes, write $N_S=\\prod _{p\\in S} p$ and denote by $\\mathcal {O}^\\times $ the group of units of $\\mathcal {O} ={\\mathbb {Z}}[1/N_S]$ .",
"In this section, we are interested to solve the $S$ -unit equation $x+y=1, \\ \\ \\ (x,y)\\in \\mathcal {O}^\\times \\times \\mathcal {O}^\\times .",
"\\qquad \\mathrm {(\\ref {eq:sunit})}$ If $\\vert {}S{}\\vert \\le 1$ then (REF ) has either no solutions or $(2,-1)$ , $(-1,2)$ and $(\\frac{1}{2},\\frac{1}{2})$ are the only solutions.",
"Hence we may and do assume that $\\vert {}S{}\\vert \\ge 2$ in this section.",
"As mentioned in the introduction, there already exists a practical method of de Weger [161] which solves (REF ).",
"De Weger [161] used his method to completely solve (REF ) in the case $S=\\lbrace 2,3,5,7,11,13\\rbrace $ .",
"Further, Wildanger [165] and afterwards Smart [138] generalized the ideas of de Weger and they obtained a practical algorithm which solves (REF ) over arbitrary number fields; see also Hajdu [75] and Evertse–Győry [52].",
"There is also the recent work of Dan-Cohen–Wewers [43], [44], with the ultimate goal to construct an algorithm [42] solving (REF ) via “explicit motivic Chabauty–Kim theory\".",
"This method is inspired by Kim's ($p$ -adic étale) approach [94], see also the discussion in [42] which mentions an additional method of Brown.",
"So far all practical approaches solving (REF ) crucially rely on the theory of logarithmic forms.",
"In the following Sections REF and REF , we present and discuss two alternative algorithms which solve $S$ -unit equations (REF ).",
"Both of our algorithms do not use the theory of logarithmic forms.",
"The first algorithm relies on Cremona's algorithm, and we refer to the beginning of Section REF for a short description of the ingredients of the second algorithm.",
"Before we begin to describe our algorithms, we discuss useful properties of symmetric solutions and we give a lower bound for the complexity of any algorithm solving (REF ).",
"Definition 3.1 Suppose that $(x,y)$ and $(x^{\\prime },y^{\\prime })$ are solutions of (REF ).",
"Then we say that $(x,y)$ and $(x^{\\prime },y^{\\prime })$ are symmetric solutions if $x^{\\prime }$ or $y^{\\prime }$ lies in $\\lbrace x,\\tfrac{1}{x},\\tfrac{1}{1-x}\\rbrace $ .",
"It turns out that this defines an equivalence relation on the set of solutions of (REF ), and hence we can consider the set $\\Sigma (S)$ of solutions of (REF ) up to symmetry.",
"Suppose that $(x,y)$ is a solution of (REF ).",
"Then one can directly determine all its symmetric solutions.",
"In fact there are exactly six solutions of (REF ) which are symmetric to $(x,y)$ provided that $(x,y)$ is not equal to $(2,-1)$ , $(-1,2)$ or $(\\tfrac{1}{2},\\tfrac{1}{2})$ .",
"In particular, we see that the number of solutions of (REF ) is either zero or $6\\cdot \\vert {}\\Sigma (S){}\\vert -3$ .",
"The following remark shows that any algorithm solving the $S$ -unit equation (REF ) can not be too fast in general.",
"Remark 3.2 (Lower bound for complexity) The result of Erdös–Stewart–Tijdeman [57], see also the more recent work of Harper, Konyagin, Lagarias, Soundararajan [101], [106], [76], implies the existence of an effective absolute constant $s_0$ with the following property.",
"For any $s\\in {\\mathbb {Z}}_{\\ge s_0}$ there exists a set $S$ with $\\vert {}S{}\\vert =s$ such that the $S$ -unit equation (REF ) has at least $\\exp ((s/\\log s)^{1/2})$ solutions.",
"Furthermore, there are infinitely many sets $S$ such that the $S$ -unit equation (REF ) has a solution $(x,y)$ with $H(x)\\ge N_S$ for $H(x)$ the multiplicative Weil height [17] of $x$ .",
"Therefore we conclude that any algorithm solving (REF ) has running time which is in general not better than linear in $\\log N_S$ and which is in general not better than $\\exp ((\\vert {}S{}\\vert /\\log \\vert {}S{}\\vert )^{1/2})$ ."
],
[
"Algorithm via modular symbols",
"To assure that our algorithm really computes all solutions of the $S$ -unit equation (REF ), we use the following observations.",
"We suppose that $(x,y)$ satisfies (REF ).",
"Then there exist nonzero $a,b,c\\in {\\mathbb {Z}},$ with $\\gcd (a,b,c)=1$ and $\\textnormal {rad}(abc)\\mid N_S$ , such that $x=\\frac{a}{c}$ , $y=\\frac{b}{c}$ and $a+b=c$ .",
"In other words, resolving (REF ) is equivalent to resolving $a+b = c, \\quad a,b,c\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace , \\quad \\gcd (a,b,c)=1,\\quad \\textnormal {rad}(abc)\\mid N_S.$ Further we observe that to find all solutions of (REF ) it suffices to consider solutions $(a,b,c)$ of (REF ) with $a,b,c$ all positive.",
"Indeed if $(a,b,c)$ satisfies (REF ) then there exists a solution $(\\alpha ,\\beta ,\\gamma )$ of (REF ) such that the sets $\\lbrace \\alpha ,\\beta ,\\gamma \\rbrace $ and $\\lbrace \\vert {}a{}\\vert ,\\vert {}b{}\\vert ,\\vert {}c{}\\vert \\rbrace $ coincide.",
"For any elliptic curve $E$ over ${\\mathbb {Q}}$ , we denote by $\\Delta _E$ the minimal discriminant of $E$ and we write $N_E$ for the conductor of $E$ .",
"A construction of Frey–Hellegouarch [64], [77] associates to any solution $(a,b,c)$ of (REF ) an elliptic curveWe warn the reader that here $E_{abc}$ is not necessarily the usual Frey–Hellegouarch curve with Weierstrass equation $y^2=x(x-a)(x+b)$ .",
"See the proof of Lemma REF in which $E_{abc}$ is denoted by $E$ .",
"$E_{abc}$ over ${\\mathbb {Q}}$ .",
"On taking the quotient of $E_{abc}$ by the “subgroup\" generated by a suitable 2-torsion point of $E_{abc}$ , one obtains an elliptic curve $E$ over ${\\mathbb {Q}}$ with the following properties (see Lemma REF ).",
"Lemma 3.3 Suppose that $(a,b,c)$ is a solution of (REF ).",
"Then there exists an elliptic curve $E$ over ${\\mathbb {Q}}$ such that $N_E$ divides $2^4N_S$ and $\\Delta _{E}=2^{8-12m}\\vert {}ab{}\\vert c^4$ with $m\\in \\lbrace 0,1,2,3\\rbrace $ .",
"In fact this lemma may be viewed (see [89]) as an explicit Paršin construction for integral points on the Legendre moduli scheme $\\mathbb {P}^1_{{\\mathbb {Z}}[1/2]}-\\lbrace 0,1,\\infty \\rbrace $ of elliptic curves, which is induced by forgetting the Legendre level structure.",
"Algorithm 3.4 ($S$ -unit equation via modular symbols) The input is a finite set of rational primes $S$ , and the output is the set of solutions $(x,y)$ of the $S$ -unit equation (REF ).",
"The algorithm: If $2\\notin S$ then output the empty set, and if $2\\in S$ then do the following.",
"(i) Use Cremona's algorithm, described in Section , to compute the set $\\mathcal {T}_\\Delta $ of minimal discriminants of modular elliptic curves over ${\\mathbb {Q}}$ of conductor dividing $2^4N_S$ .",
"(ii) Let $\\mathcal {T}=\\cup _{m}\\mathcal {T}_m$ be the union of the sets $\\mathcal {T}_m=\\lbrace 2^{12m-8}\\Delta _E\\,;\\,\\Delta _E\\in \\mathcal {T}_\\Delta \\rbrace \\cap {\\mathbb {Z}}$ where $m=0,1,2,3$ .",
"For each $d\\in \\mathcal {T}$ , factor $d$ as $d=\\prod _{p\\in S}p^{n_p}$ with $n_p\\in {\\mathbb {Z}}$ .",
"Then for each disjoint partition of the set $S=S_{\\alpha }\\dot{\\cup }S_{\\beta }\\dot{\\cup }S_{\\gamma }$ , with $S_{\\gamma }\\subseteq \\lbrace p\\in S\\,;\\,4\\mid n_p\\rbrace $ , define $\\alpha =\\prod _{p\\in S_{\\alpha }}p^{n_p}$ , $\\beta =\\prod _{p\\in S_{\\beta }}p^{n_p}$ , and $\\gamma =\\prod _{p\\in S_{\\gamma }}p^{n_p/4}$ .",
"Then output all $(x,y)$ of the form $(x,y)=(\\frac{a}{c},\\frac{b}{c})$ with $a,b,c\\in {\\mathbb {Z}}$ satisfying $a+b=c$ and $\\lbrace \\vert {}a{}\\vert ,\\vert {}b{}\\vert ,\\vert {}c{}\\vert \\rbrace =\\lbrace \\alpha ,\\beta ,\\gamma \\rbrace $ ."
],
[
"Correctness.",
"We now verify that this algorithm indeed finds all solutions of any $S$ -unit equation (REF ).",
"Let $(x,y)$ be a solution of (REF ).",
"As explained above (REF ), there is a corresponding solution $(a,b,c)$ of (REF ) with $(x,y)=(\\frac{a}{c},\\frac{b}{c})$ .",
"We write $d=\\vert {}ab{}\\vert c^4$ .",
"Lemma REF gives an elliptic curve $E$ over ${\\mathbb {Q}}$ such that $N_E\\mid 2^4N_S$ and $\\Delta _E=2^{8-12m}d$ with $m\\in \\lbrace 0,1,2,3\\rbrace $ .",
"The Shimura–Taniyama conjecture assures that $E$ is modular.",
"This proves that $\\Delta _E$ is contained in the set $\\mathcal {T}_\\Delta $ computed in step (i) and it follows that $d\\in \\mathcal {T}$ .",
"Furthermore, the number $d=\\vert {}ab{}\\vert c^4\\in \\mathcal {T}$ factors in step (ii) as $d=\\prod _{p\\in S}p^{n_p}$ with $n_p\\in {\\mathbb {Z}}$ .",
"Thus the disjoint partition $S=S_{\\alpha }\\dot{\\cup }S_{\\beta }\\dot{\\cup }S_{\\gamma }$ , with $S_{\\alpha }=\\lbrace p\\,;\\,p\\mid a\\rbrace $ , $S_{\\beta }=\\lbrace p\\,;\\,p\\mid b\\rbrace $ and $S_{\\gamma }=S-(S_{\\alpha }\\cup S_{\\beta })$ , produces in step (ii) our solution $(x,y)=(\\frac{a}{c},\\frac{b}{c})$ as desired.",
"Here we used that our coprime $a,b,c\\in {\\mathbb {Z}}$ satisfy $a+b=c$ and $\\lbrace \\vert {}a{}\\vert ,\\vert {}b{}\\vert ,\\vert {}c{}\\vert \\rbrace =\\lbrace \\alpha ,\\beta ,\\gamma \\rbrace $ ."
],
[
"Complexity.",
"In step (i) the algorithm has to compute all (modular) elliptic curves over ${\\mathbb {Q}}$ of conductor $N$ for preciselyNotice that $6\\cdot 2^{\\vert {}S{}\\vert -1}$ is the number of positive rational integers dividing $2^4N_S$ , since $2\\in S$ .",
"$6\\cdot 2^{\\vert {}S{}\\vert -1}$ positive integers $N$ .",
"Here we can exploit the fact that Cremona's algorithm proceeds by induction over the divisors of $N$ , see Section .",
"However at the time of writing it is not clear to us what is the running time of Cremona's algorithm.",
"We also mention that step (i) greatly benefits from the ongoing extension of Cremona's tables which in particular list all (modular) elliptic curves over ${\\mathbb {Q}}$ of given conductor $N$ .",
"As of August 2014, these tables are complete for all $N\\le 350 000$ .",
"The complexity of step (ii) crucially depends on the size of ${\\mathcal {T}}$ .",
"It is an open (Diophantine) problem to find a simple formula for $\\vert {}\\mathcal {T}{}\\vert $ in terms of $S$ .",
"However one can give an upper bound for $\\vert {}\\mathcal {T}{}\\vert $ in terms of $S$ .",
"For example, the work of Ellenberg, Helfgott and Venkatesh [83], [58] implies that $\\vert {}\\mathcal {T}{}\\vert \\ll N_S^{0.1689}$ .",
"In step (ii) the algorithm first needs to compute the prime factorization of each $d\\in \\mathcal {T}$ and then it needs to compute certain integers $a,b,c$ for most of the disjoint 3-partitions of the set $S$ .",
"Inequality (REF ) implies that any $d\\in \\mathcal {T}$ satisfies $\\log d=O(N_S^2)$ , and the number of disjoint 3-partitions of $S$ is $3^{\\vert {}S{}\\vert }$ .",
"Hence the above discussions show that the running time of step (ii) is at most polynomial in terms of $N_S$ .",
"This running time estimate can be considerably improved by assuming various conjectures.",
"Indeed for each $d\\in \\mathcal {T}$ the $abc$ -conjecture ($abc$ ) recalled in Remark REF would provide that $\\log d=O(\\log N_S)$ , and it follows from Brumer–Silverman [27] that the Birch–Swinnerton-Dyer conjecture (BSD) together with the General Riemann Hypothesis (GRH) would give $\\vert {}\\mathcal {T}{}\\vert =O_\\varepsilon (N_S^\\varepsilon )$ .",
"Hence the running time of step (ii) is $O_\\varepsilon (N_S^\\varepsilon )$ if the three conjectures (BSD), (GRH) and ($abc$ ) all hold.",
"Remark 3.5 (Variation) We now discuss a variation of Algorithm REF which in practice provides an improvement for large $\\vert {}S{}\\vert $ .",
"The idea is that instead of using in (ii) the curve of Lemma REF , one can work with Legendre curves as in [89].",
"(i)' Use Cremona's algorithm to compute the set $\\mathcal {T}_W$ of minimal Weierstrass models over ${\\mathbb {Z}}$ of modular elliptic curves over ${\\mathbb {Q}}$ with conductor dividing $2^4N_S$ .",
"(ii)' For each $W\\in \\mathcal {T}_W$ determine its set $\\lambda (W)=\\lbrace \\lambda ,1-\\lambda ,\\lambda ^{-1},\\cdots \\rbrace $ of Legendre parameters by computing the three roots of $f+f_2^2/4$ , where $Y^2+f_2(X)Y=f(X)$ defines $W$ .",
"If $\\lambda \\in \\lambda (W)$ and $(\\lambda ,1-\\lambda )$ satisfies (REF ) then output $(x,y)=(\\lambda ,1-\\lambda )$ .",
"This algorithm outputs the set of solutions of (REF ).",
"Indeed for any solution $(x,y)$ of (REF ) the proof of Lemma REF gives $W\\in \\mathcal {T}_W$ with $x\\in \\lambda (W)$ .",
"The running times of (i) and (i)' are essentially equal, and it follows for example from [89] that the running time of step (ii)' is polynomial in terms of $N_S$ .",
"Furthermore one can show that the running time of step (ii)' is in fact $O_\\varepsilon (N_S^\\varepsilon )$ , provided that all three conjectures (BSD), (GRH) and $(abc)$ hold.",
"If $\\vert {}S{}\\vert $ is large, then step (ii)' is in practice considerably faster than (ii) since the latter iterates in addition over many disjoint 3-partitions of $S$ .",
"However for large $\\vert {}S{}\\vert $ the bottleneck of the algorithm is the computation of $\\mathcal {T}_W$ or $\\mathcal {T}_{\\Delta }$ , and for small $\\vert {}S{}\\vert $ it turns out that (ii)' and (ii) are essentially equally fast.",
"Hence we implemented the algorithm involving (i) and (ii), since the implementation of (i) is simpler compared to (i)'."
],
[
"Applications.",
"Let $\\Sigma (S)$ be the set of solutions of (REF ) up to symmetry.",
"For any $N\\in {\\mathbb {Z}}_{\\ge 1}$ , consider the set $\\Sigma (N)=\\cup _S \\Sigma (S)$ with the union taken over all sets $S$ with $N_S\\le N$ .",
"On using an implementation in Sage of the above Algorithm REF , we computed the sets $\\Sigma (N)$ for all $N\\le 20000$ .",
"This computation was very fast for $N\\le 20000$ , since for such $N$ part (i) of our Algorithm REF can use Cremona's database which contains in particular the required data for all elliptic curves of conductor dividing $2^4N< 350 000$ .",
"On the other hand, if the required data of the involved elliptic curves is not already known, then our Algorithm REF is often not practical anymore.",
"Here the problem is the application of Cremona's algorithm (using modular symbols) in step (i) which requires a huge amount of memory in order to deal with medium sized or large conductors.",
"Remark 3.6 ($abc$ -triples) The $abc$ -conjecture $(abc)$ states that for any real number $\\varepsilon >0$ there are only a finite number of solutions of (REF ) with quality larger than $1+\\varepsilon $ , see for example [113].",
"Here the quality $q=q(a,b,c)$ of a solution $(a,b,c)$ of (REF ) is defined as $q=\\frac{\\log \\max (\\vert {}a{}\\vert ,\\vert {}b{}\\vert ,\\vert {}c{}\\vert )}{\\log \\textnormal {rad}(abc)}.$ Among all known high quality $abc$ -triples the top two, as of October 2014, can be obtained with our method: $(2,3^{10}109,23^5)$ with $q=1.6299$ due to Reyssat, and $(11^2,3^25^67^3,2^{21}23)$ with $q=1.6260$ due to de Weger; see abcathome.com."
],
[
"Algorithm via height bounds",
"In this section we use the optimized height bounds in Proposition REF to construct an algorithm which practically resolves the $S$ -unit equation (REF ).",
"The decomposition of our algorithm is inspired by de Weger's classical method whose main ingredients are as follows: (1) De Weger uses explicit height bounds for the solutions of (REF ), based on the theory of logarithmic forms, to rule out the existence of solutions with very “large\" height.",
"See Section for references and more details.",
"(2) Then de Weger tries to further reduce the height bounds in (1) by using the LLL lattice reduction algorithm applied to certain approximation lattices, which are defined using $p$ -adic logarithms.",
"If this reduction can be applied, then he can rule out in addition the existence of solutions with “large\" and “medium sized\" height.",
"(3) To find most of the solutions with “small\" height de Weger applies a sieve which we call de Weger's sieve, see Section REF .",
"He repeats this step as many times as required to make sure that the remaining solutions have “tiny” height.",
"(4) Finally de Weger checks (by hand) all potential solutions of “tiny\" height.",
"In our algorithm we replace the height bounds in (1) by the optimized height bounds which we shall work out in Section .",
"These optimized bounds are strong enough such that we can now omit the reduction step (2).",
"In Section REF we discuss a slight variation of de Weger's sieve described in (3).",
"Then in Section REF we develop a refined sieve which has considerably improved running time in practice, in particular for all sets $S$ with $\\vert {}S{}\\vert >6$ .",
"Moreover, in Section REF we construct an enumeration algorithm which is faster than the standard algorithm in (4).",
"In fact our enumeration is fast enough such that it is now beneficial to go from (3) to (4) at an earlier stage, which leads to additional running time improvements.",
"Finally we present and discuss our Algorithm REF solving the $S$ -unit equation (REF ), and we also give various applications of our algorithm.",
"Before we begin to carry out the above program, we introduce some notation which will be used throughout Section REF .",
"Let $n\\in {\\mathbb {Z}}_{\\ge 1}$ and consider two vectors $l,u\\in \\mathbb {R}^n$ .",
"We write $l\\le u$ if and only if $l_i\\le u_i$ for all $i\\in \\lbrace 1,\\cdots ,n\\rbrace $ .",
"Further by $l<u$ we mean that $l\\le u$ with $l\\ne u$ , and we use the symbol $l\\lnot \\le u$ in order to say that $l\\le u$ does not hold.",
"In other words, we use poset (partially ordered set) and not coordinate-wise notation for $``<\"$ and $``\\lnot \\le \"$ .",
"The use of poset notation for $``<\"$ and $``\\lnot \\le \"$ will simplify our exposition.",
"Next, we suppose that $(x,y)$ is a solution of the $S$ -unit equation (REF ).",
"It will be convenient for the description of the algorithm to use the following quantities associated to $(x,y)$ .",
"For any rational prime $p$ we put $m_p(x,y)=\\max (\\vert {}{\\textnormal {ord}}_p(x){}\\vert ,\\vert {}{\\textnormal {ord}}_p(y){}\\vert )$ , and then we define $m(x,y)= (m_p(x,y))_{p\\in S} \\ \\ \\textnormal { and } \\ \\ M(x,y)= \\max (m_p(x,y)\\log p\\,;\\,p\\in S).$ Proposition REF gives an upper bound $M_0$ for $M(x,y)$ .",
"Hence it remains to find the solutions $(x,y)$ of (REF ) with $M(x,y)$ between zero and $M_0$ , or between $M_{k+1}$ and $M_{k}$ for $k=0,\\cdots ,k_0$ and some convenient sequence $0=M_{k_0}<\\ldots < M_{1}<M_0$ .",
"For this purpose we shall use the following refinements of steps (3) and (4) of de Weger's method."
],
[
"De Weger's sieve",
"We now begin to describe the sieve which is used in step (3) of de Weger's method.",
"In fact we shall describe a slight variation of this sieve, see Remark REF .",
"For any given $M^{\\prime }, M^{\\prime \\prime }\\in {\\mathbb {R}}_{\\ge 0}$ , we want to enumerate all solutions $(x,y)$ of (REF ) with $M^{\\prime }<M(x,y)\\le M^{\\prime \\prime }$ .",
"For this purpose, it suffices by (REF ) to solve the following problem: For any two vectors $l,u\\in {\\mathbb {Z}}^S$ with $0\\le l\\le u$ , find all solutions $(x,y)$ of (REF ) which satisfy $m(x,y)\\lnot \\le l \\ \\ \\ \\textnormal { and } \\ \\ \\ m(x,y)\\le u.$ If a solution $(x,y)$ of (REF ) satisfies (REF ), then all its symmetric solutions satisfy (REF ) as well.",
"The condition $m(x,y)\\lnot \\le l$ means that there exists at least one “large\" exponent in the prime factorization of $x$ or $y$ .",
"We now exploit this to reduce (REF ) to a Diophantine problem whose solutions can be quickly enumerated.",
"Suppose that $(x,y)$ is a solution of (REF ) which satisfies (REF ).",
"Then there exists $q\\in S$ with $m_q(x,y)\\ge 1+l_q$ .",
"Further the discussion given above (REF ) delivers a solution $(a,b,c)$ of (REF ) with $x=\\frac{a}{c}$ and $y=\\frac{b}{c}$ .",
"After possibly replacing $(x,y)$ with a symmetric solution, we may and do assume that $q^{l_q+1}$ divides $c$ .",
"Then it holds that $a+b=0\\mod {q}^{l_q+1}$ and thus $(a/b)^2 = 1$ in $G$ for $G=({\\mathbb {Z}}/q^{l_q+1}{\\mathbb {Z}})^\\times .$ Here we squared plainly in order to get rid of the minus sign, and we used that $a,b$ are both coprime to $q$ which provides that $a,b$ are in $G$ .",
"Next we write $(a/b)^2 =\\prod _{p\\in {S\\backslash q}} p^{2\\gamma _p}$ with $\\gamma _p={\\textnormal {ord}}_p(a/b)$ .",
"Then we see that any solution $(x,y)$ of (REF ) satisfying (REF ) induces a solution $\\gamma =(\\gamma _p)\\in {\\mathbb {Z}}^{S\\backslash q}$ of the following problem: If $l,u\\in {\\mathbb {Z}}^{S}$ are given with $0\\le l\\le u$ , then for each $q\\in S$ find all $\\gamma \\in {\\mathbb {Z}}^{{S\\backslash q}}$ such that $\\vert {}\\gamma _p{}\\vert \\le u_p$ for all $p\\in {S\\backslash q}$ and such that $\\prod _{p\\in {S\\backslash q}} p^{2\\gamma _p} = 1 \\ \\textnormal { in } \\ G.$ Furthermore if $\\gamma $ is a solution of (REF ) then one can quickly reconstruct all solutions of (REF ) satisfying (REF ) which map to $\\gamma $ via the above construction.",
"Indeed one defines $a=\\prod _{\\gamma _p>0}p^{\\gamma _p}$ , $b_{+}=\\prod _{\\gamma _p<0}p^{-\\gamma _p}$ and $b_-=-b_+$ , and then one checks for each $b\\in \\lbrace b_+,b_-\\rbrace $ whether $\\textnormal {rad}(a+b)\\operatorname{|}N_S$ and whether $(x,y)=(\\frac{a}{a+b},\\frac{b}{a+b})$ satisfies (REF ).",
"In particular, we see that we can quickly enumerate all solutions of (REF ) satisfying (REF ) provided that we know all solutions of (REF ).",
"Finally it remains to enumerate all solutions of (REF ).",
"For this purpose we observe that the set of vectors $\\gamma \\in {\\mathbb {Z}}^{{S\\backslash q}}$ satisfying (REF ) is the intersection of a certain lattice $\\Gamma \\subseteq {\\mathbb {Z}}^{S\\backslash q}$ with the cube $\\lbrace \\gamma \\,;\\,|\\gamma _p|\\le u_p\\rbrace \\subset \\mathbb {R}^{S\\backslash q}$ .",
"To determine this intersection we combine the following two algorithms: The first algorithm is an application of Teske's “Minimize” [157].",
"It takes as the input a set of generators $g_1,\\ldots ,g_d$ of a finite abelian groupHere it is not required to know the group $G_0$ explicitly.",
"In fact it suffices to know a number divisible by $\\vert {}G_0{}\\vert $ and to be able to compute the following operations in $G_0$ : Multiplying elements, inverting elements and testing whether an element is the neutral element.",
"$G_0$ , and it outputs a basis for the lattice $\\Gamma \\subseteq {\\mathbb {Z}}^d$ formed by those $\\gamma \\in {\\mathbb {Z}}^d$ with $\\prod _{i=1}^dg_i^{\\gamma _i}=1$ .",
"The second algorithm is a version of the Fincke–Pohst algorithm, see Remark REF .",
"For any $d\\in {\\mathbb {Z}}_{\\ge 1}$ it takes as the input a basis of a lattice $\\Gamma \\subseteq {\\mathbb {Z}}^d$ together with an ellipsoid $\\mathcal {E}\\subset {\\mathbb {R}}^d$ centered at the origin, and it outputs the intersection $\\Gamma \\cap \\mathcal {E}$ .",
"More precisely, we apply (T) and (FP) as follows.",
"Let $G_0$ be the subgroup of $G$ which is generated by the squares of the elements in $S\\backslash q$ .",
"An application of (T) with $G_0$ gives a basis for the lattice $\\Gamma $ underlying (REF ), and then (FP) computes the intersection of $\\Gamma $ with the smallest ellipsoid $\\mathcal {E}\\subset \\mathbb {R}^{S\\backslash q}$ that contains the cube $\\lbrace \\gamma \\,;\\,|\\gamma _p|\\le u_p\\rbrace \\subset \\mathbb {R}^{S\\backslash q}$ .",
"Algorithm 3.7 (De Weger's sieve) The input consists of a finite set of rational primes $S$ together with two vectors $l,u\\in {\\mathbb {Z}}^S$ such that $0\\le l\\le u$ , and the output is the set of solutions $(x,y)$ of the $S$ -unit equation (REF ) which satisfy (REF ).",
"The algorithm: After possibly shrinking $S$ , we may and do assume that all $u_p\\ge 1$ .",
"If $2\\notin S$ then output the empty set, and if $2\\in S$ then do the following for each $q\\in S$ .",
"(i) Use the application (T) of Teske's algorithm in order to compute a basis for the lattice $\\Gamma \\subseteq {\\mathbb {Z}}^{S\\backslash q}$ of all $\\gamma \\in {\\mathbb {Z}}^{S\\backslash q}$ with $\\prod _{p\\in S\\backslash q}p^{2\\gamma _p}=1$ in $({\\mathbb {Z}}/q^{l_q+1}{\\mathbb {Z}})^\\times $ .",
"(ii) Define the ellipsoid $\\mathcal {E}=\\lbrace x\\in {\\mathbb {R}}^{S\\backslash q}\\,;\\,\\sum _{p\\in S\\backslash q} |x_p/u_p|^2 \\le |S|-1\\rbrace $ .",
"Then compute the intersection $\\Gamma \\cap \\mathcal {E}$ using the version (FP) of the Fincke–Pohst algorithm.",
"(iii) For each $\\gamma \\in \\Gamma \\cap \\mathcal {E}$ lying in the cube $\\lbrace \\gamma \\,;\\,|\\gamma _p| \\le u_p\\rbrace \\subset \\mathbb {R}^{S\\backslash q}$ , define the numbers $a=\\prod _{\\gamma _p>0}p^{\\gamma _p}$ , $b_+=\\prod _{\\gamma _p<0}p^{-\\gamma _p}$ and $b_-=-b_+$ .",
"For any $b\\in \\lbrace b_{+},b_-\\rbrace $ such that $c=a+b$ satisfies ${\\textnormal {ord}}_p(c)\\le u_p$ for all $p\\in S$ , $q^{l_q+1}\\mid c$ and $\\prod _{p\\in S} p^{{\\textnormal {ord}}_p(c)}=\\vert {}c{}\\vert $ , output the solution $(x,y)=(\\tfrac{a}{c},\\tfrac{b}{c})$ together with all its symmetric solutions."
],
[
"Correctness.",
"We now verify that this algorithm indeed finds all solutions of (REF ) satisfying (REF ).",
"Suppose that $(x,y)$ is such a solution.",
"Then our assumption $m(x,y)\\lnot \\le l$ gives $q\\in S$ together with $\\gamma \\in {\\mathbb {Z}}^{S\\backslash q}$ which is associated to $(x,y)$ via the construction given above (REF ).",
"This $\\gamma $ lies in the lattice $\\Gamma $ appearing in (i).",
"Furthermore, our assumption $m(x,y)\\le u$ implies that $\\gamma $ is also contained in the ellipsoid $\\mathcal {E}$ from step (ii).",
"In particular $\\gamma $ lies in the intersection $\\Gamma \\cap \\mathcal {E}$ computed in step (ii), and $m(x,y)\\le u$ provides that $\\vert {}\\gamma _p{}\\vert \\le u_p$ for all $p\\in S\\backslash q$ .",
"Therefore on using that $\\gamma $ is associated to $(x,y)$ via the construction described above (REF ), we see that step (iii) produces our solution $(x,y)$ as desired."
],
[
"Complexity.",
"Step (i) uses (T).",
"The algorithm (T) is reminiscent of the discrete logarithm problem in the multiplicative group $G$ , and the bottleneck for this is the prime factorization of $\\vert {}G{}\\vert $ .",
"In our situation it holds $\\vert {}G{}\\vert = q^{l_q}(q-1)$ and therefore it suffices to factor $q-1$ , which is in general much easier than factoring an arbitrary number of size $\\vert {}G{}\\vert $ .",
"In fact step (i) is in practice not the bottleneck of Algorithm REF , except in the case when $S$ consists of the prime 2 together with a few large primes.",
"Usually step (ii) is the bottleneck of Algorithm REF , and thus we shall refine this step in Section REF below.",
"Remark 3.8 To find a basis for the lattice $\\Gamma $ , de Weger used $q$ -adic logarithms instead of the application (T) of Teske's algorithm.",
"In fact (T) was already used in generalizations of de Weger's method to number fields, see Wildanger [165] and Smart [138]."
],
[
"Application.",
"For later use we now mention how we will apply Algorithm REF in order to find all solutions $(x,y)$ of (REF ) satisfying $M^{\\prime }< M(x,y)\\le M^{\\prime \\prime }$ , for some given $M^{\\prime },M^{\\prime \\prime }$ in $\\mathbb {R}_{\\ge 0}$ with $M^{\\prime }<M^{\\prime \\prime }$ .",
"Consider the two vectors $l=(l_p)$ and $u=(u_p)$ in ${\\mathbb {Z}}^S$ , with $l_p=\\left\\lfloor {M^{\\prime }/\\log p}\\right\\rfloor \\ \\ \\ \\textnormal { and } \\ \\ \\ u_p=\\left\\lfloor {M^{\\prime \\prime }/\\log p}\\right\\rfloor .$ We observe that $0\\le l\\le u$ .",
"Then an application of Algorithm REF with $l,u$ finds in particular all solutions $(x,y)$ of (REF ) satisfying $M^{\\prime }< M(x,y)\\le M^{\\prime \\prime }$ , as desired.",
"Remark 3.9 According to (REF ) below, splitting the possible candidates with respect to values of $M(x,y)$ is in particular reasonable in the case when $\\log p$ is small compared to $\\vert {}S{}\\vert $ for all $p\\in S$ .",
"In this case, the iteration over all $q\\in S$ in Algorithm REF will take about equally long for each $q\\in S$ .",
"If $S$ contains primes which are exponentially large in terms of $\\vert {}S{}\\vert $ , then one should split the initial space $0\\le M(x,y)\\le M_0$ into more general pieces of the form $\\lbrace m(x,y)\\lnot \\le u(k+1) \\textnormal { and } m(x,y)\\le u(k)\\rbrace $ for suitable $u(k)\\in {\\mathbb {Z}}^{S}$ with $k=0,1,\\cdots ,k_0$ .",
"Here the vectors $u(k)\\in {\\mathbb {Z}}^S$ should satisfy $0=u(k_0)<\\ldots <u(1)<u(0)$ and $u(0)_p=\\left\\lfloor {M_0/\\log p}\\right\\rfloor $ for $p\\in S$ .",
"In practice it is reasonable to choose greedily the next $u(k+1)<u(k)$ , such that the subsequent sieving step is as fast as possible."
],
[
"Refined sieve",
"In this section we continue our notation introduced above.",
"We begin with a short description of the geometric main idea underlying our refinement.",
"Recall that in de Weger's sieve one needs to determine the intersection of a certain lattice $\\Gamma \\subseteq {\\mathbb {Z}}^{S\\backslash q}$ with the cube $\\lbrace x\\,;\\,\\vert {}x_p{}\\vert \\le u_p\\rbrace \\subset \\mathbb {R}^{S\\backslash q}$ , and for this purpose Algorithm REF (ii) first determines $\\Gamma \\cap \\mathcal {E}$ for $\\mathcal {E}$ the smallest ellipsoid containing the cube.",
"However, in terms of the rank $s-1$ where $s=\\vert {}S{}\\vert $ , the volume of $\\mathcal {E}$ is exponentially larger than the volume of the cube.",
"In our refinement we essentially truncate from the cube some regions near the faces of codimension $2,\\ldots ,\\left\\lfloor {s/3}\\right\\rfloor $ .",
"For $s\\ge 6$ the resulting geometric object is contained in a notably smaller ellipsoid $\\mathcal {E}^{\\prime }$ , which allows to determine $\\Gamma \\cap \\mathcal {E}^{\\prime }$ considerably faster than $\\Gamma \\cap \\mathcal {E}$ .",
"To explain in more detail our refined sieve, we need to introduce additional notation.",
"Let $(x,y)$ be a solution of the $S$ -unit equation (REF ), and let $(a,b,c)$ be a solution of (REF ) with $(x,y)=(\\tfrac{a}{c},\\tfrac{b}{c})$ .",
"We take $j\\in \\lbrace 1,\\cdots ,t\\rbrace $ for $t=\\max (1,\\left\\lfloor {\\vert {}S{}\\vert /3}\\right\\rfloor )$ .",
"This choice of $t$ takes into account that any prime appearing in the prime factorization of $x$ or $y$ divides one of the three coprime integers $a,b,c$ .",
"For any $n\\in {\\mathbb {Z}}$ with $\\textnormal {rad}(n)\\operatorname{|}N_S$ , we denote by $\\mu _j(n)$ the $j$ -th largestMore precisely, $\\mu _j(n)$ is the $j$ -th largest element of the ordered multi-set of cardinality $\\vert {}S{}\\vert $ obtained by ordering the $\\vert {}S{}\\vert $ non-negative real numbers ${\\textnormal {ord}}_p(n)\\log p$ , $p\\in S$ , with respect to their absolute values.",
"of the real numbers ${\\textnormal {ord}}_p(n)\\log p$ , $p\\in S$ .",
"Then we define $\\mu _j(x,y)=\\max \\bigl (\\mu _j(a),\\mu _j(b),\\mu _j(c)\\bigl ) \\ \\ \\ \\textnormal { and } \\ \\ \\ \\mu (x,y) = \\bigl (\\mu _1(x,y),\\ldots ,\\mu _t(x,y)\\bigl ).$ We observe that $\\mu _1(x,y)=M(x,y)$ .",
"However we point out that if $j\\in \\lbrace 2,\\cdots ,t\\rbrace $ then $\\mu _j(x,y)$ is not necessarily the $j$ -th largest of the numbers $m_p(x,y)\\log p$ , $p\\in S$ .",
"Now we consider the following problem: For any given vectors $\\mu ^{\\prime }, \\mu ^{\\prime \\prime }\\in {\\mathbb {Z}}^t$ having monotonously decreasing entries such that $0\\le \\mu ^{\\prime }< \\mu ^{\\prime \\prime }$ , find all solutions $(x,y)$ of (REF ) that satisfy $\\mu (x,y) \\lnot \\le \\mu ^{\\prime } \\ \\ \\ \\textnormal { and } \\ \\ \\ \\mu (x,y) \\le \\mu ^{\\prime \\prime }.$ If a solution $(x,y)$ of (REF ) satisfies (REF ), then all its symmetric solutions satisfy (REF ) as well.",
"Further we note that the condition $\\mu (x,y)\\lnot \\le \\mu ^{\\prime }$ implies, for some $j\\in \\lbrace 1,\\cdots ,t\\rbrace $ , the existence of at least $j$ “large\" exponents in the prime factorization of $x$ or $y$ .",
"In the following algorithm we exploit this in order to work with lattices of rank $\\vert {}S{}\\vert -j$ .",
"Algorithm 3.10 (Refined sieve) The input is a finite set of rational primes $S$ , together with two vectors $\\mu ^{\\prime }, \\mu ^{\\prime \\prime }\\in {\\mathbb {Z}}^t$ having monotonously decreasing entries such that $0\\le \\mu ^{\\prime }< \\mu ^{\\prime \\prime }$ ; where $t=\\max (1,\\left\\lfloor {\\vert {}S{}\\vert /3}\\right\\rfloor )$ .",
"The output is the set of solutions $(x,y)$ of the $S$ -unit equation (REF ) which satisfy (REF ).",
"The algorithm: If $2\\notin S$ then output the empty set, and if $2\\in S$ then do the following for each non-empty subset $T\\subseteq S$ of cardinality $\\vert {}T{}\\vert \\le t$ .",
"Put $n=\\prod _{q\\in T} q^{\\left\\lfloor {\\mu ^{\\prime }_{\\vert {}T{}\\vert }/\\log q}\\right\\rfloor +1}$ , and use the application (T) of Teske's algorithm to compute a basis of the lattice $\\Gamma _T$ of all $\\gamma \\in {\\mathbb {Z}}^{S\\backslash T}$ with $\\prod _{p\\in S\\backslash T} p^{2\\gamma _p} = 1$ in $({\\mathbb {Z}}/n{\\mathbb {Z}})^\\times .$ Then use the version (FP) of the Fincke–Pohst algorithm in order to determine the intersection of the lattice $\\Gamma _T\\subseteq {\\mathbb {Z}}^{S\\backslash T}$ with the ellipsoid $\\mathcal {E}_T\\subset \\mathbb {R}^{S\\backslash T}$ defined by $\\mathcal {E}_T=\\lbrace x\\in \\mathbb {R}^{S\\backslash T}\\,;\\,\\sum _{p\\in S\\backslash T} |x_p \\log p|^2 \\le \\sum _{i=1}^{|S\\backslash T|} (\\mu ^{\\prime \\prime }_{\\min (\\left\\lceil {i/2}\\right\\rceil ,t)})^2\\rbrace .$ For each $\\gamma \\in \\Gamma _T\\cap \\mathcal {E}_T$ , define $a=\\prod _{\\gamma _p>0}p^{\\gamma _p}$ , $b_+=\\prod _{\\gamma _p<0}p^{-\\gamma _p}$ and $b_-=-b_+$ .",
"For any $b\\in \\lbrace b_{+},b_-\\rbrace $ such that $c=a+b$ satisfies $\\prod _{p\\in S}p^{{\\textnormal {ord}}_p(c)}=\\vert {}c{}\\vert $ and such that $(x,y)=(\\tfrac{a}{c},\\tfrac{b}{c})$ satisfies (REF ), output $(x,y)$ together with all its symmetric solutions."
],
[
"Correctness.",
"To see that this algorithm works correctly, we suppose that $(x,y)$ is a solution of the $S$ -unit equation (REF ) that satisfies (REF ).",
"Let $(a,b,c)$ be a solution of (REF ) with $(x,y)=(\\tfrac{a}{c},\\tfrac{b}{c})$ .",
"Our assumption $\\mu (x,y)\\lnot \\le \\mu ^{\\prime }$ then gives a subset $T\\subseteq S$ of cardinality $j=\\vert {}T{}\\vert $ in $\\lbrace 1,\\cdots ,t\\rbrace $ together with $C\\in \\lbrace a,b,c\\rbrace $ such that ${\\textnormal {ord}}_q(C)\\log q> \\mu ^{\\prime }_j$ for all $q\\in T$ .",
"Furthermore after replacing $(x,y)$ by a symmetric solution, we may and do assume that $C=c$ .",
"It holds that ${\\textnormal {ord}}_q(c)\\ge \\lfloor \\mu ^{\\prime }_{\\vert {}T{}\\vert }/\\log q\\rfloor +1$ for all $q\\in T$ .",
"Hence $n$ divides $c$ , and this implies that $a,b$ are both invertible in ${\\mathbb {Z}}/n{\\mathbb {Z}}$ since $\\gcd (a,b,c)=1$ .",
"Therefore the equation $a+b=c$ leads to $(a/b)^2=1$ in $({\\mathbb {Z}}/n{\\mathbb {Z}})^\\times $ .",
"Then on writing $(a/b)^2=\\prod _{p\\in S\\backslash T}p^{2\\gamma _p}$ with $\\gamma _p={\\textnormal {ord}}_p(a/b)$ , we see that the vector $\\gamma =(\\gamma _p)\\in {\\mathbb {Z}}^{S\\backslash T}$ lies in the lattice $\\Gamma _T\\subseteq {\\mathbb {Z}}^{S\\backslash T}$ of (i).",
"Moreover, if $i\\in \\lbrace 1,\\cdots ,\\vert {}S\\backslash T{}\\vert \\rbrace $ then the $i$ -th largest of the real numbers $\\vert {}\\gamma _p\\log p{}\\vert ,$ $p\\in S\\backslash T$ , is at most $\\max (\\mu _\\iota (a),\\mu _\\iota (b))$ for $\\iota =\\min (\\left\\lceil {i/2}\\right\\rceil ,t)$ , and our assumption $\\mu (x,y)\\le \\mu ^{\\prime \\prime }$ implies that $\\max (\\mu _\\iota (a),\\mu _\\iota (b))\\le \\mu ^{\\prime \\prime }_{\\iota }$ .",
"Hence we deduce that $\\gamma $ lies in the ellipsoid $\\mathcal {E}_T$ of (ii) and then we conclude that (iii) produces our solution $(x,y)$ as desired."
],
[
"Application.",
"For any $M\\in {\\mathbb {Z}}_{\\ge 1}$ , we would like to find all solutions $(x,y)$ of the $S$ -unit equation (REF ) with $M(x,y)\\le M$ .",
"For this purpose we can use for example Algorithm REF , which we successively apply with $\\mu ^{\\prime }(n),\\mu ^{\\prime \\prime }(n)\\in {\\mathbb {Z}}^t$ for $n=M+1, M,\\cdots ,1$ ; where $\\mu ^{\\prime }(n)=\\left\\lfloor {(n-1)\\cdot (1,1/2,\\cdots ,1/t)}\\right\\rfloor \\textnormal { for } n\\in \\lbrace 1,\\cdots , M+1\\rbrace ,$ $\\mu ^{\\prime \\prime }(M+1)=M\\cdot (1,\\cdots ,1), \\ \\ \\ \\mu ^{\\prime \\prime }(n)=\\mu ^{\\prime }(n+1) \\textnormal { for } n\\in \\lbrace 1,\\cdots , M\\rbrace .$ Here $\\left\\lfloor {v}\\right\\rfloor =(\\left\\lfloor {v_i}\\right\\rfloor )$ for $v=(v_i)$ a vector with entries $v_i\\in \\mathbb {R}$ .",
"Suppose that $\\vert {}S{}\\vert \\ge 6$ , $T=\\lbrace q\\rbrace $ and $\\mu ^{\\prime \\prime }=\\mu ^{\\prime \\prime }(n)$ with $n\\in \\lbrace 1,\\cdots ,M\\rbrace $ .",
"Then the “radius\" $R$ of the ellipsoid in (REF ) satisfies $R=\\sum _{i=1}^{\\vert {}S{}\\vert -1}(\\mu ^{\\prime \\prime }_{\\iota (i)})^2\\le n^2\\left(\\tfrac{t+1}{t^2}+4\\sum _{i=1}^{2t}\\tfrac{1}{i^2}\\right)$ for $\\iota (i)=\\min (\\left\\lceil {i/2}\\right\\rceil ,t)$ .",
"It follows that $R\\le 7n^2$ , and this uniform bound together with $R<(\\vert {}S{}\\vert -1)n^2$ shows that $R$ is considerably smaller (in particular for large $\\vert {}S{}\\vert $ ) than the “radius\" $(\\vert {}S{}\\vert -1)n^2$ of the ellipsoid in Algorithm REF with $u=(\\left\\lfloor {n/\\log p}\\right\\rfloor )$ .",
"The following complexity discussion provides some motivation for our choice of $t$ and $\\mu ^{\\prime }(n),\\mu ^{\\prime \\prime }(n)$ ."
],
[
"Complexity.",
"To discuss the improvements provided by our refined algorithm for sets $S$ with $\\vert {}S{}\\vert \\ge 6$ , we consider $M\\in {\\mathbb {Z}}_{\\ge 2}$ and we apply Algorithm REF with $\\mu ^{\\prime }(n),\\mu ^{\\prime \\prime }(n)$ for some $n\\in \\lbrace 1,\\cdots , M\\rbrace $ .",
"Recall that Algorithm REF needs to compute in particular $\\Gamma _{T}\\cap \\mathcal {E}_T$ for all $T\\subseteq S$ with $\\vert {}T{}\\vert \\in \\lbrace 1,\\cdots ,t\\rbrace $ .",
"Here the cases $T=\\lbrace q\\rbrace $ are essentially an application of Algorithm REF as in (REF ), with $M^{\\prime }=n-1$ and $M^{\\prime \\prime }=n$ .",
"However in light of the discussions surrounding (REF ), the crucial difference is that the volume of the ellipsoid $\\mathcal {E}_{\\lbrace q\\rbrace }$ of our refined Algorithm REF is considerably smaller than the volume of the corresponding ellipsoid $\\mathcal {E}$ of Algorithm REF .",
"In practice, this is the reason for the significantly improved running time of Algorithm REF for $\\vert {}S{}\\vert \\ge 6$ .",
"We note that our refinement needs to iterate in addition over certain sets $T$ with $\\vert {}T{}\\vert \\ge 2$ .",
"However these additional iterations have little influence on the running time, since the running time for $\\vert {}T{}\\vert \\ge 2$ is in practice much better than for $\\vert {}T{}\\vert =1$ .",
"Indeed if $\\vert {}T{}\\vert \\ge 2$ then the lattices $\\Gamma _T$ have smaller rank $\\vert {}S{}\\vert -\\vert {}T{}\\vert $ , which crucially improves the running time of (FP) in Algorithm REF (ii).",
"Remark 3.11 (Implementation of Fincke–Pohst) To avoid numerical issues in Algorithms REF and REF , we use our own implementation of the version of the Fincke–Pohst algorithm [63] described in (FP).",
"Our implementation only uses integer arithmetic, and in particular we do not take square roots.",
"For this purpose the coordinates and the bound in (REF ) have been scaled and rounded to integers in such a way, that (FP) will return possibly slightly more candidates $\\gamma $ than those fulfilling (REF ).",
"Furthermore we use an LLL improvement of the original Fincke–Pohst algorithm, such as for example the one in Cohen [38].",
"This is important since the original implementation [63] becomes in many instances very slow, in fact already for $|S|\\ge 10$ it is too slow for our purpose."
],
[
"Refined enumeration",
"We continue the notation introduced above and we take $u\\in {\\mathbb {Z}}^S$ with $u\\ge 0$ .",
"To find all solutions $(x,y)$ of the $S$ -unit equation (REF ) satisfying $m(x,y)\\le u$ , we use the following refined enumeration Algorithm REF .",
"In practice, our refined enumeration algorithm is considerably faster than the standard algorithm which is described in the complexity discussion below.",
"For any subset $T\\subseteq S$ , we define its weight $w(T)=\\prod _{p\\in T}(1+u_p)$ .",
"Algorithm 3.12 (Refined enumeration) The input is a finite set of rational primes $S$ , together with a vector $u\\in {\\mathbb {Z}}^{S}$ such that $u\\ge 0$ .",
"The output consists of all solutions $(x,y)$ of the $S$ -unit equation (REF ) that satisfy $m(x,y)\\le u$ .",
"The algorithm: For each subset $S_a\\subseteq S$ with $w(S_a)\\ge w(S)^{1/3}$ , do the following.",
"Split the set $S_a=S_{a_1}\\operatorname{\\dot{\\cup }}S_{a_2}$ into disjoint subsets $S_{a_1},S_{a_2}$ such that $w(S_{a_1})\\le w(S)^{1/2}$ is fulfilled as tight as possible.",
"Construct the set $X$ , implemented as a hash, of all $a_1\\in {\\mathbb {Z}}$ such that $\\vert {}a_1{}\\vert =\\prod _{p\\in S_{a_1}} p^{v_p(a_1)}$ with $0\\le v_p(a_1)\\le u_p$ for $v_p={\\textnormal {ord}}_p$ .",
"Construct the set $Y=\\cup Y(S_b,S_c)$ with the union taken over all pairs $(S_b,S_c)$ such that $S_a\\operatorname{\\dot{\\cup }}S_b\\operatorname{\\dot{\\cup }}S_c=S$ and such that either $w(S_b)<w(S_c)$ or $w(S_b)=w(S_c)$ and $\\min S_b< \\min S_c$ .",
"Here $Y(S_b,S_c)$ is the set of $(b,c,a_2)\\in {\\mathbb {Z}}^3$ such that $b=\\prod _{p\\in S_b} p^{v_p(b)}$ with $1\\le v_p(b)\\le u_p$ , $c=\\prod _{p\\in S_c} p^{v_p(c)}$ with $1\\le v_p(c)\\le u_p$ , $a_2=\\prod _{p\\in S_{a_2}} p^{v_p(a_2)}$ with $0\\le v_p(a_2)\\le u_p$ and such that $a_2$ divides $b+c$ or $b-c$ .",
"For each $(b,c,a_2)\\in Y$ , check if $a_1:=(b+\\varepsilon c)/a_2\\in X$ for some $\\varepsilon \\in \\lbrace 1,-1\\rbrace $ .",
"If so then output all $(x,y)=(\\tfrac{\\alpha }{\\gamma },\\tfrac{\\beta }{\\gamma })$ with $\\alpha +\\beta =\\gamma $ and $\\lbrace \\vert {}\\alpha {}\\vert ,\\vert {}\\beta {}\\vert ,\\vert {}\\gamma {}\\vert \\rbrace =\\lbrace \\vert {}a_1{}\\vert a_2,b,c\\rbrace $ .",
"In the implementation of the above algorithm, we simultaneously carry out steps (ii) and (iii) as follows.",
"We iterate over all $(S_b,S_c)$ and over all $(b,c)$ : For each $(b,c)$ we determine the divisors $a_2\\in {\\mathbb {Z}}_{\\ge 1}$ of $b\\pm c$ which are only divisible by primes in $S_{a_2}$ and which have exponents bounded by $u$ .",
"In the same iteration step, we also check whether $(b\\pm c)/a_2\\in X$ and if this is the case then we output the corresponding solutions."
],
[
"Correctness.",
"We now show that Algorithm REF indeed finds all solutions $(x,y)$ of (REF ) with $m(x,y)\\le u$ .",
"Suppose that $(x,y)$ is such a solution, and let $(a,b,c)$ be a solution of (REF ) with $(x,y)=(\\tfrac{a}{c},\\tfrac{b}{c})$ .",
"After replacing $(x,y)$ with a symmetric solution, we may and do assume that $w(S_a)\\ge \\max (w(S_b),w(S_c))$ for $S_b=\\lbrace p\\,;\\,p\\operatorname{|}b\\rbrace $ , $S_c=\\lbrace p\\,;\\,p\\operatorname{|}c\\rbrace $ and $S_a=S- (S_b\\cup S_c)$ .",
"This implies that $w(S_a)\\ge w(S)^{1/3}$ , since $w(S)=w(S_a)w(S_b)w(S_c)$ .",
"Let $S_{a_1},S_{a_2},X,Y$ be the sets appearing in steps (i) and (ii), and define $a_i=\\prod _{p\\in S_{a_i}}p^{{\\textnormal {ord}}_p(a)}$ for $i=1,2$ .",
"Our assumption $m(x,y)\\le u$ implies that $a_1\\in X$ and $(\\vert {}b{}\\vert ,\\vert {}c{}\\vert ,a_2)\\in Y$ since $a_2\\mid a=-(b-c)$ .",
"Further we observe that $(\\vert {}b{}\\vert +\\varepsilon \\vert {}c{}\\vert )/a_2=\\pm a/a_2=\\pm a_1\\in X$ for some $\\epsilon \\in \\lbrace 1,-1\\rbrace $ , and by construction it holds that $a+b=c$ with $\\lbrace \\vert {}a{}\\vert ,\\vert {}b{}\\vert ,\\vert {}c{}\\vert \\rbrace =\\lbrace a_1a_2,\\vert {}b{}\\vert ,\\vert {}c{}\\vert \\rbrace $ .",
"Therefore we see that step (iii) produces our solution $(x,y)$ as desired."
],
[
"Complexity.",
"To explain the improvements provided by Algorithm REF , we recall that the standard enumeration of all solutions $(x,y)$ of (REF ) with $m(x,y)\\le u$ is as follows: Consider all coprime pairs $(a,b)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ with $\\textnormal {rad}(ab)\\operatorname{|}N_S$ , and with $\\max \\bigl ({\\textnormal {ord}}_p(a),{\\textnormal {ord}}_p(b)\\bigl )\\le u_p$ for all $p\\in S$ .",
"If $c=a+b$ satisfies $\\textnormal {rad}(c)\\operatorname{|}N_S$ and ${\\textnormal {ord}}_p(c)\\le u_p$ for all $p\\in S$ , then output the solution $(x,y)=(\\tfrac{a}{c},\\tfrac{b}{c})$ .",
"Now the improved running time of Algorithm REF has the following basic reason.",
"For fixed $S_a$ , $S_b$ and $S_c$ , we iterate over all $a_1$ and over all $(b,c)$ in a subsequent way; see the remark given below Algorithm REF .",
"That is the running time of these two iterations adds, and it does not multiply as in the standard enumeration which iterates over all coprime pairs $(a,b)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ .",
"Here we note that the splitting of $S_a$ into $S_a=S_{a_1}\\operatorname{\\dot{\\cup }}S_{a_2}$ assures that the running time of (i) does not differ too much from the running time of (ii) together with (iii).",
"Indeed $S_{a_1}$ is chosen such that $w(S_{a_1})$ is approximately $w(S)^{1/2}$ and our assumption $w(S_a)\\ge w(S)^{1/3}$ provides that $w(S_b)w(S_c)\\le w(S)^{2/3}$ .",
"In particular, for bounded $\\vert {}S{}\\vert $ the complexity of our refined enumeration is asymptotically the $2/3$ -th power of the complexity of the standard enumeration algorithm.",
"Remark 3.13 Due to the hash our refinement needs more memory than the standard algorithm.",
"In practice this is not much compared to the running time, and in case this becomes an issue we can avoid creating the hash as follows.",
"Iterating over all $(b,c)$ as above, we try to factor $(b\\pm c)/a_2$ using only primes in $S_{a_1}$ and if this succeeds then we output the corresponding solutions provided the exponents are bounded by $u$ ."
],
[
"Application.",
"We shall apply Algorithm REF in order to enumerate the solutions $(x,y)$ of (REF ) with bounded $\\mu (x,y)$ .",
"More precisely, let $\\mu \\in {\\mathbb {Z}}^t$ with $\\mu \\ge 0$ and suppose that we want to enumerate all solutions $(x,y)$ of (REF ) with $\\mu (x,y)\\le \\mu $ .",
"For this purpose it suffices to apply Algorithm REF with $u\\in {\\mathbb {Z}}^S$ given by $u_p =\\left\\lfloor {\\mu _1/\\log p}\\right\\rfloor $ for $p\\in S$ .",
"Indeed any solution $(x,y)$ with $\\mu (x,y)\\le \\mu $ satisfies $m_p(x,y)\\log p\\le \\mu _1(x,y)\\le \\mu _1$ for all $p\\in S$ and thus $m(x,y)\\le u$ .",
"This application could output too many solutions $(x,y)$ , since not any $(x,y)$ with $m(x,y)\\le u$ satisfies $\\mu (x,y)\\le \\mu $ .",
"To avoid this we can directly check the latter condition at each step of the recursions building $X$ and $Y$ , and we do this in our implementation.",
"In general an improvement could come from a choice of the weight $w$ making the cardinalities of $X$ and $Y$ even more balanced; we leave this for the future."
],
[
"The algorithm",
"We continue the notation introduced above.",
"On putting everything together, we obtain the following algorithm which solves the $S$ -unit equation (REF ).",
"Algorithm 3.14 ($S$ -unit equation via height bounds) The input is a finite set of rational primes $S$ , and the output is the set of solutions $(x,y)$ of the $S$ -unit equation (REF ).",
"To rule out solutions with “large\" height, use Proposition REF in order to compute $M_0\\in {\\mathbb {Z}}_{\\ge 1}$ such that all solutions $(x,y)$ of (REF ) satisfy $M(x,y)\\le M_0$ .",
"To find the solutions of “medium sized\" and “small\" height, apply de Weger's sieve.",
"Find a small $M_1\\in {\\mathbb {Z}}_{\\ge 1}$ such that an application of Algorithm REF (ii) as in (REF ), with $M^{\\prime }=M_1$ and $M^{\\prime \\prime }=M_0$ , only returns $\\gamma =0$ .",
"Here first try $M_1=10$ .",
"If this does not work then replace $M_1$ by $\\left\\lfloor {1.3M_1}\\right\\rfloor $ , and so on.",
"Having found such an $M_1$ , successively apply Algorithm REF as in (REF ), with $M^{\\prime }=M_{k+1}$ , $M^{\\prime \\prime }=M_{k}$ and $M_{k+1}=\\left\\lfloor {M_k/1.3}\\right\\rfloor $ , for $k=1,2,\\cdots $ until either $M_k=0$ or Algorithm REF (ii) returns more than $10^3$ candidates.",
"To find the remaining solutions, combine the refined sieve with the refined enumeration.",
"The following two steps run simultaneously, or alternatingly, until the parameters $n,n^{\\prime }$ appearing in these two steps satisfy $n=n^{\\prime }$ .",
"To enumerate the remaining solutions from above, let $k_0$ be the $k$ where the above step (ii) ended.",
"Then successively apply Algorithm REF as in (REF ), with $M=M_{k_0}$ and $\\mu ^{\\prime }(n),\\mu ^{\\prime \\prime }(n)$ , for $n=M+1,M,\\cdots $ until $n=n^{\\prime }$ .",
"To enumerate the remaining solutions from below, successively apply the refined enumeration Algorithm REF with $\\mu ^{\\prime }(n^{\\prime })$ for $n^{\\prime }=1,2,\\ldots $ until $n^{\\prime }=n$ ."
],
[
"Correctness.",
"To verify that this algorithm works correctly, we let $(x,y)$ be a solution of (REF ) and we let $M$ be as in step (iii).",
"The construction of $M_0$ in step (i) shows that $M(x,y)\\le M_0$ and hence our solution $(x,y)$ is found in step (ii) if $M(x,y)>M$ .",
"On the other hand, if $M(x,y)\\le M$ then step (iii) produces our solution $(x,y)$ as desired."
],
[
"Complexity",
"We conducted some effort to optimize the running time in practice.",
"To explain our optimizations, we now discuss the above composition of Algorithm REF and we motivate our choice of the parameters appearing therein.",
"Furthermore, we also consider some additional practical aspects of our algorithm.",
"We continue the notation introduced above."
],
[
"Step (i).",
"In this step we need to find a number $M_0$ with the property that any solution $(x,y)$ of (REF ) satisfies $M(x,y)\\le M_0$ .",
"For this purpose we use Proposition REF which requires to compute the number $\\alpha (N)$ appearing therein, where $N=2^4N_S$ .",
"In case the computation of $\\alpha (N)$ takes too long, we can replace $\\alpha (N)$ by the slightly larger number $\\bar{\\alpha }(N)$ defined below (REF ) and this $\\bar{\\alpha }(N)$ can always be computed very fast.",
"Hence we see that Proposition REF allows in any case to quickly determine a relatively small number $M_0$ with the desired property.",
"In practice this $M_0$ is small enough such that we can skip de Weger's first reduction process (2) described at the beginning of Section REF , and thereby remove an uncertainty in de Weger's original method which crucially relies on (2).",
"Indeed the reduction process (2) is (a priori) an uncertainty, since it is not proved that it always works.",
"However, we should mention that in practice it is (essentially) always possible to successfully apply (2) by adapting the parameters to the specific situation at hand."
],
[
"Step (ii).",
"Here we apply Algorithm REF as in (REF ).",
"For this purpose we divided the space $0\\le M(x,y)\\le M_0$ into subspaces $M^{\\prime }< M(x,y)\\le M^{\\prime \\prime }$ , with $M^{\\prime }=M_{k+1}$ and $M^{\\prime \\prime }=M_k$ for $k=0,\\cdots ,k_0$ and for some convenient sequence $M_{k_0}<\\ldots < M_{1}<M_0$ .",
"To explain for which choices of $M^{\\prime },M^{\\prime \\prime }$ the application of Algorithm REF is fast in practice, we use the notation of Section REF .",
"The efficiency of the sieve depends on the number of points in $\\Gamma \\cap \\mathcal {E}$ .",
"In the best case, the sieve (REF ) decreases the space of candidates by a factor which is approximately $\\vert {}G{}\\vert /2= q^{l_q}(q-1)/2$ .",
"However in the worst case, when the square of each element in $S\\backslash q$ is 1 modulo $q^{l_q+1}$ , we obtain $\\Gamma ={\\mathbb {Z}}^{S\\backslash q}$ and the sieve (REF ) does not decrease the number of candidates at all.",
"In practice we are almost always close to the best case.",
"Let $V$ be the euclidean volume of the ball in ${\\mathbb {R}}^{s-1}$ of square radius $s-1$ for $s=\\vert {}S{}\\vert $ , and let $\\textnormal {covol}(\\Gamma )$ denote the covolume of $\\Gamma $ .",
"The ellipsoid $\\mathcal {E}$ has volume $\\operatorname{\\textnormal {vol}}(\\mathcal {E})=V\\prod _{p\\in S\\setminus q}u_p$ , and in the generic case the cardinality of $\\Gamma \\cap \\mathcal {E}$ can be approximated by $\\operatorname{\\textnormal {vol}}(\\mathcal {E})/\\textnormal {covol}(\\Gamma )$ .",
"Thus the sieve is efficient in practice if the ratio $\\operatorname{\\textnormal {vol}}(\\mathcal {E})/(\\vert {}G{}\\vert /2)$ is “small\".",
"For example this ratio is strictly smaller than 1 when $M^{\\prime }$ and $M^{\\prime \\prime }$ satisfy $M^{\\prime }>(s-1)\\log M^{\\prime \\prime }+\\log V+2\\log 2-\\sum _{p\\in S\\setminus q}\\log \\log p.$ Stirling's approximation leads to a simpler expression for $V$ in terms of $s$ .",
"If we now choose $M_1=(s-1)(\\log M_0+\\tfrac{1}{2}\\log (2\\pi e))$ , then (REF ) suggests that the sieve (REF ) is strong enough such that Algorithm REF (ii) only returns the trivial candidate $\\gamma =0$ .",
"A relatively small $M_1$ with this property is produced in step (a) where we start with a very optimistic choice $M_1=10$ .",
"Step (a) is fast in practice and it improves the running time of (ii) as follows: Algorithm REF (iii) is trivialThe construction of $M_1$ provides that here Algorithm REF (iii) only needs to output the trivial solutions $(2,-1)$ , $(-1,2)$ and $(\\frac{1}{2},\\frac{1}{2})$ of (REF ), which all come from the trivial candidate $\\gamma =0$ .",
"for the large space $M_1<M(x,y)\\le M_0$ , and for the space $M_2<M(x,y)\\le M_1$ the running time of Algorithm REF (iii) is considerably faster than for $M_2<M(x,y)\\le M_0$ since $M_0$ is much larger than $M_1$ .",
"Indeed Algorithm REF (ii) applied with $M_0$ can produce much larger candidates $\\gamma $ , and for large candidates $\\gamma $ the reconstruction process in Algorithm REF (iii) becomes slow.",
"We next discuss step (b).",
"According to (REF ), the choices $M_{k+1}=\\left\\lfloor {M_{k}/1.3}\\right\\rfloor $ for $k\\ge 1$ should give a strong sieve in the range where step (b) is applied and this turned out to be true in practice.",
"We apply step (b) for $k=1,2,\\cdots $ until Algorithm REF (ii) returns more than $10^3$ candidates.",
"Here the condition more than $10^3$ candidates means that our refined sieve can find these candidates considerably faster, and thus we switch at this point to step (iii)."
],
[
"Step (iii).",
"In this step we are in a situation where we can fully exploit our refinements worked out in the previous sections.",
"To explain this more precisely, we now mention two points which significantly slow down Algorithm REF in the situation of step (iii) where many solutions exist.",
"The first point is the application of (FP) in Algorithm REF (ii), which is the bottleneck of Algorithm REF .",
"The second point is that Algorithm REF (ii) repetitively enumerates the “same\" candidate $\\gamma $ in many steps $k$ with $M_k$ small.",
"This is due to the large fraction between the volume of the ellipsoid and the volume of the cube appearing in Algorithm REF ; indeed this fraction depends exponentially on the cardinality of $S$ .",
"To improve these two points we worked out the following refinements: Concerning the first and second point, we developed our refined Algorithm REF which works with smaller ellipsoids.",
"This leads to less repetitions of the candidates $\\gamma $ , and it considerably improves the running time of the step in which we apply (FP).",
"See also the complexity discussions given in Section REF .",
"Regarding the second point, we constructed the refined enumeration Algorithm REF .",
"This enumeration is fast enough such that we can now skip the final applications ($M_k=1,2,\\cdots $ ) of Algorithm REF which are very slow, and thereby we can in particular circumvent for many candidates $\\gamma $ that they get enumerated repetitively.",
"In (iii) we carry out steps (a) and (b) alternatingly, depending on which step took less time so far.",
"In some cases this leads to a significantly improved running time of (iii).",
"Further we mention that our choice $\\mu ^{\\prime \\prime }(n)=\\mu ^{\\prime }(n+1)$ is motivated by (REF ).",
"Indeed according to (REF ), one should work with step size $1=(n+1)-n$ in the situation of (iii) where usually $M(x,y)\\le (s-1)(\\log s+\\log \\log s+\\tfrac{1}{2}\\log (2\\pi e))$ .",
"See also the discussions surrounding (REF ) which provide additional motivation for our definition of $\\mu ^{\\prime }(n),\\mu ^{\\prime \\prime }(n)$ ."
],
[
"Bottleneck.",
"Despite our refinements which considerably improve the running time in practice, step (iii) still remains in general the bottleneck of Algorithm REF .",
"However, for certain special sets $S$ the location of the bottleneck can change.",
"For example if $S$ consists of the prime 2 together with a few very large primes, then Algorithm REF often finds all solutions already in step (ii).",
"In this case the bottleneck of Algorithm REF is located in step (ii) where we apply Algorithm REF (i).",
"The main reason is that here the application of (T) becomes slow since $S$ contains very large primes, and the application of (FP) in Algorithm REF (ii) becomes fast since the cardinality of $S$ is small.",
"Remark 3.15 (Parallelization) In our implementation of Algorithm REF we successfully parallelized essentially everything, except Algorithm REF (i) and Algorithm REF (i) which both involve the application of Teske's algorithm described in (T)."
],
[
"Applications",
"In this section we give some applications of Algorithm REF .",
"In particular we discuss parts of our database $\\mathcal {D}_1$ containing the solutions of the $S$ -unit equation (REF ) for many distinct sets $S$ .",
"We also use our database to motivate various Diophantine questions related to (REF ), including Baker's explicit $abc$ -conjecture and a new conjecture.",
"We continue the notation introduced above.",
"Recall that $\\Sigma (S)$ denotes the set of solutions of the $S$ -unit equation (REF ) up to symmetry, and for any $n\\in {\\mathbb {Z}}_{\\ge 1}$ we recall that $S(n)$ denotes the set of the $n$ smallest rational primes.",
"Let $N\\in {\\mathbb {Z}}_{\\ge 1}$ and define $\\Sigma (N)=\\cup \\Sigma (S)$ with the union taken over all finite sets of rational primes $S$ with $N_S\\le N$ ."
],
[
"The sets $\\Sigma (S(n))$ .",
"We determined the sets $\\Sigma (S(n))$ for all $n\\le 16$ .",
"The cardinality of these sets is given in the table of Theorem A stated in the introduction.",
"As already mentioned, our Algorithm REF substantially improves de Weger's original method in [161] which de Weger used to compute the set $\\Sigma (S(6))$ in [161].",
"To illustrate our improvements in practice, we now compare Algorithm REF with de Weger's original method.",
"For this purpose we implemented in Sage de Weger's original method in a slightly improved form (dW), which uses in addition our optimized height bounds.",
"If $S=S(6)$ then (dW) took 21 seconds, while it took 6 seconds by using Algorithm REF .",
"For larger $|S|$ our running time improvement significantly increases: For example if $S=S(10)$ then (dW) takes four days, whereas this decreases to only 25 minutes by using Algorithm REF .",
"Roughly speaking, for large $\\vert {}S{}\\vert $ our refinements should save an exponential factor with respect to $|S|$ in comparison to de Weger's original method.",
"Further if $n>10$ then (dW) becomes too slow and thus we did not try to use (dW) in order to compute $\\Sigma (S(n))$ for $n>10$ .",
"To deal with $S=S(n)$ for $10<n\\le 16$ we additionally parallelized (see Remark REF ) our Algorithm REF .",
"Then it took 8 days for $n=15$ and 34 days for $n=16$ , using 30 CPU's.",
"Remark 3.16 (Automatically choosing parameters) De Weger's original method does not specify in general how to choose the involved parameters.",
"For example de Weger's first reduction process requires to make a choice, and to efficiently apply de Weger's sieve one has to choose suitable subspaces dividing the initial space.",
"To compute the set $\\Sigma (S(6))$ , de Weger has chosen by hand the required parameters.",
"Although we can skip de Weger's first reduction process using our optimized height bounds, we still need to make many choices in Algorithm REF .",
"In particular, it is now favourable that the choices required for Algorithm REF are made by an automatism.",
"We implemented such an automatism, which takes into account (REF ) and which properly adjusts the parameters during run time.",
"In view of (REF ), our automatism chooses parameters for Algorithm REF such that the running time is considerably less than twice as long as for the optimal parameters.",
"However, we do not claim and can not prove that our automatism is optimal."
],
[
"The sets $\\Sigma (N)$ .",
"We computed the sets $\\Sigma (N)$ for all $N\\le 10^7$ in approximately 13 days.",
"For this computation it was crucial that Algorithm REF automatically chooses all required parameters, as mentioned in Remark REF .",
"Indeed to compute the sets $\\Sigma (N)$ for all $N\\le 10^7$ , we had to apply Algorithm REF with so many distinct sets $S$ such that it would have been impossible to suitably choose by hand all the involved parameters."
],
[
"Explicit $abc$ -conjecture.",
"Baker [7] proposed the following fully explicit version of the $abc$ -conjecture: If $a,b,c\\in {\\mathbb {Z}}_{\\ge 1}$ are coprime with $a+b=c$ , then it holds $c\\le \\tfrac{6}{5}N(\\log N)^{\\omega }/\\omega !$ for $\\omega $ the number of rational primes dividing the radical $N=\\textnormal {rad}(abc)$ ; here one should exclude the triple $(a,b,c)=(1,1,2)$ .",
"On using our database $\\mathcal {D}_1$ containing in particular the sets $\\Sigma (N)$ for all $N\\le 10^7$ , we verified Baker's explicit $abc$ -conjecture for all coprime $a,b,c\\in {\\mathbb {Z}}_{\\ge 1}$ with $\\textnormal {rad}(abc)\\le 10^7$ .",
"Furthermore, Algorithm REF can be used to verify other properties of all $abc$ -triples with bounded radical.",
"In particular all existing $abc$ -triples with $\\textnormal {rad}(abc)\\le 10^7$ can be directly taken from our database $\\mathcal {D}_1$ ."
],
[
"Elliptic curves with full 2-torsion.",
"For any given $N\\in {\\mathbb {Z}}_{\\ge 1}$ we denote by $\\mathcal {M}$ the set of ${\\mathbb {Q}}$ -isomorphism classes of elliptic curves over ${\\mathbb {Q}}$ of conductor $N$ , with all two torsion points defined over ${\\mathbb {Q}}$ .",
"Algorithm REF allows to efficiently compute the set $\\mathcal {M}$ .",
"To see this, we let $S$ be the set of rational primes dividing $2N$ .",
"Any elliptic curve in $\\mathcal {M}$ admits a Weierstrass equation $y^2=x(x-a)(x+b)$ such that $a,b\\in {\\mathbb {Z}}$ have the following properties: $d=\\gcd (a,b)$ divides $N_S$ and $(\\frac{a}{a+b},\\frac{b}{a+b})$ satisfies the $S$ -unit equation (REF ).",
"An application of Algorithm REF with $S$ determines all possible values of $a/d$ and $b/d$ , and this then allows to directly compute the set $\\mathcal {M}$ .",
"In particular, we see that Algorithm REF provides an alternative way to check the completeness of a small part of Cremona's database [39].",
"This application of Algorithm REF already turned out to be useful in practice."
],
[
"Conjecture and question.",
"We next use our database $\\mathcal {D}_1$ to motivate various questions concerning the solutions of $S$ -unit equations (REF ).",
"First we recall Conjecture 1 which is motivated by our data and by the construction of the refined sieve in Section REF .",
"Conjecture 1.",
"There exists $c\\in {\\mathbb {Z}}$ with the following property: If $n\\in {\\mathbb {Z}}_{\\ge 1}$ then any finite set of rational primes $S$ with $\\vert {}S{}\\vert \\le n$ satisfies $\\vert {}\\Sigma (S){}\\vert \\le \\vert {}\\Sigma (S(n)){}\\vert +c$ .",
"In other words, if $\\mathcal {T}$ denotes the collection of schemes $T$ such that $T$ can be obtained by removing $n$ closed points of $\\textnormal {Spec}({\\mathbb {Z}})$ , then Conjecture 1 means the following: Among all $T\\in \\mathcal {T}$ , the maximal number (up to some constant) of $T$ -points of $\\mathbb {P}^1_{\\mathbb {Z}}-\\lbrace 0,1,\\infty \\rbrace $ is attained at the scheme in $\\mathcal {T}$ which corresponds to the $n$ closed points of smallest norm.",
"This conjectured property of $\\mathbb {P}^1_{\\mathbb {Z}}-\\lbrace 0,1,\\infty \\rbrace $ is rather unexpected from a general Diophantine geometry perspective.",
"We further ask whether one can remove the constant.",
"Question 1.1.",
"Does Conjecture 1 hold with $c=0$ ?",
"Here the main motivation is given by our data.",
"Indeed Question 1.1 has a positive answer for all sets $S$ in our database $\\mathcal {D}_1$ .",
"In view of Theorem A listing the cardinality of $\\Sigma (S(n))$ for all $n\\in \\lbrace 1,\\cdots ,16\\rbrace $ , a positive answer to Question 1.1 would give an optimal upper bound for the number of solutions of any $S$ -unit equation (REF ) with $\\vert {}S{}\\vert $ in $\\lbrace 1,\\cdots ,16\\rbrace $ ."
],
[
"Comparison of algorithms",
"To compare Algorithm REF with our Algorithm REF using modular symbols, we continue the notation introduced above.",
"Recall that we already computed the sets $\\Sigma (N)$ for all $N\\le 20000$ by using Algorithm REF , see the examples in Section REF .",
"Here it turned out that the output of Algorithm REF agrees with the output of Algorithm REF .",
"To determine all solutions of the $S$ -unit equation (REF ), we recommend to use Algorithm REF as long as one already knows the set of elliptic curves over ${\\mathbb {Q}}$ of conductor dividing $2^4N_S$ .",
"If this set is not already known, then it is usually much more efficient to use Algorithm REF .",
"In fact, as demonstrated in the previous sections, our Algorithm REF is substantially more efficient in all aspects than the known methods which practically resolve (REF )."
],
[
"Algorithms for Mordell equations",
"Let $S$ be a finite set of rational primes, write $N_S=\\prod _{p\\in S}p$ and put $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ .",
"We take a nonzero $a\\in \\mathcal {O}$ .",
"In this section, we would like to solve the Mordell equation $y^2=x^3+a, \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}.",
"\\qquad \\mathrm {(\\ref {eq:mordell})}$ This Diophantine problem is a priori more difficult than solving the $S$ -unit equation (REF ).",
"Indeed elementary transformations reduce (REF ) to (REF ), while the known reductions of (REF ) to (REF ) require to solve (REF ) in number fields or they require a height bound for the solutions of (REF ) which is equivalent to the $abc$ -conjecture in Remark REF .",
"So far all known practical methods solving (REF ) crucially rely on the theory of logarithmic forms [33], see below for an overview.",
"In the following sections we present two alternative algorithms which allow to practically resolve (REF ).",
"They both do not use the theory of logarithmic forms.",
"In Section REF we describe the first algorithm which relies on Cremona's algorithm using modular symbols.",
"Then in Section REF we construct the second algorithm via height bounds.",
"Here we also give several applications and we discuss various questions motivated by our results, see Sections REF -REF .",
"Finally in Section REF we compare our algorithms with the actual best practical methods solving (REF )."
],
[
"Known methods.",
"We now discuss algorithms and methods in the literature which allow to solve (REF ).",
"First we consider the classical case $\\mathcal {O}={\\mathbb {Z}}$ .",
"Ellison et al [51] used the approach of Baker–Davenport [4], [13] to solve (REF ) for some $a$ .",
"Recently the latter approach was refined by Bennett–Ghadermarzi [18] who applied their algorithm to find all solutions of (REF ) in ${\\mathbb {Z}}\\times {\\mathbb {Z}}$ for any nonzero $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^7$ ; see also the work of Wildanger and Jätzschmann discussed in Fieker–Gaál–Pohst [62].",
"Alternatively, Masser [112], Lang [102], Wüstholz [166] and Zagier [167] initiated a practical approach to solve arbitrary elliptic Weierstrass equations $(W)$ over ${\\mathbb {Z}}$ via elliptic logarithms.",
"On applying this approach with David's explicit bounds [41], Stroeker–Tzanakis [141] and Gebel–Pethő–Zimmer [68] obtained independently a practical algorithm solving $(W)$ over ${\\mathbb {Z}}$ .",
"Gebel–Pethő–Zimmer [71] used this algorithm to determine all solutions of (REF ) in ${\\mathbb {Z}}\\times {\\mathbb {Z}}$ for any nonzero $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^4$ and for most $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^5$ .",
"Let $r$ be the Mordell–Weil rank of the group $E({\\mathbb {Q}})$ associated to the elliptic curve $E$ over ${\\mathbb {Q}}$ defined by $(W)$ .",
"In the important special case $r=1$ , there exists in addition a practical approach of Balakrishnan–Besser–Müller [9], [10] which is in the spirit of Kim's non-abelian Chabauty program initiated in [94].",
"We now discuss practical methods in the literature solving (REF ) over any ring $\\mathcal {O}$ as above.",
"In practice and in theory, this task is considerably more difficult than solving (REF ) in $\\mathcal {O}={\\mathbb {Z}}$ .",
"The method of Bilu [22] and Bilu–Hanrot [20] for superelliptic Diophantine equations allows in particular to solve (REF ).",
"Further, classical constructions reduce (REF ) to Thue–Mahler equations which in turn can be solved using the method of Tzanakis–de Weger [154].",
"Smart [137] extended the above mentioned elliptic logarithm approach to solve $(W)$ over $\\mathcal {O}$ .",
"His algorithm is conditional on explicit lower bounds for linear forms in $p$ -adic elliptic logarithms.",
"Rémond–Urfels [129] proved such boundsHirata-Kohno [79] recently established the general case.",
"Tzanakis [159] combined her bounds with the elliptic logarithm reduction to solve $(W)$ over $\\mathcal {O}$ , see also Hirata-Kohno–Kovács [82].",
"for $r\\le 2$ , and these bounds were then applied by Gebel–Pethő–Zimmer [69], [70] to solve (REF ) for some nonempty $S$ .",
"Furthermore, in a joint work with Herrmann [126], they obtained a variation of the elliptic logarithm approach which works also for $r\\ge 3$ , see Section REF .",
"We point out that the elliptic logarithm approach requires a basis of $E({\\mathbb {Q}})$ .",
"There are methods which often can compute such a basis in practice, in particular in our case of elliptic curves defined by (REF ).",
"However, these methods are not (yet) effective in general as discussed in Section REF .",
"For a detailed description of the elliptic logarithm approach, we refer to the excellent book of Tzanakis [159] which is devoted to this method."
],
[
"Algorithm via modular symbols",
"We continue our notation.",
"For any elliptic curve $E$ over ${\\mathbb {Q}}$ , we denote by $c_4$ and $c_6$ the usual quantities associated to a minimal Weierstrass model of $E$ over ${\\mathbb {Z}}$ ; see [150].",
"Write $N_E$ and $\\Delta _E$ for the conductor and minimal discriminant of $E$ respectively.",
"We define $a_S=1728N_S^2\\prod p^{\\min ({\\textnormal {ord}}_p(a),2)}$ with the product taken over all rational primes $p$ not in $S$ .",
"If $(x,y)$ satisfies the Mordell equation (REF ), then one can consider the elliptic curve $E$ over ${\\mathbb {Q}}$ which admits the Weierstrass equation $t^2 = s^3 - 27xs - 54y$ with “indeterminates\" $s$ and $t$ .",
"This construction leads to the following lemma which will be proven in course of the proof of Lemma REF .",
"Lemma 4.1 Suppose that $(x,y)$ is a solution of the Mordell equation (REF ).",
"Then there exists an elliptic curve $E$ over ${\\mathbb {Q}}$ such that $N_E\\mid a_S$ and such that $c_4 =u^4x$ and $c_6= u^6y$ for some $u\\in {\\mathbb {Q}}$ with $u^{12}=1728\\Delta _E\\vert {}a{}\\vert ^{-1}$ .",
"In fact this lemma may be viewed as an explicit Paršin construction for integral points on the moduli scheme of elliptic curves defined by (REF ), see [89] for details.",
"Algorithm 4.2 (Mordell equations via modular symbols) The input consists of a finite set of rational primes $S$ together with a nonzero number $a\\in \\mathcal {O}$ , and the output is the set of solutions $(x,y)$ of the Mordell equation (REF ).",
"(i) Define $a_S$ as in (REF ).",
"Then use Cremona's algorithm using modular symbols, described in Section , to compute the set $\\mathcal {T}\\subset {\\mathbb {Z}}\\times {\\mathbb {Z}}$ of quantities $(c_4,c_6)$ which are associated to some modular elliptic curve over ${\\mathbb {Q}}$ of conductor dividing $a_S$ .",
"(ii) For each $(c_4,c_6)\\in \\mathcal {T}$ , write $\\vert {}(c_4^3-c_6^2)/a{}\\vert =\\frac{m}{n}$ with coprime $m,n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"Compute the subset $\\mathcal {T}_{0}\\subseteq \\mathcal {T}$ of $(c_4,c_6)\\in \\mathcal {T}$ with $m=u_1^{12}$ and $n=u_2^{12}$ for $u_1,u_2\\in {\\mathbb {Z}}$ .",
"(iii) For any $(c_4,c_6)\\in \\mathcal {T}_{0}$ take $u=\\frac{u_1}{u_2}$ in $\\mathbb {{\\mathbb {Q}}}$ with $u^{12}=\\vert {}(c_4^3-c_6^2)/a{}\\vert $ and define $x=u^{-4}c_4$ and $y=u^{-6}c_6$ .",
"If $x$ and $y$ are both in $\\mathcal {O}$ , then output $(x,y)$ ."
],
[
"Correctness.",
"We now verify that this algorithm indeed finds all solutions of any Mordell equation (REF ).",
"Suppose that $(x,y)$ satisfies (REF ).",
"Then Lemma REF gives an elliptic curve $E$ over ${\\mathbb {Q}}$ such that $N_E\\mid a_S$ and such that $c_4 =u^4x$ and $c_6= u^6y$ for some $u\\in {\\mathbb {Q}}$ with $u^{12}=1728\\Delta _E\\vert {}a{}\\vert ^{-1}$ .",
"The Shimura–Taniyama conjecture assures that $E$ is modular.",
"This proves that $(c_4,c_6)$ is contained in the set $\\mathcal {T}$ computed in step (i).",
"Furthermore we obtain that $(c_4,c_6)\\in \\mathcal {T}_{0}$ , since it holds $u^{12}=1728\\Delta _E\\vert {}a{}\\vert ^{-1}=\\vert {}(c_4^3-c_6^2)/a{}\\vert $ by the definition of the discriminant.",
"Any $u^{\\prime }\\in {\\mathbb {Q}}$ with $u^{\\prime 12}=\\vert {}(c_4^3-c_6^2)/a{}\\vert $ satisfies $u^{\\prime 4}=u^4$ and $u^{\\prime 6}=u^6$ .",
"Therefore we see that step (iii) produces our solution $(x,y)=(u^{-4}c_4,u^{-6}c_6)$ as desired."
],
[
"Complexity.",
"The discussion of the complexity of step (i) is analogous to the discussion of the complexity of Algorithm REF (i) and thus we refer the reader to Section REF .",
"For each $(c_4,c_6)\\in \\mathcal {T}$ , step (ii) needs to check if there is $u=\\frac{u_1}{u_2}\\in {\\mathbb {Q}}$ with $u^{12}=\\vert {}(c_4^3-c_6^2)/a{}\\vert $ and then step (iii) needs to check if $u_1\\in {\\mathbb {Z}}$ is only divisible by primes in $S$ .",
"Therefore we see that the complexity discussions of Algorithm REF together with the arguments in Remark REF imply the following: The running time of step (ii) and (iii) is at most polynomial in terms of $H(a)N_S$ , and is at most $O_\\varepsilon ((H(a)N_S)^\\varepsilon )$ if all three conjectures (BSD), (GRH) and $(abc)$ hold.",
"Here $H(a)=\\exp (h(a))$ denotes the multiplicative Weil height of $a$ ."
],
[
"Applications.",
"We recall from [17] that a solution $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ of (REF ) is called primitive if $\\pm 1$ are the only $m\\in {\\mathbb {Z}}$ with $m^6\\mid \\gcd (x^3,y^2)$ .",
"For each nonzero $a\\in {\\mathbb {Z}}$ one can quickly enumerate all $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ with $y^2=x^3+a$ if one knows the primitive solutions of any Mordell equation $y^2=x^3+a^{\\prime }$ with $a^{\\prime }\\in {\\mathbb {Z}}$ satisfying $a=a^{\\prime }m^6$ for some $m\\in {\\mathbb {Z}}$ .",
"On using an implementation in Sage of Algorithm REF , we computed the set of primitive solutions of the family of Mordell equations (REF ) with parameter $a\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ satisfying $r_2(a)\\le 200$ for $r_2(a)=\\prod p^{\\min (2,{\\textnormal {ord}}_p(a))}$ with the product taken over all rational primes $p$ .",
"Here the computation of the solutions was very fast, since for each $a\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ with $r_2(a)\\le 200$ part (i) of Algorithm REF can use Cremona's database which contains in particular the required data for all elliptic curves over ${\\mathbb {Q}}$ of conductor dividing $1728r_2(a)< 350 000$ .",
"On the other hand, if the required data of the involved elliptic curves is not already known, then our Algorithm REF is usually not practical anymore.",
"Here the problem is the application of Cremona's algorithm (using modular symbols) in step (i), which requires a huge amount of memory to deal with conductors that are not small."
],
[
"Algorithm via height bounds",
"In this section we use the optimized height bounds of Proposition REF to construct Algorithm REF which allows to solve the Mordell equation (REF ).",
"We continue our notation and we work with the following setup: We may and do view $y^2=x^3+a$ as a Weierstrass equation of an elliptic curve $E_a$ over ${\\mathbb {Q}}$ , since our $a\\in \\mathcal {O}$ is nonzero.",
"A classical result of Mordell gives that the abelian group $E_a({\\mathbb {Q}})$ is finitely generated.",
"Let $P_1,\\cdots ,P_r$ be a basis of the free part of $E_a({\\mathbb {Q}})$ , and let $E_a({\\mathbb {Q}})_{\\textnormal {tor}}$ be the torsion group of $E_a({\\mathbb {Q}})$ .",
"We call $r$ the Mordell–Weil rank of $E_a({\\mathbb {Q}})$ and we say that $P_1,\\cdots ,P_r$ is a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ .",
"Let $(x,y)$ be a solution of (REF ).",
"The corresponding point $P$ in $E_a({\\mathbb {Q}})$ takes the form $P=Q+\\sum _{i=1}^r n_i P_i$ with $n_i\\in {\\mathbb {Z}}$ and $Q\\in E_a({\\mathbb {Q}})_{\\textnormal {tor}}$ , and we define $N(x,y)=\\max _{i}\\vert {}n_i{}\\vert $ ."
],
[
"Decomposition.",
"Before we describe our Algorithm REF in detail, we discuss its decomposition which is inspired by the elliptic logarithm approach introduced by Masser–Zagier.",
"A variation of the latter approach was used for example in the algorithm of Pethő–Zimmer–Gebel–Herrmann [126] whose main ingredients can be described as follows: (1) First they try to find a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ .",
"(2) On using the explicit result of Hajdu–Herendi [78] which is based on the theory of logarithmic forms [33], they determine an initial upper bound $N_0$ such that any solution $(x,y)$ of (REF ) satisfies $N(x,y)\\le N_0$ .",
"(3) Following Smart [137] they apply the elliptic logarithm reduction in order to reduce the initial upper bound $N_0$ to a bound $N_1$ which is usually much smaller.",
"(4) Finally they enumerate all solutions $(x,y)$ of (REF ) with $N(x,y)\\le N_1$ .",
"Here in the case $\\mathcal {O}={\\mathbb {Z}}$ the “inequality trick\" usually improves the enumeration process.",
"Our Algorithm REF substantially improves in all aspects the elliptic logarithm approach for (REF ).",
"More precisely, without introducing new ideas we use the known methods for (1) described in Section REF .",
"To obtain the initial upper bound for Algorithm REF , we apply in Proposition REF the optimized height bounds worked out in Section .",
"In practice our initial bound is considerably stronger than the initial bound $N_0$ in (2) based on the theory of logarithmic forms, and this leads to significant running time improvements of the reduction process as illustrated in Section REF .",
"Then to enumerate the solutions of bounded height, we apply the elliptic logarithm sieve constructed in Section .",
"This sieve is substantially more efficient than the reduction process (3) together with the subsequent enumeration (4).",
"We now discuss the ingredients of our approach in more detail."
],
[
"Torsion group and Mordell–Weil basis",
"On using the notation introduced above, we next briefly discuss methods which allow to determine the torsion group $E_a({\\mathbb {Q}})_{\\textnormal {tor}}$ of $E_a({\\mathbb {Q}})$ and a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ ."
],
[
"Torsion.",
"Fueter [67] completely determined $E_a({\\mathbb {Q}})_{\\textnormal {tor}}$ .",
"To state his result we write $a=e/d$ with coprime $e,d\\in {\\mathbb {Z}}$ .",
"If $d^5e=k\\cdot l^6$ for $k,l\\in {\\mathbb {Z}}$ with $k$ sixth power free, then $E_a({\\mathbb {Q}})_{\\textnormal {tor}}\\cong {\\left\\lbrace \\begin{array}{ll}{\\mathbb {Z}}/6{\\mathbb {Z}}& \\textnormal {if } k=1,\\\\{\\mathbb {Z}}/3{\\mathbb {Z}}& \\textnormal {if } k\\ne 1 \\textnormal { is a square, or } k=-432,\\\\{\\mathbb {Z}}/2{\\mathbb {Z}}& \\textnormal {if } k\\ne 1 \\textnormal { is a cube,}\\\\0 & \\textnormal {otherwise.}\\end{array}\\right.",
"}$ This completely determines all solutions of (REF ) in ${\\mathbb {Q}}\\times {\\mathbb {Q}}$ in the case when the Mordell–Weil rank $r$ of $E_a({\\mathbb {Q}})$ satisfies $r=0$ .",
"Therefore for the purpose of determining all solutions of the Mordell equation (REF ) we always may assume that $r\\ge 1$ ."
],
[
"Mordell–Weil basis.",
"The problem of finding a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ is usually more difficult.",
"If the elliptic curve $E_a$ satisfies the part of the Birch–Swinnerton-Dyer conjecture (BSD) predicting that $r$ coincides with the analytic rank of $E_a$ , then a result of Manin [109] leads to a practical algorithm (see for example Gebel–Zimmer [74]) which computes such a basis.",
"On applying this algorithm and an algorithm of Cremona given in [39], Gebel–Pethő–Zimmer [71] computed a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for most $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^5$ .",
"Parts of their database (Mordell$\\pm $ ) are uploaded on their homepage tnt.math.se.tmu.ac.jp/simath/MORDELL.",
"On using (PSM) we checked their data for all nonzero $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^4$ .",
"Here it turned out that for $a = 7823$ a basis was missing and that for $a = -7086$ and $a=-6789$ the given bases were not saturated.",
"In all these three cases we determined a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ using (PSM) and now the (updated) database contains a correct Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for any nonzero $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^4$ .",
"We note that in our special case given by the Mordell elliptic curve $E_a$ , one can often exploit isogenies to find a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ .",
"In fact, it turned out in practice that the known techniques implemented in (PSM) (Generators, two-descent, HeegnerPoint, etc.)",
"usually allow to quickly determine such a basis.",
"However we point out that in the case of an arbitrary nonzero $a\\in {\\mathbb {Z}}$ there is so far no unconditional method which allows to determine a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ ."
],
[
"Elliptic logarithm sieve",
"Starting with Zagier [167] and de Weger [162], many authors developed over the last decades the elliptic logarithm reduction process; see for example Stroeker–Tzanakis [141], Gebel–Pethő–Zimmer [68] and Smart [137].",
"In practice this process allows to show that the solutions in $\\mathcal {O}\\times \\mathcal {O}$ of an arbitrary elliptic Weierstrass equation have either relatively small or huge height.",
"In Section we constructed the elliptic logarithm sieve which considerably improves the elliptic logarithm reduction and the subsequent enumeration of solutions of small height.",
"In particular, for any given bound $N\\in {\\mathbb {Z}}$ , the elliptic logarithm sieve solves the problem of efficiently enumerating all solutions $(x,y)$ of (REF ) with $N(x,y)\\le N$ .",
"The sieve combines the core idea of the elliptic logarithm reduction with several conceptually new ideas.",
"We refer to Section for an overview of the new ideas introduced by the elliptic logarithm sieve and for a detailed discussion of the practical and theoretical improvements provided by these ideas."
],
[
"Initial bounds",
"We continue the notation introduced above.",
"In this section we give an initial upper bound for various heights attached to the solutions of the Mordell equation (REF ).",
"We also compare our bound with the actual best results in the literature and we explain how our result improves the running time of the reduction process in the elliptic logarithm sieve.",
"We recall that $P_1,\\cdots ,P_r$ denotes a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ , and for any solution $(x,y)$ of (REF ) we write as above $N(x,y)=\\max \\vert {}n_i{}\\vert $ for the “infinity norm\" of the corresponding point $P=Q+\\sum n_i P_i$ in $E_a({\\mathbb {Q}})$ .",
"Let $\\hat{h}$ be the canonical Néron-Tate height on $E_a({\\mathbb {Q}})$ .",
"Here we work with the natural normalization of $\\hat{h}$ which divides by the degree of the involved rational function, see for example [136]."
],
[
"Initial bounds.",
"Let $(x,y)$ be a solution of (REF ), with corresponding point $P\\in E_a({\\mathbb {Q}})$ .",
"To deduce an upper bound for $N(x,y)$ , we recall standard properties of $\\hat{h}$ .",
"One can control $\\hat{h}(P)$ in terms of the usual logarithmic Weil height $h$ as follows $\\hat{h}(P)\\le \\tfrac{1}{2}h(x)+\\tfrac{m}{6}h(a)+1.58, \\ \\ \\ \\ m ={\\left\\lbrace \\begin{array}{ll}1 & \\textnormal {if } a\\in {\\mathbb {Z}},\\\\12 & \\textnormal {otherwise.}\\end{array}\\right.",
"}$ Indeed in the integral case $a\\in {\\mathbb {Z}}$ the displayed inequality directly follows for example from Silverman [135].",
"To deal with any nonzero $a\\in \\mathcal {O}$ , we write $a=c/d$ with coprime $c,d\\in {\\mathbb {Z}}$ and we consider $E_{b}$ for $b=d^6a\\in {\\mathbb {Z}}$ .",
"There is an isomorphism $\\varphi :E_a\\rightarrow E_{b}$ induced by $x\\mapsto d^2x$ .",
"Then an application of the already verified integral case of (REF ), with $E_b$ and $b\\in {\\mathbb {Z}}$ , gives an upper bound for $\\hat{h}(P)=\\hat{h}(\\varphi (P))$ in terms of $h(d^2x)$ and $h(b)$ which proves (REF ) as desired.",
"Further it is known that $\\hat{h}$ defines a positive definite quadratic form on the real vector space $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}$ .",
"On using the basis $P_1,\\cdots ,P_r$ we identify $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}$ with ${\\mathbb {R}}^r$ and then we denote by $\\lambda $ the smallest eigenvalue of the matrix defining the binary form associated to the quadratic form $\\hat{h}$ on ${\\mathbb {R}}^r$ .",
"We now use the optimized height bound in Proposition REF , or the $abc$ -conjecture of Masser-Oesterlé [113] stated in Remark REF , in order to obtain initial bounds.",
"Proposition 4.3 Suppose that $(x,y)$ is a solution of (REF ), and denote by $P$ the corresponding point in $E_a({\\mathbb {Q}})$ .",
"Then the following statements hold.",
"(i) Let $\\alpha =\\alpha (a_S)$ be the number from Proposition REF , and recall that $m=1$ if $a\\in {\\mathbb {Z}}$ and $m=12$ otherwise.",
"It holds $\\lambda N(x,y)^2\\le \\hat{h}(P) \\le M_0$ for some $M_0\\in {\\mathbb {Z}}$ with $M_0\\le \\tfrac{m+1}{6}h(a)+2\\alpha +\\log (\\alpha +16.52)+52.12.$ (ii) Suppose that $a\\in {\\mathbb {Z}}$ , and assume that the $abc$ -conjecture holds.",
"Then for any real number $\\epsilon >0$ there exists a constant $c_\\epsilon $ depending only on $\\epsilon $ such that $\\hat{h}(P)\\le (1+\\epsilon )(\\log N_S+\\tfrac{7}{6}h(a))+c_\\epsilon .$ In case the computation of the number $\\alpha $ from Proposition REF takes too long, one can replace $\\alpha $ by the (slightly) larger number $\\bar{\\alpha }=\\bar{\\alpha }(a_S)$ which is defined below (REF ).",
"The number $\\bar{\\alpha }$ has the advantage that it can be quickly computed in all cases.",
"We further mention that the bound in Proposition REF (ii) is a direct consequence of a result in Bombieri–Gubler [17]; this bound is optimal in terms of $N_S$ .",
"[Proof of Proposition REF ] We first prove (i).",
"Linear algebra leads to $\\lambda N(x,y)^2\\le \\hat{h}(P)$ , since $\\hat{h}$ is a quadratic form on $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}$ .",
"Further, the estimate for $M_0$ in (i) follows by combining (REF ) with the upper bound for $h(x)$ given in Proposition REF .",
"To show assertion (ii) we assume that $a\\in {\\mathbb {Z}}$ and we write $x=x_1/d^2$ and $y=y_1/d^3$ with $x_1,y_1,d\\in {\\mathbb {Z}}$ satisfying $\\gcd (d,x_1y_1)=1$ and $d>0$ .",
"Let $n$ be the largest element in ${\\mathbb {Z}}$ with $n^6\\mid \\gcd (x_1^3,y_1^2)$ .",
"On dividing the equation $x_1^3-y_1^2=-ad^6$ by $n^6$ , we obtain a new equation of the form $u^3-v^2=w$ with $u,v,w\\in {\\mathbb {Z}}$ and $\\gcd (u^3,v^2)=1$ .",
"Now, on assuming the $abc$ -conjecture, we see that [17] gives estimates for $\\vert {}u{}\\vert ,\\vert {}v{}\\vert $ .",
"These estimates lead to an upper bound for $h(x)$ which together with (REF ) proves (ii)."
],
[
"Comparison with literature.",
"There are several explicit bounds for $M_0$ and $N(x,y)$ in the literature.",
"They are all based on the theory of logarithmic forms.",
"In fact this theory allows to effectively solve Diophantine equations which are considerably more general than Mordell equations (REF ), see for example [33].",
"To compare Proposition REF (i) with the actual best bounds for (REF ) in the literature, we use a simpler but weaker version of our bound.",
"On replacing in the proof of Proposition REF (i) our optimized height bounds by the simplified height bounds in Proposition REF , we obtain $\\lambda N(x,y)^2\\le M_0\\le \\tfrac{m+1}{6}h(a)+\\tfrac{1}{2}a_S\\log a_S.$ For the purpose of the following discussion, we recall that in the case $a\\in {\\mathbb {Z}}$ it holds that $a_S\\le 1728\\vert {}a{}\\vert N_S^2$ and $m=1$ .",
"The actual best effective upper bound for $N(x,y)$ and $M_0$ was established by Pethő–Zimmer–Gebel–Herrmann [126].",
"Their result is based on the work of Hajdu–Herendi [78] which in turn relies on the theory of logarithmic forms.",
"To state the rather complicated bound for $N(x,y)$ provided by Pethő et al, we need to introduce some notation.",
"As in [126] we define the constants $k_3=\\frac{32}{3}\\Delta _0^{\\frac{1}{2}}(8+\\frac{1}{2}\\log \\Delta _0)^4, \\ \\ \\ k_4=10^4\\cdot 256\\cdot \\Delta _0^{\\frac{2}{3}}, \\ \\ \\ \\Delta _0=27\\vert a\\vert ^2.$ Further, we write $s=\\vert S\\vert $ and $q=\\max S$ (with $q=1$ if $S=\\emptyset $ ).",
"Then we define $\\kappa _1=\\tfrac{7}{2}\\cdot 10^{38s+87}(s+1)^{20s+35}q^{24}\\max (1,\\log q)^{4s+2} k_3(\\log k_3)^2(k_3+20sk_3+\\log (ek_4)).$ We mention that the result in [126] is stated under the assumption that the given Weierstrass equation over ${\\mathbb {Z}}$ is minimal at all primes in $S$ .",
"However, on looking at the proof one sees that this minimality assumption is not necessary for the portion of the theorem which provides an upper bound for $N(x,y)$ .",
"We conclude that [126] provides in general that any solution $(x,y)$ of (REF ) with $a\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ satisfies $\\lambda N(x,y)^2\\le M_0\\le \\kappa _1+\\tfrac{1}{3}\\log \\vert {}4\\cdot 6^3a{}\\vert .$ In our simplified bound (REF ) the dependence on $a\\in {\\mathbb {Z}}$ is of the form $\\vert a\\vert \\log \\vert a\\vert $ , while in (REF ) it is of the weaker form $\\vert a \\vert ^2(\\log \\vert a\\vert )^{10}$ .",
"Further we see that (REF ) considerably improves (REF ) for essentially all sets $S$ of practical interest, in particular for all sets $S$ with $N_S\\le 2^{1200}$ or $s\\le 12$ and for all sets $S$ of the form $S=S(n)$ where $S(n)$ denotes the set of the first $n$ primes for some $n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"We now choose parameters $\\mathcal {A}$ and $\\mathcal {S}$ as follows: The set $\\mathcal {S}$ is given by $\\lbrace \\emptyset ,S(1),S(10)\\rbrace $ , and the set $\\mathcal {A}$ consists of 24 distinct nonzero $a\\in {\\mathbb {Z}}$ such that for each $r\\in \\lbrace 1,\\cdots ,12\\rbrace $ there are precisely two $a$ in $\\mathcal {A}$ with $E_a({\\mathbb {Q}})$ of rank $r$ ; here we triedFor $r\\le 6$ we found the “smallest\" possible $a$ .",
"Further, we note that for our purpose of illustrating the running time improvements of the reduction process it suffices to work with $r$ independent points of $E_a({\\mathbb {Q}})$ ; for $r\\ge 9$ we could not prove (unconditionally) that our $r$ independent points form a basis.",
"to choose these elements $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert $ as small as possible.",
"To illustrate that our bound leads to significant running time improvements, we computed for all parameter pairs $(a,S)\\in \\mathcal {A}\\times \\mathcal {S}$ the running times $\\rho $ and $\\rho ^*$ of the elliptic logarithm reduction in Algorithm REF (ii) using Proposition REF (i) and (REF ) respectively.",
"In the case $S=\\emptyset $ , it turned out that we obtain a running time improvement by a factor $\\rho ^*/\\rho $ which is approximately 2 for small/medium $\\vert {}a{}\\vert $ and which is close to 4 for large $\\vert {}a{}\\vert $ .",
"The running time improvements become more significant in the case $S=S(1)$ .",
"Here the factor $\\rho ^*/\\rho $ is approximately 30 for small/medium $\\vert {}a{}\\vert $ and it is approximately 60 for large $\\vert {}a{}\\vert $ .",
"Finally we achieve big running time improvements when $S=S(10)$ .",
"In this case the factor $\\rho ^*/\\rho $ is approximately 300 for small $\\vert {}a{}\\vert $ , it lies between 500 and $10^3$ for medium sized $\\vert {}a{}\\vert $ and it varies between $10^3$ and $10^4$ for large $\\vert {}a{}\\vert $ .",
"For example if $a=-2520963512$ ($r=8$ ) and $S=S(10)$ then $\\rho $ is less than 23 seconds while $\\rho ^*$ exceeds 2 days.",
"In the classical case $\\mathcal {O}={\\mathbb {Z}}$ , there is also a fully explicit estimate $N_0\\ge N(x,y)$ which was independently established by Stroeker–Tzanakis [141] and Gebel–Pethő–Zimmer [68].",
"This estimate is based on lower bounds for linear forms in elliptic logarithms (see Masser [112], Wüstholz [166], Hirata-Kohno [80] and David [41]).",
"We do not state here $N_0$ in its precise form, since $N_0$ is even more complicated than the bound in (REF ).",
"To see that our result improves $N_0$ for essentially all $a\\in {\\mathbb {Z}}$ of practical interest, it suffices to consider the following simpler lower bound $N_0\\ge \\lambda ^{-1/2}10^{3(r+2)}4^{(r+1)^2}(r+2)^{(r^2+13r+23.3)/2} \\prod _{i=1}^{r} \\max \\bigl (\\hat{h}(P_i)^{1/2},\\log (4\\vert {}a{}\\vert )\\bigl ).$ This lower bound follows for example from [126], see also the recent book of Tzanakis [159].",
"In (REF ) we may and do assume that $r\\ge 1$ by Fueter's result (REF ).",
"Then we see that our simplified bound (REF ) improves (REF ) for all nonzero $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^{40}$ .",
"In the case of arbitrary $\\mathcal {O}\\ne {\\mathbb {Z}}$ and $r=2$ , one can deduce an explicit estimate $N_0\\ge N(x,y)$ by using lower bounds of David [41] and Rémond–Urfels [129]; see also the recent work of Hirata-Kohno–Kovács [79], [82] removing the assumption $r=2$ .",
"Here the quantity $N_0$ is very complicated and its dependence on $S$ is quite involved.",
"In any case $N_0$ is larger than the lower bound in (REF ) and thus our result is better than the estimate $N_0\\ge N(x,y)$ for all pairs $(a,S)$ with $a_S\\le 10^{40}$ .",
"On the other hand, for large $a_S$ it is rather difficult (when not impossible in general) to compare Proposition REF (i) with the corresponding results based on lower bounds for linear forms in elliptic logarithms.",
"The reason is that the involved quantities are quite different.",
"However, the dependence of (REF ) on the rank $r$ means that our result leads to significant running time improvements in the notoriously difficult case when $r$ is not small.",
"To illustrate this we computed for all parameter pairs $(a,S)\\in \\mathcal {A}\\times \\mathcal {S}$ the running times $\\rho $ and $\\rho ^{\\prime }$ of the elliptic logarithm reduction in Algorithm REF (ii) using Proposition REF (i) and $N_0$ respectively.",
"We note that instead of implementing the very complicated estimate $N_0$ in its precise form, we used here the simpler lower bound in (REF ).",
"In other words the running time $\\rho ^{\\prime }$ is slightly too good, which means that our running time improvements are slightly better than illustrated by the numbers appearing in the following discussion.",
"In the case $S=\\emptyset $ , we obtain a running time improvement by a factor $\\rho ^{\\prime }/\\rho $ which is approximately 2 when $2\\le r\\le 4$ and which lies between 3 and 10 in the range $5\\le r\\le 12$ .",
"The running time improvements become more significant in the case $S=S(1)$ .",
"Here the factor $\\rho ^{\\prime }/\\rho $ lies between 2 and 20 when $2\\le r\\le 4$ , it varies between 50 and 100 in the range $5\\le r\\le 8$ , and it lies between 500 and $10^3$ for $9\\le r\\le 12$ .",
"Finally we obtain big running time improvements when $S=S(10)$ .",
"In this case the factor $\\rho ^{\\prime }/\\rho $ varies between 2 and 10 in the range $2\\le r\\le 4$ , it lies between 30 and 500 for $r\\le 5\\le 8$ , and it varies between 700 and 3000 in the range $r\\le 9\\le 12$ .",
"For example, if $S=S(10)$ then there is an $a\\in \\mathcal {A}$ with $r=12$ such that our running time $\\rho $ is less than 3 minutes while $\\rho ^{\\prime }$ is approximately 5 days."
],
[
"The algorithm",
"We continue the notation introduced above.",
"On combining the ingredients of the previous sections, we obtain an algorithm which allows to solve the Mordell equation (REF ).",
"Here we point out that our algorithm requires an explicitly given Mordell–Weil basis of $E_a({\\mathbb {Q}})$ .",
"While it is usually possible to determine such a basis in practice (see Section REF ), there is so far no unconditional method which in principle works for an arbitrary nonzero $a\\in {\\mathbb {Z}}$ .",
"In view of this we included a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ in the input.",
"Algorithm 4.4 (Mordell equations via height bounds) The inputs are a finite set of rational primes $S$ , a nonzero number $a\\in \\mathcal {O}$ and the coordinates of a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ .",
"The output is the set of solutions $(x,y)$ of (REF ).",
"(i) Use Proposition REF (i) to compute an initial bound $M_0$ such that for any solution $(x,y)$ of (REF ) the corresponding point $P\\in E_a({\\mathbb {Q}})$ satisfies $\\hat{h}(P)\\le M_0$ .",
"(ii) Apply the elliptic logarithm sieve in Algorithm REF in order to find all solutions $(x,y)$ of (REF ) with corresponding point $P\\in E_a({\\mathbb {Q}})$ satisfying $\\hat{h}(P)\\le M_0$ ."
],
[
"Correctness.",
"If $P\\in E_a({\\mathbb {Q}})$ corresponds to a solution $(x,y)$ of (REF ), then Proposition REF (i) gives that $\\hat{h}(P)\\le M_0$ .",
"Thus the application of the elliptic logarithm sieve in step (ii) produces all solutions of (REF ) as desired, see Remark REF when $a\\notin {\\mathbb {Z}}$ ."
],
[
"Complexity.",
"We now discuss various aspects which significantly influence the running time of Algorithm REF .",
"In view of the remark given below Proposition REF (i), the computation of the initial upper bound $M_0$ in step (i) is always very fast.",
"The running time of step (ii) crucially depends on the size of $M_0$ , the height $h(a)$ , the rank $r$ and the cardinality of $S$ .",
"For a complexity discussion of the elliptic logarithm sieve used in step (ii) we refer to Section .",
"Therein we explain in detail various complexity aspects and we also discuss in detail the influence of the parameters $M_0,r,h(a)$ and $\\vert {}S{}\\vert $ on the running time in practice (and in theory).",
"See also Section REF where we illustrated the improvements provided by the sharpened initial bound $M_0$ obtained in Proposition REF (i).",
"Remark 4.5 (Generalizations) Algorithm REF allows to solve more general Diophantine equations associated to a Mordell curve, that is an elliptic curve with vanishing $j$ -invariant.",
"Assume that we are given the coefficients $a_1\\cdots ,a_6\\in {\\mathbb {Q}}$ of a Weierstrass equation $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$ of an elliptic curve $E$ with $j$ -invariant $j=0$ , and suppose that we know a basis of the free part of $E({\\mathbb {Q}})$ .",
"Then Algorithm REF allows to find all solutions $(x,y)$ of (REF ) in $\\mathcal {O}\\times \\mathcal {O}$ .",
"Indeed there is an explicit isomorphism which transforms any such solution of (REF ) into a solution of (REF ) for some explicit $a\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ , and hence an application of Algorithm REF with this $a$ produces the set of solutions of (REF ) and then of (REF ).",
"In fact we implemented this slightly more general version of Algorithm REF .",
"To conclude we mention that further generalizations are possible by using the arguments in [159]."
],
[
"Elliptic curves with good reduction outside a given set of primes",
"We continue the notation introduced above.",
"Let $M(S)$ be the set of ${\\mathbb {Q}}$ -isomorphism classes of elliptic curves over ${\\mathbb {Q}}$ with good reduction outside a given finite set of rational primes $S$ .",
"In this section we apply Algorithm REF in order to compute the set $M(S)$ ."
],
[
"Known methods.",
"There are already several practical methods in the literature which allow to determine $M(S)$ .",
"Agrawal–Coates–Hunt–van der Poorten [1] computed the semi-stable locus of $M(\\lbrace 11\\rbrace )$ by using an approach via Thue–Mahler equations which ultimately relies on the theory of logarithmic forms.",
"Their work builds on Coates' effective proof [37] of Shafarevich's theorem mentioned in the introduction.",
"Alternatively one can compute $M(S)$ by using the Shimura–Taniyama conjecture and modular symbols, see Cremona [39].",
"There are also two more recent approaches which ultimately rely on the theory of logarithmic forms: The method of Cremona–Lingham [34] discussed in the introduction, and the very recent approach of Koutsianas [98] via $S$ -unit equations over number fields.",
"Furthermore, very recently Bennett–Rechnitzer [25], [26] substantially refined (in particular for $\\vert {}S{}\\vert =1$ ) the above mentioned classical Thue–Mahler approach: In the irreducible case they use the Thue–Mahler algorithm of Tzanakis–de Weger [154] and in the rational two torsion case they apply the algorithm of de Weger [162], [163] for sums of units being a square.",
"Finally, several authors used ingenious ad hoc methods to determine $M(S)$ for specific sets $S$ .",
"For an overview, see for example the discussions and references in [40] and [26]."
],
[
"The algorithm.",
"On combining Shafarevich's classical reduction to Mordell equations (REF ) with Algorithm REF , we obtain an alternative approach to determine $M(S)$ .",
"Here we do not use modular symbols or lower bounds for linear forms in logarithms.",
"To state our algorithm, we introduce some terminology.",
"We may and do identify any $[E]$ in $M(S)$ with the pair $(c_4,c_6)$ associated by Tate [150] to a minimal Weierstrass model of $E$ over ${\\mathbb {Z}}$ .",
"Further, for arbitrary $s,t\\in {\\mathbb {Q}}$ we say that an elliptic curve over ${\\mathbb {Q}}$ is given by $(s,t)$ if the equation $y^2=x^3-27sx-54t$ defines an affine model of the curve.",
"If $s,t$ are in ${\\mathbb {Q}}$ with $s^3-t^2$ nonzero and if $E$ denotes an elliptic curve over ${\\mathbb {Q}}$ given by $(s,t)$ , then Tate's algorithm [151] allows to compute the pair $(c_4,c_6)$ associated to a minimal Weierstrass model of $E$ over ${\\mathbb {Z}}$ and it allows to check whether $[E]$ lies in $M(S)$ .",
"Algorithm 4.6 The inputs are a finite set of rational primes $S$ and a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for all $a=1728w$ with $w\\in {\\mathbb {Z}}$ dividing $N_S^5$ .",
"The output is the set $M(S)$ .",
"The algorithm: For each $a=1728w$ with $w\\in {\\mathbb {Z}}$ dividing $N_S^5$ , do the following.",
"(i) Apply Algorithm REF in order to determine the set $Y_a(\\mathcal {O})$ formed by the solutions of the Mordell equation (REF ) defined by the parameter pair $(a,S)$ .",
"(ii) For each $(x,y)\\in Y_a(\\mathcal {O})$ and for any $d\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S$ , let $E$ be the elliptic curve over ${\\mathbb {Q}}$ given by $(d^2x,d^3y)$ and output the pair $(c_4,c_6)$ associated to a minimal Weierstrass model of $E$ over ${\\mathbb {Z}}$ provided that $[E]$ lies in $M(S)$ ."
],
[
"Correctness.",
"We take $[E]=(c_4,c_6)$ in $M(S)$ .",
"The minimimal discriminant $\\Delta $ of $E$ lies in $\\mathcal {O}^\\times $ .",
"Hence there are integers $u,w,d\\in \\mathcal {O}^\\times $ , with $d\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S$ and $w$ dividing $N_S^5$ , such that $\\Delta =-wd^6u^{12}$ .",
"We define $x=\\tfrac{c_4}{u^4d^2}$ and $y=\\tfrac{c_6}{u^6d^3}$ .",
"The formula $1728\\Delta =c_4^3-c_6^2$ shows that $(x,y)$ lies in the set $Y_a(\\mathcal {O})$ computed in step (i) for $a=1728w$ , and the elliptic curve $E$ is given by $(d^2x,d^3y)$ .",
"Hence we see that step (ii) produces our $[E]$ as desired."
],
[
"Complexity.",
"The running time of Algorithm REF is essentially determined by step (i).",
"Therein we compute the sets $Y_a(\\mathcal {O})$ for all $a=1728w$ with $w\\in {\\mathbb {Z}}$ dividing $N_S^5$ and for this purpose we need to apply Algorithm REF with $2\\cdot 6^{\\vert {}S{}\\vert }$ distinct parameters $a$ .",
"Hence the running time of Algorithm REF crucially depends on $\\vert {}S{}\\vert $ and on the complexity of Algorithm REF which we already discussed in the previous section.",
"In step (ii), it might be possible that one can omit to check whether $[E]$ lies in $M(S)$ .",
"In any case this check is always very quick and it has no influence on the running time in practice."
],
[
"Input obstruction and the family $\\mathcal {S}$ .",
"The input of Algorithm REF requires $2\\cdot 6^{\\vert {}S{}\\vert }$ distinct Mordell–Weil bases.",
"In fact, for large $\\vert {}S{}\\vert $ , it usually happens that one can not determine unconditionally all bases and then our Algorithm REF can not be used to compute $M(S)$ .",
"However, for small $\\vert {}S{}\\vert $ it turned out in practice that one can often efficiently compute the required bases by using the known techniques in (PSM).",
"For example, without introducing crucial new ideas, we computed the required bases for each set $S$ in $\\mathcal {S}$ .",
"Here $\\mathcal {S}$ is a family of sets which contains in particular the set $S(5)$ and all sets $S$ with $N_S\\le 10^3$ .",
"We observe that any elliptic curve over ${\\mathbb {Q}}$ with good reduction outside $S$ has conductor dividing $N_S^{\\textnormal {cond}}=\\prod _{p\\in S} p^{f_p}$ , where $(f_2,f_3)=(8,5)$ and $f_p=2$ if $p\\ge 5$ .",
"It holds that $N_S^2$ divides $N_S^{\\textnormal {\\textnormal {cond}}}$ , and thus $\\mathcal {S}$ contains in particular all sets $S$ with $N_S^{\\textnormal {\\textnormal {cond}}}\\le 10^6$ ."
],
[
"Applications.",
"On using Algorithm REF , we determined the sets $M(S)$ for all $S\\in \\mathcal {S}$ .",
"This took less than $2.5$ hours for $S=S(5)$ , and on average it took approximately 30 seconds for sets $S\\in \\mathcal {S}$ with $\\vert {}S{}\\vert =2$ , roughly 2.5 minutes for sets $S\\in \\mathcal {S}$ with $\\vert {}S{}\\vert =3$ and approximately 8 minutes for sets $S\\in \\mathcal {S}$ with $\\vert {}S{}\\vert =4$ .",
"Here we did not take into account the time required to determine the Mordell–Weil bases for the input.",
"In fact if the bases for the input are not already known, then their computation is usually the bottleneck of our approach to determine $M(S)$ via Algorithm REF .",
"Let $T$ be a nonempty open subscheme of $\\textnormal {Spec}({\\mathbb {Z}})$ .",
"Inspired by our Conjecture 1 on $T$ -points of $\\mathbb {P}^1_{\\mathbb {Z}}-\\lbrace 0,1,\\infty \\rbrace $ , we propose the following analogous conjecture on $T$ -points of the moduli stack $\\mathcal {M}_{1,1}$ of elliptic curves.",
"Conjecture 1 for $\\mathcal {M}_{1,1}$ .",
"Does there exist $c\\in {\\mathbb {Z}}$ with the following property: If $n\\in {\\mathbb {Z}}_{\\ge 1}$ then any set of rational primes $S$ with $\\vert {}S{}\\vert \\le n$ satisfies $\\vert {}M(S){}\\vert \\le \\vert {}M(S(n)){}\\vert +c?$ Our database listing the sets $M(S)$ for all $S\\in \\mathcal {S}$ shows the following: For any $n\\in {\\mathbb {Z}}_{\\ge 1}$ and for each $S\\in \\mathcal {S}$ with $\\vert {}S{}\\vert \\le n$ , it holds that $\\vert {}M(S){}\\vert $ is at most $\\vert {}M(S(n)){}\\vert $ .",
"In light of this we ask whether the above conjecture is true with the optimal constant $c=0$ ?",
"We point out that for certain sets $S\\in \\mathcal {S}$ one can compute the spaces $M(S)$ by using different methods.",
"For example Cremona–Lingham [34] determined $M(S)$ for $S=\\lbrace 2,p\\rbrace $ with $p\\le 23$ , and Koutsianas [98] moreover computed $M(S)$ for $S=\\lbrace 2,3,23\\rbrace $ and $S=\\lbrace 2,p\\rbrace $ with $p\\le 127$ .",
"Further Cremona's database [39] allows to directly determine the space $M(S)$ for all sets $S$ with $N_S^{\\textnormal {cond}}\\le 380000$ (as of February 2016).",
"We also mention that the case $\\vert {}S{}\\vert =1$ was studied by Edixhoven–Groot–Top in [50].",
"In particular they showed that $M(\\lbrace p\\rbrace )$ is empty for many rational primes $p$ , see [50] which explicitly lists such primes $p$ .",
"Furthermore very recently Bennett–Rechnitzer [26] determined $M(\\lbrace p\\rbrace )$ for all primes $p< 2\\cdot 10^9$ , and for all $p< 10^{12}$ conditional on an explicit version of Hall's conjecture with “Hall ratio\" $10^{14}$ .",
"Here to prove their unconditional results, they exploit that $\\vert {}S{}\\vert =1$ in order to reduce to Thue equations which can be solved much more efficiently than Thue–Mahler equations.",
"Their ingenious reduction uses in particular a result of Mestre–Oesterlé [116] which in turn relies (inter alia) on the geometric version of the Shimura–Taniyama conjecture."
],
[
"Comparison.",
"We now briefly discuss advantages and disadvantages of the different methods which allow to compute $M(S)$ in practice.",
"Our Algorithm REF significantly improves the method (CL) of Cremona–Lingham [34].",
"Indeed our Algorithm REF is considerably more efficient in solving (REF ) than the algorithm [126] used in (CL).",
"To illustrate that our improvements are significant, we used (CL) to determine $M(S)$ for $S=S(3)$ .",
"This took more than 35 minutesThis is a lower bound for the time required for (CL) to solve the involved equations (REF ).",
"Here we used the official Sage implementation of [126] which works with an “absolute\" reduction process.",
"This means that the running times of (CL) are in fact larger than the numbers we listed., while it took Algorithm REF less than 2 minutes.",
"Furthermore there are several sets $S\\in \\mathcal {S}$ which seem to be out of reach for (CL).",
"For example in the case $S=S(4)$ it took Algorithm REF less than 20 minutes to compute $M(S)$ , while (CL) did not terminate within 2 months.",
"In comparison with the other practical methods which allow to compute $M(S)$ , the main disadvantage of our approach and of (CL) is that they both require $2\\cdot 6^{\\vert {}S{}\\vert }$ distinct Mordell–Weil bases.",
"The modular symbols method (see Cremona [39]) can efficiently compute the curves in $M(S)$ with small conductor, while the curves of large conductor cause memory problems.",
"We note that even for relatively small sets $S$ the maximal conductor $N_S^{\\textnormal {cond}}$ can be large.",
"For example if $S$ contains $\\lbrace 2,3,p\\rbrace $ for some $p\\ge 13$ then it holds that $N_S^{\\textnormal {cond}}\\ge 10^8$ and thus the practical computation of $M(S)$ seems to be out of reach for the modular symbol method.",
"On the other hand, the modular symbol method deals much more efficiently with the important related problem of compiling a database which lists all elliptic curves over ${\\mathbb {Q}}$ of given conductor $N\\le 380000$ .",
"The efficiency of the approach of Koutsianas [98] strongly depends on the size of $\\vert {}S{}\\vert $ and on the involved number fields (quadratic, cubic, or $S_3$ -extension) in which one has to solve the unit equations.",
"We point out that (CL) and the method of Koutsianas [98] both allow to deal with more general number fields $K$ , while our approach currently only works in the considerably simpler case $K={\\mathbb {Q}}$ .",
"As already mentioned, Bennett–Rechnitzer [25], [26] substantially refined the classical Thue–Mahler approach in order to compute $M(\\lbrace p\\rbrace )$ for all primes $p<2\\cdot 10^9$ .",
"This computation is unfavorable for our method, since finding unconditionally all the required Mordell–Weil bases would (when possible) take a long time with the known techniques.",
"In general, [26] crucially depends on the algorithms [154], [163] for which we are not aware of a complexity analysis.",
"In particular, if $\\vert {}S{}\\vert \\ne 1$ then it is not clear to us how efficient is the Thue–Mahler approach of [25], [26].",
"To compare some data, we computed the space $M(S)$ for all sets $S$ considered in the papers of Cremona–Lingham [34] and Koutsianas [98] and for all sets $S$ which can be covered by Cremona's database [39] (as of February 2016).",
"In all cases it turned out that our Algorithm REF produced exactly the same number of curves."
],
[
"Integral points on moduli schemes",
"We continue our notation.",
"Many Diophantine equations can be reduced via the moduli formalism to the study of $M(S)$ .",
"To explain this more precisely, we use the notation and terminology of [89].",
"The set $M(S)$ identifies with the set $M(T)$ of isomorphism classes of elliptic curves over the open subscheme $T$ of $\\textnormal {Spec}({\\mathbb {Z}})$ given by $T=\\textnormal {Spec}({\\mathbb {Z}})-S$ .",
"Let $Y$ be a $T$ -scheme and suppose that $Y=M_\\mathcal {P}$ is a moduli scheme of elliptic curves.",
"We further assume that the Paršin construction $\\phi :Y(T)\\rightarrow M(T)$ , induced by forgetting the level structure $\\mathcal {P}$ , is effective in the sense that for each $E\\in M(T)$ one can determine the set $\\mathcal {P}(E)$ .",
"Then [89] and the discussions in [89] show that one can in principle determine $Y(T)$ .",
"Furthermore, if $S\\in \\mathcal {S}$ then one can indeed determine $Y(T)$ by applying our explicit results for $M(T)=M(S)$ .",
"This strategy allows to efficiently solve various classical Diophantine problems, including the following equations.",
"(i) One can directly solve the $S$ -unit equation (REF ) for any set $S\\in \\mathcal {S}$ .",
"Here one works with the moduli problem $\\mathcal {P}=[Legendre]$ as in the proof of [89].",
"(ii) We can directly solve any Mordell equation (REF ) defined by $(a,S)$ such that $6a$ is invertible in ${\\mathbb {Z}}[1/N_{S^{\\prime }}]$ for some $S^{\\prime }\\in \\mathcal {S}$ with $S\\subseteq S^{\\prime }$ .",
"Here one works with the moduli problem $\\mathcal {P}_b=[\\Delta =b]$ as in the proof of [89], where $1728b=-a$ .",
"(iii) One can directly solve any cubic Thue equation (REF ) defined by $(f,S,m)$ such that $6\\Delta m$ is invertible in ${\\mathbb {Z}}[1/N_{S^{\\prime }}]$ for some $S^{\\prime }\\in \\mathcal {S}$ with $S\\subseteq S^{\\prime }$ , where $\\Delta $ denotes the discriminant of $f$ .",
"Here one works with the moduli problem obtained by pulling back the problem $\\mathcal {P}_b$ , with $4b=-\\Delta m^2$ , along the morphism $\\varphi $ given in (REF ).",
"(iv) We can directly solve any cubic Thue–Mahler equation (REF ) defined by $(f,S,m)$ such that $6\\Delta m$ is invertible in ${\\mathbb {Z}}[1/N_{S^{\\prime }}]$ for some $S^{\\prime }\\in \\mathcal {S}$ with $S\\subseteq S^{\\prime }$ .",
"Here we work with the moduli problem $\\mathcal {P}$ represented by the elliptic curve $E$ over the moduli scheme $Y=\\textnormal {Spec}\\bigl (\\mathcal {O}[x,y,\\tfrac{1}{d}]\\bigl )$ for $d=6\\Delta f^2$ , where $E$ is given by the closed subscheme of $\\mathbb {P}^2_Y$ defined by $v^2w=u^3+3\\mathcal {H}uw^2+Jw^3$ with $\\mathcal {H}$ and $J$ the covariants (Hessian and Jacobian) of the cubic form $f$ normalized as in (REF ) and (REF ).",
"(v) We can directly solve any generalized Ramanujan–Nagell equation (REF ) defined by $(b,c,S)$ such that $2bc$ is invertible in ${\\mathbb {Z}}[1/N_{S^{\\prime }}]$ for some $S^{\\prime }\\in \\mathcal {S}$ with $S\\subseteq S^{\\prime }$ .",
"Here we work with the moduli problem $\\mathcal {P}$ represented by the elliptic curve $E$ over the moduli scheme $Y=\\textnormal {Spec}\\bigl ({\\mathbb {Z}}[a_2,a_4,\\tfrac{1}{\\delta }]\\bigl )$ for $\\delta =16a_4^2(a_2^2-4a_4)$ , where $E$ is given by the closed subscheme of $\\mathbb {P}^2_Y$ defined by $v^2w=u(u^2+a_2uw+a_4w^2)$ .",
"Note that $\\mathcal {P}$ is related to the classical moduli problem $[\\Gamma _1(2)]$ , see for example [96].",
"In (iv) and (v), any elliptic curve $E$ over $T$ has either no or infinitely many level $\\mathcal {P}$ -structures and Tate's formulas [150] allow to explicitly determine the set of level $\\mathcal {P}$ -structures $\\mathcal {P}(E)$ of $E$ .",
"Here, in (iv) one can proceed similarly as in (R1) of Section REF and in (v) we exploit that one can compute the two torsion of the group $E(T)$ .",
"In fact for each moduli problem used in (i)-(v), one can quickly compute the preimage of the involved Paršin construction $\\phi :Y(T)\\rightarrow M(T)$ and one can directly determine whether a given point in $Y(T)$ corresponds to a solution of the considered Diophantine problem.",
"Hence, if $M(T)$ is known for some $T$ then one can directly solve the equations in (i)-(v) defined by parameters satisfying the mentioned conditions with respect to $T$ ; for example the parameters need to be invertible in $\\mathcal {O}_T(T)$ .",
"On the other hand, if $M(T)$ is not already known, then our algorithms via height bounds are more efficient than first computing $M(T)$ and afterwards the preimage of $\\phi $ .",
"Here the main reason is that these algorithms only need to compute the image of $\\phi $ inside $M(T)$ and this image is usually much smaller than the whole space $M(T)$ .",
"We conclude by mentioning that in (i) we do not use modular symbols as in Algorithm REF or de Weger's sieve as in Algorithm REF ."
],
[
"Applications",
"In this section we present other applications of Algorithm REF .",
"We first discuss parts of our database $\\mathcal {D}_2$ containing the solutions of large classes of Mordell equations (REF ).",
"Then we use $\\mathcal {D}_2$ to motivate a conjecture and two questions on the number of solutions of (REF ).",
"Here we also construct a probabilistic model providing additional motivation.",
"We continue the notation introduced above.",
"Let $Y_a(\\mathcal {O})$ be the set of solutions of (REF ) and recall that $S(n)$ denotes the set of the first $n$ rational primes.",
"To determine a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ which is required in the input of Algorithm REF , we used the methods discussed in Section REF .",
"Further we mention that among all sets $S$ of cardinality $n$ the set $S(n)$ is usually the most difficult case to determine $Y_a(\\mathcal {O})$ .",
"In particular the following running times of Algorithm REF would be considerably better if $S(n)$ is replaced by any set $S$ of $n$ large rational primes.",
"The reason is that the elliptic logarithm sieve becomes considerably stronger for large primes.",
"In fact one would already obtain significant running time improvements by removing from $S(n)$ the notoriously difficult prime 2."
],
[
"The case $\\vert {}a{}\\vert \\le 10^4$ .",
"We solved the Mordell equation (REF ) for all pairs $(a,S)$ such that $S\\subseteq S(300)$ and such that $a\\in {\\mathbb {Z}}$ is nonzero with $\\vert {}a{}\\vert \\le 10^4$ .",
"Here the important special case $S=\\emptyset $ was already established by Gebel–Pethő–Zimmer [71] using a different algorithm.",
"Further we mention that for many $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^4$ we determined $Y_a(\\mathcal {O})$ for sets $S$ which are considerably larger than $S(300)$ .",
"For example, in the ranges $\\vert {}a{}\\vert \\le 10$ and $\\vert {}a{}\\vert \\le 100$ we computed $Y_a(\\mathcal {O})$ for all $S\\subseteq S(10^5)$ and all $S\\subseteq S(10^3)$ respectively."
],
[
"Huge $a$ and {{formula:74d24f2a-1f9e-4c8c-a8ea-cfba50b1f9d3}} .",
"In practice the most common (nontrivial) case is when the Mordell–Weil rank of $E_a({\\mathbb {Q}})$ is one, and in this case our algorithm allows to deal efficiently with huge parameters $a$ and $S$ .",
"To illustrate this feature, we have randomly chosen 100 distinct rank one curves $E_a$ with $\\vert {}a{}\\vert \\ge 10^{10}$ and for each of these curves we then determined the sets $Y_a(\\mathcal {O})$ for all $S\\subseteq S(10^5)$ .",
"On average it took Algorithm REF approximately 0.15 seconds, 6 seconds and 5 hours for $S=\\emptyset $ , $S=S(100)$ and $S=S(10^5)$ respectively."
],
[
"Small rank.",
"The efficiency of Algorithm REF crucially depends on the Mordell–Weil rank $r$ of $E_a({\\mathbb {Q}})$ .",
"We recall that Fueter's result (REF ) completely determines the set $Y_a(\\mathcal {O})$ when $r=0$ .",
"Thus we assume that $r\\ge 1$ in the following discussion.",
"In the generic case, the Mordell curve $E_a$ has small rank $r$ and then our algorithm is very fast.",
"(Rank 1).",
"As already mentioned, in this situation our algorithm can deal efficiently with huge sets $S$ .",
"In particular for each rank one curve $E_a$ with $\\vert {}a{}\\vert \\le 10^4$ we computed the set $Y_a(\\mathcal {O})$ for all $S\\subseteq S(10^4)$ .",
"There are 9546 such rank one curves and on average it took Algorithm REF approximately 20 minutes to determine $Y_a(\\mathcal {O})$ for $S=S(10^4)$ .",
"(Rank 2 and 3).",
"These cases also appear quite often in practice.",
"For example in the range $\\vert {}a{}\\vert \\le 10^4$ there are 3426 curves $E_a$ of rank two and 478 curves $E_a$ of rank three.",
"For these curves, we computed the set $Y_a(\\mathcal {O})$ for all $S\\subseteq S(300)$ and on average it took less than 5 hours and 7 hours in the case of a curve of rank two and three respectively."
],
[
"Large rank.",
"The situation $r\\ge 4$ is rather uncommon in practice.",
"However the notoriously difficult case of large rank $r$ is of particular interest, since it is the most challenging for the known methods computing $Y_a(\\mathcal {O})$ inside the Mordell–Weil group $E_a({\\mathbb {Q}})$ .",
"We mention that in the present case $r\\ge 4$ the following running times can be significantly improved by parallelizing the elliptic logarithm sieve which is used in Algorithm REF .",
"(Rank 4, 5 and 6).",
"We computed $Y_a(\\mathcal {O})$ for 18 rank four curves with $S=S(300)$ , for 12 rank five curves with $S= S(100)$ and for 2 rank six curves with $S=S(50)$ .",
"On average the corresponding running time was roughly 4 days, 2 days and 19 hours in the case of a curve of rank four, five and six respectively.",
"The running times considerably increased for larger $S$ .",
"For example on enlarging $S(100)$ to $S(150)$ and $S(50)$ to $S(75)$ , the running time was on average 6 days and 5 days in the case of a curve of rank five and six respectively.",
"(Rank 7 and 8).",
"We determined the set $Y_a(\\mathcal {O})$ for 2 rank seven curves with $S=S(40)$ and for 4 rank eight curves with $S=S(30)$ .",
"On average the corresponding running time was less than 3 days and 5 days in the case of a curve of rank seven and eight respectively.",
"Here again, the running times significantly increased for larger sets $S$ .",
"For instance on enlarging $S(40)$ to $S(50)$ and $S(30)$ to $S(40)$ , the running time was on average approximately 5 days and 14 days in the case of a curve of rank seven and eight respectively.",
"(Rank at least 9).",
"This situation is extremely rare.",
"However there exist Mordell curves $E_a$ with $E_a({\\mathbb {Q}})$ of rank at least nine.",
"Unfortunately we could not find such a curve for which we were able to determine a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ ; here we usually could only prove that our candidate “basis\" generates a subgroup of $E_a({\\mathbb {Q}})$ which has full rank."
],
[
"Conjecture and questions.",
"We next use our database $\\mathcal {D}_2$ to motivate various questions on the cardinality of the set $Y_a(\\mathcal {O})$ of solutions of (REF ).",
"First we recall Conjecture 2 which is motivated by our data and by the construction of the elliptic logarithm sieve; see also the discussion at the end of this paragraph for additional motivation.",
"Conjecture 2.",
"There are constants $c_a$ and $c_r$ , depending only on $a$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ satisfies $\\vert {}Y_a(\\mathcal {O}){}\\vert \\le c_a \\vert {}S{}\\vert ^{c_r}.$ We now discuss the exponent $c_r$ in this conjecture.",
"For any $b\\in {\\mathbb {Z}}_{\\ge 1}$ we denote by $S[b]$ the smallest set of rational primes such that for any nonzero $P\\in E_a({\\mathbb {Q}})$ with $\\hat{h}(P)\\le b$ the corresponding solution $(x,y)$ of $y^2=x^3+a$ lies in $Y_a(S[b])$ .",
"The Néron–Tate height $\\hat{h}$ defines a positive definite quadratic form on $E_a({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}\\cong {\\mathbb {R}}^r$ .",
"Therefore we obtain that $b^{r/2}=O(\\vert {}Y_a(S[b]){}\\vert )$ and we deduce that $\\vert {}S[b]{}\\vert =O(b^{r/2}\\cdot b)$ since all nonzero $P\\in E_a({\\mathbb {Q}})$ satisfy $\\vert {}\\tfrac{1}{2}h(x)-\\hat{h}(P){}\\vert =O(1)$ ; here the $O$ constants depend only on $a$ .",
"It follows that the exponent $c_r$ has to be at least $\\tfrac{r}{r+2}$ and this leads us to the following question.",
"Question 2.1.",
"What is the optimal exponent $c_r$ in Conjecture 2?",
"In addition our database $\\mathcal {D}_2$ strongly indicates that the exponent $c_r=\\tfrac{r}{r+2}$ is still far from optimal for many families of sets $S$ of interest, including the family $S(n)$ with $n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"More precisely, together with the bound (REF ), our database $\\mathcal {D}_2$ motivates the following question concerning the dependence on $q=\\max S$ .",
"Question 2.2.",
"Are there constants $c_a$ and $c_r$ , depending only on $a$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ with $q=\\max S$ satisfies $\\vert {}Y_a(\\mathcal {O}){}\\vert \\le c_a(\\log q)^{c_r} \\, \\textnormal {?",
"}$ In the case $S=S(n)$ with $n\\ge 2$ , one can replace here $q$ by $n\\log n$ without changing the question.",
"However the above discussion of Conjecture 2 shows that Question 2.2 has in general a negative answer when $q$ is replaced by any power of $\\max (2,\\vert {}S{}\\vert )$ .",
"Further, on considering again the family $S[b]$ , we see that the exponent $c_r$ of Question 2.2 has to be at least $r/2$ .",
"Now we ask whether Question 2.2 has a positive answer for the exponent $c_r=r/2 \\, \\textnormal {?",
"}$ To motivate this refined question, we may and do assume that $a\\in {\\mathbb {Z}}$ .",
"Recall that $S$ is nonempty with $q=\\max S$ .",
"Mahler's result (1933) gives that $\\vert {}Y_a(S^{\\prime }){}\\vert $ is bounded for all sets of rational primes $S^{\\prime }$ with $\\max S^{\\prime }\\le q$ .",
"Thus we may and do assume in addition that $q$ is large.",
"Now we take a nonzero point $P\\in E_a({\\mathbb {Q}})$ and we denote by $(x,y)$ the corresponding solution of $y^2=x^3+a$ .",
"We write $x=x_1/d^2$ and $y=y_1/d^3$ with $x_1,y_1,d\\in {\\mathbb {Z}}$ satisfying $\\gcd (d,x_1y_1)=1$ and $d>0$ .",
"Further we define $\\rho (0)=0$ and $\\rho (P)=\\tfrac{n(P)}{d(P)}$ , where $d(P)=d$ and $n(P)$ is the number of positive integers $n\\in \\mathcal {O}^\\times $ with $n\\le d(P)$ .",
"It holds that $d(P)\\in \\mathcal {O}^\\times $ if and only if $(x,y)$ lies in $Y_a(\\mathcal {O})$ .",
"In light of this we would like to interpret $\\rho (P)$ as the probability of the event that $P\\in E_a({\\mathbb {Q}})$ corresponds to some $(x,y)\\in Y_a(\\mathcal {O})$ .",
"More precisely, putting $\\mu _P(\\lbrace 1\\rbrace )=\\rho (P)$ defines a probability measure $\\mu _P$ on the space $\\Omega _P=(\\lbrace 0,1\\rbrace ,\\mathcal {P})$ for $\\mathcal {P}$ the power set of $\\lbrace 0,1\\rbrace $ .",
"Consider the associated product probability space $\\Omega =(\\prod \\Omega _P,\\prod \\mu _P)$ with the product taken over all nonzero points $P\\in E_a({\\mathbb {Q}})$ .",
"It follows that the random variable $\\vert {}\\tilde{Y}_a(S){}\\vert =\\sum \\omega _P$ on $\\Omega $ has expected value $\\mathbb {E}\\bigl (\\vert {}\\tilde{Y}_a(S){}\\vert \\bigl )=\\sum _{P\\in E_a({\\mathbb {Q}})} \\rho (P)$ where $\\omega _P:\\Omega \\rightarrow \\Omega _P$ denotes the coordinate function.",
"We next estimate this expected value.",
"For each $n\\in {\\mathbb {Z}}_{\\ge 1}$ we denote by $\\Psi (n,q)$ the de Bruijn function, that is the number of $q$ -smooth numbers which are at most $n$ .",
"We observe that $\\rho (P)\\le \\Psi (d(P),q)/d(P)$ and de Bruijn (1951) gives absolute constants $c_1,c_2\\in {\\mathbb {R}}_{>0}$ such that $\\tfrac{1}{n}\\Psi (n,q)\\le c_1 n^{-c_2/\\log q}$ .",
"Further, for each $\\varepsilon >0$ a classical Diophantine approximation result of Siegel (1929) implies that $\\hat{h}(P)\\le (1+\\varepsilon )d(P)+c_3$ with a constant $c_3$ depending only on $a$ and $\\varepsilon $ .",
"We also recall that $E_a({\\mathbb {Q}})_{\\textnormal {tor}}$ has bounded cardinality and that $\\hat{h}$ defines a positive definite quadratic form on $E_a({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}\\cong {\\mathbb {R}}^r$ .",
"Therefore, on combining the above observations, we see that elementary analysis gives a constant $c_a$ depending only on $a$ such that $\\mathbb {E}\\bigl (\\vert {}\\tilde{Y}_a(S){}\\vert \\bigl )\\le c_a(\\log q)^{r/2}.$ This motivates Question 2.2 and its refinement in (REF ).",
"Moreover, the above arguments allow to describe explicitly the constant $c_a$ of (REF ) in terms of $r$ , $a$ , the regulator of $E_a({\\mathbb {Q}})$ , the cardinality of $E_a({\\mathbb {Q}})_{\\textnormal {tor}}$ and a constant given by an effectively computable integral involving the Dickman function.",
"To control here the constant $c_3$ in terms of $a$ , one can use Baker's explicit abc-conjecture stated in Section REF .",
"We point out that all constructions of this paragraph do not use that $E_a$ is a Mordell curve.",
"In fact they can be directly applied to motivate the corresponding conjecture and questions for any hyperbolic genus one curve over $\\textnormal {Spec}({\\mathbb {Z}})-S$ .",
"We refer to Section REF for details."
],
[
"Comparison of algorithms",
"In this section we discuss advantages and disadvantages of Algorithms REF and REF .",
"We also compare our approach to the actual best methods solving (REF )."
],
[
"Advantages and disadvantages.",
"Our Algorithm REF via modular symbols is very fast for all parameters $S$ and $a$ which are small enough such that the image of the Paršin construction $\\phi $ is contained (see Sections REF and REF ) in a database listing all elliptic curves over ${\\mathbb {Q}}$ of given conductor.",
"Unfortunately this image is usually not contained in the actual largest known database (due to Cremona) and then the computation of the required elliptic curves via modular symbols is not efficient; here the main problem is the memory.",
"Thus in most cases Algorithm REF can presently not compete with other approaches.",
"In the generic case, our Algorithm REF considerably improves the actual best methods resolving (REF ).",
"In particular it is significantly faster than the known algorithms using the elliptic logarithm approach.",
"Indeed our optimized height bounds are sharper in practice and our elliptic logarithm sieve substantially improves in all aspects the known enumerations.",
"Furthermore, an important feature of Algorithm REF is that it allows to efficiently solve (REF ) for large sets $S$ .",
"This seems to be out of reach for approaches via logarithmic forms which usually reduce to Thue(–Mahler) equations or to $S$ -unit equations over number fields.",
"Here we point out that in the important special case $S=\\emptyset $ and varying $a\\in {\\mathbb {Z}}-0$ with $\\vert {}a{}\\vert \\le A$ for some given $A\\in {\\mathbb {Z}}_{\\ge 1}$ , the classical Baker–Davenport approach via logarithmic forms is very efficient.",
"As already mentioned, Bennett–Ghadermarzi [18] refined this approach and computed the solutions of (REF ) in ${\\mathbb {Z}}\\times {\\mathbb {Z}}$ for all nonzero $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\le 10^7$ .",
"This computation involves many distinct parameters $a$ , which is unfavorable for our approach since finding unconditionally all the required Mordell–Weil bases would (when possible) take a long time with the known techniques.",
"In particular this highlights the disadvantage of Algorithm REF which is its dependence on an explicitly given Mordell-Weil basis.",
"On the other hand, for $a\\in {\\mathbb {Z}}-0$ fixed one can usually determine a basis in practice and then Algorithm REF is very fast even when $\\vert {}a{}\\vert $ is huge; see Section REF ."
],
[
"Comparison of data.",
"Some parts of our database $\\mathcal {D}_2$ containing the solutions of large classes of Mordell equations (REF ) were already computed by other authors using different methods; see the work of Gebel–Pethő–Zimmer [69], [70], [71] and Bennett–Ghadermarzi [18].",
"On comparing the data in the overlapping cases, it turned out that our Algorithm REF never produced less solutions.",
"In particular, for all parameters in the class $\\lbrace \\vert {}a{}\\vert \\le 10^4, S=\\emptyset \\rbrace $ one verifies that our data coincides with the corresponding results in the database obtained by Bennett–Ghadermarzi [18]."
],
[
"Algorithms for Thue and Thue–Mahler equations",
"In [89] an effective finiteness proof (see Section ) for arbitrary cubic Thue equations was obtained by using inter alia an explicit reduction to a specific Mordell equation.",
"In the present section we combine the same strategy with our algorithms for Mordell equations in order to solve cubic Thue and Thue–Mahler equations.",
"We continue the notation introduced in the previous sections.",
"In particular we denote by $S$ an arbitrary finite set of rational prime numbers and we write $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ for $N_S=\\prod _{p\\in S} p$ .",
"Let $f\\in \\mathcal {O}[x,y]$ be a homogeneous polynomial of degree 3 with nonzero discriminant and let $m\\in \\mathcal {O}$ be nonzero.",
"We recall the cubic Thue equation $f(x,y)=m, \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}.",
"\\qquad \\mathrm {(\\ref {eq:thue})}$ In theory, the problem of solving cubic Thue equations (REF ) is equivalent to the problem of finding all primitive solutions of general cubic Thue–Mahler equations (REF ).",
"Definition 5.1 We say that $(x,y,z)$ is a primitive solution of the general cubic Thue–Mahler equation (REF ) if $x,y,z\\in {\\mathbb {Z}}$ satisfy the equation $f(x,y)=mz$ with $z\\in \\mathcal {O}^\\times $ and if $\\pm 1$ are the only $d\\in {\\mathbb {Z}}$ with the property that $d\\mid \\gcd (x,y)$ and $d^3\\mid z$ .",
"If $(x,y,z)$ is a solution of the cubic Thue–Mahler equation (REF ) discussed in the introduction, then $(x,y,z)$ is in particular a primitive solution in the sense of Definition REF .",
"In fact one can directly write down all solutions of the Thue–Mahler equation (REF ) if one knows all primitive solutions of the general cubic Thue–Mahler equation (REF )."
],
[
"Known methods.",
"Baker–Davenport [4], [13] obtained a practical approach (see e.g.",
"Ellison et al [51]) to solve the cubic Thue equation (REF ) in ${\\mathbb {Z}}\\times {\\mathbb {Z}}$ .",
"See also the variation of Pethő–Schulenberg [125] which uses in addition the $L^3$ algorithm.",
"Moreover, Tzanakis–de Weger [152], [154] and Bilu–Hanrot [19], [21] constructed practical algorithms solving Thue and Thue–Mahler equations of arbitrary degree by applying the theory of logarithmic forms [33].",
"We further remark that the classical $p$ -adic method of Skolem often allows to find all solutions of the cubic Thue equation (REF ) in ${\\mathbb {Z}}\\times {\\mathbb {Z}}$ .",
"In fact several authors used this method to practically resolve specific Thue equations.",
"See for instance Stroeker–Tzanakis [140] and the references therein.",
"There is also a recent algorithm for (REF ) due to Kim [95], which we shall discuss in Section REF ."
],
[
"Preliminary constructions",
"In this section we discuss various constructions which shall be used in our algorithms for Thue and Thue–Mahler equations.",
"We continue the notation introduced above."
],
[
"Invariant theory.",
"To reduce our given cubic Thue equation (REF ) to some specific Mordell equation (REF ), we use classical invariant theory for cubic binary forms going back at least to Cayley.",
"We write $\\Delta $ for the discriminant of $f$ and we denote by $\\mathcal {H}$ and $J$ the covariant polynomials of $f$ of degree two and three respectively; see Section REF for the definitions and for our normalizations.",
"Classical invariant theory gives that $u=-4\\mathcal {H}$ and $v=4J$ satisfy the relation $v^2=u^3+432\\Delta f^2$ in $\\mathcal {O}[x,y]$ .",
"This induces a morphism $\\varphi : X\\rightarrow Y$ of $\\mathcal {O}$ -schemes, where $X$ and $Y$ are the closed subschemes of $\\mathbb {A}^2_\\mathcal {O}$ associated to the Thue equation (REF ) and to the Mordell equation (REF ) with $a=432\\Delta m^2$ respectively.",
"The solution sets of (REF ) and (REF ) identify with the sets of sections $X(\\mathcal {O})$ and $Y(\\mathcal {O})$ of the $\\mathcal {O}$ -schemes $X$ and $Y$ respectively.",
"Further, the projective closure inside $\\mathbb {P}^2_{\\mathbb {Q}}$ of the generic fiber of $Y$ coincides with the elliptic curve $E_a$ over ${\\mathbb {Q}}$ appearing in previous sections."
],
[
"The preimage of $\\varphi $ .",
"The above morphism $\\varphi :X\\rightarrow Y$ is effective in the following sense: For any given $Q\\in Y(\\bar{{\\mathbb {Q}}})$ , one can determine all $P\\in X(\\bar{{\\mathbb {Q}}})$ with $\\varphi (P)=Q$ .",
"Indeed this follows for example directly from the explicit height inequality in Proposition REF .",
"Alternatively, for any given point $Q\\in Y({\\mathbb {Q}})$ one can efficiently determine all $P\\in X({\\mathbb {Q}})$ with $\\varphi (P)=Q$ by using triangular decomposition.",
"In particular if we are given all points in $Y(\\mathcal {O})$ , then we can efficiently reconstruct the set $X(\\mathcal {O})$ as follows: (R1) For any given $Q\\in Y(\\mathcal {O})$ do the following: First determine the set $Z({\\mathbb {Q}})$ by applying a function in Sage (Singular) based on triangular decomposition, where $Z$ is the spectrum of ${\\mathbb {Q}}[x,y]/I$ for $I=\\bigl (4\\mathcal {H}+u,v-4J\\bigl )$ with $(u,v)$ the solution of (REF ) corresponding to $Q$ .",
"Then output the points of $Z({\\mathbb {Q}})$ which are in $X(\\mathcal {O})$ .",
"Here one can apply triangular decomposition with the affine scheme $Z$ , since it has dimension zero.",
"Indeed it turns out (see Section ) that $\\varphi $ induces a finite morphism $\\bar{X}\\rightarrow E_a$ of degree 3, where $\\bar{X}$ is the projective closure inside $\\mathbb {P}^2_{\\mathbb {Q}}$ of the generic fiber of $X$ ."
],
[
"Reduction to Thue equations.",
"To find all primitive solutions of the general cubic Thue–Mahler equation (REF ), it suffices to solve certain cubic Thue equations (REF ).",
"We now consider an elementary standard reduction: For any $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ , we denote by $X_w$ the closed subscheme of $\\mathbb {A}^2_\\mathcal {O}$ given by $f=mw$ .",
"Suppose that $(x,y,z)$ is a primitive solution of the general cubic Thue–Mahler equation (REF ).",
"On using that the integer $z$ lies in $\\mathcal {O}^\\times $ , we may and do write $z=w\\epsilon ^3$ with an integer $\\epsilon \\in \\mathcal {O}^\\times $ and $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ .",
"Then $u=x/\\epsilon $ and $v=y/\\epsilon $ are elements in $\\mathcal {O}$ which satisfy the Thue equation $f(u,v)=mw$ .",
"In other words $(u,v)$ lies in $X_w(\\mathcal {O})$ .",
"This motivates to consider the following reconstruction: (R2) For each $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ and for any point $(u,v)$ in $X_w(\\mathcal {O})$ , define $x=lu$ , $y=lv$ and $z=l^3w$ for $l\\in {\\mathbb {Z}}_{\\ge 1}$ the least common multiple of the denominators of $u$ and $v$ and output the two primitive solutions $\\pm (x,y,z)$ .",
"Here one verifies that $\\pm (x,y,z)$ are indeed primitive solutions by using that the integer $w\\mid N_S^2$ is cube free.",
"Suppose now that we are given the sets $X_w(\\mathcal {O})$ for all $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ .",
"Then an application of (R2) produces all primitive solutions of the general cubic Thue–Mahler equation (REF ).",
"To prove this statement, we assume that $(x,y,z)$ is such a primitive solution.",
"Then the construction described above (R2) gives $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ and $(u,v)\\in X_w(\\mathcal {O})$ .",
"If $x^{\\prime },y^{\\prime },z^{\\prime }$ are the integers in (R2) associated to $w$ and $(u,v)$ , then there exists $\\delta \\in \\mathcal {O}^\\times $ such that $(x,y,z)=(\\delta x^{\\prime },\\delta y^{\\prime },\\delta ^3z^{\\prime })$ .",
"We deduce that $\\delta =\\pm 1$ , since the triples are primitive.",
"Hence (R2) produces all primitive solutions as desired."
],
[
"Algorithms via modular symbols",
"We continue the above notation.",
"Further we denote by $\\mathcal {I}(S,f,m)$ the data consisting of a finite set of rational primes $S$ , the coefficients of a homogeneous polynomial $f\\in \\mathcal {O}[x,y]$ of degree three with nonzero discriminant $\\Delta $ and a nonzero number $m\\in \\mathcal {O}$ .",
"Algorithm 5.2 (Thue equation via modular symbols) The input is the data $\\mathcal {I}(S,f,m)$ and the output is the set of solutions $(x,y)$ of the Thue equation (REF ).",
"The algorithm: First use Algorithm REF in order to compute the set $Y(\\mathcal {O})$ and then apply the reconstruction algorithm described in (R1).",
"Algorithm 5.3 (Thue–Mahler equation via modular symbols) The input consists of the data $\\mathcal {I}(S,f,m)$ and the output is the set formed by the primitive solutions $(x,y,z)$ of the general cubic Thue–Mahler equation (REF ).",
"The algorithm: First use Algorithm REF in order to determine the sets $X_w(\\mathcal {O})$ for all $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ and then apply the reconstruction described in (R2)."
],
[
"Correctness.",
"The discussions surrounding the reconstruction (R1) imply that Algorithm REF finds all solutions of the cubic Thue equation (REF ) as desired.",
"Furthermore, in view of the arguments given below the reconstruction (R2), we see that Algorithm REF indeed produces all primitive solutions of the general cubic Thue–Mahler equation (REF )."
],
[
"Complexity.",
"The set $Y(\\mathcal {O})$ appearing in Algorithm REF contains very few elements in practice and then the reconstruction (R1) is always very efficient.",
"In fact the bottleneck of Algorithm REF is usually the application of Algorithm REF whose complexity is discussed in Section REF .",
"We further mention that the running time of Algorithm REF is essentially determined by the computation of the sets $X_w(\\mathcal {O})$ for all $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ ."
],
[
"Applications.",
"To discuss practical applications, we define $a=432\\Delta m^2$ and we let $a_S$ be as in (REF ).",
"In the case $a_S\\le 350 000$ , Algorithm REF efficiently solves the cubic Thue equation (REF ) and Algorithm REF quickly finds all primitive solutions of the general cubic Thue–Mahler equation (REF ).",
"Indeed in this case the applications of Algorithm REF are very efficient, since the involved elliptic curves are given in Cremona's database (see Section REF ).",
"On the other hand, if the required data of the involved elliptic curves is not already known, then our Algorithms REF and REF are often not practical anymore.",
"Here the problem is Cremona's algorithm involving modular symbols, which is used in Algorithm REF and which requires a huge amount of memory for large parameters."
],
[
"Algorithms via height bounds",
"We continue the above notation.",
"In view of the discussions at the beginning of Section REF , we included the required Mordell–Weil bases in the input of the following algorithms.",
"We refer to Section REF for methods computing such a basis in practice.",
"Algorithm 5.4 (Thue equation via height bounds) The input is the data $\\mathcal {I}(S,f,m)$ together with the coordinates of a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for $a=432\\Delta m^2$ .",
"The output is the set of solutions $(x,y)$ of the cubic Thue equation (REF ).",
"The algorithm: First use Algorithm REF in order to compute the set $Y(\\mathcal {O})$ and then apply the reconstruction algorithm described in (R1).",
"Algorithm 5.5 (Thue–Mahler equation via height bounds) The input consists of the data $\\mathcal {I}(S,f,m)$ together with the coordinates of a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for all parameters $a=432\\Delta (m w)^2$ with $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ .",
"The output is the set of primitive solutions $(x,y,z)$ of the general cubic Thue–Mahler equation (REF ).",
"The algorithm: First use Algorithm REF in order to determine the sets $X_w(\\mathcal {O})$ for all $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ and then apply the reconstruction described in (R2)."
],
[
"Correctness.",
"On using the arguments appearing in the correctness proof of Algorithms REF and REF , we see that Algorithms REF and REF work correctly."
],
[
"Complexity.",
"We first discuss aspects influencing the running time of Algorithm REF in practice.",
"In this algorithm the reconstruction (R1) is always very fast, while the running time of the computation of $Y(\\mathcal {O})$ is determined by the efficiency of the application of Algorithm REF with $a=432\\Delta m^2$ .",
"Here the efficiency crucially depends on $\\vert {}S{}\\vert $ and on the size of the Mordell–Weil rank of $E_a({\\mathbb {Q}})$ , see the complexity discussions in Section REF .",
"The computation of $Y(\\mathcal {O})$ is usually the bottleneck of Algorithm REF .",
"We next discuss Algorithm REF .",
"The running time of this algorithm is essentially determined by the computation of the sets $X_w(\\mathcal {O})$ for all $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ .",
"For this computation we need to apply Algorithm REF with $3^{\\vert {}S{}\\vert }$ distinct inputs $\\mathcal {I}(S,f,m^{\\prime })$ , where $m^{\\prime }$ is of the form $m^{\\prime }=mw$ with $w\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ .",
"In particular the running time of Algorithm REF crucially depends on $\\vert {}S{}\\vert $ and on the aspects influencing the complexity of Algorithm REF as discussed above."
],
[
"Input obstruction.",
"The inputs of the above algorithms require a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for certain parameters $a$ .",
"In the case of Algorithm REF , one needs to determine such a basis for only one parameter $a$ and this is usually possible in practice (see Section REF ) by using the known techniques implemented in (PSM).",
"On the other hand, the input of Algorithm REF requires a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for $3^{\\vert {}S{}\\vert }$ distinct parameters $a$ .",
"Here, for large $\\vert {}S{}\\vert $ , it often happens in practice that one can not determine unconditionally all required bases in an efficient way and then our Algorithm REF can not be applied to find all primitive solutions of the general cubic Thue–Mahler equation (REF ).",
"However for small $\\vert {}S{}\\vert $ it turned out in practice that the known techniques are usually efficient enough to determine unconditionally the required bases, see Section REF ."
],
[
"Cubic forms of given discriminant",
"There are infinitely many cubic Thue and Thue–Mahler equations of some given nonzero discriminant.",
"However, to solve all these equations, it essentially suffices to consider the equations up to the equivalence relation induced by the action of $\\textnormal {GL}_2({\\mathbb {Z}})$ .",
"In this section we discuss certain aspects of this equivalence relation and we explain how to efficiently determine an explicit equation in each equivalence class.",
"We continue our notation."
],
[
"Equivalence classes.",
"We say that a polynomial in ${\\mathbb {Q}}[x,y]$ is a cubic form if it is homogeneous of degree three with nonzero discriminant.",
"The group $G=\\textnormal {GL}_2({\\mathbb {Z}})$ acts on the set of cubic forms in the usual way.",
"If $f,f^{\\prime }\\in \\mathcal {O}[x,y]$ are cubic forms with $f^{\\prime }=g\\cdot f$ for some $g\\in G$ , then their discriminants coincide and there is an explicit isomorphism between $X(\\mathcal {O})$ and $X^{\\prime }(\\mathcal {O})$ induced by $g$ ; here $X$ and $X^{\\prime }$ are the closed subschemes of $\\mathbb {A}^2_\\mathcal {O}$ given by $f-m$ and $f^{\\prime }-m$ respectively.",
"To determine the set of solutions $X(\\mathcal {O})$ of the cubic Thue equation (REF ), it now suffices to know $g$ together with the set $X^{\\prime }(\\mathcal {O})$ .",
"Similarly if one is given $g\\in G$ together with the set of primitive solutions of the general cubic Thue–Mahler equation (REF ) defined by $(f^{\\prime },S,m)$ with $f^{\\prime }=g\\cdot f$ , then one can directly write down all primitive solutions of the general cubic Thue–Mahler equation (REF ) defined by $(f,S,m)$ ."
],
[
"Reduced cubic forms.",
"The reduction theory of binary forms over ${\\mathbb {Z}}$ is well-developed.",
"See for example the recent book of Evertse–Győry [53].",
"Let $f\\in {\\mathbb {Z}}[x,y]$ be a cubic form.",
"We next discuss how to obtain a cubic form in $G\\cdot f$ which is reduced in some sense.",
"The notion of a reduced cubic form varies a lot in the literature and therefore we now explain in detail the notion which we shall use in this paper.",
"We first consider the case when $f$ is irreducible in ${\\mathbb {Q}}[x,y]$ .",
"In this case, Belabas showed in [15] that the orbit $G\\cdot f$ contains a unique cubic form $f^{\\prime }\\in {\\mathbb {Z}}[x,y]$ which is reduced in the sense of [15]; this notion of a reduced form is inspired by the work of Hermite (1848/1859) if the roots of $f(x,1)$ are all real and of Mathews (1912) otherwise.",
"Furthermore the arguments given in Belabas [15] can be transformed into a simple algorithm which allows to efficiently determine the reduced form $f^{\\prime }$ together with $g\\in G$ satisfying $f^{\\prime }=g\\cdot f$ .",
"Suppose now that $f$ is reducible in ${\\mathbb {Q}}[x,y]$ .",
"In this case we work with a notion of a reduced form which is very simple and which is convenient in the sense that one can trivially determine such a form in each equivalence class.",
"More precisely, the orbit $G\\cdot f$ contains a cubic form $\\sum a_i x^{3-i}y^i$ in ${\\mathbb {Z}}[x,y]$ which is reduced in our following sense: $0=a_3\\le a_1\\le a_2.$ To prove this statement we first observe that we may assume that $f$ is primitive, that is the greatest common divisor of the coefficients $a_i$ of our cubic form $f=\\sum a_i x^{3-i}y^i\\in {\\mathbb {Z}}[x,y]$ is one.",
"Hence we assume that $f$ is primitive.",
"We next show that one can obtain that $a_3=0$ .",
"Suppose that $a_3$ is nonzero.",
"After possibly exchanging $x$ and $y$ , we can assure that $a_0$ is nonzero.",
"Then $f(x,1)$ is reducible in ${\\mathbb {Q}}[x]$ , and thus it is reducible in ${\\mathbb {Z}}[x]$ by Gauss' lemma and by our assumption that $f$ is primitive.",
"Hence, on exploiting again that $f$ is primitive, we see that the extended Euclidean algorithm provides a transformation in $G$ which makes $a_3=0$ as desired.",
"Furthermore, after possibly replacing $x$ by $-x$ , we can assure that $a_2\\ge 0$ .",
"It follows that $a_2\\ge 1$ , since the discriminant of $f$ is nonzero and since $a_3=0$ .",
"Then on using that $a_2\\in {\\mathbb {Z}}_{\\ge 1}$ , we find $\\alpha \\in {\\mathbb {Z}}$ depending on $a_1,a_2$ such that $(x,y)\\mapsto (x,y+\\alpha x)$ leads to $-a_2<a_1\\le a_2$ .",
"Finally, after possibly replacing $y$ by $-y$ we can assure that $a_1\\ge 0$ .",
"We conclude that the orbit $G\\cdot f$ indeed contains a cubic form in ${\\mathbb {Z}}[x,y]$ which is reduced in the sense of (REF ).",
"Here the reduced form may not be unique in its $G$ -orbit, which is no disadvantage for our purpose of solving equations."
],
[
"Given discriminant.",
"For any given nonzero $\\Delta \\in {\\mathbb {Z}}$ , we now explain how we determine all reduced cubic forms in ${\\mathbb {Z}}[x,y]$ of discriminant $\\Delta $ .",
"First we apply the results [15] of Belabas in order to list all desired forms which are irreducible in ${\\mathbb {Q}}[x,y]$ .",
"Then to find the remaining cubic forms we proceed as follows: If $\\sum a_ix^{3-i}y^i$ is a reduced cubic form in ${\\mathbb {Z}}[x,y]$ which is reducible in ${\\mathbb {Q}}[x,y]$ , then the property $a_3=0$ assures that $a_2^2\\mid \\Delta $ .",
"Hence we can directly write down all possible values for $a_2$ , which together with $0\\le a_1\\le a_2$ allows to list all possible values for $a_1$ .",
"Finally we find all possible values for $a_0$ by using an explicit expression for $a_0$ in terms of $\\Delta ,a_1,a_2$ .",
"Here the explicit expression for $a_0$ can be obtained by inserting $a_3=0$ in the discriminant equation."
],
[
"Applications",
"In this section we discuss applications of Algorithms REF and REF .",
"After explaining the database $\\mathcal {D}_3$ containing the solutions of large classes of cubic Thue equations (REF ), we motivate new conjectures and questions on the number of solutions of (REF ).",
"Then we discuss the database $\\mathcal {D}_4$ listing the primitive solutions of many general cubic Thue–Mahler equations (REF ) and we consider generalized superelliptic equations studied by Darmon–Granville [46] and Bennett–Dahmen [14].",
"We continue the above notation."
],
[
"Preliminaries.",
"In our databases $\\mathcal {D}_3$ and $\\mathcal {D}_4$ we use the set $\\mathcal {F}_\\Delta $ of reduced cubic forms in ${\\mathbb {Z}}[x,y]$ of given nonzero discriminant $\\Delta $ .",
"This is sufficient to cover the general case of an arbitrary cubic form $f\\in {\\mathbb {Z}}[x,y]$ of discriminant $\\Delta $ .",
"Indeed the arguments of the previous section allow to quickly find a reduced cubic form $f^{\\prime }\\in {\\mathbb {Z}}[x,y]$ and $g\\in \\textnormal {GL}_2({\\mathbb {Z}})$ with $f^{\\prime }=g\\cdot f$ , and then one can directly write down the solutions with respect to $f$ using the solutions in $\\mathcal {D}_3$ and $\\mathcal {D}_4$ .",
"On applying the techniques described in Section REF , we computed in 3 seconds the sets $\\mathcal {F}_\\Delta $ for all nonzero $\\Delta \\in {\\mathbb {Z}}$ with $\\vert {}\\Delta {}\\vert \\le 10^4$ .",
"The database $\\mathcal {F}$ containing the 17044 reduced forms is uploaded on our homepage: There are 2683 distinct $\\textnormal {GL}_2({\\mathbb {Z}})$ -orbits of cubic forms in ${\\mathbb {Z}}[x,y]$ which are irreducible in ${\\mathbb {Q}}[x,y]$ and $\\mathcal {F}$ lists the reduced form of each such orbit.",
"In addition $\\mathcal {F}$ contains the 14361 reduced cubic forms in ${\\mathbb {Z}}[x,y]$ which are reducible in ${\\mathbb {Q}}[x,y]$ .",
"Further, we determined the Mordell–Weil bases required in the inputs of Algorithms REF and REF by using the known techniques implemented in (PSM) without introducing new ideas; see also Section REF ."
],
[
"Thue equation.",
"For any $n\\in {\\mathbb {Z}}_{\\ge 1}$ we recall that $S(n)$ denotes the set of the first $n$ rational primes.",
"Our database $\\mathcal {D}_3$ contains in particular the solutions of the cubic Thue equation (REF ) for all parameter triples $(f,S,m)$ such that $f\\in \\mathcal {F}_\\Delta $ with $1\\le \\vert {}\\Delta {}\\vert \\le d$ , $S\\subseteq S(n)$ and $m\\in {\\mathbb {Z}}-0$ with $\\vert {}m{}\\vert \\le \\mu $ , where $(d,n,\\mu )$ is as in the following discussion.",
"In the case $(d,n,\\mu )=(10^4,300,1)$ , we could quickly compute almost all of the required Mordell–Weil bases: If the rank was not one then this took a few seconds (in rare cases a few minutes), and also for most rank one curves we could instantly determine a generator.",
"However there were a few rank one curves of large regulator for which it took several hours to compute a generator by using methods in (PSM) (2, 4 and 8-descent, Heegner points).",
"On average it then took Algorithm REF approximately 5 seconds and 5 minutes in order to solve (REF ) for $S$ the empty set and $S=S(100)$ respectively.",
"In certain situations we can make $S$ huge.",
"For example in the case $(d,n,\\mu )=(100,10^3,1)$ , we could compute the required bases in less than 1 minute and on average it then took approximately 1.3 hours and 12 hours in order to solve (REF ) for $S=S(500)$ and $S=S(10^3)$ respectively.",
"Furthermore, in the case $(d,n,\\mu )=(20,10^5,1)$ , we instantly found the required Mordell–Weil bases and on average we then solved (REF ) for $S=S(10^4)$ and $S=S(10^5)$ in less than 20 minutes and 5 hours respectively.",
"Finally for the classical form $f=x^3+y^3$ we solved (REF ) for all $(S,m)$ as followsFor varying $m\\in {\\mathbb {Z}}$ with $\\vert {}m{}\\vert $ bounded, it suffices to consider the case $m\\ge 1$ .",
"Indeed the polynomial $f$ is homogeneous of odd degree and thus the equation $f(x,y)=m$ is equivalent to $f(-x,-y)=-m$ ..",
"In the case $(S(10^5),m)$ with $m\\in {\\mathbb {Z}}_{\\ge 1}$ satisfying $m\\le 15$ , we computed the required bases in less than 1 second and on average it then took roughly 2 hours to solve (REF ).",
"Further in the case $(S(10^3),m)$ with $m\\in {\\mathbb {Z}}_{\\ge 1}$ satisfying $m\\le 100$ , we computed the required bases in less than 1 second and on average it then took approximately 6 hours to solve (REF ).",
"Remark 5.6 (Dependence on rank) We created our database $\\mathcal {D}_3$ with $\\Delta $ , $S$ and $m$ in a given range.",
"These parameters directly influence the efficiency of the known methods solving (REF ).",
"However for our approach the crucial parameter is the involved Mordell–Weil rank $r$ , which does not depend on $S$ and which is usually small even for huge $\\Delta $ , $m$ .",
"Hence the discussions in Section REF , containing running times for any given $r$ , might be more meaningful than the above running times.",
"Finally we mention that in the generic situation where $r\\le 2$ , our Algorithm REF is very fast even for huge parameters $\\Delta $ , $S$ , $m$ ."
],
[
"Conjectures and questions.",
"The morphism $\\varphi :X\\rightarrow Y$ in (REF ) induces a finite morphism $\\bar{X}\\rightarrow \\bar{Y}$ of degree 3, where $\\bar{X}$ and $\\bar{Y}$ are the projective closures inside $\\mathbb {P}^2_{\\mathbb {Q}}$ of the generic fibers of $X$ and $Y$ respectively.",
"It follows that $\\vert {}X(\\mathcal {O}){}\\vert \\le 3\\vert {}Y(\\mathcal {O}){}\\vert $ .",
"Hence on applying our conjectures and questions in Section REF with $Y=Y_a$ for $a=432\\Delta m^2$ , we directly obtain the analogous conjectures and questions on upper bounds for the number of solutions of the cubic Thue equation (REF ) in terms of $S$ and the Mordell–Weil rank $r$ of $\\textnormal {Pic}^0(\\bar{X})({\\mathbb {Q}})$ .",
"Our database $\\mathcal {D}_3$ motivates these analogous conjectures and questions for cubic Thue equations (REF ).",
"Furthermore, it might be possible to obtain more precise conjectures for (REF ) by analyzing in addition the fibers of $\\varphi $ .",
"We leave this for the future."
],
[
"Thue–Mahler equation.",
"We next discuss our database $\\mathcal {D}_4$ .",
"In what follows, by solving (REF ) for $(f,S)$ we mean finding all primitive solutions of the general cubic Thue–Mahler equation (REF ) defined by $f$ , $S$ and $m=1$ ; note that any such primitive solution $(x,y,z)$ satisfies $\\gcd (x,y)=1$ provided that $f\\in {\\mathbb {Z}}[x,y]$ .",
"We solved (REF ) for all $(f,S)$ such that $f\\in \\mathcal {F}_\\Delta $ with $1\\le \\vert {}\\Delta {}\\vert \\le d$ and $S\\subseteq S(n)$ , where $(d,n)$ is of the form $(3000,2)$ , $(1000,3)$ , $(100,4)$ or $(16,5)$ .",
"Here again we could quickly determine almost all of the required Mordell–Weil bases.",
"However for increasing $\\vert {}S{}\\vert $ and $\\vert {}\\Delta {}\\vert $ there were more and more rank one curves of large regulator, and finding a generator for these rank one curves was often the bottleneck of our approach.",
"Given the input, Algorithm REF was fast in all cases.",
"To give the reader an idea of our running times, we now discuss some equations appearing in the literature.",
"We solved the equation of Tzanakis–de Weger [153], and we determined all solutions of the equation of Agraval–Coates–Hunt–van der Poorten [1].",
"Here our total running times were 3 minutes and 3 seconds, which includes the 2.5 minutes and 1.5 seconds that were required to compute the involved bases.",
"In addition we solved the equation of Tzanakis–de Weger [154] which they used to illustrate the practicality of their method.",
"Here we determined the required bases in less than a day and then it took Algorithm REF approximately 1 minute to solve the Thue–Mahler equation.",
"Further, we also solved (REF ) for all $(f,S(6))$ with $f\\in \\mathcal {F}_\\Delta $ and $\\Delta \\in \\lbrace -9,-1,3,27\\rbrace $ .",
"Here the case $\\Delta =27$ covers in particular $f=x^3+y^3$ , and the case $\\Delta =-1$ corresponds to the $S$ -unit equation (REF ) which means that our Theorem B solves in particular (REF ) for all $(f,S)$ with $f\\in \\mathcal {F}_{-1}$ and with $S$ satisfying $S\\subseteq S(16)$ or $N_S\\le 10^7$ .",
"To conclude we mention that our Algorithm REF can be used to study properties of certain generalized superelliptic equations.",
"More precisely, let $f\\in {\\mathbb {Z}}[x,y]$ be a cubic form with nonzero discriminant $\\Delta $ and take $l\\in {\\mathbb {Z}}$ with $l\\ge 4$ .",
"Darmon–Granville [46] deduced from the Mordell conjecture [60] that the generalized superelliptic equation $f(x,y)=z^l, \\ \\ \\ (x,y,z)\\in {\\mathbb {Z}}^3$ with $\\gcd (x,y)=1$ has at most finitely many solutions.",
"Moreover on using inter alia modularity of certain Galois representations, level lowering, classical invariant theory and properties of elliptic curves with isomorphic mod-$n$ Galois representations, Bennett–Dahmen [14] proved: The equation $f(x,y)=z^l$ has only finitely many solutions $(x,y,z,l)\\in {\\mathbb {Z}}^4$ with $l\\ge 4$ and $\\gcd (x,y)=1$ if the following condition $(*)$ holds.",
"$(*)$ The polynomial $f$ is irreducible and there are no solutions of the Thue–Mahler equation (REF ) defined by $f$ , $S=\\lbrace p\\,;\\,p\\mid 2\\Delta \\rbrace $ and $m=1$ .",
"Bennett–Dahmen explicitly constructed in [14] an infinite family of polynomials satisfying condition $(*)$ and they explained in [14] a heuristic indicating that “almost all\" cubic forms should satisfy $(*)$ .",
"Now, for any given cubic form $f\\in {\\mathbb {Z}}[x,y]$ with nonzero discriminant $\\Delta $ , our Algorithm REF allows to verify in practice whether condition $(*)$ holds.",
"In other words, one can check condition $(*)$ without using algorithms which ultimately rely on the theory of logarithmic forms.",
"For example, we used Algorithm REF to verify that $3x^3+2x^2y+5xy^2+3y^3$ satisfies condition $(*)$ ; note that according to [14] this is the cubic form of minimal $\\vert {}\\Delta {}\\vert $ which satisfies condition $(*)$ ."
],
[
"Comparison of algorithms",
"In this section we compare our algorithms for cubic Thue equations (REF ) and cubic Thue–Mahler equations (REF ) with the actual best practical methods in the literature."
],
[
"Advantages and disadvantages.",
"We begin by discussing Algorithms REF and REF for (REF ) and (REF ) using modular symbols (Cremona's algorithm).",
"In the recent work [95], Kim independently constructed an algorithm for cubic Thue–Mahler equations (REF ) using modular symbols and the Shimura–Taniyama conjecture.",
"Kim's method differs from our strategy in the sense that he is not using the route via Thue and Mordell equations, but directly associates to each solution of (REF ) a certain elliptic curve.",
"It turns out that his method is more efficient in terms of $S$ and our strategy is more efficient in terms of $\\Delta $ .",
"In fact both approaches are very fast for all parameters such that the involved elliptic curves are already known.",
"However, usually these curves are not already known and computing these curves via modular symbols is currently not efficient for large parameters; here the main problem is the memory.",
"Thus in most cases the algorithms via modular symbols can presently not compete with approaches solving (REF ) and (REF ) via height bounds.",
"We next compare our Algorithm REF with the actual best methods in the literature solving cubic Thue equations (REF ) using height bounds.",
"Our algorithm requires a Mordell–Weil basis in its input.",
"In practice this basis can usually be computed and then our approach is very efficient.",
"In the important special case when $S$ is empty, the already mentioned method of Tzanakis–de Weger [152] works very well in practice and it usually allows to efficiently solve (REF ).",
"Their method has the advantage of not requiring a Mordell–Weil basis in the input.",
"On the other hand, an advantage of Algorithm REF is that it efficiently deals with large sets $S$ .",
"For example it seems that already sets $S$ with $\\vert {}S{}\\vert \\ge 10$ are out of reach for the known methods solving (REF ), while in the generic case Algorithm REF allows to efficiently solve (REF ) for essentially all sets $S$ with $\\vert {}S{}\\vert \\le 10^3$ .",
"It remains to discuss our Algorithm REF for cubic Thue–Mahler equations (REF ).",
"Its input requires $3^{\\vert {}S{}\\vert }$ distinct Mordell–Weil bases and thus our approach is not practical when $\\vert {}S{}\\vert $ is large.",
"However for small sets $S$ it turned out in practice that one can usually determine the required bases and then our approach is efficient as illustrated in Section REF .",
"If $f\\in {\\mathbb {Z}}[x,y]$ is irreducible, then the above mentioned method (TW) of Tzanakis–de Weger [154] solves in particular any cubic Thue–Mahler equation (REF ).",
"We are not aware of a complexity analysis of (TW) and thus we restrict ourselves to the following comments.",
"There are several results in the literature which resolved specific equations (REF ) using (TW).",
"As far as we know, these results all involve small sets $S$ with $\\vert {}S{}\\vert \\le 4$ and (TW) is quite practical for such small sets.",
"On the other hand, sets $S$ of large cardinality are also problematic for (TW) since this method needs to enumerate points in lattices of rank at least $\\vert {}S{}\\vert $ ."
],
[
"Comparison of data.",
"We are not aware of any database in the literature which contains the solutions of large classes of cubic Thue equations (REF ) or cubic Thue–Mahler equations (REF ).",
"To compare at least some data, we solved the equations of [1], [153], [154] and in all cases it turned out that we found the same set of solutions."
],
[
"Algorithms for generalized Ramanujan–Nagell equations",
"In the present section we use our approaches for Mordell equations (REF ) in order to construct algorithms for the generalized Ramanujan–Nagell equation (REF ).",
"We continue the notation of the previous sections.",
"In particular we denote by $S$ an arbitrary finite set of rational prime numbers and we let $\\mathcal {O}^\\times $ be the group of units of $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ for $N_S=\\prod _{p\\in S} p$ .",
"Further we suppose that $b$ and $c$ are arbitrary nonzero elements of $\\mathcal {O}$ .",
"Now we recall the generalized Ramanujan–Nagell equation $x^2+b=cy, \\ \\ \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}^\\times .",
"\\qquad \\mathrm {(\\ref {eq:rana})}$ We observe that this Diophantine problem is equivalent to the a priori more general Diophantine problem obtained by replacing in (REF ) the polynomial $x^2+b$ by any given polynomial $f\\in \\mathcal {O}[x]$ of degree two with nonzero discriminant."
],
[
"Known methods.",
"As mentioned in the introduction, if $b\\in {\\mathbb {Z}}$ is nonzero and $c=1$ then Pethő–de Weger [124] already obtained a practical method to find all solutions $(x,y)$ of (REF ) with $x,y\\in {\\mathbb {Z}}_{\\ge 0}$ .",
"They use inter alia the theory of logarithmic forms and binary recurrence sequences; see also de Weger [162], [163].",
"In addition, Kim [95] and Bennett–Billerey [8] recently obtained other practical approaches for (REF ) which are briefly discussed in Sections REF and REF respectively."
],
[
"Algorithm via modular symbols",
"We continue the notation introduced above.",
"For any nonzero $a\\in \\mathcal {O}$ , we denote by $Y_a(\\mathcal {O})$ the set of solutions of the Mordell equation (REF ) defined by $(a,S)$ .",
"The following algorithm is a direct application of our Algorithm REF for Mordell equations (REF ).",
"Algorithm 6.1 (Ramanujan–Nagell equation via modular symbols) The input consists of a finite set of rational primes $S$ together with nonzero $b,c\\in \\mathcal {O}$ .",
"The output is the set of solutions $(x,y)$ of the generalized Ramanujan–Nagell equation (REF ).",
"The algorithm: For each $\\epsilon \\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ , use Algorithm REF to determine $Y_a(\\mathcal {O})$ with $a=-b(\\epsilon c)^2$ and for any $(u,v)\\in Y_a(\\mathcal {O})$ output $(\\tfrac{v}{\\epsilon c},\\tfrac{u^3}{\\epsilon ^2 c^3})$ if it satisfies (REF )."
],
[
"Correctness.",
"To show that this algorithm works correctly, we take a solution $(x,y)$ of (REF ).",
"We write $y=\\epsilon y^{\\prime 3}$ with $y^{\\prime }\\in \\mathcal {O}^\\times $ and $\\epsilon \\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ , and we define $a=-b(c\\epsilon )^2$ .",
"Further we put $u=\\epsilon c y^{\\prime }$ and $v=\\epsilon c x$ .",
"It follows that $(u,v)$ lies in $Y_a(\\mathcal {O})$ and thus we see that the above Algorithm REF indeed finds all solutions of (REF ) as desired."
],
[
"Complexity.",
"The running time of Algorithm REF is essentially determined by the applications of Algorithm REF whose complexity is discussed in Section REF ."
],
[
"Applications.",
"To discuss practical applications of Algorithm REF , we define $a=bc^2$ and we let $a_S$ be as in (REF ).",
"In the case $a_S\\le 350 000$ , our Algorithm REF allows to efficiently determine all solutions of the generalized Ramanujan–Nagell equation (REF ).",
"Indeed in this case the applications of Algorithm REF are very efficient, since the involved elliptic curves can be found in Cremona's database (see Section REF ).",
"For example, one can quickly resolve the classical Ramanujan–Nagell equation: $x^2+7=2^n$ with $x,n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"Resolving this Diophantine equation is equivalent to the problem of finding all triangular Mersenne numbers, and any solution lies in the set $\\lbrace (1,3),(3,4),(5,5),(11,7),(181,15)\\rbrace $ .",
"The latter assertion was conjectured by Ramanujan (1913) and was proven by Nagell (1948).",
"One obtains an alternative proof of Nagell's result by using Algorithm REF .",
"To conclude the discussion we mention that our Algorithm REF is often not practical anymore if the elliptic curves induced by the solutions of (REF ) need to be computed via Cremona's algorithm involving modular symbols (see Algorithm REF ).",
"Here the problem is that Cremona's algorithm requires a huge amount of memory for all parameters which are not small."
],
[
"Algorithm via height bounds",
"We continue the above notation.",
"In the next algorithm we apply Algorithm REF several times.",
"These applications require certain Mordell–Weil bases which we included in the input.",
"See Section REF for methods computing such bases in practice.",
"Algorithm 6.2 (Ramanujan–Nagell equation via height bounds) The inputs are nonzero $b,c\\in \\mathcal {O}$ , a finite set of rational primes $S$ and the coordinates of a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for all parameters $a=-b(\\epsilon c)^2$ with $\\epsilon \\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ .",
"The output is the set of solutions $(x,y)$ of the generalized Ramanujan–Nagell equation (REF ).",
"The algorithm: For each $\\epsilon \\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ , use Algorithm REF to determine $Y_a(\\mathcal {O})$ with $a=-b(\\epsilon c)^2$ and for any $(u,v)\\in Y_a(\\mathcal {O})$ output $(\\tfrac{v}{\\epsilon c},\\tfrac{u^3}{\\epsilon ^2 c^3})$ if it satisfies (REF )."
],
[
"Correctness.",
"The arguments given in the correctness proof of Algorithm REF show that the above Algorithm REF indeed finds all solutions of (REF ) as desired."
],
[
"Complexity.",
"To compute in Algorithm REF the sets $Y_a(\\mathcal {O})$ , we need to apply Algorithm REF with $3^{\\vert {}S{}\\vert }$ distinct parameters $a$ .",
"In particular the running time of Algorithm REF crucially depends on $\\vert {}S{}\\vert $ and on the complexity of Algorithm REF discussed in Section REF .",
"Here we mention that the involved Mordell–Weil ranks are usually small in practice and then our Algorithm REF is very fast even for parameters $b,c$ with huge height."
],
[
"Refinement.",
"The input of Algorithm REF requires $3^{\\vert {}S{}\\vert }$ distinct Mordell–Weil bases.",
"In practice it turned out that the computation of these bases is currently the bottleneck of our approach solving (REF ) via height bounds.",
"Consider arbitrary nonzero $b,c,d$ in ${\\mathbb {Z}}$ with $d\\ge 2$ .",
"We now work out a refinement of Algorithm REF which only requires three distinct Mordell–Weil bases to find all solutions of the classical Diophantine problem $x^2+b=c d^n, \\ \\ \\ \\ \\ (x,n)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}.\\qquad \\mathrm {(\\ref {eq:rana2})}$ This is a special case of (REF ) with $y=d^n$ and $S$ given by $S_d=\\lbrace p\\,;\\,p\\mid d\\rbrace $ .",
"In fact many authors refer by “generalized Ramanujan–Nagell equation\" to (special cases of) the Diophantine problem (REF ).",
"We obtain the following algorithm for (REF ).",
"Algorithm 6.3 (Refinement) The input consists of nonzero $b,c,d$ in ${\\mathbb {Z}}$ such that $d\\ge 2$ together with the coordinates of a Mordell–Weil basis of $E_a({\\mathbb {Q}})$ for all $a=-b(\\epsilon c)^2$ with $\\epsilon \\in \\lbrace 1,d,d^2\\rbrace $ .",
"The output is the set of solutions $(x,n)$ of (REF ).",
"The algorithm: For each $\\epsilon \\in \\lbrace 1,d,d^2\\rbrace $ , use Algorithm REF to find $Y_a(\\mathcal {O})$ with $(a,S)=(-b(\\epsilon c)^2,S_d)$ and for any $(u,v)\\in Y_a(\\mathcal {O})$ output $\\bigl (\\tfrac{v}{\\epsilon c},\\log _d(\\tfrac{u^3}{\\epsilon ^2 c^3})\\bigl )$ if it satisfies (REF ).",
"Here for any $z\\in {\\mathbb {R}}$ we define $\\log _d(z)=(\\log z)/\\log d$ if $z>0$ and $\\log _d(z)=-\\infty $ otherwise.",
"To prove that the above algorithm indeed finds all solutions of (REF ), we suppose that $(x,n)$ is such a solution.",
"We write $d^n=\\epsilon d^{3m}$ with $\\epsilon \\in \\lbrace 1,d,d^2\\rbrace $ and $m\\in {\\mathbb {Z}}$ .",
"Further we define $v=\\epsilon c x$ and $u= \\epsilon cd^m$ .",
"Then we observe that $(u,v)$ lies in $Y_a(\\mathcal {O})$ for $(a,S)=(-b(\\epsilon c)^2,S_d)$ and thus we see that Algorithm REF finds all solutions of (REF ) as desired."
],
[
"Applications",
"In this section we give some applications of Algorithms REF and REF .",
"In particular we discuss the database $\\mathcal {D}_5$ containing the solutions of many generalized Ramanujan–Nagell equations (REF ) and of many equations which are of the more classical form (REF ).",
"We also explain how to apply our approach in order to study $S$ -units $m,n\\in {\\mathbb {Z}}$ with $m+n$ a square or cube, and we provide some motivation for Terai's conjectures on Pythagorean numbers."
],
[
"Preliminaries.",
"We continue the above notation.",
"Further for any $n\\in {\\mathbb {Z}}_{\\ge 1}$ we denote by $S(n)$ the set of the first $n$ rational primes.",
"In what follows in this section, the running time $(t_1,t_2)$ of our approach via (Al) is given by the time $t_1$ which was required to compute via (PSM) the Mordell–Weil bases for the input of (Al) and the time $t_2$ which was required to solve via (Al) the discussed equation.",
"Here (Al) is either Algorithm REF or REF ."
],
[
"Generalized Ramanujan–Nagell equation.",
"Our database $\\mathcal {D}_5$ contains in particular the solutions of the generalized Ramanujan–Nagell equation (REF ) for all parameter triples $(b,c,S)$ such that $b\\in {\\mathbb {Z}}$ is nonzero with $\\vert {}b{}\\vert \\le B$ , $c=1$ and $S\\subseteq S(n)$ , where $(B,n)$ is of the form $(6,5)$ , $(35,4)$ , $(250,3)$ or $(10^3,2)$ .",
"For our algorithms via height bounds, the running time to solve the equation (REF ) defined by $(b,c,S)$ is essentially the same as the one to solve the Thue–Mahler equation (REF ) defined by $(f,S,m)$ with $m=c$ and $f\\in \\mathcal {O}[x,y]$ of discriminant $\\Delta =b$ .",
"Hence we refer to the discussions in Section REF which contain in particular our running times for many distinct Thue–Mahler equations (REF ).",
"We next discuss a problem inspired by the original Ramanujan–Nagell equation.",
"Recall that the original equation is $x^2+7=y$ with $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ and $y=2^m$ for some $m\\in {\\mathbb {Z}}$ .",
"Now we put $b=7$ , $c=1$ and $S=S(n)$ with $n\\in {\\mathbb {Z}}_{\\ge 1}$ and we consider the problem of finding all solutions $(x,y)$ of (REF ) with $x,y\\in {\\mathbb {Z}}$ .",
"Here the assumption $x,y\\in {\\mathbb {Z}}$ considerably simplifies the problem in practice.",
"Indeed we can remove all primes $p\\in S$ with $-7$ not a square modulo $p$ , and we know that ${\\textnormal {ord}}_7(y)$ is either zero or one.",
"Hence to solve the problem for $n=8$ , it suffices to find all solutions of (REF ) for $(b,c,S)=(7,c,\\lbrace 2,11\\rbrace )$ with $c=1$ and $c=7$ .",
"These solutions were computed by Pethő–de Weger [124].",
"We obtained an alternative proof of their theorem by using Algorithm REF .",
"Indeed it took our approach less than (5 sec, 4 sec) and (3 sec, 4 sec) to find all solutions in the case $c=1$ and $c=7$ respectively.",
"In particular, we solved the case $n=8$ of the problem in less than 16 seconds.",
"To settle in addition the open case $n=9$ , we need to find all solutions of (REF ) defined by $(b,c,S)=(7,c,\\lbrace 2,11,23\\rbrace )$ with $c=1$ and $c=7$ .",
"Here it took our approach less than (15 sec, 15 sec) and (12 sec, 15 sec) respectively, which means that we solved the case $n=9$ of the problem in less than 1 minute.",
"However in the cases $n\\ge 10$ the running times $t_1$ become significantly larger, since for increasing $n$ we have to deal with more and more rank one curves of large regulator.",
"For example to establish the case $n=11$ of the problem, it took our approach approximately (5 hours, 2 minutes) in total.",
"Here we needed to solve (REF ) for $(b,c,S)=(7,c,\\lbrace 2,11,23,29\\rbrace )$ with $c=1$ and $c=7$ ."
],
[
"Classical case.",
"Our database $\\mathcal {D}_5$ contains in addition the solutions of the more classical equation (REF ) for all $(b,c,d)$ of the form $(7,1,d)$ with $d\\le 888$ .",
"To give the reader an idea of the running times of our approach via Algorithm REF , we determined the solutions of various equations (REF ) of interest which were already solved by different methods.",
"For instance, we found in (1 sec, 1 sec) and (3 sec, 4 sec) all solutions of the equations appearing in the title of the papers of Leu–Li [104] and Stiller [145] respectively.",
"The Diophantine problem (REF ) was intensively studied in the literature when $d$ is prime, see for example [139] and [29] for an overview.",
"Here we solved in (8 sec, 8 sec) all four exceptional equations [139] which appear in the classification of Le initiated in [103].",
"Further, it took less than (30 sec, 20 sec) in total to solve all nine exceptional equations appearing in the classification of Bugeaud–Shorey [28].",
"In particular, this includes the two exceptional equations $x^2+19=55^n$ and $x^2+341=377^n$ of [28] which we solved in (3 sec, 4 sec) and (19 sec, 4 sec) respectively."
],
[
"Sums of units being a square or cube.",
"We now consider the problem of finding all integers $m,n\\in \\mathcal {O}^\\times $ with $\\gcd (m,n)$ square-free and $m+n$ a perfect square.",
"On using inter alia the theory of logarithmic forms and generalized recurrences, de Weger [162], [163] obtained a practical approach for this problem which he used to settle the case $S=S(4)$ .",
"Suppose that $l\\in \\lbrace 2,3\\rbrace $ .",
"In a recent work, Bennett–Billerey [8] show in particular how to practically solve the following problem (in which $l=2$ is the original problem) $m+n=z^l, \\ \\ \\ (m,n,z)\\in {\\mathbb {Z}}^3,$ where $m,n\\in \\mathcal {O}^\\times $ have $l$ -th power free $\\gcd (m,n)$ .",
"They use a different method which combines the Shimura–Taniyama conjecture, modular symbols (Cremona's algorithm) and Frey–Hellegouarch curves.",
"On using in addition congruence arguments and Cremona's database of elliptic curves of given conductor, they solved (REF ) for $S=S(4)$ and $S=\\lbrace 2,3,p\\rbrace $ with $p\\le 100$ .",
"Here for various sets $S=\\lbrace 2,3,p\\rbrace $ they moreover applied the archimedean elliptic logarithm approach in the form of [141], [68].",
"In the case $l=2$ , we directly obtain an alternative approach for (REF ) by applying Algorithm REF with $(-b,1,S)$ for all $b\\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S$ .",
"Indeed for any solution $(m,n,z)$ of (REF ) we may and do write $\\max (m,n)=bu^2$ with $b,u$ positive integers in $\\mathcal {O}^\\times $ such that $b\\mid N_S$ and then we see that $x=z/u$ and $y=\\min (m,n)/u^2$ satisfy the generalized Ramanujan–Nagell equation (REF ) defined by $(-b,1,S)$ .",
"Here $z$ and $u$ are coprime, since $\\gcd (m,n)$ is square-free.",
"Similarly in the case $l=3$ , we directly obtain an alternative approach for (REF ) by combining Algorithm REF with an elementary reduction to Mordell equations (REF ) with parameter $a\\in {\\mathbb {Z}}$ dividing $N_S^5$ ; see the proof of Corollary REF (ii).",
"To illustrate the practicality of our approach, we solved (REF ) for all $S\\subseteq S(5)$ and all $S$ with $N_S\\le 10^3$ .",
"Given the required bases, this took less than 1 day in total.",
"Here we could use our database which already contained the required bases.",
"For $l=2$ (resp.",
"$l=3$ ) we need in general $6^{\\vert {}S{}\\vert }$ (resp.",
"$2\\cdot 6^{\\vert {}S{}\\vert }$ ) distinct Mordell–Weil bases to solve (REF ), which means that our approach is not practical when $\\vert {}S{}\\vert $ is large.",
"On the other hand, for small $\\vert {}S{}\\vert $ it turned out that one can usually determine the required bases in practice and then our approach is efficient.",
"We compared our data with the known results, obtained by de Weger [162], [163] ($S=S(4)$ , $l=2$ ) and Bennett–Billerey [8] ($S=S(4)$ and $S=\\lbrace 2,3,p\\rbrace $ with $p\\le 100$ ).",
"In all cases it turned out that we found the same solutions.",
"We briefly discuss advantages and disadvantages of the different approaches.",
"De Weger's method for $l=2$ is quite involved and we are not aware of its strengths and weaknesses.",
"The strategy of Bennett–Billerey via modular symbolsIn fact on replacing in our approach the involved algorithms via height bounds by the corresponding Algorithms REF and REF via modular symbols, we would directly obtain an alternative approach for (REF ) via modular symbols.",
"However we did not include this, since the arguments of Bennett–Billerey (using a careful analysis of conductors of Frey–Hellegouarch curves) are more direct and more efficient.",
"has the usual advantages and disadvantages of an effective method involving modular symbols (Cremona's algorithm), see the analogous discussions in Section REF .",
"A weakness of our approach is its dependence on many Mordell–Weil bases, and a strength is its efficiency in the case when these bases can be determined.",
"We also mention that Bennett–Billerey used additional tools (such as for example level lowering and the theory of logarithmic forms) to moreover prove explicit finiteness results for (REF ) for all $l\\ge 4$ , see [8].",
"Without introducing crucial new ideas, such results are out of reach for our approach."
],
[
"Pythagorean numbers.",
"We next illustrate that Algorithm REF is a useful tool to study certain classical Diophantine problem on Pythagorean numbers which appear in the literature.",
"To state the first Diophantine equation we take coprime $a,b,c\\in {\\mathbb {Z}}_{\\ge 1}$ with $a$ even, and we assume that $a^2+b^2=c^2$ .",
"Inspired by the works of Jeśmanovic and Sierpiński published in 1956, Terai [155] conjectured that $(a,2,2)$ is the unique solution of $x^2+b^m=c^n, \\ \\ \\ (x,m,n)\\in {\\mathbb {Z}}^3,$ with $x,m,n$ all positive.",
"Several authors settled special cases of (REF ), see for example [156] for an overview.",
"We observe that equation (REF ) is a special case of (REF ) with $l=2$ and hence the above described approach via Algorithm REF allows to solve (REF ) for any given Pythagorean triple $(a,b,c)$ .",
"For example we verified Terai's conjecture for all triples $(a,b,c)$ with $c\\le 85$ .",
"Given the required Mordell–Weil bases, this took less than 1 minute in total.",
"In fact to deal with (REF ) we used a modified version of Algorithm REF , which exploits the special shape of (REF ) in order to reduce the number of required Mordell–Weil bases to 18.",
"More recently, Terai [156] studied the following variation of (REF ).",
"Let $d\\ge 2$ be a rational integer and consider the Diophantine equation $x^2+(2d-1)^m=d^n, \\ \\ \\ (x,m,n)\\in {\\mathbb {Z}}^3,$ with $x,m,n$ all positive.",
"Terai conjectured in [156] that $(d-1,1,2)$ is the unique solution of (REF ).",
"He verified his conjecture for certain values of $d$ , including all $d\\le 30$ except the two cases $d=12,24$ which were both settled independently by Deng [45] and Bennett–Billerey [8].",
"As above, we observe that our approach via Algorithm REF allows to solve (REF ) for any given $d$ .",
"To illustrate the utility of this strategy, we also verified [156] for all $d\\le 30$ .",
"Here it was no problem to compute the required bases and then it took the modified version of Algorithm REF less than 1 minute to solve (REF ) for all $d\\le 30$ .",
"We can also prove new cases of Terai's conjecture concerning (REF ).",
"However the running times to compute the required bases explode for larger $d$ .",
"For example, in the range $30< d< 35$ it took several days to compute the bases and then we solved all equations (REF ) in roughly 30 seconds by using the modified version of Algorithm REF .",
"It turned out that Terai's conjecture holds in the range $30< d< 35$ ."
],
[
"Comparison of algorithms",
"In this section we compare our algorithms for the generalized Ramanujan–Nagell equation (REF ) and its more classical form (REF ) with the known practical methods."
],
[
"Advantages and disadvantages.",
"Algorithms solving (REF ) via modular symbols (Cremona's algorithm) have the usual strengths and weaknesses, see the analogous discussion in Section REF .",
"We further mention that our Algorithm REF uses a reduction to Mordell equations (REF ), while Kim's approach [95] for (REF ) works with a reduction to cubic Thue–Mahler equations (REF ).",
"In fact Kim studies the special case $x^2+7=y$ with $x,y\\in {\\mathbb {Z}}$ and $y\\in \\mathcal {O}^\\times $ .",
"Similar as in the case of (REF ), Kim's method is more efficient in terms of $S$ and our Algorithm REF is more efficient in terms of $b$ .",
"We next discuss approaches via height bounds.",
"The discussion of advantages and disadvantages of Algorithm REF (resp.",
"of Algorithm REF ) is analogous to the corresponding discussion of Algorithm REF for cubic Thue–Mahler equations (REF ) (resp.",
"of Algorithm REF for cubic Thue equations (REF )).",
"Hence we refer to Section REF .",
"The approach of Pethő–de Weger [124], using inter alia the theory of logarithmic forms and binary recurrence sequences, is quite involved and we are not aware of its strengths and weaknesses."
],
[
"Comparison of data.",
"In the cases where our results were already known (see above), we compared the data.",
"In all cases it turned out that we obtained the same solutions."
],
[
"Mordell equations and almost primitive solutions",
"In this section we state and discuss Theorem REF which gives results for almost primitive solutions of Mordell equations.",
"We also deduce Corollaries REF and REF on the difference of perfect squares and perfect cubes, and we discuss how these corollaries improve old theorems of Coates [37] which are based on the theory of logarithmic forms.",
"Let $S$ be a finite set of rational primes.",
"We write $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ for $N_S=\\prod _{p\\in S} p$ and we denote by $h$ the logarithmic Weil height.",
"Let $a\\in \\mathcal {O}$ be nonzero and define $a_S=1728N_S^2r_2(a), \\ \\ \\ \\ r_2(a)=\\prod p^{\\min (2,{\\textnormal {ord}}_p(a))}$ with the product taken over all rational primes $p$ not in $S$ .",
"It holds that $\\log r_2(a)\\le h(a)$ , and if $a\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ then we observe that $r_2(a)\\le \\min (\\vert {}a{}\\vert ,\\textnormal {rad}(a)^2)$ .",
"In view of the definition of Bombieri–Gubler [17], we say that $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ is primitive if $\\pm 1$ are the only $n\\in {\\mathbb {Z}}$ with $n^{6}$ dividing $\\gcd (x^3,y^2)$ .",
"We recall the Mordell equation $y^2=x^3+a, \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}.",
"\\qquad \\mathrm {(\\ref {eq:mordell})}$ Inspired by Szpiro's small points philosophy (see e.g.",
"[148]) discussed below, we consider a certain class of solutions of (REF ) which contains all primitive solutions of (REF ).",
"For want of a better name we callWe note that (REF ) defines naturally a moduli scheme of elliptic curves.",
"Then one observes that the notions minimal and almost minimal solutions may be more appropriate (from a geometric point of view) than primitive and almost primitive solutions respectively.",
"these solutions “almost primitive\".",
"Definition 7.1 Let $\\mu :{\\mathbb {Z}}_{\\ge 1}\\rightarrow \\mathbb {R}_{\\ge 0}$ be an arbitrary function, and let $(x,y)\\in \\mathcal {O}\\times \\mathcal {O}$ .",
"We define $u_{x,y}=u=u_1/u_2$ with $u_1\\in {\\mathbb {Z}}_{\\ge 1}$ minimal such that $u_1x,u_1y$ are in ${\\mathbb {Z}}$ and with $u_2\\in {\\mathbb {Z}}$ maximal such that $u_2^6$ divides $\\gcd \\bigl ((u_1^2x)^3,(u_1^3y)^2\\bigl )$ .",
"Then $(u^2x,u^3y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ is primitive and we say that $(x,y)$ is almost primitive with respect to $\\mu $ if $h(u)\\le \\mu (a_S)$ .",
"We notice that $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ is primitive if and only if it is almost primitive with respect to all functions $\\mu :{\\mathbb {Z}}_{\\ge 1}\\rightarrow \\mathbb {R}_{\\ge 0}$ .",
"Intuitively, one may view almost primitive elements of $\\mathcal {O}\\times \\mathcal {O}$ as those elements with “non-primitive part\" bounded in terms of $a_S$ .",
"Building on the arguments of [89], we obtain the following result which depends on the quantity $a_S$ but which does not involve the height $h(a)$ of $a$ .",
"Theorem 7.2 The following statements hold.",
"(i) Let $\\mu :\\mathbb {{\\mathbb {Z}}}_{\\ge 1}\\rightarrow \\mathbb {R}_{\\ge 0}$ be an arbitrary function.",
"Assume that (REF ) has a solution which is almost primitive with respect to $\\mu $ .",
"Then any solution $(x,y)$ of (REF ) satisfies $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le \\tfrac{2}{3}a_S\\log a_S+\\tfrac{1}{4}a_S\\log \\log \\log a_S+\\tfrac{3}{5}a_S+2\\mu (a_S).$ Moreover, if $(x,y)$ is in addition almost primitive with respect to $\\mu $ then $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le \\tfrac{2}{9}a_S\\log a_S+\\tfrac{1}{12}a_S\\log \\log \\log a_S+\\tfrac{1}{4}a_S+2\\mu (a_S).$ (ii) Suppose that $a\\in {\\mathbb {Z}}$ with $\\vert {}a{}\\vert \\rightarrow \\infty $ and define $a_*=1728\\prod _{p\\mid a} p^{\\min (2,{\\textnormal {ord}}_p(a))}$ .",
"Then any primitive solution $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ of the Mordell equation (REF ) satisfies $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le \\tfrac{1}{6}a_*\\log a_* + \\frac{(\\frac{2}{9}\\log 2+o(1))}{\\log \\log a_*}a_*\\log a_*.$ We now make some remarks which complement our discussions of Theorem REF given in the introduction: Any solution $(x,y)$ of (REF ) is by definition almost primitive with respect to the constant function $\\mu =h(u_{x,y})$ on ${\\mathbb {Z}}_{\\ge 1}$ , and therefore we see that Theorem REF (i) provides in particular an explicit upper bound in terms of $a_S$ and $h(u_{x,y})$ .",
"Theorem REF and its proof fit into Szpiro's small points philosophy [148] for hyperbolic curves $X$ of genus at least two defined over number fields.",
"This philosophy says, roughly speaking, that the rational points of $X$ have smallHere small means that the Arakelov height is effectively bounded from above only in terms of the bad reduction places of $X$ , the genus of $X$ and the given base field over which $X$ is defined.",
"Arakelov height defined with the minimal regular model of $X$ .",
"In our case of the hyperbolic genus one curve (REF ), the existence of an almost primitive solution assures that (REF ) is sufficiently minimal, and then the height bound in Theorem REF (i) shows that all solutions of (REF ) are smallA solution of (REF ) is small if its Weil height, given on the subvariety of $\\mathbb {A}^2$ defined by (REF ), is effectively bounded from above only in terms of the bad reduction places of the affine curve (REF ).",
"The Weil height is more suitable in our case of the affine genus one curve (REF ), since Szpiro's result [149] implies that the Arakelov height is in fact constant on the rational points of any elliptic curve..",
"Furthermore our proof of Theorem REF uses inter alia (REF ) which is in fact a versionIt follows for example from [149] that (REF ) is indeed a version of Szpiro's small points conjecture for elliptic curves over ${\\mathbb {Q}}$ .",
"of Szpiro's small points conjecture [147] for elliptic curves over ${\\mathbb {Q}}$ ."
],
[
"The difference of perfect squares and perfect cubes",
"Let $a\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace $ and let $S$ be a finite set of rational primes.",
"Following Coates [37], we denote by $f\\in {\\mathbb {Z}}$ the largest divisor of $a$ which is only divisible by primes in $S$ .",
"Then $\\vert {}a/f{}\\vert $ is the largest divisor of $a$ which is coprime to $N_S=\\prod _{p\\in S}p$ .",
"We define $\\alpha _S=\\frac{a_S}{N_S}=1728N_Sr_2(a)$ for $a_S$ and $r_2(a)$ the quantities given in (REF ).",
"Now we can state the following corollary.",
"Corollary 7.3 If $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ with $y^2-x^3=a$ and $\\gcd (x,y,N_S)=1$ , then $\\log \\max (\\vert {}x{}\\vert ,\\vert {}y{}\\vert )\\le \\tfrac{1}{2}\\log \\vert {}a/f{}\\vert +2\\alpha _S\\log \\alpha _S+\\tfrac{3}{4}\\alpha _S\\log \\log \\log \\alpha _S+6\\alpha _S.$ It follows that $(x,y)$ is almost primitive with respect to $\\mu =\\frac{1}{6}\\log \\vert {}a/f{}\\vert $ .",
"Then Theorem REF (i) proves the desired bound, but with $a_S$ in place of $\\alpha _S$ .",
"To obtain the bound involving $\\alpha _S$ claimed by Corollary REF , one uses a version of Theorem REF (i) which takes into account that $\\gcd (x,y,N_S)=1$ .",
"See Section REF for details.",
"We write $s=\\vert {}S{}\\vert $ and we define $P=\\max (S\\cup \\lbrace 2\\rbrace )$ .",
"On using a completely different method, based on the theory of logarithmic forms, Coates [37] obtained that any $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ with $y^2-x^3=a$ and $\\gcd (x,y,N_S)=1$ satisfies $\\log \\max (\\vert {}x{}\\vert ,\\vert {}y{}\\vert )\\le 2^{10^7(s+1)^4}P^{10^9(s+1)^3}\\vert {}a/f{}\\vert ^{10^6(s+1)^2}.$ We observe that $\\alpha _S\\le 1728N_S\\vert {}a/f{}\\vert $ and it holds that $N_S\\le P^s$ .",
"Therefore we see that Corollary REF improves Coates' result [37] stated in (REF ).",
"We also obtain the following corollary on the size of the greatest rational prime divisor of the difference of (coprime) perfect squares and perfect cubes.",
"Corollary 7.4 For any real number $\\varepsilon >0$ there is an effective constant $c(\\varepsilon )$ , depending only on $\\varepsilon $ , with the following property: Suppose that $x$ and $y$ are coprime rational integers, and write $X=\\max (\\vert {}x{}\\vert ,\\vert {}y{}\\vert )$ .",
"Then the greatest rational prime factor of $y^2-x^3$ exceeds $ (1-\\varepsilon )\\log \\log X+c(\\varepsilon ).$ For example, if $\\varepsilon =\\frac{1}{10}$ then one can take here the constant $c(\\varepsilon )=-20$ .",
"Let $S$ be the set of rational primes dividing $a=y^2-x^3$ and write $q=\\max (S)$ .",
"The explicit version of the prime number theorem given in [128] shows that $\\log N_S\\le \\sum _{p\\le q}\\log p\\le q\\bigl (1+\\frac{1}{2\\log q}\\bigl )$ .",
"Thus Corollary REF implies Corollary REF .",
"We conclude with several remarks.",
"Coates obtained in [37] the weaker lower bound $10^{-3}(\\log \\log X)^{1/4}$ by using his result [37] displayed in (REF ).",
"The proofs of Corollaries REF and REF show in addition that one can weaken the assumptions $\\gcd (x,y,N_S)=1$ and $\\gcd (x,y)=1$ in Corollaries REF and REF by slightly changing the bounds.",
"Further on using the link $(abc)\\Rightarrow (\\ref {eq:mordell})$ in [17], between the $abc$ -conjecture $(abc)$ and height bounds for the solutions of (REF ), one can show that $(abc)$ gives grosso modo our inequalities with the logarithmic Weil height $h$ replaced by $\\exp (h)$ .",
"Finally we point out that the known links $(abc)\\Rightarrow (\\ref {eq:mordell})$ are not compatible with exponential versions of $(abc)$ .",
"In particular one can not combine the exponential version of $(abc)$ given in Stewart–Yu [146] with the known links $(abc)\\Rightarrow (\\ref {eq:mordell})$ in order to improve our results in this Section or Proposition REF below."
],
[
"Height bounds for Thue and Thue–Mahler equations",
"To deduce Corollary J from our results for Mordell equations, we work out explicitly the arguments of [89].",
"In fact in the present section we shall establish a more precise version of Corollary J.",
"This version provides optimized and sharper height bounds for the solutions of any cubic Thue and Thue–Mahler equation, see Corollary REF .",
"We continue the terminology and notation introduced in Section ."
],
[
"Reduction to Mordell equations",
"We begin by recalling that $X$ is given by the Thue equation $f-m=0$ , where $m\\in \\mathcal {O}$ is nonzero and $f\\in \\mathcal {O}[x,y]$ is a homogeneous polynomial of degree 3 with nonzero discriminant $\\Delta $ .",
"Further we recall that $Y$ is given by the Mordell equation $y^2-(x^3+a)=0$ with parameter $a=432\\Delta m^2$ .",
"We shall work with the following morphism $\\varphi :X\\rightarrow Y$ of $\\mathcal {O}$ -schemes, which was constructed in (REF ) using classical invariant theory.",
"For any polynomial $g$ with rational coefficients $a_\\alpha $ , we define $h(g)=\\max _\\alpha h(a_\\alpha )$ .",
"On recalling that $X$ and $Y$ are affine, we see that the usual logarithmic Weil height $h$ (see [17]) naturally defines a height function $h$ on $X(\\bar{{\\mathbb {Q}}})$ and $Y(\\bar{{\\mathbb {Q}}})$ .",
"We obtain the following result.",
"Proposition 8.1 Assume that $f,m\\in {\\mathbb {Z}}[x,y]$ .",
"Then any $P\\in X(\\bar{{\\mathbb {Q}}})$ satisfies $h(P)\\le \\tfrac{1}{3}h(\\varphi (P))+12\\bigl (h(f-m)+6h(f)+186\\bigl ).$ We point out that the first term $\\tfrac{1}{3}h(\\varphi (P))$ is optimal here.",
"The coefficients in the second term come from a recent version of the arithmetic Nullstellensatz discussed in the next paragraph.",
"These coefficients can be slightly improved in certain cases.",
"For example, the proof of Proposition REF shows in addition that one can replace the coefficient 12 by the smaller number 6 in the case when $P\\in X({\\mathbb {Z}})$ .",
"We also mention that one can directly remove the assumption $f,m\\in {\\mathbb {Z}}[x,y]$ in Proposition REF by replacing in the bound the quantity $h(f-m)+6h(f)$ by the larger expression $7h(f-m)+28h(f)$ ."
],
[
"Arithmetic Nullstellensatz and covariants of cubic forms",
"We continue our notation.",
"In this section we collect some results which shall be used in the proof of Proposition REF .",
"In particular we discuss effective versions of the arithmetic Nullstellensatz and we recall classical properties of covariants of cubic forms."
],
[
"Arithmetic Nullstellensatz.",
"An important ingredient for our proof of Proposition REF is an effective version of the Nullstellensatz.",
"Masser–Wüstholz [122] worked out a fully explicit version of the strong arithmetic Nullstellensatz.",
"Their result would be sufficient for our purpose in the sense that it would give a version of Proposition REF in which the coefficient 12 is replaced by a number exceeding $24^{15}$ .",
"To obtain the considerably smaller number 12, we shall apply a recent result of D'Andrea–Krick–Sombra [48] providing an explicit version of the strong arithmetic Nullstellensatz over an affine variety of pure dimension.",
"More precisely we shall work over the affine hypersurface $V\\subset \\mathbb {A}^3_{\\mathbb {Q}}$ defined by the polynomial $g=f-mz^3$ .",
"On using our assumptions that $m$ and the discriminant of $f$ are both nonzero, we see that $g$ is geometrically irreducible and therefore the affine variety $V_{\\bar{{\\mathbb {Q}}}}$ is of pure dimension two.",
"Write $h(V)$ for the projective logarithmic Weil height $h$ (see [17]) of the point in projective space determined by the coefficient vector of the polynomial $g$ , and let $\\hat{h}(V)$ be the canonical height (see [48]) of the projective closure $\\bar{V}$ of $V$ inside $\\mathbb {P}^3_{\\mathbb {Q}}$ .",
"It holds $\\hat{h}(V)\\le h(V)+3\\log 5.$ To verify this inequality, we temporarily write $R={\\mathbb {Z}}[x_1,\\cdots ,x_4]$ and we let $D$ be the effective Cartier/Weil divisor of $\\mathbb {P}^3_{\\mathbb {Q}}$ given by the irreducible hypersurface $\\bar{V}\\subset \\mathbb {P}^3_{\\mathbb {Q}}$ .",
"On using the terminology of [48], we denote by $f_{D}\\in R$ a primitive polynomial determined by the Cartier divisor $D$ .",
"The homogeneous polynomial $g$ is irreducible in $R_{\\mathbb {Q}}=R\\otimes _{\\mathbb {Z}}{\\mathbb {Q}}$ and it holds that $\\bar{V}\\cong \\textnormal {Proj}\\bigl (R_{\\mathbb {Q}}/(g)\\bigl )$ .",
"It follows that there exists $\\epsilon \\in {\\mathbb {Q}}^\\times $ such that $\\epsilon f_{D}$ coincides with $g$ in $R_{\\mathbb {Q}}$ , which implies that $h(f_D)=h(V)$ since $f_D$ is primitive.",
"Let $m(f_D)$ be the Mahler measure of $f_D$ defined in [48].",
"An application of [48] with $D$ gives that $\\hat{h}(V)=m(f_D)$ , and we deduce from [48] that $m(f_D)\\le h(f_{D})+3\\log 5$ .",
"Hence the equality $h(f_D)=h(V)$ proves (REF )."
],
[
"Covariants of cubic forms.",
"We next recall some properties of cubic forms which shall be used in the proof of Proposition REF .",
"Write $f(x,y)=ax^3+bx^2y+cxy^2+dy^3$ with $a,b,c,d\\in {\\mathbb {Q}}$ and denote by $\\mathcal {H}$ the covariant of $f$ of degree two.",
"The form $\\mathcal {H}$ is the Hessian of $f$ in the sense of [132]; we shall work with the normalization $\\mathcal {H}(x,y)=Ax^2+Bxy+Cy^2, \\ \\ \\ A=3ac-b^2, \\ B=9ad-bc, \\ C=3bd-c^2.$ To avoid a collision of notation, we write throughout this section $k=432\\Delta m^2$ for the parameter $a$ appearing in the equation (REF ) which defines $Y$ .",
"Here $\\Delta $ denotes the discriminant of $f$ defined in [132]; note the sign convention used in [132].",
"Further we denote by $J$ the covariant of $f$ of degree three.",
"It is a binary form $J(x,y)=\\sum a_i x^{3-i}y^{i}$ given by the Jacobian of the forms $f$ and $\\mathcal {H}$ in the sense of [132]; for our purpose it will be convenient to normalize the coefficients $a_i$ of the polynomial $J$ as follows: $a_0&= 27a^2d-9abc+2b^3, & a_1&=3(b^2c+9abd-6ac^2),\\\\a_2&= 3(6b^2d-bc^2-9acd), & a_3&=9bcd-27ad^2-2c^3.$ To determine the set $Z(f,J)\\subset \\bar{{\\mathbb {Q}}}^2$ of common zeroes of $f$ and $J$ , we suppose that $(x,y)\\in Z(f,J)$ .",
"A formula going back at least to Cayley (see [132]) shows that the polynomials $u=-4\\mathcal {H}$ and $v=4J$ satisfy the relation $v^2=u^3+432\\Delta f^2$ .",
"This implies that $(x,y)$ is a zero of $\\mathcal {H}$ .",
"In the case $a=0=d$ , we then see that our assumption $\\Delta \\ne 0$ together with (REF ) implies that $(x,y)=0$ and therefore we obtain $Z(f,J)=0.$ To prove that (REF ) holds in general, we now may and do assume that the coefficient $a$ of $f$ is nonzero.",
"Then the polynomial $f(t,1)=a\\prod (t-\\gamma _j)$ in ${\\mathbb {Q}}[t]$ has degree 3.",
"Furthermore the roots $\\gamma _j$ of $f(t,1)$ are distinct since $\\Delta \\ne 0$ .",
"Thus $\\mathcal {H}(\\gamma _j,1)$ is nonzero by the formula for the Hessian given in [132].",
"It follows that $(x,y)=0$ , since $f,\\mathcal {H}$ are homogeneous and since $a\\ne 0$ .",
"Hence the set $Z(f,J)$ is trivial.",
"Alternatively, one can use here invariant theory providing that the resultant of $f$ and $H$ is a power of the invariant $\\Delta $ up to a sign."
],
[
"Proof of Proposition ",
"We continue our notation.",
"In this section we use the above results to prove Proposition REF .",
"[Proof of Proposition REF ] To obtain the desired height inequality, we work in the projective space.",
"Let $\\bar{X}$ and $\\bar{Y}$ be the projective closures in $\\mathbb {P}^2_{\\mathbb {Q}}$ of the affine curves $X_{\\mathbb {Q}}$ and $Y_{\\mathbb {Q}}$ respectively.",
"It follows from (REF ) that $\\varphi :X\\rightarrow Y$ induces a finite morphism $\\bar{\\varphi }:\\bar{X}\\rightarrow \\bar{Y}$ of degree three, which is given by $\\varphi _1=-4x_3\\mathcal {H}(x_1,x_2)$ , $\\varphi _2=4J(x_1,x_2)$ and $\\varphi _3=x_3^3$ in terms of coordinates $x_i$ on $\\mathbb {P}^2_{\\mathbb {Q}}$ .",
"To simplify the exposition, we write $R={\\mathbb {Q}}[x_1,x_2,x_3]$ and we shall identify (when convenient) a polynomial in $R$ with its image in $\\mathcal {O}_V(V)$ .",
"We next apply the Nullstellensatz to express $x_i$ in terms of $\\varphi _j$ .",
"Let $Z\\subset V(\\bar{{\\mathbb {Q}}})$ be the set of common zeroes of the two functions $\\varphi _1,\\varphi _2\\in \\mathcal {O}_V(V)$ .",
"Suppose that $(x,y,z)\\in Z$ .",
"Then it holds that $z=0$ or $\\mathcal {H}(x,y)=0$ and thus the identity $432\\Delta f^2=\\varphi _2^2-(\\varphi _1/x_3)^3$ together with $\\Delta \\ne 0$ implies $(x,y)\\in Z(f,J)$ .",
"We conclude that $(x,y)=0$ , since $Z(f,J)$ is trivial by (REF ).",
"It follows that the functions $x_i$ vanish on $Z$ for $i=1,2$ .",
"Furthermore, our assumptions $f,m\\in {\\mathbb {Z}}[x,y]$ together with (REF ) and (REF ) show that $\\varphi _1,\\varphi _2$ have coefficients in ${\\mathbb {Z}}$ .",
"Therefore applications of the strong arithmetic Nullstellensatz over $V$ , with $\\varphi _1, \\varphi _2$ and $x_i$ , give $\\alpha _i,e_i\\in {\\mathbb {Z}}_{\\ge 1}$ and $\\rho _{ij}\\in R$ such that the two functions $\\alpha _i x_i^{e_i}$ and $\\sum \\rho _{ij}\\varphi _j$ coincide on $V(\\bar{{\\mathbb {Q}}})$ for $i=1,2$ .",
"In other words the difference of these two functions vanishes on $V(\\bar{{\\mathbb {Q}}})$ , and then $V=\\textnormal {Spec}\\bigl (R/(g)\\bigl )$ implies that this difference is divisible in $R$ by the geometrically irreducible polynomial $g$ .",
"Thus there exist $\\rho _{i0}\\in R$ such that $x_i^{e_i}=\\sum (\\rho _{ij}/\\alpha _i)\\varphi _j, \\ \\ \\ i=1,2,$ where $\\varphi _0=g$ .",
"On multiplying here both sides with $x_i^{e-e_i}$ for $e=\\max e_i$ , we may and do assume that $e_i=e$ .",
"Furthermore we may and do assume that all $\\rho _{ij}$ are homogeneous of degree $e-3$ , since the polynomials $\\varphi _j$ are all homogeneous of degree 3.",
"We now control the right hand side of (REF ) in terms of $\\varphi $ and $g$ .",
"Put $h(\\rho /\\alpha )=h(Q)$ for $Q$ the point in projective space whose coordinates are given by the coefficients of the four polynomials $\\rho _{ij}/\\alpha _i\\in R$ with $i,j\\in \\lbrace 1,2\\rbrace $ .",
"Each polynomial $\\rho _{ij}$ has at most $\\tbinom{e-1}{2}$ nonzero coefficients, and the function $\\varphi _0$ vanishes on the $\\bar{{\\mathbb {Q}}}$ -points of $\\bar{X}=\\textnormal {Proj}\\bigl (R/(\\varphi _0)\\bigl )$ .",
"Therefore on combining (REF ) with the above observations, we see that standard height arguments lead to the following statement: Any $\\bar{{\\mathbb {Q}}}$ -point $P$ of $\\bar{X}$ satisfies $3h(P)\\le h(\\bar{\\varphi }(P))+h(\\rho /\\alpha )+\\log \\bigl (2\\tbinom{e-1}{2}\\bigl ).$ In particular any $P\\in X(\\bar{{\\mathbb {Q}}})$ satisfies (REF ) with $\\bar{\\varphi }$ replaced by $\\varphi $ .",
"Now we see that the explicit version [48] of the strong arithmetic Nullstellensatz gives the following: In (REF ) one can choose $\\alpha _i$ , $e_i$ and $\\rho _{ij}$ , with $e_i\\le 54$ and $h(\\rho /\\alpha )$ explicitly bounded in terms of $\\hat{h}(V)$ and $h(\\varphi _j)$ , such that (REF ), (REF ), (REF ) and (REF ) lead to the desired height inequality.",
"This completes the proof of Proposition REF .",
"The above proof gives moreover a version of Proposition REF for projective closures inside $\\mathbb {P}^2_{\\mathbb {Q}}$ .",
"Indeed it follows from (REF ) that any $\\bar{{\\mathbb {Q}}}$ -point $P$ of $\\bar{X}$ satisfies the height inequality in Proposition REF with $\\varphi $ replaced by the morphism $\\bar{\\varphi }:\\bar{X}\\rightarrow \\bar{Y}.$"
],
[
"Optimized height bounds",
"We continue the notation introduced above.",
"In this section we give Corollary REF which provides optimized height bounds for the solutions of cubic Thue and Thue–Mahler equations.",
"To state our height bounds, we recall that $k=432\\Delta m^2$ and we denote by $\\Omega _{\\textnormal {sim}}$ the simplified height height bound given in Proposition REF with $a=k$ .",
"It holds $\\Omega _{\\textnormal {opt}}\\le \\Omega _{\\textnormal {sim}}=\\tfrac{1}{3}h(k)+\\tfrac{4}{9}k_S\\log k_S+\\tfrac{1}{6}k_S\\log \\log \\log k_S+\\tfrac{2}{5}k_S$ for $k_S$ as in (REF ) and $\\Omega _{\\textnormal {opt}}$ the optimized height bound provided by Proposition REF with $a=k$ .",
"The next result is a sharper (but more complicated) version of Corollary J. Corollary 8.2 Assume that $f,m\\in {\\mathbb {Z}}[x,y]$ .",
"Then the following statements hold.",
"(i) Suppose that $(x,y)$ is a solution of the cubic Thue equation (REF ), and put $n=1$ if $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ and $n=2$ otherwise.",
"Then $h(x)$ and $h(y)$ are at most $\\tfrac{n}{2}\\Omega _{\\textnormal {opt}}+6n\\bigl (h(f-m)+6h(f)+186\\bigl ).$ (ii) Suppose that $(x,y,z)$ is a primitive solution of the general cubic Thue–Mahler equation (REF ).",
"Then $h(x)$ , $h(y)$ and $\\tfrac{1}{3}h(z)$ are at most $2\\Omega _{\\textnormal {opt}}+51\\log N_S+24\\bigl (h(f-m)+6h(f)+186\\bigl ).$ One can directly remove the extra assumption $f,m\\in {\\mathbb {Z}}[x,y]$ by multiplying the equation $f(x,y)=m$ with a suitable integer, see (REF ) for the resulting bounds.",
"[Proof of Corollary REF ] To prove (i) we take $P\\in X(\\mathcal {O})$ .",
"Then $(u,v)=\\varphi (P)$ satisfies the Mordell equation (REF ) with parameter $a=k$ .",
"The number $k=432\\Delta m^2$ is nonzero, since $m\\Delta \\ne 0$ by assumption.",
"Hence an application of Propositions REF and REF with $(u,v)$ gives an upper bound for $h(\\varphi (P))$ which together with Proposition REF implies (i).",
"Here we used that one can replace in Proposition REF the coefficient 12 by 6 if $P\\in X({\\mathbb {Z}})$ .",
"To show (ii) we assume that $(x,y,z)$ is a primitive solution of the general Thue–Mahler equation (REF ).",
"We write $z=z_0\\epsilon ^3$ with $z_0,\\epsilon \\in {\\mathbb {Z}}$ such that $\\pm 1$ are the only $l\\in {\\mathbb {Z}}$ with $l^{3}$ dividing $z_0$ .",
"Then $u=x/\\epsilon $ and $v=y/\\epsilon $ satisfy the Thue equation $f(u,v)=m^{\\prime }$ with $m^{\\prime }=mz_0$ .",
"On exploiting our assumption that $(x,y,z)$ is primitive, one controls the absolute values of $x,y,z$ in terms of the Weil heights of $u,v$ and then (ii) follows from the height bound for $(u,v)$ obtained in (i).",
"Here we used that $k^{\\prime }=432\\Delta m^{\\prime 2}$ satisfies $k_S=k^{\\prime }_S$ and that $h(k^{\\prime })$ is at most $h(k)+4\\log N_S$ .",
"This completes the proof of Corollary REF .",
"To remove the extra assumption $f,m\\in {\\mathbb {Z}}[x,y]$ in Corollary REF , we define $f^*=lf$ and $m^*=lm$ for $l$ the least common multiple of the denominators of the coefficients of the polynomial $f-m\\in \\mathcal {O}[x,y]$ .",
"Any solution $(x,y)$ of the Thue equation (REF ) satisfies $f^*(x,y)=m^*$ , and any primitive solution $(x,y,z)$ of the general Thue–Mahler equation (REF ) satisfies $f^*(x,y)=m^*z$ .",
"Therefore an application of Corollary REF with $f^*,m^*\\in {\\mathbb {Z}}[x,y]$ implies the following: Statements (i) and (ii) of Corollary REF hold without the extra assumption $f,m\\in {\\mathbb {Z}}[x,y]$ if the bounds in (i) and (ii) are replaced by $B_1=\\Omega _{\\textnormal {opt}}+86\\bigl (4h(f)+h(m)+26\\bigl ) \\ \\ \\ \\textnormal { and } \\ \\ \\ 2B_1+51\\log N_S$ respectively.",
"Here we used that $h(f^*),h(m^*)$ are at most $4h(f)+h(m)$ and we exploited that $k^*_S=k_S$ , where $k^*=432\\Delta ^* (m^*)^2$ for $\\Delta ^*$ the discriminant of $f^*$ .",
"Note that $k^*_S=k_S$ follows from $k^*=l^6k$ and from our assumptions $f,m\\in \\mathcal {O}[x,y]$ which assure that $l\\in \\mathcal {O}^\\times $ .",
"Finally we deduce Corollary J by simplifying the bounds in Corollary REF .",
"[Proof of Corollary J] In the case $f,m\\in {\\mathbb {Z}}[x,y]$ , we see that Corollary REF together with $h(\\Delta )\\le 4h(f)+5\\log 3$ implies (i) and (ii).",
"In general, on following the proof of (REF ) we reduce to the case $f^*,m^*\\in {\\mathbb {Z}}[x,y]$ and then we apply the already established case of Corollary J with $f^*,m^*\\in {\\mathbb {Z}}[x,y]$ .",
"This completes the proof of Corollary J."
],
[
"Height bounds for Ramanujan–Nagell equations",
"In this section we give explicit height bounds for the solutions of the generalized Ramanujan–Nagell equation.",
"We also study pairs of units whose sum is a square or cube.",
"As in the previous sections let $S$ be a finite set of rational primes, write $h$ for the usual logarithmic Weil height and denote by $\\mathcal {O}^\\times $ the units of $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ for $N_S=\\prod _{p\\in S} p$ .",
"Further let $b,c\\in \\mathcal {O}$ be nonzero and recall the generalized Ramanujan–Nagell equation $x^2+b=cy, \\ \\ \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}^\\times .",
"\\qquad \\mathrm {(\\ref {eq:rana})}$ To state our height bounds for the solutions of (REF ), we define $a=bc^2$ and we denote by $\\Omega _{\\textnormal {sim}}=\\Omega _{\\textnormal {sim}}(a,S)$ the simplified height height bound given in Proposition REF .",
"It holds $\\Omega _{\\textnormal {opt}}\\le \\Omega _{\\textnormal {sim}}=\\tfrac{1}{3}h(a)+\\tfrac{4}{9}a_S\\log a_S+\\tfrac{1}{6}a_S\\log \\log \\log a_S+\\tfrac{2}{5}a_S$ for $a_S$ as in (REF ) and $\\Omega _{\\textnormal {opt}}=\\Omega _{\\textnormal {opt}}(a,S)$ the optimized height bound in Proposition REF .",
"The next result is a direct consequence of our height bounds for Mordell equations.",
"Corollary 9.1 If $(x,y)$ satisfies (REF ) then $h(x^2),h(y)\\le 3\\Omega _{\\textnormal {opt}}+3h(c)+8\\log N_S.$ We write $y=\\epsilon y^{\\prime 3}$ with $y^{\\prime }\\in \\mathcal {O}^\\times $ and $\\epsilon \\in {\\mathbb {Z}}_{\\ge 1}$ dividing $N_S^2$ .",
"Then $u=\\epsilon c y^{\\prime }$ and $v=\\epsilon c x$ satisfy the Mordell equation $v^2=u^3+a^{\\prime }$ with $a^{\\prime }=-b(\\epsilon c)^2$ .",
"It holds that $a^{\\prime }\\ne 0$ and $a^{\\prime }_S=a_S$ , since $bc\\ne 0$ and $\\epsilon \\in \\mathcal {O}^\\times $ .",
"Thus Proposition REF implies Corollary REF .",
"It holds that $\\Omega _{\\textnormal {opt}}\\le \\Omega _{\\textnormal {sim}}$ and we observe that $3\\Omega _{\\textnormal {sim}}+8\\log N_S$ is at most $2a_S+h(a)$ .",
"Therefore we see that Corollary REF proves Corollary K stated in the introduction.",
"Remark 9.2 (Generalization) Suppose that $f\\in \\mathcal {O}[x]$ is a polynomial of degree two, with nonzero discriminant $\\Delta $ .",
"Then we claim that Corollary REF gives an explicit height bound for any solution $(x,y)\\in \\mathcal {O}\\times \\mathcal {O}^\\times $ of the more general Diophantine equation $f(x)=cy.$ To prove this claim we suppose that $(x,y)\\in \\mathcal {O}\\times \\mathcal {O}^\\times $ satisfies $f(x)=cy$ .",
"We write $f(x)=a_1x^2+a_2x+a_3$ with $a_i\\in \\mathcal {O}$ and we put $x^{\\prime }=2a_1x+a_2$ .",
"It follows that $(x^{\\prime },y)$ is a solution of (REF ) with parameters $b=-\\Delta $ and $c=4a_1c$ .",
"Hence an application of Corollary REF with $(x^{\\prime },y)$ gives an explicit upper bound for $h(x)$ and $h(y)$ as claimed.",
"We next consider the problem of finding all $(u,v)\\in \\mathcal {O}^\\times \\times \\mathcal {O}^\\times $ with $u+v$ a square or cube in ${\\mathbb {Q}}$ .",
"To obtain here finiteness statements, one has to work modulo the actions of $\\mathcal {O}^\\times $ arising naturally in this context.",
"The above Corollary REF leads to the following fully explicit results which involve the quantity $\\Omega =3\\Omega _{\\textnormal {opt}}(1,S)+9\\log N_S$ .",
"Corollary 9.3 Assume that $u,v$ are in $\\mathcal {O}^\\times $ .",
"Then the following statements hold.",
"(i) If $u+v$ is a square in ${\\mathbb {Q}}$ , then there is $\\epsilon \\in \\mathcal {O}^\\times $ such that $h(\\epsilon ^2u),h(\\epsilon ^2v)\\le \\Omega $ .",
"(ii) If $u+v$ is a cube in ${\\mathbb {Q}}$ , then there is $\\delta \\in \\mathcal {O}^\\times $ such that $h(\\delta ^3u),h(\\delta ^3v)\\le \\Omega $ .",
"To prove (i) we assume that $u+v$ is a square in ${\\mathbb {Q}}$ .",
"Then there is $\\epsilon \\in \\mathcal {O}^\\times $ such that $\\epsilon ^2u$ and $\\epsilon ^2v$ are in ${\\mathbb {Z}}$ , with $\\epsilon ^2u+\\epsilon ^2v$ a perfect square and $\\gcd (\\epsilon ^2u,\\epsilon ^2v)$ square-free.",
"If $m,n$ are in ${\\mathbb {Z}}\\cap \\mathcal {O}^\\times $ with $m+n$ a perfect square and $\\gcd (m,n)$ square-free, then we claim that $h(n)\\le \\Omega .",
"$ To prove this inequality we take $l\\in {\\mathbb {Z}}$ with $l^2=m+n$ and we write $m=m^{\\prime 2}m_0$ with $m^{\\prime },m_0\\in {\\mathbb {Z}}$ such that $m_0\\mid N_S$ .",
"Further we define $x=l/m^{\\prime }$ and $y=n/m^{\\prime 2}$ .",
"Then $(x,y)$ satisfies (REF ) with $b=-m_0$ and $c=1$ .",
"Thus an application of Corollary REF with $(x,y)$ implies (REF ) as claimed.",
"Here we used that $n$ and $m^{\\prime }$ are coprime which follows from our assumption that $\\gcd (m,n)$ is square-free.",
"Now, an application of (REF ) with $m=\\epsilon ^2u$ and $n=\\epsilon ^2v$ shows that $\\epsilon $ has the desired properties.",
"This proves assertion (i).",
"The following proof of (ii) uses the arguments of (i) with some modifications.",
"We assume that $u+v$ is a cube in ${\\mathbb {Q}}$ .",
"Then there is $\\delta \\in \\mathcal {O}^\\times $ such that $\\delta ^3u$ and $\\delta ^3v$ are in ${\\mathbb {Z}}$ , with $\\delta ^3u+\\delta ^3v$ a perfect cube and $\\gcd (\\delta ^3u,\\delta ^3v)$ cube-free.",
"We now consider the following claim: If $m,n$ are in ${\\mathbb {Z}}\\cap \\mathcal {O}^\\times $ with $m+n$ a perfect cube and $\\gcd (m,n)$ cube-free, then $h(n)\\le \\Omega .",
"$ To prove this claim we take $l\\in {\\mathbb {Z}}$ with $l^3=m+n$ and we write $m=m^{\\prime 3}m_0$ with $m^{\\prime },m_0\\in {\\mathbb {Z}}$ such that $m_0\\mid N_S^2$ .",
"Further we define $x=l/m^{\\prime }$ and $y=n/m^{\\prime 3}$ .",
"Then $(x,y)$ lies in $\\mathcal {O}\\times \\mathcal {O}^\\times $ and satisfies $x^3-m_0=y$ .",
"On writing $y=w y^{\\prime 2}$ with $y^{\\prime }\\in \\mathcal {O}^\\times $ and $w\\in {\\mathbb {Z}}$ dividing $N_S$ , we see that $(xw,y^{\\prime }w^2)$ is a solution of the Mordell equation (REF ) with parameter $a=-m_0w^3$ .",
"Here $a\\in {\\mathbb {Z}}$ is nonzero.",
"Thus on using that $\\gcd (m,n)$ is cube-free, we see that Proposition REF implies our claim in (REF ).",
"Finally we deduce (ii) by applying (REF ) with $m=\\delta ^3u$ and $n=\\delta ^3v$ .",
"This completes the proof of Corollary REF .",
"We now suppose that $m,n\\in {\\mathbb {Z}}$ are coprime.",
"Then $\\gcd (m,n)$ is in particular square-free and cube-free, and $m,n$ are in $\\mathcal {O}^\\times $ for $S=\\lbrace p\\,;\\,p\\mid mn\\rbrace $ with $N_S=\\textnormal {rad}(mn)$ .",
"Therefore (REF ) and (REF ) imply Corollary L stated in the introduction."
],
[
"Height bounds for Mordell and $S$ -unit equations",
"In this section we prove the results of Section , and we establish in Proposition REF the height bounds for Mordell and $S$ -unit equations which are used in the algorithms of Sections REF and REF .",
"The plan of Section is as follows: In Section REF we give simplified versions of the height bounds.",
"Then we prove the results for Mordell and $S$ -unit equations in Sections REF and REF respectively.",
"Finally, in Section REF , we work out the height conductor inequalities for elliptic curves over ${\\mathbb {Q}}$ which are used in our proofs."
],
[
"Simplified versions",
"The precise form of our height bounds in Proposition REF is fairly complicated.",
"To give the reader an idea of the size of the used height bounds, we worked out simplified versions of our bounds for Mordell equations (see Proposition REF ) and for $S$ -unit equations (see Propositions REF and REF ).",
"These simplified versions slightly improve several estimates in the literature and they will allow us (up to some extent) to compare our optimized height bounds with results based on the theory of logarithmic forms.",
"As in the previous sections we let $S$ be a finite set of rational primes, we write $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ for $N_S=\\prod _{p\\in S} p$ , and we denote by $h$ the logarithmic Weil height."
],
[
"Mordell equation",
"Let $a\\in \\mathcal {O}$ be nonzero and let $a_S=1728N_S^2\\prod _{p\\notin S}p^{\\min (2,{\\textnormal {ord}}_p(a))}$ be as in (REF ).",
"On using and refining the arguments of [89], we obtain the following result.",
"Proposition 10.1 If $x,y\\in \\mathcal {O}$ satisfy $y^2=x^3+a$ , then $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le \\tfrac{1}{3}h(a)+\\tfrac{4}{9}a_S\\log a_S+\\tfrac{1}{6}a_S\\log \\log \\log a_S+\\tfrac{2}{5}a_S.$ To compare this result with the actual best height bounds for Mordell equations (REF ), we suppose that $x,y\\in \\mathcal {O}$ satisfy $y^2=x^3+a$ and we observe that Proposition REF implies $\\max \\bigl (h(x),h(y)\\bigl )\\le \\tfrac{1}{2}h(a)+a_S\\log a_S.$ We first consider the classical case $\\mathcal {O}={\\mathbb {Z}}$ .",
"The discussions in [89] show that (REF ) improves the results of Baker [5], Stark [144] and Juricevic [84] which are all based on the theory of logarithmic forms [33].",
"In fact Proposition REF provides the actual best height bound for Mordell equations (REF ), since it updates the inequality $\\max \\bigl (h(x),h(y)\\bigl )\\le h(a)+4\\cdot 36a_S\\log (36a_S)^2$ in [89].",
"We now discuss the case $\\mathcal {O}\\supsetneq {\\mathbb {Z}}$ .",
"On using the theory of logarithmic forms, Hajdu–Herendi [78] obtained explicit height bounds for the solutions in $\\mathcal {O}$ of arbitrary elliptic Diophantine equations.",
"The discussion in [89] shows that Proposition REF improves the case of Mordell equations (REF ) in [78] for “small” sets $S$ , in particular for any set $S$ with $N_S\\le 2^{1200}$ or $\\vert {}S{}\\vert \\le 12$ .",
"If $N_S\\gg \\vert a\\vert $ then there are sets $S$ for which Proposition REF is better than [78], and vice versa.",
"To conclude our comparisons we mention that the theory of logarithmic forms allows to deal with more general Diophantine equations over arbitrary number fields; see [33]."
],
[
"$S$ -unit equations",
"On using and refining the arguments of [120] and [89], which were discovered independently in 2011 by Murty–Pasten and by the first mentioned author, we obtain the following update of the explicit height bounds in [120] and [89]; see also Frey's remark in [66] and Proposition REF .",
"Proposition 10.2 Any solution $(x,y)$ of the $S$ -unit equation (REF ) satisfies $\\max \\bigl (h(x),h(y)\\bigl ) \\le \\tfrac{5}{2}N_S\\log N_S+9N_S \\ \\textnormal { and }$ $\\max \\bigl (h(x),h(y)\\bigl ) \\le \\tfrac{12}{5}N_S\\log N_S+\\tfrac{9}{10}N_S\\log \\log \\log (16N_S)+8.26N_S+28.$ To compare this result with the actual best height bounds for $S$ -unit equations (REF ), we suppose that $(x,y)$ satisfies (REF ) and we write $h=\\max (h(x),h(y))$ .",
"Starting with Győry [73] several authors proved explicit bounds for $h$ by using the theory of logarithmic forms [33]; see the references in [72].",
"In particular Győry–Yu [72] obtained that $h\\le 2^{10s+22}s^4q\\prod \\log p$ with the product taken over all rational primes $p\\in S-\\lbrace q\\rbrace $ for $q=\\max S$ and $s=\\vert {}S{}\\vert $ .",
"Further Proposition REF updates the inequalities $h\\le 4.8N_S\\log N_S+13N_S+25$ in Murty–Pasten [120] and $h\\le 3\\cdot 2^6N_S\\log (2^7N_S)^2+65$ in [89].",
"Hence Proposition REF establishes the actual best height bound for all sets $S$ with “small\" $N_S$ , in particular for all sets $S$ with $N_S\\le 2^{90}.$ Further we see that there are sets $S$ with arbitrarily large $N_S$ for which Proposition REF is sharper than [72], and vice versa.",
"If $N_S\\rightarrow \\infty $ then our bounds (see also (REF )) are worse than $h\\le O(N_S^{1/3}(\\log N_S)^3)$ in Stewart–Yu [146].",
"To conclude our comparison, we mention that the theory of logarithmic forms and Bombieri's refinement of the Thue–Siegel method both give effective height bounds for the solutions of $S$ -unit equations in arbitrary number fields; see for example [72], [11]."
],
[
"Notation",
"In the remaining of Section we shall use throughout the following notation.",
"Let $S$ be a finite set of rational primes.",
"We write $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ for $N_S=\\prod _{p\\in S}p$ and we denote by $h$ the usual logarithmic Weil height.",
"For any elliptic curve $E$ over ${\\mathbb {Q}}$ , we denote by $c_4$ and $c_6$ the usual quantities (see for example [150]) associated to a minimal Weierstrass equation of $E$ over ${\\mathbb {Z}}$ , and we write $N_E$ and $\\Delta _E$ for the conductor and minimal discriminant of $E$ respectively.",
"We denote by $h(E)$ the relative Faltings height of $E$ , defined for example in [89] using Faltings' original normalization [60] of the metric."
],
[
"Mordell equation",
"We use the notation introduced in Section REF above.",
"Let $a\\in \\mathcal {O}-\\lbrace 0\\rbrace $ and recall that $a_S=1728N_S^2r_2(a)$ for $r_2(a)=\\prod p^{\\min ({2,{\\textnormal {ord}}_p(a)})}$ with the product taken over all rational primes $p$ not in $S$ .",
"Further we recall the Mordell equation $y^2=x^3+a, \\ \\ \\ (x,y)\\in \\mathcal {O}\\times \\mathcal {O}.\\qquad \\mathrm {(\\ref {eq:mordell})}$ The following lemma gives an explicit Paršin construction for the solutions of this equation.",
"Lemma 10.3 Suppose that $(x,y)$ satisfies the Mordell equation (REF ).",
"Then there exists an elliptic curve $E$ over ${\\mathbb {Q}}$ with the following properties.",
"It holds that $c_4=u^4x$ and $c_6=u^6y$ for some $u\\in {\\mathbb {Q}}$ with $u^{12}=1728\\Delta _E\\vert {}a{}\\vert ^{-1}$ , the conductor $N_E$ divides $a_S$ and $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le \\tfrac{1}{3}h(a)+8h(E)+2\\log \\max (1,h(E))+36.$ We conducted here some effort to refine the weaker result $N_E\\mid 2^23^2a_S$ which follows directly from [89].",
"In fact Lemma REF is based on [89], and we mention that the height inequality in [89] relies inter alia on explicit versions of certain results of Faltings [60], [61] and Silverman [134].",
"[Proof of Lemma REF ] Suppose that $(x,y)$ is a solution of (REF ).",
"Then an application of [89] with $S=\\textnormal {Spec}(\\mathcal {O})$ and $T=\\textnormal {Spec}(\\mathcal {O}[1/(6a)])$ gives an elliptic curve over $T$ , with generic fiber $E=E_{\\mathbb {Q}}$ , such that $N_E\\mid 2^{2}3^2a_S$ and such that $h(x)$ is bounded in terms of $h(E)$ and $h(a)$ as desired.",
"Let $c_4$ and $c_6$ be the quantities associated to $E$ .",
"The proof of [89] shows in addition that $c_4=u^4x$ and $c_6=u^6y$ for some $u\\in {\\mathbb {Q}}$ with $u^{12}=1728\\Delta _E\\vert {}a{}\\vert ^{-1}$ .",
"Thus on combining [89] with [89], we deduce an upper bound for $h(y)=\\frac{1}{2}h(u^{-12}c_6^2)$ in terms of $h(E)$ and $h(a)$ as desired.",
"Furthermore, it was shown in the proof of [89] that $E$ admits a Weierstrass model $W$ over $\\mathcal {O}$ defined by the Weierstrass equation $t^2 = s^3 - 27xs - 54y$ with “indeterminates” $s$ and $t$ .",
"This Weierstrass equation has discriminant $\\Delta ^{\\prime }=-2^63^9a$ .",
"It remains to show that $N_E\\mid a_S$ .",
"For this purpose, we observe that $2^83^5=2^23^2\\cdot 1728$ and we recall that $N_E\\mid 2^23^2a_S$ .",
"Hence it suffices to consider the exponents $f_2$ and $f_3$ of $N_E$ at the primes 2 and 3 respectively.",
"They always satisfy $f_2\\le 8$ and $f_3\\le 5$ , see for example [23].",
"Thus we obtain that $f_2\\le 8= {\\textnormal {ord}}_2(a_S)$ if $2\\in S$ and $f_3\\le 5= {\\textnormal {ord}}_3(a_S)$ if $3\\in S$ , and then we see that the desired result $N_E\\mid a_S$ holds if $2\\in S$ or $3\\in S$ .",
"Hence we are left to treat the case where 2 and 3 are both not in $S$ , or equivalently that $2,3\\in \\textnormal {Spec}(\\mathcal {O})$ .",
"In this case, after localizing at 2 and 3, we may and do view the model $W$ over $\\mathcal {O}$ as an integral model of $E$ over the local rings at 2 and 3 respectively.",
"We first consider $f_2$ .",
"It holds that ${\\textnormal {ord}}_2(\\Delta _E)\\le {\\textnormal {ord}}_2(\\Delta ^{\\prime })$ , since $W$ is a Weierstrass model of $E$ over the local ring at 2 and since $\\Delta _E$ is minimal at 2.",
"Thus the formula of Ogg–Saito [131] implies that $f_2\\le {\\textnormal {ord}}_2(\\Delta ^{\\prime })=6+{\\textnormal {ord}}_2(a)$ , and then the inequality $f_2\\le 8$ proves that $f_2\\le 6+\\min (2,{\\textnormal {ord}}_2(a))={\\textnormal {ord}}_2(a_S)$ as desired.",
"We now consider $f_3$ .",
"If $W$ is not the minimal Weierstrass model of $E$ over the local ring at 3, then we get that ${\\textnormal {ord}}_3(\\Delta ^{\\prime })\\ge 12$ and hence ${\\textnormal {ord}}_3(a)\\ge 3$ .",
"This together with $f_3\\le 5$ shows the desired inequality $f_3\\le 3+\\min (2,{\\textnormal {ord}}_3(a))={\\textnormal {ord}}_3(a_S)$ when $W$ is not minimal at 3.",
"Hence we may and do assume in addition that $W$ is minimal at 3.",
"Then we claim that the “reduction\" of $E$ at 3 is of Kodaira type IV$^*$ , III$^*$ , or II$^*$ and that $f_3 = {\\textnormal {ord}}_3(\\Delta ^{\\prime })-6$ , $f_3={\\textnormal {ord}}_3(\\Delta ^{\\prime })-7$ , or $f_3={\\textnormal {ord}}_3(\\Delta ^{\\prime })-8$ respectively.",
"It follows that $f_3\\le 3+{\\textnormal {ord}}_3(a)$ and this together with the inequality $f_3\\le 5$ shows that $f_3\\le 3+\\min (2,{\\textnormal {ord}}_3(a))={\\textnormal {ord}}_3(a_S)$ as desired.",
"It remains to verify our claim.",
"For this purpose we use Tate's algorithm [151] and we let $a_1,\\ldots ,a_6$ and $b_2,\\ldots ,b_8$ denote the usual quantities associated to the Weierstrass equation $t^2 = s^3 - 27xs - 54y$ which is minimal at 3 by assumption.",
"We observe that $3\\operatorname{|}\\Delta ^{\\prime }$ , $3\\operatorname{|}a_1=0$ , $3\\operatorname{|}a_2=0$ , $3^3\\operatorname{|}a_3=0$ , $3^2\\operatorname{|}a_4=-27 x$ , $3^3\\operatorname{|}a_6=-54y$ , $3\\operatorname{|}b_2=0$ , $3^3\\operatorname{|}b_6=-216y$ , and $3^3\\operatorname{|}b_8=-729x^2$ .",
"This brings us into case 6) of Tate's algorithm.",
"Here one considers the polynomial $P(T):=T^3+a_{2,1}T^2+a_{4,2}T+a_{6,3}=T^3-3xT-2y$ .",
"In ${\\mathbb {F}}_3$ , it reduces to $T^3-2y$ which is purely inseparable with 3-fold root $2y$ .",
"Therefore 8), 9) and 10) of Tate's algorithm prove our claim.",
"This completes the proof of Lemma REF .",
"In what follows we need to control the function $\\nu (n)=n\\nu ^*(n)$ on ${\\mathbb {Z}}_{\\ge 1}$ , where $\\nu ^*$ is defined in Proposition REF (ii).",
"If $m,n$ are in ${\\mathbb {Z}}_{\\ge 1}$ with $m$ dividing $n$ , then it holds $\\nu (m)\\le \\nu (n).$ Indeed this inequality directly follows by unwinding the definitions and by taking into account rational primes $p$ with ${\\textnormal {ord}}_p(m)=1$ and ${\\textnormal {ord}}_p(n)\\ge 2$ .",
"We now combine the above Lemma REF with Proposition REF in order to prove Proposition REF .",
"[Proof of Proposition REF ] Suppose that $(x,y)$ is a solution of Mordell's equation (REF ).",
"Then Lemma REF provides an elliptic curve $E$ over ${\\mathbb {Q}}$ such that $N_E\\mid a_S$ and such that $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le \\tfrac{1}{3}h(a)+8h(E)+2\\log \\max (1,h(E))+36.$ It follows from (REF ) that $\\nu (N_E)\\le \\frac{2}{3}a_S$ .",
"Therefore on combining the displayed inequality with the explicit bound for $h(E)$ in terms of $\\nu (N_E)$ given in Proposition REF (ii), we deduce an estimate for $\\max \\bigl (h(x),\\frac{2}{3}h(y)\\bigl )$ in terms of $a_S$ and $h(a)$ as stated in Proposition REF .",
"We recall that a solution $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ of (REF ) is primitive if $\\pm 1$ are the only $n\\in {\\mathbb {Z}}$ with $n^{6}\\mid \\gcd (x^3,y^2)$ .",
"The following result refines Lemma REF for primitive solutions.",
"Lemma 10.4 Suppose that $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ is a primitive solution of (REF ).",
"If Lemma REF associates $(x,y)$ to the elliptic curve $E$ , then $E$ has in addition the following properties.",
"(i) It holds that $\\Delta _E=2^m3^n\\vert a\\vert $ with $m\\in \\lbrace -6,6\\rbrace $ and $n\\in \\lbrace -3,9\\rbrace $ .",
"(ii) The curve $E$ is semi-stable at each rational prime $p\\ge 5$ with $\\gcd (x,y,p)=1$ .",
"(iii) There is the refined height inequality $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le 4h(E)+2\\log \\max (1,h(E))+28.$ Let $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ be a primitive solution of (REF ).",
"Suppose that Lemma REF associates $(x,y)$ to the elliptic curve $E$ over ${\\mathbb {Q}}$ .",
"To prove (i) we recall from the proof of Lemma REF that $t^2 = s^3 - 27xs - 54y$ is a Weierstrass equation of $E$ with “indeterminates” $s$ and $t$ and with discriminant $\\Delta ^{\\prime }=-2^63^9a$ .",
"This Weierstrass equation has associated quantities $c_4^{\\prime }=6^4x$ and $c_6^{\\prime }=6^6y$ .",
"On using that $x,y$ are both in ${\\mathbb {Z}}$ and that $\\Delta _E$ is the discriminant of a minimal Weierstrass model of $E$ over ${\\mathbb {Z}}$ , we obtain a nonzero $u\\in {\\mathbb {Z}}$ with $u^4c_4=6^4x, \\ \\ \\ u^6c_6=6^6y, \\ \\ \\ u^{12}\\Delta _E=2^63^9\\vert {}a{}\\vert .$ It follows that $u^{12}$ divides $6^{12}\\gcd (x^3,y^2)$ and hence we deduce that $\\vert {}u{}\\vert \\in \\lbrace 1,2,3,6\\rbrace $ since $(x,y)$ is primitive by assumption.",
"Therefore the equality $u^{12}\\Delta _E=2^63^9\\vert {}a{}\\vert $ implies that $2^m3^n\\vert a\\vert =\\Delta _E$ with $m\\in \\lbrace -6,6\\rbrace $ and $n\\in \\lbrace -3,9\\rbrace $ .",
"This proves assertion (i).",
"To show the semi-stable properties of $E$ claimed in (ii), we take a rational prime $p$ .",
"If $p\\nmid \\Delta _E$ then $E$ has good (and thus semi-stable) reduction at $p$ .",
"We now assume that $p\\mid \\Delta _E$ , that $p\\ge 5$ and that $\\gcd (x,y,p)=1$ .",
"Then (REF ) together with $\\vert {}u{}\\vert \\in \\lbrace 1,2,3,6\\rbrace $ implies that $\\gcd (c_4,c_6,p)=1$ .",
"Hence we obtain that $p\\nmid c_4$ since $p\\ge 5$ divides $12^3\\Delta _E=\\vert {}c_4^3-c_6^2{}\\vert $ by assumption.",
"Therefore the semi-stable criterion in [136] shows that $E$ has semi-stable reduction at each $p\\ge 5$ with $\\gcd (x,y,p)=1$ .",
"This proves assertion (ii).",
"It remains to show the refined height inequality in (iii).",
"The identities in (REF ) together with [89] and $\\vert {}u{}\\vert \\le 6$ lead to an explicit upper bound for $\\max \\bigl (h(x),\\frac{2}{3}h(y)\\bigl )$ in terms of $h(E)$ as claimed in (iii).",
"This completes the proof of Lemma REF .",
"The above proof shows in addition that Lemma REF holds more generally for all solutions $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ of (REF ) such that $\\pm 1$ are the only $n\\in {\\mathbb {Z}}$ with $n^{12}\\mid \\gcd (x^3,y^2)$ .",
"We now use Lemma REF and Proposition REF to prove Theorem REF .",
"[Proof of Theorem REF ] We begin to prove the first part of assertion (i).",
"Let $\\mu :{\\mathbb {Z}}_{\\ge 1}\\rightarrow \\mathbb {R}_{\\ge 0}$ be an arbitrary function, and suppose that $(x,y)$ is a solution of (REF ) which is almost primitive with respect to $\\mu $ .",
"Let $u=u_{x,y}$ be as in Definition REF , and define $x^{\\prime }=u^2x$ and $y^{\\prime }=u^3y$ .",
"Then it follows that $(x^{\\prime },y^{\\prime })\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ is a primitive solution of the Mordell equation $y^{\\prime 2}=x^{\\prime 3}+a^{\\prime }$ for $a^{\\prime }=u^6a\\in {\\mathbb {Z}}$ .",
"Therefore Lemmas REF and REF give an elliptic curve $E$ over ${\\mathbb {Q}}$ together with integers $m\\in \\lbrace -6,6\\rbrace $ and $n\\in \\lbrace -3,9\\rbrace $ such that $\\Delta _E=2^{m}3^n\\vert {}a^{\\prime }{}\\vert \\ \\ \\ \\textnormal { and } \\ \\ \\ N_E\\mid a^{\\prime }_{S}.$ The construction of the number $u=u_1/u_2$ in Definition REF shows that any rational prime $p$ with ${\\textnormal {ord}}_p(a^{\\prime })>{\\textnormal {ord}}_p(a)$ satisfies $p\\mid u_1$ and thus $p\\in S$ since $x,y\\in \\mathcal {O}$ .",
"It follows that ${\\textnormal {ord}}_p(a^{\\prime })\\le {\\textnormal {ord}}_p(a)$ for all rational primes $p$ not in $S$ and therefore we find that $a^{\\prime }_S$ divides $a_S.$ Further it holds that $h(a)\\le 6h(u)+\\log \\vert {}a^{\\prime }{}\\vert $ and $h(u)\\le \\mu (a_S)$ since $(x,y)$ is almost primitive with respect to $\\mu $ .",
"Then on combining (REF ) and $N_E\\mid a^{\\prime }_S\\mid a_{S}$ with the estimate for $\\log \\Delta _E$ in terms of $N_E$ given in (REF ), we derive an explicit upper bound for $h(a)$ in terms of $a_S$ and $\\mu (a_S)$ which together with Proposition REF implies the first part of (i).",
"We now show the second part of (i).",
"Since $(x,y)$ is almost primitive with respect to $\\mu $ we obtain that $h(x)\\le 2\\mu (a_S)+h(x^{\\prime })$ and $\\frac{2}{3}h(y)\\le 2\\mu (a_S)+\\frac{2}{3}h(y^{\\prime })$ .",
"Hence an application of Lemma REF (iii) with the primitive $(x^{\\prime },y^{\\prime })\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ leads to $\\max \\bigl (h(x),\\tfrac{2}{3}h(y)\\bigl )\\le 2\\mu (a_S)+4h(E)+2\\log \\max (1,h(E))+28.$ Then on combining (REF ) and $a^{\\prime }_S\\mid a_{S}$ with the explicit estimate for $h(E)$ in terms of $N_E$ given in Proposition REF , we deduce an inequality as claimed in the second part of (i).",
"To prove (ii) we may and do assume that $\\vert {}a{}\\vert \\rightarrow \\infty $ and that $S$ is empty.",
"Let $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ be a primitive solution of (REF ), and take $(x^{\\prime },y^{\\prime })=(x,y)$ and $a^{\\prime }=a$ in the proof of (i).",
"Then (REF ) and (REF ) show that $\\log \\vert {}a{}\\vert \\le O( N_E^2)$ and thus $\\vert {}a{}\\vert \\rightarrow \\infty $ forces $N_E\\rightarrow \\infty $ .",
"Hence on using (REF ) with $\\mu =0$ and on applying the asymptotic bound for $h(E)$ in terms of $N_E$ given in Proposition REF , we see that (REF ) together with $a_S=a_*$ leads to the asymptotic estimate claimed in (ii).",
"This completes the proof of Theorem REF .",
"Next we give a proof of the inequality claimed in Corollary REF by using a version of Theorem REF (i) which takes into account Lemma REF (ii).",
"[Proof of Corollary REF ] Recall that $a\\in {\\mathbb {Z}}$ is nonzero, $S$ is an arbitrary finite set of rational primes and $f\\in {\\mathbb {Z}}$ is the largest divisor of $a$ which is only divisible by primes in $S$ .",
"We take $(x,y)\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ with $y^2-x^3=a$ and $\\gcd (x,y,N_S)=1$ as in the statement.",
"We first show that $(x,y)$ is almost primitive with respect to $\\mu =\\frac{1}{6}\\log \\vert {}a/f{}\\vert $ .",
"Let $m=u_2\\in {\\mathbb {Z}}$ be maximal such that $m^{6}\\mid \\gcd (x^3,y^2)$ , and define $x^{\\prime }=m^{-2}x$ and $y^{\\prime }=m^{-3}y$ .",
"Then $(x^{\\prime },y^{\\prime })\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ is a primitive solution of the Mordell equation $y^{\\prime 2}=x^{\\prime 3}+a^{\\prime }$ for $a^{\\prime }=m^{-6}a\\in {\\mathbb {Z}}$ .",
"Further, $\\gcd (x,y,N_S)=1$ together with $\\textnormal {rad}(f)\\mid N_S$ implies that $\\gcd (x,y,f)=1$ , and hence $\\gcd (m,f)=1$ since $m$ divides $x$ and $y$ .",
"Thus on using that $m^6\\mid a$ , we see that $m^{6}$ divides $a/f$ .",
"This shows that the number $u_{x,y}=1/m$ in Definition REF satisfies $h(u_{x,y})\\le \\mu $ as desired.",
"We now apply (a version of) Theorem REF (i).",
"Suppose that Lemma REF associates the primitive $(x^{\\prime },y^{\\prime })\\in {\\mathbb {Z}}\\times {\\mathbb {Z}}$ to the elliptic curve $E$ over ${\\mathbb {Q}}$ .",
"Lemma REF (ii) gives that $E$ is semi-stable at each $p\\in S$ with $p\\ge 5$ , since $\\gcd (x^{\\prime },y^{\\prime },N_S)=1$ .",
"Then (the proof of) Lemma REF shows that $N_E$ divides $6\\cdot 1728N_Sr_2(a^{\\prime })$ , and hence we obtain that $N_E\\mid 6\\alpha _S$ since $a^{\\prime }\\mid a$ .",
"Thus, on replacing $a_S$ by $6\\alpha _S=6a_S/N_S$ in the proof of Theorem REF (i) and on taking $\\mu =\\frac{1}{6}\\log \\vert {}a/f{}\\vert $ , we deduce Corollary REF ."
],
[
"$S$ -unit equations",
"We use the notation introduced in Section REF above.",
"The discussions in Section REF show that bounding the Weil height of the solutions of the $S$ -unit equation (REF ) is equivalent to estimating the absolute value of the integers satisfying the Diophantine equation $a+b = c, \\quad a,b,c\\in {\\mathbb {Z}}-\\lbrace 0\\rbrace , \\quad \\gcd (a,b,c)=1,\\quad \\textnormal {rad}(abc)\\mid N_S.",
"\\qquad \\mathrm {(\\ref {eq:abc})}$ We first prove Lemma REF which is used in the proof of Proposition REF given below.",
"Lemma 10.5 Suppose that $(a,b,c)$ is a solution of (REF ).",
"Then there exist ${\\mathbb {Q}}$ -isogenous elliptic curves $E$ and $E^{\\prime }$ over ${\\mathbb {Q}}$ such that $N_E$ divides $2^4N_S$ , $\\Delta _E=2^n(abc)^2$ with $n\\in \\lbrace 4,-8\\rbrace $ , and $\\Delta _{E^{\\prime }}=2^{8-12m}\\vert {}ab{}\\vert c^4$ with $m\\in \\lbrace 0,1,2,3\\rbrace $ .",
"The proof of Lemma REF uses inter alia a formula of Diamond–Kramer [47] and classical results which are given for example in [136].",
"In fact the proof consists of explicit computations with Frey–Hellegouarch elliptic curves.",
"A more conceptual proof of (parts of) Lemma REF can be given by using the point of view in [89].",
"[Proof of Lemma REF ] We now use in particular several (known) reductions.",
"To make sure that these reductions are compatible with each other we prefer to give all details.",
"We suppose that $(a,b,c)$ is a solution of (REF ).",
"Then exactly one of the numbers $a, b, c$ is even, since they are coprime and satisfy $a+b=c$ .",
"We denote this number by $B^{\\prime }$ .",
"Now we define $A^{\\prime }=a$ if $B^{\\prime }=b$ , $A^{\\prime }=b$ if $B^{\\prime }=a$ , and $A^{\\prime }=-a$ if $B^{\\prime }=c$ .",
"Further we define $(A,B,C)=(A^{\\prime },B^{\\prime },A^{\\prime }+B^{\\prime })$ if $A^{\\prime }=-1\\pmod {4}$ , and $(A,B,C)=(-A^{\\prime },-B^{\\prime },-A^{\\prime }-B^{\\prime })$ otherwise.",
"Let $E$ be the elliptic curve over ${\\mathbb {Q}}$ , defined by the Weierstrass equation $y^2=x(x-A)(x+B)$ with discriminant $\\Delta =2^4(ABC)^2$ .",
"We observe that $(A,B,C)$ has the following properties: $A,B,C$ are coprime, $A=-1\\pmod {4}$ , $B$ is even, $A+B=C$ and $(ABC)^2=(abc)^2$ .",
"Thus [47] give that the conductor $N_E$ of $E$ divides $2^4\\textnormal {rad}(abc)$ .",
"Our assumption, that $(a,b,c)$ satisfies (REF ), provides that $\\textnormal {rad}(abc)\\mid N_S$ and hence we obtain that $N_E$ divides $2^4 N_S$ .",
"It follows from [47] and [136] that the minimal discriminant $\\Delta _E$ of $E$ satisfies $\\Delta _E=\\Delta $ if ${\\textnormal {ord}}_2(abc)\\le 3$ , and $\\Delta _E=2^{-8}(abc)^2$ if ${\\textnormal {ord}}_2(abc)\\ge 4$ .",
"We conclude that $\\Delta _E=2^n(abc)^2$ with $n\\in \\lbrace 4,-8\\rbrace $ , and since $N_E$ divides $2^4N_S$ we see that the elliptic curve $E$ has all the desired properties.",
"To construct the elliptic curve $E^{\\prime }$ , we notice that $A,B,C$ are in $\\lbrace \\pm a,\\pm b,\\pm c\\rbrace $ .",
"We define $(\\alpha ,\\beta ,\\gamma ,x^{\\prime })=(A,B,C,x)$ if $C=\\pm c$ , $(\\alpha ,\\beta ,\\gamma ,x^{\\prime })=(C,-B,A,x+B)$ if $A=\\pm c$ , and $(\\alpha ,\\beta ,\\gamma ,x^{\\prime })=(-A,C,B,x-A)$ if $B=\\pm c$ .",
"It follows that $\\gamma =\\pm c$ , that $\\alpha +\\beta =\\gamma $ , that $(\\alpha ,\\beta )=1$ and that $E$ admits the Weierstrass equation $y^2=x^{\\prime }(x^{\\prime }-\\alpha )(x^{\\prime }+\\beta ).$ Then we obtain that $E$ is ${\\mathbb {Q}}$ -isogenous to the elliptic curve $E^{\\prime }$ over ${\\mathbb {Q}}$ , which is defined by the Weierstrass equation (see for example [136]) $w^2=z^3-2(\\beta -\\alpha )z^2 +\\gamma ^2z$ with discriminant $\\Delta ^{\\prime }=-2^8\\alpha \\beta \\gamma ^4$ .",
"This Weierstrass equation has associated quantities $c_4=16(\\alpha ^2-14\\alpha \\beta +\\beta ^2)$ and $c_6=64(\\alpha ^3+33\\alpha ^2\\beta -33\\alpha \\beta ^2-\\beta ^3)$ .",
"Further we observe that $\\Delta ^{\\prime }=-2^8abc^4$ .",
"To determine the minimal discriminant $\\Delta _{E^{\\prime }}$ of $E^{\\prime }$ we use a strategy inspired by [136].",
"An application of the Euclidean algorithm gives the identities $4(395\\alpha ^2-430\\alpha \\beta -13\\beta ^2 )c_4 + (181\\alpha - 13\\beta )c_6 &= 2^{12} 3^2\\alpha ^4,\\\\4(-13\\alpha ^2 - 430\\alpha \\beta + 395\\beta ^2)c_4 + (13\\alpha - 181\\beta )c_6 &= 2^{12} 3^2\\beta ^4.$ On using that $\\Delta ^{\\prime },c_4,c_6\\in {\\mathbb {Z}}$ and that $\\Delta _{E^{\\prime }}$ is the absolute value of the discriminant of a minimal Weierstrass model of $E^{\\prime }$ over ${\\mathbb {Z}}$ , we obtain $u\\in {\\mathbb {Z}}$ such that $u^{12}\\Delta _{E^{\\prime }}=\\pm \\Delta ^{\\prime }$ , $u^4\\mid c_4$ and $u^6\\mid c_6$ .",
"Thus the displayed identities together with $(\\alpha ,\\beta )=1$ imply that $u^4\\mid 2^{12} 3^2$ and hence $\\pm u\\in \\lbrace 1,2,4,8\\rbrace $ .",
"We deduce that $\\Delta _{E^{\\prime }}=2^{8-12m}\\vert {}ab{}\\vert c^4$ with $m\\in \\lbrace 0,1,2,3\\rbrace $ , and therefore $E^{\\prime }$ has the desired properties.",
"This completes the proof of Lemma REF .",
"We remark that the application of [47] in the above proof shows in addition the following: Suppose that $(a,b,c)$ is a solution of (REF ).",
"If Lemma REF associates $(a,b,c)$ to the elliptic curve $E$ over ${\\mathbb {Q}}$ , then the conductor $N_E$ of $E$ satisfies ${\\textnormal {ord}}_2(N_E)=e+1, \\ \\ \\ (e,\\lambda )={\\left\\lbrace \\begin{array}{ll}(4,12) & \\textnormal {if }{\\textnormal {ord}}_2(abc) = 1,\\\\(2,3) & \\textnormal {if }{\\textnormal {ord}}_2(abc) = 2, 3,\\\\(-1,\\frac{1}{2}) & \\textnormal {if }{\\textnormal {ord}}_2(abc) = 4,\\\\(0,1) & \\textnormal {if }{\\textnormal {ord}}_2(abc) \\ge 5.\\\\\\end{array}\\right.",
"}$ We note that it always holds that ${\\textnormal {ord}}_2(abc)\\ge 1$ , since $a+b=c$ and $(a,b,c)=1$ .",
"The following result is in fact a (slightly) more precise version of Proposition REF .",
"Proposition 10.6 Suppose that $(a,b,c)$ is a solution of (REF ), and let $(e,\\lambda )$ be the numbers in (REF ) associated to $(a,b,c)$ .",
"Then it holds $\\log \\max \\left(\\vert {}a{}\\vert ,\\vert {}b{}\\vert ,\\vert {}c{}\\vert \\right)\\le \\tfrac{\\lambda }{5}N_S\\log (2^{e}N_S)+\\tfrac{3\\lambda }{40} N_S\\log \\log \\log (2^{e}N_S)+\\tfrac{2\\lambda }{15}N_S+28.$ We begin by noting that the elliptic curves appearing in Lemma REF have the same conductor since they are ${\\mathbb {Q}}$ -isogenous.",
"Thus an application of Lemma REF with the solution $(a,b,c)$ of (REF ) gives an elliptic curve $E^{\\prime }$ over ${\\mathbb {Q}}$ , with conductor $N_{E^{\\prime }}$ and minimal discriminant $\\Delta _{E^{\\prime }}$ , such that $\\vert {}ab{}\\vert c^4\\le 2^{28}\\Delta _{E^{\\prime }}$ and such that $N_{E^{\\prime }}\\mid 2^{e}N_S$ for $e$ as in (REF ).",
"The equality $a+b=c$ proves that $\\vert {}c{}\\vert -1\\le \\vert {}ab{}\\vert $ and then we deduce $(\\vert {}c{}\\vert -1)^5\\le (\\vert {}c{}\\vert -1)c^4\\le \\vert {}ab{}\\vert c^4\\le 2^{28}\\Delta _{E^{\\prime }}.$ Further, $N_{E^{\\prime }}\\mid 2^{e}N_S$ with ${\\textnormal {ord}}_2(N_{E^{\\prime }})=e+1$ implies that $\\nu (N_{E^{\\prime }})\\le \\lambda N_S$ for $\\lambda $ as in (REF ) and for $\\nu $ the function on ${\\mathbb {Z}}_{\\ge 1}$ defined in Proposition REF (ii).",
"Then we see that (REF ) leads to Proposition REF provided that $H=\\vert {}c{}\\vert $ for $H=\\max (\\vert {}a{}\\vert ,\\vert {}b{}\\vert ,\\vert {}c{}\\vert )$ .",
"To deal with the remaining cases $H=\\vert {}a{}\\vert $ and $H=\\vert {}b{}\\vert $ , we notice that $(b,-c,-a)$ and $(a,-c,-b)$ are solutions of (REF ) as well.",
"Thus applications of the above arguments with $(b,-c,-a)$ and $(a,-c,-b)$ prove Proposition REF in the cases $H=\\vert {}a{}\\vert $ and $H=\\vert {}b{}\\vert $ respectively.",
"Let $(a,b,c)$ be a solution of (REF ).",
"Lemma REF associates to $(a,b,c)$ an elliptic $E$ over ${\\mathbb {Q}}$ with conductor $N_E$ that satisfies $N_E\\rightarrow \\infty $ if $\\textnormal {rad}(abc)\\rightarrow \\infty $ .",
"Therefore the arguments of Proposition REF together with the asymptotic bound obtained below (REF ) show that any solution $(a,b,c)$ of (REF ) with $\\textnormal {rad}(abc)=r$ satisfies $\\log \\max (|a|,|b|,|c|)\\le \\tfrac{9}{5}r\\log r + O\\Big (\\frac{r\\log r}{\\log \\log r}\\Big ) \\ \\ \\textnormal { for } r\\rightarrow \\infty .$ This (slightly) improves the bound $4r\\log r+O(r\\log \\log r)$ obtained by Murty–Pasten in [120].",
"However, our estimate displayed in (REF ) is still worse than the actual best asymptotic bound $O(r^{1/3}(\\log r)^3)$ of Stewart–Yu [146].",
"[Proof of Proposition REF ] We suppose that $(x,y)$ satisfies the $S$ -unit equation (REF ).",
"Then there exists a solution $(a,b,c)$ of (REF ) with $(x,y)=(\\frac{a}{c},\\frac{b}{c})$ .",
"The number $\\max (h(x),h(y))$ equals $\\log \\max (\\vert {}a{}\\vert ,\\vert {}b{}\\vert ,\\vert {}c{}\\vert )$ and thus Proposition REF implies Proposition REF ."
],
[
"Optimized height bounds and height conductor inequalities",
"We use the notation of Sections and REF .",
"Let $N\\ge 1$ be an integer.",
"We now define constants $\\alpha $ , $\\beta $ and $\\beta ^*$ depending on $N$ , which will appear in the optimized height bounds."
],
[
"The constants $\\alpha $ , {{formula:43f77f8d-25ee-48f5-b21b-4e10b11d36fe}} , {{formula:8e723ce6-ef17-4cb7-b9ba-e8d1b72f9342}} and optimized height bounds",
"To define $\\beta $ and $\\beta ^*$ we let $m$ be the number of newforms of level dividing $N$ , and we let $g=g(N)$ be the genus of $X_0(N)$ .",
"We write $l=\\lfloor \\frac{N}{6}\\prod (p+1)\\rfloor $ and $l^*=\\lfloor \\frac{N}{6}\\prod (1+1/p)\\rfloor $ with both products taken over all rational primes $p$ dividing $N$ , where $\\lfloor r\\rfloor =\\max (n\\in {\\mathbb {Z}}; \\, n\\le r)$ for any $r\\in \\mathbb {R}$ .",
"Let $\\tau (n)$ be the number of divisors of any $n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"We define $\\beta =\\tfrac{1}{2}m\\log m+\\max _J \\sum _{j\\in J}\\log (\\tau (j)j^{1/2}) \\ \\textnormal { and } \\ \\beta ^*=\\tfrac{1}{2}g\\log g+\\max _J \\sum _{j\\in J}\\log (\\tau (j)j^{1/2})$ with the first maximum taken over all subsets $J\\subseteq \\lbrace 1,\\cdots ,l\\rbrace $ of cardinality $m$ and with the second maximum taken over all subsets $J\\subseteq \\lbrace 1,\\cdots ,l^*\\rbrace $ of cardinality $g$ .",
"On comparing $\\beta $ with $\\beta ^*$ we notice that $\\beta $ involves the smaller number $m\\le g$ at the expense of depending on the larger parameter $l\\ge l^*$ .",
"It turns out that $\\beta \\le O(N\\log N)$ , while such an upper bound can not hold for $\\beta ^*$ since there is a constant $r\\in \\mathbb {R}$ such that infinitely many $n\\in {\\mathbb {Z}}_{\\ge 1}$ satisfy $n\\log \\log n\\le rg(n)$ .",
"On the other hand, it holds that $\\beta ^*< \\beta $ for infinitely many $N$ and thus we shall work in our algorithms with the quantity $\\alpha =\\min (\\beta ,\\beta ^*).$ We define $\\kappa =4\\pi +\\log (163/\\pi )$ and we mention that in the statement of the following Proposition REF we use the notation of Propositions REF and REF .",
"Proposition 10.7 Proposition REF holds with the bound $\\frac{6}{5}\\alpha (2^{e}N_S)+28,$ and Proposition REF holds with the bound $\\frac{1}{3}h(a)+4\\alpha (a_S)+2\\log (\\alpha (a_S)+\\kappa )+35+4\\kappa $ .",
"We observe that $\\alpha (m)\\le \\alpha (n)$ for all $m,n$ in ${\\mathbb {Z}}_{\\ge 1}$ with $m$ dividing $n$ .",
"Therefore Proposition REF follows directly by using in the above proofs of Propositions REF and REF the optimized bound given in Proposition REF (i).",
"We remark that for any given $N$ , one can practically compute $\\alpha $ by using the formulas for $m$ and $g$ in [111] and [49] respectively.",
"However, if $N$ becomes large then the precise computation of $\\alpha $ becomes slow and in this case we shall use $\\bar{\\beta }=\\tfrac{1}{2}m\\log m+\\tfrac{5}{8}m(18+\\log l) \\ \\textnormal { and } \\ \\bar{\\beta ^*}=\\tfrac{1}{2}g\\log (gl^*)+\\tfrac{1}{2}l^*\\log (4+4\\log l^*).$ Remark REF gives that $\\tau (j)\\le 45197j^{1/8}$ for all $j\\in {\\mathbb {Z}}_{\\ge 1}$ and hence $\\beta \\le \\bar{\\beta }$ .",
"To prove that $\\beta ^*\\le \\bar{\\beta ^*}$ we may and do assume that $g\\ge 1$ .",
"Let $J\\subseteq \\lbrace 1,\\cdots ,l^*\\rbrace $ be a subset of cardinality $g$ and observe that $g\\le n=\\lfloor l^*/2\\rfloor $ .",
"Then the elementary inequalities $l^*\\le 4n$ , $\\prod _{j\\in J} \\tau (j)\\le \\bigl (\\frac{1}{n}\\sum _{j=1}^{l^*}\\tau (j)\\bigl )^n$ and $\\frac{1}{l^*}\\sum _{j=1}^{l^*}\\tau (j)\\le 1+\\log l^*$ show that $\\sum _{j\\in J}\\log \\tau (j)\\le n\\log (4+4\\log l^*)$ .",
"This implies that $\\beta ^*\\le \\bar{\\beta ^*}$ as desired.",
"It follows that $\\alpha \\le \\bar{\\alpha }$ for $\\bar{\\alpha }=\\min (\\bar{\\beta },\\bar{\\beta ^*}).$ We take here the minimum since there are infinitely many $N$ for which $\\bar{\\beta ^*}<\\bar{\\beta }$ and vice versa.",
"Finally we point out that the computation of $\\bar{\\alpha }$ is very fast, even for large $N$ .",
"It remains to work out the explicit height conductor inequalities for elliptic curves over ${\\mathbb {Q}}$ which are used in our proofs and this will be done in the next section."
],
[
"Height and conductor of elliptic curves over ${\\mathbb {Q}}$",
"The geometric version of the Shimura–Taniyama conjecture gives that any elliptic curve $E$ over ${\\mathbb {Q}}$ of conductor $N$ is ${\\mathbb {Q}}$ -isogenous to $E_f$ for some rational newform $f\\in S_2(\\Gamma _0(N))$ , see Section .",
"We say that $f$ is the newform associated to $E$ .",
"Let $m_f$ be the modular degree of $f$ and let $r_f$ be the congruence number of $f$ , defined in (REF ) and (REF ) respectively.",
"On using and refining the argumentsThese arguments use an approach of Frey which involves the modular degree $m_f$ and the geometric version of the Shimura–Taniyama conjecture, see for example Frey [66].",
"of [120] and [89], which were discovered independently in 2011 by Murty–Pasten and by the first mentioned author, we obtain the following update of several results (see below) in [120], [89].",
"Proposition 10.8 Let $\\beta $ and $\\beta ^*$ be as in (REF ).",
"Suppose that $E$ is an elliptic curve over ${\\mathbb {Q}}$ of conductor $N$ , with associated newform $f$ .",
"Then the following statements hold.",
"(i) Define $\\kappa =4\\pi +\\log (163/\\pi )$ .",
"There are inequalities $2h(E)-\\kappa \\le \\log m_f \\le \\log r_f\\le \\alpha =\\min (\\beta ,\\beta ^*).$ (ii) Let $\\nu ^*$ be the multiplicative function on ${\\mathbb {Z}}_{\\ge 1}$ defined by $\\nu ^*(p)=1$ for $p$ a rational prime and $\\nu ^*(p^k)=1-1/p^{2}$ for $k\\in {\\mathbb {Z}}_{\\ge 2}$ , and put $\\nu =N\\nu ^*(N)$ .",
"It holds $\\beta \\le \\tfrac{1}{6}\\nu \\log N + \\tfrac{1}{16}\\nu \\log \\log \\log N + \\tfrac{1}{9}\\nu , \\ \\ \\ \\nu \\le N.$ (iii) If $N\\rightarrow \\infty $ then $\\beta \\le \\frac{1}{8}\\nu \\log N + \\frac{(\\frac{1}{6}\\log 2+o(1))}{\\log \\log N}\\nu \\log N.$ It is known (see e.g.",
"[65], [134]) that $h(E)$ is related to $m_f$ , and a result attributed to Ribet provides that $m_f\\mid r_f$ .",
"We now compare Proposition REF with the literature.",
"It improves $h(E)\\le (2N)^{40^2}$ inThe result [87] bounds a “naive\" height of $E$ .",
"However, Silverman's arguments in [134] lead to an explicit upper bound for $h(E)$ in terms of the “naive\" height of $E$ used in [87].",
"[87] and $h(E)\\le (25N)^{162}$ in [85] which were established by different methods: [87] uses the effective reduction theory of Evertse–Győry and [85] combines Legendre level structure with the theory of logarithmic forms.",
"Furthermore, Proposition REF updates $2h(E)\\le \\frac{1}{5}N\\log N+22$ in Murty–Pasten [120] and $2h(E)\\le \\frac{1}{2}N(\\log N)^2+18$ in [89].",
"It also updates the asymptotic bound $2h(E)\\le \\frac{1}{6}N\\log N+O(N\\log \\log N)$ in Murty–Pasten [120] and the bounds for $m_f$ and $r_f$ in [120] and [89].",
"We proved Proposition REF in our unpublished 2014 preprint “Solving S-unit and Mordell equations via Shimura–Taniyama conjecture\".",
"Hector Pasten told us that in Taipei (May 20, 2016) he will announce the following results which he obtained independently.",
"Remark 10.9 (Recent results of Hector Pasten) Let $S$ be a finite set of rational primes.",
"For all elliptic curves $E$ over ${\\mathbb {Q}}$ that are semistable outside S, we have $h(E) \\ll _S \\phi (N)\\log N.$ Furthermore, if we assume GRH then we have $h(E) \\ll _S \\phi (N)\\log \\log N.$ Here $\\phi $ denotes the Euler totient function, and the constants $\\ll _S$ are effective, small and explicit.",
"Further, it holds that $\\log m_f\\le (o(1)+\\tfrac{1}{24})N\\log N$ as $N\\rightarrow \\infty $ , and if $E$ is semistable then $\\log m_f\\le (o(1)+\\tfrac{1}{24})\\phi (N)\\log N $ as $N\\rightarrow \\infty $ ; both estimates are effective and can be made explicit with good constants.",
"We thank Hector Pasten for sending us the statements of his results."
],
[
"Proof of Proposition ",
"As already mentioned, the proof of Proposition REF uses the ideas of [120] and [89]; see also Frey [66].",
"For example the arguments of [89] directly lead to $2h(E)-\\kappa \\le \\log m_f \\le \\log r_f\\le \\beta ^*.$ Then to replace here $\\beta ^*$ with $\\beta $ we go through the proof of [89] by using the “coprime\" matrix constructed in Murty–Pasten [120].",
"To show (ii) we combine a formula of Martin [111] with classical analytic number theory.",
"The latter is used to explicitly bound the quantities $\\prod _{p\\mid n}(1+1/p)$ and $\\tau (n)$ in terms of $n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"Finally we deduce (iii) by replacing in the proof of (ii) our explicit estimate for $\\tau (n)$ by Wigert's asymptotic bound.",
"[Proof of Proposition REF ] We first prove (i).",
"To show that $\\log r_f\\le \\beta $ we denote by $\\mathcal {B}=\\lbrace f_1,\\ldots ,f_g\\rbrace $ the Atkin–Lehner basis [2] for $S_2(\\Gamma _0(N))$ , indexed in such a way such that $f_1=f$ and such that $f_1,\\ldots ,f_m$ are the newforms of level dividing $N$ .",
"We write $I=\\lbrace 1,\\cdots ,m\\rbrace $ and $J^{\\prime }=\\lbrace j\\in \\lbrace 1,\\cdots ,l\\rbrace ; (j,N)=1\\rbrace $ .",
"On using Atkin–Lehner theory, Murty–Pasten proved in [120] that the matrix $F^{\\prime }=(F_{ij})$ has full rank $m$ for $F_{ij}=a_j(f_i)$ with $i\\in I$ and $j\\in J^{\\prime }$ .",
"Hence there is a subset $J\\subseteq J^{\\prime }$ of cardinality $m$ such that the matrix $F=(F_{ij})$ , with $i\\in I$ and $j\\in J$ , is invertible.",
"The Ramanujan–Petersson bounds for Fourier coefficients imply that $\\vert F_{ij}\\vert \\le \\tau (j)j^{1/2}$ for all $i\\in I$ and $j\\in J$ .",
"Thus Hadamard's determinant inequality shows that $\\log \\det (F)\\le \\beta $ and then the claim $r_f^2\\mid \\det (F)^2\\in {\\mathbb {Z}}$ proves that $\\log r_f\\le \\beta $ as desired.",
"To verify the claim $r_f^2\\mid \\det (F)^2\\in {\\mathbb {Z}}$ we take $f_c\\in S_2(\\Gamma _0(N))$ as in (REF ).",
"Then there exists $y=(y_j)\\in {\\mathbb {Z}}^J$ such that $a_j(f_c)=a_j(f)+y_jr_f \\ \\textnormal { for all } j\\in J.$ We write $f_c=\\sum _{i=1}^g k_if_i$ with $(k_i)\\in {\\mathbb {C}}^g$ .",
"It holds that $k_1=0$ , since $\\mathcal {B}$ is an orthogonal basis and since $(f,f_c)=0$ by (REF ).",
"Therefore on comparing Fourier coefficients we deduce $a_j(f_c)=\\sum _{i=2}^{g} k_ia_j(f_i)=\\sum _{i=2}^m k_ia_j(f_i) \\ \\textnormal { for all } j\\in J.$ Here we used that $a_j(f_i)=0$ for all $i\\ge m+1$ and $j\\in J$ .",
"To see that $a_j(f_i)=0$ for all $i\\ge m+1$ and $j\\in J$ , one recalls that each $j\\in J\\subseteq J^{\\prime }$ is coprime to $N$ and any $f_i\\in \\mathcal {B}$ with $i\\ge m+1$ is of the following form: $f_i(\\tau )=f^*(n\\tau )$ with $f^*\\in S_2(\\Gamma _0(M))$ a newform, $M$ a proper divisor of $N$ , and $n\\ge 2$ a divisor of $N/M$ .",
"Now on using exactly the same arguments as in the proof of [89], we see that the formulas (REF ) and (REF ) imply the claim $r_f^2\\mid \\det (F)^2$ .",
"It follows that $\\log r_f\\le \\min (\\beta ,\\beta ^*)$ since [89] gives the upper bound $\\log r_f\\le \\beta ^*$ .",
"Further [3] implies that $\\log m_f\\le \\log r_f$ .",
"Finally, the desired lower bound for $\\log m_f$ follows for example from the explicit inequality $h(E)\\le \\frac{1}{2}\\log m_f+2\\pi +\\frac{1}{2}\\log (163/\\pi )$ which was obtained in course of the proof of [89].",
"This completes the proof of assertion (i).",
"We now prove statement (ii).",
"Martin obtained in [111] an explicit formula for $m$ in terms of $N$ and $\\nu $ .",
"This formula implies the estimate $m\\le \\frac{\\nu }{12}-\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}=\\frac{\\nu +1}{12}.$ We next work out an explicit upper bound for $l=\\lfloor \\frac{N}{6}\\prod _{p\\mid N}p(1+1/p)\\rfloor $ with the product taken over all rational primes $p$ dividing $N$ .",
"Let $\\gamma =0.577\\cdots $ denote the Euler–Mascheroni constant.",
"In a first step we show that any $n\\in {\\mathbb {Z}}_{\\ge 3}$ satisfies $\\prod _{p\\operatorname{|}n} (1+1/p) \\le \\frac{6 e^{\\gamma }}{\\pi ^2}\\left(\\log \\log n + \\frac{2}{\\log \\log n}\\right)$ with the product taken over all rational primes $p\\mid n$ .",
"To prove (REF ) we may and do assume that $n$ is of the form $n=\\prod _{p\\le x}p=e^{\\vartheta (x)}$ with the product taken over all rational primes $p$ at most $x=x(n)\\in {\\mathbb {Z}}_{\\ge 2}$ .",
"Indeed this follows by observing that the function $\\log \\log n+2/\\log \\log n$ is monotonously increasing for $n\\ge 62$ , by considering special cases such as for example the case when $n$ is a prime power and by checking (for example with Sage) all $n\\le 62$ .",
"On writing $1+1/p = (1-1/p^2)/(1-1/p)$ , we obtain $\\prod _{p\\le x}(1+1/p)=\\left(\\prod _{p}(1-\\frac{1}{p^2})\\cdot \\bigl (\\prod _{p>x}(1-\\frac{1}{p^2})\\bigl )^{-1}\\right)/\\prod _{p\\le x}(1-1/p)$ with the products taken over all rational primes $p$ satisfying the specified conditions.",
"The effective version of Merten's theorem in [128] provides $\\prod _{p\\le x}(1-1/p) > \\frac{e^{-\\gamma }}{\\log x}\\Big (1-\\dfrac{1}{2(\\log x)^2}\\Big ) \\ \\textnormal { if } x\\ge 285.$ Euler's product formula gives that $\\prod _p (1-1/p^2)=\\zeta (2)^{-1}=6/\\pi ^2$ with the product taken over all rational primes $p$ .",
"Further, we deduce the inequalities $\\log \\prod _{p>x}\\Big (1-\\frac{1}{p^2}\\Big ) \\ge -\\sum _{p>x}\\frac{1}{p^2}\\Big (1+\\frac{1}{2p^2}\\Big )\\ge -\\Big (1+\\frac{1}{2x^2}\\Big )\\int _{x}^{\\infty }\\frac{1}{t^2}\\,dt = -\\frac{1}{x}\\Big (1+\\frac{1}{2x^2}\\Big ).$ On combining the results collected above with the effective prime number theorem of the following form (see [128]) $x-\\frac{x}{2\\log x} < \\vartheta (x)=\\log n < x+\\frac{x}{2\\log x} \\ \\textnormal { if } x\\ge 563,$ we see that the claimed inequality (REF ) holds for all $n=e^{\\vartheta (x)}$ with $x>10^4$ .",
"Finally, one checks (for example with Sage) that (REF ) holds in addition in the remaining cases $n=e^{\\vartheta (x)}$ for $2\\le x\\le 10^4$ .",
"The conductor $N$ is at least 11 and hence (REF ) gives $l\\le \\frac{e^\\gamma }{\\pi ^2}N^2\\bigl (\\log \\log N + \\frac{2}{\\log \\log N}\\bigl ).$ To estimate $\\tau (n)$ we consider the real valued function $u(n)=\\tau (n)/n^{1/4}$ on ${\\mathbb {Z}}_{\\ge 1}$ .",
"This function is multiplicative and satisfies $u(n)= \\prod _{p}(n_p+1)p^{-n_p/4}$ with the product taken over all rational primes $p$ , where $n_p={\\textnormal {ord}}_p(n)$ denotes the order of $p$ in $n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"To find the maximum of $u$ we look at each factor separately.",
"It holds that $(n_p+1)p^{-n_p/4}=1$ when $n_p=0$ , and if $n_p\\ge 1$ then we observe that $(n_p+1)p^{-n_p/4}< 1$ provided that $p\\ge 17$ or $n_p\\ge 17$ .",
"Thus after checking (for example with Sage) the remaining cases we find that $\\sup (u)=8.44\\cdots $ ; in fact this supremum is attained at $n=2^5\\cdot 3^3\\cdot 5^2\\cdot 7\\cdot 11\\cdot 13$ .",
"Therefore we conclude that any $n\\in {\\mathbb {Z}}_{\\ge 1}$ satisfies the inequality $\\tau (n)\\le 8.5\\, n^{1/4}.$ To put everything together we recall that in $\\beta =\\frac{m}{2}\\log m+\\max _J \\sum _{j\\in J}\\log (\\tau (j)j^{1/2})$ the maximum is taken over all subsets $J\\subseteq \\lbrace 1,\\cdots ,l\\rbrace $ of cardinality $m$ .",
"Hence (REF ) gives $\\beta \\le \\beta ^{\\prime }=m\\left(\\tfrac{1}{2}\\log m+\\tfrac{3}{4}\\log l+\\log 8.5\\right).$ Further Euler's product formula shows that $\\nu \\ge 6N/\\pi ^2$ and then (REF ) together with (REF ) leads to $\\beta ^{\\prime }\\le \\frac{\\nu }{6}\\log N+\\frac{\\nu }{16}\\log \\log \\log N+\\frac{\\nu }{9}$ for all $N\\ge 23$ .",
"Moreover, one checks (for example with Sage) that this upper bound holds in addition for all $N$ with $11\\le N< 23$ .",
"Therefore the displayed inequality $\\beta \\le \\beta ^{\\prime }$ proves (ii).",
"To show (iii) we observe that $N\\rightarrow \\infty $ implies $l\\rightarrow \\infty $ and $m\\rightarrow \\infty $ .",
"Hence we may and do assume that $l\\rightarrow \\infty $ and $m\\rightarrow \\infty $ .",
"If $n\\in {\\mathbb {Z}}_{\\ge 1}$ with $n\\rightarrow \\infty $ , then Wigert's bound gives $\\log \\tau (n)\\le (\\log 2+o(1))\\frac{\\log n}{\\log \\log n}.$ This implies that $\\sum _{j\\in J}\\log (\\tau (j))\\le m(\\log 2+o(1))\\frac{\\log l}{\\log \\log l}$ for any subset $J\\subseteq \\lbrace 1,\\cdots ,l\\rbrace $ of cardinality $m$ .",
"Therefore on recalling the definition of $\\beta $ we obtain $\\beta \\le m\\left(\\tfrac{1}{2}\\log m+\\tfrac{1}{2}\\log l+(\\log 2+o(1))\\frac{\\log l}{\\log \\log l}\\right).$ Then we see that the above estimates for $m$ and $l$ , given in (REF ) and (REF ) respectively, lead to statement (iii).",
"This completes the proof of Proposition REF .",
"Remark 10.10 To prove (ii) we used the explicit bound $\\tau (n)\\le 8.5n^{1/4}$ obtained in (REF ), since the constant $8.5$ is reasonably small.",
"On enlarging the constant $8.5$ , one could replace the exponent $1/4$ by any other positive real number.",
"However, for small exponents the corresponding (effective) constants become quite large.",
"For example, if we take the exponent $1/8$ then the constant $8.5$ needs to be replaced by $45196.8$ .",
"We mention that Proposition REF allows to update several bounds in the literature.",
"For example, Proposition REF together with [89] directly implies that any elliptic curve $E$ over ${\\mathbb {Q}}$ of conductor $N$ and minimal discriminant $\\Delta _E$ satisfies $\\log \\Delta _E\\le \\nu \\log N+\\tfrac{3}{8}\\nu \\log \\log \\log N+\\tfrac{2}{3}\\nu +115.1$ and if $N\\rightarrow \\infty $ then $\\log \\Delta _E\\le \\frac{3}{4}\\nu \\log N+\\frac{\\log 2+o(1)}{\\log \\log N}\\nu \\log N$ .",
"Here $\\nu $ is as in Proposition REF (ii).",
"These inequalities update the discriminant conductor inequalities in Murty–Pasten [120] and in [89]; see also [90] for more general (but weaker) discriminant bounds based on the theory of logarithmic forms."
],
[
"Introduction",
"In this section we solve the problem of constructing an efficient sieve for the $S$ -integral points of bounded height on any elliptic curve $E$ over ${\\mathbb {Q}}$ with given Mordell–Weil basis of $E({\\mathbb {Q}})$ .",
"Our construction combines a geometric interpretation of the known elliptic logarithm reduction (initiated by Zagier [167]) with several conceptually new ideas.",
"The resulting “elliptic logarithm sieve\" considerably extends the class of elliptic Diophantine equations which can be solved in practice.",
"To illustrate this we solved many notoriously difficult equations by applying our sieve.",
"We also used the resulting data and our sieve to motivate new conjectures and questions on the number of $S$ -integral points of $E$ .",
"The precise construction of the elliptic logarithm sieve is rather technical.",
"We now begin to describe the underlying ideas using a geometric point of view: After fixing the setup and briefly discussing the known elliptic logarithm approach, we explain the main ingredients of our sieve and we describe the improvements provided by our new ideas."
],
[
"Setup.",
"Throughout this section we shall work with the following setup.",
"Let $E$ be an elliptic curve over ${\\mathbb {Q}}$ .",
"We suppose that we are given an arbitrary Weierstrass equation of $E$ , with coefficients $a_1,\\cdots ,a_6$ in ${\\mathbb {Z}}$ , of the following form $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$ For any field $K$ containing ${\\mathbb {Q}}$ , we often identify a nonzero point in $E(K)$ with the corresponding solution of (REF ) and vice versa.",
"We further assume that we are given a basis $P_1,\\cdots ,P_r$ of the free part of the finitely generated abelian group $E({\\mathbb {Q}})$ , see Section REF .",
"Let $S$ be a finite set of rational primes and let $\\Sigma (S)$ be the set of $(x,y)$ in $\\mathcal {O}\\times \\mathcal {O}$ satisfying (REF ), where $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ and $N_S=\\prod _{p\\in S} p$ .",
"Finally, we suppose that we are given an initial bound $M_0$ , that is $M_0\\in {\\mathbb {Z}}_{\\ge 1}$ such that any $P\\in \\Sigma (S)$ satisfies $\\hat{h}(P)\\le M_0$ for $\\hat{h}$ the canonical Néron–Tate height of $E$ .",
"In fact everything works equally well with an initial bound for the usual Weil height or for the infinity norm $\\Vert \\cdot \\Vert _\\infty $ , see Remark REF (ii)."
],
[
"Elliptic logarithm approach",
"Starting with Masser [112], Lang [102] and Wüstholz [166], many authors developed a practical approach to determine $\\Sigma (S)$ using elliptic logarithms.",
"Here a fundamental ingredient is a technique introduced by Zagier [167] which we call the elliptic logarithm reduction.",
"In practice this technique allows to considerably reduce the initial bound $N_0$ coming from transcendence theory; $\\Vert P\\Vert _\\infty \\le N_0$ for all $P\\in \\Sigma (S)$ .",
"More precisely on combining Zagier's arguments with de Weger's approach via $L^3$ [105], Stroeker–Tzanakis [141] and Gebel–Pethő–Zimmer [68], [71] showed independently the following when $\\mathcal {O}={\\mathbb {Z}}$ : The elliptic logarithm reduction produces a relatively small number $N_1<N_0$ such that any point $P\\in \\Sigma (S)$ with $N_1< \\Vert P\\Vert _\\infty \\le N_0$ has to be exceptional (Definition REF ).",
"Smart [137] extended the method to general $\\mathcal {O}$ , see also [126] and the recent book of Tzanakis [159] devoted to the elliptic logarithm approach.",
"If $N_1^r$ is small enough then $\\Sigma (S)$ can be enumerated by checking all remaining candidates $P$ with $\\Vert P\\Vert _\\infty \\le N_1$ and by finding the exceptional points.",
"However there are many situations of interest in which $N_1^r$ is usually too large to determine $\\Sigma (S)$ via the known methods.",
"In view of this, resolving the following problem would be of fundamental importance.",
"Problem.",
"Construct an efficient sieve for the points $\\Sigma (S)\\subseteq E({\\mathbb {Q}})$ of bounded height.",
"Here the known sieves are often useless in practice.",
"For example, working in the finite groups obtained by “reducing the curve $E$ mod $p$ \" for suitable primes $p$ is usually not efficient (see Smart [137]).",
"In fact, since an efficient sieve for $\\Sigma (S)$ inside $E({\\mathbb {Q}})$ was not available, various authors conducted some effort to develop other techniques to enumerate $\\Sigma (S)$ in certain cases when $N_1^r$ is not small enough; see Section REF ."
],
[
"The elliptic logarithm sieve",
"Building on the core idea of the elliptic logarithm reduction, we construct the elliptic logarithm sieve which resolves in particular the above problem.",
"Here we introduce several conceptually new ideas.",
"They all rely on a geometric point of view and to explain our ideas we now give a geometric interpretation of the known elliptic logarithm reduction: For each $v$ in $S^*=S\\cup \\lbrace \\infty \\rbrace $ one uses elliptic logarithms to construct a lattice $\\Gamma _v\\subset {\\mathbb {Z}}^d$ of rank $d$ such that any non-exceptional $P\\in \\Sigma (S)$ with $\\Vert P\\Vert _\\infty >N_1$ is essentially determined by a nonzero point in some $\\Gamma _v$ .",
"Then one tries to show via $L^3$ that a certain cube $Q_v\\subset {\\mathbb {R}}^{d}$ satisfies $\\Gamma _v\\cap Q_v=0$ proving that all $P\\in \\Sigma (S)$ with $\\Vert P\\Vert _\\infty >N_1$ are exceptional.",
"Here $d$ equals $r$ or $r+1$ , and for any $v\\in S$ the cube $Q_v\\subset {\\mathbb {R}}^d$ is given by $\\lbrace \\Vert P\\Vert _\\infty \\le N_0\\rbrace $ inside $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}\\cong {\\mathbb {R}}^d$ .",
"Further, increasing $N_1$ enlarges the co-volume of each $\\Gamma _v$ and hence $\\Gamma _v\\cap Q_v$ is usually trivial for sufficiently large $N_1$ .",
"In fact the cube $Q_v$ always contains a certain ellipsoid $\\mathcal {E}_v\\subset {\\mathbb {R}}^d$ arising from $\\hat{h}$ .",
"Now, our new ideas can be described as follows:"
],
[
"Global sieves.",
"We use $\\Gamma _v\\cap \\mathcal {E}_v$ to construct various global sieves for $\\Sigma (S)$ inside $E({\\mathbb {Q}})$ .",
"Here, instead of computing a lower bound for the length of the shortest nonzero vector in $\\Gamma _v$ , we actually determine the points in $\\Gamma _v\\cap \\mathcal {E}_v$ using Fincke–Pohst [105], [63] and we check if these points come from $\\Sigma (S)$ .",
"This has the following advantages: (i) We can further reduce $N_1$ in the crucial situation where the usual reduction is not working anymore (e.g.",
"the shortest nonzero vector of $\\Gamma _v$ actually lies in $Q_v$ ).",
"(ii) On “covering\" the set $\\Sigma (S)$ by the local sieves $\\Gamma _v\\cap \\mathcal {E}_v$ , $v\\in S^*$ , we obtain a global sieve for $\\Sigma (S)$ inside $E({\\mathbb {Q}})$ which is more efficient than the standard enumeration."
],
[
"Refined coverings.",
"On using the geometric point of view, we construct in Proposition REF refined “coverings\" of $\\Sigma (S)$ in order to improve the global-local passage in (ii).",
"This leads to a refined sieve which enhances our sieve in (ii) and which allows to reduce $N_1$ even further.",
"Here the construction is inspired by our refined sieve for $S$ -unit equations in Section REF .",
"However, in the present case of elliptic curves, the technicalities arising from $v$ -adic elliptic logarithms at $v=2$ and $v=\\infty $ are more involved."
],
[
"Height-logarithm sieve.",
"We construct a sieve for $\\Sigma (S)$ inside $E({\\mathbb {Q}})$ by exploiting that for any non-exceptional point $P\\in \\Sigma (S)$ the height $\\hat{h}(P)$ is essentially determined by the local $v$ -adic elliptic logarithms with $v\\in S^*$ .",
"The height-logarithm sieve is a crucial ingredient of our global sieves discussed above.",
"For many involved points $P$ , it allows to avoid the slow process of testing whether the coordinates of $P$ are in fact $S$ -integers."
],
[
"Ellipsoids.",
"Instead of using the infinity norm $\\Vert \\cdot \\Vert _\\infty $ as done by all other authors, we work directly with the canonical height $\\hat{h}$ .",
"Our approach using $\\hat{h}$ is more efficient than the known improvements of the elliptic logarithm reduction (see Section REF ), since the cubes $Q_v$ arising from $\\Vert \\cdot \\Vert _\\infty $ always contain our ellipsoids $\\mathcal {E}_v$ determined by $\\hat{h}$ .",
"In fact working here with ellipsoids is optimal from a geometric point of view and it is crucial for the construction of our sieves.",
"To circumvent issues with the real valued function $\\hat{h}$ , we constructed a suitable rational approximation of the quadratic form $\\hat{h}$ on $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}$ ."
],
[
"Exceptional points.",
"We conducted some effort to avoid as much as possible working with the coordinates of the points.",
"For example to deal with exceptional points, we prove the crucial Proposition REF which allows here to work entirely in the finitely generated group $E({\\mathbb {Q}})$ .",
"This considerably improves the “extra search\" for exceptional points.",
"In fact in most cases Proposition REF completely removes the “extra search\"."
],
[
"Generic situation.",
"The case $r\\le 1$ is of particular importance, since it represents the most common situation in practice; see also Katz–Sarnak [100] and Bhargava–Shankar [30].",
"Furthermore, one can efficiently verify in practice whether $r\\le 1$ by using for example the work of Kolyvagin [97] and Gross–Zagier–Zhang [169].",
"Also one can directly determine $\\Sigma (S)$ when $r=0$ .",
"In view of this we tried to further improve our sieves for $r=1$ .",
"On exploiting that $\\Gamma _v$ has rank $r=1$ for $v\\in S$ , we optimized the reduction process at $v\\in S$ and we enhanced our height-logarithm sieve for huge sets $S$ ."
],
[
"Discussion",
"We shall motivate (using geometry) the ideas and constructions underlying the elliptic logarithm sieve.",
"Further, for each of our algorithms, we conducted some effort to discuss important complexity aspects, to motivate our choice of parameters, to circumvent potential numerical issues, and to give detailed correctness proofs.",
"In particular we prove in detail that our constructions involving $v$ -adic elliptic logarithms have the required properties at the problematic places $v$ of ${\\mathbb {Q}}$ , that is $v=\\infty $ , $v=2$ and bad reduction $v$ ."
],
[
"Improvements.",
"The elliptic logarithm sieve improves in all aspects the known elliptic logarithm reduction and its subsequent enumeration.",
"In Sections REF and REF , we shall demonstrate (in theory and in practice) that our improvements are substantial.",
"In fact we obtain running time improvements by a factor which is exponential in terms of the rank $r$ , and which is exponential in terms of $\\vert {}S{}\\vert $ when $\\max \\vert {}a_i{}\\vert $ is large.",
"Furthermore in the case of a generic Weierstrass equation (REF ) we can efficiently determine all $S$ -integral solutions for huge sets $S$ .",
"For example sets $S$ with $\\vert {}S{}\\vert =10^5$ are usually no problem here.",
"Also, if $\\vert {}S{}\\vert $ is very small then our sieve allows to deal efficiently with large ranks $r$ such as $r=14,\\cdots ,19$ .",
"In particular, in the case when $\\mathcal {O}={\\mathbb {Z}}$ , the sieve is practical even for huge ranks such as $r=28$ .",
"The elliptic logarithm sieve considerably extends the class of elliptic Diophantine equations which can be solved in practice.",
"We shall demonstrate this by solving several notoriously difficult Diophantine problems which appear to be completely out of reach for the known methods, see Section REF for explicit examples."
],
[
"Applications",
"We solved large classes of elliptic Diophantine equations by applying our sieve.",
"In particular, we efficiently solved several Diophantine problems in which the involved rank $r$ is large.",
"Further, for each globally minimal Weierstrass equation (REF ) of any elliptic curve over ${\\mathbb {Q}}$ of conductor at most 100 (resp.",
"1000), we determined its set of $S$ -integral solutions with $S$ given by the first $10^4$ (resp.",
"20) primes.",
"See Section REF for more information."
],
[
"Conjecture and questions.",
"We used our data to motivate various questions on points of hyperbolic curves $Y=(X,D)$ of genus one.",
"More precisely, let $B$ be a nonempty open subscheme of $\\textnormal {Spec}({\\mathbb {Z}})$ and let $X\\rightarrow B$ be a smooth, proper and geometrically connected genus one curve.",
"Let $Y\\hookrightarrow X$ be an open immersion onto the complement $X-D$ of a nontrivial relative Cartier divisor $D\\subset X$ which is finite étale over $B$ .",
"We now state the following conjecture involving the rank $r$ of the group formed by the ${\\mathbb {Q}}$ -points of $\\textnormal {Pic}^0(X_{\\mathbb {Q}})$ .",
"Conjecture.",
"There are constants $c_Y$ and $c_r$ , depending only on $Y$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ with $T=\\textnormal {Spec}({\\mathbb {Z}})-S$ satisfies $\\vert {}Y(T){}\\vert \\le c_Y \\vert {}S{}\\vert ^{c_r}.$ Our initial motivation for making this conjecture is explained in Section REF .",
"Further, the above conjecture generalizes our conjecture for Mordell equations which we discussed and motivated in Section REF .",
"In fact our discussion and motivation given there, including the construction of our probabilistic model, can be applied in exactly the same way in the case when $Y$ is a Weierstrass curve.",
"Here $Y$ is a Weierstrass curve if the Cartier divisor $D$ is given by the image of a section of $X\\rightarrow B$ .",
"We shall also discuss and motivate various questions related to the above conjecture.",
"For example, we ask whether the above conjecture holds with $\\vert {}S{}\\vert $ replaced by the logarithm of the largest prime in $S$ ?"
],
[
"Plan.",
"In Section REF we discuss a suitable rational approximation of the Néron–Tate height on $E({\\mathbb {Q}})$ .",
"The subsequent Sections REF and REF contain our construction of the local sieves at the archimedean place and the non-archimedean places.",
"In Sections REF and REF we explain the height-logarithm sieve and the refined enumeration.",
"Then we construct the refined sieve in Section REF .",
"In Sections REF and REF , we put everything together to obtain the elliptic logarithm sieve.",
"Here we also compare our sieve with the known approach.",
"Then, after recalling in Section REF results and methods which allow to compute the required input data, we discuss applications of our sieve in Section REF .",
"Finally, we explain in Section REF computational aspects of our constructions."
],
[
"Notation.",
"Throughout this section we shall use the following conventions.",
"By $\\log $ we mean the principal value of the natural logarithm.",
"Unless mentioned otherwise, $\\vert {}z{}\\vert $ denotes the usual complex absolute value of $z\\in {\\mathbb {C}}$ and the product taken over the empty set is 1.",
"Further $\\textnormal {lcm}(a_1,\\cdots ,a_n)$ denotes the least common multiple of $a_1,\\cdots ,a_n\\in {\\mathbb {Z}}$ .",
"For any real number $x\\in \\mathbb {R}$ , we write $\\left\\lfloor {x}\\right\\rfloor =\\max (n\\in {\\mathbb {Z}}\\,;\\,n\\le x)$ and $\\left\\lceil {x}\\right\\rceil =\\min (n\\in {\\mathbb {Z}}\\,;\\,n\\ge x)$ .",
"We denote by $h(\\alpha )$ the usual absolute logarithmic Weil height of $\\alpha \\in {\\mathbb {Q}}$ , with $h(0)=0$ and $h(\\alpha )=\\log \\max (\\vert {}m{}\\vert ,\\vert {}n{}\\vert )$ if $\\alpha =m/n$ for coprime $m,n\\in {\\mathbb {Z}}$ .",
"If $\\alpha \\in {\\mathbb {Q}}$ is nonzero and if $p$ is a rational prime, then we write ${\\textnormal {ord}}_p(\\alpha )\\in {\\mathbb {Z}}$ for the order of $p$ in $\\alpha $ .",
"For any set $M$ , we denote by $\\vert M\\vert $ the (possibly infinite) number of distinct elements of $M$ .",
"Finally, for any $n\\in {\\mathbb {Z}}_{\\ge 1}$ , we say that $\\mathcal {E}\\subset \\mathbb {R}^n$ is an ellipsoid if $\\mathcal {E}=\\lbrace x\\in \\mathbb {R}^n\\,;\\,q(x)\\le c\\rbrace $ for some positive definite quadratic form $q:{\\mathbb {R}}^n\\rightarrow {\\mathbb {R}}$ and some positive real number $c$ ."
],
[
"Computer, software and algorithms.",
"We implemented all our algorithms in Sage.",
"Here a significant part of our program code is devoted to assure that the numerical aspects of the algorithms are all correct, see Section REF for certain important numerical aspects.",
"We point out that we shall use various functions of Sage [130] which in fact are direct applications of the corresponding functions of Pari [123].",
"Further, to compute the Mordell–Weil bases required for the input of the elliptic logarithm sieve, we used the techniques implemented in the computer packages Pari, Sage and Magma [107].",
"For all our computations, we used a standard personal working computer at the MPIM Bonn.",
"We shall list the running times of our algorithms for many examples.",
"In fact the listed times are always upper bounds and some of them were obtained by using older versions of our algorithms.",
"Here in many cases the running times would now be significantly better when using the most recent versions (as of February 2016) of our algorithms."
],
[
"Acknowledgements.",
"Our construction of the elliptic logarithm sieve crucially builds on ideas and techniques of the authors who developed, generalized and/or refined the elliptic logarithm reduction over the last 30 years (see [159] for an overview)."
],
[
"Heights",
"In this section we first recall useful results for heights of rational points on our given elliptic curve $E$ over ${\\mathbb {Q}}$ with Weierstrass equation (REF ).",
"Then we construct a suitable rational approximation of the canonical Néron–Tate height on $E({\\mathbb {Q}})$ , and we fix some terminology."
],
[
"Canonical height.",
"Let $\\hat{h}$ be the canonical Néron–Tate height on $E({\\mathbb {Q}})$ .",
"Here we use the natural normalization which divides by the degree of the involved rational function, see for example [136].",
"For any nonzero $P\\in E({\\mathbb {Q}})$ , it is known that the logarithmic Weil height $h(x)$ of the corresponding solution $(x,y)$ of (REF ) can be explicitly compared with $\\hat{h}(P)$ .",
"For instance Silverman [135] used an approach of Lang to obtain an explicit constant $\\mu (E)$ , depending only on the coefficients $a_i$ of (REF ), such that $-\\tfrac{1}{24}h(j_E)-\\mu \\le \\hat{h}(P)-\\tfrac{1}{2}h(x)\\le \\mu $ for $\\mu =\\mu (E)+1.07$ and $j_E$ the $j$ -invariant of $E$ .",
"Further it is known that $\\hat{h}$ defines a positive semi-definite quadratic form on the geometric points of $E$ , and for any point $P\\in E({\\mathbb {Q}})$ it holds $\\hat{h}(P)=0$ if and only if $P$ lies in the torsion subgroup $E({\\mathbb {Q}})_{\\textnormal {tor}}$ of $E({\\mathbb {Q}})$ .",
"Therefore on identifying the real vector space $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}$ with ${\\mathbb {R}}^r$ via our given basis $P_1,\\cdots ,P_r$ of the free part of $E({\\mathbb {Q}})$ , we obtain that $\\hat{h}$ extends to a positive definite quadratic form on ${\\mathbb {R}}^r$ .",
"Let $\\lambda $ be the smallest eigenvalue of the matrix $(\\hat{h}_{ij})$ in ${\\mathbb {R}}^{r\\times r}$ defining the bilinear form associated to $\\hat{h}$ .",
"Linear algebra gives that any point $P$ in $E({\\mathbb {Q}})$ satisfies $\\lambda \\Vert P\\Vert ^2_\\infty \\le \\hat{h}(P)\\le r\\lambda ^{\\prime }\\Vert P\\Vert ^2_\\infty $ for $\\lambda ^{\\prime }$ the largest eigenvalue of $(\\hat{h}_{ij})$ .",
"Here the infinity norm is defined by $\\Vert P\\Vert _\\infty =\\max \\vert {}n_i{}\\vert $ , where $P=Q+\\sum n_i P_i$ with $n_i\\in {\\mathbb {Z}}$ and $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ .",
"We point out that in practice it is always possible to quickly determine the points in $E({\\mathbb {Q}})_{\\textnormal {tor}}$ .",
"For what follows we therefore always may assume that the rank $r\\ge 1$ .",
"To avoid numerical problems with the real valued function $\\hat{h}$ , we shall work with a rational approximation of $\\hat{h}$ ."
],
[
"Rational approximation.",
"We next explain our construction of a suitable rational approximation of $\\hat{h}$ .",
"Let $k\\in {\\mathbb {Z}}_{\\ge 1}$ and define the norm $\\Vert \\hat{h}_{ij}\\Vert $ of $(\\hat{h}_{ij})$ by $\\Vert \\hat{h}_{ij}\\Vert =\\max \\vert {}\\hat{h}_{ij}{}\\vert $ .",
"On using continued fractions we obtain $f\\in {\\mathbb {Q}}$ which approximates the real number $2^{k}/\\Vert \\hat{h}_{ij}\\Vert $ up to any required precision, see Section REF .",
"We identify the rational vector space $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {Q}}$ with ${\\mathbb {Q}}^r$ via the basis $P_1,\\cdots ,P_r$ and we consider the quadratic form $\\hat{h}_k:E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {Q}}\\rightarrow {\\mathbb {Q}}$ associated to $\\tfrac{1}{f}([f\\hat{h}]-r\\cdot \\textnormal {id})\\in {\\mathbb {Q}}^{r\\times r}$ .",
"Here $[f\\hat{h}]$ denotes the symmetric matrix in ${\\mathbb {Z}}^{r\\times r}$ with $ij$ -th entry given by $[f\\hat{h}_{ij}]$ for $[\\cdot ]$ the rounding “function\" defined in Section REF .",
"The following lemma compares $\\hat{h}$ with the natural extension of $\\hat{h}_k$ to $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}={\\mathbb {R}}^r$ .",
"Lemma 11.1 If $\\Vert \\cdot \\Vert _2$ denotes the euclidean norm on ${\\mathbb {R}}^r$ , then any $x\\in {\\mathbb {R}}^r$ satisfies $\\hat{h}(x)-\\tfrac{2r}{f}\\Vert x\\Vert _2^2\\le \\hat{h}_k(x)\\le \\hat{h}(x).$ To simplify notation we write $V={\\mathbb {R}}^r$ and we denote by $q$ the quadratic form on $V$ which is associated to $(\\delta _{ij})=(\\hat{h}_{ij})-\\tfrac{1}{f}[f\\hat{h}]$ .",
"We take $x\\in V$ and we deduce $\\hat{h}_k(x)=\\hat{h}(x)-q(x)-\\tfrac{r}{f}\\Vert x\\Vert _2^2.$ It holds that $f\\vert {}\\delta _{ij}{}\\vert \\le 1$ and the Cauchy–Schwarz inequality implies that $\\sum _{ij}\\vert {}x_ix_j{}\\vert \\le r\\Vert x\\Vert _2^2$ .",
"Therefore we obtain $\\vert {}q(x){}\\vert \\le \\tfrac{r}{f}\\Vert x\\Vert _2^2$ and then we see that the displayed formula leads to the claimed inequality.",
"This completes the proof of the lemma.",
"If $k$ is sufficiently large then the above lemma implies that the quadratic form $\\hat{h}_k$ is positive definite and is close to $\\hat{h}$ .",
"For what follows we fix an element $k\\in {\\mathbb {Z}}$ such that $\\hat{h}_k$ is positive definite and is close to $\\hat{h}$ , see also the discussions in Section REF ."
],
[
"Terminology.",
"To introduce some terminology, we take $\\sigma \\in {\\mathbb {R}}_{>0}$ and we consider a place $v$ of ${\\mathbb {Q}}$ .",
"The $v$ -adic elliptic logarithm is of local nature, while $\\hat{h}$ and $\\hat{h}_k$ are global height functions.",
"In our global sieve we shall need to measure the “weight\" of the $v$ -adic norm of the $v$ -adic elliptic logarithm inside $\\hat{h}_k$ .",
"For this purpose, we shall work with the set $\\Sigma (v,\\sigma )$ formed by the nonzero points $P\\in E({\\mathbb {Q}})$ whose corresponding solution $(x,y)$ of (REF ) satisfies $\\tfrac{1}{2}\\log \\vert {}x{}\\vert _v\\ge \\tfrac{1}{\\sigma }(\\hat{h}_k(P)-\\mu ).$ Here we write $\\vert {}x{}\\vert _v=p^{-{\\textnormal {ord}}_p(x)}$ if $v$ is a finite place given by the rational prime $p$ , and if $v=\\infty $ then $\\vert {}x{}\\vert _v$ is defined by $\\vert {}x{}\\vert _v=\\vert {}x{}\\vert $ for $\\vert {}\\cdot {}\\vert $ the usual complex absolute value.",
"We note that one can define the set $\\Sigma (v,\\sigma )$ more intrinsically using (Arakelov) intersection theory.",
"However, it is not clear to us if this provides a significant advantage in practice and thus we work with (REF ) using (REF ).",
"For any rational integer $n\\ge r$ , we say that $P\\in E({\\mathbb {Q}})$ is determined modulo torsion by $\\gamma \\in {\\mathbb {Z}}^n$ if there exists $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ such that $P=Q+\\sum \\gamma _i P_i$ .",
"Further we denote by $\\Gamma _E={\\mathbb {Z}}^r$ the lattice inside ${\\mathbb {R}}^r$ given by the image of $E({\\mathbb {Q}})$ inside $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}={\\mathbb {R}}^r$ using the identification via $P_i$ ."
],
[
"Archimedean sieve",
"Building on ideas of Zagier [167], de Weger [162], Stroeker–Tzanakis [141] and Gebel–Pethő–Zimmer [68], we construct in this section our archimedean sieve.",
"We shall use this sieve to improve inter alia the known reduction process at infinity of the elliptic logarithm method, see the discussions in Sections REF and REF .",
"Throughout this Section REF we use the setup of Section REF and we continue the notation introduced above.",
"Further throughout this section we write $\\vert {}\\cdot {}\\vert =\\vert {}\\cdot {}\\vert _\\infty $ ."
],
[
"Real elliptic logarithm.",
"We shall work with the following normalization of the elliptic logarithm on the identity component $E^0({\\mathbb {R}})$ of the real Lie group $E({\\mathbb {R}})$ .",
"First we recall that the uniformization theorem for complex elliptic curves gives a lattice $\\Lambda =\\omega _1{\\mathbb {Z}}+\\omega _2{\\mathbb {Z}}$ inside ${\\mathbb {C}}$ with $\\omega _1\\in {\\mathbb {R}}_{>0}$ and an isomorphism ${\\mathbb {C}}/\\Lambda \\xrightarrow{} E({\\mathbb {C}})$ whose inverse we denote by $\\log : E({\\mathbb {C}})\\xrightarrow{} {\\mathbb {C}}/\\Lambda .$ To describe more explicitly the restriction to $E^0({\\mathbb {R}})$ of the displayed morphism, we write $x=x^{\\prime }-\\tfrac{1}{12}b_2$ with $b_2=a_1^2+4a_2$ and $y=\\tfrac{1}{2}(y^{\\prime }-a_1x-a_3)$ and we transform (REF ) into the Weierstrass equation $y^{\\prime 2}=4x^{\\prime 3}-g_2x^{\\prime }-g_3$ whose complex solutions we identify with the nonzero points of $E({\\mathbb {C}})$ .",
"We may and do assume that $g_i=g_i(\\Lambda )$ is associated to $\\Lambda $ as in [136] and then the isomorphism ${\\mathbb {C}}/\\Lambda \\xrightarrow{} E({\\mathbb {C}})$ is given outside zero by $z\\mapsto (\\wp (z),\\wp ^{\\prime }(z))$ for $\\wp =\\wp (\\Lambda )$ the Weierstrass $\\wp $ -function and $\\wp ^{\\prime }$ its derivative.",
"Hence we deduce for example from [136] that the restriction of $\\log : E({\\mathbb {C}})\\xrightarrow{} {\\mathbb {C}}/\\Lambda $ to $E^0({\\mathbb {R}})$ is of the form $E^0({\\mathbb {R}})\\xrightarrow{} {\\mathbb {R}}/(\\omega _1{\\mathbb {Z}})$ , which in turn induces a bijective map $\\log : E^0({\\mathbb {R}})\\rightarrow \\lbrace z\\in {\\mathbb {R}}\\,;\\,0\\le z<\\omega _1\\rbrace .$ Explicitly if $P\\in E^0({\\mathbb {R}})$ corresponds to a real solution $(x,y)$ of (REF ) then it holds that $\\log (P)=\\tfrac{y^{\\prime }}{\\vert {}y^{\\prime }{}\\vert } \\int ^{x^{\\prime }}_{\\infty }\\tfrac{dz}{f(z)^{1/2}}$ mod $(\\omega _1{\\mathbb {Z}})$ for $f(z)=4z^3-g_2z-g_3$ .",
"Here one can compute the real number $\\log (P)$ up to any required precision, see for example Zagier [167].",
"Further, we denote by $e_t$ the exponent of the finite group $E({\\mathbb {Q}})_{\\textnormal {tor}}$ and we define $m=\\textnormal {lcm}(e_t,\\iota )$ for $\\iota $ the index of $E^0({\\mathbb {R}})$ inside $E({\\mathbb {R}})$ .",
"It holds that $\\iota \\in \\lbrace 1,2\\rbrace $ , since $E({\\mathbb {R}})$ is either connected or isomorphic to $E^0({\\mathbb {R}})\\times ({\\mathbb {Z}}/2{\\mathbb {Z}})$ .",
"We recall that the points $P_1,\\cdots ,P_r$ form a basis of the free part of $E({\\mathbb {Q}})$ .",
"Now any $P=Q+\\sum n_i P_i$ in $E({\\mathbb {Q}})$ , with $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ and $n_i\\in {\\mathbb {Z}}$ , satisfies $mP=\\sum n_i (mP_i)\\in E^0({\\mathbb {R}})$ .",
"Next we take $\\kappa \\in {\\mathbb {Z}}_{\\ge 1}$ and we define $x_0(\\kappa )=(\\kappa +1)(\\vert {}b_2{}\\vert /12+\\max \\vert {}\\xi _i{}\\vert )$ for $\\lbrace \\xi _i\\rbrace $ the set of roots of $f(z)=4z^3-g_2z-g_3$ .",
"If $P\\in E^0({\\mathbb {R}})$ corresponds to a solution $(x,y)$ of (REF ), then the next lemma allows to control $\\log (P)$ in terms of $\\vert {}x{}\\vert $ .",
"Lemma 11.2 The following statements hold.",
"(i) Suppose that $P\\in E({\\mathbb {R}})$ corresponds to a real solution $(x,y)$ of (REF ) with $\\vert {}x{}\\vert \\ge x_0(\\kappa )$ .",
"Then $P$ lies in $E^{0}({\\mathbb {R}})$ and there is $\\epsilon \\in \\lbrace 0,-1\\rbrace $ such that any $n\\in {\\mathbb {Z}}$ satisfies $\\vert {}n\\log (P)+n\\epsilon \\omega _1{}\\vert \\le \\vert {}n{}\\vert \\bigl (1+\\tfrac{1}{\\kappa }\\bigl )^2\\vert {}x{}\\vert ^{-1/2}.$ (ii) If $P=Q+\\sum n_i P_i$ lies in $E^0({\\mathbb {R}})$ with $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ and $n_i\\in {\\mathbb {Z}}$ , then there exists $l\\in {\\mathbb {Z}}$ with $\\vert {}l{}\\vert \\le m+ \\sum \\vert {}n_i{}\\vert $ such that $m\\log (P)=\\sum n_i\\log (mP_i)+l\\omega _1$ .",
"We first prove assertion (i).",
"Our assumption implies that $x^{\\prime }=x+\\tfrac{1}{12}b_2$ is positive and that $x^{\\prime }$ strictly exceeds the largest real root of $f(z)=4z^3-g_2z-g_3$ .",
"Hence we conclude that $P\\in E^0({\\mathbb {R}})$ .",
"To verify the second statement of (i) we observe that any $z\\in {\\mathbb {R}}$ with $z\\ge (\\kappa +1)\\max \\vert {}\\xi _i{}\\vert $ satisfies $f(z)\\ge 4\\bigl (\\tfrac{\\kappa }{\\kappa +1}\\bigl )^3z^3$ .",
"It follows that $\\vert {}\\int ^{x^{\\prime }}_\\infty \\tfrac{dz}{f(z)^{1/2}}{}\\vert ^2$ is at most $\\bigl (\\tfrac{\\kappa +1}{\\kappa }\\bigl )^{3}\\vert {}x^{\\prime }{}\\vert ^{-1}$ , since our assumption gives $\\vert {}x^{\\prime }{}\\vert \\ge (\\kappa +1)\\max \\vert {}\\xi _i{}\\vert $ .",
"Furthermore our assumption provides that $\\vert {}x^{\\prime }{}\\vert \\ge \\tfrac{\\kappa }{\\kappa +1}\\vert {}x{}\\vert $ , and then we see that there exists $\\epsilon \\in \\lbrace 0,-1\\rbrace $ such that the claimed inequality holds for $n=1$ and thus for all $n\\in {\\mathbb {Z}}$ .",
"It remains to show (ii).",
"The points $mP_i$ are all in $E^0({\\mathbb {R}})$ since $\\iota $ divides $m$ , and the point $P$ is in $E^0({\\mathbb {R}})$ by assumption.",
"Thus, on exploiting that the real elliptic logarithm is induced by a group isomorphism $E^0({\\mathbb {R}})\\xrightarrow{} {\\mathbb {R}}/(\\omega _1{\\mathbb {Z}})$ , we find $l^{\\prime },l^{\\prime \\prime }\\in {\\mathbb {Z}}$ with $m\\log (P)=\\log (mP)+l^{\\prime }\\omega _1$ and $\\log (mP)=\\sum n_i\\log (mP_i)+l^{\\prime \\prime }\\omega _1$ .",
"Then on using that $\\log (P)$ , $\\log (mP)$ and $\\log (mP_i)$ are in the interval $[0,\\omega _1[$ , we deduce that $\\vert {}l^{\\prime }{}\\vert \\le m$ and $\\vert {}l^{\\prime \\prime }{}\\vert \\le \\sum \\vert {}n_i{}\\vert $ .",
"Hence the integer $l=l^{\\prime }+l^{\\prime \\prime }$ has the desired property.",
"This completes the proof of the lemma."
],
[
"Construction of $\\Gamma $ and {{formula:132e3ad3-eadc-4e1a-8ec3-279eeb293505}} .",
"Let $\\sigma >0$ be a real number, let $\\mu $ be as in (REF ) and write $\\Sigma $ for the set $\\Sigma (\\infty ,\\sigma )$ defined in (REF ).",
"Suppose that $M^{\\prime },M\\in {\\mathbb {Z}}$ with $\\mu \\le M^{\\prime }<M$ and let $\\kappa \\in {\\mathbb {Z}}_{\\ge 1}$ .",
"We would like to construct a lattice $\\Gamma \\subset {\\mathbb {Z}}^{r+1}$ and an ellipsoid $\\mathcal {E}\\subset {\\mathbb {R}}^{r+1}$ such that any $P\\in \\Sigma $ with $M^{\\prime }<\\hat{h}_k(P)\\le M$ is determined modulo torsion by a point in $\\Gamma \\cap \\mathcal {E}$ .",
"The following construction depends on a suitable choice of a parameter $c\\in {\\mathbb {Z}}_{\\ge 1}$ , which we shall explain below (REF ).",
"We write $\\alpha _i=\\log (mP_i)$ for $i\\in \\lbrace 1,\\cdots ,r\\rbrace $ and we denote by $\\Gamma \\subset {\\mathbb {Z}}^{r+1}$ the lattice formed by the elements $\\gamma \\in {\\mathbb {Z}}^{r+1}$ such that $\\gamma _{r+1}=l[c\\omega _1]+\\sum \\gamma _i [c\\alpha _i]$ for some $l\\in {\\mathbb {Z}}$ .",
"Next we choose a positive number $\\delta \\in {\\mathbb {Q}}$ as explained in the discussion surrounding (REF ) and we denote by $q$ the positive definite quadratic form on ${\\mathbb {R}}^{r+1}$ which is given by $q(z)=\\hat{h}_k(z_1,\\cdots ,z_r)+(M/\\delta ^2)z_{r+1}^2$ for any $z\\in {\\mathbb {R}}^{r+1}$ .",
"Now we define the ellipsoid $\\mathcal {E}=\\lbrace z\\in \\mathbb {R}^{r+1}\\,;\\,q(z)\\le 2M\\rbrace .$ Let $x_0=x_0(\\kappa )$ be as in (REF ) and let $\\Sigma (x_0)$ be the set of points $P$ in $\\Sigma $ with $\\vert {}x{}\\vert >x_0$ , where $(x,y)$ is the solution of (REF ) corresponding to $P$ .",
"We obtain the following lemma.",
"Lemma 11.3 Suppose that $P\\in \\Sigma (x_0)$ satisfies $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"Then the point $P$ is determined modulo torsion by some lattice point $\\gamma $ in $\\Gamma \\cap \\mathcal {E}$ .",
"Let $(x,y)$ be the solution of (REF ) corresponding to $P$ , and write $P=Q+\\sum n_i P_i$ with $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ and $n_i\\in {\\mathbb {Z}}$ .",
"It holds that $\\vert {}x{}\\vert \\ge x_0$ since $P\\in \\Sigma (x_0)$ and therefore Lemma REF (i) shows that $P\\in E^0({\\mathbb {R}})$ .",
"Hence we see that Lemma REF (ii) gives $l_0\\in {\\mathbb {Z}}$ with $\\vert {}l_0{}\\vert \\le m+\\sum \\vert {}n_i{}\\vert $ such that $m\\log (P)=\\sum n_i\\alpha _i+l_0\\omega _1$ .",
"On inserting this into the inequality in Lemma REF (i) with $n=m$ , we obtain $l\\in {\\mathbb {Z}}$ with $\\vert {}l{}\\vert \\le 2m+\\sum \\vert {}n_i{}\\vert $ such that $\\vert {}\\sum n_i\\alpha _i+l\\omega _1{}\\vert \\le m\\bigl (1+\\tfrac{1}{\\kappa }\\bigl )^2e^{-\\tfrac{1}{\\sigma }(\\hat{h}_k(P)-\\mu )}.$ Here we used our assumption $P\\in \\Sigma (x_0)$ , which provides that $\\tfrac{1}{2}\\log \\vert {}x{}\\vert \\ge \\tfrac{1}{\\sigma }(\\hat{h}_k(P)-\\mu )$ .",
"Next we define $d=l[c\\omega _1]+\\sum n_i [c\\alpha _i]$ and we observe that $\\gamma =((n_i),d)\\in {\\mathbb {Z}}^{r+1}$ lies in our lattice $\\Gamma $ .",
"To show that $\\gamma $ lies in addition in the ellipsoid $\\mathcal {E}$ , we use the mean inequality and linear algebra in order to deduce that $\\lambda _k(\\sum \\vert {}n_i{}\\vert )^2\\le r\\hat{h}_k(P)$ for $\\lambda _k\\in {\\mathbb {Q}}$ the smallest eigenvalue of the positive definite quadratic form $\\hat{h}_k$ on $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {Q}}$ .",
"Then we see that (REF ) together with our assumption $M^{\\prime }<\\hat{h}_k(P)\\le M$ implies that $\\vert {}d{}\\vert \\le \\delta $ .",
"Here $\\delta \\in {\\mathbb {Q}}$ is chosen such that $\\delta $ has “small\" height in the sense of Section REF and such that $\\delta \\ge \\delta _1+\\delta _2$ for $\\delta _1=2m+2\\bigl (r\\tfrac{M}{\\lambda _k}\\bigl )^{1/2} \\ \\ \\ \\textnormal { and } \\ \\ \\ \\delta _2=cm\\bigl (1+\\tfrac{1}{\\kappa }\\bigl )^2e^{-\\tfrac{1}{\\sigma }(M^{\\prime }-\\mu )}.$ On using again that $\\hat{h}_k(P)\\le M$ we obtain $q(\\gamma )\\le M+(M/\\delta ^2) d^2$ .",
"This together with $\\vert {}d{}\\vert \\le \\delta $ implies that $\\gamma \\in \\mathcal {E}$ and thus $P$ is determined modulo torsion by $\\gamma \\in \\Gamma \\cap \\mathcal {E}$ .",
"This lemma provides a sieve for the points $P$ in $\\Sigma (x_0)$ with $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"In the following paragraph we discuss the strength of the sieve depending on various parameters."
],
[
"Strength of the sieve.",
"To make the sieve as efficient as possible, we would like choose the parameter $c$ such that the intersection $\\Gamma \\cap \\mathcal {E}$ does not contain many points.",
"In the generic case the cardinality of $\\Gamma \\cap \\mathcal {E}$ can be approximated (for large $M$ ) by the euclidean volume of the ellipsoid $\\mathcal {E}_\\psi =\\lbrace z\\in {\\mathbb {R}}^{r+1}\\,;\\,q_\\psi (z)\\le 2M\\rbrace $ inside ${\\mathbb {R}}^{r+1}$ .",
"Here $q_\\psi $ denotes the positive definite quadratic form obtained by pulling back $q$ along the linear transformation $\\psi $ of ${\\mathbb {R}}^{r+1}$ which satisfies $\\psi ({\\mathbb {Z}}^{r+1})=\\Gamma $ and which is explicitly given by $ \\begin{pmatrix}1 & & & 0 \\\\& \\ddots & & \\vdots \\\\& & 1 & 0 \\\\[c\\alpha _1] & \\cdots & [c\\alpha _r] & [c\\omega _{1}]\\end{pmatrix}.$ To compute the euclidean volume $\\operatorname{\\textnormal {vol}}(\\mathcal {E}_\\psi )$ of $\\mathcal {E}_\\psi $ , we let $R_{E}=2^r\\det (\\hat{h}_{ij})$ be the regulator of $E({\\mathbb {Q}})$ normalized as in [136] and we denote by $V_{r+1}$ the euclidean volume of the unit ball in ${\\mathbb {R}}^{r+1}$ .",
"Then the volume $\\operatorname{\\textnormal {vol}}(\\mathcal {E}_\\psi )$ is approximately $u\\cdot M^{r/2}\\tfrac{(\\delta _1+\\delta _2)}{c}, \\ \\ \\ u=\\tfrac{2^{r+1/2}V_{r+1}}{\\omega _1R_{E}^{1/2}}.$ We note that $\\delta _2/c$ does not depend on $c$ .",
"Hence in view of (REF ) we choose $c$ such that $u\\cdot M^{r/2}\\tfrac{\\delta _1}{c}$ is smaller than $u\\cdot M^{r/2}\\tfrac{\\delta _2}{c}$ .",
"For example $c$ should always dominate $M^{(r+1)/2}$ if $M$ is large.",
"We next discuss the dependence of the sieve on $M^{\\prime }$ and $M$ .",
"For some large $M$ , we choose $c$ as indicated above and we assume for a moment that $M^{\\prime }$ dominates $\\frac{\\sigma r}{2}\\log M.$ Then it follows that $M^{r/2}\\tfrac{\\delta _2}{c}$ is close to zero and hence (REF ) implies that the volume of $\\mathcal {E}_\\psi $ is small.",
"In the generic case this assures that $\\Gamma \\cap \\mathcal {E}$ has very little points or is even trivial.",
"In particular, we see that the archimedean sieve is very efficient for such $M^{\\prime }$ and $M$ .",
"On the other hand, for small $M^{\\prime }$ our sieve is not that efficient in view of (REF ).",
"Remark 11.4 One can replace $\\mathcal {E}$ by the more balanced ellipsoid $\\mathcal {E}^*\\subset {\\mathbb {R}}^{r+1}$ of the form $\\mathcal {E}^*=\\lbrace z\\in {\\mathbb {R}}^{r+1}\\,;\\,q^*(z)\\le M\\rbrace $ for $q^*(z)=\\tfrac{r}{r+1}\\hat{h}_k(z_1,\\cdots ,z_r)+\\tfrac{1}{r+1}(M/\\delta ^2) z_{r+1}^2$ .",
"Indeed this follows by observing that $\\gamma $ appearing in the proof of Lemma REF satisfies $q^*(\\gamma )\\le M$ ."
],
[
"Archimedean sieve.",
"In the following sieve, we use the version (FP) of the Fincke–Pohst algorithm described in Section REF in order to determine all points in $\\Gamma \\cap \\mathcal {E}$ .",
"Algorithm 11.5 (Archimedean sieve) The inputs are $\\kappa \\in {\\mathbb {Z}}_{\\ge 1}$ and $M^{\\prime },M\\in {\\mathbb {Z}}$ with $\\mu \\le M^{\\prime }< M$ .",
"The output is the set of points $P\\in \\Sigma (x_0)$ with $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"(i) First choose the parameter $c\\in {\\mathbb {Z}}_{\\ge 1}$ as explained in the discussion surrounding (REF ).",
"Then compute the lattice $\\Gamma \\subset {\\mathbb {Z}}^{r+1}$ , by determining the period $\\omega _1$ and the real elliptic logarithms $\\alpha _i=\\log (mP_i)$ up to the required precision for all $i\\in \\lbrace 1,\\cdots ,r\\rbrace $ .",
"(ii) Determine $\\Gamma \\cap \\mathcal {E}$ by using the version of the Fincke–Pohst algorithm in (FP).",
"(iii) For each lattice point $\\gamma $ in $\\Gamma \\cap \\mathcal {E}$ and for each torsion point $Q$ in $E({\\mathbb {Q}})_{\\textnormal {tor}}$ , output the point $P=Q+\\sum \\gamma _i P_i$ if $M^{\\prime }<\\hat{h}_k(P)\\le M$ and if $P$ is in $\\Sigma (x_0)$ ."
],
[
"Correctness.",
"Suppose that $P\\in \\Sigma (x_0)$ satisfies $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"Lemma REF gives that $P$ is determined modulo torsion by some $\\gamma \\in \\Gamma \\cap \\mathcal {E}$ .",
"In other words, there is $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ such that $P=Q+\\sum \\gamma _i P_i$ and hence step (iii) produces our point $P$ as desired."
],
[
"Complexity.",
"We now discuss aspects of Algorithm REF which significantly influence the running time.",
"In step (i) the running time of the computation of the lattice $\\Gamma =\\psi ({\\mathbb {Z}}^{r+1})$ crucially depends on the size of $c$ .",
"For example if $c$ is approximately $M^{(r+1)/2}$ then we need to know the real logarithms $\\omega _1$ and $\\log (mP_i)$ up to a number of decimal digits which is approximately $\\tfrac{r+1}{2}\\log _{10} M$ , where $\\log _b z=(\\log z)/\\log b$ for $z,b\\in {\\mathbb {R}}_{>0}$ .",
"We shall apply the algorithm with huge parameters $M$ .",
"Therefore we need to compute $(r+1)$ real elliptic logarithms up to a very high precision and this can take a long time.",
"Step (ii) is essentially always fast in practice.",
"The reason is that the involved Mordell–Weil rank $r$ of $E({\\mathbb {Q}})$ is usually not that large and hence the application of (FP) with the lattice $\\Gamma $ of rank $r+1$ is fast.",
"Finally step (iii) needs to compute in particular the coordinates of certain points in $E({\\mathbb {Q}})$ and this can take some time if $\\hat{h}(P)$ and $r$ are not small."
],
[
"Comparison.",
"There are important differences between our approach and the known approach.",
"In particular we work with the Néron–Tate height $\\hat{h}$ , while all other authors use the inequality $\\lambda \\Vert \\cdot \\Vert _\\infty ^2\\le \\hat{h}(\\cdot )$ to work with the norm $\\Vert \\cdot \\Vert _\\infty $ .",
"Also we actually determine the intersection $\\Gamma \\cap \\mathcal {E}$ , while the known approach computes a lower bound for the length of the shortest nonzero vector in $\\Gamma $ in order to rule out non-trivial points in $\\Gamma \\cap \\mathcal {E}$ .",
"Other, more technical, differences are the following: The parameter $\\kappa $ allows us (up to a certain extent) to adapt the strength of the sieve to the given situation, and the construction of our ellipsoid $\\mathcal {E}^*$ involving the weights $\\tfrac{r}{r+1}$ and $\\tfrac{1}{r+1}$ is more balanced in particular for large $r$ .",
"In Sections REF and REF , we shall further compare the two approaches and we shall explain in detail the improvements provided by our new ideas."
],
[
"Non-archimedean sieve",
"Building on ideas of Smart [137], Pethő–Zimmer–Gebel–Herrmann [126] and Tzanakis [159], we construct in this section the non-archimedean sieve.",
"We shall use this sieve to improve inter alia the known reduction process of the elliptic logarithm method at non-archimedean primes, see the discussions in Sections REF and REF .",
"Throughout this Section REF we work with the setup of Section REF and we continue the notation introduced above.",
"Further we fix $p$ in $S$ and we assume that the Weierstrass model (REF ) of our given elliptic curve $E$ is minimal at $p$ .",
"To simplify the notation of this section, we write $v(\\cdot )={\\textnormal {ord}}_p(\\cdot )$ and $\\vert {}\\cdot {}\\vert =\\vert {}\\cdot {}\\vert _p$ with $\\vert {}x{}\\vert _p=p^{-v(x)}$ for $x\\in {\\mathbb {Q}}_p$ ."
],
[
"The $p$ -adic elliptic logarithm.",
"Let $E_1({\\mathbb {Q}}_p)$ be the subgroup of $E({\\mathbb {Q}}_p)$ formed by the points $P$ in $E({\\mathbb {Q}}_p)$ with $\\pi (P)=0$ for $\\pi :E({\\mathbb {Q}}_p)\\rightarrow E(\\mathbb {F}_p)$ the reduction map.",
"Here $E(\\mathbb {F}_p)$ denotes the set of $\\mathbb {F}_p$ -points of the special fiber of the projective closureHere we mean $\\textnormal {Proj}\\bigl ({\\mathbb {Z}}_p[x,y,z]/(f)\\bigl )$ for $f=y^2z+a_1xyz+a_3yz^2-(x^3+a_2x^2z+a_4xz^2+a_6z^3)$ .",
"of $(\\ref {eq:weieq})$ inside $\\mathbb {P}^2_{{\\mathbb {Z}}_p}$ .",
"Let $\\hat{E}$ be the formal group over ${\\mathbb {Z}}_p$ associated to $E_{{\\mathbb {Q}}_p}$ .",
"There is an isomorphism $E_1({\\mathbb {Q}}_p)\\xrightarrow{} \\hat{E}(p{\\mathbb {Z}}_p)$ of abelian groups, which is given away from zero by $(x,y)\\mapsto -\\tfrac{x}{y}$ .",
"Composing this isomorphism with the formal logarithm of $\\hat{E}$ induces a morphism $\\log :E_1({\\mathbb {Q}}_p)\\rightarrow \\mathbb {G}_a({\\mathbb {Q}}_p)$ of abelian groups.",
"We call the displayed morphism the $p$ -adic elliptic logarithm.",
"Explicitly if $P\\in E_1({\\mathbb {Q}}_p)$ is nonzero and corresponds to the solution $(x,y)$ of (REF ), then it holds that $\\log (P)=z+\\sum _{n\\ge 2} \\tfrac{b_{n}}{n} z^{n}$ with $z=-\\tfrac{x}{y}$ and $b_n\\in {\\mathbb {Z}}_p$ .",
"A priori the $p$ -adic elliptic logarithm is only defined on the subgroup $E_1({\\mathbb {Q}}_p)$ of $E({\\mathbb {Q}}_p)$ .",
"One can somehow circumvent this problem by multiplying the points in $E({\\mathbb {Q}}_p)$ with a suitable integer.",
"To construct such an integer, let $E_{\\textnormal {ns}}(\\mathbb {F}_p)$ be the group formed by the nonsingular points in $E(\\mathbb {F}_p)$ and consider the subgroup $E_0({\\mathbb {Q}}_p)=\\pi ^{-1}(E_{\\textnormal {ns}}(\\mathbb {F}_p))$ of $E({\\mathbb {Q}}_p)$ .",
"We denote by $\\iota $ the index of $E_0({\\mathbb {Q}}_p)$ in $E({\\mathbb {Q}}_p)$ , and we write $e_t$ and $e_{ns}$ for the exponents of the finite groups $E({\\mathbb {Q}})_{\\textnormal {tor}}$ and $E_{\\textnormal {ns}}(\\mathbb {F}_p)$ respectively.",
"The short exact sequence $0\\rightarrow E_1({\\mathbb {Q}}_p)\\rightarrow E_0({\\mathbb {Q}}_p)\\overset{\\pi }{\\rightarrow } E_{\\textnormal {ns}}(\\mathbb {F}_p)\\rightarrow 0$ of abelian groups shows that $(\\iota e_{ns})P\\in E_1({\\mathbb {Q}}_p)$ for all $P\\in E({\\mathbb {Q}}_p)$ .",
"We now define $m=\\textnormal {lcm}\\bigl (e_t,\\iota e_{ns}\\bigl ).$ Recall that $P_1,\\cdots ,P_r$ denotes our given basis of the free part of $E({\\mathbb {Q}})$ .",
"Any $P=Q+\\sum n_i P_i$ in $E({\\mathbb {Q}})$ , with $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ and $n_i\\in {\\mathbb {Z}}$ , satisfies $mP=\\sum n_i (mP_i)\\in E_1({\\mathbb {Q}}_p)$ .",
"The case distinction in the following lemma takes into account that in general the formal logarithm of $\\hat{E}$ is not necessarily an isomorphism of formal groups over the given base.",
"Lemma 11.6 Let $P\\in E({\\mathbb {Q}}_p)$ be nonzero, and suppose that $(x,y)$ is the solution of (REF ) corresponding to $P$ .",
"Then the following two statements hold.",
"(i) Assume that $p\\ge 3$ .",
"If $P\\notin E_1({\\mathbb {Q}}_p)$ with $mP\\ne 0$ then $\\vert {}\\log (mP){}\\vert ^2<\\vert {}x{}\\vert ^{-1}$ , and if $P\\in E_1({\\mathbb {Q}}_p)$ then $\\vert {}\\log (nP){}\\vert ^2=\\vert {}n{}\\vert ^2\\vert {}x{}\\vert ^{-1}$ for all $n\\in {\\mathbb {Z}}$ .",
"(ii) If $p=2$ and $v(x)<-2$ , then any $n\\in {\\mathbb {Z}}$ satisfies $\\vert {}\\log (nP){}\\vert ^2=\\vert {}n{}\\vert ^2\\vert {}x{}\\vert ^{-1}$ .",
"Our proof given below relies on the classical result that the formal logarithm is compatible with the valuation $v$ in the following sense: For any $l\\in {\\mathbb {Z}}$ with $l>v(p)/(p-1)$ , the restriction of the formal logarithm of $\\hat{E}$ induces an isomorphism $\\hat{E}((p{\\mathbb {Z}}_p)^l)\\cong (p{\\mathbb {Z}}_p)^l$ of abelian groups.",
"Further, we shall use below that if $P\\in E_1({\\mathbb {Q}}_p)$ then it holds that $3v(x)=2v(y)$ , thus $v(x)$ is even and the number $z=-x/y$ satisfies $2v(z)=-v(x)$ .",
"If $mP\\ne 0$ then we denote by $(x_m,y_m)$ the solution of (REF ) corresponding to $mP$ .",
"To prove (i) we may and do assume that $p\\ge 3$ .",
"Then $p$ satisfies $1>v(p)/(p-1)$ and hence the isomorphism in (REF ) exists for all $l\\ge 1$ .",
"This implies that $2v(\\log (mP))=-v(x_m)$ since $mP\\in E_1({\\mathbb {Q}}_p)$ is nonzero by assumption.",
"If $P$ is not in $E_1({\\mathbb {Q}}_p)$ then $v(x)\\ge 0$ , and $mP\\in E_1({\\mathbb {Q}}_p)$ thus shows that $v(x)\\ge 0> v(x_m)$ .",
"This together with $2v(\\log (mP))=-v(x_m)$ proves the claimed inequality if $P$ is not in $E_1({\\mathbb {Q}}_p)$ .",
"Suppose now that $P\\in E_1({\\mathbb {Q}}_p)$ .",
"Then we obtain that $n\\log (P)=\\log (nP)$ for all $n\\in {\\mathbb {Z}}$ since the formal logarithm is a morphism of abelian groups, and the isomorphisms in (REF ) provide that $2v(\\log (P))=-v(x)$ .",
"On combining these two equalities, we deduce the second statement of (i).",
"To show (ii) we may and do assume that $p=2$ and $v(x)<-2$ .",
"The latter assumption implies that $P\\in E_1({\\mathbb {Q}}_2)$ and $v(x)\\le -4$ .",
"We deduce that $v(z)\\ge 2$ and hence $z$ lies in $(2{\\mathbb {Z}}_2)^2$ .",
"Further, the isomorphism in (REF ) exists for all $l\\ge 2$ since $2>v(2)/(2-1)$ .",
"Thus we obtain that $2v(\\log (P))=-v(x)$ and then the equality $n\\log (P)=\\log (nP)$ , which holds for all $n\\in {\\mathbb {Z}}$ since $P\\in E_1({\\mathbb {Q}}_2)$ , implies (ii).",
"This completes the proof of the lemma.",
"We remark that the assumptions $P\\in E_1({\\mathbb {Q}}_p)$ and $v(x)<-2$ , in (i) and (ii) respectively, assure in particular that the point $P$ has infinite order in $E({\\mathbb {Q}}_p)$ ."
],
[
"Construction of $\\Gamma $ and {{formula:93b22614-85e2-4eaa-b888-a2189cb32400}} .",
"Let $\\sigma > 0$ be a real number and write $\\Sigma $ for the set $\\Sigma (v,\\sigma )$ defined in (REF ).",
"Suppose that we are given $M^{\\prime },M\\in {\\mathbb {Z}}$ with $\\mu \\le M^{\\prime }< M$ for $\\mu $ as in (REF ).",
"We would like to find a lattice $\\Gamma \\subset {\\mathbb {Z}}^r$ such that any $P\\in \\Sigma $ with $M^{\\prime }<\\hat{h}_k(P)\\le M$ is determined modulo torsion by some point in $\\Gamma \\cap \\mathcal {E}$ .",
"Here $\\mathcal {E}\\subset {\\mathbb {R}}^{r}$ is the ellipsoid $\\mathcal {E}=\\lbrace z\\in {\\mathbb {R}}^{r}\\,;\\,\\hat{h}_k(z)\\le M\\rbrace .$ We identify $\\alpha \\in {\\mathbb {Z}}_p$ with the corresponding element $(\\alpha ^{(1)},\\alpha ^{(2)},\\cdots )$ of the inverse limit $\\lim {\\mathbb {Z}}/(p^n{\\mathbb {Z}})$ and we set $\\alpha _i=\\log (mP_i)$ for each $i\\in \\lbrace 1,\\cdots ,r\\rbrace $ .",
"If $p\\ge 3$ then Lemma REF implies that $\\alpha _i\\in {\\mathbb {Z}}_p$ .",
"To deal with the general case, we choose $i^*\\in \\lbrace 1,\\cdots ,r\\rbrace $ with $v(\\alpha _{i^*})= \\min v(\\alpha _i)$ and then $\\beta _i=\\alpha _i/p^{v(\\alpha _{i^*})}$ lies in ${\\mathbb {Z}}_p$ .",
"Now we denote by $\\Gamma \\subset {\\mathbb {Z}}^r$ the lattice formed by the elements $\\gamma \\in {\\mathbb {Z}}^r$ with $\\sum \\gamma _i\\beta _i^{(c)}=0$ in ${\\mathbb {Z}}/(p^c{\\mathbb {Z}})$ , where $c\\in {\\mathbb {Z}}_{\\ge 0}$ will be chosen in (REF ).",
"Further we define the set $\\Sigma ^*$ by setting $\\Sigma ^*=\\Sigma $ if $p\\ge 3$ and $\\Sigma ^*=\\Sigma (4)$ if $p=2$ .",
"Here $\\Sigma (4)$ denotes the set of points $P$ in $\\Sigma $ with $\\vert {}x{}\\vert >4$ , where $(x,y)$ is the solution of (REF ) corresponding to $P$ .",
"We obtain the following lemma.",
"Lemma 11.7 Suppose that $P\\in \\Sigma ^*$ satisfies $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"Then the point $P$ is determined modulo torsion by some lattice point in $\\Gamma \\cap \\mathcal {E}$ .",
"Let $(x,y)$ be the solution of (REF ) which corresponds to $P$ , and write $P=Q+\\sum n_i P_i$ with $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ and $n_i\\in {\\mathbb {Z}}$ .",
"On using that $P$ is in $\\Sigma $ and that $\\mu \\le M^{\\prime }<\\hat{h}_k(P)$ , we deduce that $\\log \\vert {}x{}\\vert >0$ and thus our point $P$ lies in fact in $E_1({\\mathbb {Q}}_p)$ .",
"Further, if $p=2$ then our additional assumption $P\\in \\Sigma (4)$ provides that $v(x)<-2$ .",
"Hence on recalling that $P\\in \\Sigma $ , we see that Lemma REF leads to the inequality $\\vert {}\\sum n_i\\alpha _i{}\\vert \\le \\vert {}m{}\\vert e^{-\\tfrac{1}{\\sigma }(\\hat{h}_k(P)-\\mu )}.$ Here we used that $v(\\log (mP))=v(\\sum n_i \\alpha _i)$ , which in turn follows from $mP=\\sum n_i(mP_i)$ and $mP_i\\in E_1({\\mathbb {Q}}_p)$ .",
"The displayed inequality together with $M^{\\prime }<\\hat{h}_k(P)$ shows that $v(\\sum n_i\\beta _i)\\ge c$ , where $c$ is the smallest element of ${\\mathbb {Z}}_{\\ge 0}$ which exceeds $v(m)-v(\\alpha _{i^*})+\\tfrac{1}{\\sigma \\log p}(M^{\\prime }-\\mu ).$ It follows that $\\sum n_i \\beta _i^{(c)}=0$ in ${\\mathbb {Z}}/(p^c{\\mathbb {Z}})$ and therefore $\\gamma =(n_i)$ lies in $\\Gamma $ .",
"Furthermore, our assumption $\\hat{h}_k(P)\\le M$ assures that $\\gamma $ lies in $\\mathcal {E}$ and hence $P$ is determined modulo torsion by the lattice point $\\gamma \\in \\Gamma \\cap \\mathcal {E}$ .",
"This completes the proof of the lemma.",
"The above lemma provides a sieve for the points $P$ in $\\Sigma ^*$ with $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"The discussion of the strength of this sieve, depending on the parameters $M^{\\prime }$ , $M$ and $p^c$ , is analogous to the discussion of the strength of the archimedean sieve in Section REF .",
"However there are some minor differences.",
"For example, in the non-archimedean case we can work entirely in dimension $r$ and the parameter $p^c$ is uniquely determined by (REF ); note that $p^c$ plays here the role of the parameter $c$ in the archimedean sieve."
],
[
"Non-archimedean sieve.",
"The following sieve uses the version (FP) of the Fincke–Pohst algorithm described in Section REF in order to determine all points in $\\Gamma \\cap \\mathcal {E}$ .",
"Algorithm 11.8 (Non-archimedean sieve) The inputs are $M^{\\prime },M\\in {\\mathbb {Z}}$ with $\\mu \\le M^{\\prime }< M$ .",
"The output is the set of points $P$ in $\\Sigma ^*$ with $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"(i) To find the number $m$ , determine $e_{ns}$ , $e_t$ and $\\iota $ .",
"(ii) Determine the lattice $\\Gamma \\subset {\\mathbb {Z}}^{r}$ by computing the $p$ -adic elliptic logarithms $\\log (mP_i)$ up to the required precision for all $i\\in \\lbrace 1,\\cdots ,r\\rbrace $ .",
"(iii) Use (FP) to find all points in $\\Gamma \\cap \\mathcal {E}$ .",
"(iv) For each $\\gamma $ in $\\Gamma \\cap \\mathcal {E}$ and for each torsion point $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ , output the point $P=Q+\\sum \\gamma _i P_i$ if $M^{\\prime }<\\hat{h}_k(P)\\le M$ and if $P\\in \\Sigma ^*$ ."
],
[
"Correctness.",
"We take a point $P\\in \\Sigma ^*$ which satisfies $M^{\\prime }<\\hat{h}_k(P)\\le M$ and we write $P=Q+\\sum n_i P_i$ with $n_i\\in {\\mathbb {Z}}$ and $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ .",
"Lemma REF gives that $\\gamma =(n_i)$ lies in $\\Gamma \\cap \\mathcal {E}$ and hence we see that step (iv) produces our point $P$ as desired."
],
[
"Complexity.",
"We now discuss the influence of each step on the running time in practice.",
"In step (i) standard results and algorithms allow to quickly compute the numbers $e_{ns},e_t$ and $\\iota $ .",
"In fact the computation of (a suitable) $m$ is very fast in practice, even if $p$ is relatively large.",
"Step (ii) needs to compute $r$ distinct $p$ -adic elliptic logarithms up to a number of $p$ -adic digits which is approximately $c+v(\\alpha _{i^*})$ .",
"The efficiency of this computation crucially depends on the size of $c+v(\\alpha _{i^*})$ , which in turn depends in particular on the lower bound $M^{\\prime }$ .",
"If the number $M^{\\prime }$ is huge, then this step (ii) can become very slow in practice.",
"Finally we mention that a complexity analysis of steps (iii) and (iv) is contained in the complexity discussions of the analogous steps of the archimedean sieve in Algorithm REF ."
],
[
"Comparison.",
"Similarly as in the archimedean case, there are important differences between our approach and the known method.",
"For instance, we work with the ellipsoid $\\mathcal {E}$ arising from the Néron–Tate height $\\hat{h}$ and we actually determine all points in the intersection $\\Gamma \\cap \\mathcal {E}$ .",
"We refer to Sections REF and REF for a comparison of the two approaches and for a detailed discussion of the improvements provided by our approach."
],
[
"Height-logarithm sieve",
"We work with the setup of Section REF .",
"The goal of this section is to construct a sieve which allows to efficiently determine the set of $S$ -integral points in any given finite subset of $E({\\mathbb {Q}})$ .",
"The sieve exploits that the global Néron–Tate height is essentially determined by the various local elliptic logarithms and thus we call it the height-logarithm sieve.",
"Throughout this section we use the notation introduced above and we assume that the Weierstrass model (REF ) of our given elliptic curve $E$ is minimal at all $p\\in S$ ."
],
[
"Main idea.",
"To describe the main idea of the sieve, we take $P\\in E({\\mathbb {Q}})$ .",
"For any finite place $v$ of ${\\mathbb {Q}}$ , we define $\\log _v(P)=\\tfrac{1}{m_v}\\log (m_vP)$ with $\\log (\\cdot )$ and $m_v=m$ as in Section REF .",
"There are real valued functions $f$ and $f_\\infty $ on $E({\\mathbb {Q}})$ , with $f$ bounded and $f_\\infty $ determined by the real elliptic logarithm, such that any non-exceptionalHere we exclude the exceptional points (Definition REF ) in order to avoid the usual technical problems arising when working with the $v$ -adic elliptic logarithm at $v=\\infty $ and $v=2$ .",
"point $P\\in E({\\mathbb {Q}})$ satisfies $\\hat{h}(P)=f(P)+f_\\infty (P)-\\log \\prod \\vert {}\\log _v(P){}\\vert _v$ with the product taken over certain finite places $v$ of ${\\mathbb {Q}}$ .",
"Here if $P$ is an $S$ -integral point then the product ranges only over $v$ in $S$ , providing a strong condition for points in $E({\\mathbb {Q}})$ to be $S$ -integral.",
"Furthermore, one can check this condition requiring only to know the form of $P$ in $E({\\mathbb {Q}})_{\\textnormal {tor}}\\oplus {\\mathbb {Z}}^r$ .",
"For most points $P$ , this allows to circumvent the slow process of checking whether the coordinates of $P$ are $S$ -integral, that is whether $P\\in \\Sigma (S)$ ."
],
[
"Construction.",
"To transform the above idea into an efficient sieve for the set of $S$ -integral points $\\Sigma (S)$ inside $E({\\mathbb {Q}})$ , we shall work with a slightly weaker version of (REF ) which is suitable for our purpose.",
"More precisely, we shall work with an inequality of the form $\\hat{h}_k(P)\\le L(P)$ involving an efficiently computable quantity $L(P)$ which is essentially determined by the right hand side of (REF ).",
"We begin to explain how to determine $L(P)$ for any $P\\in E({\\mathbb {Q}})$ .",
"Suppose that $P=Q+\\sum n_i P_i$ with $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ and $n_i\\in {\\mathbb {Z}}$ .",
"If $v$ is a finite place of ${\\mathbb {Q}}$ and if $\\alpha _{i,v}=m_v\\log _v(P_i)$ , then we define $l_v(P)=\\log \\max \\bigl (\\vert {}\\tfrac{1}{2}{}\\vert _v,\\vert {}\\tfrac{1}{m_v}\\sum n_i \\alpha _{i,v}{}\\vert _v^{-1}\\bigl ).$ To give a similar definition at $v=\\infty $ , we take $\\kappa \\in {\\mathbb {Z}}_{\\ge 1}$ and we let $x_0=x_0(\\kappa )$ be as in (REF ).",
"Let $\\omega _1$ be the period associated to (REF ), see Section REF .",
"If $v=\\infty $ then we write $\\alpha _{i,v}=\\log (m_vP_i)$ with $\\log (\\cdot )$ and $m_v=m$ as in Section REF and for any $l\\in {\\mathbb {Z}}$ we define $l_v(P,l)=\\log \\max \\bigl ( x_0^{1/2},(1+\\tfrac{1}{\\kappa })^2\\vert {}\\tfrac{1}{m_v}\\bigl (l\\omega _1+\\sum n_i\\alpha _{i,v}\\bigl ){}\\vert _v^{-1}\\bigl ).$ Here we say that $l\\in {\\mathbb {Z}}$ is admissible for $P$ if $\\vert {}l{}\\vert \\le 2m_\\infty +\\sum \\vert {}n_i{}\\vert $ .",
"Let $\\mu $ be as in (REF ).",
"For any finite place $v$ of ${\\mathbb {Q}}$ , we denote by $G_v$ the subgroup of $E({\\mathbb {Q}})$ formed by the points whose images in $E({\\mathbb {Q}}_v)$ lie in fact in $E_1({\\mathbb {Q}}_v)$ .",
"We shall use the following lemma.",
"Lemma 11.9 If $P\\in \\Sigma (S)$ then there is an admissible $l\\in {\\mathbb {Z}}$ such that $\\hat{h}_k(P)\\le \\mu +l_\\infty (P,l)+\\sum _{v\\in S\\,;\\,P\\in G_v} l_v(P).$ The statement follows by combining (REF ) with Lemmas REF , REF and REF .",
"In the next paragraph we shall explain how to control the quantities $l_\\infty (P,l)$ and $\\sum l_v(P)$ in order to obtain a suitable upper bound $L(P)$ for the right hand side of the inequality in Lemma REF .",
"The resulting height-logarithm inequality $\\hat{h}_k(P)\\le L(P)$ is the main ingredient of the following algorithm in which we identify $E({\\mathbb {Q}})$ with $E({\\mathbb {Q}})_{\\textnormal {tor}}\\oplus \\Gamma _E$ , where $\\Gamma _E={\\mathbb {Z}}^r$ is the image of $E({\\mathbb {Q}})$ inside $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}\\cong {\\mathbb {R}}^r$ as in Section REF .",
"Algorithm 11.10 (Height-logarithm sieve) The inputs are $\\kappa \\in {\\mathbb {Z}}_{\\ge 1}$ and a finite subset $\\Sigma $ of $E({\\mathbb {Q}})_{\\textnormal {tor}}\\oplus \\Gamma _E$ .",
"The output is the set $\\Sigma \\cap \\Sigma (S)$ of $S$ -integral points inside $\\Sigma $ .",
"Determine the set $S_E$ formed by the places $v\\in S$ where the elliptic curve $E$ has bad reduction.",
"Then for each nonzero point $P\\in \\Sigma $ do the following: (i) Use the arguments of (1) below to determine an upper bound $l_\\infty (P)\\ge \\max l_\\infty (P,l)$ with the maximum taken over all $l\\in {\\mathbb {Z}}$ which are admissible for $P$ .",
"(ii) Compute the set $S_P=\\lbrace v\\in S\\,;\\,P\\in G_v\\rbrace \\cup S_E$ as described in (2) below.",
"(iii) For each $v\\in S_P$ determine $l_v(P)$ by using the arguments of (3) below, and then set $L(P)=\\mu +l_\\infty (P)+\\sum _{v\\in S_P}l_v(P)$ .",
"Output $P$ if $\\hat{h}_k(P)\\le L(P)$ and if $P\\in \\Sigma (S)$ ."
],
[
"Correctness.",
"Suppose that $P$ lies in $\\Sigma \\cap \\Sigma (S)$ .",
"For each $v\\in S_E$ we obtain that $l_v(P)\\ge 0$ and thus $\\sum _{v\\in S_P}l_v(P)$ exceeds the sum $\\sum l_v(P)$ taken over all $v\\in S$ with $P\\in G_v$ .",
"Hence Lemma REF implies that $\\hat{h}_k(P)\\le L(P)$ and thus (iii) produces our point $P$ as desired."
],
[
"Computing $L(P)$ .",
"We consider a nonzero point $P=(Q,(n_i))$ in $E({\\mathbb {Q}})_{\\textnormal {tor}}\\oplus \\Gamma _E$ ; note that $P=Q+\\sum n_i P_i$ in $E({\\mathbb {Q}})$ .",
"To compute the quantity $L(P)$ we proceed as follows: (1) To control $l_\\infty (P,l)$ for any admissible $l\\in {\\mathbb {Z}}$ , we compute the real elliptic logarithms $\\omega _1$ and $\\alpha _i=\\alpha _{i,v}$ up to a certain precision with respect to $\\vert {}\\cdot {}\\vert =\\vert {}\\cdot {}\\vert _v$ for $v=\\infty $ .",
"If the linear form $\\Lambda =l\\omega _1+\\sum n_i\\alpha _i$ is nonzero then the required precision can be obtained as follows: After choosing a sufficiently large $n\\in {\\mathbb {Z}}$ , one determines approximations $\\alpha _{i}^{\\prime }$ and $\\omega _1^{\\prime }$ of $\\alpha _{i}$ and $\\omega _1$ respectively such that $\\Lambda ^{\\prime }=l\\omega _1^{\\prime }+\\sum n_i\\alpha _i^{\\prime }$ satisfies $\\vert {}\\Lambda ^{\\prime }{}\\vert >\\epsilon =10^{-n}$ and such that the absolute differences $\\vert {}\\alpha _i-\\alpha _i^{\\prime }{}\\vert $ and $\\vert {}\\omega _1-\\omega _1^{\\prime }{}\\vert $ are at most $10^{-c}$ for some fixed integer $c\\ge n+\\log _{10} (2m_\\infty +2\\sum \\vert {}n_i{}\\vert )$ .",
"Then the proof of Lemma REF gives that $-\\log \\vert {}\\Lambda {}\\vert \\le -\\log (\\vert {}\\Lambda ^{\\prime }{}\\vert -\\epsilon )$ and hence we can practically compute an upper bound $l_\\infty (P)$ in ${\\mathbb {R}}\\cup \\lbrace \\infty \\rbrace $ for $\\max l_\\infty (P,l)$ with the maximum taken over all admissible $l\\in {\\mathbb {Z}}$ .",
"Here if $\\Lambda $ is close to zero or if $c$ is too large, then we just put $l_\\infty (P)=\\infty $ to assure that the computation of $l_\\infty (P)$ is always efficient.",
"(2) We would like to quickly compute the set $S_P=\\lbrace v\\in S\\,;\\,P\\in G_v\\rbrace \\cup S_E$ .",
"Here one can directly determine $S_E$ , since the Weierstrass model (REF ) is minimal at all $p\\in S$ .",
"It remains to deal with the places $p\\in S_P-S_E$ .",
"The elliptic curve $E$ has good reduction at $p$ and the canonical reduction map $E({\\mathbb {Q}})\\hookrightarrow E({\\mathbb {Q}}_p)\\rightarrow E(\\mathbb {F}_p)$ is a morphism of abelian groups.",
"We determine the images $\\bar{Q}$ and $\\bar{P_i}$ in $E(\\mathbb {F}_p)$ of all $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ and all $P_i$ .",
"It follows that our point $P=Q+\\sum n_i P_i$ lies in $G_v$ if and only if the point $\\bar{Q}+\\sum n_i\\bar{P_i}$ is zero in $E(\\mathbb {F}_p)$ .",
"Therefore we see that we can quickly determine the set $S_P$ provided we already know all $\\bar{Q}$ , all $\\bar{P_i}$ and the group structure of $E(\\mathbb {F}_p)$ .",
"(3) We take $v\\in S$ and we now explain how to efficiently determine $l_v(P)$ .",
"As already mentioned in Section REF one can always quickly compute the number $m_v$ in practice.",
"We write $\\alpha _i=m_v\\log _v(P_i)$ and we define $\\alpha =\\sum n_i \\alpha _i$ .",
"To compute $v(\\alpha )$ we need to know the $v$ -adic elliptic logarithms $\\alpha _i$ with a certain precision.",
"In practice it usually suffices here to know $\\alpha _i$ with a small $v$ -adic precision.",
"Indeed after computing $v(\\alpha _{i^*})=\\min v(\\alpha _i)$ , we consider $\\beta =\\sum n_i\\beta _i$ for $\\beta _i\\in {\\mathbb {Z}}_p$ of the form $\\beta _i=\\alpha _i/p^{v(\\alpha _{i^*})}$ .",
"The integer $v(\\beta )$ is almost always small in practice.",
"Hence one can usually compute $v(\\beta )$ and $v(\\alpha )$ by knowing only the first coefficients of the $v$ -adic power series of $\\alpha _i$ ."
],
[
"Huge and tiny parameters",
"To assure that Algorithm REF is still fast for huge parameters, one can slightly weaken the sieve as follows: If one of the steps (except the final check whether $P\\in \\Sigma (S)$ ) should take too long for a point $P\\in \\Sigma $ , then abort these steps and directly check whether $P\\in \\Sigma (S)$ .",
"To deal with the case of huge sets $S$ in which step (ii) becomes slow (see Remark REF ), we can always replace $S_P$ by the usually much larger set $S$ .",
"The resulting sieve is still strong for points $P$ with $\\hat{h}(P)\\gg \\log N_S$ .",
"However, replacing $S_P$ by $S$ considerably weakens the sieve for points of small height.",
"We now discuss the case when the rank $r$ is small and the height of the involved point $P$ is tiny.",
"Here one can quickly compute the coordinates of $P$ and the Weil height of these coordinates is not that large.",
"Hence in this case it is often faster to skip steps (i) and (ii) and to directly determine in (iii) whether the coordinates of $P$ are $S$ -integers.",
"In our implementation of the height-logarithm sieve we take into account the above observations to avoid that Algorithm REF is unnecessarily slow for huge or tiny parameters."
],
[
"Complexity.",
"We now discuss aspects of Algorithm REF which considerably influence the running time in practice.",
"In steps (i) and (iii) we need to compute various elliptic logarithms up to a certain precision depending on the height $\\hat{h}(P)$ of the points $P\\in \\Sigma $ .",
"In practice we will apply the height-logarithm sieve only in situations in which the heights $\\hat{h}(P)$ are not huge and in such situations steps (i) and (iii) are always very fast.",
"In step (ii) the running time of the computation of the set $S_P$ crucially depends on the number of primes in $S$ .",
"In practice it turned out that this step is fast when $\\vert {}S{}\\vert $ is small.",
"However, if $\\vert {}S{}\\vert $ becomes huge then step (ii) can take a long time as explained in the following remark.",
"Remark 11.11 (Rank $r=1$ ) For huge sets $S$ the computation of $S_P$ takes a long time, since one has to compute with many large groups $E(\\mathbb {F}_p)$ .",
"In the case $r=1$ the following observation considerably improves this process.",
"Let $v\\in S$ such that $E$ has good reduction at $v$ and write $p=v$ .",
"Let $e_v$ be the order of $P_1$ in the finite group $E(\\mathbb {F}_p)$ .",
"Consider a point $P\\in E({\\mathbb {Q}})$ with $P=Q+n_1P_1$ for $n_1\\in {\\mathbb {Z}}$ and $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ , and let $e_Q$ be the order of $Q$ in $ E({\\mathbb {Q}})_{\\textnormal {tor}}$ .",
"If $P\\in G_v$ then the points $-Q$ and $n_1P_1$ coincide in $E(\\mathbb {F}_p)$ and hence $e_v$ divides $n_1e_Q$ .",
"In other words, if $e_v$ does not divide $n_1e_Q$ then $v$ is not in $S_P$ and therefore we obtain a sufficient criterion to decide whether $v\\in S$ satisfies $v\\notin S_P$ .",
"Remark 11.12 (Inequality trick) In the case when $S$ is empty, one can use the known inequality trick [142] which tests whether a given nonzero point $P\\in E({\\mathbb {Q}})$ satisfies the inequality $\\lambda \\Vert P\\Vert _\\infty ^2\\le \\mu + l_\\infty (P)$ .",
"This inequality is weaker than $\\hat{h}(P)\\le \\mu +l_\\infty (P)$ used in our height-logarithm sieve when $S$ is empty, since $\\lambda \\Vert P\\Vert _\\infty ^2\\le \\hat{h}(P)$ .",
"Hence our height-logarithm sieve is more efficient than the inequality trick, in particular in the case of large rank $r\\ge 2$ where the function $\\lambda \\Vert \\cdot \\Vert _\\infty ^2$ is usually much smaller than $\\hat{h}(\\cdot )$ .",
"See also the examples in the next section.",
"In the case when $S$ is nonempty, one could obtain in principle an inequality trick by testing whether $P$ satisfies the inequality $\\lambda \\Vert P\\Vert _\\infty ^2\\le \\mu +\\sigma l_v(P)$ for some $v\\in S^*=S\\cup \\lbrace \\infty \\rbrace $ and $\\sigma =\\vert {}S^*{}\\vert $ .",
"However the resulting sieve is not that efficient (and often useless if $\\sigma $ is large), since an arbitrary point $P$ usually satisfies at least one of these $\\sigma $ different inequalities which are all considerably weakened by the factor $\\sigma $ ."
],
[
"Refined enumeration",
"We work with the setup of Section REF .",
"The goal of this section is to construct a refined enumeration for the set of $S$ -integral points $\\Sigma (S)\\subset E({\\mathbb {Q}})$ of bounded height which improves the standard enumeration.",
"Throughout this section we assume that the Weierstrass model (REF ) of $E$ is minimal at all $p\\in S$ and we continue the notation introduced above.",
"Recall from Section REF that $\\Gamma _E={\\mathbb {Z}}^r$ denotes the image of $E({\\mathbb {Q}})$ inside $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}\\cong {\\mathbb {R}}^r$ .",
"For any given upper bound $b\\in {\\mathbb {R}}_{\\ge 1}$ , consider the ellipsoid $\\mathcal {E}_b=\\lbrace z\\in {\\mathbb {R}}^r\\,;\\,\\hat{h}_k(z)\\le b\\rbrace $ contained in ${\\mathbb {R}}^r$ .",
"We observe that the following algorithm works correctly.",
"Algorithm 11.13 (Refined enumeration) The input consists of $\\kappa \\in {\\mathbb {Z}}_{\\ge 1}$ together with an upper bound $b\\in {\\mathbb {R}}_{\\ge 1}$ .",
"The output is the set of points $P\\in \\Sigma (S)$ with $\\hat{h}_k(P)\\le b.$ (i) Use (FP) to determine all points in the intersection $\\Gamma _E\\cap \\mathcal {E}_b$ .",
"(ii) For each $\\gamma \\in \\Gamma _E\\cap \\mathcal {E}_{b}$ and for any $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ , output the point $P=Q+\\sum \\gamma _i P_i$ if $P$ lies in the set obtained by applying Algorithm REF with $\\kappa =\\kappa $ and $\\Sigma =\\lbrace P\\rbrace $ ."
],
[
"Complexity.",
"We now discuss various aspects which influence the running time of Algorithm REF in practice.",
"As usual, the application of (FP) in step (i) crucially depends on the rank $r$ .",
"The running time of step (ii) depends on the cardinality of $\\Gamma _E\\cap \\mathcal {E}_b$ , which in turn depends on $r$ , $b$ and the regulator of $E({\\mathbb {Q}})$ .",
"Here the application of the height-logarithm sieve efficiently throws away most points in $\\Gamma _E\\cap \\mathcal {E}_b$ , in particular essentially all points in $\\Gamma _E\\cap \\mathcal {E}_b$ of large height.",
"This considerably improves the running time."
],
[
"Comparison.",
"We next compare our refined enumeration with the standard enumeration of the points $P\\in \\Sigma (S)$ with $\\Vert P\\Vert _\\infty ^2\\le b^{\\prime }$ , where $b^{\\prime }=b/\\lambda _k$ depends on the smallest eigenvalue $\\lambda _k$ of $\\hat{h}_k$ .",
"Recall that the standard enumeration proceeds as follows: For any $\\gamma \\in \\Gamma _E$ with $\\max \\vert {}\\gamma _i{}\\vert ^2\\le b^{\\prime }$ and for each $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ , output the point $P=Q+\\sum \\gamma _i P_i$ if the coordinates of $P$ are $S$ -integers.",
"In the case when $S$ is empty, one can use here in addition the known inequality trick explained in Remark REF .",
"In general we observe that our refined sieve working with the ellipsoid $\\mathcal {E}_b$ is more efficient, in particular for large rank $r$ .",
"Indeed the cube $\\lbrace \\Vert \\cdot \\Vert _\\infty ^2\\le b^{\\prime }\\rbrace \\subset {\\mathbb {R}}^r$ always contains the ellipsoid $\\mathcal {E}_b$ and then on comparing volumes we see that our refined sieve involves much fewer points.",
"Furthermore the application of the height-logarithm sieve in the refined enumeration gives significant running time improvements.",
"See Section REF for examples and tables which illustrate in particular the running time improvements provided by our refined enumeration."
],
[
"Refined sieve",
"In this section we work out a refinement of the global sieve obtained by patching together the archimedean sieve of Section REF with the various non-archimedean sieves of Section REF .",
"Throughout this section we work with the setup of Section REF and we continue the notation introduced above.",
"Furthermore we assume that the Weierstrass model (REF ) of our given elliptic curve $E$ is minimal at all primes $p\\in S$ ."
],
[
"Main idea.",
"The main ingredient of the refined sieve is Proposition REF .",
"Therein we construct a refined covering of certain subsets of the set of $S$ -integral points $\\Sigma (S)$ , which allows to improve the global-local passage required to apply the local sieves obtained in Sections REF and REF .",
"The construction of the covering is inspired by the refined sieve for $S$ -unit equations developed in Section REF .",
"However, in the present case of elliptic curves, everything is more complicated.",
"For example, one has to distinguish archimedean and non-archimedean places and one has to take care of certain exceptional points (Definition REF ) arising from technical issues of the $v$ -adic elliptic logarithm at the places $v=\\infty $ and $v=2$ .",
"To deal efficiently with the exceptional points, we conducted some effort to work entirely in the abelian group $E({\\mathbb {Q}})$ .",
"This allows here to avoid working with coordinate functions, which in turn is crucial to solve equations (REF ) with huge parameters."
],
[
"Construction of the covering.",
"For any given $M,M^{\\prime }$ in ${\\mathbb {Z}}$ with $0\\le M^{\\prime }<M$ , we would like to find the set of points $P\\in \\Sigma (S)$ which satisfy $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"For this purpose we “cover\" this set as follows.",
"Let $\\kappa ,n,\\tau $ in ${\\mathbb {Z}}_{\\ge 1}$ with $\\tau \\le n\\le s^*$ for $s^*=\\vert {}S{}\\vert +1$ and choose an admissible partition $\\lbrace S_j\\rbrace $ of $S^*=S\\cup \\lbrace \\infty \\rbrace $ into disjoint nonempty parts $S^*= S_1\\operatorname{\\dot{\\cup }}\\ldots \\operatorname{\\dot{\\cup }}S_g$ .",
"Here admissible partition means that $|S_j|\\le n$ for all $j\\in \\lbrace 1,\\cdots ,g\\rbrace $ and that $g\\le \\left\\lceil {\\tfrac{s^*}{n}}\\right\\rceil +2$ .",
"For a motivation of working with admissible partitions of $S^*$ , we refer to the efficiency discussion given below.",
"Further we choose “weights\" $w_1,\\cdots ,w_\\tau $ in ${\\mathbb {Q}}$ with $w_1=1$ and $w_1\\ge \\cdots \\ge w_\\tau >0$ and for any $t\\in \\lbrace 1,\\cdots ,\\tau \\rbrace $ we put $\\sigma _t=\\tfrac{w}{w_t}, \\ \\ \\ w=\\sum _{j=1}^g w(j), \\ \\ \\ w(j)={\\left\\lbrace \\begin{array}{ll}(|S_j|-\\tau )w_\\tau +\\sum _{t\\le \\tau } w_t & \\textnormal {if } \\tau \\le \\vert {}S_j{}\\vert ,\\\\\\sum _{t\\le \\vert {}S_j{}\\vert } w_t & \\textnormal {if } \\tau >\\vert {}S_j{}\\vert .\\end{array}\\right.",
"}$ Next we take $j\\in \\lbrace 1,\\cdots ,g\\rbrace $ and we consider a nonempty subset $T$ of $S_j$ with cardinality $\\vert {}T{}\\vert $ at most $\\tau $ .",
"Write $t=\\vert {}T{}\\vert $ and suppose that $v\\in T$ .",
"If $v\\in S$ then we denote by $\\Gamma _v\\subseteq {\\mathbb {Z}}^r$ the lattice constructed in Section REF with $\\sigma =\\sigma _t$ , and if $v=\\infty $ then $\\Gamma _v\\subseteq {\\mathbb {Z}}^{r+1}$ denotes the lattice from Section REF with $\\sigma =\\sigma _t$ and $\\kappa =\\kappa $ .",
"In the case $\\mu >M^{\\prime }$ , where $\\mu $ is as in (REF ), we set here $\\Gamma _v={\\mathbb {Z}}^r$ if $v\\in S$ and $\\Gamma _v={\\mathbb {Z}}^{r+1}$ if $v=\\infty $ .",
"Now we define $\\Gamma _T=\\bigcap _{v\\in T} \\Gamma _v.$ Here if $T$ contains $\\infty $ then for any $v\\in S\\cap T$ we identify $\\Gamma _v\\subseteq {\\mathbb {Z}}^r$ with the lattice inside ${\\mathbb {Z}}^{r+1}$ given by $\\phi (\\Gamma _v)\\oplus e{\\mathbb {Z}}$ , where $\\phi $ denotes the canonical product embedding of ${\\mathbb {Z}}^r$ into ${\\mathbb {Z}}^r\\times {\\mathbb {Z}}={\\mathbb {Z}}^{r+1}$ and $e=(0,1)\\in {\\mathbb {Z}}^{r}\\times {\\mathbb {Z}}$ .",
"Next we consider the ellipsoid $\\mathcal {E}_T$ which is defined as follows: If $T\\subseteq S$ then $\\mathcal {E}_T\\subset {\\mathbb {R}}^r$ is the ellipsoid appearing in Section REF , and if $T$ contains $\\infty $ then $\\mathcal {E}_T\\subset {\\mathbb {R}}^{r+1}$ is the ellipsoid constructed in Section REF with respect to the parameters $\\sigma =\\sigma _t$ and $\\kappa =\\kappa $ .",
"In the case $\\mu >M^{\\prime }$ we define here $\\mathcal {E}_T={\\mathbb {R}}^r$ if $T\\subseteq S$ and $\\mathcal {E}_T={\\mathbb {R}}^{r+1}$ if $T$ contains $\\infty $ .",
"We next define the exceptional points.",
"Definition 11.14 (Exceptional point) Consider a point $P\\in \\Sigma (S)$ and denote by $(x,y)$ the corresponding solution of (REF ).",
"We say that $P$ is an exceptional point if $\\vert {}x{}\\vert _2\\le 4$ or if $\\vert {}x{}\\vert _\\infty \\le x_0$ , where $x_0$ denotes the number $x_0(\\kappa )$ defined in (REF ).",
"To completely “cover\" our set of interest, we need to take into account the exceptional points.",
"For this purpose we let $b$ be the positive real number defined in (REF ), which depends inter alia on the parameters $\\kappa ,\\tau ,\\lbrace S_j\\rbrace $ and $w_t$ , and we work with the ellipsoid $\\mathcal {E}_{b}=\\lbrace z\\in {\\mathbb {R}}^r\\,;\\,\\hat{h}_k(z)\\le b\\rbrace .$ Recall from Section REF that $\\Gamma _E={\\mathbb {Z}}^r$ denotes the image of $E({\\mathbb {Q}})$ inside $E({\\mathbb {Q}})\\otimes _{\\mathbb {Z}}{\\mathbb {R}}\\cong {\\mathbb {R}}^r$ .",
"The following result shows that our set of interest can be “covered\" by the set $\\Gamma _E\\cap \\mathcal {E}_{b}$ together with the sets $\\Gamma _T\\cap \\mathcal {E}_T$ associated to some $T$ as above.",
"Proposition 11.15 Suppose that $P$ lies in $\\Sigma (S)$ and assume that $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"Then at least one of the following statements holds.",
"(i) The point $P$ is determined modulo torsion by some $\\gamma \\in \\Gamma _E\\cap \\mathcal {E}_b$ .",
"(ii) There is a nonempty set $T$ with $\\vert {}T{}\\vert \\le \\tau $ such that $T\\subseteq S_j$ for some $S_j$ in $\\lbrace S_j\\rbrace $ and such that $P$ is determined modulo torsion by an element $\\gamma \\in \\Gamma _T\\cap \\mathcal {E}_T$ .",
"If $M^{\\prime }<\\mu $ then (ii) holds for example with $T=\\lbrace \\infty \\rbrace $ .",
"Thus we may and do assume that $M^{\\prime }\\ge \\mu $ .",
"Let $(x,y)$ be the solution of (REF ) corresponding to $P$ .",
"We claim that there exists a nonempty set $T$ with $\\vert {}T{}\\vert \\le \\tau $ such that $T\\subseteq S_j$ for some $S_j$ in $\\lbrace S_j\\rbrace $ and such that $P\\in \\bigcap _{v\\in T}\\Sigma (v,\\sigma _t).$ Here $\\Sigma (v,\\sigma _t)$ is defined in (REF ) with $t=\\vert {}T{}\\vert $ .",
"To prove this claim by contradiction, we assume that (REF ) does not hold.",
"Then for each $j$ and for any nonempty subset $T\\subseteq S_j$ with $t=\\vert {}T{}\\vert \\le \\tau $ , there exists $v\\in T$ such that $P\\notin \\Sigma (v,\\sigma _t)$ .",
"In particular, for any $j$ and for each $t\\in {\\mathbb {Z}}_{\\ge 1}$ with $t\\le \\min (\\tau ,\\vert {}S_j{}\\vert )$ , it follows that the $t$ -th largest of the real numbers $\\tfrac{1}{2}\\log \\vert {}x{}\\vert _v$ , $v\\in S_j,$ is strictly smaller than $\\tfrac{1}{\\sigma _t}(\\hat{h}_k(P)-\\mu )$ .",
"We deduce that $\\tfrac{1}{2}\\sum \\max (0,\\log \\vert {}x{}\\vert _v)< \\tfrac{w(j)}{w}(\\hat{h}_k(P)-\\mu )$ with the sum taken over all $v\\in S_j$ .",
"Here we used that $\\mu \\le M^{\\prime }<\\hat{h}_k(P)$ and that the weights $w_t$ satisfy $w_1\\ge \\cdots \\ge w_\\tau >0$ .",
"Then our assumption $P\\in \\Sigma (S)$ together with $S^*=\\cup S_j$ implies that $\\tfrac{1}{2}h(x)< \\hat{h}_k(P)-\\mu $ .",
"But this contradicts the inequality $\\hat{h}_k(P)-\\mu \\le \\tfrac{1}{2}h(x)$ which follows by combining Lemma REF with (REF ).",
"Therefore we conclude that our claim (REF ) holds as desired.",
"Let $\\mathcal {T}$ be the nonempty set of all sets $T$ satisfying (REF ); put $t=\\min \\lbrace \\vert {}T{}\\vert \\,;\\,T\\in \\mathcal {T}\\rbrace $ and define $\\mathcal {T}_{\\min }=\\lbrace T\\in \\mathcal {T}\\,;\\,\\vert {}T{}\\vert =t\\rbrace $ .",
"Further, on slightly abusing terminology, we write $\\Sigma (4)$ for the subset of $\\Sigma (2,\\sigma _t)$ defined in Section REF and we denote by $\\Sigma (x_0)$ the subset of $\\Sigma (\\infty ,\\sigma _t)$ from Section REF .",
"First we consider the case $t=1$ .",
"Suppose that we can choose $T\\in \\mathcal {T}_{\\min }$ with $T\\subseteq S-2$ .",
"Then we obtain that $T=\\lbrace p\\rbrace $ with $p\\ge 3$ .",
"Thus on recalling that the Weierstrass model (REF ) is minimal at all $p\\in S$ , we see that the inequalities $\\mu \\le M^{\\prime }$ and $M^{\\prime }<\\hat{h}_k(P)\\le M$ together with (REF ) show that $P$ satisfies the assumptions of Lemma REF .",
"Hence Lemma REF implies (ii).",
"Suppose now that there is no $T\\in \\mathcal {T}_{\\min }$ with $T\\subseteq S-2$ .",
"Then $\\lbrace \\infty \\rbrace $ or $\\lbrace 2\\rbrace $ lies in $\\mathcal {T}_{\\min }$ and any $v\\in S-2$ satisfies $\\tfrac{1}{2}\\log |x|_v< \\tfrac{1}{\\sigma _1}(\\hat{h}_k(P)-\\mu ).$ To complete the proof for $t=1$ , it remains to establish (i) or (ii) in the following cases (a), (b) and (c).",
"Before we go into these cases we define the number $b$ appearing in $\\mathcal {E}_b$ : If $t^*=\\min (\\tau ,\\max _{v=2,\\infty }\\vert {}S_{j(v)}{}\\vert )$ with $S_{j(v)}$ denoting the set $S_j$ which contains $v$ , then $b=\\mu +\\tfrac{1}{2w_{t^*}}(1+s_{t^*})\\log \\max (x_0,4).$ Here $x_0=x_0(\\kappa )$ is defined in (REF ) and $s_{t^*}$ is the number of $p\\in S$ with $p^{2w_{t^*}}\\le \\max (x_0,4)$ .",
"In what follows we shall use that $P\\in \\Sigma (S)$ , that for each finite place $v$ of ${\\mathbb {Q}}$ it holds $v(x)\\le -2$ if $P\\in E_1({\\mathbb {Q}}_v)$ , that $1=w_1\\ge \\cdots \\ge w_\\tau >0$ , and that $\\hat{h}_k\\le \\hat{h}$ by Lemma REF .",
"(a) Case $\\lbrace 2\\rbrace \\in \\mathcal {T}_{\\min }$ and $\\lbrace \\infty \\rbrace \\in \\mathcal {T}_{\\min }$ .",
"If $\\vert {}x{}\\vert _\\infty >x_0$ or $\\vert {}x{}\\vert _2> 2^2$ , then (REF ) implies that $P\\in \\Sigma (x_0)$ or $P\\in \\Sigma (4)$ and thus Lemma REF or Lemma REF shows (ii) for $T=\\lbrace \\infty \\rbrace $ or $T=\\lbrace 2\\rbrace $ respectively.",
"On the other hand, if $\\vert {}x{}\\vert _\\infty \\le x_0$ and $\\vert {}x{}\\vert _2\\le 2^2$ then (REF ) and (REF ) lead to an upper bound for $h(x)$ which together with (REF ) proves (i).",
"(b) Case $\\lbrace 2\\rbrace \\in \\mathcal {T}_{\\min }$ and $\\lbrace \\infty \\rbrace \\notin \\mathcal {T}_{\\min }$ .",
"Here inequality (REF ) holds in addition for $v=\\infty $ , since $\\lbrace \\infty \\rbrace $ is not in $\\mathcal {T}_{\\min }$ .",
"Therefore, if $\\vert {}x{}\\vert _2\\le 2^2$ then we see as above that (REF ), (REF ) and (REF ) imply statement (i).",
"If $\\vert {}x{}\\vert _2> 2^ 2$ then (REF ) gives that $P\\in \\Sigma (4)$ and hence Lemma REF shows statement (ii) with $T=\\lbrace 2\\rbrace $ .",
"(c) Case $\\lbrace 2\\rbrace \\notin \\mathcal {T}_{\\min }$ and $\\lbrace \\infty \\rbrace \\in \\mathcal {T}_{\\min }$ .",
"Now (REF ) holds in addition for $v=2$ , since $\\lbrace 2\\rbrace $ is not in $\\mathcal {T}_{\\min }$ .",
"Thus as above we deduce (i) if $\\vert {}x{}\\vert _\\infty \\le x_0$ .",
"If $\\vert {}x{}\\vert _\\infty > x_0$ then (REF ) gives $P\\in \\Sigma (x_0)$ and hence Lemma REF proves (ii) with $T=\\lbrace \\infty \\rbrace $ .",
"We now establish the case $t\\ge 2$ .",
"If we can choose $T\\in \\mathcal {T}_{\\min }$ with $T\\subseteq S-2$ , then (REF ) together with Lemma REF implies (ii).",
"Suppose now that there is no $T\\in \\mathcal {T}_{\\min }$ with $T\\subseteq S-2$ .",
"Then each $T$ in $\\mathcal {T}_{\\min }$ contains $\\infty $ or 2.",
"Furthermore any $v\\in S^*$ satisfies (REF ), since $t\\ge 2$ .",
"To complete the proof it thus suffices to consider the following cases: (d) Case when each $T\\in \\mathcal {T}_{\\min }$ contains 2 and $\\infty $ .",
"If $\\vert {}x{}\\vert _\\infty > x_0$ and $\\vert {}x{}\\vert _2> 2^2$ , then (REF ) gives that $P\\in \\Sigma (x_0)\\cap \\Sigma (4)$ .",
"Thus on recalling the construction of $\\Gamma _T$ and $\\mathcal {E}_T$ , we see that Lemmas REF and REF together with (REF ) show that (ii) holds for any $T\\in \\mathcal {T}_{\\min }$ .",
"On the other hand, if $\\vert {}x{}\\vert _\\infty \\le x_0$ or $\\vert {}x{}\\vert _2\\le 2^2$ then (REF ), (REF ) and (REF ) prove (i).",
"(e) Case when there is $T\\in \\mathcal {T}_{\\min }$ with $2\\in T$ and $\\infty \\notin T$ .",
"If $\\vert {}x{}\\vert _2> 2^2$ then (REF ) gives that $P\\in \\Sigma (4)$ and thus we deduce (ii) by using $\\infty \\notin T$ , (REF ) and Lemma REF .",
"On the other hand, if $\\vert {}x{}\\vert _2\\le 2^2$ then (REF ), (REF ) and (REF ) imply (i).",
"(f) Case when there exists $T\\in \\mathcal {T}_{\\min }$ with $2\\notin T$ and $\\infty \\in T$ .",
"If $\\vert {}x{}\\vert _\\infty \\le x_0$ then (REF ), (REF ) and (REF ) imply (i).",
"Finally, if $\\vert {}x{}\\vert _\\infty > x_0$ then (REF ) gives $P\\in \\Sigma (x_0)$ .",
"Therefore on using $2\\notin T$ and (REF ), we see that Lemmas REF and REF prove (ii).",
"Hence we conclude that in all cases (i) or (ii) holds.",
"This completes the proof.",
"The arguments used to prove (REF ) show in addition that one can further refine the covering in Proposition REF by working with the ellipsoids $\\mathcal {E}_T^*$ discussed in Remark REF ."
],
[
"Refined Sieve.",
"A collection of sieve parameters $\\mathcal {P}$ consists of the following data: Parameters $\\kappa ,\\tau ,n\\in {\\mathbb {Z}}_{\\ge 1}$ with $\\tau \\le n\\le s^*$ , an admissible partition $\\lbrace S_j\\rbrace $ of $S^*=S_1\\operatorname{\\dot{\\cup }}\\cdots \\operatorname{\\dot{\\cup }}S_g$ with respect to $n$ , and weights $w_1,\\cdots ,w_\\tau $ as in (REF ).",
"We denote by $b(\\mathcal {P})$ the number associated to $\\mathcal {P}$ as in (REF ) and we obtain the following algorithm.",
"Algorithm 11.16 (Refined sieve) The input is a collection of sieve parameters $\\mathcal {P}$ together with bounds $M^{\\prime },M\\in {\\mathbb {Z}}_{\\ge 0}$ satisfying $M^{\\prime }< M$ .",
"Put $M^{\\prime }_b=\\max (b(\\mathcal {P}),M^{\\prime })$ .",
"The output is the set of points $P\\in \\Sigma (S)$ with $M^{\\prime }_b<\\hat{h}_k(P)\\le M$ .",
"For any $j\\in \\lbrace 1,\\cdots ,g\\rbrace $ and for each $T\\subseteq S_j$ with $1\\le \\vert {}T{}\\vert \\le \\tau $ , do the following: (i) Determine a basis of $\\Gamma _T$ and then compute $\\Gamma _T\\cap \\mathcal {E}_T$ by using the version of the Fincke–Pohst algorithm in (FP).",
"(ii) For each $\\gamma \\in \\Gamma _T\\cap \\mathcal {E}_T$ and for any $Q\\in E({\\mathbb {Q}})_{\\textnormal {tor}}$ , output the point $P=Q+\\sum \\gamma _i P_i$ if $P$ satisfies $M^{\\prime }_b<\\hat{h}_k(P)\\le M$ and if $P$ lies in the set produced by an application of Algorithm REF with $\\Sigma =\\lbrace P\\rbrace $ and $\\kappa =1$ ."
],
[
"Correctness.",
"Assume that $P\\in \\Sigma (S)$ satisfies $M^{\\prime }_b<\\hat{h}_k(P)\\le M$ .",
"Then it holds that $\\hat{h}_k(P)>b(\\mathcal {P})$ and hence there is no lattice point $\\gamma $ in $\\Gamma _E\\cap \\mathcal {E}_{b(\\mathcal {P})}$ such that $P$ is determined modulo torsion by $\\gamma $ .",
"Furthermore, our assumption provides that $M^{\\prime }<\\hat{h}_k(P)\\le M$ .",
"Therefore Proposition REF shows that step (ii) produces our point $P$ as desired."
],
[
"Efficiency.",
"We now discuss the efficiency of the refined sieve and we motivate several concepts appearing therein.",
"First we observe that the case $n=1$ in the refined sieve corresponds to the non-refined sieve obtained by patching together the local sieves at $v\\in S^*$ with $\\sigma =s^*$ .",
"Suppose now that $n\\ge \\tau \\ge 2$ in the refined sieve.",
"Then the iteration over the sets $T$ ranges in particular over all sets $T=\\lbrace v\\rbrace $ with $v\\in S^*$ .",
"However, compared with the non-refined sieve $n=1$ , there is the following fundamental difference: If $T=\\lbrace v\\rbrace $ then the discussions in Sections REF and REF together with $\\sigma _1\\le s^*$ show that the refined sieve involving $\\Gamma _T\\cap \\mathcal {E}_T$ with $\\sigma _1$ is usually much stronger than the non-refined sieve $n=1$ involving the local sieve at $v$ with $\\sigma =s^*$ .",
"Furthermore, if $\\vert {}T{}\\vert \\ge 2$ then the intersection $\\Gamma _T=\\cap \\Gamma _v$ is usually considerably smaller than each part $\\Gamma _v$ .",
"These observations suggest that the improvements coming from $\\vert {}T{}\\vert =1$ are significant enough to absorb the additional iterations over sets $T$ with $\\vert {}T{}\\vert \\ge 2$ .",
"In practice this turned out to be correct in many fundamental situations (see Section REF ), showing that the refined sieve provides significant running time improvements.",
"To deal with huge sets $S$ , we introduced admissible coverings of $S^*$ which allow to control the number of additional iterations over sets $T$ with $\\vert {}T{}\\vert \\ge 2$ .",
"Indeed the conditions $\\vert {}S_j{}\\vert \\le n$ and $\\vert {}T{}\\vert \\le \\tau $ assure that the number of additional iterations are controlled in terms of $n,\\tau $ .",
"Further we point out that a canonical choice for the weights $w_t$ would be $w_t=\\tfrac{1}{t}$ .",
"However in practice it turned out that for $t\\ge 2$ it would be better to choose $w_t$ slightly larger than $\\tfrac{1}{t}$ .",
"In fact this is the reason for working with the more general weights $w_t$ defined above (REF ).",
"Finally we mention that the discussion of the influence of the parameters $M^{\\prime }$ , $M$ and $\\sigma _t$ on the strength of the sieve $\\Gamma _T\\cap \\mathcal {E}_T$ with $\\vert {}T{}\\vert =t$ is similar to the corresponding discussions in Sections REF and REF ."
],
[
"Complexity.",
"We do not try to analyse the complexity of the refined sieve in general, since it depends on too many parameters.",
"However, in Sections REF and REF we shall discuss aspects influencing the complexity of the refined sieve and we shall illustrate the running improvements provided by the refined sieve in various fundamental cases.",
"Remark 11.17 (Refined ellipsoids) For any set $T$ as above with $\\vert {}T{}\\vert \\ge 2$ , the arguments used in the proof of (REF ) show in addition the following: Instead of using in Algorithm REF the ellipsoids $\\mathcal {E}_T$ , one can work with the ellipsoids $\\mathcal {E}_T^*$ obtained by replacing in the definition of $\\mathcal {E}_T$ the bound $M$ by the possibly much smaller number $M_T=\\min \\left(M,w_T(M^{\\prime }-\\mu )+\\mu \\right), \\ \\ \\ w_T=\\tfrac{1}{w}\\sum _{j=1}^g w^*(j).$ Here $w^*(j)$ is obtained by replacing in the definition of $w(j)$ the number $\\tau $ by $\\vert {}T{}\\vert -1$ .",
"Note that $\\mathcal {E}_T^*\\subseteq \\mathcal {E}_T$ and $w_T$ does not depend on $M$ .",
"Now if $M>M_T$ then $\\mathcal {E}_T^*$ is strictly contained in $\\mathcal {E}_T$ and hence using $\\mathcal {E}_T^*$ improves Algorithm REF (i).",
"In principle further refinements are possible by taking into account the part $S_{j}$ of $S^*$ which contains $T\\subseteq S_{j}$ ."
],
[
"Global sieve",
"We continue the setup, notation and assumptions of the previous section.",
"After choosing suitable collections of sieve parameters, we combine in this section the refined enumeration with the refined sieve: For any given upper bound $M_1\\in {\\mathbb {Z}}_{\\ge 1}$ , we obtain a global sieve which allows to efficiently determine all points $P\\in \\Sigma (S)$ with $\\hat{h}_k(P)\\le M_1$ ."
],
[
"Sieve parameters.",
"We shall apply our refined sieve with the following collections of sieve parameters.",
"Choose $\\kappa \\in {\\mathbb {Z}}_{\\ge 1}$ such that $\\vert {}b-10{}\\vert $ is as small as possible, where $b$ is defined in (REF ) with $\\tau =1$ and $\\kappa =\\kappa $ .",
"Then let $\\mathcal {P}(1)$ be the collection of sieve parameters determined by $\\kappa =\\kappa $ and $n=1$ .",
"For any $i\\in \\lbrace 2,3,4\\rbrace $ we define $\\mathcal {P}(i)$ as follows: We take $\\kappa =\\kappa $ , $\\tau =i$ and $n=10$ , and we choose weights $w_1,\\cdots ,w_\\tau $ as in (REF ) such that each $w_t$ is slightly larger than $\\tfrac{1}{t}$ for $t\\ge 2$ .",
"Further we use here an admissible covering $\\lbrace S_j\\rbrace $ of $S^*=S\\cup \\lbrace \\infty \\rbrace $ with $S_1=\\lbrace \\infty \\rbrace $ and with the following properties: If $2\\notin S$ then $\\vert {}S_j{}\\vert =n$ for each $j\\in \\lbrace 2,g-1\\rbrace $ , and if $2\\in S$ then $S_2=\\lbrace 2\\rbrace $ and $\\vert {}S_j{}\\vert =n$ for any $j\\in \\lbrace 3,g-1\\rbrace $ .",
"We shall motivate our choice of sieve parameters in the discussions below.",
"Remark 11.18 For each $i\\in \\lbrace 1,\\cdots ,4\\rbrace $ the number $b_i$ , associated to $\\mathcal {P}(i)$ in (REF ), satisfies $b=b_i$ .",
"Indeed on using that $\\tau =1$ in $\\mathcal {P}(1)$ and that $\\max _{v=2,\\infty }\\vert {}S_{j(v)}{}\\vert =1$ in $\\mathcal {P}(i)$ with $i\\ge 2$ , we see in all four cases that $t^*=1$ and hence we obtain that $b=b_i$ as desired."
],
[
"Global sieve.",
"For any given $M_1\\in {\\mathbb {Z}}_{\\ge 1}$ , we would like to efficiently determine the set of points $P\\in \\Sigma (S)$ with $\\hat{h}_k(P)\\le M_1$ .",
"For this purpose we enumerate these points from below and from above, using the refined enumeration in Algorithm REF and the refined sieve in Algorithm REF respectively.",
"More precisely we proceed as follows: (a) Let $b$ be as in the above paragraph, and define parameters $\\nu ^{\\prime }=b$ , $\\nu =M_1$ and $f(\\nu )=\\min (\\nu -1,\\lfloor 0.99\\nu \\rfloor )$ .",
"Then apply the following sieves $(i=1,\\cdots ,4)$ : 0.",
"Apply Algorithm REF with $b=\\nu ^{\\prime }$ and $\\kappa =1$ , and let $\\rho _0$ be the running time divided by the euclidean volume $\\operatorname{\\textnormal {vol}}(\\mathcal {E}_{\\nu ^{\\prime }})$ .",
"Put $\\nu ^{\\prime }= 4^{1/r}\\nu ^{\\prime }$ .",
"i.",
"If $\\vert {}S^*{}\\vert \\ge i$ and $\\nu >\\nu ^{\\prime }/4^{1/r}$ then apply Algorithm REF with the parameters $\\mathcal {P}=\\mathcal {P}(i)$ , $M^{\\prime }=f(\\nu )$ and $M=\\nu $ , and let $\\rho _i$ be the running time divided by the euclidean volume $\\operatorname{\\textnormal {vol}}(\\mathcal {E}_{M}\\setminus \\mathcal {E}_{M^{\\prime }})$ .",
"Put $\\nu =f(\\nu )$ .",
"(b) As long as $\\nu >\\nu ^{\\prime }/4^{1/r}$ continue with the most efficient sieve in (a), that is the sieve $i^*\\in \\lbrace 0,\\cdots ,4\\rbrace $ for which the “efficiency measure\" $1/\\rho _{i^*}$ is maximal.",
"This algorithm outputs the set of points $P\\in \\Sigma (S)$ with $\\hat{h}_k(P)\\le M_1$ .",
"Indeed Remark REF gives that $b=b_i$ for each $i\\in \\lbrace 1,\\cdots ,4\\rbrace $ , and therefore we see that the whole space $\\lbrace P\\in \\Sigma (S)\\,;\\,\\hat{h}_k(P)\\le M_1\\rbrace $ is covered by an application of steps (a) and (b)."
],
[
"Decomposition.",
"We now motivate the decomposition of the above algorithm and we explain our choice of the parameters appearing therein.",
"First we discuss the step size functions.",
"In the enumeration from below, we double the volume of the ellipsoid in each step by working with the step size function given by multiplication with $4^{1/r}$ .",
"This assures that the repeated enumerations of candidates with tiny height are not that significant for the running time, see also the remark at the end of this paragraph.",
"In the enumeration from above, we work with the step size function $f(\\nu )=\\min (\\nu -1,\\lfloor 0.99\\nu \\rfloor )$ .",
"If $\\nu $ is large then the height lower bound $M^{\\prime }=f(\\nu )$ is still large and thus the refined sieve is strong in view of the complexity discussions in Sections REF and REF .",
"Hence, to accelerate the enumeration from above, we work with the relatively big step size $\\nu -f(\\nu )\\ge 10^{-2}\\nu $ for large $\\nu $ ; here the factor $0.99$ turned out to be suitable in practice where usually $M_1\\le 10^9$ .",
"On the other hand, if $\\nu \\le 100$ is small then we work with the tiny step size $\\nu -f(\\nu )=1$ to assure that $M^{\\prime }=\\nu -1$ is relatively large making the sieves stronger.",
"We next motivate our choice of the sieve parameters $\\mathcal {P}(i)$ .",
"If the parameter $\\kappa $ becomes larger then the refined sieve becomes stronger at $v=\\infty $ .",
"On the other hand, we can not choose $\\kappa $ arbitrarily large since the “square-radius\" $b$ of the ellipsoid $\\mathcal {E}_b$ satisfies $b\\ge \\log (\\kappa )$ .",
"Now, choosing $\\kappa $ such that $\\vert {}b-10{}\\vert $ is as small as possible, assures that the refined enumeration via $\\Gamma _E\\cap \\mathcal {E}_b$ is efficient in practice where usually $r\\le 12$ .",
"To explain our choices for $n$ and $\\tau $ , we recall that the efficiency of the refined sieve depends inter alia on $\\sigma _1=\\sigma _1(n,\\tau )$ and the number of additional iterations over subsets $T$ with $\\vert {}T{}\\vert \\le \\tau $ .",
"Here it is not clear to us what are the optimal choices for $\\tau $ and $n$ .",
"In practice it turned out that for $\\tau \\ge 5$ or $n\\ge 11$ there are usually too many additional iterations and thus we only work with $\\tau \\le 4$ and $n=10$ .",
"The reason for using admissible partitions $\\lbrace S_j\\rbrace $ of $S^*$ with $S_1=\\lbrace \\infty \\rbrace $ and $S_2=\\lbrace 2\\rbrace $ if $2\\in S$ , is to assure that $b=b_i$ (see Remark REF ) which means that the ellipsoids $\\mathcal {E}_{b_i}$ are not larger than the minimal involved ellipsoid $\\mathcal {E}_{b}$ .",
"In fact controlling the ellipsoids $\\mathcal {E}_{b_i}$ is crucial for dealing efficiently with huge parameters.",
"Finally we mention that in the enumeration from below, the application of (FP) repetitively enumerates candidates.",
"Here it is not clear to us how to avoid these repeated enumerations of candidates, since (FP) is not faster for circular discs than for the whole ellipsoid.",
"In any case these repeated enumerations of candidates have a small influence on the running time in practice, since (FP) is very fast in our situations of interest where usually the rank $r$ is small.",
"Furthermore, to assure that the height-logarithm sieve is applied at most once for each candidate, we order the candidates with respect to their height $\\hat{h}_k$ in the implementation of the enumeration from below."
],
[
"Main features.",
"We now discuss the main features of the global sieve.",
"The first steps of the enumeration from above may be viewed as a reduction of $M_1$ .",
"Indeed these steps are usually very fast in practice and they often allow to considerably reduce $M_1$ .",
"Further we point out that each step of the global sieve is more efficient than the standard enumeration.",
"In fact in general it is not clear to us in which situation which sieve of (a) is the most efficient.",
"To overcome this problem, we work with the quantities $1/\\rho _i$ in order to “measure\" the efficiency of the sieves in the given situation.",
"In practice this allows our algorithm to choose a suitable sieve in each step.",
"This is very important for the running time, since the efficiency of the involved sieves strongly depends on the given situation.",
"We also mention that Section REF contains explicit examples which illustrate (up to some extent) the improvements in practice provided by our global sieve."
],
[
"Elliptic logarithm sieve",
"We work with the setup of Section REF and we continue the notation introduced above.",
"In the first part of this section we construct the elliptic logarithm sieve by putting together the sieves obtained in the previous sections.",
"In the second part we compare the elliptic logarithm sieve with the known approach and we explain in detail our improvements.",
"We recall that in the setup of Section REF we are given the following information: The coefficients $a_i\\in {\\mathbb {Z}}$ of a Weierstrass equation (REF ) of an elliptic curve $E$ over ${\\mathbb {Q}}$ , a finite set of rational primes $S$ , a basis $P_1,\\cdots ,P_r$ of the free part of $E({\\mathbb {Q}})$ and a number $M_0\\in {\\mathbb {Z}}$ such that any $P\\in \\Sigma (S)$ satisfies $\\hat{h}(P)\\le M_0$ .",
"Given this information, the following algorithm completely determines the set $\\Sigma (S)$ formed by the $S$ -integral solutions of (REF ).",
"Algorithm 11.19 (Elliptic logarithm sieve) The inputs are the coefficients $a_i$ of (REF ), the set $S$ , the basis $P_1,\\cdots ,P_r$ and the initial bound $M_0$ .",
"The output is the set $\\Sigma (S)$ .",
"(i) Compute the following additional input data.",
"(a) Determine the equation of an affine Weierstrass model $W$ of $E$ over ${\\mathbb {Z}}$ , which is minimal at all primes in $S$ , together with an isomorphism $\\varphi $ over $\\mathcal {O}={\\mathbb {Z}}[1/N_S]$ from $W$ to the affine model defined by (REF ).",
"(b) Compute a suitable rational approximation $\\hat{h}_k$ of $\\hat{h}$ .",
"(c) Determine the torsion subgroup $E({\\mathbb {Q}})_{\\textnormal {tor}}$ of $E({\\mathbb {Q}})$ , and compute the numbers $b$ and $\\kappa $ appearing in the collections of sieve parameters from Section REF .",
"(ii) To reduce locally the initial bound $M_0$ , apply the archimedean sieve and the non-archimedean sieve.",
"More precisely, work with the set $\\Sigma ^{\\prime }(S)$ formed by the $\\mathcal {O}$ -points of $W$ and for each place $v$ in $S^*=S\\cup \\lbrace \\infty \\rbrace $ do the following: (a) Find an integer $M_1(v)\\ge b$ with the following property: If $M^{\\prime }=M_1(v)$ , $M=M_0$ , $\\kappa =\\kappa $ and $\\sigma =\\vert {}S{}\\vert +1$ , then Algorithm REF (ii) outputs only 0 when $v=\\infty $ or 0 is the output of Algorithm REF (iii) when $v\\in S$ .",
"Here first try $M_1(v)=b$ .",
"If this does not work then try a slightly larger number, and so on.",
"(b) Having found such an $M_1(v)$ , try to reduce it further by repeating (a) with different parameters $M^{\\prime }$ and $M$ .",
"Let $M_1(v)$ be the final reduced bound at $v$ .",
"(iii) Determine the set $\\Sigma ^{\\prime }(S)$ by applying the global sieve from Section REF with $M_1=\\max _{v\\in S^*}M_1(v)$ .",
"Then output the set $\\varphi (\\Sigma ^{\\prime }(S))$ ."
],
[
"Correctness.",
"To prove that this algorithm works correctly, we recall that $\\varphi $ is an isomorphism of affine Weierstrass models of $E$ over $\\mathcal {O}$ .",
"This shows that $\\varphi (\\Sigma ^{\\prime }(S))=\\Sigma (S)$ and hence it remains to verify that $\\Sigma ^{\\prime }(S)$ is completely determined.",
"Lemma REF gives that $\\hat{h}_k\\le \\hat{h}$ , and any $P\\in \\Sigma ^{\\prime }(S)$ satisfies $\\hat{h}(P)\\le M_0$ since $\\hat{h}$ is invariant under isomorphisms.",
"Therefore on using that $b\\le M_1$ , we see that the arguments of Proposition REF , with $n=1$ and $\\kappa =\\kappa $ , prove that $\\hat{h}_k(P)\\le M_1$ for all $P\\in \\Sigma ^{\\prime }(S)$ .",
"It follows that the application of the global sieve with $M_1$ produces the set $\\Sigma ^{\\prime }(S)$ in step (iii) as desired."
],
[
"Complexity.",
"We now discuss various aspects which influence the running time in practice.",
"The computation of the additional input data in step (i) is always very fast.",
"More precisely, in (a) we use Tate's algorithm to transform (REF ) into a globally minimal Weierstrass model of $E$ over ${\\mathbb {Z}}$ which then can be used to directly determine a pair $(W,\\varphi )$ with the desired properties.",
"To construct the quadratic form $\\hat{h}_k$ in (b) we proceed as described in Section REF ; see also Section REF for computational aspects.",
"Finally in (c) the numbers $b$ and $\\kappa $ can be directly determined and the computation of $E({\\mathbb {Q}})_{\\textnormal {tor}}$ is always efficient.",
"The running time of step (ii) crucially depends on the initial upper bound $M_0$ , see the complexity discussions in Sections REF and REF for details.",
"Here step (a) can take a long time for huge $M_0$ , while the repetitions in step (b) are then quite fast since at this point we have already computed the involved elliptic logarithms.",
"Step (ii) usually allows to avoid the process of testing whether candidates of huge height have $S$ -integral coordinates.",
"This process is so slow that it is beneficial in (b) to make as many repetitions as required to obtain a reduced global bound $M_1$ which is as small as possible.",
"The running time of the application of the global sieve in step (iii) crucially depends on the cardinality of $S$ and the rank $r$ .",
"For example, as already explained in previous sections, the rank $r$ has a huge influence on the running time of the refined enumeration and on the refined sieve.",
"We also recall that the cardinality of $S$ significantly influences the efficiency of the height-logarithm sieve, which in turn is used in the refined sieve and the refined enumeration.",
"See also the discussions in Sections REF and REF ."
],
[
"Bottleneck.",
"In practice the bottleneck of the elliptic logarithm sieve crucially depends on the situation, in particular on the Mordell–Weil rank $r$ , the cardinality of $S$ and the size of the initial bound $M_0$ .",
"We first suppose that $M_0$ is huge, say $M_0$ is the initial bound coming from the theory of logarithmic forms (see Section REF ).",
"If $S$ is empty or when $r\\le 1$ , then either the elliptic logarithm sieve is fast or the bottleneck is step (iii).",
"Assume now that $r\\ge 2$ and that $S$ is nonempty.",
"If in addition $r\\le 4$ then the bottleneck is usually part (a) of step (ii), in particular when $\\vert {}S{}\\vert $ is large.",
"On the other hand, if in addition $r\\ge 5$ then the bottleneck is either step (iii) or part (a) of step (ii).",
"If $M_0$ is not that large then the bottleneck is usually step (iii).",
"For example, in the case when (REF ) is a Mordell equation, the initial bounds of Proposition REF are strong enough such that either the elliptic logarithm sieve is fast or the bottleneck is step (iii).",
"Remark 11.20 (Generalizations) (i) Algorithm REF allows to solve Diophantine equations which are a priori more general than (REF ).",
"For example, our algorithm can be applied to find all $S$ -integral solutions with bounded height of any Weierstrass equation (REF ) of $E$ with coefficients $a_i$ in ${\\mathbb {Q}}$ .",
"Indeed on multiplying equation (REF ) with the sixth power $u^6$ of the least common multiple $u$ of the denominators of the $a_i$ , one obtains a Weierstrass equation (REF )$^*$ with coefficients $a_i^*=u^ia_i$ in ${\\mathbb {Z}}$ and then one checks for each $(x,y)\\in \\Sigma ^*(S)$ whether $u^{-2}x$ and $u^{-3}y$ are in $\\mathcal {O}$ .",
"Here $\\Sigma ^*(S)$ denotes the set of solutions of (REF )$^*$ in $\\mathcal {O}\\times \\mathcal {O}$ obtained by applying Algorithm REF with the coefficients $a_i^*$ , with the same initial bound $M_0$ and with the transformed coordinates of the same basis $P_i$ .",
"(ii) The above Algorithm REF works equally well with any initial bound for the usual Weil height $h$ or for the infinity norm $\\Vert \\cdot \\Vert _\\infty $ .",
"Indeed the explicit inequalities (REF ) and (REF ) translate any initial bound for $h$ or $\\Vert \\cdot \\Vert _\\infty $ into an initial bound for $\\hat{h}$ .",
"(iii) We mention that various authors (including Stroeker, Tzanakis and de Weger) generalized and modified the known elliptic logarithm approach in order to efficiently solve more general Diophantine equations defining genus one curves.",
"For an overview we refer to the discussions in Stroeker–Tzanakis [143] and Tzanakis [159]."
],
[
"Comparison with the known approach",
"To discuss the improvements provided by the elliptic logarithm sieve, we now compare our sieve with the known approach via elliptic logarithms.",
"We recall that the main steps of the known method are as follows (see for example [126] or [159]): (1) As explained in Section REF , one tries to obtain a reduced bound $N_1$ which is as small as possible such that any non-exceptional point $P\\in \\Sigma (S)$ satisfies $\\Vert P\\Vert _\\infty \\le N_1$ .",
"(2) One goes through all points $P\\in E({\\mathbb {Q}})$ with $\\Vert P\\Vert _\\infty \\le N_1$ and one tests whether $P$ lies in fact in $\\Sigma (S)$ .",
"In the case $S^*=\\lbrace \\infty \\rbrace $ , one can apply in addition the known inequality trick (see Remark REF ) before one tests whether $P$ lies in $\\Sigma (S)$ .",
"(3) One makes a so-called extra search to find all exceptional points."
],
[
"Reduction.",
"Steps (i)+(ii) of our elliptic logarithm sieve may be viewed as an analogue of (1).",
"Here the running times of (1) and (i)+(ii) are essentially equal, since the aspects in which the two approaches differ are irrelevant for the running time.",
"Indeed in both approaches the running time is essentially determined by the computations of the involved elliptic logarithms and these computations are the same in both approaches.",
"On the other hand, in view of the subsequent enumeration, an important difference is that (1) uses the inequality $\\lambda \\Vert P\\Vert _\\infty ^2\\le \\hat{h}(P)$ in order to work with $\\Vert \\cdot \\Vert _\\infty $ , while our reduction in (ii) directly works with $\\hat{h}$ .",
"Geometrically, this means that (1) uses a cube which always contains the ellipsoids $\\mathcal {E}$ used in (ii).",
"In fact there exist non-trivial improvements of (1), see Stroeker–Tzanakis [142] which optimizes the Mordell–Weil basis and see Hajdu–Kovács [81] which in the case $S=\\emptyset $ intersects cubes containing $\\mathcal {E}$ .",
"Our reduction in (ii) is always as good as these improvements of (1), since we work directly with the ellipsoid $\\mathcal {E}$ which is optimal from a geometric point of view.",
"We define $N_{\\textnormal {opt}}=\\lfloor (M_1/\\lambda )^{1/2}\\rfloor $ with $\\lambda $ coming from a Mordell–Weil basis which is optimized in the sense of [142].",
"For instance, the usual elliptic logarithm reduction can not reduce anymore ([126]) the bound $N_1=17$ in the example of [126] involving an optimized Mordell–Weil basis ($r=4$ ).",
"On the other hand, our reduction in (ii) gives $M_1\\le 61$ which implies that $N_{\\textnormal {opt}}=12$ .",
"Furthermore, in many important situations, the reduction in (ii) is considerably stronger than the known improvements of (1).",
"For example if $r\\ge 2$ becomes large then the volume of $\\mathcal {E}$ becomes much smaller than the volume of any cube containing $\\mathcal {E}$ .",
"Hence (ii) is significantly more efficient when $r\\ge 2$ is large.",
"In particular, in the generic case we obtain here a running time improvement by a factor which is exponential in terms of $r$ .",
"Furthermore, in the most common nontrivial case (where $r=1$ ), we obtain huge running time improvements for large $\\vert {}S{}\\vert $ by using the following trick: The idea is that we do not need to know the involved $v$ -adic elliptic logarithms $(\\beta _{1,v}^{(c)}, v\\in S)$ appearing in Section REF .",
"Indeed on exploiting that the involved lattice has rank $r=1$ , one observes that it suffices here to know the orders $(v(\\alpha _{1,v}),v\\in S)$ .",
"These orders can always be efficiently computed in practice."
],
[
"Enumeration.",
"Step (iii) of our sieve plays the role of (2)+(3).",
"In the following comparison, we denote by $t$ and $t^*$ the running times of (iii) and (2)+(3) respectively.",
"Comparing $t$ with $t^*$ is suitable to illustrate our improvements.",
"Indeed it takes into account that the running times of (i)+(ii) and (1) are essentially equal and it makes the comparison independent of the initial bound.",
"We denote by $t_2$ the running time of our refined enumeration in Algorithm REF (i) applied with our reduced initial bound $b=M_1$ from (ii).",
"In practice the running time of (2) always exceeds $t_2$ , which means that $t^*> t_2$ .",
"In fact, in many cases of interest (e.g.",
"when $r$ , $\\vert {}S{}\\vert $ or $\\max \\vert {}a_i{}\\vert $ is not small), the time $t^*$ is often considerably larger than $t_2$ .",
"To explain the improvements provided by our sieve, we mention four situations in which the known enumeration (2)+(3) usually becomes very slow: (S1) Suppose that $r$ is not small and $S$ is nonempty.",
"Then the enumeration in (2) becomes very slow or even hopeless, since one has to consider $(2N_1+1)^r$ points of $E({\\mathbb {Q}})$ .",
"(S2) Assume that $r$ is large and $S$ is empty.",
"Then the enumeration in (2) becomes often hopeless, since one has to go through $(2N_1+1)^r$ points of $E({\\mathbb {Q}})$ .",
"(S3) Suppose that the height $\\max \\vert {}a_i{}\\vert $ of (REF ) is large.",
"Then the extra search in (3) has to test many pairs $(x,y)$ of $S$ -integers whether they satisfy (REF ).",
"Thus (3) becomes very slow when $\\max \\vert {}a_i{}\\vert $ is large, in particular if $S$ is nonempty.",
"(S4) Suppose that $\\vert {}S{}\\vert $ and $N_1$ are both large.",
"Then (2) becomes very slow, since one has to compute many rational numbers $x(P)$ of huge height and one has to test whether they are $S$ -integral.",
"Here $x(P)$ is the $x$ -coordinate of some $P\\in E({\\mathbb {Q}})$ .",
"To deal with situation (S1) we developed the global sieve which combines our refined sieve and our refined enumeration.",
"We recall from the discussions in previous sections that the first steps of the global sieve are essentially a further reduction of $M_1$ , while the subsequent steps of the global sieve are also more efficient than the corresponding enumerations in (2) and (3).",
"To illustrate our improvements in practice, we consider the Mordell curve $y^2=x^3+1358556$ of rank $r=6$ and three additional examples which shall be further discussed in Section REF below.",
"These three additional examples are given by Kretschmer's curve in [133] with $r=8$ , Mestre's curve in [133] with (conditional) $r=12$ and the curve of Fermigier with $r=14$ .",
"In the following table, the entries of the first and second row are rounded up and down respectively.",
"Table: NO_CAPTIONWe point out that here the running times of (2) would be significantly larger than the listed times $t_2$ , since $N_1\\ge N_{\\textnormal {opt}}$ and since $(2N_{\\textnormal {opt}}+1)^r$ is huge in each case.",
"In particular the enumeration (2) would be very slow in the above cases involving $r=6$ , while the cases involving $r=8,12,14$ seem to be completely out of reach for (2).",
"Furthermore our global sieve leads in addition to a significant improvement in situation (S2) where $r$ is large and $S$ is empty.",
"To illustrate our improvements in practice, we consider again Mestre's curve of rank $r=12$ , Fermigier's curve of rank $r=14$ and two curves of Elkies of rank $r=17,19$ respectively.",
"These four curves shall be discussed in more detail in Section REF below.",
"In the case when $r=12,14,17,19$ , it turned out that $t$ is less than 2 minutes, 13 minutes, 26 hours, 73 hours respectively and that $N_{\\textnormal {opt}}= 14,10,22,15$ respectively.",
"It seems that all these cases are out of reach for the enumeration (2).",
"To deal with situation (S3) where $\\max \\vert {}a_i{}\\vert $ is large, we constructed the refined covering in Proposition REF .",
"This result allows us to work entirely in the finitely generated abelian group $E({\\mathbb {Q}})$ in order to find the exceptional points.",
"We illustrate our improvements in practice by considering a rather randomly chosen Mordell equation (REF ) of rank $r=1$ with $a_6=-17817895$ .",
"This choice is suitable, since the assumption $r=1$ is satisfied in the nontrivial generic case and since our sieve does not exploit special properties of Mordell equations.",
"Furthermore, if the rank $r$ is large then $\\max \\vert {}a_i{}\\vert $ is large and the above table already contains times $t$ for $r\\ge 6$ .",
"To obtain the lower bounds for $t^*$ listed in the following table, we used the running times of the extra search (3); this search was implemented in Sage by Cremona, Mardaus and Nagel following the presentation in [126].",
"Table: NO_CAPTIONHere the entries of the first and second row are upper and lower bounds for $t$ and $t^*$ respectively.",
"We note that $t^*$ explodes when $\\max \\vert {}a_i{}\\vert $ becomes larger.",
"For example, let us consider the Mordell equation (REF ) of rank $r=1$ with $a_6=- 4211349581402184375$ .",
"Here our running times $t$ essentially coincide with the times displayed in the above table, while the extra search (3) did not terminate within 48 hours in the simplest case $S=S(1)$ .",
"We mention that our improvements in the case of large $\\max \\vert {}a_i{}\\vert $ are crucial for the computation (see Section REF ) of elliptic curves over ${\\mathbb {Q}}$ with good reduction outside $S$ .",
"Indeed these computations usually require to find all $S^{\\prime }$ -integral solutions of many distinct Mordell equations $y^2=x^3+a$ , with $a\\in {\\mathbb {Z}}$ having huge $\\vert {}a{}\\vert \\ge 10^{15}$ and $S^{\\prime }=S\\cup \\lbrace 2,3\\rbrace $ .",
"We developed various global constructions to efficiently deal with situation (S4) where $\\vert {}S{}\\vert $ and $N_1$ are large.",
"In particular, for many involved points $P\\in E({\\mathbb {Q}})$ our height-logarithm sieve allows to avoid the slow process of testing whether the coordinates of $P$ are $S$ -integral.",
"To demonstrate our improvements in practice, we considered the curves 37a1, 389a1, 5077a1 in Cremona's database with rank $r=1,2,3$ respectively.",
"They have minimal conductor among all elliptic curves over ${\\mathbb {Q}}$ of rank $r=1,2,3$ respectively.",
"Table: NO_CAPTIONHere the entries of the first row are rounded up.",
"Further, it is reasonable to expect that the first two entries in the $t_2$ row are larger than 150 hours and 900 hours respectively.",
"Indeed, it took 192 seconds (resp.",
"777 seconds) to determine whether the coordinates of the point $11000P$ (resp.",
"$16015P$ ) are $S(10^5)$ -integral (resp.",
"$S(2\\cdot 10^5)$ -integral), where $P{\\mathbb {Z}}=E({\\mathbb {Q}})$ and $E$ denotes the rank one curve 37a1 used in the above table.",
"We mention that the ability of Algorithm REF to solve (REF ) for large sets $S$ was crucial for obtaining data motivating our conjectures in Section REF and in Section REF below."
],
[
"Input data",
"We continue our notation.",
"The elliptic logarithm sieve requires an initial height bound for the points in $\\Sigma (S)$ and a Mordell–Weil basis of $E({\\mathbb {Q}})$ .",
"In this section we recall some results and techniques which allow to compute the required input data in practice."
],
[
"Mordell–Weil basis",
"The problem of finding a Mordell–Weil basis of $E({\\mathbb {Q}})$ is difficult in theory and in practice.",
"In fact in the case of an arbitrary elliptic curve $E$ over ${\\mathbb {Q}}$ there is so far no unconditional method which allows in principle to determine a Mordell–Weil basis of $E({\\mathbb {Q}})$ .",
"However, thanks to the work of many authors, it is usually possible to compute such a basis in practice.",
"In fact it turned out in practice that the methods implemented in Pari, Sage and Magma are remarkably efficient in computing such a basis, even in the case when the height $\\max \\vert {}a_i{}\\vert $ of (REF ) is large.",
"Furthermore, Cremona's database contains a Mordell–Weil basis of $E({\\mathbb {Q}})$ for each elliptic curve $E$ over ${\\mathbb {Q}}$ with conductor at most 350000.",
"Unless mentioned otherwise, we shall use below these bases of Cremona's database."
],
[
"Initial height bounds",
"Starting with the works of Baker [4], [5], [6], there is a long tradition of establishing explicit height bounds for the points in $\\Sigma (S)$ using lower bounds for linear forms in logarithms.",
"See for example Baker–Wüstholz [33] for an overview and a discussion of the state of the art.",
"Furthermore Masser [112] and Wüstholz [166] initiated an approach which provides explicit height bounds for the points in $\\Sigma (S)$ using lower bounds for linear forms in elliptic logarithms.",
"Here the actual best lower bounds can be found in the works of David [41] and Hirata-Kohno [79].",
"To obtain our results discussed in the next section, we compute two initial height bounds for the points in $\\Sigma (S)$ and then we take the minimum of these two bounds.",
"More precisely, the first initial height bound is a direct consequence of the results of Hajdu–Herendi [78] combined with a height comparison in the style of (REF ), see Pethő–Zimmer–Gebel–Herrmann [126].",
"Here the results of Hajdu–Herendi ultimately rely on lower bounds for linear forms in complex and $p$ -adic logarithms.",
"The second initial height bound depends on the above mentioned lower bounds for linear forms in elliptic logarithms due to David in the archimedean case and due toIn fact explicit lower bounds for linear forms in two nonarchimedean elliptic logarithms were established in the work of Rémond–Urfels [129].",
"Hirata-Kohno in the nonarchimedean case.",
"See for example the proof of Tzanakis [159]; here one has to take into account that in some cases the normalizations in [159] do not coincide with our corresponding normalizations used in Sections REF and REF .",
"In fact, unless mentioned otherwise, we obtained all applications in Section REF by using the first height bound."
],
[
"Applications",
"In this section we discuss additional applications of the elliptic logarithm sieve.",
"To obtain the input data required for the applications of our sieve, we used unless mentioned otherwise the results described in the previous section.",
"We continue the notation introduced above and for any $n\\in {\\mathbb {Z}}_{\\ge 1}$ we denote by $S(n)$ the set of the first $n$ rational primes."
],
[
"Elliptic curves database",
"For any elliptic curve $E$ over ${\\mathbb {Q}}$ of conductor at most 1000, Cremona's database contains in particular a minimal Weierstrass equation (REF ) of $E$ .",
"We used Algorithm REF to compute the $S$ -integral solutions of each of these equations with $S=S(20)$ .",
"Moreover, for any of these minimal equations defining an elliptic curve over ${\\mathbb {Q}}$ of conductor at most 100, we used Algorithm REF to determine its set of $S$ -integral solutions with $S=S(10^4)$ ."
],
[
"Elliptic curves of large rank",
"In addition, we used the elliptic logarithm sieve (Algorithm REF ) in order to determine the set of $S$ -integral solutions of various Weierstrass equations (REF ) for which the involved Mordell–Weil rank $r$ of $E({\\mathbb {Q}})$ is relatively large.",
"We now discuss some examples."
],
[
"Mordell curves.",
"Recall that (REF ) is called a Mordell equation if the coefficients $a_1,\\cdots ,a_5$ are all zero.",
"In Section REF we used the elliptic logarithm sieve to find all $S$ -integral solutions of two (resp.",
"four) Mordell equations with $S=S(50)$ and $r=7$ (resp.",
"$S=S(40)$ and $r=8$ ).",
"Instead of using initial bounds coming from the theory of logarithmic forms, we applied here our optimized height bound in Proposition REF which is based on the method of Faltings (Arakelov, Paršin, Szpiro) [60] combined with the Shimura–Taniyama conjecture [164], [158], [12].",
"Further, we determined here the required Mordell–Weil bases by using techniques implemented in Pari, Sage and Magma."
],
[
"Rank eight.",
"Let $E_{\\textnormal {Kr}}$ be the elliptic curve over ${\\mathbb {Q}}$ considered in [133], with $E_{\\textnormal {Kr}}({\\mathbb {Q}})$ of rank $r=8$ by Kretschmer [99].",
"Siksek [133] combined his refined descent techniques with Cremona's “mwrank\" to find a Mordell–Weil basis of $E_{\\textnormal {Kr}}({\\mathbb {Q}})$ .",
"On using Siksek's basis as an input for our sieve, we determined all $S$ -integral solutions of the minimal Weierstrass equation (REF ) of $E_{\\textnormal {Kr}}$ with $S=S(10)$ .",
"This took our sieve less than 35 seconds, 17 hours and 75 hours for $S=\\emptyset $ , $S=S(8)$ and $S=S(10)$ respectively.",
"Here we notice that our running times for $E_{\\textnormal {Kr}}$ are significantly worse than for the four Mordell curves of rank $r=8$ discussed in Section REF .",
"The reason is that in the case of $E_{\\textnormal {Kr}}$ we need to use initial height bounds based on the theory of logarithm forms.",
"These initial bounds are substantially weaker (see Section REF ) than our optimized height bound in Proposition REF which is currently only available for Mordell curves."
],
[
"Rank twelve.",
"Mestre [115] constructed an elliptic curve $E_{\\textnormal {Me}}$ over ${\\mathbb {Q}}$ of analytic rank 12, together with 12 independent points in $E_{\\textnormal {Me}}({\\mathbb {Q}})$ of infinite order.",
"Here again Siksek [133] applied his refined descent techniques to find a Mordell–Weil basis of $E_{\\textnormal {Me}}({\\mathbb {Q}})$ .",
"On using Siksek's basis as an input for our sieve, we determined the set of $S$ -integral solutions of a minimal Weierstrass equation (REF ) of $E_{\\textnormal {Me}}$ with $S=S(7)$ .",
"This took our sieve less than 2 minutes, 4 days and 16 days for $S=\\emptyset $ , $S=S(6)$ and $S=S(7)$ respectively.",
"We point out that the completeness of our solution sets are here conditional on Siksek's assumption that $E_{\\textnormal {Me}}({\\mathbb {Q}})$ has rank $r=12$ which he used in his construction of a basis of $E_{\\textnormal {Me}}({\\mathbb {Q}})$ .",
"For example, this assumption is satisfied if $r$ is at most the analytic rank of $E_{\\textnormal {Me}}$ as predicted by the rank part of the Birch–Swinnerton-Dyer conjecture."
],
[
"Rank at most 28.",
"At the time of writing, we are not aware of an elliptic curve over ${\\mathbb {Q}}$ of rank $r\\ge 13$ for which an explicit Mordell–Weil basis can be computed explicitly.",
"If $r\\le 28$ and $S$ is empty then the elliptic logarithm sieve would allow to determine $\\Sigma (S)$ for such large rank elliptic curves $E$ over ${\\mathbb {Q}}$ , provided that one knows an explicit Mordell–Weil basis of $E({\\mathbb {Q}})$ .",
"To demonstrate this feature, we work with independent points $Q_1,\\cdots ,Q_r$ generating a rank $r$ subgroup $\\Lambda $ of the free part of $E({\\mathbb {Q}})$ .",
"Dujella lists in particular such points (see web.math.pmf.unizg.hr/~duje) for three elliptic curves over ${\\mathbb {Q}}$ of rank $r=14,17,19$ respectively.",
"Here the curve of rank $r=14$ was constructed by Fermigier, while the other two curves were found by Elkies.",
"We denote by $\\Sigma _\\Lambda (S)$ the intersection of $\\Sigma (S)$ with $\\Lambda \\oplus E({\\mathbb {Q}})_{\\textnormal {tor}}$ .",
"On using the basis $Q_1,\\cdots ,Q_r$ of $\\Lambda $ in the input of Algorithm REF , we determined the set $\\Sigma _\\Lambda (S)$ .",
"If $S$ is empty then this took less than 13 minutes, 26 hours, 73 hours for $r=14,17,19$ respectively, and it took less than 18 hours when $r=14$ and $S=S(2)$ .",
"Now, given a Mordell–Weil basis, one can expect similar running times of Algorithm REF when computing the full set $\\Sigma (S)\\supseteq \\Sigma _\\Lambda (S)$ .",
"Indeed the index of $\\Lambda $ in the free part of $E({\\mathbb {Q}})$ is not that large for these three curves.",
"Finally, we consider Elkies' elliptic curve $E_{\\textnormal {El}}$ over ${\\mathbb {Q}}$ of rank $r\\ge 28$ .",
"In the case when $S$ is empty, we applied Algorithm REF with the 28 independent points $Q_i$ constructed by Elkies and we computed the intersection of $\\Sigma (S)$ with $(\\oplus _{i}Q_i{\\mathbb {Z}})\\oplus E_{\\textnormal {El}}({\\mathbb {Q}})_{\\textnormal {tor}}$ in less than 75 days."
],
[
"Conjectures and questions",
"In this section we generalize our conjectures and questions for Mordell curves (see Section REF ) to hyperbolic genus one curves.",
"We then provide some motivation by using our data and by generalizing our constructions for Mordell curves.",
"See also Section REF which contains the initial motivation for our conjectures and questions.",
"We continue our notation.",
"As in Section REF , we let $Y=(X,D)$ be a hyperbolic genus one curve over some open subscheme $B$ of $\\textnormal {Spec}({\\mathbb {Z}})$ and we denote by $r$ the rank of the group formed by the ${\\mathbb {Q}}$ -points of $\\textnormal {Pic}^0(X_{\\mathbb {Q}})$ .",
"Now we recall our conjecture.",
"Conjecture.",
"There are constants $c_{Y}$ and $c_r$ , depending only on $Y$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ with $T=\\textnormal {Spec}({\\mathbb {Z}})-S$ satisfies $\\vert {}Y(T){}\\vert \\le c_{Y} \\vert {}S{}\\vert ^{c_r}.$ If $D$ is given by a section of $X\\rightarrow B$ , then $Y$ identifies with a closed subscheme of $\\mathbb {A}^2_B$ defined by a Weierstrass equation (REF ).",
"Hence the family $S[b]$ constructed in Section REF shows that the exponent $c_r$ has to be at least $\\tfrac{r}{r+2}$ when $Y$ is a Weierstrass curve.",
"Question 1.",
"What is the optimal exponent $c_r$ in the above conjecture?",
"Our data strongly indicates that the exponent $c_r=\\tfrac{r}{r+2}$ is still far from optimal for many families of sets $S$ of interest, including the family $S(n)$ with $n\\in {\\mathbb {Z}}_{\\ge 1}$ .",
"Further, our data motivates in addition the following question on the dependence on $q=\\max S$ .",
"Question 2.",
"Are there constants $c_{Y}$ and $c_r$ , depending only on $Y$ and $r$ respectively, such that any nonempty finite set of rational primes $S$ with $T=\\textnormal {Spec}({\\mathbb {Z}})-S$ satisfies $\\vert {}Y(T){}\\vert \\le c_{Y}(\\log \\max S)^{c_r} \\, \\textnormal {?",
"}$ In the case when $S=S(n)$ with $n\\in {\\mathbb {Z}}_{\\ge 2}$ , one can replace here $q$ by $n\\log n$ without changing the content of the question.",
"However Question 2 has in general a negative answer when $q$ is replaced by any power of $\\max (2,\\vert {}S{}\\vert )$ .",
"On using again the arguments of Section REF , we see that the exponent $c_r$ of Question 2 has to be at least $r/2$ if $Y$ is a Weierstrass curve.",
"In light of this we ask whether Question 2 has a positive answer with the exponent $c_r=r/2 \\, \\textnormal {?",
"}$ Our probabilistic model constructed in the discussion surrounding (REF ) predicts that this question has a positive answer when $Y$ is a Weierstrass curve.",
"To conclude we mention that additional motivation for our conjecture is given by the theory of logarithmic forms [33], which was applied in [86] to obtain new bounds for the number of integral points on arbitrary hyperbolic genus one curves over any number field.",
"These bounds establish in particular our Conjecture for certain sets $S$ of interest, including the sets $S(n)$ ."
],
[
"Computational aspects",
"In this final section we discuss various computational aspects of our algorithms.",
"In particular we explain the numerical details of constructions used in previous sections."
],
[
"Interval arithmetic.",
"Most real numbers $x$ cannot be explicitly presented by a computer.",
"Hence we apply standard interval arithmetic which uses lower and upper bounds for $x$ .",
"This allows us to control numerical errors.",
"In particular, one can detect when the error explodes and in this case one can restart the computation with higher precision."
],
[
"Rounding real numbers.",
"We consider a real number $x\\in {\\mathbb {R}}$ .",
"In this paper the symbol $[x]$ denotes an element of ${\\mathbb {Z}}$ with $\\vert {}x-[x]{}\\vert <1$ .",
"There might be two distinct choices for $[x]\\in {\\mathbb {Z}}$ with the desired property.",
"However, for our purpose any choice will be sufficient.",
"In the implementation, real numbers $x$ are only stored up to some chosen precision.",
"Suppose that $x$ is stored with absolute precision $m\\in {\\mathbb {Z}}_{\\ge 1}$ , that is the computer stores $x$ as $x^{\\prime }\\in {\\mathbb {Q}}$ with $\\vert {}x-x^{\\prime }{}\\vert \\le \\tfrac{1}{2^{m}}$ .",
"Then we can compute $[x]$ as a closest integer to $x^{\\prime }$ ; there may be two choices, in which case we pick one.",
"This $[x]\\in {\\mathbb {Z}}$ moreover satisfies $\\vert {}x-[x]{}\\vert \\le \\tfrac{1}{2}+\\tfrac{1}{2^m}$ ."
],
[
"Construction and eigenvalues of $\\hat{h}_k$ .",
"To discuss the numerical details required for the construction of $\\hat{h}_k$ , we continue the notation and terminology of Section REF .",
"We recall that we need to find a suitable $k\\in {\\mathbb {Z}}_{\\ge 1}$ for which $\\hat{h}_k$ is close enough to $\\hat{h}$ .",
"To find such a $k$ , we start with some $k$ such as for example $k=10$ .",
"We first determine a reasonably good rational approximation $f\\in {\\mathbb {Q}}$ of $2^k/\\Vert \\hat{h}_{ij}\\Vert $ .",
"After computing a lower bound for the smallest eigenvalue $\\lambda _k$ of $\\hat{h}_k$ , we can check whether $\\lambda _k>0$ and $r<\\tfrac{1}{100}f\\lambda _k$ .",
"If both conditions are satisfied, then $k$ is suitable.",
"Otherwise we increase $k$ and we repeat the above procedure as long as required to find a $k$ with the desired properties.",
"Lemma REF implies that this procedure terminates; in fact it terminates always very quickly in practice.",
"While the condition $r<\\tfrac{1}{100}f\\lambda _k$ is in principle not required for the correctness of our algorithms, it assures via Lemma REF that $\\hat{h}_k$ is relatively close to $\\hat{h}$ , which is reasonable in practice.",
"To explain how we compute a reasonably good lower bound for $\\lambda _k$ , we recall that the quadratic form $\\hat{h}_k$ is given by $A/f$ with $A=([f\\hat{h}]-r\\cdot \\textnormal {id})\\in {\\mathbb {Z}}^{r\\times r}$ .",
"First, we apply Newton's method to compute a root of the characteristic polynomial of $A$ .",
"Here we start at minus the Cauchy bound for the largest absolute value of its roots.",
"At each step of Newton's method we stay below all eigenvalues.",
"Indeed the characteristic polynomial is a convex or concave function on the interval $(-\\infty ,f\\lambda _k)$ , since all $r$ eigenvalues of $A$ are real as $A$ is symmetric.",
"In order to remove numerical issues, we use interval arithmetic.",
"After each step of Newton's method, we moreover replace the point by its lower bound.",
"Once this value is smaller than or equal to the one in the previous step, we terminate.",
"An upper bound for the largest eigenvalue of $A$ can be obtained in an analogous way."
],
[
"Choice of $\\delta $ .",
"To explain our choice of the parameter $\\delta $ in (REF ), we continue the notation and terminology of Section REF .",
"We choose $a,b\\in {\\mathbb {Z}}$ such that $\\tfrac{a}{b}$ is a good lower approximation of $M/(\\delta _1+\\delta _2)^2$ , such that $\\tfrac{a}{b}=M/\\delta ^2$ for some positive $\\delta \\in {\\mathbb {Q}}$ and such that $\\vert {}a{}\\vert ,\\vert {}b{}\\vert $ are not too large.",
"These are two simultaneous objective functions that we try to optimize: 1.",
"Making the inequality $\\tfrac{a}{b} \\le M/(\\delta _1+\\delta _2)^2$ as sharp as possible assures that the Fincke–Pohst algorithm does not return too many additional candidates.",
"2.",
"The smaller $|a|$ and $|b|$ , the smaller are the entries of the quadratic form on which we run the Fincke–Pohst algorithm.",
"Further, this construction of $\\delta $ (that is of the integers $a,b$ ) allows us to apply our implementation of the Fincke–Pohst lattice point enumeration which works with integral quadratic forms.",
"Indeed on multiplying by the (controlled) denominator $bf$ of the quadratic form $q$ used in the definition of $\\mathcal {E}$ , one obtains an integral quadratic form."
],
[
"Lattice point enumeration.",
"To enumerate lattice points in an ellipsoid, we apply the version (FP) of Fincke–Pohst [63] which in turn uses $L^3$ [105].",
"More precisely, for any $d\\in {\\mathbb {Z}}_{\\ge 1}$ the algorithm (FP) takes as the input a basis of a lattice $\\Gamma \\subseteq {\\mathbb {Z}}^d$ together with an ellipsoid $\\mathcal {E}\\subset {\\mathbb {R}}^d$ centered at the origin, and it outputs the intersection $\\Gamma \\cap \\mathcal {E}$ .",
"In fact we use here our own implementation of (FP) which is described in Remark REF ."
],
[
"Elliptic logarithms.",
"Information on the computation of the elliptic logarithms can be found in Zagier [167] for real elliptic logarithms and in Pethő et al [126] for $p$ -adic elliptic logarithms, see also Tzanakis [159].",
"To compute the elliptic logarithms we use Sage which in turn is based on Pari.",
"Our normalizations of the real and $p$ -adic elliptic logarithm in Sections REF and REF respectively coincide with the normalizations of Pari.",
"Rafael von Känel, MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany Current affiliation: Princeton University, Mathematics, Fine Hall, NJ 08544-1000, USA E-mail address: [email protected] Benjamin Matschke, MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany Current affiliation: Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351, cours de la Libération, 33405 Talence, France E-mail address: [email protected]"
]
] | 1605.06079 |
[
[
"Drinfeld double of $GL_n$ and generalized cluster structures"
],
[
"Abstract We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward of the Poisson bracket on the dual Poisson--Lie group."
],
[
"Introduction",
"The connection between cluster algebras and Poisson structures is documented in [15].",
"Among the most important examples in which this connection has been utilized are coordinate rings of double Bruhat cells in semisimple Lie groups equipped with (the restriction of) the standard Poisson-Lie structure.",
"In [15], we applied our technique of constructing a cluster structure compatible with a given Poisson structure in this situation and recovered the cluster structure built in [2].",
"The standard Poisson-Lie structure is a particular case of a family of Poisson-Lie structures corresponding to quasi-triangular Lie bialgebras.",
"Such structures are associated with solutions to the classical Yang-Baxter equation.",
"Their complete classification was obtained by Belavin and Drinfeld in [1] using certain combinatorial data defined in terms of the corresponding root system.",
"In [16] we conjectured that any such solution gives rise to a compatible cluster structure on the Lie group and provided several examples supporting this conjecture.",
"Recently [17], [19], we constructed the cluster structure corresponding to the Cremmer–Gervais Poisson structure in $GL_n$ for any $n$ .",
"As we established in [19], the construction of cluster structures on a simple Poisson-Lie group ${\\mathcal {G}}$ relies on properties of the Drinfeld double $D({\\mathcal {G}})$ .",
"Moreover, in the Cremmer-Gervais case generalized determinantal identities on which cluster transformations are modeled can be extended to identities valid in the double.",
"It is not too far-fetched then to suspect that there exists a cluster structure on $D({\\mathcal {G}})$ compatible with the Poisson-Lie bracket induced by the Poisson-Lie bracket on ${\\mathcal {G}}$ .",
"However, an interesting phenomenon was observed even in the first nontrivial example of $D(GL_2)$ : although we were able to construct a log-canonical regular coordinate chart in terms of which all standard coordinate functions are expressed as (subtraction free) Laurent polynomials, it is not possible to define cluster transformations in such a way that all cluster variables which one expects to be mutable are transformed into regular functions.",
"This problem is resolved, however, if one is allowed to use generalized cluster transformations previously considered in [14], [15] and, more recently, axiomatized in [6].",
"In this paper, we describe such a generalized cluster structure on the Drinfeld double in the case of the standard Poisson-Lie group $GL_n$ .",
"Using this structure, one can recover the standard cluster structure on $GL_n$ .",
"Furthermore, there is a well-known map from the dual Poisson–Lie group $GL_n^*\\subset D(GL_n)$ to an open dense subset $GL_n^\\dag $ of $GL_n$ (see [20]).",
"The push-forward of the Poisson structure on $GL_n^*$ extends from $GL_n^\\dag $ to the whole $GL_n$ (the resulting Poisson structure is not Poisson–Lie).",
"We define a generalized cluster structure on $GL_n$ compatible with the latter Poisson structure, using a special seed of the generalized cluster structure on $D(GL_n)$ .",
"Note that the log-canonical basis suggested in [4] is different from the one constructed here and does not lead to a regular cluster structure.",
"Section contains the definition of a generalized cluster structure of geometric type, as well as properties of such structures in the rings of regular functions of algebraic varieties.",
"This includes several basic results on compatibility with Poisson brackets and toric actions.",
"These statements are natural extensions of the corresponding results on ordinary cluster structures, and their proofs are obtained via minor modifications.",
"Section also contains basic information on Poisson–Lie groups and the corresponding Drinfeld doubles.",
"Section contain the main results of the paper.",
"The initial log-canonical basis is described in Section REF , the corresponding quiver is presented in Section REF , and the generalized exchange relation is given in Section REF .",
"The main results are stated in Section REF and include two theorems: Theorem REF treats the generalized cluster structure on the Drinfeld double, while Theorem REF deals with the generalized cluster structure on $GL_n$ compatible with the push-forward of the bracket on the dual Poisson–Lie group.",
"Section REF explains the main steps in the proof of Theorem REF , and shows how it yields Theorem REF .",
"Section has an independent value.",
"Recall that originally upper cluster algebras were defined over the ring of Laurent polynomials in stable variables.",
"In this section we prove that upper cluster algebras over subrings of this ring retain all properties of usual upper cluster algebras, and under certain coprimality conditions coincide with the intersection of rings of Laurent polynomials in a finite collection of clusters.",
"The log-canonicity of the suggested initial basis is proved in Section .",
"The proofs rely on invariance properties of the elements of the basis.",
"The first part of Theorem REF is proved in Section .",
"We start with studying a left–right toric action by diagonal matrices, see Section REF .",
"In Section REF we prove the compatibility of our generalized cluster structure with the standard Poisson–Lie structure on the Drinfeld double.",
"To prove the regularity of the generalized cluster structure, we verify the coprimality conditions in the initial cluster and its neighbors.",
"The second part of Theorem REF is proved in Section .",
"Section contains proofs of various auxiliary statements in matrix theory that are used in other sections.",
"A short preliminary version of this paper was published in [18]."
],
[
"Generalized cluster structures of geometric type and compatible Poisson brackets",
"Let $\\widetilde{B}=(b_{ij})$ be an $N\\times (N+M)$ integer matrix whose principal part $B$ is skew-symmetrizable (recall that the principal part of a rectangular matrix is its maximal leading square submatrix).",
"Let ${\\mathcal {F}}$ be the field of rational functions in $N+M$ independent variables with rational coefficients.",
"There are $M$ distinguished variables; they are denoted $x_{N+1},\\dots ,x_{N+M}$ and called stable.",
"A stable variable $x_{j}$ is called isolated if $b_{ij}=0$ for $1\\le i\\le N$ .",
"The coefficient group is a free multiplicative abelian group of Laurent monomials in stable variables, and its integer group ring is $\\bar{{\\mathbb {A}}}={\\mathbb {Z}}[x_{N+1}^{\\pm 1},\\dots ,x_{N+M}^{\\pm 1}]$ (we write $x^{\\pm 1}$ instead of $x,x^{-1}$ ).",
"For each $i$ , $1\\le i\\le N$ , fix a factor $d_i$ of $\\gcd \\lbrace b_{ij}: 1\\le j\\le N\\rbrace $ .",
"A seed (of geometric type) in ${\\mathcal {F}}$ is a triple $\\Sigma =({\\bf x},\\widetilde{B},{\\mathcal {P}})$ , where ${\\bf x}=(x_1,\\dots ,x_N)$ is a transcendence basis of ${\\mathcal {F}}$ over the field of fractions of $\\bar{{\\mathbb {A}}}$ and ${\\mathcal {P}}$ is a set of $n$ strings.",
"The $i$ th string is a collection of monomials $p_{ir}\\in {\\mathbb {A}}={\\mathbb {Z}}[x_{N+1},\\dots ,x_{N+M}]$ , $0\\le r\\le d_i$ , such that $p_{i0}=p_{id_i}=1$ ; it is called trivial if $d_i=1$ , and hence both elements of the string are equal to one.",
"Matrices $B$ and ${\\widetilde{B}}$ are called the exchange matrix and the extended exchange matrix, respectively.",
"The $N$ -tuple ${\\bf x}$ is called a cluster, and its elements $x_1,\\dots ,x_N$ are called cluster variables.",
"The monomials $p_{ir}$ are called exchange coefficients.",
"We say that $\\widetilde{{\\bf x}}=(x_1,\\dots ,x_{N+M})$ is an extended cluster, and $\\widetilde{\\Sigma }=({\\widetilde{\\bf x}},\\widetilde{B},{\\mathcal {P}})$ is an extended seed.",
"Given a seed as above, the adjacent cluster in direction $k$ , $1\\le k\\le N$ , is defined by ${\\bf x}^{\\prime }=({\\bf x}\\setminus \\lbrace x_k\\rbrace )\\cup \\lbrace x^{\\prime }_k\\rbrace $ , where the new cluster variable $x^{\\prime }_k$ is given by the generalized exchange relation $x_kx^{\\prime }_k=\\sum _{r=0}^{d_k}p_{kr}u_{k;>}^r v_{k;>}^{[r]}u_{k;<}^{d_k-r}v_{k;<}^{[d_k-r]}$ with cluster $\\tau $ -monomials $u_{k;>}$ and $u_{k;<}$ , $1\\le k\\le N$ , defined by $\\begin{aligned}u_{k;>}&=\\prod \\lbrace x_i^{b_{ki}/d_k}: 1\\le i\\le N, b_{ki}>0\\rbrace , \\\\u_{k;<}&=\\prod \\lbrace x_i^{-b_{ki}/d_k}: 1\\le i\\le N, b_{ki}<0\\rbrace ,\\end{aligned}$ and stable $\\tau $ -monomials $v_{k;>}^{[r]}$ and $v_{k;<}^{[r]}$ , $1\\le k\\le N$ , $0\\le r\\le d_k$ , defined by $\\begin{aligned}v_{k;>}^{[r]}&=\\prod \\lbrace x_i^{\\lfloor rb_{ki}/d_k\\rfloor }: N+1\\le i\\le N+M, b_{ki}>0\\rbrace ,\\\\v_{k;<}^{[r]}&=\\prod \\lbrace x_i^{\\lfloor -rb_{ki}/d_k\\rfloor }: N+1\\le i\\le N+M, b_{ki}<0\\rbrace ;\\end{aligned}$ here, as usual, the product over the empty set is assumed to be equal to 1.",
"In what follows we write $v_{k;>}$ instead of $v_{k;>}^{[d_k]}$ and $v_{k;<}$ instead of $v_{k;<}^{[d_k]}$ .",
"The right hand side of (REF ) is called a generalized exchange polynomial.",
"We say that ${\\widetilde{B}}^{\\prime }$ is obtained from ${\\widetilde{B}}$ by a matrix mutation in direction $k$ if $b^{\\prime }_{ij}={\\left\\lbrace \\begin{array}{ll}-b_{ij}, & \\text{if $i=k$ or $j=k$;}\\\\b_{ij}+\\displaystyle \\frac{|b_{ik}|b_{kj}+b_{ik}|b_{kj}|}{2},&\\text{otherwise.}\\end{array}\\right.",
"}$ Note that $b_{ij}=0$ for $1\\le i\\le N$ implies $b^{\\prime }_{ij}=0$ for $1\\le i\\le N$ ; in other words, the set of isolated variables does not depend on the cluster.",
"Moreover, $\\gcd \\lbrace b_{ij}:1\\le j\\le N\\rbrace =\\gcd \\lbrace b^{\\prime }_{ij}:1\\le j\\le N\\rbrace $ , and hence the $N$ -tuple $(d_1,\\dots ,d_N)$ retains its property.",
"The exchange coefficient mutation in direction $k$ is given by $p^{\\prime }_{ir}={\\left\\lbrace \\begin{array}{ll}p_{i,d_i-r}, & \\text{if $i=k$;}\\\\p_{ir}, &\\text{otherwise.}\\end{array}\\right.",
"}$ Remark 2.1 The definition above is adjusted from an earlier definition of generalized cluster structures given in [6].",
"In that case a somewhat less involved construction was modeled on examples appearing in a study of triangulations of surfaces with orbifold points, and coefficients of exchange polynomials were assumed to be elements of an arbitrary tropical semifield.",
"In contrast, our main example forces us to consider a situation in which the coefficients have to be realized as regular functions on an underlying variety which results in a complicated definition above.",
"More exactly, if one defines coefficients $p_{i;l}$ in [6] as $p_{il}v_{i;>}^{[l]}v_{i;<}^{[d_i-l]}$ , then our exchange coefficient mutation rule (REF ) becomes a specialization of the general rule (2.5) in [6].",
"As we will explain in future publications, many examples of this sort arise in an investigation of exotic cluster structures on Poisson–Lie groups.",
"Consider Laurent monomials in stable variables $q_{ir}=\\frac{v_{i;>}^rv_{i;<}^{d_i-r}}{\\left(v_{i;>}^{[r]}v_{i;<}^{[d_i-r]}\\right)^{d_i}}, \\qquad 0\\le r\\le d_i, 1\\le i\\le N.$ By (REF ), $b_{ij}=b^{\\prime }_{ij}\\bmod d_i$ for $1\\le j\\le N+M$ .",
"Consequently, the mutation rule for $q_{ir}$ is the same as (REF ).",
"In what follows we will use Laurent monomials $\\hat{p}_{ir}=p_{ir}^{d_i}/q_{ir}$ .",
"One can rewrite the generalized exchange relation (REF ) in terms of the $\\hat{p}_{ir}$ as follows: $x_kx^{\\prime }_k=\\sum _{r=0}^{d_k}\\left(\\hat{p}_{kr}v_{k;>}^rv_{k;<}^{d_k-r}\\right)^{1/d_k} u_{k;>}^r u_{k;<}^{d_k-r};$ note that $\\left(\\hat{p}_{kr}v_{k;>}^rv_{k;<}^{d_k-r}\\right)^{1/d_k}$ is a monomial in ${\\mathbb {A}}$ for $0\\le r\\le d_k$ .",
"In certain cases, it is convenient to represent the data $({\\widetilde{B}}, d_1,\\dots ,d_N)$ by a quiver.",
"Assume that the modified extended exchange matrix $\\widehat{B}$ obtained from ${\\widetilde{B}}$ by replacing each $b_{ij}$ by $b_{ij}/d_i$ for $1\\le j\\le N$ and retaining it for $N+1\\le j\\le N+M$ has a skew-symmetric principal part; we say that the corresponding quiver $Q$ with vertex multiplicities $d_1,\\dots ,d_N$ represents $({\\widetilde{B}}, d_1,\\dots ,d_N)$ and write $\\Sigma =({\\bf x},Q,{\\mathcal {P}})$ .",
"Vertices that correspond to cluster variables are called mutable, those that correspond to stable variables are called frozen.",
"A mutable vertex with $d_i\\ne 1$ is called special, and $d_i$ is said to be its multiplicity.",
"A frozen vertex corresponding to an isolated variable is called isolated.",
"Given a seed $\\Sigma =({\\bf x},\\widetilde{B},{\\mathcal {P}})$ , we say that a seed $\\Sigma ^{\\prime }=({\\bf x}^{\\prime },\\widetilde{B}^{\\prime },{\\mathcal {P}}^{\\prime })$ is adjacent to $\\Sigma $ (in direction $k$ ) if ${\\bf x}^{\\prime }$ , $\\widetilde{B}^{\\prime }$ and ${\\mathcal {P}}^{\\prime }$ are as above.",
"Two seeds are mutation equivalent if they can be connected by a sequence of pairwise adjacent seeds.",
"The set of all seeds mutation equivalent to $\\Sigma $ is called the generalized cluster structure (of geometric type) in ${\\mathcal {F}}$ associated with $\\Sigma $ and denoted by ${{\\mathcal {G}}{\\mathcal {C}}}(\\Sigma )$ ; in what follows, we usually write ${{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}})$ , or even just ${{\\mathcal {G}}{\\mathcal {C}}}$ instead.",
"Clearly, by taking $d_i=1$ for $1\\le i\\le N$ , and hence making all strings trivial, we get an ordinary cluster structure.",
"Indeed, in this case the right hand side of the generalized exchange relation (REF ) contains two terms; furthermore, $u_{k;>}^0=u_{k;<}^0=v_{k;>}^{[0]}=v_{k;<}^{[0]}=1$ and $\\begin{aligned}u_{k;>}^1 v_{k;>}^{[1]}&=\\prod \\lbrace x_i^{b_{ki}} : 1\\le i\\le N+M, b_{ki}>0\\rbrace ,\\\\u_{k;<}^1 v_{k;<}^{[1]}&=\\prod \\lbrace x_i^{-b_{ki}} : 1\\le i\\le N+M, b_{ki}<0\\rbrace .\\end{aligned}$ Consequently, in this case (REF ) coincides with the ordinary exchange relation, while the exchange coefficient mutation (REF ) becomes trivial.",
"Similarly to the case of ordinary cluster structures, we will associate to ${{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}})$ a labeled $N$ -regular tree ${\\mathbb {T}}_N$ whose vertices correspond to seeds, and edges correspond to the adjacency of seeds.",
"Fix a ground ring $\\widehat{{\\mathbb {A}}}$ such that ${\\mathbb {A}}\\subseteq \\widehat{{\\mathbb {A}}}\\subseteq \\bar{{\\mathbb {A}}}$ .",
"We associate with ${{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}})$ two algebras of rank $N$ over $\\widehat{{\\mathbb {A}}}$ : the generalized cluster algebra ${\\mathcal {A}}={\\mathcal {A}}({{\\mathcal {G}}{\\mathcal {C}}})={\\mathcal {A}}({\\widetilde{B}},{\\mathcal {P}})$ , which is the $\\widehat{{\\mathbb {A}}}$ -subalgebra of ${\\mathcal {F}}$ generated by all cluster variables from all seeds in ${{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}})$ , and the generalized upper cluster algebra $\\overline{{\\mathcal {A}}}=\\overline{{\\mathcal {A}}}({{\\mathcal {G}}{\\mathcal {C}}})=\\overline{{\\mathcal {A}}}({\\widetilde{B}},{\\mathcal {P}})$ , which is the intersection of the rings of Laurent polynomials over $\\widehat{{\\mathbb {A}}}$ in cluster variables taken over all seeds in ${{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}})$ .",
"The generalized Laurent phenomenon [6] claims the inclusion ${\\mathcal {A}}({{\\mathcal {G}}{\\mathcal {C}}})\\subseteq \\overline{{\\mathcal {A}}}({{\\mathcal {G}}{\\mathcal {C}}})$ .",
"Remark 2.2 Note that our definition of the generalized cluster algebra is slightly more general than the one used in [6].",
"However, the proof in [6] utilizes the Caterpillar Lemma of Fomin and Zelevinsky (see [10]) and follows their standard pattern of reasoning; it can be repeated verbatim in our case as well.",
"Let $V$ be a quasi-affine variety over ${\\mathbb {C}}$ , ${\\mathbb {C}}(V)$ be the field of rational functions on $V$ , and ${\\mathcal {O}}(V)$ be the ring of regular functions on $V$ .",
"Let ${{\\mathcal {G}}{\\mathcal {C}}}$ be a generalized cluster structure in ${\\mathcal {F}}$ as above.",
"Assume that $\\lbrace f_1,\\dots ,f_{N+M}\\rbrace $ is a transcendence basis of ${\\mathbb {C}}(V)$ .",
"Then the map $\\theta : x_i\\mapsto f_i$ , $1\\le i\\le N+M$ , can be extended to a field isomorphism $\\theta : {\\mathcal {F}}_{\\mathbb {C}}\\rightarrow {\\mathbb {C}}(V)$ , where ${\\mathcal {F}}_{\\mathbb {C}}={\\mathcal {F}}\\otimes {\\mathbb {C}}$ is obtained from ${\\mathcal {F}}$ by extension of scalars.",
"The pair $({{\\mathcal {G}}{\\mathcal {C}}},\\theta )$ is called a generalized cluster structure in ${\\mathbb {C}}(V)$ , $\\lbrace f_1,\\dots ,f_{N+M}\\rbrace $ is called an extended cluster in $({{\\mathcal {G}}{\\mathcal {C}}},\\theta )$ .",
"Sometimes we omit direct indication of $\\theta $ and say that ${{\\mathcal {G}}{\\mathcal {C}}}$ is a generalized cluster structure on $V$ .",
"A generalized cluster structure $({{\\mathcal {G}}{\\mathcal {C}}},\\theta )$ is called regular if $\\theta (x)$ is a regular function for any cluster variable $x$ .",
"The two algebras defined above have their counterparts in ${\\mathcal {F}}_{\\mathbb {C}}$ obtained by extension of scalars; they are denoted ${\\mathcal {A}}_{\\mathbb {C}}$ and $\\overline{{\\mathcal {A}}}_{\\mathbb {C}}$ .",
"As it is explained in [15], the natural choice of the ground ring for ${\\mathcal {A}}_{\\mathbb {C}}$ and $\\overline{{\\mathcal {A}}}_{\\mathbb {C}}$ is $\\widehat{{\\mathbb {A}}}={\\mathbb {C}}[x_{N+1}^{\\pm 1},\\dots ,x_{N+M^{\\prime }}^{\\pm 1}, x_{N+M^{\\prime }+1},\\dots ,x_{N+M}],$ where $\\theta (x_{N+i})$ does not vanish on $V$ if and only if $1\\le i\\le M^{\\prime }$ .",
"If, moreover, the field isomorphism $\\theta $ can be restricted to an isomorphism of ${\\mathcal {A}}_{\\mathbb {C}}$ (or $\\overline{{\\mathcal {A}}}_{\\mathbb {C}}$ ) and ${\\mathcal {O}}(V)$ , we say that ${\\mathcal {A}}_{\\mathbb {C}}$ (or $\\overline{{\\mathcal {A}}}_{\\mathbb {C}}$ ) is naturally isomorphic to ${\\mathcal {O}}(V)$ .",
"The following statement is a direct corollary of the natural extension of the Starfish Lemma (Proposition 3.6 in [8]) to the case of generalized cluster structures.",
"The proof of this extension literally follows the proof of the Starfish Lemma.",
"Proposition 2.3 Let $V$ be a Zariski open subset in ${\\mathbb {C}}^{N+M}$ and ${{\\mathcal {G}}{\\mathcal {C}}}=({{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}}),\\theta )$ be a generalized cluster structure in ${\\mathbb {C}}(V)$ with $N$ cluster and $M$ stable variables such that (i) there exists an extended cluster ${\\widetilde{\\bf x}}=(x_1,\\dots ,x_{N+M})$ in ${{\\mathcal {G}}{\\mathcal {C}}}$ such that $\\theta (x_i)$ is regular on $V$ for $1\\le i\\le N+M$ , and $\\theta (x_i)$ and $\\theta (x_j)$ are coprime in ${\\mathcal {O}}(V)$ for $1\\le i\\ne j\\le N+M$ ; (ii) for any cluster variable $x_k^{\\prime }$ , $1\\le k\\le N$ , obtained via the generalized exchange relation (REF ) applied to ${\\widetilde{\\bf x}}$ , $\\theta (x_k^{\\prime })$ is regular on $V$ and coprime in ${\\mathcal {O}}(V)$ with $\\theta (x_k)$ .",
"Then ${{\\mathcal {G}}{\\mathcal {C}}}$ is a regular generalized cluster structure.",
"If additionally (iii) each regular function on $V$ belongs to $\\theta (\\overline{{\\mathcal {A}}}_{\\mathbb {C}}({{\\mathcal {G}}{\\mathcal {C}}}))$ , then $\\overline{{\\mathcal {A}}}_{\\mathbb {C}}({{\\mathcal {G}}{\\mathcal {C}}})$ is naturally isomorphic to ${\\mathcal {O}}(V)$ .",
"Remark 2.4 Conditions of the Starfish Lemma in our case are satisfied since ${\\mathcal {O}}(V)$ is a unique factorization domain.",
"Let ${\\lbrace \\cdot ,\\cdot \\rbrace }$ be a Poisson bracket on the ambient field ${\\mathcal {F}}$ , and ${{\\mathcal {G}}{\\mathcal {C}}}$ be a generalized cluster structure in ${\\mathcal {F}}$ .",
"We say that the bracket and the generalized cluster structure are compatible if any extended cluster $\\widetilde{{\\bf x}}=(x_1,\\dots ,x_{N+M})$ is log-canonical with respect to ${\\lbrace \\cdot ,\\cdot \\rbrace }$ , that is, $\\lbrace x_i,x_j\\rbrace =\\omega _{ij} x_ix_j$ , where $\\omega _{ij}\\in {\\mathbb {Z}}$ are constants for all $i,j$ , $1\\le i,j\\le N+M$ .",
"Let $\\Omega =(\\omega _{ij})_{i,j=1}^{N+M}$ .",
"The following proposition can be considered as a natural extension of Proposition 2.3 in [19].",
"The proof is similar to the proof of Theorem 4.5 in [15].",
"Proposition 2.5 Assume that ${\\widetilde{B}}\\Omega =[\\Delta \\;\\; 0]$ for a non-degenerate diagonal matrix $\\Delta $ , and all Laurent polynomials $\\hat{p}_{ir}$ are Casimirs of the bracket ${\\lbrace \\cdot ,\\cdot \\rbrace }$ .",
"Then $\\operatorname{rank}{\\widetilde{B}}=N$ , and the bracket ${\\lbrace \\cdot ,\\cdot \\rbrace }$ is compatible with ${{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}})$ .",
"The notion of compatibility extends to Poisson brackets on ${\\mathcal {F}}_{\\mathbb {C}}$ without any changes.",
"Fix an arbitrary extended cluster ${\\widetilde{\\bf x}}=(x_1,\\dots ,x_{N+M})$ and define a local toric action of rank $s$ as a map ${\\mathcal {T}}^W_{{\\bf q}}:{\\mathcal {F}}_{\\mathbb {C}}\\rightarrow {\\mathcal {F}}_{\\mathbb {C}}$ given on the generators of ${\\mathcal {F}}_{\\mathbb {C}}={\\mathbb {C}}(x_1,\\dots ,x_{N+M})$ by the formula ${\\mathcal {T}}^W_{{\\bf q}}({\\widetilde{\\bf x}})=\\left( x_i \\prod _{\\alpha =1}^s q_\\alpha ^{w_{i\\alpha }}\\right)_{i=1}^{N+M},\\qquad {\\bf q}=(q_1,\\dots ,q_s)\\in ({\\mathbb {C}}^*)^s,$ where $W=(w_{i\\alpha })$ is an integer $(N+M)\\times s$ weight matrix of full rank, and extended naturally to the whole ${\\mathcal {F}}_{\\mathbb {C}}$ .",
"Let ${\\widetilde{\\bf x}}^{\\prime }$ be another extended cluster in ${{\\mathcal {G}}{\\mathcal {C}}}$ , then the corresponding local toric action defined by the weight matrix $W^{\\prime }$ is compatible with the local toric action (REF ) if it commutes with the sequence of (generalized) cluster transformations that takes ${\\widetilde{\\bf x}}$ to ${\\widetilde{\\bf x}}^{\\prime }$ .",
"If local toric actions at all clusters are compatible, they define a global toric action ${\\mathcal {T}}_{{\\bf q}}$ on ${\\mathcal {F}}_{\\mathbb {C}}$ called a ${{\\mathcal {G}}{\\mathcal {C}}}$ -extension of the local toric action (REF ).",
"The following proposition can be viewed as a natural extension of Lemma 2.3 in [14] and is proved in a similar way.",
"Proposition 2.6 The local toric action (REF ) is uniquely ${{\\mathcal {G}}{\\mathcal {C}}}$ -extendable to a global action of $({\\mathbb {C}}^*)^s$ if ${\\widetilde{B}}W = 0$ and all Casimirs $\\hat{p}_{ir}$ are invariant under (REF )."
],
[
"Standard Poisson-Lie group ${\\mathcal {G}}$ and its Drinfeld double",
"A reductive complex Lie group ${\\mathcal {G}}$ equipped with a Poisson bracket ${\\lbrace \\cdot ,\\cdot \\rbrace }$ is called a Poisson–Lie group if the multiplication map ${\\mathcal {G}}\\times {\\mathcal {G}}\\ni (X,Y) \\mapsto XY \\in {\\mathcal {G}}$ is Poisson.",
"Denote by $\\langle \\ , \\ \\rangle $ an invariant nondegenerate form on $\\mathfrak {g}$ , and by $\\nabla ^R$ , $\\nabla ^L$ the right and left gradients of functions on ${\\mathcal {G}}$ with respect to this form defined by $\\left\\langle \\nabla ^R f(X),\\xi \\right\\rangle =\\left.\\frac{d}{dt}\\right|_{t=0}f(Xe^{t\\xi }), \\quad \\left\\langle \\nabla ^L f(X),\\xi \\right\\rangle =\\left.\\frac{d}{dt}\\right|_{t=0}f(e^{t\\xi }X)$ for any $\\xi \\in \\mathfrak {g}$ , $X\\in {\\mathcal {G}}$ .",
"Let $\\pi _{>0}, \\pi _{<0}$ be projections of $\\mathfrak {g}$ onto subalgebras spanned by positive and negative roots, $\\pi _0$ be the projection onto the Cartan subalgebra $\\mathfrak {h}$ , and let $R=\\pi _{>0} - \\pi _{<0}$ .",
"The standard Poisson-Lie bracket ${\\lbrace \\cdot ,\\cdot \\rbrace }_r$ on ${\\mathcal {G}}$ can be written as $\\lbrace f_1,f_2\\rbrace _r = \\frac{1}{2} \\left( \\left\\langle R(\\nabla ^L f_1), \\nabla ^L f_2 \\rangle - \\langle R(\\nabla ^R f_1), \\nabla ^R f_2 \\right\\rangle \\right).$ The standard Poisson–Lie structure is a particular case of Poisson–Lie structures corresponding to quasitriangular Lie bialgebras.",
"For a detailed exposition of these structures see, e. g., [5], [20] and [21].",
"Following [20], let us recall the construction of the Drinfeld double.",
"The double of $\\mathfrak {g}$ is $D(\\mathfrak {g})=\\mathfrak {g}\\oplus \\mathfrak {g}$ equipped with an invariant nondegenerate bilinear form $\\langle \\langle (\\xi ,\\eta ), (\\xi ^{\\prime },\\eta ^{\\prime })\\rangle \\rangle = \\langle \\xi , \\xi ^{\\prime }\\rangle - \\langle \\eta , \\eta ^{\\prime }\\rangle $ .",
"Define subalgebras ${\\mathfrak {d}}_\\pm $ of $D(\\mathfrak {g})$ by ${\\mathfrak {d}}_+=\\lbrace ( \\xi ,\\xi ) : \\xi \\in \\mathfrak {g}\\rbrace $ and ${\\mathfrak {d}}_-=\\lbrace (R_+(\\xi ),R_-(\\xi )) : \\xi \\in \\mathfrak {g}\\rbrace $ , where $R_\\pm \\in \\operatorname{End}\\mathfrak {g}$ is given by $R_\\pm =\\frac{1}{2} ( R \\pm {\\operatorname{Id}})$ .",
"The operator $R_D= \\pi _{{\\mathfrak {d}}_+} - \\pi _{{\\mathfrak {d}}_-}$ can be used to define a Poisson–Lie structure on $D({\\mathcal {G}})={\\mathcal {G}}\\times {\\mathcal {G}}$ , the double of the group ${\\mathcal {G}}$ , via $\\lbrace f_1,f_2\\rbrace _D = \\frac{1}{2}\\left(\\left\\langle \\left\\langle R_D({\\raisebox {2pt}{\\bigtriangledown }}^L f_1), {\\raisebox {2pt}{\\bigtriangledown }}{^L} f_2 \\right\\rangle \\right\\rangle - \\left\\langle \\left\\langle R_D({\\raisebox {2pt}{\\bigtriangledown }}^R f_1), {\\raisebox {2pt}{\\bigtriangledown }}^R f_2 \\right\\rangle \\right\\rangle \\right),$ where ${\\raisebox {2pt}{\\bigtriangledown }}^R$ and ${\\raisebox {2pt}{\\bigtriangledown }}^L$ are right and left gradients with respect to $\\langle \\langle \\cdot ,\\cdot \\rangle \\rangle $ .",
"The diagonal subgroup $\\lbrace (X,X)\\ : \\ X\\in {\\mathcal {G}}\\rbrace $ is a Poisson–Lie subgroup of $D({\\mathcal {G}})$ (whose Lie algebra is ${\\mathfrak {d}}_+$ ) naturally isomorphic to $({\\mathcal {G}},{\\lbrace \\cdot ,\\cdot \\rbrace }_r)$ .",
"The group ${\\mathcal {G}}^*$ whose Lie algebra is ${\\mathfrak {d}}_-$ is a Poisson-Lie subgroup of $D({\\mathcal {G}})$ called the dual Poisson-Lie group of ${\\mathcal {G}}$.",
"The map $D({\\mathcal {G}}) \\rightarrow {\\mathcal {G}}$ given by $(X,Y) \\mapsto U=X^{-1} Y$ induces another Poisson bracket on ${\\mathcal {G}}$ , see [20]; we denote this bracket ${\\lbrace \\cdot ,\\cdot \\rbrace }_*$ .",
"The image of the restriction of this map to ${\\mathcal {G}}^*$ is denoted ${\\mathcal {G}}^\\dag $ .",
"Symplectic leaves on $({\\mathcal {G}},{\\lbrace \\cdot ,\\cdot \\rbrace }_*)$ were studied in [7].",
"In this paper we only deal with the case of ${\\mathcal {G}}=GL_n$ .",
"In that case ${\\mathcal {G}}^\\dag $ is the non-vanishing locus of trailing principal minors $\\det U_{[i,n]}^{[i,n]}$ (here and in what follows we write $[a,b]$ to denote the set $\\lbrace i\\in {\\mathbb {Z}}: a\\le i\\le b\\rbrace $ ).",
"The bracket (REF ) takes the form $\\begin{split}\\lbrace f_1,f_2\\rbrace _D = &\\langle R_+(E_L f_1), E_L f_2\\rangle - \\langle R_+(E_R f_1), E_R f_2\\rangle \\\\&+ \\langle X\\nabla _X f_1, Y\\nabla _Y f_2\\rangle - \\langle \\nabla _X f_1\\cdot X, \\nabla _Y f_2 \\cdot Y\\rangle ,\\end{split}$ where $\\nabla _X f=\\left(\\frac{\\partial f}{\\partial x_{ji}}\\right)_{i,j=1}^n$ , $\\nabla _Y f=\\left(\\frac{\\partial f}{\\partial y_{ji}}\\right)_{i,j=1}^n$ , and $E_R = X \\nabla _X + Y\\nabla _Y, \\quad E_L = \\nabla _X X+ \\nabla _Y Y.$ So, (REF ) can be rewritten as $\\begin{split}\\lbrace f_1,f_2\\rbrace _D = &\\left\\langle R_+(E_L f_1), E_L f_2\\right\\rangle - \\left\\langle R_+(E_R f_1), E_R f_2\\right\\rangle \\\\&+ \\left\\langle E_R f_1, Y\\nabla _Y f_2\\right\\rangle - \\left\\langle E_L f_1, \\nabla _Y f_2 \\cdot Y\\right\\rangle .\\end{split}$ Further, if $\\varphi $ is a function of $U=X^{-1}Y$ then $E_L\\varphi =[\\nabla \\varphi ,U]$ and $E_R\\varphi =0$ , and hence for an arbitrary function $f$ on $D(GL_n)$ one has $\\lbrace \\varphi ,f\\rbrace _D = \\left\\langle R_+([\\nabla \\varphi ,U]), E_L f\\right\\rangle - \\left\\langle [\\nabla \\varphi ,U], \\nabla _Y f \\cdot Y\\right\\rangle .$"
],
[
"Log-canonical basis",
"Let $(X,Y)$ be a point in the double $D(GL_n)$ .",
"For $k,l\\ge 1$ , $k+l\\le n-1$ define a $(k+l)\\times (k+l)$ matrix $F_{kl}=F_{kl}(X,Y)=\\left[\\begin{array}{cc}X^{[n-k+1,n]} & Y^{[n-l+1,n]}\\end{array}\\right]_{[n-k-l+1,n]}.$ For $1\\le j\\le i\\le n$ define an $(n-i+1)\\times (n-i+1)$ matrix $G_{ij}=G_{ij}(X)=X_{[i,n]}^{[j,j+n-i]}.$ For $1\\le i\\le j\\le n$ define an $(n-j+1)\\times (n-j+1)$ matrix $H_{ij}=H_{ij}(Y)=Y_{[i,i+n-j]}^{[j,n]}.$ For $k,l\\ge 1$ , $k+l\\le n$ define an $n\\times n$ matrix $\\Phi _{kl}=\\Phi _{kl}(X,Y)=\\left[\\begin{array}{ccccc}(U^0)^{[n-k+1,n]}& U^{[n-l+1,n]} & (U^2)^{[n]} & \\dots & (U^{n-k-l+1})^{[n]}\\end{array}\\right]$ where $U=X^{-1}Y$ .",
"Remark 3.1 Note that the definition of $F_{kl}$ can be extended to the case $k+l=n$ yielding $F_{n-l,l}=X\\Phi _{n-l,l}$ .",
"One can also identify $F_{0 l}$ with $H_{n-l+1,n-l+1}$ and $F_{k 0}$ with $G_{n-k+1,n-k+1}$ .",
"Finally, it will be convenient, for technical reasons, to identify $G_{i,i+1}$ with $F_{n-i,1}$ .",
"Denote $f_{kl}=\\det F_{kl},\\quad g_{ij}=\\det G_{ij},\\quad h_{ij}=\\det H_{ij},\\\\\\varphi _{kl}=s_{kl}(\\det X)^{n-k-l+1}\\det \\Phi _{kl},$ $2n^2-n+1$ functions in total.",
"Here $s_{kl}$ is a sign defined as follows: $s_{kl}={\\left\\lbrace \\begin{array}{ll}(-1)^{k(l+1)}\\qquad \\qquad \\qquad \\qquad \\text{for $n$ even},\\\\(-1)^{(n-1)/2+k(k-1)/2+l(l-1)/2} \\quad \\text{for $n$ odd}.\\end{array}\\right.",
"}$ It is periodic in $k+l$ with period 4 for $n$ odd and period 2 for $n$ even; $s_{n-l,l}=1$ ; $s_{n-l-1,l}=(-1)^l$ for $n$ odd and $s_{n-l-1,l}=(-1)^{l+1}$ for $n$ even; $s_{n-l-2,l}=-1$ for $n$ odd; $s_{n-l-3,l}=(-1)^{l+1}$ for $n$ odd.",
"Note that the pre-factor in the definition of $\\varphi _{kl}$ is needed to obtain a regular function in matrix entries of $X$ and $Y$ .",
"Consider the polynomial $\\det (X+\\lambda Y)=\\sum _{i=0}^n \\lambda ^i s_ic_i(X,Y),$ where $s_i=(-1)^{i(n-1)}$ .",
"Note that $c_0(X,Y)=\\det X=g_{11}$ and $c_n(X,Y)=\\det Y=h_{11}$ .",
"Theorem 3.2 The family of functions $F_n=\\lbrace g_{ij}, h_{ij}, f_{kl}, \\varphi _{kl}, c_1,\\ldots , c_{n-1} \\rbrace $ forms a log-canonical coordinate system with respect to the Poisson-Lie bracket (REF ) on $D(GL_n)$ ."
],
[
"Initial quiver",
"The modified extended exchange matrix $\\widehat{B}$ has a skew-symmetric principal part, and hence can be represented by a quiver.",
"The quiver $Q_n$ contains $2n^2 - n +1$ vertices labeled by the functions $g_{ij}, h_{ij}, f_{kl}, \\varphi _{kl}$ in the log-canonical basis $F_n$ .",
"The functions $c_1, \\ldots , c_{n-1}$ correspond to isolated vertices.",
"They are not connected to any of the other vertices, and will be not shown on figures.",
"The vertex $\\varphi _{11}$ is the only special vertex, and its multiplicity equals $n$ .",
"The vertices $g_{i1}$ , $1\\le i\\le n$ , and $h_{1j}$ , $1\\le j\\le n$ , are frozen, so $N=2n^2-3n+1$ and $M=3n-1$ .",
"Below we describe $Q_n$ assuming that $n>2$ .",
"Vertex $\\varphi _{kl}$ , $k,l\\ne 1$ , $k+l<n$ , has degree 6.",
"The edges pointing from $\\varphi _{kl}$ are $\\varphi _{kl}\\rightarrow \\varphi _{k+1,l}$ , $\\varphi _{kl}\\rightarrow \\varphi _{k,l-1}$ , and $\\varphi _{kl}\\rightarrow \\varphi _{k-1,l+1}$ ; the edges pointing towards $\\varphi _{kl}$ are $\\varphi _{k,l+1}\\rightarrow \\varphi _{kl}$ , $\\varphi _{k+1,l-1}\\rightarrow \\varphi _{kl}$ , and $\\varphi _{k-1,l}\\rightarrow \\varphi _{kl}$ .",
"Vertex $\\varphi _{kl}$ , $k,l\\ne 1$ , $k+l=n$ , has degree 4.",
"The edges pointing from $\\varphi _{kl}$ in this case are $\\varphi _{kl}\\rightarrow \\varphi _{k,l-1}$ and $\\varphi _{kl}\\rightarrow f_{k-1,l}$ ; the edges pointing towards $\\varphi _{kl}$ are $\\varphi _{k-1,l}\\rightarrow \\varphi _{kl}$ and $f_{k,l-1}\\rightarrow \\varphi _{kl}$ .",
"Vertex $\\varphi _{k1}$ , $k\\in [2,n-2]$ , has degree 4.",
"The edges pointing from $\\varphi _{k1}$ are $\\varphi _{k1}\\rightarrow \\varphi _{k-1,2}$ and $\\varphi _{k1}\\rightarrow \\varphi _{1k}$ ; the edges pointing towards $\\varphi _{k1}$ are $\\varphi _{k2}\\rightarrow \\varphi _{k1}$ and $\\varphi _{1,k-1}\\rightarrow \\varphi _{k1}$ .",
"Note that for $k=2$ the vertices $\\varphi _{k-1,2}$ and $\\varphi _{1k}$ coincide, hence for $n>3$ there are two edges pointing from $\\varphi _{21}$ to $\\varphi _{12}$ .",
"Vertex $\\varphi _{n-1,1}$ has degree 5.",
"The edges pointing from $\\varphi _{n-1,1}$ are $\\varphi _{n-1,1}\\rightarrow \\varphi _{1,n-1}$ , $\\varphi _{n-1,1}\\rightarrow f_{n-2,1}$ , and $\\varphi _{n-1,1}\\rightarrow g_{11}$ ; the edges pointing towards $\\varphi _{n-1,1}$ are $\\varphi _{1,n-2}\\rightarrow \\varphi _{n-1,1}$ and $g_{22}\\rightarrow \\varphi _{n-1,1}$ .",
"Vertex $\\varphi _{1l}$ , $l\\in [2,n-2]$ , has degree 6.",
"The edges pointing from $\\varphi _{1l}$ are $\\varphi _{1l}\\rightarrow \\varphi _{2l}$ , $\\varphi _{1l}\\rightarrow \\varphi _{1,l-1}$ , and $\\varphi _{1l}\\rightarrow \\varphi _{l+1,1}$ ; the edges pointing towards $\\varphi _{1l}$ are $\\varphi _{1,l+1}\\rightarrow \\varphi _{1l}$ , $\\varphi _{2,l-1}\\rightarrow \\varphi _{1l}$ , and $\\varphi _{l1}\\rightarrow \\varphi _{1l}$ .",
"Vertex $\\varphi _{1,n-1}$ has degree 5.",
"The edges pointing from $\\varphi _{1,n-1}$ are $\\varphi _{1,n-1}\\rightarrow \\varphi _{1,n-2}$ and $\\varphi _{1,n-1}\\rightarrow h_{22}$ ; the edges pointing towards $\\varphi _{1,n-1}$ are $\\varphi _{n-1,1}\\rightarrow \\varphi _{1,n-1}$ , $f_{1,n-2}\\rightarrow \\varphi _{1,n-1}$ , and $h_{11}\\rightarrow \\varphi _{1,n-1}$ .",
"Finally, $\\varphi _{11}$ is the special vertex.",
"It has degree 4, and the corresponding edges are $\\varphi _{12}\\rightarrow \\varphi _{11}$ , $g_{11}\\rightarrow \\varphi _{11}$ and $\\varphi _{11}\\rightarrow \\varphi _{21}$ , $\\varphi _{11}\\rightarrow h_{11}$ .",
"Vertex $f_{kl}$ , $k+l<n$ , has degree 6.",
"The edges pointing from $f_{kl}$ are $f_{kl}\\rightarrow f_{k+1,l-1}$ , $f_{kl}\\rightarrow f_{k,l+1}$ , and $f_{kl}\\rightarrow f_{k-1,l}$ ; the edges pointing towards $f_{kl}$ are $f_{k-1,l+1}\\rightarrow f_{kl}$ , $f_{k,l-1}\\rightarrow f_{kl}$ , and $f_{k+1,l}\\rightarrow f_{kl}$ .",
"To justify this description in the extreme cases (such that $k+l=n-1$ , $k=1$ , and $l=1$ ), we use the identification of Remark REF .",
"Figure: Quiver Q 4 Q_4Vertex $g_{ij}$ , $i\\ne n$ , $j\\ne 1$ , has degree 6.",
"The edges pointing from $g_{ij}$ are $g_{ij}\\rightarrow g_{i+1,j+1}$ , $g_{ij}\\rightarrow g_{i,j-1}$ , and $g_{ij}\\rightarrow g_{i-1,j}$ ; the edges pointing towards $g_{ij}$ are $g_{i,j+1}\\rightarrow g_{ij}$ , $g_{i-1,j-1}\\rightarrow g_{ij}$ , and $g_{i+1,j}\\rightarrow g_{ij}$ (for $i=j$ we use the identification of Remark REF ).",
"Vertex $g_{nj}$ , $j\\ne 1$ , has degree 4.",
"The edges pointing from $g_{nj}$ are $g_{nj}\\rightarrow g_{n-1,j}$ and $g_{nj}\\rightarrow g_{n,j-1}$ ; the edges pointing towards $g_{nj}$ are $g_{n-1,j-1}\\rightarrow g_{nj}$ and $g_{n,j+1}\\rightarrow g_{nj}$ (for $j=n$ we use the identification of Remark REF ).",
"Vertex $g_{i1}$ , $i\\ne 1$ , $i\\ne n$ , has degree 2, and the corresponding edges are $g_{i1}\\rightarrow g_{i+1,2}$ and $g_{i2}\\rightarrow g_{i1}$ .",
"Vertex $g_{11}$ has degree 3, and the corresponding edges are $\\varphi _{n-1,1}\\rightarrow g_{11}$ and $g_{11}\\rightarrow g_{21}$ , $g_{11}\\rightarrow \\varphi _{11}$ .",
"Finally, $g_{n1}$ has degree 1, and the only edge is $g_{n2}\\rightarrow g_{n1}$ .",
"Vertex $h_{ij}$ , $i\\ne 1$ , $j\\ne n$ , $i\\ne j$ , has degree 6.",
"The edges pointing from $h_{ij}$ are $h_{ij} \\rightarrow h_{i,j-1}$ , $h_{ij} \\rightarrow h_{i-1,j}$ , and $h_{ij} \\rightarrow h_{i+1,j+1}$ ; the edges pointing towards $h_{ij}$ are $h_{i+1,j}\\rightarrow h_{ij}$ , $h_{i,j+1}\\rightarrow h_{ij}$ , and $h_{i-1,j-1}\\rightarrow h_{ij} $ .",
"Vertex $h_{ii}$ , $i\\ne 1$ , $i\\ne n$ , has degree 4.",
"The edges pointing from $h_{ii}$ are $h_{ii}\\rightarrow f_{1,n-i}$ and $h_{ii}\\rightarrow h_{i-1,i}$ ; the edges pointing to $h_{ii}$ are $f_{1,n-i+1}\\rightarrow h_{ii}$ and $h_{i,i+1}\\rightarrow h_{ii}$ .",
"Vertex $h_{in}$ , $i\\ne 1$ , $i\\ne n$ , has degree 4.",
"The edges pointing from $h_{in}$ are $h_{in}\\rightarrow h_{i,n-1}$ and $h_{in}\\rightarrow h_{i-1,n}$ ; the edges pointing towards $h_{in}$ are $h_{i+1,n}\\rightarrow h_{in}$ and $h_{i-1,n-1}\\rightarrow h_{in}$ .",
"Vertex $h_{1j}$ , $j\\ne 1$ , $j\\ne n$ , has degree 2, and the corresponding edges are $h_{1j}\\rightarrow h_{2,j+1}$ and $h_{2j}\\rightarrow h_{1j}$ .",
"Vertex $h_{nn}$ has degree 3.",
"The edges pointing from $h_{nn}$ are $h_{nn}\\rightarrow g_{nn}$ and $h_{nn}\\rightarrow h_{n-1,n}$ ; the edge pointing to $h_{nn}$ is $f_{11}\\rightarrow h_{nn}$ .",
"Vertex $h_{11}$ has degree 2, and the corresponding edges are $h_{11}\\rightarrow \\varphi _{1,n-1}$ and $\\varphi _{11}\\rightarrow h_{11}$ .",
"Finally, $h_{1n}$ has degree 1, and the only edge is $h_{2n}\\rightarrow h_{1n}$ .",
"The quiver $Q_4$ is shown in Fig.",
"REF .",
"The frozen vertices are shown as squares, the special vertex is shown as a hexagon, isolated vertices are not shown.",
"Certain arrows are dashed; this does not have any particular meaning, and just makes the picture more comprehensible.",
"One can identify in $Q_n$ four “triangular regions\" associated with four families $\\lbrace g_{ij}\\rbrace $ , $\\lbrace h_{ij}\\rbrace $ , $\\lbrace f_{kl}\\rbrace $ , $\\lbrace \\varphi _{kl}\\rbrace $ .",
"We will call vertices in these regions $g$ -, $h$ -, $f$ - and $\\varphi $ -vertices, respectively.",
"It is easy to see that $Q_4$ , as well as $Q_n$ for any $n$ , can be embedded into a torus.",
"The case $n=2$ is special.",
"In this case there are only three types of vertices: $g$ , $h$ , and $\\varphi $ .",
"The quiver $Q_2$ is shown in Fig.",
"REF .",
"Figure: Quiver Q 2 Q_2Remark 3.3 On the diagonal subgroup $\\lbrace (X,X) : X\\in GL_n\\rbrace $ of $D(GL_n)$ , $g_{ii} = h_{ii}$ for $1\\le i\\le n$ , and functions $f_{kl}$ and $\\varphi _{kl}$ vanish identically.",
"Accordingly, vertices in $Q_n$ that correspond to $f_{kl}$ and $\\varphi _{kl}$ are erased and, for $1\\le i\\le n$ , vertices corresponding to $g_{ii}$ and $h_{ii}$ are identified.",
"As a result, one recovers a seed of the cluster structure compatible with the standard Poisson-Lie structure on $GL_n$ , see [15].",
"Remark 3.4 At this point, we should emphasize a connection between the data $(F_n, Q_n)$ and particular seeds that give rise to the standard cluster structures on double Bruhat cells ${\\mathcal {G}}^{e,w_0}$ , ${\\mathcal {G}}^{w_0,e}$ for ${\\mathcal {G}}=GL_n$ and $w_0$ the longest element in the corresponding Weyl group.",
"We will frequently explore this connection in what follows.",
"Consider, in particular, the subquiver $Q_n^h$ of $Q_n$ associated with functions $h_{ij}$ in which, in addition to vertices $h_{1i}$ , we also view vertices $h_{ii}$ as frozen.",
"Restricted to upper triangular matrices, the family $\\lbrace h_{ij}\\rbrace $ together with the quiver $Q_n^h$ defines an initial seed for the cluster structure on ${\\mathcal {G}}^{e,w_0}$ .",
"This can be seen, for example, by applying the construction of Section 2.4 in [2] using a reduced word $1 2 1 3 2 1\\ldots (n-1) (n-2) \\ldots 2 1$ for $w_0$ .",
"This leads to the cluster formed by all dense minors that involve the first row.",
"The seed we are interested in is then obtained via the transformation $B \\mapsto W_0 B^T W_0$ applied to upper triangular matrices.",
"Similarly, the family $\\lbrace g_{ij}\\rbrace $ restricted to lower-triangular matrices together with the quiver $Q_n^g$ obtained from $Q_n$ in the same way as $Q_n^h$ (and isomorphic to it) defines an initial seed for ${\\mathcal {G}}^{w_0,e}$ .",
"As explained in Remark 2.20 in [2], in the case of the standard cluster structures on ${\\mathcal {G}}^{e,w_0}$ or ${\\mathcal {G}}^{w_0,e}$ , the cluster algebra and the upper cluster algebra coincide.",
"This implies, in particular, that in every cluster for this cluster structure, each matrix entry of an upper/lower triangular matrix is expressed as a Laurent polynomial in cluster variables which is polynomial in stable variables.",
"Furthermore, using similar considerations and the invariance under right multiplication by unipotent lower triangular matrices of column-dense minors that involve the first column, it is easy to conclude that each such minor has a Laurent polynomial expression in terms of dense minors involving the first column and, moreover, each leading dense principal minor enters this expression polynomially.",
"Both these properties will be utilized below."
],
[
"Exchange relations",
"We define the set ${\\mathcal {P}}_n$ of strings for $Q_n$ that contains only one nontrivial string $\\lbrace p_{1r}\\rbrace $ , $0\\le r\\le n$ .",
"It corresponds to the vertex $\\varphi _{11}$ of multiplicity $n$ , and $p_{1r}=c_r$ , $1\\le r\\le n-1$ .",
"The strings corresponding to all other vertices are trivial.",
"Consequently, the generalized exchange relation at the vertex $\\varphi _{11}$ for $n>2$ is expected to look as follows: $\\varphi _{11}\\varphi ^{\\prime }_{11}=\\sum _{r=0}^nc_r\\varphi _{21}^r\\varphi _{12}^{n-r}.$ Indeed, such a relation exists in the ring of regular functions on $D(GL_n)$ , and is given by the following proposition.",
"Proposition 3.5 For any $n>2$ , $\\det ((-1)^{n-1}\\varphi _{12}X+\\varphi _{21}Y)=\\varphi _{11}P^*_n,$ where $P^*_n$ is a polynomial in the entries of $X$ and $Y$ .",
"For $n=2$ , relation (REF ) is replaced by $\\det (-y_{22}X+x_{22}Y)=\\varphi _{11}\\left|\\begin{matrix}y_{21} & y_{22}\\cr x_{21} & x_{22}\\end{matrix}\\right|.$"
],
[
"Statement of main results",
"Let $n\\ge 2$ .",
"Theorem 3.6 (i) The extended seed $\\widetilde{\\Sigma }_n=(F_n,Q_n,{\\mathcal {P}}_n)$ defines a generalized cluster structure ${{\\mathcal {G}}{\\mathcal {C}}}_n^D$ in the ring of regular functions on $D(GL_n)$ compatible with the standard Poisson–Lie structure on $D(GL_n)$ .",
"(ii) The corresponding generalized upper cluster algebra $\\overline{{\\mathcal {A}}}({{\\mathcal {G}}{\\mathcal {C}}}_n^D)$ over $\\widehat{{\\mathbb {A}}}={\\mathbb {C}}[g_{11}^{\\pm 1},g_{21},\\dots , g_{n1},h_{11}^{\\pm 1}, h_{12},\\dots ,h_{1n}, c_1,\\dots , c_{n-1}]$ is naturally isomorphic to the ring of regular functions on $D(GL_n)$ .",
"Remark 3.7 1.",
"Since the only stable variables that do not vanish on $D(GL_n)$ are $g_{11}=\\det X$ and $h_{11}=\\det Y$ , the ground ring in (ii) above is a particular case of (REF ).",
"In fact, it follows from the proof that a stronger statement holds: (i) ${{\\mathcal {G}}{\\mathcal {C}}}_n^D$ extends to a regular generalized cluster structure on $\\operatorname{Mat}_n\\times \\operatorname{Mat}_n$ ; (ii) the generalized upper cluster algebra over $\\widehat{{\\mathbb {A}}}={\\mathbb {C}}[g_{11}^{\\pm 1},g_{21},\\dots , g_{n1},h_{11}, h_{12},\\dots ,h_{1n}, c_1,\\dots , c_{n-1}]$ is naturally isomorphic to the ring of regular function on $GL_n\\times \\operatorname{Mat}_n$ .",
"2.",
"For $n=2$ the obtained generalized cluster structure has a finite type.",
"Indeed, the principal part of the exchange matrix for the cluster shown in Fig.",
"REF has a form $\\left(\\begin{array}{rrr}0 & 2 & -2 \\\\-1 & 0 & 1 \\\\1 & -1 & 0\\\\\\end{array}\\right).$ The mutation of this matrix in direction 2 transforms it into $\\left(\\begin{array}{rrr}0 & -2 & 0 \\\\1 & 0 &-1 \\\\0 & 1 & 0\\\\\\end{array}\\right),$ and its Cartan companion is a Cartan matrix of type $B_3$ .",
"Therefore, by [6], the generalized cluster structure has type $B_3$ .",
"This implies, in particular, that its exchange graph is the 1-skeleton of the 3-dimensional cyclohedron (also known as the Bott–Taubes polytope), and its cluster complex is the 3-dimensional polytope polar dual to the cyclohedron (see [9] for further details).",
"3.",
"It follows immediately from Theorem REF (i) that the extended seed obtained from $\\widetilde{\\Sigma }_n$ by deleting functions $\\det X$ and $\\det Y$ from $F_n$ , deleting the corresponding vertices from $Q_n$ and restricting relation (REF ) to $\\det X=\\det Y=1$ defines a generalized cluster structure in the ring of regular functions on $D(SL_n)$ compatible with the standard Poisson–Lie structure on $D(SL_n)$ .",
"Moreover, by Theorem REF (ii), the corresponding generalized upper cluster algebra is naturally isomorphic to the ring of regular functions on $D(SL_n)$ .",
"Using Theorem REF , we can construct a generalized cluster structure on $GL_n^\\dag $ .",
"For $U\\in GL_n^\\dag $ , denote $\\psi _{kl}(U) = s_{kl}\\det \\Phi _{kl}(U)$ , where $s_{kl}$ are the signs defined in Section REF .",
"The initial extended cluster $F^\\dag _n$ for $GL_n^\\dag $ consists of functions $\\psi _{kl}(U)$ , $k,l\\ge 1$ , $k+l\\le n$ , $(-1)^{(n-i)(j-i)}h_{ij}(U)$ , $2\\le i\\le j\\le n$ , $h_{11}(U)=\\det U$ , and $c_i(\\mathbf {1}, U)$ , $1\\le i\\le n-1$ .",
"To obtain the initial seed for $GL_n^\\dag $ , we apply a certain sequence ${\\mathcal {S}}$ of cluster transformations to the initial seed for $D(GL_n)$ .",
"This sequence does not involve vertices associated with functions $\\psi _{kl}$ .",
"The resulting cluster ${\\mathcal {S}}(F_n)$ contains a subset $\\lbrace \\left(\\det X\\right)^{\\nu (f)}f : f \\in F^\\dag _n\\rbrace $ with $\\nu (\\psi _{kl})=n-k-l+1$ and $\\nu (h_{ij})=1$ .",
"These functions are attached to a subquiver $Q^\\dag _n$ in the resulting quiver ${\\mathcal {S}}(Q_n)$ , which is isomorphic to the subquiver of $Q_n$ formed by vertices associated with functions $\\varphi _{kl}, f_{ij}$ and $h_{ii}$ , see Fig.",
"REF .",
"Functions $h_{ii}(U)$ are declared stable variables, $c_i(\\mathbf {1},U)$ remain isolated.",
"See Theorem REF below for more details.",
"Figure: Quiver Q 4 † Q^\\dag _4All exchange relations defined by mutable vertices of $Q^\\dag _n$ are homogeneous in $\\det X$ .",
"This allows us to use $(F^\\dag _n, Q_n^\\dag ,{\\mathcal {P}}_n)$ as an initial seed for $GL_n^\\dag $ .",
"The generalized exchange relation associated with the cluster variable $\\psi _{11}$ now takes the form $\\det ((-1)^{n-1}\\psi _{12}\\mathbf {1}+\\psi _{21}U)=\\psi _{11}\\Pi ^*_n$ , where $\\Pi ^*_n$ is a polynomial in the entries of $U$ .",
"Theorem 3.8 (i) The extended seed $(F_n^\\dag ,Q_n^\\dag ,{\\mathcal {P}}_n)$ defines a generalized cluster structure ${{\\mathcal {G}}{\\mathcal {C}}}_n^{\\dag }$ in the ring of regular functions on $GL_n^\\dag $ compatible with ${\\lbrace \\cdot ,\\cdot \\rbrace }_*$ .",
"(ii) The corresponding generalized upper cluster algebra over $\\widehat{{\\mathbb {A}}}={\\mathbb {C}}[h_{11}(U)^{\\pm 1}, \\dots , h_{nn}(U)^{\\pm 1}, c_1(\\mathbf {1},U),\\dots , c_{n-1}(\\mathbf {1},U)]$ is naturally isomorphic to the ring of regular functions on $GL_n^\\dag $ .",
"Remark 3.9 1.",
"It follows from Remark REF that a stronger statement holds, similarly to Remark REF .1: (i) ${{\\mathcal {G}}{\\mathcal {C}}}_n^\\dag $ extends to a regular generalized cluster structure on $\\operatorname{Mat}_n$ ; (ii) the generalized upper cluster algebra over $\\widehat{{\\mathbb {A}}}={\\mathbb {C}}[h_{11}(U),\\dots , h_{nn}(U), c_1(\\mathbf {1},U),\\dots , c_{n-1}(\\mathbf {1},U) ]$ is naturally isomorphic to the ring of regular function on $\\operatorname{Mat}_n$ .",
"2.",
"Let $\\mathcal {V}_n$ be the intersection of $SL_n^\\dag $ with a generic conjugation orbit in $SL_n$ .",
"This variety plays a role in a rigorous mathematical description of Coulomb branches in 4D gauge theories.",
"The generalized cluster structure ${{\\mathcal {G}}{\\mathcal {C}}}_n^{\\dag }$ descends to $\\mathcal {V}_n$ if one fixes the values of $c_1(\\mathbf {1},U),\\dots ,c_{n-1}(\\mathbf {1},U)$ .",
"The existence of a cluster structure on $\\mathcal {V}_n$ was suggested by D. Gaiotto (A. Braverman, private communication)."
],
[
"The outline of the proof",
"We start with defining a local toric action of right and left multiplication by diagonal matrices, and use Proposition REF to check that this action can be extended to a global one.",
"This fact is then used in the proof of the compatibility assertion in Theorem REF (i), which is based on Proposition REF .",
"As a byproduct, we get that the extended exchange matrix of ${{\\mathcal {G}}{\\mathcal {C}}}_n^D$ is of full rank.",
"Next, we have to check conditions (i)–(iii) of Proposition REF .",
"The regularity condition in (i) follows from Theorem REF and the explicit description of the basis.",
"The coprimality condition in (i) is a corollary of the following stronger statement.",
"Theorem 3.10 All functions in $F_n$ are irreducible polynomials in matrix entries.",
"We then establish the regularity and coprimality conditions in (ii), which completes the proof of Theorem REF (i).",
"To prove Theorem REF (ii), it is left to check condition (iii) of Proposition REF .",
"The usual way to do that consists in applying Theorem 3.21 from [15] which claims that for cluster structures of geometric type with an exchange matrix of full rank, the upper cluster algebra coincides with the upper bound at any cluster.",
"It remains to choose an appropriate set of generators in ${\\mathcal {O}}(V)$ and to check that each element of this set can be represented as a Laurent polynomial in some fixed cluster and in all its neighbors.",
"We will have to extend the above result in three directions: 1) to upper cluster algebras over $\\widehat{{\\mathbb {A}}}$ , as opposed to upper cluster algebras over $\\bar{{\\mathbb {A}}}$ ; 2) to more general neighborhoods of a vertex in ${\\mathbb {T}}_N$ , as opposed to the stars of vertices; 3) to generalized cluster structures of geometric type, as opposed to ordinary cluster structures.",
"Let ${{\\mathcal {G}}{\\mathcal {C}}}={{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}})$ be a generalized cluster structure as defined in Section REF , and let $L$ be the number of isolated variables in ${{\\mathcal {G}}{\\mathcal {C}}}$ .",
"For the $i$ th nontrivial string in ${\\mathcal {P}}$ , define a $(d_i-1)\\times L$ integer matrix ${\\widetilde{B}}(i)$ : the $r$ th row of ${\\widetilde{B}}(i)$ contains the exponents of isolated variables in the exchange coefficient $p_{ir}$ (recall that $p_{ir}$ are monomials).",
"Following [11], we call a nerve an arbitrary subtree of ${\\mathbb {T}}_N$ on $N+1$ vertices such that all its edges have different labels.",
"A star of a vertex in ${\\mathbb {T}}_N$ is an example of a nerve.",
"Given a nerve $\\mathcal {N}$ , we define an upper bound $\\overline{{\\mathcal {A}}}(\\mathcal {N})$ as the intersection of the rings of Laurent polynomials $\\widehat{{\\mathbb {A}}}[{\\bf x}^{\\pm 1}]$ taken over all seeds in $\\mathcal {N}$ .",
"We prove the following theorem that seems to be interesting in its own right.",
"Theorem 3.11 Let ${\\widetilde{B}}$ be a skew-symmetrizable matrix of full rank, and let $\\operatorname{rank}{\\widetilde{B}}(i)=d_i-1$ for any nontrivial string in ${\\mathcal {P}}$ .",
"Then the upper bounds $\\overline{{\\mathcal {A}}}(\\mathcal {N})$ do not depend on the choice of $\\mathcal {N}$ , and hence coincide with the generalized upper cluster algebra $\\overline{{\\mathcal {A}}}({\\widetilde{B}},{\\mathcal {P}})$ over $\\widehat{{\\mathbb {A}}}$ .",
"We then proceed as follows.",
"First, we choose the $2n^2$ matrix entries of $X$ and $Y$ as the generating set of the ring of regular functions on $D(GL_n)$ .",
"Then we prove the following result.",
"Theorem 3.12 Each matrix entry of $X$ is either a stable variable or a cluster variable in ${{\\mathcal {G}}{\\mathcal {C}}}_n^D$ .",
"To treat the remaining part of the generating set we consider a special nerve $\\mathcal {N}_0$ in the tree ${\\mathbb {T}}_{(n-1)(2n-1)}$ .",
"First of all, we design a sequence ${\\mathcal {S}}$ of cluster transformations that takes the initial extended seed $\\widetilde{\\Sigma }_n$ to a new extended seed $\\widetilde{\\Sigma }^{\\prime }_n={\\mathcal {S}}(\\widetilde{\\Sigma }_n)=({\\mathcal {S}}(F_n),{\\mathcal {S}}(Q_n),{\\mathcal {S}}({\\mathcal {P}}_n))$ having the following properties.",
"Let $Q_n^\\dag $ and $F_n^\\dag $ be as defined in Section REF , and $U=X^{-1}Y$ .",
"Theorem 3.13 There exists a sequence ${\\mathcal {S}}$ of cluster transformations such that (i) ${\\mathcal {S}}({\\mathcal {P}}_n)={\\mathcal {P}}_n$ ; (ii) ${\\mathcal {S}}(Q_n)$ contains a subquiver $Q_n^{\\prime }$ isomorphic to $Q_n^\\dag $ ; (iii) the functions in ${\\mathcal {S}}(F_n)$ assigned to the vertices of $Q_n^{\\prime }$ constitute the set $\\left\\lbrace \\left(\\det X^{n-k-l+1}\\psi _{kl}(U)\\right)_{k,l\\ge 1, k+l\\le n}, \\left(\\det X\\cdot h_{ij}(U)\\right)_{2\\le i\\le j\\le n}, \\det X\\cdot h_{11}(U)\\right\\rbrace $ ; (iv) the only vertices in $Q^{\\prime }_n$ connected with the rest of vertices in ${\\mathcal {S}}(Q_n)$ are those associated with $\\det X\\cdot h_{ii}$ , $2\\le i\\le n$ , and $\\varphi _{11}$ .",
"As an immediate corollary we get Theorem REF (i).",
"The nerve $\\mathcal {N}_0$ contains the seed $\\widetilde{\\Sigma }^{\\prime }_n$ , a seed $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ adjacent to $\\widetilde{\\Sigma }^{\\prime }_n$ , and a seed $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ adjacent to $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ .",
"Besides, it contains $2(n-1)^2$ seeds adjacent to $\\widetilde{\\Sigma }^{\\prime }_n$ and distinct from $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ , and $n-3$ seeds adjacent to $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ and distinct from $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ .",
"A more detailed description of $\\mathcal {N}_0$ is given in Section REF below.",
"We then prove Theorem 3.14 Each matrix entry of $U=X^{-1}Y$ multiplied by an appropriate power of $\\det X$ belongs to the upper bound $\\overline{{\\mathcal {A}}}(\\mathcal {N}_0)$ .",
"Consequently each matrix entry of $Y=XU$ belongs to $\\overline{{\\mathcal {A}}}(\\mathcal {N}_0)$ .",
"It remains to note that ${\\widetilde{B}}(1)$ is the $(n-1)\\times (n-1)$ identity matrix.",
"Therefore, all conditions in Theorem REF are satisfied, and we get the proofs of Theorems REF (ii) and REF (ii)."
],
[
"Generalized upper cluster algebras of geometric type over $\\hat{{\\mathbb {A}}}$",
"Let ${{\\mathcal {G}}{\\mathcal {C}}}={{\\mathcal {G}}{\\mathcal {C}}}({\\widetilde{B}},{\\mathcal {P}})$ be a generalized cluster structure as defined in Section REF , and let ${\\mathbb {A}}\\subseteq \\widehat{{\\mathbb {A}}}\\subseteq \\bar{{\\mathbb {A}}}$ be the corresponding rings.",
"The goal of this section is to prove Theorem REF .",
"We start with the following statement, which is an extension of the standard result on the coincidence of upper bounds (see e.g.",
"[15]).",
"Theorem 4.1 If the generalized exchange polynomials are coprime in ${\\mathbb {A}}[x_1,\\dots ,x_{N}]$ for any seed in ${{\\mathcal {G}}{\\mathcal {C}}}$ , then the upper bounds $\\overline{{\\mathcal {A}}}(\\mathcal {N})$ do not depend on the choice of the nerve $\\mathcal {N}$ , and hence coincide with the upper cluster algebra $\\overline{{\\mathcal {A}}}({\\widetilde{B}},{\\mathcal {P}})$ over $\\widehat{{\\mathbb {A}}}$ .",
"Let us consider first the case $N=1$ .",
"In this case everything is exactly the same as in the standard situation.",
"Namely, we consider two adjacent clusters ${\\bf x}=\\lbrace x_1\\rbrace $ and ${\\bf x}_1=\\lbrace x_1^{\\prime }\\rbrace $ and the exchange relation $x_1x_1^{\\prime }=P_1$ , where $P_1\\in {\\mathbb {A}}$ .",
"The same reasoning as in Lemma 3.15 from [15] yields $\\widehat{{\\mathbb {A}}}[x_1^{\\pm 1}]\\cap \\widehat{{\\mathbb {A}}}[(x_1^{\\prime })^{\\pm 1}]=\\widehat{{\\mathbb {A}}}[x_1,x_1^{\\prime }].$ As a corollary, for general $N$ one gets $\\widehat{{\\mathbb {A}}}[x_1^{\\pm 1},x_2^{\\pm 1},\\dots ,x_N^{\\pm 1}]\\cap \\widehat{{\\mathbb {A}}}[(x_1^{\\prime })^{\\pm 1},x_2^{\\pm 1},\\dots ,x_N^{\\pm 1}]=\\widehat{{\\mathbb {A}}}[x_1,x_1^{\\prime },x_2^{\\pm 1},\\dots ,x_N^{\\pm 1}].$ The latter relation is obtained from the one for $N=1$ via replacing ${\\mathbb {A}}$ with ${\\mathbb {A}}[x_2,\\dots ,x_N]$ and the ground ring $\\widehat{{\\mathbb {A}}}$ with $\\widehat{{\\mathbb {A}}}[x_2^{\\pm 1},\\dots ,x_N^{\\pm 1}]$ .",
"Let now $N=2$ .",
"Note that ${\\mathbb {T}}_2$ is an infinite path, and hence all nerves are just two-pointed stars.",
"Let ${\\bf x}=\\lbrace x_1,x_2\\rbrace $ be an arbitrary cluster, ${\\bf x}_1=\\lbrace x_1^{\\prime },x_2\\rbrace $ and ${\\bf x}_2=\\lbrace x_1,x_2^{\\prime }\\rbrace $ be the two adjacent clusters obtained via generalized exchange relations $x_1x_1^{\\prime }=P_1$ and $x_2x_2^{\\prime }=P_2$ with $P_1\\in {\\mathbb {A}}[x_2]$ and $P_2\\in {\\mathbb {A}}[x_1]$ .",
"Besides, let ${\\bf x}_3=\\lbrace x_1^{\\prime },x_2^{\\prime \\prime }\\rbrace $ be the cluster obtained from ${\\bf x}_1$ via the generalized exchange relation $x_2x_2^{\\prime \\prime }=\\bar{P}_2$ with $\\bar{P}_2\\in {\\mathbb {A}}[x_1^{\\prime }]$ .",
"Let $\\mathcal {N}$ be the nerve ${\\bf x}_1$ —${\\bf x}$ —${\\bf x}_2$ , and $\\mathcal {N}_1$ be the nerve consisting of the clusters ${\\bf x}_1$ —${\\bf x}$ —${\\bf x}_3$ .",
"The following statement is an analog of Lemma 3.19 in [15].",
"Lemma 4.2 Assume that $P_1$ and $P_2$ are coprime in ${\\mathbb {A}}[x_1,x_2]$ and $P_1$ and $\\bar{P}_2$ are coprime in ${\\mathbb {A}}[x_1^{\\prime },x_2]$ .",
"Then $\\overline{{\\mathcal {A}}}(\\mathcal {N})=\\overline{{\\mathcal {A}}}(\\mathcal {N}_1)$ .",
"The proof differs substantially from the proof of Lemma 3.19, since we are not allowed to invert monomials in $\\widehat{{\\mathbb {A}}}$ .",
"It is enough to prove the inclusion $\\overline{{\\mathcal {A}}}(\\mathcal {N})\\subseteq \\overline{{\\mathcal {A}}}(\\mathcal {N}_1)$ , since the opposite inclusion is obtained by switching roles between ${\\bf x}$ and ${\\bf x}_1$ .",
"By (REF ), we have $\\overline{{\\mathcal {A}}}(\\mathcal {N})=\\widehat{{\\mathbb {A}}}[x_1,x_1^{\\prime },x_2^{\\pm 1}]\\cap \\widehat{{\\mathbb {A}}}[x_1^{\\pm 1},(x_2^{\\prime })^{\\pm 1}],\\quad \\overline{{\\mathcal {A}}}(\\mathcal {N}_1)=\\widehat{{\\mathbb {A}}}[x_1,x_1^{\\prime },x_2^{\\pm 1}]\\cap \\widehat{{\\mathbb {A}}}[(x_1^{\\prime })^{\\pm 1},(x_2^{\\prime \\prime })^{\\pm 1}].$ Let $y\\in \\widehat{{\\mathbb {A}}}[x_1,x_1^{\\prime },x_2^{\\pm 1}]$ ; expand $y$ as a Laurent polynomial in $x_2$ .",
"Each term of this expansion containing a non-negative power of $x_2$ belongs to $\\widehat{{\\mathbb {A}}}[(x_1^{\\prime })^{\\pm 1},(x_2^{\\prime \\prime })^{\\pm 1}]$ , so we have to consider only $y$ of the form $y=\\sum _{k=1}^a \\frac{Q^{\\prime }_k+Q_k}{x_2^k}, \\qquad a\\ge 1,$ with $Q_k\\in \\widehat{{\\mathbb {A}}}[x_1]$ , $Q^{\\prime }_k\\in \\widehat{{\\mathbb {A}}}[x_1^{\\prime }]$ .",
"We can treat $y$ as above in two different ways.",
"On the one hand, by substituting $x_1^{\\prime }=P_1/x_1$ , it can be considered as an element in $\\widehat{{\\mathbb {A}}}[x_1^{\\pm 1},x_2^{\\pm 1}]$ and written as $y=\\sum _{k\\le a}\\frac{R_k}{x_1^{\\delta _k}x_2^k}$ with $R_k\\in \\widehat{{\\mathbb {A}}}[x_1]$ and $\\delta _k\\ge 0$ .",
"Imposing the condition $y\\in \\widehat{{\\mathbb {A}}}[x_1^{\\pm 1},(x_2^{\\prime })^{\\pm 1}]$ , we get $R_k=P_2^kS_k$ for $k> 0$ and some $S_k\\in \\widehat{{\\mathbb {A}}}[x_1]$ .",
"Note that each summand in (REF ) with $k\\le 0$ belongs to $\\widehat{{\\mathbb {A}}}[x_1^{\\pm 1},(x_2^{\\prime })^{\\pm 1}]$ automatically.",
"On the other hand, by substituting $x_1=P_1/x_1^{\\prime }$ , $y$ can be considered as an element in $\\widehat{{\\mathbb {A}}}[(x_1^{\\prime })^{\\pm 1},x_2^{\\pm 1}]$ and written as $y=\\sum _{k\\le a}\\frac{R_k^{\\prime }}{(x_1^{\\prime })^{\\delta _k^{\\prime }}x_2^k}$ with $R_k^{\\prime }\\in \\widehat{{\\mathbb {A}}}[x_1^{\\prime }]$ and $\\delta _k^{\\prime }\\ge 0$ .",
"Note that $R_k^{\\prime }$ can be restored via $R_l$ and $\\delta _l$ , $k\\le l\\le a$ .",
"We will prove that $\\bar{P}_2^k$ divides $R_k^{\\prime }$ in $\\widehat{{\\mathbb {A}}}[x_1^{\\prime }]$ for any $k>0$ .",
"This would mean that each summand in (REF ) belongs to $\\widehat{{\\mathbb {A}}}[(x_1^{\\prime })^{\\pm 1},(x_2^{\\prime \\prime })^{\\pm 1}]$ , and hence $\\overline{{\\mathcal {A}}}(\\mathcal {N})\\subseteq \\overline{{\\mathcal {A}}}(\\mathcal {N}_1)$ as claimed above.",
"Assume first that $\\hat{b}_{12}=\\hat{b}_{21}=0$ in the modified exchange matrix $\\hat{B}$ , which means that $P_1, P_2\\in {\\mathbb {A}}$ and $\\bar{P}_2=P_2$ .",
"Rewrite an arbitrary term $T_k=R_k/(x_1^{\\delta _k}x_2^k)$ , $k> 0$ , in (REF ) as an element in $\\widehat{{\\mathbb {A}}}[(x_1^{\\prime })^{\\pm 1},x_2^{\\pm 1}]$ via substituting $x_1=P_1/x_1^{\\prime }$ .",
"Recall that $R_k$ is divisible by $P_2^k$ , hence $T_k=\\frac{(x_1^{\\prime })^{\\delta _k}P_2^k S_k\\vert _{x_1\\leftarrow P_1/x_1^{\\prime }}}{P_1^{\\delta _k}x_2^k}=\\frac{\\bar{P}_2^k \\bar{S}_k}{(x_1^{\\prime })^{\\gamma _k}{P_1^{\\delta _k}x_2^k}}$ for some $\\gamma _k\\ge 0$ and $\\bar{S}_k\\in \\widehat{{\\mathbb {A}}}[x_1^{\\prime }]$ .",
"Comparing the latter expression with (REF ), we see that $\\gamma _k=\\delta _k^{\\prime }$ and $R_k^{\\prime }=\\bar{P}_2^k \\bar{S}_k/P_1^{\\delta _k}$ .",
"Since $\\bar{P}_2$ and $P_1$ are coprime, this means that $\\bar{P}_2^k$ divides $R_k^{\\prime }$ in $\\widehat{{\\mathbb {A}}}[x_1^{\\prime }]$ .",
"Assume now that $\\hat{b}_{12}=b> 0$ (otherwise $\\hat{b}_{21}>0$ , and we proceed in the same way with $P_2$ instead of $P_1$ ).",
"Then one can rewrite $P_1$ as $P_1=P_{10}+x_2^bP_{11}$ with $P_{10}$ is a monomial in ${\\mathbb {A}}$ and $P_{11}\\in {\\mathbb {A}}[x_2]$ is not divisible by $x_2$ .",
"Consider an arbitrary term $T_k=R_k/(x_1^{\\delta _k}x_2^k)$ , $k> 0$ , in (REF ) as an element in $\\widehat{{\\mathbb {A}}}[(x_1^{\\prime })^{\\pm 1}]((x_2))$ via substituting $x_1=(P_{10}+x_2^bP_{11})/x_1^{\\prime }$ and expanding the result in the Taylor series in $x_2$ .",
"Similarly to the previous case, we get $T_k=\\sum _{j=0}^k\\frac{P_2^{k-j}\\vert _{x_1\\leftarrow P_{10}/x_1^{\\prime }}\\hat{S}_j}{j!P_{10}^{\\delta _k+j}x_2^{k-j}}+\\sum _{j>k}\\frac{\\hat{S}_jx_2^{j-k}}{j!P_{10}^{\\delta _k+j}}$ for some $\\hat{S}_j\\in \\widehat{{\\mathbb {A}}}[x_1^{\\prime }]$ .",
"Since $y\\in \\hat{{\\mathcal {A}}}[(x_1^{\\prime })^{\\pm 1},x_2^{\\pm 1}]$ , we conclude that the infinite sums above contribute only finitely many terms to $y=\\sum _{k\\le a} T_k$ .",
"By (REF ), any term of these finitely many automatically belongs to $\\hat{{\\mathcal {A}}}[(x^{\\prime }_1)^{\\pm 1},(x^{\\prime \\prime }_2)^{\\pm 1}]$ .",
"To treat the finite sum in $T_k$ we note that $\\frac{(x_1^{\\prime })^{{\\operatorname{deg}}_{x_1}P_2}P_2\\vert _{x_1\\leftarrow P_{10}/x_1^{\\prime }}}{\\bar{P}_2}$ is a monomial in ${\\mathbb {A}}$ .",
"So, the finite sum can be rewritten as $\\sum _{j=0}^k\\frac{\\bar{P}_2^{k-j}\\bar{S}_j}{j!",
"(x_1^{\\prime })^{\\gamma _j}P_{10}^{\\delta _k+j}x_2^{k-j}}$ for some $\\gamma _j\\ge 0$ and $\\bar{S}_j\\in \\widehat{{\\mathbb {A}}}[x_1^{\\prime }]$ .",
"Comparing the latter expression with (REF ), we get $\\frac{R_k^{\\prime }}{(x_1^{\\prime })^{\\delta ^{\\prime }_k}}=\\bar{P}_2^k\\sum _{j=0}^{a-k}\\frac{\\bar{S}_j}{j!",
"(x_1^{\\prime })^{\\gamma _{j+k}}P_{10}^{\\delta _{j+k}+j}}.$ Note that $\\bar{P}_2$ and $P_{10}$ are coprime, since $P_{10}$ is a monomial and $\\bar{P}_2$ does not have monomial factors, which means that $\\bar{P}_2^k$ divides $R_k^{\\prime }$ in $\\widehat{{\\mathbb {A}}}[x_1^{\\prime }]$ .",
"In the case of an arbitrary $N$ one can use Lemma REF to reshape nerves while preserving the upper bounds.",
"Namely, let $\\mathcal {N}$ be a nerve, and $v_1,v_2,v_3\\in \\mathcal {N}$ be three vertices such that $v_1$ is adjacent to $v_2$ and $v_2$ is the unique vertex adjacent to $v_3$ .",
"Consider the nerve $\\mathcal {N}^{\\prime }$ that does not contain $v_3$ , contains a new vertex $v_3^{\\prime }$ adjacent only to $v_1$ , and otherwise is identical to $\\mathcal {N}$ ; the edge between $v_1$ and $v_3^{\\prime }$ in $\\mathcal {N}^{\\prime }$ bears the same label as the edge between $v_2$ and $v_3$ in $\\mathcal {N}$ .",
"A single application of Lemma REF with $\\widehat{{\\mathbb {A}}}$ replaced by $\\widehat{{\\mathbb {A}}}[x_3^{\\pm 1},\\dots ,x_N^{\\pm 1}]$ shows that $\\overline{{\\mathcal {A}}}(\\mathcal {N})=\\overline{{\\mathcal {A}}}(\\mathcal {N}^{\\prime })$ .",
"Clearly, any two nerves can be connected via a sequence of such transformations, and the result follows.",
"To complete the proof of Theorem REF , it remains to establish the following result.",
"Lemma 4.3 Let ${\\widetilde{B}}$ be a skew-symmetrizable matrix of full rank, and let $\\operatorname{rank}{\\widetilde{B}}(i)=d_i-1$ for any nontrivial string in ${\\mathcal {P}}$ .",
"Then the generalized exchange polynomials are coprime in ${\\mathbb {A}}[x_1,\\dots ,x_{N}]$ for any seed in ${{\\mathcal {G}}{\\mathcal {C}}}$ .",
"We follow the proof of Lemma 3.24 from [15] with minor modifications.",
"Fix an arbitrary seed $\\Sigma =({\\widetilde{\\bf x}},{\\widetilde{B}},{\\mathcal {P}})$ , and let $P_i$ be the generalized exchange polynomial corresponding to the $i$ th cluster variable.",
"Assume first that there exist $j$ and $j^{\\prime }$ such that $b_{ij}>0$ and $b_{ij^{\\prime }}<0$ .",
"We want to define the weights of the variables that make $P_i$ into a quasihomogeneous polynomial.",
"Put $w(x_j)=1/b_{ij}$ , $w(x_{j^{\\prime }})=-1/b_{ij^{\\prime }}$ .",
"If $j,j^{\\prime }\\le N$ , put $w(x_k)=0$ for $k\\ne j,j^{\\prime }$ .",
"Otherwise, put $w(x_k)=0$ for all remaining cluster variables and all remaining stable non-isolated variables.",
"Finally, define the weights of isolated variables from the equations $w(\\hat{p}_{ir})=0$ , $1\\le r\\le d_i-1$ .",
"The condition on the rank of ${\\widetilde{B}}(i)$ guarantees that these equations possess a unique solution.",
"Now (REF ) shows that this weight assignment turns $P_i$ into a quasihomogeneous polynomial of weight one.",
"Let $P_i=P^{\\prime }P^{\\prime \\prime }$ for some nontrivial polynomials $P^{\\prime }$ and $P^{\\prime \\prime }$ , then they both are quasihomogeneous with respect to the weights defined above, and each one of them contains exactly one monomial in variables entering $u_{i;>}, v_{i;>}$ , and exactly one monomial in variables entering $u_{i;<}, v_{i;<}$ .",
"Consider these two monomials in $P^{\\prime }$ .",
"Let $\\delta _j$ and $\\delta _{j^{\\prime }}$ be the degrees of $x_j$ and $x_{j^{\\prime }}$ in these two monomials, respectively.",
"Then the quasihomogeneity condition implies $\\delta _j/b_{ij}=-\\delta _{j^{\\prime }}/b_{ij^{\\prime }}$ .",
"Moreover, for any $j^{\\prime \\prime }\\ne j, j^{\\prime }$ such that $b_{ij^{\\prime \\prime }}>0$ (or $b_{ij^{\\prime \\prime }}<0$ ) a similar procedure gives $\\delta _{j^{\\prime \\prime }}/b_{ij^{\\prime \\prime }}=-\\delta _{j^{\\prime }}/b_{ij^{\\prime }}$ (or $\\delta _j/b_{ij}=-\\delta _{j^{\\prime \\prime }}/b_{ij^{\\prime \\prime }}$ , respectively.",
"This means that the $i$ th row of ${\\widetilde{B}}$ can be restored from the exponents of variables entering the above two monomials by dividing them by a constant.",
"Consequently, if $P_i$ and $P_j$ possess a nontrivial common factor, the corresponding rows of ${\\widetilde{B}}$ are proportional, which contradicts the full rank assumption.",
"If all nonzero entries in the $i$ th row have the same sign, we proceed in a similar way.",
"Namely, if there exist $j,j^{\\prime }$ such that $b_{ij}, b_{ij^{\\prime }}\\ne 0$ , we put $w(x_j)=1/|b_{ij}|$ , $w(x_{j^{\\prime }})=-1/|b_{ij^{\\prime }}|$ .",
"The weights of other variables are defined in the same way as above.",
"This makes $P_i$ into a quasihomogeneous polynomial of weight zero, and the result follows.",
"The case when there exists a unique $j$ such that $b_{ij}\\ne 0$ is trivial."
],
[
"Proof of Theorem ",
"The proof exploits various invariance properties of functions in $F_n$ .",
"First, we need some preliminary lemmas.",
"Let a bilinear form $\\langle \\cdot , \\cdot \\rangle _0$ on $\\mathfrak {gl}_n$ be defined as $\\langle A ,B \\rangle _0 = \\langle \\pi _0(A) , \\pi _0(B) \\rangle $ .",
"Lemma 5.1 Let $g(X), h(Y), f(X,Y), \\varphi (X,Y)$ be functions with the following invariance properties: $\\begin{aligned}g(X)&=g(N_+X), \\quad h(Y)=h(YN_-), \\\\f(X,Y) &= f(N_+X N_- ,N_+Y N^{\\prime }_-), \\quad \\varphi (X,Y)= \\varphi (A X N_-, A Y N_-),\\end{aligned}$ where $A$ is an arbitrary element of $GL_n$ , $N_+$ is an arbitrary unipotent upper-triangular element and $N_-, N^{\\prime }_-$ are arbitrary unipotent lower-triangular elements.",
"Then $&\\lbrace g, h\\rbrace _D = \\frac{1}{2} \\langle X\\nabla _X g, Y\\nabla _Y h\\rangle _0 - \\frac{1}{2} \\langle \\nabla _X g\\cdot X, \\nabla _Y h \\cdot Y\\rangle _0, \\\\&\\lbrace f , g\\rbrace _D = \\frac{1}{2} \\langle E_L f , \\nabla _X g\\cdot X\\rangle _0 - \\frac{1}{2} \\langle E_R f , X\\nabla _X g\\rangle _0, \\\\&\\lbrace h, f\\rbrace _D = \\frac{1}{2} \\langle \\nabla _Y h \\cdot Y , E_L f\\rangle _0 - \\frac{1}{2} \\langle Y\\nabla _Y h, E_R f \\rangle _0, \\\\&\\lbrace \\varphi , f\\rbrace _D = \\frac{1}{2} \\langle E_L \\varphi , \\nabla _X f\\cdot X\\rangle _0 - \\frac{1}{2}\\langle E_L \\varphi , \\nabla _Y f\\cdot Y \\rangle _0, \\\\&\\lbrace \\varphi , g\\rbrace _D = \\frac{1}{2} \\langle E_L \\varphi , \\nabla _X g\\cdot X \\rangle _0, \\\\&\\lbrace \\varphi , h\\rbrace _D = -\\frac{1}{2} \\langle E_L \\varphi , \\nabla _Y h\\cdot Y \\rangle _0,$ where $E_L$ and $E_R$ are given by (REF ).",
"From (REF ), we obtain $ X\\nabla _X g, E_R f \\in \\mathfrak {b}_+$ , $ \\nabla _Y h\\cdot Y, \\nabla _X f\\cdot X, \\nabla _Y f\\cdot Y, E_L \\varphi \\in \\mathfrak {b}_-$ , $E_R \\varphi =0$ .",
"Taking into account that $R_+(\\xi )=\\frac{1}{2}\\pi _0(\\xi )$ for $\\xi \\in \\mathfrak {b}_-$ and that $\\mathfrak {b}_\\pm \\perp \\mathfrak {n}_\\pm $ with respect to $\\langle \\cdot ,\\cdot \\rangle $ , the result follows from (REF ),(REF ).",
"Lemma 5.2 Let $g(X), h(Y), f(X,Y), \\varphi (X,Y)$ be functions as in Lemma REF .",
"Assume, in addition, that $g$ and $h$ are homogeneous with respect to right and left multiplication of their arguments by arbitrary diagonal matrices and that $f$ and $\\varphi $ are homogeneous with respect to right and left multiplication of $X, Y$ by the same pair of diagonal matrices.",
"Then all Poisson brackets ${\\lbrace \\cdot ,\\cdot \\rbrace }_D$ among functions $\\log g$ , $\\log h$ , $\\log f$ , $\\log \\varphi $ are constant.",
"The homogeneity of $g(X)$ with respect to the left multiplication by diagonal matrices implies that there exists a diagonal element $\\xi $ such that for any diagonal $h$ and any $X$ , $g(\\exp (h) X) = \\exp \\langle h, \\xi \\rangle g(X)$ .",
"The infinitesimal version of this property reads $\\pi _0(X \\nabla \\log g(X)) = \\xi $ .",
"A similar argument shows that diagonal projections of all elements needed to compute Poisson brackets between $\\log g$ , $\\log h$ , $\\log f$ , $\\log \\varphi $ using formulas of Lemma REF are constant diagonal matrices, and the claim follows.",
"Lemmas REF , REF show that any four functions $f_{ij}, g_{kl}, h_{\\alpha \\beta }, \\varphi _{\\mu \\nu }$ are log-canonical.",
"Indeed, it is clear from definitions in Section REF that $\\begin{aligned}g_{ij}(X)&=g_{ij}(N_+X), \\quad g_{ii}(X)=g_{ii}(N_+XN_-), \\\\h_{ij}(Y)&=h_{ij}(YN_-), \\quad h_{ii}(Y)=h_{ii}(N_+YN_-),\\\\f_{kl}(X,Y) &= f_{kl}(N_+X N_- ,N_+Y N^{\\prime }_-),\\\\\\tilde{\\varphi }_{kl}(X,Y) &= \\tilde{\\varphi }_{kl}(A X N_-, A Y N_-),\\end{aligned}$ where $\\tilde{\\varphi }_{kl} = \\det \\Phi _{kl}$ , and so the corresponding invariance properties in (REF ) are satisfied for any function taken in any of these four families.",
"Besides, all these functions possess the homogeneity property as in Lemma REF as well.",
"For a generic element $X \\in GL_n$ , consider its Gauss factorization $X=X_{>0} X_0 X_{<0}$ with $X_{<0}$ unipotent lower-triangular, $X_0$ diagonal and $X_{>0}$ unipotent upper-triangular elements.",
"Sometimes it will be convenient to use notations $X_{\\le 0} = X_0 X_{<0}$ and $X_{\\ge 0} = X_{>0} X_0$ .",
"Taking $N_+=(X_{>0})^{-1}$ in the first relation in (REF ), $N_-=(Y_{<0})^{-1}$ in the second relation, and $N_+=(Y_{>0})^{-1}$ , $N_-=(X_{<0})^{-1}$ , $N^{\\prime }_-=(Y_{<0})^{-1}$ in the third relation, one gets $\\begin{aligned}g_{ij}(X)&=g_{ij}(X_{\\le 0}),\\\\h_{ij}(Y)&=h_{ij}(Y_{\\ge 0}), \\\\f_{kl}(X,Y) &= f_{kl}\\left((Y_{>0})^{-1}X_{\\ge 0},Y_0\\right)\\\\&= h_{n-l+1,n-l+1}(Y) h_{n-k-l+1,n-k+1}\\left((Y_{>0})^{-1}X_{\\ge 0}\\right).\\end{aligned}$ Next, we need to prove log-canonicity within each of the four families.",
"The following lemma is motivated by the third formula in (REF ).",
"Lemma 5.3 The almost everywhere defined map $Z : (D(GL_n), {\\lbrace \\cdot ,\\cdot \\rbrace }_D)\\rightarrow (GL_n, {\\lbrace \\cdot ,\\cdot \\rbrace }_r)$ given by $(X,Y) \\mapsto Z= Z(X,Y)=(Y_{>0})^{-1}X_{\\ge 0}$ is Poisson.",
"Denote $\\pi _{\\ge 0}=\\pi _{>0}+\\pi _0$ and $\\pi _{\\le 0}=\\pi _{<0}+\\pi _0$ .",
"We start by computing the variation $\\begin{aligned}\\delta Z &= (Y_{>0})^{-1}\\left(\\delta X_{\\ge 0} - \\delta Y_{>0} (Y_{>0})^{-1}X_{\\ge 0}\\right)=Z (X_{\\ge 0})^{-1} \\delta X_{\\ge 0} - (Y_{>0})^{-1} \\delta Y_{>0} Z\\\\&=Z \\pi _{\\ge 0}\\left( (X_{\\ge 0})^{-1} \\delta X (X_{<0})^{-1}\\right) -\\pi _{>0}\\left( (Y_{>0})^{-1} \\delta Y (Y_{\\le 0})^{-1}\\right) Z.\\end{aligned}$ Then for a smooth function $f$ on $GL_n$ we have $\\delta f(Z(X,Y))= \\left\\langle \\nabla f,\\delta Z\\right\\rangle \\\\=\\left\\langle (X_{<0})^{-1} \\pi _{\\le 0} ( \\nabla f \\cdot Z) (X_{\\ge 0})^{-1}, \\delta X \\right\\rangle - \\left\\langle (Y_{\\le 0})^{-1} \\pi _{<0} ( Z \\nabla f ) (Y_{>0})^{-1}, \\delta Y \\right\\rangle .$ Therefore, if we denote $\\tilde{f}(X,Y) = f\\circ Z(X,Y)$ then $&X \\nabla _X \\tilde{f} = \\operatorname{Ad}_{X_{\\ge 0}} \\pi _{\\le 0} (\\nabla f \\cdot Z),\\\\&Y \\nabla _Y \\tilde{f} = - \\operatorname{Ad}_{Y_{>0}} \\pi _{<0} (Z \\nabla f ),\\\\&\\nabla _X \\tilde{f}\\cdot X = \\operatorname{Ad}_{(X_{<0})^{-1}} \\pi _{\\le 0} (\\nabla f \\cdot Z) \\in \\mathfrak {b}_-,\\\\&\\nabla _Y \\tilde{f} \\cdot Y= - \\operatorname{Ad}_{(Y_{\\le 0})^{-1}} \\pi _{<0} (Z \\nabla f ) \\in \\mathfrak {n}_-,$ and $E_R \\tilde{f} &= \\operatorname{Ad}_{Y_{>0}} \\left( \\operatorname{Ad}_Z \\pi _{\\le 0} (\\nabla f \\cdot Z) - \\pi _{<0} (Z \\nabla f )\\right)\\\\&= \\operatorname{Ad}_{Y_{>0}} \\left( Z \\nabla f - \\operatorname{Ad}_Z \\pi _{>0} (\\nabla f \\cdot Z) - \\pi _{<0} (Z \\nabla f )\\right)\\\\&=\\operatorname{Ad}_{Y_{>0}} \\left(\\pi _{\\ge 0} (Z \\nabla f ) - \\operatorname{Ad}_Z \\pi _{>0} (\\nabla f \\cdot Z) \\right)\\in \\mathfrak {b}_+.$ Plugging into (REF ) we obtain $\\lbrace f_1\\circ Z, f_1\\circ Z\\rbrace _D&= \\frac{1}{2} \\langle \\nabla f_1\\cdot Z, \\nabla f_2\\cdot Z\\rangle _0- \\frac{1}{2} \\langle Z\\nabla f_1, Z\\nabla f_2\\rangle _0 + \\langle X\\nabla _X \\tilde{f}_1, Y\\nabla _Y \\tilde{f}_2\\rangle \\\\&= \\frac{1}{2} \\langle \\nabla f_1\\cdot Z, \\nabla f_2\\cdot Z\\rangle _0- \\frac{1}{2} \\langle Z\\nabla f_1, Z\\nabla f_2\\rangle _0\\\\&\\qquad - \\langle \\operatorname{Ad}_Z\\pi _{\\le 0}( \\nabla f_1 \\cdot Z), \\pi _{<0}(Z \\nabla f_2)\\rangle .$ The last term can be rewritten as $\\langle Z \\nabla f_1 , \\pi _{<0}(Z \\nabla f_2)\\rangle - \\langle \\pi _{>0} ( \\nabla f_1 \\cdot Z) , Z \\nabla f_2)\\rangle $ .",
"Comparing with (REF ), we obtain $\\lbrace f_1\\circ Z, f_1\\circ Z\\rbrace _D = \\lbrace f_1, f_2\\rbrace _r \\circ Z$ .",
"We are now ready to deal with the three families out of four.",
"Lemma 5.4 Families of functions $\\lbrace f_{ij}\\rbrace , \\lbrace g_{ij}\\rbrace , \\lbrace h_{ij}\\rbrace $ are log-canonical with respect to ${\\lbrace \\cdot ,\\cdot \\rbrace }_D$ .",
"If $\\varphi _1(X,Y) = g_{ij}(X)$ and $\\varphi _2(X,Y) = g_{\\alpha \\beta }(X)$ (or $\\varphi _1(X,Y) = h_{ij}(Y)$ and $\\varphi _2(X,Y) = h_{\\alpha \\beta }(Y)$ ) then $\\lbrace \\varphi _1,\\varphi _2\\rbrace _D = \\lbrace \\varphi _1,\\varphi _2\\rbrace _r$ .",
"Furthermore, Proposition 4.19 in [15] specialized to the $GL_n$ case implies that in both cases $\\lbrace \\log \\varphi _1,\\log \\varphi _2\\rbrace _r = \\frac{1}{2} \\langle \\xi _{1,L} , \\xi _{2,L}\\rangle _0 - \\frac{1}{2}\\langle \\xi _{1,R} , \\xi _{2,R}\\rangle _0,$ provided $i-j\\ge \\alpha -\\beta $ ($i-j\\le \\alpha -\\beta $ , respectively), where $\\xi _{\\bullet ,L}$ , $\\xi _{\\bullet ,R}$ are projections of the left and right gradients of $\\log \\varphi _\\bullet $ to the diagonal subalgebra.",
"These projections are constant due to the homogeneity of all functions involved with respect to both left and right multiplication by diagonal matrices.",
"Thus, families $\\lbrace g_{ij}\\rbrace $ , $\\lbrace h_{ij}\\rbrace $ are log-canonical.",
"The claim about the family $\\lbrace f_{ij}\\rbrace $ now follows from Lemma REF and the third equation in (REF ).",
"The remaining family $\\lbrace \\varphi _{ij}\\rbrace $ is treated separately.",
"Lemma 5.5 The family $\\lbrace \\varphi _{kl}\\rbrace $ is log-canonical with respect to ${\\lbrace \\cdot ,\\cdot \\rbrace }_D$ .",
"Since $\\det X$ is a Casimir function for ${\\lbrace \\cdot ,\\cdot \\rbrace }_D$ , we only need to show that functions $\\tilde{\\varphi }_{kl}=\\det \\Phi _{kl}$ are log-canonical with respect to the Poisson bracket $\\lbrace \\varphi _1,\\varphi _2\\rbrace _* = \\left\\langle R_+([\\nabla \\varphi _1,U]), [\\nabla \\varphi _2,U]\\right\\rangle - \\left\\langle [\\nabla \\varphi _1,U], \\nabla \\varphi _2 \\cdot U\\right\\rangle ,$ which one obtains from (REF ) by assuming that $f(X,Y)=\\varphi (X^{-1}Y)$ .",
"In other words, ${\\lbrace \\cdot ,\\cdot \\rbrace }_*$ is the push-forward of ${\\lbrace \\cdot ,\\cdot \\rbrace }_D$ under the map $(X,Y) \\mapsto U=X^{-1}Y$ .",
"Let $C=e_{21} + \\cdots + e_{n, n-1} + e_{1n}$ be the cyclic permutation matrix.",
"By Lemma REF , we write $U$ as $U = N_- B_+ C N_-^{-1},$ where $N_-$ is unipotent lower triangular and $B_+$ is upper triangular.",
"Since functions $\\tilde{\\varphi }_{kl}$ are invariant under the conjugation by unipotent lower triangular matrices, we have $\\tilde{\\varphi }_{kl}(U)=\\tilde{\\varphi }_{kl}(B_+C)$ .",
"Furthermore, $\\left((B_+C)^i\\right)^{[n]} = b_{ii} \\cdots b_{11} e_i + \\sum _{s < i} \\alpha _{is} e_s$ for $i\\le n$ , where $b_{ij}$ , $1\\le i\\le j\\le n$ , are the entries of $B_+$ .",
"It follows that $\\begin{split}\\tilde{\\varphi }_{kl}(U)&= \\pm \\left(\\prod _{s=1}^{n-k-l+1}b_{ss}^{n-k-l-s+2}\\right)\\det (B_+)_{[n-k-l+2, n-k]}^{[n-l+2, n]}\\\\&= \\pm \\det U^{n-k-l+1}\\frac{h_{n-k-l+2,n-l+2}(B_+)}{\\prod _{s=2}^{n-k-l+2}h_{ss}(B_+)}.\\end{split}$ Remark 5.6 It is easy to check that the sign in the first line of (REF ) equals $(-1)^{k(n-k)+(l-1)(n-k-l+1)}s_{kl}$ .",
"We will use this fact below in the proof of Theorem REF (ii).",
"Note that $\\det U=\\det B_+C= \\pm \\prod _{s=1}^{n}b_{ss} $ is a Casimir function for (REF ).",
"Therefore to prove Lemma REF it suffices to show that functions $\\det (B_+)_{[i, i+ n-j ]}^{[j, n]}$ , $2\\le i \\le j \\le n$ , are log-canonical with respect to ${\\lbrace \\cdot ,\\cdot \\rbrace }_*$ as functions of $U$ .",
"To this end, we first will compute the push-forward of ${\\lbrace \\cdot ,\\cdot \\rbrace }_*$ under the map $U \\mapsto B_+^{\\prime }=B_+^{\\prime }(U)=(B_+)_{[2,n]}^{[2,n]}$ of $GL_n$ to the space ${\\mathcal {B}}_{n-1}$ of $(n-1)\\times (n-1)$ invertible upper triangular matrices.",
"Let $S= e_{12} + \\cdots + e_{m-1, m}$ be the $m\\times m$ upper shift matrix.",
"For an $m\\times m$ matrix $A$ , define $\\tau (A)= S A S^T.$ Lemma 5.7 Let $f_1, f_2$ be two differentiable functions on ${\\mathcal {B}}_{n-1}$ .",
"Then $\\lbrace f_1\\circ B_+^{\\prime }, f_2\\circ B_+^{\\prime }\\rbrace _* = \\lbrace f_1, f_2\\rbrace _b\\circ B_+^{\\prime },$ where ${\\lbrace \\cdot ,\\cdot \\rbrace }_b$ is defined by $\\lbrace f_1, f_2\\rbrace _b(A) = \\lbrace f_1, f_2\\rbrace _r(A) + \\frac{1}{2} \\left( \\left\\langle A \\nabla f_1, \\tau ( (\\nabla f_2) A)\\right\\rangle _0 -\\left\\langle \\tau ( (\\nabla f_1) A), A \\nabla f_2 \\right\\rangle _0 \\right)$ for any $A\\in {\\mathcal {B}}_{n-1}$ .",
"We start by computing an infinitesimal variation of $B_+$ as a function of $U$ .",
"From (REF ), we obtain $\\left(\\operatorname{Ad}_{N_-^{-1}}\\delta U\\right)C^{-1} =[ N_-^{-1}\\delta N_-, B_+C] C^{-1} + \\delta B_+$ .",
"Then $\\pi _{< 0}\\left( \\left(\\operatorname{Ad}_{N_-^{-1}}\\delta U\\right)C^{-1} \\right) = \\pi _{< 0}\\left( [ N_-^{-1}\\delta N_-, B_+C] C^{-1}\\right).$ If we define ${\\Lambda }: \\mathfrak {n}_- \\rightarrow \\mathfrak {n}_-$ via ${\\Lambda }(\\nu _-) = - \\pi _{< 0}\\left( \\operatorname{ad}_{B_+C}(\\nu _-) C^{-1}\\right)$ for $\\nu _-\\in \\mathfrak {n}_-$ then (REF ) above implies that $N_-^{-1}\\delta N_- = {\\Lambda }^{-1} \\left( \\pi _{< 0}\\left( \\left(\\operatorname{Ad}_{N_-^{-1}}\\delta U\\right)C^{-1} \\right) \\right)$ .",
"Invertibility of ${\\Lambda }$ is easy to establish by observing that (REF ) can be written as a triangular linear system for matrix entries of $N_-^{-1}\\delta N_-$ .",
"The operator ${\\Lambda }^* : \\mathfrak {n}_+ \\rightarrow \\mathfrak {n}_+$ dual to ${\\Lambda }$ with respect to $\\langle \\cdot ,\\cdot \\rangle $ acts on $\\nu _+\\in \\mathfrak {n}_+$ as $\\begin{split}{\\Lambda }^*(\\nu _+) &= \\pi _{>0} \\left( \\operatorname{ad}_{B_+C} (C^{-1} \\nu _+) \\right)=\\pi _{>0} \\left( B_+ \\nu _+ - C^{-1} \\nu _+ B_+ C \\right)\\\\& = B_+ \\nu _+ - S \\nu _+ B_+ S^T= \\left(\\mathbf {1}- \\tau \\circ \\operatorname{Ad}_{B_+^{-1}}\\right) (B_+ \\nu _+).\\end{split}$ Note that ${\\Lambda }^*$ extends to an operator on $\\mathfrak {gl}_n$ given by the same formula and invertible due to the fact that $\\tau \\circ \\operatorname{Ad}_{B_+^{-1}}$ is nilpotent.",
"Let $f$ be a differentiable function on ${\\mathcal {B}}_n$ .",
"Denote $\\tilde{f} (U)=(f\\circ B_+)(U)$ .",
"Then $\\begin{split}\\langle \\nabla \\tilde{f}, & \\delta U\\rangle = \\langle \\nabla f, \\delta B_+\\rangle =\\langle C^{-1}\\nabla f, \\left(\\operatorname{Ad}_{N_-^{-1}}\\delta U\\right) - [ N_-^{-1}\\delta N_-, B_+C] \\rangle \\\\& =\\langle \\operatorname{Ad}_{N_-}(C^{-1} \\nabla f ), \\delta U\\rangle + \\left\\langle [C^{-1}\\nabla f, B_+C ], {\\Lambda }^{-1} \\left(\\pi _{< 0}\\left(\\left(\\operatorname{Ad}_{N_-^{-1}}\\delta U\\right)C^{-1}\\right)\\right)\\right\\rangle \\\\&=\\langle \\operatorname{Ad}_{N_-}\\left(C^{-1} \\left( \\nabla f + ({\\Lambda }^*)^{-1}\\left(\\pi _{> 0} ( [ C^{-1}\\nabla f, B_+C ] )\\right)\\right) \\right), \\delta U\\rangle ;\\end{split}$ in the last equality we have used the identities ${\\Lambda }^{-1}=\\pi _{<0}\\circ {\\Lambda }^{-1}$ and $({\\Lambda }^*)^{-1}=\\pi _{>0}\\circ ({\\Lambda }^*)^{-1}$ .",
"From now on we assume that $f$ depends only on $B_+^{\\prime }$ , that is, does not depend on the first row of $B_+$ .",
"Thus, the first column of $\\nabla f$ is zero.",
"Define $\\zeta ^f=\\pi _{> 0} ( [ C^{-1}\\nabla f, B_+C ] )$ and $\\xi ^f = \\nabla f+ ({\\Lambda }^*)^{-1}(\\zeta ^f)$ .",
"Clearly, $\\zeta ^f= \\pi _{> 0} ( C^{-1}\\nabla f \\cdot B_+C - B_+ \\nabla f )= \\pi _{> 0}( \\tau (\\nabla f\\cdot B_+)) - \\pi _{> 0} ( B_+\\nabla f )$ and $[C^{-1}\\xi ^f, B_+C] &= C^{-1}\\xi ^f B_+C - B_+ \\xi ^f = C^{-1}\\xi ^f B_+ S^T - B_+ \\xi ^f \\\\& = S\\xi ^f B_+ S^T - B_+ \\xi ^f + e_{n1} \\xi ^f B_+C \\\\&= - \\left(\\mathbf {1}- \\tau \\circ \\operatorname{Ad}_{B_+^{-1}}\\right) (B_+ \\xi ^f) + e_{n1} \\xi ^f B_+C,$ which is equivalent to $[C^{-1}\\xi ^f, B_+C]= -{\\Lambda }^*( \\xi ^f) + e_{n1} \\xi ^f B_+C$ by (REF ).",
"Furthermore, ${\\Lambda }^*( \\xi ^f) = {\\Lambda }^*(\\nabla f) + \\zeta ^f= \\pi _{\\le 0} \\left( B_+ \\nabla f- \\tau (\\nabla f \\cdot B_+)\\right).$ Consequently, $[C^{-1}\\xi ^f, B_+C]\\in \\mathfrak {b}_-$ .",
"Using this fact and the invariance of the trace form, for any $f_1, f_2$ that depend only on $B_+^{\\prime }$ we can now compute $\\lbrace f_1\\circ B_+, f_2\\circ B_+\\rbrace _*$ from (REF ) and (REF ) as $\\begin{split}\\lbrace f_1\\circ B_+, f_2\\circ B_+\\rbrace _* (U) &= \\langle R_+ \\left([C^{-1}\\xi ^1, B_+C] \\right), [C^{-1}\\xi ^2, B_+C] \\rangle \\\\& - \\langle [C^{-1}\\xi ^1, B_+C] , C^{-1}\\xi ^2 B_+C\\rangle ,\\end{split}$ where $\\xi ^i = \\xi ^{f_i}$ , $i=1,2$ .",
"Thus, $\\nabla ^{i}=\\nabla f_i$ are lower triangular matrices with zero first column, and so $\\nabla ^i B_+$ , $B_+ \\nabla ^i$ , $\\xi ^i$ , $\\xi ^i B_+$ , $B_+ \\xi ^i$ have zero first column as well, and $C^{-1}\\xi ^2 B_+C$ has zero last column.",
"We conclude that the second term in (REF ) does not affect either term in the right hand side of (REF ).",
"In particular, the first term in (REF ) becomes $\\frac{1}{2} \\left\\langle B_+ \\nabla ^1- \\tau (\\nabla ^1 B_+), B_+ \\nabla ^2- \\tau (\\nabla ^2 B_+)\\right\\rangle _0,$ while the second can be re-written as $\\nonumber \\begin{split}\\big \\langle \\pi _{\\le 0} \\left( \\tau (\\nabla ^1 B_+) - B_+ \\nabla ^1\\right),& C^{-1}\\xi ^2 B_+C\\big \\rangle \\ =\\left\\langle S^T\\pi _{\\le 0} \\left( \\tau (\\nabla ^1 B_+) - B_+ \\nabla ^1 \\right)S, \\xi ^2 B_+\\right\\rangle \\\\&= \\left\\langle \\pi _{\\le 0} \\left( \\nabla ^1 B_+\\right) - S^T\\pi _{\\le 0} \\left( B_+ \\nabla ^1 \\right) S, \\xi ^2 B_+\\right\\rangle \\\\&= \\left\\langle \\pi _{\\le 0} \\left( \\nabla ^1 B_+\\right) - S^T\\pi _{\\le 0} \\left( B_+ \\nabla ^1 \\right) S, \\nabla ^2 B_+\\right\\rangle \\\\& + \\left\\langle \\pi _{\\le 0} \\left( \\nabla ^1 B_+\\right) - S^T\\pi _{\\le 0} \\left( B_+ \\nabla ^1 \\right) S, ({\\Lambda }^*)^{-1} (\\zeta ^2)B_+\\right\\rangle ,\\end{split}$ where $\\zeta ^2=\\zeta ^{f_2}$ .",
"The last term can be transformed as $\\nonumber \\begin{split}\\langle \\operatorname{Ad}_{B_+}\\left(\\pi _{\\le 0}( \\nabla ^1 B_+) \\right.&\\left.- S^T\\pi _{\\le 0} ( B_+ \\nabla ^1) S\\right), B_+({\\Lambda }^*)^{-1} (\\zeta ^2)\\rangle \\\\& = \\langle \\left(\\mathbf {1}- \\operatorname{Ad}_{B_+}\\circ \\tau ^* \\right) \\pi _{\\le 0} ( B_+ \\nabla ^1 ), B_+({\\Lambda }^*)^{-1} (\\zeta ^2)\\rangle \\\\& = \\langle \\pi _{\\le 0} \\left( B_+ \\nabla ^1 \\right), \\left(\\mathbf {1}- \\tau \\circ \\operatorname{Ad}_{B_+^{-1}}\\right) B_+ ({\\Lambda }^*)^{-1} (\\zeta ^2)\\rangle \\\\&= \\langle \\pi _{\\le 0} \\left( B_+ \\nabla ^1 \\right), {\\Lambda }^* (({\\Lambda }^*)^{-1} (\\zeta ^2))\\rangle \\\\& = \\langle \\pi _{\\le 0} \\left( B_+ \\nabla ^1 \\right), \\pi _{> 0}( \\tau (\\nabla ^2 B_+)) - \\pi _{> 0} ( B_+\\nabla ^2 )\\rangle .\\end{split}$ Here, in the first equality we used the fact that $({\\Lambda }^*)^{-1} (\\zeta ^2) \\in \\mathfrak {n}_+$ , and that $\\langle \\operatorname{Ad}_{B_+}\\pi _{\\le 0}( \\nabla ^1 B_+),A\\rangle =\\langle \\pi _{\\le 0}\\left( \\operatorname{Ad}_{B_+}\\pi _{\\le 0}( \\nabla ^1 B_+)\\right),A\\rangle =\\langle \\pi _{\\le 0}\\left( \\operatorname{Ad}_{B_+} (\\nabla ^1 B_+)\\right),A\\rangle $ for $A\\in \\mathfrak {b}_+$ .",
"Combining our simplified expressions for two terms in the right hand side of (REF ) and taking into account that $\\langle \\tau (\\nabla ^1B_+),\\tau (\\nabla ^2B_+)\\rangle _0=\\langle \\nabla ^1B_+,\\nabla ^2B_+\\rangle _0$ we obtain $\\begin{split}\\lbrace f_1\\circ B_+, f_2\\circ &B_+\\rbrace _*(U) \\\\&= \\frac{1}{2} \\left( \\left\\langle B_+ \\nabla ^1, \\tau ( \\nabla ^2 B_+)\\right\\rangle _0 - \\left\\langle \\tau ( \\nabla ^1 B_+), B_+ \\nabla ^2 \\right\\rangle _0\\right) + \\lbrace f_1, f_2\\rbrace _r(B_+)\\end{split}$ for functions $f_1, f_2$ on ${\\mathcal {B}}_n$ that depend only on $B_+^{\\prime }$ .",
"To complete the proof of Lemma REF , it remains to observe that for such functions, the right hand side does not depend on the first row of $B_+$ and is equal to a similar expression in which $B_+$ is replaced with $B_+^{\\prime }$ and the bracket ${\\lbrace \\cdot ,\\cdot \\rbrace }_r$ and the forms $\\langle \\cdot ,\\cdot \\rangle $ , $\\langle \\cdot , \\cdot \\rangle _0$ are replaced with their counterparts for $GL_{n-1}$ .",
"Now we can finish the proof of Lemma REF .",
"Let functions $f_1, f_2$ belong to the family $\\lbrace \\log \\det (B_+)_{[i, i+ n-j ]}^{[j, n]}= \\log \\det (B_+^{\\prime })_{[i-1, i+ n-j ]}^{[j-1, n]},\\ 2\\le i \\le j \\le n\\rbrace $ .",
"Then the second term in our expression (REF ) for $\\lbrace f_1, f_2\\rbrace _b(B_+^{\\prime })$ is constant because of the homogeneity of minors of $B_+^{\\prime }$ under right and left diagonal multiplication, and the first term is constant because, as we discussed earlier, functions $\\det (B_+^{\\prime })_{[i-1, i+ n-j ]}^{[j-1, n]}$ are log-canonical with respect to ${\\lbrace \\cdot ,\\cdot \\rbrace }_r$ (see, e.g., (REF )).",
"This ends the proof of Theorem REF .",
"Remark 5.8 The bracket (REF ) can be extended to the entire $GL_{n-1}$ .",
"In fact, the right hand side makes sense for $GL_m$ for any $m\\in \\mathbb {N}$ .",
"It can be induced via the map ${\\mathbb {T}}_{m}\\times {\\mathbb {T}}_{m}\\times GL_{m} \\ni (H=\\operatorname{diag}(h_1,\\ldots , h_{m}), \\tilde{H}=\\operatorname{diag}(\\tilde{h}_1,\\ldots , \\tilde{h}_{m}), X) \\mapsto H X\\tilde{H}\\in GL_{m}$ if one equips ${\\mathbb {T}}_{m}\\times {\\mathbb {T}}_{m}$ with a Poisson bracket $\\lbrace h_i, h_j\\rbrace = \\lbrace \\tilde{h}_i, \\tilde{h}_j\\rbrace =0, \\lbrace h_i, \\tilde{h}_j\\rbrace =\\delta _{i,j-1}$ .",
"It follows from [13] that right and left diagonal multiplication generates a global toric action for the standard cluster structure on $GL_m$ (and on double Bruhat cells in $GL_m$ ), for which ${\\lbrace \\cdot ,\\cdot \\rbrace }_r$ is a compatible Poisson structure.",
"Therefore, the above extension of (REF ) to the entire group is compatible with this cluster structure as well."
],
[
"Toric action",
"Let us start from the following important statement.",
"Theorem 6.1 The action $(X,Y)\\mapsto (T_1 X T_2, T_1 Y T_2)$ of right and left multiplication by diagonal matrices is ${{\\mathcal {G}}{\\mathcal {C}}}_n^D$ -extendable to a global toric action on ${\\mathcal {F}}_{{\\mathbb {C}}}$ .",
"For an arbitrary vertex $v$ in $Q_n$ denote by $x_v$ the cluster variable attached to $v$ .",
"If $v$ is a mutable vertex, then the $y$ -variable ($\\tau $ -variable in the terminology of [14]) corresponding to $v$ is defined as $y_v = \\frac{\\prod _{(v\\rightarrow u) \\in Q} x_u}{\\prod _{(w\\rightarrow v) \\in Q} x_w}.$ Note that the product in the above formula is taken over all arrows, so, for example, $\\varphi _{21}^2$ enters the numerator of the $y$ -variable corresponding to $\\varphi _{12}$ .",
"By Proposition REF , to prove the theorem it suffices to check that $y_v$ is a homogeneous function of degree zero with respect to the action (REF ) (see [16] for details), and that the Casimirs $\\hat{p}_{1r}$ are invariant under (REF ).",
"Let us start with verifying the latter condition.",
"According to Section REF , $\\hat{p}_{1r}=c_r^ng_{11}^{r-n}h_{11}^{-r}$ , $1\\le r\\le n-1$ .",
"It is well-known that functions $c_i$ are Casimirs for the Poisson-Lie bracket (REF ) on $D(GL_n)$ , as well as $g_{11}$ and $h_{11}$ .",
"Therefore, $\\hat{p}_{1r}$ are indeed Casimirs.",
"Their invariance under (REF ) is an easy calculation.",
"Next, for a function $\\psi (X,Y)$ on $D(GL_n)$ homogeneous with respect to (REF ), define the left and right weights $\\xi _L(\\psi )$ , $\\xi _R(\\psi )$ of $\\psi $ as the constant diagonal matrices $\\pi _0(E_L \\log \\psi )$ and $\\pi _0(E_R \\log \\psi )$ .",
"Recall that all functions $g_{ij}$ , $h_{ij}$ , $f_{kl}$ , $\\varphi _{kl}$ possess this homogeneity property.",
"For $1\\le i \\le j \\le n$ , let $\\Delta (i,j)$ denote a diagonal matrix with 1's in the entries $(i,i), \\ldots , (j,j)$ and 0's everywhere else.",
"It follows directly from the definitions in Section REF that $\\begin{aligned}& \\xi _L(g_{ij})= \\Delta (j, n+j -i),\\quad \\xi _R(g_{ij})= \\Delta (i,n), \\\\& \\xi _L(h_{ij})= \\Delta (j, n),\\quad \\xi _R(h_{ij})= \\Delta (i,n+i-j),\\\\& \\xi _L(f_{kl})= \\Delta (n-k+1, n)+ \\Delta (n-l+1, n),\\quad \\xi _R(f_{kl})= \\Delta (n-k-l+1,n), \\\\& \\xi _L(\\varphi _{kl})= (n-k-l)\\left(\\mathbf {1}+ \\Delta (n,n)\\right) + \\Delta (n-k+1,n)+ \\Delta (n-l+1,n),\\\\& \\xi _R(\\varphi _{kl})= (n-k-l+1)\\mathbf {1}.\\end{aligned}$ Now the verification of the claim above becomes a straightforward calculation based on the description of $Q_n$ in Section REF and the fact that for a monomial in homogeneous functions $M=\\psi _1^{\\alpha _1}\\psi _2^{\\alpha _2}\\cdots $ the right and left weights are $\\xi _{R,L}(M) = \\alpha _1 \\xi _{R,L}(\\psi _1) + \\alpha _2 \\xi _{R,L}(\\psi _2) + \\cdots $ .",
"For example, if $v$ is the vertex associated with the function $h_{ii}$ , $i\\ne 1$ , $j\\ne n$ , $i\\ne j$ , then $\\xi _R(y_v) &= \\xi _R(h_{i-1,i}) + \\xi _R(f_{1,n-i}) -\\xi _R(h_{i,i+1}) - \\xi _R(f_{1,n-i+1})\\\\&= \\Delta (i-1, n-1)+ \\Delta (i,n)- \\Delta (i, n-1) - \\Delta (i-1,n)= 0$ and $\\xi _L(y_v) &= \\xi _L(h_{i-1,i}) + \\xi _L(f_{1,n-i}) -\\xi _L(h_{i,i+1}) - \\xi _L(f_{1,n-i+1})\\\\& = \\Delta (i, n) + \\Delta (i+1,n) + \\Delta (n,n)- \\Delta (i+1, n) -\\Delta (n,n) - \\Delta (i,n)= 0.$ Other vertices are treated in a similar way."
],
[
"Compatibility",
"Let us proceed to the proof of the compatibility statement of Theorem REF (i).",
"We have already seen above that $\\hat{p}_{1r}$ are Casimirs of the bracket ${\\lbrace \\cdot ,\\cdot \\rbrace }_D$ .",
"Therefore, by Proposition REF , it suffices to show that for every mutable vertex $v\\in Q_n$ $\\lbrace \\log x_u, \\log y_v\\rbrace _D = \\lambda \\delta _{u,v}\\quad \\text{for any $u\\in Q_n$},$ where $\\lambda $ is some nonzero rational number not depending on $v$ .",
"Let $v$ be a mutable $g$ -vertex in $Q_n$ and $u$ be a vertex in one of the other three regions of $Q_n$ .",
"Then to show that $\\lbrace \\log x_u, \\log y_v\\rbrace _D=0$ one can use (REF ), Lemma REF and the proof of Theorem REF , which implies that $\\pi _0\\left(X\\nabla _X \\log y_v\\right)& = \\pi _0\\left(E_R \\log y_v\\right) = \\xi _R(y_v)=0,\\\\\\pi _0\\left(\\nabla _X \\log y_v X\\right)& = \\pi _0\\left(E_L \\log y_v\\right) = \\xi _L(y_v)=0.$ The same argument works if $u$ and $v$ belong to any two different regions of the quiver $Q_n$ .",
"Thus, to complete the proof it remains to verify (REF ) for vertices $u, v$ in the same region of $Q_n$ .",
"In view of (REF ), for $g$ - and $h$ -vertices other than vertices corresponding to $g_{ii}$ and $h_{ii}$ , this becomes a particular case of Theorem 4.18 in [15] which establishes the compatibility of the standard Poisson–Lie structure on a simple Lie group ${\\mathcal {G}}$ with the cluster structure on double Bruhat cells in ${\\mathcal {G}}$ .",
"We just need to set ${\\mathcal {G}}=GL_n$ (a transition to $GL_n$ from a simple group $SL_n$ is straightforward), set $\\lambda $ in (REF ) to be equal to $-1$ and apply the theorem to ${\\mathcal {G}}^{id,w_0}, {\\mathcal {G}}^{w_0,id}$ in the case of $g$ - and $h$ -regions, respectively (here $w_0$ is the longest permutation of length $n-1$ ).",
"Vertices corresponding to $g_{ii}$ and $h_{ii}$ are treated separately, because in quivers for ${\\mathcal {G}}^{id,w_0}, {\\mathcal {G}}^{w_0,id}$ they would have been frozen.",
"For any such vertex $v$ we only need to check that $\\lbrace \\log x_v, \\log y_v\\rbrace _D = -1$ .",
"Using the description of $Q_n$ in Section REF , the third equation in Lemma REF , the second and the third lines in (REF ), and equation (REF ), we compute $\\begin{aligned}\\left\\lbrace \\log h_{ii},\\log \\frac{f_{1, n-i} h_{i-1,i}}{f_{1, n-i+1} h_{i,i+1}}\\right\\rbrace _D&= \\frac{1}{2} \\langle \\Delta (i,n) ,2\\Delta (i+1,n) - 3\\Delta (i,n) +\\Delta (i-1,n)\\\\&\\qquad + \\Delta (i-1,n-1) - \\Delta (i,n-1)\\rangle _0\\\\& = \\frac{1}{2} \\left\\langle \\Delta (i,n) , 2(\\Delta (i-1,i-1) -\\Delta (i,i)) \\right\\rangle _0 = -1\\end{aligned}$ for $1<i<n$ .",
"Using in addition the first equation in Lemma REF and the first line in (REF ) we get $\\begin{aligned}\\left\\lbrace \\log h_{nn},\\log \\frac{ h_{n-1,n}g_{nn}}{f_{11}}\\right\\rbrace _D&= \\frac{1}{2} \\langle \\Delta (n,n) ,\\Delta (n-1,n)- 3\\Delta (n,n) + \\Delta (n-1,n-1)\\rangle _0\\\\& = \\frac{1}{2} \\left\\langle \\Delta (n,n) , 2(\\Delta (n-1,n-1) -\\Delta (n,n)) \\right\\rangle _0 = -1.\\end{aligned}$ Vertices corresponding to $g_{ii}$ are treated in a similar way.",
"Now, let us turn to the $f$ -region.",
"Let $v$ be a vertex that corresponds to $f_{kl}$ , $k,l\\ge 1$ , $k+l< n$ , then by the last equality in (REF ), $y_v = \\frac{f_{k+1,l-1} f_{k,l+1}f_{k-1,l}}{f_{k+1,l} f_{k-1,l+1}f_{k,l-1}}(X,Y) = \\frac{h_{\\alpha ,\\beta -1} h_{\\alpha -1,\\beta }h_{\\alpha +1,\\beta +1}}{h_{\\alpha ,\\beta +1}h_{\\alpha +1,\\beta }h_{\\alpha -1,\\beta -1}}(Z),$ where $\\alpha =n-k-l+1$ , $\\beta =n-k+1$ , and $Z=(Y_{>0})^{-1}X_{\\ge 0}$ .",
"Consequently, if $u$ is a vertex that corresponds to $f_{k^{\\prime }l^{\\prime }}$ , and $\\alpha ^{\\prime }=n-k^{\\prime }-l^{\\prime }+1$ , $\\beta ^{\\prime }=n-k^{\\prime }+1$ , then $\\lbrace \\log x_u, \\log y_v\\rbrace _D = \\lbrace \\log h_{n-l^{\\prime }+1,n-l^{\\prime }+1}(Y), \\log y_v\\rbrace _D + \\lbrace \\log h_{\\alpha ^{\\prime }\\beta ^{\\prime }}, \\log y_v\\rbrace _r(Z)$ by Lemma REF .",
"The first term in the right hand side vanishes, as it was already shown above (this corresponds to the case when one vertex belongs to the $h$ -region and the other to the $f$ -region), and we are left with $\\lbrace \\log x_u, \\log y_v\\rbrace _D = \\lbrace \\log h_{\\alpha ^{\\prime }\\beta ^{\\prime }}, \\log y_v\\rbrace _r(Z).$ Consider the subquiver of $Q_n$ formed by all $f$ -vertices, as well as vertices (viewed as frozen) that correspond to functions $g_{ii}(X)$ and $\\varphi _{n-i, i}(X,Y)$ .",
"It is isomorphic, up to edges between the frozen vertices, to the $h$ -part of $Q_n$ in which vertices corresponding to $h_{ii}(Y)$ are viewed as frozen.",
"The isomorphism consists in sending the vertex occupied by $f_{kl}(X,Y)$ to the vertex occupied by $h_{\\alpha \\beta }(Z)$ , including the values of $k,l$ subject to identifications of Remark REF .",
"The latter is possible since $g_{ii}(X)=h_{ii}(Z)$ by the second equation in (REF ), and since the third equation in (REF ) can be extended to the cases $k=0$ and $l=0$ by setting $h_{i,n+1}\\equiv 1$ for any $i$ .",
"It now follows from (REF ), (REF ) that this isomorphism takes (REF ) for $f$ -vertices to the same statement for $h$ -vertices, which has been already proved.",
"We are left with the $\\varphi $ -region.",
"If $v$ is a vertex that corresponds to $\\varphi _{kl}$ , $k > 1$ , $l>1$ , $k+l < n$ , then by (REF ) $y_v = \\frac{\\varphi _{k,l-1} \\varphi _{k-1,l+1}\\varphi _{k+1,l}}{\\varphi _{k+1,l-1}\\varphi _{k,l+1}\\varphi _{k-1,l}}(X,Y) = \\frac{h_{\\alpha -1,\\gamma } h_{\\alpha +1,\\gamma +1}h_{\\alpha ,\\gamma -1}}{h_{\\alpha ,\\gamma +1} h_{\\alpha +1,\\gamma }h_{\\alpha -1,\\gamma -1}}(B_+^{\\prime }),$ where $\\alpha =n-k-l+2$ , $\\gamma =n-l+2$ , $B_+^{\\prime }=(B_+)_{[2,n]}^{[2,n]}$ (here we use the identity $h_{ij}(B_+)=h_{i-1,j-1}(B_+^{\\prime })$ for $i,j>1$ ).",
"In view of Lemma REF and Remark REF , we can establish (REF ) for $v$ by applying the same reasoning as in the case of $f$ -vertices.",
"To include the case $k=1$ it suffices to use the same convention $h_{i,n+1}\\equiv 1$ as above.",
"Therefore, the only vertices left to consider are the ones corresponding to $\\varphi _{k,n-k}$ , $1\\le k\\le n-1$ , and $\\varphi _{k1}$ , $1\\le k\\le n-2$ .",
"They are treated by straightforward, albeit lengthy, calculations based on Lemma REF , equations (REF ), (REF ), and the fourth equation in Lemma REF .",
"Note that in all the cases equation (REF ) is satisfied with $\\lambda =-1$ .",
"Remark 6.2 It follows from (REF ) and Proposition REF that the exchange matrix corresponding to the extended seed $\\widetilde{\\Sigma }_n=(F_n, Q_n, {\\mathcal {P}}_n)$ is of full rank."
],
[
"Irreducibility: the proof of Theorem ",
"The claim is clear for the functions $g_{ij}$ , $h_{ij}$ and $f_{kl}$ , since each one of them is a minor of a matrix whose entries are independent variables (see e.g. [3]).",
"Functions $c_k$ , $1\\le k\\le n-1$ , are sums of such minors.",
"Consequently, each $c_k$ is linear in any of the variables $x_{ij}$ , $y_{ij}$ .",
"Assume that $c_k=P_1P_2$ , where $P_1$ and $P_2$ are nonconstant polynomials, and that $P_1$ is linear in $y_{11}$ , hence $P_2$ does not depend on $y_{11}$ .",
"If $P_2$ is linear in any of $y_{1j}$ , $2\\le j\\le n$ , then $c_k$ includes a multiple of the product $y_{11}y_{1j}$ , a contradiction.",
"Therefore, $P_1$ is linear in any one of $y_{1j}$ , $1\\le j\\le n$ .",
"If $P_2$ is linear in $z_{ij}$ for some values of $i$ and $j$ (here $z$ is either $x$ or $y$ ) then $c_k$ includes a multiple of the product $y_{1j}z_{ij}$ , a contradiction.",
"Hence, $P_2$ is trivial, which means that $c_k$ is irreducible.",
"Our goal is to prove Proposition 6.3 $\\varphi _{kl}(X,Y)$ is an irreducible polynomial in the entries of $X$ and $Y$ .",
"We first aim at a weaker statement: Proposition 6.4 $\\varphi _{kl}(I,Y)$ is irreducible in the entries of $Y$ .",
"In this case $U=X^{-1}Y=Y$ , so we write $\\psi _{kl}(U)$ instead of $\\varphi _{kl}(I,Y)$ .",
"It will be convenient to indicate explicitly the size of $U$ and to write $\\psi _{kl}^{[n]}(U)$ for the function $\\psi _{kl}$ evaluated on an $n \\times n$ matrix $U=\\begin{pmatrix} u_{11} & u_{12}& \\dots & u_{1n}\\cr u_{21} & u_{22} & \\dots & u_{2n}\\cr \\vdots & & & \\vdots \\cr u_{n1} & u_{n2} & \\dots & u_{nn}\\end{pmatrix}.$ We start with the following observations.",
"Lemma 6.5 (i) Consider $\\psi _{kl}^{[n]}(U)$ as a polynomial in $u_{jn}$ , then its leading coefficient does not depend on the entries $u_{ji}$ , $1\\le i\\le n-1$ , nor on $u_{in}$ , $1\\le i\\le n$ , $i\\ne j$ .",
"(ii) The degree of $\\psi _{kl}^{[n]}(U)$ as a polynomial in $u_{1n}$ equals $n-k-l+1$ .",
"Explicit computation.",
"Next, let us find a specialization of variables $u_{ij}$ such that the corresponding $\\psi _{kl}(U)$ is irreducible.",
"Define a polynomial in two variables $z, t$ by $P_m(z,t)=t^{m-1}z^m+\\sum _{i=0}^{m-2}(-1)^{m-i-1}\\binom{m}{i} \\frac{t^{m-1}-t^i}{t-1}z^i.$ This is an explicit expression for the determinant of the $m\\times m$ matrix $\\begin{pmatrix} z & 1 & 1 & 1 &\\dots & 1\\cr 1 & tz & t & t &\\dots & t \\cr 1 & 1 & tz & t &\\dots & t \\cr \\ddots & & & & \\ddots & \\cr 1 & 1 & 1 & 1 & \\dots & tz\\end{pmatrix}.$ Lemma 6.6 For any $m\\ge 2$ , $P_m(z,t)$ is an irreducible polynomial.",
"It is easy to see that a polynomial in two variables is irreducible if its Newton polygon can not be represented as a Minkowski sum of two polygons.",
"The Newton polygon of $P_m(z,t)$ in the $(z,t)$ -plane has the following vertices: $(m,m-1), (0,m-2), (0,m-3),\\dots , (0,1), (0,0)$ .",
"It contains two non-vertical sides connecting $(m,m-1)$ with $(0,m-2)$ and $(0,0)$ , respectively; all the other sides are vertical.",
"Assume that two polygons as above exist.",
"Consequently, both non-vertical sides should belong to the same one out of the two.",
"However, it is not possible to complete this polygon without using all the remaining vertical sides, a contradiction.",
"Lemma 6.7 Define an $n\\times n$ matrix $M_{kl}$ via $M_{kl}=\\begin{pmatrix}0 & 0 & 1 & z \\cr t & 0 & 0 & 1 \\cr 0 & \\mathbf {1}_{n-3} & 0 & 1 \\cr 0 & 0 & 0 & 1 \\end{pmatrix} \\qquad \\text{if $l=1$},\\\\M_{kl}=\\begin{pmatrix}0 & 0 & 1 & 0 & 0 & 0 & z \\cr t & 0 & 0 & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 0 & 0 & \\mathbf {1}_{l-1} & 1 \\cr 0 & 1 & 0 & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 0 & \\mathbf {1}_{m-1} & 0 & 1 \\cr 0 & 0 & 0 & \\mathbf {1}_{l-2} & 0 & 0 & 1 \\cr 0 & 0 & 0 & 0 & 0 & 0 & 1 \\end{pmatrix}\\qquad \\text{if $k\\ge l$, $l>1$},\\\\M_{kl}=\\begin{pmatrix}0 & 0 & 0 & 0 & 1 & 0 & z \\cr t & 0 & 0 & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 0 & 0 & \\mathbf {1}_{k-1} & 1 \\cr 0 & 1 & 0 & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & \\mathbf {1}_{m-2} & 0 & 0 & 1 \\cr 0 & 0 & \\mathbf {1}_{k-1} & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 0 & 0 & 0 & 1 \\end{pmatrix}\\qquad \\text{if $k\\le l$, $l>1$}.$ where $m=\\max \\lbrace n-2l, n-2k\\rbrace $ .",
"Then $\\det M_{kl}=\\pm t$ , and $\\psi ^{[n]}_{kl}(M_{kl})=\\pm P_{n-k-l+1}(z,t).$ Explicit computation.",
"Let us proceed with the proof of Proposition REF .",
"Assume to the contrary that $\\psi _{kl}^{[n]}(U)=P_1P_2$ for some values of $n$ , $k$ and $l$ , where both $P_1$ and $P_2$ are nontrivial.",
"It follows from Lemmas REF , REF and REF (ii) that only one of $P_1$ and $P_2$ depends on $u_{1n}$ , say, $P_1$ .",
"Therefore, $P_2(M_{kl})$ is a nonzero constant.",
"Consequently, $P_2$ contains a monomial $\\prod _{j=3}^{n-1}u_{j\\sigma _{kl}(j)}^{r_j}$ , where $r_j\\ge 0$ and $\\sigma _{kl}\\in S_{n-1}$ is the permutation defined by the first $n-1$ rows and columns of $M_{kl}$ .",
"Assume there exists $j$ such that $r_j>0$ , and consider $u_{jn}$ .",
"If $P_2$ depends on $u_{jn}$ then the leading coefficient of $\\psi _{kl}^{[n]}(U)$ at $u_{1n}$ depends on $u_{jn}$ , in a contradiction to Lemma REF (i).",
"Therefore, $P_2$ does not depend on $u_{jn}$ , and hence the leading coefficient of $\\psi _{kl}^{[n]}(U)$ at $u_{jn}$ depends on $u_{j\\sigma _{kl}(j)}$ , which again contradicts Lemma REF (i).",
"We thus obtain that $r_j=0$ for $3\\le j\\le n-1$ , which means that $P_2$ is a constant, and Proposition REF holds true.",
"The next step of the proof is the following statement: Proposition 6.8 $\\varphi _{kl}(X,I)$ is irreducible in the entries of $X$ .",
"Note that $\\left[I^{[n-k+1,n]}\\;(X^{-1})^{[n-l+1,n]}\\;(X^{-2})^{[n]}\\dots (X^{k+l-n-1})^{[n]}\\right]=\\hfill \\\\X^{k+l-n-1}\\left[(X^{n-k-l+1})^{[n-k+1,n]}\\;(X^{n-k-l})^{[n-l+1,n]}\\;(X^{n-k-l-1})^{[n]}\\;\\dots I^{[n]}\\right],$ hence it suffices to prove that the functions $\\bar{\\psi }_{kl}=\\det \\left[(X^{n-k-l+1})^{[n-k+1,n]}\\;(X^{n-k-l})^{[n-l+1,n]}\\;(X^{n-k-l-1})^{[n]}\\;\\dots I^{[n]}\\right]$ are irreducible.",
"As in the proof of Proposition REF , we write $\\bar{\\psi }_{kl}^{[n]}(X)$ for the function $\\psi _{kl}$ evaluated on an $n \\times n$ matrix $X=\\begin{pmatrix} x_{11} & x_{12}& \\dots & x_{1n}\\cr x_{21} & x_{22} & \\dots & x_{2n}\\cr \\vdots & & & \\vdots \\cr x_{n1} & x_{n2} & \\dots & x_{nn}\\end{pmatrix}.$ Similarly to the case of $\\psi _{kl}^{[n]}(U)$ , we have the following observations: Lemma 6.9 (i) Consider $\\bar{\\psi }_{kl}^{[n]}(X)$ as a polynomial in $x_{jn}$ , then its leading coefficient does not depend on the entries $x_{ji}$ , $1\\le i\\le n-1$ , and $x_{in}$ , $1\\le i\\le n$ , $i\\ne j$ .",
"(ii) The degree of $\\bar{\\psi }_{kl}^{[n]}(X)$ as a polynomial in $x_{1n}$ equals $n-k-l+1$ .",
"Explicit computation.",
"As in the previous case, we find a specialization of variables $x_{ij}$ , such that the corresponding $\\bar{\\psi }_{kl}(X)$ is irreducible.",
"Lemma 6.10 Define an $n\\times n$ matrix $\\bar{M}_{kl}$ via $\\bar{M}_{kl}=\\begin{pmatrix}0 & 0 & 1 & 0 & 0 & 0 & 0 & z \\cr t & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\cr 0 & \\mathbf {1}_{m-1} & 0 & 0 & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\cr 0 & 0 & 0 & \\mathbf {1}_{l-2} & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 0 & 0 & 0 & \\mathbf {1}_{k-2} & 1 \\cr 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\end{pmatrix},$ where $m=n-k-l$ .",
"Then $\\det \\bar{M}_{kl}=\\pm t$ , and $\\bar{\\psi }^{[n]}_{kl}(\\bar{M}_{kl})=\\pm P_{n-k-l+1}(z,t).$ Similar to the proof of Lemma REF .",
"The rest of the proof of Proposition REF is parallel to the proof of Proposition REF .",
"Assume now that $\\varphi _{kl}(X,Y)=P_1P_2$ .",
"It follows from Propositions REF and REF and the quasihomogeneity of $\\varphi _{kl}(X,Y)$ that one of $P_1, P_2$ depends only on $X$ , and the other only on $Y$ ; moreover, both $P_1(I)$ and $P_2(I)$ are nonzero complex numbers.",
"Consequently, $\\varphi _{kl}(I,I)$ is a nonzero complex number, which contradicts the explicit expression for $\\varphi _{kl}(X,Y)$ .",
"The proof of Theorem REF is complete."
],
[
"Regularity",
"We now proceed with condition (ii) of Proposition REF .",
"By Theorem REF , we have to check that for any mutable vertex of $Q_n$ the new variable in the adjacent cluster is regular and not divisible by the corresponding variable in the initial cluster.",
"Let us consider the exchange relations according to the type of the vertices.",
"Regularity of the function $\\varphi _{11}^{\\prime }$ that replaces $\\varphi _{11}$ follows from Proposition REF .",
"Let us prove that this function is not divisible by $\\varphi _{11}$ .",
"Indeed, assume the contrary, and define an $n \\times n$ matrix $\\Sigma _{11}$ via $\\Sigma _{11}=\\begin{pmatrix}0 & 0 & 0 & 1 \\cr \\mathbf {1}_{n-3} & 0 & 0 & 0 \\cr 0 & t & 0 & 0 \\cr 0 & 0 & 1 & 0 \\end{pmatrix}.$ An explicit computation shows that $\\det \\Sigma _{11}=\\pm t$ and $\\varphi _{11}(I,\\Sigma _{11})=\\pm t,\\qquad \\varphi _{21}(I,\\Sigma _{11})=\\pm 1,\\qquad \\varphi _{12}(I,\\Sigma _{11})=0.$ Therefore, (REF ) yields $\\varphi _{11}(I,\\Sigma _{11})\\varphi ^{\\prime }_{11}(I,\\Sigma _{11})=\\varphi _{21}(I,\\Sigma _{11})\\det \\Sigma _{11},$ and hence $\\varphi ^{\\prime }_{11}(I,\\Sigma _{11})=\\pm 1$ , which contradicts the divisibility assumption.",
"Denote by $\\Phi _{pl}^*$ the matrix obtained from $\\Phi _{pl}$ via replacing the column $U^{[n-l+1]}$ by the column $U^{[n-l]}$ , and put $\\tilde{\\varphi }^*_{pl}=\\det \\Phi ^*_{pl}$ .",
"Clearly, $\\varphi ^*_{pl}=-s_{pl}\\tilde{\\varphi }^*_{pl}\\det X^{t_{p+l}}$ is regular (here and in what follows we set $t_s=n-s+1$ ).",
"By the short Plücker relation for the matrix $\\left[\\begin{array}{ccccc}(U^0)^{[n-p+1,n]}& U^{[n-l,n]} & (U^2)^{[n]} & \\dots & (U^{n-p-l+2})^{[n]}\\end{array}\\right]$ and columns $I^{[n-p+1]}$ , $U^{[n-l]}$ , $U^{[n-l+1]}$ , $(U^{n-p-l+2})^{[n]}$ , one has $\\tilde{\\varphi }_{pl}\\tilde{\\varphi }^*_{p-1,l}=\\tilde{\\varphi }_{pl}^*\\tilde{\\varphi }_{p-1,l}+\\tilde{\\varphi }_{p,l-1}\\tilde{\\varphi }_{p-1,l+1},$ provided $p,l>1$ and $p+l\\le n$ .",
"Multiplying the above relation by $\\det X^{t_{p+l}+t_{p+l-1}}$ and $s_{p,l-1}s_{p-1,l+1}=-s_{pl}s_{p-1,l}$ , one gets $\\varphi _{pl}\\varphi ^*_{p-1,l}=\\varphi _{pl}^*\\varphi _{p-1,l}+\\varphi _{p,l-1}\\varphi _{p-1,l+1}.$ A linear combination of (REF ) for $p=k$ and for $p=k+1$ yields $\\varphi _{kl} (\\varphi _{k+1,l}\\varphi ^*_{k-1,l}-\\varphi _{k-1,l}\\varphi ^*_{k+1,l})=\\varphi _{k+1,l} \\varphi _{k,l-1} \\varphi _{k-1,l+1}+\\varphi _{k+1,l-1} \\varphi _{k,l+1} \\psi _{k-1,l},$ which means that the function that replaces $\\varphi _{kl}$ after the transformation is regular whenever $k,l>1$ and $k+l<n$ .",
"Let us prove that $\\varphi _{k+1,l}\\varphi ^*_{k-1,l}-\\varphi _{k-1,l}\\varphi ^*_{k+1,l}$ is not divisible by $\\varphi _{kl}$ .",
"Indeed, assume the contrary, and define an $n \\times n$ matrix $\\Sigma _{kl}$ via $\\Sigma _{kl}=\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 1 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0 & J_{k-2} & 0 \\cr 1 & 0 & 0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & \\mathbf {1}_{m} & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 1 & 0 & 0 \\cr 0 & J_{k-1} & 0 & 0 & 0 & 0 & 0 \\end{pmatrix}, \\qquad \\text{if $k=l$},\\\\\\Sigma _{kl}=\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\cr 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{l-2} & 0 \\cr 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & \\mathbf {1}_{m} & 0 & 0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0 & \\mathbf {1}_{k-l-1} & 0 & 0 & 0\\cr 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\cr 0& J_{l-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\end{pmatrix}+e_{2,n-l},\\qquad \\text{if $k\\ge l+1$},\\\\\\Sigma _{kl}=\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 1 \\cr \\mathbf {1}_{m+1} & 0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & \\mathbf {1}_{l-2} & 0 \\cr 0 & 1 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 1 & 0 & 0 \\cr 0 & 0 & \\mathbf {1}_{k-1} & 0 & 0 & 0 \\end{pmatrix}+e_{m+2,n-l},\\qquad \\text{if $k< l$},$ where $m=n-k-l-1$ and $J_r$ is the $r\\times r$ unitary antidiagonal matrix.",
"An explicit computation shows that $\\det \\Sigma _{kl}=\\pm 1$ , and $\\varphi _{k-1,l}(I,\\Sigma _{kl})=\\pm 1,\\qquad \\varphi _{k+1,l}^*(I,\\Sigma _{kl})=\\pm 1,\\\\\\varphi _{k+1,l}(I,\\Sigma _{kl})=\\varphi _{k-1,l}^*(I,\\Sigma _{kl})=\\varphi _{kl}(I,\\Sigma _{kl})=0,$ which contradicts the divisibility assumption.",
"Extend the definition of $\\Phi _{pq}$ and $s_{pq}$ to the case $p=0$ , denote by $\\Phi ^{**}_{pq}$ the matrix obtained from $\\Phi _{pq}$ by deleting the last column and inserting $(U^2)^{[n-1]}$ as the $(p+q+1)$ -st column, and put $\\tilde{\\varphi }^{**}_{pq}=\\det \\Phi ^{**}_{pq}$ .",
"Clearly, $\\varphi ^{**}_{pq}=s_{q,p+2}\\tilde{\\varphi }^{**}_{pq}\\det X^{t_{p+q}-1}$ is regular.",
"It is easy to see that $U\\Phi _{p1}=\\Phi _{0p}$ and $U\\Phi _{p2}=\\Phi ^{**}_{0p}$ .",
"Therefore, by the short Plücker relation for the matrix $\\left[\\begin{array}{cccccc} I^{[n]} & U^{[n-k+1,n]} & (U^2)^{[n-1,n]} & (U^3)^{[n]} \\dots & (U^{n-k+1})^{[n]}\\end{array}\\right]$ and columns $I^{[n]}$ , $U^{[n-k+1]}$ , $(U^2)^{[n-1]}$ , $(U^{n-k+1})^{[n]}$ , one has $\\tilde{\\varphi }_{0k}\\tilde{\\varphi }^{**}_{1,k-1}=\\tilde{\\varphi }^{**}_{0,k-1}\\tilde{\\varphi }_{1k}+\\tilde{\\varphi }^{**}_{0k}\\tilde{\\varphi }_{1,k-1}.$ Multiplying the above relation by $\\det X^{2t_{k}-1}=\\det X^{t_{k-1}+t_{k+1}-1}$ and $s_{k2}s_{1,k-1}=s_{k-1,2}s_{1k}=s_{0k}s_{k-1,3}$ , taking into account that $\\varphi _{0p}=h_{11}\\varphi _{p1}$ , $\\varphi ^{**}_{0p}=h_{11}\\varphi _{p2}$ , $s_{k1}=(-1)^ns_{0k}$ and dividing the above relation by $h_{11}$ we arrive at $(-1)^n\\varphi _{k1}\\varphi ^{**}_{1,k-1}=\\varphi _{k-1,2}\\varphi _{1k}+\\varphi _{k2}\\varphi _{1,k-1},$ which means that the function that replaces $\\varphi _{k1}$ after the transformation is regular whenever $1<k<n-1$ .",
"To prove that $\\varphi ^{**}_{1,k-1}$ is not divisible by $\\varphi _{k1}$ , we assume the contrary and define an $n \\times n$ matrix $\\Sigma _{k1}$ via $\\Sigma _{k1}=\\begin{pmatrix}0 & 0 & 0 & 1 \\cr \\mathbf {1}_{n-4} & 0 & 0 & 0 \\cr 0 & 0 & 1 & 0 \\cr 0 & \\mathbf {1}_{2} & 0 & 0 \\end{pmatrix}+e_{n-1,n-l}, \\qquad \\text{if $k=2$},\\\\\\Sigma _{k1}=\\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 1 \\cr \\mathbf {1}_{m}& 0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 1 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 1 & 0 \\cr 0& 0 & 0 & \\mathbf {1}_{k-3} & 0 & 0 \\cr 0 & \\mathbf {1}_2 & 0 & 0 & 0 & 0 \\end{pmatrix},\\qquad \\text{if $k\\ge 3$},$ where $m=n-k-2$ .",
"An explicit computation shows that $\\det \\Sigma _{k1}=\\pm 1$ , and $\\varphi ^{**}_{1,k-1}(I,\\Sigma _{k1})=\\pm 1,\\qquad \\varphi _{kl}(I,\\Sigma _{k1})=0,$ which contradicts the divisibility assumption.",
"With the extended definition of $\\Phi _{pq}$ , relation (REF ) becomes valid for $p=1$ .",
"Taking a linear combination of (REF ) for $p=1$ and $p=2$ one gets $\\varphi _{1l}(\\varphi _{2l}\\varphi ^*_{0l}-\\varphi _{0l}\\varphi ^*_{2l})=\\varphi _{2l}\\varphi _{1,l-1}\\varphi _{0,l+1}+\\varphi _{2,l-1}\\varphi _{1,l+1}\\varphi _{0l}.$ Recall that $\\varphi _{0p}=h_{11}\\varphi _{p1}$ .",
"Besides, $\\Phi ^*_{0l}=U\\Phi ^\\circ _{l1}$ , where $\\Phi _{pq}^\\circ $ is the matrix obtained from $\\Phi _{pq}$ via replacing the column $I^{[n-p+1]}$ by the column $I^{[n-p]}$ .",
"Denote $\\tilde{\\varphi }^\\circ _{pq}=\\det \\Phi ^\\circ _{pq}$ ; clearly, $\\varphi ^\\circ _{pq}=-s_{q-1,p}\\tilde{\\varphi }^\\circ _{pq}\\det X^{t_{p+q}}$ is regular.",
"We thus have $\\tilde{\\varphi }^*_{0l}=\\det U\\tilde{\\varphi }^\\circ _{l1}$ , and hence $\\varphi ^*_{0l}=h_{11}\\varphi ^\\circ _{l1}$ .",
"Consequently, $\\varphi _{1l}(\\varphi _{2l}\\varphi ^\\circ _{l1}-\\varphi _{l1}\\varphi ^*_{2l})=\\varphi _{2l}\\varphi _{1,l-1}\\varphi _{l+1,1}+\\varphi _{2,l-1}\\varphi _{1,l+1}\\varphi _{l1},$ which means that the function that replaces $\\varphi _{1l}$ after the transformation is regular whenever $1<l<n-1$ .",
"To prove that $\\varphi _{2l}\\varphi ^\\circ _{l1}-\\varphi _{l1}\\varphi ^*_{2l}$ is not divisible by $\\varphi _{1l}$ , we assume the contrary and define an $n \\times n$ matrix $\\Sigma _{1l}$ via $\\Sigma _{1l}=\\begin{pmatrix}0 & 0 & 0 & 1 \\cr \\mathbf {1}_{m} & 0 & 0 & 0 \\cr 0 & 0 & \\mathbf {1}_{l-2} & 0 \\cr 0 & \\mathbf {1}_{2} & 0 & 0 \\end{pmatrix}+e_{n-l,n-l},$ where $m=n-l-1$ .",
"An explicit computation shows that $\\det \\Sigma _{1l}=\\pm 1$ , and $\\varphi _{l1}(I,\\Sigma _{1l})=\\pm 1,\\qquad \\varphi _{2l}^*(I,\\Sigma _{1l})=\\pm 1,\\\\\\varphi _{2l}(I,\\Sigma _{1l})=\\varphi _{l1}^\\circ (I,\\Sigma _{1l})=\\varphi _{1l}(I,\\Sigma _{1l})=0,$ which contradicts the divisibility assumption.",
"Define an $n\\times n$ matrix $F^\\lozenge _{k,n-k-1}$ , $0\\le k\\le n-2$ , by $F^\\lozenge _{k,n-k-1}=\\left[\\begin{array}{ccc}I^{[1]} & X^{[n-k+1,n]} & Y^{[k+2,n]}\\end{array}\\right].$ Clearly, $f_{k,n-k-1}=\\det F^\\lozenge _{k,n-k-1}$ .",
"Besides, denote by $\\Phi ^\\lozenge _{k,n-k-1}$ the matrix obtained from $X\\Phi _{k,n-k-1}$ via replacing the column $X^{[n-k+1]}$ by the column $I^{[1]}$ .",
"Denote $\\tilde{\\varphi }^\\lozenge _{k,n-k-1}=\\det \\Phi ^\\lozenge _{k,n-k-1}$ .",
"Clearly, $\\varphi ^\\lozenge _{k,n-k-1}=s_{k,n-k-1}\\tilde{\\varphi }^\\lozenge _{k,n-k-1}\\det X$ is regular.",
"Therefore, the short Plücker relation for the matrix $\\left[\\begin{array}{cccc}I^{[1]}& X^{[n-k+1,n]} & Y^{[k+1,n]} & (YX^{-1}Y)^{[n]}\\end{array}\\right]$ and columns $I^{[1]}$ , $X^{[n-k+1]}$ , $Y^{[k+1]}$ , $(YX^{-1}Y)^{[n]}$ involves submatrices $X\\Phi _{k,n-k}$ , $X\\Phi _{k,n-k-1}$ , $X\\Phi _{k-1,n-k}$ , $\\Phi ^\\lozenge _{k,n-k-1}$ , $F^\\lozenge _{k,n-k-1}$ , and $F^\\lozenge _{k-1,n-k}$ and gives $\\det X\\tilde{\\varphi }_{k,n-k}\\tilde{\\varphi }^\\lozenge _{k,n-k-1}=-\\det X\\tilde{\\varphi }_{k-1,n-k}f_{k,n-k-1}+\\det X\\tilde{\\varphi }_{k,n-k-1}f_{k-1,n-k}.$ Multiplying by $\\det X$ and $s_{k,n-k-1}=-s_{k-1,n-k}$ and taking into account that $s_{k,n-k}=1$ , we arrive at $\\varphi _{k,n-k}\\varphi ^\\lozenge _{k,n-k-1}=\\varphi _{k-1,n-k}f_{k,n-k-1}+\\varphi _{k,n-k-1}f_{k-1,n-k},$ which together with the description of the quiver $Q_n$ means that the function that replaces $\\varphi _{k,n-k}$ after the transformation is regular whenever $1<k<n-1$ .",
"For $k=1$ the latter relation is modified taking into account that $f_{0,n-1}=h_{22}$ and $\\varphi _{0,n-1}=s_{0,n-1}s_{n-1,1}h_{11}\\varphi _{n-1,1}$ , which together with $s_{0,n-1}=s_{n-1,1}=1$ gives $\\varphi _{1,n-1}\\varphi ^\\lozenge _{1,n-2}=h_{11}\\varphi _{n-1,1}f_{1,n-2}+\\varphi _{1,n-2}h_{22}.$ This means that the function that replaces $\\varphi _{1,n-1}$ after the transformation is regular.",
"To prove that $\\varphi ^{\\lozenge }_{k,n-k-1}$ is not divisible by $\\varphi _{k,n-k}$ is enough to note that $\\varphi ^{\\lozenge }_{k,n-k-1}$ is irreducible as a minor of a matrix whose entries are independent variables.",
"Define two $n \\times n$ matrices $\\Phi ^\\square _{2,n-2}$ and $F^\\square _{n-2,1}$ via replacing the first column of $\\Phi _{2,n-2}$ by $U^{[1]}$ and the first column of $F^\\lozenge _{n-2,1}$ by $X^{[1]}$ .",
"Clearly, $f^\\square _{n-2,1}=\\det F^\\square _{n-2,1}$ is regular; besides, denote $\\tilde{\\varphi }^\\square _{2,n-2}=\\det \\Phi ^\\square _{2,n-2}$ , then $\\varphi ^\\square _{2,n-2}=-s_{2,n-2}\\tilde{\\varphi }^\\square _{2,n-2}\\det X$ is regular.",
"The short Plücker relation for the matrix $\\left[\\begin{array}{cccc}U^{[1]}& I^{[n]} & U^{[2,n]} & (U^2)^{[n]}\\end{array}\\right]$ and columns $U^{[1]}$ , $I^{[n]}$ , $U^{[2]}$ , $(U^2)^{[n]}$ involves submatrices $U\\Phi _{n-1,1}$ , $\\Phi _{1,n-2}$ , $\\Phi _{1,n-1}$ , $UX^{-1}F^\\square _{n-2,1}$ , $\\Phi ^\\square _{2,n-2}$ , and $U$ and gives $\\det U\\tilde{\\varphi }_{n-1,1}\\tilde{\\varphi }^\\square _{2,n-2}=\\det U\\tilde{\\varphi }_{1,n-2}-\\det U\\tilde{\\varphi }_{1,n-1}f^\\square _{n-2,1}.$ Multiplying the above relation by $\\det X^2$ and $s_{1,n-2}=-s_{1,n-1}$ , dividing by $\\det U$ and taking into account that $s_{k,n-k}=1$ for $0\\le k \\le n$ one gets $\\varphi _{n-1,1}\\varphi ^\\square _{2,n-2}=\\varphi _{1,n-2}+\\varphi _{1,n-1}f^\\square _{n-2,1},$ which means that the function that replaces $\\varphi _{n-1,1}$ after the transformation is regular.",
"To prove that $\\varphi ^{\\square }_{2,n-2}$ is not divisible by $\\varphi _{n-1,1}$ is enough to note that $\\varphi _{n-1,1}(I,Y)=y_{1n}$ and $\\varphi ^{\\square }_{2,n-2}(I,Y)$ is a $(n-1)\\times (n-1)$ minor of $Y$ .",
"Define a $(p+q+1)\\times (p+q+1)$ matrix $F^*_{pq}$ by $F^*_{pq}=\\left[\\begin{array}{ccc}I^{[n-p-q+1]} & X^{[n-p+1,n]} & Y^{[n-q+1,n]}\\end{array}\\right]_{[n-p-q,n]},$ and put $f^*_{pq}=\\det F^*_{pq}$ .",
"The short Plücker relation for the matrix $\\left[\\begin{array}{ccc}I^{[n-p-q,n-p-q+1]}& X^{[n-p+1,n]} & Y^{[n-q,n]}\\end{array}\\right]_{[n-p-q,n]}$ and columns $I^{[n-p-q]}$ , $I^{[n-p-q+1]}$ , $X^{[n-p+1]}$ , $Y^{[n-q]}$ gives $f_{pq}f^*_{p-1,q+1}=f_{p-1,q+1}f^*_{pq}+f_{p-1,q}f_{p,q+1}$ for $p+q<n-1$ , $p>0$ , $q>0$ .",
"Applying this relation for $(p,q)=(i,j)$ and $(p,q)=(i+1,j-1)$ we get $f_{ij}(f_{i+1,j-1}f^*_{i-1,j+1}-f_{i-1,j+1}f^*_{i+1,j-1})=f_{i+1,j-1}f_{i-1,j}f_{i,j+1}+f_{i,j-1}f_{i+1,j}f_{i-1,j+1},$ which means that the function that replaces $f_{ij}$ after the transformation is regular whenever $i+j<n-1$ , $i>0$ , $j>0$ .",
"To extend the above relation to the case $i+j=n-1$ is enough to recall that $f_{k,n-k}=\\varphi _{k,n-k}$ by the identification of Remark REF .",
"To prove that $f^{\\prime }_{ij}=f_{i+1,j-1}f^*_{i-1,j+1}-f_{i-1,j+1}f^*_{i+1,j-1}$ is not divisible by $f_{ij}$ is enough to show that $f^{\\prime }_{ij}$ is irreducible.",
"Indeed, $f^{\\prime }_{ij}$ can be written as $f^{\\prime }_{ij}=y_{n-i-j,n-j}f_{i+1,j-1}f_{i-1,j}-x_{n-i-j,n-i}f_{i-1,j+1}f_{i,j-1}+R(X,Y),$ where $R$ does not depend on $y_{n-i-j,n-j}$ and $x_{n-i-j,n-i}$ .",
"Moreover, $f_{i+1,j-1}$ , $f_{i-1,j}$ , $f_{i-1,j+1}$ , and $f_{i,j-1}$ do not depend on these two variables as well.",
"Consequently, reducibility of $f^{\\prime }_{ij}$ would imply $f^{\\prime }_{ij}=\\left(y_{n-i-j,n-j}P(X,Y)+x_{n-i-j,n-i}Q(X,Y)+R^{\\prime }(X,Y)\\right)R^{\\prime \\prime }(X,Y),$ which contradicts the irreducibility of $f_{i+1,j-1}$ , $f_{i-1,j}$ , $f_{i-1,j+1}$ , $f_{i,j-1}$ .",
"Define an $(n-i+2)\\times (n-i+2)$ matrix $H^\\star _{ii}$ by $H^\\star _{ii}=\\left[\\begin{array}{ccc} X^{[n]} & Y^{[i+1,n]} & I^{[n]}\\end{array}\\right]_{[i-1,n]}$ and denote $h^\\star _{ii}=\\det H^\\star _{ii}$ .",
"Clearly, $h_{ii}^\\star = \\det \\left[\\begin{array}{cc} X^{[n]} & Y^{[i+1,n]}\\end{array} \\right]_{[i-1,n-1]};$ in particular, $h_{nn}^\\star =x_{n-1,n}$ .",
"The short Plücker relation for the matrix $\\left[\\begin{array}{cccc} I^{[i-1]} & X^{[n]} & Y^{[i,n]} & I^{[n]}\\end{array}\\right]_{[i-1,n]}$ and columns $I^{[i-1]}$ , $X^{[n]}$ , $Y^{[i]}$ and $I^{[n]}$ gives $h_{ii}h^\\star _{ii}=f_{1,n-i}h_{i-1,i}+f_{1,n-i+1}h_{i,i+1},$ which means that the function that replaces $h_{ii}$ after the transformation is regular whenever $2\\le i\\le n$ .",
"Note that for $i=n$ we use the convention $h_{n,n+1}=1$ , which was already used above.",
"To prove that $h^\\star _{ii}$ is not divisible by $h_{ii}$ is enough to note that $h^\\star _{ii}$ is irreducible as a minor of a matrix whose entries are independent variables.",
"Define an $(n-i+2)\\times (n-i+2)$ matrix $F^\\circ _{n-i+1,1}$ and an $(n-i+1)\\times (n-i+1)$ matrix $G^\\circ _{ii}$ via replacing the column $X^{[i]}$ with the column $X^{[i-1]}$ in $F_{n-i+1,1}$ and $G_{ii}$ , respectively, and denote $f^\\circ _{n-i+1,1}=\\det F^\\circ _{n-i+1,1}$ , $g^\\circ _{ii}=\\det G^\\circ _{ii}$ .",
"The short Plücker relation for the matrix $\\left[\\begin{array}{ccc} I^{[i-1]} & X^{[i-1,n]} & Y^{[n]}\\end{array}\\right]_{[i-1,n]}$ and columns $I^{[i-1]}$ , $X^{[i-1]}$ , $X^{[i]}$ , and $Y^{[n]}$ gives $g_{ii}f^\\circ _{n-i+1,1}=f_{n-i,1}g_{i-1,i-1}+f_{n-i+1,1}g^\\circ _{ii}$ for $2\\le i\\le n$ .",
"Taking into account that $g^\\circ _{nn}=g_{n,n-1}$ and the description of the quiver $Q_n$ , we see that the function that replaces $g_{nn}$ after the transformation is regular.",
"To prove that $f^\\circ _{n-i+1,1}$ is not divisible by $g_{ii}$ is enough to note that $f^\\circ _{n-i+1,1}$ is irreducible as a minor of a matrix whose entries are independent variables.",
"Next, define an $(n-i)\\times (n-i)$ matrix $G^\\circ _{i+1,i}$ via replacing the column $X^{[i]}$ with the column $X^{[i-1]}$ in $G_{i+1,i}$ , and denote $g^\\circ _{i+1,i}=\\det G^\\circ _{i+1,i}$ .",
"The short Plücker relation for the matrix $\\left[\\begin{array}{cc} I^{[i-1,i]} & X^{[i-1,n]}\\end{array}\\right]_{[i-1,n]}$ and columns $I^{[i]}$ , $X^{[i-1]}$ , $X^{[i]}$ , and $X^{[n]}$ gives $g_{i+1,i}g^\\circ _{ii}=g_{i,i-1}g_{i+1,i+1}+g_{ii}g^\\circ _{i+1,i}$ for $2\\le i\\le n-1$ .",
"Combining relations (REF ) and (REF ) one gets $g_{ii}(g_{i+1,i}f^\\circ _{n-i+1,1}-g^\\circ _{i+1,i}f_{n-i+1,1})=f_{n-i,1}g_{i-1,i-1}g_{i+1,i}+f_{n-i+1,1}g_{i,i-1}g_{i+1,i+1},$ which means that the function that replaces $g_{ii}$ after the transformation is regular whenever $2\\le i\\le n-1$ .",
"To prove that $g_{ii}^{\\prime }=g_{i+1,i}f^\\circ _{n-i+1,1}-g^\\circ _{i+1,i}f_{n-i+1,1}$ is not divisible by $g_{ii}$ , define $X^\\circ $ to be the lower bidiagonal matrix with $x^\\circ _{ii}=t$ and all other entries in the two diagonals equal to one.",
"Then $g_{i+1,i}(X^\\circ )=1$ , $f^\\circ _{n-i+1,1}(X^\\circ ,Y)=\\pm (y_{in}-y_{i-1,n})$ , $g^\\circ _{i+1,i}(X^\\circ )=0$ , and hence $g_{ii}^{\\prime }(X^\\circ ,Y)=\\pm (y_{in}-y_{i-1,n})$ is not divisible by $g_{ii}(X^\\circ )=t$ .",
"The rest of the vertices $g_{ij}$ and $h_{ij}$ do not need a separate treatment since the corresponding relations coincide with those for the standard cluster structure in $GL_n$ ."
],
[
"Proof of Theorem ",
"Functions $x_{ni}$ , $1\\le i\\le n$ , are in the initial cluster.",
"Our goal is to explicitly construct a sequence of cluster transformations that will allow us to recover all $x_{ij}$ as cluster variables.",
"For this, we will only need to work with a subquiver $\\Gamma _n^n$ of $Q_n$ whose vertices belong to lower $n$ levels of $Q_n$ and in which we view the vertices in the top row as frozen (see Fig.",
"REF for the quiver $\\Gamma _5^5$ ).",
"Figure: Quiver Γ 5 5 \\Gamma _5^5One can distinguish two oriented triangular grid subquivers of $\\Gamma _n^n$ : a “square” one on $n^2$ vertices with the clockwise orientation of triangles in the lowest row (dashed horizontally on Fig.",
"REF ), and a “triangular” one on $n(n-1)/2$ vertices with the counterclockwise orientation of triangles in the lowest row (dashed vertically).",
"They are glued together with the help of the quiver $\\Gamma _{n}$ on $3n-2$ vertices placed in three columns of size $n$ , $n-1$ , and $n-1$ .",
"The left column of $\\Gamma _{n}$ is identified with the rightmost side of the square subquiver, and the right column, with the leftmost side of the triangular subquiver, see Fig.",
"REF a) for the case $n=5$ .",
"More generally, we can define a quiver $\\Gamma _m^n$ , $m\\le n$ , by using $\\Gamma _{m}$ to glue an oriented triangular grid quiver on $m n$ vertices forming a parallelogram with a base of length $n$ and the side of length $m$ with an oriented triangular grid quiver on $m(m-1)/2$ vertices forming a triangle with sides $m-1$ (the orientations of the triangles in the lowest rows of both grids obeys the same rule as above).",
"Figure: Quivers Γ 5 \\Gamma _5 and Γ 5 * \\Gamma ^*_5Let $A$ be an $m\\times m$ matrix; for $1\\le i\\le m-1$ , denote $a_{m-i}=\\det A_{[i+1,m]}^{[i+1,m]}, \\qquad b_{m-i}=\\det A_{[i,m-1]}^{[i+1,m]}, \\qquad c_{m-i}=\\det A_{[i+1,m]}^{[1]\\cup [i+2,m]}.$ It is easy to check that $a_{m-i}$ , $b_{m-i}$ and $c_{m-i}$ are maximal minors in the $(m-i+1)\\times (m-i+3)$ matrix $\\left[\\begin{array}{ccc} I^{[i]} & A^{[1]\\cup [i+1,m]} & I^{[m]} \\end{array}\\right]_{[i,m]},$ obtained by removing columns $A^{[1]}$ and $I^{[m]}$ , $I^{[i]}$ and $A^{[1]}$ , $A^{[i+1]}$ and $I^{[m]}$ , respectively.",
"Moreover, the maximal minor obtained by removing columns $A^{[1]}$ and $A^{[i+1]}$ equals $b_{m-i-1}$ (assuming $b_0=1$ ), and the one obtained by removing columns $I^{[i]}$ and $I^{[m]}$ equals $c_{m-i+1}$ (assuming $c_m=\\det A$ ).",
"Consequently, the short Plücker relation for the matrix above and columns $ I^{[i]}$ , $A^{[1]}$ , $A^{[i+1]}$ , $I^{[m]}$ reads $a_i a^*_i = b_i c_i + b_{i-1} c_{i+1},$ where $a^*_{m-i} =\\det A^{[1]\\cup [i+2,m]}_{[i,m-1]}$ .",
"Let us assign variables $a_i$ to the vertices in the central column of $\\Gamma _m$ bottom up, variables $b_i$ to the vertices in the right column, and variables $c_i$ to the vertices in the left column.",
"It follows from the above discussion that applying commuting cluster transformations and the corresponding quiver mutations to vertices in the central column of $\\Gamma _m$ results in the quiver $\\Gamma ^*_m$ shown in Fig.",
"REF b) for $m=5$ .",
"The variables attached to the vertices of the central column bottom up are $a_i^*$ defined above.",
"Denote $X_m=X_{[1,m]}$ and $Y_m=Y_{[1,m]}^{[n-m+1,n]}$ .",
"Extending the definition of functions $g_{ij}(X)$ in Section REF to rectangular matrices, we write $g_{ij}^m=g_{ij}(X_m) = \\det X_{[i,m]}^{[j,j+m-i]}$ for $1\\le j \\le i + n -m$ .",
"Functions $f_{kl}^m=f_{kl}(X_m,Y_m)$ and $h_{ij}^m=h_{ij}(Y_m)$ are defined exactly as in Section REF .",
"Consequently, $h_{ij}^m=h_{i+1,j}^{m-1} \\qquad \\text{for $i<j$}.$ Going back to the quiver $\\Gamma _m^n$ , let us denote by $\\widetilde{\\Sigma }_m^n$ the extended seed that consists of $\\Gamma _m^n$ and the following family of functions attached to its vertices: the functions attached to the $s$ th row listed left to right are $g_{m-s+1,1}^m,\\dots , g_{m-s+1,n-s+1}^m$ followed by $f_{s-1,1}^m,\\dots ,f_{1,s-1}^m$ followed by $h_{m-s+1,m-s+1}^m,\\dots $ , $h_{1,m-s+1}^m$ (except for the top row that does not contain $h_{11}^m$ ).",
"Now, consider the sequence of $m-1$ commuting mutations of $\\Gamma _m^n$ at vertices $h_{ii}^m$ , $2\\le i\\le m$ .",
"As in (REF ), one sees that the new variables associated with these vertices are $h_{ii}^\\star = \\det \\left[ X^{[n]} \\;\\; Y^{[i+1,m]} \\right]_{[i-1,m-1]}$ .",
"In particular, $h^\\star _{mm}=x_{m-1,n}$ , and we thus have obtained $x_{m-1,n}$ as a cluster variable.",
"Moreover, $h^\\star _{mm}=g_{m-1,n}^{m-1}$ , and $h^\\star _{ii}=f_{1,m-i}^{m-1}$ for $2\\le i\\le m-1$ .",
"The cumulative effect of this sequence of transformations can be summarized as follows: (i) detach from the quiver $\\Gamma _m^n$ a quiver isomorphic to $\\Gamma _{m}$ with $h_{ii}^m$ , $i=m, \\dots , 2$ , playing the role of $a_{i}$ , $i=1, \\dots , m-1$ , and note that functions assigned to its vertices are of the form described above if one selects $A= \\left[ X_m^{[n]}\\;\\; Y_m^{[2,m]} \\right]$ ; (ii) apply cluster transformations to vertices $a_{i}$ , $1\\le i\\le m-1$ , of $\\Gamma _{m}$ ; (iii) glue the resulting quiver $\\Gamma ^*_{m}$ back into $\\Gamma _m^n$ and erase any two-cycles that may have been created in this process; (iv) note that the new variables attached to the mutated vertices are $g_{m-1,n}^{m-1}$ , $f_{1,m-i}^{m-1}$ , $i=m-1,\\dots , 2$ .",
"The resulting quiver contains another copy of $\\Gamma _{m}$ , shifted leftwards by 1, with vertices $g_{mn}^m$ , $f_{11}^m,\\ldots , f_{1,m-2}^m$ playing the role of $a_{i}$ , $i=1, \\dots , m-1$ , and the matrix $\\left[ X_m^{[n-1,n]}\\;\\; Y_m^{[3,m]} \\right]$ playing the role of $A$ .",
"Therefore, we can repeat the procedure used on the previous step to obtain a new quiver in which $g_{mn}^m$ , $f_{11}^m,\\ldots , f_{1,m-2}^m$ are replaced by $x_{m-1,n-1}$ , $\\det \\left[ X^{[n-1,n]}\\;\\; Y^{[i+2,m]} \\right]_{[i-1,m-1]}$ , $i= m-1, \\ldots , 2$ , respectively.",
"Thus, we have obtained $x_{m-1,n-1}$ as a cluster variable, and, moreover, the new variables attached to the mutated vertices are $g_{m-1,n-1}^{m-1}$ , $g_{m-2,n-1}^{m-1}$ , $f_{2,m-i}^{m-1}$ for $i=m-1,\\dots , 3$ .",
"We proceed in the same way $n-2$ more times.",
"At the $j$ th stage, $1\\le j\\le n-1$ , the copy of $\\Gamma _m$ is shifted leftwards by $j$ , $A= \\left[ X_m^{[n-j,n]}\\;\\; Y_m^{[j+2,m]} \\right]$ , the role of $a_{i}$ , $i=1, \\dots , m-1$ , is played by vertices associated with the functions $g_{m,n-j+1}^m,\\ldots {}, g_{m-j+1, n-j+1}^m$ , $f_{j1}^m,\\ldots , f_{j,m-j}^m$ , which are being replaced with $\\begin{split}&\\det X_{[m-i,m-1]}^{[n-j,n-j+i-1]},\\quad i=1,\\dots , j+1,\\\\&\\det \\left[ X^{[n-j,n]}\\;\\; Y^{[j+i+1,m]} \\right]_{[i-1,m-1]}, \\quad i= m-j, \\ldots , 2.\\end{split}$ Note that the first of the functions listed above is $x_{m-1,n-j}$ , so in the end of this process, we have restored all the entries of the $(m-1)$ -st row of $X$ .",
"Moreover, the new variables are $g_{m-i,n-j}^{m-1}$ , $i=1,\\dots , j+1$ , $f_{j+1,m-i}^{m-1}$ , $i=m-1,\\dots , j+2$ .",
"Let us freeze in the resulting quiver $\\tilde{\\Gamma }_m^n$ all $g$ -vertices and $f$ -vertices adjacent to the frozen vertices.",
"It is easy to check that the quiver obtained in this way is isomorphic to $\\Gamma _{m-1}^n$ .",
"Moreover, the above discussion together with the identity (REF ) shows that the functions assigned to its vertices are exactly those stipulated by the definition of the extended seed $\\widetilde{\\Sigma }_{m-1}^n$ .",
"Thus we establish the claim of the theorem by applying the procedure described above consecutively to $\\widetilde{\\Sigma }_{n}^n$ , $\\widetilde{\\Sigma }_{n-1}^n, \\ldots , \\widetilde{\\Sigma }_{2}^n$ ."
],
[
"Sequence ${\\mathcal {S}}$ : the proof of Theorem ",
"Consider the subquiver $\\widehat{\\Gamma }_n^n$ of $\\Gamma _n^n$ obtained by freezing the vertices corresponding to functions $h_{ii}(Y)$ , $2\\le i\\le n$ , and ignoring vertices to the right of them.",
"In other words, $\\widehat{\\Gamma }_n^n$ is the subquiver of $Q_n$ induced by all $g$ -vertices, all $f$ -vertices, $\\varphi $ -vertices with $k+l=n$ , and $h$ -vertices with $i=j\\ge 2$ .",
"The quiver $\\widehat{\\Gamma }_5^5$ is shown in Fig.",
"REF ; the vertices that are frozen in $\\widehat{\\Gamma }_5^5$ , but are mutable in $Q_5$ are shown by rounded squares.",
"Note the special edge shown by the dashed line.",
"It does not exist in $\\widehat{\\Gamma }_n^n$ (since it connects frozen vertices), but it exists in $Q_n$ .",
"Within this proof we label the vertices of $\\widehat{\\Gamma }_n^n$ by pair of indices $(i,j)$ , $1\\le i\\le n$ , $1\\le j\\le n+1$ , $(i,j)\\ne (1, n+1)$ , where $i$ increases from top to bottom and $j$ increases from left to right; thus, the special edge is $(1,n)\\rightarrow (2,n+1)$ .",
"The set of vertices with $j-i=l$ forms the $l$ th diagonal in $\\widehat{\\Gamma }_n^n$ , $1-n\\le l \\le n-1$ .",
"The function attached to the vertex $(i,j)$ is $\\chi _{ij} = {\\left\\lbrace \\begin{array}{ll} g_{ij}(X) = \\det X_{[i,n]}^{ [j, n+j-i]}& \\mbox{if}\\quad i \\ge j, \\\\f_{n-j+1,j-i}(X,Y)= \\det \\left[ X^{ [j,n]}\\;\\; Y^{[n+i-j+1,n]} \\right]_{[i,n]} & \\mbox{if} \\quad i < j.",
"\\end{array}\\right.",
"}$ We denote the extended seed thus obtained from $\\widetilde{\\Sigma }_n^n$ by $\\widehat{\\Sigma }_n^n$ .",
"Note that it is a seed of an ordinary cluster structure, since no generalized exchange relations are involved.",
"Figure: Quiver Γ ^ 5 5 \\widehat{\\Gamma }_5^5Consider a sequence of mutations ${\\mathcal {S}}_n$ which involves mutating once at every non-frozen vertex of $\\widehat{\\Gamma }_n^n$ starting with $(n,2)$ then using vertices of the $(3-n)$ -th, $(4-n)$ -th, ..., $(n-3)$ -rd, $(n-2)$ -nd diagonals.",
"Note that a similar sequence of transformations was used in the proof of Proposition 4.15 in [15] in the study of the natural cluster structure on Grassmannians.",
"The order in which vertices of each diagonal are mutated is not important, since at the moment a diagonal is reached in this sequence of transformations, there are no edges between its vertices.",
"In fact, functions $\\chi _{ij}^1$ obtained as a result of applying ${\\mathcal {S}}_n$ are subject to relations $\\chi _{ij}^1 \\chi _{ij} = \\chi ^1_{i,j-1} \\chi _{i,j+1} + \\chi ^1_{i+1,j} \\chi _{i-1,j},\\qquad 2\\le i,j \\le n,$ where we adopt a convention $\\chi ^1_{n+1,j} = x_{n1}$ , $\\chi ^1_{i1} = \\chi _{i-1,1}=g_{i-1,1}(X)$ .",
"These relations imply $\\chi ^1_{ij}= {\\left\\lbrace \\begin{array}{ll} \\det X_{[i-1,n]}^{[1] \\cup [j+1, n+j-i+1]} & \\mbox{if}\\quad i > j, \\\\\\det \\left[ X^{[1]\\cup [j+1,n]}\\;\\; Y^{[n+i-j,n]} \\right]_{[i-1,n]} & \\mbox{if} \\quad i \\le j.",
"\\end{array}\\right.",
"}$ To verify (REF ) for $i>j$ , one has to apply the short Plücker relation to $\\left[\\begin{array}{cc} I^{[i-1]} & X^{[1]\\cup [j,n+j-i+1]} \\end{array}\\right]_{[i-1,n]}$ using columns $I^{[i-1]}$ , $X^{[1]}$ , $X^{[j]}$ , $X^{[n+j-i+1]}$ .",
"In the case $i\\le j$ , the short Plücker relation is applied to $\\left[\\begin{array}{ccc} I^{[i-1]} & X^{[1]\\cup [j,n]} & Y^{[n+i-j,n]} \\end{array}\\right]_{[i-1,n]}$ using columns $I^{[i-1]}$ , $X^{[1]}$ , $X^{[j]}$ , $Y^{[n+i-j]} $ .",
"Note that $\\begin{aligned}\\chi ^1_{2j} &= \\det \\left[ X^{[1]\\cup [j+1,n]}\\;\\; Y^{[n-j+2,n]} \\right]_{[1,n]}= \\det X \\cdot (-1)^{(n-j)(j-1)}\\det U_{[2,j]}^{[n-j+2,n]}\\\\ &= \\det X\\cdot (-1)^{(n-j)(n-2)}h_{2,n-j+2}(U), \\qquad 2\\le j\\le n.\\end{aligned}$ The subquiver of ${\\mathcal {S}}_n(\\widehat{\\Gamma }_n^n)$ formed by non-frozen vertices is isomorphic to the corresponding subquiver of $\\widehat{\\Gamma }_n^n$ .",
"However, the frozen vertices are connected to non-frozen ones in a different way now: there are arrows $(i,1)\\rightarrow (i+2, 2)$ and $(i+1,n+1)\\rightarrow (i+2, n)$ for $1\\le i\\le n-2$ , $(i,2)\\rightarrow (i-1, 1)$ and $(i,n)\\rightarrow (i, n+1)$ for $2\\le i\\le n$ , $(1,j)\\rightarrow (2, j)$ for $2\\le j\\le n$ , $(n,1)\\rightarrow (n,n)$ , and $(2,j)\\rightarrow (1, j+1)$ for $2\\le j\\le n-1$ .",
"After moving frozen vertices we can make ${\\mathcal {S}}_n(\\widehat{\\Gamma }_n^n)$ look as shown in Fig.",
"REF .",
"Figure: Quiver 𝒮 n (Γ ^ 5 5 ){\\mathcal {S}}_n(\\widehat{\\Gamma }_5^5)Note that if we freeze the vertices $(2,2), \\ldots , (2,n), (3,n), \\ldots (n,n)$ in ${\\mathcal {S}}_n(\\widehat{\\Gamma }_n^n)$ (marked gray in Fig.",
"REF ) and ignore the isolated frozen vertices thus obtained, we will be left with a quiver isomorphic to $\\widehat{\\Gamma }_{n-1}^{n-1}$ whose vertices are labeled by $(i,j)$ , $2\\le i\\le n$ , $1\\le j\\le n$ , $(i,j)\\ne (2, n)$ , and have functions $\\chi ^1_{ij}$ attached to them.",
"The new special edge is $(2,n-1)\\rightarrow (3,n)$ .",
"Let us call the resulting extended seed $\\widehat{\\Sigma }_{n-1}^{n-1}$ .",
"We can now repeat the procedure described above $n-2$ more times by applying, on the $k$ th step, the sequence of mutations ${\\mathcal {S}}_{n-k+1}$ to the extended seed $\\widehat{\\Sigma }_{n-k+1}^{n-k+1}= \\left((\\chi ^{k-1}_{ij})_{k\\le i\\le n, 1\\le j\\le n-k+2,(i,j)\\ne (k, n-k+2)}, \\widehat{\\Gamma }_{n-k+1}^{n-k+1}\\right).$ The functions $\\chi ^k_{ij}$ are subject to relations $\\chi ^{k}_{ij} \\chi ^{k-1}_{ij} = \\chi ^{k}_{i,j-1} \\chi ^{k-1}_{i,j+1} + \\chi ^{k}_{i+1,j} \\chi ^{k-1}_{i-1,j},\\qquad k+1\\le i\\le n, \\quad 2\\le j\\le n-k+1,$ where we adopt the convention $\\chi ^{k}_{n+1,j} = \\chi ^{k-1}_{n1}$ , $\\chi ^{k}_{i1} = \\chi ^{k-1}_{i-1,1}$ .",
"Arguing as above, we conclude that $\\chi ^k_{ij}= {\\left\\lbrace \\begin{array}{ll} \\det X_{[i-k,n]}^{[1,k] \\cup [j+k, n+j-i+k]} & \\mbox{if}\\quad i - k+1 > j, \\\\\\det \\left[ X^{[1,k]\\cup [j+k,n]}\\;\\; Y^{[n+i-j+1-k,n]} \\right]_{[i-k,n]} & \\mbox{if} \\quad i - k+1 \\le j.",
"\\end{array}\\right.",
"}$ To verify (REF ) for $i-k+1>j$ , one has to apply the short Plücker relation to $\\left[ \\begin{array}{cc}I^{[i-k]} & X^{[1,k] \\cup [j+k-1, n+j-i+k]} \\end{array} \\right]_{[i-k,n]}$ using columns $I^{[i-k]}$ , $X^{[k]}$ , $X^{[j+k-1]}$ , $X^{[n+j-i+k]}$ .",
"In the case $i-k+1\\le j$ , the short Plücker relation is applied to $\\left[ \\begin{array}{ccc} I^{[i-k]} & X^{[1,k]\\cup [j+k-1,n]} & Y^{[n+i-j-k+1,n]}\\end{array} \\right]_{[i-k,n]}$ using columns $I^{[i-k]}$ , $X^{[k]}$ , $X^{[j+k-1]}$ , $Y^{[n+i-j-k+1]}$ .",
"Note that $\\chi ^k_{k+1,j} &= \\det \\left[ X^{[1,k]\\cup [j+k,n]}\\;\\; Y^{[n-j+2,n]} \\right]_{[1,n]} \\\\&= \\det X \\cdot (-1)^{(n-j-k+1)(j-1)}\\det U_{[k+1,j+k-1]}^{[n-j+2,n]}\\\\&= \\det X \\cdot (-1)^{(n-j-k+1)(n-k-1)}h_{k+1, n-j+2}(U),\\qquad 2\\le j\\le n-k+1.$ Define the sequence of transformations ${\\mathcal {S}}$ as the composition ${\\mathcal {S}}= {\\mathcal {S}}_2 \\circ \\cdots \\circ {\\mathcal {S}}_n$ .",
"Assertion (i) of Theorem REF follows from the fact that $\\varphi $ -vertices of $Q_n$ are not involved in any of ${\\mathcal {S}}_i$ .",
"This fact also implies that the subquiver of $Q_n$ induced by $\\varphi $ -vertices remains intact in ${\\mathcal {S}}(Q_n)$ .",
"As it was shown above, the function $\\det X\\cdot h_{ij}(U)$ is attached to the vertex $(i,n-j+2)$ .",
"It is easy to prove by induction that the last mutation at $(i,j)$ (which occurs at the $(i-1)$ -st step) creates edges $(i,j)\\rightarrow (i,j-1)$ , $(i-1,j)\\rightarrow (i,j)$ and $(i,j-1)\\rightarrow (i-1,j)$ .",
"Comparing this with the description of $Q_n$ in Section REF and the definition of $Q_n^\\dag $ in Section REF yields assertions (ii) and (iii) of Theorem REF .",
"Finally, assertion (iv) follows from the fact that the special edge $(i,n-i+1)\\rightarrow (i+1,n-i+2)$ disappears after the last mutation at $(i+1,n-i+1)$ .",
"The quiver ${\\mathcal {S}}(Q_5)$ and the subquiver $Q_5^{\\prime }$ are shown in Fig.",
"REF .",
"The vertices of $Q_5^{\\prime }$ are shadowed in dark gray.",
"The area shadowed in light gray represents the remaining part of ${\\mathcal {S}}(Q_5)$ .",
"The only vertices in this part shown in the figure are those connected to vertices of $Q_5^{\\prime }$ .",
"Figure: Quiver 𝒮(Q 5 ){\\mathcal {S}}(Q_5) and the subquiver Q 5 ' Q_5^{\\prime }"
],
[
"The nerve $\\mathcal {N}_0$",
"The nerve $\\mathcal {N}_0$ is obtained as follows: it contains the seed $\\widetilde{\\Sigma }^{\\prime }_n$ built in the proof of Theorem REF , the seed $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ adjacent to $\\widetilde{\\Sigma }^{\\prime }_n$ in direction $\\psi _{n-1,1}$ , and the seed $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ adjacent to $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ in direction $\\psi _{1,n-1}$ .",
"Besides, it contains $n-3$ seeds adjacent to $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ in directions $\\psi _{n-i,i}$ , $2\\le i\\le n-2$ , and $2(n-1)^2$ seeds adjacent to $\\widetilde{\\Sigma }^{\\prime }_n$ in all the remaining directions.",
"We subdivide $\\mathcal {N}_0$ into several disjoint components.",
"Component I contains the seed $\\widetilde{\\Sigma }^{\\prime }_n$ and its $(n-1)^2$ neighbors in directions that are not in $Q_n^{\\prime }$ .",
"Component II contains $n-3$ neighbors of $\\widetilde{\\Sigma }^{\\prime }_n$ in directions $\\psi _{i1}$ , $2\\le i\\le n-2$ .",
"Component III contains only the neighbor of $\\widetilde{\\Sigma }^{\\prime }_n$ in direction $\\psi _{11}$ .",
"Component IV contains $(n-2)(n-3)/2$ seeds adjacent to $\\widetilde{\\Sigma }^{\\prime }_n$ in directions $\\psi _{kl}$ , $k+l<n$ , $l>1$ .",
"Component V contains $n(n-1)/2$ seeds adjacent to $\\widetilde{\\Sigma }^{\\prime }_n$ in directions $h_{ij}$ , $2\\le i\\le j\\le n$ .",
"Component VI contains the seeds $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ and $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ together with all other seeds adjacent to the latter.",
"In each of the components we consider several normal forms for $U$ with respect to actions of different subgroups of $GL_n$ .",
"We show how to restore entries of these normal forms and, consequently, the entries of $U$ as Laurent expressions in corresponding clusters.",
"Recall that $\\det X$ and $\\det X^{-1}$ belong to the ground ring, so it suffices to obtain Laurent expressions in variables $\\psi _{kl}$ , $h_{ij}$ (and their neighbors), and $c_i$ instead of actual cluster variables."
],
[
"Component I",
"To restore $U$ in component I, we use two normal forms for $U$ : one under right multiplication by unipotent lower triangular matrices, and the other under conjugation by unipotent lower triangular matrices, so $U=B_+N_-=\\bar{N}_- \\bar{B}_+ C \\bar{N}_-^{-1}$ , where $B_+, \\bar{B}_+$ are upper triangular, $N_-, \\bar{N}_-$ are unipotent lower triangular, and $C=e_{21} + \\cdots + e_{n, n-1} + e_{1n}$ is the cyclic permutation matrix (cf.",
"with (REF )).",
"Note that by (REF ), $h_{ij}(U)=h_{ij}(B_+)$ and $\\psi _{kl}(U)=\\psi _{kl}(\\bar{B}_+C)$ .",
"Our goal is to restore $B_+$ and $\\bar{B}_+C$ .",
"Once this is done, the matrix $U$ itself is restored as follows.",
"We multiply the equality $B_+N_-=\\bar{N}_-\\bar{B}_+ C \\bar{N}_-^{-1}$ by $W_0$ on the left and by $\\bar{N}_-$ on the right, where $W_0$ is the matrix corresponding to the longest permutation of size $n$ , and consider the Gauss factorizations (REF ) for $W_0B_+$ in the left side and for $W_0\\bar{B}_+C$ in the right hand side.",
"This gives $(W_0B_+)_{>0}(W_0B_+)_{\\le 0}N_-\\bar{N}_-=W_0\\bar{N}_-W_0\\cdot (W_0\\bar{B}_+C)_{>0}(W_0\\bar{B}_+C)_{\\le 0},$ where $W_0\\bar{N}_-W_0$ is unipotent upper triangular.",
"Consequently, $\\bar{N}_-=W_0(W_0B_+)_{>0}(W_0\\bar{B}_+C)_{>0}^{-1}W_0.$ Recall that matrix entries in the Gauss factorization are given by Laurent expressions in the entries of the initial matrix with denominators equal to its trailing principal minors (see, e.g., [12]).",
"Clearly, the trailing principal minors of $W_0B_+$ and $W_0 \\bar{B}_+C$ are just $\\psi _{n-i,i}$ , which allows to restore $U$ in any cluster of component I.",
"Restoration of $B_+=(\\beta _{ij})$ is standard: an explicit computation shows that $\\beta _{ii}=\\pm h_{ii}/h_{i+1,i+1}$ with $h_{n+1,n+1}=1$ , and $\\beta _{ij}$ for $i<j$ is a Laurent polynomial in $h_{kl}$ , $k\\le l$ , with denominators in the range $i+1\\le k\\le n$ , $j+1\\le l\\le n$ (here $h_{1l}$ is identified up to a sign with $\\psi _{l-1, n-l+1}$ for $l>1$ ).",
"Since all $h_{ij}$ are cluster variables in the clusters of component I, we are done.",
"In order to restore $\\bar{M}=\\bar{B}_+C$ we proceed as follows.",
"Let $\\bar{B}_+=(\\bar{\\beta }_{ij})$ .",
"Clearly, $\\psi _{n-k,1}=\\pm \\prod _{i=1}^k\\bar{\\beta }_{ii}^{k-i+1}$ for $1\\le k\\le n-1$ , which yields $\\bar{\\beta }_{ii}=\\pm \\frac{\\psi _{n-i,1}\\psi _{n-i+2,1}}{\\psi _{n-i+1,1}^2}, \\qquad 1\\le i\\le n-1,$ where we assume $\\psi _{n1}=\\psi _{n+1,1}=1$ .",
"The remaining diagonal entry $\\bar{\\beta }_{nn}$ is given by $\\bar{\\beta }_{nn}=h_{11}\\prod _{i=1}^{n-1}\\bar{\\beta }_{ii}^{-1}=\\pm \\frac{h_{11}\\psi _{21}}{\\psi _{11}}.$ Remark 7.1 Note that the only diagonal entries depending on $\\psi _{11}$ are $\\bar{\\beta }_{nn}$ and $\\bar{\\beta }_{n-1,n-1}=\\pm \\psi _{11}\\psi _{31}/\\psi _{21}^2$ .",
"This fact will be important for the restoration process in component III below.",
"We proceed with the restoration process and use (REF ) to find $\\det (\\bar{M})_{[n-k-l+2,n-k]}^{[n-l+1,n-1]}=\\pm \\psi _{kl}\\prod _{i=1}^{n-k-l+1}\\bar{\\beta }_{ii}^{k+l+i-n-2},$ which together with (REF ) gives $\\det (\\bar{M})_{[n-k-l+2,n-k]}^{[n-l+1,n-1]}=\\pm \\psi _{kl}\\psi _{k+l,1}^{k+l-3}/\\psi _{k+l-1,1}^{k+l-2}$ .",
"By Remark REF , this means that all entries $\\bar{\\beta }_{ij}$ with $i>1$ are restored as Laurent polynomials in any cluster in component I.",
"Note that non-diagonal entries do not depend on $\\psi _{11}$ .",
"Remark 7.2 Using Remark REF , we can find signs in the above relations.",
"Specifically, $\\bar{\\beta }_{n-1,n}=(-1)^n\\psi _{12}/\\psi _{21}$ .",
"This fact will be used in the restoration process in component III below.",
"To restore the entries in the first row of $\\bar{M}$ , we first conjugate it by a diagonal matrix $\\Delta =\\operatorname{diag}(\\delta _1,\\dots \\delta _{n-1},h_{11}^{-1})$ so that $\\Delta \\bar{M}\\Delta ^{-1}$ has ones on the subdiagonal.",
"This condition implies $\\delta _i=\\pm \\psi _{n-i+1,1}/\\psi _{n-i,1}$ .",
"Consequently, the entries of the rows $2,\\dots ,n$ of $\\Delta \\bar{M}\\Delta ^{-1}$ remain Laurent polynomials.",
"Next, we further conjugate the obtained matrix with a unipotent upper triangular matrix $N_+$ so that $\\bar{M}^*=N_+\\Delta \\bar{M}\\Delta ^{-1}N_+^{-1}$ has the companion form $\\bar{M}^*=\\left[\\begin{array}{c} \\gamma \\\\ \\mathbf {1}_{n-1}\\;\\; 0\\end{array}\\right]$ with $\\gamma =(\\gamma _1,\\dots ,\\gamma _n)$ .",
"If we set all non-diagonal entries in the last column of $N_+$ equal to zero, all other entries of $N_+$ (and hence of $N_+^{-1}$ ) can be restored uniquely as polynomials in the entries in the rows $2,\\dots ,n$ of $\\Delta \\bar{M}\\Delta ^{-1}$ .",
"Recall that $\\bar{M}^*$ is obtained from $U$ by conjugations, and hence $U$ and $\\bar{M}^*$ are isospectral.",
"Therefore, $\\gamma _i=(-1)^{i-1} c_{n-i}$ for $1\\le i\\le n$ .",
"This allows to restore the entries in the first row of $\\bar{M}=\\Delta ^{-1}N_+^{-1}\\bar{M}^* N_+\\Delta $ as Laurent polynomials in any cluster in component I.",
"Remark 7.3 Note that although diagonal entries of $B_+$ are Laurent monomials in stable variables $h_{ii}$ , Laurent expressions for entries of $(W_0B_+)_{>0}$ depend on $h_{ii}$ polynomially.",
"This follows from the fact that these entries are Laurent polynomials in dense minors of $W_0B_+$ containing the last column; recall that such minors are cluster variables in any cluster of component I.",
"Moreover, dense minors containing the upper right corner enter these expressions polynomially, see Remark REF .",
"Consequently, stable variables $h_{ii}$ do not enter denominators of Laurent expressions for entries of $U$ by (REF ), since restoration of $\\bar{B}_+C$ does not involve division by $h_{ii}$ ."
],
[
"Component II",
"The two normal forms used in this component are given by $U=B_+N_-=\\check{N}_-\\check{B}_+ W_0 \\check{N}_-^{-1}$ , where $B_+, \\check{B}_+$ are upper triangular, $N_-, \\check{N}_-$ are unipotent lower triangular, and $W_0$ is the matrix corresponding to the longest permutation, see Lemma REF .",
"Note that by (REF ), $h_{ij}(U)=h_{ij}(B_+)$ and $\\psi _{kl}(U)=\\psi _{kl}(\\check{B}_+W_0)$ .",
"Our goal is to restore $B_+$ and $\\check{B}_+W_0$ .",
"Once this is done, the matrix $U$ itself is restored as follows.",
"We multiply the equality $B_+N_-=\\check{N}_-\\check{B}_+ W_0 N_-^{-1}$ by $W_0$ on the left and by $\\check{N}_-$ on the right and consider the Gauss factorization (REF ) for $W_0B_+$ in the left hand side.",
"This gives $(W_0B_+)_{>0}(W_0B_+)_{\\le 0}N_-\\check{N}_-=W_0\\check{N}_-W_0\\cdot W_0\\check{B}_+W_0$ where $W_0\\check{N}_-W_0$ is unipotent upper triangular and $W_0\\check{B}_+W_0$ is lower triangular.",
"Consequently, $\\check{N}_-=W_0(W_0B_+)_{>0}W_0$ .",
"Clearly, the trailing principal minors of $W_0B_+$ are just $\\psi _{n-i,i}$ , which allows to restore $U$ in $\\widetilde{\\Sigma }^{\\prime }_n$ and in any cluster of component II.",
"Restoration of $B_+$ is exactly the same as before.",
"In order to restore $\\check{M}=\\check{B}_+W_0$ we proceed as follows.",
"Let $\\check{B}_+=(\\check{\\beta }_{ij})_{1\\le i\\le j\\le n}$ .",
"We start with the diagonal entries.",
"An explicit computation immediately gives $\\check{\\beta }_{ii}=\\pm \\frac{\\psi _{n-i,i}}{\\psi _{n-i+1,i-1}}, \\qquad 1\\le i\\le n,$ with $\\psi _{n0}=1$ and $\\psi _{0n}=h_{11}$ .",
"Next, we recover the entries in the last column of $\\check{B}_+$ .",
"We find $\\psi _{n-l,l-1}=\\pm \\check{\\beta }_{11}\\check{\\beta }_{ln}\\prod _{i=1}^{l-1}\\check{\\beta }_{ii},$ which together with (REF ) gives $\\check{\\beta }_{ln}=\\pm \\frac{\\psi _{n-l,l-1}}{\\psi _{n-1,1}\\psi _{n-l+1,l-1}}, \\quad 2\\le l\\le n-1.$ Note that we have already restored the last two rows of $\\check{M}$ .",
"We restore the other rows consecutively, starting from row $n-2$ and moving upwards.",
"To this end, define an $n\\times n$ matrix $\\Psi $ via $\\Psi =\\left[\\begin{array}{ccccc} e_1 & \\check{M}e_1 & \\check{M}^{2}e_1 & \\dots & \\check{M}^{n-1}e_1\\end{array}\\right].$ Clearly, for $2\\le l\\le n-2$ , $2\\le t\\le n-l+1$ one has $\\psi _{n-t-l+2,l-1}=\\pm \\check{\\beta }_{11}^{t-1}\\prod _{i=1}^{l-1}\\check{\\beta }_{ii}\\det \\Psi _{[l,t+l-2]}^{[2,t]},$ which together with (REF ) yields $\\det \\Psi _{[l,t+l-2]}^{[2,t]}=\\pm \\frac{\\psi _{n-t-l+2,l-1}}{\\psi _{n-l+1,l-1}\\psi _{n-1,1}^{t-1}}.$ Therefore, each $\\det \\Psi _{[l,t+l-2]}^{[2,t]}$ is a Laurent polynomial in any cluster in component II.",
"Moreover, the minors $\\det \\Psi _I^{[2,t]}$ possess the same property for any index set $I\\subset [2,n]$ , $|I|=t-1$ , since they can be expressed as Laurent polynomials in $\\det \\Psi _{[l,t+l-2]}^{[2,t]}$ for $l>2$ that are polynomials in $\\det \\Psi _{[2,t]}^{[2,t]}$ , see Remark REF .",
"On the other hand, $\\Psi _{[l,t+l-2]}^{[2,t]}=\\check{M}_{[l,t+l-2]}\\Psi ^{[1,t-1]}$ , which yields a system of linear equations on the entries $\\check{\\beta }_{lj}$ : $\\begin{split}\\sum _{j=2}^{n-l}\\check{\\beta }_{lj}&\\det \\left(\\left[\\begin{array}{c} e_j^T\\\\ \\check{M}_{[l+1,t+l-2]}\\end{array}\\right]\\Psi ^{[1,t-1]}\\right)\\\\& =\\det \\Psi _{[l,t+l-2]}^{[2,t]}-\\det \\left(\\left[\\begin{array}{c} \\hat{\\beta }_l\\\\ \\check{M}_{[l+1,t+l-2]}\\end{array}\\right]\\Psi ^{[1,t-1]}\\right),\\quad 3\\le t\\le n-l+1,\\end{split}$ where $\\hat{\\beta }_{l1}=\\check{\\beta }_{1l}$ , $\\hat{\\beta }_{l,n-l+1}=\\check{\\beta }_{ll}$ , and $\\hat{\\beta }_{lj}=0$ for $j\\ne 1, n-l+1$ .",
"Rewrite the second determinant in the right hand side of (REF ) via the Binet–Cauchy formula; it involves minors $\\det \\Psi ^{[2,t]}_I$ and minors of $\\check{M}_{[l+1,t+l-2]}$ .",
"Assuming that the entries in rows $l+1,\\dots ,n$ have been already restored and taking into account (REF ), we ascertain that the right hand side can be expressed as a Laurent polynomial in any cluster in component II.",
"Clearly, the same is also true for the coefficients in the left hand side of (REF ).",
"It remains to calculate the determinant of the linear system (REF ).",
"Denote the coefficient at $\\check{\\beta }_{lj}$ in the $t$ -th equation by $a_{j,t-2}$ .",
"Then $a_{jk}=\\sum _{i=1}^k\\left(\\check{M}^i\\right)_{l1}\\det \\Psi _{[l+1,k+l]}^{[2,i+1]\\cup [i+3,k+2]}=\\sum _{i=1}^k\\left(\\check{M}^i\\right)_{l1}\\gamma _{ik}.$ Let $A=(a_{jk})$ , $2\\le j\\le n-l$ , $1\\le k\\le n-l-1$ , be the matrix of the system (REF ).",
"By the above formula we get $A=\\left[\\begin{array}{ccc} \\check{M}e_1 & \\dots &\\check{M}^{n-l-1}e_1\\end{array}\\right]_{[2,n-l]}\\left[\\begin{array}{cccc}\\gamma _{11} & \\gamma _{12} & \\dots &\\gamma _{1,n-l-1}\\\\0 & \\gamma _{22} & \\dots & \\gamma _{2,n-l-1}\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & \\dots & \\gamma _{n-l-1,n-l-1}\\end{array}\\right],$ and hence $\\begin{split}\\det A=\\det \\Psi _{[2,n-l]}^{[2,n-l]}\\prod _{i=1}^{n-l-1}\\gamma _{ii}&=\\det \\Psi _{[2,n-l]}^{[2,n-l]}\\prod _{i=1}^{n-l-1}\\det \\Psi _{[l+1,l+i]}^{[2,i+1]}\\\\&=\\pm \\frac{\\psi _{l1}}{\\psi _{n-l,l}^{n-l-1}\\psi _{n-1,1}^{(n-l)(n-l+1)/2}}\\prod _{i=1}^{n-l-1}\\psi _{n-l-i,l}.\\end{split}$ We thus restored the entries in the $l$ -th row of $\\check{M}$ as Laurent polynomials in all clusters in component II except for the neighbor of $\\widetilde{\\Sigma }^{\\prime }_n$ in direction $\\psi _{l1}$ , which we denote $\\widetilde{\\Sigma }^{\\prime }_n(l)$ .",
"In the latter cluster $\\psi _{l1}$ , which enters the expression for $\\det A$ , is no longer available.",
"It is easy to see that the factor $\\psi _{l1}$ in $\\det A$ comes from the $(n-l+1)$ -st equation in (REF ): its left hand side is defined by the expression $\\det \\check{M}_{[l,n-1]}^{[1,n-l]}\\det \\Psi _{[2,n-l]}^{[2,n-l]}=\\pm \\det \\check{M}_{[l,n-1]}^{[1,n-l]}\\frac{\\psi _{l1}}{\\psi _{n-1,1}^{n-l}}.$ In other words, each coefficient in the left hand side of this equation is proportional to $\\psi _{l1}$ .",
"To avoid this problem, we replace this equation by a different one.",
"Recall that by (REF ), in the cluster under consideration $\\psi _{l1}$ is replaced by $\\psi _{1,l-1}^{**}$ given by $\\begin{split}&\\psi _{1,l-1}^{**}=\\pm \\det \\left[e_n \\; \\check{M}^{[n-l+2,n]} \\; \\check{M}^2e_{n-1} \\; \\check{M}^2e_n \\; \\check{M}^3e_n \\dots \\check{M}^{n-l}e_n\\right]\\\\&\\quad =\\pm \\check{\\beta }_{11}^{n-l-1}\\check{\\beta }_{22} \\det \\left[\\check{M}^{[n-l+2,n]} \\; \\check{M}e_{2} \\; \\check{M}e_1 \\; \\check{M}^2e_1 \\dots \\check{M}^{n-l-1}e_1\\right]_{[1,n-1]}\\\\&\\quad =\\pm \\check{\\beta }_{11}^{n-l}\\check{\\beta }_{22}\\det \\left(\\check{M}_{[2,n-1]}^{[1,n-1]}\\left[\\mathbf {1}^{[1,2]\\cup [n-l+2,n-1]} \\; \\check{M}e_1 \\; \\check{M}^2e_1 \\dots \\check{M}^{n-l-2}e_1\\right]_{[1,n-1]}\\right);\\end{split}$ here we used relations $\\check{M}e_n=\\check{\\beta }_{11}e_1$ and $\\check{M}e_{n-1}=\\check{\\beta }_{12}e_1+\\check{\\beta }_{22}e_2$ .",
"By the Binet–Cauchy formula, the latter determinant can be written as $\\sum _{j=1}^{n-1}\\det \\check{M}_{[n-j+1,n-1]}^{[1,j-1]}\\prod _{i=2}^{n-j}\\check{\\beta }_{ii}\\det \\left[\\mathbf {1}^{[1,2]\\cup [n-l+2,n-1]} \\; \\check{M}e_1 \\; \\check{M}^2e_1 \\dots \\check{M}^{n-l-2}e_1\\right]_{[1,n-1]\\setminus \\lbrace j\\rbrace }.$ Clearly, the second factor in each summand vanishes for $j=1,2$ and $n-l+2\\le j\\le n-1$ .",
"For $3\\le j\\le n-l+1$ , the second factor equals $\\det \\Psi _{[3,n-l+1]\\setminus \\lbrace j\\rbrace }^{[2,n-l-1]}$ .",
"As it was explained above, for $3\\le j\\le n-l$ these determinants are Laurent polynomials in all clusters of component II, whereas the corresponding first factors contain only entries in rows $l+1,\\dots ,n$ of $\\check{M}$ , which are already restored as Laurent polynomials.",
"Consequently, the left hand side of the new equation for the entries in the $l$ -th row is defined by the $(n-l+1)$ -st summand $\\det \\check{M}_{[l,n-1]}^{[1,n-l]}\\prod _{i=2}^{l-1}\\check{\\beta }_{ii}\\det \\Psi _{[3,n-l]}^{[2,n-l-1]}=\\pm \\det \\check{M}_{[l,n-1]}^{[1,n-l]}\\frac{\\psi _{l-1,n-l+1}\\psi _{l2}}{\\psi _{n-2,2}\\psi _{n-1,1}^{n-l-1}};$ note that the inverse of the factor in the right hand side above is a Laurent monomial in the cluster under consideration.",
"Comparing with (REF ), we infer that the left hand side of the new equation is proportional to the left hand side of the initial one.",
"Therefore, the determinant of the new system is Laurent in the cluster $\\widetilde{\\Sigma }^{\\prime }_n(l)$ , and the entries of the $l$ -th row of $\\check{M}$ are restored as Laurent polynomials.",
"Therefore, the entries in all rows of $\\check{M}$ except for the first one are restored as Laurent polynomials in component II.",
"The entries of the first row are restored via Lemma REF as polynomials in the entries of other rows and variables $c_i$ divided by the $\\det K=\\det \\Psi =\\pm h_{11}\\psi _{11}/\\psi _{n-1,1}^n$ ; the latter equality follows from $\\check{\\beta }_{11}e_1=\\check{M} e_n$ and (REF )."
],
[
"Component III",
"In this component we use all three normal forms that have been used in components I and II.",
"Restoration of $B_+$ is exactly the same as before.",
"Restoration of $\\check{B}_+W_0$ goes through for all entries except for the entries in the first row, since the determinant $\\det K$ involves a factor $\\psi _{11}$ , which is no longer available.",
"On the other hand, restoration of $\\bar{B}_+C$ also fails at two instances: firstly, at the entry $\\bar{\\beta }_{nn}$ , see Remark REF , and secondly, at the entry $\\bar{\\beta }_{1n}$ , which gets $\\psi _{11}$ in the denominator after the conjugation by $\\Delta ^{-1}$ .",
"However, we will be able to use partial results obtained during restoration of $\\bar{B}_+C$ in order to restore the first row of $\\check{B}_+W_0$ .",
"Indeed, using Lemma REF we can write $\\check{M} = W_0\\left(W_0 U\\right)_{\\le 0} W_0 \\left(W_0 \\bar{M}\\right)_{>0}W_0$ .",
"Clearly, $W_0\\bar{M}$ is block-triangular with diagonal blocks 1 and $W_0^{\\prime }\\bar{B}_+^{\\prime }$ , where $\\bar{B}_+^{\\prime }=(\\bar{B}_+)_{[2,n]}^{[2,n]}$ and $W_0^{\\prime }$ is the matrix of the longest permutation on size $n-1$ .",
"Therefore, $(\\check{\\beta }_{11}, \\check{\\beta }_{12}, \\dots , \\check{\\beta }_{1n})=(\\bar{\\beta }_{11},\\bar{\\beta }_{1n},\\bar{\\beta }_{1,n-1},\\dots ,\\bar{\\beta }_{12})\\left[\\begin{array}{cc} 0 & (W_0^{\\prime }\\bar{B}_+^{\\prime })_{>0}W_0^{\\prime }\\\\1 & 0\\end{array}\\right],$ which can be more conveniently written as $\\begin{split}(\\check{\\beta }_{11}, \\check{\\beta }_{12}, \\dots , \\check{\\beta }_{nn})&\\left[\\begin{array}{cc} 1 & 0\\\\0 & W_0^{\\prime }(W_0^{\\prime }\\bar{B}_+^{\\prime })_{\\le 0}W_0^{\\prime }\\end{array}\\right]\\\\&=(\\bar{\\beta }_{11},\\bar{\\beta }_{1n},\\bar{\\beta }_{1,n-1},\\dots ,\\bar{\\beta }_{12})\\left[\\begin{array}{cc} 0 & W_0^{\\prime }\\bar{B}_+^{\\prime }W_0^{\\prime }\\\\1 & 0\\end{array}\\right].",
"\\end{split}$ It follows from the description of the restoration process for $\\bar{M}$ , that all entries of the matrix in the left hand side of the system (REF ) are Laurent polynomials in component III, and the determinant of this matrix equals $\\pm h_{11}/\\psi _{n-1,1}$ .",
"In the right hand side, $W_0^{\\prime }\\bar{B}_+^{\\prime }W_0^{\\prime }$ is a lower triangular matrix with $\\bar{\\beta }_{nn}$ in the upper left corner and $\\bar{\\beta }_{n-1,n-1}$ next to it along the diagonal.",
"Recall that other entries of $W_0^{\\prime }\\bar{B}_+^{\\prime }W_0^{\\prime }$ do not involve $\\psi _{11}$ ; besides, the entries $\\bar{\\beta }_{1j}$ for $j\\ne n$ may involve $\\psi _{11}$ only in the numerator, which makes them Laurent polynomials in component III.",
"Therefore, the right hand side of (REF ) involves two expressions that should be investigated: $\\bar{\\beta }_{11}\\bar{\\beta }_{nn}+\\bar{\\beta }_{1n}\\bar{\\beta }_{n-1,n}$ and $\\bar{\\beta }_{1n}\\bar{\\beta }_{n-1,n-1}$ .",
"Recall that $\\bar{\\beta }_{1n}$ is the product of a Laurent polynomial in component III by $\\delta _{n-1}/\\delta _1=\\pm \\psi _{21}\\psi _{n-1,1}/\\psi _{11}$ , whereas $\\bar{\\beta }_{n-1,n-1}=\\pm \\psi _{11}\\psi _{31}/\\psi _{21}^2$ by Remark REF , hence the second expression above is a Laurent polynomial in component III.",
"Recall further that $\\bar{M}$ is transformed to the companion form (REF ) by a conjugation first with $\\Delta $ and second with $N_+$ .",
"The latter can be written in two forms: $N_+=\\left[\\begin{array}{cc} N_+^{\\prime } & 0\\\\0 & 1\\end{array}\\right] \\qquad \\text{and}\\qquad N_+=\\left[\\begin{array}{cc} 1 & \\nu \\\\0 & N_+^{\\prime \\prime }\\end{array}\\right]$ where $N_+=(\\nu _{ij})$ and $N_+^{\\prime \\prime }$ are $(n-1)\\times (n-1)$ unipotent upper triangular matrices, and $\\nu =(\\nu _{12},\\dots ,\\nu _{1,n-1})$ .",
"Consequently, (REF ) yields $N_+^{\\prime \\prime }(\\Delta \\bar{M}\\Delta ^{-1})_{[2,n]}^{[1,n-1]}=N_+^{\\prime }.$ Note that (REF ) implies $\\left(\\frac{\\bar{\\beta }_{12}}{\\delta _1}, \\frac{\\bar{\\beta }_{13}}{\\delta _2},\\dots ,\\frac{\\bar{\\beta }_{1n}}{\\delta _{n-1}}\\right)=\\frac{1}{\\delta _1}\\left(\\gamma _1+\\bar{\\nu }_{12},\\gamma _2+\\bar{\\nu }_{13},\\dots , \\gamma _{n-2}+\\bar{\\nu }_{1,n-1},\\gamma _{n-1}\\right)N_+^{\\prime },$ where $\\bar{\\nu }_{ij}$ are the entries of $(N_+^{\\prime })^{-1}$ .",
"Taking into account that $\\delta _1=\\prod _{i=2}^n\\bar{\\beta }_{ii},\\qquad \\delta _{n-1}=\\bar{\\beta }_{nn},\\qquad \\prod _{i=1}^n\\bar{\\beta }_{ii}=(-1)^{n-1}h_{11},$ we infer that $\\bar{\\beta }_{1n}=(-1)^{n-1}\\frac{\\bar{\\beta }_{11}\\bar{\\beta }_{nn}}{h_{11}}\\left(\\sum _{i=1}^{n-1}\\gamma _i\\nu _{i,n-1}+\\sum _{i=2}^{n-1}\\bar{\\nu }_{1i}\\nu _{i-1,n-1}\\right).$ Besides, $\\bar{\\beta }_{n-1,n}=(-1)^n\\psi _{12}/\\psi _{21}$ by Remark REF .",
"Denote $\\zeta =(-1)^{n-1}\\psi _{12}/\\psi _{21}$ , Then one can write $\\bar{\\beta }_{11}\\bar{\\beta }_{nn}+\\bar{\\beta }_{1n}\\bar{\\beta }_{n-1,n}=\\frac{\\bar{\\beta }_{11}\\bar{\\beta }_{nn}}{h_{11}}\\left(h_{11}+(-1)^{n}\\zeta \\left( \\sum _{i=1}^{n-1}\\gamma _i\\nu _{i,n-1}+\\sum _{i=2}^{n-1}\\bar{\\nu }_{1i}\\nu _{i-1,n-1}\\right)\\right).$ To treat the latter expression, we consider first the last column of $(\\Delta \\bar{M}\\Delta ^{-1})_{[2,n]}^{[1,n-1]}$ .",
"It is easy to see that the first $n-3$ entries in this column equal zero modulo $\\psi _{11}$ , whereas the $(n-2)$ -nd entry equals $-\\zeta $ , and the last entry equals 1.",
"Consequently, (REF ) implies that $\\nu _{i,n-1}=(-1)^{n-i-1}\\zeta ^{n-i-1}\\mod {\\psi }_{11}$ .",
"Therefore, the first sum in the right hand side of (REF ) equals $\\sum _{i=1}^{n-1}\\gamma _i(-1)^{n-i-1}\\zeta ^{n-i-1} \\mod {\\psi }_{11}.$ By Lemma REF , $-\\bar{\\nu }_{12},\\dots ,-\\bar{\\nu }_{1,n-1},0$ form the first row of the companion form for $(\\Delta \\bar{M}\\Delta ^{-1})_{[2,n]}^{[2,n]}$ .",
"Consequently, $\\begin{split}\\sum _{i=2}^{n-1}\\bar{\\nu }_{1i}\\nu _{i-1,n-1}&=\\sum _{i=2}^{n-1}(-1)^{n-i}\\bar{\\nu }_{1i}\\zeta ^{n-i} \\mod {\\psi }_{11}\\\\&=(-1)^{n-1}\\left(\\det \\left((\\Delta \\bar{M}\\Delta ^{-1})_{[2,n]}^{[2,n]}+\\zeta \\mathbf {1}_{n-1}\\right)-\\zeta ^{n-1}\\right) \\mod {\\psi }_{11}.\\end{split}$ Note that $\\det ((\\Delta \\bar{M}\\Delta ^{-1})_{[2,n]}^{[2,n]}+\\zeta \\mathbf {1}_{n-1})=0 \\mod {\\psi }_{11}$ , since the last column of $(\\Delta \\bar{M}\\Delta ^{-1})_{[2,n]}^{[2,n]}$ is zero, and the second to last column equals $e_{n-1}-\\zeta e_{n-2} \\mod {\\psi }_{11}$ ; therefore, the second sum in the right hand side of (REF ) equals $(-1)^{n}\\zeta ^{n-1} \\mod {\\psi }_{11}$ .",
"Combining this with the previous result and taking into account that $h_{11}=(-1)^{n-1}\\gamma _n$ , we get $\\begin{split}\\bar{\\beta }_{11}\\bar{\\beta }_{nn}&+\\bar{\\beta }_{1n}\\bar{\\beta }_{n-1,n}\\\\&=\\frac{\\bar{\\beta }_{11}\\bar{\\beta }_{nn}}{h_{11}}\\left(h_{11}+(-1)^{n}\\left(\\sum _{i=1}^{n-1}\\gamma _i(-1)^{n-i-1}\\zeta ^{n-i}+(-1)^{n}\\zeta ^{n}\\right)\\right)\\mod {\\psi }_{11}\\\\&=\\frac{\\bar{\\beta }_{11}\\bar{\\beta }_{nn}}{h_{11}}\\left(\\zeta ^n+\\sum _{i=1}^n\\gamma _i(-1)^{i+1}\\zeta ^{n-i}\\right) \\mod {\\psi }_{11}\\\\&=\\pm \\frac{\\psi _{n-1,1}}{\\psi _{21}^{n-1}\\psi _{11}}\\det (\\psi _{21}\\bar{M}+(-1)^{n-1}\\psi _{12}\\mathbf {1}_n) \\mod {\\psi }_{11}.\\end{split}$ Recall that $\\det (\\psi _{21}\\bar{M}+(-1)^{n-1}\\psi _{12}\\mathbf {1}_n)/\\psi _{11}$ multiplied by an appropriate power of $\\det X$ is the new variable that replaces $\\psi _{11}$ in component III, and hence (REF ) defines the entries of the first row of $\\check{M}$ as Laurent polynomials."
],
[
"Component IV",
"In this component we use the same two normal forms as in component I.",
"Restoration of $B_+$ and of diagonal entries of $\\bar{B}_+$ is exactly the same as before.",
"Next, we apply the second line of (REF ) and (REF ) to (REF ) (or (REF )) and observe that the right hand side of the exchange relation for $\\psi _{kl}$ , $k>1$ , can be written as $\\begin{split}&\\psi _{k+l-3,1}\\psi _{k+l-2,1}\\psi _{k+l-1,1}\\\\&\\times \\left(h_{\\alpha -1,\\gamma +1}(\\bar{B}_+)h_{\\alpha ,\\gamma -1}(\\bar{B}_+)h_{\\alpha +1,\\gamma }(\\bar{B}_+)+h_{\\alpha -1,\\gamma }(\\bar{B}_+)h_{\\alpha ,\\gamma +1}(\\bar{B}_+)h_{\\alpha +1,\\gamma -1}(\\bar{B}_+)\\right),\\end{split}$ where $\\alpha =n-k-l+2$ , $\\gamma =n-l+2$ (cf.",
"with (REF )).",
"Note that for $l=2$ we have $h_{\\alpha -1,\\gamma +1}(\\bar{B}_+)=h_{\\alpha ,\\gamma +1}(\\bar{B}_+)=1$ , and hence the expression above can be rewritten as ${\\psi _{k-1,1}\\psi _{k1}\\psi _{k+1,1}}\\left(h_{\\alpha ,n-1}(\\bar{B}_+)h_{\\alpha +1,n}(\\bar{B}_+)+h_{\\alpha -1,n}(\\bar{B}_+)h_{\\alpha +1,n-1}(\\bar{B}_+)\\right).$ We conclude that the map $(k,l)\\mapsto {\\left\\lbrace \\begin{array}{ll} (\\alpha ,\\gamma )\\quad \\text{for $l>1$},\\\\(\\alpha +1,\\alpha +1)\\quad \\text{for $l=1$ }\\end{array}\\right.",
"}$ transforms exchange relations for $\\psi _{kl}$ , $l>1$ , $k+l<n$ , to exchange relations for the standard cluster structure on triangular matrices of size $(n-2)\\times (n-2)$ , up to a monomial factor in variables that are fixed in component IV, see Fig.",
"REF .",
"Consequently, the entries $\\bar{\\beta }_{ij}$ , $2\\le i <j\\le n$ , can be restored as Laurent polynomials in component IV via Remark REF .",
"Figure: Modification of the relevant part of Q 6 † Q_6^\\dag in component IVThe entries in the first row of $\\bar{B}_+$ are restored exactly as in component I."
],
[
"Component V",
"In this component we once again use the same two normal forms as in component I.",
"Restoration of $\\bar{B}_+$ is exactly the same as before.",
"To restore $B_+$ , we note that the cluster structure in component V coincides with the standard cluster structure on triangular $n\\times n$ matrices, and hence the entries of $B_+$ can be restored as Laurent polynomials via Remark REF ."
],
[
"Component VI",
"The two normal forms used in this component are given by $U=B_+N_-=\\widehat{N}_+\\widehat{N}_-\\widehat{M} S_{12}\\widehat{N}_-^{-1}\\widehat{N}_+^{-1}$ , where $B_+$ is upper triangular, $N_-, \\widehat{N}_-=(\\hat{\\nu }_{ij})$ are unipotent lower triangular with $\\hat{\\nu }_{j1}=0$ for $2\\le j\\le n$ , $\\widehat{N}_+=\\mathbf {1}_n+\\hat{\\nu }e_{12}$ , and $\\widehat{M}=(\\hat{\\mu }_{ij})$ satisfies conditions $\\hat{\\mu }_{1n}=0$ and $\\hat{\\mu }_{i,n+2-j}=0$ for $2\\le j<i\\le n$ , see Lemma REF .",
"Note that $h_{2j}(U)=h_{2j}(\\widehat{M})$ and $\\psi _{kl}(U)=\\psi _{kl}(\\widehat{M})$ .",
"Our goal is to restore $B_+$ and $\\widehat{M}$ .",
"Once this is done, the matrix $U$ itself is restored as follows.",
"First, by the proof of Lemma REF , $\\hat{\\nu }=\\beta _{1n}/\\hat{\\mu }_{2n}$ , which is a Laurent polynomial in component VI, since $\\hat{\\mu }_{2n}= h_{2n}$ is a Laurent monomial.",
"Next, we write $B_+N_-\\widehat{N}_+\\widehat{N}_-=\\widehat{N}_+\\widehat{N}_-\\widehat{M}$ .",
"Taking into account that $\\widehat{N}_+\\widehat{N}_-=\\widehat{N}_-\\widehat{N}_+$ , we arrive at $\\bar{W}_0^{\\prime } B_+\\cdot N_-\\widehat{N}_-=\\bar{W}_0^{\\prime }\\widehat{N}_- \\bar{W}_0^{\\prime }\\cdot \\bar{W}_0^{\\prime } \\widehat{N}_+\\widehat{M}\\widehat{N}_+^{-1}$ with $\\bar{W}_0^{\\prime }=\\begin{pmatrix} 1 & 0\\\\ 0 & W_0^{\\prime }\\end{pmatrix}$ .",
"Note that the second factor on the left is unipotent lower triangular, whereas the first factor on the right is unipotent upper triangular.",
"Taking the Gauss factorizations of the remaining two factors, we restore $\\bar{W}_0^{\\prime }\\widehat{N}_- \\bar{W}_0^{\\prime }=\\left( \\bar{W}_0^{\\prime } B_+\\right)_{> 0}\\left(\\bar{W}_0^{\\prime } \\widehat{N}_+\\widehat{M}\\widehat{N}_+^{-1}\\right)^{-1}_{> 0}$ .",
"Note that trailing minors needed for Gauss factorizations in the right hand side above equal $\\det \\widehat{M}_{[2,i]}^{[n-i+2,n]}= h_{2i}$ , and hence are Laurent monomials in component VI.",
"We describe the restoration process at $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ , and indicate the changes that occur at its neighbors.",
"Restoration of $B_+$ is almost the same as before.",
"The difference is that $h_{1n}$ and $h_{12}$ , which coincide up to a sign with $\\psi _{n-1,1}$ and $\\psi _{1,n-1}$ , are no longer available (at $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ only $h_{1n}$ is not available).",
"However, since they are cluster variables in other clusters, say, in any cluster in component I, they both can be written as Laurent polynomials at $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ .",
"Moreover, they never enter denominators in expressions for $\\beta _{ij}$ , and hence $B_+$ is restored.",
"At the neighbor of $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ in direction $\\psi _{n-i,i}$ we apply the same reasoning to $h_{1,n-i+1}=\\pm \\psi _{n-i,i}$ ; at $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ we apply it only to $h_{1n}$ .",
"In order to restore $\\widehat{M}$ we proceed as follows.",
"First, we note that $h_{2j}=\\pm \\prod _{i=j}^n \\hat{\\mu }_{n+2-i,i},$ and hence $\\hat{\\mu }_{n+2-i,i}=\\pm h_{2j}/h_{2,j+1}$ for $2\\le j\\le n$ with $h_{2,n+1}=1$ .",
"Next, note that by (REF ), the function $\\psi _{n-1,1}$ is replaced in component VI by $\\psi ^{\\prime }_{n-1,1}=\\pm \\det \\widehat{M}_{[1,n-1]}^{1\\cup [3,n]}$ .",
"Besides, it is easy to see that in all clusters in component VI except for $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ , $\\psi _{1,n-1}$ is replaced by $\\psi _{1,n-1}^{\\prime }=\\pm \\det \\widehat{M}_{[2,n]}^{1\\cup [3,n]}$ .",
"Consequently, $h_{11}=\\pm \\left(\\hat{\\mu }_{n1}\\psi _{1,n-1}-\\hat{\\mu }_{n2}\\psi ^{\\prime }_{n-1,1}\\right)$ yields $\\hat{\\mu }_{n1}$ as a Laurent monomial at $\\widetilde{\\Sigma }^{\\prime \\prime }_n$ , and $\\psi _{1,n-1}^{\\prime }=\\pm \\hat{\\mu }_{n1}h_{23}$ yields it as a Laurent monomial in all other clusters in component VI.",
"To proceed further, we introduce an $n \\times n$ matrix $\\widehat{\\Psi }$ similar to the matrix $\\Psi $ used in component II: $\\widehat{\\Psi }=\\left[\\begin{array}{ccccc} e_2 & \\widehat{M}e_2 & \\widehat{M}^{2}e_2 & \\dots & \\widehat{M}^{n-1}e_2\\end{array}\\right].$ Lemma 7.4 The minors $\\det \\widehat{\\Psi }_I^{[2,t]}$ , $2\\le t\\le n-1$ , are Laurent polynomials in component VI for any index set $I$ such that $2\\notin I$ , $|I|=t-1$ .",
"An easy computation shows that for $k+l<n$ one has $\\psi _{kl}=\\pm h_{2n}^{n-k-l+1}\\sum _{j\\in [1,l+1]\\setminus 2}(-1)^{j+\\chi _j}\\det \\widehat{M}_{[1,l+1]\\setminus \\lbrace 2, j\\rbrace }^{[n-l+1,n-1]}\\det \\widehat{\\Psi }_{j\\cup [l+2,n-k]}^{[2,n-k-l+1]},$ where $\\chi _1=1$ and $\\chi _j=0$ for $j\\ne 1$ .",
"It follows from the discussion above that $\\det \\widehat{M}_{[3,l+1]}^{[n-l+1,n-1]}=(-1)^{l-1}h_{2,n-l+1}/h_{2n}$ .",
"Besides, $\\begin{split}\\det \\widehat{M}_{[1,l+1]\\setminus \\lbrace 2, j\\rbrace }^{[n-l+1,n-1]}&=(-1)^{(j-1)(l-j+1)}\\det \\widehat{M}_{[1,j-1]\\setminus 2}^{[n-j+2,n-1]}\\det \\widehat{M}_{[j+1,l+1]}^{[n-l+1,n-j+1]}\\\\&=(-1)^{l}\\frac{h_{1,n-j+2}}{h_{2n}}\\frac{h_{2,n-l+1}}{h_{2,n-j+2}},\\end{split}$ and hence $\\psi _{kl}=\\pm h_{2n}^{n-k-l}h_{2,n-l+1}\\sum _{j\\in [1,l+1]\\setminus 2}\\eta _j\\det \\widehat{\\Psi }_{j\\cup [l+2,n-k]}^{[2,n-k-l+1]},$ where $\\eta _j=(-1)^{(n-j)(j-1)}{\\psi _{n-j+1,j-1}}/{h_{2,n-j+2}}$ with $\\psi _{n0}=h_{2,n+1}=1$ .",
"Now we can prove the claim of the lemma by induction on the maximal index in $I$ .",
"The minimum value of this index is $t$ .",
"In this case we use (REF ) to see that $\\det \\widehat{\\Psi }_{[1,t]\\setminus 2}^{[2,t]}=\\pm \\psi _{n-t,1}/h_{2n}^{t}$ is a Laurent monomial in component VI.",
"Assume that the value of the maximal index in $I$ equals $r>t$ .",
"We multiply the $(r-1)\\times (t-1)$ matrix $\\widehat{\\Psi }^{[2,t]}_{[1,r]\\setminus 2}$ by a $(t-1)\\times (t-1)$ block upper triangular matrix with unimodular blocks of size $t-2$ and 1, so that the upper $(t-2)\\times (t-1)$ submatrix is diagonalized with 1's on the diagonal except for the first row.",
"Clearly, this transformation does not change the values of any minors in the first $t-2$ and $t-1$ columns.",
"Consequently, each matrix entry (except for the one in the lower right corner, which we denote $z$ ) is a Laurent polynomial in component VI.",
"We then consider (REF ) with $k=n-r$ and $l=r-t+1$ and expand each minor in the right hand side by the last row.",
"Each entry in the last row distinct from $z$ enters this expansion with a coefficient that is a Laurent polynomial in component VI.",
"The entry $z$ enters the expansion with the coefficient $\\pm h_{2n}^{t-1}h_{2,n-r+t}\\sum _{j\\in [1,r-t+2]\\setminus 2}\\eta _j\\det \\widehat{\\Psi }^{[2,t-1]}_{j\\cup [r-t+3,r-1]}=\\pm \\psi _{n-r+1,r-t+1}h_{2n}.$ Thus, $z$ is a Laurent monomial in component VI, and hence any minor of $\\widehat{\\Psi }^{[2,t]}_{[1,r]\\setminus 2}$ that involves the $r$ -th row is a Laurent polynomial.",
"We can now proceed with restoring the entries of $\\widehat{M}$ .",
"Equation (REF ) for $k=n-l-1$ gives $\\psi _{n-l-1,l}=\\pm h_{2n}h_{2,n-l+1}\\sum _{j\\in [1,l+1]\\setminus 2}\\eta _j\\widehat{\\Psi }_j^2=\\pm h_{2n}^2h_{2,n-l+1}\\sum _{j\\in [1,l+1]\\setminus 2}\\eta _j\\hat{\\mu }_{j2},$ and we consecutively restore $\\hat{\\mu }_{j2}$ , $1\\le j\\le n-1$ , $j\\ne 2$ , as Laurent polynomials at all clusters in component VI except for the neighbor of $\\widetilde{\\Sigma }^{\\prime \\prime \\prime }_n$ in direction $\\psi _{n-j+1,j-1}$ .",
"An easy computation shows that in this cluster $\\psi _{n-j+1,j-1}$ is replaced by $\\psi _{n-j+1,j-1}^{\\prime }=\\pm h_{2n}\\det \\widehat{M}_{[2,j]}^{2\\cup [n-j+3,n]}=\\pm h_{2n}h_{2,n-j+3}\\hat{\\mu }_{j2}$ , and hence $\\hat{\\mu }_{j2}$ is restored there as a Laurent monomial.",
"In particular, it follows from above that $\\hat{\\mu }_{12}=\\pm \\frac{\\psi _{n-2,1}}{h_{2n}^3}$ in any cluster in component VI.",
"Recall that we have already restored the last row of $\\widehat{M}$ .",
"We restore the other rows consecutively, starting from row $n-1$ and moving upwards.",
"Matrix entries in the $l$ -th row, $3\\le l\\le n-1$ , are restored in two stages, together with the minors $\\det \\widehat{M}^{i\\cup [n-l+3,n]}_{[1,l-1]}$ for $1\\le i\\le n-l+1$ , $i\\ne 2$ .",
"First, we recover minors $\\det \\widehat{M}^{2\\cup i\\cup [n-l+3,n]}_{[1,l]}$ for $1\\le i\\le n-l+1$ , $i\\ne 2$ , as Laurent polynomials in component VI.",
"Once they are recovered, we find $\\hat{\\mu }_{li}$ and $\\det \\widehat{M}^{i\\cup [n-l+3,n]}_{[1,l-1]}$ via expanding $\\det \\widehat{M}^{2\\cup i\\cup [n-l+3,n]}_{[1,l]}$ and $\\det \\widehat{M}^{i\\cup [n-l+2,n]}_{[1,l]}$ by the last row.",
"This gives a system of two linear equations for a fixed $i$ , and its determinant equals $\\pm \\psi _{n-l,l-1}$ .",
"Note that minors $\\det \\widehat{M}^{i\\cup [n-l+2,n]}_{[1,l]}$ for $1\\le i\\le n-l$ , $i\\ne 2$ , were recovered together with the entries of the $(l+1)$ -st row, and $\\det \\widehat{M}^{[n-l+1,n]}_{[1,l]}=\\pm \\psi _{n-l,l}$ (recall that $\\psi _{n-l,l}$ are Laurent polynomials in component VI).",
"For $l=3$ , we have $\\det \\widehat{M}^{i\\cup n}_{[1,2]}=\\hat{\\mu }_{1i}h_{2n}$ , and hence the entries of the first row are recovered together with the entries of the third row.",
"The minors $\\det \\widehat{M}^{2\\cup i\\cup [n-l+3,n]}_{[1,l]}$ for $1\\le i\\le n-l+1$ , $i\\ne 2$ , are recovered together with all other minors $\\det \\widehat{M}^{i\\cup j\\cup [n-l+3,n]}_{[1,l]}$ for $1\\le i<j\\le n-l+1$ , $i,j\\ne 2$ , altogether $(n-l)(n-l+1)/2$ minors.",
"We first note that the Binet–Cauchy formula gives $\\psi _{kl}=\\sum _{\\begin{array}{c}J\\subseteq [1,n-l]\\setminus 2\\\\ |J|=n-k-l-1\\end{array}}(-1)^{\\chi _J}\\det \\widehat{M}^{2\\cup J\\cup [n-l+1,n]}_{[1,n-k]}\\det \\widehat{\\Psi }^{[2,n-k-l]}_J$ for $k+l<n-1$ , where $\\chi _J=\\sum _{j\\in J}\\chi _j$ .",
"Recall that by Lemma REF , $\\det \\widehat{\\Psi }^{[2,n-k-l]}_J$ are Laurent polynomials in component VI.",
"Writing down $\\psi _{k,l-2}$ for $1\\le k\\le n-l$ via the above formula, and expanding the minors $\\det \\widehat{M}^{2\\cup J\\cup [n-l+3,n]}_{[1,n-k]}$ for $J\\lnot \\ni n-l+2$ by the last $n-k-l$ rows, we get $n-l$ linear equations; note that the corresponding minors for $J\\ni n-l+2$ have been already restored as Laurent polynomials in component VI when we dealt with the previous rows.",
"For $l=n-1$ we get a single equation $\\psi _{1,n-3}=-\\det \\widehat{M}_{[1,n-1]}^{[1,n]\\setminus 3}\\hat{\\mu }_{12}+\\det \\widehat{M}_{[1,n-1]}^{[2,n]}\\hat{\\mu }_{32},$ and hence $\\det \\widehat{M}_{[1,n-1]}^{[1,n]\\setminus 3}$ is a Laurent polynomial in component VI, since $\\hat{\\mu }_{12}$ is a Laurent monomial by (REF ).",
"In what follows we assume that $3\\le l\\le n-2$ .",
"The remaining $(n-l)(n-l-1)/2$ linear equations are provided by short Plücker relations $\\begin{split}\\det \\widehat{M}^{i\\cup j\\cup [n-l+3,n]}_{[1,l]}\\det \\widehat{M}^{2\\cup [n-l+2,n]}_{[1,l]}&=\\det \\widehat{M}^{2\\cup j\\cup [n-l+3,n]}_{[1,l]}\\det \\widehat{M}^{i\\cup [n-l+2,n]}_{[1,l]}\\\\&+(-1)^{\\chi _i+1}\\det \\widehat{M}^{2\\cup i\\cup [n-l+3,n]}_{[1,l]}\\det \\widehat{M}^{j\\cup [n-l+2,n]}_{[1,l]},\\end{split}$ where the second factor in each term is a Laurent polynomial in component VI, and, moreover, $\\det \\widehat{M}^{2\\cup [n-l+2,n]}_{[1,l]}=\\pm \\psi _{n-l-1,l}/h_{2n}$ .",
"We can arrange the variables in such a way that the matrix of the linear system takes the form $A=\\begin{pmatrix} A_1 & A_2\\\\ A_3 & A_4 \\end{pmatrix}$ , where $A_4$ is an $(n-l)(n-l-1)/2\\times (n-l)(n-l-1)/2$ diagonal matrix with $\\det \\widehat{M}^{2\\cup [n-l+2,n]}_{[1,l]}$ on the diagonal.",
"The column $(i,j)$ of $A_2$ corresponding to the variable $\\det \\widehat{M}^{i\\cup j\\cup [n-l+3,n]}_{[1,l]}$ , $i,j\\ne 2$ , contains $\\sum _{\\begin{array}{c}I\\subseteq [1,n-l+1]\\setminus \\lbrace 2,i,j\\rbrace \\\\ |I|=t-1\\end{array}}(-1)^{\\theta (i,j,I)}\\det \\widehat{M}^{2\\cup I}_{[l+1,l+t]}\\det \\widehat{\\Psi }_{i\\cup j\\cup I}^{[2, t+2]}$ in row $n-l-t$ for $1\\le t\\le n-l-1$ , where $\\theta (i,j,I)=\\chi _{i\\cup I}+\\#\\lbrace p\\in 2\\cup I: i<p<j\\rbrace $ , and zero in row $n-l$ .",
"Similarly, the column $(2,i)$ of $A_1$ corresponding to the variable $\\det \\widehat{M}^{2\\cup i\\cup [n-l+3,n]}_{[1,l]}$ , $i\\ne 2$ , contains $\\sum _{\\begin{array}{c}J\\subseteq [1,n-l+1]\\setminus \\lbrace 2,i\\rbrace \\\\ |J|=t\\end{array}}(-1)^{\\theta (i,J)}\\det \\widehat{M}^{ J}_{[l+1,l+t]}\\det \\widehat{\\Psi }_{i\\cup J}^{[2, t+2]}$ in row $n-l-t$ for $1\\le t\\le n-l-1$ , where $\\theta (i,J)={\\left\\lbrace \\begin{array}{ll} 1 \\quad \\text{for $i=1$},\\\\\\chi _J+\\#\\lbrace p\\in J: 2<p<i\\rbrace \\quad \\text{for $i\\ne 1$},\\end{array}\\right.",
"}$ and $(-1)^{\\chi _i}\\hat{\\mu }_{i2}$ in row $n-l$ .",
"To find $\\det A$ , we multiply $A$ by a square lower triangular matrix whose column $(2,i)$ contains $\\hat{\\mu }_{l+1,2}$ in row $(2,i)$ , $\\hat{\\mu }_{l+1,j}$ in row $(j,i)$ , $1\\le j\\le i-1$ , $j\\ne 2$ , and $(-1)^{\\chi _i+1}\\hat{\\mu }_{l+1,j}$ in row $(i,j)$ , $i+1\\le j\\le n-l+1$ , $j\\ne 2$ ; column $(i,j)$ of this matrix contains a single 1 on the main diagonal, and all its other entries are equal to zero.",
"Let $A^{\\prime }=\\begin{pmatrix} A^{\\prime }_1 & A_2\\\\ A^{\\prime }_3 & A_4 \\end{pmatrix}$ be the obtained product.",
"Clearly, $\\det A=\\hat{\\mu }_{l+1,2}^{l-n}\\det A^{\\prime }$ .",
"We claim that $(A^{\\prime }_1)_{[1,n-l-1]}$ is a zero matrix, and $(A^{\\prime }_1)_{[n-l]}=\\hat{\\mu }_{l+1,2}(A_1)_{[n-l]}$ .",
"The second claim follows immediately from the fact that $(A_2)_{[n-l]}$ is a zero vector.",
"To prove the first one, we fix arbitrary $i$ , $t$ , and $J=\\lbrace j_1<j_2<\\cdots <j_t\\rbrace $ , and find the coefficient at $\\det \\widehat{\\Psi }_{i\\cup J}^{[2, t+2]}$ in the entry of $A_1^{\\prime }$ in row $n-l-t$ and column $(2,i)$ .",
"For $i=1$ this coefficient equals $(-1)^{\\theta (1,J)}\\hat{\\mu }_{l+1,2}\\det \\widehat{M}^J_{[l+1,l+t]}+\\sum _{r=1}^t(-1)^{2+\\theta (1,j_r,J\\setminus j_r)}\\hat{\\mu }_{l+1,j_r}\\det \\widehat{M}^{2\\cup J\\setminus j_r}_{[l+1,l+t]}.$ Taking into account that $\\theta (1,J)=1$ and $\\theta (1,j_r,J\\setminus j_r)=2+\\#\\lbrace p\\in J:2<p<j_r\\rbrace =1+r$ , we conclude that the signs in the above expression alternate.",
"For $i\\ne 1$ , the coefficient in question equals $\\begin{split}(-1)^{\\theta (i,J)}\\hat{\\mu }_{l+1,2}\\det \\widehat{M}^J_{[l+1,l+t]}&+\\sum _{j_r<i}(-1)^{\\theta (j_r,i,J\\setminus j_r)}\\hat{\\mu }_{l+1,j_r}\\det \\widehat{M}^{2\\cup J\\setminus j_r}_{[l+1,l+t]}\\\\&+\\sum _{j_r>i}(-1)^{1+\\theta (i,j_r,J\\setminus j_r)}\\hat{\\mu }_{l+1,j_r}\\det \\widehat{M}^{2\\cup J\\setminus j_r}_{[l+1,l+t]}.\\end{split}$ If $j_1=1$ , then $\\theta (i,J)=r^*$ , where $r^*=r^*(i,J)=\\max \\lbrace r: j_r<i\\rbrace $ .",
"Further, $\\theta (1,i,J\\setminus 1)=1+r^*$ , $\\theta (j_r,i,J\\setminus j_r)=1+r^*-r$ for $1<j_r<i$ and $\\theta (i,j_r,J\\setminus j_r)=2+r-r^*$ for $i<j_r$ , hence the signs in the above expression taken in the order $1, 2, j_2,\\dots , j_t$ alternate once again.",
"Finally, if $j_1>1$ , then $\\theta (i,J)=r^*$ , $\\theta (j_r,i,J\\setminus j_r)=r^*-r$ for $j_r<i$ and $\\theta (i,j_r,J\\setminus j_r)=1+r-r^*$ for $i<j_r$ , and again the signs taken in the corresponding order alternate.",
"Therefore, in all three cases the coefficient at $\\det \\widehat{\\Psi }_{i\\cup J}^{[2, t+2]}$ equals $\\det \\widehat{M}^{2\\cup J}_{{l+1}\\cup [l+1,l+t]}$ , and hence vanishes as the determinant of a matrix with two identical rows.",
"The entry of $A_3^{\\prime }$ in column $(2,i)$ and row $(i,j)$ is $\\begin{split}(-1)^{\\chi _i+1}\\hat{\\mu }_{l+1,j}\\det \\widehat{M}^{2\\cup [n-l+2,n]}_{[1l]}&+(-1)^{\\chi _i}\\hat{\\mu }_{l+1,2}\\det \\widehat{M}^{j\\cup [n-l+2,n]}_{[1l]}\\\\&=(-1)^{\\chi _i}\\widehat{M}^{2\\cup j\\cup [n-l+2,n]}_{[1,l+1]},\\end{split}$ and the entry in column $(2,j)$ and row $(i,j)$ is $\\hat{\\mu }_{l+1,i}\\det \\widehat{M}^{2\\cup [n-l+2,n]}_{[1l]}-\\hat{\\mu }_{l+1,2}\\det \\widehat{M}^{i\\cup [n-l+2,n]}_{[1l]}=(-1)^{\\chi _i+1}\\widehat{M}^{2\\cup i\\cup [n-l+2,n]}_{[1,l+1]}.$ By the Schur determinant lemma, $\\det A^{\\prime }=\\det A_4\\cdot \\det \\left(A_1^{\\prime }-A_2A_4^{-1}A_3^{\\prime }\\right)=\\pm \\hat{\\mu }_{l+1,2}\\left(\\frac{\\psi _{n-l-1,l}}{h_{2n}}\\right)^{(n-l)(n-l-3)/2+1}\\det A^{\\prime \\prime },$ where $A^{\\prime \\prime }$ is the $(n-l)\\times (n-l)$ matrix satisfying $A^{\\prime \\prime }_{[1,n-l-1]}=(A_2A_3^{\\prime })_{[1,n-l-1]}$ and $A^{\\prime \\prime }_{[n-l]}=(A_1^{\\prime })_{[n-l]}$ .",
"Using (REF ), (REF ), and (REF ), we compute the entry $A^{\\prime \\prime }_{n-l-t,i}$ for $1\\le t\\le n-l-1$ , $1\\le i\\le n-l+1$ , $i\\ne 2$ , as $\\begin{split}&\\sum _{j<i}\\sum _{\\begin{array}{c}I\\subseteq [1,n-l+1]\\setminus \\lbrace 2,i,j\\rbrace \\\\ |I|=t-1\\end{array}}(-1)^{\\theta (j,i,I)+\\chi _j+1}\\det \\widehat{M}^{2\\cup I}_{[l+1,l+t]}\\det \\widehat{\\Psi }_{i\\cup j\\cup I}^{[2, t+2]}\\det \\widehat{M}_{[1,l+1]}^{2\\cup j\\cup [n-l+2,n]}\\\\&+\\sum _{j>i}\\sum _{\\begin{array}{c}I\\subseteq [1,n-l+1]\\setminus \\lbrace 2,i,j\\rbrace \\\\ |I|=t-1\\end{array}}(-1)^{\\theta (i,j,I)+\\chi _i}\\det \\widehat{M}^{2\\cup I}_{[l+1,l+t]}\\det \\widehat{\\Psi }_{i\\cup j\\cup I}^{[2, t+2]}\\det \\widehat{M}_{[1,l+1]}^{2\\cup j\\cup [n-l+2,n]}\\\\&=\\sum _{\\begin{array}{c}J\\subseteq [1,n-l+1]\\setminus \\lbrace 2,i\\rbrace \\\\ |J|=t\\end{array}}\\det \\widehat{\\Psi }_{i\\cup J}^{[2, t+2]}\\\\&\\qquad \\times \\left(\\sum _{j_r<i}(-1)^{\\theta (j_r,i,J\\setminus j_r)+\\chi _{j_r}+1}\\det \\widehat{M}^{2\\cup J\\setminus j_r}_{[l+1,l+t]}\\det \\widehat{M}_{[1,l+1]}^{2\\cup j_r\\cup [n-l+2,n]}\\right.\\\\&\\qquad \\qquad +\\left.\\sum _{j_r>i} (-1)^{\\theta (i,j_r,J\\setminus j_r)+\\chi _i+1}\\det \\widehat{M}^{2\\cup J\\setminus j_r}_{[l+1,l+t]}\\det \\widehat{M}_{[1,l+1]}^{2\\cup j_r\\cup [n-l+2,n]}\\right).\\end{split}$ Analyzing the same three cases as in the computation of the entries of $A_1^{\\prime }$ , we conclude that the second factor in the above expression can be rewritten as $(-1)^{\\chi _J+\\chi _i+r^*+1}\\sum _{r=1}^t(-1)^{r+1}\\det \\widehat{M}^{2\\cup J\\setminus j_r}_{[l+1,l+t]}\\det \\widehat{M}_{[1,l+1]}^{2\\cup j_r\\cup [n-l+2,n]}.$ To evaluate the latter sum, we multiply $\\widehat{M}_{[1,l+t]}^{2\\cup J\\cup [n-l+2,n]}$ by a matrix $\\begin{pmatrix} \\mathbf {1}_{l+1} & 0\\\\ 0 & N \\end{pmatrix}$ , where $N$ is unipotent lower triangular, so that the entries of column 2 in rows $l+2,\\dots ,l+t$ vanish.",
"This operation preserves the minors involved in the above sum, and hence the latter is equal to $\\sum _{r=1}^t(-1)^{r+1+\\chi _J}\\hat{\\mu }_{l+1,2}\\det \\widehat{M}^{J\\setminus j_r}_{[l+2,l+t]}\\det \\widehat{M}_{[1,l+1]}^{2\\cup j_r\\cup [n-l+2,n]}=\\hat{\\mu }_{l+1,2}\\det \\widehat{M}_{[1,l+t]}^{2\\cup J\\cup [n-l+2,n]}.$ Thus, finally, $A^{\\prime \\prime }_{n-l-t,i}=\\hat{\\mu }_{l+1,2}\\sum _{\\begin{array}{c}J\\subseteq [1,n-l+1]\\setminus \\lbrace 2,i\\rbrace \\\\ |J|=t\\end{array}}(-1)^{\\chi _J+\\chi _i+r^*+1}\\det \\widehat{\\Psi }_{i\\cup J}^{[2, t+2]}\\det \\widehat{M}_{[1,l+t]}^{2\\cup J\\cup [n-l+2,n]}.$ To evaluate $\\det A^{\\prime \\prime }$ , we multiply $A^{\\prime \\prime }$ by the $(n-l)\\times (n-l)$ upper triangular matrix having $(-1)^i\\det \\widehat{\\Psi }_{[1,k]\\setminus \\lbrace 2,i\\rbrace }^{[2,k-1]}$ in row $i$ and column $k$ , $i,k\\in [1,n-l+1]\\setminus 2$ , $i\\le k$ ; we assume that for $i=k=1$ this expression equals 1.",
"Denote the obtained product by $A^{\\prime \\prime \\prime }$ and note that $\\det A^{\\prime \\prime \\prime }=\\pm \\det A^{\\prime \\prime }\\prod _{k=3}^{n-l+1}\\det \\widehat{\\Psi }^{[2,k-1]}_{[1,k-1]\\setminus 2}$ .",
"The entry $A^{\\prime \\prime \\prime }_{n-l-t,k}$ equals $\\begin{split}&\\hat{\\mu }_{l+1,2}\\sum _{i\\in [1,k]\\setminus 2}(-1)^i\\det \\widehat{\\Psi }_{[1,k]\\setminus \\lbrace 2,i\\rbrace }^{[2,k-1]}\\\\&\\qquad \\qquad \\times \\sum _{\\begin{array}{c}J\\subseteq [1,n-l+1]\\setminus \\lbrace 2,i\\rbrace \\\\ |J|=t\\end{array}}(-1)^{\\chi _J+\\chi _i+r^*+1}\\det \\widehat{\\Psi }_{i\\cup J}^{[2, t+2]}\\det \\widehat{M}_{[1,l+t]}^{2\\cup J\\cup [n-l+2,n]}\\\\&=\\hat{\\mu }_{l+1,2}\\sum _{\\begin{array}{c}J\\subseteq [1,n-l+1]\\setminus 2\\\\ |J|=t\\end{array}}\\det \\widehat{M}_{[1,l+t]}^{2\\cup J\\cup [n-l+2,n]}\\\\&\\qquad \\qquad \\times \\sum _{i\\in [1,k]\\setminus 2}(-1)^{\\chi _J+\\chi _i+i+r^*+1}\\det \\widehat{\\Psi }_{[1,k]\\setminus \\lbrace 2,i\\rbrace }^{[2,k-1]}\\det \\widehat{\\Psi }_{i\\cup J}^{[2, t+2]};\\end{split}$ the equality follows from the fact that $\\det \\widehat{\\Psi }_{i\\cup J}^{[2, t+2]}=0$ for $i\\in J$ .",
"Expanding the latter determinant by the $i$ -th row, we see that the entry in question equals $\\begin{split}\\hat{\\mu }_{l+1,2}\\sum _{\\begin{array}{c}J\\subseteq [1,n-l+1]\\setminus 2\\\\ |J|=t\\end{array}}&(-1)^{\\chi _J+1}\\det \\widehat{M}_{[1,l+t]}^{2\\cup J\\cup [n-l+2,n]}\\\\&\\times \\sum _{s=2}^{t+2}(-1)^s\\det \\widehat{\\Psi }_{J}^{[2, t+2]\\setminus s}\\sum _{i\\in [1,k]\\setminus 2}(-1)^{\\chi _i+i}\\hat{\\mu }_{is}\\det \\widehat{\\Psi }_{[1,k]\\setminus \\lbrace 2,i\\rbrace }^{[2,k-1]}.\\end{split}$ Note that the inner sum above equals $\\det \\widehat{\\Psi }_{[1,k]\\setminus 2}^{s\\cup [2,k-1]}={\\left\\lbrace \\begin{array}{ll}0 , \\qquad \\qquad \\quad 2\\le s \\le k-1,\\\\\\det \\widehat{\\Psi }_{[1,k]\\setminus 2}^{[2,k]}, \\quad s=k.\\end{array}\\right.",
"}$ We conclude that $A^{\\prime \\prime \\prime }_{n-l-t,k}=0$ if $t+2<k$ , and hence $\\det A^{\\prime \\prime \\prime }$ equals the product of the entries on the main antidiagonal.",
"The latter correspond to $t+2=k$ and are equal to $(-1)^{t+1}\\hat{\\mu }_{l+1,2}\\det \\widehat{\\Psi }_{[1,t+2]\\setminus 2}^{[2,t+2]}\\sum _{\\begin{array}{c}J\\subseteq [1,n-l+1]\\setminus 2\\\\ |J|=t\\end{array}}(-1)^{\\chi _J}\\det \\widehat{M}_{[1,l+t]}^{2\\cup J\\cup [n-l+2,n]}\\det \\widehat{\\Psi }_J^{[2,t+1]},$ which by (REF ) coincides with $\\pm \\hat{\\mu }_{l+1,2}\\det \\widehat{\\Psi }_{[1,t+2]\\setminus 2}^{[2,t+2]}\\psi _{n-l-t,l-1}$ .",
"Therefore, $\\det A^{\\prime \\prime \\prime }=\\pm \\hat{\\mu }_{l+1,2}^{n-l-1}\\prod _{t=1}^{n-l-1}\\psi _{n-l-t,l-1}\\prod _{t=0}^{n-l-1}\\det \\widehat{\\Psi }_{[1,t+2]\\setminus 2}^{[2,t+2]},$ and hence, taking into account (REF ) for $t=n-l+1$ , we get $\\det A=\\pm \\frac{\\psi _{n-l-1,l}^{(n-l)(n-l-3)/2+1}\\psi _{l-1,1}}{h_{2n}^{(n-l)(n-l-1)/2+2}}\\prod _{t=1}^{n-l-1}\\psi _{n-l-t,l-1}.$ So, all minors $\\det \\widehat{M}^{2\\cup i\\cup [n-l+3,n]}_{[1,l]}$ for $1\\le i\\le n-l+1$ , $i\\ne 2$ are Laurent polynomials in component VI, and hence so are all entries in the rows $1,3,4,\\dots ,n$ of $\\widetilde{M}$ .",
"The entries of the second row are restored via Lemma REF as polynomials in the entries of other rows and variables $c_i$ divided by the $\\det K=\\det \\widehat{\\Psi }=\\pm h_{11}\\psi _{11}/h_{2n}^n$ .",
"Remark 7.5 Once we have established that stable variables $h_{ii}$ do not enter denominators of Laurent expressions for entries of $U$ at $\\widetilde{\\Sigma }^{\\prime }_n$ (see Remark REF ), this remains valid for any other cluster in $\\mathcal {N}_0$ .",
"This is guaranteed by the fact that entries of $U$ are Laurent polynomials in any cluster in $\\mathcal {N}_0$ and the condition that exchange polynomials are not divisible by any of the stable variables."
],
[
"Proof of Proposition ",
"This proposition is an easy corollary of the following more general result.",
"Proposition 8.1 Let $A$ be a complex $n\\times n$ matrix.",
"For $u,v\\in {\\mathbb {C}}^n$ , define matrices $\\nonumber K(u)=\\left[ u \\; A u \\; A^2 u \\dots A^{n-1} u\\right],\\\\K_1(u,v)=\\left[ v \\; u \\; A u \\dots A^{n-2} u\\right],\\;\\;K_2(u,v)=\\left[A v \\; u \\;A u \\dots A^{n-2} u\\right].$ In addition, let $w$ be the last row of the classical adjoint of $ K_1(u,v)$ , i.e.",
"$w K_1(u,v) = \\left(\\det K_1(u,v) \\right)e_n^T$ .",
"Define $ K^*(u,v)$ to be the matrix with rows $w, w A, \\ldots , w A^{n-1}$ .",
"Then $\\det \\Big (\\det K_1(u,v) A - \\det K_2(u,v) \\mathbf {1}\\Big ) = (-1)^{\\frac{n(n-1)}{2}} \\det K(u) \\det K^*(u,v).$ It suffices to prove the identity for generic $A, u$ and $v$ .",
"In particular, we can assume that $A$ has distinct eigenvalues.",
"Then, after a change of basis, we can reduce the proof to the case of a diagonal $A=\\operatorname{diag}(a_1,\\ldots , a_n)$ and vectors $u=(u_i)_{i=1}^n$ and $v=(v_i)_{i=1}^n$ with all entries non-zero.",
"In this case, $\\det K(u) = \\operatorname{Van}A \\prod _{j=1}^n u_j, \\qquad \\det K^*(u,v) = \\operatorname{Van}A \\prod _{j=1}^n w_j,$ where $\\operatorname{Van}A$ is the $n\\times n$ Vandermonde determinant based on $a_1,\\ldots , a_n$ and $w_j$ is the $j$ th component of the row vector $w$ .",
"The the left-hand side of the identity can be rewritten as $\\prod _{j=1}^n \\left(a_j \\det K_1(u,v) - \\det K_2(u,v) \\right) =\\prod _{j=1}^n \\det \\left[ (a_j\\mathbf {1}- A)v\\ u \\ Au \\ldots \\ A^{n-2} u \\right].$ We compute the $j$ th factor in the product above as $\\begin{split}P_j&=\\sum _{\\alpha \\ne j} (-1)^{\\alpha + 1} (a_j - a_\\alpha ) {v_\\alpha } \\operatorname{Van}A_{\\lbrace \\alpha \\rbrace }\\prod _{i\\ne \\alpha } u_i \\\\&=\\left(u_j \\prod _{\\beta \\ne j} (a_j -a_\\beta )\\right) \\sum _{\\alpha \\ne j} (-1)^{\\alpha + 1}(-1)^{n-j -\\theta (\\alpha - j)} {v_\\alpha }\\operatorname{Van}A_{\\lbrace \\alpha ,j\\rbrace }\\prod _{i\\ne \\alpha ,j} u_i,\\end{split}$ where $\\operatorname{Van}A_I$ is the $(n-|I|)\\times (n-|I|)$ Vandermonde determinant based on $a_i$ , $i\\notin I$ , and $\\theta (\\alpha - j)$ is 1 if $\\alpha > j$ and 0 otherwise.",
"Note that $\\prod _{j=1}^n u_j \\prod _{\\beta \\ne j} (a_j -a_\\beta ) = (-1)^{\\frac{n(n-1)}{2}}(\\operatorname{Van}A)^2\\prod _{j=1}^n u_j = (-1)^{\\frac{n(n-1)}{2}}\\operatorname{Van}A \\det K(u),$ and that the minor of $ K(u)$ obtained by deleting the last two columns and rows $\\alpha $ and $j$ is $\\det K(u)^{[1,n-2]}_{[1,n]\\setminus \\lbrace \\alpha ,j\\rbrace } = \\operatorname{Van}A_{\\lbrace \\alpha ,j\\rbrace }\\prod _{i\\ne \\alpha ,j} u_i.$ Therefore $\\begin{aligned}w_j =& (-1)^{n+j} \\det K_1(u,v)^{[1,n-1]}_{[1,n]\\setminus \\lbrace j\\rbrace }\\\\=&(-1)^{n+j} \\sum _{\\alpha \\ne j}(-1)^{ \\alpha + 1-\\theta (\\alpha - j)}v_\\alpha \\det K(u)^{[1,n-2]}_{[1,n]\\setminus \\lbrace \\alpha ,j\\rbrace }.\\end{aligned}$ Comparing with (REF ), we obtain $P_1&\\cdots P_n \\\\&= (-1)^{\\frac{n(n-1)}{2}} \\det K(u) \\operatorname{Van}A \\prod _{j=1}^n \\sum _{\\alpha \\ne j}(-1)^{n + \\alpha - j + 1-\\theta (\\alpha - j)}v_\\alpha \\det K(u)^{[1,n-2]}_{[1,n]\\setminus \\lbrace \\alpha ,j\\rbrace }\\\\& = (-1)^{\\frac{n(n-1)}{2}} \\det K(u) \\operatorname{Van}A \\prod _{j=1}^n w_j = (-1)^{\\frac{n(n-1)}{2}} \\det K(u) \\det K^*(u,v),$ as needed.",
"Specializing Proposition REF to the case $A=X^{-1} Y$ , $u=e_n$ , $v= e_{n-1}$ , one obtains Proposition REF ."
],
[
"Normal forms",
"In this section we derive several normal forms that were used in the main body of the paper.",
"Lemma 8.2 For a generic $n\\times n$ matrix $U$ there exist a unique unipotent lower triangular matrix $N_-$ and an upper triangular matrix $B_+$ such that $U = N_- B_+ C N_-^{-1},$ where $C=e_{21} + \\cdots + e_{n, n-1} + e_{1n}$ is the cyclic permutation matrix.",
"Let $N_1= \\mathbf {1}- \\sum _{i=2}^n u_{in}/u_{1n}$ .",
"Then $U_1= N_1 U N_1^{-1}$ is a matrix whose last column has the first entry equal to $u_{1n}$ and all other entries equal to zero.",
"Next, let $N_2$ be a unipotent lower triangular matrix such that (i) off-diagonal entries in the first column of $N_2$ are zero, and (ii) $U_2=N_2 U_1 N_2^{-1}$ is in the upper Hessenberg form, that, is has zeroes below the first subdiagonal.",
"Then $U_2$ has a required form $B_+C$ , where $B_+$ is upper triangular.",
"To establish uniqueness, it is enough to show that if $B_1, B_2$ are invertible upper triangular matrices and $N$ is lower unipotent matrix, such that $N B_1 C = B_2 C N$ , then $N=\\mathbf {1}$ .",
"Comparing the last columns on both sides of the equality, it is easy to see that off-diagonal elements in the first column of $N$ are zero.",
"Then, comparison of first columns implies that the same is true for off-diagonal elements in the second column of $N$ , etc.",
"Lemma 8.3 For a generic $n\\times n$ matrix $U$ there exist a unique unipotent lower triangular matrix $N_-$ and an upper triangular matrix $B_+$ such that $U = N_- B_+ W_0 N_-^{-1},$ where $W_0$ is the matrix of the longest permutation.",
"Equality $U N_- = N_- B_+ W_0$ implies $W_0 U\\cdot N_- = W_0 N_- W_0\\cdot W_0 B_+ W_0$ .",
"Using the uniqueness of the Gauss factorization, we obtain $W_0 N_- W_0 =\\left(W_0 U\\right)_{> 0} $ and $W_0 B_+ W_0 = \\left(W_0 U\\right)_{\\le 0} N_-$ , and thus recover $N_-= W_0 \\left(W_0 U\\right)_{>0}W_0 $ and $B_+= W_0\\left(W_0 U\\right)_{\\le 0} W_0 \\left(W_0 U\\right)_{>0}$ .",
"Lemma 8.4 For a generic $n\\times n$ matrix $U$ there exists a unique representation $U = \\left(\\mathbf {1}_n + \\nu e_{12}\\right) N_- M N_-^{-1} \\left(\\mathbf {1}_n - \\nu e_{12}\\right),$ where $N_-=(\\nu _{ij})$ is a unipotent lower triangular matrix with $\\nu _{j1}=0$ for $2\\le j\\le n$ and $M=(\\mu _{ij})$ has $\\mu _{1n}=0$ and $\\mu _{i,n+2-j}=0$ for $2\\le j < i\\le n$ .",
"First, observe that if (REF ) is valid, then $\\mu _{2n}=u_{2n}$ , and $\\nu ={u_{1n}} /{u_{2n}}$ .",
"Denote $U^{\\prime }= \\left(\\mathbf {1}_n - \\nu e_{12}\\right) U\\left(\\mathbf {1}_n + \\nu e_{12}\\right)$ .",
"Then (REF ) implies that the first row and column of $M$ coincide with those of $U^{\\prime }$ , and that $M_{[2,n]}^{[2,n]}$ and $N$ are uniquely determined by applying Lemma REF to ${U^{\\prime }}_{[2,n]}^{[2,n]}$ ."
],
[
"Matrix entries via eigenvalues",
"For an $n\\times n$ matrix $A$ , denote $K= K(e_1)$ , where $ K(u)$ is defined in Proposition REF .",
"Lemma 8.5 Every matrix entry in the first row of $A$ can be expressed as ${P}/{\\det K}$ , where $P$ is a polynomial in matrix entries of the last $n-1$ rows of $A$ and coefficients of the characteristic polynomial of $A$ .",
"Let $\\det (\\lambda \\mathbf {1}- A) = \\lambda ^n + c_1 \\lambda ^{n-1} + \\cdots +c_n$ .",
"Compare two expressions for the first row $R(\\lambda )$ of the resolvent $(\\lambda \\mathbf {1}- A)^{-1}$ of $A$ : $R(\\lambda )= \\sum _{j\\ge 0} \\frac{1}{\\lambda ^{j+1}} A^j e_1= \\frac{1}{\\det (\\lambda \\mathbf {1}- A) } \\left( \\lambda ^{n-1} e_1 + \\lambda ^{n-2} Q_2 +\\cdots + Q_{n-1} \\right),$ where $Q_i$ , $2\\le i\\le n-1$ , are vectors polynomial in matrix entries of the last $n-1$ rows of $A$ .",
"Cross-multiplying by $\\det (\\lambda \\mathbf {1}- A)$ and comparing coefficients at positive degrees of $\\lambda $ on both sides, we obtain $c_i e_1+ c_{i-1} A e_1 + \\cdots + c_{1} A^{i-1} e_1 + A^{i} e_1 = Q_{i}, \\qquad 2\\le i\\le n-1.$ Let $T$ be an $n\\times n$ upper triangular unipotent Toeplitz matrix with entries of the $i$ th superdiagonal equal to $c_i$ , and let $Q$ be the matrix with columns $e_1, Q_2,\\ldots , Q_{n-1}$ .",
"Then the equations above can be re-written as a single matrix equation $K T = Q$ .",
"Note also that $A K = K C_A$ , where $C_A$ is a companion matrix of $A$ with 1s on the first subdiagonal, $-c_n,\\ldots , -c_{1}$ in the last column, and 0s everywhere else.",
"Therefore, $A K T = K T\\cdot T^{-1} C_A T = Q T^{-1} C_A T$ (note that $C^{\\prime }_A=T^{-1} C_A T= W_0 C_A^T W_0$ is an alternative companion form of $A$ ).",
"Consequently, the first row $A_{[1]}$ of $A$ satisfies the linear equation $A_{[1]} K T = Q_{[1]} C_A^{\\prime }$ , and the claim follows.",
"Lemma 8.6 Let $H$ be an $n\\times n$ upper Hessenberg matrix, $C^H=\\left[ \\begin{array}{cc} \\star & \\star \\\\ \\mathbf {1}_{n-1}& 0\\end{array} \\right]$ be its upper Hessenberg companion and $N=(\\nu _{ij})$ be a unipotent upper triangular matrix such that $N^{-1} H N = C^H$ .",
"Then the row vector $\\nu =(-\\nu _{1i})_{i=2}^n$ coincides with the first row of the companion form of $\\tilde{H}=H_{[2,n]}^{[2,n]}$ .",
"Factor $N$ as $N=\\left[ \\begin{array}{cc} 1 & 0\\\\ 0 & \\tilde{N}\\end{array} \\right] \\left[ \\begin{array}{cc} 1 & \\nu \\\\ 0 & \\mathbf {1}_{n-1}\\end{array} \\right] $ .",
"Then ${\\tilde{N}}^{-1} \\tilde{H} \\tilde{N} = \\left( \\left[ \\begin{array}{cc} \\mathbf {1}_{n-1} & 0 \\end{array} \\right] \\left[ \\begin{array}{cc} 1 & -\\nu \\\\ 0 & \\mathbf {1}_{n-1}\\end{array} \\right] \\right)^{[2,n]},$ and the claim follows."
],
[
"Acknowledgments",
"M. G. was supported in part by NSF Grant DMS #1362801.",
"M. S. was supported in part by NSF Grants DMS #1362352.",
"A. V. was supported in part by ISF Grant #162/12.",
"The authors would like to thank the following institutions for support and excellent working conditions: Max-Planck-Institut für Mathematik, Bonn (M. G., Summer 2014), Institut des Hautes Études Scientifiques (A. V., Fall 2015), Stockholm University and Higher School of Economics, Moscow (M. S., Fall 2015), Université Claude Bernard Lyon 1 and Université Paris Diderot (M. S., Spring 2016).",
"This paper was completed during the joint visit of the authors to the University of Notre Dame London Global Gateway in May 2016.",
"The authors are grateful to this institution for warm hospitality.",
"Special thanks are due to A. Berenstein, A. Braverman, Y. Greenstein, D. Rupel, G. Schrader and A. Shapiro for valuable discussions and to an anonymous referee for helpful comments."
]
] | 1605.05705 |
[
[
"Philosophy in the Face of Artificial Intelligence"
],
[
"Abstract In this article, I discuss how the AI community views concerns about the emergence of superintelligent AI and related philosophical issues."
],
[
"The idea of Artificial Intelligence has captured our collective imagination for many decades.",
"Can behavior that we think of as intelligent be replicated on a machine?",
"If so, what consequences could this have for society?",
"And what does it tell us about ourselves as human beings?",
"Besides being a long-running topic of philosophical reflection and science fiction, AI is also a well-established scientific research area.",
"Many universities have AI research labs, usually housed in computer science departments.",
"The feats accomplished in such research have generally been far more modest than those imagined in the movies.",
"But over time, the gap between reality and fiction has been closing.",
"For example, self-driving cars are now a reality.",
"And the world outside academia has taken notice.",
"The commercial opportunities are endless and technology companies are in fierce competition over the top AI talent.",
"Meanwhile, there is a growing popular worry about where this is all headed.",
"Most of the technical progress on AI is reported at scientific conferences on the subject.",
"These conferences have been running for decades and are attended by a steady community of devoted researchers.",
"But in recent years they have also started to attract a broader mix of participants.",
"At the 2016 conference in Phoenix, one speaker was more controversial than any other in recent memory: Nick Bostrom.",
"While the audience consisted mostly of computer scientists, Bostrom is a philosopher who directs the Future of Humanity Institute at Oxford.",
"He recently made waves with his book Superintelligence [1].",
"In it, he contemplates the problem that we may soon build AI that broadly exceeds human capabilities, and considers what steps we can take now to ensure that the result will be in our best interest.",
"A key concern is that of an “intelligence explosion”: if we are intelligent enough to build a machine more intelligent than ourselves, then, so the thinking goes, surely that machine in turn would be capable of building something even more intelligent, and so on.",
"The phrase “technological singularity” is sometimes also used to describe such runaway intelligence.",
"Will humanity be left in the dust?",
"Will we be wiped out?",
"Since the appearance of Bostrom's book, public figures including Elon Musk, Stephen Hawking, and Bill Gates have warned of the risks of superintelligent AI.",
"Musk even donated $10M to the Boston-based Future of Life Institute, establishing a grants program to ensure that AI remains beneficial.",
"The topic has remained in the news, with, for example, recently United Nations Chief Information Technology Officer Atefeh Riazi joining the chorus emphasizing the risks of AI.",
"These concerns have mostly been raised by people outside the core AI research community, which has not been very vocal in this debate.",
"Some in the community cautiously agree with some of the points; others dismiss them.",
"As an AI researcher myself, after Bostrom's talk I saw a number of people express their displeasure on social media, saying that giving him such a forum gives him credibility that he does not deserve.",
"Others emphasized open-mindedness, but (as far as I saw) fell short of endorsing his ideas.",
"But I assume that most in the AI community shrugged and continued with their research as usual.",
"Why?",
"Do AI researchers just not care about the future of humanity?",
"I think the real answer requires some familiarity with the history of AI research, which took off in the fifties.",
"Early research showed that computers could do things that few at the time had expected, leading to excitement, optimism, and promises of the moon.",
"But limitations of this early work soon became apparent.",
"Approaches that produced impressive results on small, toy examples simply would not scale to real-world problems.",
"Also, the real world is messy and ambiguous, and AI researchers struggle to this day with making their programs robust enough to handle this.",
"This led to what was called an “AI winter”: AI got a bad reputation in the academic community and funding was reduced.",
"In fact, this cycle repeated itself.",
"AI researchers yearned for their work to be scientifically rigorous and respected, and learned to be careful.",
"Some sought to dissociate themselves from the term “AI” altogether and instead associated with more narrowly defined technical problems.",
"For example, many researchers in the machine learning community – which focuses on having computers learn automatically from data how to make predictions and decisions – no longer wanted to be considered “AI” researchers.",
"Even most of the researchers that did stick with the term started focusing on narrower problems, not only because of perception issues but also for technical reasons: these problems seemed to be important roadblocks for AI but were not easy to solve.",
"Also, progress on their solution often led to direct beneficial impact on society.",
"For example, part of the community has focused on automated planning and scheduling systems, which have been used in a variety of applications, such as scheduling the observations of the Hubble Space Telescope.",
"The AI community has also mostly avoided the philosophical issues.",
"An introductory AI course will typically spend a little time on basic philosophical questions, such as those raised by Searle's “Chinese Room Argument” [5].",
"In this argument, someone who does not know Chinese at all sits in a room and has an incredibly detailed step-by-step manual – read, a computer program – for how to respond to Chinese characters slipped under the door, by drawing other characters and slipping them back out.",
"The manual is so good that from the outside it appears that there is someone inside who speaks Chinese, no matter how sophisticated the questions posed.",
"Now, we can ask whether there is any real understanding of Chinese in the room.",
"At first, it may appear that there is not.",
"But if not, then how could a computer, which operates similarly, ever have any real understanding?",
"While some AI professors enjoy posing such conundrums in, for example, the introductory lecture, after that the typical AI course – my own included – will quickly move on to teaching technical material that can be used to create programs that do something interesting, like playing the game of Connect Four.",
"After all, the course is generally taught in a computer science department, not a philosophy department.",
"Similarly, very little of the research presented at any major AI conference is philosophical in nature.",
"Most of it comes in the form of technical progress – a better algorithm for solving an established problem, say.",
"This is where AI researchers believe they can make useful progress and win respect in the eyes of their scientific peers, whether they feel the philosophical problems are important or not.",
"All this explains some of the reluctance of the AI community to engage with the superintelligence debate.",
"It has fought very hard to establish itself as a respected scientific discipline, overcoming outside bias and its own careless early claims.",
"The mindset is that anything perceived as unsubstantiated hype, or as being outside the realm of science, is to be avoided at all costs.",
"Tellingly, in a panel after Bostrom's talk, Oren Etzioni, director of the Allen Institute for Artificial Intelligence, drew supportive laughs from the crowd when he pointed out that Bostrom's talk was blissfully devoid of any data – even though Etzioni was quick to acknowledge that this was inherent in the problem.",
"Tom Dietterich, a computer science professor at Oregon State and President of the Association for the Advancement of Artificial Intelligence, expressed skepticism that an intelligence explosion of the kind Bostrom describes would happen, and asked what experiments we could run to test this hypothesis.",
"The AI community generally eschews speculation about the deep future and is more comfortable engaging with important problems that are more concrete and tangible at this point, such as autonomous weapons – weapon systems that can act without human intervention – or the unemployment caused by AI replacing human workers.",
"The latter was, in fact, the topic of the panel.",
"Another issue is that AI researchers, perhaps unlike the general public, generally feel that there are still quite a few needed components missing before something like the superintelligent AI of Bostrom's book could possibly emerge.",
"Many of the problems that were once thought to be great benchmark problems for AI – say, beating human champions at chess – ended up being solved using special-purpose techniques that, while impressive, could not immediately be used to solve many other problems in AI, suggesting that the “hard problems” of AI lay elsewhere.",
"(This has also led AI researchers to lament that “once we solve something, it's not considered AI anymore.”) So while recent breakthroughs, such as Google DeepMind's AI learning to play old Atari games surprisingly well, may raise concern in the general public, perhaps AI researchers have become accustomed to the idea that this just means the hard problems must lie elsewhere.",
"That being said, these results are certainly impressive to the AI community as well, not least because this time there are common techniques – now generally referred to as “deep learning” – underlying not only the Atari results, but also surprising progress in speech and image recognition.",
"(Consider the problems that Apple needs to solve to get Siri to understand what you said, or that Facebook needs to solve to automatically recognize faces in the pictures you upload.)",
"Researchers had previously attacked these problems with separate special-purpose techniques.",
"And Google DeepMind's AlphaGo program, which recently defeated Lee Sedol, possibly the best human player, in the game of Go, also has deep learning at its core.",
"The techniques used for chess had been largely ineffective on Go.",
"It is worth noting that the line of research that led to the deep learning results had been largely dismissed by most AI and machine learning researchers, before the few that tenaciously stuck with it started producing impressive results.",
"So our predictions about how AI will progress can be far off even in the very short term.",
"Accurately predicting all the way to, say, the end of the century seems humanly impossible.",
"If we go equally far into the past, we end up at a time before even Alan Turing's 1936 paper that laid the theoretical foundation for computer science [6].",
"This, too, makes it difficult for mainstream AI researchers to connect with those raising concerns about the future.",
"Some disaster scenarios, such as those related to asteroid strikes or global warming, allow for reasonable predictions over such timescales, so it is natural to want the same for AI.",
"But AI researchers and computer scientists in general tend to reason over much shorter timescales, which is already challenging given the pace of progress.",
"As one of the recipients of a Musk-funded Future of Life Institute grant, I participated on a keeping-AI-beneficial panel in a workshop at the conference in Phoenix.",
"The panel was moderated by Max Tegmark, one of the founders of the Future of Life Institute and a physics professor at MIT – again, an outsider to the AI community.",
"Besides relatively more accessible questions about autonomous weapons and technological unemployment, Tegmark also asked the panel some philosophical questions.",
"All other things being equal, would you want your artificially intelligent virtual assistant (imagine an enormously improved Siri) to be conscious?",
"Would you want it to be able to feel pain?",
"The first question had no takers; some in attendance argued that pain could be beneficial from the perspective of the AI learning to avoid bad actions.",
"The substantial philosophical literature on consciousness and qualia did not come up.",
"(In philosophy, the word “qualia” refers to subjective experiences, such as pain, and more specifically to what it is like to have the experience.",
"A famous example due to the philosopher Thomas Nagel is that presumably, there is something it is like to be a bat, though we, as a species that does not use echolocation, may never know exactly what this is like [4].",
"Is there something it is like to be an AI virtual assistant?",
"A self-driving car?)",
"Perhaps this was less due to unfamiliarity with such concepts, and more due to discomfort with how to approach these questions.",
"Even philosophers have difficulty agreeing on the meaning of these terms, and the literature ranges from the more scientifically oriented search for the “neural correlates of consciousness” (roughly: what is going on in the brain when conscious experience takes place) all the way to more esoteric studies of the subjective: how is it that my subjective experiences so appear so vividly present, while yours do not?",
"Well, surely your experiences appear similarly somewhere else.",
"Where?",
"In your brain, as opposed to mine?",
"But when we inspect a brain, we do not find any qualia, just neurons.",
"(If all this seems hopelessly obscure to you, you are not alone – but if you are intrigued, see, for example, Caspar Hare's On Myself, and Other, Less Important Subjects [3] or J.J. Valberg's Dream, Death, and the Self [7].)",
"The state of our understanding makes it difficult even to agree on what exactly Tegmark's questions mean – is objectively assessing whether an AI virtual assistant has subjective experiences a contradiction in terms?",
"– let alone give actionable advice to AI practitioners.",
"I believe philosophers do make progress on these issues, but it is slow and hard-won.",
"When discussing what philosophers are to do, Bostrom in his book suggests to postpone work on “some of the eternal questions” for a while, and instead to focus on how best to make it through the transition to a world with superintelligent AI.",
"But it is not entirely clear whether and how we can sidestep the eternal questions in this endeavor, even if we accept the premise that such a transition will take place.",
"(Of course, philosophers do not necessarily accept the premise either.)",
"So, generally, AI researchers prefer to avoid these questions and return to making progress on more tractable problems.",
"Many of us are driven to make the world a better place – by reducing the number of deaths from automobile accidents, increasing access to education, improving sustainability and healthcare, preventing terrorist attacks, etc.",
"– and are a bit frustrated to see every other article on AI in the news accompanied by an image from The Terminator.",
"Meanwhile, genuine concerns are developing outside the AI community.",
"While the AI conference in Phoenix was already underway, there was a call at a meeting of the American Association for the Advancement of Science to devote 10% of the AI research budget to the study of the societal impact of AI.",
"Now, in a climate where funding is already tight, diverting some of it may not make AI researchers look more kindly on the people raising these concerns.",
"But the AI community should take part in the debate on societal impact, because without it the debate will still take place and be less informed.",
"Fortunately, members of the community are increasingly taking an interest in short-to-medium-term policy questions, including calling for a ban on autonomous weapons.",
"Unfortunately, we have yet to figure out how to rigorously and productively engage with the more nebulous long-term philosophical issues.",
"One area where some immediate traction seems possible is the study of how (pre-superintelligence) AI can make ethical decisions – for example, when a self-driving car needs to make a decision in a scenario that is likely to kill or injure someone.",
"In fact, automated ethical decision making is the topic of a number of the Future of Life Institute grants, including my own grant with Walter Sinnott-Armstrong, a professor of practical ethics and philosophy at Duke University.",
"But at this point it is not clear to AI researchers how to usefully address the notion of superintelligence and the philosophical questions raised by it.",
"At the end of Bostrom's talk, Moshe Vardi, a computer science professor at Rice University, suggested that this all was very much as if upon Watson and Crick's discovery of the structure of DNA, the focus had immediately been on all the ways in which it could be abused.",
"I think this is an excellent point.",
"Progress in AI will unfold in unexpected ways and some of the current concerns will turn out to be unfounded, especially among those concerning the far off future.",
"But this argument cuts both ways; we can be sure that there are risks that are not currently appreciated.",
"It is not clear what exact course of action is called for, but those that know the most about AI cannot be complacent."
]
] | 1605.06048 |
[
[
"Do asteroids evaporate near pulsars? Induction heating by pulsar waves\n revisited"
],
[
"Abstract We investigate the evaporation of close-by pulsar companions, such as planets, asteroids, and white dwarfs, by induction heating.",
"Assuming that the outflow energy is dominated by a Poynting flux (or pulsar wave) at the location of the companions, we calculate their evaporation timescales, by applying the Mie theory.",
"Depending on the size of the companion compared to the incident electromagnetic wavelength, the heating regime varies and can lead to a total evaporation of the companion.",
"In particular, we find that inductive heating is mostly inefficient for small pulsar companions, although it is generally considered the dominant process.",
"Small objects like asteroids can survive induction heating for $10^4\\,$years at distances as small as $1\\,R_\\odot$ from the neutron star.",
"For degenerate companions, induction heating cannot lead to evaporation and another source of heating (likely by kinetic energy of the pulsar wind) has to be considered.",
"It was recently proposed that bodies orbiting pulsars are the cause of fast radio bursts; the present results explain how those bodies can survive in the pulsar's highly energetic environment."
],
[
"Introduction",
"Pulsars are highly magnetized, rapidly rotating neutron stars (NS) that lose their energy principally via electromagnetic cooling, which results in their spin-down.",
"The pulsar outflow comprises a low-frequency ($\\omega =2 \\pi / P$ , with $P$ the pulsar spin period) Poynting flux-dominated component (also called the pulsar wave), a relativistic wind, and high-energy radiation from the magnetosphere.",
"The energy of the outflow is believed to be dominated by Poynting flux close to the star, and by relativistic particles farther out (as observed in particular in the case of the Crab pulsar).",
"However, the location of this transition remains a puzzle to the community (see, e.g., [17] for a review).",
"Pulsars are often observed to evolve in multiple systems, and evaluating the amount of energy absorbed by the companions is paramount to understanding their evolution and emission.",
"The most common companions are white dwarfs [31], [21], [27] or M-dwarfs [29], but smaller objects such as planets, asteroids, or comets can also orbit pulsars [42], [37], [3].",
"The existence of asteroid belts around pulsars has been invoked by several authors to explain timing irregularities [32], anti-glitches [15], or burst intermittency [9], [11], [23].",
"Asteroids around pulsars could also be at the origin of fast radio bursts (FRBs).",
"These brief radio signals (typically 5 ms at a given frequency) are dispersed in frequency (as for pulsar signals), but with a dispersion corresponding to cosmological distances (e.g., [8] and references therein).",
"Going against the mainstream scenarios involving the collapse of massive objects, [25] proposed that FRBs could be emitted by bodies orbiting extragalactic pulsars (see also [10]).",
"Highly collimated waves would be produced by the magnetic wake of these objects immersed in the relativistic pulsar wind.",
"The collimation would enable their detection even at distances of hundreds of Mpc.",
"This model can naturally produce the repeating bursts reported by [35] in the presence of an asteroid belt.",
"For massive companions, the standard scenario stipulates that old neutron stars are spun up to periods $<10\\,$ ms in close binary systems by transfer of mass and angular momentum by the low-mass companion [1].",
"This accretion process leads to the production of X-rays detected as low-mass X-ray binaries.",
"In particular, the so-called black widow (dwarf companion with mass $m\\sim 0.002-0.07\\,M_\\odot $ ) and redback (M-dwarf companion with mass $m\\sim 0.1-0.4\\,M_\\odot $ ) pulsars illustrate the importance of energy absorption by companions on the evolution of binary systems.",
"These systems have very low-mass secondaries and orbital periods of less than 10 h. Early studies pointed out that these pulsars should theoretically evaporate their companion by high-energy radiation [30] and/or by Roche lobe overflow.",
"At the time of their discovery, it was assumed that black widows provided the missing link between low-mass X-ray binaries and isolated millisecond pulsars.",
"Since pulsar environments can evaporate white dwarves and M-stars orbiting them, it may seem surprising that smaller planets and asteroids could be maintained in a solid state at similar distances to the neutron star.",
"In the present paper, we propose an explanation for their survival.",
"To date, it has been confirmed that two pulsars (PSR B1257+12 and PSR B1620-26) host planets.",
"PSR B1257+12 hosts three planets at distances on the order of the astronomical unit (in the pulsar wind) with orbital periods in the range of days and weeks [42].",
"The distances are therefore greater than in close binary systems (orbital periods in the range of hours) such as black widow and redback systems, which can provide an explanation to the stability of the planets.",
"PSR B1620-26 is a neutron star-white dwarf binary, that was confirmed to host a Jovian mass companion orbiting at 23 AU [34].",
"The distant orbit makes this system ill-suited for our framework, and could explain by itself why the companion has not been evaporated.",
"On the other hand and if confirmed, the suspected belt of asteroids at a small distance from PSR 1931+24, with an orbital period ranging in minutes [9], [11], [22], [23] is puzzling and would require an alternative justification for its survival.",
"PSR J1719-1438 presents an orbiting candidate planet [3].",
"Its Jovian mass companion has a short orbital period of $\\sim 2.2\\,$ h, implying a compact orbital distance of $\\sim 0.95\\,R_{\\odot }$ , and a minimum mean density of $23.3\\,$ g cm$^{-3}$ .",
"This density suggests that it may be an ultra-low-mass carbon white dwarf rather than a planet.",
"The evaporation timescale of this companion should also be investigated.",
"At distances within a few light-cylinder radii from the neutron star, there is a general agreement that most of the energy outflowing from the pulsar is under the form of the Poynting flux associated with the pulsar wave [17].",
"The pulsar wave is the electromagnetic structure created by the rotation of the neutron star magnetic field at the pulsar spin frequency (with an inclination angle $i \\ne 0$ relative to the spin axis).",
"Outside the light-cylinder, this corresponds to a low-frequency spherical wave propagating at a velocity $ \\sim c$ and of wavelength equal to two light-cylinder radii [12].",
"On the scale of a companion radius, this wave can be considered a plane wave.",
"Because the pulsar wave constitutes the main flux of energy out of the pulsar, we consider that it is potentially the main source of heat for pulsar companions.",
"In the remaining sections, we investigate the physical parameters that control the efficiency of the pulsar companion heating by the pulsar wave."
],
[
"Companion heating by induction: model",
"All numerical quantities are denoted $Q_x\\equiv Q/10^x$ in cgs units unless specified otherwise."
],
[
"Energy flux of the pulsar wind",
"The energy loss rate (or luminosity) of a pulsar with moment of inertia $I$ , rotation period $P$ , radius $R_\\star $ , dipole magnetic field strength $B_\\star $ , and corresponding period derivative $\\dot{P}$ reads (e.g., [33]) $L_{\\rm p} \\equiv \\dot{E}_{\\rm rot} &=& \\frac{8\\pi ^4 R_\\star ^6 B_\\star ^2}{3c^3 P^4} = I(2\\pi )^2\\frac{\\dot{P}}{P^3}\\\\&\\sim & 3.9\\times 10^{35}\\,{\\rm erg\\, s}^{-1}\\,I_{45}\\dot{P}_{-20}P_{-3}^{-3}\\ \\nonumber \\\\&\\sim & 9.6\\times 10^{34}\\,{\\rm erg\\, s}^{-1}\\,I_{45}P_{-3}^{-4}B_{\\star ,8}^2R_{\\star ,6}^6\\ .\\nonumber $ The energy flux in the pulsar wind at distance $a$ , large compared to the pulsar light cylinder radius, $R_{\\rm L} = cP/(2\\pi )\\sim 4.8\\times 10^8\\,{\\rm cm}\\,P_{-3}$ , can then be written [2] $F_{\\rm w} &=&\\frac{L_{\\rm p}}{4\\pi f_{\\rm p}a^2}= { \\frac{1}{f_{\\rm p}a^2}\\frac{ 2 \\pi ^3 R_\\star ^6 B_\\star ^2}{3c^3 P^4} }=\\frac{I\\pi }{f_{\\rm p}a^2}\\frac{\\dot{P}}{P^3}\\\\&\\sim & 1.8\\times 10^{13}\\,{\\rm erg\\,s}^{-1}{\\rm cm}^{-2}\\,f_{\\rm p}^{-1}\\,\\left(\\frac{a}{R_\\odot }\\right)^{-2}I_{45}P_{-3}\\dot{P}_{-20} \\ ,\\nonumber $ where $f_{\\rm p}=\\Delta \\Omega _{\\rm p}/(4\\pi )$ is the fraction of the sky into which the pulsar wind is emitted.",
"We note that this flux can also be expressed as $F_{\\rm w} ={cB^2(1+\\sigma _B)}/({4\\pi \\sigma _B})$ ; the magnetization parameter is defined as $\\sigma _B\\equiv B^2/[4\\pi c^2(n_{\\rm i}m_{\\rm i}+\\kappa _\\pm m_\\pm n_\\pm )]$ , i.e., the ratio of the magnetic energy flux to the kinetic energy flux in the comoving wind frame.",
"Here $B$ , $n_{\\rm i}$ , $m_{\\rm i}$ , and $n_{\\pm }$ and $m_{\\pm }$ are respectively the magnetic field strength, the number density, the mass of ions, and pairs of the cold plasma, and $\\kappa _\\pm $ the pair multiplicity.",
"In this work, we assume that the wind is Poynting-flux dominated ($\\sigma _B\\gg 1$ ) in the region where the companion is located.",
"It can be demonstrated that this is valid out to the termination shock for a supersonic, radially expanding ideal magneto-hydrodynamics (MHD) wind.",
"In more realistic situations, the dissipation to kinetic energy could happen earlier, but all models predict a Poynting-flux dominated wind close to the star (e.g., [17]).",
"The companion at a distance $a$ can intercept a fraction $f$ of this flux, provided that it falls in the wind beam and absorbs a flux, $F_{\\rm abs} = f Q_{\\rm abs} F_{\\rm w}$ , which can be written in terms of absorbed luminosity: $L_{\\rm abs} = \\pi R^2 f Q_{\\rm abs} F_{\\rm w} = \\frac{2 \\pi ^4}{3c^3}Q_{\\rm abs} \\frac{ f }{f_p}\\frac{R_\\star ^6 B_\\star ^2}{P^4}.$ Here, $Q_{\\rm abs}$ is the energy absorption efficiency.",
"Using the magnetization parameter $\\sigma $ defined above, this quantity can be decomposed into two components: the absorption of energy from the pulsar wave, characterized by an efficiency $Q_{\\rm em}$ , and the absorption of kinetic energy from the pulsar wind, of efficiency $Q_{\\rm kin}$ , $Q_{\\rm abs}= \\frac{\\sigma _B}{1+\\sigma _B}\\, Q_{\\rm em} + \\frac{1}{1+\\sigma _B} \\, Q_{\\rm kin} \\ .$ The next two sections are devoted to the evaluation of $Q_{\\rm em}\\sim Q_{\\rm abs}$ for a Poynting-flux dominated case when $\\sigma _B\\gg 1$ ."
],
[
"Absorption properties of a body in an electromagnetic wave",
"The absorption properties of a spherical body in an electromagnetic wave can be modeled by the Mie theory [38].",
"Two parameters govern the regime in which absorption or scattering occurs: the ratio of the size of the body to the incident wavelength $ x\\equiv \\frac{R}{cP} = \\frac{R}{2 \\pi R_L} \\sim 3.3\\times 10^{-3}\\,R_5P_{-3}$ and the complex refractive index of the medium $N=N_{\\rm r} + N_{\\rm i}$ , where $N_{\\rm r}$ and $N_{\\rm i}$ are real."
],
[
"Refractive index of the companion",
"Companions orbiting a 10 ms pulsar at a distance $a=1\\,R_{\\odot }$ see an alternating magnetic field of amplitude $B \\le 10$ G with a 100 Hz frequencyThe wave amplitude is much smaller than in a domestic transformer (where magnetic fields of $10^4$ G are common)..",
"The electric field induced by the pulsar wind ($v \\sim c$ ) is $E \\le 10^5$ V/m.",
"This is less than the electric strength encountered in most materials (larger than $10^6$ V/m).",
"Therefore, the companions are not ionized by the electric field induced by the pulsar wave.",
"Similar conditions are met for companions orbiting standard pulsars ($P \\sim 1$ s) beyond the light cylinder.",
"Therefore, the usual approximations and textbook data can be used to estimate the electromagnetic properties of the constituents of pulsar companions.",
"The refractive index can be calculated using the Maxwell equations and the material equation $\\vec{J}= \\sigma \\vec{E}$ where $J$ is the current density and $\\sigma $ is a scalar conductivity [5].",
"The dispersion equation reads $\\mu \\epsilon \\omega ^2 - i 4 \\pi \\omega \\mu \\sigma + k^2 c^2 =0,$ where $\\omega =2\\pi /P$ , $k$ is the wave number, and $\\epsilon $ and $\\mu $ the dielectric permittivity and magnetic permeability of the medium, respectively.",
"The refractive index $N$ defined by $N^2\\equiv k^2 c^2 / \\omega ^2$ can be written $N^2 = \\mu \\left( \\epsilon - i \\frac{4\\pi \\sigma }{\\omega } \\right).$ Pulsars rarely spin faster than with a millisecond period, thus $\\omega \\lesssim 6 \\times 10^3$ s$^{-1}$ (a strict lower bound on the spin is given by [14]).",
"For low electromagnetic frequencies (up to infrared frequencies), $\\sigma $ is real in any kind of material.",
"For companions made of degenerate matter, the electrical conductivity can be estimated as $\\sigma \\sim 10^{24}\\,{\\rm s}^{-1}\\, \\rho ^{2/3}T^{-1}$ , for densities ranging from $\\rho \\sim 10^{6-12}\\,$ g cm$^{-3}$ , and $T$ the stellar temperature [7].",
"White dwarfs typically have $\\rho \\sim 10^9\\,$ g cm$^{-3}$ and $T\\sim 10^{4-7}\\,$ K. For $T=10^5$ K, $\\sigma \\sim 10^{22}$ s$^{-1}$ .",
"To date, the planets orbiting pulsars that have been discovered could be either low-mass white dwarfs made of degenerate matter, or made of ordinary matter.",
"For $T=10^5$ K, $\\sigma \\sim 10^{25}$ s$^{-1}$ .",
"Smaller objects such as asteroids could be modeled as silicate rock or as ferrous metals.",
"The conductivity of sea water is on the order of $\\sigma \\sim 10$ Mho m$^{-1} \\sim 10^{10}$ s$^{-1}$ .",
"In metals, $\\sigma \\sim 10^6 - 10^7$ Mho m$^{-1} \\sim 10^{17}$ s$^{-1}$ .",
"For solid rocks in the terrestrial crust, $\\sigma \\sim 10^{-3}$ Mho m$^{-1} \\sim 10^{6}$ s$^{-1}$ , but in some conductive belts (for instance between the African Rift and Namibia) $\\sigma \\sim 10^{-1}$ Mho m$^{-1}$ [13].",
"In all materials, the permittivity $\\epsilon $ is of order unity, rarely exceeding 100.",
"For instance, water-free iron ore has $\\epsilon \\sim 5$ and iron oxides $\\epsilon \\sim 2$ .",
"Given the above estimates, one can neglect the real part of $N^2$ in our framework.",
"The real and imaginary parts of $N$ can then be extracted from Eq.",
"(REF ): $ N_{\\rm r}=-N_{\\rm i} = \\sqrt{\\frac{2 \\pi \\sigma \\mu }{\\omega }}=\\sqrt{\\sigma \\mu P}.$ For example, for a companion made of iron (or any other ferromagnetic material), $\\mu > 10^3$ and $N_{\\rm r} \\gg 10^3$ .",
"For solid rocks in the terrestrial crust, $\\mu \\sim 1$ and $N_{\\rm r} \\sim {10^3}$ for a $P=1$ s pulsar and $N_{\\rm r} \\sim 30$ for a $P=1$ ms pulsar.",
"For white dwarfs, the very large conductivity $\\sigma $ implies $N_{\\rm r} \\gg 10^6$ ."
],
[
"Absorption coefficient",
"Here we compute the absorption coefficient $Q_{\\rm abs}$ , defined in Eq.",
"(REF ) as the proportion of energy absorbed by the companion, relative to the flux of incident energy of the wave through the section of the companion.",
"We perform a numerical computation of $Q_{\\rm abs}$ using a version of the Damie code, based on [20]http://diogenes.iwt.uni-bremen.de/vt/laser/codes/ddave.zip.",
"We have tested the program with real and complex values of $N^2$ , and compared the results with analytical approximations given in [38] for various regions of the $(x, N)$ parameter space.",
"We find that numerical and analytical methods give consistent values within an error of $<21\\%$ .",
"This level of agreement is sufficient for our purpose as we are mostly interested in order-of-magnitude estimates, given the uncertainties of our other parameters.",
"Figure REF shows the value of the absorption coefficient computed with Damie as a function of the size parameter $x=R/(cP)$ and of the refractive index $N$ , in the so-called Metallic regime, where $N=N_{\\rm r}=-N_{\\rm i}$ for values of $N$ larger than one.",
"As expected, for large objects ($x \\gtrsim 1$ ), $Q_{\\rm abs} \\sim 1$ .",
"This means that all the flux received from the pulsar wave is absorbed.",
"The values $x \\gtrsim 1$ correspond to planet-sized and white dwarf-sized objects ($R \\ge 1000$ km) in a millisecond pulsar wind ($c P =3000$ km for $P=10$ ms).",
"For such objects, taking into account the Mie theory does not modify the heating rate.",
"On the other hand, for smaller objects with $x<1$ , the level of absorption is reduced by orders of magnitude.",
"For instance, for a kilometer-sized asteroid orbiting a standard pulsar ($P=1$ s), $x=3\\times 10^{-6}$ .",
"If it is made of rocks, $N_{\\rm r} \\sim 30$ , and it can be read $Q_{\\rm abs} \\sim 10^{-8}$ .",
"A value $Q_{\\rm abs} \\sim 10^{-12}$ is even reached when $N_{\\rm r} \\sim 10^3$ .",
"For $x\\ll 1$ , [38] gives an analytical expression for the absorption coefficient (we note, however, the corrected typo in the numerator) $Q_{\\rm abs} = \\frac{3}{N_{\\rm r}}\\left[ \\frac{\\sinh (2xN_{\\rm r}) + \\sin (2xN_{\\rm r})}{\\cosh (2xN_{\\rm r}) - \\cos (2xN_{\\rm r})} -\\frac{1}{xN_{\\rm r}}\\right]\\ .$ The absorption coefficient computed with Eq.",
"(REF ) is plotted in Fig.",
"REF (white contours).",
"For small values of $x$ , this approximation is numerically unstable.",
"An approximation for the parameter space where $N_{\\rm r} x\\ll 1$ called “Region 1” was proposed [38]: $Q_{\\rm abs} = \\frac{12 x }{N_r^2} +\\frac{2 x^3 N_r^2}{15} \\ .$ As shown in Fig.",
"REF (black contours), this approximation is valid for the smallest values of $N_{\\rm r} x$ .",
"The domain corresponding to $xN_{\\rm r}\\ge 1$ was denoted “Region 2” by [38].",
"For $xN_{\\rm r}\\gg 1$ , Eq.",
"(REF ) can be approximated as $Q_{\\rm abs} = \\frac{3}{N_{\\rm r}}\\ .$ Region 2 corresponds roughly to planets and white dwarfs, while Region 1 corresponds to planetesimals, asteroids, and cometsFrom an electromagnetism point of view, one can understand these two regions as different regimes of penetration of the magnetic field inside the object.",
"In Region 2, the inductive currents do not penetrate deep under the surface (the resistive skin-depth is shallow).",
"For Region 1, the skin-depth is approximately the entire size of the object, and the magnetic field can be considered as roughly uniform over it.. White dwarfs, with $N_{\\rm r}\\gg 10^6$ , are hence highly reflective bodies, as confirmed by the Mie theory.",
"Indeed, $Q_{\\rm abs} \\rightarrow 0$ (Eq.",
"REF ).",
"If the pulsar outflow is under the form of a pulsar wave while it reaches the white dwarf, it should not absorb any energy and should not evaporate.",
"This goes against the existence of black widow systems.",
"It thus seems likely that the outflow is not Poynting-flux dominated as it reaches the white dwarf companion.",
"One possible scenario is that the pulsar wave energy is dissipated by plasma heating, in the dense plasma surrounding the evaporating white dwarf.",
"Figure: Absorption coefficient Q abs Q_{\\rm abs} as a function of the size ratio x=R/cPx=R / c P and of the refractive index N r N_{\\rm r} of the companion in the Metallic regime (N=N r =-N i N=N_{\\rm r}=-N_{\\rm i}).",
"Overlayed are the contours of the analytical approximation, in black for Region 1 (xN r ≪1xN_{\\rm r}\\ll 1, Eq. )",
"and in white for Region 2 (xN r ≫1xN_{\\rm r}\\gg 1, Eq.",
").The gray-shaded region corresponds to parameters outside the confidence range of the computation with Damie."
],
[
"Effects on the evaporation of companions",
"We discuss in this section the impact of $Q_{\\rm abs}$ on the evaporation rate of pulsar companions.",
"A simple energy balance gives the evaporation timescale of the companions as ${E_\\mathrm {abs}(t_{\\rm ev})}={[E_{\\rm g} + E_{\\rm c} - E_\\mathrm {rad}](t_{\\rm ev})} \\ ,$ where $t_{\\rm ev}$ is the minimum time of evaporation, $E_\\mathrm {abs}$ is the total energy absorbed at time $t$ , $E_\\mathrm {rad}$ the total radiated energy, $E_{\\rm g}$ the gravitational binding energy, and $E_{\\rm c}$ the cohesive energyCohesive energy is the energy required to form separated neutral atoms in their ground electronic state from the solid at a given temperature under a given pressure [18].. Two regimes will be distinguished in this paper.",
"For massive, degenerate companions the cohesive energy can be neglected (section REF ), while for low-mass companions it dominates (section REF ).",
"Gravitational and cohesive energies are of the same order of magnitude for bodies about the mass of the Earth.",
"In both cases, we neglect the radiative cooling, but we will include it in a more refined treatment in a future paper."
],
[
"Ablation and evaporation of massive companions (degenerate and non-degenerate)",
"For massive companions, the dominant process for evaporation is the gravitational escape of the gas.",
"The evaporation timescale, $t_{\\rm ev}$ , results from the balance between the energy flux input from the pulsar wave and the rate of gravitational escape of the heated material [39].",
"The luminosity available to drive the wind of a companion of mass $M$ and radius $R$ located at a distance $a$ from the pulsar is then $(1/2)\\dot{M}v^2 = Q_{\\rm abs}(f/f_{\\rm p})(R/2a)^2L_{\\rm p}$ .",
"Assuming that the wind velocity is on the order of the escape velocity $v = (2GM/R)^{1/2}$ from the surface of the companion, the gravitational binding energy of the companion reads $E_{\\rm g} = \\frac{GM^2}{R} \\sim 3.8\\times 10^{48}\\, {\\rm erg} \\,\\left(\\frac{M}{M_\\odot }\\right)^2\\frac{ R_\\odot }{R} \\ ,$ and the evaporation timescale $t_{\\rm ev} \\equiv \\frac{M}{\\dot{M}}&=&\\frac{4GM^2a^2}{R^3}\\frac{f_{\\rm p}}{f}(L_{\\rm p}Q_{\\rm abs})^{-1} \\\\&=& \\frac{3c^3}{2\\pi ^4}\\frac{P^4}{R_\\star ^6 B_\\star ^2}\\frac{GM^2}{R^3}\\frac{a^2 f_{\\rm p}}{f Q_{\\rm abs}}\\ .$ For low values of $M$ when the star is degenerate, the mass-radius relation follows roughly $R/R_\\odot = 0.013(1+X)^{5/3}(M/M_\\odot )^{-1/3}$ , where $X$ is the hydrogen fraction; the evaporation time can then be expressed $ \\frac{t_{\\rm ev,WD}}{{\\rm yr} }= 2.3 \\times 10^{12}\\, ( {1+X})^{-5}\\left( \\frac{M}{M_\\odot }\\right)^{3}\\left( \\frac{a}{R_\\odot }\\right)^{2} \\times \\nonumber \\\\\\frac{f_{\\rm p}}{f}\\frac{1}{L_{\\rm p,35}\\,{Q_{\\rm abs}}} \\ .$ The spin-down luminosity is on the order of $L_{\\rm p}\\sim 10^{35}\\,$ erg s$^{-1}$ for a typical millisecond pulsar with $R_\\star =10$ km, $P=10$ ms, $B=10^8$ G, and on the order of $L_{\\rm p}\\sim 10^{31}\\,$ erg s$^{-1}$ for a standard pulsar with $P=1000$ ms, $B=10^{12}$ G (see Eq.",
"REF ).",
"This equation is analogous to those in [39].",
"These estimates can be applied to PSR B1957+20, a benchmark black widow system suspected of ablating its companion (e.g., [26], [19], [2], [6], [16], [36], [28], [40]).",
"PSR B1957+20 has a period of $P=1.61\\,$ ms, and its dwarf companion of mass $M=0.021\\,M_\\odot $ orbits at a distance $a=2.1\\,R_\\odot $ ($P_{\\rm orb}=9.2$ hr), hence $t_{\\rm ev,WD}\\sim {\\cal O} (10^6\\,Q_{\\rm abs}^{-1})\\,$ yrs.",
"However, as argued at the end of Section REF , the high refractive index of white dwarfs implies that $Q_{\\rm abs}\\ll 1$ , thus $t_{\\rm ev,WD}\\gg 10^6\\,$ yrs.",
"The conclusions derived in [39], namely that the evaporation of the companion (strongly dependent on $R_\\star $ and $X$ ) takes only a few million years, remain valid only if the absorbed energy is principally kinetic (Eq.",
"REF ).",
"The same conclusions apply for PSR J1719-1438 (period $P=5.7\\,$ ms), if its companion is indeed an ultra-low-mass carbon white dwarf with $M= 0.015\\times 10^{-3}\\,M_\\odot $ , $R=4.2\\times 10^4\\,$ km, and $a=0.95R_\\odot $ .",
"For planet companions made of ordinary matter, the mass/radius relationship simply reads $M =(4 \\pi /3) R^3 \\rho $ , where $\\rho $ is the average density.",
"The evaporation time can then be expressed $ \\frac{t_{\\rm ev,g}}{\\rm yr}= 7.2 \\times 10^{-12} \\left(\\frac{R}{\\rm km}\\right)^3 \\left( \\frac{\\rho }{{\\rm \\,g\\,cm}^{-3}}\\right)^{2} \\left( \\frac{a}{R_\\odot }\\right)^{2}\\times \\nonumber \\\\\\frac{f_{\\rm p}}{f}\\frac{1}{L_{\\rm p,35}\\,{Q_{\\rm abs}}}\\ .$ This equation can be applied to the planets orbiting PSR 1257+12.",
"The results are given in Table REF for two values of the refractive index $N_{\\rm r}$ that correspond to planets made of rocks and to more metallic planets.",
"We find that the closest planet can survive a few million years.",
"The two other planets can reach a billion years.",
"This estimate neglects the thermal energy radiated by the planet; therefore, the evaporation time through Poynting flux absorption may be underestimated.",
"If the companion of PSR J1719-1438 is actually a non-degenerate planet with minimum mean density $\\rho =23.3\\,$ g cm$^{-3}$ , then $x\\gg 1$ , but with $N_{\\rm r}\\lesssim 100$ , implying $Q_{\\rm abs}\\gtrsim 10^{-2}$ and $t_{\\rm ev,g}={\\cal O}(10^{6-8}\\,{\\rm yrs})$ for $f=f_{\\rm p}=1$ , which is compatible with the observation of the companion.",
"Figure: Influence of companion composition on evaporation timescale.",
"The contours of t ev =10 6 t_{\\rm ev}=10^6\\,yrs (solid lines) and t ev =10 12 t_{\\rm ev}=10^{12}\\,yrs (dashed lines) are represented as a function of companion distance aa and size RR for P=10P=10\\,ms, B=10 8 B=10^8\\,G, and an indicative (and conservative) density ρ=1\\rho = 1\\,g cm -3 ^{-3}, and for various companion refractive indices as indicated in the legend (increasing from right to left): σμ=10 4-14 \\sigma \\mu =10^{4-14}\\,s -1 ^{-1}, corresponding to N r =10 1-6 N_{\\rm r}=10^{1-6}.Table: Gravitational (t ev ,g t_{\\rm ev,g}) and cohesive (t ev ,c t_{\\rm ev,c}) evaporation timescales due to induction heating for various pulsar and companion parameters.",
"We assume a mass density ρ=3\\rho =3 g cm -3 ^{-3} and μσ=10 6 ,10 7 \\mu \\sigma =10^6,10^7 s -1 ^{-1} for columns (8-98-9) and (10-1110-11), respectively.",
"Values of Q abs Q_{\\rm abs} were computed with Damie.",
"Timescale values in parentheses are only indicative as the other timescale dominates the evaporation regime."
],
[
"Evaporation of small non-degenerate companions",
"In the case of small objects like asteroids, Eq.",
"(REF ) fails because the dominant process for evaporating the body is no longer gravitational escape but heating, melting, and evaporation.",
"The cohesive energy can be expressed as $E_{\\rm c}=KM \\sim 1.6\\times 10^{44}\\,{\\rm erg}\\,\\frac{K}{K_{\\rm iron}}\\frac{M}{M_\\odot }\\ ,$ with $K$ the cohesive factor that depends on the material [18].",
"In particular, the cohesive factor of iron is $K_{\\mathrm {iron}}=7.4\\times 10^{10}$ erg/g (and the orders of magnitude are similar for rock).",
"Equating with $E_{\\rm g}$ , the cohesive energy starts to dominate for a mass-to-size ratio of $M/R\\lesssim 10^{18}\\,$ g cm$^{-1}$ .",
"For ordinary matter, this constraint can then be expressed in terms of companion size as $R\\lesssim 5\\times 10^8\\,{\\rm cm} \\,\\rho ^{-1/2}\\left(\\frac{K}{K_{\\rm iron}}\\right)^{1/2}\\ .$ In this regime, the evaporation timescale can be calculated $KM = Q_{\\rm abs}(f/f_{\\rm p})(R/2a)^2L_{\\rm p} t_{\\rm ev 3}$ which leads to $ \\frac{t_{\\rm ev,c}}{\\rm yr} &=&2.6 \\times 10^{-4} \\left(\\frac{R}{\\rm km}\\right) \\left(\\frac{\\rho }{{\\rm g\\,cm}^{-3}}\\right) \\left( \\frac{K}{K_{\\rm iron}}\\right) \\left(\\frac{a}{R_\\odot }\\right)^{2} \\times \\nonumber \\\\& &\\frac{f}{f_{\\rm p}}\\frac{1}{L_{\\rm p,35}{Q_{\\rm abs}}} \\ .$ For small objects, and when the pulsar energy is mostly in the form of Poynting flux, $Q_{\\rm abs}$ is low and has a strong influence on the evaporation time.",
"The comparison of ${t_{\\rm ev, g}}$ and ${t_{\\rm ev, c}}$ shows, as expected, that when small bodies have melted and vaporized, the gravitational escape that follows is very fast.",
"Figure REF shows the evaporation timescales of non-degenerate pulsar companions ($t_{\\rm ev,g}$ and $t_{\\rm ev,c}$ , Eqs.",
"REF and REF ), for three spin periods.",
"For small companions, on the left-hand side of the white dotted line, the cohesive energy dominates over gravitational energy for evaporation.",
"The green contours indicate evaporation timescales shorter than $\\sim 10^6\\,$ yrs.",
"A few typical examples are also given in Table REF for companions made of rock (columns $8-9$ ) and for a more metallic composition (columns $10-11$ ).",
"The table shows that small non-degenerate bodies at one solar radius from a 1 ms pulsar evaporate in less than a few years.",
"We note that for $P=1-100\\,$ ms and $N_{\\rm r}\\sim 100$ , the fastest evaporation occurs for $R \\sim 100\\,$ km, and not for the smallest companions.",
"We also note also the strong dependency of the evaporation time on the pulsar period $P$ .",
"An asteroid at the same distance of a 10 ms pulsar can survive a few $10^7$ years.",
"At 1 AU (200 $R_\\odot $ ), it is definitively stable.",
"Figure REF demonstrates that $N_{\\rm r}$ , namely the chemical composition of the companion, has a strong influence on the evaporation timescale.",
"An asteroid ($R \\sim 1$ km) made of rock ($N_{\\rm r}\\sim 100$ or $\\mu \\sigma =10^6$ s$^{-1}$ ) has an optimal lifetime.",
"More resistive ($N_{\\rm r}\\sim 10$ ), or more metallic asteroids ($ N_{\\rm r}\\gtrsim 10^3$ ) evaporate over a shorter timescale.",
"For $N_{\\rm r}\\lesssim 100$ , the distance $a$ remains constant for small size ratios $x$ (black line in Fig.",
"REF ).",
"This stems from the change in the dependency of $Q_{\\rm abs}$ over $x$ (and thus over $R$ ), as shown in Fig.",
"REF for small $x$ and low $N_{\\rm r}$ .",
"The comparison between columns ($8-9$ ) and ($10-11$ ) of Table REF confirms that kilometer-size companions are more stable against evaporation when made of rock ($\\mu \\sigma =10^6$ s$^{-1}$ ) than for a more metallic refractive index $\\mu \\sigma =10^7$ s$^{-1}$ .",
"On the other hand, the table and Figure REF show that for $R \\ge 100$ km the evaporation timescale increases with the conductivity of the companion.",
"[23] proposed that the intermittency of PSR 1931+24 can be explained by the existence of a stream of small bodies of kilometer or subkilometer sizes close to the pulsar ($a=0.14\\,R_\\odot $ ).",
"We calculate that these asteroids would evaporate in about one Gyr.",
"If the Mie theory were not taken into account, such asteroids would not survive a year.",
"A billion years being a long timescale compared to the migration time in the pulsar wind magnetic field (10,000 years, [24]), the evaporation caused by induction does not invalidate the model of [23].",
"The last lines in Table REF refers to planets discovered near PSR 1257+12 [41].",
"Because of their size, the Mie theory still has an influence on the vaporization rate, with $Q_{\\rm abs} < 0.06$ .",
"If it is made of rock, 1257+21a should evaporate in $10^5$ years.",
"With a more metallic composition, it could last a few million years.",
"The other two planets are stable for $10^8$ years, or more if they are metallic.",
"Figure: Amplitude of the pulsar wind magnetic field for (a) a surface dipole field l=1,m=1l=1, m=1, (b) quadrupole l=2,m=2,l=2, m=2, and (c) l=3,m=3l=3, m=3.Panels (a) to (c) are plotted with the same color scale.",
"On these panels, the spiral structure is the pulsar wave.",
"Comparison ratio aa, see Eq.",
"() for modes (d) l=2,m=2l=2, m=2, (e) l=3,m=2,l=3, m=2, and (f) l=3,m=3l=3, m=3.",
"Panels (d) to (f) are plotted with the same color scale."
],
[
"Effect of multipole magnetic field components",
"We have considered the case of a star with a dipole magnetic field.",
"In terms of spherical harmonics analysis, it has longitudinal and azimuthal components $l=1$ , $m=0$ , and $m=1$ .",
"Only the $m=1$ component participates in the pulsar wave and produces a monochromatic frequency spectrum, but we know that neutron star magnetic fields can have multipole components $l>1$ .",
"The components with azimuthal numbers $m>1$ induce shorter wavelengths $cP/m$ , i.e., $2 \\pi R_{\\rm L}/m$ .",
"In all the above inductive heating rates, a multipole term of azimuthal number $m$ can be accounted for by replacing the ratio of the size of the body to the incident wavelength $x$ by $x_m=m x$ .",
"If they had the same amplitude, multipole $m>1$ terms would contribute more efficiently to the companion heating than the $m=1$ wave for small bodies where $x \\sim 1$.",
"Therefore, it is important to evaluate the contribution of multipole terms of the neutron star magnetic field to the pulsar wind.",
"[4] have computed the vacuum solution of the electromagnetic field surrounding a pulsar.",
"The solutions have a complex structure at distances less than the light cylinder radius with an amplitude that decreases faster for higher values of $l$ .",
"At greater distances, it has the characteristic Parker spiral wave structure dominated by the azimuthal component $B_\\phi r^{-1}$ .",
"Beyond the light cylinder, the ratio of amplitudes of the various modes does not vary much on average.",
"To illustrate this, we have plotted in Fig.",
"REF a few vacuum field solutions corresponding to the same maximum amplitude $B_0$ (on the star surface).",
"The amplitude $B$ of the magnetic field in the equatorial plane is plotted up to distances $r=11 R_L$ for the dipole solution (panel a), for the quadrupole solution $l=2 m=2$ (panel b)A pulsar of period $P_4$ with a pure quadrupole field might be confused, from an observational point of view, with a pulsar with a dipole field and a period $P_2=2 P_4$ ., and for $l=3, m=3$ (panel c).",
"For comparison of magnetic amplitudes, we define the ratio $ a_{lm}(R,\\phi )=\\max (B_{lm}(r>R,\\phi ,\\theta =0))/ \\\\\\max (B_{\\rm dipole}(r>R,\\phi ,\\theta =0)),$ where the maximum is computed for any radius larger than $R$ .",
"This definition is intended to avoid a divergent ratio when the two functions reach their minimum for different values of $R$ that would not be pertinent to what we want to compare.",
"The ratios $a_{2,2}, a_{3,2}$ , and $a_{3,3}$ are plotted respectively in panels (d), (e), and (f).",
"We can see from $a_{2,2}$ that beyond a distance $2 R_L$ , the contributions of $l=1, m=1$ (dipole) and $l=2, m=2$ modes with the same maximum magnetic field on the star's surface are comparable, although larger with the dipole.",
"For larger values of $l$ , the dipole term is dominant by a factor exceeding $10^2$ .",
"This means that the contribution of a large amplitude quadrupole term to the heating of a pulsar companion cannot be neglected, implying that companions of radius $R/2$ twice as small can absorb an energy comparable to that of a $R$ -sized body in the dipole case.",
"Higher multipole terms should not have an important influence in our example.",
"We note that the above computation does not take into account the contribution of the plasma to the evolution of the wave amplitude into the wind; it should be considered a first-order approximation."
],
[
"Conclusion and discussion",
"In this paper, we have assessed the efficiency of the inductive heating of pulsar companions caused by the pulsar Poynting-flux electromagnetic wave.",
"Inductive heating is generally considered to be the dominant heating process, for instance when the evaporation of black widow companions is considered [39].",
"It is commonly assumed that the whole flux carried by the pulsar wave is absorbed by the companion.",
"Taking into account the Mie theory, we have shown that this assumption fails for objects with radius $R$ smaller than the pulsar wavelength $c P$ , $P$ being the pulsar spin period.",
"The rate of absorption of the wave energy decreases by many orders of magnitude for small objects like asteroids or planetesimals.",
"This is especially true with standard $P=1\\,$ s pulsars because of their long wavelength.",
"One consequence is that asteroids, considered to be objects that should quickly evaporate, do not always do so even at very close distances ($1 R_\\odot $ ) of a 1 s-period pulsar.",
"The conclusion regarding kilometer-sized asteroids at short distances from pulsars cannot be generalized to any kind of pulsar.",
"For instance, bodies orbiting a $P=1$ ms pulsar at $1 R_\\odot $ would not last a year.",
"The behavior of small objects orbiting a pulsar is important because it can explain the intermittency of their radio-emissions.",
"For instance, [9] proposed a model where small rocks (meter-sized) falling on a pulsar cause intermittency during the few seconds in which they evaporate.",
"For metric objects, $Q_{\\rm abs}$ derived from the Mie theory is so low that these rocks should not evaporate.",
"The survival of small bodies around pulsars is a pre-requisite of the theory developed by [25] to explain the origin of FRBs.",
"In that model, asteroids immersed in the pulsar relativistic wind produce collimated magnetic wakes, that can be observed at cosmological distances and lead to radio bursts similar to the observed signals.",
"An asteroid belt could also be invoked to account for the repeating FRB reported by [35] (see also [10]).",
"We have assumed in this paper that the wind was Poynting-flux dominated at the location of the companion.",
"For objects made of ordinary matter, the consistency of our evaporation timescales compared to the observations seems to validate a posteriori this hypothesis.",
"If a non-negligible fraction of the wind energy were kinetic, a heating process by high-energy particle irradiation would have to be considered.",
"This process would evaporate most close-by companions, whatever their size, as has been calculated and observed for black widow and redback systems (e.g., [26], [19], [2]).",
"In this case, the energy absorption coefficient $Q_{\\rm kin}$ can be taken as $\\sim 1$ at first-order approximation.",
"A more thorough evaluation of $Q_{\\rm kin}$ would require calculating the cascading interactions of particles, which is beyond the scope of this study.",
"Models considering simultaneously inductive heating, sputtering by the wind, and blackbody X-rays will be developed in forthcoming papers.",
"From Eqs.",
"(REF ) and (REF ), one can then infer that in order to avoid heating by particles, the fraction of kinetic energy in the wind has to be lower than $Q_{\\rm em}$ , the absorption coefficient of the Poynting flux given by the Mie Theory (denoted $Q_{\\rm abs}$ by abuse of notation throughout this paper, see values in Fig.",
"REF and Table REF ).",
"This implies a magnetization of $\\sigma _B>1/Q_{\\rm em}-1$ .",
"As $Q_{\\rm em} \\ll 1$ over a large parameter space, as is demonstrated in this paper, a very low fraction of kinetic energy in the wind and thus a very high magnetization is required for the survival of objects close to the pulsar.",
"We note that we have also argued in this work that, because of their high refractive index to the pulsar wave, the observed ablation of white dwarfs and M-stars in back widow and redback systems can only be explained if the absorbed energy is principally kinetic."
],
[
"Acknowledgement",
"This work was nurtured by Jean Heyvaerts' notes on the Mie theory and by the inspiring discussions we had with him.",
"His great insight and input were very much missed during the completion of the project.",
"KK thanks S. Phinney for fruitful discussions that led to her involvement in this project.",
"This work was supported by the Programme National des Hautes Energies.",
"KK acknowledges financial support from the PER-SU fellowship at Sorbonne Universités, from the Labex ILP (reference ANR-10-LABX-63, ANR-11-IDEX-0004-02), as well as from the NSF grant NSF PHY-1412261 and the NASA grant 11-APRA-0066 at the University of Chicago, and the grant NSF PHY-1125897 at the Kavli Institute for Cosmological Physics."
]
] | 1605.05746 |
[
[
"Generalized Galileons: instabilities of bouncing and Genesis cosmologies\n and modified Genesis"
],
[
"Abstract We study spatially flat bouncing cosmologies and models with the early-time Genesis epoch in a popular class of generalized Galileon theories.",
"We ask whether there exist solutions of these types which are free of gradient and ghost instabilities.",
"We find that irrespectively of the forms of the Lagrangian functions, the bouncing models either are plagued with these instabilities or have singularities.",
"The same result holds for the original Genesis model and its variants in which the scale factor tends to a constant as $t\\to -\\infty$.",
"The result remains valid in theories with additional matter that obeys the Null Energy Condition and interacts with the Galileon only gravitationally.",
"We propose a modified Genesis model which evades our no-go argument and give an explicit example of healthy cosmology that connects the modified Genesis epoch with kination (the epoch still driven by the Galileon field, which is a conventional massless scalar field at that stage)."
],
[
"Introduction and summary",
"Bouncing and Genesis cosmologies are interesting scenarios alternative or complementary to inflation.",
"Both require the violation of the Null Energy ConditionAn exception is bounce of a closed Universe [1].",
"(NEC), and hence fairly unconventional matter.",
"Candidates for the latter are generalized Galileons [2], [3], [4], [5], [6], [7], [8], [9], scalar fields whose Lagrangians involve second derivatives, and whose field equations are nevertheless second order (for a review see, e.g., Ref. [10]).",
"Indeed, in the original Genesis model [11] as well as in its variants [12], [13], [14], [15], [16], [17], the initial super-accelerating stage can occur without ghosts and gradient instabilities (although there is still an issue of superluminality [18], [19]).",
"Likewise, bouncing Universe models with generalized Galileons can be arranged in such a way that no ghost or gradient instabilities occur at and near the bounce [20], [21], [22], [23].",
"The situation is not so bright in more complete cosmological models.",
"Known models of the bouncing Universe, employing generalized Galileons, are in fact plagued by the gradient instabilities, provided one follows the evolution for long enough time [24], [25], [26], [27], [28].",
"Gradient instabilities occur also in the known Genesis models, once one requires that the early Genesis regime turns into more conventional expansion (inflationary or not) at later times [16], [30], [31].",
"An intriguing exception is the model [16] in which Genesis-like super-accelerated expansion starts from the de Sitter, rather than Minkowski, epoch.",
"We comment on this model in Section .",
"One way to get around the gradient instability problem is to arrange the model in such a way that the quadratic in spatial gradients, wrong sign term in the action is small, and higher derivative terms restore stability at sufficiently high spatial momenta [16], [29].",
"There is also a possibility that the strong coupling momentum scale is low enough [28].",
"In both cases the exponential growth of trustworthy perturbations does not have catastrophic consequences, provided that the time interval at which the instability operates is short enough.",
"Another option is to introduce extra terms in the action which are not invariant under general coordinate transformations [27], [30].",
"Clearly, it is of interest to understand whether gradient or ghost instabilities are inherent in all “complete” bouncing models and Genesis models with initial Minkowski space, which are based on classical generalized Galileons and General Relativity, or these instabilities are merely drawbacks of concrete models constructed so far.",
"In the latter case, it is worth designing examples in which the gradient and ghost instabilities are absent.",
"It is this set of issues we address in this paper.",
"We consider the simplest and best studied generalized Galileon theory interacting with gravity.",
"The Lagrangian is (mostly negative signature; $\\kappa = 8\\pi G$ ) $L = - \\frac{1}{2\\kappa }R + F(\\pi , X) + K (\\pi , X) \\Box \\pi \\; ,$ where $\\pi $ is the Galileon field, $F$ and $K$ are smooth Lagrangian functions, and $X = \\nabla _\\mu \\pi \\nabla ^\\mu \\pi \\; , \\;\\;\\;\\; \\Box \\pi = \\nabla _\\mu \\nabla ^\\mu \\pi \\; .$ We also allow for other types of matter, assuming that they interact with the Galileon only gravitationally and obey the NEC: $\\rho _M + p_M \\ge 0 \\; .$ To see that our observations are valid in any dimensions, we study this theory in $(d+1)$ space-time dimensions with $d \\ge 3$ ; the case of interest is of course $d=3$ .",
"We consider spatially flat FLRW Universe with the scale factor $a(t)$ where $t$ is the cosmic time, and study spatially homogeneous backgrounds $\\pi (t)$ .",
"Our framework is quite general.",
"In the Genesis case we require that neither $a(t)$ nor $\\pi (t)$ has future singularity (i.e., $a(t)$ , $\\pi (t)$ and their derivatives are finite for all $-\\infty <t < +\\infty $ ).",
"Our definition of the bouncing Universe is that the scale factor $a(t)$ either is constant in the past and future, $a (t) \\rightarrow a_{\\mp }$ as $t \\rightarrow \\mp \\infty $ , or diverges in one or both of the asymptotics (i.e., $a_- =\\infty $ or/and $a_+ = \\infty $ ), and that there is no singularity in between.",
"Somewhat surprisingly, our results for the bouncing and Genesis scenarios are quite different.",
"In the bouncing Universe case, we show that the gradient (or ghost) instability is inevitable.",
"This result is a cosmological counterpart of the observation that a static, spherically symmetric Lorentzian wormhole supported by the generalized Galileon always has the ghost or gradient instability [32] (see also Ref.",
"[33]); the technicalities involved are also similar.",
"Analogous no-go theorem does not hold in the Genesis case.",
"Yet the requirement of the absence of the gradient and ghost instabilities strongly constrains the Galileon theories (i.e., Lagrangian functions $F$ and $K$ ).",
"In particular, the gradient or ghost instability (or future singularity) does exist, if the initial stage is the original Genesis [11] or its versions in which $a(t) \\rightarrow \\mbox{const}$ as $t \\rightarrow -\\infty $ , which is the case, e.g., in the subluminal Genesis [12] as well as in the DBI [13] and generalized Genesis [17] models in which the Lagrangians have the general formThe reservation here has to do with the fact that we merely leave more complicated models aside.",
"It is worth seeng whether our result can be generalized to all Horndeski-like Lagrangians.",
"(REF ) (in the language of Ref.",
"[13], the Lagrangians from this subclass do not contain terms ${\\cal L}_4$ and ${\\cal L}_5$ ).",
"Equipped with better understanding of the instabilities in the Genesis models with generalized Galileons, we propose a modified Genesis behavior in which the space-time curvature, energy and pressure vanish as $t \\rightarrow -\\infty $ and which is not inconsistent with the absence of the gradient and ghost instabilities and the absence of future singularity.",
"The pertinent Galileon Lagrangian is similar to ones considered in Refs.",
"[15], [17]; in particular, the action is not scale-invariant.",
"Starting from this Lagrangian, we give an example of a “complete” model, with Genesis at the initial stage and kination (the epoch still driven by the Galileon field which, however, is a conventional massless scalar field at that stage) at later times.",
"This model is free of the gradient instabilities, ghosts and superluminal propagation about the homogeneous solution, while the kination stage may possibly be connected to the radiation domination epoch via, e.g., gravitational particle creation, cf.",
"Ref. [34].",
"This paper is organized as follows.",
"In Section we discuss, in general terms, the conditions for the absence of the gradient and ghost instabilities in the cosmological setting.",
"We show in Section that irrespectively of the forms of the Lagrangian functions $F(\\pi , X)$ and $K(\\pi , X)$ , these conditions cannot be satisfied in the bouncing Universe scenario as well as in the Genesis models with time-independent past asymptotics of the scale factor.",
"We propose a modified Genesis model in Section , where we first study general properties and concrete example of early Genesis-like epoch which evades the no-go argument of Section , then give an explicit example of healthy model connecting Genesis and kination and, finally, briefly discuss a spectator field whose perturbations may serve as seeds of the adiabatic perturbations.",
"We conclude in Section .",
"For completeness, the general expressions for the Galileon energy-momentum tensor and quadratic Lagrangian of the Galileon perturbations are given in Appendix."
],
[
"Generalities",
"The general expression for the Galileon energy-momentum tensor is given in Appendix, eq.",
"(REF ).",
"In the cosmological context the energy density and pressure are $\\rho &= 2F_X X - F - K_\\pi X + 2d H K_X \\dot{\\pi }^3 \\; ,\\\\p &= F - 2 K_X X \\ddot{\\pi } - K_\\pi X \\; ,$ where $H$ is the Hubble parameter and $X = \\dot{\\pi }^2 \\; .$ Hereafter $F_\\pi = \\partial F/\\partial \\pi $ , $F_X = \\partial F/\\partial X$ , etc.",
"Our main concern is the Galileon perturbations $\\chi $ of high momentum and frequency.",
"The general expression for the effective quadratic Lagrangian for perturbations is again given in Appendix, eq.",
"(REF ).",
"For homogeneous background we obtain $L^{(2)} = A \\dot{\\chi }^2 - \\frac{1}{a^2} B (\\partial _i \\chi )^2 + \\dots $ where $A &= F_X + 2F_{XX}X - K_\\pi - K_{X \\pi } X + 2dH \\dot{\\pi } (K_X + K_{XX}X) + \\frac{2d}{d-1}\\kappa K_X^2 X^2 \\; , \\\\B &= F_X - K_\\pi + 2K_X \\ddot{\\pi } + K_{X\\pi }X + 2 K_{XX} X \\ddot{\\pi }+ 2(d-1)HK_X \\dot{\\pi } - \\frac{2(d-2)}{d-1} \\kappa K_X^2 X^2 \\; ,$ and terms omitted in (REF ) do not contain second derivatives of $\\chi $ .",
"These terms are irrelevant for high momentum modes.",
"The absence of ghosts and gradient instabilities requires $A>0$ , $B \\ge 0$ .",
"In particular, if $B<0$ , there are ghosts (for $A <0$ ) or gradient instability (for $A>0$ ).",
"Our focus is on the coefficient $B$ .",
"Despite appearance, it can be cast in a simple form.",
"To this end we make use of the Friedmann and covariant conservation equations $H^2 = \\frac{2}{d(d-1)} \\kappa (\\rho + \\rho _M) \\\\\\dot{\\rho } = - d\\cdot H(\\rho +p)\\\\\\dot{\\rho }_M = - d\\cdot H (\\rho _M + p_M)$ and hence $\\dot{H} = - \\frac{1}{d-1} \\kappa [(\\rho +p) + (\\rho _M + p_M)] \\; .$ Here $\\rho $ and $p$ are Galileon energy density and pressure, while $\\rho _M$ and $p_M$ are energy and pressure of conventional matter, if any.",
"The latter obey the NEC, eq.",
"(REF ).",
"We recall that we assume that conventional matter does not interact with the Galileon directly, so the covariant conservation equations (REF ) and () have to be satisfied separately.",
"Equations (), (REF ) and (REF ) lead to a remarkable relation $2BX = \\frac{d}{dt} \\left(2K_X \\dot{\\pi }^3 - \\frac{d-1}{\\kappa } H \\right)-\\frac{2(d-2)}{d-1} \\kappa K_X \\dot{\\pi }^3 \\left(2K_X \\dot{\\pi }^3 -\\frac{d-1}{\\kappa } H \\right)- (\\rho _M+p_M) \\; .$ It is natural to introduce a combination $Q = 2K_X \\dot{\\pi }^3 - \\frac{d-1}{\\kappa } H$ and write $2BX = \\dot{Q} -\\frac{2(d-2)}{d-1} \\kappa K_X \\dot{\\pi }^3 Q- (\\rho _M+p_M) \\; .$ Another representation is in terms of the function $R = \\frac{Q}{a^{d-2}} \\; ,$ namely $\\frac{2BX}{a^{d-2}} = \\dot{R} - \\frac{d-2}{d-1} \\kappa a^{d-2} R^2 -\\frac{\\rho _M + p_M}{a^{d-2}} \\; .$ Since we assume that the conventional matter, if any, obeys the NEC, the positivity of $B$ requires $\\dot{Q} -\\frac{2(d-2)}{d-1} \\kappa K_X \\dot{\\pi }^3 Q \\ge 0$ and $\\dot{R} - \\frac{d-2}{d-1} \\kappa a^{d-2} R^2 \\ge 0 \\; .$ As we now see, these requirements are prohibitively restrictive in the bouncing Universe case and place strong constraints on the Genesis models."
],
[
"Bouncing Universe and original Genesis: no-go",
"We now show that the inequality (REF ) cannot be satisfied in the bouncing Universe scenario.",
"We write it as follows, $\\frac{\\dot{R}}{R^2} \\ge \\frac{d-2}{d-1} \\kappa a^{d-2}$ and integrate from $t_i$ to $t_f > t_i$ : $\\frac{1}{R(t_i)} - \\frac{1}{R(t_f)} \\ge \\frac{d-2}{d-1} \\kappa \\int _{t_i}^{t_f}~dt~a^{d-2} \\; .$ Suppose now that $R(t_i) > 0$ .",
"Since $\\dot{R} >0$ in view of (REF ), $R$ increases in time and remains positive.",
"We have $\\frac{1}{R(t_f)} \\le \\frac{1}{R(t_i)} - \\frac{d-2}{d-1} \\kappa \\int _{t_i}^{t_f}~dt~a^{d-2} \\; .$ Since $a(t)$ is either a constant or growing function of $t$ at large $t$ , the right hand side of the latter inequality eventually becomes negative at large $t_f$ .",
"Thus $R^{-1} (t_f)$ as function of $t_f$ starts positive (at $t_f = t_i$ ) and necessarily crosses zero.",
"At that time $R^{-1} = 0$ , and $R=\\infty $ , which means a singularity.",
"A remaining possibility is that $R(t)$ is negative at all times.",
"In particular, $R(t_f) < 0$ .",
"In that case a useful form of the inequality (REF ) is $\\frac{1}{R(t_i)} \\ge \\frac{1}{R(t_f)} + \\frac{d-2}{d-1} \\kappa \\int _{t_i}^{t_f}~dt~a^{d-2} \\; .$ Now, $a(t)$ is either a constant or tends to infinity as $t \\rightarrow -\\infty $ , so the right hand side is positive at large negative $t_i$ .",
"Hence, there is again a singularity $R =\\infty $ at $t_i < t < t_f$ .",
"This completes the argument.",
"The same argument applies to the original Genesis model [11] and many of its versions, like subluminal Genesis [12] and the DBI Genesis [13], provided the Lagrangian has the general form (REF ).",
"In these versions, the scale factor tends to a constant as $t \\rightarrow -\\infty $ and, assuming that the Universe ends up in the conventional expansion regime, the scale factor grows at large times.",
"The integral in eq.",
"(REF ) blows up at large $t_f$ or large negative $t_i$ , so the inequalities (REF ), (REF ) are impossible to satisfy without hitting the singularityNote that if there is an initial singularity, there is no argument that would forbid $Q$ to be negative and increasing towards zero at early times, cross zero at some intermediate time and continue to increase later on.",
"This is what happens in the setup [14] where the NEC is satisfied at early times and is violated later on in the Genesis phase.",
"The inequality (REF ) shows, however, that in that case there is either gradient (or ghost) instabitity or future singularity after $Q$ crosses zero.",
"$R = \\infty $ .",
"In fact, in the models of Refs.",
"[11], [12], [13], one has $Q>0$ , which is consistent with healthy behavior at early times but implies either gradient (or ghost) instabitity or singularity in future.",
"At this point let us make contact with the model of Ref.",
"[16] in which the Genesis-like super-accelerated expansion starts from the de Sitter rather than Minkowski epoch, $d=3$ , $a \\propto {\\rm e}^{\\lambda t}$ .",
"In that case the integral in (REF ) is convergent as $t_i \\rightarrow -\\infty $ .",
"Hence, our argument does not work: one can have $R<0$ at all times, leaving a room for the stable evolution.",
"In fact, in the model of Ref.",
"[16], our parameter $Q$ defined in (REF ) is constant in time and negative, while $B>0$ in full accordance with (REF ).",
"We generalize this construction in Section ."
],
[
"Early-time evolution",
"In this Section we construct a model which interpolates between a stage similar to Genesis (in the sense that space-time curvature, energy and pressure vanish as $t\\rightarrow -\\infty $ ) and kination epoch at with the Galileon behaves as a conventional massless scalar field.",
"The model is purely classical and does not have gradient or ghost instability at any time.",
"We begin with the early Genesis-like stage, having in mind the observations made in Section .",
"Since we would like the scale factor to increase at late times, and in view of the inequality (REF ), we require that at the Genesis-like stage $R<0$ and hence $Q < 0 \\; .$ This means that $H > \\frac{2\\kappa }{d-1} K_X \\dot{\\pi }^3 \\; .$ Furthermore, the second term in the right hand side of eq.",
"(REF ) must be larger than $|\\dot{Q}|$ at the Genesis-like epoch: since $H$ increases at that epoch from originally zero value, so does $|Q|$ (barring cancelations), and we have $\\dot{Q} < 0$ .",
"Thus, besides the inequality (REF ) we require $\\frac{2(d-2)}{d-1}\\kappa K_X \\dot{\\pi }^3 |Q| > |\\dot{Q}| \\; .$ Assuming power law behavior of $Q$ , we see that the simplest option is that at large negative times $K_X \\dot{\\pi }^3 \\propto (-t)^{-1} \\; , \\;\\;\\;\\;\\;\\; t \\rightarrow - \\infty .$ From eq.",
"(REF ) we deduce that $H$ cannot rapidly tend to zero as $t \\rightarrow -\\infty $ ; we can only have $H = - \\frac{h}{t} \\; , \\;\\;\\;\\;\\; a(t) \\propto \\frac{1}{(-t)^{h}} \\;, \\;\\;\\;\\;\\;\\; h =\\mbox{const} \\; , \\;\\;\\;\\;\\;\\; t \\rightarrow - \\infty \\; ,$ in contrast to the conventional Genesis, in which $H \\propto (-t)^{-3}$ .",
"Thus both energy density and pressure should behave like $t^{-2}$ as $t \\rightarrow - \\infty $ (while in the originl Genesis one has $p\\propto t^{-4}$ , $\\rho \\propto t^{-6}$ ).",
"Now, the no-go argument based on (REF ) is not valid provided that the integral in the right hand side is convergent at the lower limit of integrationFor the case of interest $d=3$ (four space-time dimensions), eq.",
"(REF ) implies that space-time is past-incomplete in the sense that past-directed null geodesics reach spatial infinity $a(t) |{\\bf x}| = \\infty $ at finite value of the affine parameter (and past-directed time-like geodesics reach spatial infinity in finite proper time), cf.",
"Ref. [36].",
"We leave this issue open in this paper., $\\int _{-\\infty }^{t}~dt~a^{d-2} < \\infty \\; .$ Thus, we require that $h > \\frac{1}{d-2} \\; .$ These are the general properties of the Genesis-like stage which is potentially consistent with the overall healthy dynamics.",
"Note that unlike in the original Genesis scenario, the scale factor does not tend to a constant as $t \\rightarrow -\\infty $ .",
"Yet the geometry tends to Minkowski at large negative times in the sense that the space-time curvature tends to zero, and so do the energy density and pressure.",
"One way to realize this scenario is to choose the Lagrangian functions in the following form $F &= -f^2 (\\partial \\pi )^2 + \\alpha _0 {\\rm e}^{-2\\pi } (\\partial \\pi )^4 \\\\K &= \\beta _0 e^{-2\\pi } (\\partial \\pi )^2 \\; .$ The resulting Lagrangian is similar to those introduced in Ref.",
"[17], although the particular exponential dependence that we have in (REF ) was not considered there.",
"There is a solution ${\\rm e}^\\pi &= - \\frac{1}{H_* t} \\\\H &= - \\frac{h}{t} \\; , \\;\\;\\;\\;\\;\\;\\;\\;\\;\\; t<0 \\; ,$ with time-independent $H_*$ and $h$ ; we relate them to the parameters $\\alpha _0$ , $\\beta _0$ and $f$ shortly.",
"For this solution $Q = - \\frac{1}{t}\\left(2 \\beta _0 H_*^2 - \\frac{d-1}{\\kappa } h \\right)$ and $2BX = \\frac{1}{ t^2 } \\left(2 \\beta _0 H_*^2 - \\frac{d-1}{\\kappa } h \\right)\\left(1- 2\\frac{d-2}{d-1} \\kappa \\beta _0 H_*^2 \\right)\\; .$ Thus, one has $Q<0$ and no gradient instability at early times ($B>0$ ), provided that $\\frac{d-1}{2\\kappa (d-2)} <\\beta _0 H_*^2 < \\frac{d-1}{2\\kappa }h \\; .$ Note that with this choice of parameters, the inequality (REF ) is satisfied, as it should.",
"The parameters $H_*$ and $h$ are related to the parameters of the Lagrangian via the field equations ().",
"Two independent combinations of these equations are $2 \\alpha _0 H_*^2 &= \\frac{d(d-1)}{\\kappa } h^2 + \\frac{(d-1)}{\\kappa } h - 2\\beta _0 H_*^2 - 2d h \\beta _0 H_*^2 \\\\f^2 &= \\frac{d(d-1)}{\\kappa } h^2 + \\frac{3(d-1)}{2\\kappa } h - \\beta _0H_*^2 - d h \\beta _0 H_*^2$ Using these relations and inequalities (REF ) one can check that $f^2 > 0$ and, importantly, the coefficient of the kinetic term in the Lagrangian for perturbations is positive, $A >0$ .",
"The modified Genesis regime is stable."
],
[
"From Genesis to kination: an example",
"A scenario for further evolution is as follows.",
"The function $\\pi (t)$ is monotonous, $\\dot{\\pi } > 0$ .",
"However, the Lagrangian functions $F$ and $K$ depend on $\\pi $ and hence on time in a non-trivial way.",
"The variable $Q$ remains negative at all times, but eventually (at $t=t_c$ ) $\\dot{Q}$ changes sign and $Q$ starts to increase towards zero.",
"Choosing $K_X > 0$ , we find that at $t>t_c$ the stability condition $B>0$ is satisfied trivially, $2BX = \\dot{Q} - \\frac{2(d-2)}{d-1} \\kappa K_X \\dot{\\pi }^3 Q > 0 \\; ,\\;\\;\\;\\;\\; t>t_c$ One should make sure, however, that at $t<t_c$ the inequality (REF ) is always satisfied.",
"Another point to check is that the coefficient of the kinetic term in the Lagrangian for perturbations is positive, $A >0$ , at all times.",
"To cook up a concrete example of a model in which the Genesis regime is smoothly connected to kination (the regime at which Galileon is a conventional massless scalar field dominating the cosmological expansion), the simplest way is to introduce a field $\\phi = \\phi (\\pi )$ in such a way that the solution is $\\phi = t \\; .$ The general formulas of Sections , remain valid, with understanding that the Lagrangian functions are now functions of $\\phi $ and $X =(\\partial \\phi )^2$ ; in particular, the combination $Q$ is the same as in (REF ) with $\\phi $ substituted for $\\pi $ .",
"We choose the Lagrangian functions in the following form: $F &= - v(\\phi ) X + \\alpha (\\phi ) X^2 \\; , \\\\K &= \\beta (\\phi ) X \\; .$ On the solution (REF ) one has $F= -v(t) + \\alpha (t) \\; , \\;\\;\\;\\;\\; F_X = -v (t) + 2\\alpha (t) \\; ,\\;\\;\\;\\;\\; K=K_X = \\beta (t) \\; .$ One way to proceed is to postulate suitable forms of $Q(t)<0$ and $\\beta (t)$ , such that the inequality (REF ) is satisfied, evaluate $H = \\frac{\\kappa }{d-1} (2\\beta -Q)$ , reconstruct $v(t)$ and $\\alpha (t)$ from the field equations and then check that $A>0$ at all times.",
"With the convention (REF ), once $v(t)$ , $\\alpha (t)$ and $\\beta (t)$ are known, the Lagrangian functions are also known, $v(\\phi ) = v(t=\\phi )$ , etc.",
"As we anticipated in eqs.",
"(REF ), (REF ), the initial behavior is $Q &= - \\frac{\\hat{q}}{t} \\; , \\;\\;\\;\\;\\;\\; \\hat{q} < 0 \\\\\\beta &= - \\frac{\\hat{\\beta }}{t} \\; , \\;\\;\\;\\;\\;\\;\\;\\; t \\rightarrow -\\infty $ with time-independent $\\hat{q}$ and $\\hat{\\beta }$ .",
"To satisfy the inequality (REF ), we impose the condition $\\hat{\\beta }> \\frac{d-1}{2\\kappa (d-2)} \\; .$ We would like the Galileon to become a conventional massless scalar field at large positive $t$ , whose equation of state is $p=\\rho $ , and require that at large $t$ the function $\\beta (t)$ rapidly vanishes, while $H=(d\\cdot t)^{-1}$ , and hence $Q = - \\frac{d-1}{d\\kappa t} \\; , \\;\\;\\;\\;\\;\\;\\;\\; t \\rightarrow +\\infty \\; .$ It is convenient to introduce rescaled variables $Q &= - \\frac{{d-1}}{d\\kappa } P \\; , \\\\\\beta &= \\frac{{d-1}}{d\\kappa } b \\; ,$ where $P$ is positive.",
"In terms of these variables the Hubble parameter is $H = \\frac{\\kappa }{d-1} (2 K_X \\dot{\\phi }^3 -Q)= \\frac{1}{d}(2b+P) \\; .$ The combinations of the field equations, analogous to eqs.",
"(REF ), give $\\alpha &= - dH\\beta + \\frac{d(d-1)}{2\\kappa } H^2 + \\frac{d-1}{2\\kappa }\\dot{H} \\nonumber \\\\&= \\frac{d-1}{d\\kappa } \\left(bP + \\frac{1}{2} P^2 + \\dot{b} +\\frac{1}{2}\\dot{P}\\right)\\; ,\\\\v &= -\\dot{\\beta } - dH\\beta + \\frac{d(d-1)}{\\kappa } H^2 +\\frac{3(d-1)}{2\\kappa } \\dot{H} \\nonumber \\\\&= \\frac{{d-1}}{d\\kappa } \\left(2b^2 + 3 bP +P^2 + 2\\dot{b} + \\frac{3}{2}\\dot{P} \\right)\\; .$ Finally, the coefficients in the Lagrangian for perturbations are $A &= \\frac{{d-1}}{d\\kappa } \\left(4b^2 + 5bP + 2P^2 + 2\\dot{b} + \\frac{3}{2}\\dot{P} \\right)\\; , \\\\B &= \\frac{{d-1}}{d\\kappa } \\left(\\frac{d-2}{d} bP - \\frac{1}{2} \\dot{P} \\right)\\; .$ The asymptotics of the solution should be $t \\rightarrow -\\infty \\;: \\;\\;\\;\\;\\; P &= - \\frac{p_0}{t} \\; , \\;\\;\\;\\; p_0 >0\\; , \\nonumber \\\\b &= - \\frac{b_0}{t} \\; , \\;\\;\\;\\; b_0 > \\frac{d}{2(d-2)} \\; ,\\\\t \\rightarrow +\\infty \\;: \\;\\;\\;\\;\\; P &= \\frac{1}{t} \\; , \\nonumber \\\\b &= 0 \\; .$ As a cross check, the late-time asymptotics () imply $\\alpha = 0$ and $v = - \\frac{d-1}{2d\\kappa } \\cdot \\frac{1}{ t^2} < 0 \\; ,$ which corresponds to the Lagrangian of free massless scalar field, albeit written in somewhat unconventional form, $L_\\phi ^{t \\rightarrow +\\infty } = \\frac{d-1}{2d\\kappa } \\frac{(\\partial \\phi )^2}{\\phi ^2} \\; .$ On the other hand, the early-time asymptotics (REF ), according to eq.",
"(REF ), give $\\alpha = \\frac{c_1}{t^2} \\; , \\;\\;\\;\\;\\;\\; c_1 > 0 \\; , \\\\v = \\frac{c_2}{t^2} \\; , \\;\\;\\;\\;\\;\\; c_2 > 0 \\; ,$ so that the Lagrangian at early times ($\\phi \\rightarrow -\\infty $ ) reads $L_\\phi ^{t \\rightarrow -\\infty } = - \\frac{c_1}{\\phi ^2} (\\partial \\phi )^2 +\\frac{c_2}{\\phi ^2} (\\partial \\phi )^4 - \\frac{\\hat{\\beta }}{\\phi } (\\partial \\phi )^2 \\Box \\phi \\; .$ Upon introducing new field at early times $\\pi = - \\mathop {\\rm ln}\\nolimits (- \\phi ) \\; ,$ one writes the Lagrangian in the following form $L_\\pi ^{t \\rightarrow -\\infty } = - c_1 (\\partial \\pi )^2 + (c_2 - \\hat{\\beta }){\\rm e}^{-2\\pi } (\\partial \\pi )^4 + \\hat{\\beta } {\\rm e}^{-2\\pi } (\\partial \\pi )^2 \\Box \\pi \\; .$ This is precisely the form (REF ).",
"One more point to note is that if the parametric form of $b (t)$ and $P(t)$ is $b = \\tau ^{-1} \\tilde{b}(t/\\tau )\\; , \\;\\;\\;\\;\\;\\; P =\\tau ^{-1}\\tilde{P}(t/\\tau ) \\; ,$ with $\\tilde{b}$ and $\\tilde{P}$ of order 1 (which is consistent with the asymptotics (REF )), then for large $\\tau $ the dynamics is sub-Planckian during entire evolution: in that case one has sub-Planckian $H \\sim \\tau ^{-1}$ , $\\alpha \\sim \\kappa ^{-1} \\tau ^{-2}$ , etc.",
"A random example is $P &= \\frac{1}{\\sqrt{t^2 + \\tau ^2}} \\; , \\\\b &= \\frac{b_0}{\\sqrt{t^2 + \\tau ^2}} \\cdot \\frac{1}{2}\\left[ 1 -\\mathop {\\rm th}\\nolimits \\left(\\mu \\frac{t}{\\tau } \\right)\\right] \\; .$ With $d=3 \\; , \\;\\;\\;\\; b_0=2 \\; , \\;\\;\\;\\; \\mu =1.1$ ($(3+1)$ -dimensional space-time) the system smoothly evolves from the modified Genesis regime to kination (free massless scalar field) regime with $A(t) > 0$ and $B(t) > 0$ for all $t$ , and also with subluminal propagation of perturbations about this background (the latter property explains the choice $\\mu =1.1$ : for $\\mu =1$ , say, there is a brief time interval in which the perturbations propagate superluminally).",
"The properties of this model are illustrated in Figs.",
"REF – REF .",
"It is worth noting that although $A(t)$ and $B(t)$ at late times appear different in Fig.",
"REF , they are actually the same, $A(t)= B(t) = \\frac{1}{3\\kappa t^2} \\; , \\;\\;\\;\\;\\; t \\rightarrow +\\infty \\; .$ The fact that these functions tend to zero as $t \\rightarrow +\\infty $ does not indicate the onset of strong coupling; this behavior rather has to do with the field redefinition from the canonical massless scalar field to the field $\\phi $ described by the Lagrangian (REF ).",
"The late-time theory is the theory of free massless scalar field, with no strong coupling or instabilities.",
"Figure: Hubble parameter in units of τ -1 \\tau ^{-1} as function of t/τt/\\tau for the model of eqs.",
"(), ().Figure: Left: the function BB in units 2 3κτ -2 \\frac{2}{3\\kappa }\\tau ^{-2} asfunction of t/τt/\\tau for the model of eqs. (),().",
"Right: same for the function AA.",
"Note the largerscale as compared to the left panel, which implies small sound speed atearly times.Figure: Sound speed squared, B/AB/A, as function of t/τt/\\tau for the modelof eqs.",
"(), ().Figure: Left: the function α\\alpha in units 2 3κτ -2 \\frac{2}{3\\kappa }\\tau ^{-2}as function of t/τt/\\tau for the model of eqs. (),().",
"Note that α\\alpha rapidly vanishes at late times.Right: the same for the function vv.",
"Negative sign of vv at late timescorresponds to conventional sign of the scalar kinetic term."
],
[
"Curvaton",
"It is unlikely that the Galileon perturbations would produce adiabatic perturbations with nearly flat power spectrum.",
"Like in Ref.",
"[11], the adiabatic perturbations may originate from perturbations of an additional scalar field, “curvaton”.",
"Let us consider this point, specifying to 4-dimensional space-time, $d=3$ .",
"We are interested in the early modified Genesis stage when the Galileon Lagrangian functions have the form (REF ).",
"The modified Galileon action is not invariant under the scale transformations $\\pi (x) \\rightarrow \\hat{\\pi } (x) = \\pi (\\lambda x) + \\mathop {\\rm ln}\\nolimits \\lambda \\;,\\;\\;\\;\\;\\; g_{\\mu \\nu } (x) \\rightarrow \\hat{g}_{\\mu \\nu } = g_{\\mu \\nu } (\\lambda x) \\;.$ One has instead $S(\\hat{\\pi } , \\hat{g}_{\\mu \\nu }) = \\lambda ^{-2} S(\\pi ,g_{\\mu \\nu })$ , just like for the Einstein–Hilbert action.",
"Let us introduce a spectator curvaton field $\\theta $ which is invariant under the scale transformations, $\\theta (x) \\rightarrow \\hat{\\theta } (x) = \\theta (\\lambda x)$ , and require that its action be scale-invariant.",
"This requirement gives $S_\\theta = \\int ~d^4x~\\sqrt{-g} {\\rm e}^{2\\pi } (\\partial \\theta )^2 \\; .$ In the backround (REF ) this action reads $S_\\theta = \\int ~dt d^3x~a^3 \\left(\\frac{1}{(H_* t)^2} \\left(\\frac{\\partial \\theta }{\\partial t} \\right)^2 - \\frac{1}{(H_* t)^2} \\frac{1}{a^2} \\left(\\frac{\\partial \\theta }{\\partial x_i} \\right)^2 \\right)\\; ,$ where the scale factor is given by (REF ).",
"Let us introduce conformal time $\\eta = \\int ~\\frac{dt}{a(t)} = - \\frac{1}{a_0 (h+1)} (-t)^{h+1} \\; .$ Then the action for $\\theta $ has the form $S_{\\theta } = \\int ~d\\eta d^3 x ~a_{eff}^2 (\\eta ) \\left[ \\left(\\frac{\\partial \\theta }{\\partial \\eta } \\right)^2 - \\left(\\frac{\\partial \\theta }{\\partial x_i} \\right)^2 \\right] \\; ,$ where $a_{eff}(\\eta ) = - \\frac{1}{H_* (h+1) \\eta } \\; ,$ which is precisely the action of the massless scalar field in de Sitter space-time.",
"We immediately deduce that the power spectrum of perturbations $\\delta \\theta $ generated at the modfied Genesis epoch is flat.",
"This is a pre-requisite for the nearly flat power spectrum of adiabatic perturbations, which may be generated from the curvaton perturbations at later stage."
],
[
"Conclusion",
"We have seen in this paper that with generalized Galileons, it is possible to construct healthy Genesis-like cosmologies, albeit with somewhat different properties as compared to the original Genesis scenario.",
"On the other hand, bouncing cosmologies with generalized Galileons are plagued by the gradient (or ghost) instability, at least at the level of second derivative Lagrangians of the form (REF ).",
"This is not particularly surprising.",
"The theory appears to protect itself [32] from having stable wormhole solutions which can be converted into time machines [35].",
"Technically the same protection mechanism, with radial coordinate and time interchanged, forbids the existence of spatially flat bouncing cosmologies.",
"It would be interesting to understand how general are these features.",
"The authors are indebted to R. Kolevatov, A. Sosnovikov, A. Starobinsky and A. Vikman for useful discussions and J.-L. Lehners and D. Pirtskhalava for helpful correspondence.",
"This work has been supported by Russian Science Foundation grant 14-22-00161."
],
[
"Galileon and its perturbations",
"The Galileon energy-momentum tensor in the theory with the Lagrangian (REF ) reads $T_{\\mu \\nu } = 2 F_X \\partial _\\mu \\pi \\partial _\\nu \\pi + 2 K_X \\Box \\pi \\cdot \\partial _\\mu \\pi \\partial _\\nu \\pi - \\partial _\\mu K \\partial _\\nu \\pi - \\partial _\\nu K \\partial _\\mu \\pi - g_{\\mu \\nu } F + g_{\\mu \\nu } g^{\\lambda \\rho } \\partial _\\lambda K \\partial _\\rho \\pi \\; ,$ We now consider Galileon perturbations, write $\\pi = \\pi _c + \\chi $ , and omit subscript $c$ in what follows.",
"We are interested in high momentum and frequency modes, so we concentrate on terms involving $\\nabla _\\mu \\chi \\nabla _\\nu \\chi $ in the quadratic effective Largangian or, equivalently, second order terms in the linearized field equation.",
"A subtlety here is that the Galileon field equation involves the second derivatives of metric, and the Einstein equations involve the second derivatives of the Galileon [7], and so do the linearized equations for perturbations.",
"The trick is to integrate the metric perturbations out of the Galileon field equation by making use of the Einstein equations [7].",
"In the cosmological setting this is equivalent to the approach adopted in Ref. [8].",
"To derive the quadratic Lagrangian for the Galileon perturbations, one writes the full Galileon field equation $\\left( -2F_X + 2 K_\\pi - 2K_{X \\pi } \\nabla _\\mu \\pi \\nabla ^\\mu \\pi - 2K_X \\Box \\pi \\right) \\Box \\pi + \\left(-4 F_{XX} + 4 K_{X \\pi } \\right)\\nabla ^\\mu \\pi \\nabla ^\\nu \\pi \\nabla _\\mu \\nabla _\\nu \\pi & \\nonumber \\\\- 4 K_{XX} \\nabla ^\\mu \\pi \\nabla ^\\nu \\pi \\nabla _\\mu \\nabla _\\nu \\pi \\Box \\pi + 4 K_{XX} \\nabla ^\\nu \\pi \\nabla ^\\lambda \\pi \\nabla _\\mu \\nabla _\\nu \\pi \\nabla ^\\mu \\nabla _\\lambda \\pi + 2 K_X \\nabla ^\\mu \\nabla ^\\nu \\pi \\nabla _\\mu \\nabla _\\nu \\pi & \\nonumber \\\\+ 2 K_X R_{\\mu \\nu } \\nabla ^\\mu \\pi \\nabla ^\\nu \\pi + \\ldots = 0 \\; ; &$ hereafter dots denote terms without second derivatives.",
"The subtle term is the last one here.",
"The linearized equation can be written in the following form $-2 [F_X + K_X \\Box \\pi - K_\\pi + \\nabla _\\nu (K_X \\nabla ^\\nu \\pi )]\\nabla _\\mu \\nabla ^\\mu \\chi & \\nonumber \\\\-2 [2 (F_{XX} + K_{XX} \\Box \\pi ) \\nabla ^\\mu \\pi \\nabla ^\\nu \\pi -2(\\nabla ^\\mu K_X) \\nabla ^\\nu \\pi - 2K_X \\nabla ^\\mu \\nabla ^\\nu \\pi ]\\nabla _\\mu \\nabla _\\nu \\chi & \\nonumber \\\\+ 2 K_X R_{\\mu \\nu }^{(1)} \\nabla ^\\mu \\pi \\nabla ^\\nu \\pi + \\ldots &= 0\\; ,$ where the terms without the second derivatives of $\\chi $ are omitted, and $R_{\\mu \\nu }^{(1)}$ is linear in metric perturbations.",
"We now make use of the Einstein equations $R_{\\mu \\nu } - \\frac{1}{2} g_{\\mu \\nu }R = \\kappa T_{\\mu \\nu }$ , or $R_{\\mu \\nu } = \\kappa \\left(T_{\\mu \\nu } - \\frac{1}{d-1} g_{\\mu \\nu }T^\\lambda _\\lambda \\right)\\; ,$ linearize the energy-momentum tensor and obtain for the last term in eq.",
"(REF ) $2 K_X R_{\\mu \\nu }^{(1)} \\nabla ^\\mu \\pi \\nabla ^\\nu \\pi = - 2 \\kappa K_X^2 \\left[-\\frac{2(d-2)}{d-1} X^2 \\Box \\chi + 4X \\nabla ^\\mu \\pi \\nabla ^\\nu \\pi \\nabla _\\mu \\nabla _\\nu \\chi \\right] + \\ldots \\; .\\nonumber $ The resulting linearized Galileon field equation is obtained from the following quadratic Lagrangian: $L^{(2)} &= [F_X + K_X \\Box \\pi - K_\\pi + \\nabla _\\nu (K_X \\nabla ^\\nu \\pi )]\\nabla _\\mu \\chi \\nabla ^\\mu \\chi \\nonumber \\\\&+ [2 (F_{XX} + K_{XX} \\Box \\pi ) \\nabla ^\\mu \\pi \\nabla ^\\nu \\pi -2(\\nabla ^\\mu K_X) \\nabla ^\\nu \\pi - 2K_X \\nabla ^\\mu \\nabla ^\\nu \\pi ]\\nabla _\\mu \\chi \\nabla _\\nu \\chi \\nonumber \\\\& -\\frac{2(d-2)}{d-1} \\kappa K_X^2 X^2 \\nabla _\\mu \\chi \\nabla ^\\mu \\chi + 4\\kappa K_X^2 X \\nabla ^\\mu \\pi \\nabla ^\\nu \\pi \\nabla _\\mu \\chi \\nabla _\\nu \\chi \\; .$ We specify to spatially homogeneous Galileon background in Section ."
]
] | 1605.05992 |
[
[
"Conditional analysis for mixed covariates, with application to feed\n intake of lactating sows"
],
[
"Abstract We propose a novel modeling framework to study the effect of covariates of various types on the conditional distribution of the response.",
"The methodology accommodates flexible model structure, allows for joint estimation of the quantiles at all levels, and involves a computationally efficient estimation algorithm.",
"Extensive numerical investigation confirms good performance of the proposed method.",
"The methodology is motivated by and applied to a lactating sow study, where the primary interest is to understand how the dynamic change of minute-by-minute temperature in the farrowing rooms within a day (functional covariate) is associated with low quantiles of feed intake of lactating sows, while accounting for other sow-specific information (vector covariate)."
],
[
"Introduction",
"Many modern applications routinely collect data on study participants comprising scalar responses and both [vector and data streams] information and the main question of interest is to examine how the covariates affect the response.",
"For example in our data application the aim is to study how the daily temperature or humidity behavior in the farrowing rooms (rooms where piglets are born and nursed by the sow until they are weaned) affect the feed intake of lactating sows during their first 21 lactation days, while accounting for other sow-specific information.",
"A popular approach in these cases is to use a semi-parametric framework and assume that the mixed covariates solely affect the mean response; see [7], [37], [21], [15], [16], [19], [30] and others.",
"While it is still important to study the average feed intake, animal scientists are often more concerned with the left tail of the feed intake distribution because low feed intake of lactating sows could lead to many serious issues, including decrease in milk production and negative impact on the sows reproductive system; see, for reference, [34], [38], [44] among others.",
"In this paper we relax the mean dependence assumption and consider that the covariates affect the entire conditional distribution of the response.",
"Our primary objective is to develop a modeling framework for a comprehensive study of mixed covariates on scalar response; the sow data application includes daily measured temperature and humidity recorded at five minute intervals (functional type) as well as sow-specific information (vector type).",
"Quantile regression models the effect of scalar/vector covariates beyond the mean response and has attracted great interest [25], [24].",
"This approach offers a more comprehensive picture of the effects of the covariates on the response distribution.",
"For pre-specified quantile levels, quantile regression models the conditional quantiles of the response as a function of the observed covariates; these approaches have been extended more recently to ensure non-crossing of quantile functions [5].",
"Quantile regression has been also extended to handle functional covariates.",
"[6] discussed quantile regression models by employing a smoothing spline modeling-based approach.",
"[23] considered the same problem and used a functional principal component (fPC) based approach.",
"Both papers mainly discussed the case of having a single functional covariate and it is not clear how to extend them to the case where there are multiple functional covariates or mixed covariates (vector and functional).",
"[14] and [8] considered a different perspective and studied the effect of a functional predictor on the quantiles of the response by first positing a model for the conditional distribution of the response and then inverting it; this approach is appealing as it jointly estimate quantiles for all the desired levels.",
"More recently, [45], [27], and [52] studied quantile regression when the covariates are of mixed types and introduced the partial functional linear quantile regression model framework.",
"The first two papers used fPC basis while the last one proposed to use partial quantile regression (PQR) basis.",
"While such approach is helpful when we are interested in studying the effect of covariate at particular quantile level, it provides an incomplete picture if we are interested in the effect at several quantile levels due to the well-known crossing-issue.",
"In this paper we propose a comprehensive study of the effect of vector and functional covariates on the distribution of the response.",
"Our approach is inspired from [8] (CM, henceforth).",
"Specifically let $Q_{Y|X}(\\tau |X)$ denote the $\\tau $ th conditional quantile of $Y$ given a functional covariate $X(\\cdot )$ , and let $F_{Y|X}(y)$ denote the conditional distribution of $Y$ given $X$ .",
"Using the relationship between $Q_{Y|X}(\\tau |X)$ and $F_{Y|X}(y)$ that $Q_{Y|X}(\\tau |X) = \\text{inf}\\lbrace y: F_{Y|X}(y) \\ge \\tau \\rbrace $ for $0 < \\tau < 1 $ , CM proposed to estimate the quantile function $Q_{Y|X}(\\tau |X)$ in two steps: 1) estimate the conditional distribution of $Y$ given $X(\\cdot )$ , $F(y) = E[\\mathbb {1}(Y < y) | X(\\cdot )]$ by positing a mean regression model for an auxiliary variable $Z(y):= \\mathbb {1}(Y \\le y)$ and the functional covariate $X(\\cdot )$ ; and 2) estimate $Q_{Y|X}(\\tau |X)$ by inverting the estimated conditional distribution function.",
"Their estimation approach is restrictive to one functional covariate and a direct extension to accommodate mixed covariates is computationally expensive.",
"We consider a similar idea and propose a modeling framework and estimation technique that easily accommodate various types of covariates in a computationally efficient manner.",
"This development represents the main contribution of our manuscript.",
"Let $X_1$ be a scalar covariate and $X_2(\\cdot )$ be a functional covariate defined on a closed domain $\\mathcal {T}$ .",
"We propose the following model for the conditional distribution of $Y$ given $X_1$ and $X_2(\\cdot )$ : $E[\\mathbb {1}(Y < y)] = g^{-1}\\big \\lbrace \\beta _0(y) + X_{1}\\beta _1(y) + \\int X_{2}(t)\\beta _2(t,y) dt\\big \\rbrace ,$ where $g(\\cdot )$ is a known, monotone link function, $\\beta _1(y)$ is unknown and smooth function and $\\beta _2(t,y)$ is unknown and smooth bi-variate function over $y$ and $t$ .",
"The parameters $\\beta _1(y)$ and $\\beta _2(\\cdot , y)$ quantify the effect of the covariates $X_1$ and $X_2(\\cdot )$ respectively onto the distribution of the response.",
"The remainder of the paper is structured as follows.",
"Section discusses the details of the proposed method and Section performs a thorough simulation study evaluating the performance of the proposed method and competitors.",
"We apply the proposed method to analyze the sow data in Section .",
"We conclude the paper with a discussion in Section ."
],
[
"Modeling Framework",
"The proposed modeling and estimation method is discussed first for the case of a scalar covariate in Section REF ; Section REF considers the extension to the case of a functional covariate and then to the case of mixed covariates; Section REF further extends the method to handle sparse and noisy functional covariates.",
"We briefly discuss the monotonization of the estimated conditional distribution in Section REF ."
],
[
"Conditional distribution of the response given scalar covariate",
"First, we focus on the case of a scalar covariate $X$ .",
"Consider the data $\\lbrace (X_i, Y_i):i=1,\\ldots , n\\rbrace $ , where $X_i$ and $Y_i$ are independent realizations of real-valued scalar random variables $X$ and $Y$ , respectively.",
"Define $Z_i(y)=\\mathbb {1}(Y_i < y)$ for $y \\in \\mathbb {R}$ , where $\\mathbb {1}(\\cdot )$ is an indicator function; for each $y$ , we view $Z_i(y)$ as a binary-valued random variable that is independent and identically distributed as $Z(y) = \\mathbb {1}(Y<y)$ .",
"It follows that the conditional distribution function $F_{Y|X}(y) = \\text{E}[Z(y)|X]$ .",
"Here we propose to model the conditional distribution, $F_{Y|X}(y)$ , using a generalized function-on-scalar regression model [18] between the `artificial' functional response $Z_i(y)$ and the scalar covariate $X_i$ .",
"Specifically, for each $y \\in \\mathbb {R}$ , consider $\\text{E}[Z(y)|X] = g^{-1}\\big \\lbrace \\beta _0(y) + X\\beta _1(y) \\big \\rbrace ,$ where $g(\\cdot )$ is a known, monotonic link function, and $\\beta _0(\\cdot )$ and $\\beta _1(\\cdot )$ are unknown, smooth coefficient functions.",
"Here we use the logit function defined as $g(x) = \\log \\lbrace x/(1+x)\\rbrace $ .",
"It is noteworty to remark that if the slope parameter $\\beta _1(\\cdot )$ is null then the covariate $X$ has no effect on the distribution of the response $Y$ , which is equivalent to $X$ having no effect on any quantile level of $Y$ .",
"We model $\\beta _0(y)$ and $\\beta _1(y)$ by using pre-specified, truncated univariate basis, $\\lbrace B_{0,d}(\\cdot ): d=1,\\ldots ,\\kappa _0\\rbrace $ and $\\lbrace B_{1,d}(\\cdot ): d=1,\\ldots ,\\kappa _1\\rbrace $ : $\\beta _0(y) = \\sum _{d=1}^{\\kappa _0} B_{0,d}(y) \\theta _{0,d}$ and $\\beta _1(y) = \\sum _{d=1}^{\\kappa _1} B_{1,d}(y) \\theta _{1,d}$ , where $\\theta _{0,d}$ 's and $\\theta _{1,d}$ 's are unknown basis coefficients.",
"Then model (REF ) can be represented as the following generalized additive model $\\text{E}[Z(y)|X] = g^{-1}\\Big \\lbrace \\sum _{d=1}^{\\kappa _0} B_{0,d}(y) \\theta _{0,d} + \\sum _{d=1}^{\\kappa _1} B_{x,d}(y) \\theta _{1,d}\\Big \\rbrace ,$ where for convenience we use the notation $B_{x,d}(y) = X B_{1,d}(y)$ .",
"The general idea is to set the basis dimensions $\\kappa _0$ and $\\kappa _1$ to be sufficiently large to capture the complexity of the coefficient functions and control the smoothness of the estimator through roughness penalties $P_0(\\theta _0)$ and $P_1(\\theta _1)$ , where $\\theta _l$ , is a vector of all basis coefficients $\\lbrace \\theta _{l,d}: l=1,\\ldots ,D_l \\rbrace $ for $l=1,2$ .",
"This approach of using roughness penalties has been widely used; see, for example, [12], [41], [47], [46] among many others.",
"In the following, we detail the estimation algorithm.",
"Let $\\lbrace y_{j}: j = 1,\\ldots , J\\rbrace $ be a set of equi-spaced points in the range of the response variable, $Y_i$ 's.",
"For each $i$ and $j$ , we define $Z_{ij} = Z_{i}(y_j)= \\mathbb {1}(Y_i < y_j)$ ; it follows that conditional on $X_i$ , the $Z_{ij}$ are independently distributed as Bernoulli distribution with mean $(\\mu _{ij})$ , where $\\mu _{ij}$ is such that $g(\\mu _{ij}) = \\mathbf {B}^T_{0,j}\\theta _0 + \\mathbf {B}^T_{x,j}\\theta _1$ .",
"Here $\\mathbf {B}^T_{0,j}$ is a $\\kappa _0 \\times 1$ vector of $\\lbrace B_{0,d}(y_j):d=1,\\ldots ,\\kappa _0\\rbrace $ and $\\mathbf {B}^T_{x,j}$ is a $\\kappa _1 \\times 1$ vector of $\\lbrace X_iB_{1,d}(y_j):d=1,\\ldots ,\\kappa _1\\rbrace $ .",
"The basis coefficients, $\\theta _0$ and $\\theta _1$ , are estimated by maximizing the penalized log likelihood criterion, $2\\mathrm {log } \\mathcal {L}(\\theta _0,\\theta _1|\\lbrace Z_i(y_j), X_i: \\forall i, j \\rbrace ) - \\lambda _0P_0(\\theta _0) - \\lambda _1P_1(\\theta _1),$ where $\\mathcal {L}$ is the likelihood function of data $\\lbrace Z_{ij}: j = 1, \\cdots , J \\rbrace _i$ , $P_0(\\theta _0)$ and $P_1(\\theta _1)$ are penalties, and $\\lambda _0$ and $\\lambda _1$ are smoothing parameters.",
"We use quadratic penalties which penalize the size of the curvature of the estimated coefficient functions.",
"Let $P_0(\\theta _0) = \\theta ^T_0 \\mathbf {D}_0 \\theta _0$ and $P_1(\\theta _1) = \\theta ^T_1 \\mathbf {D}_1 \\theta _1$ , where $\\mathbf {D}_0$ and $\\mathbf {D}_1$ are $\\kappa _0 \\times \\kappa _0$ and $\\kappa _1 \\times \\kappa _1$ dimensional matrices based on the basis used (see [46] for example; the $(s,s^{\\prime })$ element of $\\mathbf {D}_0$ is $\\int B^{\\prime \\prime }_{0,s}(y)B^{\\prime \\prime }_{0,s^{\\prime }}(y)dy$ ) and $\\mathbf {D}_1$ is defined similarly.",
"The smoothing parameters $\\lambda _0$ and $\\lambda _1$ control the trade-off between the goodness of fit and smoothness of the fit.",
"The smoothing parameters are estimated using restricted maximum likelihood (REML).",
"The criterion (REF ) can be viewed as the penalized quasi-likelihood (PQL) of the corresponding generalized linear mixed model $ Z_{ij}|\\theta _0,\\theta _1 \\sim \\text{Bernoulli}(\\mu _{ij}); \\qquad \\theta _0 \\sim N\\Big ( \\textbf {0}, \\lambda _0^{-1} \\mathbf {D}^{-}_0 \\Big );\\qquad \\theta _1 \\sim N\\Big ( \\textbf {0}, \\lambda _1^{-1} \\mathbf {D}^{-}_1 \\Big ),$ where $\\mathbf {D}^{-}_0$ is the generalized inverse matrix of $\\mathbf {D}_0$ and $\\mathbf {D}^{-}_1$ is defined similarly.",
"[46] discusses an alternative way to deal with the rank-deficient matrices, $\\mathbf {D}_0$ and $\\mathbf {D}_1$ , in the context of restricted maximum likelihood (REML) estimation.",
"See also [20] who uses the mixed model representation of a similar regression model to (REF ), but with a Gaussian functional response.",
"Let $\\lbrace \\widehat{\\theta }_{l,d}:d=1,\\ldots , \\kappa _l \\rbrace $ for $l=0,1$ be the estimated basis coefficients.",
"It follows that the estimated distribution function is $\\widehat{F}_{Y|X}(y)= g^{-1}\\lbrace \\sum _{d=1}^{\\kappa _0} B_{0,d}(y) \\widehat{\\theta }_{0,d} + \\sum _{d=1}^{\\kappa _1} B_{x,d}(y) \\widehat{\\theta }_{1,d}\\rbrace $ .",
"The $\\tau $ th conditional quantile are estimated as by inverting the estimated distribution $\\widehat{F}_{Y|X}(\\cdot )$ , $\\widehat{Q}_{Y|X}(\\tau |X) = \\underset{j}{\\text{min}}\\lbrace y_j: \\widehat{F}_{Y|X}(y_j) \\ge \\tau \\rbrace $ .",
"This approach relates the $\\tau $ th level quantile of the response in a nonlinear manner to the covariate.",
"One should note that the estimated distribution function, $\\widehat{F}_{Y|X}(y)$ , here is not a monotonic function yet.",
"However in practice one can always obtain $\\widehat{F}_{Y|X}(y)$ using a monotonization method as described in Section REF first and then invert the resulting estimated distribution to get the estimated conditional quantiles."
],
[
"Extension to mixed covariates",
"The modeling approach discussed in Section REF is quite powerful as it can accommodate covariates of various types, including functional covariates.",
"Assume that a functional covariate $X_i(\\cdot )$ is observed.",
"For convenience, we assume that $X_i(\\cdot )$ is observed at fine grid of points and without noise after all.",
"Instead of generalized function-on-scalar model as is used in Section REF , now we use generalized function-on-function regression model $ \\text{E}[Z(y)|X] = g^{-1}\\big \\lbrace \\beta _0(y) + \\int X(t)\\beta _1(t,y)dt \\big \\rbrace $ , where $\\beta _1(\\cdot ,\\cdot )$ is an unknown bi-variate coefficient function.",
"In terms of modeling and estimation, the main difference from Section REF is that the coefficient function, $\\beta _1(t,y)$ , is now bivariate and it requires appropriate pre-specified basis function and corresponding penalty term.",
"We represent $\\beta _{1}(t,y)$ using the tensor product of two univariate bases functions, $\\lbrace B^{t}_{1,d_t}(t): d_t = 1, \\ldots , \\kappa _{1,t} \\rbrace $ and $\\lbrace B^{y}_{1,d_y}(y): d_y = 1, \\ldots , \\kappa _{1,y}\\rbrace $ ; $\\beta _{1}(t,y) = \\sum _{d_t=1}^{\\kappa _{1,t}} \\sum _{d_y=1}^{\\kappa _{1,y}} $ $ B^{t}_{1,d_t}(t)B^{y}_{1,d_y}(y) \\theta _{1,d_t, d_y}$ .",
"Subsequently the previous penalty matrix $\\mathbf {D}_1$ should be also appropriately modified to control the smoothness of $\\beta _1(t,y)$ in directions of both $t$ and $y$ .",
"There are several choices to define the penalty matrix in nonparametric regression (see [28], [49]).",
"For bivariate smoothing we use $\\mathbf {D}_1 = \\lbrace \\mathbf {P}_t \\otimes \\mathbf {I}_{\\kappa _{1,y}} + \\mathbf {I}_{\\kappa _{1,t}} \\otimes \\mathbf {P}_y \\rbrace $ , where $(s,s^{\\prime })$ element of $\\mathbf {P}_t$ is $\\int \\lbrace \\partial ^2 X(t)B^{t}_{1,s}(t) / \\partial t^2\\rbrace \\lbrace \\partial ^2 X(t)B^{t}_{1,s^{\\prime }}(t) / \\partial t^2\\rbrace dt$ and $(s,s^{\\prime })$ element of $\\mathbf {P}_y$ is $\\int \\lbrace \\partial ^2 B^{y}_{1,s}(y) / \\partial y^2 \\rbrace \\lbrace \\partial ^2 B^{y}_{1,s^{\\prime }}(y) / \\partial y^2\\rbrace dy $ introduced by [46], [48].",
"In practice the integration term $\\int X(t)\\beta _1(t,y)dt$ is approximated by Riemann integration $\\int X(t)\\beta _1(t,y) dt = \\sum _{l=1}^{L} X(t_l)\\beta _1(t_l,y)(t_{l+1}-t_{l})$ , but other numerical approximation can be also used.",
"The estimation of parameters proceeds similarly to Section REF .",
"The ideas can be further extended to accommodate multiple covariates, scalar or functional, and varied types of effects.",
"For example, assume that we have one vector covariate, one scalar covariate and one functional covariate.",
"We posit a model that considers that the vector covariate has constant effect on the response while both the scalar and the functional covariates have varying effect on the response.",
"Consider the model: $ F_{Y|X}(y) = g^{-1}\\big \\lbrace \\beta _0(y) + \\mathbf {X}_1^T\\beta _1 + X_2 \\beta _2(y) + \\int X_3(t)\\beta _3(t,y)dt \\big \\rbrace ,$ for covariate vector $\\mathbf {X}_1$ , scalar covariate $X_2$ , and functional covariate $X_3(\\cdot )$ .",
"Here $\\mathbf {X}_1$ is assumed to have the $y$ -invariant linear effect, and $X_2$ and $X_3$ are assumed to have the $y$ -variant linear effects.",
"It is easy to see that a null effect, say $\\beta _3(\\cdot , \\cdot ) \\equiv 0$ is equivalent to the fact that the corresponding covariate, in this case $X_3(\\cdot )$ has no effect on any quantile levels of the response.",
"Fitting models given in (REF ) and (REF ) can be done by extending the ideas of [20], which provide general modeling and estimation methods for penalized function-on-function regression for the case of Gaussian functional response and implement their method in R [35] (namely, the pffr function in refund package [11]).",
"The extension of the model to the non-Gaussian response has recently been studied and implemented by [42].",
"Based on the existing function we implement the proposed method and provide a wrapper function in R. Furthermore the proposed method can be easily extended to relax the linearity assumption and allow more flexible model structures.",
"For example, instead of a functional linear model such as $F_{Y|X}(y) = g^{-1}\\big \\lbrace \\beta _0(y) + X_1\\beta _1(y) + \\int X_2(t)\\beta _2(t,y)dt \\big \\rbrace $ , we can model the conditional distribution as $F_{Y|X}(y) = g^{-1}\\big \\lbrace \\beta _0(y) + h_1(X_1) + \\int h_2(X_2(t), t, y)dt \\big \\rbrace $ , where $h_1(\\cdot )$ and $h_2(\\cdot , \\cdot , \\cdot )$ are unknown univariate and trivariate smooth functions, respectively.",
"We illustrate the nonlinear model in the simulation study for the case when there is a single scalar covariate and the corresponding results are presented in Section S1.1 of the Supplementary Materials.",
"The results show excellent prediction performance as the competitive nonlinear quantile regression method, namely Constrained B-Spline Smoothing [33], denoted by COBS .",
"It is important to emphasize that even in the case of a single functional covariate, our methodology differs from [8] (CM) in few directions: 1) Our proposed method is based on modeling the unknown smooth coefficient functions using pre-specified basis function expansion and using penalties to control their roughness.",
"In contrast, CM uses data-driven basis, chooses the number of basis functions through the percentage of explained variance (PVE) of the functional predictors.",
"This key difference allows us to accommodate covariates of different types.",
"2) Our estimation approach is based on a single step penalized criterion while CM uses pointwise estimation based on the residual sum of square criterion and thus requires fitting multiple generalized regressions.",
"This is an important advantage in terms of computational efficiency."
],
[
"Extension to sparse and noisy functional covariates",
"In practice the functional covariates are often observed at irregular times across the units and also are possibly corrupted with measurement errors.",
"In such case, one needs to first smooth and de-noise the trajectories before fitting.",
"When the sampling design of the functional covariate is dense, then the common approach is to take each trajectory and smooth it using spline or local polynomial smoothing, as proposed in [36] and [53].",
"When the design is sparse, the smoothing is done by pooling all the subjects and using the method proposed in [51].",
"A method following [51] (fpca.sc) is implemented in R package refund [11].",
"In our simulation study we use fpca.sc for this step irrespective of a sampling design (dense or sparse); alternatively, fpca.face [50] in refund or PACE [51] in MATLAB [29] can also be used."
],
[
"Monotonization",
"While a conditional quantile function is nondecreasing, the resulting estimated quantiles may not be.",
"We consider monotonization as proposed by [10].",
"[10] showed that in this way the monotonized estimator gives the same or better fit than the original estimator.",
"Two approaches are widely used; one is to monotonize the estimated conditional distribution function $\\widehat{F}_{Y|X}(y)$ , and the other is to monotonize the estimated conditional quantile function $\\widehat{Q}_{Y|X}(\\tau )$ .",
"We choose the former as we already have $\\widehat{F}_{Y|X}(y)$ evaluated at dense grid points $y_j$ 's and there is no need to obtain the estimated conditional quantile at fine grid points of the quantile level $\\tau \\in [0,1]$ .",
"We use an isotonic regression model [1] for monotonization as it makes no structural assumption and gives an ordered fit; It fits a nonparametric model with an order restriction.",
"The isotonic regression model is fitted through $\\lbrace (y_j, \\widehat{F}_{Y|X}(y_j)): j=1,\\ldots ,J\\rbrace $ using the isoreg function in R [35].",
"This idea was also employed in [23]."
],
[
"Simulation study",
"In this section we evaluate numerically the performance of the proposed method.",
"We present results for the case when we have both functional and scalar covariates; additional results when there is only single functional or single scalar covariate are discussed in the Supplementary Materials, Section S1.",
"We adapt the simulation settings of [8] for the cases that involve a functional covariate.",
"Suppose the observed data for the $i$ th subject is $[Y_i, X_{1i}, \\lbrace (W_{i1},t_{i1}), \\cdots , (W_{i m_i},t_{i m_i})\\rbrace ]$ , $t_{ij} \\in [0,10]$ , where $X_{1i} \\overset{i.i.d}{\\sim }Unif(-16,16)$ , $W_{ij} = X_{2i}(t_{ij}) + \\epsilon _{ij} = \\mu (t_{ij}) + \\sum _{k=1}^4 \\xi _{ik} \\phi _k(t_{ij}) + \\epsilon _{ij}$ , $1 \\le i \\le n, 1 \\le j \\le m_i$ .",
"Set the mean function $\\mu (t) = t + \\sin (t)$ , and the eigenfunctions $\\phi _k (t) = \\cos \\lbrace (k + 1) \\pi t / 10 \\rbrace / \\sqrt{5}$ for odd values of $k$ , $\\phi _k (t) = \\sin \\lbrace k \\pi t / 10 \\rbrace / \\sqrt{5}$ for even values of $k$ .",
"Here, assume that scores $\\xi _{ik} \\stackrel{iid}{\\sim } N(0, \\lambda _k)$ , where $(\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4) = \\lbrace 16, 9, 7.56, 5.06 \\rbrace $ , and $\\epsilon _{ij} \\stackrel{iid}{\\sim } N(0, \\sigma ^2_{\\epsilon })$ .",
"We assume two cases: normal distribution $Y_i|X_{1i},X_{2i}(\\cdot ) \\sim N(2\\int X_{2i}(t) \\beta (t) dt + 2 X_{1i}, 5^2)$ ; this yields the quantile regression model $Q_{Y|X_{1},X_{2}(\\cdot )}(\\tau ) = 2 \\int X_{2i}(t) \\beta (t) dt + 2 X_{1i} + 5 \\Phi ^{-1}(\\tau )$ , where $\\Phi (\\cdot )$ is the distribution function of the standard normal and $\\beta (t) = \\sum _{k=1}^4 \\beta _{k} \\phi _k(t)$ ; mixture of normal distributions $Y_i|X_{1i},X_{2i}(\\cdot ) \\sim 0.5N(\\int X_{2i}(t) \\beta (t) dt+ X_{1i}, 1^2) + 0.5N(3\\int X_{2i}(t) \\beta (t) dt + 3 X_{1i}, 4^2)$ ; this yields the quantile regression model $Q_{Y|X_{1},X_{2}(\\cdot )}(\\tau ) = 2 \\int X_{2i}(t) \\beta (t) dt +2 X_{1i} +$ $\\sqrt{(\\int X_{2i}(t) \\beta (t) dt + X_{1i})^2 + 8.5}\\Phi ^{-1}(\\tau )$ , where $\\beta (t) = \\sum _{k=1}^4 \\beta _{k} \\phi _k(t)$ .",
"Three noise levels are considered: low ($\\sigma _{\\epsilon }=0.50$ ), moderate ($\\sigma _{\\epsilon }=4.33$ ), and high ($\\sigma _{\\epsilon }=6.13$ ).",
"The three levels are such that the signal to noise ratio (SNR), which are calculated as $SNR = \\sqrt{\\sum _{k=1}^4\\lambda _k} / \\sigma _{\\epsilon }$ , are equal to $SNR = 150$ , 2, and 1, respectively.",
"Results are presented for sample sizes, $n=100$ (small) and $n=1000$ (large).",
"The performance is evaluated on a test set of size 100.",
"Two sampling designs are considered: (i) dense design, where the sampling points $\\lbrace t_{ij}: j=1, \\ldots , m_i\\rbrace $ are a set of $m_i=30$ equi-spaced time points in $[0,10]$ ; and (ii) sparse design, where $\\lbrace t_{ij}: j=1, \\ldots , m_i\\rbrace $ are $m_i=15$ randomly selected points from a set of 30 equi-spaced grids in $[0,10]$ .",
"The quantile functions with our approach are estimated as described in Section 2 by first creating an artificial binary response $Z_i(\\cdot )$ and then fitting a penalized function-on-function regression model and using the logit link function; we use the pffr function [20], [42] in the refund package [11] in R [35] for binomial responses, denoted by Joint QR.",
"We compare our method with three alternative approaches: (1) a variant of our proposed approach using pointwise fitting, denoted by Pointwise QR, and hence fitting multiple regression models with binomial link function as implemented by the penalized functional regression pfr of the refund package for generalized scalar responses, developed by [18]; (2) a modified version of the CM method, denoted by Mod CM, that we developed to account for additional scalar covariates, and which estimates pointwise using multiple generalized linear models; (3) a linear quantile regression approach using the quantile loss function and the partial quantile regression bases for functional covariates, proposed by [52] and denoted by PQR.",
"Notice that (1) and (2) account for a varying effect of the covariates on the response distribution, but do not ensure that this effect is smooth.",
"The R function pfr can incorporate both scalar/vector and functional predictors by adopting a mixed effects model framework.",
"The functional covariates are pre-smoothed by fPC analysis and truncation is done based on the number of fPCs determined by a percentage of explained variance equal to $99\\%$ for all estimation methods; pre-smoothing the functional covariates before fitting the regression model has been also considered by [18] and [20].",
"The performance is evaluated in terms of mean absolute error (MAE) for quantile levels $\\tau = 0.05, 0.1, 0.25,$ and $0.5$ .",
"Numerical results are provided in Tables REF , REF and REF .",
"Table REF gives results for the two settings (normal and mixture) when the functional covariate is observed on dense design and the sample size is $n=100$ ; Table REF shows the corresponding results for $n=1000$ .",
"We obtained similar findings for the sparse scenario and hence are not reported for brevity.",
"Consider first the case when conditional distribution of the response is normal.",
"When sample size is large ($n = 1000$ ) the proposed method (Joint QR) yields the best MAE for the SNR and the quantile levels considered.",
"Even with low-moderate sample size ($n = 100$ ) the Joint QR remains performing the best for extreme quantiles ($\\tau =0.05, 0.10$ and $0.25$ ) and relatively large noises ($\\sigma _\\epsilon =4.33$ and $6.13$ ).",
"When sample size is small-moderate the PQR method also performs very well for the small noise ($\\sigma _\\epsilon =0.50$ ) and for middle quantiles ($\\tau =0.05$ or $\\tau = 0.10$ ).",
"The Pointwise QR and Mod CM methods perform similarly, where the Pointwise QR tends to do better for low-moderate sample sizes ($n = 100$ ) while the Mod CM tends to do better for larger one ($n = 1000$ ).",
"All of the four methods are affected by the level of SNR; the higher it is, the better MAE is.",
"When the conditional distribution of the response follows the mixture of normals, there is no uniformly best method across quantiles levels or SNR levels whe sample size is large ($n = 1000$ ).",
"It seems that all four methods have similar performance with some being the best for some situations while others for other situations.",
"Overall the Joint QR method tends to perform better for extreme quantiles ($\\tau =0.05$ or $\\tau = 0.10$ ) while the other three methods tend to predict better the middle quantiles ($\\tau = 0.25$ or $\\tau = 0.50$ ).",
"Other findings are relatively similar to the ones for the normal case.",
"Table REF compares the three methods that involve estimating the conditional distribution - Joint QR, Pointwise QR and Mod CM - in terms of the computational time required for fitting; the times correspond to using a desktop computer with a $2.3$ GHz CPU and 8 GB of RAM.",
"Not surprisingly by fitting the model a single time, Joint QR is the fastest, in some cases being order of magnitude faster than the rest.",
"Pointwise QR can be up to twice as fast as Mod CM.",
"For completeness, we also compare our proposed method to the appropriate competitive methods for the cases $(1)$ when there is a single scalar covariate and $(2)$ when there is a single functional covariate.",
"The Supplementary Materials, Section S1.1 discusses the former case and compares Joint QR and Pointwise QR with the linear quantile regression and the nonlinear quantile regression (as implemented by the cobs function in the R package COBS [33]) in an extensive simulation experiment that involves both linear quantile settings and nonlinear quantile settings.",
"Overall the results show that the proposed methods have similar behavior as LQR; see Table S1.",
"Furthermore we consider the proposed methods with nonlinear modeling of the conditional distribution as discussed in Section , which we denote with Joint QR (NL) for joint fitting and Pointwise QR (NL) for pointwise fitting.",
"Nonlinear versions of the proposed methods have an excellent MAE performance, which is comparable to or better than that of the COBS method.",
"Finally, Section S1.2 in the Supplementary Materials discusses the simulation study for the case of having a single functional covariate and compares the proposed methods with CM in terms of MAE as well as computational time; see results displayed in Tables S2 and S3.",
"The results show that the proposed Joint QR is superior to CM both in terms of the prediction accuracy and computation efficiency.",
"In our simulation study we also consider the joint fitting of the model by treating the binary response as normal and use pffr [20] with Gaussian link, denoted by Joint QR (G).",
"Table: Simulation results: Average MAE of the predicted τ\\tau -level quantile for sample size n=100n=100 (standard error in parentheses).",
"Results are based on 500 replications.Table: Simulation results: Average MAE of the predicted τ\\tau -level quantile for sample size n=1000n=1000 (standard error in parentheses).",
"Results are based on 500 replications.Table: Comparison of the average computing time (seconds) for the three approaches that involve estimating the conditional distribution."
],
[
"Sow Data Application",
"Our motivating application is an experimental study carried out at a commercial farm in Oklahoma from July 21, 2013 to August 19, 2013 [40].",
"The study comprises of 480 lactating sows of different parities (i.e.",
"the number of previous pregnancies, which serves as a surrogate for age and body weight) that were observed during their first 21 lactation days; their feed intake was recorded daily as the difference between the feed offer and the feed refusal.",
"In addition the study contains information on the temperature and humidity of the farrowing rooms, each recorded at five minute intervals.",
"The final dataset we used for the analysis consists of 475 sows after five sows with unreliable measurements were removed by the experimenters.",
"The experiment was conducted to gain better insights into the way that the ambient temperature and humidity of the farrowing room affects the feed intake of lactating sows.",
"Previous studies seem to suggest a reduction in the sow's feed intake due to heat stress: above $29^{\\circ }$ C sows decrease feed intake by 0.5 kg per additional degree in temperature [34].",
"Studying the effect of heat stress on lactating sows is a very important scientific question because of a couple of reasons.",
"First, the reduction of feed intake of the lactating sows is associated with a decrease in both their bodyweight (BW) and milk production, as well as the weight gain of their litter [22], [38], [39].",
"Sows with poor feed intake during lactation continue the subsequent reproductive period with negative energy balance [3], which acts as negative feedback to prevent the onset of a new reproductive cycle.",
"Second, heat stress reduced farrowing rate (the number of sows that deliver a new litter) and reduced the number of piglets born [4]; The reduction in reproductivity due to seasonality is estimated to cost 300 million dollars per year for the swine industry [44].",
"Economic losses are estimated to increase [32] because very high temperatures are likely to occur more frequently due to global warming [31].",
"Our primary goal is to understand the thermal needs of the lactating sows for proper feeding behavior during the lactation time.",
"Specifically, we are interested in how the interplay between the temperature and humidity of the farrowing room affects the feed intake demeanor of lactating sows of different parities.",
"We focus on three specific times during the lactation period: beginning (lactation day 4), middle (day 11) and end (lactation day 18) and consider two types of responses that are meant to assess the feed intake behavior using the current and the previous lactation day.",
"The first one quantifies the absolute change in the feed intake over two consecutive days and the second one quantifies the relative change and takes into account the usual sow's feed intake.",
"We define them formally after introducing some notation.",
"Let $FI_{ij}$ be the $j$ th measurement of the feed intake observed for the $i$ th sow and denote by $LD_{ij}$ the lactation day when $FI_{ij}$ is measured; here $j=1, \\ldots , n_{i}$ .",
"Most sows are observed for every day within the first 21 lactation days and thus have $n_i=21$ .",
"First define the absolute change in the feed intake between two consecutive days as $\\Delta ^{(1)}_{i(j+1)} = FI_{i(j+1)} - FI_{ij}$ for $j$ that satisfies $LD_{i(j+1)} - LD_{ij} = 1$ .",
"For instance $\\Delta ^{(1)}_{i(j+1)}=0$ means there was no change in feed intake of sow $i$ between the current day and the previous day, while $\\Delta ^{(1)}_{i(j+1)} < 0$ means that the feed intake consumed by the $i$ th sow in the current day is smaller than the feed intake consumed in the previous day.",
"However, the same amount of change in the feed intake may reflect some stress level for a sow who typically eats a lot and a more serious stress level for a sow that usually has a lower appetite.",
"For this, we define the relative change in the feed intake by $\\Delta ^{(2)}_{i(j+1)}= (FI_{i(j+1)} - FI_{ij})/$ $\\lbrace (LD_{i(j+1)} - LD_{ij}) \\cdot TA_{i}\\rbrace $ , where $TA_{i}$ is the trimmed average of feed intake of $i$ th sow calculated as the average feed intake after removing the lowest $20\\%$ and highest $20\\%$ of the feed intake measurements taken for the corresponding sow, $FI_{i1}, \\ldots , FI_{in_i}$ .",
"Here $TA_{i}$ is surrogate for the usual amount of feed intake of the $i$ th sow.",
"Trimmed average is used instead of the common average, to remove outliers of very low feed intakes in first few lactating days.",
"For example, consider the situation of two sows: sow $i$ that typically consumes 10lb food per day and sow $i^{\\prime }$ that consumes 5lb food per day.",
"A reduction of 5lb in the feed intake over two consecutive days corresponds to $\\Delta ^{(2)}_{i(j+1)} = - 10\\%$ for the $i$ th sow and $\\Delta ^{(2)}_{i^{\\prime }(j+1)} = - 20\\%$ for the $i^{\\prime }$ th sow.",
"Clearly both sows are stressed (negative value) but the second sow is stressed more, as its absolute value is larger; in view of this we refer to the second response as the stress index.",
"Due to the definition of the two types of responses, the data size varies, so for the first response, $\\Delta ^{(1)}_{i(j+1)}$ , we have sample sizes of 233, 350, and 278 for lactation days 4 ($j=3$ ), 11 ($j=10$ ), and 18 ($j=17$ ), respectively, whereas for $\\Delta ^{(2)}_{i(j+1)}$ the sample sizes are 362, 373, and 336 respectively.",
"In this analysis we center the attention on the effect of the ambient temperature and humidity on the 1st quartile of the proxy stress measures and gain more understanding of the food consumption of sows that are most susceptible to heat stress.",
"While the association between the feed intake of lactating sows and the ambient conditions of the farrowing room has been an active research area for some time, accounting for the temperature daily profile has not been considered yet hitherto.",
"Figure REF displays the temperature and humidity profiles for three days.",
"Preliminary investigation reveals that temperature is negatively correlated with humidity at each time; this phenomenon is caused because the farm uses cool cell panels and fans to control the ambient temperature.",
"Furthermore, it appears that there is strong pointwise correlation between temperature and humidity.",
"In view of these observations we consider the mean summary of the humidity in our analysis.",
"Exploratory analysis of the feed intake behavior of the sows suggest similarities for the sows with parity greater than one (who are at their third pregnancy or higher); thus we use a parity indicator instead of the actual parity of the sow.",
"The parity indicator $P_i$ is defined as one, if the $i$ th sow has parity one and zero otherwise.",
"As emphasized throughout the paper, the existing literature on quantile regression is not suitable to incorporate covariates of different types, as it is the case here.",
"Figure: Temperature ( ∘ ^{\\circ }C) and humidity (%\\%) profiles for three randomly selected days.",
"The x-axis begins at 14H (2PM).For the analysis we smooth daily temperature measurements of each sow using univariate smoother with 15 cubic regression bases and quadratic penalty; REML is used to estimate smoothing parameter.",
"The smoothed temperature curve for sow $i$ 's $j$ th repeated measure is denoted by $T_{ij}(t)$ , $t\\in [0,24)$ .",
"In addition the corresponding daily average humidity is denoted by $AH_{ij}$ .",
"Both temperature and average humidity are centered before being used in the analysis.",
"In the following we detail estimation procedure.",
"Since the procedure is identical for both responses here we remove superscript in notation and use $\\Delta _{ij}$ as our response.",
"We first estimate the conditional distribution of $\\Delta _{ij}$ given temperature $T_{ij}(t)$ , average humidity $AH_{ij}$ , parity $P_{i}$ , and interaction $AH_{ij} \\cdot T_{ij}(t)$ .",
"Specifically for each of $j = 3, 10$ and 17 we create a set of 100 equi-spaced grid of points between the fifth smallest and fifth largest values of $\\Delta _{ij}$ 's and denote the grids with $\\mathcal {D} = \\lbrace d_{\\ell } : \\ell = 1, \\ldots , 100\\rbrace $ .",
"Then we create artificial binary response, $\\lbrace \\mathbb {1}\\left(\\Delta _{ij} < d_{\\ell }\\right): \\ell = 1, \\ldots , 100\\rbrace $ , and fit the following model for $F_{ij}(d_{\\ell }) = E\\left[\\mathbb {1}\\left(\\Delta _{ij} < d_{\\ell }\\right) \\big | T_{ij}(t), AH_{ij}, P_{i}\\right]$ : $E\\left[\\mathbb {1}\\left(\\Delta _{ij} < d_{\\ell }\\right) \\big | T_{ij}(t), AH_{ij}, P_{i}\\right] \\nonumber = g^{-1} \\Big \\lbrace \\beta _0(d_{\\ell }) + \\beta _1(d_{\\ell }) P_i + \\beta _2(d_{\\ell }) AH_{ij} \\\\+ \\int \\beta _3(d_{\\ell },t)T_{ij}(t)dt + \\int \\beta _4(d_{\\ell },t)T_{ij}(t)AH_{ij}dt \\Big \\rbrace , \\nonumber $ where $\\beta _0(\\cdot )$ is a smooth intercept, $\\beta _1(\\cdot )$ quantifies the smooth effect of young sows, $\\beta _2(\\cdot )$ describes the effect of the humidity, and $\\beta _3(\\cdot , t)$ and $\\beta _4(\\cdot ,t)$ quantify the effect of the temperature at time $t$ as well as the interaction between the temperature at time $t$ and average humidity.",
"We model $\\beta _0(\\cdot )$ using 20 univariate basis functions, $\\beta _1(\\cdot )$ and $\\beta _2(\\cdot )$ using five univariate basis functions, $\\beta _3(\\cdot ,\\cdot )$ and $\\beta _4(\\cdot ,\\cdot )$ using tensor product of two univariate bases functions (total of 25 functions).",
"Throughout analysis cubic B-spline bases are used and REML is used for estimating smoothing parameters.",
"The estimated conditional distribution, denoted by $\\widehat{F}_{ij}(d)$ , is monotonized by fitting isotonic regression to $\\lbrace (d_{\\ell }, \\widehat{F}_{ij}(d_{\\ell })):\\ell = 10, \\ldots , 90\\rbrace $ ; ten smallest and ten largest $d_{\\ell }$ and the corresponding values of $\\widehat{F}_{j}(d_\\ell )$ are removed to avoid boundary effects.",
"By abuse of notation we again denote the resulting monotonized estimated distribution with $\\widehat{F}_{ij}(d)$ .",
"Finally we obtain estimated quantiles at $\\tau =0.25$ level by inverting the estimated distribution function $\\widehat{F}_{ij}(d)$ , namely $\\widehat{Q}\\left(\\tau = 0.25 \\; | \\; T_{ij}(t), AH_{ij}, P_{i} \\right) = \\text{inf}\\lbrace d : \\widehat{F}_{ij}(d) > 0.25\\rbrace $ .",
"Figure: Temperature curves with which prediction of quantiles is made.",
"Dashed black line is pointwise average of temperature curves and solid lines are pointwise quartiles; all curves are smoothed.Figure: Distribution of responses by ParityTo study and interpret the effect of each covariate we predict quantiles at combinations of different values of covariates and investigate the predicted values against the covariates.",
"For each of three lactation days ($ j = 3, 10, 17$ ) we consider three values of average humidity (first quartile, median, and third quartile) and the two levels of parity (0 for older sows and 1 for younger sows).",
"For the functional covariate $T_{ij}(t)$ we create seven smooth temperature curves given in Figure REF by first obtaining pointwise quantiles of temperature and then smoothing them for each of quantiles levels $\\eta = 0.2, 0.3, \\ldots ,$ and $0.8$ .",
"In summary for each of three lactation days we make prediction of distribution and the first quartile of two responses for 42 ($3 \\times 2 \\times 7$ ) different combinations of covariates values.",
"To avoid extrapolation we ascertain that there are reasonably many observed measurements at each of the combinations and bottom $25\\%$ of the responses are not dominantly from one of the parity group; see distribution of each response by the parity in Figure REF .",
"The resulting predicted quantiles are shown in Figure REF .",
"Here we focus our discussion on predicted quantile of $\\Delta ^{(2)}_{i(j+1)}$ at quantile level $\\tau = 0.25$ for lactation day 4 ($j=3$ ) - the first plot of the second row in Figure REF .",
"The results suggest that the feed intake of older sows (parity indicator equal to zero) are less affected by high temperatures than that of younger sows; this finding is in agreement with [4], who also found that younger sows (parity equal to one) are more sensitive to ambient changes than sows with higher parity.",
"We also observe that humidity plays an important role in the effect of temperature on feed intake change.",
"Similar to a previous study [2] our results also imply that high humidity is related to a negative impact of high temperature on feed intake while low humidity alleviates it.",
"On the contrary, when temperature is low, high humidity leads to better feed intake than low humidity.",
"For instance on lactation day 4 ($j=3$ ) regardless of the parity group, when temperature increases the predicted first quartile of $\\Delta ^{(2)}_{i(j+1)}$ increases for low humidity (solid line) whereas it decreases for high humidity (dotted line).",
"The result seems to suggest to keep low humidity levels in order to maintain healthy feed intake behavior, when ambient temperature is above 60th percentile; high humidity levels are desirable for cooler ambient temperature.",
"The results corresponding to other lactation days can also be interpreted similarly.",
"While the effects of covariates on feed intake are less apparent toward the end of lactation period, we still observe similar pattern across all three lactation days.",
"For lactation day 11 ($j=10$ ) we observe that when temperature is above 40th percentile the predicted first quartile starts increasing with low humidity while it continues decreasing with high humidity.",
"Similarly for lactation day 18 ($j=17$ ) when temperature is above 60th percentile the predicted first quartile increases with low humidity while it decreases with high humidity.",
"The effect of temperature on feed intake seems less obvious for lactation days 11 and 18 than for day 4; while the effect may be due to lactation day it may also be a result of other factors, such as more fluctuation and variability in temperature curves on day 4 than other two days (see Figure REF ).",
"Overall we conclude that high humidity and temperature affect sows' feed intake behavior negatively and young sows (parity one) are more sensitive to heat stress than older sows (higher parity), especially in the beginning of lactation period.",
"Figure: Displayed are the predicted quantiles of Δ i(j+1) (1) \\Delta ^{(1)}_{i(j+1)} and Δ i(j+1) (2) \\Delta ^{(2)}_{i(j+1)} for different parities, average humidity, and temperature levels.",
"In each of all six panels, black thick lines correspond to the young sows (P i =1P_i = 1) and grey thin lines correspond to the old sows (P i =0P_{i} = 0).",
"Line types indicate different average humidity levels; solid, dashed, and dotted correspond to low, medium, and high average humidity levels (given by the first quartile, median, and the third quartiles of AH ij AH_{ij}), respectively.",
"The seven grids in xx-axis of each panel correspond to the 7 temperature curves given in the respective panel of Figure ."
],
[
"Discussion",
"In this paper we proposed a novel framework for a comprehensive study of covariates of mixed types on the conditional distribution of the response.",
"Extensive simulation study showed very good prediction performance in terms of estimating quantiles of various levels.",
"Additionally, the modeling approach leads to computationally efficient estimation algorithm.",
"The proposed method is flexible and easy to implement using existent software, pffr [42].",
"This modeling framework opens up a couple of future research directions.",
"A first research avenue is to develop significance tests of null covariate effect.",
"Testing for the null effect of a covariate on the conditional distribution of the response is equivalent to testing that the corresponding regression coefficient function is equal to zero in the associated function-on-function mean regression model.",
"Such significance tests have been studied when the functional response is continuous [43], [53]; however their study for binary-valued functional responses is an open problem in functional data literature.",
"One possible alternative is to construct confidence bands for the corresponding coefficient function, say using bootstrap, and examine whether the confidence band includes zero for its entire domain.",
"Another research avenue is to do variable selection in the setting where there are many scalar covariates and functional covariates.",
"Many current applications collect data with increasing number of mixed covariates and selecting the ones that have an effect on the conditional distribution of the response is very important.",
"This problem is an active research area in functional mean regression where the response is normal [17], [9].",
"The proposed modeling framework has the potential to facilitate studying such problem."
],
[
"Supplementary Material",
"Section S1 provides additional simulation settings and results for the cases of having either a single scalar covariate or a single functional covariate.",
"Section S2 presents additional data analysis done using the proposed method on the bike sharing dataset [13], [26].",
"Lastly the R function implementing the proposed method is available at http://www4.ncsu.edu/~spark13/software/QRFD_Rcode.zip/ .",
"Staicu's research was supported partially by National Science Foundation DMS 0454942 and National Institutes of Health grants R01 NS085211 and R01 MH086633.",
"The data used originated from work supported in part by the North Carolina Agricultural Foundation, Raleigh, NC."
]
] | 1605.05779 |
[
[
"Computation of the Similarity Class of the p-Curvature"
],
[
"Abstract The $p$-curvature of a system of linear differential equations in positive characteristic $p$ is a matrix that measures how far the system is from having a basis of polynomial solutions.",
"We show that the similarity class of the $p$-curvature can be determined without computing the $p$-curvature itself.",
"More precisely, we design an algorithm that computes the invariant factors of the $p$-curvature in time quasi-linear in $\\sqrt p$.",
"This is much less than the size of the $p$-curvature, which is generally linear in $p$.",
"The new algorithm allows to answer a question originating from the study of the Ising model in statistical physics."
],
[
"Introduction",
"Differential equations in positive characteristic $p$ are important and well-studied objects in mathematics [22], [32], [33].",
"The main reason is arguably one of Grothendieck’s (still unsolved) conjectures [26], [27], [1], stating that a linear differential equation with coefficients in $\\mathbb {Q}(x)$ admits a basis of algebraic solutions if and only if its reductions modulo (almost) all primes $p$ admit a basis of polynomial solutions modulo $p$ .",
"Another motivation stems from the fact that the reductions modulo prime numbers yield useful information about the factorization of differential operators in characteristic zero.",
"To a linear differential equation in fixed characteristic $p$ , or more generally to a system of such equations, is attached a simple yet very useful object, the $p$ -curvature.",
"Let $\\mathbb {F}_q$ be the finite field with $q=p^a$ elements.",
"The $p$ -curvature of a system of linear differential equations with coefficients in $\\mathbb {F}_q(x)$ is a matrix with entries in $\\mathbb {F}_q(x)$ that measures the obstructions for the given system to possess a fundamental matrix of polynomial solutions in $\\mathbb {F}_q[x]$ .",
"Its definition is remarkably simple, especially at a higher level of generality: the $p$ -curvature of a differential module $(M,\\partial )$ of dimension $r$ over $\\mathbb {F}_q(x)$ is the “differential-Frobenius-map” $\\partial ^p = \\partial \\circ \\cdots \\circ \\partial $ ($p$ times).",
"When applied to the differential module canonically attached with the system $Y^{\\prime } = A(x)Y$ , the $p$ -curvature materializes into the $p$ -th iterate $\\partial _A^p$ of the map $\\partial _A :\\mathbb {F}_q(x)^r \\rightarrow \\mathbb {F}_q(x)^r$ that sends $v$ to $v^{\\prime } - A v$ , or more concretely, into the matrix $A_p(x)$ of this map with respect to the canonical basis of $\\mathbb {F}_q(x)^r$ .",
"It is given as the term $A_p$ of the sequence $(A_i)_i$ of matrices in ${M}_{r}(\\mathbb {F}_q(x))$ defined by $A_1 = -A \\quad \\text{and} \\quad A_{i+1} = A^{\\prime }_i - A \\cdot A_i \\quad \\text{for} \\quad i \\ge 1.$ From a computer algebra perspective, many effectivity questions naturally arise.",
"They primarily concern the algorithmic complexity of various operations and properties related to the $p$ -curvature: How fast can one compute $A_p$ ?",
"How fast can one decide its nullity?",
"How fast can one determine its minimal and characteristic polynomial?",
"Apart the fundamental nature of these questions from the algebraic complexity theory viewpoint, there are concrete motivations for the efficient computation of the $p$ -curvature, coming from various applications, notably in enumerative combinatorics and statistical physics [7], [8], [2].",
"We pursue the algorithmic study of the $p$ -curvature, initiated in [9], [3], [4].",
"In those articles, several questions were answered satisfactorily, but a few other problems were left open.",
"In summary, the current state of affairs is as follows.",
"First, the $p$ -curvature $A_p$ can be computed in time $O(\\log p)$ when $r=1$ and $O\\tilde{~}(p)$ when $r>1$ .",
"The soft-O notation $O\\tilde{~}(\\,)$ indicates that polylogarithmic factors in the argument of $O(\\,)$ are deliberately not displayed.",
"These complexities match, up to polylogarithmic factors, the generic size of $A_p$ .",
"Secondly, one can decide the nullity of $A_p$ in time $O\\tilde{~}(p)$ and compute its characteristic polynomial in time $O\\tilde{~}(\\sqrt{p})$ .",
"It is not known whether the exponent $1/2$ is optimal for the last problem.",
"In all these estimates, the complexity (“time”) measure is the number of arithmetic operations $(\\pm ,\\times , \\div )$ in the ground field $\\mathbb {F}_q$ , and the dependence is expressed in the main parameter $p$ only.",
"Nevertheless, precise estimates are also available in terms of the other parameters of the input.",
"In the present work, we focus on the computation of all the invariant factors of the $p$ -curvature, and show that they can also be determined in time $O\\tilde{~}(\\sqrt{p})$ .",
"Previously, this was unknown even for the minimal polynomial of $A_p$ or for testing the nullity of $A_p$ .",
"The fact that a sublinear cost could in principle be achievable, although $A_p$ itself has a total arithmetic size linear in $p$ , comes from the observation that the coefficients of the invariant factors of $A_p$ lie in the subfield $\\mathbb {F}_q(x^p)$ of $\\mathbb {F}_q(x)$ , in other words they are very sparse.",
"To achieve our objective, we blend the methods used in our previous works [3] and [4].",
"The first key ingredient is the construction, for any point $a$ in the algebraic closure of $\\mathbb {F}_q$ that is not a pole of $A(x)$ , of a matrix $Y_a$ with entries in $\\ell = \\mathbb {F}_q(a)$ which is similar to the evaluation $A_p(a)$ of the $p$ -curvature at the point $a$ .",
"This construction comes from [4] and ultimately relies on the existence of a well-suited ring, of so-called Hurwitz series in $x-a$, for which an analogue of the Cauchy–Lipschitz theorem holds for the system $Y^{\\prime }=A(x)Y$ around the (ordinary) point $x=a$ .",
"The matrix $Y_a$ is the $p$ -th coefficient of the fundamental matrix of Hurwitz series solutions of $Y^{\\prime }=A(x)Y$ at $x=a$ .",
"The second key ingredient is a baby step / giant step algorithm that computes $Y_a$ in $O\\tilde{~}(\\sqrt{p})$ operations in $\\ell $ via fast matrix factorials.",
"Finally, we recover the invariant factors of $A_p$ from those of the matrices $Y_a$ , for a suitable number of values $a$ .",
"The main difficulty in this interpolation process is that there exist badly behaved points $a$ for which the invariant factors of $A_p(a)$ are not the evaluations at $a$ of the invariant factors of $A_p(x)$ .",
"The remaining task is then to bound the number of unlucky evaluation points $a$ .",
"The key feature allowing a good control on these points, independent of $p$ , is the fact that the invariant factors of $A_p(x)$ have coefficients in $\\mathbb {F}_q(x^p)$ .",
"Relationship to previous work.",
"There exists a large body of work concerning the computation of so-called Frobenius forms of matrices (that is, the list of their invariant factors, possibly with corresponding transformation matrices), and the related problem of Smith forms of polynomial matrices.",
"The specificities of our problem prevent us from applying these methods directly; however, our work is related to several of these previous results.",
"Let $\\omega $ be a feasible exponent for matrix multiplication.",
"The best deterministic algorithm known so far for the computation of the Frobenius form of an $n \\times n$ matrix over a field $k$ is due to Storjohann [30].",
"This algorithm has running time $O(n^\\omega \\log (n)\\log \\log (n))$ operations in $k$ .",
"We will use it to compute the invariant factors of the matrices $Y_a$ above.",
"Las Vegas algorithms were given by Giesbrecht [19], Eberly [14] and Pernet and Storjohann [28], the latter having expected running time $O(n^\\omega )$ over sufficiently large fields.",
"The case of matrices with integer or rational entries has attracted a lot of attention; this situation is close to ours, with the bit size of integers playing a role similar to the degree of the entries in the $p$ -curvature.",
"Early work goes back to algorithms of Kaltofen et al.",
"[23], [24] for the Smith form of matrices over $\\mathbb {Q}[x]$ , which introduced techniques used in several further algorithms, such as the Las Vegas algorithm by Storjohann and Labahn [31].",
"Giesbrecht's PhD thesis [18] gives a Las Vegas algorithm with expected cost $O\\tilde{~}(n^{\\omega +2} d)$ for the Frobenius normal form of an $n\\times n$ matrix with integer entries of bit size $d$ ; Storjohann and Giesbrecht substantially improved this result in [20], with an algorithm of expected cost $O\\tilde{~}(n^4 d + n^3 d^2)$ .",
"The best Monte Carlo running time known to us is $O\\tilde{~}(n^{2.698} d)$ , by Kaltofen and Villard [25].",
"In the latter case of matrices with integer coefficients, a common technique relies on reduction modulo primes, and a main source of difficulty is to control the number of “unlucky” reductions.",
"We pointed out above that this is the case in our algorithm as well.",
"In general, the number of unlucky primes is showed to be $O\\tilde{~}(n^2d)$ in [18]; in our case, the degree $d$ of the entries grows linearly with $p$ , but as we said above, we can alleviate this issue by exploiting the properties of the $p$ -curvature.",
"Storjohann and Giesbrecht proved in [20] that a candidate for the Frobenius form of an integer matrix can be verified using only $O\\tilde{~}(nd)$ primes; it would be most interesting to adapt this idea to our situation.",
"Structure of the paper.",
"In Section , we recall the main theoretical properties of the invariant factors of a polynomial matrix, and study their behavior under specialization.",
"We obtain bounds on bad evaluation points, and use them to design (deterministic and probabilistic) evaluation-interpolation algorithms for computing the invariant factors of a polynomial matrix.",
"Section is devoted to the design of our main algorithms for the similarity class of the $p$ -curvature, with deterministic and probabilistic versions for both the system case and the scalar case.",
"Finally, Section presents an application of our algorithm, that allows to answer a question coming from theoretical physics.",
"Complexity basics.",
"We use standard complexity notation, such as $\\omega $ for the exponent of matrix multiplication.",
"The best known upper bound is $\\omega <2.3729$ from [15].",
"Many arithmetic operations on univariate polynomials of degree $d$ in $k[x]$ can be performed in $O\\tilde{~}(d)$ operations in the field $k$ : addition, multiplication, shift, interpolation, etc, the key to these results being fast polynomial multiplication [29], [11], [21].",
"A general reference for these questions in [17].",
"We recall here some basic facts about invariant factors of matrices defined over a field.",
"We fix for now a field $K$ , and a matrix $M \\in {M}_n(K)$ .",
"For a monic polynomial $P = T^d - \\sum _{i=0}^{d-1} a_iT^i \\in K[T]$ , let $M_P$ denote its companion matrix: $M_P = \\left(\\begin{matrix}& & & a_0 \\\\1 & & & a_1 \\\\& \\ddots & & \\vdots \\\\& & 1 & a_{d-1}\\end{matrix}\\right).$ A well-known theorem [16] asserts that there exist a unique sequence of monic polynomials $I_1, \\ldots , I_n$ for which $I_j$ divides $I_{j+1}$ for all $j$ and $M$ is similar to a block diagonal matrix whose diagonal entries are $M_{I_1}, \\ldots , M_{I_n}$ .",
"The $I_j$ 's are called the invariant factors of $M$ .",
"We emphasize that, with our convention, there are always $n$ invariant factors but some of them may be equal to 1, in which case the corresponding companion matrix is the empty one.",
"Under this normalization, the $j$ -th invariant factor $I_j$ can be obtained as $I_j = G_j/G_{j-1}$ , where $G_j$ is the gcd of the minors of size $j$ of the matrix $T \\text{I}_n - M$ , where $\\text{I}_n$ stands for the identity matrix of size $n$ .",
"The invariant factors are closely related to the characteristic polynomial; indeed, we have $I_1 \\cdot I_2 \\cdots I_n = G_n = \\det (T \\text{I}_n - M).$ Given some irreducible polynomial $P$ in $K[T]$ , we consider the sequence (of integers): $e \\mapsto d_{P,e} = \\frac{\\dim _K \\ker P^e(M)}{\\deg P}.$ It turns out that this sequence completely determines the $P$ -adic valuation of the invariant factors.",
"Indeed, denoting by $v_j$ the $P$ -adic valuation of $I_j$ , we have the relations: $d_{P,e} & = \\sum _{j=1}^n \\min (e, v_j), \\smallskip \\\\d_{P,e} - d_{P,e-1} & = \\text{Card} \\lbrace j \\: | \\: v_j\\:{\\ge }\\:e \\rbrace $ from which the $v_j$ 's can be recovered without ambiguity since they form a nondecreasing sequence.",
"It also follows from the above formula that the sequence $e \\mapsto d_{P,e}$ is concave and eventually constant.",
"Its final value is the dimension of the characteristic subspace associated to $P$ and it is reached as soon as $e$ is greater than or equal to $v_n$ ."
],
[
"Behaviour under specialization",
"Let $k$ be a perfect field of characteristic $p$ .",
"We consider a matrix $M(x)$ with coefficients in $k[x]$ .",
"For an element $a$ lying in a finite extension $\\ell $ of $k$ , we denote by $M(a)$ the image of $M(x)$ under the mapping $k[x] \\rightarrow \\ell $ , $x \\mapsto a$ .",
"Our aim is to compare the invariant factors of $M(x)$ and those of $M(a)$ .",
"We introduce some notation.",
"Let $I_1(x,T), \\ldots , I_n(x,T)$ be the invariant factors of $M(x)$ .",
"It follows from the relation (REF ) that they all lie in $k[x,T]$ .",
"We can therefore evaluate them at $x=a$ for each element $a \\in \\ell $ as above and get this way univariate polynomials with coefficients in $\\ell $ .",
"Let $I_1(a,T), \\ldots , I_n(a,T)$ be these evaluations.",
"We also consider the invariant factors of $M(a)$ and call them $I_{1,a}(T), \\ldots , I_{n,a}(T)$ .",
"We furthemore define $G_j(x,T) & = I_1(x,T) \\cdot I_2(x,T) \\cdots I_j(x,T) \\\\\\text{and}\\quad G_{j,a}(T) & = I_{1,a}(T) \\cdot I_{2,a}(T) \\cdots I_{j,a}(T).$ The characterization of the $G_j$ 's in term of minors yields: Lemma 1 For all $a \\in \\ell $ and all $j \\in \\lbrace 1, \\ldots , n\\rbrace $ , the polynomial $G_j(a,T)$ divides $G_{j,a}(T)$ in $\\ell [T]$ .",
"Let $P_1(x,T), \\ldots , P_s(x,T)$ be the irreducible factors of the characteristic polynomial $\\chi (x,T)$ of $M(x)$ , and let us write $\\chi ^\\textrm {sep}(x,T)$ for $P_1(x,T) \\cdots P_s(x,T)$ .",
"For all $1\\le i\\le s$ and $1\\le j\\le n$ , let $e_{i,j}$ be the multiplicity of $P_i(x,T)$ in $I_j(x,T)$ .",
"Proposition 2 We assume $\\chi ^\\textrm {\\emph {sep}}(a,T)$ is separable and $\\dim _{k(x)} \\ker P_i(x,M(x))^{e_{i,j}+1}= \\dim _{\\ell } \\ker P_i(a,M(a))^{e_{i,j}+1}$ for all $i$ and for all $j<n$ .",
"Then $I_j(a,T) = I_{j,a}(T)$ for all $j$ .",
"The equality of dimensions is also true for $j=n$ , as their sum on both sides equals $n$ (using separability) and these dimensions can only increase by specialization.",
"Let $d_{P_i,e}$ be the sequence defined by Eq.",
"(REF ) with respect to the irreducible polynomial $P_i(x,T)$ and the matrix $M(x)$ .",
"We define similarly for each irreducible factor $P(T)$ of $P_i(a,T)$ the sequence $d_{P,e}$ corresponding to the polynomial $P(T)$ and the matrix $M(a)$ .",
"We claim that it is enough to prove that $d_{P_i,e} = d_{P,e}$ for all $e$ , $i$ and all irreducible divisors $P(T)$ of $P_i(a,T)$ .",
"Indeed, by Eq.",
"(), such an equality would imply: $v_{P(T)}(I_{j,a}(T)) = e_{i,j}$ provided that $P(T)$ is an irreducible divisor of $P_i(a,T)$ , and where $v_{P(T)}$ denotes the $P(T)$ -adic valuation.",
"On the other hand, still assuming that $P(T)$ is an irreducible divisor of $P_i(a,T)$ , it follows from the definition of the $e_{i,j}$ 's that: $v_{P(T)}(I_j(a,T)) \\ge e_{i,j}$ and that the equality holds if and only if $P(T)$ does not divide any of the $P_{i^{\\prime }}(a,T)$ for $i^{\\prime } \\ne i$ .",
"Comparing characteristic polynomials, we know moreover that $\\sum _{j=1}^n v_{P(T)}(I_{j,a}(T)) = \\sum _{j=1}^n v_{P(T)}(I_j(a,T))$ .",
"Combining this with (REF ) and (REF ), we find that the $P_i(a,T)$ 's are pairwise coprime and finally get $I_j(a,T) = I_{j,a}(T)$ for $1 \\le j \\le n$ , as wanted.",
"Until the end of the proof, we fix the index $i$ and reserve the letter $P$ to denote an irreducible divisor of $P_i(a,T)$ .",
"For a fixed integer $e$ , denote by $j_0$ the greatest index $j$ for which $v_{P(T)}(I_{j,a}(T)) < e$ and observe that Eq.",
"(REF ) can be rewritten $d_{P,e} = e \\cdot \\big (n-j_0\\big ) + v_{P(T)}\\big (G_{j_0,a}(T)\\big )$ .",
"Using Lemma REF , we derive $d_{P,e} \\ge e \\cdot \\big (n-j_0\\big ) + v_{P(T)}\\big (G_{j_0}(a,T)\\big )\\ge d_{P_i,e}$ for all $P$ and $e$ .",
"Eq.",
"() now implies that the indices $e$ for which $d_{P_i,e} - d_{P_i,e-1} >d_{P_i,e+1} - d_{P_i,e}$ are exactly the $e_{i,j}$ 's ($1 \\le j \\le n$ ).",
"Using concavity, we then observe that it is enough to check that $d_{P_i,e} = d_{P,e}$ for indices $e$ of the form $e_{i,j} + 1$ .",
"For those $e$ , we have by assumption: $\\begin{array}{r@{\\,\\,}}\\sum _P \\deg P \\cdot d_{P,e} & = \\dim _\\ell \\ker P_i(a,M(a))^e \\smallskip \\\\& = \\dim _{k(x)} \\ker P_i(x,M(x))^e \\smallskip \\\\& = \\deg _T P_i \\cdot d_{P_i,e}= \\sum _P \\deg P \\cdot d_{P_i,e}\\end{array}$ and thus $d_{P,e} = d_{P_i,e}$ for all $P$ because the inequalities $d_{P,e} \\ge d_{P_i,e}$ are already known."
],
[
"A bound on bad evaluation points",
"Let $M(x)$ be a square matrix of size $n$ with coefficients in $k[x]$ .",
"We set $X = x^p$ and assume that: (i) the entries of $M(x)$ have degree at most $pm$ (for a $m\\in \\mathbb {N}$ ), (ii) $M(x)$ is similar to a matrix with coefficients in $k(X)$ .",
"We are going to bound the number of values of $a$ for which the invariant factors of $M(x)$ do not specialize correctly at $x=a$ .",
"Similar discussions appear is Section 4 of Giesbrecht's thesis [18] in the (more complicated) case of integer matrices.",
"Our treatment is nevertheless rather different in many places.",
"The basic bound.",
"By assumption (ii), the characteristic polynomial $\\chi (x,T)$ lies in the subring $k[X,T]$ of $k[x,T]$ .",
"Lemma 3 The invariant factors $I_j(x,T)$ all belong to $k[X,T]$ .",
"Their degree with respect to $X$ is at most $mn$ .",
"By assumption (i), $\\chi (x,T)$ is a polynomial in $x$ of degree at most $pmn$ .",
"It then follows from Eq.",
"(REF ) that the $I_j(x,T)$ 's are polynomials in $x$ of degree at most $pmn$ as well.",
"Now, the assumption (ii) ensures that the $I_j(x,T)$ 's actually lie in $k(X)[T]$ .",
"This completes the proof.",
"Lemma 4 We assume that $p>n$ .",
"There are at most $\\deg _X \\chi (x,T) \\cdot (2n-1)$ points $a\\in k$ such that at least one of the $P_i(a,T)$ 's is not separable.",
"We have that $\\deg _X \\chi ^\\textrm {sep}(x,T) \\le \\deg _X \\chi (x,T)$ and $\\deg _T \\chi ^\\textrm {sep}(x,T) \\le n$ , since $\\chi ^\\textrm {sep}$ divides $\\chi $ .",
"Denote by $D(x)$ the discriminant of $\\chi ^\\textrm {sep}(x,T)$ with respect to $T$ .",
"Its degree in $X$ is at most $\\deg _X \\chi (x,T)\\cdot (2n-1)$ , and the assumption $p > n$ implies that $D(x)$ is not identically zero.",
"For any $a\\in k$ such that $D(a^p) \\ne 0$ , the polynomial $\\chi ^\\textrm {sep}(a^p,T)$ is separable, and the same holds for the $P_i(a^p,T)$ 's.",
"Noting that $k$ is perfect, the conclusion holds.",
"Proposition 5 We assume $p > n$ .",
"Let $a_1, \\ldots , a_N$ be elements in a separable closure of $k$ which are pairwise non conjugate over $k$ .",
"We assume that for each $i \\in \\lbrace 1, \\ldots , N\\rbrace $ , there exists $j \\in \\lbrace 1, \\ldots ,n\\rbrace $ with $I_j(a_i,T) \\ne I_{a_i,j}(T)$ .",
"Then: $\\sum _{i=1}^N \\deg (a_i) \\le 4mn\\cdot (n-1) + mn\\cdot (2n-1)$ where $\\deg (a_i)$ denotes the algebraicity degree of $a_i$ over $k$ .",
"We use the criteria of Proposition REF .",
"We start by putting away the values of $a$ for which at least one of the $P_i(a,T)$ 's is not separable.",
"By Lemma REF , there are at most $mn\\cdot (2n-1)$ such values.",
"We then have to bound from above the values of $a$ such that the equalities: $\\dim _{k(x)} \\ker P_i(x,M(x))^e= \\dim _{\\ell } \\ker P_i(a,M(a))^e$ may fail for some $i$ and some exponent $e=e_{i,j} + 1$ for some $j$ .",
"Let us fix such a pair $(i,e)$ .",
"Set $N(x) = P_i(x,M(x))^e$ for simplicity.",
"By assumption (i), the entries of $N(x)$ have degree at most $p m_{i,e}$ with $m_{i,e} = e \\cdot \\big ( m \\deg _T P_i + \\deg _X P_i \\big )$ .",
"On the other hand, we deduce from assumption (ii) that the $P_i(x,T)$ 's all lie in $k[X,T]$ and, as a consequence, that $N(x)$ is similar to a matrix with coefficients in $k(X)$ .",
"Define $d = \\dim _{k(x)} \\ker N(x)$ .",
"The equality $\\dim _{\\ell } \\ker N(a) = d$ then fails if and only if the minors of $N(x)$ of size $n-d$ all vanish at $x=a$ , i.e., if and only if the gcd $\\Delta (x)$ of these minors is divisible by the minimal polynomial of $a$ over $k$ , say $\\pi _a(x)$ .",
"Noting that $\\Delta (x) \\in k[X]$ , the latter condition is also equivalent to the fact that $\\pi _a(x)^p$ divides $\\Delta (x)$ in the ring $k[X]$ .",
"This can be possible for at most $\\deg _X \\Delta (x) \\le (n-d) m_{i,e} \\le (n-1) m_{i,e}$ values of $a$ .",
"Therefore, if $a_1, \\ldots , a_N$ are pairwise non-conjugate “unlucky values” of $a$ , the sum appearing in the statement of the proposition is bounded from above by: $\\textstyle (n-1) \\sum _{i,e} m_{i,e}& \\textstyle = m(n-1) \\sum _{i,e} e \\deg _T P_i \\\\& \\hspace{28.45274pt}\\textstyle + (n-1) \\sum _{i,e} e \\deg _X P_i.$ We notice that, when $i$ remains fixed, the number of exponents of the form $e_{i,j}+1$ ($1 \\le j < n$ ) is bounded from above by $e_{i,n} + 1$ .",
"The sum of these exponents is then at most: $\\textstyle \\big (\\sum _{j=1}^{n-1} e_{i,j}\\big ) + e_{i,n} + 1 =e_i + 1 \\le 2 e_i,$ where $e_i$ denotes the multiplicity of the factor $P_i(x,T)$ in the characteristic polynomial $\\chi (x,T)$ .",
"Our bound then becomes $2m(n-1) \\deg _T \\chi + 2(n-1) \\deg _X \\chi $ .",
"Using $\\deg _T \\chi = n$ and $\\deg _X \\chi \\le mn$ yields the bound.",
"A refinement.",
"For the applications we have in mind, we shall need a refinement of Proposition REF under the following hypothesis depending on a parameter $\\mu \\in \\mathbb {N}$ : $(\\mathbf {H}_\\mu )$ : the polynomial $\\chi $ has degree at most $p\\mu $ w.r.t $x$ .",
"We observe that $(\\mathbf {H}_\\mu )$ is fulfilled when $M(x)$ is a companion matrix whose entries are polynomials of degree at most $p\\mu $ .",
"Proposition 6 Under the assumptions of Prop.",
"REF and the additional hypothesis $(\\mathbf {H}_\\mu )$ , we have: $\\sum _{i=1}^N \\deg (a_i) \\le 2\\mu \\cdot (2n-1) + \\mu \\cdot (2n-1).$ Let $P(x,T)$ be any bivariate polynomial with coefficients in $k$ .",
"Set $N(x) = P(x,M(x))$ and let $\\delta (x)$ denote the gcd of the minors of size $s$ (for some integer $s$ ) of $N(x)$ .",
"We claim that: $\\deg _x \\delta (x) \\le p \\mu \\cdot \\deg _T P + s \\cdot \\deg _x P$ To prove the claim, we consider the Frobenius normal form $\\tilde{M}(x)$ of $M(x)$ and set $\\tilde{N}(x) = P(x, \\tilde{N}(x))$ .",
"Observe that any minor of $\\tilde{M}(x)$ vanishes or has the shape $\\pm c_1(x) \\cdots c_n(x)$ where $c_j(x)$ is a coefficient of $I_j(x,T)$ for all $j$ .",
"Noting that $\\deg _x I_1 + \\cdots + \\deg _x I_n = \\deg _x \\chi \\le p\\mu $ , we derive that all the minors of $\\tilde{M}(x)$ have degree at most $p\\mu $ .",
"Now write $P(x,T) = \\sum _{j=0}^{\\deg _T\\!P}a_j(x) T^j$ where the $a_i(x)$ 's lie in $k[x]$ .",
"Let $\\tilde{f}$ denote the $k[x]$ -linear endomorphism of $k[x]^n$ attached to the matrix $\\tilde{M}(x)$ .",
"Set $\\tilde{g} = P(x,\\tilde{f})$ ; it clearly corresponds to $\\tilde{N}(x)$ .",
"Given a vector space $E$ and $s$ linear endomorphisms $u_1, \\ldots , u_s$ of $E$ , let us agree to define $u_1 \\wedge \\cdots \\wedge u_s$ as $\\begin{array}{rc}\\quad E^{\\otimes s} & \\rightarrow & {\\textstyle \\bigwedge ^s} E \\\\x_1 \\otimes \\cdots \\otimes x_s & \\mapsto & u_1(x_1) \\wedge \\cdots \\wedge u_s(x_s).\\end{array}$ where $\\bigwedge ^s E$ is here defined as a quotient of $E^{\\otimes s}$ .",
"Expanding the exterior product $\\bigwedge ^s \\tilde{g}$ , we get: ${\\textstyle \\bigwedge ^s \\tilde{g}} = \\sum _{i_1, \\ldots , i_s = 0}^{\\deg _T P}a_{i_1}(x) \\cdots a_{i_s}(x) \\cdot \\tilde{f}^{i_1} \\wedge \\cdots \\wedge \\tilde{f}^{i_s}.$ Moreover, assuming for simplicity that $i_1 \\le i_2 \\le \\cdots \\le i_s$ and letting $i_0 = 0$ by convention, we can write: $\\tilde{f}^{i_1} \\otimes \\cdots \\otimes \\tilde{f}^{i_s} =\\bigcirc _{j=0}^s\\big [({\\textstyle \\bigotimes ^j \\text{id}}) \\otimes ({\\textstyle \\bigotimes ^{s-j} \\tilde{f}})^{i_j-i_{j-1}} \\big ],$ where $\\bigcirc $ denotes the composition of the above (pairwise commuting) maps.",
"We get that the entries of the matrix (in the canonical basis) of $\\tilde{f}^{i_1} \\wedge \\cdots \\wedge \\tilde{f}^{i_s}$ all have degree at most $p \\mu \\cdot i_s$ .",
"The same argument demonstrates that the degrees of the entries of the above matrix are not greater than: $p\\mu \\cdot \\max (i_1, \\ldots , i_s) \\le p \\mu \\cdot \\deg _T P$ when we no longer assume that the $i_j$ 's are sorted by nondecreasing order.",
"Therefore, back to Eq.",
"(REF ), we find that the entries of $\\bigwedge ^s \\tilde{N}(x)$ have degree at most $p \\mu \\cdot \\deg _T P + s \\cdot \\deg _x P$ .",
"It is then also the case of its trace, which is the same as the trace of $\\bigwedge ^sN(x)$ since $N(x)$ and $\\tilde{N}(x)$ are similar.",
"This finally implies the claimed inequality (REF ) because $\\delta (x)$ has to divide this trace.",
"The Proposition now follows by inserting the above input in the proof of Proposition REF ."
],
[
"Algorithms",
"We keep the matrix $M(x)$ satisfying the assumptions (i) and (ii) of §REF .",
"From now on, we assume that the only access we have to the matrix $M(x)$ passes through a black box invariant_factors_at${}_{M(x)}$ that takes as input an element $a$ lying in a finite extension $\\ell $ of $k$ and outputs instantly the invariant factors $I_{j,a}(T)$ of the matrix $M(a)$ .",
"Our aim is to compute the invariant factors of $M(x)$ .",
"We will propose two possible approaches: the first one is deterministic but rather slow although the second one is faster but probabilistic and comes up with a Monte-Carlo algorithm which may sometimes output wrong answers.",
"Throughout this section, the letter $D$ refers to a priori upper bound on the $X$ -degree of the characteristic polynomial of $M(x)$ .",
"One can of course always take $D = mn$ but better bounds might be available in particular cases.",
"Similarly we reserve the letter $F$ for an upper bound on the sum of degrees of “unlucky evaluation points”.",
"Proposition REF tells us that $mn(6n-5)$ is always an acceptable value for $F$ .",
"Remember however that this value can be lowered to $3 \\mu (2n-1)$ under the hypothesis $(\\mathbf {H}_\\mu )$ thanks to Proposition REF .",
"We will always assume that $F \\ge D$ .",
"For simplicity of exposition, we assume from now on that $k = \\mathbb {F}_q$ is a finite field of cardinality $q$ (it is more difficult and the case of most interest for us).",
"Deterministic.",
"The discussion of §REF suggests the following algorithm whose correctness follows directly from the definition of $F$ together with the assumption $F \\ge D$ .",
"Algorithm invariant_factors_deterministic Input: $M(x)$ satisfying (i) and (ii), $D$ , $F$ with $F \\ge D$ Output: The invariant factors of $M(x)$ 1.",
"Construct an extension $\\ell $ of $\\mathbb {F}_q$ of degree $F+1$ 1. and pick an element $a \\in \\ell $ such that $\\ell = \\mathbb {F}_q[a]$ 1.",
"Cost: $O\\tilde{~}(F)$ operations in $\\mathbb {F}_q$ 2.",
"$I_{1,a}(T), \\ldots , I_{n,a}(T) = \\texttt {invariant\\_factors\\_at}{}_{M(x)}(a)$ 3. for $j=1,\\ldots ,n$ 4.",
"Find $I_j(x,T)$ of degree $\\le D$ s.t.",
"$I_j(a,T) = I_{j,a}(T)$ 5. return $I_1(x,T), \\ldots , I_n(x,T)$ Proposition 7 The algorithm above requires only one call to the black box invariant_factors_at${}_{M(x)}$ with an input of degree exactly $F+1$ .",
"Probabilistic.",
"We now present a Monte-Carlo algorithm: Algorithm invariant_factors_montecarlo Input: $M(x)$ s.t.",
"(i) and (ii), $\\varepsilon \\in (0,1)$ , $D$ , $F$ with $F \\ge D$ Output: The invariant factors of $M(x)$ 1.",
"Find the smallest integer $s$ such that: $2 \\cdot \\frac{(D{+}s{+}1)^2}{s (q^s - 2F)} +\\frac{1}{2} \\cdot \\Big (\\frac{4F}{q^s} \\Big )^{\\!",
"(D{-}2)/s} \\:\\le \\:\\varepsilon $ 1. and set $K = \\lceil \\frac{3D}{s} \\rceil $ and $k = \\lceil \\frac{D+1}{s} \\rceil $ .",
"2. for $i=1,\\ldots ,K$ 3. pick at random $a_i \\in \\mathbb {F}_{q^s}$ s.t.",
"$\\mathbb {F}_{q^s} = \\mathbb {F}_q[a_i]$ 3.",
"Cost: $O\\tilde{~}(s)$ operations in $\\mathbb {F}_q$ 4.",
"$I_{1,i}(T), \\ldots , I_{n,i}(T) = \\texttt {invariant\\_factors\\_at}{}_{M(x)}(a_i)$ 5. for $j=1,\\ldots ,n$ 6.",
"$d_j = \\max _i \\deg (I_{1,i} (T) \\cdot I_{2,i} (T) \\cdots I_{j,i}(T))$ 7. select $I \\subset \\lbrace 1, \\ldots , K\\rbrace $ of cardinality $k$ s.t.",
"7.",
"(i) $\\deg (I_{1,i}(T)\\cdot I_{2,i}(T) \\cdots I_{j,i}(T)) = d_j$ for all $i \\in I$ 7.",
"(ii) the $a_i$ are pairwise non conjugate for $i \\in I$ 7.",
"Remark: if such $I$ does not exist, raise an error 8. compute $I_j \\in \\mathbb {F}_q[X,T]$ of $X$ -degree $\\le D$ s.t.",
"8.",
"$I_j(a_i,T) = I_{j,i}(T)$ for all $i \\in I$ 7.",
"Cost: $O\\tilde{~}(D)$ operations in $\\mathbb {F}_q$ 9. return $I_1(x,T), \\ldots , I_n(x,T)$ Proposition 8 We have $s \\in O(\\log \\frac{FD}{\\varepsilon })$ .",
"Moreover: $\\bullet $ Correctness: Algorithm invariant_factors_montecarlo fails or returns a wrong answer with probability at most $\\varepsilon $ .",
"$\\bullet $ Complexity: It performs $\\lceil \\frac{3D}{s} \\rceil $ calls to the black box with inputs of degree $s$ and $O\\tilde{~}(n(D + \\log \\frac{F}{\\varepsilon }))$ operations in $\\mathbb {F}_q$ .",
"The first assertion is left to the reader.",
"Let $\\mathcal {A}$ be the set of elements $a$ of $\\mathbb {F}_{q^s}$ such that $\\mathbb {F}_q[a] = \\mathbb {F}_{q^s}$ .",
"It is an easy exercise to prove that $\\mathcal {A}$ has at least $\\frac{q^s}{2}$ elements (the bound is not sharp).",
"Let $\\mathcal {C}_1, \\ldots , \\mathcal {C}_C$ be the conjugacy classes (under the Galois action) in $\\mathcal {A}$ .",
"Remark that each $\\mathcal {C}_i$ has by definition $s$ elements, so that $C \\ge \\frac{q^s}{2s}$ .",
"We say that a conjugacy class is bad if it contains one element $a$ for which $I_j(a,T) \\ne I_{a,j}(T)$ for some $j$ .",
"Otherwise, we say that it is good.",
"Let $B$ (resp.",
"$G$ ) be the number of bad (resp.",
"good) classes.",
"We have $B+G = C$ and $B \\le \\frac{F}{s}$ by definition of $F$ .",
"The algorithm invariant_factors_montecarlo succeeds if there exist at least $k$ indices $i$ for which the corresponding $a_i$ 's lie in pairwise distinct good classes.",
"This happens with probability at least: $\\frac{1}{C^K} \\cdot \\sum _{j=k}^K {\\textstyle \\binom{K}{j}} \\cdot G (G-1) \\cdots (G-k+1) \\cdot G^{j-k} \\cdot B^{K-j}.$ (The above formula gives the probability that the first $k$ good classes are pairwise distinct, which is actually stronger than what we need.)",
"The above quantity is at least equal to $\\left(1 - \\frac{k}{G}\\right)^{\\!k}\\: \\cdot \\left(1 \\: - \\:\\sum _{j=0}^{k-1} {\\textstyle \\binom{K}{j}} \\cdot \\Big (\\frac{G}{C}\\Big )^j\\cdot \\Big (\\frac{B}{C}\\Big )^{K-j}\\right).$ Moreover for $j \\le k-1$ , we have: $\\Big (\\frac{G}{C}\\Big )^j\\cdot \\Big (\\frac{B}{C}\\Big )^{K-j}& \\le \\Big (\\frac{BG}{C^2}\\Big )^j \\!\\cdot \\!",
"\\Big (\\frac{B}{C}\\Big )^{K-2j}\\le \\frac{1}{2^{2j}} \\cdot \\Big (\\frac{2F}{q^s} \\Big )^{\\!K{-}2j}\\\\& \\le \\frac{1}{2^K} \\cdot \\Big (\\frac{4F}{q^s} \\Big )^{\\!K{-}2j}\\le \\frac{1}{2^K} \\cdot \\Big (\\frac{4F}{q^s} \\Big )^{\\!",
"(D{-}2)/s}.$ Therefore the probability of success is at least: $\\left(1 - \\frac{k}{G}\\right)^{\\!k}\\: \\!\\cdot \\!\\left(1 \\: - \\:\\frac{1}{2} \\cdot \\Big (\\frac{4F}{q^s} \\Big )^{\\!",
"(D{-}2)/s}\\right).$ Using $k \\le \\frac{D+s+1}{s}$ and $G \\ge \\frac{q^2-2F}{2s}$ , we find that the probability of failure is at most the LHS of Eq.",
"(REF ).",
"The correctness is proved.",
"As for the complexity, the results are obvious."
],
[
"Computing invariant factors of the p-curvature",
"Throughout this section, we fix a finite field $k = \\mathbb {F}_q$ of cardinality $q$ and characteristic $p$ .",
"We endow the field of rational functions $k(x)$ with the natural derivation $f\\mapsto f^{\\prime }$ ."
],
[
"The case of differential modules",
"We recall that a differential module over $k(x)$ is $k(x)$ -vector space $M$ endowed with an additive map $\\partial : M \\rightarrow M$ satisfying the following Leibniz rule: $\\forall f \\in k(x), \\, \\forall m \\in M, \\quad \\partial (fm) = f^{\\prime } \\cdot m + f \\cdot \\partial (m).$ The $p$ -curvature of a differential module $M$ is the mapping $\\partial ^p = \\partial \\circ \\cdots \\circ \\partial $ ($p$ times).",
"Using the fact that the $p$ -th derivative of any $f \\in k(x)$ vanishes, we derive from the Leibniz relation above that $\\partial ^p$ is $k(x)$ -linear endomorphism of $M$ .",
"It follows moreover from [4] that $\\partial ^p$ is defined over $k(x^p)$ , in the sense that there exists a $k(x)$ -basis of $M$ in which the matrix of $\\partial ^p$ has coefficients in $k(x^p)$ .",
"In particular, all the invariant factors of the $p$ -curvature have their coefficients in $k(x^p)$ .",
"Statement of the main Theorem.",
"From now on, we fix a differential module $(M, \\partial )$ .",
"We assume that $M$ is finite dimensional over $k(x)$ and let $r$ denote its dimension.",
"We pick $(e_1, \\ldots , e_r)$ a basis of $M$ and let $A$ denote the matrix of $\\partial $ with respect to this basis.",
"We write $A = \\frac{1}{f_A} \\tilde{A}$ where $f_A$ and the entries of $\\tilde{A}$ all lie in $k[x]$ .",
"Let $d$ be an upper bound on the degrees of all these polynomials.",
"The aim of this section is to design fast deterministic and probabilistic algorithms for computing the invariant factors of the $p$ -curvature of $(M, \\partial )$ .",
"The following Theorem summarizes our results.",
"Theorem 9 We assume $p > r$ .",
"1.",
"There exists a deterministic algorithm that computes the invariant factors of the $p$ -curvature of $(M, \\partial )$ within $O\\tilde{~}\\big (d^{\\hspace{0.56905pt}\\omega +\\frac{3}{2}} r^{\\omega +2} \\sqrt{p}\\big )$ operations in $k = \\mathbb {F}_q$ .",
"2.",
"Let $\\varepsilon \\in (0,1)$ .",
"There exists a Monte-Carlo algorithm computing the invariant factors of the $p$ -curvature of $(M, \\partial )$ in $O\\tilde{~}\\big (d^{\\hspace{0.56905pt}\\omega + \\frac{1}{2}} r^{\\omega } \\cdot (dr - \\log \\varepsilon ) \\cdot \\sqrt{p}\\big )$ operations in $k = \\mathbb {F}_q$ .",
"This algorithm returns a wrong answer with probability at most $\\varepsilon $ .",
"In what follows, we will use the notation $A_p(x)$ for the matrix of the $p$ -curvature of $(M, \\partial )$ with respect to the distinguished basis $(e_1, \\ldots , e_r)$ .",
"Given an element $a$ lying in a finite extension $\\ell $ of $k$ , we denote by $A_p(a) \\in {M}_r(\\ell )$ the matrix deduced from $A_p$ by evaluating it at $x=a$ .",
"The similarity class of $A_p(a)$ .",
"Let $S$ be an irreducible polynomial over $k$ .",
"Set $\\ell = k[u]/S$ and let $a$ denote the image of the variable $u$ in $\\ell $ .",
"We assume that $S$ does not divide $f_A$ , i.e., $f_A(a) \\ne 0$ .",
"The first ingredient we need is the construction of an auxiliary matrix which is similar to $A_p(a)$ .",
"This construction comes from our previous paper [4].",
"Let us recall it briefly.",
"We define the ring $\\ell [[t]]^{\\mathrm {dp}}$ of Hurwitz series whose elements are formal infinite sums of the shape: $a_0 + a_1 \\gamma _1(t) + a_2 \\gamma _2(t) + \\cdots + a_n \\gamma _n(t)+ \\cdots $ and on which the addition is straightforward and the multiplication is governed by the rule $\\gamma _i(t) \\cdot \\gamma _j(t) = \\binom{i+j}{i}\\gamma _{i+j}(t)$ .",
"(The symbol $\\gamma _i(t)$ should be thought of as $\\frac{t^i}{i!",
"}$ .)",
"We moreover endow $\\ell [[t]]^{\\mathrm {dp}}$ with the derivation defined by $\\gamma _i(t)^{\\prime } =\\gamma _{i-1}(t)$ (with the convention that $\\gamma _0(t) = 1$ ) and the projection map $\\text{pr} : \\ell [[t]]^{\\mathrm {dp}}\\rightarrow \\ell $ sending the series given by Eq.",
"(REF ) to its constant coefficient $a_0$ .",
"We shall often use the alternative notation $f(0)$ for $\\text{pr}(f)$ .",
"If $f \\in \\ell [[t]]^{\\mathrm {dp}}$ is given by the series (REF ), we then have $a_n = f^{(n)}(0)$ for all nonnegative integers $n$ .",
"We have a homomorphism of rings: $\\psi _S : k[x][{\\textstyle \\frac{1}{f_A}}] \\rightarrow \\ell [[t]]^{\\mathrm {dp}}, \\quad f(x) \\mapsto \\sum _{i=0}^{p-1} f^{(i)}(a) \\gamma _i(t).$ It is easily checked that $\\psi _S$ commutes with the derivation.",
"We can then consider the differential module over $\\ell [[t]]^{\\mathrm {dp}}$ obtained from $(M,\\partial )$ by scalar extension.",
"By definition, it corresponds to the differential system $Y^{\\prime } = \\psi _S(A) \\cdot Y$ .",
"The benefit of working over $\\ell [[t]]^{\\mathrm {dp}}$ is the existence of an analogue of the well-known Cauchy–Lipschitz Theorem [4].",
"This notably implies the existence of a fundamental matrix of solutions, i.e., an $r\\times r$ matrix $Y_S$ with entries in $\\ell [[t]]^{\\mathrm {dp}}$ , and satisfying: $Y^{\\prime }_S = \\psi _S(A) \\cdot Y_S\\quad \\text{and} \\quad Y_S(0) = \\text{I}_r$ with $\\text{I}_r$ the identity matrix of size $r$ .",
"Moreover, as explained in more details later, the construction of $Y_S$ is effective.",
"For any integer $n \\ge 0$ , we let $Y_S^{(n)}$ denote the matrix obtained from $Y_S$ by taking the $n$ -th derivative entry-wise.",
"The next proposition is a consequence of [4].",
"Proposition 10 The matrices $A_p(a)$ and $-Y_S^{(p)}(0)$ are similar over $\\ell $ .",
"Fast computation of $Y_S^{(p)}(0)$ .",
"We recall that $Y_S$ is defined as the solution of the system (REF ).",
"Remembering that we have written $A= \\frac{1}{f_A} \\tilde{A}$ , we obtain the relation: $\\psi _S(f_A) \\cdot Y_S^{\\prime } = \\psi _S(\\tilde{A}) \\cdot Y_S.$ Write $f_A = \\sum _{i=0}^d f_i \\cdot (x-a)^i$ and $\\tilde{A} = \\sum _{i=0}^d \\tilde{A}_i \\cdot (x-a)^i$ where the $f_i$ 's lie in $\\ell $ and the $A_i$ 's are square matrices of size $r$ with entries in $\\ell $ .",
"Remark that $f_0$ does not vanish because it is equal to $f_A(a)$ .",
"Note moreover that the $f_i$ 's can be computed for a cost of $O\\tilde{~}(d)$ operations in $k$ using divide-and-conquer techniques.",
"Given a fixed pair of indices $(i^{\\prime },j^{\\prime })$ , the same discussion applies to the collection of the $(i^{\\prime },j^{\\prime })$ -entries of the $A_i$ 's.",
"The total cost for computing the decompositions of $f_A$ and $\\tilde{A}$ is then $O\\tilde{~}(dr^2)$ .",
"Now, coming back to the definitions, we find that $\\psi _S(f_A) =\\sum _{i=0}^d i!",
"\\: f_i \\cdot \\gamma _i(t)$ and $\\psi _S(\\tilde{A}) = \\sum _{i=0}^d i!",
"\\: \\tilde{A}_i \\cdot \\gamma _i(t)$ .",
"Eq.",
"(REF ) yields the recurrence: $Y_S^{(n+1)}(0) = \\sum _{i=0}^{\\min (n,d)} B_i(n) \\cdot Y_S^{(n-i)}(0)$ where the $B_i \\in M_r(\\ell [u])$ are defined by: $f_0 B_i = u (u{-}1) \\cdots (u{-}i{+}1) \\cdot \\big ( \\tilde{A}_i - (u{-}i) f_{i+1} \\cdot \\text{I}_r \\big )$ with the convention that $f_{d+1} = 0$ .",
"Now setting: $Z_n = \\left(\\begin{matrix}Y_S^{(n-d)}(0) \\\\Y_S^{(n-d+1)}(0) \\\\\\vdots \\\\Y_S^{(n)}(0)\\end{matrix}\\right), \\quad B = \\left(\\begin{matrix}& \\text{I}_r \\\\& & \\text{I}_r \\\\& & & \\ddots \\\\& & & & \\text{I}_r \\\\B_d & \\cdots & \\cdots & \\cdots & B_0\\end{matrix} \\right)$ (with the convention $Y_s^{(i)}(0) = 0$ when $i < 0$ ), the recurrence (REF ) becomes $Z_{n+1} = B(n)\\cdot Z_n$ .",
"Hence, we obtain $Z_p = B(p-1) \\cdot B(p-2) \\cdots B(0) \\cdot Z_0$ from what we finally get that $Y_S^{(p)}(0)$ is the $(r \\times r)$ -matrix located at the bottom right corner of $B(p-1) \\cdot B(p-2) \\cdots B(0)$ .",
"The computation of the former matrix factorial can be performed efficiently using a variation of the Chudnovskys' algorithm [12], [6].",
"Combining this with Proposition REF , we end up with the following.",
"Proposition 11 The invariant factors of $A_p(a)$ can be computed in $O\\tilde{~}(d^\\omega r^\\omega \\sqrt{dp})$ operations in the field $\\ell $ .",
"Note that $B$ is a square matrix of size $(d{+}1)r$ .",
"Moreover coming back to (REF ), we observe that the entries of $B$ all have degree at most $d$ .",
"By [5] the matrix factorial $- B(p-1) \\cdot B(p-2) \\cdots B(0)$ can then be computed for the cost of $O\\tilde{~}(d^\\omega r^\\omega \\sqrt{dp})$ operations in $\\ell $ .",
"By [30], the invariant factors of its submatrix $-Y_S^{(p)}(0)$ can be obtained for an extra cost of $O\\tilde{~}(r^\\omega )$ operations in $\\ell $ (which is negligible compared to the previous one).",
"Using Proposition REF these invariant factors are also those of $A_p(a)$ , and we are done.",
"Conclusion.",
"Proposition REF yields an acceptable primitive invariant_factors_at${}_{A_p(x)}$ .",
"Plugging it in the algorithm invariant_factors_deterministic and using the parameters $D = dr$ and $F = 6dr(r-1)$ , we end up with an algorithm that computes the invariant factors of $A_p(x)$ for the cost of one unique call to invariant_factors_at${}_{A_p(x)}$ with an input lying in an extension $\\ell /k$ of degree $F+1$ (cf.",
"Proposition REF ).",
"By Proposition REF , we find that the total complexity of the obtained algorithm is $O\\tilde{~}\\big (d^{\\hspace{0.56905pt}\\omega +\\frac{3}{2}} r^{\\omega +2} \\sqrt{p}\\big )$ operations in $\\mathbb {F}_q$ .",
"The first part of Theorem REF is then established.",
"The second part is obtained in a similar fashion using the algorithm invariant_factors_montecarlo together with Proposition REF for correctness and complexity results."
],
[
"The case of differential operators",
"The ring of differential operators $k(x)\\!\\left<\\partial \\right>$ is the ring of usual polynomials over $k(x)$ in the variable $\\partial $ except that the multiplication is ruled by the relation $\\partial \\cdot f = f \\cdot \\partial + f^{\\prime }$ .",
"We define similarly the ring $k[x]\\!\\left<\\partial \\right>$ .",
"We say that $L \\in k[x]\\!\\left<\\partial \\right>$ has bidegree $(d,r)$ if it has degree $d$ with respect to $x$ and degree $r$ with respect to $\\partial $ .",
"If $L$ is a differential operator in $k(x)\\!\\left<\\partial \\right>$ , one easily checks that the set $k(x)\\!\\left<\\partial \\right> L$ of left multiples of $L$ is a left ideal of $k(x)\\!\\left<\\partial \\right>$ .",
"The quotient $M_L = k(x)\\!\\left<\\partial \\right>/k(x)\\!\\left<\\partial \\right>L$ is then a vector space over $k(x)$ .",
"It is moreover endowed with a map $\\partial : M_L \\rightarrow M_L$ given by the left multiplication by $\\partial $ .",
"This map turns $M_L$ into a differential module.",
"We shall prove in this section that the complexities announced in Theorem REF can be improved in the case of differential modules coming from differential operators.",
"Below is the statement of our precise result.",
"Theorem 12 Let $L \\in k[x]\\!\\left<\\partial \\right>$ be a differential operator of bidegree $(d,r)$ .",
"We assume $p > r$ .",
"1.",
"There exists a deterministic algorithm that computes the invariant factors of the $p$ -curvature of $M_L$ within $O\\tilde{~}\\big ((d+r)^{\\omega +1} d^{\\frac{1}{2}} r \\cdot \\sqrt{p}\\big )$ operations in $k = \\mathbb {F}_q$ .",
"2.",
"Let $\\varepsilon \\in (0,1)$ .",
"There exists a Monte-Carlo algorithm that computes the invariant factors of the $p$ -curvature of $M_L$ in $O\\tilde{~}\\big ((d+r)^\\omega d^{\\frac{1}{2}} \\cdot (d - \\log \\varepsilon ) \\cdot \\sqrt{p}\\big )$ operations in $k = \\mathbb {F}_q$ .",
"This algorithm returns a wrong answer with probability at most $\\varepsilon $ .",
"Better bounds.",
"From now on, we fix a differential operator $L \\in k(x)\\!\\left<\\partial \\right>$ of bidegree $(d,r)$ .",
"We denote by $A_p(x)$ the matrix of the $p$ -curvature of $M_L$ with respect to the canonical basis $(1, \\partial , \\ldots , \\partial ^{r-1})$ .",
"If $a_r(x)$ is the leading coefficient of $L$ (with respect to $\\partial $ ), it follows from [13] that $A_p(x)$ has the form $A_p(x) = \\frac{1}{a_r(x)^p} \\cdot \\tilde{A}_p(x)$ where $\\tilde{A}_p(x)$ is a matrix with polynomial entries of degree at most $pd$ .",
"Proposition 13 The matrix $\\tilde{A}_p(x)$ satisfies the hypothesis $(\\textbf {H}_{r+d})$ (introduced just before Proposition REF ).",
"This is a direct consequence of Lemma 3.9 and Theorem 3.11 of [3].",
"The similarity class of $A_p(a)$ .",
"We now revisit Proposition REF when the differential module comes from the differential operator $L$ .",
"We fix an irreducible polynomial $S \\in k[x]$ and assume that $S$ is coprime with the leading coefficient $a_r(x)$ of $L$ .",
"We set $\\ell = k[x]/S$ and let $a$ denote the image of $x$ is $\\ell $ .",
"We define $t = x - a \\in \\ell [x]$ and consider the ring of differential operators $\\ell [x]\\!\\left<\\partial \\right>$ .",
"The latter acts on $\\ell [[t]]^{\\mathrm {dp}}$ by letting $\\partial $ act as the derivation.",
"Let $Y_S$ be the fundamental system of solutions of the equation $Y_S^{\\prime } = \\psi _S(A) \\cdot Y_S$ where $A$ is the companion matrix which gives the action of $\\partial $ on $M_L$ .",
"It takes the form: $Y_S = \\left(\\begin{matrix}y_0 & y_1 & \\cdots & y_{r-1} \\\\y^{\\prime }_0 & y^{\\prime }_1 & \\cdots & y^{\\prime }_{r-1} \\\\\\vdots & \\vdots & & \\vdots \\\\y_0^{(r-1)} & y_1^{(r-1)} & \\cdots & y_{r-1}^{(r-1)}\\end{matrix}\\right)$ where $y_j \\in \\ell [[t]]^{\\mathrm {dp}}$ is the unique solution of the differential equation $L y_j = 0$ with initial conditions $y_j^{(n)}(0) = \\delta _{j,n}$ (where $\\delta _{\\cdot ,\\cdot }$ is the Kronecker symbol) for $0 \\le n < r$ .",
"We introduce the Euler operator $\\theta = t \\cdot \\partial \\in \\ell [x]\\!\\left<\\partial \\right>$ .",
"Using the techniques of [3], one can write $L \\cdot \\partial ^d = \\sum _{i=0}^{d+r} b_i(\\theta ) \\partial ^i$ within $O\\tilde{~}((r+d)d)$ operations in $\\ell $ .",
"Here the $b_i$ 's are polynomials with coefficients in $\\ell $ of degree at most $d$ .",
"One can check moreover that the polynomial $b_{d+r}$ is constant equal to $a_r(a)$ ; in particular, it does not vanish thanks to our assumption on $S$ .",
"For all $j$ , define $z_j = \\sum _{n=0}^\\infty y_j^{(n)}(0)\\gamma _{n+d}(t)$ .",
"Clearly $\\partial ^d z_j = y_j$ , so that we have $\\left(\\sum _{i=0}^{d+r} b_i(\\theta ) \\partial ^i\\right) \\cdot z_i = 0$ for all $i$ .",
"Noting that $\\theta $ acts on $\\gamma _n(t)$ by multiplication by $n$ , we get the recurrence relation: $\\forall n \\ge 0, \\quad \\sum _{i=0}^{d+r} b_i(n) \\cdot y_j^{(n+i-d)}(0) = 0$ with the convention that $y_j^{(n)} = 0$ when $n < 0$ .",
"Letting: $Z_n & = \\left(\\begin{matrix}y_0^{(n-d)}(0) & \\cdots & y_{r-1}^{(n-d)}(0) \\smallskip \\\\y_0^{(n-d+1)}(0) & \\cdots & y_{r-1}^{(n-d+1)}(0) \\\\\\vdots & & \\vdots \\\\Y_0^{(n+r-1)}(0) & \\cdots & y_{r-1}^{(n+r-1)}(0)\\end{matrix}\\right) \\in {M}_{d+r,r}(\\ell ) \\medskip \\\\\\text{and} \\quad B & =\\frac{-1}{a_r(a)} \\cdot \\left(\\begin{matrix}& 1 \\\\& & \\ddots \\\\& & & 1 \\\\b_0 & b_1 & \\cdots & b_{d+r-1}\\end{matrix} \\right) \\in {M}_{d+r,d+r}(\\ell )$ the above recurrence rewrites $Z_{n+1} = B(n) Z_n$ .",
"Solving the recurrence, we get $Z_p = B(p-1) \\cdots B(0) \\cdot Z_0$ , and we derive that $Y_S^{(p)}(0)$ is the $(r\\times r)$ matrix located at the bottom right corner of $B(p-1) \\cdot B(p-2) \\cdots B(0)$ .",
"Using Proposition REF and [5], we end up with the following proposition (compare with Proposition REF ).",
"Proposition 14 The invariant factors of $A_p(a)$ can be computed in $O\\tilde{~}((d+r)^\\omega \\sqrt{dp})$ operations in the field $\\ell $ .",
"Conclusion.",
"The final discussion is now similar to the one we had in the case of differential modules.",
"Proposition REF provides the primitive invariant_factors_at${}_{A_p(x)}$ .",
"Using it in the algorithms invariant_factors_deterministic and invariant_factors_montecarlo with the parameters $D = d$ and $F = 3d(2r-1)$ (coming from the combination of Propositions REF and REF ), we respectively end up with deterministic and Monte-Carlo algorithms whose complexities agree with the ones announced in Theorem REF .",
"It is instructive to compare the methods and results of this section with those of our previous paper [3].",
"We remark that the matrix factorial considered above is nothing but the specialization at $\\theta = 0$ of the matrix factorial in [3].",
"Although the theoretical approaches of the two papers are definitively different, they lead to very similar computations.",
"However, each of them has its own advantages and disadvantages.",
"On the one hand, the methods of [3] deal with characteristic polynomials only and cannot see invariant factors.",
"On the other hand, they do not require the assumption $a_r(a) \\ne 0$ (that is why we always took $a = 0$ in [3]) and can handle at the same time the local computations at the point $a$ and around it, i.e., they provide roughly speaking a framework which allows to work modulo $(x{-}a)^{pn}$ for some integer $n$ fixed in advance (not just $n=1$ ) without increasing the cost with respect to $p$ .",
"The practical consequence is that the methods of the current paper end up with algorithms whose cost is weakened by a factor $\\sqrt{d}$ compared to what we might have expected at first.",
"It would be interesting to find a general theoretical setting unifying the two approaches discussed above and allowing the benefits of both of them."
],
[
"Solving a physical application",
"In [10], a globally nilpotent differential operator $\\phi _H^{(6)}$ was introduced in order to model the 6-particle contribution to the square lattice Ising model.",
"As shown in loc.",
"cit., this operator factors as a product of differential operators of smaller orders.",
"The factor which is the least understood is called $L_{23}$ and has order 23.",
"Actually $L_{23}$ has not been computed so far because its size is too large.",
"Nevertheless there exists a multiple of $L_{23}$ which has a more reasonable size: its bidegree is $(140,77)$ .",
"It turns out that this multiple, say $L_{77}$ , has been determined modulo several prime numbers.",
"Based on this computation and using the strategy developed in this paper, we were able to study a bit further the factorization of $L_{23}$ , answering a question of the authors of [10].",
"Proposition 15 The operator $L_{23}$ cannot be factorized as a product $L_{21}\\cdot L_2$ where $L_2$ is an operator of order 2, and $L_{21}$ is an operator of order 21 whose differential module is isomorphic to a symmetric product $\\text{Sym}^n M$ for an integer $n > 1$ and a differential module $M$ .",
"We argue by contradiction by assuming that such a factorization exists.",
"This would imply that, for all $p$ the matrix $A_{23,p}$ of the $p$ -curvature of $L_{23} \\text{ mod } p$ decomposes: $A_{23,p} = \\left(\\begin{matrix} A_{2,p} & \\star \\\\ 0 & A_{21,p} \\end{matrix}\\right)$ where $A_{2,p}$ (resp.",
"$A_{21,p}$ ) is the square matrix of size 2 (resp.",
"21) and $A_{21,p}$ is similar to a symmetric product of a $d \\times d$ matrix $A_{d,p}$ .",
"We now pick $p = 32647$ for which $L_{77} \\text{ mod } p$ is known.",
"Using Proposition REF , we were able to determine the invariant factors of the $p$ -curvature of $A_{77,p}(15)$ .",
"The computation ran actually rather fast: just a few minutes.",
"We observed that the generalized eigenspace of $A_{77,p}(15)$ for the eigenvalue 0 has dimension 23.",
"Combining this with that fact that $L_{23}$ is a factor of $L_{77}$ whose $p$ -curvature is nilpotent, we deduce that the restriction of $A_{77,p}(15)$ to this characteristic space is similar to $A_{23,p}(15)$ .",
"Arguing similarly, we determine the Jordan type of $A_{23,p}(15)$ : Table: NO_CAPTIONMoreover the writing (REF ) would imply that for all $m$ : $\\begin{array}{r}& 0 \\le \\text{rank}\\big (A_{23,p}(15)^m\\big ) - \\text{rank}\\big (A_{21,p}(15)^m\\big )\\le 2 \\smallskip \\\\\\text{and} &\\text{rank}\\big (A_{21,p}(15)^m\\big ) =\\displaystyle \\binom{n-1 + \\text{rank}\\big (A_{d,p}(15)^m\\big )}{n}.\\end{array}$ There is only one way to satisfy these numerical constraints which consists in taking $n=2$ and: Table: NO_CAPTIONSince the sequence $\\text{rank}\\big (A_{d,p}(15)^m\\big ) - \\text{rank}\\big (A_{d,p}(15)^{m+1}\\big )$ has to be non-increasing, this is impossible.",
"=1"
],
[
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]
] | 1605.06126 |
[
[
"Exploring Wake Interaction for Frequency Control in Wind Farms"
],
[
"Abstract The increasing integration of wind generation is accompanied with a growing concern about secure and reliable power system operation.",
"Due to the intermittent nature of wind, the base-load units need to cycle significantly more than they were designed for, resulting in reduced life cycle and increased costs.",
"Therefore, it is becoming necessary for wind turbines to take part in frequency control, and reduce the need for additional ancillary services provided by conventional generators.",
"In this paper, we propose an optimised operation strategy for the wind farms.",
"In this strategy, we maximise the kinetic energy of wind turbines by an optimal combination of the rotor speed and the pitch angle.",
"We exploit the wake interaction in a wind farm, and de-load some of the up-wind turbines.",
"We show that the kinetic energy accumulated in the rotating masses of the WTs can be increased compared to the base case without compromising efficiency of the wind farm.",
"In a specific system, we show that by implementing this strategy, and injecting the stored kinetic energy of the WTs' rotors into the system during a frequency dip, we can delay the system frequency nadir up to 30 s."
],
[
"Introduction",
"The dwindling fossil fuel resources, and their associated greenhouse gas emissions that significantly contribute to global warming, are encouraging nations to move toward renewable energy sources (RES).",
"Among the RES, wind is one of the most economically viable options.",
"In 2012, with 19% growth, wind's annual growth was the highest among all RES[1].",
"USA, the world's largest electricity consumer, is planning to produce 20% of its electricity from wind by 2030 [2].",
"Denmark, one of the pioneers of wind technology and the country with the highest penetration of wind in its electricity system, set the target of achieving 50% of its electricity from wind power by 2020, and 100% renewable by 2035 [1].",
"In Australia, wind is anticipated to play a major role in reaching the 20% renewable energy target (RET) by 2020[1].",
"It has been predicted that from 2014 to 2035, under the 450ppm scenario, US$3,027 billion will be invested in wind generation technologies, which is the highest investment among all renewable and conventional generation technologies [3].",
"It appears that penetration of wind energy in the power system of most countries will keep increasing in the foreseeable future[1].",
"Although financially wind is one of the most viable RES, and can compete even with some of the conventional resources [4], technically, it is considered a less reliable resource because of its intermittent nature.",
"Nowadays, variable speed wind turbines (VSWT), including Type III, using doubly fed induction generators (DFIG), and Type IV, with a generator connected through a fully rated converter (FRC), are considered the most promising technologies because of their ability to optimise the power extraction for over a wide wind speed range, as well as for being able to comply with grid codes' connection requirements.",
"Therefore, approximately 95% of wind turbines (WTs) installed all around the world are either Type III or IV[5].",
"Nonetheless, compared to the fixed speed wind turbines (FSWT), VSWTs do not have inherent inertial response [6].",
"A study in [7] has suggested that high penetration of Type III WTs can change the system frequency behaviour, which is characterized by the rate of change of frequency (ROCOF) and frequency nadir, so it is necessary for system operators to address these issues.",
"Therefore, to increase integration of wind generation in power systems, wind will need to offer frequency control ancillary services, which will in turn reduce the pressure on conventional generators [7], [8], [9].",
"According to [10], kinetic energy released by a WT exceeds that released by a synchronous generator (SG), which makes wind generation even more attractive to use for inertial contribution, including primary frequency control.",
"The wind speed namely doesn't change significantly in short durations while the primary control is active, so the kinetic energy stored in rotors of partially loaded WTs can be used in frequency regulation.",
"Due to the effective decoupling between the mechanical and the electrical systems of a WT, this capability needs to be emulated through appropriate control.",
"An obvious issue in participation of WTs in frequency control is that it requires a WT to operate below its optimum power output for a period of time, which negatively impacts the efficiency of the WT.",
"To partly overcome the negative financial impacts associated with spilling the wind energy, we propose to take advantage of the wake interaction within a wind farm (WF).",
"The extraction of energy from wind by a WT namely results in a disturbed wind flow behind the WT, which can cause fatigue to the down-WTs, and in turn increases the maintenance cost as well as shortens the WT's life-cycle.",
"An interesting approach how to deal with these issues has been proposed in [11].",
"Using a stationary wake model, it has been shown that partially operating up-WTs not only reduces turbulence levels, but also improves the row efficiency of the WF.",
"As a result, partial de-loading of the up-WTs in a WF not only offers frequency control ancillary services, but also reduces turbulence levels for the down-WTs.",
"Therefore, we propose an optimised operation strategy for a WF, in which we de-load some up-WTs to maximise the overall kinetic energy of the WF.",
"We do this by optimising the rotor speed $\\omega $ and the pitch angle $\\beta $ of a WT.",
"We implement the optimised control approach initially proposed in [12], and the wake model developed in [11].",
"For particular de-loadings, we show that not only can the kinetic energy accumulated in the rotating masses of the WTs be increased, but also the overall output power of the WF does not change significantly in a wide range of wind speeds compared to the base case.",
"We cast the problem as a constrained non-linear optimisation problem, and solve it using the pattern search algorithm.",
"Although the optimisation problem has multiple optima, we show that a good-quality solution can be found which can be readily implemented in a control algorithm.",
"The paper is organised as follows.",
"In Section II, we briefly review the conventional strategies for participation of wind power in frequency control.",
"Wake models are introduced in Section III.",
"Section IV presents the proposed optimised operation strategy for frequency control of a WF.",
"In Section V, the proposed control approach is implemented in a simple test system and results are evaluated.",
"Section VI concludes the paper."
],
[
"Participation of wind power in frequency control",
"There are mainly two options for control of frequency by WTs: inertial response and de-loaded operation[13].",
"Since the stator and the rotor of a VSWT are decoupled by the power electronic converter, an additional control loop is required to make the WT inertia available to the system, which is referred to as synthetic inertia in some references[14].",
"The addition of this new loop can provide up to 20% extra power to the system for up to 10 during a frequency dip [15].",
"Nonetheless, this inertial response decreases the rotor speed which consequently reduces the coefficient of performance.",
"To recover the coefficient of performance, the kinetic energy of the rotor should be restored, which can result in another frequency event[15].",
"Unlike synthetic inertia where no wind is spilled, in de-loading strategy a WT needs to be permanently de-loaded for frequency control, and operate with a lower coefficient of performance[16].",
"Between the cut-in wind speed and the wind speed where the rotor speed reaches its maximum value, de-loading can be achieved by changing the rotor speed in proportion to the de-loading margin ($DM$ ).",
"Once the maximum rotor speed is reached, de-loading is possible only by changing the pitch angle.",
"Fig.",
"REF shows the performance coefficient $C_{p}$ as a function of the tip speed ratio $\\lambda $ defined as $\\lambda =\\frac{R\\omega }{v}$ , where $R$ is the radius of the WT blade, $\\omega $ is the rotor speed of the WT and $v$ is wind speed.",
"This strategy is valid only for those wind speeds where $\\omega <\\omega ^{max}$ .",
"As shown in Fig.",
"REF , to reduce $C_{p}$ , we have to either increase $\\lambda $ or decrease it.",
"The aim is to increase the kinetic energy; therefore, we have to select $\\lambda ^{sub}_{high}$ which corresponds to $\\omega ^{sub}_{high} > \\omega ^{opt}$ .",
"There is another de-loading method which is based on the pitch control only in all wind speeds $v$ where $ v_{cut-in} <v<v_{cut-out}$ .",
"In this paper, the optimised control approach initially proposed in [12], which uses both $\\beta $ and $\\omega $ to de-load a VSWT is implemented.",
"Figure: Power coefficient characteristic and de-Loading Strategy of Type III WT."
],
[
"Wake models",
"Accurate modelling of wake effects in a WF is a challenging task.",
"Some of the factors which are considered for wake modelling are: the distance between the WTs, the radius of the WTs, the geography of the site and the operating points of the WTs.",
"Wake models fall into two broad categories: (i) experimental and (ii) analytical.",
"Experimental wake models are based on measurements from the WFs, and are specific to these WFs.",
"Analytical wake models are based on the laws of fluid dynamics, and are mainly classified into three subclasses [17]: (i) kinematic models, (ii) field models and (iii) roughness element models.",
"Kinematic models are based on conservation of momentum and start by modelling a single wake for a WF.",
"Although these models are suitable for large WFs, some of them assume a constant value for the coefficient of thrust $C_T$ , which makes them unreliable for accurate wake models because $C_T$ of a WT changes in every operation points[11].",
"Some well-known kinematic models are: Jensen's model [18] and Frandsen's model [19].",
"Field models give wind speed at every point behind a WT [17], so computationally they are more complicated to implement.",
"Roughness element models are subdivided into infinite cluster models and finite cluster models.",
"In infinite cluster models, the WF is considered as a single element, and the effect of individual WTs are lost.",
"Whereas, the finite cluster models give the wind speed on each row of the WTs.",
"Most of these wake models are not suitable for control purpose because either they are too complicated or unreliable.",
"A Stationary wake model which was recently developed for control purposes is thus used in this paper.",
"This wake model requires minimum data, and its parameters have clear representation, which makes it suitable for control purposes[11].",
"Figure: Row of WTs in a WF.The stationary wake model is a suitable interaction model for a single row of WTs with the wind speed parallel to the row of the WTs.",
"It maps the coefficient of thrust, which is directly linked to the wind speed deficit, the wind speed and the turbulence level of the up-WTs to identify their effect on the down-WTs[11].",
"Considering the configuration of Fig.",
"REF , we can calculate the wind speed of $WT_{i+1}$ : $v_{i+1}=v_i + k^{^{\\prime }}(v_1 - v_i) - kv_1C_{Ti}$ where $0<k^{^{\\prime }}<1$ corresponds to the recovered wind speed, $0<k<k^{^{\\prime }}$ accounts the effect of the previous WT and $0<C_T(\\lambda ,\\beta )<1$ is the thrust coefficient of the nearest up-WT.",
"Distance parameters $k^{^{\\prime }}$ and $k$ are selected based on actual WF data.",
"In the simplest approach, we need the data from three WTs.",
"$k$ can be set based on $C_{T1}$ and $v_{2}$ , and then $k^{^{\\prime }}$ can be set by having $k$ , $C_{T2}$ and $v_{3}$ .",
"It is assumed that $k^{^{\\prime }}=0.35$ and $k=0.1$ .",
"The coefficient of performance $C_{p}(\\lambda ,\\beta )$ and the coefficient of thrust $C_{T}(\\lambda ,\\beta )$ are defined as: $C_p={\\frac{2P_{mech}}{{\\rho }{\\pi }R^{2}v^{3}}}\\\\C_T={\\frac{2T_{F}}{{\\rho }{\\pi }R^{2}v^{2}}}$ where $P_{mech}$ is the wind power transferred into mechanical power in the WT's rotor, $\\rho $ is the air density, $R$ is the radius of the WT's rotor, wind speed is $v$ and $T_{F}$ is the rotor thrust.",
"Considering (2) and (3), and $P=\\omega T$ a direct relationship between $C_{p}(\\lambda ,\\beta )$ and $C_{T}(\\lambda ,\\beta )$ in all operation regions where $C_{T}<{\\frac{8}{9}}$ can be derived as follows [20]: $&C_p={\\frac{1}{2}}(1 + \\sqrt{1-C_T})C_{T}$ where $C_p$ can either be given in a look-up table [21], or it can be approximated using curve-fitting[22].",
"The above equations show that any changes in $C_{p}(\\lambda ,\\beta )$ cause $C_{T}(\\lambda ,\\beta )$ to change as well.",
"Reducing $C_{p}(\\lambda ,\\beta )$ for de-loaded operation results in a lower value of $C_{T}(\\lambda ,\\beta )$ , which results in less energy being extracted from the wind, which in turn increases the wind downstream and reduces the turbulence."
],
[
"Frequency control strategy",
"Fig.",
"REF illustrates the rotor speed, the pitch angle and the power characteristic of a Type III WT under the optimal and the sub-optimal (de-loaded) operation mode in four different zones.",
"Under the optimal operation mode, the rotor speed is constant in Zones 1 and 3, and $C_p$ is not optimal.",
"In Zone 2, any changes in the wind speed cause the rotor speed to change in order to maximise $C_p$ .",
"In Zone 4, the rotor speed is at its limit.",
"Therefore, the pitch control operates to limit input the mechanical power to its rated value.",
"Figure: Power, rotor speed and pitch angle characteristic of a Type III WT.Under the optimised sub-optimal operation mode[12] in Zones 1 and 2, both the rotor speed $\\omega $ and the pitch angle $\\beta $ vary to optimise the kinetic energy of the rotor, defined as: $E_{k}=H{\\omega }^{2}$ where $H$ is the normalized inertia of the WT, and $\\omega $ is the rotor speed in the sub-optimal operation mode.",
"In Zones 3 and 4, since the rotor speed is maximum $\\omega ^{max}$ , the only control variable is $\\beta $ , and no additional kinetic energy can be achieved.",
"In Zone 2 under the sub-optimal mode, $\\omega _{high}^{sub} > \\omega ^{opt}$ , so Zone 2 becomes narrower as the $DM$ increases, and this limits the available $DM$[12].",
"Figure: Pitch angle and rotor speed of the Type III WTs.",
"a) DM=0%DM=0\\%, b) DM=5%DM=5\\%, c) DM=10%DM=10\\%."
],
[
"The optimisation problem",
"In this paper, we optimise the total kinetic energy of a WF by de-loading some of the up-WTs using a combination of the rotor speed $\\omega $ and the pitch angle $\\beta $ .",
"We consider a WF with a single row of identical Type III WTs, so the wake effect of the neighbouring rows is not considered.",
"Additionally, we assume that the wind direction is parallel to the string of WTs.",
"Optimisation problem is formulated as follows: $\\mbox{max} &\\quad {\\sum \\limits _{i=1}^n E_{k,i}},\\nonumber \\\\ \\mbox{s.t.}",
"&\\quad P_i\\le {P_i^{max}} \\quad \\quad \\qquad \\quad \\quad \\quad \\qquad \\;\\;\\;\\;\\;\\,\\, \\forall i\\in \\left\\lbrace 1 \\ldots n \\right\\rbrace \\nonumber \\\\ &\\quad 0\\le \\beta _i^{sub}\\le \\beta _i^{max} \\quad \\quad \\quad \\qquad \\qquad \\quad \\, \\forall i\\in \\left\\lbrace 1 \\ldots n \\right\\rbrace \\nonumber \\\\ &\\quad P_i^{sub}=(1-DM)P_i^{opt} \\quad \\quad \\quad \\quad \\;\\;\\, \\forall i\\in \\left\\lbrace 1 \\ldots n-1 \\right\\rbrace \\nonumber \\\\ &\\quad \\omega _i^{min}\\le \\omega _i^{opt}\\le \\omega _i^{sub}\\le \\omega _i^{max} \\;\\;\\qquad \\;\\;\\, \\forall i\\in \\left\\lbrace 1 \\ldots n \\right\\rbrace \\nonumber \\\\ &\\quad v_{i+1}=v_{i}+k^{^{\\prime }}(v_1-v_{i})-kv_1C_{Ti} \\; \\forall i\\in \\left\\lbrace 1 \\ldots n-1 \\right\\rbrace $ The optimisation variables are $\\omega _i$ and $\\beta _i$ , so the number of variables is $2n$ , where $n$ is the number of WTs in the WF.",
"In a Type III WT, $\\omega ^{min}$ and $\\omega ^{max}$ are limited by the size of the power electronic converter.",
"Furthermore, the minimum rotor speed in the sub-optimal operation mode is further limited by the minimum rotor speed in the optimal mode.",
"As shown in Fig.",
"REF , to increase the stored kinetic energy of the rotor, its speed in the sub-optimal mode should be higher than its speed in the optimal mode $\\omega ^{sub}\\ge \\omega ^{opt}$ .",
"The resulting constrained non-linear optimisation problem is non-convex and global, having several local optima.",
"We solved it with the pattern search algorithm using MATLAB Global Optimisation Toolbox [23].",
"Although the optimisation problem has multiple optima, we were able to solve it efficiently with a good-quality solution.",
"For a small system used in this paper, the computational efficiency was not an issue.",
"For larger system, the optimisation can be performed off-line and solutions stored in a look-up table, which can then be readily implemented in a control algorithm."
],
[
"Case study",
"We test the proposed strategy on a small test system with high penetration of wind generation shown in Fig.",
"REF .",
"The system consists of three SGs, a WF with 5 rows of WTs, each consisting of five 5 Type III WTs, and a 130 constant load.",
"We assume that the distance between the rows is large enough so we can ignore the wake interaction between the neighbouring rows, which enables us to perform the optimisation for each row independently.",
"We consider a 5 NREL reference WT [21].",
"Using the pattern-search algorithm with the above constraints, we can optimise the total kinetic energy of a WF with $n$ WTs using the combination of $\\omega $ and $\\beta $ in Zones 1 and 2.",
"In Zones 3 and 4, the rotor speed is at its maximum limit; therefore, the only control variable is $\\beta $ , so the kinetic energy cannot be increased further."
],
[
"Optimisation results",
"The optimisation of the kinetic energy was performed for a WF with five Type III WTs in a row.",
"$DM$ for $WT_1-WT_4$ was set to 0%, 5%, and 10% for Cases I, II, and III, respectively.",
"$WT_5$ is the last WT in the WF, so we maximise its power production without de-loading.",
"For all three cases, the optimisation was performed for a wind speed range of $[per-mode=fraction]{7}{}\\le {v}\\le [per-mode=fraction]{12}{}$ , where the rotor speed varies between $\\omega ^{min}\\le \\omega \\le \\omega ^{max}$ .",
"Fig.",
"REF shows the optimal rotor speeds and the pitch angles of $WT_1-WT_5$ in all cases.",
"The optimised kinetic energy and the total power of the WF is shown in Fig.",
"REF .",
"In Case I, all WTs operate in the optimal mode, and no additional kinetic energy can be stored as shown in Figs.",
"REF a and 5a.",
"In Cases II and III, the results are similar for lower wind speeds (Zones 1 and 2), and we can store considerable amount of kinetic energy by varying both the rotor speed and the pitch angle as shown in Figs.",
"REF b and c. Since both $C_p$ and $C_T$ are functions of the rotor speed and the pitch angle, and the operation of the down-WTs is linked to the operation of the up-WTs by the wake equation (1), we can vary both the rotor speed and the pitch angle of the WTs and not affect the total power of the WF, but still be able to store considerable amount of kinetic energy in the rotors as shown in Fig.",
"REF a.",
"Figure: The optimised kinetic energy in p.u, based on the ratings of the WT, and the power of the WF with different DMsDMs.In Cases II and III, the rotor speed and the pitch angles of $WT_1-WT_4$ follow similar trends until the rated rotor speed is reached (Zones 1 and 2).",
"In these regions, setting the pitch angle approximately between 1 and 2, and maintaining the total power of the WF close to optimal, the rotor speed increases more rapidly, thus increasing the total amount of stored kinetic energy.",
"As shown in Fig.",
"REF a, we can increase the total kinetic energy of the WF by increasing the $DM$ .",
"However, once the rated rotor speed is reached, no further kinetic energy can be stored.",
"Fig.",
"REF a shows that for wind speeds $v\\ge [per-mode=fraction]{9.5}{}$ the rotor speeds are at their respective maximal limits in both cases.",
"Therefore, no additional kinetic energy can be stored in Case III, and in this region the optimisation of the kinetic energy is independent of the $DM$ .",
"For wind speeds $v\\le [per-mode=fraction]{9.5}{}$ on the other hand, the available kinetic energy of the rotor is directly proportional to the $DM$ as shown in Fig.",
"REF a and Fig.",
"REF .",
"Observe that for lower wind speeds the cumulative output power of the WF is almost the same in all three cases.",
"In higher wind speeds, on the other hand, as the $DM$ increases, the total output power decreases.",
"In lower wind speeds, we can take advantage of the wake interaction to partly recuperate the power which we lose by de-loading.",
"For higher wind speeds, we are unable to do this because the power that we lose by de-loading of the up-WTs is comparatively higher, which negatively impacts the efficiency of the WF, as shown in Fig.",
"REF b.",
"A better strategy for higher wind speeds would be to de-load fewer WTs with a higher DM, which would allow down-WTs not only to recuperate the power which we lose by the de-loading, but would also possibly improve the overall efficiency of the WF.",
"Indeed, it has been shown in [11] that in a WF with 10 WTs, de-loading the first WT by 10% improves the row efficiency of the WF by 3%."
],
[
"Time domain simulations",
"To validate the optimisation results, we use time-domain simulation to simulate a frequency disturbance in the simple test system.",
"Unlike voltage, frequency is a global variable so we can use a copper-plate model.",
"We assume that the free wind speed $v_1$ reaching $WT_{1,1}-WT_{1,5}$ is uniform and $v_1=[per-mode=fraction]{8}{}$ , which results in an overall output power of 39.5 for the WF in all three cases.",
"The droop is set to 1% for all WTs.",
"The characteristics and the settings of the SGs are given in Table REF .",
"Figure: One-line diagram of the power system for simulation.Table: Settings and Characteristics of Synchronous GeneratorsSimulation was performed for the three cases described in Section V.A.",
"For all cases, we disconnect SG-3 from the system at $t={10}{}$ .",
"Meanwhile, in Cases II and III, we change the operation mode of $WT_1-WT_4$ form sub-optimal to optimal to release the stored kinetic energy into the system.",
"The system frequency response is shown in Fig.",
"REF and the rotor speeds are shown in Fig.",
"REF .",
"Since all rows of WTs in the WF are identical, only the rotor speeds of the first row is shown.",
"In Case I, all WTs operate in the optimal mode, so there is no additional kinetic energy.",
"Therefore, after disconnecting SG-3, the system frequency drops rapidly and the frequency bottom of 49.52 is achieved after 6.",
"It takes about 13 for the frequency to plateau as shown in Fig.",
"REF .",
"The rotor speeds are shown in Fig.",
"REF a.",
"Because the wind speed is constant, and the WTs operate in the optimal mode, the rotor speeds remain constant.",
"In Cases II and III, considerable amount of kinetic energy is available as shown in Fig.",
"REF .",
"Therefore, after disconnecting SG-3, we inject this kinetic energy to the system.",
"As a result, for the first few seconds, the system frequency not only doesn't drop, but it also rises, as shown in Fig.",
"REF .",
"Since in Case III the $DM$ is higher than in Case II, WF contribution on the primary frequency control is even higher.",
"Fig.",
"REF shows that in these two cases too much of kinetic energy is released too soon, which results in a frequency over-shoot.",
"A better strategy would be to inject this energy into the system gradually, either by using the control strategy developed in [13], or by reducing the rate of change of the rotor speed.",
"Figs.",
"REF b and c illustrate the rotor speeds of WTs in Cases II and III.",
"Observe that there is a significant discrepancy between the rotor speeds of $WT_1-WT_4$ in the sub-optimal and the optimal operation modes, which results in a positive contribution of the WF to the primary frequency control.",
"On the other hand, $WT_5$ that is not de-loaded, doesn't participate in the frequency control, and has a constant rotor speed.",
"Figure: System frequency during generation loss.Figure: Type III WTs rotor speed during generation loss a) DM=0%DM=0\\% (Case I), b) DM=5%DM=5\\% (Case II), c) DM=10%DM=10\\% (Case III)."
],
[
"Conclusion",
"In this paper, we have proposed an optimised operation strategy for de-loading operation of WFs.",
"In this strategy, we maximise the total stored kinetic energy of the WTs' rotors, which can be released into the system during a frequency dip.",
"We do this by optimising the rotor speeds and the pitch angles of some up-WTs.",
"In contrast to the traditional WF operation based on optimally operating every individual WT, we consider the whole WF as a single unit by taking advantage of the wake interaction within the WF.",
"We have shown that not only a WF can provide primary frequency control, but also it can possibly deliver more power.",
"It has been shown that by using WTs' rotor inertia, we can delay the system frequency nadir for up to 30.",
"We have however observed that in some scenarios too much kinetic energy could be released into the system too soon, which indicates that more intricate control strategies might be needed to achieve an optimal system performance."
]
] | 1605.05823 |
[
[
"Quantum nonlocality, and the end of classical space-time"
],
[
"Abstract Quantum non-local correlations and the acausal, spooky action at a distance suggest a discord between quantum theory and special relativity.",
"We propose a resolution for this discord by first observing that there is a problem of time in quantum theory.",
"There should exist a reformulation of quantum theory which does not refer to classical time.",
"Such a reformulation is obtained by suggesting that space-time is fundamentally non-commutative.",
"Quantum theory without classical time is the equilibrium statistical thermodynamics of the underlying non-commutative relativity.",
"Stochastic fluctuations about equilibrium give rise to the classical limit and ordinary space-time geometry.",
"However, measurement on an entangled state can be correctly described only in the underlying non-commutative space-time, where there is no causality violation, nor a spooky action at a distance."
],
[
"Quantum Nonlocality, and the End of Classical Spacetime Shreya Banerjee, Sayantani Bera and Tejinder P. Singh Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India [email protected], [email protected], [email protected] ABSTRACT Quantum non-local correlations and the acausal, spooky action at a distance suggest a discord between quantum theory and special relativity.",
"We propose a resolution for this discord by first observing that there is a problem of time in quantum theory.",
"There should exist a reformulation of quantum theory which does not refer to classical time.",
"Such a reformulation is obtained by suggesting that space-time is fundamentally non-commutative.",
"Quantum theory without classical time is the equilibrium statistical thermodynamics of the underlying non-commutative relativity.",
"Stochastic fluctuations about equilibrium give rise to the classical limit and ordinary space-time geometry.",
"However, measurement on an entangled state can be correctly described only in the underlying non-commutative space-time, where there is no causality violation, nor a spooky action at a distance.",
"March 20, 2016 Essay written for the Gravity Research Foundation 2016 Awards for Essays on Gravitation Corresponding Author: Tejinder P. Singh This essay received an honorable mention in the Gravity Research Foundation 2016 Essay Contest 1.4 “It may be that a real synthesis of quantum and relativity theories requires not just technical developments but radical conceptual renewal”.",
"- J. S. Bell (1986) Measurements on entangled quantum states demonstrate non-local correlations and suggest the existence of an acausal action at a distance across space-like separated regions.",
"This is confirmed by ever more precise loophole free tests of violation of Bell's inequalities by quantum systems [1], [2].",
"Even though such correlations cannot be used for superluminal signaling [3], the acausal nature of the influence suggests a conflict with special relativity and Lorentz covariance [4].",
"Numerous investigations over decades have not provided a satisfactory resolution of the problem.",
"Further support for the conflict comes from a remarkable experiment showing that even if the influence was assumed to travel causally in a hypothetical privileged frame of reference, its speed would have be at least four orders of magnitude greater than the speed of light [5]!",
"A measurement causes the wave function to collapse instantaneously all over space.",
"A dynamical theory such as Continuous Spontaneous Localization [CSL] explains collapse as a spontaneous physical process mediated by a stochastic nonlinear modification of the Schrödinger equation [6].",
"Nonetheless, CSL is a nonrelativistic theory, and a satisfactory relativistic extension of CSL has not yet been found.",
"We take the above-mentioned facts as evidence that non-local quantum correlations and collapse of the wave function during a quantum measurement are not compatible with special relativity.",
"In this essay we demonstrate how the resolution of the so-called problem of time in quantum theory enforces a non-commutative structure on space-time.",
"We then show that this non-commutative structure naturally explains entanglement and instantaneous wave function collapse across space-like separations.",
"The same process appears unphysical and violates causality, when seen from the [incorrect] vantage point of Minkowski spacetime in special relativity.",
"The time that appears in quantum theory is part of a classical space-time, whose geometry is determined by classical matter fields in accordance with the laws of the general theory of relativity.",
"It is a consequence of the Einstein hole argument that in the absence of these classical fields, the manifold structure of space-time does not exist [7].",
"Thus, in its need for time, quantum theory has to depend on its own classical limit, and that is unsatisfactory from a fundamental viewpoint.",
"There ought to exist an equivalent reformulation of quantum theory which does not refer to an external classical time [8].",
"We have proposed that such a reformulation can be arrived at by generalizing the theory known as Trace Dynamics.",
"The goal of Trace Dynamcs [TD] is to derive quantum theory from a deeper underlying theory [9].",
"This is more satisfactory, as compared to arriving at quantum theory by imposing ad hoc canonical commutation relations on a previously known classical mechanics.",
"TD is the classical dynamics of $N\\times N$ matrices $q_r$ whose elements are either odd grade [fermionic sector F] or even grade [bosonic sector B] elements of Grassmann numbers.",
"The Lagrangian of the theory is defined as the trace of a polynomial function of the matrices and their time derivatives, and then Lagrangian and Hamiltonian dynamics can be developed in the usual way.",
"The configuration variables $q_r$ and their conjugate momenta $p_r$ all obey arbitrary commutation relations amongst each other.",
"Nonetheless, as a consequence of a global unitary invariance there occurs in the theory a remarkable conserved charge, known as the Adler-Millard charge $\\tilde{C} = \\sum _B [q_r,p_r] -\\sum _F \\lbrace q_r,p_r\\rbrace $ whose existence is central to the subsequent development [10].",
"Next, assuming that one is not examining the dynamics at this level of precision, one develops an equilibrium statistical thermodynamics of the classical dynamics described by TD.",
"The equipartition of the Adler-Millard charge implies certain Ward identities, which lead to the important result that thermal averages of canonical variables obey quantum dynamics and quantum commutation relations [9].",
"In particular, the emergent $q$ operators commute with each other, and so do the $p$ operators.",
"Furthermore, if one considers the inevitable statistical fluctuations of the Adler-Millard charge about equilibrium, this leads to a CSL type modification of the nonrelativistic Schrödinger equation.",
"The said modification, negligible for microscopic systems but significant for large objects, solves the quantum measurement problem and leads to emergent classical behavior in macroscopic systems [11].",
"The fluctuations of the conserved charge about its equilibrium value carry information about the arbitrary commutation relations amongst the configuration and momentum variables in the underlying TD.",
"In order to arrive at a formulation of quantum theory without classical time, we first generalized Trace Dynamics so as to make space-time coordinates also into operators [12].",
"Associated with every degree of freedom there are coordinate operators $(\\hat{t}, \\hat{\\bf x})$ with arbitrary commutation relations amongst them.",
"These define a Lorentz invariant line-element $d\\hat{s}^2$ , and the important notion of Trace time $s$ as follows: $ds^2 = Tr d\\hat{s}^2 \\equiv Tr[d\\hat{t}^2 - d\\hat{x}^2 - d\\hat{y}^2 - d\\hat{z}^2]$ A Poincaré invariant dynamics can be constructed, in analogy with special relativity, and in analogy with TD, but with evolution now defined with respect to trace time $s$ .",
"The theory continues to admit a conserved Adler-Millard charge, and the degrees of freedom now involve bosonic and fermionic components of space-time operators as well.",
"Because the space-time operators have arbitrary commutation relations, there is no point structure or light-cone structure, nor a notion of causality, despite the line-element being Lorentz invariant [12].",
"Given this generalized TD, we construct its equilibrium statistical thermodynamics, as before.",
"The equipartition of the Adler-Millard charge leads to the emergence of a generalized quantum dynamics [GQD] in which evolution is with respect to the trace time $s$ , and the thermally averaged space-time operators $(\\hat{t}, \\hat{\\bf x})$ are now a subset of the configuration variables of the system [13].",
"It is important to note that these averaged operators commute with each other.",
"This is the sought after reformulation of quantum theory which does not refer to classical time.",
"In the non-relativistic limit one recovers the generalized Schrödinger equation $i\\hbar \\frac{d\\Psi (s)}{ds} = H\\Psi (s)$ Before we demonstrate the equivalence of the reformulation with standard quantum theory, we must explain how the classical Universe, with its classical matter fields and ordinary space-time, emerges from the GQD in the macroscopic approximation.",
"Like in TD, one next allows for inclusion of stochastic fluctuations of the Adler-Millard charge, in the Ward identity.",
"This again results in a non-linear stochastic Schrödinger equation, but now with important additional consequences.",
"Consider the situation where matter begins to form macroscopic clumps (for instance in the very early universe, soon after the Big Bang).",
"The stochastic fluctuations become increasingly significant as the number of degrees of freedom in the clumped system is increased.",
"These fluctuations induce macroscopic objects to be localized, but now not only in space, but also in time!",
"This means that the time operator associated with every object becomes classical (a $c$ -number times a unit matrix) [14].",
"The localization of macroscopic objects is accompanied by the emergence of a classical space-time, in accordance with the Einstein hole argument.",
"If, and only if, the Universe is dominated by macroscopic objects, as the Universe today is, can one also talk of the existence of a classical space-time.",
"When this happens, the proper time $s$ may be identified with classical proper time.",
"Once the Universe reaches this classical state, it sustains itself therein, by virtue of the continuous action of stochastic fluctuations on macroscopic objects, thereby achieving also the existence of a classical space-time geometry [14].",
"Because the underlying generalized TD is Lorentz invariant, the emergent classical space-time is locally Lorentz invariant too.",
"However there is a key difference: unlike in the underlying theory, now light-cone structure and causality emerge, because the space-time coordinates are now $c$ -numbers.",
"[We note that in conventional studies of the very early Universe, gravity is assumed to become classical at the Planck scale, while matter fields are assumed to become classical later.",
"Clearly, such a scenario is feasible only if the semiclassical gravity approximation is valid, e.g.",
"$G_{\\mu \\nu }=\\kappa \\langle \\Psi |T_{\\mu \\nu }|\\Psi \\rangle $ , which typically requires matter to be in a highly coherent, nearly classical state.",
"Something like this is implicitly assumed, for example, during the inflationary epoch, with the inflaton field being composed of a dominant classical part, and a sub-dominant quantum perturbation].",
"Independent of this pre-existing classical background, a microscopic system in the laboratory is fundamentally described, on its own non-commutative space-time (REF ), by the associated generalized TD.",
"Upon coarse-graining, this leads to the system's GQD (REF ) with its trace time.",
"Under the assumption that stochastic fluctuations can be ignored, this GQD has commuting $\\hat{t}$ and $\\hat{\\bf x}$ operators.",
"These, by virtue of their commutativity, can be mapped to the $c$ -number $t$ and ${\\bf x}$ of the pre-existing classical universe, and trace time can be mapped to ordinary proper time.",
"This is a map to ordinary special relativity, and hence one recovers standard relativistic quantum mechanics, and its non-relativistic limit.",
"This shows how standard quantum theory is recovered from the reformulation which does not have classical time [14].",
"We now have at hand all the ammunition needed to attack the spooky action at a distance, when a measurement is made on an entangled quantum state over space-like separations.",
"Prior to the measurement, the stochastic fluctuations of the Adler-Millard charge can be neglected for the quantum system, and as we saw above, its GQD can be mapped to ordinary quantum theory.",
"However, when the measurement is made, the collapse inducing stochastic fluctuations in the space-time operators $\\hat{t}, \\hat{\\bf x}$ associated with the quantum system come into play.",
"These operators now carry information about the arbitrary commutation relations of the underlying TD and no longer commute with each other.",
"Hence they cannot be mapped to the ordinary space-time of special relativity.",
"Simultaneity can only be defined with respect to the trace time $s$ , and there is no special relativistic theory of collapse.",
"Collapse and the so-called non-local quantum correlation truly takes place only in the non-commutative space-time (REF ), which is devoid of point structure, devoid of light-cone structure, and devoid of the notion of distance.",
"Hence one can only say that collapse takes place at a particular trace time, which is Lorentz invariant, and it is not meaningful to talk of an influence that has travelled, nor can one call the correlation non-local.",
"If one tries to view and describe the measurement on the entangled quantum state from the vantage point of the Minkowski space-time of special relativity, the process naturally appears acausal and non-local.",
"However, such a description is invalid, because there is no map from the fluctuating and noncommuting ${\\hat{t}}, \\hat{\\bf x}$ to the commuting $t$ and ${\\bf x}$ of ordinary special relativity.",
"There is no such map in the non-relativistic case either.",
"However, in the non-relativistic case, since there is an absolute time, it is possible to model the fluctuations as a stochastic field on a given space-time background, like in CSL, and collapse is instantaneous in this absolute time, but does not violate causality.",
"We conclude that the problem of time in quantum theory is intimately connected with the vexing issue of quantum non-locality and acausality in entangled states.",
"Addressing the former compels us to revise our notions of space-time structure, which in turn provides a resolution for the latter.",
"And it compels us to think about quantum gravity in a new way.",
"REFERENCES"
]
] | 1605.06022 |
[
[
"On the UV dimensions of Loop Quantum Gravity"
],
[
"Abstract Planck-scale dynamical dimensional reduction is attracting more and more interest in the quantum-gravity literature since it seems to be a model independent effect.",
"However different studies base their results on different concepts of spacetime dimensionality.",
"Most of them rely on the \\textit{spectral} dimension, others refer to the \\textit{Hausdorff} dimension and, very recently, it has been introduced also the \\textit{thermal} dimension.",
"We here show that all these distinct definitions of dimension give the same outcome in the case of the effective regime of Loop Quantum Gravity (LQG).",
"This is achieved by deriving a modified dispersion relation from the hypersurface-deformation algebra with quantum corrections.",
"Moreover we also observe that the number of UV dimensions can be used to constrain the ambiguities in the choice of these LQG-based modifications of the Dirac spacetime algebra.",
"In this regard, introducing the \\textit{polymerization} of connections i.e.",
"$K \\rightarrow \\frac{\\sin(\\delta K)}{\\delta}$, we find that the leading quantum correction gives $d_{UV} = 2.5$.",
"This result may indicate that the running to the expected value of two dimensions is ongoing, but it has not been completed yet.",
"Finding $d_{UV}$ at ultra-short distances would require to go beyond the effective approach we here present."
],
[
"INTRODUCTION",
"There is an increasing interest in the quantum-gravity literature about the effect of dynamical dimensional reduction of spacetime.",
"It consists in a scale dependence of the dimension $d$ that runs from the standard IR value of four spacetime dimensions to the lower value $d \\simeq 2$ at Planckian energies.",
"Remarkably, despite the fact that quantum-gravity approaches start from different conceptual premises and adopt different formalisms, this dimensional running has been found in the majority of them, such as Causal Dynamical Triangulations (CDT) [1], Horava-Lifshitz gravity [2], Causal Sets [3], Asymptotic Safety [4], Spacetime Noncommutativity [5] and LQG [6], [7], [8], which is here of interest.",
"However, in quantum gravity even the concept of spacetime dimension is a troublesome issue and it requires some carefulness.",
"In fact, non-perturbative, background independent approaches (e.g.",
"LQG [9], [10] and CDT [11]) generally rely on non-geometric quantities and they have discreteness as their core feature.",
"For this reason, in order to extract phenomenological predictions, it would be necessary a coarse-graining process aimed at deriving a more manageable effective description from the fundamental discrete blocks, that characterize the Planckian realm.",
"It is a common expectation that this procedure should leave some traces in a semiclassical regime where the emerging picture would be given in terms of a quantum spacetime.",
"This reduction has the advantage of allowing us to recover at least some of our more familiar physical observables or, when it would not be possible, analogous ones with potential departures from their classical counterparts.",
"The dimension belongs to this latter set of semiclassical observables because the usual Hausdorff dimension is ill-defined for a quantum spacetime [13].",
"In the CDT approach [1], [11] it was recognized for the first time that a proper \"quantum analogue\" could be the spectral dimension $d_{S}$ , which is the scaling of the heat-kernel trace and it reproduces the standard Hausdorff dimension when the classical smooth spacetime is recovered.",
"What is more, it was found that in the UV $d_S \\simeq 2$ (see however Ref.",
"[12] for recent CDT simulations favouring a smaller value of the dimension), which is now a recurring number in the literature [14], [15], [16], [17].",
"In the asymptotic safety program such a value is also intimately connected to the hope of having a fixed point in the UV.",
"In fact, it has been proven that renormalizability is accomplished only if the dimension runs to two [18].",
"Furthermore, this prediction finds support in a recently developed approach [19] that has the advantage of relying on a minimal set of assumptions.",
"Provided that quantum gravity will host an effective limit characterized by the presence of a minimum allowed length (identified with the Planck length), then it is shown [19] that the Euclidean volume becomes two-dimensional near the Planck scale The author is grateful to Thanu Padmanabhan for pointing this out..",
"However, in a recent paper [20] the physical significance of $d_S$ has been questioned.",
"Such a concern is based on two observations: the computation of $d_S$ requires a preliminary Euclideanization of the spacetime and also it turns out to be invariant under diffeomorphisms on momentum space.",
"Both these features are regarded as an evidence of the fact that $d_S$ is an unphysical quantity [20].",
"Given that, it has been proposed to describe the phenomenon of dimensional reduction in terms of the thermal (or thermodynamical) dimension $d_T$ , which can be defined as the exponent of the Stefan-Boltzmann law.",
"Then, the UV flowing of $d_T$ is realized through a modified dispersion relation (MDR) that affects the partition function used to compute the energy density (see Ref.",
"[20] or Section III for further details).",
"Thus, the value of $d_T$ near the Planck scale depends crucially on the specific form of the MDR.",
"Furthermore, it has been recently noticed (see Refs.",
"[21], [22]) that form the MDR is also possible to infer the Hausdorff dimension $d_H$ of energy-momentum space.",
"If the duality between spacetime and momentum space is preserved in quantum gravity, this framework should provide another alternative characterization of the UV running.",
"In this way we are in presence of a proliferation of distinct descriptions of the UV dimensionality of a quantum spacetime.",
"These pictures make use of very different definitions of the dimension and, in principle, there is no reason why they should give the same outcome.",
"On the other hand, they all coincide in the IR-low-energy regime where they reduce to 4 and, thus, we could expect that this should happen also in the UV.",
"In this paper we show that this advisable convergence can be achieved in the semi-classical limit of LQG under rather general assumptions.",
"The insight we gain is based on the recently proposed quantum modifications of the hypersurface deformation algebra (or the algebra of smeared constraints) [23], [24], [25], [26], [27], which reduces to a correspondingly deformed Poincaré algebra in the asymptotic region, as shown in Ref.",
"[28].",
"These Planckian deformations of the relativistic symmetries are a key feature of the Deformed Special Relativity scenario [29], [30], as already pointed out in Ref.",
"[28], and, what is more, it has been recently shown in Ref.",
"[31] that they are consistent with a $\\kappa $ -Minkowski noncommutativity of the spacetime coordinates [32], [33].",
"We here exploit them to compute the MDR, thereby linking the LQG-based quantum corrections to the deformation of the dispersion relation.",
"The general form of the MDR we derive allows us to find that both the spectral, the thermal and the HausdorffNotice that we are always referring to the Hausdorff dimension of momentum space since, as we mentioned, that of a quantum spacetime cannot be defined.",
"dimensions follow the same UV flowing, i.e.",
"$d_S = d_T = d_H$ .",
"Another significant observation we make is that, following our analysis \"in reverse\", we can get information about the LQG quantum-geometric deformations, that affect the Dirac algebra, from the value of the UV dimension.",
"The importance of this recognition resides on the fact that these modifications are subjected to many sources of possible ambiguities [34], [35], [36] coming e.g.",
"from the regularization techniques used to formally quantize the Hamiltonian constraint.",
"These ambiguities are far from being resolved and it is still no clear if they may affect potentially physical outcomes [37].",
"Thus, they are usually addressed only on the basis of mere theoretical arguments or being guided by a principle of technical simplification.",
"The main sources of ambiguities are the spin representations of the quantum states of geometry as well as the choice of the space lattice and, in effective models we will here consider, they correspond respectively to holonomy and inverse-triad corrections.",
"We here partially fix them with what is believed to be a phenomenological prediction i.e.",
"the number of UV dimensions.",
"Notably, we notice that the leading order correction provided by the holonomy corrections of homogeneous connections [38], [39], which are often implemented by simply taking the expression $\\frac{\\sin (\\delta K)}{\\delta }$ instead of $K$ , e.g.",
"in Loop Quantum Cosmology (LQC) [40], [41] (see also [42] for a recent review on symmetry reduced models of LQG), is compatible with $d_{UV} = 2.5$ .",
"Thus, as we would have expected, the number of dimensions is correctly flowing to lower values even if it has not already reached the value of 2, a value which is favoured in the quantum-gravity literature for the aforementioned reasons.",
"On the basis of the steps we sketch out in the analysis we are here reporting it should be possible to exclude all the deformation functions $f(K)$ that are not consistent with $d_{UV} = 2$ .",
"Remarkably, those quantum corrections, which are related to LQC as well as to the semi-classical limit of the theory, seem to point toward the right UV flowing.",
"As already stated, the prediction of Planckian dimensional reduction has been also confirmed by previous LQG analyses [6], [7] but more refined computations of Ref.",
"[8] have revealed that the \"magic\" number of 2 can be reproduced only focusing on a specific superposition of kinematical spin-networks states (see [8] for the details).",
"Relying on the recently developed effective methods for LQG, we here provide further support to the idea that the effective spacetime of LQG maybe two dimensional at ultra-Planckian scales."
],
[
"MODIFIED DISPERSION RELATION",
"We start considering the classical hypersurface deformation algebra (HDA), which was first introduced by Dirac [43].",
"It is the set of Poisson brackets closed by the smeared constraints of the Arnowitt-Deser-Misner formulation of General Relativity (GR) [44].",
"The Dirac algebra of constraints is the way in which general covariance is implemented once the spacetime manifold has been split into the time direction and the spatial three surfaces i.e.",
"$\\mathcal {M} = \\mathbb {R}\\times \\Sigma $ .",
"It is given by: $\\begin{split}\\lbrace D[M^{k}],D[N^{j}]\\rbrace =D[\\mathcal {L}_{\\vec{M}}N^{k}],\\\\\\lbrace D[N^{k}],H[M]\\rbrace =H[\\mathcal {L}_{\\vec{N}}M],\\\\\\lbrace H[N],H[M]\\rbrace =D[h^{jk}(N\\partial _{j} M-M\\partial _{j} N)],\\end{split}$ where $H[N]$ is the Hamiltonian (or scalar) constraint, while $D[N^{k}]$ is the momentum (or vector) constraint.",
"The function $N$ is called the lapse and it is needed to implement time diffeomorphisms, while $N^{i}$ is the shift vector necessary to move along a given hyper-surface and, finally, $h^{ij}$ is the inverse metric induced on $\\Sigma $ .",
"Thus, $H[N]$ and $D[N^{k}]$ have to be understood as the generators of gauge transformations which, in the case of GR, are space-time diffeomorphisms.",
"For the purposes of our analysis, it is relevant the well established fact (see Ref.",
"[45]) that, when the spatial metric is flat $h_{ij} = \\delta _{ij}$ , if we take $N = \\bigtriangleup t+v_{a}x^{a}$ (where $v_a$ is the infinitesimal boost parameter) and $N^{k} = \\bigtriangleup x^{k}+R^{k}_{l}x^{l}$ (where $R^{k}_{l}$ is the matrix that generates infinitesimal rotations), we can infer the Poincaré algebra from the Dirac algebra [45].",
"This classical relation is expected to hold also at the quantum level.",
"One of the open issues in the LQG research is the search for fully quantized versions of the constraints $H[N], D[N^{k}]$ on a Hilbert space.",
"While it is known how to treat spatial diffeomorphisms and also how to solve the momentum constraint [46], [47] thereby obtaining the kinematical Hilbert space of the theory, the finding of a Hamiltonian operator is far from being completed.",
"However, over the last fifteen years several techniques have been developed, using both effective methods and discrete operator computations.",
"In this way some candidates for an effective scalar constraint $H^{Q}[N]$ have been identified [48], [49], [50].",
"For the analysis we are here reporting, the interesting fact is that the semi-classical corrections introduced in the Hamiltonian leave trace in the algebra of constraints .",
"Remarkably, even if these calculations use different formalisms and they are based on different assumptions, the general form of the modified HDA turns out to be the same in all these studies, i.e.",
"only the Poisson bracket between two scalar constraints is affected by quantum effects [39], [51]: $\\lbrace H^{Q}[M],H^{Q}[N]\\rbrace =D[\\beta h^{ij}(M\\partial _{j}N-N\\partial _{j}M)]$ where the specific form of the deformation function $\\beta $ as well as its dependence on the phase space variables, which are $(h_{ij},\\pi _{ij})$ if we use the metric formulation or $(A^{a}_{i}, E^{i}_{a})$ if we use the Ashtekar's one, varies with the quantum corrections considered to define $H^{Q}[N]$ .",
"One of the causes of these quantum modifications of the scalar constraint is the fact that LQG cannot be quantized directly in terms of the Ashtekar variables $A^{a}_{i}$ , which have to be replaced with their parallel propagators (or holonomies) [9], [10], [38], [39] $h_{\\alpha }(A) = \\mathcal {P}e^{-\\int _{\\alpha }t^{i}A^{a}_{i}\\tau _{a}}$ (where $\\mathcal {P}$ is the path-ordering operator, $t^{i}$ the tangent vector to the curve $\\alpha $ and $\\tau _a = -\\frac{i}{2}\\sigma _a$ the generators of $SU(2)$ ).",
"If $a$ is the spatial index of a direction along which spacetime is homogeneous, then one has to consider just the local point-wise holonomies $h_{i}(A) = \\cos (\\frac{\\delta A}{2})\\mathbb {I}+\\sin ( \\frac{\\delta A}{2})\\sigma _i$ (where $\\delta \\propto l_P = \\sqrt{G} \\approx 10^{-35} m$ is connected to the square root of the minimum eigenvalue of the area operator [41]).",
"These are the kinds of quantum effects considered in effective (semi-classical) LQG theories as well as both in spherically reduced models and in cosmological contexts [42].",
"In particular, for spherically symmetric LQG (see e.g.",
"Refs.",
"[28], [50], [51]) the deformation function depends on the homogeneous angular connection $K_{\\phi }$ and it is directly related to the second derivative of the square of the holonomy correction $f(K_{\\phi })$ , i.e.",
"$\\beta = \\frac{1}{2}\\frac{d^{2}f^{2}(K_{\\phi })}{dK^{2}_{\\phi }}$ .",
"Then, the important contribution of Ref.",
"[28] has been to establish a link between these LQG-inspired quantum corrections and DSR-like deformations of the relativistic symmetries, thanks to the recognition that the angular connection $K_\\phi \\propto P_r$ is proportional to the Brown-York radial momentum [52] that generates spatial translations at infinity (see [28], [31] for the details).",
"In fact, it has been shown that, taking the Minkowski limit of Eq.",
"(REF ) as we sketched above for the classical case, the LQG-deformed HDA produces a corresponding Planckian deformation of the Poincaré algebra: $[B_{r}, P_{0}] = iP_{r}\\beta (l_{P} P_{r})$ where the other commutation relations remain unmodified.",
"Here $B_r$ is the generator of radial boosts and $P_0$ the energy.",
"The explicit form of $\\beta $ is unknown and, as we already stressed, it is affected by ambiguities.",
"In light of this, for our analysis we assume a rather general form: $\\beta (\\lambda P_{r}) = 1+\\alpha l_{P}^{\\gamma } P^{\\gamma }_{r}$ which is motivated by the above considerations and, obviously, satisfy the necessary requirement: $\\lim _{l_P\\rightarrow 0}\\beta (l_P P_r) = 1$ , i.e.",
"we want to recover the standard Poincaré algebra in the continuum limit.",
"We leave unspecified the constants of order one $\\alpha $ and $\\gamma $ that parametrize the aforementioned ambiguities.",
"These parameters should encode at least the leading-order quantum correction to the Poincaré algebra.",
"Using Eqs.",
"(REF )(REF ) and taking into account that $[B_{r}, P_{r}] = i P_0, [P_{r}, P_{0}] = 0$ , a straightforward computation gives us the following MDR: $E^{2} = p^{2}+\\frac{2\\alpha }{\\gamma +2}l_{P}^{\\gamma } p^{\\gamma +2}$ This completes the analysis started in Ref.",
"[28] and carried on in Ref.",
"[31], that aimed at building a bridge between the formal structures of loop quantization to the more manageable DSR scenario with the objective to enhance to possibilities to link mathematical constructions to observable quantities.",
"The remarkable fact of having derived Eq.",
"(REF ) form the LQG-deformed algebra of constraints (REF ) is that it will give us the opportunity to constrain experimentally the formal ambiguities of the LQG approach exploiting the ever-increasing phenomenological implications of MDRs (see Ref.",
"[53] and references therein).",
"We are often in the situation in which quantum-gravity phenomenology misses a clear derivation from full-fledged developed approaches to quantum gravity or, on the contrary, the high complexity of these formalisms does not allow to infer testable effects.",
"Following the work initiated in Ref.",
"[31], we are here giving a further contribution to fill this gap.",
"We find also interesting to notice that our MDR confirms a property of two previously proposed MDRs (see Refs.",
"[54], [55]), i.e.",
"LQG corrections affect only the momentum sector of the dispersion relation leaving untouched the energy dependence.",
"Therefore, this property, that has a rigorous justification in the spherically symmetric framework [28], [50], [51] we are here adopting, seems to be a recurring feature of LQG.",
"Moreover, all the precedent analyses were confined to the kinematical Hilbert space of LQG, while we have here obtained Eq.",
"(REF ) from the flat-spacetime limit of the full HDA including also the semi-classical Hamiltonian constraint (REF ).",
"Thus, even if we are working off-shell (i.e.",
"we do not solve the constraint equations), the MDR (REF ) should contain at least part of the dynamical content of LQG.",
"In the next section, we shall see that the form of Eq.",
"(REF ) is crucial to prove that the running of dimensions does not depend on the chosen definition of the dimension."
],
[
"DIMENSIONS AND QUANTUM CORRECTIONS",
"Our next task is to use the MDR (REF ) we derived in Section II in order to show that, regardless of the value of the unknown parameters $\\alpha $ and $\\gamma $ , the different characterizations of the UV flowing introduced in the literature predict the same number of dimensions if we consider the effective regime of LQG in the sense introduced in Refs.",
"[28], [50], [51] and sketched in the previous section.",
"To see this we start by the computation of the spectral dimension, which is defined as follows: $d_S = -2\\lim _{s\\rightarrow 0}\\frac{d \\log P(s)}{d \\log (s)}$ where $P(s)$ is the average return probability of a diffusion process in a Euclidean spacetime with fictitious time $s$ .",
"Following Refs.",
"[20], [56], [57], [58], [59], [60], we compute $d_S$ from the Euclidean version of our MDR (REF ) which is a d'Alembertian operator on momentum space: $\\bigtriangleup ^E = E^2+p^2+\\frac{2\\alpha }{\\gamma +2}l_{P}^{\\gamma } p^{\\gamma +2}$ Then, a lengthy but straightforward computation (see [58], [59], [60]) leads to the following result: $d_S = 1 + \\frac{6}{2+\\gamma }$ Notice that the value of $d_S$ does not depend on $\\alpha $ but only on $\\gamma $ , i.e.",
"only on the order of Planckian correction to the dispersion relation (see Eq.",
"(REF )).",
"We will use this fact later on.",
"Now we want to show that also the thermal dimension $d_T$ is given by Eq.",
"(REF ).",
"To this end we remind the definition of $d_T$ introduced in Ref.",
"[20].",
"If you have a deformed Lorentzian d'Alembertian $\\bigtriangleup ^L_{\\gamma _{t}\\gamma _{x}} = E^2-p^2+l^{2\\gamma _t}_t E^{2(1+\\gamma _t)}-l^{2\\gamma _x}_x p^{2(1+\\gamma _x)}$ , then $d_T$ is the exponent of the temperature $T$ in the modified Stefan-Boltzmann law: $\\rho _{\\gamma _{t}\\gamma _{x}} \\propto T^{1+3\\frac{1+\\gamma _t}{1+\\gamma _x}}$ which can be obtained as usual deriving the logarithm of the thermodynamical partition function [20] with respect to the temperature $T$ .",
"Evidently, in our case we have that $\\gamma _t = 0,\\gamma _x = \\frac{\\gamma }{2}$ and, thus, we find $d_T = 1 + \\frac{6}{2+\\gamma }$ , i.e.",
"the thermal dimension agrees with the spectral dimension $d_T \\equiv d_S$ .",
"Finally, we can calculate also the Hausdorff dimension of momentum space that, if the duality with spacetime is not broken by quantum effects, should agree with both $d_S$ and $d_T$ .",
"As pointed out in Ref.",
"[21], a way to compute $d_H$ is to find a set of momenta that \"linearize\" the MDR.",
"Given Eq.",
"(REF ), a possible choice is given by: $k = \\sqrt{p^2+\\frac{2\\alpha }{\\gamma +2}l_{P}^{\\gamma } p^{\\gamma +2}}$ In terms of these new variables $(E,k)$ the UV measure on momentum space becomes: $p^2 dpdE \\longrightarrow k^{\\frac{4-\\gamma }{\\gamma +2}} dk dE$ From the above equation (REF ) we can read off $d_H$ : $d_H = 2 + \\frac{4-\\gamma }{\\gamma +2} = 1 + \\frac{6}{2+\\gamma }$ that, evidently, coincides with both $d_S$ and $d_T$ .",
"Thus, no matter which definition of dimensionality is used, in the semi-classical limit (or in symmetry reduced models) of LQG the UV running is free of ambiguities since we have found that $d_S \\equiv d_T \\equiv d_H$ .",
"The last consideration we want to make concerns what the number of UV dimensions can teach us about LQG.",
"In the analysis we here reported, the value of $\\gamma $ should be provided by the LQG corrections used to build $H^Q [N]$ , which, though, are far from being unique.",
"On the other hand, we mentioned that the number of dimensions runs to two in the UV, a prediction that seems to be model independent.",
"Moreover, support in LQG has been also found by the studies of Refs.",
"[6], [8] under certain assumptions.",
"It is evident from Eqs.",
"(REF )(REF ) that in order to reproduce such a shared expectation we should take $\\gamma = 4$ .",
"A recurring form of holonomy corrections both in spherically symmetric LQG [28], [24], [39], [61] as well as in LQC [40], [41] is represented by the choice: $f(K) = \\frac{\\sin (\\delta K)}{\\delta }$ that implies $\\beta = \\cos (2\\delta K)$ .",
"In light of the above arguments, if we restrict to mesoscopic scales where the MDR is well approximated by the first-order correction, it is easy to realize that this implies $\\gamma = 2$ (see Ref.",
"[31] for the explicit computation).",
"In this way we would have $d = 2.5$ at scales near but below the Planck scale.",
"We find rather encouraging the fact that we obtain a value which is greater than 2, since such an outcome may signal that the descent from the classical value of 4 to the UV value is in progress but it has not been completed yet.",
"In fact, at ultra-Planckian energies the Taylor expansion of the correction function $f(K)$ in series of powers of $\\delta $ is no more reliable and, as a consequence, it can not fully capture the flowing of $d$ .",
"Therefore, we are led to conclude that the much-used polymerization of homogeneous connections, which is a direct consequence of evaluating the holonomies in the fundamental $j=\\frac{1}{2}$ representation of $SU(2)$ , realizes at least partially the expected running of dimensions.",
"Polymerizing configuration variables is used not only in LQC and in symmetry reduced contexts but also in the definition of the semi-classical regime of LQG, which is based on the introduction of spin-network states peaked around a single representation, in the majority of cases $j = \\frac{1}{2}$ .",
"We have here shown that these quantum modifications can be related to the phenomenon of dimensional reduction.",
"This link we have established gives us the possibility to constrain part of quantization ambiguities in LQG.",
"In fact, the value of $d_{UV}$ fixes a specific choice of the parameter $\\gamma $ in Eq.",
"(REF ), thereby restricting the form of the allowed deformation functions $K \\rightarrow f(K)$ .",
"In light of our analysis one should select only those modifications which are compatible with the prediction of a two-dimensional spacetime at ultra-short distances, i.e.",
"those giving $2\\lesssim d < 4$ (or equivalently $0 < \\gamma \\lesssim 4$ ) to a first approximation."
],
[
"OUTLOOK",
"In the recent quantum-gravity literature there have been proposed basically three different definitions of dimension with the aim to generalize this notion for a quantum spacetime.",
"It is well known that they all provide a possible characterization of the UV running but, in the majority of cases, they also give different outcomes for the value of the dimension.",
"This clashes with the growing consensus on the fact that the phenomenon of dynamical dimensional reduction is a model independent feature of quantum gravity that gives the unique predictions $d_{UV} \\simeq 2$ .",
"Thus, if both $d_S,d_T$ and $d_H$ are really proper definitions of the UV dimension, we would like to have that $d_S \\equiv d_T \\equiv d_H$ .",
"In this paper we showed that this striking convergence can be accomplished in the case of the LQG approach.",
"To achieve such a result, we relied on the deformations of the constraint algebra recently proposed in the framework of both effective spherically symmetric LQG and LQC.",
"Remarkably, in the former case it has been proven that these quantum corrections leave traces in the Minkowski limit in terms of a DSR-like Poincaré algebra where the relevant deformations are functions of the spatial momentum.",
"We here exploited these LQG-motivated deformations of the relativistic symmetries to infer the generic form of the MDR.",
"Interestingly our MDR is qualitatively of the same type of two previously proposed MDRs for LQG [54], [55], namely the modifications affect only the momentum sector.",
"From the MDR we derived both the spectral, the thermal and the Hausdorff dimensions proving that they all agree.",
"Thus, in the top-down approach of LQG the desirable convergence of different characterization of the concept of dimensionality is accomplished.",
"On the other had, the analysis we here reported may provide a guiding principle for the construction of bottom-up approaches.",
"Moreover, our analysis led us to give a contribution toward enforcing the fecund bond between theoretical formalisms and phenomenological predictions.",
"In fact, we found that the simple polymerization of connections, which is much-used both in midi-superspace models and in LQC where semi-classical states are exploited to compute effective constraints, is able to generate the running of the dimension.",
"In this way we have provided further evidence that the phenomenon of UV dimensional reduction can be realized also in the LQG approach, thereby confirming the results of previous studies.",
"Remarkably, the value of $d_{UV}$ is sensible to the specific choice of quantum corrections which are considered in the model.",
"Therefore, we pointed out that there is an observable whose value can be used to select a particular form for the quantum correction functions, thereby reducing the LQG quantization ambiguities.",
"In particular, we showed that the evaluation of the Hamiltonian constraint over semi-classical states peaked at $j=\\frac{1}{2}$ (that can be also implemented with the substitution $K \\rightarrow \\frac{\\sin (\\delta K)}{\\delta }$ at an effective level) corresponds to $d_{UV} = 2.5$ .",
"Since we inferred such a value from the parametrization of the first LQG correction (REF ) (which corresponds to a second order correction in the Planck length $\\sim l_p^2$ ), then it is reasonable to regard our result as a first approximation that can not capture the ending outcome of the dimensional running.",
"From this perspective we observed that obtaining a dimension grater than 2 at energies near but below the Planck scale might be significant, because it can be read as a hint that the dimension is flowing to the \"magic\" number of 2 that we can expect to be reached in the deep UV.",
"Developing the off-shell constraint algebra in full generality without any symmetrical reduction or semi-classical approximation would be of fundamental importance to extend our observations to the full LQG framework."
],
[
"Acknowledgments.",
"The author is grateful to Giovanni Amelino-Camelia and Giorgio Immirzi for useful and critical discussions during the first stages of the elaboration of this work.",
"He also wants to thank Gianluca Calcagni for carefully reading the manuscript and giving many helpful comments that improved this work.",
"Finally, he acknowledges referees for their constructive suggestions."
]
] | 1605.05979 |
[
[
"Resource Provisioning and Profit Maximization for Transcoding in\n Information Centric Networking"
],
[
"Abstract Adaptive bitrate streaming (ABR) has been widely adopted to support video streaming services over heterogeneous devices and varying network conditions.",
"With ABR, each video content is transcoded into multiple representations in different bitrates and resolutions.",
"However, video transcoding is computing intensive, which requires the transcoding service providers to deploy a large number of servers for transcoding the video contents published by the content producers.",
"As such, a natural question for the transcoding service provider is how to provision the computing resource for transcoding the video contents while maximizing service profit.",
"To address this problem, we design a cloud video transcoding system by taking the advantage of cloud computing technology to elastically allocate computing resource.",
"We propose a method for jointly considering the task scheduling and resource provisioning problem in two timescales, and formulate the service profit maximization as a two-timescale stochastic optimization problem.",
"We derive some approximate policies for the task scheduling and resource provisioning.",
"Based on our proposed methods, we implement our open source cloud video transcoding system Morph and evaluate its performance in a real environment.",
"The experiment results demonstrate that our proposed method can reduce the resource consumption and achieve a higher profit compared with the baseline schemes."
],
[
"Introduction",
"With the emergence of Information-Centric Networking (ICN), the future Internet could shift away from a point-to-point paradigm to a more content-centric one [1].",
"In this condition, the content producer can focus on providing the information object of the video content.",
"Nowadays, adaptive bitrate streaming has been widely adopted to deliver the video content in different bitrates and resolutions to adopt varying network conditions and heterogeneous devices [2].",
"Video transcoding is used to pre-transcode the original contents into multiple representations.",
"For each video content published by the content producers, the transcoding service provider will transcode it into multiple representations in different bitrates and resolutions, which will be further stored in the original servers and the other caching nodes in the network.",
"As such, the viewers can request the multiple representations of the video contents in ABR.",
"Some transcoding function placed in the network can decide and dynamically adjust the optimal number of representations for the video contents and the places for caching to be accessed in high quality.",
"With the named function networking (NFN) framework, the video content can also be transcoded on the fly [3].",
"Implementing ABR yields a cost for both transcoding service provider and content producer.",
"First, video transcoding is computing intensive, consuming a huge amount of computing resource.",
"The transcoding service providers need to purchase and maintain a large number of servers for transcoding to meet the peak workload.",
"However, the servers can only be set in the idle state when no transcoding tasks are performed, wasting too much computing resource.",
"Second, video transcoding takes excessive time and the content producers need to wait for a long time for content publishing, which is intolerable for the contents needing timely delivery.",
"The emergence of cloud computing technology introduces an opportunity to improve video transcoding [2].",
"By leveraging the cloud infrastructure, it can transcode video contents more efficiently compared with the traditional transcoding methods.",
"Specifically, it can perform multiple transcoding in parallel using a large number of virtual machine (VM) instances or containers, which can greatly reduce the transcoding time.",
"Meanwhile, it can elastically provision the computing resource according to the transcoding workload, which can avoid the resource wastage and reduce monetary costs.",
"The current research on the video transcoding system mainly focuses on how to reduce the energy consumption and processing delays [4], [5], [6], without much attention on the resource provisioning problem in the cloud.",
"In this paper, we consider how to provision the computing resource for transcoding video contents while maximizing the profit for the transcoding service provider by taking the advantage of the cloud computing paradigm.",
"To this end, we mainly cope with the following system management problems in video transcoding system.",
"First, as the system workload is time-varying, we assess how to dynamically provision the resource for the transcoding service.",
"Second, with a certain amount of provisioned computing resource and a number of transcoding tasks to be processed, we consider how to schedule the tasks to maximize the revenue while meeting the service requirements.",
"Moreover, the complexity of the video data makes it hard to precisely estimate the transcoding time for a task, which is essential for the task scheduling and resource provisioning.",
"To solve the above problems, we first propose a neural network method for precisely estimating the transcoding time of the tasks.",
"We then propose a method for jointly considering the task scheduling and resource provisioning problem for the system management, and formulate it as a two-timescale Markov Decision Process [7].",
"We derive the policies for task scheduling and resource provisioning for maximizing the overall profit.",
"Based on our design, we implement the system and present the code of the practical implementation in Github [8].",
"We evaluate the system performance in a real environment.",
"We illustrate the system architecture in Fig.",
"REF .",
"The functionalities of each module are detailed as follows: Service Interface Module: It serves as the interface for processing the transcoding requests from the content producers.",
"The content producer uploads the video content and submits the transcoding request as a transcoding task.",
"For each transcoding task, the system will estimate the required computing resource and divides the original video content into independent video blocks according to the GOP structure.",
"Resource Provisioning Module: It provisions a number of homogenous VM instances in the transcoding clusterIt can also adopt the Container technology for resource virtualization.. Each VM instance runs a transcoding worker.",
"The resource provisioning module dynamically adjusts the number of active VM instances in the transcoding cluster according to the system workload and the resource provisioning policy.",
"Task Scheduling Module: It maintains a queue for the pending transcoding tasks and adjusts their transcoding order by reordering the tasks according to the task scheduling policy.",
"When there is a request for a video block from the transcoding worker, the task scheduler will pick a video block of the head-of-queue task to dispatch.",
"Once a transcoding task has been started to perform, i.e., the first video block of the transcoding task has been dispatched for transcoding, the task will stay at the head-of-queue until all of its video blocks have been dispatched to the transcoding workers.",
"Then, the next request for a video block from the transcoding worker will be replied with the video block of the next head-of-queue task."
],
[
"System Workflows",
"The workflows for fulfilling the transcoding requests from the content producers are as follows.",
"First, the content producer uploads a video file and submits the transcoding task by specifying the service requirements.",
"The system will estimate the required transcoding time for the new task and split the video file into video blocks.",
"Each of the video block consumes the same amount of computing time.",
"Second, for a number of pending transcoding tasks in the queue, the task scheduler reorders the tasks periodically according to the task scheduling policy.",
"Third, when a transcoding worker becomes idle, it will request a video block from the task scheduler.",
"After transcoding the video block into the target representation, the transcoding worker will send the transcoded video block back to the task scheduler.",
"The task scheduler will concentrate the transcoded video blocks into one video file after receiving all of the transcoded video blocks of a transcoding task from the transcoding workers.",
"The transcoded video contents can be stored in the content network for delivery."
],
[
"Task Arrival Model",
"We adopt a discrete time model and divide the time horizon into two timescales.",
"We denote the time in the fast timescale as $t \\ (t=0,1,2,...)$ , and the time in the slow timescale as $N_{k} \\ (k =0,1,2,...)$ .",
"Each $N_k$ consists of $T$ time slots in the fast timescale, and the time duration of $N_{k}$ in the slow timescale is from the time slot $kT$ to the time slot $(k+1)T-1$ .",
"The typical length of one time slot is $1\\sim 10$ seconds and $T$ is $1800 \\sim 3600$ seconds.",
"We model the task arrival as a non-stationary Poisson process with different arrival rates across the slow timescales.",
"For each time slot within $N_k$ , we assume the task arrival distribution is homogenous.",
"We denote the task arrival rate in each time slot $t$ of $N_k$ as $\\lambda _k$ , $\\ kT \\le t < (k+1)T$ ."
],
[
"Transcoding Time Estimation Model",
"For each of the transcoding tasks, we need to estimate the required computing resource (i.e., transcoding time of a task) for the task scheduling.",
"We adopt the neural network method for learning the non-linear relationship among the transcoding time of a task and other dependent factors.",
"We use a 3-layer feedforward neural network, which consists of one input layer, one hidden layer, and one output layer.",
"We denote the input feature vector of the input layer as $\\xi $ .",
"To construct $\\xi _i$ for the transcoding task $i$ , we use the duration, bitrate, frame rate, and resolution of the original video file, and the resolution of the target video file as inputs.",
"The output layer generates the estimated transcoding time for this task.",
"The nonlinear relationship between the estimated transcoding time and the input feature vector is $D_i = \\Gamma (\\xi _i),\\\\$ where $\\Gamma $ is the neural network trained offline before running the system and $D_i$ is the estimated transcoding time of task $i$ ."
],
[
"Service Revenue Model",
"We adopt a pricing mechanism which involves the task consumed computing time, task completion time, and the service level.",
"The valuation function of a transcoding task is given in Eq.",
"REF .",
"For the transcoding task $i$ , which arrives at $a_i$ , the revenue gained from this task if completed at time $t$ is $U_i(t) = \\alpha ^{t-a_i} R_i D_i, 0 < \\alpha < 1, t \\ge a_i, $ where $\\alpha $ is the price discounting factor, $R_i$ is the initial marginal price for one unit of computing time required for transcoding, and $D_i$ represents the amount of computing resource to be provided by the service provider.",
"The form of the valuation function will affect the derivation of the task scheduling policy in the fast timescale.",
"Our method can also be applicable to some other functions, e.g., linear functions and step functions.",
"The valuation function of linear form is: ${U^{\\prime }}_i(t) = w_i - \\beta _i (t-a_i), t \\ge a_i, \\beta _i > 0, \\qquad \\mathrm {(3.a)}$ where $w_i$ is initial revenue for task $i$ and $\\beta _i$ is the discounting factor.",
"The values for ${U^{\\prime }}_i(t)$ which are less than zero can be seen as service penalty for the processing delay.",
"The step valuation function is: ${U^{\\prime \\prime }}_i(t) = {\\left\\lbrace \\begin{array}{ll}w_i, & a_i \\le t \\le a_i + \\tau _i, \\\\0, & t \\ge a_i + \\tau _i,\\end{array}\\right.",
"}\\qquad \\mathrm {(4.b)}$ where $\\tau _i$ is the service deadline for task $i$ .",
"If the task miss the deadline, the revenue will be zero."
],
[
"VM Instance Cost Model",
"The system needs to dynamically adjust the number of provisioned VM instances for resource provisioning.",
"Nevertheless, provisioning a new VM instance consumes substantial time, thus the operation should not be done too frequently.",
"As such, we scale the transcoding cluster each $T$ time slots at the beginning of each $N_k$ , $k=1,2, ...$ , and also $T$ is not too large to ensure that the task arrival rate is relatively constant over the $T$ time slots.",
"We assume that the VM instances in the transcoding cluster are homogeneous and the overall cost is proportional to the number of provisioned VM instances.",
"We denote the number of provisioned VM instances in the transcoding cluster during $N_k$ as $M(N_k)$ .",
"As such, the total cost for provisioning the VM instances at $N_k$ is $C^v(N_k) = M(N_k)C_v,\\\\$ where $C_v$ is the cost for one VM instance over $T$ time slots."
],
[
"Transcoding Service Profit Maximization Problem",
"Our objective is to maximize the overall profit of the transcoding service.",
"The dynamics of our system can be characterized in the separable slow and fast timescale.",
"The resource provisioning operation, acting at the lower frequency, has a relative long-term effect on the system performance.",
"The task scheduling operation takes the resource provisioning operation as input and tracks the revenue maximization of the tasks at the higher frequency.",
"We jointly consider the task scheduling and resource provisioning in two timescales.",
"We define the system state at the beginning of each $N_k$ in the slow timescale as $\\psi ^s_{k} = \\lbrace \\lambda _k, m_{k-1}\\rbrace $ , where $\\lambda _k$ is the discrete task arrival rate for each time slot of $N_k$ and $m_{k-1}$ is the number of active VM instances before taking the resource provisioning operation.",
"The state space for the system state in the slow timescale is denoted as $\\Psi ^s$ .",
"We define the system state in the fast timescale at the time slot $t$ as $\\psi ^f_{t} = \\lbrace x_t\\rbrace $ , where $x_t$ is the set of pending tasks at time slot $t$ .",
"For each pending task, it has the information of the elapsed time from its submission and its estimated required computing time.",
"The state space for the system state in the fast timescale is denoted as $\\Psi ^f$ .",
"We assume that the system state space in the slow timescale and the fast timescale are both finite.",
"The resource management policy determines the number of VM instances to be shut down or activated at the beginning of each $N_k$ according to the system state in the slow and fast timescale.",
"We denote the resource provisioning policy in the slow timescale as $\\pi ^s$ and the resource provisioning operation at $N_k$ as $\\nu _k$ .",
"Specifically, $\\nu _k > 0$ represents the number of new activated VM instances and $\\nu _k < 0$ represents the number of shutdown VM instances.",
"The finite action space for the resource provisioning operation is denoted as $\\wedge $ .",
"The mapping from the system state in the slow and fast timescale to the resource provisioning operation by applying the policy $\\pi ^s$ is $\\pi ^s: (\\psi ^s_{k}, \\psi ^f_{kT}) \\rightarrow \\nu _k, k = 0,1,2,...$ The number of active VM instances after taking the resource provisioning operation is $m_k$ , where $m_k = m_{k-1} + \\nu _k \\ge 0$ .",
"In our system, the task scheduling policy in the fast timescale determines the transcoding order of the pending transcoding tasks to maximize the overall revenue.",
"The system state in the fast timescale evolves over $T$ time slots until the system state in the slow timescale changes.",
"The system dynamic in the fast timescale is an MDP over finite $T$ -horizon and the task scheduling policy is a sequence of $T$ -horizon nonstationary policies.",
"We define the sequence of task scheduling policies over the finite $T$ -horizon as $\\pi ^f_T$ , and $\\pi ^f_{T}(t)$ is the task scheduling policy at the time slot $t$ .",
"At each time slot $t$ , we define the mapping from the system state and operation in the slow timescale and the system state in the fast timescale to the task scheduling operation for the pending tasks as $\\pi ^f_T(t): (\\psi ^s_{k}, \\nu _k, \\psi ^f_{t}) \\rightarrow \\ell _t, \\ kT \\le t < (k+1)T,$ where $\\ell _t$ is the task scheduling operation for the pending tasks.",
"Specifically, in our system, $\\ell _t$ is the scheduled transcoding order of the pending transcoding tasks in the queue.",
"At the beginning of $N_k$ , given the slow timescale state $\\psi ^s_{k}$ , the resource provisioning operation $\\nu _k$ , the fast timescale state $\\psi ^f_{kT}$ , and the $T$ -horizon task scheduling policy $\\pi ^f_T$ , the total expected service profit over $T$ time slots in $N_k$ is $R^s_k(\\psi ^s_{k}, \\psi ^f_{kT}, \\nu _k, \\pi ^f_T) & = \\mathbb {E}_{\\psi ^f_{t}} \\Bigg \\lbrace \\sum _{t=kT}^{(k+1)T-1} P(t) - C^v(N_k) \\Bigg \\rbrace , \\nonumber $ where $P(t)$ is service revenue at the time slot $t$ .",
"We aim to maximize the overall future discounted profit by applying appropriate task scheduling policy $\\pi ^f_T$ and resource provisioning policy $\\pi ^s$ .",
"Mathematically, we present the Service Profit Maximization (SPM) Problem as follow $\\mathcal {P}1:\\ \\ \\max _{\\pi ^s \\in \\Pi ^s} \\max _{\\pi ^f_T \\in \\Pi ^f_T} \\mathbb {E}_{\\psi ^s_{k}, \\psi ^f_{t}} \\left\\lbrace \\sum _{k=0}^{\\infty } \\gamma ^k R^s_k (\\psi ^s_{k}, \\psi ^f_{kT}, \\nu _k, \\pi ^f_T) \\right\\rbrace , \\nonumber $ where $\\gamma $ is the discounting factor, and $\\Pi ^s$ and $\\Pi ^f_T$ are the finite set of possible resource provisioning policies and task scheduling policies, respectively.",
"To derive the optimal policies for maximizing the overall profit, one can in principle derive the offline policies with the method of value iteration.",
"In a practical system, however, the system state space is large and the state transition probability is difficult to obtain exactly.",
"We will discuss the approximate methods to derive the task scheduling policy and resource provisioning policy."
],
[
"Approximate Policies for Service Profit Maximization Problem",
"In this section, we present some approximate policies for task scheduling and resource provisioning."
],
[
"Value-based Task Scheduling Policy",
"Our approximation for the task scheduling policy is to assume that the number of active VM instances is unchanging for the current set of pending tasks, and the task scheduler determines the task transcoding order $\\ell _t$ of the pending tasks by maximizing the overall revenue of the existing tasks, mathematically, we present it as $\\mathcal {P}2: \\max _{\\ell _t} \\sum _{i \\in x_t} U_i(f_i),\\\\$ where $f_i$ is the completion time of task $i$ given the transcoding order $\\ell _t$ of the pending tasks.",
"To solve this problem, we first introduce a method for estimating the completion time of the pending tasks when given a transcoding order of the pending tasks and the current number of active VM instances.",
"Based on that, we present a method for deriving the solution of $\\mathcal {P}2$ .",
"The original video files are partitioned into video blocks to be dispatched to many VM instances for parallel transcoding and each of the video blocks consumes the same amount of computing time.",
"We assume that each video block consumes $F$ time slots for transcoding, and for a task $i \\in x_t$ , it is divided into $b_i$ video blocks.",
"We assume that the order of task $i$ in the queue is $o_i$ .",
"Given the order of the pending tasks in the queue, we denote the total number of video blocks of the tasks which order is not larger than $o_i$ as $g_i$ .",
"We have the following proposition for estimating the finish time of the $g_i$ -th video block (i.e., task $i$ ).",
"Proposition 1 Suppose that the transcoding cluster has $m_k$ active VM instances and the transcoding progresses of the current video blocks on these VM instances are unknown.",
"At the time slot $t_0$ , one transcoding worker becomes idle and requests the first video block in the queue.",
"Then, the expected completion time of the $g_i$ -th video block (i.e., the task $i$ ) is $E\\lbrace f_i\\rbrace = t_0 + \\frac{F}{m_k}(g_i - 1) + F.$ Please see Appendix for the detailed proof."
],
[
"Task Scheduling Policy",
"Given the set of the pending tasks $x_{t}$ , we assume task $i$ , $j$ , $k$ are successive in the queue and the order is denoted as $(...,i,j,k,...)$ .",
"We denote the durations between the completion time of task $i$ and $j$ , task $j$ and $k$ as $d_{ij} = f_j - f_i,\\ d_{jk} = f_k - f_j,$ where $f_i$ , $f_j$ , $f_k$ are the completion time of task $i$ , $j$ , $k$ , respectively.",
"Given the order $(...,i,j,k,...)$ , let the expected revenue gained from task $j$ and $k$ based on Eq.",
"(REF ) be $R_{jk}$ .",
"We exchange the order of task $j$ and $k$ while keeping the order of the other tasks unchanged, let the expected revenue gained from task $j$ and $k$ in the order of $(...,i,k,j,...)$ be $R_{kj}$ .",
"If $R_{jk} > R_{kj}$ , it can be deduced that $\\frac{\\alpha ^{d_j - a_j} R_j D_j}{1 - \\alpha ^{d_j}} \\ge \\frac{\\alpha ^{d_k - a_k} R_k D_k}{1 - \\alpha ^{d_k}}, $ where $d_j = d_{ij} = d_{kj} = \\frac{F}{m_k} b_j$ and $d_k = d_{jk} = d_{ik} = \\frac{F}{m_k} b_k$ .",
"As such, we have the following proposition for task scheduling.",
"Proposition 2 If the current set of pending tasks $x_t$ are conducted in the decreasing order of the weight $P_i$ , we can maximize the overall revenue gained from these pending tasks, $P_i = \\frac{\\alpha ^{d_i - a_i} R_i D_i}{1 - \\alpha ^{d_i}}, i \\in x_t \\ and \\ d_i = \\frac{F}{m_k} b_i, $ We assume that the current set of pending tasks has been arranged by the decreasing order of $P_i$ .",
"The transcoding order of the task $i$ is $o_i$ .",
"If we move the task $i$ from $o_i$ to $o_i^{\\prime }$ , it can be done by iteratively interchanging task $i$ with its neighboring task until it reaches $o_i^{\\prime }$ .",
"Since the tasks have been in the order of decreasing $P$ , each interchanging will incur a loss on the revenue according to Eq.",
"(REF ).",
"Hence, it's the optimal scheduling by the decreasing order of $P$ for maximizing the revenue of the current set of the pending tasks.",
"The system in the fast timescale works as follow: in each time slot $t$ , the task scheduler calculates the value $P_i$ for each task, and sorts the tasks in decreasing order of $P_i$ .",
"The value of $P_i$ does not depend on $t$ , therefore, the task scheduler only needs to resort the pending tasks when new tasks come in or the number of VM instances has changed.",
"This method can also be applied to the valuation function Eq.",
"(REF ), and we can have $P_i = \\beta _i/d_i$ for sequencing the tasks in decreasing order.",
"With the valuation function Eq.",
"(REF ), the problem is known to be NP-hard.",
"In this case, however, sequencing the tasks in the decreasing order of $P_i = w_i/d_i$ is still shown to be a popular and effective approximate solution.",
"In this subsection, we introduce the method for deriving the resource provisioning policy.",
"The system dynamic in the slow timescale is an MDP with the system reward defined as the service profit over $T$ time slots if given the task scheduling policy $\\pi ^f_T$ in the fast timescale.",
"As such, we can write the system dynamic in the slow timescale as $\\mathcal {P}3: \\hat{V}^* & (\\psi ^s_{k}, \\psi ^f_{kT}) = \\max _{\\nu _k} \\Big \\lbrace R^s_k(\\psi ^s_{k}, \\psi ^f_{kT}, \\nu _k, \\pi ^f_T) \\nonumber \\\\& + \\gamma \\mathbb {E}_{\\psi ^s_{k+1}, \\psi ^f_{(k+1)T}} \\big \\lbrace \\hat{V}^*(\\psi ^s_{k+1}, \\psi ^f_{(k+1)T}) \\big \\rbrace \\Big \\rbrace .$ We leverage the Q-Learning method, which is a model-free Reinforcement Learning technique, to find the action-selection policy for the given MDP in the slow timescale.",
"The learning procedures are as follows: at the time $N_k$ , the system observes the state $(\\psi ^s_{k}, \\psi ^f_{kT})$ in the two timescales, and selects the action $\\nu _k$ according to a certain resource provisioning policy $\\pi ^s$ .",
"After $T$ time slots, the system observes the service profit $R^s_k$ gained over the $T$ time slots in the fast timescale and the new system state $(\\psi ^s_{k+1}, \\psi ^f_{(k+1)T})$ at the time $N_{k+1}$ .",
"Then, the new estimated discounted overall future profit starting from the state $(\\psi ^s_{k}, \\psi ^f_{kT})$ by taking the action $\\nu _k$ can be calculated as $Q^{^{\\prime }}((\\psi ^s_{k}, \\psi ^f_{kT}), \\nu _k) & = R^s_k(\\psi ^s_{k}, \\psi ^f_{kT}, \\nu _k, \\pi ^f) \\nonumber \\\\& + \\gamma \\max _{\\nu _{k+1}} Q((\\psi ^s_{k+1}, \\psi ^f_{(k+1)T}), \\nu _{k+1}), $ where $Q((\\psi ^s_{k}, \\psi ^f_{kT}), \\nu _k)$ is the action-value and $\\nu _{k+1}$ is the optimal action that can maximize the expected overall discounted profit starting from the state $(\\psi ^s_{k+1}, \\psi ^f_{(k+1)T})$ .",
"As such, the action-value $Q((\\psi ^s_{k}, \\psi ^f_{kT}), \\nu _k)$ can be updated based on the new estimation according to following equation, $Q((\\psi ^s_{k}, & \\psi ^f_{kT}), \\nu _k) = Q((\\psi ^s_{k}, \\psi ^f_{kT}), \\nu _k) \\nonumber \\\\& + \\delta _k \\lbrace Q^{^{\\prime }}((\\psi ^s_{k}, \\psi ^f_{kT}), \\nu _k) - Q((\\psi ^s_{k}, \\psi ^f_{kT}), \\nu _k)\\rbrace , $ where $\\delta _k$ is the learning rate.",
"After visiting each state-value enough times, the Q-learning algorithm will converge to the optimal policy.",
"To reduce the dimensionality of the system space, we adopt the feature extraction method to obtain a compact representation of the state space, which are considered as the important characteristics of the original space.",
"We denote the compact state space as $\\Phi $ , the original state $(\\psi ^s_{k}, \\psi ^f_{kT})$ can be compacted as the state $\\phi _k = (\\omega _{kT}, m_{k}, \\lambda _k)$ , where $\\omega _{kT}$ is the summation of the valuation of the pending task at the time $kT$ , $m_{k}$ is the number of active VM instances, and $\\lambda _k$ is the average task arrival rate.",
"We replace the original system state $(\\psi ^s_{k}, \\psi ^f_{kT})$ with $\\phi _k$ in Eq.",
"(REF ) and (REF ).",
"We adopt the $\\varepsilon $ -$greedy$ method to balance the exploring and exploiting when selecting action for learning.",
"Based on the above discussions, our method for learning the resource provisioning policy in the slow timescale is illustrated in Algorithm REF .",
"Learning-based Resource Provisioning Policy [1] Initialize $Q(\\phi , \\nu ) = C$ , $\\forall \\phi $ , $\\nu $ .",
"Task scheduling policy $\\pi ^f_T$ in the fast timescale.",
"Set the maximum number of loops, M. Set k = 0.",
"The optimal action-value $Q^*(\\phi , \\nu )$ .",
"Obtain the current system state $\\phi _k$ at the beginning of $N_k$ .",
"Select $\\nu _k$ based on $\\phi _k$ and $Q(\\phi , \\nu )$ using $\\varepsilon $ -$greedy$ .",
"Take action $\\nu _k$ , observe the overall service profit $R^s_k$ over $T$ time slots in $N_k$ under task scheduling policy $\\pi ^f$ in the fast timescale.",
"Observe system state $\\phi _{k+1}$ at the beginning of $N_{k+1}$ .",
"Update action-value $Q(\\phi _{k}, \\nu _k)$ according to Eq.",
"(REF ) $k \\leftarrow k+1$ $k<M$"
],
[
"System Implementation and Experiment Settings",
"We implement our cloud video transcoding system Morph in Python.",
"The source code of our project is released in Github, which can be found in [8].",
"We run the transcoding system in a cloud environment built with Docker.",
"We use the ffmpeg for the video transcoding operation.",
"The price for renting a VM instance is $0.252 per hour.",
"Each of the VM instances in our home-built cloud has 4 CPU cores, the CPU frequency is 2.10 GHz, and the memory size is 2GB.",
"The VM instances are connected with Gigabit Ethernet, and the data transmission speed can achieve 1000 Mbit/s among the VM instances.",
"Our system provides three levels of services, denoted as Level I, Level II, and Level III, respectively.",
"The initial marginal price $R_i$ is $0.018 per minute for service level I, $0.012 per minute for service level II, and $0.006 per minute for service level III.",
"The price discounting factor $\\alpha $ is 0.999 per second.",
"The consumed computing time for each video block is 180 seconds."
],
[
"Transcoding Time Estimation Accuracy",
"We measure the time for transcoding 2020 video files of different original bitrates and resolutions into three target resolutions, namely, 854x480, 640x360, 426x240.",
"We obtain 3850 instances of the measured transcoding time.",
"We select 75% of the data for training the neural network offline, 15% of the data for model validation, and the other 15% of the data for testing.",
"The hidden layer of the neural network consists of 20 neurons.",
"The inputs of the neural network include the video bitrate, resolution, frame rate, duration of the original video file, and the target video resolution.",
"We multiply the width and height of the video resolution and use the product as one input.",
"We compare the neural network method for transcoding time estimation with the linear approximation method which estimates transcoding time as a linear function of the video duration.",
"We normalize the prediction error of the test using the following equation for comparison $Normalized \\ Error = \\frac{Predicted \\ Time - Real \\ Time}{Real \\ Time}.",
"\\nonumber $ The transcoding time is measured in seconds in our experiments.",
"The error histogram of the neural network method for transcoding time prediction is illustrated in Fig.",
"REF .",
"The normalized prediction error of most of the testing instances are within the range from -0.08 to 0.08.",
"The error histogram of the linear approximation method is illustrated in Fig.",
"REF , the normalized prediction error of the testing instances ranges from -0.58 to 1.42.",
"From the comparisons, we can observe that the neural network method can predict the transcoding time much more precisely.",
"Figure: Error histogram of linear approximation method."
],
[
"Service Profit under Real Trace Data",
"We measure the service profit under a real trace data.",
"The trace data captures the video requests to a CDN node.",
"We extract the user requests in the trace data as the transcoding requests for our service.",
"We divide the time of each day into 24 hours and average the request number during one hour of the days as the average task arrival rate in the system state for this hour of a day.",
"We scale down the average request rate in the real trace to the range of 0.1-0.7 request per minute.",
"Figure: The number of provisioned VM instances in each hour.We measure the service profit in the real environment over 24 hours under different resource provisioning policies and task scheduling policies.",
"We refer to the value-based task scheduling policy as VBS.",
"We compare the service profit under our proposed Learning-based Resource Provisioning policy and Value-based Task Scheduling (LRP-VBS) policy with the following methods: 1) The Fixed Policy (FP) which runs a fixed number of VM instances under the task scheduling policy of VBS.",
"We select two representative numbers of VM instances for the FP methods to illustrate, i.e., 10 VM instances and 15 VM instances.",
"2) The Arrival Rate based Policy (ARP), i.e., the number of provisioned VM instances is proportional to task arrival rate.",
"This method estimates the number of VM instances to satisfy the computing resource requirements of the transcoding tasks in each period based on the task arrival rate.",
"In our test, we set the number of provisioned VM instances as 30 times of the task arrival rate per minute.",
"3) The learning-based resource provisioning policy combined with other task scheduling policy.",
"We select the learning-based resource provisioning policy combined with the task scheduling policy of Highest Value First (LRP-HVF).",
"The HVF policy always selects the transcoding task which has the current highest valuation to perform.",
"The current valuation of the tasks are calculated by Eq.",
"(REF ).",
"We illustrate the cumulative profit under different methods over 24 hours in Fig.",
"REF .",
"We can observe that the cumulative service profit under our proposed LRP-VBS is larger than the baseline methods.",
"As demonstrated in Fig.",
"REF , the LRP method can effectively adjust the number of provisioned VM instances in the slow timescale given the task scheduling policy HVF or VBS in the fast timescale.",
"The task scheduling policy in the fast timescale will also affect the resource provisioning operation in the slow timescale.",
"As such, LRP-VBS can gain a higher profit than the LRP-HVF since that the VBS can outperform the HVF for task scheduling in the fast timescale.",
"In constrast, the FP method may waste much computing resource when the system workload is low and deteriorate the system performance when the system workload exceeds the processing capacity of the current number of VM instances.",
"The ARP method can dynamically adjust the number of provisioned VM instances according to the task arrival rate and system workload, however, it is hard for this method to model the decreasing of the revenue for the processing delay.",
"Moreover, the service revenue for the transcoding tasks is also affected by the task scheduling policy, but the ARP method does not take the task scheduling scheme into consideration.",
"Therefore, the ARP method cannot effectively determine the optimal policy for maximizing the service profit.",
"The LRP is more suitable for such hard-to-model problem, and it can work effectively in the system dynamic by learning the optimal policy during the training stage."
],
[
"Conclusions",
"We consider the problem of how to provision the computing resource for transcoding the video contents while maximizing the service profit for the transcoding service provider.",
"We design a practical transcoding system by leveraging the cloud computing infrastructure.",
"We jointly consider the task scheduling and resource provisioning problem in the two timescales and formulate the service profit maximization problem as a two-timescale MDP.",
"We derive the approximate solutions for task scheduling and resource provisioning.",
"Based on our proposed methods, we implement the system and conduct extensive experiments to evaluate the system performance in a real environment.",
"The experiment results show our method can work effectively for the task scheduling and resource provisioning in the cloud video transcoding system."
],
[
"Proof of Proposition ",
"We denote the remaining time to complete transcoding the current video block on the VM instance $i$ as $y_i$ .",
"We assume that at $t_0$ , the transcoding worker 1 becomes idle and requests the first video block in the queue, and therefore $y_1=F$ .",
"The waiting time for the next video block to be requested is $Y = \\min \\lbrace y_2, y_3, ..., y_{m_k}\\rbrace , \\nonumber $ where $y_2, y_3,...,y_{m_k}$ are unknown and randomly distributed in $[0,\\ F]$ and $0 \\le Y \\le F$ .",
"The transcoding progresses on the VM instances are independent and the CDF of $Y$ is $F_Y(t)& = P(0 \\le Y \\le t) \\nonumber \\\\& = 1 - P(y_2>t)P(y_3>t)...P(y_{m_k}>t).",
"\\nonumber $ Hence, the expected waiting time for the next block to be requested can be calculated as $E\\lbrace Y\\rbrace = \\frac{F}{{m_k}}$ .",
"We can deduce that the total waiting time for the $g_i$ -th video block to be requested by a transcoding worker is $\\frac{F}{{m_k}}(g_i - 1)$ .",
"As such, the estimated completion time of the $g_i$ -th video block is $E\\lbrace f_i\\rbrace = t_0 + \\frac{F}{{m_k}}(g_i - 1) + F.$"
]
] | 1605.05758 |
[
[
"High-temperature asymptotics of the 4d superconformal index"
],
[
"Abstract The superconformal index of a typical Lagrangian 4d SCFT is given by a special function known as an elliptic hypergeometric integral (EHI).",
"The high-temperature limit of the index corresponds to the hyperbolic limit of the EHI.",
"The hyperbolic limit of certain special EHIs has been analyzed by Eric Rains around 2006; extending Rains's techniques, we discover a surprisingly rich structure in the high-temperature limit of a (rather large) class of EHIs that arise as the superconformal index of unitary Lagrangian 4d SCFTs with non-chiral matter content.",
"Our result has implications for $\\mathcal{N}=1$ dualities, the AdS/CFT correspondence, and supersymmetric gauge dynamics on $R^3\\times S^1$.",
"We also investigate the high-temperature asymptotics of the large-N limit of the superconformal index of a class of holographic 4d SCFTs (described by toric quiver gauge theories with SU(N) nodes).",
"We show that from this study a rather general solution to the problem of holographic Weyl anomaly in AdS$_5$/CFT$_4$ at the subleading order (in the 1/N expansion) emerges.",
"Most of this dissertation is based on published works by Jim Liu, Phil Szepietowski, and the author.",
"We include here a few previously unpublished results as well, one of which is the high-temperature asymptotics of the superconformal index of puncture-less SU(2) class-$\\mathcal{S}$ theories."
],
[
"=1 Chapter : appendicesloaList of Appendices Appendix : top=2.5in High-temperature asymptotics of the 4d superconformal index by Arash Arabi Ardehali A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in the University of Michigan 2016 Doctoral Committee: Professor James T. Liu, Chair Professor Igor Kriz Professor Finn Larsen Professor Roberto D. Merlin Professor Leopoldo A. Pando Zayas tocchapterAbstract top=2in Abstract This dissertation contains a study of certain four-dimensional superconformal field theories (4d SCFTs).",
"Any 4d SCFT has a spectrum of local operators.",
"Some of these operators sit in short representations of the 4d $\\mathcal {N}=1$ superconformal group SU($2,2|1$ ), and can be quantified using a partition function known as the 4d superconformal index.",
"The superconformal index $\\mathcal {I}(b,\\beta )$ is a function of two positive real parameters: the squashing parameter $b$ , and the inverse temperature $\\beta $ .",
"Our study in the present dissertation is focused on the temperature- (or $\\beta $ -) dependence of the superconformal index of 4d SCFTs.",
"The superconformal index of a typical Lagrangian 4d SCFT is given by a special function known as an elliptic hypergeometric integral (EHI).",
"The high-temperature limit of the index corresponds to the hyperbolic limit of the EHI.",
"The hyperbolic limit of certain special EHIs has been analyzed by Eric Rains around 2006; extending Rains's techniques, we discover a surprisingly rich structure in the high-temperature limit of a (rather large) class of EHIs that arise as the superconformal index of unitary Lagrangian 4d SCFTs with non-chiral matter content.",
"Our result has implications for $\\mathcal {N}=1$ dualities, the AdS/CFT correspondence, and supersymmetric gauge dynamics on $R^3\\times S^1$ .",
"We also investigate the high-temperature asymptotics of the large-$N$ limit of the superconformal index of a class of holographic 4d SCFTs (described by toric quiver gauge theories with SU($N$ ) nodes).",
"We show that from this study a rather general solution to the problem of holographic Weyl anomaly in AdS$_5$ /CFT$_4$ at the subleading order (in the $1/N$ expansion) emerges.",
"Most of this dissertation is based on published works by Jim Liu, Phil Szepietowski, and the author.",
"We include here a few previously unpublished results as well, one of which is the high-temperature asymptotics of the superconformal index of puncture-less SU(2) class-$\\mathcal {S}$ theories.",
"tocchapterAcknowledgements top=2in Acknowledgements I have been very privileged to have Jim Liu as my thesis advisor.",
"The work described in this dissertation is the result of a long and enjoyable collaboration with him and Phil Szepietowski.",
"I would like to express my deepest gratitude to both of them, for the wonderful and formative time I spent with them.",
"I am thankful to Finn Larsen and Leo Pando Zayas, with whom I had shorter collaborations, which were nevertheless memorably delightful, and very educational.",
"I am indebted to Finn and Leo also for their continual encouragement and support during my phd.",
"While carrying out the analysis reported in Chapter 3 of this dissertation, I have benefited from ideas and suggestions of Peter Miller and Eric Rains; I am grateful to both of them for helpful discussions on the subject.",
"I would like to thank Ratindranath Akhoury, Henriette Elvang, Finn Larsen, Roberto Merlin, Leo Pando Zayas, and Len Sander for the numerous enjoyable scientific conversations we have had, and also for the physics that I have learned from them in their unforgettably nice classes.",
"I am grateful to Lydia Bieri and Igor Kriz for their inspiring and memorable courses on geometry and topology, through which I learned to appreciate the exciting depth of modern mathematical ideas.",
"Thanks to all of the graduate students in the physics department at University of Michigan for the fun time we had together.",
"I wish to acknowledge in particular the fruitful scientific interactions with Anthony Charles, Mahdis Ghodrati, Marios Hadjiantonis, Jack Kearney, Gino Knodel, Pedro Lisbao, Alejandro Lopez, Tim Olson, Uttam Paudel, Vimal Rathee, Sam Roland, and Bob Zheng.",
"Finally, I would especially like to thank all of my friends and family for their unending and unconditional support.",
"tocchapterList of Figures tocchapterList of Appendices tocchapter Quantum Field Theory (QFT) is, among other things, the theoretical framework for understanding fundamental particles and their interactions.",
"The particles in a given QFTA specific quantum system may be described by a specific model in quantum field theory; with an abuse of terminology, we will refer to different “model”s in the QFT framework as different “QFT”s.",
"Also, we will have in mind only conventional QFT models, consisting of fields with spin $\\le 1$ .",
"are divided into bosons carrying spin zero or one, and fermions carrying spin one-half; matter—in the form common on Earth—typically consists of the fermions, and the bosons mediate interactions between the matter particles.",
"Despite theoretical successes in the regime where the QFT particles interact weakly and perturbation theory accurately describes a wide range of observed phenomena, lack of progress on long-standing strong-interaction problems in QFT (such as the problem of quark confinement) indicates that more powerful non-perturbative techniques are needed.",
"A promising arena wherein to uncover non-perturbative structures in QFT is the realm of Supersymmetric (SUSY) QFTs.",
"These are theories enjoying a powerful symmetry that, roughly speaking, exchanges their fermions and bosons.",
"Many of the properties of supersymmetric theories are under better analytic control, sometimes even in the strong-interaction regime, thanks to their large symmetry group.",
"A subset of SUSY QFTs are yet much more symmetric, and it is natural to start with them in the quest for non-perturbative understanding of QFTs.",
"These are Conformal SUSY QFTs, also known as SuperConformal Field Theories (SCFTs).",
"An SCFT has a Hilbert space that is invariant not only under supersymmetry, but also under the action of the conformal group, which in four space-time dimensions can be describedFor brevity of exposition, we will not distinguish here between the symmetry group and its universal cover.",
"as SU($2,2$ ); with minimal supersymmetry (i.e.",
"four Poincare and four conformal supercharges) added, the conformal group extends to the $\\mathcal {N}=1$ superconformal group SU($2,2|1$ ).",
"Besides serving as toy models of the richer non-conformal SUSY QFTs, SCFTs play a conceptually important role in the renormalization group (RG) approach to SUSY QFT: one can often think of non-conformal SUSY QFTs as describing “flows” between some SCFT in the ultraviolet (UV), and some SCFT in the infrared (IR) regime of energies.",
"Thus, SCFTs can serve as signposts on the landscape of SUSY QFTs.",
"The main piece of data of an SCFT is the spectrum—of the various quantum numbers—of the states in its Hilbert space.",
"The states of a conformal field theory are in one-to-one correspondence with the local operators in the theory [1].",
"Therefore an important objective when studying a given SCFT is to classify and count the local operators/states of the theory.",
"Since the Hilbert space of an SCFT is invariant under the 4d superconformal group, the states/operators in it are labeled by quantum numbers associated to the generators of the maximal compact bosonic subgroup of SU($2,2|1$ ).",
"A subset of the states/operators (known as BPS states/operators), which have specific relations between their various quantum numbers, sit in short representations of the superconformal algebra.",
"This sector is expected to be protected against “smooth deformations” arising from RG flows or interactions, because the quantum numbers of the short representations do not undergo smooth changesThis is true modulo multiplet recombination.",
"The possibilities for recombination are limited though, by the fact that the R-charges of local operators (in a Lagrangian 4d SCFT) are algebraic numbers; c.f.",
"[2].",
"; the states/operators in this sector are hence under better analytic control thanks to this “topological” structure that the superconformal symmetry induces on the Hilbert space.",
"The topological sector consisting of the short representations is sometimes referred to as the BPS sector of the SCFT.",
"The superconformal index [3], [4] is a particular partition function which efficiently quantifies this controllable sector of an SCFT.",
"As a partition function, it depends on an inverse-temperature parameterThe index also depends on a squashing parameter $b$ which for simplicity we suppress (i.e.",
"set to unity) in the present chapter.",
"$\\beta $ used to weigh various states with Boltzmann-type factors.",
"Investigating the $\\beta $ -dependence of the superconformal index—or the index—of various interesting SCFTs is the main goal of this dissertation.",
"More precisely, we would like to understand how the index behaves as the temperature is taken to infinity—or $\\beta $ is taken to zero.",
"(The low-temperature ($\\beta \\rightarrow \\infty $ ) asymptotics of the index is rather trivial; see [5].)",
"It is worth emphasizing that the “temperature” parameter in the index does not admit a thermal interpretation: in a path-integral picture, the index is computed as the partition function on $S^3\\times S^1$ with periodic boundary conditions around the $S^1$ [6], while the more familiar thermal partition functions are computed with fermions having anti-periodic boundary conditions around the Euclidean time circle.",
"Nevertheless, the Boltzmann-type factors entering the definition of the index [see for instance Eq.",
"REF below] suggest this stretch of terminology, and thus we will keep referring to $\\beta $ as `inverse temperature'.",
"The reader without prior familiarity with superconformal indices may wonder if investigating the index, which is only a certain measure of a certain sector of an SCFT, is a worthwhile endeavor.",
"The following two remarkable applications of the index respond to this question in the positive.",
"Application to supersymmetric duality: the so-called IR dualities in SUSY QFT imply that two differently formulated SCFTs (e.g.",
"two SCFTs with different field contents) are exactly equivalent, even at the non-perturbative (!)",
"level; the superconformal index can serve as a probe of this equivalence, since a proposal for duality of two formulations may be valid only if the indices computed using the different formulations are equal.",
"The great power of the index in probing supersymmetric dualities was demonstrated in the seminal paper of Dolan and Osborn [8] around 2008.",
"Application to holography: according to the AdS/CFT correspondence, the states encoded in (or “counted by”) the superconformal index of a holographic SCFT (such as the $\\mathcal {N}=4$ SYM) correspond to states of a quantum gravity theory in Anti-de Sitter (AdS) space.",
"In particular, it is expected that the index will help the microscopic counting of high-energy quantum gravity states, such as Giant Gravitons [9].",
"Application to supersymmetric duality: the so-called IR dualities in SUSY QFT imply that two differently formulated SCFTs (e.g.",
"two SCFTs with different field contents) are exactly equivalent, even at the non-perturbative (!)",
"level; the superconformal index can serve as a probe of this equivalence, since a proposal for duality of two formulations may be valid only if the indices computed using the different formulations are equal.",
"The great power of the index in probing supersymmetric dualities was demonstrated in the seminal paper of Dolan and Osborn [8] around 2008.",
"Application to holography: according to the AdS/CFT correspondence, the states encoded in (or “counted by”) the superconformal index of a holographic SCFT (such as the $\\mathcal {N}=4$ SYM) correspond to states of a quantum gravity theory in Anti-de Sitter (AdS) space.",
"In particular, it is expected that the index will help the microscopic counting of high-energy quantum gravity states, such as Giant Gravitons [9].",
"We will see that understanding the high-temperature asymptotics of the index not only leads to advances in both of the directions itemized above, but also opens up new prospects for understanding the non-perturbative (!)",
"low-energy dynamics of 4d supersymmetric gauge theories compactified on a circle.",
"(More precisely, the high-temperature asymptotics of the index of an SCFT formulated in the UV as a gauge theory on $R^4$ , seems to encode information on the Coulomb branch dynamics of the gauge theory on $R^3\\times S^1$ ; see subsection REF .)",
"In the remaining parts of this chapter, we first introduce the 4d superconformal index, along with its famous precedent, the quantum mechanical Witten index, more elaborately in section REF .",
"Then we proceed in section REF to highlight our main result: the high-temperature asymptotics of the superconformal index of finite-rank Lagrangian unitary non-chiral 4d SCFTs.",
"In section REF we discuss the high-temperature asymptotics of the large-rank limit of the indices of a class of holographic SCFTs, and explain how our results address (for the class of theories under study) the computation of the Holographic Weyl Anomaly at the subleading order in the $1/N$ expansion.",
"Consider a unitary quantum mechanical system enjoying supersymmetry.",
"That is to say, there exists a fermionic operator $Q$ , referred to as the supercharge operator, acting on the Hilbert space of the system, and satisfying $\\lbrace Q,Q^{\\dagger }\\rbrace =H\\quad \\quad \\text{and}\\quad \\quad Q^2=0,$ with $H$ the Hamiltonian operator.",
"Existence of $Q$ implies that states of nonzero energy are paired in the system: it can be easily checked using (REF ) that $Q+Q^{\\dagger }$ provides a one-to-one mapping from the set of bosonic states $|\\mathrm {b}\\rangle $ with $H|\\mathrm {b}\\rangle \\ne 0$ , to the set of fermionic states $|\\mathrm {f}\\rangle $ with $H|\\mathrm {f}\\rangle \\ne 0$ .",
"But the zero-energy states are not necessarily paired: $\\begin{split}0=\\langle \\mathrm {b}| H |\\mathrm {b}\\rangle =\\ & \\langle \\mathrm {b}|\\lbrace Q,Q^\\dagger \\rbrace |\\mathrm {b}\\rangle =\\langle \\mathrm {b}| Q\\ Q^\\dagger |\\mathrm {b}\\rangle +\\langle \\mathrm {b}| Q^\\dagger \\ Q|\\mathrm {b}\\rangle \\\\&\\Rightarrow (Q+Q^\\dagger )|\\mathrm {b}\\rangle =0,\\end{split}$ with a similar argument applying to the fermionic zero-energy states.",
"Since the states with zero energy are annihilated by $Q$ and $Q^\\dagger $ , we say that the zero-energy states are “supersymmetric”.",
"Since (REF ) implies that $H$ necessarily has non-negative eigenvalues, we can further say that the zero-energy states are supersymmetric ground states of the theory.",
"Therefore in unitary supersymmetric quantum mechanics, the Witten index [7] $\\begin{split}\\mathcal {I}^W:=&\\sum _i (-1)^Fe^{-\\hat{\\beta }E_i}\\\\=&n^b_{\\mathrm {z.e.",
"}}-n^f_{\\mathrm {z.e.",
"}}\\end{split}$ (with $F$ the fermion-number operator, evaluating to 1 on fermionic states, and to 0 on bosonic states), receives contributions only from unpaired zero-energy (or supersymmetric ground-) states; the index is thus obviously independent of $\\hat{\\beta }$ .",
"Less obviously, $\\mathcal {I}^W$ is also independentModulo subtleties (see [7]) that are not relevant for our discussion.",
"of the interaction strength in the system!",
"The reason is that any continuous deformation of the system, such as that induced by RG flows or by variation of the interaction couplings, should cause the supersymmetric ground-states to acquire nonzero energy only in pairs; similarly, any nonzero-energy state that as a result of the continuous deformation becomes a supersymmetric ground-state, should be accompanied by a partner state all along.",
"This independence of the index from continuous deformations gives it a topological character—hence the well-deserved title “index”.",
"Since $\\mathcal {I}^W$ is independent of the couplings, it can be computed even in strongly interacting theories.",
"It can thus provide information that would otherwise be inaccessible through (the more conventional) perturbative means.",
"The original work of Witten used this index to probe the ground-state(s) of non-abelian supersymmetric gauge theories.",
"The idea was that a nonzero index would mean that either $n^b_{\\mathrm {z.e.",
"}}\\ne 0$ or $n^f_{\\mathrm {z.e.",
"}}\\ne 0$ , and thus it would imply the existence of supersymmetric ground-states, and hence the absence of spontaneous supersymmetry breaking.",
"It is worth emphasizing the remarkable fact that a weak-coupling calculation of $\\mathcal {I}^W$ can yield nontrivial information about the (strongly-interacting) ground-state of a non-abelian gauge theory!",
"The above discussion in the context of supersymmetric quantum mechanics can now help us to extend the concept of an index to unitary 4d SCFTs.",
"In any 4d SCFT there exists a supercharge operatorIn fact any 4d SCFT has at least four such operators.",
"The $Q$ that we consider here is one [it doesn't matter which] of the two that transform inside a $(0,1/2)$ representation of the (complexified) Lorentz group.",
"$Q$ , such that $\\lbrace Q,Q^{\\dagger }\\rbrace =H-2J_2^z-\\frac{3}{2}R\\quad \\quad \\text{and}\\quad \\quad Q^2=0,$ with $H$ the Hamiltonian in the radial quantization, $J_2^z$ the third generator of the right-handedHad we chosen a $Q$ operator transforming inside a $(1/2,0)$ representation of the Lorentz group, $J_2^z$ would be replaced with $J_1^z$ .",
"Lorentz SU(2), and $R$ the generator of the U(1)$_R$ inside the $\\mathcal {N}=1$ superconformal group SU($2,2|1$ ).",
"Therefore, in analogy with the above quantum mechanical discussion, we can define the following Witten index for unitary 4d SCFTs: $\\mathcal {I}^W:=\\mathrm {Tr}(-1)^{F}e^{-\\hat{\\beta }(E-2j_2-\\frac{3}{2}r)},$ with $E,j_2,r$ the eigenvalues of $H,J_2^z,R$ , and with the trace taken over the Hilbert space in the radial quantization.",
"Similarly to the quantum mechanical case above, the dependence on $\\hat{\\beta }$ drops out, since only states with vanishing $E-2j_2-\\frac{3}{2}r$ (hence sitting in short representations of SU($2,2|1$ )) have a chance of surviving bose-fermi cancelations.",
"It turns out that the combination $H-R/2$ commutes with the supercharge $Q$ used above.",
"We can hence refine $\\mathcal {I}^W$ with a fugacity $e^{-\\beta }$ for the combination $E-r/2$ , without ruining the cancelations underlying its topological character.",
"We refer to this refined Witten index as the 4d superconformal index [3], [4]: $\\mathcal {I}(\\beta ):=\\mathrm {Tr}(-1)^{F}e^{-\\beta (E-\\frac{r}{2})}e^{-\\hat{\\beta }(E-2j_2-\\frac{3}{2}r)}.$ In fact without the refinement with $\\beta $ , the index—as defined in (REF )—is divergent in interesting SCFTs; thus the Boltzmann-type factor $e^{-\\beta (E-\\frac{r}{2})}$ is actually necessary as a regulator.",
"Since the superconformal index does not depend on $\\hat{\\beta }$ , we can write $\\mathcal {I}(\\beta )=\\sum (-1)^{F}e^{-\\beta (E-\\frac{r}{2})},$ with the sum taken either over the local operators in the SCFT, or equivalently (via the CFT state/operator correspondence) over the states in the radial quantization.",
"The index (REF ) can be computed in closed form for a wide variety of interesting SCFTs.",
"For instance the index of a free chiral multiplet is given by an elliptic gamma function (see appendix REF for the definition of the elliptic gamma): $\\mathcal {I}_{\\chi }(\\beta )=\\Gamma (e^{-2\\beta /3};e^{-\\beta },e^{-\\beta }).$ In theories with several decoupled chiral multiplets, the index would be given by a product of the corresponding elliptic gamma functions.",
"In gauge theories with several chiral multiplets, the index would be given by a product of several chiral-multiplet gamma functions and several vector-multiplet gamma function, integrated (roughly speaking) over the gauge group so as to project the result onto the gauge-singlet sector.",
"We will explain this more carefully in Chapter REF .",
"We now summarize the rich structure we find in the high-temperature limit of the superconformal index.",
"Throughout this dissertation we focus on the index of unitary 4d SCFTs that admit a gauge theory description with non-chiral matter content; the SU($N$ ) $\\mathcal {N}=4$ SYM and the SU($N$ ) SQCD fixed points are the examples that we ask the reader to keep in mind while reading the somewhat abstract discussion below.",
"We emphasize that in the present section we are considering gauge theories with a finite rank; the large-$N$ limit of superconformal indices will be discussed in the next section of the present chapter.",
"The index of a Lagrangian SCFT [by that we mean an SCFT admitting a gauge theory description] is given by an elliptic hypergeometric integral (EHI) [8].",
"This is an expression of the form $\\int f(\\beta ;x_1,\\dots ,x_{r_G})\\ \\mathrm {d}^{r_G}x$ , with $r_G$ the rank of the gauge group $G$ of the Lagrangian SCFT, $\\beta \\ (>0)$ the inverse temperature, and $x_i\\ (\\in [-1/2,1/2])$ the integration variables.",
"The function $f$ is a complicated special function of its arguments, given explicitly as a product of several elliptic gamma functions; moreover, when the SCFT is non-chiral, $f$ is real and positive semi-definite.",
"The integral over $-1/2<x_i<1/2$ roughly projects onto the gauge-singlet sector; colloquially speaking, it washes out the contribution of non-gauge-invariant operators to the indexThis is analogous to how the zeroth (or the “singlet”) Fourier component of a periodic real function is obtained by integrating that function..",
"The high-temperature ($\\beta \\rightarrow 0$ ) limit of the index corresponds to the hyperbolic limit of the EHI.",
"This limit has been rigorously analyzed by Eric Rains [10] (around 2006) in certain special EHIs.",
"We put the EHIs studied by Rains in the wider context of the EHIs arising from non-chiral unitary Lagrangian 4d SCFTs.",
"In this generalized framework, the methods of Rains can be extended to uncover a surprisingly rich structure.",
"We find (using, in particular, appropriate uniform estimates (derived in appendix REF ) for the elliptic gamma function) that in the $\\beta \\rightarrow 0$ limit $\\mathcal {I}(\\beta )$ simplifies as $\\begin{split}\\mathcal {I}(\\beta )=\\int f(\\beta ;\\mathbf {x})\\ \\mathrm {d}^{r_G}x\\overset{\\beta \\rightarrow 0}{\\longrightarrow }\\int e^{-(\\mathcal {E}_0^{DK}(\\beta )+V^{\\mathrm {eff}}(x_1,\\dots ,x_{r_G};\\beta ))}\\ \\mathrm {d}^{r_G}x,\\end{split}$ with $\\begin{split}\\mathcal {E}^{DK}_0(\\beta )=-\\frac{16\\pi ^2}{3\\beta }(c-a),\\end{split}$ where $c$ and $a$ are the central chargesThe central charges ($\\in \\mathbb {R}^{>0}$ ) are measures of the number of degrees of freedom in the SCFT.",
"For an SCFT described by an SU($N$ ) gauge theory, $c$ and $a$ are typically of order $N^2$ at large $N$ ; for example, for the SU($N$ ) $\\mathcal {N}=4$ SYM we have $c=a=(N^2-1)/4$ .",
"See Chapter 3 for a precise expression for $c-a$ in terms of the matter content.",
"of the SCFT.",
"We have given a superscript DK to $\\mathcal {E}_0$ , because a proposal of Di Pietro and Komargodski [11] implies the high-temperature asymptotics $\\mathcal {I}(\\beta )\\approx e^{-\\mathcal {E}^{DK}_0(\\beta )}$ (see [12] for an earlier hint of this asymptotic formula).",
"We observe from (REF ) that an effective potential $V^{\\mathrm {eff}}(\\mathbf {x};\\beta )$ dictates the high-temperature asymptotics of $\\mathcal {I}(\\beta )$ .",
"It turns out that $V^{\\mathrm {eff}}(\\mathbf {x};\\beta )=\\frac{4\\pi ^2}{\\beta }L_h(\\mathbf {x}),$ with $L_h$ a continuous, real, piecewise linear function of the $x_i$ , which is determined by the matter content of the SCFT (examples can be found in the Figures REF , REF , and REF below).",
"We will refer to $L_h$ as the Rains function of the SCFT.",
"The relations (REF ) and (REF ) imply that the index localizes in the $\\beta \\rightarrow 0$ limit to the locus of minima of $L_h$ .",
"We thus find $\\begin{split}\\mathcal {I}(\\beta )\\approx e^{-(\\mathcal {E}_0^{DK}(\\beta )+V^{\\mathrm {eff}}_{\\mathrm {min}}(\\beta ))}.\\end{split}$ Taking the logarithm of the two sides, we can write this as [the subleading term and the error estimate will be justified in Chapter 3] $\\ln \\mathcal {I}(\\beta )= \\frac{16\\pi ^2}{3\\beta }(c-a-\\frac{3}{4}L_{h\\ min})+\\mathrm {dim}\\mathfrak {h}_{qu}\\ln (\\frac{2\\pi }{\\beta })+O(\\beta ^0),$ with $L_{h\\ \\mathrm {min}}$ (which we will prove to be $\\le 0$ ) the minimum of the Rains function over $-1/2\\le x_i\\le 1/2$ , and $\\mathrm {dim}\\mathfrak {h}_{qu}$ the dimension of the locus of minima of $L_h$ .",
"The minimization problem for $L_h(\\mathbf {x})$ can often be analytically solved on a case by case basis (as in [10]) using certain generalized triangle inequalities (GTIs); for most SCFTs of interest to us, the required GTI is obtained as a corollary of Rains's GTI, which can be found in appendix REF .",
"Note that the leading piece in (REF ) takes the same form as the Di Pietro-Komargodski formula $\\ln \\mathcal {I}(\\beta )\\approx -\\mathcal {E}^{DK}_0(\\beta )=\\frac{16\\pi ^2}{3\\beta }(c-a)$ , but with the “shifted $c-a$ ” defined as $(c-a)_{\\mathrm {shifted}}:= c-a-\\frac{3}{4}L_{h\\ \\mathrm {min}}.$ This last relation appears to be analogous to the equation $c_{\\mathrm {eff}}= c-24h_{\\mathrm {min}},$ frequently discussed in the context of non-unitary 2d CFTs (see e.g.",
"[13]).",
"One application of the result (REF ) is to supersymmetric dualities.",
"Dual SCFTs must have identical partition functions.",
"Comparison of the indices provides one of the strongest tests of any proposed duality between $\\mathcal {N}=1$ SCFTs [8], [14].",
"The full comparison of the multiple-integrals computing superconformal indices is, however, extremely challenging, except for the few cases (corresponding to various SQCD-type theories [8], [14], [15]) already established in the mathematics literature (e.g.",
"in the celebrated work of Rains [16] (from around 2005) on “transformations” of elliptic hypergeometric integrals).",
"Rather, known dualities are frequently used to conjecture new identities between multi-variable integrals of elliptic hypergeometric type [8], [14], [15], [17].",
"We propose comparison of the high-temperature asymptotics of the indices.",
"Since dual SCFTs have equal central charges, the relation (REF ) implies that dual SCFTs must also have equal $L_{h\\ \\mathrm {min}}$ and $\\mathrm {dim}\\mathfrak {h}_{qu}$ ; these new non-trivial tests of supersymmetric dualities were checked in [5] for several specific cases, validating well-known duality conjectures.",
"A few examples of the applications of these tests can be found also in subsection REF below.",
"We emphasize that these tests are independent of `t Hooft anomaly matchings (see [5] for a more detailed discussion).",
"Another application of (REF ) is to holography.",
"For the specific case of the SU($N$ ) $\\mathcal {N}=4$ SYM, we have $c-a=L_{h\\ \\mathrm {min}}=0$ and $\\mathrm {dim}\\mathfrak {h}_{qu}=N-1$ .",
"This means that the asymptotic growth of the index of this SCFT is power-law.",
"Producing this power-law asymptotics from the holographic dual seems to require the state counting of supersymmetric (more precisely, $1/16$ -BPS) giant gravitons [9]; this appears to be a very interesting objective within reach of current technology.",
"See subsection REF for a more detailed discussion.",
"Only two previous works attacked the problem of the high-temperature asymptotics of the 4d superconformal index of rather general Lagrangian SCFTs.",
"(Other papers have considered this problem in special theories; see subsection 1.2 of [5] for references to such papers.)",
"In the 2013 work of Aharony et.",
"al.",
"[12] an EHI-type expression for the index was considered.",
"Then, assuming that at high temperatures the integrand of the EHI is localized around the unit element of the gauge group, the relation $\\mathcal {I}(\\beta )\\approx e^{-\\mathcal {E}^{DK}_0(\\beta )}Z_{S^3}$ (or more precisely, an equivariant generalization thereof) was arrived at; $Z_{S^3}$ stands for the three-sphere partition function of the dimensionally reduced daughter of the 4d SCFT (see subsection REF below for a matrix-integral expression for $Z_{S^3}$ ).",
"The authors of [12] pointed out, however, that the result can not be trusted in general, as $Z_{S^3}$ may be divergent (as the cut-off of the matrix-integral computing it is taken to infinity) due to an unlifted Coulomb branch in the 3d theory; see [18] for an explicit discussion of unlifted Coulomb branches.",
"In the 2014 work of Di Pietro and Komargodski [11] no explicit form for the index was assumed.",
"But it was assumed that the 4d SCFT is Lagrangian, and that $Z_{S^3}$ is at most power-law divergent with respect to the cut-off ($\\propto 1/\\beta $ ) of the high-temperature effective field theory describing the massless sector of the circle-compactified theory living on $S^3$ .",
"It was then intuitively argued that such power-law divergences would modify the asymptotics $\\mathcal {I}(\\beta )\\approx e^{-\\mathcal {E}^{DK}_0(\\beta )}$ only at the (generically) subleading order in a small-$\\beta $ expansion, such that $\\mathcal {I}(\\beta )\\approx (\\frac{1}{\\beta })^{n_m}e^{-\\mathcal {E}^{DK}_0(\\beta )}$ , with $n_m$ related to the number of unlifted moduli.",
"In the present dissertation (following [5]) we show that Rains's rigorous machinery in [10] can be adapted for a definitive general analysis of the high-temperature asymptotics of the superconformal indices of non-chiral unitary 4d Lagrangian SCFTs.",
"We derive results that clarify the following points: $[$ explicit study of various examples leads to the conjecture that$]$ in theories where $Z_{S^3}$ is power-law divergent, the (generically) subleading power-law asymptotics of $\\mathcal {I}(\\beta )$ can be most nicely associated with a “Coulomb branch” picture in the crossed channel (see subsection REF ); in some of the most interesting SCFTs (more specifically, in certain interacting $\\mathcal {N}=1$ SCFTs with $c<a$ ), $Z_{S^3}$ is exponentially divergent, and as a result even the leading asymptotics $\\mathcal {I}(\\beta )\\approx e^{-\\mathcal {E}^{DK}_0(\\beta )}$ receives a modification, with the correct asymptotics reading $\\mathcal {I}(\\beta )\\approx e^{-(\\mathcal {E}^{DK}_0(\\beta )+V^{\\mathrm {eff}}_{\\mathrm {min}})}$ (see section REF , and subsections REF and REF ).",
"$[$ explicit study of various examples leads to the conjecture that$]$ in theories where $Z_{S^3}$ is power-law divergent, the (generically) subleading power-law asymptotics of $\\mathcal {I}(\\beta )$ can be most nicely associated with a “Coulomb branch” picture in the crossed channel (see subsection REF ); in some of the most interesting SCFTs (more specifically, in certain interacting $\\mathcal {N}=1$ SCFTs with $c<a$ ), $Z_{S^3}$ is exponentially divergent, and as a result even the leading asymptotics $\\mathcal {I}(\\beta )\\approx e^{-\\mathcal {E}^{DK}_0(\\beta )}$ receives a modification, with the correct asymptotics reading $\\mathcal {I}(\\beta )\\approx e^{-(\\mathcal {E}^{DK}_0(\\beta )+V^{\\mathrm {eff}}_{\\mathrm {min}})}$ (see section REF , and subsections REF and REF ).",
"In holography, or more specifically in the AdS/CFT correspondence, the large-$N$ limit of gauge theories plays an important role.",
"We will focus on a certain class of holographic SCFTs when discussing the large-$N$ limit; these are SCFTs arising from toric quiver gauges theories.",
"One of their important features is that they are dual to IIB string theory on AdS$_5\\times $ SE$_5$ , with SE$_5$ a toric Sasaki-Einstein 5-manifold.",
"Taking the large-$N$ limit of the index of these theories one obtains the multi-trace index of the SCFT [19]; this is the index of the multi-trace operators of the SCFT in the planar limit.",
"This index is holographically dual to the multi-particle index of the gravity side; the multi-particle index receives contributions from multi-particle Kaluza-Klein (KK) states in the bulk.",
"The multi-particle index can be related through simple combinatorial procedures (namely via plethystic exponentials/logarithms [20]) to the single-particle index of the gravity theory, which receives contributions only from the bulk single-particle KK states.",
"In a series of papers written by Jim Liu, Phil Szepietowski, and the author, it was discovered that the high-temperature asymptotics of the single-particle index encodes the bulk KK fields' contribution to the subleading holographic Weyl anomaly [21], [22], [23].",
"The problem of holographic Weyl anomaly is to reproduce the central charges $a$ and $c$ of a holographic SCFT from its gravitational dualThe expression “Weyl anomaly” is used because the central charges determine, among other things, the anomalous behavior of the SCFT partition function under Weyl re-scalings of the spacetime metric; see e.g.",
"[24]..",
"In a large-$N$ expansion, the leading ($O(N^2)$ ) piece of the central charges can be holographically obtained using Einstein gravity on the AdS side; this was done around 1998 [24].",
"Obtaining the subleading ($O(1)$ ) piece of the central charges from the gravity side was more challenging, until the relation with the superconformal index was understood [21], [22], [23].",
"The holographic connection between the subleading central charges and the single-particle index is derived roughly as follows.",
"First of all, long multiplets of SU($2,2|1$ ) in the bulk KK spectrum do not contribute to either the single-particle index or the holographic central charges.",
"Next, for short multiplets, irrespective of the type (which could be chiral, anti-chiral, conserved, semi-long I, or semi-long II), the holographic contribution to the central charges takes a simple form, determined by the high-temperature asymptotics of the contribution of the multiplet to the single-particle index.",
"Summing up the contributions of all the KK particles in the bulk, one concludes that the high-temperature asymptotics of the single-particle index is related to the subleading holographic Weyl anomaly.",
"This relation will be discussed further in Chapter 4; there we will explain how the relation leads to a solution to the problem of Holographic Weyl Anomaly in toric quiver SCFTs.",
"A. A.",
"Ardehali, J. T. Liu, and P. Szepietowski, $c-a$ from the $\\mathcal {N}=1$ superconformal index, JHEP 1412, 145 (2014) [arXiv:1407.6024 [hep-th]].",
"(Listed as reference [21].)",
"This work established a holographic relation between the difference of the central charges (i.e.",
"$c-a$ ) and the single-particle index, in the context of 4d SCFTs dual to IIB theory on AdS$_5\\times $ SE$_5$ (with SE$_5$ a Sasaki-Einstein 5-manifold).",
"The relation was then checked explicitly for toric quiver SCFTs (with SU($N$ ) nodes) without adjoint matter and with a smooth dual SE$_5$ ; this successful check can be considered a test of AdS/CFT at the subleading order (in $1/N$ ) for an infinite class of holographic SCFTs.",
"The paper also conjectured the holographically derived relation between $c-a$ and the index to hold for all (not necessarily holographic) 4d SCFTs; this conjecture was ruled out later in [5].",
"A. A.",
"Ardehali, J. T. Liu, and P. Szepietowski, Central charges from the $\\mathcal {N}=1$ superconformal index, Phys.",
"Rev.",
"Lett.",
"114, 091603 (2015) [arXiv:1411.5028 [hep-th]].",
"(Listed as reference [22].)",
"This work extended the holographic result of the previous paper to expressions for the $O(N^0)$ pieces of $a$ and $c$ separately.",
"(Note that for SCFTs dual to AdS$_5\\times $ SE$_5$ we always have $c-a=O(N^0)$ .)",
"The relations were then explicitly checked for toric quiver SCFTs (with SU($N$ ) nodes) without adjoint matter and dual to smooth SE$_5$ ; this check constitutes a very strong and general test of AdS/CFT at the subleading order in the $1/N$ expansion.",
"The paper also presented general conjectures for extracting the central charges of any (finite rank, not necessarily holographic) 4d SCFT from its index, but those conjectures were later ruled out in [5].",
"A. A.",
"Ardehali, J. T. Liu, and P. Szepietowski, High-temperature expansion of supersymmetric partition functions, JHEP 1507, 113 (2015) [arXiv:1502.07737 [hep-th]].",
"(Listed as reference [23].)",
"This work generalized the above-mentioned AdS/CFT matching of the subleading central charges to all toric quivers with SU($N$ ) nodes (even to quivers with adjoint matter fields and/or with singular dual SE$_5$ ; there were two extra assumptions made though, as explained in Chapter 4 below).",
"This paper contains also the first correct calculation of the SUSY Casimir energy in the literature; it thereby clarified the connection between the 4d superconformal index, and its corresponding SUSY partition function computed by path-integration over $S_b^3\\times S_\\beta ^1$ (with $S_b^3$ the unit three-sphere with squashing parameter $b$ , and $\\beta $ the circumference of the circle).",
"This result appeared shortly afterwards also in the independent work of Assel et.",
"al.",
"[25].",
"The paper [23] also proposed a conjecture for the high-temperature asymptotics of the indices of general (finite-rank) 4d SCFTs; that conjecture was ruled out later in [5].",
"A. A.",
"Ardehali, High-temperature asymptotics of supersymetric partition functions, [arXiv:1512.03376 [hep-th]].",
"(Listed as reference [5].)",
"This paper extended Rains's analysis [10] to study the high- (and low-) temperature asymptotics of the index of Lagrangian SCFTs with a semi-simple gauge group (under some extra simplifying assumptions spelled out at the beginning of the Discussion section in [5]).",
"A. A.",
"Ardehali, J. T. Liu, and P. Szepietowski, $c-a$ from the $\\mathcal {N}=1$ superconformal index, JHEP 1412, 145 (2014) [arXiv:1407.6024 [hep-th]].",
"(Listed as reference [21].)",
"This work established a holographic relation between the difference of the central charges (i.e.",
"$c-a$ ) and the single-particle index, in the context of 4d SCFTs dual to IIB theory on AdS$_5\\times $ SE$_5$ (with SE$_5$ a Sasaki-Einstein 5-manifold).",
"The relation was then checked explicitly for toric quiver SCFTs (with SU($N$ ) nodes) without adjoint matter and with a smooth dual SE$_5$ ; this successful check can be considered a test of AdS/CFT at the subleading order (in $1/N$ ) for an infinite class of holographic SCFTs.",
"The paper also conjectured the holographically derived relation between $c-a$ and the index to hold for all (not necessarily holographic) 4d SCFTs; this conjecture was ruled out later in [5].",
"A. A.",
"Ardehali, J. T. Liu, and P. Szepietowski, Central charges from the $\\mathcal {N}=1$ superconformal index, Phys.",
"Rev.",
"Lett.",
"114, 091603 (2015) [arXiv:1411.5028 [hep-th]].",
"(Listed as reference [22].)",
"This work extended the holographic result of the previous paper to expressions for the $O(N^0)$ pieces of $a$ and $c$ separately.",
"(Note that for SCFTs dual to AdS$_5\\times $ SE$_5$ we always have $c-a=O(N^0)$ .)",
"The relations were then explicitly checked for toric quiver SCFTs (with SU($N$ ) nodes) without adjoint matter and dual to smooth SE$_5$ ; this check constitutes a very strong and general test of AdS/CFT at the subleading order in the $1/N$ expansion.",
"The paper also presented general conjectures for extracting the central charges of any (finite rank, not necessarily holographic) 4d SCFT from its index, but those conjectures were later ruled out in [5].",
"A. A.",
"Ardehali, J. T. Liu, and P. Szepietowski, High-temperature expansion of supersymmetric partition functions, JHEP 1507, 113 (2015) [arXiv:1502.07737 [hep-th]].",
"(Listed as reference [23].)",
"This work generalized the above-mentioned AdS/CFT matching of the subleading central charges to all toric quivers with SU($N$ ) nodes (even to quivers with adjoint matter fields and/or with singular dual SE$_5$ ; there were two extra assumptions made though, as explained in Chapter 4 below).",
"This paper contains also the first correct calculation of the SUSY Casimir energy in the literature; it thereby clarified the connection between the 4d superconformal index, and its corresponding SUSY partition function computed by path-integration over $S_b^3\\times S_\\beta ^1$ (with $S_b^3$ the unit three-sphere with squashing parameter $b$ , and $\\beta $ the circumference of the circle).",
"This result appeared shortly afterwards also in the independent work of Assel et.",
"al.",
"[25].",
"The paper [23] also proposed a conjecture for the high-temperature asymptotics of the indices of general (finite-rank) 4d SCFTs; that conjecture was ruled out later in [5].",
"A. A.",
"Ardehali, High-temperature asymptotics of supersymetric partition functions, [arXiv:1512.03376 [hep-th]].",
"(Listed as reference [5].)",
"This paper extended Rains's analysis [10] to study the high- (and low-) temperature asymptotics of the index of Lagrangian SCFTs with a semi-simple gauge group (under some extra simplifying assumptions spelled out at the beginning of the Discussion section in [5]).",
"There are three previously unpublished results in the present dissertation.",
"The first is an improved derivation (compared to the original one in [5]) of the asymptotics of the indices of non-chiral SCFTs.",
"This derivation is given in Chapter 3, and leads to Eq.",
"(REF ), which is our main result.",
"The original derivation (reported in [5]) of Eq.",
"(REF ) was based on the physically expected—but mathematically unjustified—assumption that certain cancelations do not occur in the high-temperature limit of the EHIs arising from SUSY gauge theories (see the comments below Eq.",
"(3.15) of [5]).",
"The second previously unpublished result is the asymptotics, shown in (REF ), of the index of the puncture-less SU(2) class-$\\mathcal {S}$ theories of genus $g\\ge 2$ ; the result is interesting: these $\\mathcal {N}=2$ SCFTs satisfy the Di Pietro-Komargodski formula, even though they famously have the unusual balance $c<a$ between their central charges.",
"These theories are thus to be contrasted with the $\\mathcal {N}=1$ SCFTs with $c<a$ discussed in subsections REF and REF , which do not satisfy the Di Pietro-Komargodski formula.",
"The third novel result is the relation between the high-temperature asymptotics of the single-trace and multi-trace indices, shown in (REF ), and its corollary in Eq.",
"(REF ).",
"Part of the relation (REF ) was given as an ansatz in [23]; we not only prove that ansatz in appendix REF , but also derive a piece of it that was left undetermined in [23].",
"The goal of this chapter is to write down—and to explain—the explicit expression for the elliptic hypergeometric integral (EHI) whose high-temperature asymptotics we will analyze (under certain simplifying conditions) in the next chapter.",
"This expression can be found in Eq.",
"(REF ) below.",
"In the physical context, the EHI in Eq.",
"(REF ) may arise as the superconformal index of a 4d Lagrangian SCFT.",
"Elaborating on the physical context is the purpose of the following two sections.",
"The reader not interested in—or already familiar with—this physical context can skip directly to the third section below (i.e.",
"section REF ) where the EHI of our interest is spelled out.",
"The Hilbert space of a 4d SCFT is invariant under the action of the 4d $\\mathcal {N}=1$ superconformal group SU($2,2|1$ ).",
"The generators of this group constitute the 4d superconformal algebra.",
"The bosonic part of the 4d superconformal algebra consists of the 4d conformal algebra and a U(1) automorphism referred to as the U(1)$_R$ .",
"We denote the charge of a state under $H$ (the generator of dilations, which in the radial quantization becomes the Hamiltonian) by $E$ , the charge under $R$ (the generator of the U(1)$_R$ ) by $r$ , and the charges under $J^z_1$ and $J^z_2$ (the Cartan generators of the left and right SU(2) spins of the Lorentz group) by $j_1$ and $j_2$ .",
"All these charges are real numbers, $j_1$ and $j_2$ are half-integers, and unitarity implies $E\\ge 0$ .",
"The fermionic part the 4d superconformal algebra consists of the supercharges $Q_\\alpha $ , $\\bar{Q}_{\\dot{\\alpha }}$ , and their conformal partners $S^\\alpha $ , $\\bar{S}^{\\dot{\\alpha }}$ .",
"Importantly, we have $\\lbrace Q_\\alpha ,\\bar{Q}_{\\dot{\\alpha }}\\rbrace =2P_{\\alpha \\dot{\\alpha }}$ , and $\\lbrace S^\\alpha ,\\bar{S}^{\\dot{\\alpha }}\\rbrace =2K^{\\alpha \\dot{\\alpha }}$ , with $P$ and $K$ respectively the generators of translations and special conformal transformations.",
"A computationally efficient description of a Lagrangian SCFT is provided, however, not through the Hilbert space perspective, but by the field content of the gauge theory that flows to it.",
"The field content of a supersymmetric gauge theory is organized inside supermultiplets.",
"Focusing on interacting unitary 4d Lagrangian SCFTs with fields of spin $\\le 1$ , we are left with two possible supermultiplets: chiral multiplets and vector multiplets.",
"A chiral multiplet consists of a complex scalar and a Weyl fermion, whereas a vector multiplet consists of a vector boson and a Weyl fermion.",
"(Note that since we are interested in SCFTs, we are restricting our attention to QFTs with massless field content.)",
"The scalar inside a chiral multiplet $\\chi $ has an R-charge that we denote by $r_\\chi $ ; the R-charge of the supersymmetric partner (the Weyl fermion in the same multiplet) is $r_\\chi -1$ .",
"On the other hand, a vector boson has zero R-charge, and its superpartner (the Weyl fermion in the same multiplet, referred to as the gaugino) has R-charge 1.",
"The interaction of massless vector bosons is described by a gauge theory.",
"This means, among other things, that the vector boson field transforms in the adjoint representation of a gauge group $G$ (which we take to be a compact matrix Lie group with a semi-simple algebra).",
"A chiral multiplet $\\chi $ in the theory may transform in a representation $\\mathcal {R}_\\chi $ of the gauge group $G$ .",
"With the above background in mind, and for our purposes below, we take the following as the defining data of a unitary 4d Lagrangian SCFT: $i)$ a gauge group $G$ , which we take to be a compact semi-simple matrix Lie group of rank $r_G$ , denote its typical root vector by $\\alpha :=(\\alpha _1,\\dots ,\\alpha _{r_G})$ , and denote the set of all the roots by $\\Delta _G$ ; $ii)$ a finite number of chiral multiplets $\\chi _j:=\\lbrace \\mathcal {R}_j,r_j\\rbrace $ , with $j=1,\\dots ,n_\\chi $ , where $\\mathcal {R}_j$ is a finite-dimensional irreducible representation of $G$ , whose typical weight vector we denote by $\\rho ^j:=(\\rho ^j_1,\\dots ,\\rho ^j_{r_G})$ , and the set of all the weights of $\\mathcal {R}_j$ we denote by $\\Delta _{j}$ , while $r_j\\ (\\in ]0,2[)$ is the R-charge of the chiral multiplet $\\chi _j$ .",
"We further demand that the following anomaly cancelation conditions be satisfied by the $\\mathcal {R}_{j}$ and the $r_{j}$ : $\\sum _j\\sum _{\\rho ^j\\in \\Delta _{j}}\\rho ^j_l\\rho ^j_m\\rho ^j_n=0,\\quad (\\mathrm {for\\ all\\ }l,m,n)$ $\\sum _j\\sum _{\\rho ^j\\in \\Delta _j}\\rho ^j_l=0, \\quad (\\mathrm {for\\ all\\ }l)$ $\\sum _j(r_j-1)\\sum _{\\rho ^j\\in \\Delta _j}\\rho ^j_l\\rho ^j_m+\\sum _{\\alpha \\in \\Delta _G}\\alpha _{l}\\alpha _{m}=0,\\quad (\\mathrm {for\\ all\\ }l,m)$ $\\sum _j(r_j-1)^2\\sum _{\\rho ^j\\in \\Delta _j}\\rho ^j_l=0 \\quad (\\mathrm {for\\ all\\ }l).$ These relations correspond respectively to cancelation of the following anomalies: $i)$ the gauge$^3$ anomaly; $ii)$ the gauge-gravitational-gravitational anomaly; $iii)$ the U(1)$_R$ -gauge-gauge anomaly; and $iv)$ the gauge-U(1)$_R$ -U(1)$_R$ anomaly.",
"Note that it would be more appropriate to say that the above data defines `a SUSY gauge theory with a U(1) R-symmetry', and not necessarily an SCFT.",
"In particular, the above conditions on the data do not guarantee that the $r_\\chi $ are the superconformal R-charges of the chiral multiplets in the IR fixed point of the SUSY gauge theory defined by the above data; for instance, the SU($N_c$ ) SQCD with R-charge assignment $r_\\chi =1-N_c/N_f$ , for $N_f>N_c$ but outside the conformal window, does satisfy the above conditions, even though its IR fixed point is free, with emergent accidental symmetries mixing with its U(1)$_R$ in the infrared.",
"Therefore we keep in mind that only a subset of the SUSY gauge theories defined by the above data lead to SCFTs with the chiral multiplets in the IR having superconformal R-charges $r_\\chi $ .",
"On the other hand, any SUSY gauge theory with U(1) R-symmetry—as defined by the above data—can be assigned an EHI via Eq.",
"(REF ) below; for non-conformal theories the resulting EHI can be thought of as arising from path-integration (c.f.",
"[6]), rather than from a “superconformal” index calculation.",
"The superconformal index is defined as $\\mathcal {I}(b,\\beta )=\\mathrm {Tr}\\left[(-1)^Fe^{-\\hat{\\beta }(E-2j_2-{\\frac{3}{2}}r)}p^{j_1+j_2+{\\frac{1}{2}}r}q^{-j_1+j_2+{\\frac{1}{2}}r}\\right],$ with $p=e^{-b\\beta }$ and $q=e^{-b^{-1}\\beta }$ ; we take $b,\\beta >0$ , and refer to $b$ as the squashing parameter, and $\\beta $ as the inverse temperature (the reason for these names will become clear shortly); the special case with $b=1$ corresponds to the index introduced in Chapter 1.",
"The trace in the above relation is over the Hilbert space of the theory on $S^3\\times \\mathbb {R}$ , with $S^3$ the round unit three-sphere, and $\\mathbb {R}$ the time direction.",
"The index is independent of $\\hat{\\beta }$ because it only receives uncanceled contributions from states with $E-2j_2-{\\frac{3}{2}}r=0$ .",
"In a superconformal theory, these states correspond to operators that sit in short representations of the superconformal algebra.",
"The index of an SCFT thus encodes exact (non-perturbative) information about the operator spectrum of the underlying theory.",
"The exponents of $p$ and $q$ correspond to operators that commute with the supercharge used in the definition of the index: the expression $E-2j_2-{\\frac{3}{2}}r$ is $Q$ -exact for a particular supercharge $Q$ , and the combinations $J^z_1+J^z_2+R/2$ and $-J^z_1+J^z_1+R/2$ both commute with that $Q$ .",
"Therefore $p$ and $q$ refine the Witten index $\\mathcal {I}^W=\\mathrm {Tr}[(-1)^Fe^{-\\hat{\\beta }(E-2j_2-{\\frac{3}{2}}r)}]$ without ruining the cancelations underlying its topological character.",
"In fact without refinement with $p$ and $q$ , the index $\\mathcal {I}^W$ is often divergent, and thus $p$ and $q$ are necessary as regulators.",
"There are two ways to compute the index of a Lagrangian SCFT.",
"The Hamiltonian route goes through the so-called Romelsberger prescription [26].",
"The Lagrangian route uses the supersymmetric localization of the path-integral on $S_b^3\\times S^1_\\beta $ , where $S^3_b$ is the unit three-sphere with squashing parameter $b>0$ , and $\\beta >0$ is the circumference of the circle [6].",
"Originally, the indices of Lagrangian SCFTs were computed using the Romelsberger prescription; see for instance the work of Dolan and Osborn from 2008 [8].",
"Later on, supersymmetric localization caught up, and not only reproduced the correct expression for the index, but also gave an extra Casimir-type factor which is of physical significance; see [27] for the localization computation for the $\\mathcal {N}=4$ theory, and the 2014 paper of Assel et.",
"al.",
"[6] for the result for the case with more general matter content (the correct evaluation of the Casimir-type factor was done later in [23] and [25]).",
"In the present section we evaluate the index of a general unitary Lagrangian 4d SCFT (defined as in section REF ) using the Romelsberger prescription (see [26], [8]).",
"According to the prescription, one starts with adding up the single-letter indices of various multiplets, and then plethystically exponentiates the result.",
"To project onto the gauge-singlet sector though, one should $i)$ make the single-letter indices character-valued, and $ii)$ integrate the result of the plethystic exponentiation against the Haar measure of the gauge group.",
"A chiral multiplet $\\chi =(\\phi _r,\\psi _{r-1})$ , along with its CP-conjugate multiplet $\\bar{\\chi }=(\\bar{\\phi }_{-r},\\bar{\\psi }_{-r+1})$ , contributes $i_\\chi (z;p,q)=\\sum _{\\rho ^{\\chi }\\in \\Delta _\\chi }\\frac{(pq)^{r_\\chi /2}z^{\\rho ^\\chi }-(pq)^{1-r_\\chi /2}z^{-\\rho ^\\chi }}{(1-p)(1-q)},$ to the total single-letter index.",
"Recall that the set $\\Delta _\\chi $ consists of as many weights $\\rho ^{\\chi }$ as the dimension of the representation $\\mathcal {R}_\\chi $ .",
"Also, our symbolic notation $z^{\\rho ^{\\chi }}$ should be understood as $z_1^{\\rho ^{\\chi }_1}\\times \\dots \\times z_{r_G}^{\\rho ^{\\chi }_{r_G}}$ , where $\\rho ^{\\chi }\\equiv (\\rho ^{\\chi }_1,\\dots ,\\rho ^{\\chi }_{r_G})$ , with $r_G$ the rank of the gauge group.",
"The first term in the numerator of (REF ) is the contribution $(pq)^{r_\\chi /2}$ that $\\phi _r$ makes to the index, multiplied by the character $\\sum _{\\rho ^{\\chi }\\in \\Delta _\\chi }z^{\\rho ^{\\chi }}$ of the representation $\\mathcal {R}_\\chi $ of $G$ under which $\\chi $ transforms.",
"The second term in the numerator of (REF ) is the contribution $(pq)^{1-r_\\chi /2}$ of $\\bar{\\psi }_{-r+1}$ to the index, multiplied by the character of the representation $\\bar{\\mathcal {R}}_\\chi $ of $G$ under which $\\bar{\\chi }$ transforms.",
"The denominator of (REF ) comes from summing up the geometric series arising from adding the contributions of the conformal descendants of $\\phi _r$ and $\\bar{\\psi }_{-r+1}$ (see section 2 of [8] for the details).",
"The plethystic exponential of $i_\\chi (z;p,q)$ is given by a product of several elliptic gamma functions: $\\mathcal {I}_\\chi (z;p,q):=\\exp (\\sum _{n=1}^\\infty \\frac{i_\\chi (z^n;p^n,q^n)}{n})=\\prod _{\\rho ^{\\chi }\\in \\Delta _\\chi }\\Gamma ((pq)^{r_\\chi /2} z^{\\rho ^{\\chi }}).$ The elliptic gamma function $\\Gamma (\\ast )$ is a special function explained in appendix REF .",
"The vector multiplets in the theory contribute to the total single-letter index as $\\begin{split}i_v(z;p,q)&=\\left(-\\frac{p}{(1-p)(1-q)}-\\frac{q}{1-q}+\\frac{pq}{(1-p)(1-q)}\\right)[r_G+\\sum _{\\alpha _+}(z^{\\alpha _+}+z^{-\\alpha _+})]\\\\&=\\frac{2pq-p-q}{(1-p)(1-q)}[r_G+\\sum _{\\alpha _+}(z^{\\alpha _+}+z^{-\\alpha _+})].\\end{split}$ The $\\alpha _+$ are the positive roots of $G$ .",
"By $z^{\\alpha _+}$ we mean $z_1^{\\alpha _1}\\times \\dots \\times z_{r_G}^{\\alpha _{r_G}}$ , where $\\alpha _+\\equiv (\\alpha _1,\\dots ,\\alpha _{r_G})$ .",
"Inside the brackets on the RHS of the first line of (REF ) we have the character of the adjoint representation of $G$ .",
"Inside the parentheses on the RHS of the first line of (REF ) we have respectively the contribution of the first gaugino, the second gaugino, and the gauge field, along with their conformal descendants; the $p$ -descendants of the second gaugino are not taken into account because the equation of motion relates them to the $q$ -descendants of the first gaugino (see section 2 of [8] for the details).",
"The plethystic exponential of $i_v(z;p,q)$ yields a product of Pochhammer symbols and elliptic gamma functions: $\\mathcal {I}_v(z;p,q):=\\exp (\\sum _{n=1}^\\infty \\frac{i_v(z^n;p^n,q^n)}{n})=\\frac{(p;p)^{r_G}(q;q)^{r_G}}{\\prod _{\\alpha _+}(1-z^{+\\alpha _+})(1-z^{-\\alpha _+})\\Gamma (z^{\\pm \\alpha _+})}.$ The Pochhammer symbol $(\\ast ;\\ast )$ is a special function explained in appendix REF .",
"Multiplying the contribution of the various chiral multiplets $\\prod _\\chi \\mathcal {I}_\\chi (z;p,q)$ by the contribution of the vector multiplet(s) $\\mathcal {I}_v(z;p,q)$ we obtain [alternatively we could have summed up the character-valued single-letter indices of various multiplets, and then plethystically exponentiated the result] $\\begin{split}\\mathcal {I}(z;p,q)=(p;p)^{r_G}(q;q)^{r_G} \\frac{\\prod _\\chi \\prod _{\\rho ^{\\chi } \\in \\Delta _\\chi }\\Gamma ((pq)^{r_\\chi /2}z^{\\rho ^{\\chi }})}{\\prod _{\\alpha _+}(1-z^{+\\alpha _+})(1-z^{-\\alpha _+})\\Gamma (z^{\\pm \\alpha _+})}.\\end{split}$ The above index receives contributions from non-gauge-invariant operators.",
"By integrating it against the Haar measure of the gauge group $\\mathrm {d}\\mu =\\frac{1}{|W|}\\mathrm {d}^{r_G}x\\prod _{\\alpha _+}(1-z^{+\\alpha _+})(1-z^{-\\alpha _+}),$ we arrive at the contribution of only the gauge-singlet sector.",
"On the RHS of the above relation, $|W|$ is the order of the Weyl group of $G$ , and $z_j=e^{2\\pi i x_j}$ .",
"The end result is the following elliptic hypergeometric integral [for comparison with [10] note that $\\omega _{1\\ \\mathrm {there}}=ib_{\\mathrm {here}}$ , $\\omega _{2\\ \\mathrm {there}}=ib_{\\mathrm {here}}^{-1}$ , and $v_{\\mathrm {there}}=\\frac{\\beta _{\\mathrm {here}}}{2\\pi }$ ]: $\\boxed{\\begin{split}\\mathcal {I}(b,\\beta )=\\frac{(p;p)^{r_G}(q;q)^{r_G}}{|W|}\\int \\mathrm {d}^{r_G}x \\frac{\\prod _\\chi \\prod _{\\rho ^{\\chi }\\in \\Delta _\\chi }\\Gamma ((pq)^{r_\\chi /2}z^{\\rho ^{\\chi }})}{\\prod _{\\alpha _+}\\Gamma (z^{\\pm \\alpha _+})}.\\end{split}}$ The integral is over the unit hypercube $x_j\\in [-1/2,1/2]$ in the Cartan subalgebra (or alternatively, over the maximal torus of $G$ in the space of $z_j$ ).",
"Since the expression in Eq.",
"(REF ) might seem a bit complicated, let us specialize it to a very simple case: the SU(2) SQCD with three flavors.",
"The gauge group SU(2) has rank $r_G=1$ .",
"The Weyl group of SU($N$ ) is the permutation group of $N$ elements, so it has order $N!$ , which for SU(2) becomes 2.",
"We have three chiral quark multiplets with $\\rho ^{\\chi _1}_1,\\rho ^{\\chi _2}_1,\\rho ^{\\chi _3}_1=\\pm 1$ , and three chiral anti-quark multiplets with $\\rho ^{\\chi _{4}}_1,\\rho ^{\\chi _{5}}_1,\\rho ^{\\chi _{6}}_1=\\mp 1$ (each of the chiral multiplets has two weights ($\\pm 1$ ), because they sit in two-dimensional representations of the gauge group).",
"All the chiral multiplets have R-charge $r_\\chi =1/3$ .",
"Finally, the group SU(2) has two roots, corresponding to the raising and lowering operators of the 3d angular momentum, and the positive root (the raising operator) has $\\alpha _+=2$ .",
"All in all, we get for this simple example $\\begin{split}\\mathcal {I}_{N_c=2,N_f=3}(b,\\beta )=\\frac{(p;p)(q;q)}{2}\\int _{-1/2}^{1/2}\\mathrm {d}x \\frac{\\Gamma ^6((pq)^{1/6} z^{\\pm 1} )}{\\Gamma ( z^{\\pm 2})}.\\end{split}$ Many explicit expressions for the index $\\mathcal {I}(b,\\beta )$ of specific 4d SCFTs can be found in [14], [15], [5].",
"A few specific examples will be spelled out in the next chapter as well.",
"To further clarify the notation we are using for the roots and weights, we add that with our notation the three-dimensional representation of SU(3) has weights $(\\rho _1,\\rho _2)=(1,0),(0,1),(-1,-1)$ , and the positive roots of SU(3) are $\\alpha _+=(1,-1),(2,1),(1,2)$ .",
"If a Lagrangian 4d SCFT has emergent accidental symmetries mixing with its ultraviolet U(1)$_R$ to give the superconformal U(1)$_R$ in the infrared, the Romelsberger prescription can not be applied to it.",
"For such SCFTs, the EHI in (REF ) can be interpreted as arising from path-integration of the UV gauge theory using the ultraviolet U(1)$_R$ , but the EHI would not coincide with the superconformal index of the IR SCFT.",
"In this dissertation we do not discuss the superconformal index of such SCFTs.",
"The EHIs studied by Rains in [10] correspond to the Sp($2N$ ) and SU($N$ ) supersymmetric quantum chromodynamics theories [8].",
"(Note that for simplicity we are focusing on the special case where all the $u_r$ and $v_r$ in [10] are equal.)",
"For a mathematically oriented introduction to the EHIs studied in [10] see [29].",
"We now focus on non-chiral SCFTs: those in which nonzero $\\rho ^\\chi $ come in pairs with opposite signs.",
"With this restriction, the hyperbolic limit of the EHI shown in (REF ) can be analyzed completely reliably, as described below.",
"The high-temperature asymptotics of the index (REF ) of a non-chiral SCFT is found as follows.",
"Using (REF ), the Pochhammer symbols in the prefactor of (REF ) can be immediately replaced with their asymptotic expressions.",
"We have $(p;p)^{r_G}(q;q)^{r_G}\\simeq e^{-\\pi ^2(b+b^{-1})r_G/6\\beta }\\times \\left(\\frac{2\\pi }{\\beta }\\right)^{r_G}\\times e^{\\beta (b+b^{-1})r_G/24},\\quad \\quad (\\text{as$\\beta \\rightarrow 0$})$ with the symbol $\\simeq $ as defined in appendix REF .",
"The asymptotics of the integrand of (REF ) can be obtained from the estimates in (REF ).",
"With the aid of (REF ) and (REF ) we find the $\\beta \\rightarrow 0$ asymptotics of $\\mathcal {I}$ asCompared to the expression in (3.9) of [5], the RHS of (REF ) lacks a phase $i\\Theta $ in the exponent because (as explained in [5]) in non-chiral theories $\\Theta =0$ .",
"Also, the RHS of (REF ) has the extra factors $1/|W|$ , $W_0(b)$ , $e^{\\beta E_{\\mathrm {susy}}(b)}$ , and $W(\\mathbf {x};b,\\beta )$ which were absent in [5]; these arise here because in the analysis below we are using estimates that are stronger than the estimates used in [5].",
"$\\begin{split}\\mathcal {I}(b,\\beta )\\simeq \\frac{1}{|W|}\\left(\\frac{2\\pi }{\\beta }\\right)^{r_G}e^{-\\mathcal {E}^{DK}_0(b,\\beta )}W_0(b)e^{\\beta E_{\\mathrm {susy}}(b)} \\int _{\\mathfrak {h}_{cl}} \\mathrm {d}^{r_G}x\\ e^{-V^{\\mathrm {eff}}(\\mathbf {x};b,\\beta )}W(\\mathbf {x};b,\\beta ),\\end{split}$ with $\\mathfrak {h}_{cl}$ —which in the path-integral picture can be interpreted [5] as the “classical” moduli-space of the holonomies around $S^1_\\beta $ —denoting the unit hypercube $x_i\\in [-1/2,1/2]$ , and with $\\begin{split}\\mathcal {E}^{DK}_0(b,\\beta )=\\frac{\\pi ^2}{3\\beta }(\\frac{b+b^{-1}}{2})\\mathrm {Tr}R,\\end{split}$ $\\begin{split}V^{\\mathrm {eff}}(\\mathbf {x};b,\\beta )=\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})L_h(\\mathbf {x}),\\end{split}$ $\\begin{split}E_{\\mathrm {susy}}(b)=\\frac{1}{6}(\\frac{b+b^{-1}}{2})^3\\mathrm {Tr}R^3-(\\frac{b+b^{-1}}{2})(\\frac{b^2+b^{-2}}{24})\\mathrm {Tr}R.\\end{split}$ The `t Hooft anomalies in the above relations are given by $\\begin{split}\\mathrm {Tr}R&:=\\mathrm {dim}G+\\sum _\\chi (r_\\chi -1)\\mathrm {dim}\\mathcal {R}_\\chi =-16(c-a),\\\\\\mathrm {Tr}R^3&:=\\mathrm {dim}G+\\sum _\\chi (r_\\chi -1)^3\\mathrm {dim}\\mathcal {R}_\\chi =\\frac{16}{9}(5a-3c).\\end{split}$ We have also defined $W_0(b)$ , and the real functions $L_h(\\mathbf {x})$ and $W(\\mathbf {x};b,\\beta )$ via $\\begin{split}L_h(\\mathbf {x}):= \\frac{1}{2}\\sum _{\\chi }(1-r_\\chi )\\sum _{\\rho ^{\\chi }\\in \\Delta _\\chi }\\vartheta (\\langle \\rho ^{\\chi }\\cdot \\mathbf {x}\\rangle )-\\sum _{\\alpha _+}\\vartheta (\\langle \\alpha _+\\cdot \\mathbf {x}\\rangle ),\\end{split}$ $\\begin{split}W_0(b)=\\prod _\\chi \\prod _{\\rho ^\\chi =0}\\Gamma _h(r_\\chi \\omega ),\\end{split}$ $\\begin{split}W(\\mathbf {x};b,\\beta )=\\prod _\\chi \\prod _{\\rho ^\\chi _+}\\frac{\\psi _b(-\\frac{2\\pi i}{\\beta }\\lbrace \\langle \\rho ^\\chi _+\\cdot \\mathbf {x}\\rangle \\rbrace +(r_\\chi -1)\\frac{b+b^{-1}}{2})}{\\psi _b(-\\frac{2\\pi i}{\\beta }\\lbrace \\langle \\rho ^\\chi _+\\cdot \\mathbf {x}\\rangle \\rbrace -(r_\\chi -1)\\frac{b+b^{-1}}{2})}\\prod _{\\alpha _+}\\frac{\\psi _b(-\\frac{2\\pi i}{\\beta }\\lbrace \\langle \\alpha _+\\cdot \\mathbf {x}\\rangle \\rbrace +\\frac{b+b^{-1}}{2})}{\\psi _b(-\\frac{2\\pi i}{\\beta }\\lbrace \\langle \\alpha _+\\cdot \\mathbf {x}\\rangle \\rbrace -\\frac{b+b^{-1}}{2})}.\\end{split}$ The function $\\vartheta (x)$ in (REF ) is defined as $\\vartheta (x):=\\lbrace x\\rbrace (1-\\lbrace x\\rbrace )$ (with the fractional part function defined as $\\lbrace x\\rbrace :=x-\\lfloor x\\rfloor $ ).",
"In (REF ), the second product is over the zero weights of $\\mathcal {R}_\\chi $ (the adjoint representation, for instance, has $r_G$ such weights), and $\\omega $ is defined as $\\omega :=i(b+b^{-1})/2$ .",
"The $\\rho ^\\chi _+$ in (REF ) denote the positive weights of $\\mathcal {R}_\\chi $ .",
"The non-compact quantum dilogarithm $\\psi _b(\\ast )$ and the hyperbolic gamma $\\Gamma _h(\\ast )$ are special functions explained in appendix REF .",
"That $L_h(\\mathbf {x})$ is real should be obvious from the definition of $\\vartheta (x)$ ; that $W(\\mathbf {x};b,\\beta )$ is real follows from (REF ) and (REF ).",
"Note that in (REF ) we are claiming that the matrix-integral is approximated well with the integral of its approximate integrand.",
"This is true because the estimates we have used inside the integrand are $i)$ uniform, and $ii)$ accurate up to exponentially small corrections of the type $e^{-1/\\beta }$ ; these two strong conditions—on the integrand estimates—were not satisfied in the treatment of [5].",
"Now, from (REF ) it follows that $W_0(b)$ is a real number; it is moreover nonzero and finite, as we are assuming $r_\\chi \\in ]0,2[$ (the zeros and poles of the hyperbolic gamma function are described in appendix REF ).",
"We would thus make an $O(\\beta ^0)$ error in the asymptotics of $\\ln \\mathcal {I}(b,\\beta )$ by setting $W_0(b)$ , along with $|W|$ and $e^{\\beta E_{\\mathrm {susy}}(b)}$ , to unity.",
"In other words, $\\begin{split}\\mathcal {I}(b,\\beta )\\approx \\left(\\frac{2\\pi }{\\beta }\\right)^{r_G}e^{-\\mathcal {E}^{DK}_0(b,\\beta )}\\int _{\\mathfrak {h}_{cl}} \\mathrm {d}^{r_G}x\\ e^{-V^{\\mathrm {eff}}(\\mathbf {x};b,\\beta )}W(\\mathbf {x};b,\\beta ),\\end{split}$ with an $O(\\beta ^0)$ error upon taking the logarithm of the two sides.",
"We are hence left with the asymptotic analysis of the integral $\\int _{\\mathfrak {h}_{cl}} e^{-V}W$ .",
"From here, standard methods of asymptotic analysis can be employed.",
"Before continuing our asymptotic analysis further, we note that the star of our show, the real function $L_h$ which determines the effective potentialSomewhat surprisingly, $L_h$ also appears in the $n\\rightarrow 1$ limit of the zero-point energy associated to nonzero spatial holonomies on $S^1\\times S^3/\\mathbb {Z}_n$ ; c.f.",
"Eq.",
"(29) of the arXiv preprint of [30] (with $\\nu ,a$ in there set to zero).",
"It might be possible to clarify this coincidence by analytically continuing the results of [30] (see also [31], [32]) to non-integer $n$ , and then using modular properties of the generalized elliptic gamma functions employed in that work.",
"$V^{\\mathrm {eff}}(\\mathbf {x};b,\\beta )$ , is piecewise linear; the quadratic terms in it cancel because of the ABJ U(1)$_R$ -gauge-gauge anomaly cancelation: $\\frac{\\partial ^2 L_h(\\mathbf {x})}{\\partial x_i\\partial x_j}=\\sum _\\chi (r_\\chi -1)\\sum _{\\rho ^\\chi \\in \\Delta _\\chi }\\rho ^\\chi _i\\rho ^\\chi _j+\\sum _{\\alpha }\\alpha _{i}\\alpha _{j}=0.$ Also, $L_h$ is continuous, is even under $\\mathbf {x}\\rightarrow -\\mathbf {x}$ , and vanishes at $\\mathbf {x}=0$ ; these properties follow from the properties of the function $\\vartheta (x)$ defined above.",
"We refer to $L_h(\\mathbf {x})$ as the Rains function of the SCFT.",
"This function has been analyzed by Rains [10] in the special cases of the elliptic hypergeometric integrals associated to SU($N$ ) and Sp($N$ ) SQCD theories.",
"Writing $V^{\\mathrm {eff}}$ in terms of the Rains function $L_h$ , (REF ) simplifies to $\\begin{split}\\mathcal {I}(b,\\beta )\\approx \\left(\\frac{2\\pi }{\\beta }\\right)^{r_G}e^{-\\mathcal {E}^{DK}_0(b,\\beta )}\\int _{\\mathfrak {h}_{cl}}\\mathrm {d}^{r_G}x\\ e^{-\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})L_h(\\mathbf {x})}W(\\mathbf {x};b,\\beta ).\\end{split}$ It will be useful for us to know that $W(\\mathbf {x};b,\\beta )$ is a positive semi-definite function of $\\mathbf {x}$ ; this follows from (REF ) and (REF ).",
"To analyze the integral in (REF ), first note that the integrand is not smooth over $\\mathfrak {h}_{cl}$ .",
"We hence break $\\mathfrak {h}_{cl}$ into sets on which $L_h$ is linear.",
"These sets can be obtained as follows.",
"Define $\\begin{split}\\mathcal {S}_g:=\\bigcup _{\\alpha _+}\\lbrace \\mathbf {x}\\in \\mathfrak {h}_{cl}|\\langle \\alpha _+\\cdot \\mathbf {x}\\rangle \\in &\\mathbb {Z}\\rbrace ,\\quad \\quad \\mathcal {S}_\\chi :=\\bigcup _{\\rho ^\\chi _+}\\lbrace \\mathbf {x}\\in \\mathfrak {h}_{cl}|\\langle \\rho ^{\\chi }_+\\cdot \\mathbf {x}\\rangle \\in \\mathbb {Z}\\rbrace ,\\\\&\\mathcal {S}:=\\bigcup _{\\chi }\\mathcal {S}_\\chi \\cup \\mathcal {S}_g.\\end{split}$ It should be clear that everywhere in $\\mathfrak {h}_{cl}$ , except on $\\mathcal {S}$ , the function $L_h$ is guaranteed to be linear—and therefore smooth.",
"The set $\\mathcal {S}$ consists of a union of codimension one affine hyperplanes inside the space of the $x_i$ .",
"These hyperplanes chop $\\mathfrak {h}_{cl}$ into (finitely many, convex) polytopes $\\mathcal {P}_n$ .",
"The integral in (REF ) then decomposes to $\\begin{split}\\mathcal {I}(b,\\beta )\\approx e^{-\\mathcal {E}^{DK}_0(b,\\beta )}\\sum _n\\left(\\frac{2\\pi }{\\beta }\\right)^{r_G} \\int _{\\mathcal {P}_n}\\mathrm {d}^{r_G}x\\ e^{-\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})L_h(\\mathbf {x})}W(\\mathbf {x};b,\\beta ).\\end{split}$ Let $\\mathcal {S}^{(\\beta )}_g$ denote the set of all points in $\\mathfrak {h}_{cl}$ that are at a distance less than $N_0\\beta $ from $\\mathcal {S}_g$ , with some fixed $N_0>0$ .",
"We divide $\\mathcal {P}_n$ into $i)$ $\\mathcal {P}_n\\cap \\mathcal {S}^{(\\beta )}_g$ , and $ii)$ the rest of $\\mathcal {P}_n$ , which we denote by $\\mathcal {P}^{\\prime }_n$ .",
"Now, by taking $N_0$ to be large enough, we can push $\\mathcal {P}^{\\prime }_n$ away from the zeros of $\\psi _b$ , and thus make $w_i<W(\\mathbf {x};b,\\beta )<w_s$ over $\\mathcal {P}^{\\prime }_n$ (with some $0<w_i$ and some $w_s<\\infty $ ).",
"Therefore the contribution that the $n$ th summand in (REF ) receives from $\\mathcal {P}^{\\prime }_n$ is well approximated (with an $O(\\beta ^0)$ error upon taking the logs) by $\\begin{split}J_n:=\\left(\\frac{2\\pi }{\\beta }\\right)^{r_G}\\int _{\\mathcal {P}^{\\prime }_n}\\mathrm {d}^{r_G}x\\ e^{-\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})L_h(\\mathbf {x})}.\\end{split}$ Let's further replace $\\mathcal {P}^{\\prime }_n$ in (REF ) with $\\mathcal {P}_n$ ; we will shortly see that this replacement introduces a negligible error.",
"We would hence like to estimate $\\begin{split}I_n:=\\left(\\frac{2\\pi }{\\beta }\\right)^{r_G}\\int _{\\mathcal {P}_n}\\mathrm {d}^{r_G}x\\ e^{-\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})L_h(\\mathbf {x})}.\\end{split}$ Since $L_h$ is linear on each $\\mathcal {P}_n$ , its minimum over $\\mathcal {P}_n$ is guaranteed to be realized on $\\partial \\mathcal {P}_n$ .",
"Let us assume that this minimum occurs on the $k$ th $j$ -face of $\\mathcal {P}_n$ , which we denote by $j_n$ -$\\mathcal {F}^k_n$ .",
"We denote the value of $L_h$ on this $j$ -face by $L_{h\\ \\mathrm {min}}^n$ .",
"Equipped with this notation, we can write (REF ) as $\\begin{split}I_n= \\left(\\frac{2\\pi }{\\beta }\\right)^{r_G}e^{-\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})L_{h\\ \\mathrm {min}}^n}\\int _{\\mathcal {P}_n} \\mathrm {d}^{r_G}x\\ e^{-\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})\\Delta L^n_h(\\mathbf {x})},\\end{split}$ where $\\Delta L^n_h(\\mathbf {x}):=L_h(\\mathbf {x})-L_{h\\ \\mathrm {min}}^n$ is a linear function on $\\mathcal {P}_n$ .",
"Note that $\\Delta L^n_h(\\mathbf {x})$ vanishes on $j_n$ -$\\mathcal {F}^k_n$ , and it increases as we go away from $j_n$ -$\\mathcal {F}^k_n$ and into the interior of $\\mathcal {P}_n$ .",
"[The last sentence, as well as the rest of the discussion leading to (REF ), would receive a trivial modification if $j_n=r_G$ (corresponding to constant $L_h$ over $\\mathcal {P}_n$ ).]",
"Therefore as $\\beta \\rightarrow 0$ , the integral in (REF ) localizes around $j_n$ -$\\mathcal {F}^k_n$ .",
"To further simplify (REF ), we now adopt a set of new coordinates—affinely related to $x_i$ and with unit Jacobian—that are convenient on $\\mathcal {P}_n$ .",
"We pick a point on $j_n$ -$\\mathcal {F}^k_n$ as the new origin, and parameterize $j_n$ -$\\mathcal {F}^k_n$ with $\\bar{x}_1,...,\\bar{x}_{j_n}$ .",
"We take $x_{\\mathrm {in}}$ to parameterize a direction perpendicular to all the $\\bar{x}$ s, and to increase as we go away from $j_n$ -$\\mathcal {F}^k_n$ and into the interior of $\\mathcal {P}_n$ .",
"Finally, we pick $\\tilde{x}_1,...,\\tilde{x}_{r_G-j_n-1}$ to parameterize the perpendicular directions to $x_{\\mathrm {in}}$ and the $\\bar{x}$ s. Note that, because $\\Delta L^n_h$ is linear on $\\mathcal {P}_n$ , it does not depend on the $\\bar{x}$ s; they parameterize its flat directions.",
"By re-scaling $\\bar{x},x_{\\mathrm {in}},\\tilde{x}\\mapsto \\frac{\\beta }{2\\pi }\\bar{x},\\frac{\\beta }{2\\pi }x_{\\mathrm {in}},\\frac{\\beta }{2\\pi }\\tilde{x}$ , we can absorb the $(\\frac{2\\pi }{\\beta })^{r_G}$ factor in (REF ) into the integral, and write the result as $\\begin{split}I_n=\\int _{\\frac{2\\pi }{\\beta }\\mathcal {P}_n} \\mathrm {d}^{j_n}\\bar{x}\\ \\mathrm {d}x_{\\mathrm {in}}\\ \\mathrm {d}^{r_G-j_n-1}\\tilde{x}\\ e^{-2\\pi (\\frac{b+b^{-1}}{2})\\Delta L^n_h(x_{\\mathrm {in}},\\mathbf {\\tilde{x}})}.\\end{split}$ To eliminate $\\beta $ from the exponent, we have used the fact that $\\Delta L^n_h$ depends homogenously on the new coordinates.",
"We are also denoting the re-scaled polytope schematically by $\\frac{2\\pi }{\\beta }\\mathcal {P}_n$ .",
"Instead of integrating over all of $\\frac{2\\pi }{\\beta }\\mathcal {P}_n$ though, we can restrict to $x_{\\mathrm {in}}<\\epsilon /\\beta $ with some small $\\epsilon >0$ .",
"The reason is that the integrand of (REF ) is exponentially suppressed (as $\\beta \\rightarrow 0$ ) for $x_{\\mathrm {in}}>\\epsilon /\\beta $ .",
"We take $\\epsilon >0$ to be small enough such that a hyperplane at $x_{\\mathrm {in}}=\\epsilon /\\beta $ , and parallel to $j_n$ -$\\mathcal {F}^k_n$ , cuts off a prismatoid $P^n_{\\epsilon /\\beta }$ from $\\frac{2\\pi }{\\beta }\\mathcal {P}_n$ .",
"After restricting the integral in (REF ) to $P^n_{\\epsilon /\\beta }$ , the integration over the $\\bar{x}$ s is easy to perform.",
"The only potential difficulty is that the range of the $\\bar{x}$ coordinates may depend on $x_{\\mathrm {in}}$ and the $\\tilde{x}s$ .",
"But since we are dealing with a prismatoid, the dependence is linear, and by the time the range is modified significantly (compared to its $O(1/\\beta )$ size on the re-scaled $j$ -face $\\frac{2\\pi }{\\beta }(j_n$ -$\\mathcal {F}^k_n)$ ), the integrand is exponentially suppressed.",
"Therefore we can neglect the dependence of the range of the $\\bar{x}$ s on the other coordinates in (REF ).",
"The integral then simplifies to $\\begin{split}I_n\\approx \\left(\\frac{2\\pi }{\\beta }\\right)^{j_n}\\ \\mathrm {vol}(j_n\\text{-}\\mathcal {F}^k_n)\\int _{\\hat{P}^n_{\\epsilon /\\beta }}\\mathrm {d}x_{\\mathrm {in}}\\ \\mathrm {d}^{r_G-j_n-1}\\tilde{x}\\ e^{-2\\pi (\\frac{b+b^{-1}}{2})\\Delta L^n_h(x_{\\mathrm {in}},\\mathbf {\\tilde{x}})},\\end{split}$ where $\\hat{P}^n_{\\epsilon /\\beta }$ is the pyramid obtained by restricting $P^n_{\\epsilon /\\beta }$ to $\\bar{x}_1=...=\\bar{x}_{j_n}=0$ .",
"The logarithms of the two sides of (REF ) differ by $O(\\beta )$ , with the error mainly arising from our neglect of the possible dependence of the range of the $\\bar{x}$ coordinates in (REF ) on $x_{\\mathrm {in}}$ and the $\\tilde{x}s$ .",
"(Recall that the other error, arising from restricting the integral in (REF ) to $P^n_{\\epsilon /\\beta }$ , is exponentially small.)",
"We now take $\\epsilon \\rightarrow \\infty $ in (REF ).",
"This introduces an exponentially small error, as the integrand is exponentially suppressed (as $\\beta \\rightarrow 0$ ) for $x_{\\mathrm {in}}>\\epsilon /\\beta $ .",
"The resulting integral is strictly positive, because it is the integral of a strictly positive function.",
"We denote by $A_n$ the result of the integral multiplied by $\\mathrm {vol}(j_n$ -$\\mathcal {F}^k_n)$ .",
"Then $I_n$ can be approximated as $\\begin{split}I_n\\approx e^{-\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})L_{h\\ \\mathrm {min}}^n}\\left(\\frac{2\\pi }{\\beta }\\right)^{j_n}A_n.\\end{split}$ We are now in a position to argue $J_n\\approx I_n$ .",
"If we had integrated over $\\mathcal {P}^{\\prime }_n$ , then we would end up with an expression similar to (REF ), in which $L_{h\\ \\mathrm {min}}^n$ would be replaced with the minimum of $L_h$ over $\\mathcal {P}^{\\prime }_n$ ; but since $L_h$ is piecewise linear, the difference between the new minimum and $L_{h\\ \\mathrm {min}}^n$ would be $O(\\beta )$ , which translates to an $O(\\beta ^0)$ multiplicative difference between $J_n$ and $I_n$ .",
"Other sources of difference between $J_n$ and $I_n$ similarly introduce negligible error; more precisely, we have $\\ln I_n=\\ln J_n+O(\\beta ^0)$ .",
"The dominant contribution to $\\mathcal {I}(b,\\beta )$ comes, of course, from the terms/polytopes whose $L_{h\\ \\mathrm {min}}^n$ is smallest.",
"If these terms are labeled by $n=n_\\ast ^1,n_\\ast ^2,...$ , we can introduce $\\mathfrak {h}_{qu}$ and $\\mathrm {dim}\\mathfrak {h}_{qu}$ via $\\mathfrak {h}_{qu}:=\\bigcup _{n_\\ast }j_{n_\\ast }\\text{-}\\mathcal {F}^k_{n_\\ast },\\quad \\mathrm {dim}\\mathfrak {h}_{qu}:=\\mathrm {max}(j_{n_\\ast }).$ Put colloquially, if $\\mathfrak {h}_{qu}$ has multiple connected components, by $\\mathrm {dim}\\mathfrak {h}_{qu}$ we mean the dimension of the component(s) with greatest dimension, while if a connected component consists of several intersecting flat elements inside $\\mathfrak {h}_{cl}$ , by its dimension we mean the dimension of the flat element(s) of maximal dimension.",
"Our final estimate for the contribution to $\\mathcal {I}(b,\\beta )$ from $\\cup _n\\mathcal {P}^{\\prime }_n$ is thus $\\begin{split}Be^{-\\mathcal {E}^{DK}_0(b,\\beta )-\\frac{4\\pi ^2}{\\beta }(\\frac{b+b^{-1}}{2})L_{h\\ \\mathrm {min}}}\\left(\\frac{2\\pi }{\\beta }\\right)^{\\mathrm {dim}\\mathfrak {h}_{qu}},\\end{split}$ where $L_{h\\ \\mathrm {min}}:=L_{h\\ \\mathrm {min}}^{n_\\ast }$ , and $B$ is some positive real number.",
"We are left with determining the contribution to $\\mathcal {I}(b,\\beta )$ coming from $\\mathcal {S}^{(\\beta )}_g$ .",
"Over $\\mathcal {P}_n\\cap \\mathcal {S}^{(\\beta )}_g$ , the simple estimate $W(\\mathbf {x};b,\\beta )= O(1)$ (which follows from the fact that $W(\\mathbf {x};b,\\beta )$ is uniformly bounded on $\\mathcal {S}^{(\\beta )}_g$ ) suffices for our purposes; we thus learn that the contribution that the integral (REF ) receives from $\\mathcal {P}_n\\cap \\mathcal {S}^{(\\beta )}_g$ is not only positive, but also $\\begin{split}O\\left(\\int _{\\frac{2\\pi }{\\beta }(\\mathcal {P}_n\\cap \\mathcal {S}^{(\\beta )}_g)}\\mathrm {d}^{j_n}\\bar{x}\\ \\mathrm {d}x_{\\mathrm {in}}\\ \\mathrm {d}^{r_G-j_n-1}\\tilde{x}\\ e^{-2\\pi (\\frac{b+b^{-1}}{2})\\Delta L^n_h(x_{\\mathrm {in}},\\mathbf {\\tilde{x}})}\\right).\\end{split}$ Now, the argument of the $O$ above is nothing but the difference between $I_n$ and $J_n$ , which we already argued to be negligible.",
"Thus the contribution to $\\mathcal {I}(b,\\beta )$ coming from $\\mathcal {S}^{(\\beta )}_g$ is negligible.",
"Using the explicit expression (REF ) for $\\mathcal {E}^{DK}_0(b,\\beta )$ , and noting that (REF ) is an accurate estimate for $\\mathcal {I}(b,\\beta )$ up to a multiplicative factor of order $\\beta ^0$ , we arrive at our main result: $\\boxed{\\begin{split} \\ln \\mathcal {I}(b,\\beta )=-\\frac{\\pi ^2}{3\\beta }(\\frac{b+b^{-1}}{2})(\\mathrm {Tr}R+12L_{h\\ \\mathrm {min}})+\\mathrm {dim}\\mathfrak {h}_{qu}\\ln (\\frac{2\\pi }{\\beta })+O(\\beta ^0).\\end{split}}$ In this subsection we comment on the connection between the asymptotics of the index of a 4d SCFT, and the divergence of the $S^3$ partition function $Z_{S^3}$ of the dimensionally reduced daughter of the 4d theory.",
"We will show below that the degree of divergence of $Z_{S^3}$ (as the cut-off of the matrix-integral computing it is taken to infinity) is determined by the behavior of the Rains function $L_h$ near the origin of $\\mathfrak {h}_{cl}$ .",
"In particular if the origin is an isolated local minimum of $L_h$ , then $Z_{S^3}$ is finite; if the origin is part of an extended locus where $L_h$ is locally minimized, then $Z_{S^3}$ is power-law divergent; if the origin is not a local minimum of $L_h$ , then $Z_{S^3}$ is exponentially divergent.",
"if the origin is an isolated local minimum of $L_h$ , then $Z_{S^3}$ is finite; if the origin is part of an extended locus where $L_h$ is locally minimized, then $Z_{S^3}$ is power-law divergent; if the origin is not a local minimum of $L_h$ , then $Z_{S^3}$ is exponentially divergent.",
"Note that it is the local behavior of $L_h$ near the origin that determines the degree of divergence of $Z_{S^3}$ .",
"On the other hand, according to (REF ), the asymptotics of the 4d index is determined by the global properties of $L_h$ .",
"Therefore, at least until theorems relating the local and global properties of $L_h$ are established, the asymptotics of the 4d index is not as tightly connected to the divergence of $Z_{S^3}$ as one may have wished.",
"For instance, we can not say (in absence of theorems of the kind discussed in the previous paragraph) that `the Di Pietro-Komargodski asymptotics applies to the index if $Z_{S^3}$ is finite'; it may happen that in a (non-chiral unitary Lagrangian) 4d SCFT (with $r_\\chi \\in ]0,2[$ ) the origin is an isolated local, but not global, minimum of $L_h$ ; that the origin is an isolated local minimum would imply that $Z_{S^3}$ is finite; that $L_h$ is minimized somewhere else would imply—according to (REF )—that the Di Pietro-Komargodski formula receives a modification.",
"However, we can say with certainty that (in a non-chiral unitary 4d Lagrangian SCFT with $r_\\chi \\in ]0,2[$ ) `if $Z_{S^3}$ is exponentially divergent, then the Di Pietro-Komargodski formula receives a modification'; this is simply because if $L_h$ is not locally minimized at the origin, it is certainly not globally minimized there either.",
"We now demonstrate the three propositions itemized above.",
"The starting point is the observation that the function $\\vartheta (x)$ featuring in $L_h$ simplifies if its argument is “small enough”: $\\vartheta (x)= |x|-x^2 \\quad \\quad \\text{for$x\\in [-1,1]$}.$ Using the above simplification in the expression (REF ) for $L_h$ , we learn that for small enough $|\\mathbf {x}|$ the Rains function simplifies to the following homogenous functionInterestingly, on a discrete subset of its domain (corresponding to the cocharacter lattice of the gauge group $G$ ), the function $\\tilde{L}_{S^3}$ coincides (up to normalization) with the $S^2\\times S^1$ Casimir energy $\\epsilon _0$ [33] associated to monopole sectors of the 3d $\\mathcal {N}=2$ theory obtained from dimensional reduction of the 4d $\\mathcal {N}=1$ gauge theory.",
"In the context of 3d $\\mathcal {N}=4$ theories, a different connection between $\\tilde{L}_{S^3}$ and 3d monopoles was discussed in [34].",
": $\\begin{split}\\tilde{L}_{S^3}(\\mathbf {x})=\\frac{1}{2}\\sum _{\\chi }(1-r_\\chi )\\sum _{\\rho ^{\\chi }\\in \\Delta _\\chi }|\\langle \\rho ^{\\chi }\\cdot \\mathbf {x}\\rangle |-\\sum _{\\alpha _+}|\\langle \\alpha _+\\cdot \\mathbf {x}\\rangle |.\\end{split}$ Note that there is no quadratic term in $\\tilde{L}_{S^3}$ , thanks to the cancelation of the U(1)$_R$ -gauge-gauge anomaly.",
"Next, we consider (recall $\\omega :=i(b+b^{-1})/2$ ) $Z_{S^3}(b;\\Lambda ):=\\frac{1}{|W|}\\int _{\\Lambda } \\mathrm {d}^{r_G}x\\ \\frac{\\prod _\\chi \\prod _{\\rho ^{\\chi }\\in \\Delta _\\chi }\\Gamma _h(r_\\chi \\omega +\\langle \\rho ^{\\chi }\\cdot \\mathbf {x}\\rangle )}{\\prod _{\\alpha _+}\\Gamma _h(\\pm \\langle \\alpha _+\\cdot \\mathbf {x}\\rangle )}, $ which is the matrix-integral computing the squashed-three-sphere partition function of the dimensionally reduced daughter (c.f.",
"Eq.",
"(5.23) of [12]), assuming the same R-charge assignments as those directly descending from the parent 4d theory.",
"We are keeping the cut-off $\\Lambda $ explicit, emphasizing that the integration is over the hypercube $|x_i|<\\Lambda $ .",
"To study the convergence/divergence of $Z_{S^3}(b;\\Lambda )$ as $\\Lambda $ is taken to infinity, we use the estimate (REF ) for the hyperbolic gamma functions in the integrand of (REF ).",
"We find that the integrand of $Z_{S^3}(b;\\Lambda )$ can be estimated, as $|\\mathbf {x}|\\rightarrow \\infty $ , by $\\frac{\\prod _\\chi \\prod _{\\rho ^{\\chi }\\in \\Delta _\\chi }\\Gamma _h(r_\\chi \\omega +\\langle \\rho ^{\\chi }\\cdot \\mathbf {x}\\rangle )}{\\prod _{\\alpha _+}\\Gamma _h(\\pm \\langle \\alpha _+\\cdot \\mathbf {x}\\rangle )}\\approx e^{-2\\pi (\\frac{b+b^{-1}}{2})\\tilde{L}_{S^3}(\\mathbf {x})},$ with $\\tilde{L}_{S^3}$ the homogeneous function defined above.",
"Note that whether the integrand of $Z_{S^3}$ decays or grows at large $|\\mathbf {x}|$ , is determined by the behavior $\\tilde{L}_{S^3}(\\mathbf {x})$ , and does not depend on $b$ (recall that we take $b>0$ ).",
"Here comes the crucial point: since $\\tilde{L}_{S^3}(\\mathbf {x})$ is homogenous, its sign at large $|\\mathbf {x}|$ is the same as its sign at small $|\\mathbf {x}|$ .",
"Since at small enough $|\\mathbf {x}|$ , the two functions $\\tilde{L}_{S^3}(\\mathbf {x})$ and $L_{h}(\\mathbf {x})$ coincide, the large-$|\\mathbf {x}|$ behavior of the integrand of $Z_{S^3}$ is connected to the behavior of the Rains function near the origin of $\\mathfrak {h}_{cl}$ .",
"Therefore, if the origin is an isolated local minimum of $L_h$ , then $L_h$ , and hence $\\tilde{L}_{S^3}$ , is positive near the origin, and since $\\tilde{L}_{S^3}(\\mathbf {x})$ is homogeneous, it is positive also for large $|\\mathbf {x}|$ , leading in combination with (REF ) to the conclusion that the integrand of $Z_{S^3}$ decays exponentially at large $|\\mathbf {x}|$ , and implying that $Z_{S^3}$ is finite as $\\Lambda \\rightarrow \\infty $ ; if the origin is part of an extended locus where $L_h$ is locally minimized, then $\\tilde{L}_{S^3}(\\mathbf {x})$ has flat directions near the origin, and hence at large $|\\mathbf {x}|$ , and therefore the integrand of $Z_{S^3}$ does not decay in certain directions, leading to the conclusion that $Z_{S^3}$ is power-law divergent in $\\Lambda $ as $\\Lambda \\rightarrow \\infty $ ; if the origin is not a local minimum of $L_h$ , then $L_h$ , and hence $\\tilde{L}_{S^3}$ , is negative somewhere near the origin, and since $\\tilde{L}_{S^3}(\\mathbf {x})$ is homogeneous, it is negative also for large $|\\mathbf {x}|$ in certain directions, leading in combination with (REF ) to the conclusion that the integrand of $Z_{S^3}$ grows exponentially at large $|\\mathbf {x}|$ in certain directions, and implying that $Z_{S^3}$ is exponentially divergent in $\\Lambda $ as $\\Lambda \\rightarrow \\infty $ .",
"if the origin is an isolated local minimum of $L_h$ , then $L_h$ , and hence $\\tilde{L}_{S^3}$ , is positive near the origin, and since $\\tilde{L}_{S^3}(\\mathbf {x})$ is homogeneous, it is positive also for large $|\\mathbf {x}|$ , leading in combination with (REF ) to the conclusion that the integrand of $Z_{S^3}$ decays exponentially at large $|\\mathbf {x}|$ , and implying that $Z_{S^3}$ is finite as $\\Lambda \\rightarrow \\infty $ ; if the origin is part of an extended locus where $L_h$ is locally minimized, then $\\tilde{L}_{S^3}(\\mathbf {x})$ has flat directions near the origin, and hence at large $|\\mathbf {x}|$ , and therefore the integrand of $Z_{S^3}$ does not decay in certain directions, leading to the conclusion that $Z_{S^3}$ is power-law divergent in $\\Lambda $ as $\\Lambda \\rightarrow \\infty $ ; if the origin is not a local minimum of $L_h$ , then $L_h$ , and hence $\\tilde{L}_{S^3}$ , is negative somewhere near the origin, and since $\\tilde{L}_{S^3}(\\mathbf {x})$ is homogeneous, it is negative also for large $|\\mathbf {x}|$ in certain directions, leading in combination with (REF ) to the conclusion that the integrand of $Z_{S^3}$ grows exponentially at large $|\\mathbf {x}|$ in certain directions, and implying that $Z_{S^3}$ is exponentially divergent in $\\Lambda $ as $\\Lambda \\rightarrow \\infty $ .",
"Take now the example of $A_k$ SQCD with SU($N$ ) gauge group.",
"This theory has a chiral multiplet with R-charge $r_a=\\frac{2}{k+1}$ in the adjoint, $N_f$ flavors in the fundamental with R-charge $r_f=1-\\frac{2}{k+1}\\frac{N}{N_f}$ , and $N_f$ flavors in the anti-fundamental with R-charge $r_{\\bar{f}}=r_f$ .",
"For $r_f$ to be positive we must have $N_f>2N/(k+1)$ .",
"We also assume that we are in the right range of parameters, so we are inside the conformal window of this theory.",
"The superconformal index of this theory is (c.f.",
"[8]) $\\begin{split}\\mathcal {I}_{A_k}(b,\\beta )=&\\frac{(p;p)^{N-1}(q;q)^{N-1}}{N!",
"}\\Gamma ^{N-1}((pq)^{r_a/2})\\int \\mathrm {d}^{N-1}x\\\\ &\\left(\\prod _{1\\le i<j\\le N}\\frac{\\Gamma ((pq)^{r_a/2}(z_i/z_j)^{\\pm 1})}{\\Gamma ((z_i/z_j)^{\\pm 1})}\\right) \\prod _{i=1}^{N}\\Gamma ^{N_f}((pq)^{r_f/2} z_i^{\\pm 1}),\\end{split}$ with $\\prod _{i=1}^{N} z_i=1$ .",
"The Rains function of the theory is $\\begin{split}L_h^{A_k}(x_1,\\dots ,x_{N-1})&=N_f(1-r_f)\\sum _{i=1}^{N}\\vartheta (x_i)+(1-r_a)\\sum _{1\\le i<j\\le N}\\vartheta (x_i-x_j)-\\sum _{1\\le i<j\\le N}\\vartheta (x_i-x_j)\\\\&=\\frac{2}{k+1}(N\\sum _{i}\\vartheta (x_i)-\\sum _{1\\le i<j\\le N}\\vartheta (x_i-x_j)).\\end{split}$ The $x_N$ in the above expression is constrained by $\\sum _{i=1}^{N}x_i\\in \\mathbb {Z}$ , although since $\\vartheta (x)$ is periodic with period one we can simply replace $x_N\\rightarrow -x_1-\\dots -x_{N-1}$ .",
"For $k=1$ and $N=3$ , the resulting function is illustrated in Figure REF .",
"Figure: The Rains function of the A 1 A_1 SU(3) theory—also knownas SU(3) SQCD.We recommend that the reader convince herself that the Rains function in (REF ) can be easily written down by examining the integrand of (REF ).",
"Whenever the index of a theory is available in the literature, a similar examination of the integrand quickly yields the theory's $L_h$ function.",
"Using Rains's generalized triangle inequality (REF ), in the special case where $d_i=0$ , we find that the above function is minimized when all $x_i$ are zero.",
"This establishes that the integrand of (REF ) is localized around $x_i=0$ , and is exponentially suppressed everywhere else, as $\\beta \\rightarrow 0$ .",
"Therefore $L^{A_k}_{h\\ \\mathrm {min}}=0$ and $\\mathrm {dim}\\mathfrak {h}^{A_k}_{qu}=0$ .",
"We thus arrive at $\\begin{split} \\ln \\mathcal {I}_{A_k}(b,\\beta )=-\\frac{\\pi ^2}{3\\beta }(\\frac{b+b^{-1}}{2})(\\mathrm {Tr}R) +O(\\beta ^0).\\end{split}$ A more careful study shows [5] (see appendix REF for the definition of the symbol $\\sim $ ) $\\begin{split} \\ln \\mathcal {I}_{A_k}(b,\\beta )\\sim -\\frac{\\pi ^2}{3\\beta }(\\frac{b+b^{-1}}{2})(\\mathrm {Tr}R)+\\ln Z^{A_k}_{S^3}(b) +\\beta E_{\\mathrm {susy}}(b), \\end{split}$ where $\\begin{split}Z^{A_k}_{S^3}(b)=\\frac{\\Gamma _h^{N-1}(r_a \\omega )}{N!}",
"\\int \\mathrm {d}^{N-1}x \\left(\\prod _{1\\le i<j\\le N}\\frac{\\Gamma _h(r_a\\omega \\pm (x_i-x_j) )}{\\Gamma _h(\\pm (x_i-x_j) )}\\right)\\prod _{i=1}^{N}\\Gamma _h^{N_f}(r_f \\omega \\pm x_i),\\end{split}$ with the integral over $-\\infty <x_i<\\infty $ .",
"Consider the SO($n$ ) SQCD theories with $N_f$ chiral matter multiplets of R-charge $r=1-\\frac{n-2}{N_f}$ in the vector representation.",
"For the R-charges to be greater than zero, and the gauge group to be semi-simple, we must have $0<n-2<N_f$ .",
"We also assume that we are in the right range of parameters, so we are inside the conformal window of this theory.",
"We perform the analysis for odd $n$ ; the analysis for even $n$ is completely analogous, and the result is similar.",
"The index of SO($2N+1$ ) SQCD is given by (c.f.",
"[8]) $\\begin{split}\\mathcal {I}_{SO(2N+1)}(b,\\beta )=&\\frac{(p;p)^N(q;q)^N}{2^NN!",
"}\\Gamma ^{N_f}((pq)^{r/2})\\\\&\\times \\int \\mathrm {d}^N x\\frac{\\prod _{j=1}^{N}\\Gamma ^{N_f}((pq)^{r/2}z_j^{\\pm 1})}{\\prod _{j=1}^{N}\\Gamma (z_j^{\\pm 1})\\prod _{i<j}(\\Gamma ((z_i z_j)^{\\pm 1})\\Gamma ((z_i/z_j)^{\\pm 1}))}.\\end{split}$ The Rains function of the theory is $\\begin{split}L_h^{SO(2N+1)}(\\mathbf {x})=(2N-2)\\sum _{j=1}^{N}\\vartheta (x_j)-\\sum _{1\\le i<j\\le N}\\vartheta (x_i+x_j)-\\sum _{1\\le i<j\\le N}\\vartheta (x_i-x_j).\\end{split}$ For the case $N=2$ , corresponding to the SO(5) theory, this function is illustrated in Figure REF .",
"Figure: The Rains function of the SO(5) SQCD.To find the minima of the above function, we need the following result.",
"For $-1/2\\le x_i\\le 1/2$ $\\begin{split}(2N-2)\\sum _{1\\le j\\le N}\\vartheta (x_j)&-\\sum _{1\\le i<j\\le N}\\vartheta (x_i+x_j)-\\sum _{1\\le i<j\\le N}\\vartheta (x_i-x_j)=2\\sum _{1\\le i<j\\le N}\\mathrm {min}(|x_i|,|x_j|)\\\\&=2(N-1)\\mathrm {min}(|x_i|)+2(N-2)\\mathrm {min}_2(|x_i|)+\\cdots +2\\mathrm {min}_{N-1}(|x_i|),\\end{split}$ where $\\mathrm {min}(|x_i|)$ stands for the smallest of $|x_1|,\\dots ,|x_N|$ , while $\\mathrm {min}_2(|x_i|)$ stands for the next to smallest element, and so on.",
"To prove (REF ), one can first verify it for $N=2$ , and then use induction for $N>2$ .",
"Applying (REF ) we find that the Rains function in (REF ) is minimized to zero when one (and only one) of the $x_j$ is nonzero, and the rest are zero.",
"This follows from the fact that $\\mathrm {max}(|x_i|)$ does not show up on the RHS of (REF ).",
"Therefore, unlike for the theories of the previous subsection, here the matrix-integral is not localized around the origin of the $x_i$ space, but localized around the axes.",
"Equation (REF ) thus simplifies to $\\begin{split}\\ln \\mathcal {I}_{SO(2N+1)}(b,\\beta )= -\\mathcal {E}^{DK}_0(b,\\beta )+\\ln \\left(\\frac{2\\pi }{\\beta }\\right)+O(1)\\quad \\quad (\\text{as$\\beta \\rightarrow 0$}).\\end{split}$ The discussion in subsection REF implies that the three-sphere partition function $Z_{S^3}$ of the dimensionally reduced daughter of this theory diverges as $Z_{S^3}\\approx \\Lambda $ (as the cut-off $\\Lambda $ of the corresponding matrix-integral is taken to infinity); this power-law divergence is closely related to the (generically) subleading logarithmic term on the RHS of (REF ).",
"See subsection 3.2 of [5] for a more detailed discussion of the relation between the power-law divergence of $Z_{S^3}$ and the subleading asymptotics of the index.",
"Luckily, for the special case of $N=1,N_f=2,b=1$ , the asymptotic expansion in (REF ) can be completed to all orders, with the result reading [5] (see appendix REF for the definition of the symbol $\\sim $ used below) $\\begin{split}\\ln \\mathcal {I}_{SO(3)}(\\beta )\\sim \\ln (\\frac{\\pi }{2\\beta }-\\frac{1}{2\\pi })+\\frac{3}{8}\\beta \\quad (\\text{as $\\beta \\rightarrow 0$}).\\end{split}$ The SU($N$ ) $\\mathcal {N}=4$ theory has the following index [35]: $\\begin{split}\\mathcal {I}_{\\mathcal {N}=4}(b,\\beta )=&\\frac{(p;p)^{N-1}(q;q)^{N-1}}{N!",
"}\\Gamma ^{3(N-1)}((pq)^{1/3})\\\\&\\times \\int \\mathrm {d}^{N-1} x \\prod _{1\\le i<j\\le N}\\frac{\\Gamma ^{3}((pq)^{1/3}(z_i/z_j)^{\\pm 1})}{\\Gamma ((z_i/z_j)^{\\pm 1})},\\end{split}$ with $\\prod _{i=1}^{N}z_i=1$ .",
"Recall that for the $A_k$ SQCD theories the integrand of the matrix-integral was everywhere exponentially smaller than in the origin of the $x_i$ space; in other words, the integral localized at a point.",
"We will shortly find that for the $\\mathcal {N}=4$ theory the matrix-integral does not localize at all.",
"The Rains function of the theory is $\\begin{split}L_h^{\\mathcal {N}=4}=3(1-\\frac{2}{3})\\sum _{1\\le i<j\\le N}\\vartheta (x_i-x_j)-\\sum _{1\\le i<j\\le N}\\vartheta (x_i-x_j)=0.\\end{split}$ In other words, there is no effective potential, and the matrix-integral does not localize: $\\mathfrak {h}_{qu}=\\mathfrak {h}_{cl}$ .",
"Eq.",
"(REF ) thus dictates $\\begin{split}\\ln \\mathcal {I}_{\\mathcal {N}=4}(b,\\beta )=(N-1)\\ln (\\frac{2\\pi }{\\beta })+O(\\beta ^0).\\end{split}$ There is no $O(1/\\beta )$ term on the RHS, because $\\mathrm {Tr}R=0$ for the $\\mathcal {N}=4$ theory (and also $L^{\\mathcal {N}=4}_{h\\ \\mathrm {min}}=0$ ).",
"The discussion in subsection REF implies that the three-sphere partition function $Z_{S^3}$ of the dimensionally reduced daughter of this theory diverges as $Z_{S^3}\\approx \\Lambda ^{N-1}$ (as the cut-off $\\Lambda $ of the corresponding matrix-integral is taken to infinity); this power-law divergence is closely related to the logarithmic term on the RHS of (REF ).",
"See subsection 3.2 of [5] for more details.",
"A more careful treatment shows that [5] $\\begin{split}\\ln \\mathcal {I}_{\\mathcal {N}=4}(b,\\beta )=(N-1)\\ln (\\frac{2\\pi }{\\beta })+3(N-1)\\ln \\Gamma _h(\\frac{2}{3}\\omega )-\\ln N!+o(1)\\quad (\\text{as $\\beta \\rightarrow 0$}).\\end{split}$ We now study a quiver gauge theory, to illustrate how easily Rains's method generalizes to theories with more than one simple factor in their gauge group.",
"Consider the $\\mathbb {Z}_2$ orbifold of the $\\mathcal {N}=4$ SYM with SU($N$ ) gauge group.",
"The theory consists of two SU($N$ ) gauge groups, with one chiral multiplet in the adjoint of each, and one doublet of bifundamental chiral multiplets from each gauge group to the other.",
"All the chiral multiplets have R-charge $r=2/3$ .",
"The superconformal index is given by (c.f.",
"[19]) $\\begin{split}\\mathcal {I}_{\\mathbb {Z}_2}(b,\\beta )=\\ &(\\prod _{k=1,2}[\\frac{(p;p)^{N-1}(q;q)^{N-1}}{N!",
"}\\Gamma ^{N-1}((pq)^{1/3})\\int \\mathrm {d}^{N-1}x^{(k)}\\\\ &\\left(\\prod _{1\\le i<j\\le N}\\frac{\\Gamma ((pq)^{1/3}(z^{(k)}_i/z^{(k)}_j)^{\\pm 1})}{\\Gamma ((z^{(k)}_i/z^{(k)}_j)^{\\pm 1})}\\right)]) \\times \\prod _{i,j=1}^{N}\\left(\\Gamma ((pq)^{1/3} (z^{(1)}_i/z^{(2)}_j)^{\\pm 1})\\right),\\end{split}$ with $\\prod _{i=1}^{N} z^{(1)}_i=\\prod _{i=1}^{N} z^{(2)}_i=1$ .",
"The Rains function of the theory is $\\begin{split}L_h^{\\mathbb {Z}_2}(\\mathbf {x}^{(1)},\\mathbf {x}^{(2)})&=-\\frac{2}{3}\\sum _{1\\le i<j\\le N}\\vartheta (x^{(1)}_i-x^{(1)}_j)-\\frac{2}{3}\\sum _{1\\le i<j\\le N}\\vartheta (x^{(2)}_i-x^{(2)}_j)+\\frac{2}{3}\\sum _{i,j=1}^{N}\\vartheta (x^{(1)}_i-x^{(2)}_j).\\end{split}$ For the case $N=2$ , corresponding to the SU(2)$\\times $ SU(2) theory, this function is illustrated in Figure REF .",
"Figure: The Rains function of the SU(2)×\\times SU(2) orbifoldtheory.The generalized triangle inequality (REF ) applies with $c=x^{(1)},d=x^{(2)}$ , and implies that $L_h^{\\mathbb {Z}_2}$ is positive semi-definite.",
"It moreover shows that $L_h^{\\mathbb {Z}_2}$ vanishes if the $x^{(1)}_i,x^{(2)}_j$ can be permuted such that either of (REF ) or (REF ) holds.",
"For simplicity we consider all $x^{(1)}_i$ to be positive and very small, except for $x^{(1)}_N=-x^{(1)}_1-\\dots -x^{(1)}_{N-1}$ being negative and very small, and similarly for $x^{(2)}_j$ .",
"Assuming either (REF ) or (REF ), we conclude that $x^{(1)}_i=x^{(2)}_i$ .",
"Based on this result, and also the $N=2$ case whose Rains function is displayed in Figure REF , we conjecture that for the $\\mathbb {Z}_2$ orbifold theory $\\mathrm {dim}\\mathfrak {h}_{qu}=N-1$ , and thereby $\\begin{split}\\ln \\mathcal {I}_{\\mathbb {Z}_2}(b,\\beta )=-\\mathcal {E}^{DK}_0(b,\\beta )+(N-1)\\ln \\left(\\frac{2\\pi }{\\beta }\\right)+O(1)\\quad \\quad (\\text{as$\\beta \\rightarrow 0$}).\\end{split}$ The discussion in subsection REF implies that the three-sphere partition function $Z_{S^3}$ of the dimensionally reduced daughter of this theory diverges as $Z_{S^3}\\approx \\Lambda ^{N-1}$ (as the cut-off $\\Lambda $ of the corresponding matrix-integral is taken to infinity); this power-law divergence is related to the subleading logarithmic term on the RHS of (REF ).",
"See subsection 3.2 of [5] for more details.",
"There are two famous interacting Lagrangian SCFTs with $c<a$ .",
"The first is the Intriligator-Seiberg-Shenker (ISS) model of dynamical SUSY breaking [36].",
"The theory is formulated in the UV as an SU(2) vector multiplet with a single chiral multiplet in the four-dimensional representation of the gauge group.",
"Although originally suspected to confine (and to break supersymmetry upon addition of a tree-level superpotential) [36], the theory is currently believed to flow to an interacting SCFT in the IR [37], [38], where the chiral multiplet has R-charge $3/5$ .",
"The IR SCFT has $c-a=-7/80$ .",
"The index of this theory is (c.f.",
"[39]) $\\begin{split}\\mathcal {I}_{ISS}(b,\\beta )=\\frac{(p;p)(q;q)}{2}\\int \\mathrm {d}x\\frac{\\Gamma ((pq)^{3/10}z^{\\pm 1})\\Gamma ((pq)^{3/10}z^{\\pm 3})}{\\Gamma (z^{\\pm 2})}.\\end{split}$ The Rains function of the theory is $\\begin{split}L^{ISS}_h(x)=\\frac{2}{5}\\vartheta (x)+\\frac{2}{5}\\vartheta (3x)-\\vartheta (2x).\\end{split}$ This function is plotted in Figure REF .",
"Figure: The Rains function of the SU(2) ISS theory.A direct examination reveals that $L^{ISS}_h(x)$ is minimized at $x=\\pm 1/3$ , and $L^{ISS}_h(\\pm 1/3)=-2/15$ .",
"The asymptotics of $\\mathcal {I}_{ISS}$ is hence given according to (REF ) by $\\begin{split}\\ln \\mathcal {I}_{ISS}(b,\\beta )=\\frac{\\pi ^2}{15\\beta }(\\frac{b+b^{-1}}{2})+O(\\beta ^0).\\end{split}$ In other words we have $(c-a)_{\\mathrm {shifted}}=c-a+1/10=1/80$ .",
"The discussion in subsection REF implies that the three-sphere partition function of the dimensionally reduced daughter of this theory is exponentially divergent; this severe divergence is related to the modification that the Di Pietro-Komargodski formula receives in this case.",
"A more careful study shows [5] (see appendix REF for the definition of the symbol $\\sim $ ) $\\begin{split}\\ln \\mathcal {I}_{ISS}(b,\\beta )\\sim \\frac{16\\pi ^2}{3\\beta }(c-a)_{\\mathrm {shifted}}(\\frac{b+b^{-1}}{2})+\\ln Y^{ISS}_{S^3}(b)+\\beta E_{\\mathrm {susy}}(b),\\quad (\\text{as$\\beta \\rightarrow 0$})\\end{split}$ with $Y^{ISS}_{S^3}(b)=\\int _{-\\infty }^{\\infty }\\mathrm {d}x^{\\prime }e^{-\\frac{4\\pi }{5}(b+b^{-1})x^{\\prime }}\\times \\Gamma _h(3x^{\\prime }+(3/5)\\omega )\\Gamma _h(-3x^{\\prime }+(3/5)\\omega ),$ and $(c-a)_{\\mathrm {shifted}}=(c-a)+1/10=1/80$ .",
"A numerical evaluation using $\\begin{split}\\ln \\Gamma _h(ix;i,i)=(x-1)\\ln (1-e^{-2\\pi i x})-\\frac{1}{2\\pi i}Li_2(e^{-2\\pi ix})+\\frac{i\\pi }{2}(x-1)^2-\\frac{i\\pi }{12},\\end{split}$ yields $Y^{ISS}_{S^3}(b=1)\\approx .423$ .",
"The second famous example of interacting SCFTs with $c<a$ is provided by the “misleading” SO($n$ ) theory of Brodie, Cho, and Intriligator [40].",
"This is an $\\mathcal {N}=1$ SO($n$ ) gauge theory with a single chiral multiplet in the two-index symmetric traceless tensor representation of the gauge group.",
"The theory is asymptotically free if $n\\ge 5$ .",
"For $5\\le n<11$ the corresponding interacting IR SCFT has $c-a=-(n-1)/16$ (for greater values of $n$ the R-symmetry of the IR SCFT is believed to mix with an emergent accidental symmetry, and thus more care is called for; c.f.",
"[41]).",
"For the SO($2N+1$ ) theory (with $1<N<5$ ) we have (c.f.",
"[39]) $\\begin{split}\\mathcal {I}_{BCI}(b,\\beta )=&\\frac{(p;p)^N(q;q)^N}{2^NN!",
"}\\Gamma ^N((pq)^{2/(2N+3)})\\int \\mathrm {d}^N x\\\\&\\prod _{i<j}\\frac{\\Gamma ((pq)^{2/(2N+3)}z_i^{\\pm 1}z_j^{\\pm 1})}{\\Gamma (z_i^{\\pm 1}z_j^{\\pm 1})}\\prod _{j=1}^N\\frac{\\Gamma ((pq)^{2/(2N+3)}z_j^{\\pm 1},(pq)^{2/(2N+3)}z_j^{\\pm 2})}{\\Gamma (z_j^{\\pm 1})}.\\end{split}$ The Rains function of the theory is $\\begin{split}L_h^{BCI}(x)=\\frac{4}{2N+3}\\left((\\frac{2N-1}{4})\\sum _{j}\\vartheta (2x_j)-\\sum _j\\vartheta (x_j)-\\sum _{i<j}\\vartheta (x_i+x_j)-\\sum _{i<j}\\vartheta (x_i-x_j)\\right).\\end{split}$ For $N=2$ , corresponding to the SO(5) theory, this function is plotted in Figure REF .",
"Figure: The Rains function of the SO(5) BCI theory.To find the minima of the above function, we need the following result, valid for $-1/2\\le x_i\\le 1/2$ : $\\begin{split}&(\\frac{2N-1}{4})\\sum _{1\\le j\\le N}\\vartheta (2x_j)-\\sum _{1\\le j\\le N}\\vartheta (x_j)-\\sum _{1\\le i<j\\le N}\\vartheta (x_i+x_j)-\\sum _{1\\le i<j\\le N}\\vartheta (x_i-x_j)=\\\\&-\\frac{3}{2}\\sum _{ i<j}\\mathrm {max}(|x_i|,|x_j|) +\\frac{1}{2}\\sum _{i<j}\\mathrm {min}(|x_i|,|x_j|)=\\sum _{j}(-\\frac{3N}{2}+2j-\\frac{1}{2})\\mathrm {min}_{N-j+1}(|x_i|),\\end{split}$ with $\\mathrm {min}_N(|x_i|):=\\mathrm {max}(|x_i|)$ .",
"The proof of (REF ) is similar to that of (REF ).",
"Note that the coefficient of the $j$ th term on the RHS of (REF ) is negative if $j<\\frac{3N+1}{4}$ , and positive otherwise.",
"This implies that the Rains function (REF ) is minimized when $\\lfloor \\frac{3N+1}{4}\\rfloor $ of the $|x_i|$ are maximized (i.e.",
"$x_{i}=\\pm 1/2$ ), and the rest of the $|x_i|$ are minimized (i.e.",
"$x_{i}=0$ ).",
"Consequently, the minimum of the Rains function is $\\begin{split}L_{h\\ \\mathrm {min}}^{BCI}=-\\frac{1}{2N+3}\\sum _{1\\le j\\le \\lfloor \\frac{3N+1}{4}\\rfloor }(3N+1-4j).\\end{split}$ This is less than zero for any $N>1$ .",
"Therefore the Di Pietro-Komargodski formula needs to be modified in the SO($2N+1$ ) BCI model with $1<N<5$ .",
"The discussion in subsection REF implies that the three-sphere partition function of the dimensionally reduced daughter of this model (with $1<N<5$ ) is exponentially divergent; this severe divergence is related to the modification that the Di Pietro-Komargodski formula receives in this case.",
"See subsection 3.3 of [5] for more details.",
"Consider now the concrete case of the SO(5) theory corresponding to $N=2$ .",
"This theory has $c-a=-1/4$ .",
"From Eq.",
"(REF ) we have in this case $L_{h\\ min}^{BCI}(x)=-3/7$ .",
"The asymptotics of $\\mathcal {I}$ is therefore given according to (REF ) by $\\begin{split}\\ln \\mathcal {I}_{BCI_5}(b,\\beta )=\\frac{8\\pi ^2}{21\\beta }(\\frac{b+b^{-1}}{2})+O(\\beta ^0).\\end{split}$ In other words $(c-a)_{\\mathrm {shifted}}=c-a+9/28=1/14$ .",
"A more careful treatment shows [5] $\\begin{split}\\ln \\mathcal {I}_{BCI_5}(b,\\beta )\\sim \\frac{16\\pi ^2}{3\\beta }(c-a)_{\\mathrm {shifted}}(\\frac{b+b^{-1}}{2})+\\ln Y^{BCI_5}_{S^3}(b)+\\beta E_{\\mathrm {susy}}(b),\\quad (\\text{as$\\beta \\rightarrow 0$})\\end{split}$ with $\\begin{split}Y^{BCI_5}_{S^3}(b)=&\\frac{1}{2}\\int _{-\\infty }^{\\infty }\\mathrm {d}x_1^{\\prime } \\ \\Gamma _h((4/7)\\omega \\pm 2x_1^{\\prime })\\times \\\\&\\frac{\\Gamma _h^2((4/7)\\omega )}{2}\\int _{-\\infty }^{\\infty } \\mathrm {d}x_2\\ \\frac{\\Gamma _h((4/7)\\omega \\pm x_2)\\Gamma _h((4/7)\\omega \\pm 2x_2)}{\\Gamma _h(\\pm x_2)},\\end{split}$ and $(c-a)_{\\mathrm {shifted}}=(c-a)+9/28=1/14$ .",
"A numerical evaluation using (REF ) yields $Y^{BCI_5}_{S^3}(b=1)\\approx .026$ .",
"An interesting class of Lagrangian $\\mathcal {N}=2$ SCFTs arise from quiver gauge theories associated to Riemann surfaces of genus $g\\ge 2$ , without punctures (see e.g.",
"[42] for a discussion of the indices of these theories).",
"These quivers can be constructed from fundamental blocks of the kind shown in Figure REF .",
"The triangle in Figure REF represents eight chiral multiplets of R-charge $2/3$ transforming in the tri-fundamental representation of the three SU(2) gauge groupsWe focus on class-$\\mathcal {S}$ theories constructed from $T_2$ , and leave the study of higher-rank theories constructed from $T_{N>2}$ to future work.",
"represented by the (semi-circular) nodes; more precisely, when two semi-circular nodes are connected together to form a circle, they represent an $\\mathcal {N}=2$ SU(2) vector multiplet.",
"A class-$\\mathcal {S}$ theory of genus $g$ arises when $2g-2$ of these blocks are glued back-to-back (and forth-to-forth) along a straight line, with the leftmost and the rightmost blocks having two of their half-circular nodes glued together; see Figure REF for an example.",
"Figure: The building block of the puncture-less Lagrangianclass-𝒮\\mathcal {S} theories.An $\\mathcal {N}=2$ SU(2) vector multiplet contributes to the Rains function of the SCFT as $\\begin{split}L_h^{\\mathcal {N}=2\\ v}(x)=-\\frac{2}{3}\\vartheta (2x).\\end{split}$ A semi-circular node contributes half as much, and thus the three semi-circular nodes in Figure REF contribute together as $\\begin{split}L_h^{\\mathrm {semi-nodes}}(x,y,z)=-\\frac{1}{3}\\left(\\vartheta (2x)+\\vartheta (2y)+\\vartheta (2z)\\right).\\end{split}$ The eight chiral multiplets represented by the triangle in Figure REF contribute to the Rains function of the theory as $\\begin{split}L_h^{T_2}(x,y,z)=\\frac{1}{3}\\left(\\vartheta (x+y+z)+\\vartheta (x+y-z)+\\vartheta (x-y+z)+\\vartheta (-x+y+z)\\right).\\end{split}$ Adding up (REF ) and (REF ) we obtain the contribution of a single block to the Rains function: $\\begin{split}L_h^{\\mathrm {block}}(x,y,z)=&\\frac{1}{3}[\\vartheta (x+y+z)+\\vartheta (x+y-z)+\\vartheta (x-y+z)+\\vartheta (-x+y+z)\\\\&-\\vartheta (2x)-\\vartheta (2y)-\\vartheta (2z)].\\end{split}$ With the Rains function of the block at hand, we can now write down the Rains function of genus $g$ class-$\\mathcal {S}$ theories.",
"For example, the Rains function of the $g=2$ theory is given by $\\begin{split}L_h^{\\mathcal {S}_{g=2}}(x_1,x_2,x_3)=L_h^{\\mathrm {block}}(x_1,x_1,x_2)+L_h^{\\mathrm {block}}(x_2,x_3,x_3),\\end{split}$ and the Rains function of the $g=3$ theory (illustrated in Figure REF ) is obtained as $\\begin{split}L_h^{\\mathcal {S}_{g=3}}(x_1,x_2,x_3,x_4,x_5,x_6)=&L_h^{\\mathrm {block}}(x_1,x_1,x_2)+L_h^{\\mathrm {block}}(x_2,x_3,x_4)\\\\&+L_h^{\\mathrm {block}}(x_3,x_4,x_5)+L_h^{\\mathrm {block}}(x_5,x_6,x_6).\\\\\\end{split}$ Figure: The quiver diagram of the g=3g=3 class-𝒮\\mathcal {S} theory.Importantly, Rains's GTI (REF ), with $c_1=x+y,\\ c_2=x-y,\\ d_1=z,\\ d_2=-z$ , implies that $\\begin{split}L_h^{\\mathrm {block}}(x,y,z)\\ge 0.\\end{split}$ It is not difficult to show that the equality holds in a finite-volume subspace of the $x,y,z$ space; take for instance $x,y,z\\approx .1$ within $.01$ of each other, and use the fact that for small argument $L_h$ reduces to $\\tilde{L}_{S^3}$ to show that $L_h$ vanishes in the domain just described.",
"Since the Rains function of a $g\\ge 2$ class-$\\mathcal {S}$ theory is the sum of several block Rains functions, the positive semi-definiteness of $L_h^{\\mathrm {block}}$ guarantees the positive semi-definiteness of $L_h^{\\mathcal {S}_{g\\ge 2}}(x_i)$ ; moreover, taking all $x_i$ to be around $.1$ and within $.01$ of each other we can easily conclude (as in the previous paragraph) that for the genus $g$ theory $\\mathrm {dim}\\mathfrak {h}_{qu}=3(g-1)$ .",
"The relation (REF ) thus yields $\\begin{split}\\ln \\mathcal {I}_{\\mathcal {S}_{g\\ge 2}}(b,\\beta )=\\frac{16\\pi ^2}{3\\beta }(c-a)(\\frac{b+b^{-1}}{2})+3(g-1)\\ln (\\frac{2\\pi }{\\beta })+O(\\beta ^0),\\end{split}$ with $c-a=-(g-1)/24$ .",
"Dual QFTs must have equal partition functions.",
"As a trivial corollary, the high-temperature asymptotics of the index of dual 4d SCFTs must match.",
"Assume now that both sides of the duality are non-chiral 4d Lagrangian SCFTs with a semi-simple gauge group.",
"The relation (REF ) then yields two quantities to be matched between the theories: $L_{h\\ \\mathrm {min}}$ and $\\mathrm {dim}\\mathfrak {h}_{qu}$ .",
"Comparison of $L_{h\\ \\mathrm {min}}$ can rule out for instance the confinement scenario for the SU(2) ISS model: on the gauge theory (UV) side, as discussed above, we have $L_{h\\ \\mathrm {min}}=-2/15$ , while on the mesonic (IR) sideFollowing [39], we are assuming that an index can be consistently assigned to the proposed IR theory, even though the IR chiral multiplet would have R-charge $12/5\\notin ]0,2[$ .",
"This assignment requires an analytic continuation of the kind discussed in [5]; the small-$\\beta $ asymptotics of the resulting function can be obtained as in [23].",
"See [43] for an alternative take on this problem.",
"we have no gauge group and thus $L_{h}=0$ .",
"As another example, consider the recent $E_7$ SQCD duality of [17], [44].",
"In that case a direct examination reveals that $L_{h\\ \\mathrm {min}}=\\mathrm {dim}\\mathfrak {h}_{qu}=0$ , both on the electric and the magnetic side.",
"Their proposal hence passes both our tests.",
"The case of the interacting $\\mathcal {N}=1$ SCFTs with $c<a$ (namely the IR fixed points of the ISS model, and the BCI$_{2N+1}$ model with $1<N<5$ ) is particularly interesting.",
"A dual description for these theories is currently lacking.",
"Our results for $L_{h\\ \\mathrm {min}}$ and $\\mathrm {dim}\\mathfrak {h}_{qu}$ on the electric side might help to test future proposals for magnetic duals of these theories.",
"In the present chapter we analyzed the high-temperature asymptotics of the indices of various gauge theories at finite $N$.",
"The finite-$N$ indices of holographic SCFTs are expected to encode information about micro-states of the supersymmetric Giant Gravitons of the dual string theories [9].",
"Take for instance the SU($N$ ) $\\mathcal {N}=4$ SYM.",
"One of the novel results of [5] is the following high-temperature asymptotics for the superconformal index of this theory (see Eqs.",
"(REF ) and (REF ) above): $\\mathcal {I}(b=1,\\beta )=\\sum _{\\mathrm {operators}}(-1)^Fe^{-\\beta (\\Delta -{\\frac{1}{2}}r)}\\approx (\\frac{1}{\\beta })^{N-1}.$ The above canonical relation can be transformed to the micro-canonical ensemble to yield the asymptotic (fermion-number weighted) degeneracy of the protected high-energy operators in the $\\mathcal {N}=4$ theory: $N(E)\\approx E^{N-2},$ with $E=\\Delta -r/2$ .",
"This result should presumably be reproduced by geometric quantization of the $1/16$ BPS Giant Gravitons of IIB theory on AdS$_5\\times S^5$ , along the lines of [45].",
"It would be interesting to see if this expectation pans out.",
"Take a non-chiral 4d SCFT with a semi-simple gauge group, and with $r_\\chi \\in ]0,2[$ .",
"Its superconformal index $\\mathcal {I}(b,\\beta )$ can be computed by a path-integral on $S_b^3\\times S_\\beta ^1$ , with $S_b^3$ the unit-radius squashed three-sphere.",
"We now replace the $S_b^3$ with the round three-sphere $S_{r_3}^3$ of arbitrary radius $r_3>0$ .",
"The path-integral on the new space gives $\\mathcal {I}(\\beta ;r_3)=\\mathcal {I}(b=1,\\beta /r_3)$ ; i.e.",
"the resulting partition function only depends on the ratio $\\beta /r_3$ , as the theory is conformal.",
"Thus, as far as $\\mathcal {I}(\\beta ;r_3)$ is concerned, shrinking the $S^1$ (i.e.",
"the high-temperature limit) is equivalent to decompactifying the $S^3$ .",
"We hence fix $\\beta $ , and send $r_3$ to infinity.",
"In this limit we expect the unlifted zero-modes on $S_{r_3}^3\\times S_\\beta ^1$ to roughly correspond to the quantum zero-modes on $R^3\\times S^1$ .",
"Therefore at high temperatures the unlifted holonomies of the theory on $S_{r_3}^3\\times S_\\beta ^1$ should be in correspondence with (a real section of) the quantum Coulomb branch of the 3d $\\mathcal {N}=2$ theory obtained from compactifying the 4d theory on the circle of $R^3\\times S^1$ .",
"In particular, we expect $\\mathrm {dim}\\mathfrak {h}_{qu}$ to be equal to the (complex-) dimension of the quantum Coulomb branch of the 3d theory.",
"(Recall that the Coulomb branch of the circle-compactified theory living on $R^3$ consists not just of the holonomies around the $S^1$ , but also of the dual 3d photons; hence our references above to “a real section” and “complex-dimension”.)",
"We do not expect to recover the $R^3\\times S^1$ Higgs branch from the zero-modes on $S_{r_3}^3\\times S_\\beta ^1$ : for any (arbitrarily small) curvature on the $S^3$ , curvature couplings presumably lift the Higgs-type zero-modes on $S_{r_3}^3\\times S_\\beta ^1$ .",
"From the point of view of $R^3\\times S^1$ , picking one of the $R^3$ directions as timeThe following discussion is in the spirit of the arguments in [46], though our treatment is not as precise.",
"We are approaching $R^3$ from $S^3$ , rather than from $T^3$ (as in [46]).",
"While on $T^3$ each of the circles can be picked as the time direction, picking a time direction along the $S^3$ makes the spatial sections time-dependent, rendering our arguments in the paragraph of this footnote somewhat hand-wavy.",
"I thank E. Shaghoulian for several helpful conversations related to the subject of the present subsection., we can relate $\\mathcal {E}^{DK}_0$ to the Casimir energy associated to the spatial manifold $R^2\\times S^1$ : we reintroduce $r_3$ in $\\mathcal {E}^{DK}_0$ (by replacing its $\\beta $ with $\\beta /r_3$ ), set in it $b=1$ , interpret $\\tilde{\\beta }:=2\\pi r_3$ as the circumference of the crossed channel thermal circle, and write $\\mathcal {E}_0^{DK}(\\beta ;r_3)=\\tilde{\\beta }E_0^{R^2\\times S^1}(\\beta ),\\quad \\text{with}\\quad E_0^{R^2\\times S^1}(\\beta )=\\frac{\\pi }{6\\beta }\\mathrm {Tr}R.$ Now $E_0^{R^2\\times S^1}(\\beta )$ admits an interpretation as the (regularized) Casimir energy associated to the spatial $R^2\\times S_\\beta ^1$ .",
"Similarly, resurrecting the $r_3$ in $V^{\\mathrm {eff}}$ , and setting in it $b=1$ , we obtain what can be loosely regarded as $\\tilde{\\beta }$ times the quantum effective potential on (a real section of) the crossed channel Coulomb branch.",
"From this perspective, the two tests we advocated in subsection REF would not really be new, but would correspond to the comparison of low-energy properties on $R^3\\times S^1$ .",
"The discussion in the previous three paragraphs is rather intuitive, and should be considered suggestive at best.",
"It is desirable to have it made more precise.",
"Nevertheless, in the examples of the SU($N$ ), Sp($2N$ ), and SO($2N+1$ ) SQCD theories, and the SU($N$ ) $\\mathcal {N}=4$ SYM, it turns out [5] that (upon quotienting by the Weyl group) $\\mathfrak {h}_{qu}$ does indeed resemble (a real section of) the $R^3\\times S^1$ quantum Coulomb branch; see [12], [18] and [47].",
"We therefore conjecture that the relation between $\\mathfrak {h}_{qu}$ and the unlifted Coulomb branch on $R^3\\times S^1$ continues to remain valid, at least for all the theories with a positive semi-definite Rains function.",
"In particular, we predict that, when placed on $R^3\\times S^1$ , the SU($N$ ) $A_k$ SQCD theories (in the appropriate range of their parameters such that all their $r_\\chi $ are in $]0,2[$ ) have no quantum Coulomb branch, and the $\\mathbb {Z}_2$ orbifold of the SU($N$ ) $\\mathcal {N}=4$ theory has an $(N-1)$ -dimensional unlifted Coulomb branch.",
"For theories whose Rains function is not positive semi-definite, on the other hand, it seems like this connection with $R^3\\times S^1$ fails.",
"The Rains function of the SU(2) ISS model does not have a flat direction, and appears to suggest a Higgs vacuum for the theory on $R^3\\times S^1$ .",
"However, the study of [38] indicates that this theory possesses an unlifted Coulomb branch on $R^3\\times S^1$ , and in particular does not necessarily break the gauge group at low energies.",
"It would be nice to understand if this conflict is only a manifestation of the sloppiness of our intuitive arguments above, or it has a more interesting origin.",
"It is often the case that asymptotically at large $N$ a hierarchy appears in the spectrum of local operators of an SCFT.",
"This hierarchy is expected to be reflected in the superconformal index.",
"Take for instance the U($N$ ) $\\mathcal {N}=4$ SYM, which has the following Schur index [9] (see [48] for the definition of the Schur index; in the present chapter we discuss the Schur index only as a toy model of the superconformal index): $\\mathcal {I}_{Schur}(\\beta )=\\frac{(q;q)}{(q^{1/2};q^{1/2})^2}\\sum _{n=0}^{\\infty }(-1)^n\\left[\\begin{pmatrix}N+n\\\\ N\\end{pmatrix}+\\begin{pmatrix}N+n-1\\\\ N\\end{pmatrix}\\right]q^{(n N+n^2)/2},$ with $q=e^{-\\beta }$ .",
"In the above expression, $n$ clearly has an interpretation as a soliton-counting number.",
"Of course, these solitons are naturally interpreted in the gravity dual to the U($N$ ) $\\mathcal {N}=4$ SYM.",
"They presumably correspond to Giant Gravitons of the IIB theory on AdS$_5\\times S^5$ [9].",
"The large-$N$ limit suppresses (energetically) all the $n\\ne 0$ terms in (REF ), and yields $\\mathcal {I}^{N\\rightarrow \\infty }_{Schur}(\\beta )=\\frac{(q;q)}{(q^{1/2};q^{1/2})^2}.$ This is the “multi-trace” Schur index of the U($N$ ) $\\mathcal {N}=4$ SYM; it can be obtained by summing over multi-trace operators of the gauge theory in the planar limit.",
"Another general point that our toy model can help illustrate is that the high-temperature asymptotics of the multi-trace index may be (and generically is) very different from the asymptotics of the finite-$N$ index (which we focused on in the previous chapter).",
"Our toy model index (REF ) has the high-temperature asymptotics (see [5] for the similar asymptotic analysis of the Schur index of the SU($N$ ) $\\mathcal {N}=4$ SYM) $\\ln \\mathcal {I}_{Schur}(\\beta )= N\\ln (\\frac{2\\pi }{\\beta })+O(\\beta ).$ Taking the $N\\rightarrow \\infty $ limit before the $\\beta \\rightarrow 0$ limit, changes the high-temperature asymptotics drastically.",
"The high-temperature asymptotics of the large-$N$ Schur index (in (REF )) is found as (see appendix REF for the definition of the symbol $\\sim $ used below) $\\ln \\mathcal {I}_{Schur}^{N\\rightarrow \\infty }(\\beta )=\\ln (q;q)-2\\ln (q^{1/2};q^{1/2})\\sim \\frac{\\pi ^2}{2\\beta }+\\frac{1}{2}\\ln (\\frac{\\beta }{8\\pi }),\\quad \\quad (\\text{as $\\beta \\rightarrow 0$})$ differing significantly from the asymptotics of the finite-$N$ index in (REF ).",
"The purpose of the above discussion was to help orient the reader for the following analysis of toric quiver gauge theories with SU($N$ ) nodes.",
"Toric quiver theories are a much-studied subset of supersymmetric gauge theories whose field content can be efficiently summarized using quiver diagrams.",
"The latter are directed graphs with nodes representing vector multiplets and edges representing chiral multiplets.",
"The nodes at the ends of an edge represent vector multiplets under which the chiral multiplet (represented by the edge) is charged.",
"The direction of the edge encodes further information about the representation of the gauge group according to which the chiral multiplet transforms.",
"The toric condition puts further constraints on the theory, thereby guaranteeing some nice properties such as existence of a non-trivial IR fixed point with a holographic dual describable by “toric geometry” (see for instance [49]).",
"A canonical example is the $\\mathcal {N}=4$ SYM with SU($N$ ) gauge group, which can be represented by one node (standing for the SU($N$ ) vector multiplet), and three directed edges (standing for the three chiral multiplets in the adjoint) that both emanate from and end on that one node.",
"Similarly to the case of the Schur index discussed above, in the large-$N$ limit the index of toric quivers simplifies to the multi-trace index.",
"The multi-trace index is the one obtained by summing over multi-trace operators in the SCFT.",
"These operators correspond to the multi-particle KK states in the gravity dual.",
"The superconformal index of these SCFTs is studied in [19], [50], [51], [23], and its large-$N$ limit is found to be $\\mathcal {I}_{quiver}^{N\\rightarrow \\infty }(b,\\beta )=\\frac{1}{\\prod _{i=1}^{n_z}((pq)^{r_i/2};(pq)^{r_i/2})\\prod _{adj}\\Gamma ((pq)^{R_{adj}/2};p,q)},$ with the first product in the denominator being over the $n_z$ extremal BPS mesons (with R-charge $r_i$ ), and the second product over the chiral multiplets (with R-charge $R_{adj}$ ) in the adjoints of various nodes.",
"For example, the SU($N$ ) $\\mathcal {N}=4$ SYM has $n_z=3$ , and $r_{1,2,3}=R_{adj}=2/3$ , with three adjoint chiral multiplets in total.",
"Just as in the case of the Schur index discussed above, the high-temperature asymptotics of $\\mathcal {I}_{quiver}^{N\\rightarrow \\infty }(b,\\beta )$ is quite different from the asymptotics of the same quiver at finite $N$ ; the latter asymptotics can be obtained (for non-chiral quivers) from the results of the previous section.",
"Finding the asymptotics of $\\mathcal {I}_{quiver}^{N\\rightarrow \\infty }(b,\\beta )$ , on the other hand, requires separate calculations (see (REF ) below).",
"The single-trace index is defined as the plethystic log [20] of the multi-trace index $I_{s.t.",
"}(b,\\beta )\\equiv \\sum _{n=1}^{\\infty }\\frac{\\mu (n)}{n}\\ln \\mathcal {I}^{N\\rightarrow \\infty }(b,n\\beta ),$ where $\\mu (n)$ is the Möbius function.",
"The adjective “single-trace” is particularly appropriate for theories that admit a planar limit in which single-trace operators are weakly interacting.",
"For such cases if in the definition of the index in (REF ) one restricts the trace to the “single-trace states” in the Hilbert space, one obtains the single-trace index as defined above.",
"In AdS/CFT, the weakly interacting mesons of the large-$N$ SCFT at large 't Hooft coupling map to the KK supergravity modes in the bulk.",
"Therefore the boundary single-trace index is (according to AdS/CFT) equal to the bulk single-particle index, with the latter receiving contributions only from the bulk single-particle KK states.",
"The single-trace index of SU($N$ ) toric quiver SCFTs can be easily computed by taking the plethystic logarithm of the two sides of (REF ); the result is $I_{s.t.\\ quiver}(b,\\beta )=\\sum _{i=1}^{n_z}\\frac{(pq)^{r_i/2}}{1-(pq)^{r_i/2}}-\\sum _{adj}\\frac{(pq)^{R_{adj}/2}-(pq)^{1-R_{adj}/2}}{(1-p)(1-q)}.$ In the next section we will see that in its high-temperature asymptotics, the above index encodes the subleading Weyl anomaly of the underlying SU($N$ ) toric quiver SCFT.",
"An interesting problem, which is not relevant to the main discussion of the present chapter, is the connection between the small-$\\beta $ asymptotics of $I_{s.t.",
"}(b,\\beta )$ and $\\mathcal {I}^{N\\rightarrow \\infty }(b,\\beta )$ ; this problem is addressed in appendix REF .",
"In this section we present holographic results implying that the subleading central charges of a holographic SCFT are encoded in the high-temperature asymptotics of its large-$N$ index.",
"We focus on SCFTs whose dual geometry is of the form AdS$_5\\times $ SE$_5$ , with SE$_5$ a Sasaki-Einstein 5-manifold.",
"The KK spectrum of the IIB theory on AdS$_5\\times $ SE$_5$ organizes itself into representations of the 4d $\\mathcal {N}=1$ superconformal group SU$(2,2|1)$ .",
"The shortened multiplets of SU$(2,2|1)$ are listed in Table REF , along with their contributions to the single-trace index.",
"For convenience, we have introduced $t\\equiv 1/\\sqrt{pq}$ , and $y\\equiv \\sqrt{p/q}$ .",
"The chiral and SLII multiplets (on the 2nd and the last row, respectively) contribute to the right-handed indexThe index defined in (REF ) is the right-handed index.",
"One can also define the left-handed index in which one replaces $r$ with $-r$ and swaps $j_1$ and $j_2$ in the definition of the index in (REF ).",
"The index in Table REF is defined as $ I^+_{s.t.}",
"\\equiv \\frac{1}{2}( I^R_{s.t.",
"}+ I^L_{s.t.",
"})$ , in terms of the left and right single-trace indices.",
"For toric quivers $I^+_{s.t.}",
"=I^R_{s.t.",
"}= I^L_{s.t.",
"}$ ., while the CP-conjugate multiplets, namely the anti-chiral and SLI multiplets (on the 3nd and the 4th row, respectively), contribute to the left-handed index.",
"Conserved multiplets (on the 1st row), which are CP self-conjugate, contribute to both.",
"Table: Contributions to the superconformal index from the variousshortened multiplets.We begin with relating $c-a$ to the index.",
"First consider the chiral and SLII multiplets.",
"The contribution to $c-a$ from a generic chiral multiplet $\\mathcal {D}(E_0,j_1,0;r)$ in the bulk KK spectrum is given by the following holographically derived expression [52], [21] $(c-a)\\big |_{\\text{chiral}}= -{\\frac{1}{192}}(-1)^{2j_1}(2E_0-3)(2j_1+1)\\left(1-8j_1(j_1+1)\\right).",
"$ Similarly, a generic SLII multiplet $\\mathcal {D}(E_0,j_1,j_2;r)$ in the bulk spectrum contributes [52], [21] $(c-a)\\big |_{\\text{SLII}}={\\frac{1}{192}}(-1)^{2j_1+2j_2}(2E_0+2j_2-1)(2j_1+1)\\left(1-8j_1(j_1+1)\\right).$ It is now possible to see how these expressions may be obtained from the contributions to the right-handed index given in Table REF .",
"Since the SU(2) character $\\chi _j(y)$ is given by $\\chi _j(y)={\\frac{y^{2j+1}-y^{-(2j+1)}}{y-y^{-1}}},$ the differential operator $(6(y\\partial _y)^2-1)$ acting on the contributions to the index gives $(2j+1)[8j(j+1)-1]$ when $y$ is set to one.",
"The operator $(t\\partial _t+1)$ then produces the $E_0$ -dependent factors in (REF ) and (REF ).",
"The CP conjugate multiplets (anti-chiral and SLI) contribute similarly to (REF ) and (REF ) with the appropriate replacement of quantum numbers, and are accounted for in the left-handed index.",
"Finally, since conserved multiplets contribute as the sum of one SLI and one SLII multiplet, they are implicitly included in both the left- and right-handed indices.",
"Our key observation is that the contribution to $c-a$ has a uniform expression for every single bulk multiplet.",
"Hence a single differential operator acting on the index can yield the appropriate contribution to $c-a$ regardless of the shortening condition.",
"Summing over all the bulk KK multiplets, one finally arrives at $c-a &= &\\lim _{t\\rightarrow 1}-\\frac{1}{32}\\left(t\\partial _t+1\\right)\\left(6(y\\partial _y)^2-1\\right)\\nonumber \\\\&&\\times \\left[(1-t^{-1} y)(1-t^{-1} y^{-1}) I^+_{s.t.",
"}(t,y)\\right]\\Big |^{\\mbox{\\scriptsize {finite}}}_{y=1},$ where the fugacities are set to one after acting with the differential operator on the index.",
"Note that the factor $(1-t^{-1}y)(1-t^{-1} y^{-1})$ multiplying the single-trace index removes the contribution from descendant states.",
"The result obtained is often divergent, as we are working in the large-$N$ limit, so the prescription is that the finite term in an expansion about $t=1$ yields the value of $c-a$ .",
"A few remarks are now in order.",
"The $t\\rightarrow 1$ limit corresponds to the high-temperature limit $\\beta \\rightarrow 0$ .",
"Therefore the prescription (REF ) extracts $c-a$ from the high-temperature asymptotics of $I^+_{s.t.",
"}$ .",
"The index in (REF ) is the single-particle supergravity index, which is—according to the AdS/CFT conjecture—equal to the single-trace index of the SCFT.",
"In the prescription (REF ), the index provides a natural regulator for the Kaluza-Klein sums encountered in the holographic $c-a$ calculations of [55], [54], [53].",
"The $t\\rightarrow 1$ limit corresponds to the high-temperature limit $\\beta \\rightarrow 0$ .",
"Therefore the prescription (REF ) extracts $c-a$ from the high-temperature asymptotics of $I^+_{s.t.",
"}$ .",
"The index in (REF ) is the single-particle supergravity index, which is—according to the AdS/CFT conjecture—equal to the single-trace index of the SCFT.",
"In the prescription (REF ), the index provides a natural regulator for the Kaluza-Klein sums encountered in the holographic $c-a$ calculations of [55], [54], [53].",
"Following a similar approach, but using holographic expressions for the individual central charges, one arrives at [52], [22] $\\begin{split}\\delta a&={\\frac{1}{32}}(t\\partial _t+1)(-{\\textstyle \\frac{9}{2}}t\\partial _t(t\\partial _t+2)+{\\textstyle \\frac{9}{2}}(y\\partial _y)^2-3)\\hat{I}(t,y),\\\\\\delta c&={\\frac{1}{32}}(t\\partial _t+1)(-{\\textstyle \\frac{9}{2}}t\\partial _t(t\\partial _t+2)-{\\textstyle \\frac{3}{2}}(y\\partial _y)^2-2)\\hat{I}(t,y), \\end{split}$ where $\\hat{I}=(1-yt^{-1})(1-y^{-1}t^{-1}) I^+_{s.t.",
"}$ is the single-trace index with descendants removed, and $\\delta $ indicates that we are referring to the $O(N^0)$ part of the central charges (and not their leading $O(N^2)$ piece).",
"The fugacities are set to one after acting with the differential operators on $\\hat{I}$ ; we are thus again dealing with the high-temperature limit of the index.",
"In principle, a successful application of Eq.",
"(REF ) to a holographic SCFT can be viewed as a one-loop test of AdS/CFT.",
"This can be easily done for arbitrary SU($N$ ) toric quiver SCFTs without adjoint matter that are dual to smooth Sasaki-Einstein 5-manifolds.",
"The single-trace index of such a toric theory is [50] $I_{s.t.",
"}=\\sum _i \\frac{1}{t^{r_i/3}-1},$ where $r_i$ are the $R$ -charges of extremal BPS mesons.",
"Applying (REF ) to (REF ) gives $\\delta a = -\\frac{27}{32(t-1)^2}\\sum _{i=1}^{n_v} \\frac{1}{r_i} -\\frac{1}{32}\\sum _{i=1}^{n_v}r_i+\\cdots $ in an expansion about $t=1$ .",
"Noting that $\\sum r_i=6(\\mbox{\\# nodesin the quiver})$ , and keeping only the finite part, we obtain $\\delta a=-{\\frac{3}{16}}(\\mbox{\\# nodes in thequiver}).$ This matches the expected result for the $O(1)$ part of $a$ based on the decoupling of a U(1) at each node in the quiver; since there are no adjoint matter fields in the quiver, there are no additional $O(1)$ contributions to $a$ in the field theoretical computation through $a=\\frac{1}{32}(9\\mathrm {Tr}R^3-3\\mathrm {Tr}R)$ .",
"The successful matching for the $O(1)$ part of $c$ can be deduced from a similar application of the second relation in Eq.",
"(REF ) to (REF ).",
"The prescriptions in Eq.",
"(REF ) can also be successfully applied to the single-trace index of an arbitrary SU($N$ ) toric quiver, as given in (REF ).",
"However, the result would count as a successful test of AdS/CFT only up to the following two assumptions: $i)$ a combinatorial conjecture [51] that has gone into the derivation of (REF ), which although strongly supported in [51], is not yet proven; $ii)$ the assumption that the index (REF ) (which is derived as the single-trace index of the SCFT) equals the single-particle index of the gravity dual (which is what goes on the RHS of the prescriptions in Eq.",
"(REF )).",
"The equality in the assumption $ii$ is not yet proven [50] when the toric quiver has adjoint matter or is dual to singular toric SE$_5$ .",
"Finally, the study of [23] shows that the validity of the prescriptions in Eq.",
"(REF ) is guaranteed if the high-temperature asymptotics of the single-trace index of SCFTs dual to AdS$_5\\times $ SE$_5$ has the following form (see appendix REF for the definition of the symbol $\\sim $ used below): $\\begin{split}I_{s.t.",
"}\\sim &\\,\\frac{2H}{\\beta \\left(b+b^{-1}\\right)}+\\frac{G\\left(b+b^{-1}\\right)}{2\\beta }+C\\\\&\\,-\\beta \\left(\\frac{4}{27}(b+b^{-1})^3\\left(3\\delta c-2\\delta a\\right)+\\frac{4}{3}(b+b^{-1})\\left(\\delta a-\\delta c\\right)\\right),\\end{split}$ with $G$ ,$H$ ,$C$ constants that are insignificant for the formulas in Eq.",
"(REF ), except that $H$ determines the pole terms that according to the prescription of [22] one should drop.",
"The above asymptotics was explicitly verified in [23] for the single-trace index of the SU($N$ ) toric quivers, shown in (REF ).",
"We have shown that the high-temperature expansion of the superconformal index of finite-rank non-chiral SCFTs (having all their $r_\\chi $ inside $]0,2[$ ) looks like $\\ln \\mathcal {I}(b,\\beta )=\\frac{A(b)}{\\beta }+B\\ln (\\frac{2\\pi }{\\beta })+C(b)+o(\\beta ^0),\\quad \\quad (\\text{as$\\beta \\rightarrow 0$})$ with $A(b)=\\frac{16\\pi ^2}{3}(\\frac{b+b^{-1}}{2})(c-a-\\frac{3}{4}L_{h\\ min}),$ $B=\\mathrm {dim}\\mathfrak {h}_{qu},$ and $C(b)$ some real function of $b$ that we have not found a general expression for.",
"Based on various examples that we have looked at, it seems like whenever $L_{h\\ min}=0$ , the (complex-) dimension of the quantum Coulomb branch of the theory on $R^3\\times S^1$ coincides with $\\mathrm {dim}\\mathfrak {h}_{qu}$ .",
"Thus we can say the following.",
"For theories with positive semi-definite Rains function, the high-temperature expansion of $\\ln \\mathcal {I}(b,\\beta )$ encodes $i)$ in its order-$1/\\beta $ term the difference of the central charges $c-a$ , and $ii)$ in its order-$\\ln (1/\\beta )$ term the (complex-) dimension of the quantum Coulomb branch of the theory on $R^3\\times S^1$ .",
"Note that while we have proven item $i$ above, item $ii$ is only a conjecture based on various examples studied in [5].",
"Moving on to the subleading terms, the following statement was demonstrated in [5] for $C(b)$ .",
"(We define $Z_{S^3}(b):=Z_{S^3}(b;\\infty )$ ; see subsection REF for the definition of $Z_{S^3}(b;\\Lambda )$ .)",
"For theories whose Rains function is minimized only at the origin of $\\mathfrak {h}_{cl}$ (hence have $L_{h\\ min}=\\mathrm {dim}\\mathfrak {h}_{qu}=0$ ), the high-temperature expansion of $\\ln \\mathcal {I}(b,\\beta )$ encodes in its order-$\\beta ^0$ term the logarithm of the squashed three-sphere partition function $Z_{S^3}(b)$ of the dimensionally reduced theory; in other words, for the said theories $C(b)=\\ln Z_{S^3}(b)$ .",
"[The above statement was claimed in [5] to hold even for chiral theories; however, while for non-chiral theories it is straightforward to show $Z_{S^3}(b)\\ne 0$ , for chiral theories we have not been able to show that $Z_{S^3}(b)$ is non-zero; we thus emphasize that the above statement is demonstrated in [5] for chiral theories assuming $Z_{S^3}(b)\\ne 0$ .]",
"Although we have not been able to make general statements about the $o(\\beta ^0)$ terms on the RHS of (REF ), based on the examples studied in [5] it seems that $\\ln \\mathcal {I}(b,\\beta )=\\frac{A(b)}{\\beta }+B\\ln (\\frac{2\\pi }{\\beta })+C(b)+D(b)\\beta +O(\\beta ^2)\\quad \\quad (\\text{as $\\beta \\rightarrow 0$}).$ For theories whose Rains function is minimized on a set of isolated points, the above asymptotics can actually be demonstrated (with $B=0$ , of course); it can moreover be shown that the error term is not just $O(\\beta ^2)$ , but beyond all orders (and of the type $e^{-1/\\beta }$ ) [5].",
"Furthermore, in those theories $D(b)$ coincides with the SUSY Casimir energyThe SUSY Casimir energy relates the superconformal index $\\mathcal {I}(b,\\beta )$ to its corresponding partition function $Z^{\\mathrm {SUSY}}(b,\\beta )$ computed via path-integration on $S_b^3\\times S^1_\\beta $ [23], [25]: $Z^{\\mathrm {SUSY}}(b,\\beta )=e^{-\\beta E_{\\mathrm {susy}}(b)}\\mathcal {I}(b,\\beta )$ .",
"(encountered also in (REF ) above) $\\begin{split}E_{\\mathrm {susy}}(b)=\\frac{2}{27}(b+b^{-1})^3(3c-2a)+\\frac{2}{3}(b+b^{-1})(a-c).\\end{split}$ Therefore we can say the following [5].",
"For theories whose Rains function is minimized on a set of isolated points in $\\mathfrak {h}_{cl}$ , the high-temperature expansion of $\\ln \\mathcal {I}(b,\\beta )$ takes the form shown in (REF ), with $B=0$ , and with the error being not only $O(\\beta ^2)$ but also exponentially small.",
"Moreover, the order-$\\beta $ term encodes the SUSY Casimir energy; in other words, for the said theories $D(b)=E_{\\mathrm {susy}}(b)$ .",
"The above statement implies that (whenever the Rains function is minimized on a set of isolated points) the central charges $a$ and $c$ —and hence the `t Hooft anomalies $\\mathrm {Tr}R$ and $\\mathrm {Tr}R^3$ —are both encoded in the order-$\\beta $ term in the high-temperature expansion of $\\ln \\mathcal {I}(b,\\beta )$ .",
"It can actually be shown that introducing flavor fugacities $u_a=e^{i\\beta m_a}$ in the superconformal index, the relation $D(b)=E_{\\mathrm {susy}}(b)$ generalizes to $D(b;m_a)=E_{\\mathrm {susy}}(b;m_a)$ , with $E_{\\mathrm {susy}}(b;m_a)$ the equivariant SUSY Casimir energy (which encodes all the `t Hooft anomalies in the theory [56]); thus (whenever the Rains function is minimized on a set of isolated points) all the `t Hooft anomalies are encoded in the order-$\\beta $ term in the high-temperature expansion of $\\ln \\mathcal {I}(b,\\beta ;m_a)$ .",
"This statement is related (but not equivalent) to some of the claims in [28], which were made there in the context of SU($N$ ) SQCD.",
"It was shown in [23] that for SU($N$ ) toric quiver SCFTs (see appendix REF for the definition of the symbol $\\sim $ used below) $\\begin{aligned} \\ln \\mathcal {I}^{N\\rightarrow \\infty }_{quiver}(b,\\beta )\\sim &\\,\\frac{\\pi ^2}{6\\beta (\\frac{b+b^{-1}}{2})}\\sum _{i=1}^{n_z}\\frac{1}{r_i}+\\frac{16\\pi ^2(\\frac{b+b^{-1}}{2})}{3\\beta }\\sum _{adj}(\\delta c_{adj}-\\delta a_{adj})+\\frac{n_z}{2}\\ln (\\beta /2\\pi )+\\ln Y_b \\\\&\\,+\\beta \\left(\\frac{2}{27}(b+b^{-1})^3(3\\delta c-2\\delta a)+\\frac{2}{3}(b+b^{-1})(\\delta a-\\delta c)\\right),\\end{aligned}$ where the notation is similar to that in (REF ), except for $\\ln Y_b=\\frac{1}{2}\\sum _{i=1}^{n_z}\\ln (r_i(\\frac{b+b^{-1}}{2}))-\\sum _{adj}\\ln \\Gamma _h(iR_{adj}(\\frac{b+b^{-1}}{2}))$ , with $\\Gamma _h(\\ast )$ a special function explained in appendix REF .",
"Based on various specific examples, it was conjectured in [21], [23] that $\\sum _{i=1}^{n_z}\\frac{1}{r_i}=\\frac{3}{16\\pi ^3}\\left(19\\mathrm {vol}(SE)+\\frac{1}{8}\\mathrm {Riem}^2(SE)\\right),$ where $SE$ denotes the Sasaki-Einstein 5-manifold dual to the quiver gauge theory.",
"The above conjecture was motivated by the finding in [57] that one can “hear the shape of the dual geometry” in the asymptotics of the Hilbert series of mesonic operators in the SCFT.",
"We note that the leading high-temperature behavior of the index of the quivers is contained in the first two terms of (REF ).",
"The first term, according to (REF ), is dictated by the geometry of the dual internal manifold, while the second is given by the $O(1)$ part of the contribution of adjoint matter to $c-a$ .",
"The latter is hence the only part of the finite-$N$ Di Pietro-Komargodski formula that escapes metamorphosis into “geometry” in the planar limit.",
"Interestingly, the order-$\\beta $ term on the RHS of (REF ) is $\\beta $ times $\\delta E_{\\mathrm {susy}}(b)$ , where by the latter we mean $E_{\\mathrm {susy}}(b)$ as in (REF ) but with the central charges in it replaced with their $O(N^0)$ pieces.",
"Therefore the order-$\\beta $ term in the high-temperature expansion of $\\ln \\mathcal {I}^{N\\rightarrow \\infty }_{quiver}(b,\\beta )$ is somewhat similar to the corresponding term for the finite-$N$ non-chiral theories whose Rains function is minimized on a set of isolated points (see the previous subsection).",
"Note that the discussion below (REF ) (combined with relation (REF )) implies that the holographic computation of the subleading central charges can be thought of as extracting $\\delta c$ and $\\delta a$ from the order-$\\beta $ term of the high-temperature expansion of $\\ln \\mathcal {I}^{N\\rightarrow \\infty }(b,\\beta )$ .",
"In the case of the toric quivers, the holographic prescriptions in (REF ) thus extract $\\delta c$ and $\\delta a$ from $\\delta E_{\\mathrm {susy}}(b)$ .",
"The main result of this dissertation is the high-temperature asymptotics of the EHIs arising as the superconformal index of unitary non-chiral 4d Lagrangian SCFTs.",
"The most important extension of our work would be to chiral SCFTs; the preliminary investigation of [5] seems to indicate that the extension would not be straightforward.",
"A particularly interesting outcome of our work has been the connection between the high-temperature asymptotics of the index, and the Coulomb branch dynamics on $R^3\\times S^1$ ; see subsection REF .",
"This is undoubtedly worth pursuing more carefully.",
"Even before aiming at establishing the connection in a general context, it would be nice to validate it by examining the Coulomb branch dynamics of the $A_k$ SQCD (with SU($N$ ) gauge group) and the $Z_2$ orbifold theory (with SU($N$ )$\\times $ SU($N$ ) gauge group), to see if the (complex-) dimension of their quantum Coulomb branch on $R^3\\times S^1$ coincides with their $\\mathrm {dim}\\mathfrak {h}_{qu}$ , which we have found to be respectively zero and $N-1$ .",
"tocchapter The Pochhammer symbol ($|q|\\in ]0,1[$ ) $(a;q):=\\prod _{k=0}^{\\infty }(1-a q^k),$ is related to the more familiar Dedekind eta function via $\\eta (\\tau )=q^{1/24}(q;q), $ with $q=e^{2\\pi i\\tau }.$ The eta function has an SL$(2,\\mathbb {Z})$ modular property that will be useful for us: $\\eta (-1/\\tau )=\\sqrt{-i\\tau }\\eta (\\tau )$ .",
"The Pochhammer symbol $(q;q)$ equals the inverse of the generating function of integer partitions.",
"It also appears in the index of 4d SUSY gauge theories that contain vector multiplets.",
"The elliptic gamma function is defined as ($\\mathrm {Im}(\\tau ),\\mathrm {Im}(\\sigma ) >0$ ) $\\Gamma (x;\\sigma ,\\tau ):=\\prod _{j,k\\ge 0}\\frac{1-z^{-1}p^{j+1}q^{k+1}}{1-z p^{j}q^{k}},$ with $z:=e^{2\\pi i x}$ , $p:=e^{2\\pi i \\sigma }=e^{-\\beta b}$ , and $q:=e^{2\\pi i \\tau }=e^{-\\beta b^{-1}}$ .",
"The above expression gives a meromorphic function of $x\\in \\mathbb {C}$ .",
"For generic choice of $\\tau $ and $\\sigma $ , the elliptic gamma has simple poles at $x=l-m\\sigma -n\\tau $ , with $m,n\\in \\mathbb {Z}^{\\ge 0}$ , $l\\in \\mathbb {Z}$ .",
"We sometimes write $\\Gamma (x;\\sigma ,\\tau )$ as $\\Gamma (z;p,q)$ , or simply as $\\Gamma (z)$ .",
"Also, the arguments of elliptic gamma functions are frequently written with “ambiguous” signs (as in $\\Gamma (\\pm x;\\sigma ,\\tau )$ ); by that one means a multiplication of several gamma functions each with a “possible” sign of the argument (as in $\\Gamma (+x;\\sigma ,\\tau )\\times \\Gamma (-x;\\sigma ,\\tau )$ ).",
"Similarly $\\Gamma (z^{\\pm 1}):=\\Gamma (z;p,q)\\times \\Gamma (z^{-1};p,q)$ .",
"The elliptic gamma function appears in the exact solution of some important 2d integrable lattice models.",
"It also features in the index of 4d Lagrangian SUSY QFTs that contain chiral multiplets.",
"Following Rains [10], we define the hyperbolic gamma function by $\\Gamma _h(x;\\omega _1,\\omega _2):=\\exp \\left(\\mathrm {PV}\\int _{\\mathbb {R}}\\frac{e^{2\\pi i x w}}{(e^{2\\pi i\\omega _1 w}-1)(e^{2\\pi i\\omega _2w}-1)}\\frac{\\mathrm {d}w}{w}\\right).$ The above expression makes sense only for $0<\\mathrm {Im}(x)<2\\mathrm {Im}(\\omega )$ , with $\\omega :=(\\omega _1+\\omega _2)/2$ .",
"In that domain, the function defined by (REF ) satisfies $\\Gamma _h(x+\\omega _2;\\omega _1,\\omega _2)=2\\sin (\\frac{\\pi x}{\\omega _1})\\Gamma _h(x;\\omega _1,\\omega _2).$ This relation can then be used for an inductive meromorphic continuation of the hyperbolic gamma function to all $x\\in \\mathbb {C}$ .",
"For generic $\\omega _1,\\omega _2$ in the upper half plane, the resulting meromorphic function $\\Gamma _h(x;\\omega _1,\\omega _2)$ has simple zeros at $x=\\omega _1\\mathbb {Z}^{\\ge 1}+\\omega _2\\mathbb {Z}^{\\ge 1}$ and simple poles at $x=\\omega _1\\mathbb {Z}^{\\le 0}+\\omega _2\\mathbb {Z}^{\\le 0}$ .",
"For convenience, we will frequently write $\\Gamma _h(x)$ instead of $\\Gamma _h(x;\\omega _1,\\omega _2)$ , and $\\Gamma _h(x\\pm y)$ instead of $\\Gamma _h(x+y)\\Gamma _h(x-y)$ .",
"The hyperbolic gamma function has an important property that can be easily derived from the definition (REF ): $\\Gamma _h(-\\mathrm {Re}(x)+i\\mathrm {Im}(x);\\omega _1,\\omega _2)=(\\Gamma _h(\\mathrm {Re}(x)+i\\mathrm {Im}(x);\\omega _1,\\omega _2))^\\ast ,$ with $\\ast $ denoting complex conjugation.",
"We also define the non-compact quantum dilogarithm $\\psi _b$ (c.f.",
"the function $e_b(x)$ in [58]; $\\psi _b(x)=e_b(-i x)$ ) via $\\psi _b(x):=e^{-i\\pi x^2/2+i\\pi (b^2+b^{-2})/24}\\Gamma _h(ix+\\omega ;\\omega _1,\\omega _2),$ where $\\omega _1:=i b,\\quad \\omega _2:=i b^{-1},\\quad \\text{and}\\quad \\omega :=(\\omega _1+\\omega _2)/2.$ For generic choice of $b$ , the zeros of $\\psi _b(x)^{\\pm 1}$ are of first order, and lie at $\\pm ((b+b^{-1})/2+b\\mathbb {Z}^{\\ge 0}+b^{-1}\\mathbb {Z}^{\\ge 0})$ .",
"Upon setting $b=1$ we get the function $\\psi (x)$ of [59]; i.e.",
"$\\psi _{b=1}(x)=\\psi (x)$ .",
"An identity due to Narukawa [60] implies the following important relation between $\\psi _{b}(x)$ and the elliptic gamma function (see also Appendix A of [23]) $\\begin{split}\\Gamma (z;\\sigma ,\\tau )&=\\frac{e^{2i\\pi Q_{-}(x;\\sigma ,\\tau )}}{\\psi _b(\\frac{2\\pi ix}{\\beta }+\\frac{b+b^{-1}}{2})}\\prod _{n=1}^{\\infty }\\frac{\\psi _b(-\\frac{2\\pi in}{\\beta }-\\frac{2\\pi ix}{\\beta }-\\frac{b+b^{-1}}{2})}{\\psi _b(-\\frac{2\\pi in}{\\beta }+\\frac{2\\pi ix}{\\beta }+\\frac{b+b^{-1}}{2})}\\\\&=e^{2i\\pi Q_{+}(x;\\sigma ,\\tau )}\\psi _b(\\text{\\footnotesize {$-\\frac{2\\pi ix}{\\beta }-\\frac{b+b^{-1}}{2}$}})\\prod _{n=1}^{\\infty }\\frac{\\psi _b(-\\frac{2\\pi in}{\\beta }-\\frac{2\\pi ix}{\\beta }-\\frac{b+b^{-1}}{2})}{\\psi _b(-\\frac{2\\pi in}{\\beta }+\\frac{2\\pi ix}{\\beta }+\\frac{b+b^{-1}}{2})},\\end{split}$ where $\\begin{split}Q_{-}(x;\\sigma ,\\tau )=&-\\frac{x^3}{6\\tau \\sigma }+\\frac{\\tau +\\sigma -1}{4\\tau \\sigma }x^2-\\frac{\\tau ^2+\\sigma ^2+3\\tau \\sigma -3\\tau -3\\sigma +1}{12\\tau \\sigma }x\\\\&-\\frac{1}{24}(\\tau +\\sigma -1)(\\tau ^{-1}+\\sigma ^{-1}-1),\\\\Q_{+}(x;\\sigma ,\\tau )=&Q_{-}(x;\\sigma ,\\tau )+(x-\\frac{\\tau +\\sigma }{2})^2/2\\tau \\sigma -(\\tau ^2+\\sigma ^2)/24\\tau \\sigma ,\\end{split}$ We say $f(\\beta )=O(g(\\beta ))$ as $\\beta \\rightarrow 0$ , if there exist positive real numbers $C,\\beta _0$ such that for all $\\beta <\\beta _0$ we have $|f(\\beta )|< C|g(\\beta )|$ .",
"We say $f(x,\\beta )=O(g(x,\\beta ))$ uniformly over $S$ as $\\beta \\rightarrow 0$ , if there exist positive real numbers $C,\\beta _0$ such that for all $\\beta <\\beta _0$ and all $x\\in S$ we have $|f(x,\\beta )|< C|g(x,\\beta )|$ .",
"We will write $f(\\beta )=o(g(\\beta ))$ , if $f(\\beta )/g(\\beta )\\rightarrow 0$ as $\\beta \\rightarrow 0$ .",
"We use the symbol $\\sim $ when writing the all-orders asymptotics of a function.",
"For example, we have $\\ln (\\beta +e^{-1/\\beta })\\sim \\ln \\beta ,\\quad \\quad (\\text{as $\\beta \\rightarrow 0$})$ because we can write the LHS as the sum of $\\ln \\beta $ and $\\ln (1+e^{-1/\\beta }/\\beta )$ , and the latter is beyond all-orders in $\\beta $ .",
"More precisely, we say $f(\\beta )\\sim g(\\beta )$ as $\\beta \\rightarrow 0$ , if we have $f(\\beta )- g(\\beta )=O(\\beta ^n)$ for any (arbitrarily large) natural $n$ .",
"We will write $f(\\beta )\\simeq g(\\beta )$ if $\\ln f(\\beta )\\sim \\ln g(\\beta )$ (with an appropriate choice of branch for the logarithms).",
"By writing $f(x,\\beta )\\simeq g(x,\\beta )$ we mean that $\\ln f(x,\\beta )\\sim \\ln g(x,\\beta )$ for all $x$ on which $f(x,\\beta ),g(x,\\beta )\\ne 0$ , and that $f(x,\\beta )= g(x,\\beta )=0$ for all $x$ on which either $f(x,\\beta )=0$ or $g(x,\\beta )=0$ .",
"With the above notations at hand, we can asymptotically analyze the Pochhammer symbol as follows.",
"The low-temperature ($T\\rightarrow 0$ , with $q=e^{-1/T}$ ) behavior is trivial: $(q;q)\\simeq 1\\quad \\quad (\\text{as $1/\\beta \\rightarrow 0$}).$ The high-temperature ($\\beta \\rightarrow 0$ , with $q=e^{-\\beta }$ ) asymptotics is nontrivial.",
"It can be obtained using the SL($2,\\mathbb {Z}$ ) modular property of the eta function, which yields $\\ln \\eta (\\tau =\\frac{i\\beta }{2\\pi })\\sim -\\frac{\\pi ^2}{6\\beta }+\\frac{1}{2}\\ln (\\frac{2\\pi }{\\beta })\\quad \\quad (\\text{as$\\beta \\rightarrow 0$}).$ The above relation, when combined with (REF ), implies $\\ln (q;q)\\sim -\\frac{\\pi ^2}{6\\beta }+\\frac{1}{2}\\ln (\\frac{2\\pi }{\\beta })+\\frac{\\beta }{24}\\quad \\quad (\\text{as$\\beta \\rightarrow 0$}).\\\\$ For the hyperbolic gamma function, Corollary 2.3 of [10] implies that when $x\\in \\mathbb {R}$ $\\begin{split}\\ln \\Gamma _h(x+r\\omega ;\\omega _1,\\omega _2)= -\\frac{i\\pi }{2} x|x|-i\\pi (r-1)\\omega |x|+O(1), \\quad \\quad (\\text{as$|x|\\rightarrow \\infty $})\\end{split}$ for any fixed real $r$ , and fixed $b>0$ .",
"From the asymptotics of the hyperbolic gamma function, it follows that for fixed $\\mathrm {Re}(x)$ and fixed $b>0$ $\\ln \\psi _b(x)\\sim 0,\\quad \\quad \\quad (\\text{as } \\beta \\rightarrow 0, \\text{for } \\mathrm {Im}(x)= -1/\\beta )$ with a transcendentally small error, of the type $e^{-1/\\beta }$ .",
"The above estimate can be combined with (REF ) to yield the small-$\\beta $ estimates $\\begin{split}\\Gamma (x;\\sigma ,\\tau )&\\simeq \\frac{e^{2i\\pi Q_{-}(x;\\sigma ,\\tau )}}{\\psi _b(\\frac{2\\pi ix}{\\beta }+\\frac{b+b^{-1}}{2})},\\quad \\quad (\\text{for } -1<\\mathrm {Re}(x)\\le 0 )\\\\&\\simeq e^{2i\\pi Q_{+}(x;\\sigma ,\\tau )}\\psi _b(\\text{\\footnotesize {$-\\frac{2\\pi ix}{\\beta }-\\frac{b+b^{-1}}{2}$}}),\\quad \\quad (\\text{for }0\\le \\mathrm {Re}(x)<1 ) \\end{split}$ with the range of $\\mathrm {Re}(x)$ explaining our subscript notations for $Q_+$ and $Q_-$ .",
"As a result of (REF ) we have for $x\\in \\mathbb {R}$ , as $\\beta \\rightarrow 0$ : $\\begin{split}\\Gamma (-x+(\\frac{\\tau +\\sigma }{2})r;\\sigma ,\\tau )&\\simeq \\frac{e^{2i\\pi Q_{-}(-\\lbrace x\\rbrace +(\\frac{\\tau +\\sigma }{2})r;\\sigma ,\\tau )}}{\\psi _b(-\\frac{2\\pi i \\lbrace x\\rbrace }{\\beta }-(r-1)\\frac{b+b^{-1}}{2})},\\\\\\Gamma (x+(\\frac{\\tau +\\sigma }{2})r;\\sigma ,\\tau )&\\simeq e^{2i\\pi Q_{+}(\\lbrace x\\rbrace +(\\frac{\\tau +\\sigma }{2})r;\\sigma ,\\tau )}\\psi _b(\\text{\\footnotesize {$-\\frac{2\\pi i \\lbrace x\\rbrace }{\\beta }+(r-1)\\frac{b+b^{-1}}{2}$}}),\\end{split}$ with $\\lbrace x\\rbrace :=x-\\lfloor x\\rfloor $ .",
"The above estimates are first obtained in the range $0\\le x<1$ , and then extended to $x\\in \\mathbb {R}$ using the periodicity of the LHS under $x\\rightarrow x+1$ .",
"Define $\\vartheta (x):=\\lbrace x\\rbrace (1-\\lbrace x\\rbrace )$ .",
"The Lemma 3.2 of [10] says that for any sequence of real numbers $c_1,\\dots , c_n$ , $d_1,\\dots , d_n$ , the following inequality holds: $\\sum _{1\\le i,j\\le n}\\vartheta (c_i-d_j)-\\sum _{1\\le i<j\\le n}\\vartheta (c_i-c_j)-\\sum _{1\\le i<j\\le n}\\vartheta (d_i-d_j)\\ge \\vartheta (\\sum _{1\\le i\\le n}(c_i-d_i)),$ with equality iff the sequence can be permuted so that either $\\lbrace c_1\\rbrace \\le \\lbrace d_1\\rbrace \\le \\lbrace c_2\\rbrace \\le \\cdots \\le \\lbrace d_{n-1}\\rbrace \\le \\lbrace c_n\\rbrace \\le \\lbrace d_n\\rbrace ,$ or $\\lbrace d_1\\rbrace \\le \\lbrace c_1\\rbrace \\le \\lbrace d_2\\rbrace \\le \\cdots \\le \\lbrace c_{n-1}\\rbrace \\le \\lbrace d_n\\rbrace \\le \\lbrace c_n\\rbrace .$ The proof can be found in [10].",
"Re-scaling with $c_i,d_i\\mapsto vc_i,vd_i$ , taking $v\\rightarrow 0^+$ , and using the relation $\\vartheta (vx)=v|x|-v^2 x^2$ (which holds for small enough $v$ ), Rains obtains the following corollary of (REF ): $\\sum _{1\\le i,j\\le n}|c_i-d_j|-\\sum _{1\\le i<j\\le n}|c_i-c_j|-\\sum _{1\\le i<j\\le n}|d_i-d_j|\\ge |\\sum _{1\\le i\\le n}(c_i-d_i)|,$ with equality iff the sequence can be permuted so that either $c_1\\le d_1 \\le c_2 \\le \\cdots \\le d_{n-1} \\le c_n \\le d_n,$ or $d_1 \\le c_1 \\le d_2 \\le \\cdots \\le c_{n-1} \\le d_n \\le c_n.$ The fact that the inequality (REF ) arise as a corollary of (REF ) justifies the name “generalized triangle inequality” for the latter.",
"Various generalized triangle inequalities (GTIs) allow us to analytically address the minimization problems for the piecewise linear functions $L_h$ arising in Chapter REF .",
"In several physically interesting cases, the required GTI is a corollary of Rains's GTI shown in (REF ) above.",
"The universal property of large-$N$ SCFTs that allows a systematic study of their high-temperature asymptotics is the large-$N$ factorization.",
"The factorization implies that the index of large-$N$ theories is conveniently expressed in terms of the single-trace index as $\\ln \\mathcal {I}^{N\\rightarrow \\infty }(\\beta ,b)=\\sum _{n=1}^{\\infty }\\frac{1}{n}I_{s.t.",
"}(n\\beta ,b).$ Let us now review a useful technique in asymptotic analysis, which we will find useful when studying large-$N$ indices expressible as in (REF ).",
"Say we are interested in the small-$\\beta $ asymptotics of a real function $F(\\beta )$ that can be written in the form $F(\\beta )=\\sum _{m=1}^{\\infty }f(m\\beta ),$ with $f(\\beta )$ a real function having the $\\beta \\rightarrow 0$ asymptotic development $f(\\beta )\\sim \\sum _{\\lambda \\ge -1}^{\\infty }b_{\\lambda }\\beta ^{\\lambda }.$ Assume moreover that $f(\\beta )$ and all its derivatives decay faster than $1/\\beta ^{1+\\varepsilon }$ as $\\beta \\rightarrow \\infty $ , for some $\\varepsilon >0$ .",
"Then, according to Zagier [61], the $\\beta \\rightarrow 0$ asymptotics of $F(\\beta )$ is given by $F(\\beta )\\sim \\frac{1}{\\beta }\\left(b_{-1}\\ln (\\frac{1}{\\beta })+I^{\\ast }_f\\right)+\\sum _{\\lambda >-1}b_\\lambda \\zeta (-\\lambda )\\ \\beta ^{\\lambda },$ with $I^{\\ast }_{f}:=\\int _{0}^{\\infty }(f(x)-b_{-1}e^{-x}/x)\\mathrm {d}x$ .",
"Equation (REF ) has a remarkable resemblance to the sums to which Zagier's method applies.",
"In fact, dividing both sides of (REF ) by $\\beta $ , we arrive at $\\frac{\\ln \\mathcal {I}^{N\\rightarrow \\infty }(\\beta )}{\\beta }=\\sum _{n=1}^{\\infty }f(n\\beta ),$ with $f(\\beta )=I_{s.t.",
"}(\\beta )/\\beta $ .",
"In all the examples we are aware of, $I_{s.t.",
"}(\\beta )$ has a leading asymptotics of the form $I_{-1}/\\beta $ .",
"Therefore $f(\\beta )$ defined above has a leading asymptotics of the form $I_{-1}/\\beta ^2$ , and thus Zagier's formula (REF ) does not immediately apply to it.",
"However, defining $\\tilde{f}(\\beta ):=f(\\beta )-I_{-1}/\\beta ^2$ , we obtain $\\frac{\\ln \\mathcal {I}^{N\\rightarrow \\infty }(\\beta )}{\\beta }=\\frac{\\pi ^2}{6\\beta ^2}I_{-1}+\\sum _{n=1}^{\\infty }\\tilde{f}(n\\beta ),$ and now Zagier's method can be applied to find the asymptotics of the sum on the RHS of the above relation.",
"The result is $\\frac{\\ln \\mathcal {I}^{N\\rightarrow \\infty }(\\beta )}{\\beta }\\sim \\frac{\\pi ^2}{6\\beta ^2}I_{-1}+\\frac{1}{\\beta }(I_0\\ln (\\frac{1}{\\beta })+I^{\\ast }_{\\tilde{f}})+\\sum _{m=0}^{\\infty }\\tilde{f}_m\\zeta (-m)\\beta ^{m},$ where $I^{\\ast }_{\\tilde{f}}:=\\int _0^{\\infty }(\\frac{I_{s.t.",
"}(x)-I_{-1}/x}{x}-I_0\\frac{e^{-x}}{x})\\mathrm {d}x$ , and $\\tilde{f}_m$ is the coefficient of $\\beta ^m$ in the asymptotics of $\\tilde{f}(\\beta )$ .",
"Also $I_{0}$ is the $\\beta $ -independent term in the asymptotic expansion of $I_{s.t.",
"}$ .",
"Let $I_{n}$ be the coefficient of $\\beta ^n$ in the asymptotics of the single-trace index $I_{s.t.",
"}$ .",
"From $\\tilde{f}(\\beta )=I_{s.t.",
"}(\\beta )/\\beta -I_{-1}/\\beta ^2,$ we obtain $\\tilde{f}_n=I_{n+1}$ for $n=0,1,\\dots $ .",
"Therefore we can write (REF ) as $\\ln \\mathcal {I}^{N\\rightarrow \\infty }(\\beta )\\sim \\frac{\\pi ^2}{6\\beta }I_{-1}+I_0\\ln (\\frac{1}{\\beta })+I^{\\ast }_{\\tilde{f}}+\\sum _{m=1}^{\\infty }I_{m}\\zeta (-m+1)\\beta ^{m}.$ This relation is the main result of this appendix.",
"It expresses the all-orders small-$\\beta $ asymptotics of $\\ln \\mathcal {I}^{N\\rightarrow \\infty }$ in terms of data that can be found from the single-trace index.",
"As an application of the result (REF ), we derive the ansatz given in [23] for the asymptotics of the index of the $A_k$ SQCD in the Veneziano limit.",
"The single-trace index of the $A_k$ SQCD in the Veneziano limit is given by [21] $I^{A_k}_{s.t.}",
"=\\frac{\\tau ^{-\\frac{2}{k+1}}}{1-\\tau ^{-\\frac{2}{k+1}}} +\\frac{\\tau ^{-\\frac{4k}{k+1}}}{1-\\tau ^{-\\frac{4k}{k+1}}} -\\frac{\\tau ^{-\\frac{2k}{k+1}}}{1-\\tau ^{-\\frac{2k}{k+1}}} - \\frac{\\big (\\tau ^{-\\frac{2}{k+1}} - \\tau ^{-\\frac{2k}{k+1}}\\big ) - N_f^2\\frac{\\big (\\tau ^{\\frac{2N_c}{(k+1)N_f}} -\\tau ^{-\\frac{2N_c}{(k+1)N_f}}\\big )^2}{\\tau ^2\\big (1-\\tau ^{-\\frac{2}{k+1}}\\big )\\big (1+\\tau ^{-\\frac{2k}{k+1}}\\big )}}{(1-p)(1-q)},$ where $\\tau :=(pq)^{-1/2}$ .",
"Expanding $I^{A_k}_{s.t.",
"}$ at high temperatures we find a series of the form $I^{A_k}_{s.t.",
"}(\\beta ) =\\frac{I_{-1}}{\\beta }+I_0+\\sum _{m\\ odd>0}I_m\\beta ^m.$ Note that no positive even powers of $\\beta $ show up in the expansion.",
"This is because $I^{A_k}_{s.t.",
"}(\\beta )$ is “almost” an odd function of $\\beta $ : one can directly check from (REF ) that $I^{A_k}_{s.t.",
"}(\\beta )+I^{A_k}_{s.t.",
"}(-\\beta )=-1$ .",
"Plugging (REF ) in (REF ) we find that $\\ln \\mathcal {I}^{N\\rightarrow \\infty }_{A_k}(\\beta )\\sim \\frac{\\pi ^2}{\\beta }I_{-1}+I_0\\ln (\\frac{1}{\\beta })+I^{\\ast }_{\\tilde{f}}+I_1\\beta .$ Using the actual values $\\begin{split}I_{-1}&=\\frac{2k^3+3k^2-1}{4k(1+k)}\\left(\\frac{1}{(\\frac{b+b^{-1}}{2})}\\right)+\\frac{16kN_c^2-8k^2+8k}{4k(1+k)}\\left(\\frac{b+b^{-1}}{2}\\right)\\\\I_0&=-\\frac{1}{2}\\\\I_1&=-\\left(\\frac{4}{27}(b+b^{-1})^3(3c-2a)+\\frac{4}{3}(b+b^{-1})(a-c)\\right).\\end{split}$ we obtain $\\begin{split}\\ln \\mathcal {I}^{N_c\\rightarrow \\infty }_{A_k}(\\beta ,b)\\sim &\\frac{2k^3+3k^2-1}{4k(1+k)}\\left(\\frac{\\pi ^2}{6\\beta (\\frac{b+b^{-1}}{2})}\\right)+\\frac{16kN_c^2-8k^2+8k}{4k(1+k)}\\left(\\frac{\\pi ^2(\\frac{b+b^{-1}}{2})}{6\\beta }\\right)\\\\&-\\frac{1}{2}\\ln (\\frac{1}{\\beta })+I^{\\ast }_{\\tilde{f}}(b)+\\beta \\left(\\frac{2}{27}(b+b^{-1})^3(3c-2a)+\\frac{2}{3}(b+b^{-1})(a-c)\\right).\\end{split}$ This is the ansatz of [23], now rigorously derived, and supplemented with the $\\beta $ -independent term $I^{\\ast }_{\\tilde{f}}(b)$ which was left undetermined in that work.",
"tocchapterBibliography"
]
] | 1605.06100 |
[
[
"Towards information based spatiotemporal patterns as a foundation for\n agent representation in dynamical systems"
],
[
"Abstract We present some arguments why existing methods for representing agents fall short in applications crucial to artificial life.",
"Using a thought experiment involving a fictitious dynamical systems model of the biosphere we argue that the metabolism, motility, and the concept of counterfactual variation should be compatible with any agent representation in dynamical systems.",
"We then propose an information-theoretic notion of \\emph{integrated spatiotemporal patterns} which we believe can serve as the basic building block of an agent definition.",
"We argue that these patterns are capable of solving the problems mentioned before.",
"We also test this in some preliminary experiments."
],
[
"Introduction",
"Within artificial life the concept of an agent is fundamental.",
"While studying life-as-it-could-be [18], we also study agents-as-they-could-be.",
"An intuitive approach to agents is possibly to say that while not reproducing, i.e.",
"during their individual lifetime, living organisms are agents.",
"The concept of an agent in this way generalizes the concept of living organisms by de-emphasizing reproduction and with it Darwinian evolution.",
"This point of view is also in line with the common practice of referring to robots or software programs as agents.",
"To give some more background [5], there are a few properties that seem universally acknowledged as necessary for something to be referred to as an agent.",
"The first of those is probably the capacity to act [23].",
"However, [5] notice that this already presupposes a form of individuality i.e.",
"an “entity” that this capacity can be attributed to.",
"Consequently they put the individuality criterion first.",
"Having perception is another fairly uncontroversial requirement [22].",
"The last concept which is often alluded to is that of some form of goal-directedness of the agent.",
"The goals agents should strive to achieve are usually required to be in the agents' own interest/intrinsic (e.g.",
"preservation) and not the goals of some other agent (or programmer).",
"For a thorough treatment on the latter point see [14].",
"We broadly agree on the three (or four) main requirements of individuality, perception and action, as well as goal-directedness.",
"However we are not satisfied with the lack of formal definitions of the notions themselves.",
"We therefore take a different and particularly formal approach to the problem of defining agents.",
"From the start we limit ourselves to a mathematically well-defined class of systems i.e.",
"dynamical systems and their generalization to stochastic processes (we will refer to dynamical systems only, inclusion of stochastic processes is implied).",
"We want to define agents as entities that can exist within a dynamical system.",
"In other words, we are looking for a representation of agents within dynamical systems.",
"While there is no guarantee that such a representation even exists, we believe that even if we fail, there might be some insights into why we fail.",
"This would also help the community to understand the concept of agents better.",
"At the same time we expect that dynamical systems are actually a powerful enough class of systems to consider and that they will turn out to be able to contain convincing examples of agents.",
"This optimism stems from the fact that dynamical systems have been extremely successful in modeling systems from physics through chemistry to biology.",
"Compelling recent examples of dynamical systems which directly suggest they can contain agents can be found in [28], [6].",
"If we are successful, then we would obtain a definition of agents as features of dynamical systems and eventually even of life as a feature of such systems.",
"This would be a step towards defining life as a natural kind as required by [9].",
"Finally our hope is to reveal the formal counterparts of the intuitions about living systems formulated by [20].",
"In order to make it more clear what we mean by agents within a dynamical system, consider the following example, to which we will come back throughout this paper.",
"Say we had a dynamical system that is a sufficiently exact approximation of the entire biosphere including the influence of incoming (from the sun) and outgoing radiation.",
"During individual runs of this dynamical system, given the right initial conditions, things should occur that correspond to living organisms in the real biosphere.",
"In this case we would say that within this dynamical system agents occur.",
"Our goal is to find a mathematical representation of these agents.",
"Since agents are a generalization of living organisms, we expect that agent representations can at least in principle exhibit the full range of phenomena exhibited by living organisms.",
"Limitations should only be due to the chosen dynamical system and not inherent to the agent representation.",
"This paper is a contribution to the discussion of the foundations of artificial life.",
"It does not present a solution of how to represent agents in dynamical systems.",
"Rather it defines a notion that can identify intrinsically distinguished spatiotemporal patterns that we believe can act as the basic building block on which a theory of agents can be built.",
"The strategy we have in mind here is the following.",
"First, define the spatiotemporal patterns which are suitable to represent both living (bacteria, animals, plants) and non-living (rocks, crystals) persistent objects.",
"Then further classify those patterns into classes exhibiting features of agents such as perception, action and goal-directedness.",
"Spatiotemporal patterns that satisfy all criteria will represent agents.",
"Also note that for the formal definition we here restrict ourselves to finite discrete-time distributedDistributed means that the state of the system is given as a set of values of multiple variables or degrees of freedom.",
"dynamical systems with an already given “space-like” and “time-like” structure.",
"Examples of this include cellular automata.",
"The restriction to finiteness is due to the improved clarity this choice brings with it.",
"The notions we present are well-defined in various more general settings.",
"However, currently the spatiotemporal-like structure seems necessary to us.",
"The rest of this paper is structured as follows.",
"The next section will present three challenges to representations of agents in distributed dynamical systems.",
"Then we look at the literature and discuss ways to represent agents formally and in how far they succeed or fail to meet our expectations.",
"We will then quickly introduce the setting of distributed dynamical systems and formally introduce a notion that we believe is able to identify the spatiotemporal patterns.",
"We give the intuition behind this notion and discuss it in the light of the three requirements mentioned before.",
"Finally, we present some preliminary results in the setting of the game of life.",
"As mentioned before we expect the agent representation to be able to deal with all features associated to living organisms in the biosphere.",
"Two such features, their metabolism and their motility present a major challenge to the representation of agents.",
"These two features both make it hard from a formal standpoint to “keep track” of the living organism within a trajectory of the system.",
"A third feature, we call it counterfactual variation, that we attribute to the biosphere makes it hard to represent agents reliably across different initial conditions.",
"This list of three features makes no claim to be complete, obtaining a complete list is ongoing research however.",
"The three features in more detail:"
],
[
"Metabolism",
"All known living organisms are metabolic [25] and the metabolism is also in the discussion for its possible role in the origins of life [10], [16].",
"This highlights its fundamental role and any final agent representation must accommodate for this.",
"The difficulty is the following.",
"Assume that the sufficiently accurate biosphere model from the introduction is particle-based, i.e.",
"it describes the time evolution of the degrees of freedom of all the particles in the biosphere.",
"Say at a time $t_1$ we are given all the particles (and their degrees of freedom) that pertain to some bacterium.",
"Then a naive way to represent this bacterium would be to just track the time evolution of each of those particles.",
"This we could (in principle) easily do in our model as well.",
"However the particles that the bacterium is made up of at a later time $t_2$ are not the same as those at time $t_1$ because of the bacterium's metabolism.",
"We would end up with particles floating around in the environment of the bacterium and not the bacterium itself.",
"At the same time there would be particles that now pertain to the bacterium that we would not be tracking.",
"An agent representation therefore would need to be able to track the bacterium itself and not just a specific set of degrees of freedom.",
"One way this could be solved is by constantly readjusting or refocusing on the degrees of freedom pertaining to the agent.",
"Note that we cannot be entirely sure that there is no coordinate transformation which would let us track living organisms (i.e.",
"their corresponding structures in a model) by just following a particular set of degrees of freedom.",
"However we are not aware of such a transformation.",
"Any criterion however that can be used to refocus on an agent should be related to any coordinate transformation which results in the “agents' own” coordinate system."
],
[
"Motility",
"Living organisms can be motile and like the metabolism motility is in the discussion for its role in the origins of life [13].",
"A representation of agents must therefore be capable of dealing with motile agents.",
"Motility plays a similar role for field theory models of the biosphere as the metabolism plays for particle based models.",
"The degrees of freedom of a field theory are the field amplitudes at each point in space so that tracking those degrees of freedom over time only means to track the field in a specific region of space.",
"However motility demands that agents are not bound to a fixed region in space.",
"Then we again need to adjust (track) the degrees of freedom that constitute the agent as time passes."
],
[
"Counterfactual variation",
"A third feature concerns another kind of variation of the degrees of freedom that can represent agents in a dynamical system.",
"Namely variation under different initial conditions.",
"We attribute to the biosphere a large variety of possible counterfactual histories that also support living organisms.",
"Think of a biosphere where the continents are shifted a bit for example.",
"This would not seem to necessarily destroy the possibility of the biosphere (geosphere) to contain living organism.",
"Furthermore, we attribute to agents and living organisms the capability to behave differently under different environmental situations.",
"The agent should be able to “take a decision” i.e.",
"to walk either right or left, or eat the apple or the pear.",
"Depending on these “decisions” the agent will again pertain to different degrees of freedom.",
"The counterfactual histories can be studied in the dynamical system setting by studying multiple trajectories through state space.",
"Each trajectory corresponds to a different history (and possibly future).",
"If the “same” agent occurs in two different trajectories it can behave differently in one from the other.",
"This can be associated with different decisions [15].",
"The existence of benign counterfactual histories in our biosphere is an assumption and not possible to prove.",
"However it is in line with the successful way physics models systems [28], [6] and therefore in line with our general approach.",
"Now given a set of counterfactual histories containing living organisms we expect that the degrees of freedom which in one history pertain to a living system at time $t$ need not pertain to a living system within another such history at $t$ (or in fact ever).",
"More specifically, the degrees of freedom pertaining to a bacterium in one history need not pertain to any living organism in another.",
"If the biosphere can contain living organisms within various counterfactual histories, then the dynamical systems model of the biosphere must be able to contain agents under various initial conditions.",
"In that case the agent representation must be able to represent all the agents in all the trajectories where they occur.",
"If for two different initial conditions the degrees of freedom pertaining to agents at time $t$ are different as well, then the agent representation must be able to exhibit this difference."
],
[
"Related work",
"We should stress that we are only interested in work that relies on the intrinsic properties of the dynamical systems itself to represent agents.",
"References to concepts like action, perception, and goal-directedness, if they are not defined in terms of the dynamical system are not acceptable in this case.",
"The publication that most directly tackles the problem of agent representation that we are aware of is the insightful paper of [17].",
"They solve the problems of metabolism and motility by evaluating informational measures of closure and autonomy of sets of random variables.",
"Given a system represented by a set of random variables at each point in time (i.e.",
"represented by a dynamical Bayesian network as also defined below) they propose an algorithm that decides whether to include a random variable at a specific time step into the set representing the agent or leave it in the set representing the environment.",
"This decision is made according to whether the inclusion into the agent contributes to the closure or autonomy of the agent.",
"What this approach lacks however is the capability to deal with counterfactual variation.",
"Since they use measures like mutual information and mutual conditional information that average over all states of the random variables in order to decide whether they belong to the agent or not, the partition of the random variables at each time step is fixed for all possible trajectories of the system.",
"In order to deal with counterfactual variation it must be possible to have one partition into agent and environment for one trajectory and another partition for another trajectory.",
"The same argument remains true for any approach that results in a fixed partition of the nodes in a dynamical Bayesian network.",
"This includes the work of [3] which results in a coarse-grained version of the network.",
"The effective information that the glider contains about past states of the game of life, which was revealed in this work, should however be related the intrinsic spatiotemporal patterns that we investigate here.",
"Another very relevant and inspiring work is the work on the cognitive domain of the glider and autopoiesis in the game of life by [8], [7].",
"This approach is capable of dealing with metabolism, motility, as well as counterfactual variation as it analyses spatiotemporal patterns and their internal mechanisms.",
"The spatiotemporal patterns may have finite extension and can therefore occur or not occur within multiple trajectories at multiple times.",
"The internal mechanisms are analyzed with respect to their production of the next spatial pattern inside the spatiotemporal pattern.",
"The only caveat seems to be that the analysis is quite time consuming and does not have formal expressions of all the involved notions.",
"We use the notion of spatiotemporal patterns as presented by Beer and hope that the measure we propose contributes to the formalization of the notions in his work.",
"An approach that seems to solve the problem of metabolism and counterfactual variation is the Markov blanket-based clustering used by [12].",
"As the interacting degrees of freedom vary over time in a particle based system, it is possible to define a time dependent adjacency (or interaction) matrix.",
"From this matrix Friston derives a Markov blanket matrix which can be used to classify the degrees of freedom into hidden, sensory, active, and internal states.",
"This nicely defines an agent like structure within the degrees of freedom and through the time dependence of the adjacency and therefore also the Markov blanket matrix allows for the degrees of freedom to vary within a single trajectory and across initial conditions.",
"In the case of a field theoretical model where the adjacency of the degrees of freedom does not vary it is not directly obvious to us how to translate this.",
"This means that motility could be a problem for the approach in such a model.",
"However, it is definitely an alternative to our more information theory-based approach.",
"Methodologically, the framework of [19] for distributed computation is very closely related to ours.",
"They investigate localized versions of mutual information and conditional mutual information to track and highlight information transfer, storage, and modification in dynamical Bayesian networks.",
"This reveals spatiotemporal patterns very similar to ours.",
"The main formal difference is in fact that instead of localizing the (conditional) mutual information we localize multi-information in the same way.",
"In this way our work is just a trivial extension of this work.",
"The focus of our work however is different as we are not so much interested in phenomena that are related to computation and more interested in revealing spatiotemporal entities or objects which might form the basis of an agent definition.",
"Related work on spatiotemporal filtering [24], [11] of cellular automata differs from ours in a similar way.",
"While “interesting” phenomena in the time evolution of single trajectories are revealed, the focus is not on connecting the interesting phenomena together in order to obtain entities.",
"Conceptually our work is also closely related to the integrated information theory due to [27] Originally [27] this involved measurements of multi-information whose localized (in the sense of [19]) version we also employ as an estimate of integration.",
"Newer versions [21], [1] involve a more elaborate construction which, importantly, is also localized in a certain way.",
"The latter detect distinguished integrated spatial patterns which are constructed to resolve “what a system `is' from its own intrinsic perspective” [1].",
"How these spatial structures are connected in time however is not treated.",
"Our approach also aims at revealing intrinsic structure but crucially looks for spatiotemporal patterns i.e.",
"patterns with a temporal as well as a spatial extension or compositional structure.",
"In [26], [4] temporal integration is mentioned with respect to optimal spatial and temporal scale or “grain size” detection.",
"Our goal is different since we don't want to find a coarse-graining here.",
"We want to reveal the complete lifetimes of agents as a single spatiotemporal pattern.",
"Finite discrete-time distributed dynamical systems and their stochastic counterparts can be represented by dynamical Bayesian networks.",
"Dynamical here just means that there is an interpretation of time in those networks.",
"Distributed means that, at each time step, there are multiple given random variables whose states together define the state of the entire network at that time step.",
"More formally, a (dynamical) Bayesian network is a directed acyclic graph $G=(V,E)$ with nodes $V$ and edges $E$ .",
"Each node $i$ has an associated random variable $X_i$ with state space $\\mathcal {X}$ taking values $x_i \\in \\mathcal {X}$ (for simplicity we assume that all nodes have identical state spaces but this is not necessary for the definitions to hold).",
"Furthermore each node is equipped with a mechanism $p_i(x_i|x_{\\operatorname{pa}(i)})$ which gives the conditional probability distribution of $X_i$ given the parents $\\operatorname{pa}(i)$ of node $i$ in $G$ .",
"Note that for any set $A \\subseteq V$ we write $X_A: = (\\lbrace X_i|i \\in A\\rbrace )$ for the random variable composed of the random variables in $A$ .",
"We assume that our network has a set $V_0$ of nodes without parents.",
"As [2] note we can then define a partition of $V$ into $(V_0,V_1,V_2,...)$ (called time slices) where $V_{t+1}:=\\lbrace i \\in V \\mid \\exists j \\in V_t, \\operatorname{pa}(i)=j\\rbrace $ .",
"In general $\\operatorname{pa}(V_{t+1}) \\subseteq V_t$ since some nodes might not have any children.",
"Here we assume $\\operatorname{pa}(V_{t+1}) = V_t$ .",
"This allows us to interpret the various nodes in each $V_t$ as those nodes representing the state of the distributed system at time $t$ .",
"We can also interpret the cardinality of the set $V_t$ as the spatial extension of the state.",
"In this paper this cardinality does not change over time just as e.g.",
"in cellular automata.",
"The defining property of Bayesian networks (including dynamic ones) is that the joint probability distributionFor a set of nodes $A$ we write $p_A$ for the probability distribution $p_A:\\mathcal {X}^A \\rightarrow [0,1]$ .",
"$p_V$ can be factorized in a way compatible with the structure of the graph $G$ i.e.",
": $p_V(x_V)=\\prod _{i \\in V} p_i(x_i|x_{\\operatorname{pa}(i)}).$ To relate the dynamical Bayesian network to dynamical systems note that the role of the dynamical law is played by the product of all mechanisms in $V_t$ : $p_{V_{t+1}}(x_{V_{t+1}})=\\sum _{x_{V_t}} \\prod _{i \\in V_{t+1}} p_i(x_i|x_{\\operatorname{pa}(i)}) p_{V_t}(x_{V_t}).$ Recall that $\\bigcup _{i \\in V_{t+1}} \\operatorname{pa}(i) = V_t$ by definition.",
"We can also write the above in terms of the Markov matrix: $p(x_{V_{t+1}}|x_{V_t})= \\prod _{i \\in V_{t+1}} p_i(x_i|x_{\\operatorname{pa}(i)}).$ In order to equip the dynamical Bayesian network with a join probability distribution $p_V$ we then only have to define an initial probability distribution $p_{V_0}$ and propagate it throughout the network according to Eq.",
"REF .",
"Here we formally define the notion of trajectories and spatiotemporal patterns.",
"The class of the spatiotemporal patterns is very large and includes patterns that are of no specific interest.",
"How we distinguish between those and more important patterns will be defined in the next sections.",
"A spatiotemporal pattern $x_O$ of a dynamical Bayesian network on graph $G=(V,E)$ is a set of nodes $O \\subseteq V$ together a set of particular values $\\lbrace x_i \\in \\mathcal {X}|i \\in O\\rbrace $ .",
"A trajectory $x_V$ of a dynamical Bayesian network on graph $G=(V,E)$ is a spatiotemporal pattern with $p(x_V)>0$ .",
"In our setting this also means that there is an initial condition $x_{V_0}$ such that $x_V$ is possible under the time evolution induced by the Markov matrix or dynamical law.",
"We say that the spatiotemporal pattern $x_O$ occurs in a trajectory $x_V$ of the network iff $x_O \\subseteq x_V$ .",
"Employing the time slices $V_t$ of the network we can also look at the time slices $x_{O_t} := x_O \\cap x_{V_t}$ of any spatiotemporal pattern $x_O$ .",
"This section defines the notion of an integrated spatiotemporal pattern.",
"Such patterns obey a condition which distinguishes them within the class of all spatiotemporal patterns.",
"First we fix some further terminology.",
"We define the the evidence for integration of an object $O$ with respect to a partition $\\pi $ of $O$ as the local mutual information $\\operatorname{mi}_\\pi (x_O):= {\\left\\lbrace \\begin{array}{ll}0 & \\text{ if } p_O(x_O)=0, \\\\\\log \\frac{p_O(x_O)}{\\prod _{b_j \\in \\pi } p_{b_j}(x_{b_j})} & \\text{ else}.\\end{array}\\right.",
"}$ Then, we say a spatiotemporal pattern $x_O$ is integrated iff for all possible partitions $\\pi $ of the set $O$ of random variables the evidence for integration of $O$ with respect to $\\pi $ is positive.",
"Considering all possible partitions is also done by [1].",
"The interpretation of this is the following.",
"The joint probability $p_O(x_O)$ is the probability that all the $x_i$ with $i \\in O$ occur together within single trajectories.",
"I.e.",
"among all trajectories there are those in which $O$ occurs and their probability contributes to this joint probability.",
"The probability $\\prod _{b_j \\in \\pi } p_{b_j}(x_{b_j})$ however is the product of the probabilities that each part $x_{b_j}$ occurs by itself in any trajectory including as part of $x_O$ .",
"If a part of $x_O$ often occurs by itself without the rest of $x_O$ occurring then this reduces the evidence for integration of $O$ .",
"This makes sense if we want to interpret the integrated spatiotemporal patterns as persistent objects like rocks, crystals, but also living organisms.",
"If we consider for example a rock, the probability that a rock occurs at some point in time without a rock occurring at the previous and next time step in close vicinity is quite low, whereas the probability that where there was a rock before there will be a rock shortly after is quite high.",
"In fact anytime that a spatial pattern (a time slice of a spatiotemporal pattern) causes (in an intuitive sense) another spatial pattern at the next time step their joint probability will rise and especially if the first spatial pattern is among the only causes of the second, their evidence for integration will be high.",
"What about the spatial integration however?",
"The existence of rock in one place probably does increase the probability for more rock to be around it, but not extremely.",
"It is perfectly possible and occurs frequently, that the rock ends, also that it is just a small piece of rock.",
"So the evidence for spatial integration might not be so strong.",
"Now if we turn to living organisms, the evidence for temporal integration should also be high since they are autopoietic.",
"Their spatial integration will probably be higher than that of rocks (and crystals) as half a bacterium is much less likely than a whole whereas half a rock is still a rock and those are not so uncommon.",
"This reasoning scales up to larger living organisms.",
"We note that the evidence can also be interpreted in more information-theoretic terms.",
"For example as the superfluous length of a codeword for the sequence $x_O$ when we base the encoding on the product probability distribution $\\prod _{b_j \\in \\pi } p_{b_j}(x_{b_j})$ instead of on the joint probability.",
"This will be discussed in more detail in future work.",
"We also intend to investigate in how far the integrated spatiotemporal patterns are independent of (possibly moving) frames of reference.",
"Since they are not only integrated across time slices but instead across any partition we are optimistic in this regard.",
"The spatiotemporal patterns can solve the tracking problem.",
"To see this take the perspective of time slices and say that at some time $t$ a living organism is a configuration of degrees of freedom which increases the probability of a particular configuration of other degrees of freedom at a subsequent time $t+\\epsilon $ that is again a living organism.",
"More specifically, a living organism will lack certain molecules before absorbing them, conversely there will be a surplus of other molecules before they are ejected from the living organism.",
"Therefore the probability for molecular exchange will be higher than for maintaining the same composition.",
"This means the spatiotemporal patterns traversing the degrees of freedom associated to the molecules will have a higher evidence for integration over time.",
"Similarly, in the field-theoretic setting the field configuration represented by the spatial pattern will increase the probability of the neighboring degrees of freedom to assume a certain configuration.",
"This leads to more evidence for the integration of the moving pattern.",
"With respect to the problem of counterfactual variation we can see the following.",
"Integration is calculated directly for spatiotemporal patterns within a trajectory and the local mutual information vanishes for all spatiotemporal patterns that do not occur in this trajectory (see Eq.",
"REF ).",
"Then, if the spatiotemporal patterns that occur in different trajectories are different, the integrated spatiotemporal patterns will also be different.",
"Thus, if integrated spatiotemporal patterns represent agents, these can occur on some degrees of freedom in one trajectory and not occur on those in another.",
"This means counterfactual variation won't be a problem.",
"We present here the results of three preliminary experiments.",
"The first is conceived to hint at the kind of trajectories that show high evidence of integration.",
"The second experiment suggests that traversal of degrees of freedom or motility/metabolism can at least in principle be detected by integration.",
"Similarly the third experiment shows that in principle counterfactual variation is no obstacle for integration.",
"All experiments use a $4 \\times 4$ grid with game-of-life dynamics and toroidal boundary conditions as the distributed dynamical system.",
"As the initial distribution $p_{V_0}$ we use the uniform distribution in order to explore the whole range of possible trajectories.",
"We investigate only patterns covering three time steps $t=8,9,10$ and thereby neglect a lot of transient patterns that are difficult to interpret.",
"In principle, however, our method does apply to transient patterns as well.",
"Instead of integration we calculated only the evidence for integration with respect to the finest possible partition (EVIFPP).",
"The finest possible partition of a set $A$ of nodes is just the partition where each block is a set containing exactly one node in $A$ .",
"A positive EVIFPP is a necessary condition for integration and therefore a crude indication for it.",
"For the first experiment we looked at all trajectories that differ at time steps $t=8,9,10$ (a lot of trajectories end up with all cells white at those times).",
"For each of those trajectories we calculated the EVIFPP for the spatiotemporal pattern $x_O=(x_{V_8},x_{V_9},x_{V_{10}})$ .",
"So the time slices of $x_O$ are global states in this case.",
"Since the $x_O$ are global states this is more an evaluation of the integration of the trajectories that result from the different initial conditions.",
"In Fig.",
"REF five such different global three-time-step patterns with high values of integration are shown including the completely blank spatiotemporal pattern and the spatiotemporal patterns (ignoring symmetric versions) with the highest EVIFPP.",
"We can see that the blank spatiotemporal pattern has positive but much lower EVIFPP than some other patterns.",
"Figure: Three three-time step spatiotemporal patterns.",
"Each row shows the three global spatial patterns that make up the spatiotemporal pattern x O x_O.",
"The first row shows the blank spatiotemporal pattern and the others show the two patterns with the highest EVIFPP.",
"The EVIFPP values are (from top to bottom) 4.9, 81.9, and 85.4 respectively.For the second experiment we chose a specific trajectory shown in the first row in Fig.",
"REF which exhibits a moving pattern and searched through all patterns covering time steps $t=8,9,10$ and fixing $n=14$ cells (i.e.",
"nodes of the dynamic Bayesian network) in each time slice $x_{O_t}$ .",
"We can see that the degrees of freedom (i.e.",
"the cells or nodes) making up both the spatiotemporal pattern with minimal EVIFPP (second row in Fig.",
"REF ) as well as that with maximal EVIFPP (third row in Fig.",
"REF ) vary over the three time steps and adapt to the configuration of the global state.",
"Note that the patterns with minimal and maximal EVIFPP are not unique.",
"Figure: A three-time step part of a trajectory (can also be seen as a global spatiotemporal pattern) in the first row and two local spatiotemporal patterns on this trajectory in the second and third row.",
"Both spatiotemporal patterns in rows two and three have n=14n=14 specified cells per time slice.",
"The second (third) row shows a pattern attaining the minimal (maximal) EVIFPP of 32.5 (54.4) among all patterns with n=14n=14 on the trajectory of row one.",
"The global spatiotemporal pattern of row one has EVIFPP of 55.0.For the third experiment changed the initial condition of the trajectory by shifting all values of the initial condition of the second experiment “down” one cell.",
"This results in a different trajectory shown in the first row of Fig.",
"REF .",
"We then evaluated the spatiotemporal pattern that results from fixing the same nodes as in the spatiotemporal pattern with maximal EVIFPP found in the second experiment on the changed trajectory (see row two in Fig.",
"REF ).",
"We also evaluated the EVIFPP of the spatiotemporal pattern that results from shifting the fixed nodes of maximal EVIFPP pattern in the same way as the initial condition (see row three in Fig.",
"REF ).",
"The pattern with the same fixed nodes as the pattern that formerly had maximal EVIFPP now has lower EVIFPP than the pattern with the nodes adapted to the new initial condition.",
"The first experiment shows that the completely blank trajectory has low spatiotemporal EVIFPP and that more “interesting” trajectories have higher EVIFPP (Fig.",
"REF ).",
"This can also be done with other methods e.g.",
"counting black cells.",
"However, our method is general and doesn't use any prior knowledge e.g.",
"which color of cells to count.",
"For us this result is a necessary condition for further investigation.",
"The second experiment shows that the degrees of freedom pertaining to spatiotemporal patterns with high EVIFPP adapt over time to the changing configurations of the system.",
"This shows that EVIFPP is capable of solving the metabolism and motility problems.",
"We expect that the same holds true for evidence of integration with respect to any partition and therefore also for integration itself.",
"The third experiment shows that under a variation of the initial condition the degrees of freedom pertaining to spatiotemporal patterns with high EVIFPP change accordingly.",
"Since the different trajectories generated from changed initial conditions correspond to counterfactual histories this shows that the EVIFPP solves the problem of counterfactual variation.",
"Again we expect this to carry over to integration.",
"We note that larger grids become hard to evaluate computationally very fast.",
"For square grids the size of the Markov matrix grows with $2^{a^2}$ where $a$ is the number of rows and columns of the grid.",
"We also note that due to the very limited grid size we are studying any pair of cells is just separated by maximally one neighborhood cell.",
"This leads to strong dependencies which might make it irrelevant to place unspecified cells around patterns like the blinker (as for example suggested by [7]).",
"We had hoped to reveal such well known patterns and their extensions.",
"Turning to larger grids is a next step in our research.",
"We have presented our current approach to representing agents in dynamical systems.",
"Three criteria that we expect from such an agent representation were motivated with a thought experiment involving a dynamical systems model of the biosphere.",
"The literature was reviewed in the light of these criteria.",
"We also introduced our current candidate measure for identifying intrinsic spatiotemporal patterns in dynamical Bayesian networks.",
"These patterns form the basic building blocks of our approach to representing agents.",
"We argued that this approach can deal with the three criteria for agent representations that we have put forward.",
"Experimentally we verified this for a crude approximation to our more involved concept of integration.",
"However experimental results are currently inconclusive with respect to the tracking of structures that are actually relevant for agents.",
"Therefore we see the value of this work mostly as a contribution to the discussion of the foundations of artificial life.",
"Future work will bring more decisive results.",
"Figure: The three-time step part of the trajectory that results from shifting the values of the initial condition of the trajectory in the first row of Fig.",
"“down” by one cell.",
"The second row shows the spatiotemporal pattern with the same fixed nodes as the pattern that had maximal EVIFPP on the trajectory of Fig.",
"but now on the shifted trajectory.",
"The EVIFPP of this is 39.8.",
"The third row shows the spatiotemporal pattern with the fixed nodes shifted “down” in the same way as the initial condition.",
"This pattern has EVIFPP of 54.4 and is the maximal EVIFPP for patterns with n=2n=2.",
"As expected this is the same value we found for the pattern with the non-shifted nodes on the non-shifted trajectory.We thank Christoph Salge, Olaf Witkowski, Nicola Catenacci-Volpi, Julien Hubert, Nathaniel Virgo, and Nicholas Guttenberg for discussions on this topic.",
"Part of this research was performed during Martin Biehl's time as an International Research Fellow of the Japan Society for the Promotion of Science.",
"The third author was supported in part by the H2020-641321 socSMCs FET Proactive project."
]
] | 1605.05676 |
[
[
"Performance Analysis of Joint Time Delay and Doppler-Stretch Estimation\n with Random Stepped-Frequency Signals"
],
[
"Abstract This paper investigates the performance of joint time delay and Doppler-stretch estimation with the random stepp ed-frequency (RSF) signal.",
"Applying the ambiguity function (AF) to implement the estimation, we derive the compact expressions of the theoretical mean square errors (MSEs) under high signal-to-noise ratios (SNRs).",
"The obtained MSEs are shown consistent with the corresponding Cramer-Rao lower bounds (CRLBs), implying that the AF-based estimation is approximately efficient.",
"Waveform parameters including higher carrier frequencies, wider bandwidth covered by the carrier frequencies, and frequency shifting codewords with larger variance are expected for a better estimation performance.",
"As a synthetic wideband signal, the RSF signal achieves the same estimation performance as the OFDM signal within an identical bandwidth.",
"Due to its instantaneous narrowband character, requirement for the bandwidth of the receiver is much reduced."
],
[
"Introduction",
"The stepped-frequency (SF) signal has been widely adopted in modern wideband radar and sonar systems.",
"Compared to the conventional narrowband signal, the SF signal achieves higher range resolution, and the multiple scatterers of the target can be thoroughly distinguished.",
"Since its energy is dispersed to the whole bandwidth covered by the carrier frequencies, the SF signal attains a lower probability of interception [1].",
"Meanwhile, classified as one of the synthetic wideband signals, the SF signal only takes up a narrow bandwidth at any time instant, while the whole carrier frequency bandwidth can be occupied if an instantaneous wideband signal (e.g.",
"the OFDM signal) is employed.",
"From this scope, the SF signal could largely reduce the requirements for the bandwidth of the receiver [2].",
"The linear stepped-frequency (LSF) signal uses a fixed frequency shifting step, which introduces a “ridge” in its ambiguity function (AF).",
"This causes a coupling problem between the range and Doppler dimensions [4].",
"For the random stepped-frequency (RSF) signal, however, with the carrier frequencies of the pulses randomly distributed over a given bandwidth, its AF appears in a thumbtack alike shape, where the range and Doppler dimensions are completely decoupled [3].",
"The resolutions in both dimensions thus meet further improvements, and the range ambiguity is efficiently suppressed.",
"Besides, as the frequency shifting codeword of the RSF signal is usually highly self-correlated and hard to track, the interference between adjacent radars can be largely reduced, whereas the electronic counter-countermeasures (ECCM) capabilities can be also acquired [4].",
"As a result, parameter estimation with the RSF signal is of great practical significance and is gaining increasing research interests.",
"The joint estimation of time delay and Doppler-stretch is a fundamental problem that facilitates target tracking and localization in radar and sonar systems, upon which the location and velocity information of the target is managed to be attained [5].",
"One of the standard methods for joint delay-Doppler estimation is to adopt the AF [6], [7].",
"By locating the peak of the AF, the joint estimation was primarily implemented [8].",
"So as to evaluate the performance of the estimation, the Cramer-Rao lower bound (CRLB) is commonly employed, for the reason that it is regarded as a theoretical lower bound for the variance of any unbiased estimation and is usually easy to calculate.",
"Based on a wideband signal model, the CRLBs of time delay and Doppler-stretch were derived, under the assumption that the scattering coefficient of the target was known a priori [8].",
"Derivation for a more realistic case was performed by [9], where the scattering coefficient was supposed to be unavailable at the receiver.",
"The estimation problem of an extended target with multiple scatterers was even considered in [10].",
"Nevertheless, the CRLB is only reliable for accurately presenting the estimation performance when it is approached by the corresponding mean square error (MSE), i.e.",
"when the estimation is (asymptotically) efficient [11].",
"By directly calculating the theoretical MSE, the estimation performance can be straightforwardly revealed.",
"However, in most cases it is difficult to evaluate the MSE in a theoretical manner.",
"Among the limited number of related works, the MSEs of the AF-based joint estimation were calculated in [8], whereas a correction followed in [12].",
"However, the calculating results are not accurate since too many approximations were made in both works.",
"All the approaches [8], [12], [9], [10] were built on a general wideband signal model, which failed to reveal the waveform parameters that influence the estimation performance with any specific signal.",
"Based on the LSF signal model, the CRLBs of time delay and Doppler-stretch were derived in [13], whereas those of the high resolution range (HRR) profiles of an extended target were provided by [14].",
"Similar range and Doppler estimation problems with the OFDM signal were also considered in [15], [16], [17].",
"However, performance analysis on the parameter estimation with the RSF signal is quite limited in previous works.",
"In this paper, we investigate the performance of the AF-based joint delay-Dppler estimation with the RSF signal.",
"Under high signal-to-noise ratio (SNR) assumption, compact expressions of the MSEs are obtained through a novel and strict derivation.",
"The MSEs are shown consistent with their corresponding CRLBs, revealing that the AF-based estimation is approximately efficient.",
"As illustrated by the derivations and simulations, three waveform parameters of the RSF signal, namely, the central carrier frequency, the bandwidth covered by the carrier frequencies, and the variance of the frequency shifting codeword mainly influence the estimation performance.",
"In specific, by increasing either the bandwidth for the carrier frequencies or the variance of the frequency shifting codeword the performance of delay estimation can be improved, while better performance of Doppler-stretch estimation calls for higher central carrier frequency.",
"As one of the synthetic wideband signals, the RSF signal only takes up a narrow bandwidth at any time instant, whereas it achieves the similar estimation performance as the OFDM signal does, contributing to a much reduced requirement for the bandwidth of the receiver.",
"The rest of the paper is organized as follows.",
"Section II describes the model of the RSF signal and makes necessary preliminaries.",
"Section III gives the main results of this paper, while Section IV provides the derivations for the main results.",
"Section V verifies the main results by numerical examples, followed by a conclusion in Section VI."
],
[
"Modeling and Preliminaries",
"Consider an RSF signal with $K$ pulses.",
"Let $\\beta (t)$ denote the envelope of each pulse.",
"The transmitted signal is then modeled as $s(t) = \\sum _{k = 0} ^{K - 1} \\beta (t - k T_r) e^{j 2 \\pi f_k (t - k T_r)},$ where $f_k$ denotes the carrier frequency of the $k$ -th pulse, and $T_r$ is the pulse repetition interval (PRI).",
"Assume that the carrier frequency remains constant within each pulse and shifts randomly over the pulses within a given bandwidth.",
"Then the carrier frequencies of the pulses can be represented as $f_k = f_0 + d_k \\delta _f, \\quad k = 0, 1, \\ldots , K - 1,$ where $f_0$ denotes the central carrier frequency, $\\delta _f$ is the minimum frequency shifting step, $[ d_0, d_1, \\ldots , d_{K - 1} ]^T$ is the frequency shifting codeword, in which $d_k$ for each pulse is randomly selected from $[-M, M]$ , $M \\in \\mathbb {N^+}$ .",
"Suppose that a target is moving along the line of sight (LOS) with a constant radial velocity $v$ relative to the sensor (e.g.",
"a radar or a sonar).",
"The echo reflected from the moving target is then given by [8] $s_r(t) = x s(\\gamma _0 (t - \\tau _0)),$ where $x$ is the scattering coefficient of the target, accounting for the attenuation and reflection, $\\tau _0$ and $\\gamma _0 = \\frac{c - v}{c + v}$ represent the time delay and Doppler-stretch, respectively, with the wave propagation velocity denoted as $c$ .",
"The signal received by the sensor is contaminated by a white Gaussian noise (WGN) $w(t)$ with power spectral density $N_0$ .",
"The received signal is then expressed as $\\begin{aligned}y(t) & = x s(\\gamma _0 (t - \\tau _0)) + w(t).\\end{aligned}$ Sampled at the rate of $1 / \\Delta $ , the received signal (REF ) turns into $y(n \\Delta ) = x s(\\gamma _0 (n \\Delta - \\tau _0)) + w(n \\Delta ),$ $n = 0, 1, \\ldots , N-1$ , where $N$ is the total number of sampling points, $w(n \\Delta )$ is distributed as $\\mathbb {C} N (0, \\sigma ^2)$ , and $\\sigma ^2 \\Delta = N_0$ [11]."
],
[
"Ambiguity Function",
"As mentioned in Section I, the AF $A_{y s}(\\tau , \\gamma ) = \\sum _{n = 0}^{N - 1} y(n \\Delta ) s^*(\\gamma (n \\Delta - \\tau ))$ is generally applied for implementation of the joint delay-Doppler estimation [8], [11].",
"Denoted as $(\\hat{\\tau }, \\hat{\\gamma }) = \\arg \\max _{ \\tau \\in [\\tau _{\\text{min}}, \\tau _{\\text{max}}], \\gamma \\in [\\gamma _{\\text{min}}, \\gamma _{\\text{max}}] } \\left| A_{y s}(\\tau , \\gamma ) \\right|,$ the AF-based estimation is implemented by locating the peak of the AF [8], [12], where $\\hat{\\tau }$ and $\\hat{\\gamma }$ respectively denote the estimations of time delay and Doppler-stretch, and $[\\tau _{\\text{min}}, \\tau _{\\text{max}}] \\times [\\gamma _{\\text{min}}, \\gamma _{\\text{max}}]$ regulates the searching area for the estimations of the parameters.",
"Note that both $\\hat{\\tau }$ and $\\hat{\\gamma }$ are continuous random variables that vary with $\\sigma $ .",
"In addition, we also assume $(\\tau _0, \\gamma _0) \\in [\\tau _{\\text{min}}, \\tau _{\\text{max}}] \\times [\\gamma _{\\text{min}}, \\gamma _{\\text{max}}]$ ."
],
[
"Preliminaries",
"For the ease of problem statement and derivations in the following sections, we make the necessary assumptions and definitions.",
"Assumption 2.1 The envelope of each pulse $\\beta (t)$ is time-limited within $[0, T]$ , $T < \\frac{T_r}{2}$ .",
"Assumption REF guarantees that each pulse does not overlap with any other ones in time domain.",
"Assumption 2.2 $s \\in \\mathbb {C}^{(2)}(\\mathbb {R})$ , i.e.",
"$s: \\mathbb {R} \\rightarrow \\mathbb {R}$ has continuous derivatives up to order 2 inclusive.",
"Assumption 2.3 $\\begin{aligned}& \\textstyle \\frac{\\partial ^2}{\\partial \\tau ^2} \\left| A_{s_r s} \\right|^2 (\\tau _0, \\gamma _0) \\cdot \\frac{\\partial ^2}{\\partial \\gamma ^2} \\left| A_{s_r s} \\right|^2 (\\tau _0, \\gamma _0) - \\\\& \\textstyle \\left( \\frac{\\partial ^2}{\\partial \\tau \\partial \\gamma } \\left| A_{s_r s} \\right|^2 (\\tau _0, \\gamma _0) \\right)^2 \\ne 0.\\end{aligned}$ Assumption 2.4 $ | A_{s_r s}(\\tau , \\gamma ) | $ has the unique maximizer $(\\tau _0, \\gamma _0)$ , which holds almost surely if the sampling rate $1 / \\Delta $ is greater than the bandwidth of $s(t)$ .",
"Definition 2.5 For any sequences $\\lbrace a_k; k = 0, 1, \\ldots , K - 1 \\rbrace $ and $\\lbrace b_k; k = 0, 1, \\ldots , K - 1 \\rbrace $ , define $\\nonumber & \\text{Var}\\left\\lbrace a_k \\right\\rbrace := \\left( \\text{Std}\\left\\lbrace a_k \\right\\rbrace \\right)^2 := \\frac{1}{K} \\sum _{k = 0}^{K - 1} a_k^2 - \\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} a_k \\right)^2, \\\\\\nonumber & \\text{Cov}\\left\\lbrace a_k, b_k \\right\\rbrace := \\frac{1}{K} \\sum _{k = 0}^{K - 1} a_k b_k - \\frac{1}{K} \\sum _{k = 0}^{K - 1} a_k \\frac{1}{K} \\sum _{k = 0}^{K - 1} b_k, \\\\& \\rho (a_k, b_k) := \\frac{ \\text{Cov}\\left\\lbrace a_k, b_k \\right\\rbrace }{ \\text{Std}\\left\\lbrace a_k \\right\\rbrace \\text{Std}\\left\\lbrace b_k \\right\\rbrace }.$ Assumption 2.6 For RSF signals, $\\text{Cov}\\left\\lbrace k^i, d_k^j \\right\\rbrace = 0, \\quad \\forall i, j = 1, 2.$ Definition 2.7 $\\forall i = 0, 2$ , $\\begin{aligned}& \\textstyle S_i^{(0)} := \\int _{0}^{T} \\left( t - \\frac{T}{2} - \\text{Std} \\left\\lbrace T_k \\right\\rbrace \\right)^i \\beta ^{2}(t) dt, \\\\& \\textstyle S_i^{(1)} := \\int _{0}^{T} \\left( t - \\frac{T}{2} - \\text{Std} \\left\\lbrace T_k \\right\\rbrace \\right)^i \\dot{\\beta }^{2}(t) dt,\\end{aligned}$ where $T_k := k T_r + \\frac{T}{2}$ , and $\\dot{\\beta }(t) := \\frac{d}{d t} \\beta (t)$ ."
],
[
"Main Results",
"In this section, the main results of the paper are provided.",
"Under high SNRs, i.e.",
"when $\\sigma $ is sufficiently small compared to the amplitude of the echo, the unbiasedness of the AF-based joint estimation is shown as: Theorem 3.1 For each $\\epsilon > 0$ , $P \\left\\lbrace \\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\Vert _2 > \\epsilon \\right\\rbrace \\rightarrow 0 \\text{ as } \\sigma \\rightarrow 0,$ indicating that $\\hat{\\tau } \\stackrel{P}{\\longrightarrow } \\tau _0 \\text{ and } \\hat{\\gamma } \\stackrel{P}{\\longrightarrow } \\gamma _0$ as $\\sigma \\rightarrow 0$ , where $\\Vert (x, y) \\Vert _2 = \\left( x^2 + y^2 \\right)^{\\frac{1}{2}}$ , and “$x \\stackrel{P}{\\longrightarrow } y$ ” denotes that $x$ converges to $y$ in probability.",
"See Appendix REF .",
"The estimations of time delay and Doppler-stretch respectively converge to their true values as the noise level gets weaker, implying that the AF-based joint estimation is asymptotically unbiased when SNR is sufficiently large [18].",
"Building on Theorem REF , we then evaluate the MSEs of the AF-based estimation.",
"Define $\\nonumber & \\begin{array}{ll}\\displaystyle B := \\int _{-\\infty }^{\\infty } \\left| \\dot{s}(t) \\right|^2 dt, & \\displaystyle E := \\int _{-\\infty }^{\\infty } \\left| s(t) \\right|^2 dt, \\\\\\displaystyle C := \\int _{-\\infty }^{\\infty } t \\left| \\dot{s}(t) \\right|^2 dt, & \\displaystyle F := \\text{Im} \\left\\lbrace \\int _{-\\infty }^{\\infty } s(t) \\dot{s}^*(t) dt \\right\\rbrace , \\\\\\displaystyle D := \\int _{-\\infty }^{\\infty } t^2 \\left| \\dot{s}(t) \\right|^2 dt, & \\displaystyle G := \\text{Im} \\left\\lbrace \\int _{-\\infty }^{\\infty } s(t) t \\dot{s}^*(t) dt \\right\\rbrace ,\\end{array} \\\\\\nonumber & \\begin{array}{l}\\displaystyle \\Pi := \\left( E B - F^2 \\right) \\left( E D - G^2 \\right) - \\left( E C - F G \\right)^2,\\end{array} \\\\& \\begin{array}{l}\\displaystyle \\Pi _0 := \\Pi - \\frac{5}{4} E^2 (E B - F^2).\\end{array}$ The theoretical MSEs of time delay and Doppler-stretch are formally given by: Theorem 3.2 In the AF-based estimation, the theoretical MSEs of time delay and Doppler-stretch satisfy $\\nonumber & \\left[ \\begin{array}{cc}\\displaystyle \\lim _{N_0 \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{\\left| \\hat{\\tau } - \\tau _0 \\right|^2}{N_0} \\right\\rbrace ,& \\displaystyle \\lim _{N_0 \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{\\left| \\hat{\\gamma } - \\gamma _0 \\right|^2}{N_0} \\right\\rbrace \\end{array} \\right]^T \\\\ = & \\frac{E}{2 |x|^2 \\Pi _0} \\left[ \\begin{aligned}& \\frac{1}{\\gamma _0} \\left(\\begin{array}{l}\\left( E D - G^2 - \\frac{5}{4} E^2 \\right)^2 \\left( E B - F^2 \\right) + \\\\\\left( E C - F G \\right)^2 \\left( E D - G^2 + \\frac{3}{4} E^2 \\right) - \\\\2 \\left( E C - F G \\right)^2 \\left( E D - G^2 - \\frac{5}{4} E^2 \\right)\\end{array}\\right) \\\\& \\gamma _0^3 \\left(\\begin{array}{l}\\left( E B - F^2 \\right)^2 \\left( E D - G^2 + \\frac{3}{4} E^2 \\right) + \\\\\\left( E C - F G \\right)^2 \\left( E B - F^2 \\right) - \\\\2 \\left( E B - F^2 \\right) \\left( E C - F G \\right)^2\\end{array}\\right)\\end{aligned} \\right] \\\\ \\approx & \\frac{E}{2 |x|^2 \\Pi } \\left[ \\begin{array}{cc}\\frac{1}{\\gamma _0} \\left(E D - G^2 \\right),& \\gamma _0^3 \\left(E B - F^2 \\right)\\end{array} \\right]^T,$ where the approximate in () holds if $\\beta (t) = \\beta (T - t)$ .",
"See Section REF .",
"In Theorem REF , (REF ) precisely describes the MSEs of the AF-based estimation under high SNRs.",
"If the additional condition is involved, we obtain the approximated, but much simplified forms of the MSEs, which are presented by ().",
"The condition $\\beta (t) = \\beta (T - t)$ suggests that the envelope $\\beta (t)$ is a symmetric function with an axis of symmetry $t = T / 2$ .",
"This is yielded by most of the radar signals.",
"In this sense, Theorem REF reliably presents the MSEs of most cases.",
"Moreover, note that the MSEs of time delay and Doppler-stretch are represented with integrations, corresponding to the case where the sampling rate is sufficiently large, i.e.",
"$\\Delta \\rightarrow 0$ , such that the summations can be replaced by integrations.",
"The effectiveness of the AF-based estimation is judged by evaluating the gaps between the MSEs and their corresponding CRLBs, since the latters represent the minimum achievable variances of any unbiased estimation.",
"In [8] and [9], the CRLBs for joint delay-Doppler estimation with known and unknown scattering coefficient were respectively derived.",
"In the AF-based joint estimation as introduced in Section II-A, only the magnitude information of the AF is utilized.",
"However, the phase information contained in the scattering coefficient $x$ is ignored.",
"Therefore, regarding $x$ as one of the unknown parameters and according to the calculation of CRLBs in [9], [10], we have Theorem 3.3 The proof of Theorem REF is omitted due to limited space.",
"The CRLBs of time delay and Doppler-stretch are given by $\\left[ \\begin{array}{c}\\text{CRLB}_{\\tau } \\\\\\text{CRLB}_{\\gamma }\\end{array} \\right] = \\frac{N_0 E}{2 |x|^2 \\Pi } \\left[ \\begin{aligned}& \\frac{1}{\\gamma _0} (E D - G^2) \\\\& \\gamma _0^3 (E B - F^2)\\end{aligned} \\right].$ Clearly, the MSEs in () are confirmed to be consistent with the corresponding CRLBs.",
"By Theorems REF –REF , we describe the AF-based estimation as an approximately efficient estimation [18].",
"We should also notice that Theorem REF and REF in fact apply to estimations with arbitrary wideband signals modeled by (REF ) (including the RSF signals).",
"However, the specific waveform parameters determining the estimation performance are not able to be revealed by (REF ) or (REF ).",
"So as to explore the relationship between the estimation performance and the waveform parameters of the RSF signal, we substitute the signal model (REF ) into () and obtain the compact expressions of the MSEs: Theorem 3.4 For an RSF signal, the theoretical MSEs of joint delay-Doppler estimation under high SNRs are specifically expressed as $\\nonumber & \\lim _{N_0 \\rightarrow 0} \\mathbb {E} \\textstyle \\left\\lbrace \\frac{\\left| \\hat{\\tau } - \\tau _0 \\right|^2}{N_0} \\right\\rbrace \\\\ & \\approx \\textstyle \\frac{1}{2 \\gamma _0 |x|^2 K} \\left\\lbrace \\left[ S_0^{(1)} + 4 \\pi ^2 \\text{Var}\\left\\lbrace f_k \\right\\rbrace S_0^{(0)} \\right]^{-1} + \\right.",
"\\\\\\nonumber & \\quad \\textstyle \\left.",
"\\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k \\right)^2 \\textstyle \\left[ S_2^{(1)} + 4 \\pi ^2 \\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 \\right) S_2^{(0)} \\right]^{-1} \\right\\rbrace , \\\\& \\begin{aligned}& \\lim _{N_0 \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{\\left| \\hat{\\gamma } - \\gamma _0 \\right|^2}{N_0} \\right\\rbrace \\\\& \\approx \\textstyle \\frac{\\gamma _0^3}{2 |x|^2 K} \\left[ S_2^{(1)} + 4 \\pi ^2 \\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 \\right) S_2^{(0)} \\right]^{-1}.\\end{aligned}$ See Section REF .",
"As revealed by (REF ), the estimation performance with an RSF signal is mainly dominated by three factors, namely, the central carrier frequency, the bandwidth covered by the carrier frequencies, and the frequency shifting pattern.",
"Specifically, i) In both the expressions of MSEs above, there exists a component $\\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2$ .",
"Since $f_0 \\gg \\delta _f$ , $\\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 \\approx f_0^2$ .",
"Therefore, both the performances of time delay and Doppler-stretch estimations can be improved if the central carrier frequency $f_0$ increases.",
"ii) The component $\\text{Var}\\left\\lbrace f_k \\right\\rbrace $ in (REF ) can be rewritten as $\\delta _f^2 \\text{Var}\\left\\lbrace d_k \\right\\rbrace $ , where $\\delta _f$ determines the available bandwidth for the carrier frequencies, while $\\text{Var}\\left\\lbrace d_k \\right\\rbrace $ is related to the frequency shifting pattern.",
"Beyond the revealings by the expressions of the MSEs in Theorem REF , simulation results in Section V show that the performance of delay estimation is only slightly improved as the central carrier frequency increases.",
"This indicates that the performance of delay estimation is mainly influenced by the covered bandwidth of the carrier frequencies and the variance of the frequency shifting codeword.",
"With the analyses above, the theoretical MSEs given by Theorem REF could serve as a guidance for waveform design, which aims at properly configuring the waveform parameters and improving the estimation performance.",
"By increasing the central carrier frequency the performance of Doppler-stretch estimation can be significantly improved, while the performance of delay estimation can be improved by increasing the bandwidth covered by carrier frequencies and by adopting the frequency shifting codewords with large $\\text{Var}\\left\\lbrace d_k \\right\\rbrace $ ."
],
[
"Comparison with Other Waveforms",
"The estimation performance with the RSF signal is then fairly compared with those with a monotone signal and an OFDM signal, where the three signals are comprised of the same number of pulses with the same amount of energy.",
"The monotone signal fixes its carrier frequency to $f_0$ for all pulses, whereas the OFDM signal simultaneously uses $L$ orthogonal subcarriers within each pulse.",
"i) The estimation performance with the monotone signal is directly known from (REF ), since the signal can be considered as a special case of RSF signal with $\\delta _f = 0$ .",
"The MSEs of time delay and Doppler-stretch then readily reduce to $& \\begin{aligned}& \\lim _{N_0 \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{\\left| \\hat{\\tau } - \\tau _0 \\right|^2}{N_0} \\right\\rbrace \\approx \\textstyle \\frac{1}{2 \\gamma _0 |x|^2 K} \\left\\lbrace \\left( S_0^{(1)} \\right)^{-1} + \\right.",
"\\\\& \\quad \\textstyle \\left.",
"\\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k \\right)^2 \\textstyle \\left[ S_2^{(1)} + 4 \\pi ^2 f_0^2 S_2^{(0)} \\right]^{-1} \\right\\rbrace ,\\end{aligned} \\\\ & \\lim _{N_0 \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{\\left| \\hat{\\gamma } - \\gamma _0 \\right|^2}{N_0} \\right\\rbrace \\approx \\textstyle \\frac{\\gamma _0^3}{2 |x|^2 K} \\left[ S_2^{(1)} + 4 \\pi ^2 f_0^2 S_2^{(0)} \\right]^{-1},$ with $\\text{Var} \\left\\lbrace f_k \\right\\rbrace = 0$ and $\\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 = f_0^2$ .",
"Recalling that $\\text{Var} \\left\\lbrace f_k \\right\\rbrace $ mainly determines the MSE of time delay, and that $\\text{Var} \\left\\lbrace f_k \\right\\rbrace > 0$ always holds in (REF ), we discover that the RSF signal leads to a better performance of delay estimation than the monotone signal.",
"While due to the reason that $\\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 \\approx f_0^2$ holds in (), the MSEs of Doppler-stretch with the two signals are approximately the same.",
"ii) The OFDM signal can be considered as the sum of $L$ “monotone signals” regulated by $1 / \\sqrt{L}$ , which is denoted as $s(t) = \\frac{1}{\\sqrt{L}} \\sum _{k = 0}^{K - 1} \\sum _{l = 0}^{L - 1} \\beta (t - k T_r) e^{j 2 \\pi f_l (t - k T_r)},$ where $f_l = f_0 + d_l \\delta _f$ [16], [17].",
"All the available carrier frequencies within the given bandwidth for the RSF signal are simultaneously used by the OFDM signal as its subcarriers.",
"Substituting (REF ) into () and following the derivation in Section REF , we obtain $\\nonumber & \\lim _{N_0 \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{\\left| \\hat{\\tau } - \\tau _0 \\right|^2}{N_0} \\right\\rbrace \\\\& \\approx \\textstyle \\frac{1}{2 \\gamma _0 |x|^2 K} \\left\\lbrace \\left[ S_0^{(1)} + 4 \\pi ^2 \\text{Var}\\left\\lbrace f_l \\right\\rbrace S_0^{(0)} \\right]^{-1} + \\right.",
"\\\\\\nonumber & \\quad \\textstyle \\left.",
"\\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k \\right)^2 \\left[ S_2^{(1)} + 4 \\pi ^2 \\left( \\frac{1}{L} \\sum _{l = 0}^{L - 1} f_l^2 \\right) S_2^{(0)} \\right]^{-1} \\right\\rbrace , \\\\ & \\begin{aligned}& \\lim _{N_0 \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{\\left| \\hat{\\gamma } - \\gamma _0 \\right|^2}{N_0} \\right\\rbrace \\\\& \\approx \\textstyle \\frac{\\gamma _0^3}{2 |x|^2 K} \\left[ S_2^{(1)} + 4 \\pi ^2 \\left( \\frac{1}{L} \\sum _{l = 0}^{L - 1} f_l^2 \\right) S_2^{(0)} \\right]^{-1}.\\end{aligned}$ By comparing (REF ) with (REF ), it is obvious that the estimation performances with the OFDM and RSF signals are the same under high SNRs.",
"Nevertheless, as one of the synthetic wideband signals, the RSF signal only takes up a narrow instantaneous bandwidth, while on the contrary, the whole bandwidth is occupied by all the subcarriers when transmitting an OFDM signal.",
"From this perspective, the requirements for the receiver can be much reduced if the RSF signal is employed for parameter estimation.",
"Summarily, as for the performance of delay estimation, the RSF and OFDM signals have the identical performance under high SNRs.",
"Due to their wideband character, they both outperform the narrowband monotone signal.",
"While for the performance of Doppler-stretch estimation, which is mostly dependent on the central carrier frequency, the three signals have approximately the same performance."
],
[
"Derivations and Proofs",
"In this section, we provide the detailed derivations of the main results.",
"Since the AF-based joint estimation can be interpreted as searching for the maximum point of the AF, we start our derivation by focusing on the properties of $\\left| A_{y s}(\\tau , \\gamma ) \\right|^2$ at its maximum point.",
"Obviously, the partial derivatives of $\\left| A_{y s}(\\tau , \\gamma ) \\right|^2$ with respect to $\\tau $ and $\\gamma $ both reach zero at $(\\hat{\\tau }, \\hat{\\gamma })$ [8], [12], i.e., $\\begin{aligned}& \\textstyle \\frac{\\partial }{\\partial \\tau } \\left| A_{y s} \\right|^2 (\\hat{\\tau }, \\hat{\\gamma }) = 2 \\text{Re} \\left\\lbrace A^*_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\tau } A_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace = 0, \\\\& \\textstyle \\frac{\\partial }{\\partial \\gamma } \\left| A_{y s} \\right|^2 (\\hat{\\tau }, \\hat{\\gamma }) = 2 \\text{Re} \\left\\lbrace A^*_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\gamma } A_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace = 0.\\end{aligned}$ Letting $\\nonumber & \\textstyle A_{s_r s}(\\tau , \\gamma ) := \\sum _{n = 0}^{N - 1} x s(\\gamma _0 (n \\Delta - \\tau _0)) s^*(\\gamma (n \\Delta - \\tau )), \\\\& \\textstyle A_{n s}(\\tau , \\gamma ) := \\sum _{n = 0}^{N - 1} w(n \\Delta ) s^*(\\gamma (n \\Delta - \\tau )),$ we have $A_{y s}(\\hat{\\tau }, \\hat{\\gamma }) = A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) + A_{n s}(\\hat{\\tau }, \\hat{\\gamma })$ .",
"Therefore, (REF ) can be reorganized as $ & \\textstyle X = \\frac{1}{2} \\frac{\\partial }{\\partial \\tau } \\left| A_{y s} \\right|^2 (\\hat{\\tau }, \\hat{\\gamma }) = \\text{Re} \\left\\lbrace A^*_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace , \\\\ & \\textstyle Y = \\frac{1}{2} \\frac{\\partial }{\\partial \\gamma } \\left| A_{y s} \\right|^2 (\\hat{\\tau }, \\hat{\\gamma }) = \\text{Re} \\left\\lbrace A^*_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\gamma } A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace ,$ where we define $\\begin{aligned}& \\textstyle X_1 := -\\text{Re} \\left\\lbrace A^*_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace , \\\\& \\textstyle X_2 := -\\text{Re} \\left\\lbrace A^*_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\tau } A_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace , \\\\& \\textstyle X_3 := -\\text{Re} \\left\\lbrace A^*_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\tau } A_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace , \\\\& \\textstyle Y_1 := -\\text{Re} \\left\\lbrace A^*_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\gamma } A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace , \\\\& \\textstyle Y_2 := -\\text{Re} \\left\\lbrace A^*_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\gamma } A_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace , \\\\& \\textstyle Y_3 := -\\text{Re} \\left\\lbrace A^*_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\gamma } A_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace , \\\\& \\textstyle X := X_1 + X_2 + X_3, \\\\& \\textstyle Y := Y_1 + Y_2 + Y_3.\\end{aligned}$ As shown by Theorem REF , $\\hat{\\tau }$ and $\\hat{\\gamma }$ respectively converge to their true values as $\\sigma \\rightarrow 0$ .",
"This indicates that under high SNRs, $(\\hat{\\tau }, \\hat{\\gamma })$ is distributed in a neighborhood of $(\\tau _0, \\gamma _0)$ .",
"Thus we expand the right hand sides (RHSs) of () in Taylor series around $(\\tau _0, \\gamma _0)$ , respectively, $ & \\begin{aligned}& \\textstyle X = \\frac{1}{2} \\left[ (\\hat{\\tau } - \\tau _0) a(\\xi ) + (\\hat{\\gamma } - \\gamma _0) b(\\xi ) \\right], \\\\\\end{aligned} \\\\ & \\begin{aligned}& \\textstyle Y = \\frac{1}{2} \\left[ (\\hat{\\tau } - \\tau _0) c(\\eta ) + (\\hat{\\gamma } - \\gamma _0) d(\\eta ) \\right].",
"\\\\\\end{aligned}$ In () we respectively define $\\begin{aligned}& \\textstyle a(\\xi ) := \\frac{\\partial ^2}{\\partial \\tau ^2} \\left| A_{s_r s} \\right|^2 (\\tau (\\xi ), \\gamma (\\xi )), \\\\& \\textstyle b(\\xi ) := \\frac{\\partial ^2}{\\partial \\tau \\partial \\gamma } \\left| A_{s_r s} \\right|^2 (\\tau (\\xi ), \\gamma (\\xi )), \\\\& \\textstyle c(\\eta ) := \\frac{\\partial ^2}{\\partial \\tau \\partial \\gamma } \\left| A_{s_r s} \\right|^2 (\\tau (\\eta ), \\gamma (\\eta )), \\\\& \\textstyle d(\\eta ) := \\frac{\\partial ^2}{\\partial \\gamma ^2} \\left| A_{s_r s} \\right|^2 (\\tau (\\eta ), \\gamma (\\eta )),\\end{aligned}$ where $\\tau (\\xi ) = \\tau _0 + \\xi (\\hat{\\tau } - \\tau _0)$ , $\\gamma (\\xi ) = \\gamma _0 + \\xi (\\hat{\\gamma } - \\gamma _0)$ , $0 \\le \\xi \\le 1$ , and $\\tau (\\eta ) = \\tau _0 + \\eta (\\hat{\\tau } - \\tau _0)$ , $\\gamma (\\eta ) = \\gamma _0 + \\eta (\\hat{\\gamma } - \\gamma _0)$ , $0 \\le \\eta \\le 1$ .",
"Since $| A_{s_r s}(\\tau , \\gamma ) |^2$ reaches its unique maximum at $(\\tau _0, \\gamma _0)$ according to Assumption REF , both $\\text{Re} \\lbrace A^*_{s_r s}(\\tau _0, \\gamma _0) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\tau _0, \\gamma _0) \\rbrace $ and $\\text{Re} \\lbrace A^*_{s_r s}(\\tau _0, \\gamma _0) \\frac{\\partial }{\\partial \\gamma } A_{s_r s}(\\tau _0, \\gamma _0) \\rbrace $ equal zero and thus we have directly excluded them from ().",
"Then we convert () into the following forms: $ & \\textstyle d(\\eta ) \\frac{X}{\\sigma } - b(\\xi ) \\frac{Y}{\\sigma } = \\frac{1}{2} \\Lambda (\\xi , \\eta ) \\frac{\\hat{\\tau } - \\tau _0}{\\sigma }, \\\\ & \\textstyle a(\\xi ) \\frac{Y}{\\sigma } - c(\\eta ) \\frac{X}{\\sigma } = \\frac{1}{2} \\Lambda (\\xi , \\eta ) \\frac{\\hat{\\gamma } - \\gamma _0}{\\sigma },$ where $\\Lambda (\\xi , \\eta ) := a(\\xi ) d(\\eta ) - b(\\xi ) c(\\eta ).$ With MSE employed to evaluate the performance of estimation, () readily turns into $& \\begin{aligned}& \\textstyle \\mathbb {E} \\left\\lbrace \\left| d(\\eta ) \\frac{X}{\\sigma } - b(\\xi ) \\frac{Y}{\\sigma } \\right|^2 \\right\\rbrace = \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace , \\\\\\end{aligned} \\\\& \\begin{aligned} & \\textstyle \\mathbb {E} \\left\\lbrace \\left| a(\\xi ) \\frac{Y}{\\sigma } - c(\\eta ) \\frac{X}{\\sigma } \\right|^2 \\right\\rbrace = \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\gamma } - \\gamma _0}{\\sigma } \\right|^2 \\right\\rbrace .\\end{aligned}$ For each equation in (), we next calculate the limits of both sides as $\\sigma \\rightarrow 0$ , so as to investigate the MSEs of the joint estimation.",
"Based on (REF ), we focus on the further derivations for the MSE of time delay, while the MSE of Doppler-stretch can be obtained following the similar technique."
],
[
"Calculations for LHS of (",
"With the definitions in (REF ), we expand the LHS of (REF ) into the following form: $\\begin{aligned}& \\textstyle \\mathbb {E} \\left\\lbrace \\left| d(\\eta ) \\frac{X}{\\sigma } - b(\\xi ) \\frac{Y}{\\sigma } \\right|^2 \\right\\rbrace = \\sum _{i, j = 1}^3 \\mathbb {E} \\left\\lbrace d(\\eta ) \\frac{X_i X_j}{\\sigma ^2} \\right\\rbrace + \\\\& \\quad \\textstyle \\sum _{i, j = 1}^3 \\mathbb {E} \\left\\lbrace b(\\xi ) \\frac{Y_i Y_j}{\\sigma ^2} \\right\\rbrace - 2 \\sum _{i, j = 1}^3 \\mathbb {E} \\left\\lbrace b(\\xi ) d(\\eta ) \\frac{X_i Y_j}{\\sigma ^2} \\right\\rbrace .\\end{aligned}$ Hence the value of $ \\mathbb {E} \\lbrace | d(\\eta ) \\frac{X}{\\sigma } - b(\\xi ) \\frac{Y}{\\sigma } |^2 \\rbrace $ under high SNR is evaluated by successively calculating the limit of each resultant expectation on the RHS of (REF ) as $\\sigma \\rightarrow 0$ .",
"Before starting the calculations, it is worth mentioning that since $w(n \\Delta )$ follows $\\mathbb {C} N (0, \\sigma ^2)$ , the random variable $w(n \\Delta ) / \\sigma $ is thus distributed as $\\mathbb {C} N (0, 1)$ , which is independent of $\\sigma $ .",
"We firstly calculate the limit of $\\mathbb {E} \\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\rbrace $ as $\\sigma \\rightarrow 0$ .",
"For each $\\epsilon > 0$ , we have $\\nonumber & \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace = \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 > \\epsilon \\right\\rbrace } \\right\\rbrace \\\\ & ~~~~~~~~ + \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 \\le \\epsilon \\right\\rbrace } \\right\\rbrace ,$ where $I_{ \\lbrace \\cdot \\rbrace }$ denotes the indicator function [19].",
"To calculate the first term on the RHS of (REF ), we know from Theorem REF that $\\begin{aligned}& \\textstyle \\quad d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 > \\epsilon \\right\\rbrace } \\stackrel{P}{\\longrightarrow } 0 \\\\\\end{aligned}$ as $\\sigma \\rightarrow 0$ , and $\\begin{aligned}& \\textstyle \\left| d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 > \\epsilon \\right\\rbrace } \\right| \\\\& \\quad \\le \\textstyle C_0^2 \\frac{X_1^2}{\\sigma ^2} = C_0^2 \\left( \\text{Re} \\left\\lbrace A_{\\frac{n}{\\sigma }, s}^* (\\hat{\\tau }, \\hat{\\gamma }) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) \\right\\rbrace \\right)^2 \\\\& \\quad \\le \\textstyle C_0^4 \\left( \\max _t \\left| s(t) \\right| \\sum _{n = 0}^{N - 1} \\left| \\frac{w(n \\Delta )}{\\sigma } \\right| \\right)^2,\\end{aligned}$ where $\\textstyle A_{\\frac{n}{\\sigma }, s} (\\tau , \\gamma ) := \\sum _{n = 0}^{N - 1} \\frac{w(n \\Delta )}{\\sigma } s^*(\\gamma (n \\Delta - \\tau ))$ and $\\nonumber C_0 := & \\max \\left\\lbrace \\max _{ \\tau \\in [\\tau _{\\text{min}}, \\tau _{\\text{max}}], \\gamma \\in [\\gamma _{\\text{min}}, \\gamma _{\\text{max}}] } \\textstyle \\left| \\frac{\\partial }{\\partial \\tau } A_{s_r s} (\\tau , \\gamma ) \\right|, \\right.",
"\\\\& \\left.",
"\\max _{ \\tau \\in [\\tau _{\\text{min}}, \\tau _{\\text{max}}], \\gamma \\in [\\gamma _{\\text{min}}, \\gamma _{\\text{max}}] } \\textstyle \\left| \\frac{\\partial ^2}{\\partial \\gamma ^2} \\left| A_{s_r s} \\right|^2 (\\tau , \\gamma ) \\right| \\right\\rbrace .$ For (REF ), also note that $\\mathbb {E} \\lbrace C_0^4 ( \\max _t | s(t) | \\sum _{n = 0}^{N - 1} | \\frac{w(n \\Delta )}{\\sigma } | )^2 \\rbrace < + \\infty $ .",
"Therefore, according to Lebesgue's dominated convergence theorem [19], we obtain $\\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 > \\epsilon \\right\\rbrace } \\right\\rbrace = 0.$ For the second term on the RHS of (REF ), on one hand, $& \\liminf _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 \\le \\epsilon \\right\\rbrace } \\right\\rbrace \\\\\\nonumber & \\quad \\ge \\min _{\\Vert (\\tau , \\gamma ) - (\\tau _0, \\gamma _0) \\Vert _2 \\le \\epsilon } \\textstyle \\left( \\frac{\\partial ^2}{\\partial \\gamma ^2} \\left| A_{s_r s}(\\tau , \\gamma ) \\right|^2 \\right)^2 \\displaystyle \\liminf _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace .$ Letting $\\epsilon \\rightarrow 0$ and with (REF ), we have $\\liminf _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace \\ge d^2(0) \\displaystyle \\liminf _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace .$ On the other hand, it can be also derived that $\\limsup _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace \\le d^2(0) \\displaystyle \\limsup _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace .$ Combining (REF ) and (REF ), we obtain $\\begin{aligned}& d^2(0) \\liminf _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace \\le \\displaystyle \\liminf _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace \\\\& \\quad \\le \\limsup _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace \\le d^2(0) \\displaystyle \\limsup _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace .\\end{aligned}$ It is clear that if $\\lim _{\\sigma \\rightarrow 0} \\mathbb {E} \\lbrace \\frac{X_1^2}{\\sigma ^2} \\rbrace $ exists, the existence of $\\lim _{\\sigma \\rightarrow 0} \\mathbb {E} \\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\rbrace $ can be also confirmed.",
"With Theorem REF and (REF ), $\\begin{aligned}\\textstyle \\frac{X_1^2}{\\sigma ^2} \\stackrel{P}{\\longrightarrow } \\left( \\text{Re} \\left\\lbrace A_{\\frac{n}{\\sigma }, s}^* (\\tau _0, \\gamma _0) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\tau _0, \\gamma _0) \\right\\rbrace \\right)^2\\end{aligned}$ as $\\sigma \\rightarrow 0$ .",
"Again, following Lebesgue's dominated convergence theorem, $\\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace = \\mathbb {E} \\left\\lbrace \\left( \\text{Re} \\left\\lbrace A_{\\frac{n}{\\sigma }, s}^* (\\tau _0, \\gamma _0) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\tau _0, \\gamma _0) \\right\\rbrace \\right)^2 \\right\\rbrace .$ Recall that $w(n \\Delta ) / \\sigma \\sim \\mathbb {C} N (0, 1)$ .",
"As a result, the random variable $A_{\\frac{n}{\\sigma }, s}^* (\\tau _0, \\gamma _0) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\tau _0, \\gamma _0)$ in (REF ) follows $\\mathbb {C}N (0, \\sum _{n \\in \\mathcal {N}}| s(\\gamma _0 (n \\Delta - \\tau _0)) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\tau _0, \\gamma _0) |^2)$ .",
"Then (REF ) can be calculated as $\\nonumber \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace = & \\textstyle \\frac{1}{2} \\sum _{n \\in \\mathcal {N}} \\left| s(\\gamma _0 (n \\Delta - \\tau _0)) \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\tau _0, \\gamma _0) \\right|^2 \\\\= & \\textstyle \\frac{1}{2} E_s \\left| \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\tau _0, \\gamma _0) \\right|^2,$ where we respectively define $E_s := \\sum _{n = 0}^{N - 1} | s(\\gamma _0 (n \\Delta - \\tau _0)) |^2$ , and $\\mathcal {N} := \\lbrace n | s(\\gamma _0 (n \\Delta - \\tau _0)) \\ne 0, 0 \\le n \\le N - 1, n \\in \\mathbb {N} \\rbrace $ .",
"The indices collected by $\\mathcal {N}$ in fact corresponds to all the non-zero samples of $s(t)$ .",
"Totally, together with (REF ) and (REF ), we conclude that $\\lim _{\\sigma \\rightarrow 0} \\mathbb {E} \\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\rbrace $ exists and $\\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta ) \\frac{X_1^2}{\\sigma ^2} \\right\\rbrace = \\frac{1}{2} d^2(0) E_s \\left| \\frac{\\partial }{\\partial \\tau } A_{s_r s}(\\tau _0, \\gamma _0) \\right|^2.$ The calculations for the rest terms in (REF ) can be performed similarly.",
"Due to limited space, we eliminate the tedious and repetive calculating processes as presented above, whereas the corresponding results can be found in Appendix C."
],
[
"Calculations for RHS of (",
"As for the RHS of (REF ), given $\\epsilon > 0$ , we have $& \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace \\\\\\nonumber & = \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 > \\epsilon \\right\\rbrace } \\right\\rbrace + \\\\\\nonumber & \\quad ~ \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 \\le \\epsilon \\right\\rbrace } \\right\\rbrace ,$ where the two terms on the RHS can be calculated in accordance to the following lemma: Lemma 4.1 For any random variable $H(\\omega )$ which is bounded almost everywhere and any $\\epsilon > 0$ , $ & \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace H(\\omega ) \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 I_{\\lbrace \\omega : \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 > \\epsilon \\rbrace } \\right\\rbrace = 0, \\\\& \\begin{aligned}& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace H(\\omega ) \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 I_{\\lbrace \\omega : \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 \\le \\epsilon \\rbrace } \\right\\rbrace \\\\& = \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace H(\\omega ) \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace ,\\end{aligned}$ where the limit symbols on both sides of () are replaced by superior or inferior limits if the limit on either side of () does not exist.",
"See Appendix REF .",
"By Lemma REF , the first term on the RHS of (REF ) directly equals zero.",
"While for the second term, on one hand, $\\begin{aligned}& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 I_{ \\left\\lbrace \\left\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right\\Vert _2 \\le \\epsilon \\right\\rbrace } \\right\\rbrace \\\\& \\ge \\frac{1}{4} \\min _{ \\left\\Vert (\\tau _1, \\gamma _1) - (\\tau _0, \\gamma _0) \\right\\Vert _2 \\le \\epsilon , \\left\\Vert (\\tau _2, \\gamma _2) - (\\tau _0, \\gamma _0) \\right\\Vert _2 \\le \\epsilon } \\\\& \\textstyle \\quad \\left| \\frac{\\partial ^2}{\\partial \\tau ^2} \\left| A_{s_r s} \\right|^2 (\\tau _1, \\gamma _1) \\frac{\\partial ^2}{\\partial \\gamma ^2} \\left| A_{s_r s} \\right|^2 (\\tau _2, \\gamma _2) - \\right.",
"\\\\& \\textstyle \\quad \\left.",
"\\frac{\\partial ^2}{\\partial \\tau \\partial \\gamma } \\left| A_{s_r s} \\right|^2 (\\tau _1, \\gamma _1) \\frac{\\partial ^2}{\\partial \\tau \\partial \\gamma } \\left| A_{s_r s} \\right|^2 (\\tau _2, \\gamma _2) \\right|^2 \\times \\\\& \\quad \\limsup _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace .\\end{aligned}$ Since (REF ) holds for all $\\epsilon > 0$ , letting $\\epsilon \\rightarrow 0$ , we have $\\begin{aligned}& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace \\\\& \\ge \\textstyle \\frac{1}{4} \\Lambda ^2(0, 0) \\displaystyle \\limsup _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace .\\end{aligned}$ On the other hand, we derive $\\begin{aligned}& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace \\\\& \\le \\textstyle \\frac{1}{4} \\Lambda ^2(0, 0) \\displaystyle \\liminf _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace .\\end{aligned}$ By (REF ) and (REF ), it can be concluded that $\\begin{aligned}\\lim _{\\sigma \\rightarrow 0} & \\textstyle \\frac{1}{4} \\mathbb {E} \\left\\lbrace \\left| \\Lambda (\\xi , \\eta ) \\right|^2 \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace = \\textstyle \\frac{1}{4} \\Lambda ^2(0, 0) \\displaystyle \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace .\\end{aligned}$ Based on Assumption REF , the MSE of time delay can be obtained with the results in Section REF and (REF ), i.e., $& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace \\left| \\frac{\\hat{\\tau } - \\tau _0}{\\sigma } \\right|^2 \\right\\rbrace \\\\\\nonumber & = \\lim _{\\sigma \\rightarrow 0} \\textstyle \\frac{ \\mathbb {E} \\left\\lbrace d^2(\\eta (\\omega )) \\frac{X^2}{\\sigma ^2} \\right\\rbrace + \\mathbb {E} \\left\\lbrace b^2(\\xi (\\omega )) \\frac{Y^2}{\\sigma ^2} \\right\\rbrace - 2 \\mathbb {E} \\left\\lbrace b(\\xi (\\omega )) d(\\eta (\\omega )) \\frac{X Y}{\\sigma ^2} \\right\\rbrace }{ \\textstyle \\frac{1}{4} \\Lambda ^2(0, 0) }.$ Lastly, letting the sampling rate become sufficiently large and employing the definitions in (REF ), we eventually derive the integration representation of the MSE in (REF )."
],
[
"Approximation in (",
"We next show that the approximation conducted in () is a proper one.",
"Based on Assumption REF , we substitute (REF ) into $D$ and $E$ , respectively, resulting in $& \\textstyle D = \\sum _{k = 0}^{K - 1} \\int _{0}^{T} (t + k T_r)^2 \\left[ \\dot{\\beta }^2 (t) + 4 \\pi ^2 f_k^2 \\beta ^2(t) \\right] dt, \\\\&\\textstyle E = K \\int _{0}^{T} \\beta ^2(t) dt.$ Assuming that the envelope $\\beta (t)$ is a symmetric function, i.e.",
"$\\beta (t) = \\beta (T - t)$ , we perform the derivation as follows: $\\begin{aligned}\\frac{D}{E} & \\ge \\frac{ 4 \\pi ^2 \\sum _{k = 0}^{K - 1} f_k^2 \\int _{0}^{T} (t + k T_r)^2 \\beta ^2(t) dt }{ K \\int _{0}^{T} \\beta ^2(t) dt } \\\\& \\ge 4 \\pi ^2 \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 \\frac{ \\int _{0}^{T} t^2 \\beta ^2(t) dt }{ \\int _{0}^{T} \\beta ^2(t) dt } \\\\& \\approx 4 \\pi ^2 f_0^2 \\frac{ \\int _{0}^{T/2} t^2 \\beta ^2(t) dt + \\int _{0}^{T/2} (T - t)^2 \\beta ^2(t) dt }{ 2 \\int _{0}^{T/2} \\beta ^2(t) dt } \\\\& \\ge 4 \\pi ^2 f_0^2 \\frac{ \\frac{T^2}{2} \\int _{0}^{T/2} \\beta ^2(t) dt }{ 2 \\int _{0}^{T/2} \\beta ^2(t) dt } = \\pi ^2 T^2 f_0^2 \\gg 1.\\end{aligned}$ The last line of (REF ) is generally satisfied for a radar signal, since the central carrier frequency $f_0$ is usually several orders of magnitude larger than $1 / T$ .",
"Therefore, with the fact that $D \\gg E$ , the terms with $E^2$ in (REF ) are much smaller compared to those with $E D$ and thus can be eliminated.",
"The approximation made in () is hence a proper one, and the performance of the AF-based estimation can be evaluated by the much simplified forms of MSEs as given by ()."
],
[
"Proof of Theorem ",
"We utilize the symmetry property of the envelope $\\beta (t)$ .",
"As $\\beta (t) = \\beta (T - t)$ is satisfied in most cases, we can also derive that $\\dot{\\beta }(t) = -\\dot{\\beta }(T - t)$ .",
"As a result, the following relations are readily satisfied: $& \\textstyle \\int _{0}^{T} \\left(t - \\frac{T}{2} \\right) \\beta ^2(t) dt = 0, \\\\& \\textstyle \\int _{0}^{T} \\left(t - \\frac{T}{2} \\right) \\dot{\\beta }^2(t) dt = 0.$ With (REF ), we respectively obtain $\\begin{aligned}& \\textstyle B - \\frac{F^2}{E} = K \\left[ S_0^{(1)} + 4 \\pi ^2 \\text{Var}\\left\\lbrace f_k \\right\\rbrace S_0^{(0)} \\right],\\end{aligned}$ $\\nonumber & \\textstyle D - \\frac{G^2}{E} = \\textstyle K S_2^{(1)} + 4 \\pi ^2 \\sum _{k = 0}^{K - 1} f_k^2 \\int _{0}^{T} \\left(t - \\frac{T}{2} \\right)^2 \\beta ^2(t) dt + \\\\\\nonumber & \\textstyle \\quad 4 \\pi ^2 K \\Omega _1 S_0^{(0)} + K \\left[ \\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k \\right)^2 S_0^{(1)} + \\right.",
"\\\\ & \\textstyle \\quad \\left.",
"4 \\pi ^2 \\frac{ 1 }{ \\text{Var} \\left\\lbrace f_k \\right\\rbrace } \\text{Cov} \\left\\lbrace f_k, f_k T_k \\right\\rbrace S_0^{(0)} \\right],$ $\\nonumber & \\textstyle \\frac{\\Pi }{E^2} = \\textstyle K \\left(B - \\frac{F^2}{E} \\right) \\left[ S_2^{(1)} + 4 \\pi ^2 \\Omega _1 S_0^{(0)} + \\right.",
"\\\\ & \\textstyle \\left.",
"4 \\pi ^2 \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 \\int _{0}^{T} \\left(t - \\frac{T}{2} \\right)^2 \\beta ^2(t) dt \\right] + 4 \\pi ^2 K^2 \\Omega _2^2 S_0^{(1)} S_0^{(0)},$ where $& \\textstyle \\Omega _1 = \\left( 1 - \\rho ^2(f_k, f_k T_k) \\right) \\text{Var}\\left\\lbrace f_k T_k \\right\\rbrace , \\\\& \\textstyle \\Omega _2 = \\text{Std}\\left\\lbrace f_k \\right\\rbrace \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k - \\rho (f_k, f_k T_k) \\text{Std}\\left\\lbrace f_k T_k \\right\\rbrace .$ For (REF ) and (REF ), further simplifications can be conducted.",
"According to Assumption REF , we derive that $\\text{Cov}\\left\\lbrace f_k, T_k \\right\\rbrace = \\text{Cov}\\left\\lbrace f_k^2, T_k \\right\\rbrace = \\text{Cov}\\left\\lbrace f_k^2, T_k^2 \\right\\rbrace = 0.$ As a result, $\\nonumber & \\Omega _2 = \\textstyle \\frac{1}{ \\text{Std}\\left\\lbrace f_k \\right\\rbrace } \\left[ \\text{Var}\\left\\lbrace f_k \\right\\rbrace \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k - \\text{Cov}\\left\\lbrace f_k, f_k T_k \\right\\rbrace \\right] \\\\& = \\textstyle \\frac{1}{ \\text{Std}\\left\\lbrace f_k \\right\\rbrace } \\left[ \\text{Cov}\\left\\lbrace f_k, T_k \\right\\rbrace \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k - \\text{Cov}\\left\\lbrace f_k^2, T_k \\right\\rbrace \\right] = 0.$ The second term on the RHS of (REF ) thus equals zero.",
"On the other hand, since $\\text{Var}\\left\\lbrace f_k \\right\\rbrace \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k = \\text{Cov}\\left\\lbrace f_k, f_k T_k \\right\\rbrace $ (as revealed by the first line of (REF )), and together with (REF ), $\\nonumber & \\Omega _1 = \\textstyle \\text{Var}\\left\\lbrace f_k T_k \\right\\rbrace - \\text{Var}\\left\\lbrace f_k \\right\\rbrace \\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k \\right)^2 \\\\\\nonumber & = \\textstyle \\text{Cov}\\left\\lbrace f_k^2, T_k^2 \\right\\rbrace + \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 \\text{Var}\\left\\lbrace T_k \\right\\rbrace - \\text{Cov}\\left\\lbrace f_k, T_k \\right\\rbrace \\times \\\\\\nonumber & \\quad \\textstyle \\left[ \\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k \\right) \\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k \\right) + \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k T_k \\right] \\\\& = \\textstyle \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 \\text{Var}\\left\\lbrace T_k \\right\\rbrace .$ Substituting (REF )–(REF ) into (REF ) and (REF ), we eventually have $\\nonumber & \\textstyle D - \\frac{G^2}{E} = \\textstyle K \\left[ S_2^{(1)} + 4 \\pi ^2 \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 S_2^{(0)} \\right] + \\\\& ~~~~~~ \\textstyle \\left( \\frac{1}{K} \\sum _{k = 0}^{K - 1} T_k \\right)^2 \\left( B - \\frac{F^2}{E} \\right), \\\\& \\textstyle \\frac{\\Pi }{E^2} = \\textstyle K \\left( B - \\frac{F^2}{E} \\right) \\left[ S_2^{(1)} + 4 \\pi ^2 \\frac{1}{K} \\sum _{k = 0}^{K - 1} f_k^2 S_2^{(0)} \\right].$ Lastly, by substituting (REF ) and (REF ) into (), the compact expressions of MSEs as presented by Theorem REF are concluded."
],
[
"Numerical Results",
"In this section, we verify the correctness of our main results by numerical examples.",
"Set the envelope of the RSF signal $\\beta (t)$ as $\\beta (t) = \\left\\lbrace \\begin{array}{ll}t^3 (T - t)^3, & \\text{if } t \\in [0, T], \\\\0, & \\text{otherwise},\\end{array}\\right.$ which not only attains a rectangle-like shape, but also satisfies all the assumptions in Section II-B.",
"The SNR at the receiver is defined as $\\text{SNR} := \\frac{1}{N_0} \\int _{-\\infty }^{\\infty } \\left| x s(\\gamma _0 (t - \\tau _0)) \\right|^2 dt$ throughout the simulations [8].",
"Some main parameters of the simulation environment are configured in TABLE REF .",
"We first compare the MSEs of the AF-based estimation (REF ) with their theoretical counterparts (as given by Theorem REF or REF ) by simulations.",
"Under each of the SNRs from 5 dB to 40 dB, the simulated and theoretical MSEs are calculated on the basis of 200 independent Monte Carlo simulations.",
"In each trial, a Costas frequency shifting codeword is randomly generated from a set $\\mathcal {C} = \\lbrace -2.5, -1.5, -0.5, 0.5, 1.5, 2.5 \\rbrace $ [20].",
"The central carrier frequency $f_0$ and the minimum frequency shifting step $\\delta _f$ are respectively set to 20 MHz and 2 MHz.",
"As depicted in Figs.",
"REF and REF , when $\\text{SNR} \\ge 15$ dB, both the simulated MSEs of time delay and Doppler-stretch perfectly converge to their theoretical counterparts, indicating that the performance of the AF-based estimation with the RSF signal under high SNRs can be accurately described by Theorems REF –REF .",
"Table: Parameter settings for simulations.Figure: Theoretical and simulated MSEs of time delay.Figure: Theoretical and simulated MSEs of Doppler-stretch."
],
[
"Influence of Waveform Parameters",
"Next, we investigate the influence of the waveform parameters on the estimation performance with the RSF signal.",
"In order to reveal the relation between the estimation performance and the bandwidth of the carrier frequencies, we fix $f_0$ to 20 MHz and set $\\delta _f$ to $[2, 4, 6]$ MHz, respectively, in three sets of simulations.",
"The Costas codeword is still adopted.",
"Figs.",
"REF and REF illustrate the estimation performances with the three different bandwidths.",
"Just as predicted by Theorem REF , under each SNR value, with the increasing of $\\delta _f$ , the MSE of time delay decreases, while there is no significant variation found in the MSEs of Doppler-stretch.",
"Figure: MSEs of time delay with different bandwidths.Figure: MSEs of Doppler-stretch with different bandwidths.Within a fixed bandwidth covered by the carrier frequencies, we explore the impact of different frequency shifting patterns on the estimation performance.",
"Besides the Costas codeword as generated above, we also involve a Dumbbell codeword for comparison, with which only the lowest and the highest frequencies (i.e.",
"$f_0 - 2.5 \\delta _f$ and $f_0 + 2.5 \\delta _f$ ) are available for using.",
"Set $f_0$ and $\\delta _f$ to 20 MHz and 2 MHz, respectively.",
"The estimation performances with the two frequency shifting codewords are shown in Figs.",
"REF and REF .",
"The Dumbell codeword always outperforms the Costas codeword in delay estimating, while the performances of Doppler-stretch estimation with the two codewords are almost the same.",
"This is consistent with our conclusion that a randomized codeword with larger variance leads to a better performance of delay estimation.",
"Figure: MSEs of time delay with different codewords.Figure: MSEs of Doppler-stretch with different codewords.Then we compare the estimation performances with different central carrier frequencies.",
"Set $f_0$ to $[10, 20, 30]$ MHz, respectively, while fix the minimum frequency shifting step $\\delta _f$ to 2 MHz.",
"The estimation performances with the Costas encoded RSF signal are shown in Figs.",
"REF and REF .",
"As $f_0$ increases, the MSE of Doppler-stretch evidently descends, while that of time delay only slightly decreases.",
"Beyond the revealings by Theorem REF , we summarise from the simulation results that the central carrier frequency mainly determines the performance of Doppler-stretch estimation, while its impact on the performance of delay estimation is negligible.",
"Instead, it is the bandwidth covered by the carrier frequencies that dominates the performance of delay estimation.",
"Figure: MSEs of time delay with different central carrier frequencies.Figure: MSEs of Doppler-stretch with different central carrier frequencies."
],
[
"Comparison with Other Waveforms",
"Figs.",
"REF and REF compare the estimation performances with the RSF, OFDM and monotone signals.",
"For the RSF signal, reset $f_0$ and $\\delta _f$ to 20 MHz and 2 MHz, respectively, and still adopt the Costas codeword.",
"The subcarriers of the OFDM signal satisfy $\\lbrace d_l; l = 0, 1, \\ldots , L - 1 \\rbrace = \\mathcal {C}$ , while the carrier frequency of the monotone signal is 20 MHz.",
"In terms of delay estimation as shown in Fig.",
"REF , due to the wideband character, the RSF signal and the OFDM signal both outperform the narrowband monotone signal, leading by up to 15 dB of performance gain.",
"It is also observed that the performance with the RSF signal is almost the same as that with the OFDM signal, since the two signals take up the same bandwidth of the carrier frequencies.",
"For the Doppler-stretch estimation as illustrated by Fig.",
"REF , the three signals achieve the similar estimation performances.",
"This can be explained by (), (), and (REF ), where the identical central carrier frequencies of the three signals contribute to the similar performances of Doppler-stretch estimation.",
"Figure: MSEs of time delay with different waveforms.Figure: MSEs of Doppler-stretch with different waveforms."
],
[
"Concluding Remarks",
"We investigated the performance of joint delay-Doppler estimation with the RSF signals.",
"Compact expressions of MSEs with respect to time delay and Doppler-stretch were obtained, revealing the major waveform parameters that influence the estimation performance.",
"Since the derived theoretical MSEs were consistent with the corresponding CRLBs, the AF-based estimation was shown approximately efficient.",
"So as to improve the estimation performance with the RSF signal, higher carrier frequencies, wider bandwidth covered by the carrier frequencies, as well as codewords with larger variance are expected.",
"The RSF signal achieves the same estimation performance as the OFDM signal with only a narrow instantaneous bandwidth.",
"Requirement for the bandwidth of the receiver is thus largely reduced in practical radar systems."
],
[
"Proof of Theorem ",
"We first investigate the gap between $\\left| A_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\right|$ and $\\left| A_{s_r s}(\\tau _0, \\gamma _0) \\right|$ in the $L^p$ sense.",
"$\\begin{aligned}& \\max _{ \\tau \\in [\\tau _{\\text{min}}, \\tau _{\\text{max}}], \\gamma \\in [\\gamma _{\\text{min}}, \\gamma _{\\text{max}}] } \\left| A_{y s}(\\tau , \\gamma ) \\right| \\ge \\left| A_{y s}(\\tau _0, \\gamma _0) \\right| \\\\& \\ge \\textstyle \\left| A_{s_r s}(\\tau _0, \\gamma _0) \\right| - \\left| A_{n s}(\\tau _0, \\gamma _0) \\right| \\\\& = \\textstyle \\left| A_{s_r s}(\\tau _0, \\gamma _0) \\right| - \\left| \\sum _{n = 0} ^ {N - 1} w(n \\Delta ) s^*(\\gamma _0 (n \\Delta - \\tau _0)) \\right|.",
"\\\\\\end{aligned}$ $\\begin{aligned}& \\textstyle \\left| A_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\right| \\\\= & \\textstyle \\left| A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma }) + \\sum _{n = 0} ^ {N - 1} w(n \\Delta ) s^*(\\hat{\\gamma } (n \\Delta - \\hat{\\tau })) \\right| \\\\\\le & \\textstyle \\left| A_{s_r s}(\\tau _0, \\gamma _0) \\right| + \\left| \\sum _{n = 0} ^ {N - 1} w(n \\Delta ) s^*(\\hat{\\gamma } (n \\Delta - \\hat{\\tau })) \\right|.",
"\\\\\\end{aligned}$ Thus, $\\begin{aligned}& \\textstyle \\left| \\left| A_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\right| - \\left| A_{s_r s}(\\tau _0, \\gamma _0) \\right| \\right| \\\\& \\le \\textstyle \\left| \\sum _{n = 0} ^ {N - 1} w(n \\Delta ) s^*(\\hat{\\gamma } (n \\Delta - \\hat{\\tau })) \\right|.\\end{aligned}$ This also implies that $\\begin{aligned}& \\textstyle \\mathbb {E} \\left\\lbrace \\left| \\left| A_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\right| - \\left| A_{s_r s}(\\tau _0, \\gamma _0) \\right| \\right|^p \\right\\rbrace \\\\& \\le \\textstyle \\mathbb {E} \\left\\lbrace \\left( \\sum _{n = 0} ^ {N - 1} \\left| w(n \\Delta ) s^*(\\hat{\\gamma } (n \\Delta - \\hat{\\tau })) \\right| \\right)^p \\right\\rbrace \\\\& \\le \\textstyle N^{p - 1} \\left( \\max _t \\left| s(t) \\right| \\right)^p \\mathbb {E} \\left\\lbrace \\sum _{n = 0} ^ {N - 1} \\left| w(n \\Delta ) \\right|^p \\right\\rbrace .",
"\\\\\\end{aligned}$ Recall that $w(n \\Delta ) \\sim \\mathbb {C} N (0, \\sigma ^2)$ .",
"The PDF of $\\left| w(n \\Delta ) \\right|$ is then $p(\\eta ) = \\frac{2 \\eta }{\\sigma ^2} e^{-\\frac{\\eta ^2}{\\sigma ^2}}, \\quad \\eta > 0.$ As a result, $\\textstyle \\mathbb {E} \\left\\lbrace \\left| w(n \\Delta ) \\right|^p \\right\\rbrace = \\sigma ^p \\Gamma (\\frac{p}{2} + 1).$ For $p \\ge 1$ , $\\begin{aligned}& \\textstyle \\mathbb {E} \\left\\lbrace \\left| \\left| A_{y s}(\\hat{\\tau }, \\hat{\\gamma }) \\right| - \\left| A_{s_r s}(\\tau _0, \\gamma _0) \\right| \\right|^p \\right\\rbrace \\\\& \\le \\textstyle N^{p} \\sigma ^p \\Gamma (\\frac{p}{2} + 1) \\left( \\max _t \\left| s(t) \\right| \\right)^p.\\end{aligned}$ Next, we investigate the gap between $| A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })|$ and $|A_{s_r s}(\\tau _0, \\gamma _0)|$ .",
"$\\begin{aligned}& \\mathbb {E} \\left\\lbrace \\left| | A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right|^p \\right\\rbrace \\\\& \\le \\mathbb {E} \\left\\lbrace \\left( \\left| |A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{y s}(\\hat{\\tau }, \\hat{\\gamma })| \\right| + \\right.",
"\\right.",
"\\\\& \\quad \\left.",
"\\left.",
"\\left| |A_{y s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right| \\right)^p \\right\\rbrace \\\\& \\le 2^{p - 1} \\left( \\mathbb {E} \\left\\lbrace \\left| |A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{y s}(\\hat{\\tau }, \\hat{\\gamma })| \\right|^p \\right\\rbrace + \\right.",
"\\\\& \\quad \\left.",
"\\mathbb {E} \\left\\lbrace \\left| |A_{y s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right|^p \\right\\rbrace \\right),\\end{aligned}$ where $\\begin{aligned}& \\mathbb {E} \\left\\lbrace \\left| |A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{y s}(\\hat{\\tau }, \\hat{\\gamma })| \\right|^p \\right\\rbrace \\le \\mathbb {E} \\left\\lbrace \\left| A_{n s}(\\hat{\\tau }, \\hat{\\gamma }) \\right|^p \\right\\rbrace \\\\& \\le \\textstyle \\left( \\max _t \\left| s(t) \\right| \\right)^p \\mathbb {E} \\left\\lbrace \\left( \\sum _{n = 0}^{N - 1} \\left| w(n \\Delta ) \\right| \\right)^p \\right\\rbrace \\\\& \\le \\textstyle N^{p - 1} \\left( \\max _t \\left| s(t) \\right| \\right)^p \\mathbb {E} \\left\\lbrace \\sum _{n = 0} ^ {N - 1} \\left| w(n \\Delta ) \\right|^p \\right\\rbrace \\\\& = \\textstyle N^{p} \\sigma ^p \\Gamma (\\frac{p}{2} + 1) \\left( \\max _t \\left| s(t) \\right| \\right)^p.\\end{aligned}$ Thus, we conclude that for $p \\ge 1$ , $\\begin{aligned}& \\textstyle \\mathbb {E} \\left\\lbrace \\left| | A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right|^p \\right\\rbrace \\\\& \\le \\textstyle 2^p N^{p} \\sigma ^p \\Gamma (\\frac{p}{2} + 1) \\left( \\max _t \\left| s(t) \\right| \\right)^p.\\end{aligned}$ Then we prove Theorem REF by contradiction.",
"Suppose $(\\hat{\\tau }, \\hat{\\gamma })$ does not converge to $(\\tau _0, \\gamma _0)$ in probability as $\\sigma \\rightarrow 0$ , then there exists $\\epsilon _1 > 0$ such that $P(\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\Vert _2 > \\epsilon _1) \\nrightarrow 0$ as $\\sigma \\rightarrow 0$ .",
"Therefore, there exist an $\\epsilon _2 > 0$ and a sequence $\\left\\lbrace \\sigma _n \\right\\rbrace _{n \\ge 1}$ satisfying $\\sigma _n \\rightarrow 0$ as $n \\rightarrow +\\infty $ , such that $P(\\Vert (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\Vert _2 > \\epsilon _1) \\ge \\epsilon _2$ holds for all $n \\ge 1$ .",
"Then, for the aforementioned $\\epsilon _1$ , there exists $C(\\epsilon _1) > 0$ , such that for all $(\\tau , \\gamma )$ satisfying $\\Vert (\\tau , \\gamma ) - (\\tau _0, \\gamma _0) \\Vert _2 \\ge \\epsilon _1$ , $\\left| | A_{s_r s}(\\tau , \\gamma )| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right| \\ge C(\\epsilon _1).$ Thus, $\\nonumber & \\mathbb {E} \\left\\lbrace \\left| | A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right|^p \\right\\rbrace \\\\\\nonumber & \\ge \\mathbb {E} \\left\\lbrace \\left| | A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right|^p I_{\\lbrace \\left| (\\hat{\\tau }, \\hat{\\gamma }) - (\\tau _0, \\gamma _0) \\right| > \\epsilon _1 \\rbrace } \\right\\rbrace \\\\& \\ge \\left[ C(\\epsilon _1) \\right]^p \\epsilon _2,$ implying that $\\mathbb {E} \\left\\lbrace \\left| | A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right|^p \\right\\rbrace \\nrightarrow 0$ as $n \\rightarrow +\\infty $ , which contradicts (REF ).",
"Theorem REF is thus proved."
],
[
"Proof of Lemma ",
" For $p > 4$ , according to (REF ), we have $\\nonumber & \\lim _{\\sigma \\rightarrow 0} \\textstyle \\frac{1}{\\sigma ^2} \\left[ P \\left( \\omega : \\left| | A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right| > C_{\\epsilon } \\right) \\right]^{\\frac{1}{2}} \\\\& \\le \\lim _{\\sigma \\rightarrow 0} \\textstyle \\frac{1}{\\sigma ^2} \\left[ \\mathbb {E} \\left\\lbrace \\left| | A_{s_r s}(\\hat{\\tau }, \\hat{\\gamma })| - |A_{s_r s}(\\tau _0, \\gamma _0)| \\right|^p \\right\\rbrace C_{\\epsilon }^{-p} \\right]^{\\frac{1}{2}} \\\\\\nonumber & \\le \\lim _{\\sigma \\rightarrow 0} \\textstyle \\sigma ^{\\frac{p}{2} - 2} \\left[ 2^p N^{p} \\Gamma (\\frac{p}{2} + 1) \\left( \\max _t \\left| s(t) \\right| \\right)^p C_{\\epsilon }^{-p} \\right]^{\\frac{1}{2}} = 0.$ Therefore, (REF ) and () are concluded."
],
[
"Calculations for (",
"Imitating the calculation for $\\mathbb {E} \\lbrace d^2(\\eta (\\omega )) \\frac{X_1^2}{\\sigma ^2} \\rbrace $ in Section IV-A, we obtain the following results: $\\begin{aligned}& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta (\\omega )) \\frac{X_3^2}{\\sigma ^2} \\right\\rbrace \\\\& = \\textstyle d^2(0) \\frac{E_s^2 \\gamma _0^2 \\left| x \\right|^2}{2} \\sum _{n \\in \\mathcal {N}} \\left| \\dot{s}(\\gamma _0 (n \\Delta - \\tau _0)) \\right|^2, \\\\\\end{aligned}$ $\\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace b^2(\\xi (\\omega )) \\frac{Y_1^2}{\\sigma ^2} \\right\\rbrace = b^2(0) \\frac{E_s}{2} \\left| \\frac{\\partial }{\\partial \\gamma } A_{s_r s}(\\tau _0, \\gamma _0) \\right|^2,$ $\\begin{aligned}& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace b^2(\\xi (\\omega )) \\frac{Y_3^2}{\\sigma ^2} \\right\\rbrace \\\\& = \\textstyle b^2(0) \\frac{E_s^2 |x|^2}{2} \\sum _{n \\in \\mathcal {N}} \\left| (n \\Delta - \\tau _0) \\dot{s}(\\gamma _0 (n \\Delta - \\tau _0)) \\right|^2,\\end{aligned}$ $\\begin{aligned}& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace d^2(\\eta (\\omega )) \\frac{X_1 X_3}{\\sigma ^2} \\right\\rbrace = d^2(0) \\frac{\\gamma _0^2 \\left| x \\right|^2 E_s}{2} \\times \\\\& \\textstyle \\text{Re} \\left\\lbrace \\left( \\sum _{n \\in \\mathcal {N}} s(\\gamma _0 (n \\Delta - \\tau _0)) \\dot{s}^*(\\gamma _0 (n \\Delta - \\tau _0)) \\right)^2 \\right\\rbrace , \\\\\\end{aligned}$ $& \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace b^2(\\xi (\\omega )) \\frac{Y_1 Y_3}{\\sigma ^2} \\right\\rbrace = b^2(0) \\frac{\\left| x \\right|^2 E_s}{2} \\times \\\\\\nonumber & \\textstyle \\text{Re} \\left\\lbrace \\left( \\sum _{n \\in \\mathcal {N}} s(\\gamma _0 (n \\Delta - \\tau _0)) (n \\Delta - \\tau _0) \\dot{s}^*(\\gamma _0 (n \\Delta - \\tau _0)) \\right)^2 \\right\\rbrace ,$ $\\nonumber & \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace b(\\xi (\\omega )) d(\\eta (\\omega )) \\frac{X_1 Y_1}{\\sigma ^2} \\right\\rbrace = -b(0) d(0) \\frac{\\gamma _0 \\left| x \\right|^2 E_s}{2} \\times \\\\& \\textstyle \\text{Re} \\left\\lbrace \\left[ \\sum _{n \\in \\mathcal {N}} s(\\gamma _0 (n \\Delta - \\tau _0)) \\dot{s}^*(\\gamma _0 (n \\Delta - \\tau _0)) \\right] \\times \\right.",
"\\\\\\nonumber & \\textstyle \\left.",
"\\left[ \\sum _{n \\in \\mathcal {N}} s^*(\\gamma _0 (n \\Delta - \\tau _0)) (n \\Delta - \\tau _0) \\dot{s}(\\gamma _0 (n \\Delta - \\tau _0)) \\right] \\right\\rbrace ,$ $\\nonumber & \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace b(\\xi (\\omega )) d(\\eta (\\omega )) \\frac{X_1 Y_3}{\\sigma ^2} \\right\\rbrace = -b(0) d(0) \\frac{\\gamma _0 \\left| x \\right|^2 E_s}{2} \\times \\\\& \\textstyle \\text{Re} \\left\\lbrace \\left[ \\sum _{n \\in \\mathcal {N}} s(\\gamma _0 (n \\Delta - \\tau _0)) \\dot{s}^*(\\gamma _0 (n \\Delta - \\tau _0)) \\right] \\times \\right.",
"\\\\\\nonumber & \\textstyle \\left.",
"\\left[ \\sum _{n \\in \\mathcal {N}} s(\\gamma _0 (n \\Delta - \\tau _0)) (n \\Delta - \\tau _0) \\dot{s}^*(\\gamma _0 (n \\Delta - \\tau _0)) \\right] \\right\\rbrace ,$ $\\nonumber & \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace b(\\xi (\\omega )) d(\\eta (\\omega )) \\frac{X_3 Y_1}{\\sigma ^2} \\right\\rbrace = -b(0) d(0) \\frac{\\gamma _0 \\left| x \\right|^2 E_s}{2} \\times \\\\& \\textstyle \\text{Re} \\left\\lbrace \\left[ \\sum _{n \\in \\mathcal {N}} s(\\gamma _0 (n \\Delta - \\tau _0)) \\dot{s}^*(\\gamma _0 (n \\Delta - \\tau _0)) \\right] \\times \\right.",
"\\\\\\nonumber & \\textstyle \\left.",
"\\left[ \\sum _{n \\in \\mathcal {N}} s(\\gamma _0 (n \\Delta - \\tau _0)) (n \\Delta - \\tau _0) \\dot{s}^*(\\gamma _0 (n \\Delta - \\tau _0)) \\right] \\right\\rbrace ,$ $\\nonumber & \\lim _{\\sigma \\rightarrow 0} \\textstyle \\mathbb {E} \\left\\lbrace b(\\xi (\\omega )) d(\\eta (\\omega )) \\frac{X_3 Y_3}{\\sigma ^2} \\right\\rbrace = -b(0) d(0) \\frac{\\gamma _0 \\left| x \\right|^2 E_s^2}{2} \\times \\\\& \\textstyle \\sum _{n \\in \\mathcal {N}} (n \\Delta - \\tau _0) \\left| \\dot{s}(\\gamma _0(n \\Delta - \\tau _0)) \\right|^2,$ Specifically, in deriving the results given by (REF )–(REF ), it is necessary to apply the relation $\\text{Re} \\left\\lbrace R \\right\\rbrace \\text{Re} \\left\\lbrace U \\right\\rbrace = \\frac{1}{4} \\left( R U + R^* U + R U^* + R^* U^* \\right),$ which always holds for any two complex terms $R$ and $U$ .",
"Moreover, two properties of the complex Gaussian noise are also utilized [8], [12]: $& \\mathbb {E} \\left\\lbrace \\frac{w(n \\Delta )}{\\sigma } \\frac{w^*(m \\Delta )}{\\sigma } \\right\\rbrace = \\left\\lbrace \\begin{array}{ll}1, & \\text{if } n = m, \\\\0, & \\text{if } n \\ne m,\\end{array}\\right.",
"\\\\& \\mathbb {E} \\left\\lbrace \\frac{w(n \\Delta )}{\\sigma } \\frac{w(m \\Delta )}{\\sigma } \\right\\rbrace = 0, \\quad \\forall n, \\forall m.$ The limits of the rest terms on the RHS of (REF ) are all shown to equal zero as $\\sigma \\rightarrow 0$ ."
]
] | 1605.05990 |
[
[
"Adsorption structures and energetics of molecules on metal surfaces:\n Bridging experiment and theory"
],
[
"Abstract Adsorption geometry and stability of organic molecules on surfaces are key parameters that determine the observable properties and functions of hybrid inorganic/organic systems (HIOSs).",
"Despite many recent advances in precise experimental characterization and improvements in first-principles electronic structure methods, reliable databases of structures and energetics for large adsorbed molecules are largely amiss.",
"In this review, we present such a database for a range of molecules adsorbed on metal single-crystal surfaces.",
"The systems we analyze include noble-gas atoms, conjugated aromatic molecules, carbon nanostructures, and heteroaromatic compounds adsorbed on five different metal surfaces.",
"The overall objective is to establish a diverse benchmark dataset that enables an assessment of current and future electronic structure methods, and motivates further experimental studies that provide ever more reliable data.",
"Specifically, the benchmark structures and energetics from experiment are here compared with the recently developed van der Waals (vdW) inclusive density-functional theory (DFT) method, DFT+vdW$^{\\mathrm{surf}}$.",
"In comparison to 23 adsorption heights and 17 adsorption energies from experiment we find a mean average deviation of 0.06 \\AA{} and 0.16 eV, respectively.",
"This confirms the DFT+vdW$^{\\mathrm{surf}}$ method as an accurate and efficient approach to treat HIOSs.",
"A detailed discussion identifies remaining challenges to be addressed in future development of electronic structure methods, for which the here presented benchmark database may serve as an important reference."
],
[
"Introduction",
"The interaction of organic materials and molecules with metal surfaces is of widespread interest to both fundamental science and technology.",
"The eventual control of the functionality of the formed hybrid inorganic-organic systems (HIOSs) has potential applications to a variety of fields ranging from functionalized surfaces, to organic solar cells [1], molecular electronics [2], nanotechnology [3], [4], and medical implantology [5].",
"A bottom-up approach of molecular nanotechnology promises a potential route to overcome size limitations of nanoscale devices constructed with traditional top-down approaches such as lithography based device design.",
"An important prerequisite to such an approach is the ability to control and manipulate the structure and interactions of individual molecular building blocks mounted on well-defined surfaces.",
"This can only be achieved with the expertise to fully characterize the adsorption geometry and fully understand the, sometimes subtle, interplay of interactions that lead to a particular molecule-substrate binding strength.",
"This fundamental aspect has seen a rapid development over the last few years in the surface science context, i.e.",
"for the adsorption of large and complex organic adsorbates at close to ideal single-crystal surfaces and under the well-defined conditions of ultrahigh vacuum and low temperature [6], [7], [8], [9].",
"On the one hand, this development has been driven by significant methodological advancements in individual experimental techniques, such as improvements in the resolution of Scanning Tunneling Microscopy-based (STM) imaging techniques [10] or surface-enhanced Raman spectroscopy [11].",
"Many of these approaches are used complementary to each other or are being combined to enhance resolution, such as is the case for STM tip-enhanced local Raman experiments [12], [13].",
"On the other hand, theoretical developments have been driven by advancements in predictive-quality first-principles calculations.",
"The most important of which in the context of HIOSs is the efficient incorporation of long-range dispersion interactions into semi-local or hybrid Density-Functional Theory (DFT) calculations [14], [15].",
"When applied to HIOSs, first-principles calculations face a multitude of challenges, even for single-molecules with simple adsorbate geometry.",
"Localized molecular states of the adsorbate and the delocalized metal band structure have to be described on equal footing, whereas most currently used (and computationally tractable) approximations to the exchange-correlation (xc) functional in DFT are optimized to perform well for either one or the other.",
"This is aggravated by the need to additionally describe dispersive interactions, which are generally not contained in such lower-rung functionals, but can easily play a dominant role e.g.",
"in the adsorption of conjugated or aromatic molecules.",
"The resulting interplay of dispersion interactions, wave-function hybridization, Pauli repulsion, and charge-transfer at such interfaces demands an efficient and accurate electronic structure description that is able to provide a well-balanced account of this wide range of interactions.",
"Figure: Adsorption energy curve of PTCDA adsorbed at Ag(111).",
"Shown are the results of different DFT methods and dispersion correction approaches.",
"Experimental results from X-ray standing wave measurements and estimated from TPD data of the smaller analogue molecule NTCDA , are shown as blue bars.",
"A detailed analysis of this figure can be found in section .",
"Based on Fig.",
"1 of Ref.",
"Ruiz2012.A number of strong contenders to this goal have been suggested, such as the recently developed DFT+vdW$^{\\mathrm {surf}}$ method [17], [18].",
"However, there is still room for further improvement.",
"This situation is nicely illustrated in Fig.",
"REF , which compiles the predictions of the adsorption height and adsorption energy for 3,4,9,10-perylene-tetracarboxylic acid (PTCDA) adsorbed on Ag(111) as obtained by a variety of pure and dispersion-corrected xc-functionals ranging from local xc-approximations to non-local correlation based on the Random Phase Approximation (RPA) [19].",
"Quite symptomatic for HIOSs in general, a wide spread in the predicted quantities is obtained.",
"Without an independent experimental reference, we would thus not be able to unambiguously identify the merits and deficiencies of the individual approximations and models that underlie the methods in Fig.",
"REF .",
"Reliable experimental data on geometric structure and energetics of organic-inorganic interfaces are thus urgently needed as a reference for validation and benchmarking of new and improved electronic structure methods.",
"This reference data must thereby properly match what is calculated.",
"Many experimental techniques to characterize the structure and energetics of HIOSs are based on probing the statistics of an ensemble of adsorbates.",
"It is the very complex mixture of adsorbate-surface and lateral adsorbate-adsorbate interactions that drives the aspired self-assembly of HIOSs that also gives rise to a rich structural phase behavior as a function of temperature and coverage [20], [21], [22], [23].",
"Care has to be taken to compare consistent arrangements at the surface in experiment and theory.",
"The targeted molecules can furthermore exhibit a pronounced element of flexibility, disorder, and structural anharmonicity [24], [25], [26].",
"This gives rise to significant finite-temperature effects that also need to be carefully disentangled [27], [26].",
"A correct interpretation of experiments is therefore of vital importance.",
"If accomplished, the formulation of well-balanced, diverse sets of benchmark systems can then facilitate methodological improvements as has been shown by the success of the S22 dataset for intermolecular interactions of gas phase molecules [28], or the C21/X16/X23 databases of molecular crystals [29], [30], [31].",
"With the motivation of formulating such a set for HIOSs, we review a set of well-characterized systems of molecules adsorbed on metal surfaces for which detailed information from experiment and calculations exists.",
"We start out with an overview of the major experimental techniques which provide reference data regarding molecular geometry and energetics of adsorbates at surfaces.",
"We then proceed with an overview of the recent advances in DFT-based methodologies with a special focus on density-dependent dispersion-inclusive approaches such as the DFT+vdW$^{\\mathrm {surf}}$ method.",
"Following this overview we exemplify interpretation of electronic structure results using PTCDA on Ag(111) data as shown in Fig.",
"REF .",
"In the following we review and analyze the experimental data for the different systems, which range in complexity from rare-gas adatoms to large conjugated aromatics and carbon nanostructures.",
"Based on this data we attempt a first assessment on the current level of accuracy in first-principles calculations of HIOSs, by using the DFT+vdW$^{\\mathrm {surf}}$ method."
],
[
"Experimental Methods",
"Experimental characterization of HIOSs in ultra-high vacuum is performed with a vast set of surface science techniques (see Tab.",
"REF ) based on topographic surface imaging, spectroscopy, surface scattering, and thermodynamic measurements, all of which complement each other.",
"Imaging techniques such as Transmission Electron Microscopy (TEM) [32], [33], Scanning Tunneling Microscopy (STM) [34], [35], [36] and Atomic Force Microscopy (AFM) [37] play an important role in the initial characterization of adsorbate overlayers at surfaces.",
"In the case of TEM the growth mode, surface reconstruction, the approximate thickness of and phase boundaries between adsorbate overlayers can be measured.",
"Whereas atomic-level resolution in TEM can be achieved by now [38], structural analysis on the single molecule level is still difficult.",
"Such resolution can, however, be achieved for molecular adsorbates using STM and AFM.",
"This gives the ability to define a model of single molecule adsorbate structure and, in combination with low-energy electron diffraction (LEED) [39], [40], also to construct a model of the periodic surface structure and surface unit cell.",
"Recent advances due to tip-functionalization with small molecules [41] have significantly increased the applicability and lateral resolution of AFM [42], which is sometimes even referred to as \"sub-atomic resolution AFM\" [43].",
"Complementary to the information given by these diffraction and imaging techniques, surface spectroscopy methods such as x-ray photoemission (XPS) [44], x-ray absorption spectroscopy (XAS), ultraviolet photoemission (UPS), and Two-Photon Photoemission (2PPE) yield information about the changes in electronic structure along with the nature and extent of the interaction between adsorbate and substrate.",
"Table: List of experimental techniques and their abbreviations (Abbrev.)",
"from which interface structure, adsorbate geometry, and interaction energies can be extracted.The above methods can produce the basic information of overlayer surface unit cell, lateral adsorbate arrangement, and surface chemical shift that often serve as input to construct first-principles models of HIOSs.",
"Using methods based on DFT and many-body perturbation theory (MBPT), stable adsorption geometries can then be calculated, which may or may not support the initial experimental model.",
"Such calculations yield additional detailed information on the individual atomic positions and the interaction strength between adsorbate and substrate.",
"Whereas the above mentioned experimental techniques do not directly give access to geometry, validation can be achieved indirectly through e.g.",
"the comparison of spectroscopic signatures with chemical core-level shifts and diffraction patterns obtained from an accurate atomistic model of geometry and energetics.",
"In turn, a relatively novel and powerful tool for validation of atomistic models are experimental techniques that enable an accurate determination of individual atomic adsorption heights and intramolecular structure based on model fitting.",
"The most common such techniques are Near-Incidence X-Ray Standing Wave measurements (NIXSW) [45], [46], [47], Normal Emission Photoelectron Diffraction (PhD) [48], and Angular-Resolved Near Edge X-Ray Absorption Fine-Structure (NEXAFS) [49].",
"In NIXSW an x-ray wave at normal incidence forms a standing wave pattern with its reflection that has the same periodicity as the Bragg planes of the underlying substrate.",
"By tuning the photon energy of incoming x-ray light, the standing wave pattern shifts and the photoelectron spectra can be measured as a function of the x-ray incidence energy.",
"Using Fourier vector analysis, the corresponding atomic positions and the coherence of the signal can be matched with model geometries [50].",
"The corresponding structural models are highly accurate with an experimental uncertainty in the range of 0.10 Å in the determination of the vertical heights.",
"The disadvantages of NIXSW, similar to PhD, are the immense complexity of the spectral model fits and the need for near perfect adsorbate overlayer order.",
"While NIXSW and PhD both yield accurate determination of vertical heights, they also provide limited information on internal molecular degrees of freedom, except for inferred information from the model structures used in the spectral fit.",
"In this respect, angular-resolved NEXAFS serves as a powerful complementary technique that enables determination of the relative orientation and angle of chemically-distinct molecular sub-domains with respect to the surface.",
"NEXAFS-based structural characterization has been successfully used for complex systems such as metal-adsorbed porphyrine [51], [52] and azobenzene [53], [54], [55] derivatives.",
"The dominant technique employed to determine the interaction energy of adsorbates on surfaces is temperature programmed desorption spectroscopy (TPD) [56].",
"In a TPD experiment, a sample is slowly heated at a constant rate while monitoring, at the same time, the rate of appearance of gases desorbed from the surface.",
"The measured desorption temperature of a sufficiently diluted adsorbate overlayer ideally reflects the interaction strength of a single molecule on the surface.",
"A variety of different techniques exist to analyze the corresponding desorption spectra, all of which are based on the Polanyi-Wigner equation (PWE).",
"Integral methods such as the ones proposed by Redhead [57] or Chan, Aris, and Weinberg [58] impose an assumption on the order of the desorption reaction and additionally assume coverage-independence of both the desorption energy and the entropy-related pre-exponential factor.",
"These methods can therefore not be applied to adsorbates exhibiting lateral interactions [59].",
"Differential techniques to extract interaction energies such as the one proposed by [56] or Habenschaden and Küppers [60] utilize a large set of desorption measurements and make no assumptions on prefactors as function of temperature and coverage.",
"All methods share that they have been devised for the study of small adsorbates of interest at that time, therefore not addressing the characteristics of complex, extended adsorbates exhibiting strong lateral interactions and large configurational freedom [61], [62], [63], [64], [65].",
"Single crystal adsorption microcalorimetry (SCAM) [66], [67], [68] is an alternative approach that does not suffer from many of the difficulties that TPD analysis faces.",
"It has the advantage that the observed radiated heat from the surface is directly connected to the adsorption energy by knowledge of the heat capacity.",
"The main disadvantage is the complexity of its experimental setup and calibration, resulting in the operation of only few microcalorimeters at the moment.",
"One important additional technique to measure the interaction strength between adsorbate and substrate is represented by AFM force pulling experiments of single molecules [69], [70].",
"The measured force law of the single molecule pulling event can be related to a model potential with a well-depth that corresponds to a free energy of desorption.",
"This technique also opens the possibility to model different independent interaction contributions that contribute to the force pulling signal.",
"However, there are some unresolved difficulties in relating the integrated binding energy of the force pulling event with interaction energies from first-principles calculations.",
"A number of spectroscopic techniques could be used as fingerprint methods to the structure and interaction strength of molecules on surfaces.",
"Photoelectron spectroscopy, pump-probe spectroscopy such as 2PPE, and Scanning Tunneling Spectroscopy yield the positions of adsorbate molecular resonances with respect to the Fermi level of the substrate [71].",
"In combination with MBPT simulations, the image-charge potential-induced state renormalization of molecular level alignment [72] can be used to gauge on the adsorption height of the adsorbate.",
"Surface-enhanced Raman [73], [74] and Sum-Frequency Generation (SFG) spectroscopy are surface science techniques that have recently gained popularity and have been used to determine the layer thickness and structure of molecules on surfaces [75], [76].",
"The corresponding spectral shapes furthermore enable insight into which molecular moieties are strongly or loosely bound to the surface [77].",
"High resolution electron-energy loss spectroscopy (HREELS) provides insight into the vibrational and electronic properties of adsorbed molecules and has been used extensively in combination with simulations to determine adsorbate structure.",
"[78], [79] A significant portion of future method development in surface science will be geared towards combination of existing methods, such as STM tip-enhanced Raman spectroscopy [80], four-wave mixing spectroscopy [81], or transient spectroscopy during molecular scattering [82]."
],
[
"Theoretical Methods",
"The PTCDA molecule in Fig.",
"REF serves as an ideal benchmark candidate exhibiting all relevant aspects of adsorbate-substrate interaction.",
"The terminal anhydride-oxygens are chemically bound to the metal surface, whereas the conjugated aromatic core of the molecule induces attractive dispersion interactions between adsorbate and substrate.",
"The relatively small adsorption height also leads to an increased Pauli repulsion of the closed-shell molecule core on the substrate.",
"Finally, the level-alignment of molecular resonances with respect to the Fermi level of the surface determines the amount of charge transfer between adsorbate and substrate.",
"An accurate description of the surface-induced molecular distortion, adsorption height, and interaction strength can only be achieved by accounting for all the effects that we have mentioned above.",
"If we also consider the system size at hand, high computational efficiency becomes an equally important factor.",
"DFT, as the electronic-structure method of choice in condensed matter physics, represents a good compromise between accuracy and computational efficiency.",
"However, the above discussed interactions are often not described with sufficient accuracy using current (semi-)local and hybrid xc approximations to the exact density functional.",
"Chemical interactions between adsorbate species and metallic substrates are captured relatively well at the level of the Generalized Gradient Approximation (GGA) of which the functional developed by Perdew, Burke, and Ernzerhof (PBE) [83] is a popular variant, albeit at a tendency to overestimate adsorption energies in strongly interacting systems [84], [85].",
"An accurate description of Pauli repulsion effects and molecular level alignment requires a more sophisticated description of exchange.",
"The simple admixture of exact exchange on the Hartree-Fock level may simply not be sufficient [86] and in some cases can even lead to an overestimation of the substrate bandwidth and exchange splitting [87].",
"Several recent works have developed correlation descriptions for solids and surfaces based on the RPA [19], [88] and beyond [89], [90], [91] in combination with different variants of (screened) exact exchange [89] that promise chemical accuracy for short-range interactions between adsorbates and surfaces [92].",
"Admitting that several challenges remain on the level of short-range correlation and exchange, the biggest challenge in simulating HIOSs is the accurate treatment of non-local correlation effects such as dispersion interactions.",
"Whereas the most straightforward treatment of dispersion interactions follows incorporation of non-local correlation into DFT via the Adiabatic-Connection Fluctuation Dissipation Theorem [93], [94], [95], it is certainly still limited in terms of efficiency from the computational perspective.",
"A hierarchy of different and more efficient approaches to incorporate long-range dispersion into Density-Functional Approximations (DFAs) exists [96].",
"These can be grouped into three major categories: (1) empirical a posteriori dispersion correction approaches, (2) density-dependent dispersion functionals, (3) and the aforementioned correlation functionals directly based on RPA.",
"The first category is prominently represented by the series of methods proposed by [97], [98], [99].",
"In this case, an existing DFA is complemented by a pairwise-additive correction to the total energy which exhibits the $R^{-6}$ behavior of the leading-order dispersion term based on empirical pretabulated parameters for atomic polarizabilities, dispersion coefficients $C_6$ , and van der Waals radii.",
"The connection is achieved by a damping function acting on the vdW contribution at short-range.",
"This pragmatic approach has been applied in the description of intermolecular interactions in gas phase complexes [97] and molecule-surface adsorption [100].",
"Despite its low computational cost and fair accuracy for small molecules, its insufficient response to the local chemical environment and collective response effects along with the absence of higher-order dispersion terms lead to a significant overestimation of interaction energies for molecules at surfaces [100].",
"Van-der-Waals functionals (vdW-DF) are representatives of the second category of density-derived methods [101], [102].",
"An additional vdW contribution to the Hamiltonian is computed as non-local functional of the electron density by a two-point integral and a given integration kernel.",
"This approach bears considerably more computational demand.",
"However, recent improvements in computational efficiency [103] and performance [104], [102] lead to a widespread use of approaches such as the vdW-DF-cx [105], [106] and optPBE-vdW [102] functionals.",
"Compared to the first category of methods, recent vdW-DF methods yield a considerably improved description of adsorbate structure and energetics for a number of HIOSs [107], [108].",
"Several efficient approaches are based on a connection between a pairwise dispersion model and the electron density, namely by constructing vdW parameters such as atomic C$_6$ coefficients, vdW radii $R$ , and static atomic polarizabilities $\\alpha _0$ as functionals of the chemical environment and the electron density.",
"Such methods include the DFT+XDM approach originally developed by Becke and Johnson [109], [110], [111] and the DFT+vdW approach of Tkatchenko and Scheffler [112].",
"In the latter approach the vdW parameters C$_6^a$ , $\\alpha _0^a$ , and $R_{\\mathrm {0}}^a$ for an atom $a$ are constructed from free-atom reference data and renormalized by the change in effective volume of the atom in the molecule.",
"The latter effect accounts for the changes in local polarizability and chemical environment of the atomic species (see Fig.",
"REF , top).",
"The resulting C$_6$ coefficients show a mean absolute relative error (MARE) of 5.5% for intermolecular C$_6$ coefficients between a variety of atoms and molecules in gas phase.",
"[112] The effective atomic volumes are directly derived from the density using the atoms-in-molecules scheme proposed by Hirshfeld [113].",
"This approach has been recently modified to extract dispersion parameters directly from charge analysis enabling its use for semi-empirical and tight-binding approximations to DFT [114].",
"Furthermore, it should be emphasized that the methods based on the Tkatchenko-Scheffler approach (DFT+vdW, DFT+vdW$^{\\mathrm {surf}}$ , and DFT+MBD) are proper functionals of the electron density and, hence, should and have been implemented self-consistently in the context of DFT [115].",
"While the self-consistency of the vdW energy can modify electronic properties of solids and surfaces [115], its impact on structures and stabilities is typically minimal (on the order of 0.001 Å and few meV, respectively).",
"For this reason, the calculations in this manuscript were carried out without accounting for self-consistency in the vdW energy.",
"Figure: Flowchart explaining the link between Lifshitz-Zaremba-Kohn (LZK) theory and the DFT+vdW method leading to the DFT+vdW surf ^{\\mathrm {surf}} method.However, direct application of the above mentioned approaches to molecules at metal surfaces leads to a significant overestimation of the adsorbate-substrate interaction as has been observed for example for azobenzene and its derivatives adsorbed at coinage metal surfaces [116], [117], [118].",
"The recipe behind the DFT+vdW method of rescaling accurate free-atom reference vdW parameters according to the chemical environment of each atomic component often yields a highly accurate description of dispersion interactions between atoms and molecules in gas phase.",
"However, the non-local correlation interaction between adsorbate atoms and an extended metal surface requires account of the collective many-body substrate response rather than only the local atom-atom response of individual metal atoms with the atoms of the adsorbate.",
"[119] The DFT+vdW$^{\\mathrm {surf}}$ method [17] accounts for this by modelling screened vdW interactions in the adsorption of atoms and molecules on metal surfaces.",
"On the basis of the Lifshitz-Zaremba-Kohn (LZK) theory [120], [119] and its equivalent formulation in terms of interatomic pairwise potentials, [121], [122] the collective effects of the atom-substrate interaction are projected onto renormalized C$_6^{aS}$ coefficients that describe the dispersion interaction between adsorbate atoms $a$ and substrate atoms $S$ .",
"These coefficients are expressed in terms of an integral over the frequency-dependent polarizability $\\alpha _1^a(i\\omega )$ of the adsorbate atom and the dielectric function $\\epsilon _S$ of the substrate (see Fig.",
"REF , bottom).",
"With this formulation, the C$_6^{aS}$ coefficients effectively “inherit” the collective effects contained in the many-body response of the solid.",
"Obviously, not all many-body effects of the extended surface can be treated in this effective way.",
"While the vdW$^{\\mathrm {surf}}$ method exactly reproduces the long-range vdW energy limit by construction, many-body effects closer to the surface are included approximately utilizing the electron density.",
"Accurately treating all many-body effects in the vdW energy would require fully non-local microscopic approaches to the correlation energy, such as those based on the adiabatic connection formalism [93], [94], [95].",
"Finally, the vdW$^{\\mathrm {surf}}$ parameters for a given substrate species are calculated using the combination rule of the vdW method and solving C$_6^{SS}$ and $\\alpha _0^S$ with a linear set of equations for a number of different adsorbate species (see Fig.",
"REF , bottom).",
"[112] The resulting effective DFT+vdW$^{\\mathrm {surf}}$ scheme has been applied to numerous HIOSs [123], [27], [17], [124], [125], [126], [15] yielding adsorption geometries that are in good agreement with experiment as we will detail in the following chapters.",
"The DFT+vdW$^{\\mathrm {surf}}$ method introduced above is generally applicable to model adsorption on solids, independent of whether they are insulators, semiconductors, or metals.",
"The LZK theory is an exact asymptotic theory for any polarizable material and the firm foundation of vdW$^{\\mathrm {surf}}$ on the LZK theory ensures its transferability.",
"The vdW$^{\\mathrm {surf}}$ approach necessitates the dielectric function of the bulk solid as an input, which can be calculated from time-dependent DFT, RPA, or taken from experimental measurements.",
"However, the method is applicable to surfaces with any termination, defects, and other imperfections, because the vdW parameters depend on the electron density at the interface.",
"The transferability of the vdW$^{\\mathrm {surf}}$ method to different surface terminations has been recently demonstrated for adsorption on (111), (110), and (100) metallic surfaces [127].",
"Despite this success, several challenges remain to model dispersion interactions in HIOSs accurately.",
"While the description of adsorption geometries seems adequate at the level of effective pairwise interactions, adsorption energies still appear systematically overestimated.",
"This is due to the missing beyond-pairwise interactions and the neglect of the full many-body response of the combined adsorbate-substrate system [128], [129].",
"The recently developed DFT+MBD method tackles this problem by calculating the full long-range many-body response in the dipole limit [128], [130], [95].",
"In short, the MBD method makes an approximation to the density-density response function, consisting of a set of atom-centered interacting quantum harmonic oscillators.",
"Under this employed assumption, the MBD method is equivalent to RPA.",
"Initial results for molecules on metal surfaces, which include the Xe atom, benzene [18], PTCDA, and graphene on metal surfaces [129], are promising.",
"However, the current MBD approach has been developed to describe the correlation problem for molecules and finite-band gap materials with atom-centered quantum harmonic oscillators, which would not fully account for the delocalized plasmonic response of free electrons in the metal substrate.",
"Despite several remaining questions, a systematic improvement of the current MBD scheme is possible, potentially opening a path towards the exact treatment of dispersion energy at drastically reduced computational cost.",
"All the DFT calculations presented herein are performed employing the PBE+vdW$^{\\mathrm {surf}}$ functional, by means of the full-potential all-electron code fhi-aims [131], [132] and the periodic plane wave code CASTEP [133]."
],
[
"Interpretation of Electronic Structure Calculation Results",
"Having summarized the ingredients of many existing dispersion-inclusive electronic structure methods (see Table REF ), we revisit the adsorption of PTCDA on Ag(111), as depicted in Fig.",
"REF .",
"We do this to illustrate the wide range of interactions that need to be accounted for by a first-principles method to accurately describe HIOSs and to establish the merits of different types of methods in direct comparison to experiment.",
"In the case of PTCDA on Ag(111) an accurate measurement of the adsorption height from NIXSW exists, with 2.86 Å for the average height of the carbon backbone [16].",
"The value of adsorption energy remains to be directly measured.",
"Several disputed estimates for the single-molecule and monolayer adsorption energy are given in literature, ranging from 1.40 to 3.46 eV [17], [129], [106].",
"The value given in Fig.",
"REF , extrapolated from desorption measurements of the smaller homologous NTCDA molecule [17], coincidentally corresponds to the median of these estimates.",
"A DFT calculation based on a local description of exchange and correlation effects (LDA) underestimates the binding distance, but leads to a seemingly good description of adsorption energy.",
"It is important to note that this apparent agreement stems from an incorrect balance between short-range kinetic, electrostatic, and xc contributions [135], and that LDA does not include any long-range dispersive interactions.",
"DFAs based on a semi-local xc-description, namely GGAs, such as PBE [136], result in the opposite extreme case: The functional also lacks any description of long-range correlation, but offers a better description of covalent bonding contributions.",
"The result is negligible binding to the surface as indicated by a large overestimation of adsorption height and underestimation of adsorption energy.",
"This insufficient description of long-range correlation makes GGAs an ideal starting point to incorporate dispersion interactions.",
"Pairwise dispersion interaction methods built on-top of GGAs (PBE-D and PBE+vdW) yield bound structures in the range of 2.9 to 3.2 Å, but systematically overestimate the binding energy.",
"Long-range correlation functionals of the vdW-DF family (Fig.",
"REF shows a so-called vdW-DF1 [101], [106]) yield a wide range of results depending on the treatment of long-range correlation, long-range exchange and short-range exchange.",
"Depending on the construction, in many cases vdW-DF can yield a good description of either adsorption geometry or adsorption energy.",
"PBE+vdW$^{\\mathrm {surf}}$ introduces substrate-screening effects into the PBE+vdW functional and thereby improves the adsorption properties considerably.",
"However, the screened interactions also reduce the effective vdW radii and result in reduced adsorption heights that, as will be shown in the remainder of this work, are in excellent agreement with experiment for a variety of systems.",
"The PBE+vdW$^{\\mathrm {surf}}$ enables this at negligible computational overhead compared to PBE.",
"For some of the larger molecules we discuss in this work, PBE+vdW$^{\\mathrm {surf}}$ seems to overestimate the adsorption energy with respect to the limited available experimental reference data.",
"As discussed above, this can be remedied with methods that explicitly account for the many-body nature of dispersion interactions, such as the PBE+MBD method [128], [95], [129] or exact correlation treatment in the Random Phase Approximation (EX+cRPA) [19].",
"Both of these methods yield further reduced adsorption energies and in the case of PBE+MBD excellent agreement with experiment was recently established for benzene [18] and azobenzene [26] on Ag(111).",
"Published EX+cRPA results for PTCDA on Ag(111) yield a larger deviation from the experimental reference, however, they did not account for geometrical relaxation and a full numerical convergence of RPA remains a challenging issue [19].",
"The presented case of PTCDA on Ag(111) remains somewhat disputed due to the lack of a directly measured experimental adsorption energy.",
"However, this example nicely illustrates the importance of unambiguous experimental reference values in the development of improved electronic structure methodologies and motivates the development of the here presented benchmark dataset."
],
[
"Overview of Benchmark Systems",
"In the following we will review the adsorption properties of different classes of organic compounds (see Table REF ) on metal surfaces.",
"From each of these classes we will select a number of test cases representing different limiting cases of adsorbate-substrate interaction.",
"Fig.",
"REF (a) depicts different metal substrates for which the interaction with exemplary adsorbates (Fig.",
"REF (b) to (j)) is considered.",
"In each case we will review the experimental reference data and the calculated geometry and energetics as predicted by DFT+vdW$^{\\mathrm {surf}}$ .",
"Figure: Summary of surfaces and molecules incuded in the benchmark set: (a) 7 close-packed transition metal surfaces, (b) Xenon, (c) benzene (Bz), (d) naphthalene (Np), (e) Diindenoperylene (DIP), (f) C 60 _{60} Buckminster-Fullerene, (g) 3,4,9,10-perylene-tetracarboxylic acid (PTCDA), (h) Thiophene (Thp), (i) E-Azobenzene (AB), (j) E-3,3',5,5'-tetra-tert-butyl-Azobenzene (TBA)Table: List of molecules and their abbreviations (Abbrev.)",
"included in the benchmark set.We group the set of hybrid organic-metal benchmark systems into rare-gas adsorption at coinage metals on the example of Xe atom, aromatic compounds adsorbed at metal systems, extended and compacted carbon nanostructures, Sulfur-containing compounds represented by thiophene on Cu(111), Ag(111), and Au(111), Oxygen-containing compounds represented by 3,4,9,10-perylene-tetracarboxylic acid (PTCDA) adsorbed at coinage metal surfaces, and Nitrogen-containing compounds represented by E-azobenzene and E-3,3',5,5'-tetra-tert-butyl-azobenzene (E-TBA) adsorbed at Ag(111) and Au(111)."
],
[
"Experimental Data",
"The adsorption of noble gases on metal surfaces has been extensively studied as prototypical example of physisorption.",
"A historical perspective of these studies can be found in the works, for example, by [137] and [138].",
"An exhaustive historical survey is out of the scope of this work.",
"We will restrict ourselves to experimental data on structures and adsorption energetics.",
"Moreover, we focus here exclusively on the adsorption of Xe on transition-metal surfaces.",
"From this perspective, the most important fact is the paradigm shift that occurred 25 years ago with respect to the preferred adsorption site of Xe.",
"The general assumption prior to 1990 was that the adsorption potential of noble gases on surfaces would be more attractive in high-coordination sites than in those with lower coordination.",
"In the case of Xe, for instance, experimental studies using spin-polarized LEED suggested the hollow site as the preferred adsorption site on close-packed metal surfaces [139], [140].",
"This changed with the dynamical LEED studies of adsorbed Xe on Ru(0001) [141], Cu(111) [142], Pt(111) [143], and Pd(111) [144]; which showed that Xe atoms reside on top of the substrate atoms instead of in higher-coordination sites [137].",
"The other important experimental finding is that Xe adopts a $ (\\sqrt{3} \\times \\sqrt{3})\\mathrm {R30}^{\\circ }$ structure on Cu(111), Pt(111), and Pd(111).",
"In the case of Xe on Cu(110), a $(12 \\times 2)$ structure is formed at low temperature, which consists of rows of adatoms that are commensurate with the substrate, having higher-order commensurate periodicity along the substrate rows of the surface and a spacing between the rows that is equal to the Cu row spacing [145], [137].",
"Most importantly, the LEED studies by [145] indicate that the Xe rows are located on top of the Cu substrate rows.",
"For this work, we take the review papers by [137] and [146] as our guidelines for the experimental data on the adsorption of Xe on transition-metal surfaces.",
"The overview of these data is shown in Tab.",
"REF while the details of each experiment can be found in the original references.",
"The experimental adsorption distances in these systems were mainly obtained using the LEED technique.",
"The experimental adsorption energies are mostly a result of TPD.",
"These experiments report exponential prefactors of desorption of the order of $10^{12}-10^{13}$ s$^{-1}$ , which are in the expected range for simple adsorbates and small molecules [61], [147].",
"Table: Adsorption distances and energies for Xe on transition-metal surfaces in Å and eV respectively.Both adsorption distances and energies correspond to the system after relaxation.",
"The values of dd and E ad E_{\\mathrm {ad}} for Ag(111) correspond to the best estimates in Ref. Vidali:Ihm:Kim:etalSSRep.1991.",
"The experimental data is taken from Refs.",
"Diehl:Seyller:Caragiu:etalJPCondensMatter2004,Vidali:Ihm:Kim:etalSSRep.1991,Seyller:Caragiu:Diehl:etalCPL1998,Caragiu:Seyller:DiehlSurfSci2003,Pouthier:Ramseyer:Girardet:etalPRB1998,Zhu:Ellmer:Malissa:etalPRB2003,Caragiu:Seyller:DiehlPRB2002,Hilgers:etal:SS1995,Widdra:Trischberger:etalPRB1997,Seyller:Caragiu:etalPRB1999,Bruch:Graham:Toennies:MolPhys1998,Hall:Mills:etalPRB1989,Braun:Fuhrmann:etalPRL1998,Zeppenfeld:Buechel:etalPRB1994,Ramseyer:Pouthier:Girardet:etalPRB1997, and Gibson:SibenerJChemPhys1988."
],
[
"Theoretical Data",
"Even if the experiments have identified the low-coordination top site as the preferred adsorption site for Xe on transition-metal surfaces, they have not been able to identify the origin of this preference.",
"It is in this regard that first-principles calculations have the potential to contribute to the atomistic understanding of the origin of this preference.",
"For an extensive review of first-principles simulation of rare gas adsorption, we point the interested reader to the works of [137], [138], and [157].",
"The predominant physisorptive character of the binding makes rare-gases on metal surfaces ideal test systems for dispersion-inclusive DFT.",
"In general, before the advent of several vdW-inclusive DFT based methods in the last years, the LDA has been used extensively to study the adsorption of Xe on metals [158], [159], [160], where it has been found that the top site is energetically more stable by, at most, 50 meV with respect to the hollow adsorption site.",
"In addition, [161] also studied the adsorption of additional rare gases on metal surfaces using GGAs as xc functional, where they found that these other rare gases also prefer the top adsorption site with the exception of Ar and Ne on Pd(111) [160], [157].",
"In general, [157] mention that GGA xc functionals tend to underestimate the adsortion energy of these systems by a great margin whereas LDA yields equilibrium adsorption distances that are too short in comparison to experiments.",
"In this respect, it is currently well established that GGA functionals such as PBE cannot describe systems that are dominated by vdW interactions in an accurate manner.",
"The studies performed by [162], [157] report the performance of several vdW-inclusive DFT methods, such as vdW-DF, vdW-DF2, and DFT-D2, on the adsorption of noble gases on metal surfaces.",
"We have also analyzed the structure and stability of the adsorption of Xe on selected transition metal surfaces with the PBE+vdW and PBE+vdW$^{\\mathrm {surf}}$ methods taking into consideration that the latter includes the collective response of the substrate electrons in the determination of the vdW contribution [127].",
"We reproduce the PBE+vdW$^{\\mathrm {surf}}$ results for the top adsorption site of Xe on transition metal surfaces in Tab.",
"REF .",
"Before proceeding to discuss these results, we define here the adsorption distance, $d$ , as the distance between the vertical coordinate of the Xe atom with respect to the position of the unrelaxed topmost metal layer.",
"This definition allows us to compare our data with results from NIXSW experiments.",
"In case of other experimental sources, e.g.",
"LEED, $d$ is computed considering the relaxed metal surface.",
"The adsorption energy, $E_{\\rm ad}$ , of a system is computed via the general definition: $E_{\\rm ad}=\\frac{1}{N} \\left[ E_{\\rm AdSys}-(E_{\\rm Me}+E_{\\rm Mol}) \\right]\\,,$ where $E_{\\rm AdSys}$ denotes the energy of the system, while $E_{\\rm Me}$ and $E_{\\rm Mol}$ refer respectively to the energies of the clean substrate and the molecule in gas phase.",
"$N$ corresponds to the total number of molecules in the unit cell.",
"With respect to adsorption site preference, we have found that both adsorption sites, top and fcc-hollow, are almost energetically equivalent using the PBE+vdW$^{\\mathrm {surf}}$ method.",
"The top adsorption site is energetically favored by approximately 5 meV for Pd(111) and Ag(111), and 10 meV for Cu(110).",
"Both adsorption sites are virtually degenerate within our calculation settings in the cases of Pt(111) and Cu(111).",
"Nevertheless, the differences in energy between adsorption sites are too small, just a few meV, to regard them as definitive.",
"This same fact has also been found recently by [162], who reported a few meV difference in their vdW-DF2 calculations between top and fcc-hollow adsorption sites.",
"They have suggested that experimental results cannot be explained by energy differences between top and fcc-hollow adsorption sites.",
"Instead, by examining the two-dimensional potential energy surface (PES) of Xe on Pt(111), they found that the fcc-hollow adsorption sites correspond to local saddle points in the PES, while top sites correspond to a true minimum.",
"Hence, fcc-hollow sites are transient states and thus not easily observed in experiments.",
"[162], [157], [121] This result is general, according to their calculations, for the adsorption of noble gases on transition metal surfaces.",
"This result is further reported to be independent of the underlying xc functional.",
"In Tab.",
"REF we show the adsorption distances and energies with the PBE+vdW$^{\\mathrm {surf}}$ method for the top adsorption site along with available experimental results.",
"In general, the calculated adsorption distances are within 0.10 Å of the experimental results except for Xe on Cu(111), in which the agreement is within 0.14 Å of the experimental value.",
"We have not found significant differences in the adsorption distance of these systems between PBE+vdW and PBE+vdW$^{\\mathrm {surf}}$ calculations with the exception of Xe on Cu(110), in which the distance predicted by the PBE+vdW method is 0.12 Å shorter than the PBE+vdW$^{\\mathrm {surf}}$ result (see Ref.",
"Ruiz:Liu:Tkatchenko2016 for more details).",
"On the other hand, we have also found that the PBE+vdW$^{\\mathrm {surf}}$ results are in closer agreement (within 0.10 Å) to experimental results than those calculated with other vdW-inclusive DFT methods [162], [157].",
"Tab.",
"REF also shows that the PBE+vdW$^{\\mathrm {surf}}$ adsorption energies are in good agreement with experimental results.",
"These calculations slightly underestimate the adsorption energy in the cases of Pt(111) and Pd(111), while slightly overestimating it in the case of both Cu surfaces and Ag(111).",
"Nevertheless, these discrepancies amount to about 50 meV out of the range of experimental results in the worst case.",
"The influence of screening is more noticeable in the computation of the adsorption energy.",
"Neglecting the effects of the collective response of the solid leads to an overestimation of the adsorption energy as a result of the inexact magnitude of the energetic contribution originated in vdW interactions.",
"We have exemplified this effect in the adsorption of Xe on metal surfaces with PBE+vdW calculations in Ref Ruiz:Liu:Tkatchenko2016."
],
[
"Experimental Data",
"From HREELS and NEXAFS studies it was concluded that Bz binds parallel to the Cu(111) surface [163].",
"A flat-lying geometry was also reported for Bz on Ag(111) using NEXAFS [164].",
"By means of angle-resolved photoemission spectroscopy (ARPES) in combination with LEED, [165] concluded that Bz molecules are centered around the three-fold hollow sites of the Ag(111) surface, however, they could not clearly identify whether these are fcc or hcp hollow sites.",
"The interactions of Bz with coinage metal surfaces are significantly weaker than with other transition metals, such as Pt, Pd, Ir, and Rh, since the d-band centers of Cu(111), Ag(111), and Au(111) are well below the Fermi level [166].",
"STM experiments observed that Bz molecules can easily diffuse over the Cu(111) and Au(111) surface at low temperatures, suggesting a flat PES in both cases [167], [168], [169].",
"TPD experiments revealed that at a heating rate of 4 K/s, Bz desorbs from Cu(111) at a low temperature of 225 K [163], from Ag(111) at 220 K [170], and from Au(111) at low coverage of 0.1 ML at 239 K [171].",
"In the case of Bz/Ag(111), a combined NIXSW and TPD study recently reported a vertical adsorption height of benzene of $3.04 \\pm 0.02$ Å and an adsorption energy of $0.68 \\pm 0.05$ eV [18].",
"TPD-basd experimental adsorption energies for Cu(111) and Au(111) are 0.71 [163] and 0.76 eV [172], respectively.",
"More recent results based on the complete analysis method are $0.69 \\pm 0.04$ eV and $0.65 \\pm 0.03$ eV for Bz/Cu(111) and Bz/Au(111), respectively.",
"[18]"
],
[
"Theoretical Data",
"Using a (3 $\\times $ 3) supercell, we have explored the PES of a single Bz molecule on close-packed coinage metal surfaces [166].",
"For Bz on Cu(111), PBE+vdW$\\rm ^{surf}$ predicts the most stable adsorption site to be the hcp site where the molecule is rotated by $30^{\\circ }$ with respect to the high symmetry directions (hcp30$^{\\circ }$ site).",
"However, we also only find a small energy corrugation between all adsorption sites.",
"The average C–Cu adsorption height with PBE+vdW$\\rm ^{surf}$ at this site is 2.79 Å.",
"The calculated adsorption energy is 0.86 eV, which deviates by 0.15 eV from experiment [163], [173].",
"The relaxed Bz molecule adsorbs in a flat-lying geometry on the Ag(111) surface, which is consistent with the observations and conclusions from NEXAFS, EELS, and Raman spectroscopy [165], [164].",
"The carbon–metal distance is larger for Ag than for Cu, which suggests a weaker interaction for the former.",
"This is in agreement with TPD experiments which showed that the Bz molecule desorbs at a lower temperature from Ag(111) than from Cu(111) [170], [163], [173].",
"PBE+vdW$\\rm ^{surf}$ predicts a Bz/Ag(111) adsorption energy of 0.75 eV, which is in good agreement with TPD experiments (0.69 eV) [170].",
"The flatness of the PES for Bz on Au(111) (see the structure in Fig.",
"REF (a)) which results from our calculations with the PBE+vdW$\\rm ^{surf}$ method confirms the STM observations that Bz molecules are mobile over the Au(111) terraces even at 4 K [168].",
"An almost identical adsorption energy is found for all sites, which indicates a small barrier for surface diffusion of Bz on the Au(111) surface.",
"The PBE+vdW$\\rm ^{surf}$ adsorption energy for Bz/Au(111) (0.74 eV) is in excellent agreement with the TPD experiments (0.76 eV) [171].",
"The reduced agreement in the case of Bz/Cu(111) compared to the fair agreement for the other substrates might point to remaining discrepancies in the structural model of Bz surface adsorption on Cu(111)."
],
[
"Experimental Data",
"Bz adsorbed at the Pt(111) surface, including its adsorption and dehydrogenation reactions, is the best studied system among the Bz adsorption systems.",
"Nevertheless, even the preferred adsorption site remains controversial in experiments.",
"The diffuse LEED intensity analysis suggested that the bridge site with the molecule $30^{\\circ }$ tilted with respect to the high symmetry sites (bri30$^{\\circ }$ ) is the most stable site for Bz chemisorbed on the Pt(111) surface [174]; whilst nuclear magnetic resonance (NMR) results revealed that Bz molecules are located at the atop site [175].",
"Inferred from the orientations of the STM images, the coexistence of Bz molecules at both the hcp and fcc sites was concluded [176].",
"Despite the ambiguous adsorption site, all experiments clearly concluded that the adsorbate lies flat on the Pt(111) surface, binding with the Bz $\\pi $ orbitals to the Pt d bands.",
"STM topographs suggest immobile Bz molecules adsorbed on Pt(111), which points to strong binding at this surface [176], [177].",
"The Bz molecules are found to adsorb as intact molecule on the Pt(111) surface at 300 K [172].",
"However, for coverages below 0.6 ML, Bz dissociates completely into hydrogen gas and adsorbed graphitic carbon upon heating and fragment desorption is observed [172].",
"Therefore, microcalorimetric measurements, rather than desorption-based methods (such as TPD, molecular beam relaxation spectroscopy (MBRS), and equilibrium adsorption isotherms), are required to determine the heat of adsorption for Bz on the Pt(111) surface.",
"The corresponding single crystal adsorption calorimetry results by [172] report a zero-coverage extrapolated adsorption energy of 2.04 eV."
],
[
"Theoretical Data",
"For Bz on Pt(111), the PBE+vdW$\\rm ^{surf}$ method predicts the bri30$^{\\circ }$ site as the most favorable site with an adsorption energy of 1.96 eV.",
"The second and third preferred sites are the hcp0$^{\\circ }$ and fcc0$^{\\circ }$ site, respectively.",
"The calculated adsorption height from PBE+vdW$\\rm ^{surf}$ is in excellent agreement with the adsorption height as derived from LEED analysis (2.09 vs. $2.0 \\pm 0.02$ Å) [174].",
"We also constructed a larger supercell of ($4 \\times 4$ ) for Pt(111), and in this lower coverage case the adsorption energy is determined to be 2.18 eV from PBE+vdW$\\rm ^{surf}$ .",
"This lies within the uncertainty of calorimetry measurements in the limit of zero coverage (1.84–2.25 eV) [172]."
],
[
"Experimental Data",
"Naphthalene (Np) adsorbed on Ag(111) has been studied using LEED [178], [179], [180], NEXAFS [164], 2PPE [181], and TPD [178], [182].",
"Upon deposition at 90 K, the molecule is found to desorb completely at about 325 K for low initial coverages.",
"This corresponds to an adsorption energy of $0.88 \\pm 0.05$ eV assuming a pre-exponential of $10^{13}~$ s$^{-1}~$ [178].",
"Recently, a factor of $10^{15.2}~$ s$^{-1}~$ was suggested [183] to be more accurate using the Campbell-Sellers method [184], [65] to estimate pre-exponential factors.",
"The corresponding adsorption energy which we report in Tab.",
"REF is 1.04 eV.",
"For the molecule adsorbed in the monolayer a ($3 \\times 3$ ) surface overlayer structure has been found from LEED [178], [179], whereas also different non-primitive overlayers have been observed at higher temperatures [180].",
"Using 2PPE, [181] found that unoccupied states in the Np monolayer are mixed with image potential states of the interface.",
"In mono- and multilayer arrangements, NEXAFS measurements found Np to adsorb flat on Ag(111) [164]."
],
[
"Theoretical Data",
"We modelled the Np/Ag(111) interface with PBE+vdW$^{\\mathrm {surf}}$ as a ($4 \\times 4$ ) unit cell with six substrate layers.",
"In the optimal geometry of the molecule the phenyl rings are situated close to hollow sites or, equivalently, the central C–C bond is situated above a top site.",
"The molecule adsorbs flat on the surface with no sign of hybridization or displacement of hydrogen atoms above or below the molecular plane.",
"The average C–Ag vertical adsorption distance is 2.99 Å at an adsorption energy of 1.22 eV."
],
[
"Experimental Data",
"Np on Cu(111) shows a variety of overlayer structures that are both commensurate and incommensurate, the latter giving rise to Moiré patterns [185], [186].",
"The overlayer structures are all non-primitive structures in surface area between (4x3),(5x3), and larger superstructures.",
"All studies that performed TPD measurements [187], [188], [173], notwithstanding differences in absolute desorption temperatures and heating rates, identified a broad desorption feature believed to be associated with desorption from terraces and a sharp feature at lower temperatures associated with desorption from step edges [189].",
"[173] report an adsorption energy of 0.81 eV for desorption from (111) terraces.",
"Similar to the case of Np/Ag(111) we re-evaluate the desorption energy using a pre-exponential factor of $10^{15.2}$ as 1.07 eV."
],
[
"Theoretical Data",
"Equally as in the case of Np on Ag(111), we model Np/Cu(111) in a (4x4) unit cell with the molecule adsorbed flat on the surface and the conjugated phenyl rings situated above hollow sites.",
"We find a PBE+vdW$^{\\mathrm {surf}}$ adsorption energy of 1.41 eV and an average C–Cu vertical adsorption height of 2.73 Å."
],
[
"Experimental Data",
"Np and other aromatic molecules on Pt(111) have been studied with LEED by different groups.",
"An initial study proposed the overlayer structure of Np/Pt(111) to be a ($6 \\times 6$ ) surface unit cell containing 4 molecules with alternating $90^{\\circ }$ tilt angle with respect to each other [190].",
"Later, Dahlgren and coworkers proposed a ($6 \\times 3$ ) overlayer containing 2 molecules and satisfying a glide-plane symmetry [191].",
"In their model Np is adsorbed with its center at a top site and the 2 molecules are rotated by $60^{\\circ }$ with respect to each other.",
"The authors later also found that Np molecules on Pt(111) fully dehydrogenate above 200 K [192].",
"The proposed overlayer structure was also later confirmed by STM and LEED [193].",
"The adsorption energy of Np/Pt(111) has been measured by [194], [195] using SCAM at 300 K. The authors identify heats of adsorption associated with adsorption at step edges and at terraces.",
"Modelling the latter, they arrive at an adsorption energy of 3.11 eV for the zero-coverage limit.",
"Higher molecule packing significantly decreases the adsorption energy.",
"At a packing density of 0.59 ML corresponding to a (4x4) surface overlayer the adsorption energy is 2.19 eV (as reported in Tab.",
"REF )."
],
[
"Theoretical Data",
"A number of semi-empirical calculations have been performed by Gavezotti et al for naphthalene on Pt(111) [196], [197].",
"The authors conclude on a bonding distance of 2.1 Å.",
"[198] report non-dispersion corrected DFT calculations of Np/Pt(111) using the PW91 functional.",
"The authors predict a binding distance of 2.25 Å and an adsorption energy of 0.53 eV for the experimental adsorption site, but find others to be more stable.",
"As before we model the Np/Pt(111) interface in a (4x4) overlayer using PBE+vdW$^{\\mathrm {surf}}$ .",
"Contrary to Np adsorption at Cu(111) and Ag(111), we find significant distortion of the molecule upon adsorption in the most favorable adsorption site.",
"As found in experiment, the most stable adsorption site is the molecule centered above a top site.",
"We find the hydrogen atoms distorted away from the surface and out of the molecular plane.",
"As a result the vertical adsorption height of Np at Pt(111) is 2.03 Å for carbon atoms and 2.58 Å for hydrogen atoms (see Tab.",
"REF ).",
"We find the corresponding adsorption energy to be 2.92 eV which lies 0.73 eV above the experimental adsorption energy at the same coverage and 0.19 eV below the zero-coverage extrapolated adsorption energy.",
"Table: Adsorption energies (E ad \\rm _{ad}) and perpendicular heights (d) for Bz and Np on (111) metal surfaces.The values are in eV and Å, respectively."
],
[
"Experimental Data",
"The adsorption properties of Diindenoperylene (DIP, C$_{32}$ H$_{16}$ ), a $\\pi $ -conjugated molecule, on noble metals has been extensively studied because of its excellent optoelectronic device performance and the ability to form exceptionally ordered films [199], [200], [201].",
"Also, the rather simple DIP structure and chemical composition, a planar hydrocarbon with no heteroatoms, make DIP/metal interfaces suitable to be used as model systems in the context of HIOSs.",
"The deposition of DIP on clean substrates, either metals or semiconductors, and the formation of an ordered monolayer has been observed in different studies using a variety of experimental techniques.",
"The cleanliness of the metal substrate, the coverages, and the quality of the deposition has been studied using x-ray spectroscopy techniques such as XPS, NEXAFS and x-ray reflectivity techniques [202], [203], [200].",
"The morphology and the electronic properties have been investigated with several methods, e.g.",
"TEM, STM, LEED and UPS spectroscopy [204], [205], [206], [201], [199].",
"The adsorption of DIP on clean Au(111), Ag(111), and Cu(111) has been carefully monitored employing the experiments listed above and different possible interface structures are observed.",
"In general, the DIP surface density and arrangement can be influenced by the presence of step edges, terraces, or the substrate temperature during growth [205].",
"DIP/Ag(111) has been investigated with low temperature STM and LEED, revealing different closed-packed monolayer configurations, namely a brick-wall and a herringbone arrangement [206].",
"Similarly to DIP/Ag(111), STM experiments found that DIP/Au(111) assumes a brick-wall configuration [201].",
"However, no experimental data are available for a thorough comparison of the cohesive energies.",
"Finally, recent NIXSW measurements extended the characterization of these systems, providing accurate average bonding distances of DIP on all three noble metals [125].",
"The measurements are listed in Tab.",
"REF and follow the trend: $d$ (Cu)$< d$ (Ag)$< d$ (Au)."
],
[
"Theoretical Data",
"Electronic structure calculations for all the three metals were performed using a ($7 \\times 7$ ) unit cell composed of three metal layers and one DIP molecule.",
"First we computed the adsorption energy curve by rigidly tuning the surface–molecule distance $d$ (see supplemental material).",
"As a second step, we relaxed the geometry with lowest $E_\\mathrm {ad}$ for each system.",
"The average bonding distances obtained from our simulations show a remarkably good agreement with the experimental data, see Tab.",
"REF , with a discrepancy of less than 0.1 Å for all the three systems.",
"Moreover, in accordance with the bonding distances, the adsorption energies show the trend: $|E_{\\rm ad}{\\rm (Cu)}|>|E_{\\rm ad}{\\rm (Ag)}|>|E_{\\rm ad}{\\rm (Au)}|$ .",
"In the case of DIP/Ag(111), in addition to the structures considered above, we take into account the two densely-packed and well-ordered configurations of the monolayer: the brick-wall and the herringbone arrangements.",
"Remarkably, the bonding distance obtained from the relaxed geometries is $d=2.99$ Å in both cases.",
"This equilibrium distance is in almost perfect agreement with the XSW measurement, even improving the result reported in Tab.",
"REF .",
"Furthermore, the relaxed structure obtained using the brick-wall arrangement for DIP/Au(111) yields a binding distance of 3.15 Å, also in better agreement with the experiment than the ($7 \\times 7$ ) structure."
],
[
"Experimental Data",
"The fullerene C$_{60}$ molecule has been intensively studied since its discovery, for its interesting properties such as superconductivity [207] or metal-insulator transition [208].",
"In the same spirit, the deposition of thin films of C$_{60}$ on noble metal surfaces opens up numerous possible applications, e.g.",
"lubrication and molecular switching.",
"These properties, combined with the ability of C$_{60}$ to form ordered monolayers on surfaces, motivated a large number of experiments performed with several different techniques, e.g.",
"UHV-STM, LEED, Auger electron spectroscopy (AES).",
"It was found that C$_{60}$ adsorbed on Ag(111) and Au(111) is adsorbed in a well-ordered closed-packed monolayer and displays a commensurate $(2\\sqrt{3} \\times 2\\sqrt{3})R30^{\\circ }$ unit cell [209], [210].",
"C$_{60}$ adsorbs in different sites and it is also possible to manipulate the facet of the molecule exposed to the surface, e.g.",
"two distinct orientations are found by tuning the level of potassium doping [211].",
"A UHV-STM and LEED study concluded that the molecule adsorbs, on both Ag and Au(111), preferably on top sites with a pentacene ring facing down [212].",
"On the other hand, for C$_{60}$ /Ag(111), a STM experiment revealed that the adsorption took place on hollow sites, with a hexagonal ring facing the surface [213].",
"Moreover, a mix of hexagonal face-down and C-C bond down was found in an x-ray photoelectron diffraction study [211].",
"Within the different possible monolayer configurations, recent LEED experiments of C$_{60}$ on Ag(111) confirm the $(2\\sqrt{3} \\times 2\\sqrt{3})R30^{\\circ }$ unit cell and suggest that the most stable configuration is with the molecule situated above a vacancy site of the metal surface.",
"The C$_{60}$ molecule is adsorbed on a vacant-top site with a hexagon face-down orientation [214].",
"During the adsorption, the surface relaxes and the silver atoms close to the C$_{60}$ are slightly compressed into the surface.",
"This is indicated with the distance measure $\\Delta $ in Fig.",
"REF panel b.",
"This interlayer buckling amplitude and the distance between the molecule and the topmost layer are computed by fitting LEED measurements.",
"Similarly, STM images for C$_{60}$ on Au(111) illustrate the presence of a vacancy, confirming that the molecule is adsorbed above the vacancy with the hexagon face-down configuration [215].",
"To the best of our knowledge, no experimental data are available for the energetics of these two systems."
],
[
"Theoretical Data",
"In order to reproduce the experimentally observed structure, we considered, for both silver and gold, a $(2\\sqrt{3} \\times 2\\sqrt{3})R30^{\\circ }$ unit cell with six metal layers.",
"This unit cell contains one C$_{60}$ molecule, as shown in Fig.",
"REF a, placed in the center of the cell and adsorbed in correspondence of a top site.",
"Further, this particular top atom is removed creating a single vacancy site on the topmost metal layer.",
"We performed a relaxation involving all the six metal layers in order to capture also the intra-layer rearrangement and obtain a better comparison between our simulations and the experimental findings.",
"Notably, different molecular orientations have been taken into consideration as a starting point of our simulations.",
"However, during the relaxation procedure, the molecule rearranges to a hexagonal face-down orientation, confirming that the latter structure represents the most stable geometry in the presence of a vacancy.",
"The binding energies for both systems are computed taking into account the formation energy of the vacancy (see supplemental material) and are reported in Tab.",
"REF .",
"We find that C$_{60}$ binds stronger to Au(111) than to Ag(111).",
"The predicted binding distances, for both Ag(111) and Au(111), are in excellent agreement with the experimental data, as reported in Tab.",
"REF .",
"Considering the buckling amplitude $\\Delta $ , the adsorption of C$_{60}$ on Ag(111) produces a $\\Delta $ of 0.02 Å and 0.03 Å for the first and second layer respectively [214].",
"From our simulations we find $\\Delta =0.015$ Å for the first layer and a larger $\\Delta =0.03$ Å for the second, confirming the experimental trend.",
"In the case of C$_{60}$ on Au(111), the experimental measurements indicate a decrease in the amplitude from 0.05 Å to 0.02 Å within the first two metal layers [216].",
"The $\\Delta $ obtained from the theoretical calculations reproduce the same trend, with slightly larger values: 0.08 Å and 0.045 Å respectively.",
"Table: Adsorption energies (in eV) and perpendicular heights (in Å) for DIP and C 60 _{60} on (111) metal surfaces."
],
[
"Experimental Data",
"Thiophene (Thp) is one of the smallest heteroaromatic molecules for which metal-surface adsorption has been studied.",
"Thp has been adsorbed on Au(111) surfaces from vacuum [217], [218] and from solution [219], [220], whereas only vacuum adsorption leads to successful adsorption of pristine Thp molecules.",
"Adsorption from ethanol solution leads to decomposition of Thp molecules as evidenced by infrared absorption spectroscopy, XPS, and NEXAFS measurements [220] Using TPD, XPS, and NEXAFS experiments [218] and [217] have studied Thp adsorbed on Au(111).",
"Thp adsorbs below 100 K and desorbs fully already around 330 K. Both above mentioned TPD studies find slightly different desorption temperatures, due to different experimental conditions on heating rate (2 K/s vs. 3 K/s) and adsorption temperature.",
"However, both studies find a desorption peak at low coverage at 215 K (255 K) that is associated with a flat-lying adsorbate structure and related to average adsorption energies of 0.60 eV [217] (0.70 eV [218]).",
"STM experiments support the assessment that this phase corresponds to flat-lying geometries [221].",
"A second TPD peak around 180 K appears at higher coverages and saturates at a nominal coverage of 2.3 ML.",
"This signature is associated with a compressed monolayer structure of tilted Thp molecules and an effective adsorption energy of 0.47 eV by both studies.",
"At higher coverage a third signature at 140-150 K is found that does not saturate with coverage, consistent with desorption from a physisorbed multilayer.",
"The geometry change from a flat-lying to a tilted compressed monolayer is corroborated by XPS and NEXAFS experiments.",
"With increasing coverage above 1 ML the flat-lying phase gradually transforms to a tilted compressed phase with a tilt angle of $55^{\\circ }$ with respect to the surface parallel [217].",
"The LEED pattern associated with this phase is a $(\\sqrt{3}\\times \\sqrt{3})R30^{\\circ }$ .",
"Unfortunately no experimental adsorption heights have been reported for Thp/Au(111).",
"STM experiments suggest that preferential adsorption occurs at the hcp hollow site [222], however due to paired-row structures at high coverage, Thp can coexist at different adsorption sites [223].",
"A notable feature of Thp on Au surfaces is an X-ray induced polymerization reaction to polythiophene films [224]."
],
[
"Theoretical Data",
"The Thp/Au(111) interface has been modelled using pure semi-local DFT and variants of the DFT-D method [100].",
"Using the PBE functional [83] to model the flat-lying Thp molecule [100] find a vanishingly small adsorption energy at a surface-sulfur distance of 3.4 Å.",
"Incorporating dispersion interactions in the form of two variants of DFT-D, the authors find adsorption energies of 1.24 and 1.73 eV and surface-sulfur distances of 2.75 and 2.88 Å, respectively.",
"In these geometries, the sulfur atom lies closer to the surface than the rest of the aromatic ring.",
"Using DFT+vdW$^{\\mathrm {surf}}$ , we have modelled a single Thp molecule at a Au(111) surface in a ($3 \\times 3$ ) unit cell with the molecule lying flat on the surface, as shown in Fig.",
"REF (b).",
"We find the most stable adsorption geometry for the sulfur atom situated at a top site and the aromatic ring centered closely above a hcp hollow site.",
"This is the most stable adsorption geometry on Cu(111), Ag(111), and Au(111).",
"The corresponding adsorption height of Thp/Au(111) as measured from the sulfur atom is 2.95 Å (see Tab.",
"REF ) with an adsorption energy of 0.77 eV.",
"The binding energy as described by DFT+vdW$^{\\mathrm {surf}}$ is significantly reduced compared to the results of Tonigold et al., as is expected from the inclusion of substrate screening effects.",
"Analysing the desorption temperatures of experiment using the Redhead equation [57] and preexponential factors [63], [184] in the range of $10^{12}$ to $10^{15}$ s$^{-1}$ we find adsorption energies of 0.53 to 0.66 eV [217] (0.62 to 0.78 eV [218]), which deviate from the DFT+vdW$^{\\mathrm {surf}}$ by 0.01 eV for the experimental upper limit."
],
[
"Experimental Data",
"TPD measurements of Thp on Ag(111) from 2 different studies [225], [226] identified three desorption peaks corresponding to different overlayer structures: initial desorption at 128 K (140 K) is associated with Thp molecules in the multilayer, a desorption peak at 148 K (162 K) is associated with Thp molecules directly adsorbed on Ag(111) in a densely packed tilted arrangement, and desorption at 190 K (204 K) is believed to arise from molecules that are adsorbed in a flat arrangement on the surface.",
"Discrepancies between the absolute desorption temperatures of the two studies are believed to arise from differing approaches to temperature measurement in the experimental setup [226].",
"C K-edge NEXAFS measurements support the assessment that the high-temperature feature corresponds to molecules adsorbed in an almost flat arrangement [226].",
"[227] identified three monolayer structures of Thp/Ag(111) from STM experiments upon deposition of 40, 90, and 150ML, respectively: a c($2\\sqrt{3}\\times 4$ )rect structure, a ($2\\sqrt{7}\\times 2\\sqrt{7}$ ) structure, and a herringbone structure.",
"These overlayer structures show largely different packing densities.",
"The authors infer from the area per molecule that the c($2\\sqrt{3}\\times 4$ )rect structure corresponds to a flat-lying molecular geometry and the others to more strongly tilted adsorbates.",
"On the basis of the measured desorption temperatures of the above mentioned studies we estimate the adsorption energy for the flat-lying phase using the Redhead equation to be between 0.48 and 0.64 eV for preexponential factors between $10^{12}~s^{-1}$ to $10^{15}~s^{-1}$ ."
],
[
"Theoretical Data",
"We have studied Thp on Ag(111) in a ($3 \\times 3$ ) surface unit cell using DFT+vdW$^{\\mathrm {surf}}$ .",
"As in the case of Thp/Au(111), we find as optimal geometry a flat-lying molecule with a S–Ag distance of 3.17 Å and an adsorption energy of 0.72 eV, which overestimates the experimental regime by 0.08 eV."
],
[
"Experimental Data",
"Milligan and coworkers performed a combined NIXSW, NEXAFS, and TPD study of Thp on Cu(111) and identified two different monolayer phases [228], [229].",
"At low coverage Thp adsorbs in an almost flat-lying geometry, ranging from 12 to $25 \\pm 5^{\\circ }$ for 0.03 to 0.1 ML coverage with respect to the surface and a Cu–S bond distance of $2.62 \\pm 0.03$ Å.",
"The corresponding desorption maximum lies at a temperature of 234 K. At higher coverages a second coexisting phase with higher packing density forms.",
"The Thp molecules are tilted more strongly with a surface angle of $44 \\pm 6^{\\circ }$ and a S–Cu distance of $2.83 \\pm 0.03$ Å.",
"The corresponding desorption peak is found at 173 K. The authors further find that the sulfur atom of Thp predominantly adsorbs at atop sites [228].",
"A number of other studies support the finding of a flat-lying adsorbate phase [230], [231].",
"With the use of the Redhead equation [57] and the experimental heating rate of 0.5 K/s we can estimate the experimental binding energy of the flat-lying phase of Thp from the data of [229].",
"Assuming a preexponential factor in the range of $10^{12}$ to $10^{15}~s^{-1}$ , the adsorption energy is 0.61 to 0.75 eV."
],
[
"Theoretical Data",
"On Cu(111), Thp in a ($3 \\times 3$ ) overlayer unit cell also adsorbs preferentially with sulfur situated at an atop–site.",
"The corresponding S–Cu distance is 2.78 Å, which is in good agreement with experiment.",
"We find the molecule adsorbed with a tilt angle of $6^{\\circ }$ , which is significantly less than the experimentally found angle of $12 \\pm 5^{\\circ }$ for 0.03 ML.",
"The reason for this discrepancy could be an increased tilt angle at finite temperature due to anharmonicity of the adsorbate-substrate bond.",
"The calculated adsorption energy of 0.82 eV overestimates the experimental range by 0.07 eV.",
"Table: Adsorption energies (in eV) and average perpendicular heights of Carbon (C) and Sulfur (S) atoms (in Å) for Thp on Au, Ag, and Cu (111) surfaces."
],
[
"Experimental Data",
"Up to this point we have mostly addressed the performance of the DFT+vdW$^{\\mathrm {surf}}$ method in the adsorption of atoms and molecules on close-packed (111) transition-metal surfaces, but we are also interested in the performance and sensitivity of the DFT+vdW$^{\\mathrm {surf}}$ method when the adsorption occurs on non-close-packed surfaces.",
"We address this aspect by reviewing a comparative analysis of the adsorption of PTCDA on a surface with different orientations: PTCDA on Ag(111), Ag(100), and Ag(110) (see Ref. Ruiz:Liu:Tkatchenko2016).",
"PTCDA is a chemical compound formed by an aromatic perylene core (C$_{\\mathrm {peryl}}$ ) terminated with two anhydride functional groups, each of them containing two carbon atoms (C$_{\\mathrm {func}}$ ), two carboxylic oxygens (O$_{\\mathrm {carb}}$ ) and one anhydride oxygen (O$_{\\mathrm {anhyd}}$ ) [232].",
"The adsorption geometries of these systems have been investigated using the NIXSW technique.",
"[16], [232], [233] A novel feature in the studies including PTCDA on Ag(100) and Ag(110) is their higher chemical resolution resulting in the extraction of the adsorption positions of each of the chemically inequivalent atoms in PTCDA.",
"PTCDA forms a commensurate monolayer structure on silver surfaces.",
"On Ag(111), it forms a herringbone structure with two molecules per unit cell in non-equivalent adsorption configurations.",
"[234], [235] Both molecules are adsorbed above a bridge site, molecule A is aligned with the substrate in the $[10\\bar{1}]$ direction with its carboxylic oxygen atoms on top position and the anhydride oxygen atoms located on bridge sites.",
"Molecule B on the other hand is rotated with respect to the $[01\\bar{1}]$ direction, with most atoms in its functional groups located closely to adsorption bridge positions.",
"[235], [232] On Ag(100), a T-shape arrangement with two adsorbed molecules per unit cell can be observed.",
"[236] Both molecules are aligned with the $[110]$ direction of the substrate with the center of each molecule adsorbed on top position.",
"Finally, in the case of Ag(110), PTCDA forms a brick-wall adsorption pattern with one molecule adsorbed per surface unit cell.",
"[234] The long axis of the molecule is located parallel to the $[001]$ direction, while the center of the molecule is located on the bridge site between the close-packed atomic rows parallel to the $[\\bar{1}10]$ direction.",
"[237]"
],
[
"Theoretical Data",
"In the following we discuss results of Refs.",
"Ruiz:Liu:Tkatchenko2016 and VRuizPhDThesis.",
"Tab.",
"REF shows that the PBE+vdW$^{\\mathrm {surf}}$ results for the vertical adsorption distance agree very well with experimental results.",
"With the exception of the anhydride oxygen in Ag(111), the calculated distances for all atoms that form the molecule lie within 0.1 Å of the experimental results for all three surfaces.",
"These results also reveal that our calculations reproduce the experimental trends observed in the sequence of Ag(111), Ag(100), and Ag(110).",
"[232], [233] The overall vertical adsorption height given by the calculations, taken as an average over all carbon atoms (see $d_{\\mathrm {Th}}$ for `C total' in Tab.",
"REF ), decreases in the sequence of Ag(111), Ag(100), and Ag(110) by 0.26 Å in comparison to the decrease of 0.30 Å obtained in experiments.",
"The calculations reproduce the transition from a saddle-like adsorption geometry of PTCDA on Ag(111) to the arch-like adsorption geometry that can be found in the more open surfaces according to experiments (see Ref. Ruiz:Liu:Tkatchenko2016).",
"Finally, for the above mentioned sequence, we find an increase in the C backbone distortion and a decrease in the O difference ($\\Delta \\mathrm {C}$ and $\\Delta \\mathrm {O}$ defined in Tab.",
"REF ).",
"For $\\Delta \\mathrm {C}$ , the calculations yield 0.02, 0.09, and 0.13 Å for Ag(111), Ag(100), and Ag(110), respectively, values which are in excellent agreement with experiments.",
"[232], [233] In the case of Ag(111), the C backbone distortion has not been determined experimentally, [16] but the saddle-like adsorption geometry suggests a minimum distortion of the C backbone [16], [232], which we observe in our calculations as well.",
"The C backbone distortion in Ag(100) and Ag(110) is then remarkably well reproduced by the calculations.",
"With respect to the oxygen difference ($\\Delta \\mathrm {O}$ ), the resulting values are 0.15 Å for Ag(111) and Ag(100), and 0.10 Å for Ag(110).",
"These values reproduce the decrease in the sequence observed by experiments but underestimate the difference by 0.17 Å in Ag(111) and 0.10 Å in Ag(100).",
"This underestimation lies in the fact that the adsorption distances for the anhydride oxygen obtained with the calculations are also underestimated in the cases of Ag(111) and Ag(100).",
"On the other hand, the calculated distance of the anhydride oxygen to the other oxygen atoms agrees very well with the experimental result of $0.08 \\pm 0.05$ Å for PTCDA on Ag(110) [232].",
"We summarize calculated PBE+vdW$^{\\mathrm {surf}}$ adsorption energies in Tab.",
"REF .",
"The binding strength increases in the sequence $2.86$ , $2.93$ , and $3.39$ eV for Ag(111), Ag(100), and Ag(110), respectively.",
"The vdW interactions are essential in these systems as they yield the largest contribution to the adsorption energy (see Ref.",
"Ruiz:Liu:Tkatchenko2016 for details).",
"We note that $E_{\\mathrm {ad}}$ is calculated with respect to the PTCDA monolayer.",
"The binding strength becomes even larger when calculated with respect to the molecule in gas phase due to the additional contribution of the monolayer formation energy.",
"The accuracy of these results confirm the sensitivity to surface termination that the DFT+vdW$^{\\mathrm {surf}}$ scheme correctly reproduces."
],
[
"Experimental Data",
"In the case of the Au(111) surface, PTCDA does not form a commensurate monolayer but rather exhibits a situation very close to a point–on–line correspondence with the $(22 \\times \\sqrt{3})$ reconstructed surface [239], [240], [241], [242].",
"[240] reported an adsorbate structure at equilibrium conditions (grown at high substrate temperatures and small deposition rates) that suggests an optimal point–on–line relation on each of the three reconstruction domains of the substrate, which results in azimuthal domain boundaries (with an angular misfit of around $2.5^\\circ $ ) in the PTCDA monolayer.",
"The adsorbate structure consists of a rectangular unit cell with an area of approximately 232 Å$^2$ and two PTCDA molecules per surface unit cell.",
"PTCDA is physisorbed on Au(111) and its bonding interaction is governed mainly by vdW forces.",
"[243], [8], [69], [244] [69] studied the system based on single molecule manipulation experiments.",
"By combining STM and frequency-modulated AFM, they reported an adsorption energy of about $2.5$ eV per molecule of PTCDA and an adsorption distance of approximately 3.25 Å.",
"On the other hand, TPD experiments report an adsorption energy of approximately $-1.93 \\pm 0.04$ eV per molecule in the low coverage limit [245].",
"The case of PTCDA on Au(111) has also been measured using the NIXSW technique [243], [16] where they found an adsorption distance of $3.34 \\pm 0.02$ Å for the PTCDA monolayer.",
"[243] reported, however, an adsorption height that corresponds most probably to the square phase of PTCDA on Au(111) [241], [242], which does not conform the (majority) herringbone type phases observed by LEED [239], [241].",
"Accounting for an estimated outward relaxation of the topmost metal layer by 3%, the authors report an adsorption height of $3.27 \\pm 0.02$ Å."
],
[
"Theoretical Data",
"Tab.",
"REF shows the average vertical distance of each species in the PTCDA molecule.",
"Experimental results [243] are also shown for comparison.",
"Fig.",
"REF depicts the structure of the monolayer after relaxation showing the position of each of the two inequivalent molecules in the unit cell.",
"We model the system using a $\\bigl (\\begin{smallmatrix} 6 & 1 \\\\ -3 & 5 \\end{smallmatrix} \\bigr )$ surface unit cell which has an area of 247 Å$^2$ (See supplemental material).",
"Table: Experimental and theoretical results for the adsorption geometry of PTCDA on Au(111).",
"d Th / Exp d_{\\rm Th/Exp} denotes the averaged vertical adsorption heights (in Å) obtained with PBE+vdW surf ^{\\mathrm {surf}} calculations and NIXSW studies.",
"The atom nomenclature is given in Fig. (c).",
"Experimental results for the adsorption distance can be found in Ref. Henze:Bauer:etalSS2007.",
"An estimated experimental adsorption height, which takes into account an outward relaxation of the topmost metal layer by 3%, yields an adsorption height of the C backbone of 3.27 Å.Figure: (a) Structure of PTCDA adsorbed on Au(111) where the equilibrium distances dd for each chemically inequivalent atom calculated with the PBE+vdW surf ^{\\mathrm {surf}} method and measured by experiment are displayed.",
"(b) Top view of the relaxed structure of PTCDA on Au(111).",
"Both inequivalent molecules of the structure are labeled as A and B.",
"(c) Chemical structure of PTCDA.",
"Images of the structures were produced using the visualization software VESTA .Calculations with the PBE+vdW$^{\\mathrm {surf}}$ method result in an adsorption height of 3.19 Å for the C backbone, underestimating the experimental result [243] of 3.34 Å by approximately 0.15 Å.",
"The positions of the O atoms were not measured in experiment due to an overlap of Au Auger lines with the O 1s core level.",
"The results suggest a minor distortion of the C backbone as shown by $\\Delta \\mathrm {C}=-0.05$ Å.",
"The negative sign in $\\Delta \\mathrm {C}$ indicates that the C atoms belonging to the functional groups are located at a higher position than the C atoms of the perylene core.",
"This fact is also reflected in the average position of the O atoms which is around 0.4 Å higher than the average C backbone position.",
"The anhydride O atoms are located around 0.09 Å higher than the C backbone.",
"Overall, the large adsorption height of the monolayer confirms a relatively weak interaction of the molecule with the surface in comparison to adsorption on Ag surfaces.",
"The discrepancy of around 0.15 Å between the PBE+vdW$^{\\mathrm {surf}}$ results and experiment can be attributed to several factors related to both approaches.",
"First of all, the area of the surface unit cell here studied is 247 Å$^2$ , which is larger than the area of the experimental one by more than 6%.",
"On the other hand, [243] reported an adsorption height that corresponds most probably to the square phase of PTCDA on Au(111) [241], [242], which does not conform the (majority) herringbone type phases observed by LEED [239], [241].",
"Finally, neither theory nor experiments take initially into consideration the surface relaxation in the determination of the adsorption height of the C backbone.",
"Taking the estimated experimental [243] adsorption height of $3.27 \\pm 0.02$ Å as reference, which takes into account an estimated outward relaxation of the topmost metal layer by 3%, the difference between theory and experiment is reduced to less than 0.1 Å.",
"Although the correct superstructure of the monolayer including the domain boundaries cannot be achieved by any state-of-the-art modeling, the good agreement between theory and experiment suggests that the lateral arrangement of the molecule is strong due to the intermolecular interactions; thus the effect of the exact superstructure of the monolayer on the adsorption height should be minimal.",
"This fact has also been indicated in experimental studies [243], [240].",
"We have previously estimated the PBE+vdW$^{\\mathrm {surf}}$ adsorption energy of PTCDA on Au(111) to be approximately 2.4 eV per molecule for the case of the adsorbed monolayer [17] and a value between 2.23 and 2.17 eV for the case of a single adsorbed molecule [127].",
"In this work, we have calculated the adsorption energy per molecule for two different coverages $\\Theta $ of 1.0 and 0.5 ML using the above-mentioned $\\bigl (\\begin{smallmatrix} 6 & 1 \\\\ -3 & 5 \\end{smallmatrix} \\bigr )$ surface unit cell (see Tab.",
"REF ).",
"The quantity $E_{\\mathrm {ad}}^{\\Theta (\\mathrm {ML})}$ does not consider the formation of the monolayer in its definition of adsorption energy, whereas $E_{\\mathrm {ad}}^{\\Theta (\\mathrm {gas})}$ does (see the supplemental material for details on the adsorption model).",
"Table: Adsorption energy E ad Θ E_{\\mathrm {ad}}^\\Theta , given in eV, for PTCDA on Au(111) at a coverage Θ=1.0\\Theta = 1.0 ML and Θ=0.5\\Theta = 0.5 ML using the PBE+vdW surf ^{\\mathrm {surf}} method.",
"E ad ( ML ) E^{\\mathrm {(ML)}}_{\\mathrm {ad}} is the adsorption energy calculated with the PTCDA monolayer as reference whereas E ad ( gas ) E^{\\mathrm {(gas)}}_{\\mathrm {ad}} is the adsorption energy calculated with respect the molecule in gas phase as reference (see the supplemental material for details on the adsorption model).TPD analysis retrieves the adsorption energy in the limit of low coverage.",
"With this in mind, we have calculated PTCDA on Au(111) with $\\Theta $ of 0.60, 0.45, 0.30, and 0.15 ML in order to compare the calculated value of the adsorption energy in the limit of low coverage with the experimental result (see Tab.",
"REF and also Ref. VRuizPhDThesis).",
"For these results, we have modelled the system using a larger unit cell with an area of 824 Å$^2$ and a slab with three Au layers, as described in the supplemental material.",
"Notably, at $\\Theta = 0.15$ , the difference between $E_{\\mathrm {ad}}^{\\Theta (\\mathrm {ML})}$ and $E_{\\mathrm {ad}}^{\\Theta (\\mathrm {gas})}$ amounts to only 0.04 eV.",
"We take this coverage value as the limit of low coverage for our calculations.",
"Taking the average value between $E_{\\mathrm {ad}}^{\\Theta (\\mathrm {ML})}$ and $E_{\\mathrm {ad}}^{\\Theta (\\mathrm {gas})}$ at $\\Theta = 0.15$ , the adsorption energy at the limit of the single molecule is $2.15$ eV.",
"This value will be slightly increased if we consider a small correction due to the number of layers in the surface slab.",
"In comparison to the experimental result [245], the PBE+vdW$^{\\mathrm {surf}}$ adsorption energy is overestimated by approximately 0.20 eV.",
"Our current research indicates that this overestimation is related to the absence of many-body dispersion effects (see Ref. Maurer2015).",
"The inclusion of many-body dispersion effects will reduce the overbinding found in pairwise vdW-inclusive methods, yielding an improvement, for instance, in the adsorption energy.",
"Table: Adsorption energy E ad Θ E_{\\mathrm {ad}}^\\Theta , given in eV, for PTCDA on Au(111) at a coverage Θ\\Theta of 1.00, 0.60, 0.45, 0.30, 0.15 ML, and the limit of residual coverage with the PBE+vdW surf ^{\\mathrm {surf}} method.",
"Details of the adsorption model can be found in the supplemental material and Ref.",
"VRuizPhDThesis."
],
[
"Experimental Data",
"The experimental information of PTCDA on Cu(111) is not as extensive as in the cases of Ag and Au.",
"The adsorption unit cell of the system is larger in comparison to Ag and Au due to the smaller lattice constant of Cu.",
"It was characterized by [247] with STM experiments, where they found two coexisting ordered structures.",
"One corresponds to a $(4 \\times 5)$ superstructure and the other one is commensurate with respect to the substrate with two molecules per surface unit cell.",
"The lateral arrangement of the molecules in the monolayer, nevertheless, is not yet fully understood.",
"The bonding distance of the monolayer on Cu(111) was studied by [248] using the NIXSW technique.",
"They found the monolayer, in terms of the carbon backbone of the molecule, located at a distance of $2.66 \\pm 0.02$ Å with respect to the substrate.",
"Their studies also include the adsorption distances of the chemically inequivalent oxygen atoms in PTCDA.",
"The most striking fact of these studies is that the oxygen atoms are located above the carbon backbone of the molecule."
],
[
"Theoretical Data",
"We have investigated three possible adsorption structures with the PBE+vdW$^{\\mathrm {surf}}$ method, derived by [249], which are based on experimental data [247].",
"These structures correspond to different surface unit cells and lateral placement of the molecules within the monolayer structure.",
"Structure 1 corresponds to a smaller surface unit cell than the one proposed by [247].",
"Structure 2 corresponds to the experimental surface unit cell [247], whereas structure 3 corresponds to a different, plausible surface unit cell with the same area as structure 2.",
"The features of these structures are summarized in the supplemental material.",
"Table: PTCDA on Cu(111).",
"Experimental results are also shown for comparison .",
"We use d Th / Exp d_{\\rm Th/Exp} to denote the averaged vertical adsorption heights of the specific atoms obtained with PBE+vdW surf ^{\\mathrm {surf}} calculations and NIXSW studies respectively.",
"The adsorption height is given in Å.Adsorption energies are given in eV.Tab.",
"REF shows the average vertical distance of each species in the PTCDA molecule with respect to the topmost unrelaxed substrate layer.",
"The maximum difference in the adsorption distance of the carbon backbone among the three structures is approximately 0.07 Å, which is found between structures 1 and 3; while a similar difference of 0.06 Å is found between structures 2 and 3.",
"The calculations show, however, that the adsorption distance in structure 3 agrees remarkably better (deviation of 0.02 Å) with the NIXSW results.",
"On the other hand, the final positions of the oxygen atoms disagree with the experimental results regardless of the structure of the substrate.",
"The averaged position of the carboxylic oxygen atoms are below the carbon backbone, which is in contrast to the findings of [248].",
"We have also investigated the stability of each structure by calculating the adsorption energy for each case at monolayer coverage.",
"We have considered both the monolayer and the gas phase molecule as reference.",
"The results are summarized in Tab.",
"REF , showing that structure 3 is the most favorable with respect to the adsorption energy per molecule $E^{\\mathrm {(ML)}}_{\\mathrm {ad}}$ with the monolayer as reference; while structures 1 and 2 are nearly degenerate.",
"The fact that the formation of the monolayer from gas phase brings the adsorption energy closer together in the three cases and the structural differences observed in the structures investigated are evidence that the influence of the lateral placement of the molecules and its relation to the surface unit cell cannot be ignored.",
"A more in-depth research in this regard is part of ongoing efforts.",
"The structural and energetic results also suggest that effects beyond the atomic scale might be at play in the monolayer formation of PTCDA on Cu(111), for example, the statistical average of ordered structures which have subtle structural differences."
],
[
"Experimental Data",
"Azobenzene (AB, see Fig.",
"REF (i)) and its derivatives adsorbed at metal surfaces have been extensively studied experimentally because of the potential use of its photo-isomerization ability in molecular switching devices [250], [251], [252], [253].",
"STM studies have shown that AB molecules adsorbed at Au(111) preferentially adsorb in stripe patterns with molecules stacked orthogonal to the molecular axis [252], [250], [254].",
"These parallel stripes can be transformed to a zig-zag phase using STM bias scanning [254].",
"In doing so, a fully packed zig-zag monolayer at higher coverages can be created.",
"TPD measurements reveal desorption temperatures of about 400 and 440 K for AB adsorbed at low coverage on Ag(111) and Au(111), respectively.",
"On both surfaces the molecule forms multilayers and can be desorbed without fragmentation.",
"Using King's method [56] of TPD analysis this amounts to $1.02 \\pm 0.06$ and $1.00 \\pm 0.15$ eV adsorption energy per molecule on Ag(111) and Au(111) [117], [118].",
"A detailed analysis of adsorption structure and vertical height of AB on Ag(111) has been performed by Mercurio et al.",
"using NIXSW measurements [117], [27].",
"Using a Fourier vector analysis in conjunction with DFT [50] it was found that the vertical adsorption height of the central nitrogen atoms (at 210 K) is $2.97 \\pm 0.05$ Å and the central dihedral angle between the nitrogen bridge and the neighboring carbon atoms deviates $-0.7 \\pm 2.2^{\\circ }$ from a flat arrangement.",
"The outer phenyl rings are tilted by $17.7 \\pm 2.4^{\\circ }$ with respect to the surface as a result of the tilted, stacked AB arrangement.",
"Although no such thorough analysis exists for AB on Au(111), STM topographs suggest a similar arrangement."
],
[
"Theoretical Data",
"AB adsorbed at Ag(111) and Au(111) has been studied in the low coverage limit using non-dispersion corrected PBE [255], different dispersion-inclusive functionals incl.",
"PBE+vdW$^{\\mathrm {surf}}$ [116], [117], [27], [50], [256], and vdW-DF [107].",
"Adsorption on both surfaces is largely governed by dispersion interactions and description with a pure PBE functional yields almost no interaction energy.",
"Employing pairwise dispersion-correction schemes such as the PBE-D2 scheme [98] leads to a stable adsorption geometry, however at largely overestimated adsorption height and strong overbinding.",
"In the case of AB at Ag(111) described with vdW-DF, the interaction energy is found to be close to experiment (0.98 eV) [107], however the vertical adsorption height is 0.5 Å larger than what is found in experiment.",
"The most stable adsorption of AB occurs with the azobridge positioned above a bridge site of the (111)-facet [255], the structure is shown in Fig.",
"REF (a) and (b).",
"The adsorption energy and adsorption height as described by PBE+vdW$^{\\mathrm {surf}}$ is summarized in Tab.",
"REF .",
"The calculations have been performed in a ($6 \\times 4$ ) surface unit cell using a 4-layer metal slab [116].",
"For both AB at Ag(111) and Au(111), even with effectively included substrate screening via vdW$^{\\mathrm {surf}}$ , the overbinding is sizeable, as reflected in an underestimation of adsorption height and overestimation of adsorption energies when compared to experiment.",
"The extent of this overestimation is significantly larger for adsorpion on Ag(111) than on Au(111).",
"Mercurio et al.",
"have shown that equilibrium geometries at low coverage are not sufficient to model the finite-temperature high-coverage situation in the NIXSW and TPD measurements of AB on Ag(111) [27].",
"When considering higher coverages and correcting for anharmonic changes to the geometry at higher temperature, the agreement between experiment and theory is drastically improved (see Tab.",
"REF line AB/Ag(111) T=210 K).",
"Its molecular flexibility and sizable changes in geometry and interaction energy as a function of coverage make AB an especially challenging benchmark system to an accurate description of dispersion interactions.",
"At the PBE+vdW$^{\\mathrm {surf}}$ level, the adsorption energies are overestimated by about 70 to 40% when compared to TPD experiments.",
"When accounting for finite-temperature effects in the case of AB/Ag(111) the deviations in the description of adsorption energies on both substrates are very similar (33 vs. 40%).",
"An accurate description of the binding energy as observed in TPD experiments has recently been achieved using explicit ab initio molecular dynamics simulation of the desorption process [26].",
"Figure: Adsorption geometries of AB adsorbed on Ag(111) at the PBE+vdW surf ^{\\mathrm {surf}} level in side (a) and top view (b).TBA adsorbed at Ag(111) in side (c) and top (d) view."
],
[
"Experimental Data",
"3,3',5,5'-tetra-tert-butyl-azobenzene (TBA) is an AB derivative substituted with four tert-butyl legs (TB legs) (see Fig.",
"REF (j) and Fig.",
"REF (c) and (d)) for which successful photo-induced molecular switching has been reported when adsorbed at the Au(111) surface [252], [257].",
"STM topographs show distinct island formation suggesting a dominance of lateral interactions over adsorbate-substrate interactions [251], [252].",
"The STM topograph of TBA consists of four distinct protrusions representing the TB legs and a depression at the position of the central Azo-bridge.",
"TPD measurements reveal desorption temperatures of about 500 and 540 K corresponding to binding energies of $1.30 \\pm 0.20$ and $1.69 \\pm 0.15$ eV when adsorbed at Ag(111) and Au(111), respectively [118].",
"The adsorption geometry of TBA on Ag(111) at room temperature has been determined with NIXSW [258], [259].",
"The resulting adsorption height of the central nitrogen atoms is found to be $3.10 \\pm 0.06$ Å and the average adsorption height of all carbon atoms is determined as $3.34 \\pm 0.15$ Å.",
"From a Fourier vector analysis of the NIXSW signal a flat adsorption geometry with a minimally bent molecular plane is proposed.",
"The extent of signal incoherence suggests that on average 50% of TB legs are oriented with two methyl groups towards the surface, and 50% are oriented into the opposite direction [259]."
],
[
"Theoretical Data",
"TBA adsorbed at Ag(111) and Au(111) has been modelled in a ($6 \\times 5$ ) surface unit cell with 4 layers of metal [256].",
"The most stable lateral adsorption site, similar as in the AB case is the bridge site, with the molecular axis significantly bent around the central azo-bridge (25 and 14$^\\circ $ torsion away from a flat plane on Ag(111) and Au(111)).",
"The TB legs are simulated facing two methyl groups towards the surface.",
"The resulting adsorption energies and geometries are shown in Tab.",
"REF .",
"Whereas the adsorption height (experimentally only measured for TBA/Ag(111)) is in good agreement, adsorption energies are overestimated by about 70 and 30% for adsorption at Ag(111) and Au(111), which is almost identical to what is found for AB adsorbed at these surfaces.",
"The significantly larger overestimation on Ag(111) suggests, similarly as in the case of AB/Ag(111) an increased relevance of finite-temperature effects leading to an apparent adsorption geometry at larger distances from the surface.",
"The higher relevance of finite-temperature effects on Ag(111) surfaces can be understood by the coexistence of dispersion interactions between the TB legs, the phenyl groups and the substrate and small covalent interaction contributions between the central azo bridge and the metal substrate.",
"Table: Adsorption Energy (in eV) and perpendicular heights of Nitrogen atoms(in Å) for AB and TBA adsorbated on (111) metal surfaces.AB/Ag(111) T=210 K refers to simulation data that has been corrected for finite-temperature effects ."
],
[
"General Discussion",
"The adsorption geometry and adsorption energy of an adsorbate on a surface is determined by a number of different contributions: (1) long-range correlations such as dispersion interaction between adsorbate and surface, (2) covalent contributions between chemically active groups at closer distance, (3) the energetic penalty due to geometrical distortion upon adsorption, (4) the Pauli exchange repulsion due to overlapping electron densities, and (5) the attractive or repulsive lateral interactions between adsorbates.",
"All of the above contributions have to be accounted for to arrive at an accurate electronic structure description of HIOSs.",
"The set of systems we have presented in this work covers a large spectrum of interactions, ranging from pure dispersion interactions in the case of rare-gas atoms to highly flexible island-forming heteroaromatic compounds such as azobenzene.",
"In the following we will quantify the reliability of the DFT+vdW$^{\\mathrm {surf}}$ method in describing these systems in terms of the mean absolute deviation (MAD) from the above presented experimental reference values (see supplemental material).",
"Metal surface-adsorbed rare-gas atoms, such as xenon provide the case of pure dispersion interactions of a single atom with a polarizable surface.",
"This is the case for which the vdW$^{\\mathrm {surf}}$ method is set out to perform well, which it in fact does throughout the studied surfaces.",
"Adsorption heights are within a MAD of 0.06 Å from experiment, with the closest agreement in the case of Cu(110) and the largest deviation (0.14 Å) in the case of Cu(111).",
"The same holds for adsorption energies with a MAD of 0.03 eV or 0.8 kcal/mol and the largest error (0.06 eV), again, in the case of Cu(111).",
"This corresponds to a relative error of 12 % with respect to the magnitude of adsorption energies.",
"Aromatic adsorbates such as benzene and naphthalene introduce additional aspects such as covalent contributions and geometrical distortions on reactive surfaces such as Pt(111) and a more spatially polarizable electron distribution through the aromatic $\\pi $ -conjugation of the molecule.",
"Based on the few cases, where experimental references on the adsorption geometry exist (Bz/Ag(111) and Bz/Pt(111)), DFT+vdW$^{\\mathrm {surf}}$ appears to correctly describe these systems.",
"The corresponding MAD of 0.06 Å is comparable to the accuracy achieved for xenon adsorption on metals.",
"However, the description of the adsorption energy with DFT+vdW$^{\\mathrm {surf}}$ is significantly worse (MAD of 0.23 eV).",
"Throughout all systems DFT+vdW$^{\\mathrm {surf}}$ leads to overbinding of the adsorbate, which is more pronounced for the larger Np than for Bz.",
"For the latter the MAD is only 0.09 eV.",
"This may be an indication that the complex long-range interaction of the polarizable adsorbate/substrate complex cannot be mapped onto effective pairwise-additive contributions.",
"The overbinding as observed for small and intermediate aromatic molecules seems to be somewhat decoupled from the description of the adsorption geometry.",
"This becomes more evident for the adsorption of extended and compacted nanostructures such as DIP and C$_{60}$ .",
"Unfortunately, no experimentally measured adsorption energies exist, however the accuracy of DFT+vdW$^{\\mathrm {surf}}$ in describing the adsorption height of these systems is equivalent to the above systems and is reflected in an MAD in adsorption height of 0.05 Å. Heteroaromatic adsorbates with functional groups that feature lone pairs and unsaturated bonds between carbon and oxygen, sulfur, or nitrogen introduce additional complexity.",
"Upon adsorption these heteroatoms will bond differently to the surface than the carbon backbone.",
"Functional groups with sulfur and oxygen for example may engage in surface bonds that are stronger than the average C–metal interaction.",
"On the other hand nitrogen atoms may exhibit an overall weaker interaction.",
"For thiophene–metal adsorption we find the adsorbate structure to be dominated by the strong S–metal bond which has a shorter distance to the surface than the ideal vertical adsorption height of the aromatic backbone of the molecule.",
"The result is a tilted, flexible adsorption geometry that may vary considerably as a function of coverage and temperature.",
"For the low-coverage cases with little tilt angle that we reviewed here, DFT+vdW$^{\\mathrm {surf}}$ , performs similarly as for benzene and naphthalene.",
"The deviation in adsorption height for Thp on Cu(111) is 0.16 Å and the MAD in adsorption energy for the three coinage metal substrates is 0.14 eV when compared to experimental adsorption energies extracted from TPD via the Redhead technique.",
"At this point it should be mentioned that this method of calculating the adsorption energy from the desorption temperature and a range of assumed pre-exponential factors is highly disputed and only used due to lack of more recent measurements.",
"More accurate experimental reference values are in dire need.",
"In the case of PTCDA, a molecule with a conjugated backbone and terminal di-oxo anhydride groups, adsorbed on different facets of coinage metal surfaces, we find two strongly different spatially separated chemical moieties.",
"The terminal oxo groups engage with the surface in a covalent bond, whereas the interaction of backbone and surface is mainly dominated by dispersion interactions.",
"The active terminal groups in addition lead to lateral interactions that enable a multitude of different stable overlayer phases.",
"However, even for this rather complicated case DFT+vdW$^{\\mathrm {surf}}$ yields an MAD in adsorption height of 0.08 Å that is comparable to the above systems featuring considerably simpler chemistry.",
"The largest deviation can be found for the O–Cu distance in PTCDA/Cu(111) with a discrepancy of 0.19 Å when compared to experiment.",
"An experimental estimate for the adsorption energy only exists for PTCDA/Au(111), which is overestimated by DFT+vdW$^{\\mathrm {surf}}$ by 0.22 eV.",
"The adsorption energy of PTCDA/Ag(111) has recently been estimated from TPD measurements of a smaller analogue called NTCDA [129] to be in the range of 1.4 to 2.1 eV.",
"The DFT+vdW$^{\\mathrm {surf}}$ value for dilute coverage is larger than this estimate.",
"However, there are several different disputed estimates in literatute [129], [17], [106].",
"An accurate experimental reference value for the adsorption energy of PTCDA on Ag(111) that can settle this dispute remains to be measured.",
"Another important aspect of functional groups on the adsorption of aromatic molecules is the emergence of lateral intermolecular interactions such as hydrogen bonds.",
"In the case of PTCDA, the carboxylic oxygens in the molecule form hydrogen bonds with the neighboring molecules and thereby generate the herringbone pattern that is characteristic of the monolayer formation of PTCDA on coinage metals.",
"This fact is initially independent of the interaction with the substrate since PTCDA forms crystals with layered molecular stacks in which the ordering pattern of the molecules in each layer is that of a herringbone arrangement closely related to the one found in the monolayer.",
"There is, nevertheless, an effect on the ordering of the monolayer on each substrate which depends on the adsorbate-substrate interaction.",
"This can be related to the degree of commensurability between the lattice parameters of the substrate bulk and those of the stacking layer of the organic crystal.",
"This leads to the formation of a commensurate monolayer on Ag(111) and a point-on-line coincidence on Au(111).",
"In the case of Cu(111), the mismatch leads to different adsorption sites for oxygen atoms yielding a different distortion in each molecule forming the monolayer, which is reflected in the large discrepancy between experiment and calculation and the low coherent fraction for the carboxylic oxygens found in the NIXSW experiments [248].",
"Metal-adsorbed PTCDA also nicely exemplifies the importance of intermolecular interactions and their contribution to the total adsorption energy.",
"As shown above the adsorption energy varies strongly as a function of coverage and a correct simulation model of the experimental overlayer structure is extremely important to arrive at an accurate first-principles adsorption energy.",
"Contrary to the cases of Thp and PTCDA, the diazo groups in AB and TBA do not strongly interact with Ag(111) and Au(111) surfaces and also contribute to the $\\pi $ -conjugation of the molecule.",
"More importantly, the central azo bridge in the molecule makes it inherently flexible, which translates into a strongly anharmonic binding component in the adsorbate/substrate complex [27].",
"The result is a strong temperature dependence of adsorbate geometry and energetics and a large discrepancy in both height and adsorption energy.",
"MADs for AB and TBA 0 K adsorption models are 0.57 Å and 0.68 eV, respectively, when compared to finite-temperature experimental results.",
"Whereas anharmonic finite-temperature correction of the adsorbate geometry [27] leads to an excellent agreement of simulated and measured adsorption height, the adsorption energy remains 0.31 eV overestimated – a deviation that is in line with the above studied systems.",
"When we combine the DFT+vdW$^{\\mathrm {surf}}$ results for the whole dataset of vertical adsorption heights and adsorption energies we arrive at MADs of 0.11 Å and 0.26 eV.",
"However, if we exclude the 0 K models of metal-adsorbed AB and TBA and only include the finite-temperature corrected AB/Ag(111) the MADs are 0.06 Å for 23 different vertical adsorption heights and 0.16 eV for 17 different adsorption energies (see supplemental material for details).",
"The latter results reflect the same accuracy throughout the dataset and support the assessment that DFT+vdW$^{\\mathrm {surf}}$ yields a reliable description of adsorption geometries, however, at the same time appears to systematically overestimate adsorption energetics of systems more complex than rare-gas atoms.",
"The here presented dataset of 23 vertical adsorption heights and 17 adsorption energies of different HIOSs establishes a comprehensive benchmark and may serve as a tool for the assessment of current and future electronic structure methods."
],
[
"Conclusions and Outlook",
"We have presented results for the structure and stability of a series of HIOSs using a dispersion-inclusive DFT-based method that can reliably describe a wide range of interactions including covalent bonding, electrostatic interactions, Pauli repulsion, and vdW interactions.",
"The noticeable improvement in the calculation of adsorption distances and energies that we have observed with the DFT+vdW$^{\\mathrm {surf}}$ method compared to other pairwise-additive dispersion-corrected methods indicates the importance of the inclusion of the collective screening effects present in the substrate for the calculation of vdW interactions, with particular importance in the case of organic-metal interfaces.",
"From a general perspective, however, there are still many important aspects left to consider in order to achieve both quantitative and predictive power in the simulation of the structure and stability of complex interfaces.",
"Throughout the here discussed dataset, DFT+vdW$^{\\mathrm {surf}}$ yields a slight overestimation of adsorption energies.",
"We relate this to the fact that the complex adsorbate/substrate interactions in systems beyond simple rare-gas atoms cannot be effectively captured in a pairwise-additive dispersion scheme and a more explicit account of many-body long-range correlation contributions is necessary.",
"This could be achieved by incorporating the collective response of the combined adsorbate/substrate system, rather than the effective inclusion of the substrate response alone.",
"[260] High-level quantum-chemistry methods or many-body methods such as the Random Phase Approximation (RPA) for the correlation energy could be used for this purpose.",
"These approaches, however, either perform well for isolated molecules or periodic surfaces, but are rarely applicable for the combined system.",
"Recent results on Xe and PTCDA adsorption at Ag(111) [129] suggest that the many-body description of dispersion interactions at the level of the DFT+MBD method [128], [95] may be an efficient approach to remedy the intrinsic overbinding in DFT+vdW$^{\\mathrm {surf}}$ .",
"The DFT+MBD method may thereby significantly improve the description of adsorption energies, geometries of flexible systems, and vibrational properties of HIOSs.",
"Another equally important challenge for future methods will be a more sophisticated account of local correlation and exchange effects beyond a semi-local exchange-correlation treatment.",
"Range-separated hybrid functionals [261], screened second order exchange methods [262], and recent many-body perturbation theory approaches [89], [263], [264] in combination with methods that provide long-range dispersion interactions may provide the necessary accuracy to correctly reproduce the charge-distribution at the molecule/surface interface, the molecular level alignment, and the correct surface potential decay.",
"The here reviewed representative set of 23 adsorption heights and 17 adsorption energies for HIOSs may provide a useful tool in evaluation and assessment of methodological improvements and may also be extended to measurable electronic, vibrational, and spectroscopic properties.",
"For this dataset, the PBE+vdW$^{\\mathrm {surf}}$ method yields a MAD in adsorption height of 0.06 Å and a MAD in adsorption energy of 0.16 eV (3.7 kcal/mol) and represents the current state-of-the-art in electronic structure description of HIOSs."
],
[
"Acknowledgements",
"Support from the DFG project RE1509/16-02, the DFG-SFB/951 HIOS project A10, the European Research Council (ERC-StG VDW-CMAT), and the DoE - Basic Energy Sciences grant no.",
"DE-FG02-05ER15677 is acknowledged for this work.",
"The authors thank Petra Tegeder for helpful comments on section ."
]
] | 1605.05781 |
[
[
"Antibiotic resistance: a physicist's view"
],
[
"Abstract The problem of antibiotic resistance poses challenges across many disciplines.",
"One such challenge is to understand the fundamental science of how antibiotics work, and how resistance to them can emerge.",
"This is an area where physicists can make important contributions.",
"Here, we highlight cases where this is already happening, and suggest directions for further physics involvement in antimicrobial research."
],
[
"Antibiotic resistance: a physicist's view Rosalind Allen and Bartłomiej Waclaw SUPA, School of Physics and Astronomy, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom and Centre for Synthetic and Systems Biology, The University of Edinburgh The problem of antibiotic resistance poses challenges across many disciplines.",
"One such challenge is to understand the fundamental science of how antibiotics work, and how resistance to them can emerge.",
"This is an area where physicists can make important contributions.",
"Here, we highlight cases where this is already happening, and suggest directions for further physics involvement in antimicrobial research.",
"The emergence and spread of bacterial infections that are resistant to antibiotic treatment, combined with the lack of development of new antibiotics, pose a global health problem that is now well recognised [1], [2], [3], [4], [5].",
"Tackling antimicrobial resistance (AMR) requires coordinated, cross-disciplinary effort: relevant themes include clinical medicine, microbiology, diagnostics, drug discovery, epidemiology, evolutionary biology, global public health policy, veterinary science and agriculture.",
"What role can physicists play in this spectrum of activities?",
"Some of these themes obviously require physics tools, for example to aid the development of new experimental methods and devices.",
"We argue here that a `physics-inspired' approach to basic science also has a prominent role to play in the effort to tackle AMR.",
"For a new, antibiotic-resistant, infectious bacterial strain to become a clinical problem, three events must occur.",
"First, an individual pathogenic bacterium must acquire resistance to the antibiotic in question.",
"This could happen via a spontaneous mutation in one of its genes, which might for example render a target protein less susceptible to the antibiotic by modification of the antibiotic binding site.",
"Alternatively, the pathogenic bacterium could gain a gene encoding antibiotic resistance via horizontal transfer of DNA from a different bacterial strain.",
"Second, the newly resistant bacterium must proliferate such that its resistance-encoding gene spreads in the local bacterial population and cannot be wiped out through random fluctuations in the number of organisms carrying this gene.",
"Third, the resistant strain must spread beyond the local bacterial population where it originated, until it infects a significant number of humans and becomes clinically relevant.",
"These events occur on widely varying length and time scales, from those of molecules (e.g.",
"a mutational event in a DNA strand) to those of macroscopic objects (bacterial biofilms, host animals, or even whole ecosystems), and they involve processes that relate directly to the realms of soft matter, chemical and statistical physics.",
"On the molecular level, physical scientists are already contributing to our understanding of how antibiotics bind to their cellular targets, using both computer simulations and novel imaging techniques [6], [7], [8].",
"At the level of a bacterial cell, questions arise as to whether an antibiotic kills, or inhibits, a bacterial cell, via direct inhibition of its target (e.g.",
"the cell wall synthesis machinery for beta-lactam antibiotics or the protein synthesis machinery for macrolide antibiotics), or via other, downstream effects [9], [10], [11].",
"Here, physicists can contribute by developing simple models for how the complex network of reactions that constitutes bacterial physiology response to the antibiotic-induced stress [12], [13].",
"At the level of a bacterial population, physical interactions between cells and their environment shape the self-assembly of spatially-structured bacterial conglomerates such as biofilms that form on medical implants [14].",
"From a physics point of view, the interplay between biological phenomena such as growth and physical phenomena such as chemical diffusion and physical forces provides many interesting questions.",
"For example, biofilms are often surrounded by a secreted polymer matrix whose physical properties (e.g.",
"viscosity) may affect how the biofilm assembles and how it responds to drug treatment [15].",
"Moreoever nutrient and drug gradients can emerge in biofilms due to the interplay between growth and chemical transport; these can affect biofilm structure [16], [17], [18] and potentially also the rate of evolution of resistant bacteria [19], [20], [21].",
"Other population-level phenomena of interest to physicists include stochastic differences in the behaviour of individual cells, caused by noise in gene expression [22], which can have drastic consequences for the response of the population to antibiotic treatment [23].",
"“A physics-like” approach thus has a role to play in many aspects of AMR if we define such approach as a belief that biological processes can be explained by a combination of simple, yet quantitative experiments and mathematical modelling.",
"In the remainder of this article, we highlight three areas where such physics-like approaches are already proving successful, and we also comment on promising directions for future research.",
"Figure: (A) Example of a resistance landscape with three mutations.",
"“+”/“-” denote the presence/absence of a particular mutation, hence the sensitive strain is represented by ------ and the most-resistant mutant strain by ++++++.",
"The level of resistance is indicated by the height of the bars, and permitted mutations are marked by gray lines.",
"The mutant ++++++ can be reached by many paths, for example (---)→(--+)→(-++)→(+++)(---)\\rightarrow (--+)\\rightarrow (-++)\\rightarrow (+++) along which resistance increases monotonously (blue lines), or (---)→(+--)→(++-)→(+++)(---)\\rightarrow (+--)\\rightarrow (++-)\\rightarrow (+++) which has a “valley” at +--+-- (black lines).",
"(B) Evolution in the presence of an antibiotic gradient for an expanding population.",
"The fate of mutant red cells depends on whether they arise in the “bulk” or at the front of the population wave.",
"In the first case there is no selective pressure due to low drug concentration and the mutant does not spread.",
"However, if the mutant arises at the front, it benefits from the access to nutrients/higher growth rate, and it spreads quickly.",
"(C) If resistant cells arise in microbial colonies, they form spatial “sectors”.",
"The number and the size of such sectors depends on the roughness of the colony's frontier, which in turn is affected by physical interactions between the cells and their environment.",
"(D) A microbial population often contains “persister” cells which grow slowly but are resistant to drugs.",
"If the population is treated with antibiotics, persisters can survive the treatment, switch back to the growing state and cause regrowth.Pathways to resistance An active area of current research focuses on how antibiotic resistance evolves “de novo”, i.e.",
"by genetic mutation in bacterial strains that are not initially resistant (as opposed to via gene transfer from an already resistant strain).",
"Typically, the process of resistance evolution involves not just one genetic mutation but a sequence of mutations.",
"This mutational “pathway to resistance” is one of many possible sequences of mutations in a hugely multidimensional space made up of all the possible genetic variants (genotypes) of the organism (Fig.",
"1A).",
"To understand how resistance evolves we must therefore understand the structure of this “resistance landscape”.",
"Two alternative models represent extreme limits of the resistance landscape.",
"In the first model, the level of resistance changes randomly with each mutation (known as a “maximally rugged”, or “House of Cards” landscape [24]), while in the second, mutations are additive, such that the total resistance is the sum of the contributions of the various mutations that a bacterium has acquired (this corresponds to a smooth fitness landscape).",
"Using a statistical physics approach, an analysis of measured resistance landscapes for several different antibiotics suggests that these landscapes are neither fully random nor fully additive, but that they can be approximated by a “rough Mount Fuji” model, which essentially corresponds to a superposition of the rugged and smooth models [25], [26].",
"Further computer modelling [27] has shown that the pathways followed by evolution on such landscapes are most predictable in intermediate-size populations ($N\\mu ^2\\ll 1 \\ll N\\mu $ , where $N$ is the population size and $\\mu $ is the mutation probability) whereas the evolution of resistance is predicted to be less reproducible in either very small or very large populations.",
"The predictability of the evolution of resistance has recently been tested experimentally by growing bacteria in antibiotic under constant selective pressure, using a “morbidostat” [28], [29].",
"This is a device in which the population density and growth rate of a bacterial population are constantly monitored, and the dosage of antibiotic to which it is subjected is adjusted to maintain (on average) a fixed growth rate.",
"Thus, as the population evolves resistance, the antibiotic dosage increases.",
"By sequencing the DNA of bacteria sampled from the morbidostat as the population evolves, qualitative differences in the pathways by which resistance to different drugs emerges have been revealed.",
"In particular, the evolution of resistance to the antibiotic trimethoprim has been found to be rather reproducible, occurring by a well-defined and universal sequence of mutations, but the evolution of resistance to the antibiotics chloramphenicol and doxycycline was found to show a less predictable pattern, occurring via different sequences of mutations in replicate experiments [28].",
"Another “simple yet smart” experimental design has led to a different insight into the pathways leading to antibiotic resistance evolution.",
"Here, bacteria are subjected to repeated brief exposures to antibiotic, alternating with periods of growth without antibiotic [30].",
"Remarkably the bacteria become tolerant to the antibiotic by matching the duration of their “lag time”, or the dormancy period before they start to grow when exposed to fresh nutrient medium, to the duration of antibiotic exposure [30].",
"This is an important finding because it suggests that tolerance to antibiotic exposure can be achieved much more easily than full resistance, by tinkering with the cell's existing gene regulatory components, and that this may be a first step in the pathway to full resistance.",
"Thus, to fully understand the pathways by which bacteria become resistant to antibiotics, we may need to study not just genetic mutations but also phenotypic changes that arise via gene regulation [31], [32], [33].",
"Evolution in spatially structured environments Most studies of antibiotic resistance evolution assume that it happens in a well-mixed, spatially homogeneous environment.",
"However, in infections, as well as more widely in the natural environment, bacterial populations are often highly spatially structured.",
"For example, in biofilm infections bacteria deep within the biofilm are likely to experience quite different local concentrations of nutrients and, possibly, of antibiotic, compared with bacteria on the outside of the biofilm.",
"Indeed, confocal microscopy images of lab-grown biofilms exposed to antibiotics show that some antibiotics act selectively on cells on the outside of the biofilm, while others selectively kill those on the inside [34].",
"Could the spatial gradients of antibiotic that arise during treatment of infections influence the evolution of antibiotic resistance?",
"Experimental and theoretical physicists have recently addressed this topic by investigating the growth and evolution of bacterial populations in the presence of controlled spatial gradients of antibiotic.",
"In particular, Zhang et al constructed a microfluidic device which allowed antibiotic gradients to be imposed across a spatially expanding bacterial population [19].",
"In this device, the evolution of E. coli bacteria which were resistant to the antibiotic ciprofloxacin occurred much more rapidly in the presence of an antibiotic gradient than in a uniform concentration of antibiotic.",
"This work inspired the development of several theoretical models in which an expanding bacterial population invades a one-dimensional habitat containing a gradient of antibiotic [21], [20] (Fig.",
"1B).",
"These models suggest that the presence of an antibiotic gradient can accelerate the evolution of resistance because mutants that emerge at the tip of the expanding population wave are exposed to high antibiotic concentrations, for which they have a strong selective advantage, and because of the low population density at the tip of the population wave, they do not have to compete with less resistant genotypes.",
"However this is not always the case; too steep a gradient can actually slow down the evolution of resistance, and the process also depends on the rate of bacterial migration [20] and the mutational pathway to resistance [21].",
"Other important recent work shows that, even in the absence of an antibiotic gradient, evolution can work very differently in a spatially structured population compared to a spatially well-mixed one.",
"Using colonies of bacteria and yeast cells growing on the surface of a semi-solid agar gel, Hallatschek et al, and others, showed that population fluctuations at the front of an expanding microbial colony can lead to some cell lineages randomly spreading in the population while others die out [35], [36], [37], [38].",
"In these experiments, spreading lineages are strikingly visualised as sectors of differently-coloured fluorescent cells (Fig.",
"1C).",
"Further work investigating the probability of fixation of mutants in these expanding populations suggests that the roughness of the growing front plays a crucial role [39].",
"However, much more work is needed before we can translate these principles into a full understanding of how the spatial structure of a bacterial population, for example in a biofilm infection, links to its propensity to generate and fix harmful antibiotic-resistant mutants.",
"Response of individual cells to antibiotics Over the past decades, physicists have played a leading role in establishing the existence, and importance, of variation between individual cells within bacterial populations [40], [41], [42].",
"This phenotypic heterogeneity arises from stochasticity in the molecular processes involved in gene expression, modulated by the networks of interactions between proteins and genes that regulate gene expression.",
"From the perspective of antibiotic resistance, a particularly important example of phenotypic heterogeneity is the existence within bacterial populations of a subpopulation of “persister cells” which can survive antibiotic treatment (Fig.",
"1D).",
"Classic experiments using microfluidic devices by Balaban et al visualised E. coli cells switching into and out of the non-growing persister state [23]; yet the molecular mechanisms controlling this switch are still the topic of active research [43], [44].",
"Evidence is also now emerging that bacterial population heterogeneity in the response to antibiotics may be a more general phenomenon.",
"Several different theoretical models for the effects of antibiotics on bacterial growth predict the existence of bistable regimes, in which some cells in the population show fast growth while others grow only very slowly, if at all [45], [12].",
"The models suggest that this growth bistability can have a number of origins, including the interplay between bacterial growth and drug dilution [45] and an irreversible “annihilation” reaction between an antibiotic and its target [12].",
"Joint experimental and theoretical work has also shown that growth bistability can arise from the partitioning of cellular resources between growth and production of proteins leading to antibiotic resistance [46].",
"Whether, and how, these bistable growth regimes are related to the persister phenomenon, remains unclear.",
"From the clinical point of view, we would also like to understand which drugs produce a heterogeneous response and how this impacts both on treatment strategies and on the evolution of resistance.",
"From an experimental point of view, investigations of bacterial phenotypic heterogeneity often rely on microfluidic technology which combines physical, chemical, and engineering knowledge to image and manipulate individual bacterial cells.",
"Devices such as the “mother machine” [47] allow the proliferation of individual cells to be tracked under constant conditions over long times in the microscope, and these devices are beginning to reveal interesting information about the response to antibiotics [48].",
"Another recently-developed microfluidic technique, Microscopy Assisted Cell Screening, provides a way to rapidly image thousands of cells immediately after sampling them from a growth device [49].",
"This approach can be used, for example, to measure population heterogeneity in processes like DNA damage, which involve small numbers of molecules per cell.",
"Perspective The topics discussed above highlight important contributions that are being made by physicists to our understanding of antibiotic action and to AMR.",
"We believe that this is just the beginning.",
"Many questions remain to be addressed about how bacteria respond (and become resistant) to antibiotics, and physicists have an important role to play in this effort.",
"As a first example, it is imperative to gain better understanding of how bacterial cells interact mechanically with one another and with their environment.",
"Mechanical interactions appear to be very important in bacterial self-assembly [50], [51], [52], [53], yet our limited knowledge of these interactions prevents us from building accurate models of how spatially structured infections like bacterial biofilms form.",
"As a second example, horizontal gene transfer – the transmission of genes encoding antibiotic resistance between (potentially) unrelated bacteria by direct transfer of DNA – has been little studied in a “physics” context [54], [55], yet it is very important in clinically relevant antibiotic-resistant infections.",
"It would be very interesting to investigate how physical factors such as the forces existing between adjacent bacterial cells in a colony or biofilm affect the rate of gene transfer [56].",
"We also believe that AMR can provide a rich source of more abstract problems in areas of physics from non-equilibrium statistical mechanics to soft matter physics and fluid mechanics.",
"For example, the growth of bacterial biofilms shows a non-equilibrium phase transition (or fingering instability) between rough and smooth modes of interface growth [16].",
"Other examples include random walk models inspired by the dynamics of mutant sectors in expanding bacterial populations [35], [36], and stochastic differential equation models to describe the emergence of waves of resistant mutants in an antibiotic gradient [21].",
"Physics models that are inspired by AMR may also be transferable to other biological fields - for example the evolution of bacterial antibiotic resistance has important analogies with the evolution of drug resistance in cancer tumours [57], while models for the dynamics of bacterial and viral infections also have many similarities.",
"Finally, we believe that an important goal of future research in AMR, by physicists and others, must be to make a link between the findings of simple laboratory experiments and theoretical models, and what happens in real infections, in clinical settings [58].",
"Existing areas of progress in this direction include using observations on how antibiotics interact with the physiology of bacterial cells to suggest more effective clinical treatment strategies [59], [60], [12], using systematic measurements of thousands of bacterial growth curves, combined with simple theoretical models, to predict the clinical effectiveness of multi-drug therapies [61], [62], and tracking mutational pathways in in vivo infections [63].",
"Future challenges will include understanding the role of spatially structured infections in the evolution of clinical antibiotic resistance and the impact of bacterial population heterogeneity on clinical responses to treatment.",
"Conclusion Achieving a better understanding of how antibiotic resistance emerges and spreads could help us to design strategies to prevent this from happening, for both current and future antibiotics.",
"This is a goal to which physicists can contribute, not just by developing new machines and software tools, but also by designing simple yet insightful experimental and theoretical models to test basic principles.",
"This work should not be carried out in isolation, but in coordination with the broad spectrum of other efforts that are being made to tackle the problem of antibiotic resistance.",
"To make this possible, however, support is needed in the form of funding.",
"This requires a breakdown of traditional discipline barriers.",
"Indeed, for the breakthroughs highlighted in this article it is not relevant to ask “is it physics”?",
"or “is it biology”, but simply “is it ground-breaking science”?",
"It also requires a breakdown of national barriers, to allow the most talented scientists, within what is still a rather small international field, to work productively together without restrictions imposed by funding regulations.",
"Acknowledgements This article was inspired by a Royal Society International Scientific Seminar on “Antimicrobial resistance: how can physicists help?”, which took place at Chicheley Hall, UK, on October 28th and 29th 2015.",
"RJA is supported by a Royal Society University Research Fellowship and by ERC Consolidator Grant 682237-`EVOSTRUC'.",
"BW is supported by a Royal Society of Edinburgh Research Fellowship."
]
] | 1605.06086 |
[
[
"Chaos in AdS$_2$ holography"
],
[
"Abstract We revisit AdS$_2$ holography with the Sachdev-Ye-Kitaev models in mind.",
"Our main result is to rewrite a generic theory of gravity near an AdS$_2$ throat as a novel hydrodynamics coupled to the correlation functions of a conformal quantum mechanics.",
"This gives a prescription for the computation of $n$-point functions in the dual quantum mechanics.",
"We thereby find that the dual is maximally chaotic."
],
[
"A. Conformal symmetry in one dimension",
"Let us briefly recap some basic features of conformal invariance in one dimension.",
"It is helpful to keep in mind that, in interacting systems like the SYK models, this conformal invariance is only an approximate symmetry of certain large $N$ theories.",
"Consider coupling a CQM to an external metric $h_{\\mu \\nu }$ and a source $\\lambda $ for a scalar operator $\\mathcal {O}_{\\Delta }$ of dimension $\\Delta $ .",
"Define the generating functional of connected correlation functions, $W = - i \\ln \\mathcal {Z}\\,.$ The connected one-point functions of the stress tensor $t^{\\mu \\nu }$ and $\\mathcal {O}_{\\Delta }$ are defined by functional variation, $\\delta W = \\int dt \\sqrt{-h} \\left( \\frac{1}{2}\\langle t^{\\mu \\nu } \\rangle \\delta h_{\\mu \\nu } - \\langle \\mathcal {O}_{\\Delta }\\rangle \\delta \\lambda \\right)\\,.$ By assumption, $W$ is invariant under infinitesimal reparameterizations, under which $h_{\\mu \\nu }$ and $\\lambda $ vary as $\\delta _{\\xi } h_{\\mu \\nu } =D_{\\mu }\\xi _{\\nu }+D_{\\nu }\\xi _{\\mu }\\,, \\qquad \\delta _{\\xi }\\lambda = \\xi ^{\\mu }D_{\\mu }\\lambda \\,,$ with $D_{\\mu }$ the covariant derivative.",
"Plugging these variations into (REF ) and demanding $\\delta _{\\xi }W=0$ leads to the diffeomorphism Ward identity $D_{\\nu }\\langle t^{\\mu \\nu } \\rangle = - (D^{\\mu }\\lambda )\\langle \\mathcal {O}_{\\Delta }\\rangle \\,.$ In one dimension with $h = - dt^2$ , this becomes equation (REF ) from the main text, $\\dot{E} = \\dot{\\lambda } \\langle \\mathcal {O}_{\\Delta }\\rangle \\,.$ On the other hand, we ought to couple the CQM to the metric in a Weyl-invariant way.",
"Then $W$ is invariant under an infinitesimal Weyl rescaling under which the metric and source $\\lambda $ vary as $\\delta _{\\omega } h_{\\mu \\nu } = 2 \\omega h_{\\mu \\nu }\\,, \\qquad \\delta _{\\omega }\\lambda = (\\Delta - 1)\\omega \\lambda \\,,$ and $\\delta _{\\omega }W=0$ gives the Weyl Ward identity (in terms of $E = - h_{\\mu \\nu }\\langle t^{\\mu \\nu }$ ) $E = (1-\\Delta )\\lambda \\langle \\mathcal {O}_{\\Delta }\\rangle \\,.$ One way of stating Polcinski's paradox [12] is that the diffeomorphism Ward identity (REF ) is not compatible with the Weyl Ward identity (REF ) in one dimension.",
"Actually there is a loophole: the two are compatible if the correlation functions of $\\mathcal {O}$ are topological, as they are for operators in the theory of $2N$ free Majorana fermions.",
"So there is a conflict between reparameterization invariance and conformal symmetry for interacting systems in one dimension.",
"Before going on, we should reiterate what is in some sense the main point of this Letter, namely how gravity solves this paradox near AdS$_2$ .",
"The boundary dual is not a CQM on its own, but instead a particular hydrodynamics coupled to the correlation functions of a CQM.",
"Those conformal correlators are strongly constrained by the conformal symmetry, as we now discuss.",
"In any dimension, conformal transformations are the combination of a coordinate transformation $x^{\\mu }=x^{\\mu }(y^{\\nu })$ and Weyl rescaling $h_{\\mu \\nu } \\rightarrow e^{2\\Omega }h_{\\mu \\nu }$ which leave the metric invariant.",
"In one dimension with the flat metric $h=-dt^2$ , any coordinate transformation gives a conformal transformation: the combined action of $t = t(w)\\, \\qquad \\Omega = - \\ln t^{\\prime }(w)\\,,$ sends the metric to itself, $-dt^2 \\rightarrow -dw^2$ .",
"Wick-rotating to Euclidean signature and compactifying Euclidean time, the modes of $t(w)$ generate a Virasoro algebra with $c=0$ .",
"There are two ways to think about the vanishing central charge.",
"The first is simply that the stress tensor vanishes by the Weyl Ward identity.",
"The second is that there is no Weyl anomaly possible in one dimension.",
"As is familiar from two-dimensional CFT, the conformal transformations which are regular everywhere generate the global conformal group $SL(2;\\mathbb {R})$ .",
"There is a single special conformal generator $K$ , and in the usual way one defines primary operators as those annihilated by $K$ .",
"The vacuum two-point function of a primary operator $\\mathcal {O}_{\\Delta }$ of dimension $\\Delta $ is, in Euclidean signature $\\langle \\mathcal {O}_{\\Delta }(\\tau )\\mathcal {O}_{\\Delta }(0)\\rangle = \\frac{1}{|\\tau |^{2\\Delta }}\\,,$ and similarly for three-point functions of $\\mathcal {O}_{\\Delta }$ .",
"We conclude this Subsection with a few comments on thermodynamics.",
"Consider the thermal partition function of a CQM $\\mathcal {Z}_E = \\text{tr}\\left( e^{-\\beta H}\\right)\\,,$ which as usual is the partition function of the Euclidean theory on a circle of size $\\beta $ with thermal boundary conditions.",
"There is only one local counterterm which can be used to redefine the theory, $\\ln \\mathcal {Z}_E \\rightarrow \\ln \\mathcal {Z}_E + m\\int d\\tau \\sqrt{h_E}\\,,$ where $\\tau $ is Euclidean time, $h_E$ the Euclidean metric, and $m$ is some mass scale.",
"This counterterm is not scale-invariant.",
"Thus, in a CQM, the thermal partition function $\\mathcal {Z}_{CQM}$ is an unambiguous observable.",
"This should not be a surprise; in odd dimension, the logarithm of the partition function of a CFT on a Euclidean sphere – the “sphere free energy” – is a useful and unambiguous (up to a quantized imaginary ambiguity which may exist in $d=4k-1$ dimensions) CFT observable.",
"In one dimension, this partition function is the extremal entropy, $S_0 = \\frac{\\partial (T\\ln \\mathcal {Z}_{CQM})}{\\partial T} = \\ln \\mathcal {Z}_{CQM}\\,.$ Now consider a non-conformal QM which realizes an emergent conformal invariance at low energies.",
"Suppose that the low-energy description is a(n approximate) CQM with extremal entropy $S_0$ deformed by a dimension $\\Delta $ operator.",
"The low-temperature partition function is $\\ln \\mathcal {Z}_E = - \\beta E_0 + S_0 + P_1 T^{\\Delta -1} + \\hdots \\,,$ where $E_0$ is the ground state energy.",
"Observe that $E_0$ is unphysical: it is redefined by the counterterm (REF ).",
"This partition function leads to a low-temperature entropy $S = S_0 +\\Delta P_1 T^{\\Delta -1} + \\hdots \\,.$ Comparing this with the entropy (REF ) of near-extremal black holes in dilaton gravity, we see that the black hole entropy is that of a CQM deformed by a $\\Delta = 2$ operator."
],
[
"B. Holographic renormalization",
"In this Appendix we fill in various details on the near-AdS$_2$ solutions described in the main text.",
"We begin with the perturbative solution near the endpoint of a holographic RG flow, with a small amount of infalling dust $T_{ww}(w)$ .",
"The full perturbative solution is $\\nonumber \\varphi & = \\varphi _0 + \\ell r + \\mathcal {O}(\\ell ^2 r^2)\\,,\\\\\\nonumber g & = - \\left(r^2 + 2 \\lbrace t(w),w\\rbrace + \\frac{\\ell r^3}{6}U^{\\prime \\prime }[\\varphi _0] + \\mathcal {O}(\\ell ^2r^2)\\right)dw^2\\\\& \\qquad + 2dw dr\\,,$ with $\\ell \\partial _w \\lbrace f(w),w\\rbrace = -T_{ww}(w)\\,.$ We account for the dust with a massless scalar field $\\chi $ $S_{\\rm matter} = - \\frac{1}{2}\\int d^2x \\sqrt{-g} Z_0[\\varphi ] (\\partial \\chi )^2\\,,$ with $Z[\\varphi _0]=1$ .",
"The infalling solutions are $\\chi (w) = \\lambda (w)$ , and we take the perturbative limit $\\lambda ^2 \\sim \\ell $ .",
"To first order in $\\ell $ , the stress tensor evaluated on this solution is $T_{ww} = \\kappa ^2\\dot{\\lambda }^2$ and the dilaton source vanishes $\\Phi = 0$ .",
"We proceed to holographically renormalize the dilaton gravity (REF ) on backgrounds of the form (REF ).",
"To proceed we define the dual theory on a constant-$r$ slice at asymptotically large $r$ , subject to the constraint that $\\ell r$ is still asymptotically small.",
"In physical terms, we are taking the almost zero energy limit.",
"In practice we drop the corrections to the background (REF ) with two or more powers of $\\ell $ and then proceed in the usual way.",
"The bulk action is divergent as one integrates to large $r$ , so we introduce a cutoff slice at $r=\\Lambda $ , add covariant boundary terms on the cutoff slice to eliminate divergences, and then remove the cutoff by sending $\\Lambda \\rightarrow \\infty $ .",
"The authors of [19] have studied the problem of holographic renormalization in a general dilaton gravity.",
"For the case at hand, the renormalized action is simply $S_{ren} &= \\lim _{\\Lambda \\rightarrow \\infty } \\frac{1}{2\\kappa ^2}\\left\\lbrace \\int d^2x \\sqrt{-g} \\left( \\varphi R + U\\right)\\right.\\\\\\nonumber & \\qquad \\qquad + \\left.",
"\\int dt \\sqrt{-\\gamma } \\left( 2\\varphi K - U \\right) \\right\\rbrace + S_{\\rm matter} + \\mathcal {O}(\\ell ^2)\\,,$ where $\\gamma $ is the induced metric on the cutoff slice and $K$ the trace of its extrinsic curvature.",
"The variation of with on-shell action with respect to the metric and scalar is $\\begin{split}\\delta S_{ren} =& \\lim _{\\Lambda \\rightarrow \\infty } \\int dt \\sqrt{-\\gamma }\\left\\lbrace - (n^{\\mu }\\partial _{\\mu }\\chi )\\delta \\chi \\right.\\\\& \\qquad \\quad \\left.",
"+\\frac{1}{2\\kappa ^2} \\left( n^{\\rho }\\partial _{\\rho }\\varphi - \\frac{U}{2}\\right)\\gamma ^{\\mu \\nu }\\delta g_{\\mu \\nu }\\right\\rbrace \\,,\\end{split}$ with $n^{\\mu }$ the outward pointing normal vector to the cutoff slice.",
"Both $n^{\\mu }\\partial _{\\mu }\\varphi $ and $U$ are $\\mathcal {O}(\\ell )$ for the perturbative solution (REF ), so we require the on-shell variation of the $g_{\\mu \\nu }$ in response to a variation of the boundary metric to $\\mathcal {O}(\\ell ^0)$ , which is simply $\\delta g_{\\mu \\nu } = r^2 \\delta h_{\\mu \\nu } + \\mathcal {O}(r^0,\\ell )$ .",
"Plugging this variation into $\\delta S_{ren}$ and evaluating it on the solution (REF ) gives the boundary stress tensor $\\langle t^{\\mu \\nu }\\rangle = \\frac{2}{\\sqrt{-h}}\\frac{\\delta S_{ren}}{\\delta h_{\\mu \\nu }} = \\frac{\\ell h^{\\mu \\nu }}{\\kappa ^2}\\lbrace t(w),w\\rbrace \\,,$ so that the energy is $E = - h_{\\mu \\nu } \\langle t^{\\mu \\nu }\\rangle = - (\\ell /\\kappa ^2) \\lbrace t(w),w\\rbrace $ , which derives the result (REF ) in the main text.",
"We also obtain the expectation value of the dimension$-1$ operator $\\mathcal {O}$ dual to $\\chi $ , $\\langle \\mathcal {O}\\rangle = - \\frac{1}{\\sqrt{-h}}\\frac{\\delta S_{ren}}{\\delta \\lambda } = \\dot{\\lambda }\\,,$ so that (REF ) becomes $\\dot{E} = \\dot{\\lambda }\\langle \\mathcal {O}\\rangle \\,.$ Now we consider the perturbative solution in the presence of massive matter, $S_{\\rm matter} = - \\frac{1}{2}\\int d^2x \\sqrt{-g} \\left( Z_0[\\varphi ] (\\partial \\chi )^2 + Z_1[\\varphi ]m^2 \\chi ^2\\right)\\,,$ with $Z_0[\\varphi _0]=Z_1[\\varphi _0]=1$ .",
"The field $\\chi $ is now dual to an operator $\\mathcal {O}_{\\Delta }$ of dimension $\\Delta (\\Delta - 1)=m^2$ .",
"We take $\\chi $ to be a free field, but it is easy to allow for self-interactions.",
"As above we take the perturbative limit with $\\chi \\ll \\ell $ .",
"In this limit, $\\begin{split}T_{\\mu \\nu } & = \\kappa ^2 \\left( \\partial _{\\mu }\\chi \\partial _{\\nu }\\chi - \\frac{ (\\partial \\chi )^2 +m^2}{2}g_{\\mu \\nu }\\right)\\,,\\\\\\Phi & = \\kappa ^2 \\left( Z_0^{\\prime }[\\varphi _0](\\partial \\phi )^2 + Z_1^{\\prime }[\\varphi _0]m^2 \\chi ^2\\right)\\,.\\end{split}$ The dilaton and metric are perturbed as $\\nonumber \\varphi &= \\varphi _0 + \\ell \\psi (w,r) + \\mathcal {O}(\\ell ^2 r^2)\\,,\\\\\\nonumber g & = -\\Big (r^2 + 2 \\lbrace t(w),w\\rbrace + \\ell f(w,r)+ \\mathcal {O}(\\ell ^2 r^2)\\Big )dw^2\\\\& \\qquad + 2 dw dr \\,.$ To leading order in $\\ell $ , the solution for $\\chi $ is given by the most general solution of $(\\Box - m^2) \\chi = 0\\,,$ on the AdS$_2$ background (REF ), setting all of the $\\mathcal {O}(\\ell )$ corrections to vanish.",
"We take $\\Delta $ to be general but not half-integer, and pick the standard quantization for $\\chi $ so that $\\Delta >1/2$ .",
"Then near the boundary $r\\rightarrow \\infty $ $\\chi $ is $\\chi = r^{\\Delta - 1} \\sum _{n=0}\\frac{a_n(w)}{r^n}+ r^{-\\Delta } \\sum _{m=0}\\frac{b_m(w)}{r^m}\\,,$ where the $a_{n}$ with $n>0$ are determined by $a_0$ and similarly for the $b_n$ , for example $a_1 = \\dot{a}_0\\,, \\qquad b_1 = \\dot{b}_0\\,.$ We identify the source for the dual operator as $\\lambda = \\lim _{r\\rightarrow \\infty } r^{1-\\Delta } \\chi = a_0(t)\\,.$ The stress tensor and dilaton source have three distinct sets of terms.",
"Near the boundary, they are of the schematic form $T_{\\mu \\nu } = r^{2\\Delta }\\Sigma ^{a}_{\\mu \\nu }(r,a^2) + \\Sigma _{\\mu \\nu }(r,ab) + r^{-2\\Delta } \\Sigma ^b_{\\mu \\nu }(r,b^2)\\,,$ where $\\Sigma ^a_{\\mu \\nu }(r,a^2)$ is some power series in $1/r$ with coefficients built out of two powers of the $a_n$ and their derivatives, and similarly for the other two series.",
"Because $\\Delta \\mathrel {{\\hfil \\in \\hfil \\cr \\hfil /\\hfil \\cr }}\\mathbb {Z}/2$ by assumption, these three series never mix.",
"Solving the $rr$ and $rw$ components of Einstein's equations, we obtain the correction to the dilaton to be $&\\psi = r + \\frac{\\kappa ^2}{\\ell }r^{2(\\Delta -1)}\\left( \\frac{(\\Delta -1)a_0^2}{2(3-2\\Delta )} + \\mathcal {O}(r^{-1})\\right)\\\\\\nonumber &+ \\frac{\\kappa ^2}{\\ell r}\\left(\\Delta (\\Delta -1) a_0 b_0 + \\frac{(\\Delta ^2-1)\\dot{(a_0b_0)}-\\Delta \\dot{a}_0b_0}{3r} + \\mathcal {O}(r^{-2})\\right)\\\\\\nonumber & \\qquad - \\frac{\\kappa ^2}{\\ell ^2}r^{-2\\Delta }\\left( \\frac{\\Delta b_0^2}{2(2\\Delta +1)} + \\mathcal {O}(r^{-1})\\right)\\,,$ and, while the correction to the metric is calculable, we do not require it for what follows.",
"The $ww$ component imposes exactly one condition, $\\frac{\\ell }{\\kappa ^2} \\partial _w \\left( \\lbrace t(w),w\\rbrace \\right) =(2\\Delta -1) \\left( \\Delta \\dot{\\left(a_0 b_0\\right)} -a_0\\dot{b}_0\\right)\\,.$ As for a massless $\\chi $ , we have succeeded in rewriting the dilaton equations of motion in terms of an equation (REF ) which only involves the boundary time.",
"Let us rewrite it in terms of boundary variables.",
"There are additional $\\mathcal {O}(\\chi ^2)$ boundary counterterms required to renormalize the action.",
"The leading divergence is removed by the counterterm, $S_{\\rm CT} = \\frac{\\Delta -1}{2}\\int dt \\sqrt{-\\gamma }\\,\\chi ^2 + \\mathcal {O}(\\partial ^2 \\chi ^2)\\,,$ and there are subleading counterterms with at least two boundary derivatives which remove subleading divergences.",
"Varying the on-shell action with respect to the boundary metric $h$ and source $\\lambda $ gives $\\begin{split}\\langle \\mathcal {O}_{\\Delta }\\rangle &= (1- 2\\Delta ) b_0(t)\\,,\\\\E & = - \\frac{\\ell }{\\kappa ^2}\\lbrace t(w),w\\rbrace + (2\\Delta -1)(\\Delta -1)a_0b_0\\,,\\\\& = - \\frac{\\ell }{\\kappa ^2}\\lbrace t(w),w\\rbrace + (1-\\Delta )\\lambda \\langle \\mathcal {O}_{\\Delta }\\rangle \\,.\\end{split}$ (The additional $a_0b_0$ terms in the energy come from (i.)",
"the $\\mathcal {O}(1/r)$ correction to the dilaton (REF ), inserted into the metric variation of $S_{ren}$ in (REF ), and (ii.)",
"the metric variation of the leading counterterm (REF ).)",
"In terms of $E$ , $\\langle \\mathcal {O}_{\\Delta }\\rangle $ , and $\\lambda =a_0$ , the equation of motion (REF ) is simply $\\dot{E} = \\dot{\\lambda }\\langle \\mathcal {O}_{\\Delta }\\rangle \\,.$ It is straightforward to allow for half-integer $\\Delta $ .",
"When $\\Delta \\in \\mathbb {Z}/2$ there are logarithmic terms in the near-boundary solution for bulk fields as well as logarithmic boundary counterterms.",
"It is similarly straightforward to consider self-interacting matter, e.g.",
"$\\chi ^4$ theory.",
"The final result is the same: the Einstein's equations can be rewritten as the diffeomorphism Ward identity in the dual quantum mechanics for any value of $\\Delta $ .",
"As in the main text, we have shown that the gravitational dynamics near AdS$_2$ can be rewritten as the diffeomorphism Ward identity (REF ) with a “constitutive relation” for the energy given by (REF ).",
"This equation of motion follows from the same effective action (REF ) presented in the main text.",
"We see that the hydrodynamic effective action (REF ) (equivalently, (REF )) encodes the gravitational dynamics for dilaton gravity coupled to general scalar matter."
],
[
"C. Electric AdS$_2$",
"In this main text, we considered dilaton gravities with AdS$_2$ vacua at the roots of the dilaton potential.",
"There is another way to realize AdS$_2$ solutions: the AdS$_2$ may be supported by an electric field.",
"In this Appendix we work out various aspects of these “electric” AdS$_2$ vacua of dilaton gravities with a Maxwell field, Einstein-Maxwell-Dilaton gravity.",
"Our motivation for studying this problem is somewhat indirect.",
"Hartman and Strominger [32] have claimed that the dual to gravity on electric AdS$_2$ vacua is invariant under a Virasoro algebra with nonzero central charge, a claim which was supported by way of holographic renormalization in [33] (although see also [34]).",
"In the setting of two-dimensional CFT, Roberts and Stanford [31] have shown that large central charge, a large gap for higher-spin Virasoro primary operators, and a “low” density of light states is enough to guarantee a maximal Lyapunov exponent $\\lambda _L = 2\\pi T$ .",
"Their argument would go through more or less unaltered for a CQM invariant under a Virasoro symmetry with large central charge, assuming that such a thing existed.",
"Thus we revisit electric AdS$_2$ vacua and the claim of a Virasoro algebra with $c \\ne 0$ .",
"We will find that the central charge of the Virasoro symmetry vanishes.",
"Consider the most general Einstein-Maxwell-Dilaton gravity, $S_{\\rm bulk} = \\frac{1}{2\\kappa ^2}\\int d^2x \\sqrt{-g} \\left( \\varphi R + U[\\varphi ] - \\frac{W[\\varphi ]}{4}F^2\\right)\\,.$ The authors of [33] consider the case $U=8\\varphi \\,, \\qquad W = 1\\,.$ The equations of motion are now $\\nonumber 0 & = D_{\\mu }D_{\\nu }\\varphi - g_{\\mu \\nu }\\Box \\varphi +\\frac{g_{\\mu \\nu }}{2} \\left(U - \\frac{W}{4}F^2\\right) + \\frac{W}{2}F_{\\mu \\rho }F_{\\nu }{}^{\\rho }\\,,\\\\0 & = D_{\\nu }\\left( WF^{\\mu \\nu }\\right)\\,,\\\\\\nonumber 0 & = R + U^{\\prime } - \\frac{W^{\\prime }}{4}F^2\\,.$ They admit “electric” AdS$_2$ solutions with constant dilaton and constant electric field, $\\varphi = \\varphi _0\\,, \\qquad F_{\\mu \\nu } = E \\varepsilon _{\\mu \\nu }\\,, \\qquad R = - \\frac{2}{L^2}\\,,$ provided the dilaton $\\varphi _0$ , electric field $E$ , and AdS radius $L$ satisfy the two relations $U[\\varphi _0] = \\frac{W[\\varphi _0]}{2}E^2\\,, \\quad \\frac{2}{L^2} = U^{\\prime }[\\varphi _0] + \\frac{W^{\\prime }[\\varphi _0]E^2}{2}\\,.$ So, adjusting the electric field simultaneously adjusts the dilaton and AdS radius.",
"Observe that, for smooth potentials $U$ and $W$ , there is generally a one-parameter family of AdS$_2$ solutions controlled by the strength of the electric field.",
"The most general such solution is, in a radial gauge, $\\begin{split}\\varphi & = \\varphi _0\\,,\\\\A & = - EL^2 r \\left( f_1(t) - \\frac{f_2(t)}{r^2}\\right)dt + a(t) dt\\,,\\\\g & = L^2 \\left( - r^2 \\left( f_1(t) + \\frac{f_2(t)}{r^2}\\right)^2 dt^2 + \\frac{dr^2}{r^2}\\right)\\,.\\end{split}$ Using the defining function $1/(r^2L^2)$ the conformal boundary $r\\rightarrow \\infty $ is endowed with a metric $h = \\lim _{r\\rightarrow \\infty } \\frac{\\gamma }{r^2L^2} = - f_1(t)dt^2\\,,$ where $\\gamma $ is the induced metric on a constant-$r$ slice.",
"Now let us holographically renormalize the bulk theory.",
"Consider the following definition of the renormalized theory: $\\begin{split}S_{1} &= \\lim _{\\Lambda \\rightarrow \\infty } \\frac{1}{2\\kappa ^2}\\left\\lbrace \\int d^2x \\sqrt{-g} \\Big ( \\varphi R + U - \\frac{W}{4}F^2\\Big ) \\right.\\\\& \\qquad \\quad \\quad +\\left.",
"\\int dt \\sqrt{-\\gamma } \\Big ( 2\\varphi K - LU + \\frac{W}{2L}A^2\\Big )\\right\\rbrace ,\\end{split}$ where $A^2 = \\gamma ^{\\mu \\nu }A_{\\mu }A_{\\nu }$ .",
"This matches the renormalization scheme of [33] for the particular dilaton theory they study.",
"It is easy to verify that this renormalized action is finite on-shell.",
"Under a variation of the metric and gauge field, keeping the dilaton fixed and using that $\\partial _{\\mu }\\varphi = 0$ for the solutions at hand, $S_1$ varies as $\\nonumber \\delta S_1 =& \\lim _{\\Lambda \\rightarrow \\infty } \\frac{1}{2\\kappa ^2}\\int dt \\sqrt{-\\gamma } \\left\\lbrace -\\frac{L}{2}\\left( U+\\frac{W}{2L^2}A^2\\right)\\gamma ^{\\mu \\nu } \\delta g_{\\mu \\nu } \\right.\\\\& \\qquad \\qquad \\left.",
"+ W \\left( F^{\\mu \\nu } n_{\\nu } + \\frac{\\gamma ^{\\mu \\nu }}{L}A_{\\nu }\\right) \\delta A_{\\mu }\\right\\rbrace \\,.$ On-shell, a variation of the boundary metric $h = - f_1(t)^2 dt^2$ induces a variation of both the bulk metric and gauge field (REF ) according to $\\delta f_1 = - \\frac{\\delta h_{tt}}{2f_1}$ , $\\begin{split}\\delta g_{tt} &= r^2 L^2 \\delta h_{tt} + \\mathcal {O}(r^0) \\,,\\\\\\delta A_t &= r \\frac{EL^2}{2}\\frac{\\delta h_{tt}}{f_1} + \\mathcal {O}(r^0)\\,.\\end{split}$ Inserting this variation into (REF ) and evaluating it on an AdS$_2$ solution (REF ) gives the dual stress tensor.",
"This however identically vanishes, $\\langle t^{tt}\\rangle \\equiv \\frac{2}{\\sqrt{-h}}\\frac{\\delta S_1}{\\delta h_{tt}} = 0\\,,$ so the Virasoro symmetry at play has zero central charge.",
"However, this is not the end of the story.",
"First, it is more natural to work in an alternative quantization, as one commonly does for Maxwell theory in AdS$_3$ [35], [36].",
"The growing mode of $A_t$ ought to be fixed as a boundary condition, allowing the constant term to fluctuate.",
"Second, the Cvetic and Papadimitriou [37] have argued that the renormalization scheme (REF ) is not quite correct.",
"They advocate for the prescription $\\begin{split}S_{ren} &= \\lim _{\\Lambda \\rightarrow \\infty } \\frac{1}{2\\kappa ^2}\\left\\lbrace \\int d^2x \\sqrt{-g} \\Big ( \\varphi R + U - \\frac{W}{4}F^2\\Big ) \\right.\\\\& \\qquad \\quad + \\int dt \\sqrt{-\\gamma } \\Big ( 2\\varphi K -L U \\Big )\\\\& \\qquad \\quad - \\left.\\int dt\\sqrt{-\\gamma }\\,W\\Big ( A_{\\mu }F^{\\mu \\nu }n_{\\nu }+ \\frac{L}{4}F^2\\Big )\\right\\rbrace .\\end{split}$ In alternate quantization we fix the canonical momentum conjugate to $A_t$ , which in the bulk is $\\pi ^t \\equiv \\frac{\\delta S_{bulk}}{\\delta (\\partial _r A_t)} = - \\frac{WE}{2\\kappa ^2}\\,.$ We identify this conjugate momentum as a fixed, external charge $\\mathcal {Q}$ in the dual CQM.",
"This definition (REF ) has the virtue that it conserves the symplectic structure on the boundary.",
"Varying it with respect to the boundary metric gives $\\langle t^{tt}\\rangle = 0\\,.$ The charge $\\mathcal {Q}$ is conjugate to a $U(1)$ gauge field $\\mathcal {A}$ , $\\langle \\mathcal {A}\\rangle =- \\frac{\\delta S_{ren}}{\\delta \\mathcal {Q}}\\,.$ The charge $\\mathcal {Q}$ is time-independent, and so $\\langle \\mathcal {A}\\rangle $ is determined up to a total derivative, as is appropriate for a gauge field.",
"That is, $\\langle \\mathcal {A}\\rangle $ contains no local information.",
"However, its holonomy around the Euclidean time circle is physical, encoding the chemical potential $\\mu $ .",
"To deduce the holographic formula for $\\langle \\mathcal {A}\\rangle $ we need to deduce the variation of $S_{ren}$ with respect to $W[\\varphi _0]E$ .",
"In order to remain on-shell, this variation induces a variation in the AdS radius and dilaton in accordance with (REF ), with $\\delta \\mathcal {Q} = - \\frac{1}{\\kappa ^2 E L^2}\\delta \\varphi \\,.$ The final result is that $\\langle \\mathcal {A}(t)\\rangle = a(t)\\,,$ for $a(t)$ the $\\mathcal {O}(r^0)$ term in the Maxwell field (REF ).",
"Now consider an electric AdS$_2$ black hole: $\\begin{split}\\varphi &= \\varphi _0\\,,\\\\A &= - EL^2 \\left(r - 2r_h + \\frac{r_h^2}{r}\\right)dt\\,,\\\\g & = L^2 \\left( - r^2 \\left( 1 - \\frac{r_h^2}{r^2}\\right)^2dt^2 + \\frac{dr^2}{r^2}\\right)\\,.\\end{split}$ We have chosen the $\\mathcal {O}(1)$ term of the gauge field so that $A$ is regular at the Euclideanized horizon.",
"The dual CQM is at a temperature $T = r_h/\\pi $ and charge $Q = - WE/(2\\kappa ^2)$ .",
"The chemical potential is given by (REF ) to be $\\mu = 2\\pi T E L^2$ .",
"The Bekenstein-Hawking entropy is $S =\\frac{2 \\pi \\varphi _0}{\\kappa ^2}\\,.$ It can also be computed from the on-shell, Euclidean, holographically renormalized action, which is $S_{E} = \\frac{2\\pi \\varphi _0}{\\kappa ^2} \\,.$ The Euclidean action computes the canonical ensemble free energy $G(T,Q) = - T S_{E}$ , whose variation gives the entropy and chemical potential as $\\delta G = - S \\delta T + \\mu \\delta Q\\,.$ Using (REF ) and taking thermodynamic derivatives, we indeed recover the Bekenstein-Hawking entropy and chemical potential $\\mu = 2\\pi T E L^2$ ."
]
] | 1605.06098 |
[
[
"Multiquark Hadrons - A New Facet of QCD"
],
[
"Abstract I review some selected aspects of the phenomenology of multiquark states discovered in high energy experiments.",
"They have four valence quarks (called tetraquarks) and two of them are found to have five valence quarks (called pentaquarks), extending the conventional hadron spectrum which consists of quark-antiquark $(q\\bar{q})$ mesons and $qqq$ baryons.",
"Multiquark states represent a new facet of QCD and their dynamics is both challenging and currently poorly understood.",
"I discuss various approaches put forward to accommodate them, with emphasis on the diquark model."
],
[
"Introduction",
"Ever since the discovery of the state $X(3870)$ by Belle in 2003 [1], a large number of multiquark states has been discovered in particle physics experiments (see recent reviews [2], [3], [4], [5], [6]).",
"Most of them are quarkonium-like states, in that they have a $(c\\bar{c})$ or a $(b\\bar{b})$ component in their Fock space.",
"A good fraction of them is electrically neutral but some are singly-charged.",
"Examples are $X(3872) (J^{PC}=1^{++})$ , $Y(4260) (J^{PC}=1^{--})$ , $Z(3900)^\\pm (J^P=1^+)$ , $P_c(4450)^\\pm (J^P=5/2^+)$ , in the hidden charm sector, and $Y_b(10890) (J^{PC}=1^{--})$ , $Z_b(10610)^\\pm (J^P=1^+)$ and $Z_b(10650)^\\pm (J^P=1^+)$ , in the hidden bottom sector.",
"The numbers in the parentheses are their masses in MeV.",
"Of these, $P_c(4450)^\\pm (J^P=5/2^+)$ is a pentaquark state, as its discovery mode $P_c(4450)^+ \\rightarrow J/\\psi p$ requires a minimal valence quark content $c \\bar{c} u u d$ .",
"The others are tetraquark states, with characteristic decays, such as $X(3872) \\rightarrow J/\\psi \\pi ^+ \\pi ^-$ , $Y(4260) \\rightarrow J/\\psi \\pi ^+ \\pi ^-$ , $Z(3900)^+ \\rightarrow J/\\psi \\pi ^+$ , $Y_b(10890) \\rightarrow \\Upsilon (1S,2S,3S) \\pi ^+ \\pi ^-$ , and $Z_b(10610)^+ \\rightarrow h_b(1P,2P) \\pi ^\\pm ,\\Upsilon (1S,2S) \\pi ^+$ .",
"No doubly-charged multiquark hadron has been seen so far, though some are expected, such as $[\\bar{c}\\bar{u}][sd] \\rightarrow D_s^- \\pi ^-$ , in the tetraquark scenario discussed below.",
"Deciphering the underlying dynamics of the multiquark states is a formidable challenge and several models have been proposed to accommodate them.",
"They include, among others, cusps [7], [8], which assume that the final state rescatterings are enough to describe data, and as such there is no need for poles in the scattering matrix.",
"This is the minimalist approach, in particular, invoked to explain the origin of the charged states $Z_c(3900)$ and $Z(4025)$ .",
"If proven correct, one would have to admit that all this excitement about new frontiers of QCD is “much ado about nothing”.",
"A good majority of the interested hadron physics community obviously does not share this agnostic point of view, and dynamical mechanisms have been devised to accommodate the new spectroscopy.",
"One such model put forward to accommodate the exotic hadrons is hadroquarkonium, in which a $Q\\bar{Q}$ $(Q=c, b)$ forms the hard core surrounded by light matter (light $q\\bar{q}$ states).",
"For example, the hadrocharmonium core may consist of $J/\\psi , \\psi ^\\prime , \\chi _c$ states, and the light $q\\bar{q}$ degrees of freedom can be combined to accommodate the observed hadrons [9].",
"This is motivated by analogy with the good old hydrogen atom which explained a lot of atomic physics.",
"A variation on this theme is that the hard core quarkonium could be in a color-adjoint representation, in which case the light degrees of freedom are also a color-octet to form an overall singlet.",
"Next are hybrid models, the basic idea of which dates back to circa 1994 [10] based on the QCD-inspired flux-tubes, which predict exotic $J^{\\rm PC}$ states of both the light and heavy quarks.",
"Hybrids are hadrons formed from the valence quarks and gluons, for example, consisting of $q\\bar{q}g$ .",
"In the context of the $X,Y,Z$ hadrons, hybrids have been advanced as a model for the $J^{\\rm PC}=1^{--}$ state $Y(4260)$ , which has a small $e^+e^-$ annihilation cross section[11], [12], [13], But, hybrids have been offered as templates for other exotic hadrons as well [14], [15].",
"Another popular approach assumes that the multiquark states are meson-meson and meson-baryon bound states, with an attractive residual van der Waals force generated by mesonic exchanges [16].",
"This hypothesis is in part supported by the closeness of the observed exotic hadron masses to the respective meson-meson (meson-baryon) thresholds.",
"In many cases, this leads to very small binding energy, which imparts them a very large hadronic radius.",
"This is best illustrated by $X(3872)$ , which has an S-wave coupling to $D^* \\bar{D}$ (and its conjugate) and has a binding energy ${\\cal E}_X=M_{X(3872)}-M_{D^{*0}} -M_{\\bar{D}^0}=-0.3 \\pm 0.4$ MeV.",
"Such a hadron molecule will have a large mean square separation of the constituents $\\langle r_X \\rangle \\propto 1/\\sqrt{{\\cal E}_X} \\simeq 5$ fm, where the quoted radius corresponds to a binding energy ${\\cal E}_X=0.3$ MeV.",
"This would lead to small production cross-sections in hadronic collisions [17], contrary to what has been observed in a number of experiments at the Tevatron and the LHC.",
"In some theoretical constructs, this problem is mitigated by making the hadron molecules complicated by invoking a hard (point-like) core.",
"In that sense, such models resemble hadroquarkonium models, discussed above.",
"In yet others, rescattering effects are invoked to substantially increase the cross-sections [18].",
"Theoretical interest in hadron molecules has remained unabated, and there exists a vast and growing literature on this topic with ever increasing sophistication, a sampling of which is referenced here [19], [20], [25], [21], [22], [23], [24].",
"Last, but by no means least, on this list are QCD-based interpretations in which tetraquarks and pentaquarks are genuinely new hadron species [26], [27], [28].",
"In the large $N_c$ limit of QCD, tetraquarks are shown to exist [29], [30], [31] as poles in the S-matrix, and they may have narrow widths in this approximation, and hence they are reasonable candidates for multiquark states.",
"First attempts using Lattice QCD have been undertaken [32], [33] in which correlations involving four-quark operators are studied numerically.",
"Evidence of tetraquark states in the sense of S-matrix poles using these methods is still lacking.",
"Establishing the signal of a resonance requires good control of the background.",
"In the lattice QCD simulations of multiquark states, this is currently not the case.",
"This may be traced back to the presence of a number of nearby hadronic thresholds and to lattice-specific issues, such as an unrealistic pion mass.",
"More powerful analytic and computational techniques are needed to draw firm conclusions.",
"In the absence of reliable first principle calculations, approximate phenomenological methods are the only way forward.",
"In that spirit, an effective Hamiltonian approach has been often used [26], [27], [34], [35], [36], [37], in which tetraquarks are assumed to be diquark-antidiquark objects, bound by gluonic exchanges (pentaquarks are diquark-diquark-antiquark objects).",
"This allows one to work out the spectroscopy and some aspects of tetraquark decays.",
"Heavy quark symmetry is a help in that it can be used for the heavy-light diquarks relating the charmonia-like states to the bottomonium-like counterparts.",
"I will be mainly discussing interpretations of the current data based on the phenomenological diquark picture to test how far such models go in describing the observed exotic hadrons and other properties measured in current experiments."
],
[
"The Diquark Model",
"The basic assumption of this model is that diquarks are tightly bound colored objects and they are the building blocks for forming tetraquark mesons and pentaquark baryons.",
"The diquarks, for which we use the notation $[qq]_c$ , and interchangeably ${\\mathcal {Q}}$ , have two possible SU(3)-color representations.",
"Since quarks transform as a triplet $\\tt 3$ of color SU(3), the diquarks resulting from the direct product $\\tt 3 \\otimes 3=\\bar{3} \\oplus 6$ , are thus either a color anti-triplet $\\tt \\bar{3}$ or a color sextet $\\tt 6$ .",
"The leading diagram based on one-gluon exchange is shown below.",
"Figure: One-gluon exchange diagram for diquarks.The product of the SU(3)-matrices in Fig.",
"REF can be decomposed as $\\vspace{-8.5359pt}t^a_{ij}t^a_{kl}=-\\frac{2}{3}\\underbrace{(\\delta _{ij}\\delta _{kl}-\\delta _{il}\\delta _{kj})/2}_{\\rm {antisymmetric:\\; projects}\\hspace{5.0pt}{ \\bar{\\mathbf {3}} }}+\\frac{1}{3}\\underbrace{(\\delta _{ij}\\delta _{kl}+\\delta _{il}\\delta _{kj})/2}_{\\rm {symmetric:\\; projects}\\hspace{5.0pt}{ \\mathbf {6}}}~.\\nonumber $ The coefficient of the antisymmetric $\\tt \\bar{3}$ representation is $-2/3$ , reflecting that the two diquarks bind with a strength half as strong as between a quark and an antiquark, in which case the corresponding coefficient is $-4/3$ .",
"The symmetric $\\tt 6$ on the other hand has a positive coefficient, +1/3, reflecting a repulsion.",
"This perturbative argument is in agreement with lattice QCD simulations [38].",
"Thus, in working out the phenomenology, a diquark is assumed to be an $SU(3)_c$ -antitriplet, with the antidiquark a color-triplet.",
"With this, we have two color-triplet fields, quark $q_3$ and anti-diquark $\\overline{\\mathcal {Q}}$ or $[\\bar{q}\\bar{q}]_{3}$ , and two color-antitriplet fields, antiquark $\\bar{q}_{\\bar{3}}$ and diquark ${\\mathcal {Q}}$ or $[qq]_{\\bar{3}}$ , from which the spectroscopy of the conventional and exotic hadrons is built.",
"Since quarks are spin-1/2 objects, a diquark has two possible spin-configurations, spin-0, with the two quarks in a diquark having their spin-vectors anti-parallel, and spin-1, in which case the two quark spins are aligned, as shown in Fig.",
"REF .",
"Figure: Quark and diquark spins.They were given the names “good diquarks” and “bad diquarks”, respectively, by Jaffe [39], implying that in the former case, the two quarks bind, and in the latter, the binding is not as strong.",
"There is some support of this pattern from lattice simulations for light diquarks [38].",
"However, as the spin-degree of freedom decouples in the heavy quark systems, as can be shown explicitly in heavy quark effective theory context for heavy mesons and baryons, we expect that this decoupling will also hold for heavy-light diquarks $[Q_i q_j]_{\\bar{3}}$ with $Q_i=c, b; q_j=u,d,s$ .",
"So, for the heavy-light diquarks, both the spin-1 and spin-0 configurations are present.",
"Also, what concerns the diquarks in heavy baryons (such as $\\Lambda _b$ and $\\Omega _b$ ), consisting of a heavy quark and a light diquark, both $j^p=0^+$ and $j^p=1^+$ quantum numbers of the diquark are needed to accommodate the observed baryon spectrum.",
"In this lecture, we will be mostly discussing heavy-light diquarks, and following the discussion above, we construct the interpolating diquark operators for the two spin-states of such diquarks (here $Q=c,b$ ) [27]: Table: NO_CAPTIONIn the non-relativistic (NR) limit, these states are parametrized by Pauli matrices: $ \\Gamma ^0 =\\frac{\\sigma _2}{\\sqrt{2}} (\\textnormal {Scalar}~0^+),$ and $\\;\\;\\vec{\\Gamma } =\\frac{\\sigma _2\\vec{\\sigma }}{\\sqrt{2}} (\\textnormal {Axial-Vector }1^+).$ We will characterize a tetraquark state with total angular momentum $J$ by the state vector $\\left|Y_{[bq]}\\right\\rangle =\\left|s_{\\mathcal {Q}},s_{\\bar{\\mathcal {Q}}};~J\\right\\rangle $ showing the diquark spin $s_{\\mathcal {Q}}$ and the antidiquark spin $s_{\\bar{\\mathcal {Q}}}$ .",
"Thus, the tetraquarks with the following diquark-spin and angular momentum $J$ have the Pauli forms: $\\left|0_{\\mathcal {Q}},0_{\\bar{\\mathcal {Q}}};~0_{J}\\right\\rangle &=&\\Gamma ^0 \\otimes \\Gamma ^0 , \\\\\\left|1_{\\mathcal {Q}},1_{\\bar{\\mathcal {Q}}};~0_{J}\\right\\rangle &=&\\frac{1}{\\sqrt{3}}\\Gamma ^i \\otimes \\Gamma _i \\ldots , \\\\\\left|0_{\\mathcal {Q}},1_{\\bar{\\mathcal {Q}}};~1_{J}\\right\\rangle &=&\\Gamma ^0 \\otimes \\Gamma ^i , \\\\\\left|1_{\\mathcal {Q}},0_{\\bar{\\mathcal {Q}}};~1_{J}\\right\\rangle &=&\\Gamma ^i \\otimes \\Gamma ^0 , \\\\\\left|1_{\\mathcal {Q}},1_{\\bar{\\mathcal {Q}}};~1_{J}\\right\\rangle &=&\\frac{1}{\\sqrt{2}}\\varepsilon ^{ijk}\\Gamma _j \\otimes \\Gamma _k.",
"$"
],
[
"NR Hamiltonian for Tetraquarks with hidden charm",
"For the heavy quarkonium-like exotic hadrons, we work in the non-relativistic limit and use the following effective Hamiltonian to calculate the tetraquark mass spectrum [27], [34] $H_{\\rm eff}=2m_{\\mathcal {Q}}+H_{SS}^{(qq)}+H_{SS}^{(q\\bar{q})}+H_{SL}+H_{LL},$ where $m_{\\mathcal {Q}}$ is the diquark mass, the second term above is the spin-spin interaction involving the quarks (or antiquarks) in a diquark (or anti-diquark), the third term depicts spin-spin interactions involving a quark and an antiquark in two different shells, with the fourth and fifth terms being the spin-orbit and the orbit-orbit interactions, involving the quantum numbers of the tetraquark, respectively.",
"For the $S$ -states, these last two terms are absent.",
"For illustration, we consider the case $Q=c$ and display the individual terms in $H_{\\rm eff}$ : $H_{SS}^{(qq)}= 2(\\mathcal {K}_{cq})_{\\bar{3}}[(\\mathbf {S}_{c}\\cdot \\mathbf {S}_{q})+(\\mathbf {S}_{\\bar{c}}\\cdot \\mathbf {S}_{\\bar{q}})],\\\\&\\hspace{-165.02606pt} H_{SS}^{(q\\bar{q})}=2(\\mathcal {K}_{c\\bar{q}})(\\mathbf {S}_{c}\\cdot \\mathbf {S}_{\\bar{q}}+\\mathbf {S}_{\\bar{c}}\\cdot \\mathbf {S}_{q})+2 \\mathcal {K}_{c\\bar{c}} (\\mathbf {S}_{c}\\cdot \\mathbf {S}_{\\bar{c}})+2 \\mathcal {K}_{q\\bar{q}} (\\mathbf {S}_{q}\\cdot \\mathbf {S}_{\\bar{q}}),\\\\&\\hspace{-321.51622pt} H_{SL} = 2 A_{\\mathcal {Q}} (\\mathbf {S}_{\\mathcal {Q}}\\cdot \\mathbf {L}+\\mathbf {S}_{\\mathcal {\\bar{Q}} }\\cdot \\mathbf {L}),\\\\&\\hspace{-341.43306pt} H_{LL} = B_{\\mathcal {Q}} \\frac{L_{\\mathcal {Q}\\bar{\\mathcal {Q}}}(L_{\\mathcal {Q}\\bar{\\mathcal {Q}}}+1)}{2}.$ Figure: Schematic diagram of a tetraquark in the diquark-antidiquark picture.The usual angular momentum algebra then yields the following form: Heff =2mQ + BQ2 L2 - 2a LS + 2qc [ sqsc + sqsc ] =2mQ - a J(J+1) + ( BQ2 + a ) L(L+1) + a S(S+1)- 3qc + qc[ sqc(sqc+1) + sqc(sqc+1) ]."
],
[
" Low-lying $S$ and {{formula:74c47dec-956c-4ab3-a628-d8c486fcf419}} -wave tetraquark states in the {{formula:4285cb98-73da-4d4d-9355-aa3e9321d42c}} and {{formula:e3100761-a088-4269-8a37-04f8dd3f9974}} sectors",
"The states in the diquark-antidiquark basis $| s_{qQ}, s_{\\bar{q}\\bar{Q}}; S, L \\rangle _J$ and in the $Q\\bar{Q}$ and $q\\bar{q}$ basis $| s_{q\\bar{q}}, s_{Q\\bar{Q}}; S^{\\prime }, L^{\\prime } \\rangle _J$ are related by Fierz transformation.",
"The positive parity $S$ -wave tetraquarks are given in terms of the six states listed in Table REF (charge conjugation is defined for neutral states).",
"These states are characterized by the quantum number $L=0$ , hence their masses depend on just two parameters $M_{00}$ and $\\kappa _{qQ}$ , leading to several predictions to be tested against experiments.",
"The $P$ -wave states are listed in Table REF .",
"The first four of them have $L=1$ , and the fifth has $L=3$ , and hence is expected to be significantly heavier.",
"Table: NO_CAPTIONTable: NO_CAPTIONThe parameters appearing on the r.h. columns of Tables REF and REF can be determined using the masses of some of the observed $X,Y,Z$ states, and their numerical values are given in Table REF .",
"Some parameters in the $c\\bar{c}$ and $b\\bar{b}$ sectors can also be related using the heavy quark mass scaling [40].",
"Table: NO_CAPTION Table: NO_CAPTIONTypical errors on the masses due to parametric uncertainties are estimated to be about 30 MeV.",
"As we see from table REF , there are lot more $X,Y,Z$ hadrons observed in experiments in the charmonium-like sector than in the bottomonium-like sector, with essentially three entries $Z_b^+(10610)$ , $Z_b^+(10650)$ and $Y_b(10891)$ in the latter case.",
"There are several predictions in the charmonium-like sector, which, with the values of the parameters given in the tables above, are in the right ball-park I thank Satoshi Mishima for providing these estimates..",
"It should be remarked that these input values, in particular for the quark-quark couplings in a diquark, $\\kappa _{qQ}$ , are larger than in the earlier determinations of the same by Maiani et al. [27].",
"Better agreement is reached with experiments assuming that diquarks are more tightly bound than suggested from the analysis of the baryons in the diquark-quark picture, and the spectrum shown here is in agreement with the one in the modified scheme[34].",
"Alternative calculations of the tetraquark spectrum based on diquark-antidiquark model have been carried out [41].",
"The exotic bottomonium-like states are currently rather sparse.",
"The reason for this is that quite a few exotic charmonium-like states were observed in the decays of $B$ -hadrons.",
"This mode is obviously not available for the hidden $b\\bar{b}$ states.",
"They can only be produced in hadro- and electroweak high energy processes.",
"Tetraquark states with a single $b$ quark can, in principle, also be produced in the decays of the $B_c$ mesons, as pointed out recently [42].",
"As the $c\\bar{c}$ and $b\\bar{b}$ cross-section at the LHC are very large, we anticipate that the exotic spectroscopy involving the open and hidden heavy quarks is an area where significant new results will be reported by all the LHC experiments.",
"Measurements of the production and decays of exotica, such as transverse-momentum distributions and polarization information, will go a long way in understanding the underlying dynamics.",
"As a side remark, we mention that recently there has been a lot of excitement due to the D0 observation [43] of a narrow structure $X(5568)$ , consisting of four different quark flavors ${\\tt {\\it (b d u s)}}$ , found through the $B_s^0 \\pi ^\\pm $ decay mode.",
"However, this has not been confirmed by the LHCb collaboration [44], despite the fact that LHCb has 20 times higher $B_s^0$ sample than that of D0.",
"This would have been the first discovery of an open $b$ -quark tetraquark state.",
"They are anticipated in the compact tetraquark picture [42], and also in the hadron molecule framework [45].",
"We wait for more data from the LHC experiments.",
"We now discuss the three observed exotic states in the bottomonium sector in detail.",
"The hidden $b \\bar{b}$ state $Y_b(10890)$ with $J^{\\rm P}=1^{--}$ was discovered by Belle in 2007 [46] in the process $e^+e^- \\rightarrow Y_b(10890) \\rightarrow (\\Upsilon (1S), \\Upsilon (2S), \\Upsilon (3S)) \\pi ^+ \\pi ^-$ just above the $\\Upsilon (5S)$ .",
"The branching ratios measured are about two orders of magnitude larger than anticipated from similar dipionic transitions in the lower $\\Upsilon (nS)$ states and $\\psi ^\\prime $ (for a review and references to earlier work, see Brambilla et al [47].).",
"Also the dipion invariant mass distributions in the decays of $Y_b$ are marked by the presence of the resonances $f_0(980)$ and $f_2(1270)$ .",
"This state was interpreted as a $J^{\\rm PC}=1^{--}$ P-wave tetraquark [35], [36].",
"Subsequent to this, a Van Royen-Weiskopf formalism was used [37] in which direct electromagnetic couplings with the diquark-antidiquark pair of the $Y_b$ was assumed.",
"Due to the $P$ -wave nature of the $Y_b(10890)$ , with a commensurate small overlap function, the observed small production cross-section in $e^+e^- \\rightarrow b\\bar{b}$ was explained.",
"In the tetraquark picture, $Y_b(10890)$ is the $b\\bar{b}$ analogue of the $c\\bar{c}$ state $Y_c(4260)$ , also a $P$ -wave, which is likewise found to have a very small production cross-section, but decays readily into $J/\\psi \\pi ^+\\pi ^-$ .",
"Hence, the two have very similar production and decay characteristics, and, in all likelihood, they have similar compositions.",
"The current status of $Y_b(10890)$ is unclear.",
"Subsequent to the discovery of $Y_b(10890)$ , Belle undertook high-statistics scans for the ratio $R_{b\\bar{b}}=\\sigma (e^+e^- \\rightarrow b\\bar{b})/\\sigma (e^+ e^- \\rightarrow \\mu ^+ \\mu ^-)$ , and also measured more precisely the ratios $R_{\\Upsilon (nS) \\pi ^+\\pi ^-}$ .",
"No results are available on $R_{\\Upsilon (nS) \\pi ^+\\pi ^-}$ from BaBar, so we discuss the analysis reported by Belle.",
"The two masses, $M(5S)_{b\\bar{b}}$ measured through $R_{b\\bar{b}}$ , and $M(Y_b)$ , measured through $R_{\\Upsilon (nS) \\pi ^+\\pi ^-}$ , now differ by slightly more than 2$\\sigma $ , yielding $M(5S)_{b\\bar{b}} -M(Y_b)= -9 \\pm 4$ MeV.",
"From the mass difference alone, these two could very well be just one and the same state, namely the canonical $\\Upsilon (5S)$ - an interpretation adopted by the Belle collaboration [48].",
"On the other hand, it is now the book keeping of the branching ratios measured at or near the $\\Upsilon (5S)$ , which is puzzling.",
"This is reflected in the paradox that direct production of the $B^{(*)}\\bar{B}^{(*)}$ as well as of $B_s\\bar{B}_s^{(*)}$ states have essentially no place in the Belle counting [48], as the branching ratios of the $\\Upsilon (5S)$ are already saturated by the exotic states $(\\Upsilon (nS) \\pi ^+\\pi ^-, h_b(mP) \\pi ^+\\pi ^-,Z_b(10610)^\\pm \\pi ^\\mp , Z_b(10650)^\\pm \\pi ^\\mp $ and their isospin partners).",
"In our opinion, an interpretation of the Belle data based on two resonances $\\Upsilon (5S)$ and $Y_b(10890)$ is more natural, with $\\Upsilon (5S)$ having the decays expected for the bottomonium $S$ -state above the $B^{(*)}\\bar{B}^{(*)}$ threshold, and the decays of $Y_b(10890)$ , a tetraquark, being the source of the exotic states seen.",
"As data taking starts in a couple of years in the form of a new and expanded collaboration, Belle-II, cleaning up the current analysis in the $\\Upsilon (5S)$ and $\\Upsilon (6S)$ region should be one of their top priorities.",
"In the meanwhile, the 2007 discovery of $Y_b(10890)$ stands, not having been retracted by Belle, at least as far as I know.",
"Thus, there is a good case that $\\Upsilon (5S)$ and $Y_b(10890)$ , while having the same $J^{\\rm PC}=1^{--}$ quantum numbers and almost the same mass, are different states.",
"As already mesntioned, this is hinted by the drastically different decay characteristics of the dipionic transitions involving the lower quarkonia $S$ -states, such as $\\Upsilon (4S) \\rightarrow \\Upsilon (1S) \\pi ^+\\pi ^-$ , on one hand, and similar decays of the $Y_b$ , on the other.",
"These anomalies are seen both in the decay rates and in the dipion invariant mas spectra in the $\\Upsilon (nS)\\pi ^+\\pi ^-$ modes.",
"The large branching ratios of $Y_b \\rightarrow \\Upsilon (nS) \\pi ^+\\pi ^-$ , as well as of $Y(4260) \\rightarrow J/\\psi \\pi ^+\\pi ^-$ , are due to the Zweig-allowed nature of these transitions, as the initial and final states have the same valence quarks.",
"The final state $\\Upsilon (nS) \\pi ^+\\pi ^-$ in $Y_b$ decays requires the excitation of a $q\\bar{q}$ pair from the vacuum.",
"Since, the light scalars $\\sigma _0$ , $f_0(980)$ are themselves tetraquark candidates [49], [50], they are expected to show up in the $\\pi ^+\\pi ^-$ invariant mass distributions, as opposed to the corresponding spectrum in the transition $\\Upsilon (4S) \\rightarrow \\Upsilon (1S) \\pi ^+\\pi ^-$ (see Fig.",
"REF ).",
"Subsequent discoveries [51] of the charged states $Z_b^+(10610)$ and $Z_b^+(10650)$ , found in the decays $\\Upsilon (5s)/Y_b \\rightarrow Z_b^+(10610) \\pi ^-, Z_b^+(10650) \\pi ^-$ , leading to the final states $\\Upsilon (1S) \\pi ^+\\pi ^-$ , $\\Upsilon (2S) \\pi ^+\\pi ^-$ , $\\Upsilon (3S) \\pi ^+\\pi ^-$ , $h_b(1P)\\pi ^+\\pi ^-$ and $h_b(2P) \\pi ^+\\pi ^-$ , give credence to the tetraquark interpretation, as discussed below.",
"Figure: Dipion invariant mass distribution in Υ(10890)→Υ(1S)π 0 π 0 \\Upsilon (10890) \\rightarrow \\Upsilon (1S) \\pi ^0 \\pi ^0 (upper left frame);the resonances indicatedin the dipion spectrum correspond to the f 0 (980)f_0(980) and f 2 (1270)f_2(1270);the resonances Z(10610)Z(10610) and Z(10650)Z(10650) are indicated inthe Υ(2S)π + \\Upsilon (2S) \\pi ^+ invariant mass distribution from Υ(10890)→Υ(2S)π + π - \\Upsilon (10890) \\rightarrow \\Upsilon (2S) \\pi ^+ \\pi ^- (lower left frame).The data are from the Belle collaboration .The upper right hand frame shows the dipion invariant mass distribution in Υ(4S)→Υ(1S)π + π - \\Upsilon (4S) \\rightarrow \\Upsilon (1S) \\pi ^+\\pi ^-,and the theoretical curve (with the references) is based on the Zweig-forbidden process shown below.",
"The measured decay widthsfrom Υ(nS)→Υ(1S)π + π - \\Upsilon (nS) \\rightarrow \\Upsilon (1S)\\pi ^+ \\pi ^- nS=2S,3S,4SnS=2S,3S,4S and Υ(10890)→Υ(1S)π + π - \\Upsilon (10890) \\rightarrow \\Upsilon (1S) \\pi ^+\\pi ^-are also shown.This figure serves to underscore the drastically different underlying mechanisms for dipionictransitions in Υ(nS)\\Upsilon (nS) and Υ(10890)\\Upsilon (10890) decays."
],
[
" Heavy-Quark-Spin Flip in $\\Upsilon (10890) \\rightarrow h_b(1P, 2P) \\pi \\pi $",
"We summarize the relative rates and strong phases measured by Belle [51] in the process $\\Upsilon (10890) \\rightarrow \\Upsilon (nS) \\pi ^+\\pi ^-, h_b(mP) \\pi ^+\\pi ^-$ , with $n=1,2,3$ and $m=1,2$ in Table REF .",
"For ease of writing we shall use the notation $Z_b$ and $Z_b^\\prime $ for the two charged $Z_b$ states.",
"Here no assumption is made about the nature of $\\Upsilon (10890)$ , it can be either $\\Upsilon (5S)$ or $Y_b$ .",
"Of these, the decay $\\Upsilon (10890)\\rightarrow \\Upsilon (1S) \\pi ^+\\pi ^-$ involves both a resonant (i.e., via $Z/Z^\\prime $ ) and a direct component, but the other four are dominated by the resonant contribution.",
"One notices that the relative normalizations are very similar and the phases of the $(\\Upsilon (2S), \\Upsilon (3S)) \\pi ^+\\pi ^-$ differ by about $180^\\circ $ compared to the ones in $(h_b(1P, h_b(2P))\\pi ^+\\pi ^-$ .",
"At the first sight this seems to violate the heavy-quark-spin conservation, as in the initial state $s_{b\\bar{b}}=1$ , which remains unchanged for the $\\Upsilon (nS)$ in the final state, i.e., it involves an $s_{b\\bar{b}}=1 \\rightarrow s_{b\\bar{b}}=1$ transition, but as $s_{b\\bar{b}}=0$ for the $h_b(mP)$ , this involves an $s_{b\\bar{b}}=1 \\rightarrow s_{b\\bar{b}}=0$ transition, which should have been suppressed, but is not supported by data.",
"Table: NO_CAPTIONIt has been shown that this contradiction is only apparent [40].",
"Expressing the states $Z_b$ and $Z_b^\\prime $ in the basis of definite $b\\bar{b}$ and light quark $q\\bar{q}$ spins, it becomes evident that both the $Z_b$ and $Z_b^\\prime $ have $s_{b\\bar{b}}=1$ and $s_{b\\bar{b}}=0$ components, $&&|Z_b\\rangle =\\frac{|1_{q\\bar{q}},0_{b\\bar{b}}\\rangle -|0_{q\\bar{q}},1_{b\\bar{b}}\\rangle }{\\sqrt{2}},~~|Z_b^\\prime \\rangle =\\frac{|1_{q\\bar{q}},0_{b\\bar{b}}\\rangle +|0_{q\\bar{q}},1_{b\\bar{b}}\\rangle }{\\sqrt{2}}\\nonumber .$ Defining ($g$ is the effective couplings at the vertices $\\Upsilon \\, Z_b\\, \\pi $ and $Z_b\\, h_b\\, \\pi $ ) $&&g_{Z}\\equiv g(\\Upsilon \\rightarrow Z_b\\pi )g(Z_b\\rightarrow h_b\\pi )\\propto -\\alpha \\beta \\langle h_b|Z_b\\rangle \\langle Z_b|\\Upsilon \\rangle ,\\nonumber \\\\&&g_{Z^\\prime }\\equiv g(\\Upsilon \\rightarrow Z_b^\\prime \\pi )g(Z_b^\\prime \\rightarrow h_b\\pi )\\propto \\alpha \\beta \\langle h_b|Z^\\prime _b\\rangle \\langle Z^\\prime _b|\\Upsilon \\rangle \\nonumber ,$ we note that within errors, Belle data is consistent with the heavy quark spin conservation, which requires $g_Z=-g_{Z^\\prime }$ .",
"The two-component nature of the $Z_b$ and $Z_b^\\prime $ is also the feature which was pointed out earlier for $Y_b$ in the context of the direct transition $Y_b(10890) \\rightarrow \\Upsilon (1S) \\pi ^+\\pi ^-$ .",
"To determine the coefficients $\\alpha $ and $\\beta $ , one has to resort to $s_{b\\bar{b}}$ : $1 \\rightarrow 1 $ transitions $\\Upsilon (10890)\\rightarrow Z_b/Z_b^\\prime +\\pi \\rightarrow \\Upsilon (nS)\\pi \\pi ~(n=1,2,3).\\nonumber $ The analogous effective couplings are $&&f_{Z}=f(\\Upsilon \\rightarrow Z_b\\pi )f(Z_b\\rightarrow \\Upsilon (nS)\\pi )\\propto |\\beta |^2 \\langle \\Upsilon (nS)|0_{q\\bar{q}},1_{b\\bar{b}}\\rangle \\langle 0_{q\\bar{q}},1_{b\\bar{b}}|\\Upsilon \\rangle ,\\\\&&f_{Z^\\prime }=f(\\Upsilon \\rightarrow Z_b^\\prime \\pi )f(Z_b^\\prime \\rightarrow \\Upsilon (nS)\\pi )\\propto |\\alpha |^2 \\langle \\Upsilon (nS)|0_{q\\bar{q}},1_{b\\bar{b}}\\rangle \\langle 0_{q\\bar{q}},1_{b\\bar{b}} |\\Upsilon \\rangle .$ Dalitz analysis indicates that $\\Upsilon (10890)\\rightarrow Z_b/Z_b^\\prime +\\pi \\rightarrow \\Upsilon (nS)\\pi \\pi ~(n=1,2,3)$ proceed mainly through the resonances $Z_b$ and $Z_b^\\prime $ , though $\\Upsilon (10890) \\rightarrow \\Upsilon (1S)\\pi \\pi $ has a significant direct component, expected in tetraquark interpretation of $\\Upsilon (10890)$ [37].",
"A comprehensive analysis of the Belle data including the direct and resonant components is required to test the underlying dynamics, which yet to be carried out.",
"However, parametrizing the amplitudes in terms of two Breit-Wigners, one can determine the ratio $\\alpha /\\beta $ from $\\Upsilon (10890)\\rightarrow Z_b/Z_b^\\prime +\\pi \\rightarrow \\Upsilon (nS)\\pi \\pi ~(n=1,2,3)$ .",
"For the $s_{b\\bar{b}}:1\\rightarrow 1~{\\rm transition}$ , we get for the averaged quantities: $\\overline{{\\rm Rel.",
"Norm.",
"}}= 0.85\\pm 0.08=|\\alpha |^2/|\\beta |^2;~~\\overline{{\\rm Rel.",
"Phase}}= (-8\\pm 10)^\\circ .\\nonumber $ For the $s_{b\\bar{b}}:1\\rightarrow 0~{\\rm transition}$ , we get $\\overline{{\\rm Rel.",
"Norm.",
"}}= 1.4\\pm 0.3;~~\\overline{{\\rm Rel.",
"Phase}}= (185 \\pm 42)^\\circ .\\nonumber $ Within errors, the tetraquark assignment with $\\alpha =\\beta =1$ is supported, i.e., $&&|Z_b\\rangle =\\frac{|1_{bq},0_{\\bar{b}\\bar{q}}\\rangle -|0_{bq},1_{\\bar{b}\\bar{q}}\\rangle }{\\sqrt{2}},~~|Z_b^\\prime \\rangle =|1_{b q},1_{\\bar{b}\\bar{q}}\\rangle _{J=1},\\nonumber $ and $&&|Z_b\\rangle =\\frac{|1_{q\\bar{q}},0_{b\\bar{b}}\\rangle -|0_{q\\bar{q}},1_{b\\bar{b}}\\rangle }{\\sqrt{2}},~~|Z_b^\\prime \\rangle =\\frac{|1_{q\\bar{q}},0_{b\\bar{b}}\\rangle +|0_{q\\bar{q}},1_{b\\bar{b}}\\rangle }{\\sqrt{2}}.\\nonumber $ It is interesting that similar conclusion was drawn in the `molecular' interpretation [52] of the $Z_b$ and $Z_b^\\prime $ .",
"The Fierz rearrangement used in obtaining second of the above relations would put together the $b\\bar{q}$ and $q\\bar{b}$ fields, yielding $&&|Z_b\\rangle =|1_{b \\bar{q}},1_{\\bar{b} q}\\rangle _{J=1},~~|Z_b^\\prime \\rangle =\\frac{|1_{b\\bar{q}},0_{q \\bar{b}}\\rangle +|0_{b\\bar{q}},1_{q \\bar{b}}\\rangle }{\\sqrt{2}}\\nonumber .$ Here, the labels $0_{b\\bar{q}}$ and $1_{\\bar{q}b}$ could be viewed as indicating $B$ and $B^*$ mesons, respectively, leading to the prediction $Z_b \\rightarrow B^* \\bar{B}^*$ and $Z_b^\\prime \\rightarrow B \\bar{B}^*$ , which is not in agreement with the Belle data [51].",
"However, this argument rests on the conservation of the light quark spin, for which there is no theoretical foundation.",
"Hence, this last relation is not reliable.",
"Since $Y_b(10890)$ and $\\Upsilon (5S)$ are rather close in mass, and there is an issue with the unaccounted direct production of the $B^* \\bar{B}^*$ and $B \\bar{B}^*$ states in the Belle data collected in their vicinity, we conclude that the experimental situation is still in a state of flux and look forward to its resolution with the consolidated Belle-II data."
],
[
"Drell-Yan mechanism for vector exotica production at the LHC and Tevatron",
"The exotic hadrons having $J^{\\rm PC}=1^{--}$ can be produced at the Tevatron and LHC via the Drell-Yan process [53] $pp (\\bar{p}) \\rightarrow \\gamma ^* \\rightarrow V +...$ .",
"The cases $V=\\phi (2170), Y(4260), Y_b(10890)$ have been studied.",
"With the other two hadrons already discussed earlier, we recall that $\\phi (2170)$ was first observed in the ISR process $e^+e^- \\rightarrow \\gamma _{\\rm ISR} f_0(980) \\phi (1020)$ by BaBaR [54] and later confirmed by BESII [55] and Belle [56].",
"Drenska et al.",
"[57] interpreted $\\phi (1270)$ as a P-wave tetraquark $[sq][\\bar{s} \\bar{q}]$ .",
"Thus, all three vector exotica are assumed to be the first orbital excitation of diquark-antidiquark states with a hidden $s\\bar{s}$ , $c\\bar{c}$ and $b\\bar{b}$ quark content, respectively.",
"As all three have very small branching ratios in a dilepton pair, they should be searched for in the decay modes in which they have been discovered, and these are $\\phi (2170) \\rightarrow f_0(980)\\phi (1020)\\rightarrow \\pi ^+\\pi ^- K^+ K^-$ , $Y(4260) \\rightarrow J/\\psi \\pi ^+\\pi ^- \\rightarrow \\mu ^+\\mu ^- \\pi ^+ \\pi ^-$ and $Y_b(10890) \\rightarrow \\Upsilon (nS) \\pi ^+\\pi ^- \\rightarrow \\mu ^+\\mu ^- \\pi ^+\\pi ^-$ .",
"Thus, they involve four charged particles, which can be detected at hadron colliders.",
"With their masses, total and partial decay widths taken from the PDG [58], the cross sections for the processes $p \\bar{p}(p) \\rightarrow \\phi (2170) (\\rightarrow \\phi (1020) f_0(980) \\rightarrow K^+K^- \\pi ^+\\pi ^-)$ , $p \\bar{p}(p) \\rightarrow Y(4260)(\\rightarrow J/\\psi \\pi ^+\\pi ^- \\rightarrow \\mu ^+\\mu ^- \\pi ^+\\pi ^-)$ , and $ p \\bar{p}(p) \\rightarrow Y_b(10890) (\\rightarrow \\Upsilon (1S,2S,3S)\\pi ^+\\pi ^- \\rightarrow \\mu ^+\\mu ^- \\pi ^+\\pi ^-)$ , at the Tevatron ($\\sqrt{s}=$ 1.96 TeV) and the LHC are given in Table REF , with the indicated rapidity ranges.",
"All these processes have measurable rates, and they should be searched for, in particular, at the LHC.",
"Table: NO_CAPTIONSummarizing this discussion, we note that there are several puzzles in the $X,Y,Z$ sector.",
"These involve the nature of the $J^{PC}=1^{--}$ states, $Y(4260)$ and $Y(10890)$ , and whether they are related with each other.",
"Also, whether $Y(10890)$ and $\\Upsilon (5S)$ are one and the same particle is still an open issue.",
"In principle, both $Y(4260)$ and $Y(10890)$ can be produced at the LHC and measured through the $J\\psi \\pi ^+\\pi ^-$ and $\\Upsilon (nS) \\pi ^+\\pi ^-$ $(nS=1S,2S,3S)$ modes, respectively.",
"Their hadroproduction cross-sections are unfortunately uncertain, but their (normalized) transverse momentum distributions will be quite revealing.",
"As they are both $J^{PC}=1^{--}$ hadrons, they can also be produced via the Drell-Yan mechanism and detected through their signature decay modes.",
"We have argued that the tetraquark interpretation of the charged exotics $Z_b$ and $Z_b^\\prime $ leads to a straight forward understanding of the relative rates and strong phases of the heavy quark spin non-flip and spin-flip transitions in the decays $\\Upsilon (10890) \\rightarrow \\Upsilon (nS) \\pi ^+\\pi ^-$ and $\\Upsilon (10890) \\rightarrow h_b(mP) \\pi ^+\\pi ^-$ , respectively.",
"In the tetraquark picture, the corresponding hadrons in the charm sector $Z_c$ and $Z_c^\\prime $ are related to their $b\\bar{b}$ counterparts.",
"We look forward to the higher luminosity data at Bell-II and LHC to resolve some of these issues."
],
[
"Pentaquarks",
"Pentaquarks remained cursed for almost a decade under the shadow of the botched discoveries of $\\Theta (1540),\\,\\, \\Phi (1860),\\,\\,\\Theta _c(3100)$ .",
"The sentiment of the particle physics community is reflected in the terse 2014 PDG review [59]: There are two or three recent experiments that find weak evidence for signals near the nominal masses, but there is simply no point in tabulating them in view of the overwhelming evidence that the claimed pentaquarks do not exist.",
"The only advance in particle physics thought worthy of mention in the American Institute of Physics “Physics News in 2003” was a false alarm.",
"The whole story — is a curious episode in the history of science.",
"This seems to have changed by the observation of $J/\\psi p$ resonances consistent with pentaquark states in $\\Lambda _b^0 \\rightarrow J/\\psi K^- p$ decays by the LHCb collaboration [60].",
"The discovery channel ($\\sqrt{s}=7$ and 8 TeV, $\\int Ldt= 3$ fb$^{-1}$ ) is $&& pp \\rightarrow b\\bar{b} \\rightarrow \\Lambda _b X; \\Lambda _b \\rightarrow K^- J/\\psi p. \\nonumber $ A statistically good fit of the $m_{J/\\psi p}$ -distribution is consistent with the presence of two resonant states, henceforth called $P_c(4450)^+$ and $P_c(4380)^+$ , with the following characteristics $&& M=4449.8 \\pm 1.7 \\pm 2.5~{\\rm MeV};~~\\Gamma =39 \\pm 5 \\pm 19~{\\rm MeV}, \\nonumber $ and $&& M=4380 \\pm 8 \\pm 29~{\\rm MeV};~~\\Gamma =205 \\pm 18 \\pm 86~{\\rm MeV}, \\nonumber $ having the statistical significance of 12$\\sigma $ and 9$\\sigma $ , respectively.",
"Both of them carry a unit of baryonic number and have the valence quarks $ P_c^+ = \\bar{c} c u u d$ .",
"The preferred $J^{\\rm P}$ assignments are $5/2^+$ for the $P_c(4450)^+$ and $3/2^-$ for the $P_c(4380)^+$ .",
"Doing an Argand-diagram analysis in the (${\\rm Im}~A^{P_c}$ - ${\\rm Re}~A^{P_c}$ ) plane, the phase change in the amplitude is consistent with a resonance for the $P_c(4450)^+$ , but less so for the $P_c(4380)^+$ .",
"Following a pattern seen for the tetraquark candidates, namely their proximity to respective thresholds, such as $D\\bar{D}^*$ for the $X(3872)$ , $B\\bar{B}^*$ and $B^*\\bar{B}^*$ for the $Z_b(10610)$ and $Z_b(10650)$ , respectively, also the two pentaquark candidates $P_c(4380)$ and $P_c(4450)$ lie close to several charm meson-baryon thresholds [61].",
"The $\\Sigma _c^{*+} \\bar{D}^0$ has a threshold of $4382.3 \\pm 2.4$ MeV, tantalizingly close to the mass of $P_c(4380)^+$ .",
"In the case of $P_c(4450)^+$ , there are several thresholds within striking distance, $\\chi _{c1}p (4448.93 \\pm 0.07), \\Lambda _c^{*+} \\bar{D}^0 (4457.09 \\pm 0.35),\\Sigma _c^+ \\bar{D}^{*0} (4459.9 \\pm 0.9)$ , and $\\Sigma _c^+ \\bar{D}^0 \\pi ^0 (4452.7 \\pm 0.5)$ , where the masses are in units of MeV.",
"This has led to a number of hypotheses to accommodate the two $P_c$ states, which can be classified under four different mechanisms: Rescattering-induced kinematic effects [62], [63], [64], [65].",
"$P_c(4380)$ and $P_c(4450)$ as baryocharmonia [66].",
"Open charm-baryons and charm-meson bound states [67], [68], [69], [70], [71].",
"Compact pentaquarks [72], [73], [74], [75], [76], [77], [78], [79] We discuss the first three briefly and the compact pentaquarks in somewhat more detail subsequently.",
"Kinematic effects can result in a narrow structure around the $\\chi _{c1}p$ threshold.",
"Two possible mechanisms shown in Fig.",
"REF are: (a) 2-point loop with a 3-body production $\\Lambda _b^0 \\rightarrow K^- \\,\\chi _{c1}\\, p $ followed by the rescattering process $\\chi _{c1}\\, p \\rightarrow J/\\psi \\, p$ , and (b) in which $K^-\\,p $ is produced from an intermediate $\\Lambda ^*$ and the proton rescatters with the $\\chi _{c1}$ into a $J/\\psi \\, p$ , as shown below.",
"Figure: The two scattering diagrams discussed in the text .In the baryocharmonium picture, the $P_c$ states are hadroquarkonium-type composites of $J/\\psi $ and excited nucleon states similar to the known resonances $N(1440)$ and $N(1520)$ .",
"Photoproduction of the $P_c$ states in $\\gamma + p$ collisions is advocated as sensitive probe of this mechanism [66].",
"In the hadronic molecular interpretation, one identifies $P_c(4380)^+$ with $\\Sigma _c(2455)\\bar{D}^*$ and $P_c(4450)^+$ with $\\Sigma _c(2520)\\bar{D}^*$ , which are bound by a pion exchange.",
"This can be expressed in terms of the effective Lagrangians: LP =ig Tr [Ha(Q) Aab 5 Hb(Q)], LS = -32 g1 vTr [S A S].",
"Here $H_a^{(\\bar{Q})} = [P_a^{*(\\bar{Q})\\,\\mu }\\gamma _\\mu - P_a^{(\\bar{Q})} \\gamma _5](1-{v})/2$ is a pseudoscalar and vector charmed meson multiplet $(D,D^*)$ , $v$ being the four-velocity vector $v=(0,\\vec{1})$ , ${\\cal S}_\\mu = 1/\\sqrt{3}(\\gamma _\\mu + v_\\mu ) \\gamma ^5 {\\cal B}_6 + {\\cal B}^*_{6\\mu }$ stands for the charmed baryon multiplet, with ${\\cal B}_6 $ and ${\\cal B}^*_{6\\mu } $ corresponding to the $J^P=1/2^+ $ and $J^P=3/2^+$ in $6_F$ flavor representation, respectively, and $A_\\mu $ is an axial-vector current, containing a pion chiral multiplet.",
"This interaction lagrangian is used to work out effective potentials, energy levels and wave-functions of the $ \\Sigma _c^{(*)} \\bar{D}^*$ systems.",
"In this picture, $P_c(4380)^+$ is a $\\Sigma _c \\bar{D}^*$ $(I=1/2, \\, J=3/2)$ molecule, and $P_c(4450)^+$ is a $\\Sigma ^*_c \\bar{D}^*$ $(I=1/2, \\, J=5/2)$ molecule.",
"Apart from accommodating the two observed pentaquarks, this framework predicts two additional hidden-charm molecular pentaquark states, $\\Sigma _c \\bar{D}^*$ $(I=3/2,\\, J=1/2)$ and $\\Sigma ^*_c \\bar{D}^*$ $(I=3/2,\\, J=1/2)$ , which are isospin partners of $P_c(4380)^+$ and $P_c(4450)^+$ , respectively, decaying into $\\Delta (1232) J/\\psi $ and $\\Delta (1232) \\eta _c$ .",
"In addition, a rich pentaquark spectrum of states for the hidden-bottom $ (\\Sigma _b B^*, \\Sigma ^*_b B^* )$ , $B_c$ -like $ (\\Sigma _c B^*, \\Sigma ^*_c B^*)$ and $(\\Sigma _b \\bar{D}^*, \\Sigma _b^* \\bar{D}^*)$ with well-defined $(I,J)$ are predicted."
],
[
"$P_c(4380)^+$ and {{formula:57a7c8d6-c9a0-4c95-ad13-9ffec7f4c26e}} as compact pentaquarks",
"This hypothesis has been put forward in a number of papers; to be specific we shall concentrate here on the description by Maiani et al.",
"[72], in which the two $P_c$ states (also denoted by the symbols $\\mathbb {P}^+(3/2^-)$ and $\\mathbb {P}^+(5/2^+)$ ) have the valence structure diquark-diquark-antiquark, as shown schematically in Fig.",
"REF below.",
"The assumed assignments are [72]: Pc(4380)+= P+(3/2-) = {c [cq]s=1 [qq]s=1, L=0}, Pc(4450)+= P+(5/2+)={c [cq]s=1 [qq]s=0, L=1}.",
"The observed mass difference $P_c(4450)^+ -P_c(4380)^+ \\simeq 70$ MeV is accounted for as follows: The level spacing for $\\Delta L=1$ is set using the light baryons $\\Lambda (1405)- \\Lambda (1116) \\sim 290$ MeV.",
"The light-light diquark $[q^\\prime q^{\\prime \\prime }]$ spin-dependent mass difference ($\\Delta S=1$ ) is determined from the diquark-quark interpretation of the charm baryons $[qq^{\\prime }]_{s=1} - [qq^{\\prime }]_{s=0} =\\Sigma _c(2455) - \\Lambda _c(2286) \\simeq 170$ MeV.",
"Thus, the orbital mass gap $\\mathbb {P}^+(3/2^-) - \\mathbb {P}^+(5/2^+) $ is thereby reduced to 120 MeV, in approximate agreement with the data.",
"Figure: NO_CAPTION Two possible mechanisms of pentaquark production in $\\Lambda _b^0 \\rightarrow K^- J/\\psi p$ have been proposed [72].",
"In the first, the $b$ -quark spin is shared between the $K^-$ , the $\\bar{c}$ and the $[cu]$ components, and the $[ud]$ diquark in the final state retains its spin, i.e.",
"it has spin-0, (Fig.",
"REF A below).",
"This is the decay mechanism compatible with heavy-quark-spin conservation, which implies that the spin of the light diquark in $\\Lambda _b^0$ decay is also conserved.",
"In the second, the $[ud]$ diquark is formed from the original $d$ quark, and the $u$ quark from the vacuum $u\\bar{u}$ .",
"In this case, angular momentum is shared among all components, and the diquark $[ud]$ may have both spins, $s=0, 1$ (Fig.",
"REF B below).",
"Which of the two diagrams dominate is a dynamical question; entries in the PDG on the decays of $\\Lambda _b$ hint that the mechanism in Fig.",
"B is dynamically suppressed, as also anticipated by the heavy-quark-spin conservation.",
"Figure: Two mechanisms for the decays Λ b 0 →J/ψK - p\\Lambda _b^0 \\rightarrow J/\\psi K^- p in the pentaquark picture ."
],
[
"$SU(3)_F$ structure of pentaquarks",
"Concentrating on the quark flavor of the pentaquarks $\\mathbb {P}_c^+=\\bar{c}cuud$ , they are of two different types: Pu = c [cu], s=0,1 [u d],s=0,1, Pd = c [cd], s=0,1 [u u],s=1, the difference being that the $\\mathbb {P}_d$ involves a $[uu]$ diquark, and the Pauli exclusion principle implies that this diquark has to be in an $SU(3)_F$ -symmetric representation.",
"This leads to two distinct $SU(3)_F$ series of pentaquarks PA = {c [cq], s=0, 1 [qq], s=0, L}= 3 3= 1 8, PS = {c [cq], s=0, 1 [qq], s=1, L}= 3 6= 8 10.",
"For $S$ waves, the first and the second series have the angular momenta PA(L=0) : J=1/2(2), 3/2(1), PS(L=0) : J=1/2(3), 3/2(3), 5/2(1), where the multiplicities are given in parentheses.",
"One assigns $\\mathbb {P}(3/2^-)$ to the $\\mathbb {P}_A$ and $\\mathbb {P}(5/2^+)$ to the $\\mathbb {P}_S$ series of pentaquarks [72].",
"The $SU(3)_F$ based analysis of the decays $\\Lambda _b \\rightarrow \\mathbb {P}^+ K^- \\rightarrow (J/\\psi \\, p) K^-$ goes as follows.",
"With respect to $SU(3)_F$ , $\\Lambda _b(bud)~\\sim ~\\bar{3}$ and it is an isosinglet $I=0$ .",
"Thus, the weak non-leptonic Hamiltonian for $b\\rightarrow c\\bar{c}s$ decays is: Heff(3) (I=0, S=-1).",
"The explicit form of the weak Hamiltonian is given by Heff(3) = GF2 [ Vcb Vcs* (c1 O1 + c2 O2)], where $G_F$ is the Fermi coupling constant, $V_{cs}$ is the CKM matrix element, $c_1$ and $c_2$ are the Wilson coefficients of the operators $O_1$ and $O_2$ , respectively, with the operators defined as ($i,j$ are color indices) O1= (si cj)V-A(ci bj)V-A, O2= (s c)V-A (c b)V-A, and the penguin amplitudes are ignored.",
"With $M$ a nonet of $SU(3)$ light mesons $(\\pi , K, \\eta , \\eta ^\\prime )$ , the weak transitions $ \\langle \\mathbb {P}, M | H_{\\rm W} |\\Lambda _b\\rangle $ requires $\\mathbb { P}+ M$ to be in $8 \\oplus 1$ representation.",
"Recalling the $SU(3)$ group multiplication rule 8 8 = 1 8 8 10 10 27, 8 10 = 8 10 27 35, the decay $ \\langle \\mathbb { P}, M| H_{\\rm W} |\\Lambda _b\\rangle $ can be realized with $\\mathbb {P}$ in either an octet ${\\tt 8}$ or a decuplet ${\\tt 10} $ .",
"The discovery channel $\\Lambda _b \\rightarrow \\mathbb {P}^+ K^- \\rightarrow J/\\psi p K^-$ corresponds to $\\mathbb {P}$ in an octet ${\\tt 8}$ ."
],
[
"Weak decays with $ \\mathbb {P}$ in Decuplet representation",
"Decays involving the decuplet ${\\tt 10} $ pentaquarks may also occur, if the light diquark pair having spin-0 $[ud]_{s=0}$ in $\\Lambda _b$ gets broken to produce a spin-1 light diquark $[ud]_{s=1}$ .",
"In this case, one would also observe the decays of $\\Lambda _b$ , such as b P10(S=-1) (J/(1385)), b K+ P10(S=-2) K+ (J/-(1530)).",
"These decays are, however, disfavored by the heavy-quark-spin-conservation selection rules.",
"The extent to which this rule is compatible with the existing data on $B$ -meson and $\\Lambda _b$ decays can be seen in the PDG entries.",
"Whether the decays of the pentaquarks are also subject to the same selection rules is yet to be checked, but on symmetry grounds, we do expect it to hold.",
"Hence, the observation (or not) of these decays will be quite instructive.",
"Apart from $\\Lambda _b(bud)$ , several other $b$ -baryons, such as $\\Xi _b^0(usb)$ , $\\Xi _b^-(dsb)$ and $\\Omega _b^-(ssb)$ undergo weak decays.",
"These $b$ -baryons are characterized by the spin of the light diquark, as shown below, making their isospin ($I$ ) and strangeness ($S$ ) quantum numbers explicit as well as their light diquark $j^{\\rm P}$ quantum numbers.",
"Figure: bb-baryons with the light diquark spins j p =0 + j^p=0^+ (left) and j p =1 + j^p=1^+ (right).The $c$ -baryons are likewise characterized similarly.",
"Examples of bottom-strange b-baryon in various charge combinations, respecting $\\Delta I=0,\\, \\Delta S=-1$ are: b0(5794) K (J/(1385)), which corresponds to the formation of the pentaquarks with the spin configuration $ \\mathbb {P}_{10} (\\bar{c}\\, [cq]_{s=0,1}\\, [q^\\prime s]_{s=0,1})$ with $(q,q^\\prime =u,d)$ .",
"Above considerations have been extended involving the entire $SU(3)_F$ multiplets entering the generic decay amplitude $ \\langle {\\cal P} {\\cal M}|H_{\\rm eff} | {\\cal B}\\rangle ,$ where ${\\cal B}$ is the $SU(3)_F$ antitriplet $b$ -baryon, shown in the left frame of Fig.",
"REF , ${\\cal M}$ is the $3\\times 3$ pseudoscalar meson matrix $\\qquad \\qquad \\mathcal {M}_{i}^{j}=\\left(\\begin{array}{ccc}\\frac{\\pi ^{0}}{\\sqrt{2}}+\\frac{\\eta _{8}}{\\sqrt{6}} & \\pi ^{+} & K^{+} \\\\\\pi ^{-} & -\\frac{\\pi ^{0}}{\\sqrt{2}}+\\frac{\\eta _{8}}{\\sqrt{6}} & K^{0} \\\\K^{-} & \\bar{K}^{0} & -\\frac{2\\eta _{8}}{\\sqrt{6}}\\end{array}\\right)$ , and ${\\cal P}$ is a pentaquark state belonging to an octet with definite $J^P$ , denoted as a $3\\times 3$ matrix $J^P$ , ${\\cal P}^i_j(J^{\\rm P})$ , $\\qquad \\mathcal {P}_{i}^{j}\\left( J^{P}\\right) =\\left(\\begin{array}{ccc}\\frac{P_{\\Sigma ^{0}}}{\\sqrt{2}}+\\frac{P_{\\Lambda }}{\\sqrt{6}} & P_{\\Sigma ^{+}} & P_{p} \\\\P_{\\Sigma ^{-}} & -\\frac{P_{\\Sigma ^{0}}}{\\sqrt{2}}+\\frac{P_{\\Lambda }}{\\sqrt{6}} & P_{n} \\\\P_{\\Xi ^{-}} & P_{\\Xi ^{0}} & -\\frac{P_{\\Lambda }}{\\sqrt{6}}\\end{array}\\right) ,$ or a decuplet ${\\cal P}_{ijk}$ (symmetric in the indices), with ${\\cal P}_{111}= \\Delta ^{++}_{10},...,{\\cal P}_{333}=\\Omega _{10}^-$ .",
"(see Guan-Nan Li et al.",
"[74] for a detailed list of the component fields and $SU(3)_F$ -based relations among decay widths).",
"The two observed pentaquarks are denoted as $P_p(3/2^-)$ and $P_p(5/2^+)$ .",
"Estimates of the $SU(3)$ amplitudes require a dynamical model, which will be lot more complex to develop than the factorization-based models for the two-body $B$ -meson decays, but, as argued in the literature, SU(3) symmetry can be used to relate different decay modes.",
"Examples of the weak decays in which the initial $b$ -baryon has a spin-1 light diquark, i.e.",
"$j^{\\rm P}=1^+$ , which is retained in the transition, are provided by the $\\Omega _b$ decays.",
"The $s\\bar{s}$ pair in $\\Omega _b$ is in the symmetric ${\\tt 6}$ representation of $SU(3)_F$ with spin 1 and is expected to produce decuplet pentaquarks in association with a $\\phi $ or a Kaon [72] b(6049) (J/ -(1672)), K (J/ (1387)).",
"These correspond, respectively, to the formation of the following pentaquarks ($q=u,\\, d $ ) P-10 (c [cs]s=0,1 [ss]s=1), P10 (c [cq]s=0,1 [ss]s=1).",
"These transitions are expected on firmer theoretical footings, as the initial $[ss]$ diquark in $\\Omega _b$ is left unbroken.",
"Again, lot more transitions can be found relaxing this condition, which would involve a $j^{\\rm P}=1^+ \\rightarrow 0^+$ light diquark, but they are anticipated to be suppressed.",
"In summary, with the discoveries of the $X,Y,Z$ and $P_c$ states a new era of hadron spectroscopy has dawned.",
"In addition to the well-known $q\\bar{q}$ mesons and $qqq$ baryons, we have convincing evidence that the hadronic world is multi-layered, in the form of tetraquark mesons, pentaquark baryons, and likely also the hexaquarks (or $H$ dibaryons) [80].",
"However, the underlying dynamics is far from being understood.",
"It has taken almost fifty years since the advent of QCD to develop quantitative understanding of the conventional hadronic physics.",
"A very long road lies ahead of us before we can realistically expect to achieve a comparable understanding of multiquark hadrons.",
"Existence proof of tetra- and pentaquarks on the lattice would be a breakthrough.",
"In the meanwhile, phenomenological models built within constrained theoretical frameworks are unavoidable.",
"They and experiments will guide us how to navigate through this uncharted territory.",
"The case of diquark models in this context was reviewed here.",
"More data and poweful theoretical techniques are needed to make further progress on this front."
],
[
"Acknowledgements",
"I would like to thank the organizers of the 14th.",
"regional meeting on mathematical physics in Islamabad, in particular, M. Jamil Aslam and Khalid Saifullah, for inviting me and for their warm hospitality.",
"Generous travel support offered by Dr. Shaukat Hameed Khan and COMSTECH, Islamabad, is also gratefully acknowledged."
]
] | 1605.05954 |
[
[
"Algebraic approach to electro-optic modulation of light: Exactly\n solvable multimode quantum model"
],
[
"Abstract We theoretically study electro-optic light modulation based on the quantum model where the linear electro-optic effect and the externally applied microwave field result in the interaction between optical cavity modes.",
"The model assumes that the number of interacting modes is finite and effects of the mode overlapping coefficient on the strength of the intermode interaction can be taken into account through dependence of the coupling coefficient on the mode characteristics.",
"We show that, under certain conditions, the model is exactly solvable and, in the semiclassical approximation where the microwave field is treated as a classical mode, can be analyzed using the technique of the Jordan mappings for the su(2) Lie algebra.",
"Analytical results are applied to study effects of light modulation on the frequency dependence of the photon counting rate.",
"We also establish the conditions of validity of the semiclassical approximation by applying the methods of polynomially deformed Lie algebras for analysis of the model with quantized microwave field."
],
[
"Introduction",
"The quantum information science being a new rapidly developing branch of the modern informatics that analyzes how quantum systems may be used to store, transmit and process information heavily relies on quantum optical information technologies where units of information are represented by photons [1].",
"Quantum optics is at the heart of quantum communication methods such as quantum cryptography, quantum teleportation, and dense coding [2], [3], [4], [5], [6].",
"Quantum photonics and the optical information technology provide opportunities for manipulating the properties of single photons and using them in many fields [7], [8], [9], [10], [11].",
"For instance, quantum encoding of information underlies a variety of practical schemes developed for quantum cryptography.",
"These include the schemes based on polarization [12], [13], [14] and phase [15], [16] coding, implementations of quantum cryptography that use entangled photons [17], [18], [19], [20] and quantum cryptography protocols based on continuous variables [21], [22].",
"Electro-optical modulators are key devices for proper operation of the above schemes of quantum information processing.",
"These devices have also been used in quantum information methods to monitor and control different photon quantum states [16], [23], [24], [25], [26], [27], [28].",
"In solid-state and soft condensed matter physics, there is a wealth of electro-optic effects [29], [30], [31], [32], [33], [34], [35] that underlie the mode of operation of a number of optical devices such as modulators, tunable spectral filters, polarizing converters and optical switches.",
"The linear electro-optic effect (the Pockels effect) that occurs in noncentrosymmetric nonlinear crystals such as lithium niobate (LiNbO$_3$ ) crystals will be of our primary interest.",
"Though the classical physics of this effect is well understood [29], [30], the current and emerging applications of the modulators in the field of quantum information and processing systems necessitate developing quantum approaches to frequency and phase modulation [36], [37].",
"A consistent quantum theory of phase modulation requires the quantum description of the phase.",
"The problems related to the quantum phase operator and phase measurements have an almost century long history dating back to the original paper by Dirac [38] and have been the subject of intense studies (a collection of important papers can be found, e.g., in [39]).",
"In particular, the quantum theory of phase and instantaneous frequency along with the interferometry methods of measurements are described in Refs.",
"[36], [37].",
"In these studies, quantization of spectrally limited optical fields was performed by identifying a slowly varying envelope of the creation operator and limiting its spectrum to a narrow band around the carrier frequency.",
"A quantum scattering theory based black-box approach to electro-optic modulators is developed in Refs.",
"[40], [41].",
"In this method, the modulators are regarded as the scattering devices producing a multimode output from a single-mode input.",
"An alternative approach to phase modulation elaborated in early studies [42], [30] uses the method of coupled classical modes of radiation field (the classical wave coupling theory of the electro-optic effect is also discussed in Refs [43], [44] ).",
"According to this approach, phase modulation of laser radiation results from the interaction of cavity eigenmodes caused by time periodic modulation of the dielectric constant of the nonlinear crystal placed in the resonator.",
"In Ref.",
"[45], a quantum theory of the electro-optic phase modulator is formulated in terms of the Hamiltonian describing the intermode interaction in the subspace of single photon states.",
"The common feature of the theoretical considerations presented in [40], [41], [45] is that the number of modes is assumed to be infinitely large whereas the strength of interaction (the coupling coefficient) between the modes is independent of the mode characteristics.",
"Though these assumptions greatly simplify theoretical analysis, they introduce the difficulties related to the unitarity of the scattering matrix [40], [41] and are inapplicable to the case where the modulator is based on ultra-high quality whispering gallery mode microresonators made out of electro-optically active materials [46], [47], [48], [49].",
"Such resonators are characterized by the non-equidistant spectrum of the eigenmodes, so that only a small number of modes are involved in the interaction induced by the externally applied microwave field.",
"The case of three interacting modes was theoretically studied in Refs.",
"[47], [50], [51], [52].",
"An important result of these studies is that dependence of the intensity of sidebands on the power of the microwave pump shows the saturation effect which cannot be explained by the models where the number of interacting modes is indefinitely large.",
"In this paper our goal is to examine the case bridging the gap between the above mentioned models of electro-optic modulators.",
"For this purpose, we formulate an exactly solvable model in terms of the Hamiltonian describing the parametric process where the number of interacting optical modes is finite and the strength of interaction varies depending on the mode characteristics such as the mode number related to the mode frequency.",
"Owing to algebraic properties of this model, theoretical analysis can be performed using the generalized Jordan mapping technique and the results can be further extended with the help of the mathematical methods developed in [53].",
"An important point is that, within our approach, the modulator is explicitly treated as a multiport device (multiport beam splitter) that may produce and manipulate multiphoton states.",
"Such devices are known to be promising for a variety of applications [54].",
"In particular, the modulator generated multiphoton states are used as carriers of information in the frequency-coded [15], [55], [23] and subcarrier multiplexing [56] quantum key distribution systems.",
"The paper is organized as follows.",
"In Sec.",
", we introduce our model and show that the mode number dependence of the coupling coefficient can be reasonably modeled so as to generate the theory where the algebraic structure of the multimode operators is represented by the generators of the $su(2)$ Lie algebra.",
"In the semiclassical approximation where the microwave is assumed to be a classical field, analytical expressions for the evolution operator and the quasienergy spectrum are obtained in Subsection .REF .",
"In Subsection .REF , we apply the theoretical results to study the effect of light modulation on the photon counting rate and present the results of numerical analysis.",
"The limiting case where the number of interacting modes increases indefinitely (the large $S$ limit) is studied in Subsection .REF .",
"The theory is applied to analyze different regimes of two-modulator transmission in Subsection .REF .",
"Finally, in Sec.",
", we draw the results together and make some concluding remarks.",
"Details on the Jordarn mapping technique are relegated to Appendix .",
"In Appendix , we show how the method of polynomially deformed algebras can be applied to study the quantum model with quantized microwave field and derive the applicability conditions for the semiclassical approximation."
],
[
"Model",
"As an electro-optical modulator we consider a nonlinear crystal of the length $L$ placed between the metal electrodes parallel to the direction of propagation (the $z$ axis).",
"Radio frequency wave (microwave) excited between the electrodes propagates through the crystal along the $z$ axis.",
"The microwave mode is characterized by the wavenumber $k_{\\mathrm {MW}}=2\\pi /L$ and the frequency $\\Omega _{\\mathrm {MW}}=k_{\\mathrm {MW}}v_{\\mathrm {MW}}$ , where $v_{\\mathrm {MW}}$ is the phase velocity of the mode.",
"As is illustrated in Fig.",
"REF , the crystal can be regarded as the reflectionless electro-optic cavity (resonator) and we assume that all the traveling optical modes are subject to the periodic boundary conditions.",
"Then the longitudinal wavenumber (the $z$ component of the wave vector) of the modes takes the quantized values: $k_m=\\frac{2\\pi m}{L},\\quad m\\in \\mathbb {Z}.$ The frequency of the central (carrier) optical mode which is typically excited by the laser pulse is given by $\\omega _{\\mathrm {opt}}=|k_{\\mathrm {opt}}| v_{\\mathrm {opt}}=\\Omega |m_{\\mathrm {opt}}|,\\quad \\Omega =\\frac{2\\pi }{L}v_{\\mathrm {opt}},$ where $k_{\\mathrm {opt}}=\\dfrac{2\\pi m_{\\mathrm {opt}}}{L}$ , $v_{\\mathrm {opt}}$ is the phase velocity of light in the ambient dielectric medium and $m_{\\mathrm {opt}}$ stands for the mode number of this operational optical mode.",
"Note that the magnitude of the optical mode number $m_{\\mathrm {opt}}$ is typically of the order $10^4-10^6$ and, owing to mismatch between the phase velocities $v_{\\mathrm {MW}}$ and $v_{\\mathrm {opt}}$ the frequency of the microwave mode may generally differ from $\\Omega $ , $\\Omega _{\\mathrm {MW}}=\\Omega \\, v_{\\mathrm {MW}}/v_{\\mathrm {opt}}\\ne \\Omega $ .",
"Figure: Modulated light emerging afterthe modulator driven by the microwave field (MW)passes througha Fabry-Perot filter (FP) andis collected by a photodetector (D).Anti-reflective (AR) coating is applied toboth faces of the electro-optic cavity.In classical optics, the well-known picture suggests that, owing to the linear electro-optic effect in the nonlinear crystal, the externally applied microwave field modulates the phase of the optical wave producing a multimode output observed as the multiple sidebands that a single optical carrier develops after modulation [29], [30].",
"The modulation process thus involves interaction of different optical modes mediated by the microwave field and the traveling modes appear to be coupled.",
"The strength of the microwave-field-induced intermode coupling is mainly determined by the two factors: (a) the electro-optic coefficient and (b) the overlapping coefficients represented by the averages of a product of the spatial distributions of the modes and the microwave field over the volume of interaction.",
"These factors may severely constrain the number of efficiently interacting modes.",
"For instance, in Refs.",
"[47], [23], [50], [51], [52], theoretical considerations of electro-optic modulation are based on the quantum models with three interacting optical modes.",
"Our model can be regarded as a generalization of these results.",
"The electro-optically induced interactions generally falls within the realms of the parametric processes in nonlinear quantum systems and the theoretical technique developed in early studies on this subject [42], [57], [58], [59], [60] can be invoked to model them.",
"Our starting point is the Hamiltonian of the photons written in the following form: $&H/\\hbar =\\Omega _{\\mathrm {MW}} {b}^{\\dagger } b+\\Omega A_0+\\frac{\\gamma _0}{f_{\\mathrm {max}}}\\left\\lbrace A_{+} b +A_{-} {b}^{\\dagger }\\right\\rbrace ,\\\\&A_0=\\sum _{m} m {a}^{\\dagger }_m a_m,\\quad A_{-}=\\sum _{m}f(m) {a}^{\\dagger }_m a_{m+1},\\quad A_{+}={A}^{\\dagger }_{-},$ where a dagger will denote Hermitian conjugation, ${b}^{\\dagger }$ ($b$ ) is the creation (annihilation) operator of the photons in the microwave mode, ${a}^{\\dagger }_m$ ($a_m$ ) is the creation (annihilation) operator of the optical photons numbered by the mode number $m$ , $\\gamma _0$ is the bare intermode coupling constant (interaction parameter) and $f(m)/f_{\\mathrm {max}}$ is the normalized function describing the mode number dependence of the intermode interaction strength, $\\displaystyle f_{\\mathrm {max}}=\\max _{m}f(m)$ .",
"Note that in our notations the polarization index has been dropped.",
"It implies that all the modes are assumed to be linearly polarized along a principal axis of the crystal and we shall restrict our considerations to the geometry where the state of polarization remains intact."
],
[
"Hamiltonian and $su(2)$ algebra",
"Formula (REF ) presents the Hamiltonian with the three-boson interaction written in the rotating wave approximation.",
"This approximation assumes that the intermode interaction is dominated by the quasiresonant terms that commute with the operator of the linear momentum: $\\displaystyle K/\\hbar =k_{\\mathrm {MW}}{b}^{\\dagger } b + \\sum _{m} k_m {a}^{\\dagger }_m a_m$ and are slowly varying in the representation of interaction (they are proportional to $\\exp (\\pm i[\\Omega -\\Omega _{\\mathrm {MW}}]t)$ ), whereas the non-resonant terms are of minor importance.",
"They produce negligibly small effects and hence can be disregarded.",
"Another key assumption taken in our model is that the intensity of the microwave mode is sufficiently high for its quantum properties to be ignored.",
"So, it can be described as the classical wavefield.",
"In this semiclassical approximation, the creation (annihilation) operator $b$ (${b}^{\\dagger }$ ) is replaced with the c-number amplitude $B\\exp [-i\\Omega _{\\mathrm {MW}} t]$ (${B}^{\\ast }\\exp [i\\Omega _{\\mathrm {MW}} t]\\equiv |B|\\exp [i(\\Omega _{\\mathrm {MW}} t+\\phi )]$ ), where an asterisk will indicate complex conjugation and $\\phi $ is the phase of the amplitude ${B}^{\\ast }$ , and the Hamiltonian (REF ) can be recast into the form: $&H/\\hbar =\\Omega A_0+\\frac{\\gamma }{f_{\\mathrm {max}}}\\Bigl \\lbrace A_{+}\\exp [-i(\\Omega _{\\mathrm {MW}} t+\\phi )]\\\\&+A_{-} \\exp [i(\\Omega _{\\mathrm {MW}} t+\\phi )]\\Bigr \\rbrace ,$ where $\\gamma =\\gamma _0 |B|$ .",
"Applicability of the semiclassical approach is discussed in Appendix where the method of polynomially deformed algebras developed in Ref.",
"[53] is used to analyze the model with quantized microwave mode.",
"From the above discussion, the number of interacting modes is finite and the mode number range for these modes can generally be defined by the inequality of the form: $m_{\\mathrm {min}}<m\\le m_{\\mathrm {max}},$ where $m_{\\mathrm {min}}$ and $m_{\\mathrm {max}}$ are both the positive integers.",
"The optical central (operational) mode (REF ) is in the middle of the interval (REF ) with the mode number given by $m_{\\mathrm {opt}}=\\frac{m_{\\mathrm {max}}+m_{\\mathrm {min}}+1}{2}$ and we assume that the most efficient intermode coupling occurs in the vicinity of $m_{\\mathrm {opt}}$ , whereas the strength of interaction decays to the limit of non-interacting modes at the boundaries of the interaction interval (REF ).",
"Such behavior can be modeled through the function $f(m)=\\sqrt{(m-m_{\\mathrm {min}})(m_{\\mathrm {max}}-m)}$ describing dependence of the intermode coupling on the mode number.",
"The interacting modes can be conveniently relabeled by the shifted mode number $\\mu $ as follows $&m=m_{\\mathrm {opt}}+\\mu ,\\quad -S \\le \\mu \\le S,\\quad S=\\frac{m_{\\mathrm {max}}-m_{\\mathrm {min}}-1}{2},\\\\&a_{m_{\\mathrm {opt}}+\\mu }\\equiv a_\\mu ,\\quad {a}^{\\dagger }_{m_{\\mathrm {opt}}+\\mu }\\equiv {a}^{\\dagger }_\\mu ,$ where $2S+1=m_{\\mathrm {max}}-m_{\\mathrm {min}}=2f_{\\mathrm {max}}$ is the number of interacting modes.",
"The operators that enter the Hamiltonian (REF ) of our model can now be written in the following form $&A_0=m_{\\mathrm {opt}} N + J_0,\\quad N=\\sum _{\\mu =-S}^{S}{a}^{\\dagger }_\\mu a_\\mu ,\\quad J_0=\\sum _{\\mu =-S}^{S}\\mu {a}^{\\dagger }_\\mu a_\\mu ,\\\\&A_{-}\\equiv J_{-}=\\sum _{\\mu =-S}^{S-1}\\sqrt{(S+\\mu +1)(S-\\mu )}{a}^{\\dagger }_\\mu a_{\\mu +1},\\\\&A_{+}\\equiv J_{+}={J}^{\\dagger }_{-}=\\sum _{\\mu =-S}^{S-1}\\sqrt{(S+\\mu +1)(S-\\mu )}{a}^{\\dagger }_{\\mu +1} a_{\\mu },$ where $N$ is the total photon number operator.",
"The important point is that the operators $J_{0}$ and $J_{\\pm }$ given in Eqs.",
"() and () meet the well-known commutation relations for the generators of the $su(2)$ Lie algebra: $[J_0,J_{\\pm }]=\\pm J_{\\pm },\\quad [J_{+},J_{-}]=2 J_{0}.$ Note that the operator of total photon number $N$ and the Casimir operator given by $&J^2=J_{0}^2+(J_{+} J_{-}+J_{-}J_{+})/2=J_x^2+J_y^2+J_z^2,\\\\&J_z\\equiv J_0,\\quad J_{\\pm }=J_x\\pm i J_y$ both commute with the generators of $su(2)$ algebra.",
"Mathematically, the technique of the generalized Jordan mappings for bosons [61] can be applied to derive the relations (REF ).",
"This technique is briefly reviewed in Appendix .",
"The operators (REF )–() can now be substituted into Eq.",
"(REF ) with $f_{\\mathrm {max}}=(2S+1)/2$ to yield the resulting expression for the time-dependent Hamiltonian of our semiclassical model $&H(t)/\\hbar =\\omega _{\\mathrm {opt}} N+\\Omega J_z+\\frac{2 \\gamma }{2S+1}\\Bigl \\lbrace J_{+}\\exp [-i(\\Omega _{\\mathrm {MW}} t+\\phi )]\\\\&+J_{-} \\exp [i(\\Omega _{\\mathrm {MW}} t+\\phi )]\\Bigr \\rbrace .$"
],
[
"Operator of evolution and quasienergy spectrum",
"Now we show how the algebraic structure of the model can be used to evaluate the operator of evolution and the quasienergy spectrum of the time-periodic Hamiltonian (REF ): $H(t+T_{\\mathrm {MW}})=H(t)$ , where $T_{\\mathrm {MW}}=2\\pi /\\Omega _{\\mathrm {MW}}$ is the period of the microwave mode.",
"The operator of evolution (propagator) can be found by solving the initial value problem: $i \\hbar \\frac{\\mathrm {d}}{\\mathrm {d}t} U(t)=H(t) U(t),\\quad U(0)=I,$ where $I$ is the identity operator.",
"In the Floquet representation, the propagator takes the form of a product: $U(t)=U_P(t)\\exp (-i Q t/\\hbar ),$ where $U_P(t+T_{\\mathrm {MW}})=U_P(t)$ is the time-periodic operator and $Q$ is the quasienergy operator.",
"In our case, equation (REF ) can be regarded as the rotating wave ansatz $U(t)=U_P(t) U_R(t)$ with the unitary operator $U_P(t)=\\exp [-i (m_{\\mathrm {opt}} N + J_z)\\Omega _{\\mathrm {MW}} t]$ performing the transformation to the “rotating coordinate system” and the quasienergy operator is given by $&Q/\\hbar =m_{\\mathrm {opt}}\\omega N + \\omega J_z+\\frac{2 \\gamma }{2S+1}\\Bigl \\lbrace J_{+}\\exp (-i\\phi )\\\\&+J_{-} \\exp (i\\phi )\\Bigr \\rbrace ,\\quad \\omega =\\Omega -\\Omega _{\\mathrm {MW}}.$ The rotation operator $R(\\phi ,\\beta )=\\exp (-i\\phi J_z)\\exp (-i\\beta J_y)$ can now be used to transform the quasienergy operator into the diagonal form $&Q_d/\\hbar ={R}^{\\dagger }(\\phi ,\\beta ) Q R(\\phi ,\\beta )= m_{\\mathrm {opt}} \\omega N + \\Gamma J_z,\\\\&\\omega +i \\frac{4\\gamma }{2S+1} = \\Gamma \\exp [i\\beta ],\\quad \\Gamma =\\sqrt{\\omega ^2+[4\\gamma /(2S+1)]^2},$ so that the Fock states $|{n_{-S},\\ldots ,n_{\\mu },\\ldots n_{S}}\\rangle $ characterized by the mode occupation numbers are the eigenstates of the quasienergy operator (REF ) with the quasienergies given by $E(n,m_z)/\\hbar =n\\, m_{\\mathrm {opt}} \\omega + m_z \\Gamma ,$ where $n=\\sum _{\\mu =-S}^{S}n_{\\mu }$ is the total number of photons and $m_z=\\sum _{\\mu =-S}^{S}\\mu n_{\\mu }$ is the azimuthal quantum number, $-nS \\le m_z\\le nS$ .",
"Note that a set of the Fock states characterized by the quantum numbers $n$ and $m_z$ forms a vector space $\\mathcal {F}_{n,\\,m_z}$ which can further be divided into subspaces $\\mathcal {F}_{n,\\,m_z}^{j}$ classified by the eigenvalues of the Casimir operator (REF ): $J^2|{n,j,m_z,\\kappa }\\rangle =j(j+1)|{n,j,m_z,\\zeta }\\rangle $ , where $|m_z|\\le j\\le nS$ is the angular momentum number and $\\zeta $ is the integer enumerating the basis eigenstates of $\\mathcal {F}_{n,\\,m_z}^{j}$ , $\\kappa \\in \\lbrace 1,\\ldots , \\dim \\mathcal {F}_{n,\\,m_z}^{j}\\rbrace $ .",
"In other words, the space $\\mathcal {F}_{n,\\,m_z}$ can generally be decomposed into the direct sum of subspaces $\\mathcal {F}_{n,\\,m_z}^{j}$ : $\\mathcal {F}_{n,\\,m_z}=\\underset{j}{\\oplus }{\\mathcal {F}_{n,\\,m_z}^{j}}$ .",
"For example, at $S=1$ , it can be shown that $0\\le j= n-2k\\le n$ with $k\\in \\lbrace 0,1,\\ldots \\rbrace $ and all the subspaces $\\mathcal {F}_{n,\\,m_z}^{j}$ are one-dimensional, $\\dim \\mathcal {F}_{n,\\,m_z}^{j}=1$ .",
"Given the total photon number $n$ , the dimension of the space $\\mathcal {F}_n=\\underset{m_z}{\\oplus }{\\mathcal {F}_{n,\\,m_z}}$ is known to be $\\dim \\mathcal {F}_n=(n+2S)![(2S)!",
"n!",
"]^{-1}$ and, at $S>1$ , the subspaces $\\mathcal {F}_{n,\\,m_z}^{j}$ are not necessarily one-dimensional.",
"Equations (REF )– (REF ) can now be substituted into the Floquet representation (REF ) to yield the operator of evolution in the final form: $U(t)=\\mathrm {e}^{-i\\Omega _{\\mathrm {MW}}t J_z}R(\\phi ,\\beta )\\mathrm {e}^{-i\\Gamma t J_z}{R}^{\\dagger }(\\phi ,\\beta )\\mathrm {e}^{-i\\omega _{\\mathrm {opt}}t N}.$ Using this formula in combination with the identity for the rotation of the annihilation operator (REF ) derived in Appendix , we can describe the temporal evolution of the photon annihilation operator $a_{\\mu }$ as follows $&a_{\\mu }(T)={U}^{\\dagger }(T) a_{\\mu } U(T)=\\sum _{\\nu =-S}^{S}M_{\\mu \\nu } a_{\\nu },\\\\&M_{\\mu \\nu }=\\mathrm {e}^{-i(\\omega _{\\mathrm {opt}}+\\mu \\Omega _{\\mathrm {MW}}) T}\\mathrm {e}^{-i(\\mu -\\nu )\\phi }U_{\\mu \\nu }^{S}(T),\\\\&U_{\\mu \\nu }^{S}(T)=\\sum _{\\mu ^{\\prime }=-S}^{S}d_{\\mu \\mu ^{\\prime }}^{S}(\\beta )d_{\\nu \\mu ^{\\prime }}^{S}(\\beta )\\mathrm {e}^{-i\\mu ^{\\prime }\\Gamma T}=(-1)^{\\nu }\\mathrm {e}^{-i(\\mu +\\nu )\\tilde{\\alpha }}d_{\\mu \\nu }^{S}(\\tilde{\\beta }),$ where $T$ is the duration of intermode interaction which is the time an optical wavefield takes to propagate through the electro-optic modulator and the expression for $U_{\\mu \\nu }^{S}(T)$ is simplified by using the addition formula for the Wigner $D$ functions [62], $D_{\\mu \\nu }^{S}(\\alpha ,\\beta ,\\gamma )=\\exp [-i\\mu \\alpha ]d_{\\mu \\nu }^{S}(\\beta )\\exp [-i\\nu \\gamma ]$ , with the angles $\\tilde{\\alpha }$ and $\\tilde{\\beta }$ defined through the following relations: $&2\\tilde{\\alpha }=\\pi +\\arg \\lbrace \\sin ^2\\beta +(1+\\cos ^2\\beta )\\cos (\\Gamma T)+ 2i\\cos \\beta \\sin (\\Gamma T)\\rbrace ,\\\\&\\cos \\tilde{\\beta }=\\cos ^2\\beta +\\sin ^2\\beta \\cos (\\Gamma T),\\\\&\\sin \\tilde{\\beta }=\\sin \\beta \\bigr [\\cos \\tilde{\\alpha }\\cos \\beta (1-\\cos (\\Gamma T))-\\sin \\tilde{\\alpha }\\,\\sin (\\Gamma T)\\bigr ].$ Details on derivation of these relations are relegated to Appendix ."
],
[
"Results",
"The model described in the previous section represents the electro-optic modulator as a multiport device that may generally be used to manipulate multimode photonic states.",
"Quantum dynamics of such states involves different frequency modes and lie at the heart of applications based on frequency encoding [16], [23], [28] and quantum effects dealing with the frequency entangled photons [27], [63].",
"For these applications and effects an important fisrt step is to study the effect of light modulation on the photon counting rate.",
"For this purpose, in this section, we use the analytical results of Sec.",
"to evaluate the counting rate as the one-electron photodetection probability per unit time.",
"We find that it can be written in the factorized form with the modulation form-factor expressed in terms of the matrix () and present a number of numerical results for this form-factor.",
"We also discuss what happen in the limiting case where the number of interacting modes increases indefinitely and $S\\rightarrow \\infty $ (the large $S$ limit) and apply the theory to the problem of two-modulator transmission."
],
[
"Photon counting rate",
"The cavity modes of the modulator are excited by the photons with the carrier frequency $\\omega _{\\mathrm {opt}}$ that propagate through the electro-optic device.",
"Owing to the electro-optic effect, the traveling microwave field inside the cavity gives rise to the intermode interaction.",
"For the modes initially prepared in the state described by the density matrix of the radiation field $\\rho _F(0)$ , at the instant of time $T$ , the density matrix is given by $\\rho _F(T)=U(T)\\rho _F(0){U}^{\\dagger }(T),$ where $U(t)$ is the operator of evolution (REF ) (the losses are neglected).",
"Note that, similar to Eq.",
"(REF ), $T$ is the duration of interaction (the time it takes for a light wave to travel through the region where electro-optic modulation occurs) and Eq.",
"(REF ) assumes the lossless dynamics of the density matrix $\\rho _F$ .",
"An important characteristics of the radiation field is the averaged photon number of the mode with the mode number $\\mu $ : $\\langle {N_{\\mu }}\\rangle (T)=\\mathop {\\rm Tr}\\nolimits _F\\lbrace {a}^{\\dagger }_{\\mu }a_{\\mu }\\rho _F(T)\\rbrace =\\mathop {\\rm Tr}\\nolimits _p\\lbrace {a}^{\\dagger }_{\\mu }(T)a_{\\mu }(T)\\rho _F(0)\\rbrace .$ We can now use the relation (REF ) to derive the explicit expression for the average (REF ): $\\langle {N_{\\mu }}\\rangle (T)=\\sum _{\\nu ^{\\prime },\\nu =-S}^{S}U_{\\mu \\nu ^{\\prime }}^{S\\,*}(T)U_{\\mu \\nu }^{S}(T)\\mathrm {e}^{i(\\nu -\\nu ^{\\prime })\\phi }\\langle {{a}^{\\dagger }_{\\nu ^{\\prime }}a_{\\nu }}\\rangle (0)$ where $\\langle {{a}^{\\dagger }_{\\nu ^{\\prime }}a_{\\nu }}\\rangle (0)=\\mathop {\\rm Tr}\\nolimits _F\\lbrace {a}^{\\dagger }_{\\nu ^{\\prime }}a_{\\nu }\\rho _F(0)\\rbrace $ .",
"For the special case of single mode pumping where the optical mode with the mode number $\\nu $ is the only mode initially excited in the resonator and $\\langle {{a}^{\\dagger }_{\\nu ^{\\prime }}a_{\\nu }}\\rangle (0)=\\delta _{\\nu \\nu ^{\\prime }} \\langle {N_{\\nu }}\\rangle (0)$ , we have $\\langle {N_{\\mu }}\\rangle (T)\\equiv \\langle {N_{\\mu }}\\rangle =|U_{\\mu \\nu }^{S}(T)|^2\\langle {N_\\nu }\\rangle (0).$ Typically, this is the central mode which is excited and $\\nu =0$ .",
"We consider an experimental setup depicted in Fig.",
"REF .",
"In such a setup, the output of the electro-optic modulator is connected to a Fabry-Perot filter via the optical fiber channel.",
"Then the light wave passed through the filter is collected by a photodetector with sufficiently wide bandwidth.",
"The wavefield at the exit of the modulator is characterized by the density matrix $\\rho _{F}(T)$ given in Eq.",
"(REF ).",
"An important point is that the modes of the output light field should be matched to the modes of the optical fiber.",
"In what follows we shall assume that they are perfectly correlated (see, e.g., Chapter 1.4 in the book [64]).",
"So, in the interaction picture, the electric field operator, $\\mathbf {E}(\\mathbf {r},t)$ , of light normally incident on the filter can be decomposed into its positive and negative frequency parts, $\\mathbf {E}_{+}(\\mathbf {r},t)$ and $\\mathbf {E}_{-}(\\mathbf {r},t)$ , as follows $&\\mathbf {E}(\\mathbf {r},t)=\\mathbf {E}_{+}(\\mathbf {r},t)+\\mathbf {E}_{-}(\\mathbf {r},t),\\\\&\\mathbf {E}_{+}(\\mathbf {r},t)={\\mathbf {E}}^{\\dagger }_{-}(\\mathbf {r},t)=\\sum _{\\mu =-S}^{S}\\mathbf {E}_{\\mu }^{(+)}(\\mathbf {r}) a_{\\mu }(t),$ where $a_{\\mu }(t)=a_{\\mu }(T)\\exp [-i\\omega _{\\mu } (t-T)]$ , $a_{\\mu }(T)$ is given in Eq.",
"(REF ), $\\omega _{\\mu }=\\omega _{\\mathrm {opt}}+\\mu \\Omega $ is the mode frequency and $\\mathbf {E}_{\\mu }^{(+)}(\\mathbf {r})$ is the complex valued vector amplitude that generally depends on a number of characteristics of the mode such as the wavevector, the frequency and the state of polarization.",
"Throughout this paper we have used notations where the index describing the state of polarization of the modes is suppressed by assuming that all the modes are linearly polarized along the unit vector $\\hat{\\mathbf {e}}$ , so that $\\mathbf {E}_{\\mu }^{(+)}(\\mathbf {r})={E}_{\\mu }^{(+)}(\\mathbf {r})\\hat{\\mathbf {e}}$ .",
"For the light transmitted through the filter, $\\mathbf {E}_{\\mathrm {f}}=\\mathbf {E}_{+}^{(\\mathrm {f})}+\\mathbf {E}_{-}^{(\\mathrm {f})}$ , we have $\\mathbf {E}_{+}^{(\\mathrm {f})}(\\mathbf {r},t)=\\sum _{\\mu =-S}^{S}\\tilde{\\mathbf {E}}_{\\mu }^{(+)}(\\mathbf {r})a_{\\mu }(t),\\:\\tilde{\\mathbf {E}}_{\\mu }^{(+)}(\\mathbf {r})=\\mathbf {T}_{\\mathrm {f}}(\\omega _f,\\omega _{\\mu })\\mathbf {E}_{\\mu }^{(+)}(\\mathbf {r}),$ where $\\mathbf {T}_{\\mathrm {f}}(\\omega _f,\\omega _{\\mu })$ is the transmission matrix of the filter and $\\omega _f$ is the filter frequency.",
"The filter is also characterized by the bandwidth $\\Delta \\omega _f$ , so that the filter transmission is negligibly small provided that $|\\omega _f-\\omega _{\\mu }|>\\Delta \\omega _f$ .",
"We now briefly discuss the process of photoelectric detection of the light transmitted through the filter based on the model of an idealized photodetector described in the monograph [65] (Chapter 14).",
"Our task is to compute the photon counting rate as the one-electron photodetection probability per unit time.",
"For this purpose, we take the assumption of a narrowband optically isotropic filter with $\\Delta \\omega _f<\\Omega $ and $\\mathbf {T}_{\\mathrm {f}}(\\omega _f,\\omega _{\\mu })=T_{\\mathrm {f}}(\\omega _f,\\omega _{\\mu })\\mathbf {I}_3$ , where $\\mathbf {I}_3$ is the $3\\times 3$ identity matrix.",
"So, the result can now be obtained by using the quasi-monochromatic approximation (see Chapter 14.2.2 in the book [65]).",
"For an atom located at $\\mathbf {r}=\\mathbf {r}_0$ , the energy of interaction between the atom and the radiation field in the dipole approximation is $V=-\\bigl ({\\mathbf {d}}\\cdot {\\mathbf {E}_{\\mathrm {f}}(\\mathbf {r}_0)}\\bigr ),$ where $\\mathbf {d}$ is the operator of the electric dipole moment.",
"We can now closely follow the line of reasoning presented in Ref.",
"[65] and obtain the one-electron detection probability rate $p(\\omega _f)=\\sum _{\\mu =-S}^{S}|T_{\\mathrm {f}}(\\omega _f,\\omega _{\\mu })|^2\\langle {N_{\\mu }}\\rangle K(\\omega _{\\mu })$ expressed in terms of the frequency response function of the photodetector $K(\\omega _{\\mu })=&H(\\omega _{\\mu }-\\omega _g)\\int \\sigma (\\omega _{\\mu }-\\omega _g,\\kappa ) g(\\omega _{\\mu }-\\omega _g,\\kappa )\\\\&\\times |\\langle {\\omega _{\\mu }-\\omega _g,\\kappa }|\\bigl ({\\mathbf {d}}\\cdot {\\hat{\\mathbf {e}}}\\bigr )|{G}\\rangle {E}_{\\mu }^{(+)}(\\mathbf {r}_0)|^2\\mathrm {d}\\kappa ,$ where $H(x)$ is the Heaviside unit step function, $|{G}\\rangle $ is the ground (bounded) state of the atom with the negative energy equal to $-\\hbar \\omega _g$ (it is the eigenstate of the Hamiltonian of the atomic system $H_A$ : $H_A|{G}\\rangle =-\\hbar \\omega _g|{G}\\rangle $ ), $|{\\omega _e,\\kappa }\\rangle $ is the excited free electron (unbound) state characterized by the positive energy $\\hbar \\omega _e$ ($H_A|{\\omega _e,\\kappa }\\rangle =\\hbar \\omega _e|{\\omega _e,\\kappa }\\rangle $ ) and possibly by other variables represented by $\\kappa $ ; $\\sigma (\\omega _e,\\kappa )$ is the density of the excited states and $g(\\omega _e,\\kappa )$ is the probability for the electron in the state $|{\\omega _e,\\kappa }\\rangle $ to be collected and registered by the detector.",
"For the broadband detector with $K(\\omega _{\\mu })\\approx K(\\omega _{\\mathrm {opt}})$ , the expression for the counting rate (REF ) can be further simplified giving the result in the factorized form: $&p(\\omega _f)\\approx p_0(\\omega _{\\mathrm {opt}}) p_{\\mathrm {mod}}(\\omega _f,T),\\\\&p_0(\\omega _{\\mathrm {opt}})=K(\\omega _{\\mathrm {opt}}) \\langle {N_\\nu }\\rangle (0),\\\\&p_{\\mathrm {mod}}(\\omega _f,T)=\\sum _{\\mu =-S}^{S}|T_{\\mathrm {f}}(\\omega _f,\\omega _{\\mu })U_{\\mu \\nu }^{S}(T)|^2,\\\\&|U_{\\mu \\nu }^{S}(T)|^2=|d_{\\mu \\nu }^{S}(\\tilde{\\beta })|^2,$ where we have used formulas (REF ) and () for the averaged photon number $\\langle {N_{\\mu }}\\rangle $ and $U_{\\mu \\nu }^{S}(T)$ , respectively.",
"From Eqs.",
"(REF ) and (), it is clear that the photon count form-factor $p_{\\mathrm {mod}}(\\omega _f,T)$ accounts for the combined effect of the modulator and the filter, whereas the factor $p_0(\\omega _{\\mathrm {opt}})$ gives the counting rate without filtering and modulation.",
"The form-factor $p_{\\mathrm {mod}}(\\omega _f,T)$ thus might be called the light modulation form-factor of the photon count rate.",
"Figure: (Color online)The photon counting rate form-factor p mod p_{\\mathrm {mod}} as a function ofthe filter frequency detuningcomputed from Eq.",
"()at the intermode coupling parameterγ/Ω=0.1\\gamma /\\Omega =0.1(the regime of weak intermode coupling).The other parameters are:T max (f) =1T_{\\mathrm {max}}^{(f)}=1is the maximal transmittance of the filter;σ f /Ω=0.15\\sigma _f/\\Omega =0.15is the bandwidth of the filter;T=2π/ΩT= 2\\pi /\\Omega is the time of intermode interactionandω/Ω=(Ω-Ω MW )/Ω=0.01\\omega /\\Omega =(\\Omega -\\Omega _{\\mathrm {MW}})/\\Omega =0.01.Figure: (Color online)The photon counting rate form-factor p mod p_{\\mathrm {mod}} as a function ofthe filter frequency detuningat the intermode coupling parameterγ/Ω=0.4\\gamma /\\Omega =0.4(the regime of intermediate intermode coupling).Other parameters are listed in the captionof Fig.",
".Figure: (Color online)The photon counting rate form-factor p mod p_{\\mathrm {mod}} as a function ofthe filter frequency detuningat the intermode coupling parameterγ/Ω=0.9\\gamma /\\Omega =0.9(the regime of strong intermode coupling).Other parameters are listed in the captionof Fig.",
".Figure: (Color online)The photon counting rate form-factor p mod p_{\\mathrm {mod}} as a function ofthe coupling parameter γ/Ω\\gamma /\\Omega at ω f =ω opt \\omega _f=\\omega _{\\mathrm {opt}}.Other parameters are listed in the captionof Fig.",
".Figure: (Color online)The photon counting rate form-factor p mod p_{\\mathrm {mod}} as a function ofthe coupling parameter γ/Ω\\gamma /\\Omega at ω f =ω 2 =ω opt +2Ω\\omega _f=\\omega _2=\\omega _{\\mathrm {opt}}+2\\Omega .Other parameters are listed in the captionof Fig.",
".In our calculations, the frequency dependence of the filter transmittance is modeled by the Gaussian shaped curve $|T_{\\mathrm {f}}(\\omega _f,\\omega )|^2=T_{\\mathrm {max}}^{(f)}\\exp [-(\\omega _f-\\omega )^2/\\sigma _f^2],$ where $\\sqrt{\\ln 2}\\, \\sigma _f$ is the Gaussian half width at half maximum that determines the bandwidth of the filter $\\Delta \\omega _f=\\sigma _f$ and $T_{\\mathrm {max}}^{(f)}$ is the maximal transmittance of the filter at the peak $\\omega _f=\\omega $ .",
"Figures REF –REF show the photon count light modulation form-factor $p_{\\mathrm {mod}}(\\omega _f,T)$ computed from Eq.",
"() either as a function of the dimensionless filter frequency detuning $(\\omega _{f}-\\omega _{\\mathrm {opt}})/\\Omega $ (Figs.",
"REF –REF ) or in relation to the intermode coupling parameter $\\gamma /\\Omega $ (Figs.",
"REF and REF ).",
"In these figures, the mode initially excited in the electro-optic cavity is central with $\\nu =0$ and the parameters are: $T_{\\mathrm {max}}^{(f)}=1$ , $\\sigma _f/\\Omega =0.15$ , $\\omega /\\Omega =0.01$ and $T= 2\\pi /\\Omega $ ."
],
[
"\nRegime of large number of interacting modes:\nthe large $S$ limit",
"In our model, the operator of evolution (REF ) describing the effect of electro-optically induced light modulation is represented by the matrix $U_{\\mu \\nu }^{S}(T)$ given by Eq. ().",
"The latter is the $(2S+1)\\times (2S+1)$ matrix, where $2S+1$ is the number of interacting modes.",
"In this section, we discuss the limiting case where the number of interacting modes is large and $S\\rightarrow \\infty $ .",
"From Eq.",
"(), the elements of the matrix $U_{\\mu \\nu }^{S}(T)$ are determined by the Wigner $D$ functions $D_{\\mu \\nu }^{S}(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\alpha }+\\pi )$ with relations for the angles $\\tilde{\\alpha }$ and $\\tilde{\\beta }$ given by Eq.",
"(REF ).",
"Equation () can be used in combination with the relations (REF ) to derive the large $S$ asymptotics for the angles $&\\sin \\beta \\sim \\frac{2\\gamma }{|\\omega |} S^{-1},\\\\&\\tilde{\\alpha }\\xrightarrow[S\\rightarrow \\infty ]{} \\frac{\\pi +\\omega T}{2},\\quad \\omega =\\Omega -\\Omega _{\\mathrm {MW}},\\\\&\\tilde{\\beta }\\sim -\\frac{g}{S},\\quad g=\\frac{4\\gamma }{|\\omega |}\\sin (|\\omega | T/2).$ Our next step starts with the well-known expression for the Wigner $d$ -functions [61] $&d_{\\mu \\nu }^{S}(\\tilde{\\beta })=\\sqrt{\\frac{(S+\\nu )!",
"(S-\\nu )!",
"}{(S+\\mu )!",
"(S-\\mu )!",
"}}\\sin ^{\\nu -\\mu }\\left(\\frac{\\tilde{\\beta }}{2}\\right)\\\\&\\times \\cos ^{\\nu +\\mu }\\left(\\frac{\\tilde{\\beta }}{2}\\right)P_{S-\\nu }^{(\\nu -\\mu ,\\nu +\\mu )}(\\cos \\tilde{\\beta }),$ where $P_n^{(\\alpha ,\\beta )}(x)$ denotes the Jacobi polynomial [66], and uses asymptotics of the Jacobi polynomials given by the Mehler-Heine formula [67]: $&\\lim _{n\\rightarrow \\infty } n^{-\\alpha } P_n^{(\\alpha ,\\beta )}(\\cos (z/n))=\\lim _{n\\rightarrow \\infty } n^{-\\alpha } P_n^{(\\alpha ,\\beta )}(1-z^2n^{-2}/2)\\\\&=\\left[\\frac{z}{2}\\right]^{-\\alpha } J_\\alpha (z),$ where $J_\\alpha (z)$ is the Bessel function of the first kind [66] (outside this subsection symbols $J_\\mu $ without arguments denote the generators of $su(2)$ ).",
"From Eqs.",
"(REF )– (REF ), it is rather straightforward to find that the asymptotic behavior of the Wigner $d$ functions and the matrix $U_{\\mu \\nu }^{S}$ is given by $&d_{\\mu \\nu }^{S}(\\tilde{\\beta })\\xrightarrow[S\\rightarrow \\infty ]{}J_{\\mu -\\nu }(g),\\\\&U_{\\mu \\nu }^{S}(T)\\xrightarrow[S\\rightarrow \\infty ]{}(-i)^{\\mu -\\nu }\\mathrm {e}^{-i(\\mu -\\nu )\\omega T/2} J_{\\mu -\\nu }(g) \\mathrm {e}^{-i\\nu \\omega T}.$ Now we apply the asymptotic relation () to describe, in the large $S$ limit, the effect of electro-optic modulation on temporal evolution of light after passing through the modulator at $t>T$ .",
"From Eq.",
"(REF ), the averaged positive frequency part of the electric field can be written as follows $\\langle {\\mathbf {E}_{+}(\\mathbf {r},t)}\\rangle =\\sum _{\\mu =-S}^{S}\\mathbf {E}_{\\mu }^{(+)}(\\mathbf {r}) \\mathrm {e}^{-i\\omega _{\\mu }(t-T)}\\langle {a_{\\mu }(T)}\\rangle _0,$ where $\\omega _{\\mu }=\\omega _{\\mathrm {opt}}+\\mu \\Omega $ and $\\langle {a_{\\mu }(T)}\\rangle _0\\equiv \\mathop {\\rm Tr}\\nolimits _F\\lbrace a_{\\mu }(T)\\rho _F(0)\\rbrace $ .",
"$&\\langle {\\mathbf {E}_{+}(\\mathbf {r},t)}\\rangle \\xrightarrow[S\\rightarrow \\infty ]{}\\sum _{\\mu ,\\nu }\\mathbf {E}_{\\mu }^{(+)}(\\mathbf {r})\\mathrm {e}^{-i(\\mu -\\nu )(\\Omega t-\\psi )}J_{\\mu -\\nu }(g)\\mathrm {e}^{-i\\omega _{\\nu } t}\\langle {a_{\\nu }}\\rangle _0\\\\&\\approx \\sum _{\\mu ,\\nu }\\mathrm {e}^{-i(\\mu -\\nu )(\\Omega t-\\psi )}J_{\\mu -\\nu }(g)\\mathbf {E}_{\\nu }^{(+)}(\\mathbf {r})\\mathrm {e}^{-i\\omega _{\\nu } t}\\langle {a_{\\nu }}\\rangle _0\\\\&=\\mathrm {e}^{-i g\\cos (\\Omega t - \\psi )}\\langle {\\mathbf {E}_{+}(\\mathbf {r},t)}\\rangle _0,$ where $\\psi =\\omega T/2-\\phi $ and $\\langle {\\mathbf {E}_{+}(\\mathbf {r},t)}\\rangle _0=\\sum _{\\nu }\\mathbf {E}_{\\nu }^{(+)}(\\mathbf {r}) \\mathrm {e}^{-i\\omega _{\\nu }t}\\langle {a_{\\nu }}\\rangle _0$ is the average of the radiation field propagating in the free space without light modulation.",
"The result (REF ) is obtained with the help of the equality [66] $\\exp [-ig\\cos (\\Omega t)]=\\sum _{\\mu =-\\infty }^{\\infty }(-i)^{\\mu }\\mathrm {e}^{-i\\mu \\Omega t} J_{\\mu }(g)$ by assuming that the modes are linearly polarized $\\mathbf {E}_{\\mu }^{(+)} =E_{\\mu }^{(+)} \\hat{\\mathbf {e}}$ and the approximation $E_{\\mu }^{(+)}\\approx E_{\\nu }^{(+)}$ may break only in the region where $|\\mu -\\nu |$ is sufficiently large for $|J_{\\mu -\\nu }(g)|$ to be negligibly small.",
"The phase factor $\\exp [-i g\\cos (\\Omega t - \\psi )]$ on the right hand side of Eq.",
"(REF ) implies that the wave after the modulator becomes phase modulated and $g$ plays the role of the phase modulation index (the modulation depth).",
"This is the well known result of the simple classical model [30] which in our model is recovered in the large $S$ limit.",
"In Figs.",
"REF –REF , the filter frequency dependence of the photon count modulation form-factor computed in the large $S$ limit is compared with $p_{\\mathrm {mod}}$ evaluated at $S=3$ (the number of interacting modes equals 7).",
"As is illustrated in Fig.",
"REF , in the case of weak intermode interaction where the coupling constant is small, the differences between the curves are negligible.",
"Referring to Figs.",
"REF and REF , the latter is no longer the case in the regimes of intermediate and strong coupling.",
"The effect of electro-optically induced intermode interaction can be clearly seen in Figs.",
"REF and REF where the form-factor of the mode with the frequency $\\omega _{\\mu }$ selected by the filter at $\\omega _f=\\omega _{\\mu }$ is plotted as a function of the coupling constant $\\gamma $ .",
"For the central mode with $\\mu =0$ , the results are presented in Fig.",
"REF .",
"Clearly, in the large $S$ limit, the coupling constant dependence of $p_{\\mathrm {mod}}$ shown in Fig.",
"REF demonstrates that the initially pumped mode becomes depleted as the strength of interaction increases, so that the photons spread over the (infinitely) large number of modes.",
"By contrast, the model with $S=3$ predicts qualitatively different behavior of the form-factor characterized by oscillations with $p_{\\mathrm {mod}}$ being close to a periodic function of $\\gamma $ .",
"Mathematically, the oscillating behavior of $p_{\\mathrm {mod}}$ is determined by the elements of the matrix () where $|U_{\\mu \\nu }^{S}|=|d_{\\mu \\nu }^{S}(\\tilde{\\beta })|$ is a $2\\pi $ periodic even function of $\\tilde{\\beta }$ with $|d_{\\mu \\nu }^{S}(0)|=\\delta _{\\mu \\nu }$ and $|d_{\\mu \\nu }^{S}(\\pi )|=\\delta _{\\mu ,-\\nu }$ .",
"In addition, from Eq.",
"(REF ), it can be shown that, given the angle $\\beta $ , the angle $\\tilde{\\beta }$ can be regarded as a function of $\\Gamma T$ and $|\\tilde{\\beta }(\\beta ,\\Gamma T)|=|\\tilde{\\beta }(\\beta ,2\\pi \\pm \\Gamma T)|$ .",
"This implies that $|U_{\\mu \\nu }^{S}|=|d_{\\mu \\nu }^{S}(\\tilde{\\beta })|$ is a periodic function of $\\Gamma T$ .",
"From Eq.",
"(), it can be inferred that, in general, the parameters $\\beta $ and $\\Gamma $ both depend on the coupling constant $\\gamma $ .",
"At $\\gamma \\gg |\\omega |$ , $\\beta \\approx \\pi /2$ and $\\Gamma $ is linearly proportional to $\\gamma $ .",
"So, at sufficiently strong coupling $|U_{\\mu \\nu }^{S}|$ (and thus $p_{\\mathrm {mod}}$ ) will be a periodic function of $\\gamma $ .",
"In particular, when the phase velocities of microwave and optical fields are matched and $\\omega =\\Omega -\\Omega _{\\mathrm {MW}}=0$ , we deal with the resonance case where $\\beta =\\pi /2$ , $\\tilde{\\alpha }=-\\pi /2$ and $\\tilde{\\beta }=\\Gamma T=4\\gamma T/(2S+1)$ (see Eq.",
"(REF ) in Appendix ).",
"In the large $S$ limit, it is not difficult to show that $U_{\\mu \\nu }^{S}\\rightarrow (-i)^{\\mu -\\nu } J_{\\mu -\\nu }(2\\gamma T)$ and we obtain the result in the form of Eq.",
"(REF ) with $\\psi =-\\phi $ and $g=2\\gamma T$ .",
"For finite number of modes, the $\\gamma $ dependence of the photon counting rate is dictated by the coupling dependent factor $|d_{\\mu \\nu }^{S}|^2$ .",
"This factor is an even $2\\pi $ periodic function of the coupling parameter $4\\gamma T/(2S+1)$ .",
"In contrast, oscillations of the factor $|J_{\\mu -\\nu }(2\\gamma T)|^2$ rapidly decay in magnitude as $\\gamma $ increases.",
"Figure REF illustrates that similar effects occur when the detuning $\\omega $ is small and $\\mu =2$ .",
"Figures REF –REF present the results obtained by assuming that the mode excited in the cavity is central with $\\nu =0$ .",
"In this case the model and its large $S$ limit both predict that $|U_{\\mu , 0}^S|^2=|U_{-\\mu , 0}^S|^2$ and contributions to the photon counting rate coming from symmetrically arranged sideband modes, $\\mu $ and $-\\mu $ , are equal.",
"This symmetry is evident from the curves shown in Figs.",
"REF –REF .",
"Figure: (Color online)The photon counting rate form-factor p mod p_{\\mathrm {mod}} as a function ofthe coupling parameter γ/Ω\\gamma /\\Omega for 〈N ν 〉(0)=δ ν,1 〈N 1 〉(0)\\langle {N_\\nu }\\rangle (0)=\\delta _{\\nu ,1}\\langle {N_1}\\rangle (0)at (a) ω f =ω 0 =ω opt \\omega _f=\\omega _0=\\omega _{\\mathrm {opt}}and (b) ω f =ω 2 =ω opt +2Ω\\omega _f=\\omega _2=\\omega _{\\mathrm {opt}}+2\\Omega .When the pumped mode is not central and $\\nu \\ne 0$ , the symmetry between the blue-detuned and red-detuned modes with frequencies $\\omega _{\\nu }+ k\\Omega $ and $\\omega _{\\nu }- k\\Omega $ appears to be broken provided the number of interacting modes is finite.",
"Mathematically, the reason is the difference between the magnitudes of the matrix elements $|U_{\\nu +k, \\nu }^S|$ and $|U_{\\nu -k, \\nu }^S|$ .",
"By contrast, the symmetry retains in the large $S$ limit where $|U_{\\nu \\pm k, \\nu }^S|\\rightarrow |J_{\\pm k}(g)|=|J_{|k|}(g)k|$ .",
"The results computed at $\\nu =k=1$ are shown in Fig.",
"REF .",
"They clearly demonstrate pronounced asymmetry between the modes with $\\mu =\\nu +1=2$ and $\\mu =\\nu -1=0$ that occurs at $S=3$ , whereas the curves evaluated in the large $S$ limit are clearly identical."
],
[
"Two-modulator transmission",
"In conclusion of this section we briefly discuss how our results can be extended to the important case where the input state after being transformed by a modulator of a sender (Alice) is transmitted through the optical fiber to a receiver (Bob) that sends the incoming state through of a second modulator.",
"This is a simplified scheme representing the key elements used in frequency-coded setups [15], [23].",
"We characterize the evolution operator of the system in terms of the matrix $\\mathbf {M}$ [see Eq.",
"()] that enter the right hand side of Eq.",
"(REF ).",
"In our case, this matrix can be written as the product of three matrices $&\\mathbf {M}=\\mathbf {M}_{2}\\mathbf {M}_0\\mathbf {M}_{1},\\\\&M^{(0)}_{\\mu \\nu }=\\delta _{\\mu \\nu }\\mathrm {e}^{-i\\Phi _\\mu },\\quad M^{(i)}_{\\mu \\nu }=\\mathrm {e}^{-i\\Phi _{\\mu \\nu }^{(i)}} d_{\\mu \\nu }^{S}(\\beta _i),$ where the phase shift $\\Phi _{\\mu }=\\Phi _0+\\mu \\phi _0$ represents the effect of propagation in the optical fiber and the elements of the Alice's(Bob's) modulator matrix, $\\mathbf {M}_1$ ($\\mathbf {M}_2$ ), are expressed in terms of the phase given by $&\\Phi _{\\mu \\nu }^{(i)}=\\Phi _{00}^{(i)}+\\mu (\\Omega _{\\mathrm {MW}}T_i+\\alpha _i+\\phi _i)+\\nu (\\pi +\\alpha _i-\\phi _i),$ where $\\Phi _{00}^{(i)}=\\omega _{\\mathrm {opt}}T_i$ .",
"We assume that the only difference between otherwise identical modulators is the phase of the microwave field, $\\phi _1=\\phi _A$ and $\\phi _2=\\phi _B$ , that plays the role of the tuning parameter.",
"Other parameters of the modulators are: $T_1=T_2=T$ , $\\alpha _1=\\alpha _2=\\alpha _m$ and $\\beta _1=\\beta _2=\\beta _m$ .",
"Similar to Eq.",
"(), we can use the relation $&\\sum _{\\mu ^{\\prime }=-S}^{S}d_{\\mu \\mu ^{\\prime }}^{S}(\\beta _m)d_{\\nu \\mu ^{\\prime }}^{S}(\\beta _m)\\mathrm {e}^{-i\\mu ^{\\prime }\\phi _{AB}}=(-1)^{\\nu }\\mathrm {e}^{-i(\\mu +\\nu )\\tilde{\\alpha }}d_{\\mu \\nu }^{S}(\\tilde{\\beta }),\\\\&\\phi _{AB}=\\phi _{A}-\\phi _B+\\Delta ,\\quad \\Delta =\\phi _0+\\Omega _{\\mathrm {MW}}T+2\\alpha _m$ to derive the expression for the elements of the matrix (REF ) in the final form: $&M_{\\mu \\nu }=\\mathrm {e}^{-i\\Psi _{\\mu \\nu }} d_{\\mu \\nu }^{S}(\\tilde{\\beta }),\\\\&\\Psi _{\\mu \\nu }=\\psi _0+\\mu (\\Omega _{\\mathrm {MW}}T+\\alpha _m+\\tilde{\\alpha }+\\phi _B)+\\nu (\\pi +\\alpha _m+\\tilde{\\alpha }-\\phi _A),$ where $\\psi _0=\\Phi _0+2 \\omega _{\\mathrm {opt}}T$ .",
"The angles $\\tilde{\\alpha }$ and $\\tilde{\\beta }$ are determined by Eq.",
"(REF ) with the set of parameters $\\lbrace \\Gamma T, \\beta \\rbrace $ replaced by $\\lbrace \\phi _{AB}, \\beta _m\\rbrace $ .",
"In particular, from the suitably modified relation () it follows that $\\cos \\tilde{\\beta }=1$ provided that $\\cos \\phi _{AB}=1$ .",
"At these values of the tuning parameter $\\phi _{AB}$ , the modulators compensate each other and $M_{\\mu \\nu }=\\delta _{\\mu \\nu }$ .",
"It implies that, in this regime, for the input light field without sidebands, no sidebands will be detected by Bob's photodetector.",
"Another limiting case is represented by the regime where the central optical mode is suppressed after passing through Bob's modulator.",
"This regime takes place when the condition $d_{00}^{S}(\\tilde{\\beta })\\propto P_S(\\cos \\tilde{\\beta })=0,$ where $P_S(x)$ is the Legendre polynomial, is satisfied.",
"The intermode coupling should be sufficiently strong, $\\gamma >\\gamma _c$ , for the condition (REF ) to be met.",
"To show this, we note that, the value of $\\cos \\tilde{\\beta }$ varies from unity to $\\cos (2\\beta _m)$ as the phase $\\phi _{AB}$ changes from zero to $\\pi $ .",
"It implies that the condition (REF ) cannot be fulfilled if the value of $\\cos (2\\beta _m)$ is above the largest root of the Legendre polynomial $P_S$ on the interval between zero and unity: $[0,1]$ .",
"By making a simplifying assumption that $\\omega =0$ ($\\Omega _{MW}=\\Omega $ ) and $\\beta =\\pi /2$ , we find that $\\beta _m=\\Gamma T$ [$\\Gamma $ is given by Eq. ()].",
"Then, for $T=2\\pi /\\Omega $ and $S=3$ , the critical coupling ratio $\\gamma _c/\\Omega $ can be numerically estimated to be at about $0.0954$ .",
"Interestingly, when the number of modes increases, the critical coupling ratio approaches the estimate $\\gamma _c/\\Omega \\approx 0.0957$ obtained from the asymptotic form of the condition (REF ): $J_0(2 g)=0$ , where $g$ is defined in Eq.",
"()."
],
[
"Conclusions and discussion",
"In this paper, we have formulated a quantum multimode model of the electro-optic modulator, where the intermode interaction is induced by the microwave field via the linear electro-optic effect (the Pockels effect).",
"This model is shown to be exactly solvable when the strength of coupling between the interacting modes depends on the mode number characterizing its detuning from the central optical mode and the operators [see Eqs.",
"() and ()] describing the electro-optically induced interaction form the $su(2)$ Lie algebra with the commutation relations given by Eq.",
"(REF ).",
"Within the framework of the semiclassical approach where the microwave field is treated as a classical signal (the validity of this approximation is justified in Appendix ), we have used the analytical expressions for the quasienergy spectrum (REF ) and the evolution operator (REF ) in combination with the method of generalized Jordan mappings (see Appendix ) to describe the temporal evolution of the photonic annihilation (creation) operators in terms of the Wigner $D$ functions [see Eqs.",
"(REF )–(REF )].",
"These results are then employed for theoretical investigation into the effects of light modulation on the photon counting rate.",
"Based on the well-known Mandel-Wolf model of an idealized photodetector [65], we have found that the count rate computed as the one-electron photodetection probability per unit time can be written in the factorized form (REF ) with the light modulation form-factor given by Eq. ().",
"Figures REF – REF present the numerical results for the counting rate form-factor evaluated as a function of the frequency and the coupling constant.",
"In particular, the theoretical predictions for the case where $S=3$ (the number of interacting modes equals $2S+1$ ) are compared with the large $S$ limit where $S$ increases indefinitely, $S\\rightarrow \\infty $ (this limiting case is discussed in Subsection .REF ).",
"It is found that the differences between these two cases are negligible at small values of the coupling constant (see Fig.",
"REF ) and become pronounced as the strength of intermode interaction increases (see Figs.",
"REF – REF ).",
"In the large $S$ limit, coupling constant dependence of the intensities of sidebands shows that the photons spread over available photonic states leading to depletion of the pumped mode (solid lines in Figs.",
"REF –REF ).",
"This is a consequence of asymptotic behavior in the large $S$ limit where, similar to the classical optics, the effect of electro-optic light modulation is shown to be determined by the modulating phase factor given by Eq.",
"(REF ) [see also Eq.",
"(REF )].",
"By contrast, the intensities of sidebands computed as a function of the coupling coefficient at $S=3$ (dashed lines in Figs.",
"REF –REF ) appear to be nearly periodic.",
"Another interesting effect which disappears in the large $S$ limit can be described as the asymmetry in intensity between the sidebands with the frequencies symmetrically arranged with respect to the pumped mode (e.g.",
"red shifted Stokes and blue shifted anti-Stokes modes).",
"As is illustrated in Fig.",
"REF , this asymmetry arises when the pumped mode differs from the central one (the case of detuned pumping).",
"Analytical results are also employed to describe the two-modulator transmission depending on the microwave phase difference.",
"We have studied the two important limiting regimes where either the modulators compensate each other or have a destructive effect on the central optical mode.",
"The latter is found to occur only if the intermode interaction strength is sufficiently strond and exceeds its critical value.",
"We now try to place our results into a more general physical context.",
"Generally, an exactly solvable model where the electro-optic modulator is viewed as a multiport device can be employed as a theoretical tool for investigation into numerous effects coming from the complicated quantum dynamics of multimode systems.",
"In addition, this model deals with parametric processes that play important part in the so-called resonator optomechanics [68], [69] representing a new branch of quantum information science that rapidly evolves at the interface of the nanophysics and the quantum theory of light.",
"Making progress in studies of the Casimir effect, new protocols of quantum communication, quantum computing and quantum memory will require further insight into the theory of such parametric processes.",
"Mathematically, we have demonstrated in Appendix that it is feasible to apply the methods of polynomially deformed algebras [53] to extend our considerations to the case of quantized microwave field.",
"This case, however, requires a more comprehensive study which is beyond the scope of this paper.",
"On the other hand, our approach provides a useful tool for investigation of high-frequency light modulation in liquid crystal modulators driven by the orientational Kerr effect [35], [33], [34], [32].",
"In particular, the model can be generalized to take into account effects of non-trivial polarization dependent quantum dynamics.",
"This work is now in progress.",
"We acknowledge partial financial support from the Government of the Russian Federation (Grant No.",
"074-U01)."
],
[
"APPENDIX 5: 0.5em []"
],
[
"Jordan mapping technique",
"These mappings are defined as follows $\\mathbf {J}_{\\alpha }\\mapsto J_{\\alpha }=\\sum _{\\nu ,\\mu =-S}^{S}{a}^{\\dagger }_{\\nu }J_{\\nu \\mu }^{(\\alpha )}a_{\\mu }\\equiv {\\mathbf {a}}^{\\dagger }\\mathbf {J}_{\\alpha } \\mathbf {a},$ where $\\mathbf {J}_{\\alpha }$ is the $(2S+1)\\times (2S+1)$ matrix.",
"The elements $J_{\\nu \\mu }^{(\\alpha )}$ ($\\equiv [\\mathbf {J}_{\\alpha }]_{\\nu \\mu }$ ) of the matrices $\\mathbf {J}_{\\alpha }$ with $\\alpha \\in \\lbrace 0,\\pm \\rbrace $ are given by $J_{\\nu \\mu }^{(\\pm )}=\\sqrt{(S\\mp \\mu )(S\\pm \\mu +1)}\\delta _{\\nu \\mu \\pm 1},\\quad J_{\\nu \\mu }^{(0)}=\\mu \\delta _{\\nu \\mu },$ where $\\delta _{\\nu \\mu }$ is the Kronecker symbol.",
"Using the standard bosonic commutation relations $[a_{\\nu },{a}^{\\dagger }_{\\mu }]=\\delta _{\\nu \\mu },\\quad [{a}^{\\dagger }_{\\nu },{a}^{\\dagger }_{\\mu }]=[a_{\\nu },a_{\\mu }]=0$ it is not difficult to check the key useful property of the Jordan construction: $[J_{\\alpha },J_{\\beta }]={\\mathbf {a}}^{\\dagger }[\\mathbf {J}_{\\alpha },\\mathbf {J}_{\\beta }] \\mathbf {a}.$ The result (REF ) follows because the matrices $\\mathbf {J}_{\\pm }$ and $\\mathbf {J}_{0}$ with the elements given in Eq.",
"(REF ) satisfy the commutation relations for $su(2)$ algebra.",
"Another useful relation can be derived for the Baker-Campbell-Haussdorf formula $\\exp (i\\beta J_{\\alpha }) a_{\\mu } \\exp (-i\\beta J_{\\alpha })=\\sum _{k=0}^{\\infty }\\frac{i^k\\beta ^k}{k!",
"}[J_{\\alpha },a_{\\mu }]_{(k)},$ where $[J_{\\alpha },a_{\\mu }]_{(k)}$ stands for the multiple commutator $&[J_{\\alpha },a_{\\mu }]_{(k)}=[J_{\\alpha },[J_{\\alpha },a_{\\mu }]_{(k-1)}],\\\\&[J_{\\alpha },a_{\\mu }]_{(1)}= [J_{\\alpha },a_{\\mu }],\\quad [J_{\\alpha },a_{\\mu }]_{(0)}=a_{\\mu }.$ From Eqs.",
"(REF ) and (REF ) we have $[J_{\\alpha },a_{\\mu }]=-\\sum _{\\nu =-S}^{S} J_{\\mu \\nu }^{(\\alpha )} a_{\\nu },$ and formula (REF ) can be recast into the final form $\\exp (i\\beta J_{\\alpha }) a_{\\mu } \\exp (-i\\beta J_{\\alpha })=\\sum _{\\nu =-S}^{S} [\\exp (-i\\beta \\mathbf {J}_{\\alpha })]_{\\mu \\nu }a_{\\nu }.$ An important consequence of Eq.",
"(REF ) is the identity $\\mathrm {e}^{i\\gamma J_z}\\mathrm {e}^{i\\beta J_y}\\mathrm {e}^{i\\alpha J_z} a_{\\mu }\\mathrm {e}^{-i\\alpha J_z}\\mathrm {e}^{-i\\beta J_y}\\mathrm {e}^{-i\\gamma J_z}=&\\sum _{\\nu =-S}^{S}[\\mathrm {e}^{-i\\alpha \\mathbf {J}_z}\\mathrm {e}^{-i\\beta \\mathbf {J}_y}\\mathrm {e}^{-i\\gamma \\mathbf {J}_z}]_{\\mu \\nu } a_{\\nu }\\\\&=\\sum _{\\nu =-S}^{S} D_{\\mu \\nu }^{S}(\\alpha ,\\beta ,\\gamma ) a_{\\nu }$ for the rotated annihilation operator expressed in terms of the Wigner $D$ functions: $D_{\\mu \\nu }^{S}(\\alpha ,\\beta ,\\gamma )=\\exp [-i\\mu \\alpha ]d_{\\mu \\nu }^{S}(\\beta )\\exp [-i\\nu \\gamma ]$ that, for the irreducible representation of the rotation group with the angular number $S$ , give the elements of the rotation matrix parametrized by the three Euler angles [61], [62]: $\\alpha $ , $\\beta $ and $\\gamma $ .",
"We conclude this section with details on derivation of the expression for the matrix elements of the operator $\\mathbf {U}_S(t)$ given in Eq. ().",
"This operator can be written in the form $\\mathbf {U}_S(t)=\\mathrm {e}^{-i\\beta \\mathbf {J}_y}\\mathrm {e}^{-i\\Gamma t \\mathbf {J}_z}\\mathrm {e}^{i\\beta \\mathbf {J}_y}=\\mathrm {e}^{-i\\Gamma t (\\sin \\beta \\mathbf {J}_x+\\cos \\beta \\mathbf {J}_z)}.$ More generally, we consider the rotation operator $R(\\psi ,\\hat{\\mathbf {m}})=\\exp [-i \\psi \\bigl ({\\hat{\\mathbf {m}}}\\cdot {\\mathbf {J}}\\bigr )],$ where $\\mathbf {J}=(J_x,J_y,J_z)$ , $\\psi =\\Gamma t$ and $\\hat{\\mathbf {m}}=(m_x,m_y,m_z)\\equiv (m_1,m_2,m_3)=(\\sin \\beta ,0,\\cos \\beta )$ is the unit vector directed along the rotation axis.",
"Equation (REF ) defines rotation about the rotation axis $\\hat{\\mathbf {m}}$ by the rotation angle $\\psi =\\Gamma t$ .",
"Alternatively, this rotation can be parametrized by the Euler angles as follows $R(\\psi ,\\hat{\\mathbf {m}})=R(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\gamma })=\\mathrm {e}^{-i\\tilde{\\alpha }J_z}\\mathrm {e}^{-i\\tilde{\\beta }J_y}\\mathrm {e}^{-i\\tilde{\\gamma }J_z}.$ Our task is to express the Euler angles $\\tilde{\\alpha }$ , $\\tilde{\\beta }$ and $\\tilde{\\gamma }$ in terms of the rotation angle $\\psi =\\Gamma t$ and the angle of the rotation axis $\\beta $ .",
"To this end, we begin with the relations $&R(\\psi ,\\hat{\\mathbf {m}})\\bigl ({\\mathbf {n}}\\cdot {\\mathbf {J}}\\bigr ){R}^{\\dagger }(\\psi ,\\hat{\\mathbf {m}})=\\bigl ({\\mathbf {R}(\\psi ,\\hat{\\mathbf {m}})\\mathbf {n}}\\cdot {\\mathbf {J}}\\bigr ),\\\\&R(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\gamma })\\bigl ({\\mathbf {n}}\\cdot {\\mathbf {J}}\\bigr ){R}^{\\dagger }(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\gamma })=\\bigl ({\\mathbf {R}(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\gamma })\\mathbf {n}}\\cdot {\\mathbf {J}}\\bigr ),$ where $\\mathbf {R}(\\psi ,\\hat{\\mathbf {m}})$ and $\\mathbf {R}(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\gamma })$ are the $3\\times 3$ rotation matrices, that hold for arbitrary vector $\\mathbf {n}$ .",
"$\\mathbf {R}(\\psi ,\\hat{\\mathbf {m}})=\\mathbf {I}_3\\cos \\psi +\\hat{\\mathbf {m}}\\otimes \\hat{\\mathbf {m}}(1-\\cos \\psi )+\\mathbf {M}\\sin \\psi ,$ where $\\mathbf {I}_3$ is the $3\\times 3$ identity matrix and $\\mathbf {M}$ is the antisymmetric matrix with the elements $\\displaystyle M_{ij}=-\\sum _{k=1}^3\\epsilon _{ijk}m_k$ defined using the unit vector $\\hat{\\mathbf {m}}$ and the antisymmetric tensor $\\epsilon _{ijk}$ ($\\epsilon _{123}=1$ ).",
"In our case, we have $\\mathbf {M}=\\begin{pmatrix}0&\\cos \\beta &0\\\\\\cos \\beta &0&-\\sin \\beta \\\\0&\\sin \\beta &0\\end{pmatrix}.$ From the other hand, the rotation matrix $\\mathbf {R}(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\gamma })$ is given by $\\mathbf {R}(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\gamma })=\\mathbf {R}_z(\\tilde{\\alpha }) \\mathbf {R}_y(\\tilde{\\beta })\\mathbf {R}_z(\\tilde{\\gamma })$ a product of the rotation matrices of the form: $\\mathbf {R}_z(\\tilde{\\alpha })=\\begin{pmatrix}\\cos \\tilde{\\alpha }&-\\sin \\tilde{\\alpha }&0\\\\\\sin \\tilde{\\alpha }&\\cos \\tilde{\\alpha }&0\\\\0&0&1\\end{pmatrix},\\:\\mathbf {R}_y(\\tilde{\\beta })=\\begin{pmatrix}\\cos \\tilde{\\beta }&0&\\sin \\tilde{\\beta }\\\\0&1&0\\\\-\\sin \\tilde{\\beta }&0&\\cos \\tilde{\\beta }\\end{pmatrix}.$ The relations linking different parametrizations can now be obtained from the condition: $\\mathbf {R}(\\psi ,\\hat{\\mathbf {m}})=\\mathbf {R}(\\tilde{\\alpha },\\tilde{\\beta },\\tilde{\\gamma })\\equiv \\mathbf {R}.$ Since, for the matrix $\\mathbf {R}(\\psi ,\\hat{\\mathbf {m}})$ , $R_{13}=R_{31}$ , $R_{21}=-R_{12}$ and $R_{23}=-R_{32}$ , we have $\\tilde{\\gamma }=\\tilde{\\alpha }+\\pi $ and the condition (REF ) gives the following relations: $&-\\sin (2\\tilde{\\alpha })(1+\\cos \\tilde{\\beta })=2\\sin \\beta \\sin \\psi =R_{21},\\\\&-\\cos (2\\tilde{\\alpha })(1+\\cos \\tilde{\\beta })=\\sin ^2\\beta +(1+\\cos ^2\\beta )\\cos \\psi =R_{11}+R_{22},\\\\&\\cos \\tilde{\\beta }=\\cos ^2\\beta +\\sin ^2\\beta \\cos \\psi =R_{33},\\\\&\\cos \\tilde{\\alpha }\\sin \\tilde{\\beta }=\\sin \\beta \\cos \\beta (1-\\cos \\psi )=R_{13},\\\\&\\sin \\tilde{\\alpha }\\sin \\tilde{\\beta }=-\\sin \\beta \\sin \\psi =R_{23}.$ From Eqs.",
"(REF ) and (), we derive the expression for the angle $\\tilde{\\alpha }$ given in Eq.",
"(REF whereas the angle $\\tilde{\\beta }$ is described by formulas () and () that can be easily obtained from Eqs. ()–().",
"Our concluding remarks concern two special cases where either $\\sin \\beta =0$ or $\\cos \\beta =0$ .",
"When $\\sin \\beta =0$ and $\\cos \\beta =\\pm 1$ , the operator (REF ) describes rotations about the $z$ axis by the angle $\\pm \\psi $ and the angles $\\tilde{\\alpha }$ , $\\tilde{\\beta }$ and $\\tilde{\\gamma }$ are given by $\\tilde{\\beta }=0,\\quad \\tilde{\\alpha }+\\tilde{\\gamma }=\\pm \\psi .$ At $\\cos \\beta =0$ and $\\sin \\beta =\\pm 1$ , the rotation axis is parallel to the $x$ axis and we have $\\tilde{\\beta }=\\pm \\psi ,\\quad \\tilde{\\gamma }=-\\tilde{\\alpha }=\\pi /2.$"
],
[
"Quantized microwave field and polynomially deformed algebras",
"In the model with the Hamiltonian (REF ) the microwave field is treated as a classical field characterized by the c-number amplitude $B$ .",
"In this appendix we briefly discuss how this model can be extended to the case where, similar to the optical modes, the microwave field is quantized.",
"In our analysis we employ the technique of polynomially deformed algebras to study applicability of the semiclassical approach.",
"For full quantum description of the modes, we begin with the Hamiltonian (REF ) rewritten as follows $H/\\hbar = \\Omega _{MW} N_b + \\omega _{\\mathrm {opt}} N +\\Omega J_z + \\frac{2 \\gamma _0}{2S+1}\\left({J_{+} b + J_{-}{b}^{\\dagger }}\\right),$ where $N_b={b}^{\\dagger } b$ , the operators $J_z$ and $J_{\\pm }$ given by Eqs.",
"(REF )–() meet the commutation relations for generators of $su(2)$ algebra (REF ), whereas the creation and annihilation operators of the microwave mode, ${b}^{\\dagger }$ and $b$ , obey the commutation relation of the Heisenberg-Weyl algebra: $[b,{b}^{\\dagger }]=1$ .",
"A set of operators that commute with the Hamiltonian (REF ) contains three operators: (a) the operator of the total photon number for the optical modes $N$ given in Eq.",
"(REF ); (b) the Casimir operator of $su(2)$ algebra $J^2$ given by Eq.",
"(REF ); and (c) the additional operator $R=N_b+J_z$ related to the non-negative excitation number operator $M=N_b+J_z+j I$ , where $I$ is the identity operator and $j$ is the angular momentum quantum number [$j(j+1)$ is the eigenvalue of $J^2$ ].",
"The Fock states for the model under consideration are represented by a direct product of the microwave and optical Fock states: $|{n_b}\\rangle _b\\otimes |{\\psi }\\rangle _a$ , where $n_b$ is the photon number of the microwave mode.",
"The Fock space can be conveniently divided into subspaces $\\mathcal {F}_{n,\\,m,\\,j}$ classified by the quantum numbers $m$ , $n$ and $j$ , where $m$ , $n$ and $j(j+1)$ are the eigenvalues of the operators $M$ , $N$ and $J^2$ , respectively.",
"The basis of $\\mathcal {F}_{n,\\,m,\\,j}$ can be formed from the eigenstates of the operator $J_z$ $|{m,n,j,m_z}\\rangle =|{m-m_z-j}\\rangle _b\\otimes |{n,j,m_z}\\rangle _a,$ where $-j\\le m_z\\le \\min \\lbrace j,m-j\\rbrace $ is the azimuthal quantum number [the microwave photon number $n_b=m-m_z-j$ is a nonnegative integer] and $J_z|{m,n,j,m_z}\\rangle =m_z|{m,n,j,m_z}\\rangle $ .",
"Clearly, the quantum numbers $m$ and $j$ determine dimension of $\\mathcal {F}_{n,\\,m,\\,j}$ .",
"At $m\\ge 2j$ , the quantum number $m_z$ is ranged from $-j$ to $j$ and $\\dim \\mathcal {F}_{n,\\,m,\\,j}=2j+1$ .",
"In the opposite case with $m< 2j$ , we have $-j\\le m_z\\le m-j$ and $\\dim \\mathcal {F}_{n,\\,m,\\,j}=m+1$ .",
"In the subspace $\\mathcal {F}_{n,\\,m,\\,j}$ , the Hamiltonian (REF ) is reduced to the following form: $H/\\hbar = n\\,\\omega _{\\mathrm {opt}} + r \\tilde{\\Omega } -\\omega M_0 +\\frac{2 \\gamma _0}{2S+1}\\left(M_{+} + M_{-}\\right),$ where $\\tilde{\\Omega }=(\\Omega +\\Omega _{MW})/2$ , $r=m-j$ is the eigenvalue of the operator $R=N_b+J_z$ and the operators $M_0$ and $M_{\\pm }$ are given by $M_{-} = b J_{+}, \\quad M_{+} = {b}^{\\dagger } J_{-}, \\quad M_{0} = \\frac{N_b - J_z}{2}.$ We can now closely follow the line of reasoning described in Ref.",
"[53] and apply the methods of deformed (quantum) Lie algebras to solve the spectral problem for the Hamiltonian (REF ).",
"For this purpose, we note that the operators (REF ) can be regarded as the generators of polynomial algebra of excitations (PAE).",
"This algebra is generally defined through the algebraic relations: $\\left[ M_0, M_{\\pm }\\right] = \\pm M_{\\pm }, \\quad M_{+} M_{-} = p_{\\kappa }(M_0),$ where $p_{\\kappa }(q)$ is the structure polynomial of degree $\\kappa $ characterizing PAE of order $\\kappa $ .",
"In our case, we have $&M_{+}M_{-} = N_b \\left( {J^2 - J_z^2 - J_z}\\right) =p_3(M_0),\\\\&p_3(q) = -(q-q_1)(q-q_2)(q-q_3),$ where the roots of the polynomial $p_3$ are given by $q_1 = \\frac{j-m}{2}, \\quad q_2 = \\frac{m-3j}{2}, \\quad q_3 = \\frac{m+j}{2} + 1.$ The structure polynomial () defines PAE of third order that will be denoted by $\\mathcal {M}_{m,j}$ .",
"Since $m\\ge 0$ , the largest root is $q_3$ , whereas relation between $q_1$ and $q_2$ depends on the values of $m$ and $j$ : $q_1 > q_2$ at $m < 2j$ and $q_2 > q_1$ at $m > 2j$ .",
"When differences $d_1 = q_3 - q_1 = m + 1$ and $d_2 = q_3 - q_2 = 2j + 1$ are natural numbers, $d_i\\in \\mathbb {N}$ , algebra $\\mathcal {M}_{m,j}$ is known to have finite-dimensional self-adjoint representations that correspond to the positive spectrum of $p_3(M_0)$ .",
"When $m>2j$ [$r>j$ ], the finite-dimensional irreducible representation of $\\mathcal {M}_{m,j}$ will be referred to as the high-excitation zone.",
"Its dimension equals $2j+1$ and the corresponding spectrum of $p_3(M_0)$ is ranged from $q_2$ to $q_3$ .",
"In the opposite case with $m<2j$ [$r<j$ ], the positive part of the spectrum lies in the interval $[q_1,q_3]$ and the dimension of the representation — the so-called low-excitation zone — is equal to $m+1$ .",
"In the method of Ref.",
"[53], the technique of polynomially deformed algebra is used to construct the transformations that map one polynomial algebra of operators onto another.",
"More specifically, the representation of algebra $\\mathcal {M}_{m,j}$ with the generators $\\lbrace M_0,M_{+},M_{-}\\rbrace $ is related to a simpler algebra of second order with the generators $\\lbrace S_0,S_{+},S_{-}\\rbrace $ that meet the commutation relations of $su(2)$ algebra (REF ) and its irreducible representation is characterized by the angular quantum number $s$ .",
"The number $s$ is fixed by the requirement for two representations to be of the same dimension.",
"Mathematical details on the method and a more accurate formulation of the key statements can be found in Ref.",
"[53]."
],
[
"High-excitation zone",
"First we consider the important case of the high-excitation zone, where $s=j$ and the operators $\\lbrace M_0,M_{+},M_{-}\\rbrace $ are expressed in terms of $\\lbrace S_0,S_{+},S_{-}\\rbrace $ as follows [53] $&M_0 = \\frac{r}{2} - S_0,\\quad M_{+} = \\sqrt{r - S_0}\\, S_-,\\\\&M_{-}= {[M_{+}]}^{\\dagger }=S_+\\,\\sqrt{r - S_0},$ where $r=m-j$ .",
"It is also not difficult to obtain the relations $S_0 = J_z,\\quad S_+ = \\frac{1}{\\sqrt{N_b+1}} a J_+,\\quad S_- = J_- {a}^{\\dagger } \\frac{1}{\\sqrt{N_b+1}}$ linking $\\lbrace S_0,S_{+},S_{-}\\rbrace $ and the operators that enter the Hamiltonian (REF ).",
"We can now substitute relations (REF ) into Eq.",
"(REF ) to obtain the Hamiltonian expressed in terms of the operators $\\lbrace S_0,S_{+},S_{-}\\rbrace $ .",
"In the zero-order approximation, we have $M_0 = r/2 - S_0,\\quad M_{\\pm } \\approx \\sqrt{r + 1/2}\\,S_{\\mp },$ so that the approximate structure polynomial $p_2^{(s)}(M_0) = (r + 1/2) S_{-} S_{+} =- (r+1/2)(M_0 - q_2 )(M_0 - q_3)$ is quadratic.",
"The corresponding zero-order Hamiltonian is given by $H_{0}^{(s)}/\\hbar = n\\,\\omega _{\\mathrm {opt}} + r \\Omega _{MW} +\\omega S_0 + \\frac{2 \\gamma _0}{2S+1} \\sqrt{r+1/2} \\left({S_+ + S_-}\\right).$ A comparison between $H_{0}^{(s)}$ and the quasienergy operator for the semiclassical model (REF ) shows that these operators are similar in algebraic structure.",
"In particular, similar to formula (REF ), the quantum number $j$ that determines the dimension of the representation does not enter the expression for $H_{0}^{(s)}$ .",
"So, when $\\gamma $ is replaced by $\\gamma _0\\sqrt{r+1/2}$ , the spectra of these operators are identical up to the additive constant.",
"We thus may conclude that the zero-order approximation for the high-excitation zone of the model with quantized microwave field reproduces the results of semiclassical approach.",
"Note that the condition $r>n_{\\mathrm {max}}S\\equiv j_{\\mathrm {max}}$ ensures applicability of the semiclassical approximation for the Fock states of the optical modes whose total photon numbers are below $n_{\\mathrm {max}}$ ."
],
[
"Low-excitation zone",
"In conclusion, we briefly review the results for the low-excitation zone where $m< 2j$ .",
"The dimension of the representation is now equal to $m+1$ , so that $s=m/2$ .",
"The corresponding positive part of $p_3(M_0)$ spectrum is ranged from $q_1=(j-m)/2$ to $q_3=(m+j)/2+1$ and relations linking $\\lbrace M_0,M_{+},M_{-}\\rbrace $ and $\\lbrace S_0,S_{+},S_{-}\\rbrace $ are given by $&M_0 = \\frac{j}{2} - S_0,\\\\&M_+ = \\sqrt{ 2j - m/2 - S_0}\\,S_{-},\\quad M_- = S_+\\sqrt{2j - m/2 - S_0},\\\\&S_0 = \\frac{m}{2}-N_b,\\quad S_+ = J_+ \\frac{1}{\\sqrt{j-J_0}}a,\\quad S_- = {a}^{\\dagger } \\frac{1}{\\sqrt{j-J_0}} J_-.$ In the zero-order approximation, the operators (REF ) are simplified as follows $M_0 = -S_0 + \\frac{j}{2},\\quad M_{\\pm } \\approx \\sqrt{(1-m)/2+2j}\\, S_{\\mp }$ and the approximate structure polynomial is given by $p_2^{(w)}(M_0) = -[(1-m)/2+2j](M_0-q_1)(M_0-q_3).$ Finally, substituting relations (REF ) into formula (REF ) yields the expression for the zero-order Hamiltonian in the low-excitation zone $&H_{0}^{(w)}/\\hbar =n\\, \\omega _{\\mathrm {opt}} + \\frac{m}{2} \\Omega _{MW}+\\frac{r}{2} \\Omega + \\omega S_0\\\\&+\\frac{2 \\gamma _0}{2S+1} \\sqrt{(1-m)/2+2j} \\left({S_+ + S_-} \\right).$ In contrast to the case of the high-excitation zone, the parameters of the Hamiltonian (REF ) and the dimension of the representation both depend on the quantum numbers $r$ and $j$ .",
"So, the semiclassical approximation breaks down in the low-excitation zone and quantum effects become essential for description of this regime even in the zero-order approximation."
]
] | 1605.05770 |
[
[
"Characterizations of $A_2$ Matrix Power Weights"
],
[
"Abstract In the scalar setting, the power functions $|x|^{\\gamma}$, for $-1 < \\gamma<1$, are the canonical examples of $A_2$ weights.",
"In this paper, we study two types of power functions in the matrix setting, with the goal of obtaining canonical examples of $A_2$ matrix weights.",
"We first study Type 1 matrix power functions, which are $n\\times n$ matrix functions whose entries are of the form $a|x|^{\\gamma}.$ Our main result characterizes when these power functions are $A_2$ matrix weights and has two extensions to Type $1$ power functions of several variables.",
"We also study Type 2 matrix power functions, which are $n\\times n$ matrix functions whose eigenvalues are of the form $a|x|^{\\gamma}.$ We find necessary conditions for these to be $A_2$ matrix weights and give an example showing that even nice functions of this form can fail to be $A_2$ matrix weights."
],
[
"Background.",
"Let $W: \\mathbb {R} \\rightarrow M_n(\\mathbb {C})$ be an $n \\times n$ matrix-valued function.",
"Then $W$ is a matrix weight if $W(x)$ is positive definite for a.e.",
"$x \\in \\mathbb {R}$ and if each entry of $W$ is a locally integrable function.",
"Such matrix weights are natural objects arising in a variety of settings, for example from spectral measures of stationary processes, from matrix Toeplitz operators, and in the multivariate case, in the study of systems of certain elliptic PDEs [16], [25].",
"Given a matrix weight $W$ , one can naturally consider the weighted space $L^2(W),$ which is the set of vector-valued functions $f: \\mathbb {R} \\rightarrow \\mathbb {C}^n$ such that $\\Vert f\\Vert ^2_{L^2(W)} \\equiv \\int _{\\mathbb {R}} \\left\\langle W(x) f(x), f(x) \\right\\rangle _{\\mathbb {C}^n} dx = \\int _{\\mathbb {R}} \\left\\Vert W(x)^{\\frac{1}{2}} f(x) \\right\\Vert ^2_{\\mathbb {C}^n} dx <\\infty .$ A standard question to ask about such spaces is: If an operator $T$ is bounded on $L^2(\\mathbb {R}, \\mathbb {C}^n)$ , when is $T$ also bounded on $L^2(W)$ ?",
"In [25], Treil and Volberg answered this question for the Hilbert transform and in [20], [27], [8], Nazarov and Treil, Volberg, and Christ and Goldberg separately answered the question for many important operators, including classes of Calderón-Zgymund operators and maximal functions.",
"In all cases, the required condition is that $W$ be an $A_2$ matrix weight, namely $ \\big [ W \\big ] _{A_2} \\equiv \\sup _{I} \\left\\Vert \\langle W \\rangle _I^{\\frac{1}{2}}\\langle W^{-1} \\rangle _I^{\\frac{1}{2}} \\right\\Vert ^2 \\approx \\sup _{I} \\text{Trace} \\left( \\langle W \\rangle _I\\langle W^{-1} \\rangle _I \\right) < \\infty ,$ where the supremum is taken over all intervals $I$ , $\\langle W \\rangle _I$ denotes the average $\\frac{1}{|I|} \\int _I W(x) dx$ , and $\\left\\Vert \\cdot \\right\\Vert $ denotes the norm of the matrix acting on $\\mathbb {C}^n$ .",
"It is worth observing that if $W$ is an $A_2$ matrix weight, then $W^{-1}$ is also an $A_2$ matrix weight.",
"These boundedness results provided nontrivial matrix analogues of classical scalar results and spurred a wave of interest in harmonic analysis in the matrix-weighted and more general settings.",
"See, for example [2], [4], [7], [10], [11], [15], [17], [18], [19], [21], [23], [24], [26].",
"Recall that in the scalar setting, $A_2$ weights $w$ are positive a.e., locally integrable functions satisfying $ [w]_{A_2} = \\sup _{I} \\left( \\langle w \\rangle _I \\langle w^{-1} \\rangle _I \\right) < \\infty .$ where $\\langle w \\rangle _I=\\frac{1}{|I|} \\int _I w(x) dx$ and the supremum is over all intervals $I$ .",
"While many scalar results involving such $A_2$ weights have been extended to or disproved in the matrix weighted setting, there are still many open questions.",
"Arguably the most famous is the Matrix $A_2$ conjecture.",
"Namely, a celebrated result by T. Hytönen [14] proved the scalar $A_2$ conjecture, i.e.",
"that $ \\Vert T \\Vert _{L^2(w) \\rightarrow L^2(w)} \\lesssim [w]_{A_2}$ for all Calderón-Zgymund operators $T$ , where the implied constant does not depend on $w$ .",
"However in the matrix setting, despite substantial work, the sharp dependence of the operator norms on $[W]_{A_2}$ is unknown.",
"Currently, the best known bound for any non-trivial operator is the bound $[W]^{\\frac{3}{2}}_{A_2}$ for sparse operators, with proofs appearing in both [3], [15].",
"In this paper, we study particular examples of $A_2$ matrix weights.",
"These examples are motivated by the scalar power weights, which are weights of the form $w(x) = a |x|^{\\gamma }$ , where $a, \\gamma \\in \\mathbb {R}.$ It is well known that such a $w(x)$ is an $A_2$ weight if and only if $a$ is positive and $-1 < \\gamma <1$ .",
"See e.g.",
"Example $9.1.7$ in [9].",
"In the scalar setting, these are useful both as simple examples to check conjectures and because they provide the example showing that the linear bound (REF ) is sharp [22].",
"Our goal is to provide important examples of $A_2$ weights in the matrix setting, which may act as a testing ground for conjectures or be used to build interesting objects, such as systems of PDEs like the ones studied in [16]."
],
[
"Outline of Paper.",
"In this paper, we study two generalizations of power functions in the matrix setting.",
"To do this, we require additional preliminary definitions and results related to the structure of matrices.",
"These are presented in Section .",
"Then in Section , we examine matrix functions where each entry is a power of $|x|$ .",
"Specifically, we define Type 1 matrix power functions to be matrix functions of the form $ W(x) = \\begin{bmatrix}a_{11} |x|^{\\gamma _{11}} & \\dots & a_{1n} |x|^{\\gamma _{1n}} \\\\\\vdots &\\ddots & \\vdots \\\\a_{n1} |x|^{\\gamma _{n1}} & \\dots & a_{nn} |x|^{\\gamma _{nn}}\\end{bmatrix},$ where each $a_{ij} \\in \\mathbb {C}$ and each $\\gamma _{ij} \\in \\mathbb {R}.$ Given a Type 1 matrix power function, we define its coefficient matrix $A$ to be the matrix $ A \\equiv \\begin{bmatrix}a_{11} & \\dots & a_{1n} \\\\\\vdots &\\ddots & \\vdots \\\\a_{n1} & \\dots & a_{nn}\\end{bmatrix}.$ In Subsection REF , we characterize when Type 1 matrix functions $W(x)$ are positive definite for a.e.",
"$x$ .",
"In Subsection REF , we establish our main result Theorem REF , which characterizes when Type 1 matrix power functions are $A_2$ matrix weights.",
"We use this result to generate several nontrivial $A_2$ matrix weights in Example REF and in Remark REF , point out that our arguments and results generalize to two kinds of Type 1 matrix power functions in several variables.",
"In Section , we consider matrix functions whose eigenvalues are powers of $|x|$ .",
"Specifically, we define Type 2 matrix power functions to be matrix functions of the form $ W(x) = U(x)\\begin{bmatrix}\\alpha _1|x|^{\\gamma _1} & \\dots & 0 \\\\\\vdots & \\ddots & \\vdots \\\\0 & \\dots & \\alpha _n|x|^{\\gamma _n}\\end{bmatrix}U^\\ast (x),$ where each $\\alpha _i \\in \\mathbb {C}$ , each $\\gamma _{i} \\in \\mathbb {R},$ the interior matrix is diagonal, and $U(x)$ is unitary for a.e.",
"$x$ .",
"In Subsection REF , we provide necessary and sufficient conditions for these matrix functions to be locally integrable.",
"It is clear that they are positive definite a.e.",
"precisely when each $\\alpha _i$ is positive.",
"However, any necessary and sufficient conditions for Type 2 matrix power functions to be $A_2$ matrix weights will likely be complicated and depend heavily on the structure of the unitary $U(x)$ .",
"Indeed, in Example REF , we show that the supposedly-simple matrix weight $W(x) = \\begin{bmatrix}\\cos x & -\\sin x \\\\\\sin x & \\cos x \\\\\\end{bmatrix} \\begin{bmatrix}|x|^{\\gamma _1} & 0 \\\\0 & |x|^{\\gamma _2} \\\\\\end{bmatrix}\\begin{bmatrix}\\cos x & \\sin x \\\\-\\sin x & \\cos x \\\\\\end{bmatrix}$ is almost never an $A_2$ matrix weight.",
"A necessary requirement is that $\\gamma _1=\\gamma _2$ , which turns $W(x)$ into the trivial weight $|x|^{\\gamma _1} I_{2\\times 2}.$ In [5], [6], Bloom studied a variant of this example for the case where $\\gamma _2 = -\\gamma _1$ and concluded that this weight is a “good weight.” As our result shows that even these supposedly-nice weights are almost never $A_2$ matrix weights, we conjecture that Type 2 power functions are rarely $A_2$ matrix weights.",
"Lastly, in Remark REF , we point out that this (non)example generalizes to higher dimensions."
],
[
"Acknowledgements",
"The first author would like to thank Brett Wick for many valuable discussions about $A_2$ matrix weights and Joshua Isralowitz for insightful conversations related to $A_2$ matrix weights in several variables and their applications."
],
[
"Preliminary Matrix Theory",
"In order to study matrix power functions, we require standard facts about matrices, including when a matrix is positive definite (i.e.",
"self-adjoint with positive eigenvalues) and methods to compute the inverse of a matrix.",
"The needed definitions and facts are presented below and can be found in the monographs [1], [12].",
"Throughout this section, let $A$ be an $n \\times n$ matrix with entries $a_{ij} \\in \\mathbb {C}.$ The needed characterization of positive definiteness requires submatrices.",
"A submatrix of $A$ is a matrix composed of the entries of $A$ that lie in a subcollection of its rows and columns.",
"Specifically, let $\\alpha \\subseteq \\lbrace 1, \\dots , n\\rbrace $ and $\\beta \\subseteq \\lbrace 1, \\dots , n\\rbrace $ be index sets.",
"The notation $A [\\alpha , \\beta ]$ indicates the submatrix of $A$ composed of the entries of $A$ from the rows of $A$ indexed by $\\alpha $ and the columns of $A$ indexed by $\\beta .$ A submatrix $A [\\alpha , \\beta ]$ is a principal submatrix if $\\alpha =\\beta $ .",
"A principle submatrix $A[\\alpha , \\alpha ]$ is a leading principal submatrix if there is a $k \\in \\lbrace 1, \\dots , n\\rbrace $ such that $\\alpha =\\lbrace 1, \\dots , k\\rbrace .$ Finally, the shorthand notation $A_{ij} = A[ \\lbrace i\\rbrace ^c, \\lbrace j\\rbrace ^c]$ represents the submatrix of $A$ obtained by removing the ith row and jth column from $A$ .",
"The determinant of a (principle, leading principle) submatrix of $A$ is called a (principle, leading principle) minor of $A$ .",
"One can determine if a self-adjoint matrix is positive definite using its principle and leading principle minors, via a characterization called Sylvester's Criterion: Theorem 2.1 Let $A$ be an $n \\times n$ self-adjoint matrix.",
"Then a.",
"$A$ is positive definite if and only if every principal minor of $A$ is positive.",
"b.",
"$A$ is positive definite if and only if every leading principal minor of $A$ is positive.",
"To study $A_2$ matrix weights, we also need information about matrix inverses.",
"This requires formulas for determinants, which require permutations.",
"A permutation of length $n$ is a one-to-one function $\\sigma : \\lbrace 1,\\dots , n\\rbrace \\rightarrow \\lbrace 1, \\dots , n\\rbrace ,$ and the sign of $\\sigma ,$ denoted $\\text{sgn}(\\sigma ),$ is $+1$ or $-1$ depending on whether the minimum number of transpositions needed to turn $\\lbrace 1,\\dots , n\\rbrace $ into $\\sigma $ is even or odd.",
"Lastly, $S_n$ denotes the set of all permutations of length $n$ .",
"Then the determinant of $A$ can be computed using the formula $ \\det A = \\sum _{\\sigma \\in S_n} \\left( \\text{sgn}(\\sigma ) \\prod _{k=1}^n a_{k \\sigma (k)} \\right).$ If $A$ is invertible, we can compute $A^{-1}$ using minors and determinants as follows $A^{-1} = \\frac{1}{\\det A}\\left[\\begin{array}{ccc}C_{11} & \\dots & C_{n1} \\\\\\vdots & \\ddots & \\vdots \\\\C_{1n} & \\dots & C_{nn} \\\\\\end{array} \\right],$ where $C_{ij}$ is the $ij^{th}$ cofactor of $A$ , meaning $C_{ij} = (-1)^{i+j} \\det A_{ij}$ , where $A_{ij}$ was defined earlier.",
"Note that $A^{-1}$ is $\\frac{1}{\\det A}$ times the transpose of the matrix of cofactors of $A$ ."
],
[
"Type 1 Matrix Power Functions",
"In this section, we study Type 1 matrix power functions, i.e.",
"matrix functions of form (REF ) with coefficient matrix $A$ ."
],
[
"Positive Matrix Power Functions.",
"We will first characterize when a Type 1 matrix power function $W(x)$ is positive definite a.e.",
"Theorem 3.1 An $n\\times n$ Type 1 matrix power function $W(x)$ is positive definite a.e.",
"if and only if its coefficient matrix $A$ is positive definite and the powers $\\gamma _{ij}$ satisfy $ \\gamma _{ij} = \\frac{\\gamma _{ii} + \\gamma _{jj}}{2} \\ \\text{ for } 1 \\le i,j \\le n.$ This characterization requires the following lemma about determinant formulas of matrix power functions, which is proved in Subsection REF .",
"Lemma 3.2 Let $W$ be an $n\\times n$ Type 1 matrix power function with coefficient matrix $A$ and powers $\\gamma _{ij}$ satisfying $\\gamma _{ij} = \\frac{\\gamma _{ii} + \\gamma _{jj}}{2} \\ \\text{ for } 1 \\le i,j \\le n$ .",
"Let $W(x)_{ij}$ be the matrix obtained by removing the $i^{th}$ row and $j^{th}$ column from $W(x)$ for each $x$ .",
"Then $ \\det W(x) = |x|^{\\sum _{k=1}^n \\gamma _{kk}} \\det A \\ \\ \\text{ and } \\ \\ \\det W(x)_{ij} = |x|^{ - \\gamma _{ij} +\\sum _{k=1}^n \\gamma _{kk}} \\det A_{ij}.$ Here is the proof of Theorem REF : We will prove this by induction on the size of $W$ .",
"For the base case, let $n=1.$ Then $W(x) = [a_{11}|x|^{\\gamma _{11}}]$ .",
"First assume $W(x)$ is positive definite a.e.",
"Then since $|x|^{\\gamma _{11}}> 0$ for $x\\ne 0$ , we can conclude $a_{11} > 0$ .",
"Thus, $A \\equiv [a_{11}]$ is positive definite.",
"In addition, condition (REF ) is trivial since $\\frac{\\gamma _{11} + \\gamma _{11}}{2} = \\gamma _{11}.$ Now assume $A = [a_{11}]$ is positive definite.",
"As $|x|^{\\gamma _{11}}> 0$ for $x\\ne 0$ , this immediately implies $W(x) = [a_{11}|x|^{\\gamma _{11}}]$ is positive definite.",
"For the inductive step, assume the theorem holds for every $(n-1) \\times (n-1)$ Type 1 matrix power function, and let $W$ be an $n\\times n$ Type 1 matrix power function with coefficient matrix $A$ and powers $\\gamma _{ij}.$ ($\\Rightarrow $ ) For the forward direction, assume $W(x)$ is positive definite a.e.",
"We will show $A$ is positive definite and the $\\gamma _{ij}$ powers satisfy condition (REF ).",
"First, since $W(x)$ is positive definite $a.e.$ , it is self-adjoint a.e., i.e.",
"$a_{ij}|x|^{\\gamma _{ij}} = \\overline{a_{ji}|x|^{\\gamma _{ji}}} = \\bar{a}_{ji} |x|^{\\gamma _{ji}}, \\qquad \\text{ for } 1 \\le i,j \\le n.$ This implies each $\\gamma _{ij}=\\gamma _{ji}$ and $a_{ij} = \\bar{a}_{ji}$ , so $A$ is self-adjoint.",
"Furthermore, by Theorem REF , all of the leading principal minors of $W(x)$ are positive a.e.",
"Specifically, for each $k=1, \\dots , n-1$ , $ \\det \\left[\\begin{array}{ccc}a_{11} |x|^{\\gamma _{11}} & \\dots & a_{1k} |x|^{\\gamma _{1k}} \\\\\\vdots & \\ddots & \\vdots \\\\a_{k1} |x|^{\\gamma _{k1}} & \\dots & a_{kk} |x|^{\\gamma _{kk}}\\end{array} \\right]>0.",
"$ Thus, by Theorem REF , we can conclude that the $(n-1) \\times (n-1)$ Type 1 matrix power function $ W(x)_{nn} \\equiv \\left[\\begin{array}{ccc}a_{11} |x|^{\\gamma _{11}} & \\dots & a_{1(n-1)} |x|^{\\gamma _{1(n-1)}} \\\\\\vdots & \\ddots & \\vdots \\\\a_{(n-1)1} |x|^{\\gamma _{(n-1)1}} & \\dots & a_{(n-1)(n-1)} |x|^{\\gamma _{(n-1)(n-1)}}\\end{array} \\right], $ obtained by removing the $n^{th}$ row and $n^{th}$ column from $W(x),$ is also positive definite a.e.",
"Now, we establish condition (REF ).",
"First, applying the inductive hypothesis to $W(x)_{nn}$ gives $\\gamma _{ij} = \\frac{\\gamma _{ii}+\\gamma _{jj}}{2}, \\qquad \\text{ for }1 \\le i,j \\le n-1.$ To conclude (REF ), we only need $\\gamma _{ni} =\\gamma _{in} = \\frac{\\gamma _{ii} + \\gamma _{nn}}{2}, \\qquad \\text{ for } 1 \\le i \\le n,$ where the first equality was already established.",
"Also, it is clear that $\\gamma _{nn} = \\frac{\\gamma _{nn} + \\gamma _{nn}}{2}$ .",
"Thus, we must show that $\\gamma _{in} = \\frac{\\gamma _{ii} + \\gamma _{nn}}{2}$ for $1 \\le i \\le n-1$ .",
"Fix such an $i$ .",
"Then the following determinant $ \\det \\left[ \\begin{array}{cc}a_{ii}|x|^{\\gamma _{ii}} & a_{in}|x|^{\\gamma _{in}} \\\\a_{ni}|x|^{\\gamma _{ni}} & a_{nn}|x|^{\\gamma _{nn}}\\end{array} \\right] $ is a principal minor of $W(x)$ and so by Theorem REF , is positive a.e.",
"Computing this determinant and using the fact that $A$ is self-adjoint gives $ a_{ii}a_{nn}|x|^{\\gamma _{ii} + \\gamma _{nn}} - |a_{in}|^2|x|^{2\\gamma _{in}} > 0.$ Here we can assume $a_{in} \\ne 0$ , because otherwise we could trivially choose $\\gamma _{in} = \\frac{\\gamma _{ii}+\\gamma _{nn}}{2}.$ Now by looking at both $x$ values near zero and $x$ values arbitrarily large, one can see that (REF ) holds a.e.",
"if and only if $ \\gamma _{ii} + \\gamma _{nn} = 2 \\gamma _{in},$ which is the desired equality.",
"Now we show $A$ is positive definite.",
"By the inductive hypothesis, the coefficient matrix of $W(x)_{nn}$ $ A_{nn} \\equiv \\left[\\begin{array}{ccc}a_{11} & \\dots & a_{1(n-1)} \\\\\\vdots & \\ddots & \\vdots \\\\a_{(n-1)1} & \\dots & a_{(n-1)(n-1)}\\end{array} \\right] $ is positive definite a.e.",
"Then Theorem REF implies that the leading principal minors of $A_{nn}$ are positive.",
"Namely, $ \\det \\left[\\begin{array}{ccc}a_{11} & \\dots & a_{1k} \\\\\\vdots & \\ddots & \\vdots \\\\a_{k1} & \\dots & a_{kk}\\end{array} \\right] >0,$ for $k=1, \\dots , n-1.$ Now, we show that $\\det A > 0$ .",
"As $W(x)$ is positive definite a.e., $\\det W(x) >0$ a.e.",
"As Lemma REF gives $\\det W(x) = |x|^{\\sum _{k=1}^n \\gamma _{kk}} \\det A$ , we can conclude $\\det A >0.$ This shows that all of the leading principal minors of $A$ are positive, so $A$ is positive definite.",
"($\\Leftarrow $ ) Now assume $A$ is positive definite and condition (REF ) holds.",
"As $A$ is self-adjoint, $a_{ij} = \\bar{a}_{ji}$ for $1 \\le i,j \\le n$ .",
"As condition (REF ) implies that each $\\gamma _{ij} = \\gamma _{ji}$ , $W(x)$ is also self-adjoint.",
"Using Theorem REF and the positive definiteness of $A$ , one can easily show that the $(n-1) \\times (n-1)$ matrix $A_{nn} \\equiv \\left[\\begin{array}{ccc}a_{11} & \\dots & a_{1(n-1)} \\\\\\vdots & \\ddots & \\vdots \\\\a_{(n-1)1} & \\dots & a_{(n-1)(n-1)}\\end{array} \\right] $ is positive definite.",
"Then by the inductive hypothesis, the $(n-1) \\times (n-1)$ Type 1 matrix power function $ W(x)_{nn} \\equiv \\left[\\begin{array}{ccc}a_{11} |x|^{\\gamma _{11}} & \\dots & a_{1(n-1)} |x|^{\\gamma _{1(n-1)}} \\\\\\vdots & \\ddots & \\vdots \\\\a_{n1} |x|^{\\gamma _{(n-1)1}} & \\dots & a_{(n-1)(n-1)} |x|^{\\gamma _{(n-1)(n-1)}}\\end{array} \\right] $ is positive definite a.e.",
"Again by Theorem REF , this implies that all of the leading principal minors of $W(x)$ of the form $ \\det \\left[\\begin{array}{ccc}a_{11} |x|^{\\gamma _{11}} & \\dots & a_{1k} |x|^{\\gamma _{1k}} \\\\\\vdots & \\ddots & \\vdots \\\\a_{k1} |x|^{\\gamma _{k1}} & \\dots & a_{kk} |x|^{\\gamma _{kk}}\\end{array} \\right] >0, $ a.e.",
"for $k=1, \\dots , n-1.$ Furthermore, since $\\det A >0$ and by Lemma REF , $\\det W(x) = |x|^{\\sum _{k=1}^n \\gamma _{kk}} \\det A$ , we also have $\\det W(x)> 0$ a.e.",
"Thus, all leading principal minors of $W(x)$ are positive a.e., which implies $W(x)$ is positive definite a.e.",
"and completes the proof."
],
[
"Matrix $A_2$ Power Weights.",
"Now, we characterize when Type 1 matrix power functions are actually $A_2$ matrix weights.",
"Theorem 3.3 Let $W$ be a Type 1 $n\\times n$ matrix power function with coefficient matrix $A$ and powers $\\gamma _{ij}.$ Then $W$ is an $A_2$ matrix weight if and only if $A$ is positive definite and the $\\gamma _{ij}$ satisfy $ \\gamma _{ij} = \\frac{\\gamma _{ii} + \\gamma _{jj}}{2} \\ \\text{ and } \\ -1 < \\gamma _{ii} < 1 \\qquad \\text{ for } 1 \\le i,j \\le n.$ To establish this, we require the following lemma, which is proved in Subsection REF .",
"Lemma 3.4 Let $W$ be a Type 1 $n\\times n$ matrix power function with coefficient matrix $A$ and powers $\\gamma _{ij}$ that is positive definite a.e.",
"Then $ W(x)^{-1} = \\frac{1}{\\det A}\\left[\\begin{array}{ccc}c_{11} |x|^{-\\gamma _{11}} & \\dots & c_{n1} |x|^{-\\gamma _{1n}}\\\\\\vdots & \\ddots & \\vdots \\\\c_{1n} |x|^{-\\gamma _{n1}} & \\dots & c_{nn} |x|^{-\\gamma _{nn}}\\\\\\end{array} \\right],$ where each $c_{ij} = (-1)^{i+j} \\det A_{ij}.$ Now we prove Theorem REF : ($\\Rightarrow $ ) First, assume that $W$ is an $A_2$ matrix weight.",
"Then $W(x)$ is positive definite a.e.",
"and so by Theorem REF , $A$ is positive definite and $ \\gamma _{ij} = \\frac{\\gamma _{ii} + \\gamma _{jj}}{2} \\qquad \\text{ for } 1 \\le i,j \\le n.$ To complete this direction of the proof, we just need to establish the bounds on the $\\gamma _{ii}.$ Observe that the diagonal entries $a_{ii}|x|^{\\gamma _{ii}}$ of $W(x)$ are exactly the $1 \\times 1$ principle minors of $W(x).$ Then, as $W(x)$ is positive definite a.e., Theorem REF implies that each $a_{ii}|x|^{\\gamma _{ii}}$ is positive $a.e.$ Thus, each $a_{ii}$ is nonzero.",
"Furthermore, by Lemma REF , $ W(x)^{-1} = \\frac{1}{\\det A}\\left[\\begin{array}{ccc}c_{11} |x|^{-\\gamma _{11}} & \\dots & c_{n1} |x|^{-\\gamma _{1n}}\\\\\\vdots & \\ddots & \\vdots \\\\c_{1n} |x|^{-\\gamma _{n1}} & \\dots & c_{nn} |x|^{-\\gamma _{nn}}\\\\\\end{array} \\right],$ where each $c_{ij} = (-1)^{i+j} \\det A_{ij}.$ As $W(x)^{-1}$ is positive definite a.e., we can apply Theorem REF to conclude that each $c_{ii}$ is nonzero.",
"Since $W$ and hence, $W^{-1}$ are $A_2$ matrix weights, their entries are locally integrable.",
"In particular, the diagonal entries $a_{ii} |x|^{\\gamma _{ii}}$ and $c_{ii} |x|^{-\\gamma _{ii}}$ are integrable on intervals containing the origin.",
"As each $a_{ii}$ and $c_{ii}$ are nonzero, this implies $ -1 < \\gamma _{ii} <1,$ as desired.",
"($\\Leftarrow $ ) Now, assume that $A$ is positive definite and the $\\gamma _{ij}$ satisfy the given conditions.",
"Then by Theorem REF , $W(x)$ is positive definite a.e.",
"Moreover, the conditions on the $\\gamma _{ij}$ imply that $-1 < \\gamma _{ij} < 1$ for $1 \\le i,j\\le n,$ and so each entry of $W$ is locally integrable.",
"Thus, $W$ is a matrix weight.",
"To show $W$ is an $A_2$ matrix weight, we need both $W$ and $W^{-1}.$ Using the given formula for $W$ and the one for $W^{-1}$ in Lemma REF , we can compute their averages over intervals componentwise.",
"Indeed for any interval $I$ , $ \\left\\langle W \\right\\rangle _I = \\left[\\begin{array}{ccc}a_{11} \\langle |x|^{\\gamma _{11}} \\rangle _I & \\dots & a_{1n} \\langle |x|^{\\gamma _{1n}} \\rangle _I \\\\\\vdots & \\ddots & \\vdots \\\\a_{n1} \\langle |x|^{\\gamma _{n1}} \\rangle _I & \\dots &a_{nn} \\langle |x|^{\\gamma _{nn}} \\rangle _I\\\\\\end{array} \\right] $ and similarly $ \\left\\langle W^{-1} \\right\\rangle _I = \\frac{1}{\\det A}\\left[\\begin{array}{ccc}c_{11} \\langle |x|^{-\\gamma _{11}} \\rangle _I & \\dots & c_{n1} \\langle |x|^{-\\gamma _{1n}}\\rangle _I \\\\\\vdots & \\ddots & \\vdots \\\\c_{1n} \\langle |x|^{-\\gamma _{n1}} \\rangle _I & \\dots & c_{nn} \\langle |x|^{-\\gamma _{nn}} \\rangle _I \\\\\\end{array} \\right].$ To show that $W$ is an $A_2$ matrix weight, we simply need to verify that $ [W]_{A_2} \\approx \\sup _I \\text{Tr} \\left(\\langle W \\rangle _I \\langle W^{-1} \\rangle _I \\right)< \\infty .$ To see this, fix an interval $I$ and observe that $\\text{Tr}(\\langle W \\rangle _I \\langle W^{-1} \\rangle _I) = \\sum _{i,j=1}^n \\left( \\langle W \\rangle _I\\right)_{ij} \\left( \\langle W^{-1} \\rangle _I\\right)_{ji}\\le \\displaystyle \\frac{1}{|\\det A|} \\sum _{i,j = 1}^n |a_{ij}c_{ij}| \\langle |x|^{\\gamma _{ij}} \\rangle _I \\langle |x|^{-\\gamma _{ij}} \\rangle _I.$ Notice that each $\\gamma _{ij}$ satisfies $-1 < \\gamma _{ij}<1,$ so $|x|^{\\gamma _{ij}}$ is a scalar $A_2$ weight.",
"Thus, we can conclude that $ \\text{Tr}(\\langle W \\rangle _I \\langle W^{-1} \\rangle _I) \\le \\sup _{i,j} \\left( \\frac{|a_{ij} \\det A_{ij}|}{|\\det A|} \\right) \\displaystyle \\sum _{i,j=1}^n \\big [|x|^{\\gamma _{ij}} \\big ]_{A_2},$ which is bounded independent of $I$ .",
"Hence, $W$ is an $A_2$ matrix weight.",
"Alternately, one could prove that these positive definite power functions are matrix $A_2$ weights using the sufficient conditions from Theorem $ 4.3$ in [21] or by proving they are “almost diagonal” and using Proposition $4.2$ in [5].",
"Now, let us apply this theorem to generate nontrivial examples of $A_2$ matrix weights.",
"Example 3.5 To obtain an $n \\times n$ Type 1 matrix $A_2$ power weight, one need only choose a positive coefficient matrix $A$ and diagonal powers $-1< \\gamma _{11}, \\dots , \\gamma _{nn}< 1$ .",
"For example, consider the positive definite matrix $ A = \\begin{bmatrix*}[r] 5 & 3 \\\\ 3 & 2 \\end{bmatrix*}\\ \\ \\text{ and powers } \\ \\ \\gamma _{11} = \\frac{1}{2}, \\gamma _{22} = -\\frac{2}{3}.",
"$ These generate the $A_2$ matrix weight $ W(x) = \\begin{bmatrix*}[c] 5 |x|^{1/2} & 3|x|^{-1/12} \\\\ 3|x|^{-1/12} & 2 |x|^{-2/3} \\end{bmatrix*}.$ Similarly, we can use the positive definite matrix $B = \\begin{bmatrix*}[r] 4 & 1 & 2 \\\\1 & 2 & -1 \\\\2 & -1 & 3 \\end{bmatrix*} \\ \\ \\text{ and powers } \\ \\ \\gamma _{11} = \\frac{3}{4}, \\gamma _{22} = {-\\frac{3}{4}}, \\gamma _{33} = \\frac{1}{2} $ to generate the $A_2$ matrix weight $ W(x) = \\begin{bmatrix*}[c] 4 |x|^{3/4} & 1 & 2|x|^{5/8} \\\\1 & 2|x|^{-3/4} & -|x|^{-1/8} \\\\2|x|^{5/8} & -|x|^{-1/8} & 3|x|^{1/2} \\end{bmatrix*}.$ Many of our arguments generalize almost immediately to two kinds of Type 1 matrix power weights in several variables, as detailed below.",
"Remark 3.6 Here, we consider the $d$ -variable setting, where $x = (x_1, \\dots , x_d) \\in \\mathbb {R}^d.$ First, a matrix function $W$ in $d$ variables is an $A_2$ matrix weight if its entries are locally integrable and if it is positive definite a.e.",
"and satisfies $\\big [ W \\big ] _{A_2} \\equiv \\sup _{Q} \\left\\Vert \\langle W \\rangle _Q^{\\frac{1}{2}}\\langle W^{-1} \\rangle _Q^{\\frac{1}{2}} \\right\\Vert ^2 < \\infty ,$ where the supremum is over cubes $Q = I_1 \\times \\cdots \\times I_d$ , with each pair of intervals $I_i$ , $I_j$ satisfying $|I_i| = |I_j|.$ Now, we restrict to two variables to simplify notation.",
"In two variables, define the Type $1.a$ matrix power functions to be matrix functions of the form $ W(x) = \\begin{bmatrix}a_{11} |x_1|^{\\gamma _{11}} |x_2|^{\\beta _{11}} & \\dots & a_{1n} |x_1|^{\\gamma _{1n}} |x_2|^{\\beta _{1n}} \\\\\\vdots &\\ddots & \\vdots \\\\a_{n1} |x_1|^{\\gamma _{n1}} |x_2|^{\\beta _{n1}} & \\dots & a_{nn} |x_1|^{\\gamma _{nn}} |x_2|^{\\beta _{nn}}\\end{bmatrix},$ with coefficient matrix $A$ and powers $\\gamma _{ij}$ .",
"Without too much effort, one can generalize Lemmas REF and REF as well as Theorems REF and REF to this two variable setting.",
"In this setting, the main result states: a Type 1.a $n\\times n$ matrix power function $W$ with coefficient matrix $A$ and powers $\\gamma _{ij}$ is an $A_2$ matrix weight if and only if $A$ is positive definite and the $\\gamma _{ij}$ and $\\beta _{ij}$ satisfy $ \\gamma _{ij} = \\frac{\\gamma _{ii} + \\gamma _{jj}}{2}, \\beta _{ij} = \\frac{\\beta _{ii} + \\beta _{jj}}{2} \\ \\text{ and } \\ -1 < \\gamma _{ii}, \\beta _{ii} < 1 \\qquad \\text{ for } 1 \\le i,j \\le n.$ The obvious generalization holds in $d$ variables.",
"Similarly, define $d$ variable Type $1.b$ matrix power functions to be matrix functions of the form $ W(x) = \\begin{bmatrix}a_{11} \\Vert x\\Vert ^{\\gamma _{11}} & \\dots & a_{1n} \\Vert x\\Vert ^{\\gamma _{1n}} \\\\\\vdots &\\ddots & \\vdots \\\\a_{n1} \\Vert x\\Vert ^{\\gamma _{n1}} & \\dots & a_{nn} \\Vert x\\Vert ^{\\gamma _{nn}}\\end{bmatrix},$ with coefficient matrix $A$ , where $ \\Vert x \\Vert = \\sqrt{ x_1^2 + \\cdots + x_d^2}.$ One can also generalize Lemmas REF and REF as well as Theorems REF and REF to these types of matrix functions.",
"Here the generalization of Theorem REF is slightly different, as $\\Vert x \\Vert ^{\\gamma }$ is locally integrable if and only if $\\gamma > -d$ and $\\Vert x \\Vert ^{\\gamma }$ is an $A_2$ weight if and only if $-d < \\gamma < d.$ For this fact, see Example $9.1.7$ in [9].",
"Then in this setting, the main result states: a Type 1.b $n\\times n$ matrix power function $W$ with coefficient matrix $A$ and powers $\\gamma _{ij}$ is an $A_2$ matrix weight if and only if $A$ is positive definite and the $\\gamma _{ij}$ satisfy $ \\gamma _{ij} = \\frac{\\gamma _{ii} + \\gamma _{jj}}{2} \\ \\text{ and } \\ -d < \\gamma _{ii} < d \\qquad \\text{ for } 1 \\le i,j \\le n.$"
],
[
"Proofs of Lemmas ",
"In this subsection, we include the proofs of the lemmas used earlier.",
"Here is the proof of Lemma REF : First we show $\\det W(x) = |x|^{\\sum _{k=1}^n \\gamma _{kk}} \\det A$ .",
"Let $(W(x))_{ij}$ denote the $ij^{th}$ entry of $W(x).$ Then by (REF ), we have $ \\det W(x) = \\sum _{\\sigma \\in S_n} \\left(\\text{sgn}(\\sigma ) \\prod _{k=1}^n (W(x))_{k\\sigma (k)}\\right).$ Fix $\\sigma \\in S_n$ and consider $\\prod _{k=1}^n (W(x))_{k\\sigma (k)}$ .",
"As $k = 1,...,n$ and $\\sigma $ is injective, the product has one entry from each row and column of $W(x)$ .",
"Then $\\prod _{k=1}^n (W(x))_{k\\sigma (k)} &= \\prod _{k=1}^n a_{k\\sigma (k)}|x|^{\\gamma _{k\\sigma (k)}}\\\\&= \\prod _{k=1}^n a_{k\\sigma (k)}|x|^{(\\gamma _{kk} + \\gamma _{\\sigma (k)\\sigma (k)})/2}\\\\&= \\prod _{k=1}^n a_{k\\sigma (k)} \\prod _{k=1}^n |x|^{\\gamma _{kk}/2} \\prod _{\\sigma (k)=1}^n |x|^{\\gamma _{\\sigma (k)\\sigma (k)}/2}\\\\&= \\prod _{k=1}^n a_{k\\sigma (k)} |x|^{\\sum _{k=1}^n \\gamma _{kk}}.$ By substituting this into (REF ) and factoring $|x|^{\\sum _{k=1}^n \\gamma _{kk}} $ out of the sum, we obtain $\\det W(x) = \\sum _{\\sigma \\in S_n} \\left(\\text{sgn}(\\sigma ) |x|^{\\sum _{k=1}^n \\gamma _{kk}} \\prod _{k=1}^n a_{k\\sigma (k)}\\right) = |x|^{\\sum _{k=1}^n \\gamma _{kk}}\\det A,$ which gives the first equation.",
"Now we fix $i,j$ and show $ \\det W(x)_{ij} = |x|^{ - \\gamma _{ij} +\\sum _{k=1}^n \\gamma _{kk}}\\det A_{ij}.$ First by (REF ), we have $ \\det W(x)_{ij} = \\sum _{\\sigma \\in S_{n-1}} \\left(\\text{sgn}(\\sigma ) \\prod _{k=1}^{n-1}(W(x)_{ij})_{k\\sigma (k)} \\right).$ Fix $\\sigma \\in S_{n-1}$ and consider $\\prod _{k=1}^{n-1} (W(x)_{ij})_{k\\sigma (k)}$ .",
"As $k = 1,...,n-1$ and $\\sigma $ is injective, the product has one entry from each row and column of $W(x)_{ij}$ .",
"Define the indices $\\beta (k) \\equiv \\left\\lbrace \\begin{array}{ll} k & \\text{ if } k < i, \\\\k+1 & \\text{ if } k \\ge i, \\end{array} \\right.\\ \\ \\text{and} \\ \\ \\phi (k) \\equiv \\left\\lbrace \\begin{array}{ll} \\sigma (k) & \\text{ if } \\sigma (k) < j, \\\\\\sigma (k)+1 & \\text{ if } \\sigma (k) \\ge j.",
"\\end{array} \\right.$ Then using arguments similar to those used to obtain the previous determinant formula, $ \\prod _{k=1}^{n-1} (W(x)_{ij})_{k\\sigma (k)} = \\prod _{k=1}^{n-1} (A_{ij})_{k\\sigma (k)}|x|^{\\gamma _{\\beta (k)\\phi (k)}}= |x|^{ - \\gamma _{ij} +\\sum _{k=1}^n \\gamma _{kk}} \\prod _{k=1}^{n-1} (A_{ij})_{k\\sigma (k)}.$ Substituting that into (REF ) gives the desired formula.",
"Here is the proof of Lemma REF : Using the inverse formula (REF ), we have $ W(x)^{-1} \\equiv \\frac{1}{\\det W(x)}\\left[\\begin{array}{ccc}C_{11}(x) & \\dots & C_{n1}(x) \\\\\\vdots & \\ddots & \\vdots \\\\C_{1n}(x) & \\dots & C_{nn}(x) \\\\\\end{array} \\right],$ where $C_{ij}(x) = (-1)^{i+j} \\det W(x)_{ij}$ is the $ij^{th}$ cofactor of $W(x)$ .",
"As $W(x)$ is positive definite a.e., we can use Theorem REF to conclude that $W$ satisfies the conditions of Lemma REF .",
"Then by Lemma REF , we know that each $ C_{ij}(x) = (-1)^{i+j} |x|^{ -\\gamma _{ij} + \\sum _{k=1}^n \\gamma _{kk}} \\det A_{ij} \\ \\ \\text{ and} \\ \\ \\det W(x) = |x|^{\\sum _{k=1}^n \\gamma _{kk}} \\det A.$ To simplify notation, define $c_{ij} = (-1)^{i+j} \\det A_{ij}$ for each $ 1 \\le i,j\\le n.$ Then combining formulas gives $\\begin{aligned}W(x)^{-1} &\\equiv \\frac{{|x|^{-\\sum _{k = 1}^n \\gamma _{kk}}}}{ \\det A}\\left[\\begin{array}{ccc}c_{11} |x|^{-\\gamma _{11} + \\sum _{k=1}^n \\gamma _{kk}} & \\dots & c_{n1} |x|^{-\\gamma _{n1} + \\sum _{k=1}^n \\gamma _{kk}} \\\\\\vdots & \\ddots & \\vdots \\\\c_{1n} |x|^{-\\gamma _{1n} + \\sum _{k=1}^n \\gamma _{kk}} & \\dots & c_{nn} |x|^{-\\gamma _{nn} + \\sum _{k=1}^n \\gamma _{kk}} \\\\\\end{array} \\right] \\\\&= \\frac{1}{\\det A}\\left[\\begin{array}{ccc}c_{11} |x|^{-\\gamma _{11}} & \\dots & c_{n1} |x|^{-\\gamma _{1n}}\\\\\\vdots & \\ddots & \\vdots \\\\c_{1n} |x|^{-\\gamma _{n1}} & \\dots & c_{nn} |x|^{-\\gamma _{nn}}\\\\\\end{array} \\right],\\end{aligned}$ as desired."
],
[
"Type 2 Matrix Power Functions",
"In this section, we consider Type 2 matrix power functions, which have form (REF ).",
"Our original goal was to characterize when such $W$ are $A_2$ matrix weights, but this appears to depend very closely on the structure of the unitary $U(x)$ .",
"Nevertheless, we have determined some necessary conditions for these matrix functions to be $A_2$ matrix weights."
],
[
"Locally Integrable Matrix Power Functions.",
"If $W$ is an $A_2$ matrix weight, then $W(x)$ and $W(x)^{-1}$ must be positive definite a.e.",
"and have locally integrable entries.",
"For Type 2 matrix power functions, it is straightforward to characterize when this occurs.",
"Theorem 4.1 Let $W$ be an $n \\times n$ Type 2 matrix power function with unitary $U(x)$ and associated eigenvalues $\\alpha _i |x|^{\\gamma _i}$ .",
"Then $W(x)$ is positive definite a.e.",
"and has locally integrable entries if and only if $\\gamma _i > -1$ and $\\alpha _i >0$ for $ 1 \\le i \\le n$ .",
"Let $u_{ij}(x)$ denote the $ij^{th}$ entry of $U(x)$ and let $w_{ij}(x)$ denote the $ij^{th}$ entry of $W(x)$ .",
"Then multiplying out the matrices in (REF ) gives $w_{ij}(x) = \\sum _{k=1}^n \\alpha _k|x|^{\\gamma _k} u_{ik}(x)\\overline{u_{jk}(x)}.$ ($\\Rightarrow $ ) Suppose $W(x)$ is positive definite a.e.",
"with locally integrable entries.",
"Because the eigenvalues of $W(x)$ are $\\alpha _i |x|^{\\gamma _i}$ , it follows that each $\\alpha _i$ is positive.",
"By assumption, each $w_{ii}(x)$ is locally integrable, so the following sum is locally integrable $\\begin{aligned}\\sum _{i=1}^n w_{ii}(x) &= \\sum _{i=1}^n\\sum _{k=1}^n \\alpha _k|x|^{\\gamma _k} u_{ik}(x)\\overline{u_{ik}(x)}\\\\&= \\sum _{k=1}^n\\alpha _k|x|^{\\gamma _k} \\left( \\sum _{i=1}^n |u_{ik}(x)|^2 \\right) \\\\&= \\sum _{k=1}^n \\alpha _k|x|^{\\gamma _k}.\\end{aligned}$ Local integrability implies that $ \\int _I \\left(\\sum _{k=1}^n \\alpha _k|x|^{\\gamma _k} \\right) dx = \\sum _{k=1}^n \\alpha _k\\int _I|x|^{\\gamma _k}dx < \\infty ,$ for all finite intervals $I$ .",
"Since each $\\alpha _k$ is positive, we can conclude that each $\\gamma _k> -1$ .",
"Otherwise, inequality (REF ) would fail for intervals containing zero.",
"($\\Leftarrow $ ) Now, assume that each $ \\gamma _i>-1$ and each $\\alpha _i >0$ .",
"As $W(x)$ is self-adjoint and the eigenvalues of $W(x)$ are $\\alpha _i |x|^{\\gamma _i}$ , it follows that $W(x)$ is positive definite a.e.",
"To see that each $w_{ij}(x)$ is locally integrable, observe that $ |w_{ij}(x)| = \\left|\\sum _{k=1}^n \\alpha _k|x|^{\\gamma _k} u_{ik}(x)\\overline{u_{jk}(x)} \\right| \\le \\ \\sum _{k=1}^n\\alpha _k |x|^{\\gamma _k}, $ where we used the fact that since $U(x)$ is unitary, each $|u_{ik}(x) |\\le 1.$ As each $ \\gamma _i>-1$ implies each function in the final sum is locally integrable, each $w_{ij}(x)$ is locally integrable as well.",
"This result has implications for characterizing when Type 2 matrix power functions are also $A_2$ matrix weights.",
"Remark 4.2 If $W$ is both a Type 2 matrix power function and an $A_2$ matrix weight, then both $W$ and $W^{-1}$ are positive definite a.e.",
"with locally integrable entries.",
"Then Theorem REF implies that the coefficients $\\alpha _i$ are positive and the powers $\\gamma _i$ satisfy $-1< \\gamma _i < 1.$ It is worth noting that results similar to Theorem REF , with more restrictive unitary matrices and more general eigenvalues, are discussed by Bloom in [5].",
"Now we consider a specific example that shows that Type 2 matrix power functions can satisfy the necessary conditions from Remark REF but fail to be $A_2$ matrix weights.",
"In this example, the unitary $U(x)$ is the standard two dimensional rotation matrix.",
"In [5], [6], Bloom studied a variant of this example when $\\gamma _2 = -\\gamma _1$ and determined that, by certain criteria, the weight is well behaved.",
"In contrast, we show that this matrix weight almost always fails to be an $A_2$ matrix weight.",
"Example 4.3 Define the matrix weight $W(x) = \\begin{bmatrix}\\cos x & -\\sin x \\\\\\sin x & \\cos x \\\\\\end{bmatrix} \\begin{bmatrix}|x|^{\\gamma _1} & 0 \\\\0 & |x|^{\\gamma _2} \\\\\\end{bmatrix}\\begin{bmatrix}\\cos x & \\sin x \\\\-\\sin x & \\cos x \\\\\\end{bmatrix}.$ Then $W$ is an $A_2$ matrix weight if and only if $\\gamma _1 = \\gamma _2$ and $-1 < \\gamma _1 < 1$ .",
"($\\Rightarrow $ ) We prove this by contrapositive and assume $\\gamma _1 \\ne \\gamma _2$ or $\\gamma _1$ does not satisfy the given inequality.",
"Clearly, if $\\gamma _1\\ge 1$ or $\\gamma _1 \\le -1$ , then Theorem REF implies that $W$ does not have locally integrable entries and so, is not an $A_2$ matrix weight.",
"Thus, we need only show that if $\\gamma _1 \\ne \\gamma _2$ , then $W$ is not an $A_2$ matrix weight.",
"By Lemma $3.6$ in [25], if $W$ is an $A_2$ matrix weight, then the diagonal entries of $W$ are scalar $A_2$ weights.",
"So, to show that $W$ is not an $A_2$ matrix weight, we need only show that one of its diagonal entries, either $w_{11}(x)$ or $w_{22}(x)$ , fails to be a scalar $A_2$ weight.",
"Without loss of generality, assume $\\gamma _2 > \\gamma _1.$ In this situation, we claim $w_{11}(x) = |x|^{\\gamma _1}\\cos ^2x+|x|^{\\gamma _2}\\sin ^2x$ is not a scalar $A_2$ weight.",
"If instead $\\gamma _1> \\gamma _2$ , one can apply identical arguments to show $w_{22}(x)$ is not a scalar $A_2$ weight.",
"To show $w_{11}(x)$ is not an $A_2$ weight, define the interval $I_n = [2\\pi n, 2\\pi n + \\pi ]$ for each $n \\in \\mathbb {N}.$ We will show $\\lim _{n \\rightarrow \\infty } \\left(\\frac{1}{|I_n|}\\int _{I_n} w_{11}(x) \\ dx\\right) \\left(\\frac{1}{|I_n|}\\int _{I_n} \\frac{1}{w_{11}(x)} \\ dx \\right)= \\infty .$ Choosing $n$ large enough, using the definition of $I_n$ , and restricting to an interval where $\\sin x \\ge \\frac{1}{\\sqrt{2}}$ gives $\\begin{aligned}\\frac{1}{|I_n|}\\int _{I_n} w_{11}(x) \\ dx &= \\frac{1}{\\pi }\\int _{2\\pi n}^{2\\pi n + \\pi } w_{11}(x) \\ dx \\\\&\\ge \\frac{1}{\\pi }\\int _{2\\pi n + \\frac{\\pi }{4}}^{2\\pi n + \\frac{3\\pi }{4}} |x|^{\\gamma _2}\\sin ^2x \\ dx\\\\&\\ge \\frac{1}{2\\pi }\\int _{2\\pi n + \\frac{\\pi }{4}}^{2\\pi n + \\frac{3\\pi }{4}} |x|^{\\gamma _2} \\ dx\\\\&\\gtrsim n^{\\gamma _2},\\end{aligned}$ where the implied constant does not depend on $n$ .",
"Define $\\epsilon _n = n^{(\\gamma _1-\\gamma _2)/2}$ , where $n$ is large enough so $\\epsilon _n < \\pi $ .",
"For $n$ large enough, we always have $\\epsilon _n<\\pi $ because $\\gamma _1 - \\gamma _2 < 0$ .",
"Furthermore, by the periodicity of $\\sin x$ and the fact that $|\\sin x | \\le |x|$ , we have $w_{11}(x) = |x|^{\\gamma _1}\\cos ^2x+|x|^{\\gamma _2}\\sin ^2x \\le |x|^{\\gamma _1} + |x|^{\\gamma _2}|x-2\\pi n|^2.$ Restricting $I_n$ to a smaller interval, we have $\\begin{aligned}\\frac{1}{|I_n|}\\int _{I_n} \\frac{1}{w_{11}(x)} \\ dx &\\ge \\frac{1}{\\pi }\\int _{2\\pi n}^{2\\pi n + \\epsilon _n} \\frac{dx}{|x|^{\\gamma _1}\\cos ^2x+|x|^{\\gamma _2}\\sin ^2x}\\\\&\\ge \\frac{1}{\\pi }\\int _{2\\pi n}^{2\\pi n + \\epsilon _n} \\frac{dx}{|x|^{\\gamma _1} + |x|^{\\gamma _2}|x-2\\pi n|^2}\\\\&\\approx \\frac{1}{\\pi }\\int _{2\\pi n}^{2\\pi n + \\epsilon _n} \\frac{dx}{|2\\pi n|^{\\gamma _1} + |2\\pi n |^{\\gamma _2}\\epsilon _n^2}\\\\&\\gtrsim \\int _{2\\pi n}^{2\\pi n + \\epsilon _n} \\frac{dx}{n^{\\gamma _1}+n^{\\gamma _2}n^{\\gamma _1-\\gamma _2}}\\\\&\\approx n^{-\\gamma _1}\\epsilon _n\\\\&= n^{\\frac{-\\gamma _1-\\gamma _2}{2}},\\end{aligned}$ where the implied constant depends on $\\gamma _1$ and $\\gamma _2$ but not on $n$ .",
"Thus for large $n$ , $\\left(\\frac{1}{|I_n|}\\int _{I_n} w_{11}(x) \\ dx\\right) \\left(\\frac{1}{|I_n|}\\int _{I_n} \\frac{1}{w_{11}(x)} \\ dx\\right) \\gtrsim (n^{\\gamma _2})(n^{\\frac{-\\gamma _1-\\gamma _2}{2}}) = n^{\\frac{\\gamma _2-\\gamma _1}{2}}.$ As $\\gamma _2 - \\gamma _1>0$ , we can conclude $\\lim _{n\\rightarrow \\infty } n^{\\frac{\\gamma _2-\\gamma _1}{2}} = \\infty ,$ and so $w_{11}(x)$ is not a scalar $A_2$ weight.",
"Thus, $W$ is not an $A_2$ matrix weight.",
"($\\Leftarrow $ ) For the other direction, assume $\\gamma _1=\\gamma _2$ and $-1 < \\gamma _1 <1.$ Then $|x|^{\\gamma _1}$ is a scalar $A_2$ weight and $ W(x) = |x|^{\\gamma _1} I_{n\\times n}$ is a matrix weight.",
"Furthermore, for each interval $I$ , $ \\text{Tr }\\left( \\left\\langle W \\right\\rangle _I \\left\\langle W^{-1} \\right\\rangle _I \\right) = \\text{Tr}\\left( \\langle |x|^{\\gamma _1} \\rangle _I \\langle |x|^{-\\gamma _1} \\rangle _I I_{n \\times n} \\right) \\le n [|x|^{\\gamma _1}]_{A_2}.$ As this bound is independent of the interval $I$ , $W$ is an $A_2$ matrix weight.",
"Remark 4.4 One can extend Example REF to higher dimensions by observing that $W$ is built using a unitary matrix, which is a rotation matrix with angle $x$ , and a diagonal matrix, whose entries are powers of $|x|$ .",
"To extend to three dimensions, recall that the basic three dimensional rotation matrices are the following $ R_x(\\theta ) =\\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & \\cos \\theta & \\sin \\theta \\\\0 &- \\sin \\theta & \\cos \\theta \\end{bmatrix} \\ \\ R_y(\\theta ) =\\begin{bmatrix} \\cos \\theta & 0 & -\\sin \\theta \\\\ 0 & 1 & 0 \\\\\\sin \\theta & 0 & \\cos \\theta \\end{bmatrix}\\ \\ R_z(\\theta ) =\\begin{bmatrix} \\cos \\theta & \\sin \\theta & 0 \\\\ -\\sin \\theta & \\cos \\theta & 0 \\\\0 & 0 & 1 \\end{bmatrix}$ where $R_{x}(\\theta )$ , $R_{y}(\\theta )$ and $R_z(\\theta )$ respectively rotate a vector around the $x$ -, $y$ -, or $z$ -axis by an angle $\\theta $ .",
"Every three dimensional rotation can be written as a product $R(\\alpha , \\beta , \\gamma )\\equiv R_x(\\alpha )R_y(\\beta ) R_z(\\gamma ) $ , where $\\gamma , \\beta , \\alpha $ are the Euler angles of the rotation.",
"More specifically, a general rotation matrix in three dimensions can be represented as $R(\\alpha , \\beta , \\gamma ) = \\begin{bmatrix}\\cos \\beta \\cos \\gamma & \\cos \\beta \\sin \\gamma & -\\sin \\beta & \\\\\\sin \\alpha \\sin \\beta \\cos \\gamma -\\cos \\alpha \\sin \\gamma & \\sin \\alpha \\sin \\beta \\sin \\gamma + \\cos \\alpha \\cos \\gamma & \\sin \\alpha \\cos \\beta \\\\\\cos \\alpha \\sin \\beta \\cos \\gamma + \\sin \\alpha \\sin \\gamma &\\cos \\alpha \\sin \\beta \\sin \\gamma - \\sin \\alpha \\cos \\gamma & \\cos \\alpha \\cos \\beta \\end{bmatrix}.$ For details, see [13], pp.",
"$59.$ To define a matrix function $W$ , we need to specify a unitary and diagonal matrix.",
"By setting $\\alpha = \\beta = \\gamma = x$ , we obtain the following three dimensional rotation matrix: $ U(x) \\equiv \\begin{bmatrix}\\cos ^2x &\\cos x \\sin x & -\\sin x \\\\\\cos x \\sin ^2 x - \\cos x \\sin x & \\cos ^2x + \\sin ^3 x & \\cos x \\sin x \\\\\\cos ^2 \\sin x + \\sin ^2 x &\\cos x \\sin ^2x - \\cos x\\sin x & \\cos ^2 x\\end{bmatrix}.$ Mirroring Example REF , define the Type 2 matrix power function $ W(x) = U(x) \\begin{bmatrix} |x|^{\\gamma _1} & 0 & 0 \\\\ 0 & |x|^{\\gamma _2} & 0 \\\\ 0 & 0 & |x|^{\\gamma _3} \\end{bmatrix} U^*(x).",
"$ Then, one can use arguments very similar to those in Example REF to show that $W$ is an $A_2$ matrix weight if and only if $\\gamma _2 = \\gamma _2 = \\gamma _3$ and $-1 < \\gamma _1 <1.$"
]
] | 1605.06071 |
[
[
"Multifractal analysis of three-dimensional grayscale images:\n Characterization of natural porous structures"
],
[
"Abstract A multifractal analysis (MFA) is performed on three-dimensional grayscale images associated with natural porous structures (soil samples).",
"First, computed tomography (CT) scans are carried out on the samples to generate 3D grayscale images.",
"Then, a preliminary analysis is conducted to evaluate key quantities associated with the porosity, such as void fraction, pore volume, connectivity, and surface area.",
"Finally, the samples are successfully identified and separated into two different structure families by using the MFA.",
"A new software (Munari) to carry out the MFA of 3D grayscale images is also presented."
],
[
"Multifractal analysis of three-dimensional grayscale images: Characterization of natural porous structures Lorenzo Milazzo$^1$ and Radoslaw Pajor$^2$ $^1$Edinburgh, UK email: [email protected] $^2$Nottingham, UK email: [email protected] Abstract A multifractal analysis (MFA) is performed on three-dimensional grayscale images associated with natural porous structures (soil samples).",
"First, computed tomography (CT) scans are carried out on the samples to generate 3D grayscale images.",
"Then, a preliminary analysis is conducted to evaluate key quantities associated with the porosity, such as void fraction, pore volume, connectivity, and surface area.",
"Finally, the samples are successfully identified and separated into two different structure families by using the MFA.",
"A new software (Munari) to carry out the MFA of 3D grayscale images is also presented.",
"1.",
"Introduction Several techniques can be used to obtain representations of complex structures through 3D images.",
"Examples of 3D images are the computed tomography (CT) images.",
"Typically, a 3D image is constituted by a stack of 2D images.",
"Characterization of natural porous structures can contribute to better understand the physical processes in soil, in particular the transport processes.",
"In general, the properties of these structures and the variables associated with them exhibit spatial irregularity.",
"If the irregularity on the distribution of a given variable remains statistically similar at different scales, the variable is assumed to be self-similar.",
"Thus, exploring self-similarity and scale invariance in natural porous structures can provide insights into the nature of the spatial variability of the soil properties.",
"The aim of the present paper is to present the multifractal analysis (MFA) of 3D grayscale images of soil samples and, more generally, to explore the possibility to use this method for the classification of 2D/3D complex structures.",
"2.",
"Multifractal Analysis (MFA): Theory and Methods Geometric (mono)fractals are self-similar sets of points.",
"Multifractals (or multifractal measures) are self-similar measures defined on specific set of points [Baveye et al.",
"2008; Evertsz et al.",
"1992; Falconer 2003].",
"In general, the former can be generated by additive processes and the latter by multiplicative cascade of random processes.",
"The generalized fractal dimension $D_{q}$ , which is closely related to the Rényi entropy [Rényi 1961], provides a direct measurement of the fractal properties of an object – several values of the momentum order $q$ correspond to well-known generalized dimensions, such as the capacity dimension (box-counting dimension) $D_{0}$ , the information dimension $D_{1}$ , and the correlation dimension $D_{2}$ .",
"The singularity spectrum $f(\\alpha )$ provides information about the scaling properties of the structure [Halsey et al.",
"1986, Hentschel et al.",
"1983; Lévy Véhel 1998; Lowen et al.",
"2005; G. Paladin et al.",
"1987; Theiler 1990].",
"The generalized fractal dimension is defined by: $D_{q} = \\lim _{l \\rightarrow 0} \\frac{1}{q-1} \\frac{\\ln {\\sum _{i=1}^{N(l)} p_{i}^{q}(l)}}{\\ln {l}} $ where $p_{i}(l)$ is the integrated measure associated with the i-th box, $q$ is the momentum order, and $N(l)$ is the number of boxes of linear size $l$ .",
"The integrate measure is the concentration of the variable of interest in a given box relative to the whole system and it represents a probability – e.g.",
"in the case of the mass, the integrate measure is the mass probability.",
"Once $D_{q}$ is known, the singularity spectrum $f(\\alpha )$ can be evaluated via a Legendre transformation: $f(\\alpha (q)) = q \\alpha (q) - \\tau (q), \\hspace*{28.45274pt} \\alpha (q) = \\frac{d\\tau (q)}{dq}$ where $\\alpha $ is the singularity strength and $\\tau (q) = (q-1)D_{q}$ [Halsey et al.",
"1986].",
"The singularity spectrum can also be directly (without knowing $D_{q}$ ) evaluated by using the method proposed by Chhabra et al.",
"[1989].",
"The first step of this approach consists of defining a family of normalized measures $\\mu (q)$ : $\\mu _{i}(q,l) = \\frac{[p_{i}(l)]^{q}}{\\sum _{j=1}^{N(l)}[p_{j}(l)]^{q}} $ For each box i, the normalized measure $\\mu _{i}(q,l)$ depends on the order of the statistical moment and on the box size and it takes values in the range [0,1] for any value of $q$ .",
"Then, the two functions $f(q)$ and $\\alpha (q)$ are evaluated: $f(q) & = & \\lim _{l \\rightarrow 0} \\frac{\\sum _{i=1}^{N(l)}\\mu _{i}(q,l) \\ln \\mu _{i}(q,l)}{\\ln l} \\\\ \\alpha (q) & = & \\lim _{l \\rightarrow 0} \\frac{\\sum _{i=1}^{N(l)}\\mu _{i}(q,l) \\ln p_{i}(l)}{\\ln l} $ where $\\alpha (q)$ is the average value of the singularity strength $\\alpha _{i} = \\ln p_{i}(l) / \\ln l$ .",
"For each $q$ , values of $f(q)$ and $\\alpha (q)$ are obtained from the slope of plots of $\\sum _{i=1}^{N(l)}\\mu _{i}(q,l) \\ln \\mu _{i}(q,l)$ versus $(\\ln l)$ and $\\sum _{i=1}^{N(l)}\\mu _{i}(q,l) \\ln p_{i}(l)$ versus $(\\ln l)$ over the entire range of box size values under consideration.",
"Finally, the singularity spectrum $f(\\alpha )$ is constructed from these two data sets.",
"3.",
"Materials and Methods The samples considered in this study are part of a larger set used by Harris et al.",
"[2003] and Pajor et al.",
"[2010] to analyse fungal colony spread through soil pore space.",
"The soil used to prepare the samples was a sandy loam soil from an experimental site at the Scottish Crop Research Institute (now James Hutton Institute), Invergowrie Dundee UK.",
"First, the material was air-dried and sieved to obtain aggregates with diameter of 1-2 mm; then, it was sterilized and packed into PVC rings with densities ranging from 1.2 g/cm3 to 1.6 g/cm3.",
"The 3D volumes representing the soil samples were obtained by using the X-Tek (Metris) X-ray microtomography system.",
"All samples were scanned with the same settings at 160 kV, 201 $\\mu $ A, with 0.1 mm Al filter in front of the X-ray gun with tungsten target.",
"In total, 3003 angular projections (2D radiographs) were collected at 4 frames per second.",
"The 2D radiographs were reconstructed into a 3D volume by using the CT Pro software; to perform this operation, a voxel size of 30 $\\mu $ m was chosen.",
"The reconstructed volumes were then rendered and converted to a stack of grayscale TIFF images by using the VG Studio Max software.",
"By using the Fiji software [Schindelin et al.",
"2012], the 3D images (stacks of single-voxel thick, 8-bit images) were first cropped to $128\\times 128\\times 128$ voxels and then treated with median filter (radius = 2.0) and thresholded via a procedure based on the Ridler-Calvard method [Ridler et al.",
"1978].",
"Finally, 3D grayscale images in ASCII format were generated from the stacks of TIFF images.",
"The measurements of void fraction, pore volume, connectivity, and surface area of the material were carried out by using the Fiji/ImageJ software and its plugin BoneJ [Doube et al.",
"2010].",
"If an image is in grayscale format, the pixel values lay in the range [0,255] and can be considered as measures of mass (0 = black = no mass; 255 = white = mass).",
"Since the porous structures under analysis are represented by grayscale images, we also introduced the following metric $\\phi ^{gs}$ to characterize the porosity of the system: $\\phi ^{gs} = 1-((\\text{sum of pixel values})/\\text{sum of pixel values if no void in the sample})) $ It is important to highlight the limitations of this quantity: $\\phi ^{gs}$ is expected to be more effective in the case of grayscale images associated with high heterogeneous structures containing large pores.",
"A program – Munari – [Milazzo 2010] has been developed to perform MFA of 2D and 3D grayscale images and, in particular, to directly evaluate the singularity spectrum $f(\\alpha )$ by using the Chhabra method.",
"The application is written in C++ and, at this stage, it processes grayscale images in ASCII format.",
"The algorithms within Munari have been designed to be general and, as a result, they are independent from the resolution of the images and from the values of box sizes and moment orders used within the Chhabra method.",
"Several validation tests have been implemented to ensure reliability and stability of the application; one of them is based on the analysis of 3D synthetic images representing multifractal lattices generated by using a random multiplicative process [Milazzo 2013].",
"In this study, the Munari software was also used to evaluate the porosity-related metric $\\phi ^{gs}$ .",
"4.",
"Results and Discussion In this study, based on the analysis carried out by Pajor et al.",
"[2010], we focus on a set of eight samples.",
"The samples S3DXX can be grouped into two families: four of them (`family 1': S3D01, S3D02, S3D03, S3D04) have a density equal to 1.3 g/cm3, the other four (`family 2': S3D05, S3D06, S3D07, S3D08) a density equal to 1.6 g/cm3.",
"There are significant differences in characteristics of pore geometry between the two families.",
"Samples from `family 1' have a higher volume of pores that are thicker and better connected than those present in the structures grouped in `family 2', whereas samples from `family 2' have a higher surface area and higher number of micro and mesopores.",
"The measured values of porosity ($\\phi $ , [decimal fraction]), porosity-related metric ($\\phi ^{gs}$ , [decimal fraction]), pore volume ($V_{V}$ , [mm3]), connectivity ($C$ , [%]), and surface area ($S$ , [mm2]) for the samples S3DXX are shown in Table REF and REF .",
"Note that, because of how it is defined, $\\phi ^{gs}$ is more effective in characterizing the porous structures of the `family 1' than those of the `family 2'.",
"Table: Key quantities associated with the porosity for the samples S3D01-04.",
"Table: Key quantities associated with the porosity for the samples S3D05-08.",
"MFA is one of the methods used to characterize and, in particular, to classify natural porous structures.",
"The characterization can be performed on either binary or grayscale images of the system [Dathe et al.",
"2006; Lafond et al.",
"2012; Posadas et al.",
"2003; San José Martínez et al.",
"2010; Tarquis et al.",
"2009; Zhou et al.",
"2011].",
"The classification can be achieved by analysing the curves associated with either the singularity spectrum $f(\\alpha )$ or the generalized fractal dimension $D_{q}$ .",
"In general, by using this approach, two or more groups of complex structures can be classified according the following main features: a) the width of the spectrum ($\\alpha _{max} −- \\alpha _{min}$ ); b) the position of the maximum of the spectrum; and c) the ratio between the information dimension and the capacity dimension, $D_{1}/D_{0}$ [Biswas et al.",
"2012; Reljin et al.",
"2002].",
"In this study, we focus on the singularity spectrum and on its width.",
"The singularity strength $\\alpha $ is a local scaling index and the singularity spectrum $f(\\alpha )$ represents the frequency of the occurrence of a certain value of the singularity strength – in other words, $f(\\alpha )$ captures how frequently a value of the local scaling index is found [Halsey et al.",
"1986; G. Paladin et al.",
"1987; Theiler 1990].",
"Moreover, $\\alpha $ corresponds to the asymptotic behaviour of the coarse singular exponent (coarse Hölder exponent) $\\alpha = \\ln \\mu / \\ln l$ [Evertsz et al.",
"1992]; this exponent represents the crowding or the degree of concentration of the measure: the greater this value is, the smaller is the concentration of the measure and vice versa [San José Martínez et al.",
"2010].",
"The width of the spectrum ($\\alpha _{max} −- \\alpha _{min}$ ) is related to the heterogeneity of the local scaling index $\\alpha $ ; in the case of porous structures, it provides information on the scaling diversity associated with the distribution of the pores.",
"Each 3D image (image stack) represents a finite volume of linear size $L=128$ [pixels].",
"The image stacks were processed by using six values of box sizes: $l = \\lbrace 2, 4, 8, 16, 32, 64\\rbrace $ .",
"For these complex structures, we evaluated the singularity spectrum $\\alpha (q)$ by adopting the set of 21 values of momentum orders: $q~=~\\lbrace -5.0, -4.5, -4.0, \\dots ,$ $4.0, 4.5, 5.0\\rbrace $ .",
"Note that, for $q > 1$ , the normalized measure $\\mu (q)$ (Eq.",
"REF ) amplifies the more singular regions of the measure; while for $q < 1$ , it amplifies the less singular regions [Chhabra et al.",
"1989; San José Martínez et al.",
"2010].",
"We also analysed the scaling behaviour for various values of $q$ .",
"Fig.",
"REF and REF show the quantity $\\sum _{i=1}^{N(l)}\\mu _{i}(q,l) \\ln \\mu _{i}(q,l)$ versus $(\\ln l)$ for $q=-3$ .",
"In the case of the samples S3D01-04 (Fig.",
"REF ), most of the normalized measures $\\mu (q)$ are undefined for $q<0$ ; as a result, a fractal behaviour is not detectable by using this method and the singularity spectra were not evaluated.",
"Conversely, fractality is observed in the case of the samples S3D05-08 (Fig.",
"REF ).",
"In Fig.",
"REF , a degradation of the scaling can be noted for the sample S3D07; this effect, especially for negative values of $q$ , has been already observed by other authors [Chhabra et al.",
"1989; Dathe et al.",
"2006].",
"Thus, for this sample, we reduced the scaling range under analysis in order to reduce the standard error of the regression (Chhabra method) and to improve the reliability of the singularity spectrum.",
"The singularity spectra for the samples S3D05-08 are shown in Fig.",
"REF .",
"As expected, the curves $f(\\alpha )$ are convex with a single maximum at $q=0$ ($\\alpha =3.01$ ) and with infinite slope at $q=\\mp \\infty $ [Halsey et al.",
"1986].",
"The four samples have almost the same width of the spectrum ($\\alpha _{max} −- \\alpha _{min}$ ), suggesting that there are similarities in their distributions of the pores.",
"Thus, the initial classification (`family 1' and `2') is confirmed by the MFA: a) the two groups of samples are different because fractality is observed only for the second group; b) the samples from family 2 are characterized by similar singularity spectra.",
"5.",
"Conclusions The aim of this study has been to investigate the use of the MFA as a tool for characterization of complex structures.",
"To our knowledge this is the first MFA performed to 3D porous structures, based on 3D grayscale images.",
"In the case of grayscale images, all the studies that we are aware of have been conducted on 2D cross-sections of the samples (2D images).",
"Eight soil samples have been selected for CT scan imaging.",
"Afterward, by using the MFA, they have been successfully identified and separated into two different structure families.",
"In order to carry out the MFA of 3D grayscale images, a new software for image processing (Munari) has been developed.",
"Several key implications have emerged from the analysis.",
"In general, a degradation of the scaling is expected to be observed, especially for negative values of the momentum order.",
"If this is the case, it is recommended to carefully choose the scaling range used for the calculations.",
"A better study of the scaling could be performed and more reliable singularity spectra could be obtained if 3D images at higher resolution are processed and analysed – however, note that this would introduce considerable challenges for the data management due to the number and size of the image files.",
"Finally, the choice of a larger number of different types of structures to be compared would allow to further explore the potentialities of the MFA as a classification method.",
"Acknowledgements The soil samples were collected from an experimental site at the Scottish Crop Research Institute (now James Hutton Institute), Invergowrie Dundee UK.",
"The 3D grayscale images were acquired by using the CT scanner at the SIMBIOS Centre, Abertay University, Dundee UK.",
"Author Contributions Conceived and designed the study: LM and RP.",
"Prepared the samples: RP.",
"Performed the CT scanning and image generation: RP.",
"Analyzed the data: LM and RP.",
"Designed and developed the Munari software used for the MFA: LM.",
"Performed the MFA: LM.",
"Wrote the paper: LM and RP.",
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"A 7, 1055–1073 (1990) H. Zhou et al., Multifractal analyses of grayscale and binary soil thin section images, Fractals 19(3), 299–309 (2011) VG Studio Max (software), Volume Graphics GmbH, Heidelberg, Germany"
]
] | 1605.06009 |
[
[
"J1216+0709 : A radio galaxy with three episodes of AGN jet activity"
],
[
"Abstract We report the discovery of a `Triple-Double Radio Galaxy (TDRG)' J1216+0709 detected in deep low-frequency Giant Metrewave Radio Telescope (GMRT) observations.",
"J1216+0709 is only the third radio galaxy, after B0925+420 and Speca, with three pairs of lobes resulting from three different episodes of AGN jet activity.",
"The 610 MHz GMRT image clearly displays an inner pair of lobes, a nearly co-axial middle pair of lobes and a pair of outer lobes that is bent w.r.t.",
"the axis of inner pair of lobes.",
"The total end-to-end projected sizes of the inner, middle, and outer lobes are 40$^{{\\prime}{\\prime}}$ ($\\sim$ 95 kpc), 1$^{\\prime}$.65 ($\\sim$ 235 kpc) and 5$^{\\prime}$.7 ($\\sim$ 814 kpc), respectively.",
"Unlike the outer pair of lobes both the inner and middle pairs of lobes exhibit asymmetries in arm-lengths and flux densities, but in opposite sense, i.e., the eastern sides are farther and also brighter that the western sides, thus suggesting the possibility of jet being intrinsically asymmetric rather than due to relativistic beaming effect.",
"The host galaxy is a bright elliptical (m$_{\\rm r}$ $\\sim$ 16.56) with M$_{\\rm SMBH}$ $\\sim$ 3.9 $\\times$ 10$^{9}$ M$\\odot$ and star-formation rate of $\\sim$ 4.66$_{\\rm -1.61}^{\\rm +4.65}$ M$_{\\odot}$ yr$^{-1}$.",
"The host galaxy resides is a small group of three galaxies (m$_{\\rm r}$ $\\leq$ 17.77) and is possibly going through the interaction with faint, dwarf galaxies in the neighbourhood, which may have triggered the recent episodes of AGN activity."
],
[
"Introduction",
"Radio galaxies, a subclass of Active Galactic Nuclei (AGN), are powerful radio emitters and typically exhibit radio morphology that consists of a core producing a pair of bipolar collimated jets terminating in the form of lobes.",
"Therefore, detection of `core-jet-lobe' radio morphology is a clear indication of AGN activity.",
"In fact, radio morphological structure and spectral properties can be used to probe the history of AGN activity.",
"Morphological studies of radio galaxies have been useful in understanding the precession or change of jet axis in X-shaped radio galaxies, effect of the motion of host galaxies in bent radio galaxies, and intermittent AGN activity in radio galaxies showing two pairs of lobes [11], [32].",
"In recent years, there have been attempts to understand the details of recurrent AGN activity in galaxies by studying radio galaxies exhibiting two pairs of lobes that are formed during two different phases of AGN activity [16], [22].",
"These galaxies are generally termed as `Double-Double Radio Galaxies (DDRGs)' in which a new pair of radio lobes is seen closer to nucleus, before distant and old pair of radio lobes fades away [33].",
"The new pair of lobes are edge brightened, and therefore, it can easily be distinguished from knots in the jets.",
"Interestingly, despite the identification of thousands of radio galaxies in various radio surveys only few dozens are confirmed DDRGs [28], possibly due to lack of sensitivity and resolution such that radio structures at different spatial scales remain undetected.",
"Low-frequency sensitive GMRT observations with the resolution of few arcsecond are well suited to detect both steep spectrum, low-surface-brightness radio emission from the old pair of lobes as well as kpc-scale emission from the new pair of lobes.",
"In this paper we report the discovery of rare `Triple-Double Radio Galaxy (TDRG)' named as J1216+0709 that displays three pairs of lobes in the 610 MHz GMRT image.",
"This is only the third TDRG reported so far after B0925+420 [5] and Speca [14].",
"This radio galaxy is either undetected or poorly detected in previous radio surveys ($e.g.,\\;$ Faint Images of the Radio Sky at Twenty-cm [4], NRAO VLA Sky Survey [8], VLA Low-frequency Sky Survey [9], and TIFR-GMRT Sky Survey [15] due to the lack of optimum sensitivity and resolution.",
"The host galaxy has been identified in the Sloan Digital Sky Survey [1] as an early type galaxy at RA (J2000) = 12$^{\\rm h}$ 16$^{\\rm m}$ 32$^{\\rm s}$ .42 and DEC (J2000) = +07$^{\\circ }$ 09$$ 55$$ .8 with a spectroscopically measured redshift (z) = 0.136.",
"The cosmological parameters that we adopt are H$_{\\rm 0}$ = 71 km s$^{-1}$ Mpc$^{-1}$ , ${\\Omega }_{\\rm M}$ = 0.27 and ${\\Omega }_{\\rm vac}$ = 0.73.",
"Using this cosmology, 1 arcsec corresponds to 2.381 kpc at the luminosity distance of $\\sim $ 633.8 Mpc for J1216+0709."
],
[
"GMRT observations and data reduction",
"The Radio galaxy J1216+0709 was observed with the GMRT at 610 MHz on 20 June 2012 and at 325 MHz on 23 April 2016.",
"During our GMRT observations, we used the full array of 30 antennas and the software backend with receiver bandwidth of 32 MHz subdivided into 256 channels.",
"During the 610 MHz observations the target field centered at NGC 4235 (RA (J2000) = 12$^{\\rm h}$ 17$^{\\rm m}$ 09$^{\\rm s}$ .9 and DEC = +07$^{\\circ }$ 11$$ 30$$ ) was observed for nearly four hours.",
"While, in 325 MHz observations the field was centered on the target source J1216+0709 and was observed for nearly 3.5 hours.",
"The amplitude calibrators were observed for $\\sim $ 20 minutes at the start and/or end of each run, and the phase calibrators were observed for $\\sim $ 5 minutes in every $\\sim $ 40 minutes.",
"GMRT data were reduced and analysed in the standard way using the NRAO Astronomical Image Processing System (AIPS).",
"Calibrated visibilities were Fourier transformed to create radio images by using the `IMAGR' task, and robust weighing scheme, where robust parameter was set to `0' (between uniform and natural weighing) at both frequencies.",
"All the images were self-calibrated and primary beam corrected.",
"Our final maps have noise-rms $\\sim $ 40 $\\mu $ Jy beam$^{-1}$ with synthesized beam-size $\\sim $ 6$$ .2 $\\times $ 4$$ .5 at 610 MHz, and $\\sim $ 160 $\\mu $ Jy beam$^{-1}$ with synthesized beam-size $\\sim $ 11$$ .4 $\\times $ 8$$ .5 at 325 MHz.",
"More details on the data reduction are presented in [20].",
"The total flux densities of different components were measured using the AIPS task `TVSTAT' that allows us to choose an area of any shape.",
"Error on the flux density of a component was obtained by multiplying the average noise-rms to the total area of the component measured in the units of synthesized beams.",
"Flux density of an individual component was obtained by considering the area shown by contours overlaid on to grey-scale image."
],
[
"Radio Morphology",
"J1216+0709 is best imaged at 610 MHz in GMRT observations, while it is either undetected or poorly detected in existing radio surveys.",
"The 610 MHz GMRT image displays an inner pair of lobes, a nearly co-axial middle pair of lobes and a pair of outer lobes that are bent w.r.t.",
"the inner pair of lobes (see figure REF ).",
"The overall radio morphology of the source appears bent in a `C' shaped like structure and resembles somewhat to a Wide Angle Tail (WAT) radio galaxy.",
"The total end-to-end projected sizes of the inner lobes, middle lobes and outer lobes are 40$$ ($\\sim $ 95 kpc), 1$$ .65 ($\\sim $ 235.7 kpc) and 5$$ .7 ($\\sim $ 814 kpc), respectively (see figure REF ).",
"The three distinct pairs of lobes can be interpreted as evidence for the three different episodes of AGN jet activity.",
"The inner and middle pair of lobes show clear edge-brightened structures which distinguish them from knots in the jets.",
"In the 610 MHz image, there is relatively faint bridge-like radio emission connecting the two successive lobes, which tentatively indicates that the newly formed jets propagate outwards possibly through the jet-cocoon structure formed by the previous episode of activity rather than through intergalactic medium.",
"Radio flux densities of different components at different frequencies are given in Table REF .",
"Based on the 610 MHz image, Table REF lists total radio sizes, ratios of flux densities and arm-lengths of the eastern and western side lobes and the luminosities of the three pairs of lobes.",
"We note that for outer pair of lobes, both eastern and western lobes have similar flux densities and distance from the center ($i.e.,\\;$ arm-lengths), which suggests that the outer pair of lobes is lying nearly in the plane of sky.",
"For both the inner and middle pairs of lobes, the eastern side is nearly 2.3 - 2.5 times stronger than the western side $i.e.,\\;$ ratios of flux densities of eastern-to-western lobe for inner and middle pairs are (R$_{\\rm f,~in}$ ) $\\sim $ 2.3 and (R$_{\\rm f,~mid}$ ) $\\sim $ 2.5, respectively (see table REF ).",
"While arm-lengths of the eastern sides are larger in comparison to the western sides $i.e.,\\;$ the ratios of eastern to western sides are (R$_{\\rm l,~in}$ ) $\\sim $ 2.0 to (R$_{\\rm l,~mid}$ ) $\\sim $ 1.6, for the inner and middle pair, respectively.",
"We note that the asymmetry shown by inner and middle doubles in our source is consistent with the general trend found in DDRGs in which the inner doubles tend to be more asymmetric in both its arm-length and its flux density ratios compared to the outer doubles [34].",
"It is worth to note that for both inner and middle doubles the asymmetries in arm-lengths and flux densities are in opposite sense $i.e.,\\;$ the eastern lobes being farther and also brighter that the western lobes.",
"The opposite asymmetry is difficult to explain by simple version of relativistic beaming effect.",
"Also, in radio galaxies the viewing angle is relatively large ($>$ 45$^{\\circ }$ ; [40]), and therefore, relativistic beaming effect is unlikely to be dominant.",
"This strengthens the possibility of jet being intrinsically asymmetric.",
"Indeed, asymmetric jet have been suggested for some of DDRGs reported in the literature [16].",
"In our TDRG the outer pair of lobes are bent w.r.t.",
"the axis of the inner pair of lobes and this can be understood if the outer lobes are entraining into a medium having large-scale density gradients or host galaxy has moved during the two cycles of AGN activity [18].",
"The 610 MHz radio luminosity of the inner, middle and outer pair of lobes are 3.5 $\\times $ 10$^{23}$ W Hz$^{-1}$ , 5.4 $\\times $ 10$^{23}$ W Hz$^{-1}$ and 4.8 $\\times $ 10$^{24}$ W Hz$^{-1}$ , respectively (see table REF ).",
"The higher luminosity of the outer pair of lobes in comparison to the inner pair of lobes is consistent with other cases of TDRGs and DDRGs [35].",
"The total 610 MHz radio luminosity (5.8 $\\times $ 10$^{24}$ W Hz$^{-1}$ ) of our TDRG (with r-band absolute magnitude (M$_{\\rm R}$ ) $\\sim $ -20.69$\\pm $ 0.03 [36]) is close to the separation line between FR I and FR II types [12].",
"Interestingly, the western outer lobe with edge-brightening is similar to FR II, while eastern one bears resemblance to a FR I type.",
"We note that there is no clear detection of the AGN core in both the 610 MHz and 325 MHz images.",
"While core is marginally detected in FIRST image at 2${\\sigma }$ level ($\\sim $ 0.45 mJy).",
"Since our 610 MHz images have typical noise-rms of 0.04 mJy and this gives an upper limit of core flux density of 0.12 mJy at 3$\\sigma $ level.",
"Therefore, AGN radio core exhibits inverted spectral index between 610 MHz to 1.4 GHz (see table REF ).",
"The compact inverted-spectrum AGN core is similar to Giga-Hertz Peaked-spectrum Sources (GPSs) that exhibit peak in their radio Spectra between 1 GHz to 5 GHz and an inverted-shape spectrum at lower frequencies [13].",
"GPSs are interpreted as radio AGN in early phase of their evolution, and therefore, compact inverted-spectrum radio core of TDRG may be considered as an indication of recent AGN activity [31], [29].",
"Indeed, some DDRGs are known to show mildly inverted spectrum of the core [25]."
],
[
"Radio spectrum",
"Figure REF shows spectral index maps between 610 MHz and 1.4 GHz, and between 325 MHz and 610 MHz.",
"We use the task `COMB' in AIPS to create spectral index images after considering flux density values above 2.5$\\sigma $ at both frequencies.",
"The resolutions of the images at two frequencies were matched by convolving the higher resolution image with a Gaussian equivalent to the beam-size of lower resolution image.",
"Spectral index map between 610 MHz (GMRT) and 1.4 GHz (NVSS) shows that the outer lobes have steeper spectral index ($\\alpha $ $\\sim $ - 1.0, where S$_{\\rm {\\nu }}$ $\\propto $ ${\\nu }^{\\alpha }$ ), while the inner region (covering core and inner lobes) have relatively less steep spectral index ($\\alpha $ $>$ - 0.5).",
"325 MHz - 610 MHz spectral index map is of higher spatial resolution and shows all three lobes.",
"We note that, in 325 MHz - 610 MHz spectral index map, the outer lobes have steeper spectral index ($\\alpha $ $\\le $ - 1.0) in compared to the inner and middle lobes.",
"Also, outer edges of the outer lobes have very steep spectral index ($\\alpha $ $\\le $ - 2.0) which is typically seen in relic plasma.",
"There is no signature of relatively compact hotspot-like structures with less steep spectral index, and therefore, suggesting that the supply of jet material has stopped long ago and hotspots in the outer lobes have completely faded away, if they existed."
],
[
"Kinematic age estimates",
"It is important to estimate the time-scales of active and quiescent phase of AGN activity to understand the cause of episodic AGN activity and its duty cycle.",
"Given the lack of multi-frequency radio data for our TDRG we can only constrain the lower limit to the active and quiescent phase time-scales by using kinematic age estimates based on projected radio sizes and a reasonable assumed value for the jet speed.",
"We note that, in general, for a constant jet power the advancement speed of the head of lobe is higher at an early phase of evolution and it decreases (via interaction with the surrounding medium) as the source size increases [2].",
"Based on the previous studies of young as well as evolved radio galaxies, we adopt the average speed of the advancement of the head of outer, middle and inner lobes of our source to be 0.01c, 0.05c and 0.1c, respectively, (where `c' is the speed of light) [24], [25], [3].",
"With these assumed speeds and the projected linear sizes of $\\sim $ 814 kpc, $\\sim $ 235 kpc and $\\sim $ 95 kpc for the outer, middle and inner doubles, respectively, we obtain kinematic ages $\\sim $ 1.3 $\\times $ 10$^{8}$ years, $\\sim $ 7.6 $\\times $ 10$^{6}$ years and $\\sim $ 1.5 $\\times $ 10$^{6}$ years for the outer, middle and inner doubles, respectively.",
"However, we caution that our estimates of kinematic ages are only order-of-magnitude approximation under the simplified assumption that the advancement speed of lobe remains constant.",
"We also attempt to put constraint on the time-scale of quiescent phase between the two AGN episodes.",
"The duration of quiescent phase is equivalent to the time interval between the last jet material being injected in to the outer lobes and the first jet material being ejected from AGN in the next episode.",
"The time elapsed since the last injection of relativistic particle in the jet can be estimated by synchrotron ageing method, which requires modelling of radio spectrum to obtain break frequency that is related to the age of synchrotron emitting plasma.",
"Due to the lack of multi-frequency radio data necessary for spectral modelling we can only put a lower limit to the quiescent phase time-scale (t$_{\\rm q}$ ) by subtracting the kinematic ages of outer (t$_{\\rm out-lobe}$ ) and inner doubles (t$_{\\rm in-lobe}$ ) $i.e.,\\;$ t$_{\\rm q}$ $\\le $ t$_{\\rm out-lobe}$ - t$_{\\rm in-lobe}$ .",
"Therefore, the quiescent phase time-scales between outer (first episode) and middle (second episode) doubles, and the middle (second episode) and inner (third episode) are, $\\le $ 1.2 $\\times $ 10$^{8}$ years and $\\le $ 6.1 $\\times $ 10$^{6}$ years, respectively.",
"These limits on the quiescent phase time-scales are consistent with the time-scale estimates in DDRGs [22].",
"Our proposed multi-frequency GMRT and VLA observations will allow us to put much stronger constraints on the time-scales via spectral aging measurements.",
"Figure: Top panel: The 610 MHz image displaying three pairs of lobes.For better visualization of different lobes the contours represent only bright emission with the lowest contour at >> 6σ\\sigma level.Bottom left panel: Zoom in view of the 610 MHz contours of the inner lobes and 1.4 GHz FIRST--detected core (in Green) overlaid on the SDSS r-band image.Bottom right panel: 325 MHz GMRT image with contours overlaid on to the Grey-scale.",
"Contours are spaced in logarithmic scale withthe lowest contour at 5σ\\sigma level.",
"The synthesized elliptical beam enclosed in a rectangular box is shown at the bottom left corner.Table: Radio propertiesTable: Radio sizes and luminositiesFigure: Top panel : Spectral index map between 610 MHz (GMRT) and 1.4 GHz (NVSS) withNVSS contours overlaid on to it.",
"Bottom panel : Spectral index map between 325 MHz and 610 MHz with 610 MHz contours overlaid on to it."
],
[
"Host galaxy and its large-scale environment",
"Host galaxy and its surrounding large-scale environment can play crucial role in triggering episodic AGN activity, therefore, we examine the nature of host galaxy and its environment.",
"Radio contours overlaid on the SDSS optical image identify the host galaxy as an elliptical galaxy at the redshift of $\\sim $ 0.136 (see figure REF ).",
"Host galaxy is fairly bright (r-band magnitude $\\simeq $ 16.56) and redder in colour (u - r = 2.89).",
"The SDSS optical spectrum is dominated by a red continuum with noticeable 4000 Å break, a characteristic of early-type galaxies.",
"The optical emission line flux ratio diagnostic classifies the host galaxy as a Low Excitation Radio Galaxy (LERG).",
"The Wide-field Infrared Survey Explorer [41] colors of the host galaxy ([3.4] - [4.6] = 0.12 $<$ 0.8 in Vega magnitudes) indicate that the mid-IR emission is dominated by star formation and AGN contamination is not significant [37].",
"We estimate star formation rate (SFR) of $\\sim $ 4.66$_{\\rm -1.61}^{\\rm +4.65}$ M$_{\\odot }$ yr$^{-1}$ in the host galaxy using [19] empirical relation (SFR (M$_{\\odot }$ yr$^{-1}$ ) = 4.5 $\\times $ 10$^{-44}$ L$_{\\rm 8~-~1000~{\\mu }m}$ (erg s$^{-1}$ )) based on IR luminosity.",
"The total IR (8 - 1000 $\\mu $ m) luminosity is estimated from the WISE 22 $\\mu $ m luminosity using the full range of templates in the libraries of [6] and [10].",
"For our TDRG we estimate the mass of the super-massive black hole (M$_{\\rm SMBH}$ ) to be $\\sim $ 3.87 $\\times $ 10$^{9}$ M$\\odot $ by using the `black hole mass -bulge luminosity' relation for early type galaxies given in [26].",
"The absolute bulge magnitude of our source is taken from [36] who present bulge$-$ disc decomposition for SDSS DR7 galaxies.",
"To examine if our TDRG is associated with any cluster or group we use the catalogue of [39] who identified galaxy groups and clusters based on a modified friends-of-friends method and present a flux (m$_{\\rm r}$ $\\le $ 17.77) and volume-limited catalogue using SDSS DR 10 data.",
"The value of flux limit is based on the fact that the SDSS data are incomplete for fainter sources [38].",
"According to [39] catalog, the host galaxy is part of a small group of three galaxies (other two galaxies located at RA (J2000) = 12$^{\\rm h}$ 16$^{\\rm m}$ 39$^{\\rm s}$ , DEC (J2000) = +07$^{\\circ }$ 10${}$ 25${}$ , and RA (J2000) = 12$^{\\rm h}$ 16$^{\\rm m}$ 30$^{\\rm s}$ , DEC (J2000) = +07$^{\\circ }$ 06${}$ 14${}$ ) with a total estimated mass of $\\sim $ 1.99 $\\times $ 10$^{13}$ M$\\odot $ and virial radius of $\\sim $ 0.28 Mpc.",
"There is no apparent disturbance in the host galaxy morphology, and thus, we can rule out any recent major merger.",
"However, a minor merger or a strong interaction with a dwarf galaxy is plausible without resulting any prominent disturbance.",
"In the SDSS image, two faint, blue-color, dwarf galaxy like objects (one at RA (J2000) = 12$^{\\rm h}$ 16$^{\\rm m}$ 31$^{\\rm s}$ .97, DEC (J2000) = +07$^{\\circ }$ 09${}$ 47${}$ .28 with m$_{\\rm r}$ $\\sim $ 23.31$\\pm $ 0.27 but without any estimate of redshift due to faintness, and second one at RA (J2000) = 12$^{\\rm h}$ 16$^{\\rm m}$ 32$^{\\rm s}$ .25, DEC (J2000) = +07$^{\\circ }$ 09${}$ 40${}$ .92 with m$_{\\rm r}$ $\\sim $ 19.67$\\pm $ 0.02 and z$_{\\rm phot}$ $\\sim $ 0.118$\\pm $ 0.0402) are seen close to the southern side of the TDRG host galaxy.",
"The apparent fuzziness and blue color may be an indication of strong interaction that might have triggered a recent star-formation.",
"So, it is possible that both these dwarf galaxies are interacting with TDRG host galaxy which can bring sufficient matter close to the SMBH to trigger AGN activity.",
"However, more sensitive optical observations are required to obtain spectroscopic redshifts of these dwarf galaxies, and to confirm this possibility."
],
[
"Summary",
"We report the discovery of a rare `Triple-Double Radio Galaxy (TDRG)' J1216+0709 that exhibits three distinct pairs of lobes in the 610 MHz GMRT image.",
"This TDRG is only the third such source reported after B0925+420, and Speca, where three pairs of lobes are result of three different episodes of AGN jet activity.",
"The 610 MHz GMRT image exhibits an inner pair of lobes, a nearly co-axial middle pair of lobes and a pair of outer lobes that are bent w.r.t.",
"the inner pair of lobes.",
"The total end-to-end projected sizes of the inner double, middle double and outer double are 40$$ ($\\sim $ 95 kpc), 1$$ .65 ($\\sim $ 235.7 kpc) and 5$$ .7 ($\\sim $ 814 kpc), respectively.",
"We note that unlike the outer pair of lobes both the inner and middle doubles exhibit asymmetries in arm-lengths and flux densities but in opposite sense $i.e.,\\;$ eastern sides are farther and also brighter that the western sides.",
"The opposite asymmetry is difficult to explain by a simple version of relativistic beaming effect and suggests the possibility of jet being intrinsically asymmetric.",
"Also, all three pairs of lobes bear edge-brightened resemblance with FR II type radio galaxies, while their total radio luminosities are lower than that for classical FR II radio galaxies.",
"Spectral index map between 325 MHz and 610 MHz shows that the outer lobes exhibit steeper spectral index ($\\alpha $ $\\le $ -1) in compared to the middle and inner lobes.",
"The lack of hotspots and very steep spectral index in the outer edges of outer lobes indicate the presence of relic plasma.",
"Kinematic age estimates based on assumed advancement speed of the head of the lobes to be 0.01c, 0.05c and 0.1c, for the outer, middle and inner doubles, respectively, are $\\sim $ 1.3 $\\times $ 10$^{8}$ years, 7.6 $\\times $ 10$^{6}$ years, and 1.5 $\\times $ 10$^{6}$ years for the outer, middle and inner pair of lobes, respectively.",
"The kinematic age estimates allow us to put a lower limit on the quiescent phase time-scales between outer (first episode) and middle (second episode) doubles, and the middle (second episode) and inner (third episode) doubles to be $\\le $ 1.2 $\\times $ 10$^{8}$ years and $\\le $ 6.1 $\\times $ 10$^{6}$ years, respectively.",
"The host galaxy is found to be a bright elliptical (r-band magnitude $\\sim $ 16.56) for which the optical spectrum is dominated by a red continuum and emission line ratios suggest AGN emission to be of low excitation.",
"The host galaxy contains SMBH with the mass of $\\sim $ 3.87 $\\times $ 10$^{9}$ and exhibit SFR $\\sim $ 4.66$_{\\rm -1.61}^{\\rm +4.65}$ M$_{\\odot }$ yr$^{-1}$ .",
"Also, host galaxy belongs to only a small group of three galaxies with the total estimated mass of the group to be $\\sim $ 1.99 $\\times $ 10$^{13}$ M$\\odot $ and virial radius of $\\sim $ 0.28 Mpc.",
"There is no apparent disturbance in the morphology of host galaxy, however, it may be interacting with two nearby dwarf galaxies.",
"So, the AGN activity might have been triggered by the interaction with neighboring dwarf galaxies.",
"Although, more sensitive optical data are required to confirm this plausibility.",
"The GMRT is a national facility operated by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.",
"We thank the staff at NCRA and GMRT for their support.",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"This publication makes use of data products from the WISE, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.",
"Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the US Department of Energy Office of Science.",
"The SDSS-III web site is http://www.sdss3.org/.",
"SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University."
]
] | 1605.05821 |
[
[
"Phonon spectral functions of photo-generated hot carrier plasmas:\n effects of carrier screening and plasmon-phonon coupling"
],
[
"Abstract We investigate spectral behavior of phonon spectral functions in an interacting multi-component hot carrier plasma.",
"Dielectric polarization functions are formulated so that they satisfy Dyson equations of the effective interactions among plasma components.",
"We find a useful sum rule giving simple relation between plasma-species resolved dielectric functions.",
"Spectral analysis of various phonon spectral functions is performed considering carrier-phonon channels of polar and nonpolar optical phonons, acoustic deformation-potential, and piezoelectric Coulomb couplings.",
"Effects of phonon self-energy corrections are examined at finite temperature within a random phase approximation extended to include the effects of dynamic screening, plasmon--phonon coupling, and local-field corrections of the plasma species.",
"We provide numerical data for the case of a photo-generated electron-hole plasma formed in a wurtzite GaN.",
"Our result shows clear significance of the multiplicity of the plasma species in the dielectric response and phonon spectral functions of a multi-component plasma."
],
[
"Introduction",
"Photo-generated hot electrons and holes in semiconductors form a multi-component plasma (mcp) separated in the reciprocal space and have recently gained renewed interest because of their practical importance in application to short-wavelength optoelectronic devices [1], [2], [3], [4], [5].",
"In solid state plasmas formed in doped polar semiconductors, longitudinal phonons and plasmons are strongly coupled through their macroscopic electric fields, and if the frequency of plasma is comparable to that of the phonon, the mode coupling of these excitations is maximized leading to the formation of coupled plasmon–phonon modes [6].",
"The coupling problems of these excitations between the longitudinal optic (LO) phonons and plasmons in a single-component plasma (scp) such as of doped semiconductors have been investigated extensively both theoretically and experimentally [7] including experimental confirmation in Raman scattering measurements [8], [9], [10].",
"In a mcp, different plasma species responds differently to an external disturbance resulting in nonidentical multiple polarization functions.",
"In a multi-component electron–hole plasma, in addition to the well-known compressive optic plasmon–LO phonon coupling of the scp [11], low-frequency acoustic (AC) collective oscillations also occur [12].",
"The acoustic plasmon mode is a low-frequency plasma density oscillation due to in-phase motion of electrons and holes similar to the case of AC phonons [15], [13], [14].",
"Since the number of plasma components modifies the response functions of the system, the mcp would support a great variety of spectral behavior giving rise to novel features (absent in the scp) such as the coupling of individual plasmons to polar optical phonons.",
"The mode coupling properties are expected to depend on the plasma species, because the dispersive behavior of self-sustaining oscillations varies for different plasma species [5].",
"Phonon spectral function of a solid state plasma is characterized by the behavior of coupled plasmon–phonon modes through carrier-phonon interaction (in addition to the individual properties of phonons and carriers) and depends strongly on the nature of the screening and collective behavior of the carriers.",
"For example, Jain and co-workers proposed a coupling of LO phonons to quasiparticle excitations (QPE) and studied its effect on the hot electron energy loss rate at low temperature in a single-component electron plasma [16].",
"However, little is known about the nature of the low energy coupled phonon–QPE modes, and the spectral behavior of the QPE modes.",
"Since there are several polarization functions in the case of a mcp, a careful examination of the spectral behavior is required.",
"The spectral behavior in a mcp is expected to be quite involved and different from that of a single-component one [17].",
"The effects of dynamic screening of plasma species on collisional broadening of the phonon spectral function need be clarified in a systematic way in a mcp–phonon coupled system.",
"Table: Material parameters used for a wurtzite GaNThe purpose of this paper is to map out various phonon spectral functions of a mcp in the $\\omega $ –$q$ plane and analyze their behavior in detail to elucidate the effects of dynamic screening and various plasmon-phonon coupling at finite temperature.",
"The dynamic screening of the carrier–phonon interactions is included by extending the random phase approximation (rpa) to take into account local-field corrections (LFCs) of carriers, and carrier–phonon coupling channels of polar and nonpolar optical phonons are considered.",
"We present numerical results applied to the case of electrons and holes optically generated in a solid focusing our study on the influence of dynamic screening and plasmon–phonon coupling on various phonon spectral functions.",
"A brief account of some preliminary analysis on dielectric responses in a mcp has recently been reported [18].",
"We briefly review the formulation of dielectric responses and phonon spectral function of a mcp in Sec.",
"II and discuss results in Sec.",
"III for the cases of optically excited wurtzite GaN.",
"The main conclusion of our study is in Sec.",
"IV.",
"Table I shows the physical parameters of a wurtzite GaN used in our numerical calculation of polarization and dielectric response functions for an idealized mcp.",
"In GaN, the formation of hot electron-hole plasma is expected for carrier densities larger than $10^{18}-10^{19} \\rm cm^{-3}$ [29].",
"In a wurtzite GaN, the band extrema of conduction and valence bands are located at the center of the Brillouin zone, the former being of $\\Gamma _7$ symmetry and the latter splitting into heavy-hole, light-hole, and split-off bands [30].",
"As one increases the degree of photo excitations, the number of plasma species can be tuned from a two-component (conduction electron–heavy hole) plasma at weak excitation and a three-component (conduction electron–heavy hole–light hole) one at strong excitation."
],
[
"Formulation",
"We consider a mcp produced in an intrinsic semiconductor, by optical excitations, generating electrons in the conduction band and various holes in the valence bands.",
"Here we describe the formulation of dielectric response functions of a mcp and their application to a study of phonon spectral functions of the system.",
"The phonon spectral behavior of a solid state plasma would dependent on the carrier screening and scattering channels (distinguished by a parameter $j$ ) of various carrier–phonon interaction $H_{\\rm c-ph}^{(j)}$ via a specific coupling of carrier $(\\rm c)$ and phonon (ph).",
"Here $j$ denotes various types of phonons with either polar or nonpolar character.",
"Specific scattering channel dependent carrier–phonon interaction is written as [33], [35], [34] $H_{\\rm c-ph}^{(j)} = \\sum _{k q\\nu } M_{q,\\nu }^{(j)} {\\hat{n}}_{k+q,\\nu }\\left(a_{q}^{(j)}+ a_{-q}^{(j)\\dagger } \\right), $ where $\\hat{n}_{k+q,\\nu } (=c_{k+q,\\nu }^\\dagger c_{k\\nu })$ and $M_{q,\\nu }^{(j)} ( =M_{-q,\\nu }^{(j)*}$ ) are the Fourier transform of the carrier density operator $\\hat{n} (r)$ and the matrix element of carrier–phonon coupling for the phonon mode $\\omega _{qj}$ with distinct polarization, respectively.",
"(See Appendix for further discussion on various matrix element $M_{q,\\nu }^{(j)}$ used in our computations.)",
"Here $q$ and $k$ denote vectors $\\vec{q}$ and $\\vec{k}$ , respectively, and $c_{k\\nu }$ ($c_{k\\nu }^\\dagger $ ) is the ordinary annihilation (creation) operator of a carrier in the band $\\nu $ and $a_{q}^{(j)}$ ($a_{q}^{(j)\\dagger }$ ), the annihilation (creation) operator for a phonon with frequency $\\omega _{qj}$ [36].",
"Carrier–phonon interaction induces transition $|i\\rangle \\rightarrow |f\\rangle $ of energy $E_i=E_i^{\\rm c}+E_i^{\\rm ph}$ and $E_f=E_f^{\\rm c}+E_f^{\\rm ph}$ .",
"We consider the case that carriers (electrons and holes) and the lattice are weakly coupled so that carrier–phonon interaction $H_{\\rm c-ph}^{(j)}$ would be treated as a perturbation."
],
[
"Dielectric response functions",
"The carrier-carrier interaction introduces dielectric screening and would weaken the carrier-phonon coupling.",
"As an illustration of linear response of a mcp, we consider an electron–hole plasma, a two-component plasma (2cp), subject to a weak external potential field $\\phi _{\\rm ext}(q,\\omega )$ caused by some external test charge distribution $\\rho _{\\rm ext}$ .",
"Electrons and holes respond differently to an external disturbance resulting in nonidentical polarization functions $\\Pi _{\\nu \\nu }^{\\rm rpa} (q,\\omega )$ ($\\nu = \\rm e ~ or ~ hh$ ).",
"The potential field $\\phi _{\\rm ext}(q,\\omega )$ gives rise to corresponding potential energy $V_{\\rm ext}^i$ for a carrier of type $i$ .",
"Here $i$ denotes different species of the plasma and we limit our consideration to the simplest case of a nondegenerate valence band in order to simplify our discussion, i.e., $i=1$ (2) for electrons (holes) of the plasma.",
"We note that $V_{\\rm ext}^i(q,\\omega )=e_i \\phi _{\\rm ext}(q,\\omega )$ for the carriers of electric charge $e_i=\\mp e$ , where $-e$ is the elementary charge of an electron.",
"The external potential will cause changes in densities of each plasma species, and the charge density fluctuation $\\delta \\rho _i$ in the $i^{\\rm th}$ component is written as $\\delta \\rho _i(q,\\omega ) = \\sum _{ij}\\Pi _{ij}(q,\\omega )V_{\\rm ext}^j(q,\\omega ), $ where $\\lbrace V_{\\rm ext}^1, V_{\\rm ext}^2\\rbrace $ are the carrier density probe and $\\Pi _{ij}(q,\\omega )$ is the reducible polarization propagator – the Fourier transform of the density-density response function of a mcp.",
"That is, $\\mathcal {R}e~ \\Pi _{ij}(q,\\omega )$ is a measure of the response of an electron liquid to a bare external disturbance.",
"The imaginary part of the retarded polarization function, $\\mathcal {I}m~ \\Pi _{ij} (q,\\omega )$ , is directly linked to the real part of the conductivity – a measure of dissipative processes, in which quanta of wave number $q$ and frequency $\\omega $ are absorbed by the carriers in the plasma.",
"Now, the density fluctuation $\\delta \\rho _i(q,\\omega )$ introduces polarization field to the carriers in the plasma, and the effective (self-consistent) potential energy of a carrier in the $i^{\\rm th}$ plasma component is given by $V_{\\rm et}^i(q,\\omega )=V_{\\rm ext}^i(q,\\omega )+\\sum _j \\psi _{ij}(q,\\omega )\\delta \\rho _j(q,\\omega ).",
"$ Here the second term on the right hand side denotes the additional potential due to the polarization of the system, and $\\psi _{ij}$ ($\\ne \\psi _{ji}$ in general) is the effective interaction between carriers of the $i^{\\rm th}$ and $j^{\\rm th}$ components including exchange and correlation effects.",
"In general, $\\psi _{ij}$ is nonlocal and need be determined self-consistently because exact expression for $\\psi _{ij}$ is not available.",
"Within the local density functional scheme of Kohn and Sham [37], [38], $\\psi _{ij}$ can be represented as $\\psi _{ij}=v_{ij}[1-G_{ij}(q)]$ with $v_{ij}=\\frac{4\\pi e_i e_j}{\\kappa q^2}$ and generalized local-field corrections $G_{ij}(q)$ as of Hubbard [39], [40].",
"Here $\\kappa $ is the background dielectric constant of the material.",
"Our discussion on the exchange–correlation effect of carriers parallels that of Kukkonen and Overhauser [38] and Hedin and Lundqvist [41] but with extension to the case of mcp [42].",
"(For our convenience, we will minimize reminding the wave-vector and frequency dependences of the quantities explicitly in each expression now on.",
"For example, $\\Pi _{ij}(q,\\omega )$ will be written as $\\Pi _{ij}$ .)",
"The density fluctuation $\\delta \\rho _i$ can also be determined by treating the plasma as a noninteracting system responding to the self-consistent effective potential $V_{\\rm et}^i$ such as $\\delta \\rho _i(q,\\omega ) = \\tilde{\\Pi }_{i}(q,\\omega )V_{\\rm et}^i(q,\\omega ).",
"$ Here $\\tilde{\\Pi }_{i}$ denotes the temperature-dependent proper polarization function of the multi-component many carrier system and describes the response of a plasma component of type $i$ to the effective potential $V_{\\rm et}^i$ , the sum of the external and polarization potentials.",
"Hence $\\tilde{\\Pi }_{i}$ is a measure of the density fluctuation induced by a screened external charge $\\tilde{\\rho }_{\\rm ext}^i (=\\rho _{\\rm ext}^i /\\tilde{\\varepsilon })$ .",
"Combining Eqs.",
"(REF ) and (REF ) gives rise to coupled equations of $\\delta \\rho _i$ and $\\delta \\rho _j$ .",
"One can easily solve the equations for $\\delta \\rho _i$ and $\\delta \\rho _j$ in terms of $V_{\\rm ext}^i$ and $V_{\\rm ext}^j$ to write $\\left( \\begin{array}{c}\\delta \\rho _1(q,\\omega ) \\\\ \\delta \\rho _2(q,\\omega )\\end{array}\\right)= \\left( \\begin{array}{cc}\\Pi _{11} & \\Pi _{12} \\\\\\Pi _{21} & \\Pi _{22}\\end{array} \\right)\\left(\\begin{array}{c}V_{\\rm ext}^1(q,\\omega ) \\\\ V_{\\rm ext}^2(q,\\omega )\\end{array} \\right), $ where individual components of the polarization propagators $\\Pi _{ij}(q,\\omega )$ are now given by $\\Pi _{ii}={\\tilde{\\Pi }_{i}(1- \\tilde{\\Pi }_{j}\\psi _{jj})}/{\\Delta }\\mbox{ } $ for the intra-species (diagonal) parts [$(i,j)=(1 \\mbox{ or } 2)$ ] and $\\Pi _{ij}={\\tilde{\\Pi }_{i}\\tilde{\\Pi }_{j}\\psi _{ij}}/{\\Delta }\\mbox{ } $ for the inter-species (off-diagonal) parts ($i\\ne j$ ).",
"Here $\\Delta $ is defined by $\\Delta =1-\\tilde{\\Pi }_{1}\\psi _{11} -\\tilde{\\Pi }_{2}\\psi _{22} +\\tilde{\\Pi }_{1}\\tilde{\\Pi }_{2}(\\psi _{11}\\psi _{22}-\\psi _{12}\\psi _{21})$ and plays the role of the effective dielectric function in a mcp.",
"(See Eq.",
"(REF ) below.)",
"In the mean-field rpa, the proper polarization function $\\tilde{\\Pi }_{i}$ is replaced with the noninteracting expression commonly known as Lindhard polarizability ${\\Pi }_{ii}^0$ ($\\equiv \\Pi _{i}^{0}$ for simplicity of notation).",
"It means that the effects of many-body correlations are neglected to let the effective pair interaction $\\psi _{ij}$ become the bare Coulomb interaction $v_{ij}(=v_{ji})=\\mp v$ with $v=4\\pi e^2/(\\kappa q^2)$ .",
"The generalized noninteracting polarization function $\\Pi _{ij}^{0}$ is given by [43], [44] $\\Pi _{ij}^0(q,\\omega ) = 2\\sum _{k}\\frac{f_{k+q,i}^{(0)}-f_{k,j}^{(0)}}{\\varepsilon _{k+q,i}-\\varepsilon _{k,j}-\\hbar \\omega -i\\eta },$ where $f_{k,j}^{(0)}$ is the temperature dependent carrier distribution function.",
"The Lindhard-type expression of Eq.",
"(REF ) is analogous to the case of spin-resolved expression in a multi-component spin system [42].",
"The intra and inter-species components of the polarization propagator ${\\Pi }_{ij}$ are written, in the rpa, as $\\Pi _{ii}^{\\rm rpa}=\\Pi _{i}^0(1- v\\Pi _{j}^0) /\\left[1- v(\\Pi _{i}^0+\\Pi _{j}^0)\\right] $ and $\\Pi _{ij}^{\\rm rpa}=-v\\Pi _{i}^0 \\Pi _{j}^0 /\\left[1- v(\\Pi _{i}^0+\\Pi _{j}^0)\\right].",
"$ We note that $\\Delta ^{\\rm rpa}=1- v(\\Pi _{1}^0+\\Pi _{2}^0)$ is the effective macroscopic dielectric function $\\epsilon _{\\rm eff}^{\\rm rpa}$ in a mcp.",
"(See further discussion on $\\epsilon _{\\rm eff}$ below.)",
"Further reducing to the case of single-component system such as in doped semiconductors, we have $\\Pi _{ii}^{\\rm rpa}=\\Pi _{i}^0/(1- v\\Pi _i^0)$ , the well known expression of mean-field polarizability [43].",
"Since $\\Pi _{ii}^0(q,\\omega ,T)$ is a complex function, the real and imaginary parts $ \\mathcal {R}e ~\\Pi _{ii}(q,\\omega ,T)$ and $\\mathcal {I}m ~\\Pi _{ii}(q,\\omega ,T)$ can be expressed in terms of $\\mathcal {R}e ~\\Pi ^0_{ii}$ and $\\mathcal {I}m ~\\Pi ^0_{ii}$ , which are given, respectively, by [45], [46] $\\mathcal {R}e ~\\Pi _{ii}^0 (q,\\omega ;T) =& -g_{i} \\int _0^\\infty dx \\frac{F(x,T)}{q/k_{Fi}} \\nonumber \\\\\\times \\frac{1}{2}&\\left[\\ln \\left|\\frac{x-\\eta _{i-}}{x+\\eta _{i-}}\\right|-\\ln \\left|\\frac{x-\\eta _{i+}}{x+\\eta _{i+}}\\right| \\right]$ and $\\mathcal {I}m ~\\Pi _{ii}^0 &(q,\\omega ;T) \\nonumber \\\\=-\\pi g_{i} &\\left[\\frac{\\omega }{v_{Fi} q}+\\frac{k_{\\rm B} T}{\\hbar v_{Fi} q}\\ln \\frac{1+e^{\\beta [\\eta _{i-}^2\\varepsilon _{Fi} - \\mu _{i} (T)]}}{1+e^{\\beta [\\eta _{i+}^2\\varepsilon _{Fi} - \\mu _{i}(T)]}}\\right].$ Here $\\beta =1/k_{\\rm B}T$ , $F(x,T)=x[e^{\\beta \\left(x^2\\varepsilon _{Fi} - \\mu _{i}(T)\\right)}+1]^{-1}$ , $g_{i}=\\frac{m_{i}k_{Fi}}{\\pi ^2\\hbar ^2}$ , $k_{Fi}=(3\\pi ^2 n_i)^{1/3}$ , and $\\eta _{i\\pm }=\\frac{\\omega }{qv_{Fi}} \\pm \\frac{q}{2k_{Fi}}$ .",
"In Eqs.",
"(REF ) and (REF ), $T$ is the effective temperature of the corresponding carriers in quasi-equilibrium and conventional notations are used such as $m_{i}$ , $k_{Fi}$ , $v_{Fi}$ , $\\varepsilon _{Fi}$ , and $\\mu _{i}$ denoting effective mass, Fermi wave number, Fermi velocity, Fermi energy, and chemical potential, respectively, of a carrier (either electrons or holes) indicated by subscript $i$ .",
"Figure: (Color Online) Real and imaginary parts of noninteracting polarization functions Π c 0 (q,ω)\\Pi _c^0 (q,\\omega ) of conduction electron plasma for the carrier density of 2×10 19 cm -3 2 \\times 10^{19} \\rm cm^{-3} at effective temperature T eff T_{\\rm eff}=25 K.Insets illustrate the wave-number dependence of Π c 0 (q,ω)\\Pi _c^0 (q,\\omega ) for representative frequencies at 25 K and 300 K.Pair of dashed lines denotes the region of allowed single-particle excitations in a wurtzite GaN.In Fig.REF real and imaginary parts of $\\Pi _{ii}^0 (q,\\omega )$ ($i=e$ ) of a conduction electron plasma is illustrated in the $\\omega $ –$q$ plane for the carrier density of $2 \\times 10^{19} \\rm cm^{-3}$ at effective temperature $T_{\\rm eff}$ =25 K. Each inset illustrates the wave number dependence of $\\mathcal {R}e ~\\Pi _{ii}^0 (q,\\omega )$ and $\\mathcal {I}m ~\\Pi _{ii}^0 (q,\\omega )$ , respectively, for representative frequencies at 25 K and 300 K. Carriers within the Fermi sea can be excited to states outside the Fermi sea, and single-particle excitations of free electrons or free holes are the only processes for the energy and momentum dissipation, since the effects of carrier screening is completely neglected in $\\Pi _{ii}^0 (q,\\omega )$ [17].",
"Real part of the noninteracting polarization function $\\mathcal {R}e ~\\Pi _{ii}^0 (q,\\omega )$ shows a broad valley structure within the single-particle excitation continuum at small frequency $\\omega \\ll \\omega _{\\rm LO}$ for $q < q_{\\rm sc}$ and a peak structure of moderate width along the upper boundary of the continuum.",
"The line of $\\mathcal {R}e ~\\Pi _{ii}^0 (q,\\omega )=0$ lies in the continuum region of the single-particle excitations and, at small $q$ , $\\mathcal {R}e ~\\Pi _{ii}^0 (q,\\omega )$ changes sign from negative to positive as $\\omega $ increases sweeping across the continuum region.",
"On the other hand, $\\mathcal {I}m ~\\Pi _{ii}^0 (q,\\omega )$ is finite and negative inside the continuum region of the single-particle excitations showing a sharp dipped structure along the zero line of $\\mathcal {R}e ~\\Pi _{ii}^0 (q,\\omega )$ .",
"Since we now have all the components of the polarization propagator $\\Pi _{ij}$ , the self-consistent interaction of Eq.",
"(REF ) can now be written, with $\\delta \\rho _i$ 's given by Eq.",
"(REF ), as $\\left( \\begin{array}{c}V_{\\rm et}^1 \\\\ V_{\\rm et}^2\\end{array}\\right)&=& \\frac{1}{\\Delta }\\left( \\begin{array}{cc}1-\\Pi _2^0\\psi _{22} & \\Pi _2^0\\psi _{12} \\\\\\Pi _1^0\\psi _{21} & 1-\\Pi _1^0\\psi _{11}\\end{array} \\right)\\left(\\begin{array}{c}V_{\\rm ext}^1 \\\\ V_{\\rm ext}^2 \\nonumber \\end{array} \\right) \\\\&\\equiv &\\left( \\begin{array}{cc}\\tilde{\\epsilon }_{11}^{-1} & \\tilde{\\epsilon }_{12}^{-1} \\\\\\tilde{\\epsilon }_{21}^{-1} & \\tilde{\\epsilon }_{22}^{-1}\\end{array} \\right)\\left(\\begin{array}{c}V_{\\rm ext}^1 \\\\ V_{\\rm ext}^2\\end{array} \\right).",
"$ Here the $2 \\times 2$ matrix multiplied to the column of external probe potentials $(V_{\\rm ext}^1, V_{\\rm ext}^2)$ on the right hand side is just the inverse of `plasma–test charge' dielectric tensor ${\\underline{\\tilde{\\epsilon }}}$ in the $2\\times 2$ space of plasma species and each component is given by $\\tilde{\\epsilon }_{ij}^{-1} =\\delta _{ij} + \\sum _{\\ell =1,2}\\psi _{i\\ell }\\Pi _{\\ell j}.",
"$ (Similar description corresponding to the case of spin-polarized electrons was investigated earlier by one of us [44].)",
"The intra-species `electron–test charge' dielectric function $\\tilde{\\epsilon }_{ii}=1-\\Pi _i^0\\psi _{ii}-\\Pi _i^0\\psi _{ij}\\Pi _j^0\\psi _{ji}/(1-\\Pi _j^0\\psi _{jj})$ reduces, in the rpa, to $\\tilde{\\epsilon }_{ii}^{\\rm rpa}=1-v\\Pi _i^0/(1-v\\Pi _j^0)$ and to $\\tilde{\\epsilon }^{\\rm rpa}=1-v\\Pi _i^0$ in the case of single species electron liquid [41].",
"On the other hand, the inter-species component $\\tilde{\\epsilon }_{ij}=\\Delta /(\\psi _{ij}\\Pi _j^0)$ reduces to $\\tilde{\\epsilon }_{ij}^{\\rm rpa}=1-(1-v\\Pi _i^0)/(v\\Pi _j^0)$ in the rpa and is undefined in the case of single-species electron system.",
"We find that $\\sum _{ij} (\\tilde{\\epsilon }_{ij}^{-1}-&\\delta _{ij})=[(\\psi _{11}+\\psi _{21})\\Pi _1^0 +(\\psi _{22}+\\psi _{12})\\Pi _2^0 \\nonumber \\\\&-2(\\psi _{11}\\psi _{22}-\\psi _{12}\\psi _{21})\\Pi _1^0\\Pi _2^0]/\\Delta $ and, hence, that $\\sum _{ij} (\\tilde{\\epsilon }_{ij}^{-1}-\\delta _{ij})|_{\\rm rpa} =0$ since $\\psi _{ii}=-\\psi _{ij} (i\\ne j)$ in the rpa.",
"In response to the potential field $\\phi _{\\rm ext}$ due to an external test charge $\\rho _{\\rm ext}^i$ ($\\rho _{\\rm ext}^j$ ) of electrical charge $e_i$ ($e_j$ ), another probing test charge would experience the `test charge–test charge' interaction $V_{\\rm tt}^i$ ($V_{\\rm tt}^j$ ) written as $V_{\\rm tt}^{i}(q,\\omega )= v_{ii}(\\rho _{\\rm ext}^{i}+\\delta \\rho _{i} ) +v_{ij}\\delta \\rho _{j}, $ and, similarly, $V_{\\rm tt}^{j}(q,\\omega )$ with the indices $i$ and $j$ interchanged in $V_{\\rm tt}^{i}$ .",
"In general, the dielectric function $\\epsilon (q,\\omega )$ of a material is defined, in terms of `test charge–test charge' interaction, by $V_{\\rm tt}(q,\\omega )= V_{\\rm ext} (q,\\omega )/\\epsilon (q,\\omega )$ [38], and can be extended to a mcp as follows.",
"On substituting the density fluctuations $\\delta \\rho _i$ 's of Eq.",
"(REF ) into Eq.",
"(REF ), $V_{\\rm tt}^i$ and $V_{\\rm tt}^j$ are written, in terms of $V_{\\rm ext}^i$ and $V_{\\rm ext}^j$ , as $\\left( \\begin{array}{c}V_{\\rm tt}^1 \\\\ V_{\\rm tt}^2\\end{array}\\right)&=& \\left( \\begin{array}{cc}\\epsilon _{11}^{-1} & \\epsilon _{12}^{-1} \\\\\\epsilon _{21}^{-1} & \\epsilon _{22}^{-1}\\end{array} \\right)\\left(\\begin{array}{c}V_{\\rm ext}^1 \\\\ V_{\\rm ext}^2\\end{array} \\right).",
"$ Here $\\epsilon _{ij}^{-1}$ 's are the components of the inverse dielectric tensor ${\\underline{\\epsilon }}^{-1}$ of a mcp and are given in terms of the polarization propagators.",
"The intra- and inter-species components are expressed, respectively, as $\\epsilon _{ii}^{-1}=1+v(\\Pi _{ii}-\\Pi _{ji})$ and $\\epsilon _{ij}^{-1}=v(\\Pi _{ij}-\\Pi _{jj})$ and can be combined to be written, in general, as $\\epsilon _{ij}^{-1} =\\delta _{ij} + (-1)^{i+j}\\sum _{\\ell =1,2}v_{j\\ell }\\Pi _{\\ell j}.$ On substituting $\\Pi _{ij}$ 's of Eqs.",
"(REF ) and (REF ) into Eq.",
"(REF ), one can obtain, in a mcp, an identity of $\\sum _{ij} (\\epsilon _{ij}^{-1}-\\delta _{ij}) =0.",
"$ In the rpa, $V_{\\rm et}^{j}=V_{\\rm tt}^{j}$ and, hence, $\\tilde{\\epsilon }_{ij}$ and $\\epsilon _{ij}$ are identical.",
"If we neglect inter-species correlations (i.e., $G_{ij}=G\\delta _{ij}$ as of Hubbard's local-field correction [39]), we have $\\epsilon _{ii}=1-\\frac{\\Lambda _i v \\Pi _{i}^0}{1-\\Lambda _j v\\Pi _j^0}$ and $\\epsilon _{ij}=1-\\frac{1-\\Lambda _i v \\Pi _{i}^0}{\\Lambda _j v\\Pi _j^0}.$ Here the vertex function $\\Lambda _i$ is defined by $\\Lambda _i =[1-(\\psi _{ii}-v)\\Pi _i^0]^{-1}$ generally known as $1/(1+vG_{ii}\\Pi _{i}^0)$ [43].",
"For the case of a single species system, one resumes $\\epsilon _{ii}^{-1}=1+v\\Pi _{i}^0/(1-\\psi _{ii}\\Pi _{i}^0)$ giving rise to $\\epsilon _{ii}=1-\\Lambda _i v\\Pi _{i}^0$ .",
"Since $\\Lambda _i\\rightarrow 1$ in the rpa, we have $\\epsilon _{ij}^{\\rm rpa}=1-{(1-v \\Pi _{i}^0)}/{(v\\Pi _j^0)}$ and $\\epsilon _{ii}^{\\rm rpa} =1-v\\Pi _{i}^0/(1-v\\Pi _{j}^0)$ further reducing to $\\epsilon _{ii}^{\\rm rpa}=1-v\\Pi _{i}^0$ in a single species electron gas.",
"Figure: Dyson equations for the effective Coulomb interaction V ˜ ij \\tilde{V}_{ij} (i=e or hi=e {\\rm ~or~} h) between carriers in a multi-component plasmaLet us introduce the macroscopic effective dielectric constant $\\epsilon _{\\rm eff}$ of a mcp by equating $[\\epsilon _{\\rm eff}^{-1}(q,\\omega ) -1]\\rho _{\\rm ext}$ phenomenologically in the presence of the external test charge $\\rho _{\\rm ext}$ to the net charge density induced in the plasma as follows [15] $[\\epsilon _{\\rm eff}^{-1}(q,\\omega ) -1]\\rho _{\\rm ext}\\equiv \\delta \\rho _1(q,\\omega )+\\delta \\rho _2(q,\\omega ).",
"$ Substitution of $\\delta \\rho _i$ given by Eq.",
"(REF ) into Eq.",
"(REF ) along with Eqs.",
"(REF ) and (REF ) leads us to $\\epsilon _{\\rm eff}^{-1}(q,\\omega )-1=v[(\\Pi _1^0+\\Pi _2^0-\\Pi _1^0\\Pi _2^0\\sum _{ij=1,2}\\psi _{ij}]/\\Delta ~~~ \\\\=(\\epsilon _{11}^{-1}-1) +(\\epsilon _{22}^{-1}-1) =-(\\epsilon _{12}^{-1}+\\epsilon _{21}^{-1}), \\nonumber $ where the identity of Eq.",
"(REF ) has been observed in writing the last equality.",
"In the rpa, $\\epsilon _{\\rm eff}^{\\rm rpa}=1-v(\\Pi _1^0+\\Pi _2^0)$ , which is the same as that suggested by Vashishta et al.",
"and others [40], [47].",
"However, we note that Vashishta et al.",
"introduced their effective dielectric constant $\\epsilon _{\\rm eff}$ differently.",
"(They defined their $\\epsilon _{\\rm eff}^{-1}(q,\\omega ) -1$ as $\\sum _{ij} (\\epsilon _{ij}^{-1}-\\delta _{ij})$ , which vanishes according to the identity given by Eq.",
"(REF ).)",
"Our description of $\\epsilon _{\\rm eff}$ is consistent with that of the Dyson equation approach.",
"Figure: (Color Online) Effective dielectric functions ℐmϵ eff (q,ω)\\mathcal {I}m ~\\epsilon _{\\rm eff}(q,\\omega ) of two-component plasma formed in a wurtzite GaN with conduction electron density of 2×10 19 cm -3 2 \\times 10^{19} \\rm cm^{-3} at effective carrier temperature 25 K: (a) ℐmϵ eff (q,ω)\\mathcal {I}m ~\\epsilon _{\\rm eff}(q,\\omega ).",
"(b) ℐmϵ eff rpa (q,ω)\\mathcal {I}m ~\\epsilon _{\\rm eff}^{\\rm rpa}(q,\\omega ).",
"(c) ℐm[ϵ eff (q,ω)-ϵ eff rpa (q,ω)]\\mathcal {I}m~[\\epsilon _{\\rm eff}(q,\\omega )-\\epsilon _{\\rm eff}^{\\rm rpa}(q,\\omega )], and (d) ℐmϵ eff (q,ω)\\mathcal {I}m~\\epsilon _{\\rm eff}(q,\\omega ) and ℐmϵ eff rpa (q,ω)\\mathcal {I}m ~\\epsilon _{\\rm eff}^{\\rm rpa}(q,\\omega ) for ω=0.1ω LO \\omega = 0.1 \\omega _{\\rm LO}.Pairs of dashed lines denote the corresponding boundaries of allowed single-particle excitation continuum.In a multi-component many carrier system (consisting of electrons and various holes in the present case), the effective (dressed) interactions $\\tilde{V}_{ij}$ between carriers of types $i$ and $j$ are the solutions of Dyson equations.",
"For example, in terms of full retarded (proper) polarization function $\\tilde{\\Pi }_{\\ell }$ and bare Coulomb interaction $v_{ij}$ , it is written as $\\tilde{V}_{ij}=v_{ij}+ \\sum _{\\ell =e,h} v_{i\\ell }\\tilde{\\Pi }_{\\ell }\\tilde{V}_{\\ell j}.$ Here the bare Coulomb interaction $v_{ij}$ is $v_q (\\equiv \\frac{4\\pi e^2}{q^2})$ for $i=j$ or $-v_q$ for $i\\ne j$ .",
"The internal Coulomb interactions of the carriers in the plasma are renormalized in exactly the same way as the external potential fields.",
"Dyson equations for dressed Coulomb interactions $\\tilde{V}_{ee}$ between conduction electrons (denoted by $e$ 's) and $\\tilde{V}_{eh}$ between an electron and a hole (denoted by $h$ ) are illustrated in Fig.",
"REF .",
"One can solve the coupled equations of Eq.",
"(REF ) explicitly for $\\tilde{V}_{ee}$ , $\\tilde{V}_{hh}$ , and $\\tilde{V}_{eh}$ to write $\\tilde{V}_{ij}=v_{ij}/\\epsilon _{\\rm eff}$ confirming $\\epsilon _{\\rm eff}^{\\rm rpa} (q,\\omega ) = 1-v_q \\sum _{i=e,h} {\\Pi }_{i}^0(q,\\omega )$ .",
"In Fig.",
"REF (a) and (b), dispersive behaviors of the real part of effective dielectric function $\\epsilon _{\\rm eff}(q,\\omega )$ of Eq.",
"(REF ) and $\\epsilon _{\\rm eff}^{\\rm rpa}(q,\\omega )$ are shown, respectively, for a 2cp formed of conduction electrons and heavy holes each with concentration $2 \\times 10^{19} \\rm cm^{-3}$ at effective carrier temperature 25 K. The zero value contours of $\\mathcal {R}e ~\\epsilon _{\\rm eff}(q,\\omega )=0$ are indicated with dark solid lines, each denoting the dispersion curves of high-frequency `optic' and low-frequency `acoustic' plasmon modes.",
"Pairs of dashed and dotted lines indicate the boundaries of allowed single-particle excitation continua for electrons and heavy holes.",
"In the region of long wavelength and high frequency, $\\mathcal {I}m ~\\Pi (q,\\omega )$ vanishes (See Fig.",
"REF below.)",
"and, hence, $\\mathcal {I}m ~\\epsilon _{\\rm eff}(q,\\omega )=0$ allowing well-defined dissipationless self-sustaining collective oscillations.",
"The difference of $\\mathcal {R}e ~[\\epsilon _{\\rm eff}(q,\\omega )-\\epsilon _{\\rm eff}^{\\rm rpa}(q,\\omega )]$ is illustrated in panel (c), and the wave-number dependences of $\\mathcal {R}e~\\epsilon _{\\rm eff}(q,\\omega )$ and $\\mathcal {R}e~\\epsilon _{\\rm eff}^{\\rm rpa}(q,\\omega )$ are compared in panel (d) for $\\omega = 0.1 \\omega _{\\rm LO}$ .",
"In panel (b), the contour of $\\mathcal {R}e ~\\epsilon _{\\rm eff}(q,\\omega )=0$ is indicated by green dotted line for comparison.",
"In Fig.",
"REF (a), we find that a pair of plasmon branches are observed in a 2cp, and that optic and acoustic branches are well separated within the rpa, each damped through single-particle excitations of electrons and holes, respectively.",
"However, the local-field corrections of carriers modify the plasmon branches slightly reducing plasmon frequencies of both the optical and acoustical branches for given values of wave number $q$ .",
"We note that, in the 2cp of $n_e=2 \\times 10^{19} \\rm cm^{-3}$ , the higher frequency electron plasmon mode of bare frequency $\\omega _{\\rm p,e}\\simeq 1.8 \\omega _{\\rm LO}$ is almost intact and well defined but that the lower frequency plasmon mode of heavier species (heavy holes) at $\\omega _{\\rm p,hh} \\simeq 0.69 \\omega _{\\rm LO}$ is screened by the lighter conduction electrons giving rise to an acoustic branch, the latter mode being subject to Landau damping by the lighter species, since the branch is located well inside the electron excitation continuum [18].",
"Dispersive behaviors of $\\mathcal {I}m ~\\epsilon _{\\rm eff}(q,\\omega )$ and $\\mathcal {I}m ~\\epsilon _{\\rm eff}^{\\rm rpa}(q,\\omega )$ are shown in Fig.",
"REF for a 2cp of conduction electrons and heavy holes each with concentration $2 \\times 10^{19} \\rm cm^{-3}$ at effective carrier temperature 25 K. While $\\mathcal {I}m ~\\epsilon _{\\rm eff}(q,\\omega )$ has a single peaked structure in a $\\omega -q$ plane for a scp [17], it reveals double peaked structure each appearing inside the single-particle excitation continua of electrons and holes in a 2cp, respectively, right below the upper boundaries of each continuum in the region of low frequency and long wavelength.",
"The effects of local-field corrections of the carriers are appreciable only in the region of low frequency and long wavelength as illustrated in panels (c) and (d) of Fig.",
"REF ."
],
[
"Phonon spectral functions",
"The dielectric screening in many carrier system gives rise to renormalized electron–phonon coupling and thus to dressed phonon propagator modifying the phonon dispersion relations along with phonon spectral function.",
"In compound semiconductors, charge carriers couple to phonons via various channels such as couplings to LO and TO phonons, and also through acoustic deformation potential and piezoacoustic (AP) couplings.",
"The interaction of specific phonon–carrier coupling (designated by a parameter $j$ ) represented by $H_{\\rm c-ph}^{(j)}$ is given in Eq.",
"(REF ).",
"The phonon spectral function $\\mathcal {A}(q,\\omega )$ of a material describes the probability distribution of having phonons with wave number $q$ and frequency $\\omega $ .",
"The bare phonon modes in a solid would be modified due to carrier screening and phonon–plasmon coupling, and the poles of the dressed phonon propagator determine the renormalized phonon dispersion relations, and the effective phonon spectral function of a solid is, in general, the sum of contributions from each individual electron–phonon coupling channel such that $\\mathcal {A}(q,\\omega )=\\sum _{j\\nu } \\mathcal {A}_{j\\nu }(q,\\omega ).$ Here $\\mathcal {A}_{j\\nu }(q,\\omega )$ describes the probability distribution of having phonons (of $j^{\\rm th}$ type) dressed by plasma species $\\nu $ and is given, in terms of retarded phonon propagator $D_{j\\nu } (q,\\omega )$ for the individual phonon mode, as [43] $\\mathcal {A}_{j\\nu }(q,\\omega )= -\\frac{1}{\\pi }\\mathcal {I}m ~D_{j\\nu } (q,\\omega ).$ The dielectric screening in many carrier system gives rise to renormalized electron–phonon coupling and thus to dressed phonon propagator [48].",
"This collisional broadening modifies the phonon dispersion relations along with phonon spectral function.",
"The dressed phonon propagator $D_{j\\nu }$ with collisional broadening is written, in general, as [33], [35] $D_{j\\nu }(q,\\omega )= & \\frac{2\\omega _{q j}}{\\omega ^2- \\omega _{q j}^2 - 2 \\omega _{q j}\\mid M_{q}^{(j)}\\mid ^2 \\Pi _{\\nu \\nu }(q,\\omega )/\\hbar },$ where $\\omega _{q j}$ is the bare (undoped crystal) phonon frequency of mode $j$ .",
"Ignoring the phonon renormalization, $D_{j\\nu }$ reduces, with an infinitesimal positive $\\eta $ , to [43] $D_j^{(0)}(q,\\omega )= & \\frac{2\\omega _{q j}}{\\omega ^2- \\omega _{q j}^2 +i\\eta },$ giving rise to the well-known bare phonon spectral function $\\mathcal {A}_j^{(0)}(q,\\omega )= [\\delta (\\omega -\\omega _{q j})- \\delta (\\omega +\\omega _{q j})]$ .",
"In Eq.",
"(REF ), $\\Pi _{\\nu \\nu }(q,\\omega )$ is the full retarded polarization propagator of each plasma component (distinguished by $\\nu $ ) given by Eq.",
"(REF ), and $\\mid M_{q,\\nu }^{(j)}\\mid ^2 \\Pi _{\\nu \\nu }(q,\\omega )/\\hbar $ in the denominator represents the (complex) phonon self-energy correction $\\mathcal {P}_{j\\nu }(q,\\omega )(\\equiv \\Delta _{j\\nu }-i\\Gamma _{j\\nu }/2)$ via polarization function of plasma species $\\nu $ , which is known to introduce low energy quantum interference branch in the phonon spectral function [16].",
"The real and imaginary parts of the phonon self-energy, $\\Delta _{j\\nu }$ and $\\Gamma _{j\\nu }$ , are given, respectively, by $\\Delta _{j\\nu }(q,\\omega )= \\mid M_{q,\\nu }^{(j)}\\mid ^2\\mathcal {R}e ~\\Pi _{\\nu \\nu }(q,\\omega )/\\hbar $ and $\\Gamma _{j\\nu }(q,\\omega )=-2\\mid M_{q}^{(j)}\\mid ^2\\mathcal {I}m~\\Pi _{\\nu \\nu }(q,\\omega )/\\hbar $ , the former describing the frequency renormalization correction due to the electronic screening of the long-ranged Coulomb fields associated with the phonons and the latter being a measure of phonon lifetime $\\tau _{q j}$ or the width of the spectral function due to collisional broadening.",
"Phonon self-energy and, hence, $\\Delta _{j\\nu }$ and $\\Gamma _{j\\nu }$ are also functions of $\\omega $ and $q$ , and $\\mathcal {I}m~\\Pi _{\\nu \\nu }(q,\\omega ) \\le 0$ , in general.",
"Now, the phonon spectral function $\\mathcal {A}_{j\\nu }(q,\\omega )$ is written as $\\mathcal {A}_{j\\nu }& (q,\\omega ) \\nonumber \\\\= &\\frac{\\omega _{q j}^2 \\Gamma _{j\\nu }(q,\\omega )}{[\\omega ^2- \\omega _{q j}^2 -2 \\omega _{q j}\\Delta _{j\\nu }(q,\\omega )]^2 + [\\omega _{q j}\\Gamma _{j\\nu } (q,\\omega )]^2 }.$ The denominator of $\\mathcal {A}_{j\\nu }(q,\\omega )$ can be rewritten, in terms of phenomenological renormalized phonon frequency $\\tilde{\\omega }_{q j}$ and phonon lifetime $\\tau _{q j}$ , as $\\omega ^2- (\\tilde{\\omega }_{q j} -\\frac{i}{2\\tau _{q j}})^2 $ , where $\\tilde{\\omega }_{q j}$ satisfies a quadratic equation given by $\\tilde{\\omega }_{q j}^4 - \\omega _{q j}^2 (1+{ 2 \\Delta _{j\\nu }}/{\\omega _{q j}})\\tilde{\\omega }_{q j}^2 -\\omega _{q j}^2\\Gamma _{j\\nu }^2 =0$ with $\\tau _{q j}^{-1}=\\frac{\\omega _{q j}}{\\tilde{\\omega }_{q j}} \\Gamma _{j\\nu }$ .",
"For each phonon mode $j$ , the electron–phonon interaction introduces the phonon self-energy to change the bare phonon frequencies $\\omega _{q j}$ to the new frequencies $\\tilde{\\omega }_{q j}$ with finite lifetime $\\tau _{q j}$ .",
"Equation (REF ) gives rise to, along with the renormalized primary mode close to ${\\tilde{\\omega }}_{j\\nu }^+(q,\\omega )\\simeq \\omega _{q j} (1+\\frac{ 2 \\Delta _{j\\nu }}{\\omega _{q j}}+\\frac{\\Gamma _{j\\nu }^2/\\omega _{q j}^2}{1+ 2 \\Delta _{j\\nu }/\\omega _{q j}})^{1/2}$ for $ 2 \\Delta _{j\\nu } \\ge -{\\omega _{q j}}$ , a secondary (low energy) mode ${\\tilde{\\omega }}_{j\\nu }^-(q,\\omega )\\simeq \\frac{\\Gamma _{j\\nu }}{{\\sqrt{|1+\\frac{ 2 \\Delta _{j\\nu }}{\\omega _{q j}}|}}}$ for $2\\Delta _{j\\nu } < -\\omega _{q j}$ with negative self-energy correction $\\Delta _{j\\nu } (<0)$ .",
"We find that the latter mode ${\\tilde{\\omega }}_{j\\nu }^-$ would be well-defined only with finite values of $\\Gamma _{j\\nu }(q,\\omega )$ in the region of $\\mathcal {R}e ~\\Pi _{\\nu \\nu }(q,\\omega )<0$ in the $\\omega -q$ space.",
"$\\Gamma _{j\\nu }(q,\\omega )$ is finite only in the $\\omega -q$ plane of finite $\\mathcal {I}m ~\\Pi _{\\nu \\nu }(q,\\omega )$ , which occurs in the presence of single-particle excitations as is shown in Fig.",
"REF below.",
"Deep valley with negative $\\mathcal {R}e ~\\Pi _{\\nu \\nu }(q,\\omega )$ occurs in the (very) low energy region of high damping inside the single-particle excitation continuum on the $\\omega -q$ plane.",
"(See Fig.",
"REF (a) and (b).)"
],
[
"Results and Discussion",
"In order to illustrate numerical results for the spectral behaviors of dielectric response and phonon spectral functions in an ideal mcp, use has been made of effective masses $m_{\\rm e}= 0.22 m_0$ , $m_{\\rm hh}= 1.3 m_0$ , and $m_{\\rm lh}= 0.30 m_0$ of a (simplified) wurtzite GaN with parabolic bands.",
"The bare plasma frequencies for carrier concentration of $2\\times 10^{19} \\rm cm^{-3}$ are $\\omega _{\\rm p,e}(\\epsilon _\\infty ) \\simeq 153 ~\\rm meV ~(= 1.68\\omega _{\\rm LO})$ , $\\omega _{\\rm p,hh}(\\epsilon _\\infty ) \\simeq 63 ~ \\rm meV ~(=0.69 \\omega _{\\rm LO})$ , and $\\omega _{\\rm p,lh}(\\epsilon _\\infty ) \\simeq 131~ \\rm meV ~(= 1.44 \\omega _{\\rm LO})$ for scp plasmas of electrons, heavy holes, and light holes, respectively.",
"In the present work, frequencies and wave numbers are scaled by the longitudinal bare phonon frequency $\\omega _{\\rm LO}$ and the Thomas–Fermi screening wave number $q_{\\rm sc}$ , respectively.",
"We consider a simplified nondispersive model of Einstein for bare optical phonons with $\\omega _{\\rm LO}=92~ \\rm meV$ and $\\omega _{\\rm TO}=66~ \\rm meV (\\simeq 0.72 \\omega _{\\rm LO})$ and a Debye-type model for bare acoustical phonons of $\\omega _{q j}=s q$ in a undoped GaN.",
"Here $s$ is the speed of sound wave in the material.",
"For a scp, $q_{\\rm sc}$ is given, in terms of particle number density $n$ and the chemical potential $\\mu $ , by $q_{\\rm sc}^2 = \\frac{4\\pi e^2}{\\epsilon _{\\infty }}\\frac{\\partial n}{\\partial \\mu }$ [49], and we have $q_{\\rm sc}^{(\\rm e)}=9.2 \\times 10^6 \\rm cm^{-1}$ and $q_{\\rm sc}^{(\\rm hh)}=2.2 \\times 10^7 \\rm cm^{-1}$ at 25 K and $q_{\\rm sc}^{(\\rm e)}=8.6 \\times 10^6 \\rm cm^{-1}$ and $q_{\\rm sc}^{(\\rm hh)}=2.1 \\times 10^7 \\rm cm^{-1}$ at 300 K, respectively, for carrier concentration of $2 \\times 10^{19} \\rm cm^{-3}$ .",
"For a mcp, $q_{\\rm sc}$ is written as [18] $q_{\\rm sc}^2 = \\frac{4\\pi e^2}{\\epsilon _{\\infty }}\\sum _{\\nu }\\frac{\\partial n_\\nu }{\\partial \\mu _\\nu },$ where $n_\\nu $ and $\\mu _\\nu $ are, respectively, the particle number density and the quasi chemical potential of the band occupied by the plasma species $\\nu $ .",
"For a 2cp consisting (of conduction electrons and heavy holes) with electron concentration of $2 \\times 10^{19} \\rm cm^{-3}$ , we have $q_{\\rm sc}=2.94 \\times 10^7 \\rm cm^{-1}$ at 25 K and $q_{\\rm sc}=2.54 \\times 10^7 \\rm cm^{-1}$ at 300 K, respectively.",
"Figure: (Color Online) Real and imaginary parts of dressed polarization functions, Π νν (q,ω)\\Pi _{\\nu \\nu } (q,\\omega ), for conduction electrons (ν=e\\nu = \\rm e) and heavy holes (ν= hh \\nu = \\rm hh) of concentration 2×10 19 cm -3 2 \\times 10^{19} \\rm cm^{-3} and effective carrier temperature 25 K. Insets illustrate the frequency or wave-number dependences of ℛeΠ νν (q,ω)\\mathcal {R}e ~\\Pi _{\\nu \\nu } (q,\\omega ) for representative values of wave number or frequencies.Pairs of dashed and dotted lines denote the allowed regions of single-particle excitations for conduction electrons and heavy holes, respectively, in a wurtzite GaN.For a 2pc consisting of conduction electrons and heavy holes, each species of carrier concentration $2 \\times 10^{19} \\rm cm^{-3}$ , the spectral behavior of the dressed polarization function $\\Pi _{\\nu \\nu }^{\\rm rpa} (q,\\omega )$ ($\\nu = \\rm e ~ or ~ hh$ ) shown in Fig.",
"REF reveals the multi-component character of the high-frequency optic and low-frequency acoustic plasmon branches.",
"Boundaries of a pair of single-particle excitation continua for electrons and heavy holes are indicated with steeper dashed (for electrons) and slower dotted (for holes) lines, respectively, and branches of the optic and acoustic plasmon excitations are clearly distinguished in strong color intensities.",
"Each inset illustrates the frequency or wave-number dependence of $\\Pi _{\\nu \\nu } (q,\\omega )$ at 25 K and 300 K for representative wave number or frequencies, respectively.",
"In a mcp, in addition to the contribution from the conventional optic plasmon-LO phonon coupling of the scp such as of doped semiconductors [11], the contribution from the low-frequency acoustic collective oscillations occurs [12], [7], as is illustrated in panels (a) and (c) of Fig.",
"REF for $\\mathcal {R}e~\\Pi _{\\rm e}(q,\\omega )$ and $\\mathcal {I}m~\\Pi _{\\rm e}(q,\\omega )$ , and in (b) and (d) for $\\mathcal {R}e~\\Pi _{\\rm hh}(q,\\omega )$ and $\\mathcal {I}m ~\\Pi _{\\rm hh}(q,\\omega )$ .",
"The mode of acoustic collective oscillation is expected to be Landau damped due to single-particle excitations of the lighter mass species (conduction electrons), since the acoustic branch is lying within the single-particle excitation regime of the lighter species of the mcp as clearly demonstrated in Fig.REF .",
"The well defined optic modes dominated by the lighter species are intact to be clearly seen in $\\mathcal {R}e~\\Pi _{e}(q,\\omega )$ and $\\mathcal {I}m~\\Pi _{e}(q,\\omega )$ occurring at frequencies higher than that of the bare LO phonons well outside the single-particle excitation continuum.",
"The longitudinal low-frequency (optic) bare modes of the heavier species are drastically screened by the lighter mass species to become acoustic [7].",
"The low-frequency acoustic plasmon branch is located outside single-particle excitations of heavy holes but is more broadened compared to the optic one.",
"The contribution of the heavier mass species to dressed $\\Pi _{hh} (q,\\omega )$ [panels (b) and (d) of Fig.",
"REF ] is very similar to that of bare polarization functions $\\Pi _c^0 (q,\\omega )$ given by Eq.",
"(REF ) and shown in Fig.",
"REF .",
"Figure: (Color Online) Phonon spectral function 𝒜 jν scp (q,ω)\\mathcal {A}_{j\\nu }^{\\rm scp}(q,\\omega ) of a single-component plasma formed by conduction electrons [(a) and (c)] and heavy holes [(b) and (d)] of concentration 2×10 19 cm -3 2 \\times 10^{19} \\rm cm^{-3} at effective carrier temperature 25 K.(a) and (b): Contributions from polar optical (LO) phonon and nonpolar acoustic deformation (AD) potential.",
"(c) and (d): Contributions from nonpolar optical (TO) and polar piezoacoustic (AP) deformation potentials.The peak structure of the phonon spectral function $\\mathcal {A}_{j\\nu }(q,\\omega )$ in the $\\omega $ -$q$ plane signifies the spectral behavior of renormalized phonon–plasmon coupled branches with collisional broadening.",
"In the presence of plasmon–phonon coupling, multiple peak structure is expected in the phonon spectral function of the material.",
"The mode coupling of LO phonons and plasmons introduces a pair of branches named $L^{(+)}(\\omega ,q)$ and $L^{(-)}(\\omega ,q)$ , the former (latter) representing high-frequency (low-frequency) coupled mode.",
"Detailed analyses of the longitudinal coupled modes have been reported extensively in the past for a single-component solid state plasma [50], [51].",
"In Fig.",
"REF , the spectral behaviors of the phonon spectral functions $\\mathcal {A}_{j\\nu }^{\\rm scp}(q,\\omega )$ in an electron plasma are illustrated for carrier concentration $2 \\times 10^{19} \\rm cm^{-3}$ at effective temperature 25 K. The contributions of conduction electrons or of heavy holes to LO phonon spectral functions [$\\mathcal {A}_{\\rm LOe}^{\\rm scp}(q,\\omega )$ and $\\mathcal {A}_{\\rm LOhh}^{\\rm scp}(q,\\omega )$ ] and to acoustic phonon spectral functions [$\\mathcal {A}_{\\rm ADe}^{\\rm scp}(\\omega )$ and $\\mathcal {A}_{\\rm ADhh}^{\\rm scp}(\\omega )$ ] are displayed in panels (a) and (b).",
"Contributions to those of TO phonons [$\\mathcal {A}_{\\rm TOe}^{\\rm scp}(q,\\omega )$ and $\\mathcal {A}_{\\rm TOhh}^{\\rm scp}(q,\\omega )$ ] and of piezoacoustic deformation potentials [$\\mathcal {A}_{\\rm APe}^{\\rm scp}(\\omega )$ and $\\mathcal {A}_{\\rm APhh}^{\\rm scp}(\\omega )$ ] are illustrated in (c) and (d).",
"We note that latter phonon modes are negligibly sensitive to the presence of longitudinal plasmons resulting in narrow peaks near the corresponding bare phonon frequencies $\\omega _{\\rm TO}$ and $\\omega _{\\rm AC} (\\sim sq)$ due to the absence of density fluctuations accompanied by the macroscopic electric fields associated with these nonpolar TO and acoustic modes.",
"For the case of highly doped light-mass conduction electron plasma (of $n_{\\rm e} =2 \\times 10^{19} \\rm cm^{-3}$ and $\\omega _{\\rm p,e}\\gg \\omega _{\\rm LO}$ ) coupled with LO phonons [panel (a)], the $L^{(+)}(\\omega ,q)$ branch of $\\tilde{\\omega }_+(q) (\\ge \\omega _{\\rm p,e})$ is highly plasmon-like with pronounced dispersion, but the $L^{(-)}(\\omega ,q)$ mode $\\tilde{\\omega }_-(q)$ shows strong phonon-like behavior with broad peaks for frequencies between $\\omega _{\\rm LO} > \\omega >\\omega _{\\rm TO}$ .",
"Near $q=0$ , the coupled phonon–electron plasma branches begin with frequencies $\\tilde{\\omega }_-\\approx ~0.7\\omega _{\\rm LO}$ and $\\tilde{\\omega }_+\\approx ~1.7\\omega _{\\rm LO}$ .",
"We observe that, when $\\omega _{\\rm p,e} \\gg \\omega _{\\rm LO}$ , long-ranged Coulomb part of the LO mode is completely screened, in the long wave length limit, by the light-mass species (conduction electrons) becoming $\\tilde{\\omega }_-(q) \\rightarrow \\omega _{\\rm TO} (\\simeq 0.72\\omega _{\\rm LO})$ .",
"The maximum in $\\mathcal {A}_{j\\nu }^{\\rm scp}(q,\\omega )$ of the phonon-like coupled branch of $\\tilde{\\omega }_-(q)$ appears at frequencies between $\\omega _{\\rm TO}$ and $\\omega _{\\rm LO}$ approaching asymptotically to $\\omega _{\\rm LO}$ subject to Landau damping as the wave number is increased [7].",
"The higher frequency plasmon-like coupled branch of $\\tilde{\\omega }_+(q)$ starts at $q=0$ with $\\omega _{\\rm p,e} (\\sim 1.7\\omega _{\\rm LO})$ showing strong dispersive behavior of relatively weaker strength.",
"Figure: (Color Online) Effective phonon spectral function 𝒜(q,ω)\\mathcal {A}(q,\\omega ) of a two-component plasma with conduction electron concentration of 2×10 19 cm -3 2 \\times 10^{19} \\rm cm^{-3}.",
"(a) and (b): Dressed phonon spectral function at effective carrier temperatures 25 K and 300 K.(c) and (d): Cross-sectional view of species-resolved phonon spectral function 𝒜 ν (q,ω)\\mathcal {A}_{\\nu }(q,\\omega ) (ν=e, hh \\nu =\\rm e, hh) at four different values of wave number qq.Contributions via electron-phonon coupling channels of polar (LO) and nonpolar (TO) optical phonons, and of piezoacoustic (AP) and nonpolar acoustic deformation (AD) potentials are summed.On the other hand, for the case of heavy-mass hole plasma (of $n_{\\rm hh} =2 \\times 10^{19} \\rm cm^{-3}$ and $\\omega _{\\rm p,hh}\\ll \\omega _{\\rm LO}$ ) coupled with LO phonons [panel (b)], the $L^{(+)}(\\omega ,q)$ branch is highly phonon-like starting $\\tilde{\\omega }_+(q\\rightarrow 0)\\sim \\omega _{\\rm LO}$ , while the plasmon-like $L^{(-)}(\\omega ,q)$ coupled branch starts at $q=0$ with $\\tilde{\\omega }_-(0) \\sim 0.4~\\omega _{\\rm LO}~(\\ll \\omega _{\\rm TO})$ and approaches asymptotically to $\\omega _{\\rm LO}$ .",
"The latter heavy-mass species–phonon coupled mode shows larger broadening compared to the case of light-mass (electron) plasma [panel (a)].",
"The maximum in $\\mathcal {A}_{j\\nu }^{\\rm scp}(q,\\omega )$ of the phonon-like $L^{(+)}(\\omega ,q)$ coupled branch shows non-dispersive behavior of frequencies $\\sim \\omega _{\\rm LO}$ with very minor strength.",
"The behavior of $L^{(+)}(\\omega ,q)$ branch reflects the fact that, since $\\omega _{\\rm p,hh} < \\omega _{\\rm LO}$ , the heavier mass hole plasma of low bare frequency is not fast enough to screen effectively the long-ranged Coulombic part of the LO oscillation, while the phonon gas is fast enough to screen the hole plasma depressing the plasmon oscillations [35].",
"The $L^{(-)}(\\omega ,q)$ plasmonic branch is located in the region of quasi-static screening due to $\\omega _{\\rm p,hh}\\ll \\omega _{\\rm LO}$ .",
"Within the continuum region of single-particle excitations, modes are broadened and ill-defined because they are subject to Landau damping.",
"For large wave numbers outside the single-particle excitation continuum, the screening is ineffective and, hence, effectively bare modes are resumed with sharp peaks.",
"The frequency of the coupled LO modes tends to $\\omega _{\\rm LO}$ for $q \\gg q_{\\rm sc}$ .",
"(See panels (a) and (b) of Fig.",
"REF .)",
"Figure: (Color Online) Spectral behaviors of effective phonon spectral functions 𝒜(q,ω)\\mathcal {A}(q,\\omega ) of a two-component plasma (2cp) having conduction electron concentration of 3×10 19 cm -3 3 \\times 10^{19} \\rm cm^{-3} and of a three-component plasma (3cp) having conduction electron concentration of 5×10 19 cm -3 5 \\times 10^{19} \\rm cm^{-3} at effective carrier temperature 25 K.(a) and (b): Phonon spectral functions of 𝒜(q,ω)\\mathcal {A}(q,\\omega ).",
"(c) and (d): Cross-sectional view of species-resolved 𝒜 ν 2 cp (q,ω)\\mathcal {A}_{\\nu }^{\\rm 2cp}(q,\\omega ) (ν=e, hh \\nu = \\rm e,~hh) and 𝒜 ν 3 cp (q,ω)\\mathcal {A}_{\\nu }^{\\rm 3cp}(q,\\omega ) (ν=e, hh , lh \\nu = \\rm e,~hh,~lh) at three different values of qq.Contributions via electron–phonon coupling channels of polar (LO) and nonpolar (TO) optical phonons, and of piezoacoustic (AP) and nonpolar acoustic deformation (AD) potentials are added.In a multi-component solid state plasma, the phonon spectral function would reveal multi-component features of the plasma – multiple sets of phonon–plasmon coupled modes – and their spectral behavior would become so involved to be resolved in the same way as the case of the scp.",
"For optically generated plasmas of weak excitation having not too much carrier concentration, for example, $\\sim 10^{19} \\rm cm^{-3}$ , one can have a 2cp consisting of conduction electrons and heavy holes.",
"In Fig.",
"REF the effective phonon spectral function $\\mathcal {A}(q,\\omega )$ of a 2cp is shown for conduction electron concentration of $2 \\times 10^{19} \\rm cm^{-3}$ at effective carrier temperatures 25 K and 300 K, respectively.",
"Species-resolved frequency dependences are displayed separately in panels (c) and (d) at 25 K and 300 K for the purpose of spectral analysis.",
"The cross-sectional view of $\\mathcal {A}_{\\rm e}(q,\\omega )$ and $\\mathcal {A}_{\\rm hh}(q,\\omega )$ are illustrated for four different values of $q$ .",
"Two sets of LO phonon–plasmon coupled modes ($\\tilde{\\omega }_{\\pm }$ ) are seen in addition to the TO modes of $\\omega _{\\rm TO} \\simeq 0.72 \\omega _{\\rm LO}$ .",
"Boundaries of allowed single-particle excitations for electrons (holes) are designated by a pair of dashed (dotted) lines in each figure.",
"At $n_{\\rm e} =2 \\times 10^{19} \\rm cm^{-3}$ , the bare plasmon frequencies are $\\omega _{\\rm p,e}(\\epsilon _\\infty ) = 1.68\\, \\omega _{\\rm LO}$ and $\\omega _{\\rm p,hh}(\\epsilon _\\infty ) = 0.69 \\,\\omega _{\\rm LO}$ for the electrons and heavy holes, respectively, well separated from both optical phonon modes $L^{(+)}$ and $L^{(-)}$ .",
"In a 2cp, there occur both optic and acoustic plasmons, and thus one can expect not only the individual LO phonon–optic plasmon coupled spectral behaviors of lighter and heavier species as illustrated in panels (a) and (b) of Fig.",
"REF , but also the contribution from the acoustic plasmon modes [15], [7].",
"Our result shows that the multiple plasma species give substantial influence on the spectral behavior of the phonon spectral function.",
"Couplings of conduction electrons and heavy holes to both optical and acoustical phonon spectral functions appear simultaneously in the case of 2cp.",
"In panels (a) and (b), contributions to the spectral functions $\\mathcal {A}_{\\rm AD\\nu }(q,\\omega )$ of the deformation-potential induced acoustic phonon and $\\mathcal {A}_{\\rm AP\\nu }(q,\\omega )$ of piezoacoustic Coulomb interaction are also illustrated showing sharp peaks around the low-frequency bare $\\omega _{\\rm AC} (q)$ .",
"For the case of piezoelectricity induced Coulomb interaction, $A_{\\rm AP\\nu }(q,\\omega )$ shows little but slightly enhanced collisional broadening compared to $\\mathcal {A}_{\\rm AD\\nu }(q,\\omega )$ of acoustic deformation-potential mechanism.",
"The spectral intensity for the high-frequency plasmon-like $\\tilde{\\omega }_+^{\\rm (e)} (q)$ conduction electron–phonon couples modes shows strong dispersive behavior starting with $\\omega _{\\rm p,e} (\\sim 1.7\\omega _{\\rm LO})$ at $q=0$ and approaching the boundary of electronic single-particle excitation continuum.",
"The spectral intensities for phonon-like coupled branches of $\\tilde{\\omega }_-^{(\\rm e)} (q)$ and $\\tilde{\\omega }_+^{(\\rm hh)} (q)$ are of very weak dispersion with peaks at frequencies close to $\\omega _{\\rm LO}$ and $\\omega _{\\rm TO}$ .",
"Both branches show very sharp peak structures and vanish for the wave numbers beyond $q\\simeq 1.5\\,q_{\\rm sc}$ .",
"For the low-frequency plasmon-like $\\tilde{\\omega }_-^{(\\rm hh)} (q)$ phonon–heavy hole plasma coupled mode, we understand that the spectral intensity of the mode is strongly suppressed because the low-frequency heavy-mass species plasma oscillation is so heavily screened by the light-mass species (conduction electrons) that the self-sustaining oscillations of hole plasmons are not permitted any more in this quasi-static region [52].",
"We conjecture that additional little spectral strength with some broadening near the zero frequency peaked at around $q/q_{\\rm sc}\\sim 0.2$ is the so-called QPE-like branch [11], which appears in the region of finite $\\mathcal {I}m~\\Pi _{\\nu \\nu }$ and $\\mathcal {R}e~\\Pi _{\\nu \\nu } <0$ inside the continuum of single-particle excitations.",
"We observe that individual dressed branches are well resolved with slight overlap as seen in the panels (c) and (d).",
"Figure: (Color Online) Species-resolved phonon spectral function 𝒜 TO ν (q,ω)\\mathcal {A}_{\\rm TO\\nu }(q,\\omega ) (ν=e, hh , lh \\nu = \\rm e,~hh,~lh) of a three-component plasma formed by conduction electrons of concentration 5×10 19 cm -3 5 \\times 10^{19} \\rm cm^{-3}, heavy holes of concentration 4.968×10 19 cm -3 4.968 \\times 10^{19} \\rm cm^{-3}, and light holes of concentration 3.2×10 17 cm -3 3.2 \\times 10^{17} \\rm cm^{-3} at effective carrier temperature 25 K.(a) Contributions of conduction electron plasmon–TO phonon coupling.",
"(b) Contributions of heavy hole plasmon–TO phonon coupling.",
"(c) Contributions of light hole plasmon–TO phonon coupling.In Fig.",
"REF , spectral behaviors of effective phonon spectral functions $\\mathcal {A}(q,\\omega )$ are compared for the cases of 2cp having conduction electron concentration $3 \\times 10^{19} \\rm cm^{-3}$ [panels (a) and (c)] and of three-component plasma (3cp) having conduction electron concentration $5 \\times 10^{19} \\rm cm^{-3}$ [panels (b) and (d)] at effective carrier temperature 25 K. The contributions via electron–phonon coupling channels of polar LO and nonpolar TO phonons, and of piezoacoustic and nonpolar acoustic deformation potentials are summed.",
"Cross-sectional views of $\\mathcal {A}_{\\nu }^{\\rm 2cp}(q,\\omega )$ ($\\nu =\\rm e,~ hh$ ) for $3 \\times 10^{19} \\rm cm^{-3}$ and $\\mathcal {A}_{\\nu }^{\\rm 3cp}(q,\\omega )$ ($\\nu =\\rm e, ~hh, ~lh$ ) for $5 \\times 10^{19} \\rm cm^{-3}$ are illustrated in panels (c) and (d), respectively, at four different values of wave number $q$ .",
"In a mcp of increased carrier concentrations, the compressive (optical) plasmon-like coupled modes begin with higher frequencies (for example, $\\omega \\ge 3 \\omega _{\\rm LO}$ for $3 \\times 10^{19} \\rm cm^{-3}$ ) at $q=0$ , and the corresponding branches of very weak strengths are seen at $\\omega \\sim 3.2 \\omega _{\\rm LO}$ for $3 \\times 10^{19} \\rm cm^{-3}$ [panels (a) and (c)] and $\\omega \\sim 4.2 \\omega _{\\rm LO}$ for $5 \\times 10^{19} \\rm cm^{-3}$ [panel (d)] at $q\\simeq 0.2q_{\\rm sc}$ .",
"At 25 K, $q_{\\rm sc} = 3.15\\times 10^7 \\rm cm^{-1}$ ($3.49\\times 10^7 \\rm cm^{-1}$ ) for an electron-hole plasma of conduction electron concentrations $3 \\times 10^{19} \\rm cm^{-3}$ ($5 \\times 10^{19} \\rm cm^{-3}$ ).",
"The spectral behavior shown in panels (a) and (c) for $3 \\times 10^{19} \\rm cm^{-3}$ is very similar to a weakly excited case of $2 \\times 10^{19} \\rm cm^{-3}$ shown in Fig.",
"REF (a) and (c), except the fact that the branch $\\tilde{\\omega }_+^{\\rm (e)} (q)$ occurs at much increased frequency, for example, $\\omega \\sim 3.2~\\omega _{\\rm LO}$ at $q\\simeq 0.2 q_{\\rm sc}$ .",
"For a photo-generated 3cp having conduction electrons of $5 \\times 10^{19} \\rm cm^{-3}$ at effective carrier temperature 25 K, the charge carriers of the plasma can be resolved into conduction electrons of concentration $5 \\times 10^{19} \\rm cm^{-3}$ , heavy holes of concentration $4.968 \\times 10^{19} \\rm cm^{-3}$ , and light holes of concentration $3.2 \\times 10^{17} \\rm cm^{-3}$ .",
"Species-resolved phonon spectral function $\\mathcal {A}_{\\rm LO\\nu }(q,\\omega )$ and $\\mathcal {A}_{\\rm TO\\nu }(q,\\omega )$ of a 3cp ($\\nu $ =e, hh, and lh) with conduction electron concentration $5 \\times 10^{19} \\rm cm^{-3}$ are displayed in Figs.",
"REF and REF .",
"Boundaries of each individual allowed single-particle excitations are designated by a pair of dashed lines in each panel.",
"It is seen that, with the present choice of sample parameters, the effects of collisional broadening through carrier screening are not significant on the nonpolar phonon spectral functions $\\mathcal {A}_{\\rm TO\\nu }(q,\\omega )$ ($\\nu =\\rm e,~ hh,~ lh$ ) and $\\mathcal {A}_{\\rm LOlh}(q,\\omega )$ because of very low concentration of light hole carriers.",
"Figure: (Color Online) Phonon self-energy correction 𝒫 LOe (q,ω)(≡Δ LOe -iΓ LOe /2)\\mathcal {P}_{\\rm LOe}(q,\\omega )(\\equiv \\Delta _{\\rm LOe}-i\\Gamma _{\\rm LOe}/2) in a two-component plasma via LO phonon–conduction electron coupling for concentration of 2×10 19 cm -3 2 \\times 10^{19} \\rm cm^{-3} for each species.",
"(a) and (b): Real part of the self-energy correction Δ LOe (q,ω)\\Delta _{\\rm LOe}(q,\\omega ) at 25 K and 300 K, respectively.",
"(c) and (d): Imaginary parts of the self-energy correction Γ(q,ω)\\Gamma (q,\\omega ) at 25 K and 300 K, respectively.Figure: (Color Online) Cross-sectional view of renormalized phonon modes ω ˜ LOe + (q,ω){\\tilde{\\omega }}_{\\rm LOe}^+(q,\\omega ) and ω ˜ LOe - (q,ω){\\tilde{\\omega }}_{\\rm LOe}^-(q,\\omega ) for electron concentrations of 2×10 19 cm -3 2 \\times 10^{19} \\rm cm^{-3}(a) and (b): Behavior at representative values of frequency ω\\omega at 25 K and 300 K, respectively.",
"(c) and (d): Behavior at representative values of wave number qq at 25 K and 300 K, respectively.In Fig.REF , the frequency and wave-number dependences of phonon self-energy correction $\\mathcal {P}_{\\rm LOe}(q,\\omega )[\\equiv \\Delta _{\\rm LOe}(q,\\omega )-i\\Gamma _{\\rm LOe}(q,\\omega )/2]$ are shown at 25 K and 300 K, respectively, for a photo-generated 2cp formed of conduction electrons and heavy holes each with concentration $2 \\times 10^{19} \\rm cm^{-3}$ .",
"Contributions of the light-mass species (conduction electrons) $\\Delta _{\\rm LOe}(q,\\omega )$ and $\\Gamma _{\\rm LOe}(q,\\omega )$ are illustrated, respectively, in the $\\omega $ -$q$ space.",
"The contours of $\\Delta _{\\rm LOe}(q,\\omega )=0$ are indicated by thin solid lines in panels (a) and (b).",
"The spectral behaviors of $\\mathcal {P}_{j\\nu }(q,\\omega )$ reveals very much similar structure as that of the dressed polarization function $\\Pi _{\\nu \\nu }(q,\\omega )$ of the plasma species, since $\\Delta _{\\rm j\\nu }$ and $\\Gamma _{\\rm j\\nu }$ are proportional to $\\mathcal {R}e~\\Pi _{\\nu \\nu }$ and $\\mathcal {I}m~\\Pi _{\\nu \\nu }$ , respectively.",
"[See Fig.",
"REF (a) and (c).]",
"The sign of $\\Delta _{j\\nu }(q,\\omega )$ is the same as that of ${R}e~\\Pi _{\\nu \\nu }(q,\\omega )$ .",
"The real part of the self-energy correction $\\Delta _{\\rm LOe}(q,\\omega )$ also changes signs crossing the zeros of $\\mathcal {R}e~ \\Pi _{\\nu \\nu }(q,\\omega )$ with peaks just above and below the branches of high-frequency optic and low-frequency acoustic plasmon modes.",
"On the other hand, $\\Gamma _{j\\nu }(q,\\omega )$ is defined to be nonnegative and shows peak structure along the well-defined high-frequency compressive optic and low-frequency acoustic plasmon branches near the plasma cut-offs.",
"We observe that the effect of plasma temperature on the phonon frequency renormalization [panels (a) and (b)] is moderate, but the broadening in the self-energy correction [panels (c) and (d)] is more pronounced at higher temperature as was also seen in the case of the spectral function $\\mathcal {A}(q,\\omega )$ illustrated in Fig.",
"REF (a) and (b).",
"The phonon self-energy corrections $\\mathcal {P}_{\\rm LOe}(q,\\omega )$ of a photo-generated mcp for two different conduction electron concentrations $n_{\\rm e}=3 \\times 10^{19} \\rm cm^{-3}$ and $5 \\times 10^{19} \\rm cm^{-3}$ at 25 K are given in Appendix B.",
"(See Fig.",
"REF .)",
"In the case of conduction electron concentration $3 \\times 10^{19} \\rm cm^{-3}$ [panels (a) and (c) of Fig.",
"REF ], the plasma consists of conduction electrons and heavy holes forming a 2cp and the spectral behavior of phonon self-energy correction $\\mathcal {P}_{\\rm LOe}(q,\\omega )$ is very close to that of conduction electron concentration $2 \\times 10^{19} \\rm cm^{-3}$ shown in Fig.",
"REF .",
"Compelling difference is that the bare frequencies of both optic and acoustic plasmon modes are increased accordingly in a mcp as the carrier concentration is increased.",
"For a plasma of conduction electron concentration $5 \\times 10^{19} \\rm cm^{-3}$ , the light hole band is also occupied becoming a 3cp of $n_{\\rm hh}=4.968 \\times 10^{19} \\rm cm^{-3}$ and $n_{\\rm lh}=3.2 \\times 10^{17} \\rm cm^{-3}$ .",
"Panels (b) and (d) of Fig.",
"REF illustrate the spectral behavior of $\\Delta _{\\rm j\\nu }$ and $\\Gamma _{\\rm j\\nu }$ , respectively, of a 3cp with conduction electron concentration $n_{\\rm e}=5 \\times 10^{19} \\rm cm^{-3}$ .",
"The effect of light hole carriers on $\\mathcal {R}e~\\Pi _{\\rm ee}$ and $\\mathcal {I}m~\\Pi _{\\rm ee}$ and, thus, to the LO phonon self-energy corrections $\\Delta _{\\rm LOe}$ and $\\Gamma _{\\rm LOe}$ are found to be negligible due to relatively too small concentration of light holes.",
"In Fig.",
"REF , spectral behaviors of the renormalized LO phonon frequencies ${\\tilde{\\omega }}_{\\rm LOe}^+(q,\\omega )$ and ${\\tilde{\\omega }}_{\\rm LOe}^-(q,\\omega )$ are shown for a photo-generated 2cp of electron concentration $2 \\times 10^{19} \\rm cm^{-3}$ at 25 K and 300 K, respectively.",
"The primary mode ${\\tilde{\\omega }}_{\\rm LOe}^+(q,\\omega )$ occurs in the domain of $2\\Delta _{\\rm LOe} > -\\omega _{\\rm LO}$ , while the mode ${\\tilde{\\omega }}_{\\rm LOe}^-(q,\\omega )$ in the domain of $2\\Delta _{\\rm LOe} < -\\omega _{\\rm LO}$ only with finite values of $\\Gamma _{\\rm LOe}$ and negative $\\Delta _{\\rm LOe}$ .",
"The latter mode is expected to be observable in the case of large $\\Gamma _{\\rm LOe}$ .",
"In Fig.",
"REF , renormalized phonon modes $\\tilde{\\omega }_{\\rm LOe}^+(q,\\omega )$ and $\\tilde{\\omega }_{\\rm LOe}^-(q,\\omega )$ shown in Fig.",
"REF are resolved for representative values of frequency $\\omega $ [panel (a) and (b)] and of wave number $q$ [panel (c) and (d)], respectively.",
"The frequency and wave number are scaled by the longitudinal bare phonon frequency $\\omega _{\\rm LO}$ and Thomas–Fermi screening wave number $q_{\\rm sc}$ , respectively.",
"We note that $\\tilde{\\omega }_{\\rm LOe}^+(q,\\omega )\\simeq \\omega _{\\rm LO}$ over most of the domain satisfying the condition $2 \\Delta _{\\rm LOe}(q,\\omega ) > -{\\omega _{\\rm LO}}$ in the $\\omega $ –$q$ plane except near the regions of well-defined plasmonic collective modes becoming $\\tilde{\\omega }_{\\rm LOe}^+(q,\\omega ) >> \\omega _{\\rm LO}$ .",
"Below the small opening gap [as indicated by white blank in panel (a)] of $2 \\Delta _{\\rm LOe}(q,\\omega ) < -{\\omega _{\\rm LO}}$ , we observe the secondary mode $\\tilde{\\omega }_{\\rm LOe}^-(q,\\omega ) < \\omega _{\\rm LO}$ over the domain designated in the insets.",
"The mode $\\tilde{\\omega }_{\\rm LOe}^-(q,\\omega )$ is expected in the region of $2 \\Delta _{\\rm LOe}(q,\\omega ) < -{\\omega _{\\rm LO}}$ with finite values of $\\Gamma _{\\rm LOe}(q,\\omega )$ , as shown in Fig.",
"REF (c) and (d).",
"Spectral behaviors of the renormalized phonon frequencies ${\\tilde{\\omega }}_{\\rm LOe}^+(q,\\omega )$ and ${\\tilde{\\omega }}_{\\rm LOe}^-(q,\\omega )$ are compared in Appendix B for a 2cp of conduction electron concentrations $3 \\times 10^{19} \\rm cm^{-3}$ and a 3cp of $5 \\times 10^{19} \\rm cm^{-3}$ , respectively.",
"(See Fig.",
"REF .)",
"The results illustrated in Fig.",
"REF (a) and (b) are very similar to that of electron concentration $2 \\times 10^{19} \\rm cm^{-3}$ shown in Fig.",
"REF (a) except the shifts in locations of the extrema along the frequency axis at a given effective carrier temperature.",
"Cross-sectional views of the renormalized phonon modes ${\\tilde{\\omega }}_{\\rm LOe}^+(q,\\omega )$ and ${\\tilde{\\omega }}_{\\rm LOe}^-(q,\\omega )$ are given in Appendix B for representative values of frequency and wave number.",
"(See Fig.",
"REF .)",
"At increased conduction electron concentration of $5 \\times 10^{19} \\rm cm^{-3}$ , the frequencies of collective modes of plasmon-phonon coupling shift to higher values, but the presence of light hole carriers of minor concentration is not seen to give observable effects in the spectral behavior of the phonon renormalization."
],
[
"Summary and Conclusion",
"We have investigated spectral behavior of the dielectric response and phonon spectral functions of a mcp in an extended random phase approximation.",
"The effects of dynamic screening, plasmon–phonon coupling, and exchange–correlations of the plasma species are included.",
"We have applied the formulation to the case of a photo-generated electron–hole plasma formed in an ideal wurtzite GaN, and scattering channels of various carrier-phonon couplings are considered.",
"Clear significance of the multiplicity of the plasma species is shown in the dielectric response and phonon spectral behaviors of a mcp.",
"From the comparative study of the responses and phonon spectral functions of a mcp with that of a scp, we find following conclusions: Dynamic screening and plasmon–phonon coupling are essential in understanding the spectral behavior of phonon spectral functions in a mcp.",
"The effects of exchange and correlations among the carriers of the plasma are not seen significant for carrier concentrations $\\sim 10^{19} \\rm cm^{-3}$ shifting slightly down the frequencies of the optic and acoustic plasmon oscillations in the plasma.",
"By extending linear response calculation to a mcp, we have detailed the spectral behavior of the dressed polarization functions $\\Pi _{\\mu \\nu }$ ($\\nu $ =e, hh) and examined plasma species-resolved dielectric functions $\\epsilon _{\\mu \\nu }$ .",
"A sum rule of $\\sum _{\\mu \\nu } (\\epsilon _{\\mu \\nu }^{-1}-\\delta _{\\mu \\nu }) =0$ is found and multi-component character of the plasmonic oscillations was investigated.",
"From the zeros of the effective dielectric function $\\mathcal {R}e ~\\epsilon _{\\rm eff}(q,\\omega )$ of a 2cp, it is found that optic and acoustic plasmon branches are well separated from each other.",
"Hubbard-like local-field corrections of carriers are found to shift both branches slightly to reduced plasmon frequencies for given values of wave number.",
"In a two-component electron–hole plasma, the higher frequency electron plasmon mode is almost intact and well defined near the bare plasma frequency $\\omega _{\\rm p,e}$ .",
"However, the lower frequency bare plasmon mode of heavier species (heavy holes) at $\\omega _{\\rm p,hh} $ is heavily screened by the lighter mass species (conduction electrons) resulting in an acoustic branch positioned inside the electron excitation continuum, the latter mode being subject to Landau damping by the light-mass species.",
"$\\mathcal {I}m ~\\epsilon _{\\rm eff}(q,\\omega )$ reveals double peak structure in a 2cp each peak appearing inside the single-particle excitation continua of electrons and holes of the plasma, unlike the case of a scp.",
"The effects of local-field corrections of the carriers are appreciable only in the region of low frequency and long wavelength.",
"Dressed phonon propagators are evaluated in the $\\omega $ –$q$ plane and we have demonstrated that the dielectric screening in many carrier system gives rise to renormalized electron–phonon coupling modifying the phonon dispersion relations along with phonon spectral function.",
"The phonon frequency renormalization $\\Delta _{j\\nu }(q,\\omega )$ and the phonon broadening $\\Gamma _{j\\nu }(q,\\omega )$ are analyzed in terms of $\\mathcal {R}e~\\Pi _{\\nu \\nu }(q,\\omega )$ and $\\mathcal {I}m~\\Pi _{\\nu \\nu }(q,\\omega )$ , respectively, and it is found that the effect of plasma temperature on the phonon frequency renormalization is moderate, but the phonon broadening is more pronounced at higher temperature.",
"We have demonstrated that the plasmon–LO phonon coupling gives a pair of branches $L^{(+)}(\\omega ,q)$ and $L^{(-)}(\\omega ,q)$ and, in a 2cp, two sets of LO phonon–plasmon coupled modes ($\\tilde{\\omega }_{\\pm }$ ) are seen.",
"The spectral intensity for the high-frequency plasmon-like coupled mode $\\tilde{\\omega }_+^{\\rm (e)} (q)$ shows strong dispersive behavior starting with $\\omega _{\\rm p,e}$ at $q=0$ and approaching the boundary of electronic single-particle excitation continuum.",
"The spectral intensities for phonon-like coupled branches of $\\tilde{\\omega }_-^{(\\rm e)} (q)$ and $\\tilde{\\omega }_+^{(\\rm hh)} (q)$ are of very weak dispersion with peaks at frequencies close to $\\omega _{\\rm LO}$ and $\\omega _{\\rm TO}$ .",
"The low-frequency plasmon-like coupled mode $\\tilde{\\omega }_-^{(\\rm hh)}(q)$ is strongly suppressed so that the self-sustaining oscillations of hole plasmons are not permitted any more in this quasi-static region.",
"Moreover, we have observed an additional peak of little spectral strength at $q\\sim 0.2~q_{\\rm sc}$ near the zero frequency in the domain of finite $\\mathcal {I}m ~\\Pi _{\\nu \\nu }$ and $\\mathcal {R}e ~\\Pi _{\\nu \\nu } <0$ inside the continuum of single-particle excitations.",
"The origin for the little peak is conjectured to be the so-called QPE-like branch [11].",
"In a 3cp of conduction electron concentration $5 \\times 10^{19} \\rm cm^{-3}$ , the effects of light hole carriers on $\\mathcal {R}e~\\Pi _{\\rm ee}$ and $\\mathcal {I}m~\\Pi _{\\rm ee}$ and, thus, to the LO phonon self-energy corrections $\\Delta _{\\rm LOe}$ and $\\Gamma _{\\rm LOe}$ are found to be negligible due to relatively too small concentration of light holes.",
"The spectral behaviors of the phonon self-energy correction reveals very much similar structure as that of the dressed polarization function $\\Pi _{\\nu \\nu }(q,\\omega )$ of the plasma species.",
"In conclusion, in a multi-component solid state plasma, multiple plasma species give substantial influences on the spectral behavior of the phonon spectral function, which is very distinct from that in a scp commonly seen in doped semiconductors.",
"The results of our computations are suitable for experimental observation, and we hope that the spectral behaviors demonstrated in the present work would be confirmed with various energy loss scattering measurements and hot carrier spectroscopies on photo-generated electron–hole plasmas in polar semiconductors.",
"Meaningful informations on the spectral behavior of a mcp with thermal and collisional broadening effects would be obtained by comparing experimental spectra with the results presented in the present paper.",
"The acoustic plasmon-like coupled modes of low frequency at long wavelength may be tested by ultrasonic measurements.",
"The authors acknowledge the support in part by Basic Science Research Program (NRF-2013R1A1A4A01004433) through the NRF of Korea.",
"One of the authors (HJK) acknowledges support by Priority Research Centers Program (2009-0093818) through the NRF of Korea.",
"We are grateful for stimulating discussions with E.H. Hwang on many-body correlations in a solid.",
"Figure: (Color Online) Phonon self-energy corrections 𝒫 LOe (q,ω)(≡Δ LOe -iΓ LOe /2)\\mathcal {P}_{\\rm LOe}(q,\\omega )(\\equiv \\Delta _{\\rm LOe}-i\\Gamma _{\\rm LOe}/2) for a two-component plasma and a three-component plasma at 25 K.(a) and (c): Real and imaginary parts of the self-energy correction for a two-component plasma with conduction electron concentration 3×10 19 cm -3 3 \\times 10^{19} \\rm cm^{-3}.",
"(b) and (d): Real and imaginary parts of the self-energy correction for a three-component plasma with conduction electron concentration 5×10 19 cm -3 5 \\times 10^{19} \\rm cm^{-3}."
],
[
"Matrix elements of carrier-phonon coupling",
"Here we briefly summarize matrix elements of carrier–phonon coupling channels employed in our discussion of spectral behavior of phonon spectral functions.",
"Of various channels of carrier–phonon scattering, we consider four distinct carrier-phonon coupling channels would-be important in compound semiconductors such as wurtzite GaN material [53]: couplings to polar longitudinal optical (LO) and nonpolar transverse optical (TO) phonons and couplings through acoustic deformation potential and piezoelectricity.",
"In polar crystals, the longitudinal modes due to Fröhlich-type Coulombic interaction induces a long ranged electrical polarization field.",
"For the case of carrier–polar optical phonon coupling, the squared coupling matrix element is given by [43] $|M_{q,\\nu }^{\\mathrm {LO}}(q)|^2 = \\frac{4\\pi \\hbar }{q^2} \\sqrt{\\frac{\\hbar ^3\\omega _{\\rm LO}^3}{2m_\\nu }} \\alpha ,$ where $\\alpha $ is the dimensionless Fröhlich coupling constant $\\alpha =\\frac{e^2}{\\hbar }\\sqrt{m_{\\nu }/(\\hbar \\omega _{\\rm LO})}\\left(\\frac{1}{\\epsilon _\\infty }-\\frac{1}{\\epsilon _0}\\right)$ , and $\\epsilon _\\infty $ and $\\epsilon _0$ are, respectively, the optical and static dielectric constants of the material.",
"In GaN, much enhanced carrier–polar phonon interaction is expected due to its higher ionicity giving, for example, the Fröhlich coupling constant $\\alpha (\\rm GaN) \\sim 6 \\alpha (\\rm GaAs)$ with $\\omega _{\\rm LO} (\\rm GaN) \\sim 3 \\omega _{\\rm LO}(\\rm GaAs)$ [54].",
"For crystals lacking a center of symmetry, acoustic phonons also induce an electric polarization field giving rise to piezoelectricity through Coulombic interaction.",
"The piezoelectricity is most commonly found in the wurtzite structure, and the matrix element $M_{q,\\nu }^{\\mathrm {AP}}(q)$ for piezoelectric acoustic (AP) phonon coupling with frequency $\\omega (q)$ and velocity $s(q)$ is given by [35] $\\mid M_{q,\\nu }^{\\mathrm {AP}}(q)\\mid ^2 = \\frac{2\\hbar s(q)}{\\rho _{\\nu } \\omega (q) q^2 \\epsilon ^2 (q)}\\biggl [\\sum _{ijk} q_k e_{k,ij} \\xi _i(\\lambda ,q ) q_j \\biggr ]^2.$ Here $\\rho _{\\nu }$ , $\\epsilon (q)$ , $\\xi (\\lambda ,q)$ , and $e_{k,ij}$ denote the average mass density, dielectric constant of the material, the unit vector of the acoustic lattice polarization $\\lambda $ (= TA or LA), and the piezoelectric coupling constants of a third rank tensor, respectively [35].",
"Because strain modifies local band structure of the material, the deformation potentials can also induce carrier–nonpolar TO and acoustical phonon couplings.",
"The matrix element $M_{q,\\nu }^{\\mathrm {TO}}(q)$ for nonpolar optical phonon coupling is given, in terms of optical deformation potential, by [53] $\\mid M_{q,\\nu }^{\\mathrm {TO}}(q)\\mid ^2 = \\frac{\\hbar \\mathcal {D}^2}{ 2 \\rho _{\\nu } \\omega _{\\mathrm {TO}}\\mathcal {V}}.$ Here $\\mathcal {D}$ and $\\mathcal {V}$ are the optical deformation potential constant and the volume of the sample, respectively.",
"On the other hand, the matrix element $M_{q,\\nu }^{\\mathrm {AD}}(q)$ for acoustical deformation potential scattering is written as [53], [55] $\\mid M_{q,\\nu }^{\\mathrm {AD}}(q)\\mid ^2 = \\frac{\\hbar \\mathcal {E}_\\nu ^2 q^2}{ 2 \\rho _{\\nu } \\omega _q\\mathcal {V}},.$ Here $\\mathcal {E}_\\nu $ is the deformation potential constants of the longitudinal acoustical waves for the carriers in the conduction and valence bands."
],
[
"Spectral behavior of phonon self-energy corrections and renormalized phonon modes of two-component and three-component plasmas",
"In a photo-generated electron–hole plasma, the number of plasma species can be modulated as a function of the carrier concentration $n_{\\rm e}$ in the conduction band.",
"For higher values of $n_{\\rm e}$ , both valence bands of heavy holes and light holes can be occupied giving rise to 3cp.",
"For a plasma of conduction electron concentration $5 \\times 10^{19} \\rm cm^{-3}$ , the light hole band is also occupied becoming a 3cp of $n_{\\rm hh}=4.968 \\times 10^{19} \\rm cm^{-3}$ and $n_{\\rm lh}=3.2 \\times 10^{17} \\rm cm^{-3}$ .",
"Phonon self-energy corrections $\\mathcal {P}_{\\rm LOe}(q,\\omega )$ of a photo-generated mcp for two different conduction electron concentrations $n_{\\rm e}=3 \\times 10^{19} \\rm cm^{-3}$ and $5 \\times 10^{19} \\rm cm^{-3}$ at 25 K are given in Fig.",
"REF .",
"In the case of conduction electron concentration $3 \\times 10^{19} \\rm cm^{-3}$ [panels (a) and (c) of Fig.",
"REF ], the plasma consists of conduction electrons and heavy holes forming a 2cp and the spectral behavior of phonon self-energy correction $\\mathcal {P}_{\\rm LOe}(q,\\omega )$ is very close to that of conduction electron concentration $2 \\times 10^{19} \\rm cm^{-3}$ shown in Fig.",
"REF .",
"Noticeable difference is that the bare frequencies of both optic and acoustic plasmon modes are increased accordingly in a mcp as the carrier concentration is increased.",
"Panels (b) and (d) of Fig.",
"REF illustrate the spectral behavior of $\\Delta _{\\rm j\\nu }$ and $\\Gamma _{\\rm j\\nu }$ , respectively, of a 3cp with conduction electron concentration $n_{\\rm e}=5 \\times 10^{19} \\rm cm^{-3}$ .",
"The effect of light hole carriers on $\\mathcal {R}e~\\Pi _{\\rm ee}$ and $\\mathcal {I}m~\\Pi _{\\rm ee}$ and, thus, to the LO phonon self-energy corrections $\\Delta _{\\rm LOe}$ and $\\Gamma _{\\rm LOe}$ are found to be negligible due to relatively too small concentration of light holes.",
"Spectral behaviors of renormalized phonon frequencies ${\\tilde{\\omega }}_{\\rm LOe}^+(q,\\omega )$ and ${\\tilde{\\omega }}_{\\rm LOe}^-(q,\\omega )$ are compared in Fig.",
"REF for a 2cp of conduction electron concentrations $3 \\times 10^{19} \\rm cm^{-3}$ and a 3cp of $5 \\times 10^{19} \\rm cm^{-3}$ , respectively.",
"In panel (b), boundaries of the continuum of single-particle excitations for light holes are indicated by a pair of dash-dotted lines.",
"Figure: (Color Online) Spectral behavior of the renormalized phonon frequencies ω ˜ LOe ± (q,ω){\\tilde{\\omega }}_{\\rm LOe}^{\\pm }(q,\\omega ) in a photo-generated plasma at 25 K of conduction electron concentrations (a) 3×10 19 cm -3 3 \\times 10^{19} \\rm cm^{-3} and (b) 5×10 19 cm -3 5 \\times 10^{19} \\rm cm^{-3}.Figure: (Color Online) Wave-number and frequency dependences of renormalized phonon modes ω ˜ LOe ± (q,ω){\\tilde{\\omega }}_{\\rm LOe}^{\\pm }(q,\\omega ) at 25 K.(a) and (b): Behavior for representative values of frequency ω\\omega for electron concentrations of 3×10 19 cm -3 3 \\times 10^{19} \\rm cm^{-3} and 5×10 19 cm -3 5 \\times 10^{19} \\rm cm^{-3}, respectively.",
"(c) and (d): Behavior for representative values of wave number qq for electron concentrations of 3×10 19 cm -3 3 \\times 10^{19} \\rm cm^{-3} and 5×10 19 cm -3 5 \\times 10^{19} \\rm cm^{-3}, respectively.The frequency and wave number are scaled by the longitudinal bare phonon frequency $\\omega _{\\rm LO}$ and Thomas–Fermi screening wave number $q_{\\rm sc}$ , respectively.",
"For a photo-generated plasma with electron concentration $5 \\times 10^{19} \\rm cm^{-3}$ , the presence of additional carriers of concentration $3.2 \\times 10^{17} \\rm cm^{-3}$ in the light hole band gives rise to the third species making the plasma a three-component one.",
"The results illustrated in Fig.",
"REF (a) and (b) are very similar to that of electron concentration $2 \\times 10^{19} \\rm cm^{-3}$ shown in Fig.",
"REF (a), the locations of the extrema are shifted along the frequency axis at a given effective carrier temperature.",
"In Fig.",
"REF , cross-sectional views of the renormalized phonon modes ${\\tilde{\\omega }}_{\\rm LOe}^+(q,\\omega )$ and ${\\tilde{\\omega }}_{\\rm LOe}^-(q,\\omega )$ shown in Fig.",
"REF are illustrated for representative values of frequency and wave number.",
"In panels (a) and (b), the wave-number dependences of ${\\tilde{\\omega }}_{\\rm LOe}^{\\pm }(q,\\omega )$ are shown for constant values of frequency $\\omega $ at 25 K, while the frequency dependences at representative values of wavenumber $q$ are shown in panels (c) and (d) for electron concentrations of $3 \\times 10^{19} \\rm cm^{-3}$ and $5 \\times 10^{19} \\rm cm^{-3}$ , respectively.",
"For conduction electron concentration of $5 \\times 10^{19} \\rm cm^{-3}$ , the frequencies of plasmon–phonon coupled modes shift to higher values, but the presence of light hole carriers of minor concentration is not seen to give observable effects in the spectral behavior of the phonon renormalization."
]
] | 1605.05817 |
[
[
"Observational challenges in Ly-alpha intensity mapping"
],
[
"Abstract Intensity mapping (IM) is sensitive to the cumulative line emission of galaxies.",
"As such it represents a promising technique for statistical studies of galaxies fainter than the limiting magnitude of traditional galaxy surveys.",
"The strong hydrogen Ly-alpha line is the primary target for such an experiment, as its intensity is linked to star formation activity and the physical state of the interstellar (ISM) and intergalactic (IGM) medium.",
"However, to extract the meaningful information one has to solve the confusion problems caused by interloping lines from foreground galaxies.",
"We discuss here the challenges for a Ly-alpha IM experiment targeting z > 4 sources.",
"We find that the Ly-alpha power spectrum can be in principle easily (marginally) obtained with a 40 cm space telescope in a few days of observing time up to z < 8 (z = 10) assuming that the interloping lines (e.g.",
"H-alpha, [O II], [O III] lines) can be efficiently removed.",
"We show that interlopers can be removed by using an ancillary photometric galaxy survey with limiting AB mag 26 in the NIR bands (Y, J, H, or K).",
"This would enable detection of the Ly-alpha signal from 5 < z < 9 faint sources.",
"However, if a [C II] IM experiment is feasible, by cross-correlating the Ly-alpha with the [C II] signal the required depth of the galaxy survey can be decreased to AB mag 24.",
"This would bring the detection at reach of future facilities working in close synergy."
],
[
"Introduction",
"One of the key open problems in cosmology is the origin and evolution of galaxies and their stars.",
"In the last decade astonishing technological progresses have allowed to probe galaxies located within less than one billion year from the Big Bang [4], [37], [38], [40], [39], [35].",
"These searches reveal an early Universe in which complex phenomena were simultaneously taking place, ranging from the formation of supermassive black holes [58] to the reionization process, [1], along with the metal enrichment by the first stars [21].",
"High redshift sources are very faint and their detection is remarkably challenging: up to now, less than 1000 galaxies have been detected at $z $ > $$ 8$, and among them only a handful are at $ z 10$ (e.g.",
"\\cite {2014arXiv1403.4295B,2016arXiv160205199M,2016ApJ...817..120C}).",
"Moreover, it is believed that low-mass galaxies have a dominant role \\cite {2011MNRAS.414..847S} in driving reionization, while the most-luminous ones appear to be only rare outliers.",
"Such ultra-faint galaxies are likely to remain undetected even by the next generation observatories, such as JWST\\footnote {\\url {http://www.jwst.nasa.gov}}, TMT\\footnote {\\url {http://www.tmt.org}} or E-ELT\\footnote {\\url {https://www.eso.org/sci/facilities/eelt/}}.$ A novel approach has been proposed to overcome the problem and study, at least statistically, the early faint galaxy population.",
"Basically the idea is to trade the ability to resolve individual sources, with a statistical analysis of the cumulative signal produced by the entire population [29], [14].",
"Intensity mapping (IM, see e.g.",
"[56], [57]) is one implementation of such concept and aims at detecting 3D large scale emission line fluctuations.",
"In the last years this concept has become very popular and several lines have been proposed as candidates.",
"Among these are the HI 21cm [22], CO [33], [45], [7] , C $\\scriptstyle \\rm II$ [24], [49], [61], H$_2$ [23], HeII [55] and Ly$\\alpha $ [44], [50], [12] emission lines.",
"Although IM experiments seem indeed promising, their reliability has not yet been convincingly demonstrated.",
"In particular, continuum foregrounds dominate over line intensity by several orders of magnitude: cleaning algorithms have been developed for 21cm radiation [59], [11], [60], but not comparably well understood for other lines [61].",
"Moreover, some lines (such as Ly$\\alpha $ and FIR emission lines) suffer from line confusion: for example the H$_\\alpha $ line ($\\lambda _{{\\rm H}_\\alpha } = 0.6563~\\mu $ m) if emitted at $z = 0.48$ can be misclassified as a Ly$\\alpha $ line emitted at $z=7$ [25].",
"We will refer to such intervening sources as interlopers.",
"Considering that the first generation of instruments devoted to IM are starting to be proposed or funded [18], [15], [16], it is essential to gain a deeper understanding of the difficulties implied by an IM experiment.",
"This forms the motivation of this work and we will pay particular attention to the Ly$\\alpha $ emission line which is the most luminous UV line and one of the most promising candidates for an IM survey in the near infrared (NIR) spectral region.",
"Ly$\\alpha $ emission is associated with UV and ionizing radiation and therefore is strongly correlated with the star formation rate (SFR) in galaxies.",
"Moreover, the reprocessing of UV photons by neutral hydrogen in the IGM also produces Ly$\\alpha $ photons.",
"Some recent works have predicted the power spectrum (PS) of the target line and assessed its observability.",
"[44] and [50] developed analytical models for the Ly$\\alpha $ PS and showed that it is at reach of a small space instrument.",
"[25] used the model developed by [50] to study the problem of line confusion, finding that masking bright voxels can represent a viable strategy.",
"In a similar attempt, [8] pointed out that masking bright voxels is an effective strategy for the removal of the interlopers, but it might jeopardize the recovered line PS, causing loss of astrophysical information.",
"A realistic Ly$\\alpha $ model has to deal with all the astrophysics processes (e.g.",
"star formation, radiative transfer) self-consistently.",
"This is rather challenging even for high resolution hydrodynamic simulations.",
"Alternatively, a viable strategy for studying such complex processes is to develop an analytical model that includes all the theoretical uncertainties represented by a few parameters: in this way it is possible to understand easily how the results depends on the unknowns and what is the available parameter space of the problem yielding solution compatible with existing observations.",
"[12] (hereafter CF16) developed an analytical model for diffuse Ly$\\alpha $ intensity and its PS, with a focus on IM at the epoch of reionization (EoR).",
"The model is observation-driven and it includes the most recent determinations both for galaxies and IGM.",
"They associated dust-corrected UV luminosity to dark matter halos by the abundance matching technique [13], [3], [51], using the LF from the Hubble legacy fields [6], and the UV luminosity spectral slope in [4].",
"Then using a template spectral energy distribution (SED) from starburst99http://www.stsci.edu/science/starburst99/docs/default.htm [32], [54], [31] and the Calzetti extinction law [10] they were able to model self-consistently the interaction of ionizing photons with the interstellar medium (ISM) and the IGM, calibrating the poorly constrained parameters in order to have a realistic reionization history [42], [20].",
"CF16 found that for Ly$\\alpha $ absolute intensity is dominated by recombinations in ISM, and Lyman continuum absorption and relaxation in the IGM, with the latter being about a factor 2 stronger.",
"However, intensity fluctuations are mostly contributed by the ISM emission on all scales $<100~h^{-1}$ Mpc.",
"Such scale essentially corresponds to the distance at which UV photons emitted by galaxies are redshifted into Ly$\\alpha $ resonance.",
"We present in the following a feasibility study of a Ly$\\alpha $ IM survey based on CF16 results.",
"In particular, we tackle the problem of (i) required sensitivity; (ii) suppression of line confusion through interlopers removal; (iii) detectability of the cross-correlation with the C $\\scriptstyle \\rm II$ line.",
"The paper is organized as follows: in Sec.",
"we compute in a general way the signal-to-noise ratio (S/N) of an IM observation; in Sec.",
"we model the sensitivity of an intensity mapper and compute the S/N of an observation; in Sec.",
"we analyse the problem of line confusion.",
"Sec.",
"contains a study of the cross-correlation between Ly$\\alpha $ and C $\\scriptstyle \\rm II$ emission and of the S/N of a realistic observationWe assume a flat $\\Lambda $ CDM cosmology compatible with the latest Planck results: $h = 0.677$ , $\\Omega _m = 0.31$ , $\\Omega _b = 0.049$ , $\\Omega _\\Lambda = 1 - \\Omega _m$ , $n = 0.97$ , $\\sigma _8 = 0.82$ [42].."
],
[
"Signal Power spectrum",
"In this Section we derive the PS (auto-correlation PS and cross-correlation PS) of the measured intensity fluctuations and its variance, with an approach similar to [56].",
"For simplicity we assume that the detected intensity includes three components: (i) the signal; (ii) the instrumental white noise; (iii) the interloping lines which are redshifted to the same frequency as the signal line, namely $I(\\Omega , \\nu ) = I_\\alpha (\\Omega , \\nu ) + I_{\\rm N} + \\sum _i I^i_f(\\Omega , \\nu ).$ Throughout work we will neglect the possible presence of continuum foregrounds, assuming that they can be easily removed thanks to the smoothness of the frequency spectrum [59], [11], [60].",
"Note that comoving coordinates are related to angle and frequency displacement from an arbitrary origin, $\\mathbf {x}^0$ , as follows: $x_1, x_2 = \\chi (z_\\alpha ) \\Delta \\theta + x^0_{1},x^0_2\\\\x_3 = \\frac{d\\chi }{d\\nu }\\Delta \\nu + x^0_{3}$ where $\\chi (z_\\alpha )$ is the comoving distance from the observer to the signal, $d\\chi /d\\nu = c(1+z_\\alpha )[H(z_\\alpha )\\nu ]^{-1}$ , $(\\Delta \\theta , \\Delta \\nu )$ are the displacements in angle and frequency from the origin ${\\bf x}^0$ (center of the survey).",
"In this process a subtlety arises [56], [25] because $I^i_f$ is not emitted at $z_\\alpha $ .",
"Therefore, in that term we should consider coordinates that are the projection at $z_\\alpha $ of the real coordinates at $z_i$ : $I({\\bf x}) = I_\\alpha ({\\bf x}, z_\\alpha ) + I_{\\rm N} + \\\\+ \\sum _i I^i_f\\left(x_1\\frac{\\chi (z_i)}{\\chi (z_\\alpha )}, x_2\\frac{\\chi (z_i)}{\\chi (z_\\alpha )}, x_3 \\frac{(1+z_i)H(z_\\alpha )}{(1+z_\\alpha )H(z_i)}, z_i\\right)$ where $1+ z_i = (1+z_\\alpha )\\lambda _\\alpha /\\lambda _i$ .",
"When considering the Fourier transform of the fluctuations, this projection introduces (i) a global extra factor that multiplies the PS; (ii) anisotropies due to the different projection of modes along and across the line of sight; (iii) a loss of correspondence between comoving and observed $k$ -modes: $\\delta I({\\bf k}) = \\overline{I}_\\alpha \\langle b \\rangle _\\alpha \\delta ^\\alpha _{\\bf k} + \\delta ^{N}_{\\bf k} + \\sum _i C(z_i) \\overline{I}_f^i \\langle b \\rangle _i \\delta ^{i}_{{\\bf k^{\\prime }}({\\bf k})};$ where $\\overline{I}$ and $\\langle b \\rangle $ with each subscript are the mean intensity and halo luminosity weighted mean bias of each line; $\\delta ^{N}_{\\bf k}$ is the instrumental noise (see Sec.",
").",
"The global extra factor is $C(z_i) = \\left( \\frac{\\chi (z_\\alpha )}{\\chi (z_i)} \\right)^2 \\frac{(1+z_\\alpha )H(z_i)}{(1+z_i)H(z_\\alpha )}; \\\\{\\bf k^{\\prime }}({\\bf k}) = \\left( k_1 \\frac{\\chi (z_\\alpha )}{\\chi (z_i)}, k_2 \\frac{\\chi (z_\\alpha )}{\\chi (z_i)}, k_3 \\frac{(1+z_\\alpha )H(z_i)}{(1+z_i)H(z_\\alpha )} \\right).$ From the above equations, the PS of the measured intensity fluctuations becomes $P({\\bf k}) = \\langle \\delta I({\\bf k}) \\delta I^*({\\bf k}) \\rangle = P_\\alpha ({\\bf k}) + P_{\\rm N} + \\sum _i P_f^i$ where $P_\\alpha ({\\bf k}) = {\\overline{I}_\\alpha }^2\\langle b \\rangle _\\alpha ^2 P_{\\rm dm}({\\bf k},z_\\alpha ),\\nonumber \\\\P_{\\rm N} = \\langle \\delta ^{\\rm N} \\delta ^{\\rm N} \\rangle ,\\nonumber \\\\P_f^i({\\bf k}) = C(z_i) ({\\overline{I}^i_f})^2 \\langle b \\rangle ^2_i P_{\\rm dm}({\\bf k^{\\prime }},z_i)\\nonumber ,$ in deriving the last line the relation $\\langle \\delta _{{\\bf k^{\\prime }}({\\bf k})} \\delta _{{\\bf p^{\\prime }}({\\bf p})} \\rangle = C^{-1} P_{\\rm dm}({\\bf k^{\\prime }}) \\delta ^3({\\bf k} - {\\bf p})$ is used and $P_{\\rm dm}$ is the dark matter PS.",
"The noise component $P_{\\rm N}$ is well known and easily subtracted; the interlopers power spectrum, $P_f^i({\\bf k})$ , is however unknown and yet must be removed in order to extract the astrophysical PS signal.",
"The variance of $P(\\bf k)$ is $\\sigma ^2_{P}({\\bf k}) =\\delta P^2({\\bf k})= \\langle \\left( \\delta I({\\bf k}) \\delta I^*({\\bf k}) \\right)^2 \\rangle - \\langle \\delta I({\\bf k}) \\delta I^*({\\bf k}) \\rangle ^2.$ Using that fact that noise and interloping lines only correlate with themselves, and that $\\langle \\mid \\delta _{\\bf k}\\mid \\rangle ^4 = 2 \\sigma _{\\bf k}^4$ and $\\langle \\vert \\delta ^{\\rm N} \\vert ^4 \\rangle = 2 P^2_{\\rm N}$ , it is easy to prove (see Appendix for the full calculation) $\\sigma ^2_{P}({\\bf k}) = \\left[{P_{\\alpha }({\\bf k})} + P_{\\rm N} + \\sum _i P_f^i({\\bf k})\\right]^2.$ From this equation we can see that the variance depends strongly on the detector noise and on the PS of the interloping lines.",
"In case the PS is isotropic, $P({\\bf k}) = P(k)$ , several independent modes can be combined to reduce the PS variance at given $k$ : $P(k) = \\left(\\sum _{\\bf k} \\frac{1}{\\sigma ^2_{P}({\\bf k})} \\right)^{-1} \\sum _{\\bf k} \\frac{P({\\bf k})}{\\sigma ^2_{P}({\\bf k})},$ where the sum is over all the modes with $\\vert {\\bf k}\\vert = k$ .",
"In order to estimate the S/N we have to consider the PS variance due to the finite survey volume and resolution.",
"In this case the probed $k$ -modes are discrete and multiples of $(2\\pi / L_1, 2\\pi /L_2, 2\\pi /L_3)$ , where $L_1, L_2, L_3$ are the dimensions of the survey volume.",
"Suppose the survey has a resolution $l_\\parallel $ and $l_\\perp $ along and perpendicular to the line-of-sight (generally $l_\\parallel \\gg l_\\perp $ ), respectively.",
"Then only modes satisfying $2\\pi /L_1,2\\pi /L_2<k_1, k_2 < 2\\pi / l_\\perp $ and $2\\pi /L_3<k_3 < 2 \\pi / l_\\parallel $ are accounted.",
"Sometimes it is useful to estimate the total PS variance and S/N for all modes with $k_{\\rm min} <k<k_{\\rm max}$ [44]: $\\langle \\sigma _{P}^2 \\rangle = \\left( \\int \\frac{d^3k}{\\Delta k^3} \\frac{1}{\\sigma ^2_{P}({\\bf k})}\\right)^{-1}; \\\\\\langle (S/N)^2 \\rangle = \\int \\frac{d^3k}{\\Delta k^3} \\left(\\frac{P({\\bf k})}{\\sigma _{P}({\\bf k})} \\right)^2,$ where $\\Delta k^3 = (2\\pi )^3/V_{\\rm s}$ is the $k$ -space volume occupied by each discrete mode and the integral is over all wavenumbers with $ k_{\\rm min} < \\mid {\\bf k} \\mid < k_{\\rm max}$ , $k_1, k_2 < 2\\pi / l_\\perp $ and $k_3 < 2 \\pi / l_\\parallel $ .",
"The contamination in the auto-correlation PS (Eq.",
"(REF )) could be suppressed by cross-correlating different measurements targeting two different signals, $\\alpha $ and $\\beta $ , that are contaminated by uncorrelated interloping lines [56].",
"The cross-correlation PS is [56]: $P_{\\alpha ,\\beta }({\\bf k}) = \\langle I_\\alpha \\rangle \\langle b \\rangle _\\alpha \\langle I_\\beta \\rangle \\langle b \\rangle _\\beta P_{\\rm dm}({\\bf k}),$ where only the signal term is left as noise and interloping terms are uncorrelated for $\\alpha $ and $\\beta $ .",
"Nevertheless, noise and interloping lines increase the variance: $\\sigma ^2_{P\\alpha \\beta } = \\frac{1}{2}\\left[ P^2_{\\alpha \\beta } + \\right.\\\\+ \\left.",
"\\left(P_\\alpha + P_{N,1} + \\sum _i {P^i_{f,1}} \\right)\\left( P_\\beta + P_{N,2} + \\sum _i {P^i_{f,2}}\\right) \\right],$ where the subscripts $1, 2$ represent the qualities in the two measurements respectively.",
"We will apply this suppression method to our model and discuss more specific details in Sec.",
"."
],
[
"Line Detectability",
"We start by assessing first the detectability of the Ly$\\alpha $ PS without considering the interlopers contamination.",
"Our discussions are based on different setup parameters of a small space telescope that can map efficiently a large sky area in the visible (corresponding to $2.2 < z_\\alpha < 4.8$ ) and NIR ($z>4.8$ ) spectral bands.",
"We do not aim at proposing a optimal setup of such instrument, but rather at understanding to what extent the Ly$\\alpha $ IM is a viable tool for studying high-$z$ galaxies.",
"The size of the voxel is one of the most relevant factors for detectability.",
"The voxel size along the line-of-sight is given by $l_\\parallel =\\frac{dl}{dz}\\Delta z = \\frac{c (1+z)}{H(z) R} $ ; in the perpendicular direction it is instead $l_\\perp = \\chi (z) \\theta _\\mathrm {min}$ .",
"As such, it depends on the spectral resolution, $R$ , and angular resolution, $\\theta _{\\rm min}$ , of the telescope.",
"The choice of an optimal $l_\\parallel $ and $l_\\perp $ is crucial: a small voxel results in a smaller volume loss following interlopers removal but requires a longer time to complete a survey for a given area; large voxels suffer from the opposite problem.",
"Moreover, as we will discuss in Sec.",
", there are additional limitations imposed by the ancillary imaging-survey used to identify the interlopers.",
"The latter sets the minimum voxel size to the precision of the redshift measurement (i.e.",
"typically $\\approx 0.05(1+z)$ for photometric surveys) along the line-of-sight.",
"It is necessary to find a balanced choice that is specific to the IM experiment configuration and goals.",
"Our fiducial instrument has a $\\theta _\\mathrm {min} = 6 ~$ arcsec beam FWHM (full width at half maximum), a spectrometer with resolution $R = \\lambda /\\Delta \\lambda = 100$ and a survey area of $ 250~\\mathrm {deg}^2$ [44], [50], [18], [15].",
"Therefore the sample space has voxels with $\\Delta l_\\parallel =$ 35.3 and 28.1 Mpc, and $\\Delta l_\\perp =$ 214 and 257 kpc, for $z=4$ and 7 respectively.",
"In our setup, the voxel size is always larger than the galaxy correlation length (typically $\\approx 1$ Mpc$^3$ ); therefore we expect that each voxel contains several galaxies.",
"Also, as $\\Delta l_\\parallel \\gg \\Delta l_\\perp $ (typically $\\approx 20-30$ Mpc vs. $\\approx 200-300$ kpc), only transversal modes contribute to the PS measurement at $k > 0.1~h {\\rm Mpc}^{-1}$ .",
"Another relevant crucial point is the instrumental noise.",
"A space telescope is usually background limited, i.e.",
"the noise levelWe ignore dark current and readout noise as they depend strongly on survey implementation; This approximation is safe at least for instruments similar to SPHEREx (M. Zemcov, private communication) is set by Poisson fluctuations of the background light.",
"For for Ly$\\alpha $ observations the most important background is the Zodiacal Light (ZL).",
"In this case, $\\sigma _N \\approx 1.37 \\, \\epsilon ^{-1/2} \\, {\\rm nW m}^{-2}{\\rm sr}^{-1} \\times \\\\\\left[\\frac{\\mu {\\rm m}}{\\lambda }\\frac{R}{100}\\frac{ I_{\\rm ZL} }{10^3 {\\rm nW m}^{-2}{\\rm sr}^{-1}} \\frac{0.126 {\\rm m}^2}{\\pi D^2}\\frac{8.5\\times 10^{-10}{\\rm sr}}{\\Omega _{\\rm pix}} \\frac{10^5{\\rm s}}{t_{obs}}\\right]^{1/2},$ where $I_{\\rm ZL}=\\nu I_\\nu \\approx 10^{2-3} {\\rm nW m}^{-2}{\\rm sr}^{-1} $ is the typical ZL flux in the relevant frequency range [29], [18].",
"An efficiency $\\epsilon $ is added to account for the photons loss by mirrors and integral field unit (IFU), here assumed conservatively to be $\\epsilon =0.25$ ."
],
[
"Power spectrum observations",
"The first generation of intensity mappers are likely to have a limited S/N that will allow only to probe EoR Ly$\\alpha $ fluctuations power-spectrum.",
"In this Section we discuss an instrument designed for this aim.",
"Nevertheless in the future more powerful instruments could undertake tomographic observations and we will discuss such possibility in Sec.",
"REF .",
"The PS of the instrumental noise is $P_N(z) = \\sigma _N^2 V_\\mathrm {pix};$ where $\\sigma _N$ is from Eq.",
"(REF ) and $V_{\\rm pix}$ is the comoving voxel volume.",
"We then compute the S/N using $\\sigma ^2_P =(P_\\alpha +P_N)^2$ and Eq.",
"(REF ) (we use $\\Delta z = 1$ for the survey volume $V_s$ and divide $k$ -space in $k$ -bins with $\\Delta k= 1.2 k$ ).",
"A telescope with a FOV of $\\approx 20-30 ~{\\rm deg}^2$ (similar to the proposed SPHEREx, see [18]) can observe a field of $200-250 ~{\\rm deg}^2$ in two years, with exposure time $t_{\\rm obs} \\approx 10^7$ s per pointing.",
"Here we consider a more conservative setup with $t_{\\rm obs} = 10^5$ s, it is more feasible as it only takes several days to complete the survey.",
"Figure: Top: Predicted Lyα\\alpha power-spectrum from z=7z=7 with errorbars; Bottom: S/N in each kk-space bin.",
"The S/N is computed for a background-limited NIR telescope with diameter D=0.4D=0.4 m, angular and spectral resolution (δθ=6 arcsec ,R=100)(\\delta \\theta = 6~{\\rm arcsec}, R=100), exposure time 10 5 10^5 s per pointing and a survey area of 250 deg 2 250~{\\rm deg}^2.",
"Each bin has a width Δk=1.2k\\Delta k = 1.2 k.Fig.",
"REF shows the Ly$\\alpha $ PS from $z=7$ , and the corresponding S/N assuming $10^5$ s exposure time per pointing and a $250~{\\rm deg}^2$ survey areaThis survey set-up is rather conservative; a deeper survey should be possible..",
"The S/N is proportional to the number of probed modes: it scales as $k^3$ for bins with $k $ < $$ 0.1 hMpc-1$ and as $ k2$ for smaller scales, due to the limited spectroscopic resolution.",
"This transition generates a decreasing S/N for $ 0.1 hMpc-1 $\\; < \\over \\sim \\;$ k $\\; < \\over \\sim \\;$ 1 hMpc-1$, where the PS is steeper than $ k-2$.",
"Above $ k 1 hMpc-1$ the S/N increases again because shot noise dominates and PS is constant.",
"However, as discussed in CF16, shot noise on the $$Mpc scale might be suppressed by Ly$$~diffusion in the IGM, and therefore the S/N can be overestimated in that range.",
"We conclude that Ly$$~intensity mapping is best suited to study fluctuations in the linear regime on scales $ 10$~Mpc.",
"These results are encouraging because they show that Ly$$~IM from the late EoR can be detected, provided that continuum foregrounds and low redshift interlopers can be efficiently removed.$ Fig.",
"REF shows a more general dependence of S/N on wavenumber $k$ for Ly$\\alpha $ signals coming from different redshifts.",
"The observational setup is the same as in Fig.",
"REF .",
"We find that the Ly$\\alpha $ PS is accessible to this kind of observations at least for the late EoR (i.e.",
"S/N $\\approx 5$ at $k = 0.1$ $h$ Mpc$^{-1}$ for $z \\sim 7$ ).",
"Figure: The S/N of the observed PS as function of redshift zz and wave-number kk; the observational setup is the same of Fig.",
".We then investigate how the detectability depends on varying exposure time $t_{\\rm obs}$ , using the total S/N, computed using Eq.",
"(REF ), in the range $ 5\\times 10^{-3}~h{\\rm Mpc}^{-1} < k < 2~h{\\rm Mpc}^{-1}$ as the indicator.",
"The results are plotted in Fig.",
"REF .",
"From there we see that a detection of Ly$\\alpha $ PS with low S/N is at reach even at $z > 7$ .",
"This formalism also allows us to find the best observational strategy for a PS observation: given a fixed total observing time we want to find the optimal exposure time per pointing.",
"Considering only the instrumental noise, from Eq.",
"(REF ), (REF ) and (REF ) we have ${\\rm (S/N)}^2 \\propto \\frac{A_{\\rm surv}}{\\sigma _N^2}.$ Since $A_{\\rm surv} \\propto t_{\\rm obs}^{-1}$ and $\\sigma _N^2 \\propto t_{\\rm obs}^{-1}$ , the S/N does not depend on the depth of the survey as long as the cosmic variance term negligibly appears in Eq.",
"(REF ).",
"In other words the best strategy for an IM experiment is to carry out a shallow, however large area survey.",
"Figure: Total S/N for detection of Lyα\\alpha fluctuations; the observational setup is the same of Fig.",
".In practice, though, the optimal $t_{\\rm obs}$ is set by the technical implementation of the survey, which should take into account the following limitations: (i) $t_{\\rm obs}$ cannot be shorter than, or even comparable to, the instrumental pointing time; (ii) with a large survey area it is impossible to avoid sky regions with higher foregrounds; (iii) as we will discuss in Sec.",
", the IM survey might need deep ancillary galaxy surveys for interloper removal, and therefore the data available for final analysis is limited to the overlapping sky regions."
],
[
"Tomography",
"Alternatively, an IM experiment allows us to make tomographic maps of the Ly$\\alpha $ intensity, although only the low-$z$ part of the signal is accessible to fiducial space telescope design introduced above.",
"In CF16 we found that the mean Ly$\\alpha $ intensity at $z=4$ is $I_\\alpha \\approx 0.1~{\\rm nW m}^{-2}{\\rm sr}^{-1}$ .",
"At the same redshift the dark matter field has a fluctuations level of $\\sigma _{\\rm dm} \\approx 0.23$ on 10 Mpc scales, and the mean Ly$\\alpha $ bias $\\langle b \\rangle _\\alpha \\approx 3$ .",
"Therefore if the survey has voxels of volume $(10~{\\rm Mpc})^3$ , corresponding to $R \\approx 350$ and $\\Delta \\theta _{\\rm pix} = 4.7$ arcmin at $z=4$ , the $1\\sigma $ Ly$\\alpha $ fluctuations level is $\\sigma _\\alpha = I_\\alpha \\langle b \\rangle _\\alpha \\sigma _{\\rm dm} \\approx 0.07~ {\\rm nW}/{\\rm m}^2/{\\rm sr},$ which is larger than the noise level $\\sigma _N\\approx 0.04$ ${\\rm nW m}^{-2}{\\rm sr}^{-1}$ in Eq.",
"(REF ) for $t_{\\rm obs} = 10^6$ s. Therefore even this small intensity mapper can observe directly the spatial fluctuations of Ly$\\alpha $ emission from low redshift galaxies, although with a modest S/N.",
"The tomographic observation of the Ly$\\alpha $ signal from the EoR is more challenging, as the Ly$\\alpha $ intensity drops by one order of magnitude.",
"Thus a tomographic map of the EoR signal requires a more powerful instrument.",
"Fig.",
"REF shows the ${\\rm S/N}=\\sigma _\\alpha /\\sigma _{\\rm N}$ as a function of $z$ and $t_{\\rm obs}$ for a 2 m space telescope and same voxels of $(10~{\\rm Mpc})^3$ volume.",
"The observation requires an integration time of at least few months and even so it will be only feasible for the late stages of the EoR.",
"The experiment can be even more challenging once the confusion by interloping lines such as H$_\\alpha $ and [O $\\scriptstyle \\rm II$ ] that dominate over Ly$\\alpha $ emission are accounted for.",
"Figure: The S/N of tomographic observations for a voxel size (10 Mpc) 3 ^3.",
"The S/N is computed for a background-limited NIR telescope with diameter D=2D=2 m, angular and spectral resolution (δθ=4.7 arcsec ,R=350)(\\delta \\theta = 4.7~{\\rm arcsec}, R=350)."
],
[
"Interloping lines",
"Low redshift emission lines could significantly contribute to the observed intensity fluctuations (see Eq.",
"(REF )).",
"Particularly important for Ly$\\alpha $ experiments are the H$\\alpha $ ($0.6563~\\mu m$ ), [O $\\scriptstyle \\rm III$ ] ($0.5007~\\mu m$ ) and [O $\\scriptstyle \\rm II$ ] ($0.3727~\\mu m$ ) [25], [44] lines.",
"Their power spectra may dominate the Ly$\\alpha $ signal, and are distorted and amplified due to coordinate projection effects.",
"Their contribution must therefore be accurately removed from the received flux.",
"In what follows we investigate the power spectra of these interloping lines and suggest a technique to remove them."
],
[
"Power spectra of interloping lines",
"The abundance matching technique required to compute the power-spectrum of interloping lines involves the knowledge of the line LF, which is not as easy as the continuum LF to measure.",
"Fortunately our Ly$\\alpha $ signal is only contaminated by interlopers at low redshift ($z < 2$ ), where observations are more easily available.",
"We use the Schechter LF parameterization [47] in [34] and [19] (see Tab.",
"REF ).",
"The intrinsic intensity of interlopers is not relevant in our work, therefore we use the unprocessed LF, i.e.",
"without dust correction.",
"Currently the observed interloper LFs are not complete enough to derive a redshift evolution.",
"This forces us to use the variance-weighted mean Schechter parameters to construct the $L=L(M)$ relations at the variance-weighted mean redshift.",
"The same relation is then applied to all redshifts (see CF16).",
"The mean Schechter parameters and redshifts are listed in Tab.",
"REF .",
"In this scenario the redshift evolution of the LF is purely attributed to the halo mass function evolution.",
"Although this might seem a strong assumption, the redshift intervalsThe emission redshift is $1 + z_\\mathrm {em} = (1+z)(\\nu _\\mathrm {int}/\\nu _{\\alpha })$ , therefore at $z_{\\alpha } = 6,7,8$ the corresponding emission redshifts for the interlopers are $z_{H_\\alpha } = 0.30,0.48,0.67$ , $z_{\\rm OII} = 1.28,1.61,1.94$ and $z_{\\rm OIII}= 0.70,0.94,1.19$ .",
"of the interloper lines that we need to consider are relatively small.",
"For example, for the H$_\\alpha $ line (the strongest contributor) the relevant interval is $0.30<z<0.67$ .",
"As a result, we believe that the assumption does not affect our conclusions.",
"Figure: Left: Comparison of the interlopers and Lyα\\alpha power spectra at z=7z=7.",
"Right: Same after bright interlopers removal.Fig.",
"REF shows the PS of interlopers compared with Ly$\\alpha $ from $z=7$ .",
"As we discuss in Sec.",
", incorrectly projecting the interlopers to higher redshift introduces distortions that can amplify their PS.",
"Since the projected interlopers PS is anisotropic, we average it over the solid angle, $P(k) = \\frac{1}{4\\pi } \\int d\\Omega P({\\bf k})$ .",
"However, the anisotropy information can be used to assess the quality of the removal procedure [25].",
"We find that interlopers dominate the PS by 1-2 orders of magnitude on all scales, and that H$_\\alpha $ is the dominant confusion source.",
"Therefore an appropriate removal of the interloping PS from Ly$\\alpha $ signal, discussed in the following, is crucial."
],
[
"Interlopers removal",
"Removing the interloping lines requires a strategy that is different from that used to deal with continuum foregrounds.",
"A possible strategy is to mask the contaminated pixels [25], [44], [8].",
"This is feasible because the galaxy population emitting the interloping lines is very different from the signal sources at EoR: bright galaxies are very rare at high redshift because they are exponentially suppressed in the LF.",
"Hence, if we remove the most luminous pixels from the survey, most of them would be occupied by low-$z$ galaxies and the intensity of interloping lines could be reduced significantly.",
"However, although straightforward this approach has two drawbacks: (i) if the S/N of the observation is not high, bright voxels can result from noise or foreground fluctuations; (ii) it removes also Ly$\\alpha $ flux [8].",
"For this reason in this work we will use a different approach relying on ancillary galaxy surveys for the identification of the interlopers [44], [48], [61].",
"This strategy would affect only weakly the Ly-$\\alpha $ PS; however, ancillary surveys have to be sufficiently deep, wide and galaxy redshifts have to be estimated precisely.",
"To demonstrate the feasibility of such approach, we first perform a calculation similar to that shown in the left panel of Fig.",
"REF but imposing an upper limit to the mass of the interloping galaxies.",
"We assume that the pixels containing galaxies larger than this upper limit are removed from the survey.",
"Fig.",
"REF (right) shows the PS of Ly$\\alpha $ signal at $z = 7$ and interlopers, normalizing all power spectra at $k = 0.1~h{\\rm Mpc}^{-1}$ .",
"Both the mean intensity, the mean bias and the shot noise depend on the upper limit ($2 \\times 10^{11}~M_\\odot $ for H$_\\alpha $ , $4.4 \\times 10^{11}~M_\\odot $ for [O $\\scriptstyle \\rm II$ ], and $2 \\times 10^{12} ~M_\\odot $ for [O $\\scriptstyle \\rm III$ ]).",
"Removing massive galaxies suppresses very efficiently the PS of the interlopers.",
"We find that the removed voxels occupy only 2% of the survey volume.",
"In the left panel of Fig.",
"REF we show the minimum mass of halos that have to be removed from the survey to reach a interloper-to-signal ratio $r$ (defined as the PS ratio at scale $k = 0.1~h{\\rm Mpc}^{-1}$ ) for PS of Ly$\\alpha $ from redshift $z$ .",
"We find that an effective interloper removal requires to resolve galaxies hosted by halos with $M $ > $$ 1011M$ and line flux $ f $\\; > \\over \\sim \\;$ 10-16 erg cm-2$.",
"This can be challenging for a large area survey.",
"The fraction of the volume loss can be substantial, as shown by the right panel of Fig.",
"\\ref {fig:frem} when considering a 5\\% ($ R = 20$) redshift uncertainty in the ancillary galaxy survey, resulting in more than one voxels discarded per galaxy.",
"We remind that if more than $ 30%$ of the survey volume is masked, the PS reconstruction can be unfeasible \\cite {2005Natur.438...45K}.",
"From the right panel of Fig.",
"\\ref {fig:frem} we conclude that cleaning a Ly$$~IM survey can be intrinsically difficult at $ z > 12$, while the volume loss is not problematic for observations at later epochs.$ Figure: Left: Maximum mass of the galaxies contributing to the interlopers PS.",
"The sharp discontinuity at z=4.4z = 4.4 is due to the H α _\\alpha line entering the survey.",
"Right: Fraction of voxels that has to be removed to obtain a ratio rr between the interlopers and Lyα\\alpha PS on scale k=0.1h Mpc -1 k = 0.1~h{\\rm Mpc}^{-1} at redshift zz.",
"A redshift uncertainty in the ancillary galaxy survey of 5% (R=20R = 20) has been assumed, thus multiple voxels are discarded for each interloper.We can then translate the above constraints on a limiting apparent magnitude at which interloper galaxies must be removed.",
"To this aim we use the optical and NIR rest frame LFs in [28] and assign luminosities to DM halos using the abundance matching technique.",
"The apparent AB magnitude at a specified wavelength is obtained from linear interpolation between two neighboring bands in [28].",
"Fig.",
"REF shows the maximum depth needed by a survey to remove interlopers as a function of $r$ and signa redshift in the Y, J, H and K bands.",
"To access the signal from late EoR the ancillary survey must reach an AB mag $$ > $$ 26$.",
"Compared with the designed sensitivity of future photometric surveys this is rather challenging.",
"For example the EUCLID\\footnote {\\url {http://sci.esa.int/euclid/}} wide survey will reach a limiting magnitude of $ 24$ in bands $ Y$, $ J$ and $ H$: this can be enough only to clean the Ly$$~PS at $ z < 4.4$ (without the H$$ line).",
"Observing the EoR signal and reaching AB mag $ m = 27-28$ is extremely challenging and is at the edge of the capabilities of future instruments, such as WFIRST\\footnote {\\url {http://wfirst.gsfc.nasa.gov}} or FLARE.$ Figure: Limiting magnitude in different spectral bands (Y, J, H, K) required for a survey to clean the Lyα\\alpha PS signal from interlopers; rr is the ratio between the interlopers and Lyα\\alpha PS on scale k=0.1h Mpc -1 k = 0.1~h{\\rm Mpc}^{-1} at redshift zz.",
"The sharp horizontal feature at z=4.4z = 4.4 is due to the H α _\\alpha PS from z=0z = 0; it formally diverges (see Eq.",
"())."
],
[
"Cross-power spectra",
"In Sec.",
"REF we have discussed an interloper removal method based on ancillary surveys.",
"In spite of the optimistic assumptions (for example, we have neglected the scatter in the line luminosity, SFR and halo mass relations) the required masking depth is relatively demanding.",
"An alternative strategy would be to use the cross-correlation between two different intensity mapping experiments contaminated by different interloping lines.",
"The $157.7~\\mu $ m [C $\\scriptstyle \\rm II$ ] fine structure line is the brightest of all the metal lines, contributing generally up to $\\sim 1\\%$ of the total galaxy IR luminosity.",
"Its line luminosity scales tightly with the SFR, but is affected also by the ISM metallicity [52], [53].",
"The removal of continuum foreground and interloping lines for [C $\\scriptstyle \\rm II$ ] auto-correlation PS measurements was investigated in [61].",
"In this section we investigate its cross-correlation with the Ly$\\alpha $ line.",
"The interloping lines for these two signals are not correlated with each other because they are produced in non-overlapping redshift intervals."
],
[
"[C $\\scriptstyle \\rm II$ ] line intensity and cross power",
"The mean [C $\\scriptstyle \\rm II$ ] intensity can be directly obtained from the galaxy line luminosity [12]: $I_{\\rm CII}(z)=\\frac{c}{4\\pi \\nu _{\\rm CII} H(z)} \\int d{M} \\frac{d{n}}{d{M}} L_{\\rm CII}(M, z).$ It spatially fluctuates following the large scale DM density field multiplied by a line luminosity-weighted mean bias, $\\langle b \\rangle _{\\rm CII}$ : $\\delta I_{\\rm CII} = I_{\\rm CII} \\langle b \\rangle _{\\rm CII} \\delta ;$ where $\\delta $ is the DM density contrast, $\\langle b \\rangle _{\\rm CII} = \\frac{1}{ \\rho _{\\rm CII} } \\int d{M} \\frac{d{n}}{d{M}} b(M, z) L_{\\rm CII}(M, z).$ The cross-correlation PS includes three main terms: Large scale DM fluctuations originating from the Ly$\\alpha $ and [C $\\scriptstyle \\rm II$ ] lines, both emitted by the ISM.",
"This component dominates the PS on scales $$ > $$ 1$~Mpc.",
"It can be written as\\begin{equation}P_{{\\rm CII}, \\alpha }^{\\rm s,s}(k, z) = I_{\\rm CII}(z) I_{\\alpha }^{\\rm s}(z) \\langle b \\rangle _{\\alpha } \\langle b \\rangle _{\\rm CII} P_{\\rm dm}(k,z),\\end{equation}where $ Is$ is the Ly$$~emission from the ISM;\\item {\\em Fluctuations from UV continuum emission} resulting from the correlation between Ly$$~emission in the IGM and [\\hbox{C~$\\scriptstyle \\rm II$}]~ emission in the ISM.",
"Ly$$~fluctuations are produced by (i) UV emission from the galaxies, and (ii) Lyman absorption followed by relaxation in the IGM.",
"We can express the spatial intensity fluctuations as\\begin{multline}\\delta I^\\alpha _{\\mathrm {cont}}(z) = \\frac{c h_P \\nu _{\\alpha }}{4 \\pi (1+z)} \\sum _{n = 2}^{\\infty } P_{abs}(n, z) f(n) \\times \\\\\\times \\int dz^{\\prime } \\frac{\\dot{n}_\\nu (\\nu ^{\\prime }, z^{\\prime })}{H(z^{\\prime })} \\prod _{n^{\\prime } = n+1}^{n^{\\prime }_{max}} T(n^{\\prime }, z_{n^{\\prime }}) [\\langle b(z^{\\prime })\\rangle _{\\nu ^{\\prime }} \\delta ]= \\\\=\\frac{1}{4\\pi } \\int d\\Omega \\int _z^{+\\infty }dz^{\\prime } A(z, z^{\\prime }) \\dot{n}_{\\nu }(\\nu ^{\\prime }, z^{\\prime })[\\langle b(z^{\\prime })\\rangle _{\\nu ^{\\prime }} \\delta ],\\end{multline}where $ Pabs(n, z)$ is the IGM absorption probability of a Lyman-$ n$ photon at redshift $ z$, $ f(n)$ is the fraction of Ly$$~photons emitted by an HI atom during the decay from the $ n$-th energy level, $ n$ is the number of UV photons emitted per unit time, volume and frequency, $ T(n, z) = 1 - Pabs(n,z)$ is the transmission probability; we refer to CF16 for details.",
"The associated cross-correlation PS is\\begin{multline}P_{{\\rm CII}, \\alpha }^{s,c}(k, z) =\\\\\\left[I_{\\rm CII}(z) \\langle b \\rangle _{\\rm CII} \\int _z^{+\\infty } d{z^{\\prime }} A(z, z^{\\prime }) \\dot{n}_\\nu ^{\\prime }D^{\\prime }\\frac{\\sin (kl^{\\prime })}{kl^{\\prime }}\\right] P_{\\rm dm}(k,z),\\end{multline}where $ l'(z, z') = c zz' dx H(x)-1$.",
"It becomes important only on scales $$\\; > \\over \\sim \\;$ 100$~Mpc as fluctuations on scales smaller than the typical mean free path of a photon with energy between the Ly$$~and the Lyman-limit are washed out.\\item {\\em Shot noise} due to the discrete nature of the sources dominates on small scales:\\begin{equation}P_{{\\rm CII}, \\alpha }^{\\rm SN}(k, z) = I_{\\rm CII} I_{\\alpha }^{\\rm h} \\frac{1}{ \\rho _\\alpha ^{\\rm h} \\rho _{\\rm CII} } \\int dM \\frac{d{n}}{d{M}} L_\\alpha (M) L_{\\rm CII}(M);\\end{equation}$ Fig.",
"REF shows the Ly$\\alpha $ -[C $\\scriptstyle \\rm II$ ] cross-correlation PS at $z=6$ with the three main components plotted separately.",
"As expected the PS is largely dominated by the ISM emission, and only on scales $$ > $$ 100$~Mpc the IGM becomes important.",
"We plot also the S/N of an hypothetical observation, using the same [\\hbox{C~$\\scriptstyle \\rm II$}] survey proposed in \\cite {mappingCII}.",
"For consistency, we adopt a spectral resolution $ R=100$ and angular resolution $ = 42$ arcsec for both [\\hbox{C~$\\scriptstyle \\rm II$}] and Ly$$~observations.",
"The total survey area is $ 250 deg2$, corresponding to about $ 100$ pointings, each with exposure time of $ 105$~s (total observing time $ 107$s, or about 4 months).$ Figure: Top: Lyα\\alpha -[C II \\scriptstyle \\rm II] cross-correlation power spectrum at z=6z = 6.",
"We show the total PS (solid), large scale dark matter fluctuations (dashed), fluctuations from UV continuum emission (dot-dashed), and shot noise (see text).",
"The error bars are computed considering the same [C II \\scriptstyle \\rm II] instrumental setup in and in Sec.",
".",
"Bottom: S/N of the observation as a function of wavenumber.The left panel of Fig.",
"REF shows the S/N as a function of $k$ and $z$ (assuming $\\Delta k = 1.2 k$ ) for the most optimistic case where all interlopers are cleanly removed (Eq.",
"(REF ) with $P^i_{f,1} = P^i_{f,2} = 0$ ).",
"These results are encouraging, because they show that in principle a Ly$\\alpha $ -[C $\\scriptstyle \\rm II$ ] intensity mapping observation of the late EoR is feasible.",
"However, as we showed in Sec.",
"REF , interlopers can increase the PS variance well beyond the instrumental noise.",
"For [C $\\scriptstyle \\rm II$ ] IM, CO rotational lines (see [26], [2], [43] for detailed CO emission line studies) are the most important interlopers.",
"They have PS amplitude comparable or even larger than the [C $\\scriptstyle \\rm II$ ] one, and therefore they must be removed (see [61], [48]).",
"We then added the CO lines, H$_\\alpha $ , [O $\\scriptstyle \\rm III$ ] and [O $\\scriptstyle \\rm II$ ] lines to the variance of the cross-correlation PS (see Eq.",
"REF ).",
"The right panel of Fig.",
"REF shows the S/N with interlopers (the strong features at $z \\approx 4.5$ and $7.2$ are due to H$_\\alpha $ , CO 2-1 and CO 3-2 lines entering the survey, respectively).",
"The effect of the interlopers is to decrease the S/N significantly; without an efficient removal the EoR signal is inaccessible.",
"However, compared to the Ly$\\alpha $ auto-correlation PS discussed in Sec.",
"REF , the Ly$\\alpha $ -[C $\\scriptstyle \\rm II$ ] cross-correlation spectrum can be more easily recovered by using a shallower ancillary survey within the capability of a near future instrument.",
"To support this statement we recompute the S/N however removing all the interlopers with $m_{\\rm AB} < 24$ (vs. $\\sim 26$ for recovering the Ly$\\alpha $ auto-correlation PS) in the EUCLID NIR bands (Y, J and H), finding that the recovered signal matches almost perfectly the model without interlopers.",
"This approach, even though promising, is more difficult to interpret.",
"The information recovered by the cross-PS is degenerate and it is not possible to recover information about Ly$\\alpha $ or C $\\scriptstyle \\rm II$ lines individually.",
"It is necessary to rely on ancillary data to extract the relevant astrophysical information, such as a PS measurement of one of the two lines or a combination of several cross-correlations.",
"Another possibility is to cross-correlate with resorved sources, such as QSOs [17] or LAE (Comaschi & Ferrara in prep.).",
"Figure: Left: S/N of the Lyα\\alpha -[C II \\scriptstyle \\rm II] cross-correlation spectrum vs. wavenumber after interloping lines have been removed; Right: Same as left panel, before interloper removal.",
"The observational setup is the same as in Fig."
],
[
"Conclusions",
"We have investigated the feasibility of a Ly$\\alpha $ intensity mapping experiment targeting the collective signal from galaxies located at $z > 4$ .",
"We have used a recently developed analytical model to predict the Ly$\\alpha $ power spectrum, and carefully studied the main observational challenges.",
"These are ultimately quantified by the expected S/N for various observational strategies.",
"We found that in principle the Ly$\\alpha $ PS for $z<8$ is well at reach of a small space telescope (40 cm in diameter); detections with low S/N are possible only in some optimistic cases up to $z\\sim 10$ .",
"However, the foreground from interloping lines represent a serious source of confusion and must be removed.",
"The host galaxies of these interloping lines can be resolved via an ancillary photometric galaxy survey in the NIR bands (Y, J, H, K).",
"If the hosts are removed down to AB mag $\\sim 26$ , then the Ly$\\alpha $ PS for $5 < z < 9$ can be recovered with good S/N.",
"We further found that, by cross-correlating the Ly$\\alpha $ emission with [C $\\scriptstyle \\rm II$ ] emission from the same redshift, the required depth of the ancillary galaxy survey could be is within reach of Euclid (AB mag $\\sim 24$ ).",
"The results of this work show the yet unexplored, remarkable potential of Ly$\\alpha $ IM experiments.",
"By using a small space telescope and a few days observing time it is possible to probe galaxies hosted by DM halos with $M \\approx 10^{10}~M_\\odot $ well into the EoR.",
"Such galaxies emit the bulk of the collective Ly$\\alpha $ radiation.",
"However, the technical difficulty is represented by the interloping lines removal, which sets demanding requirements to the ancillary survey: the combination of very large survey areas ($\\sim 250$ deg$^2$ ) and significant depth (AB mag $\\sim 26$ ) appear to be challenging also for the next generation telescopes.",
"We have suggested however, that such problem can be overcome by cross-correlating the Ly$\\alpha $ IM with other lines (as the $157.7~\\mu $ m [C $\\scriptstyle \\rm II$ ] fine structure line), thus making a strong synergy between programs targeting different bands almost mandatory."
],
[
"Power Spectrum variance",
"In this Appendix we discuss the derivation of the deviate of Eq.",
"(REF ).",
"For simplicity we consider only two components: the line intensity $I_\\alpha $ and the detector noise $I_{\\rm N}$ ; we will work in $\\bf k$ -space: $\\delta I = \\delta I_\\alpha + \\delta I_{\\rm N}.$ Since other components do not correlate with $I_\\alpha $ and with $I_{\\rm N}$ , adding them to the results is trivial.",
"In this paper we use the Fourier convention from [41]: $\\delta _{\\bf k} = \\int _V \\delta ({\\bf x}) e^{-i {\\bf k}\\cdot {\\bf x}} d^3 {\\bf x};$ $\\delta _{\\bf k}$ has a Gaussian Probability Distribution function (PDF).",
"If $\\delta _{\\bf k} $ is written in polar coordinates, $\\delta _{\\bf k} = r_{\\bf k} \\exp {i\\phi _{\\bf k}}$ , the PDF assumes the form $g_{\\bf k}(r_{\\bf k}, \\phi _{\\bf k}) d{r_{\\bf k}} d{\\phi _{\\bf k}} = \\frac{2 r_{\\bf k} d{r_{\\bf k}}}{\\sigma _{\\bf k}^2} \\left( \\frac{d{\\phi _{\\bf k}}}{2\\pi } \\right)e^{-\\frac{r_{\\bf k}^2}{\\sigma _{\\bf k}^2}}.$ With Eq.",
"(REF ) it is easy to prove that $\\langle \\delta _{\\bf k}\\delta ^*_{\\bf p} \\rangle = \\eta _{\\bf kp}\\sigma _{\\bf k}^2$ (where $\\eta _{\\bf kp}$ is the Kronecker delta function).",
"The cases for $\\delta I_\\alpha $ and for $\\delta I_N$ are similar with the only exception that the variance of $\\sigma _N$ does not depend on ${\\bf k}$ .",
"Expanding the first term in Eq.",
"(REF ), we get $\\langle ( \\delta I \\delta I^*)^2 \\rangle &= \\langle \\mid \\delta I_\\alpha \\mid ^4 \\rangle + \\langle \\mid \\delta _{\\rm N} \\mid ^4 \\rangle + {\\langle \\delta I_\\alpha ^2 (\\delta I^*_{\\rm N})^2 \\rangle } + \\nonumber \\\\&+ { \\langle (\\delta I^*_\\alpha )^2 \\delta I_{\\rm N}^2 \\rangle } + 4\\langle \\mid \\delta I_\\alpha \\mid ^2 \\mid \\delta I_{\\rm N} \\mid ^2 \\rangle + \\nonumber \\\\&+ {2 \\langle \\mid \\delta I_\\alpha \\mid ^2 \\delta I_\\alpha \\delta I_{\\rm N}^* \\rangle } + {2\\langle \\mid \\delta I_\\alpha \\mid ^2 \\delta I_\\alpha ^* \\delta I_{\\rm N} \\rangle } \\nonumber \\\\&+ {2 \\langle \\delta I_\\alpha \\mid \\delta I_{\\rm N} \\mid ^2 \\delta I_{\\rm N}^* \\rangle } +{2 \\langle \\delta I_\\alpha ^* \\mid \\delta I_{\\rm N} \\mid ^2 \\delta I_{\\rm N} \\rangle } \\nonumber \\\\&= \\langle \\mid \\delta I_\\alpha \\mid ^4 \\rangle + \\langle \\mid \\delta _{\\rm N} \\mid ^4 \\rangle +4\\langle \\mid \\delta I_\\alpha \\mid ^2 \\mid \\delta I_{\\rm N} \\mid ^2 \\rangle $ where terms like $\\langle (\\delta I^*_\\alpha )^2 \\delta I_{\\rm N}^2 \\rangle $ are null because of the averaging over the phase $\\phi $ in Eq.",
"(REF ) $\\langle (\\delta I^*_\\alpha )^2 \\delta I_{\\rm N}^2 \\rangle \\propto \\int d \\phi _\\alpha d \\phi _{\\rm N} e^{-2 i \\phi _\\alpha } e^{2i \\phi _{\\rm N}} = 0.$ The second term in Eq.",
"(REF ) is $\\langle ( \\delta I \\delta I^*) \\rangle ^2 = \\left( \\langle \\mid \\delta I_\\alpha \\mid ^2 \\rangle + \\langle \\mid \\delta _{\\rm N} \\mid ^2 \\rangle \\right)^2.$ Using the fact that both $\\delta I_\\alpha $ and $\\delta I_{\\rm N}$ are Gaussian we have $\\langle \\mid \\delta _{\\bf k} \\mid ^4 \\rangle = \\int r_{\\bf k}^4 \\frac{d r_{\\bf k}^2}{\\sigma _k^2} e^{-\\frac{r_{\\bf k}^2}{\\sigma _k^2}} = 2 \\sigma _k^4,$ and finally $\\langle ( \\delta I \\delta I^*)^2 \\rangle - \\langle ( \\delta I \\delta I^*) \\rangle ^2 = \\left( \\langle \\mid \\delta I_\\alpha \\mid ^2 \\rangle + \\langle \\mid \\delta _{\\rm N} \\mid ^2 \\rangle \\right)^2.$"
]
] | 1605.05733 |
[
[
"Photoelectric effect for twist-deformed space-time"
],
[
"Abstract In this article, we investigate the impact of twisted space-time on the photoelectric effect, i.e., we derive the $\\theta$-deformed threshold frequency.",
"In such a way we indicate that the space-time noncommutativity strongly enhances the photoelectric process."
],
[
"Acknowledgments",
"The author would like to thank J. Lukierski for valuable discussions.",
"This paper has been financially supported by Polish NCN grant No 2014/13/B/ST2/04043.",
"$~~~~~~~~~~~~~~~~~~$ Figure: The shape of the threshold frequency ω tr (θ)\\omega _{tr}(\\theta ) for the three different values ofthe parameter mm: m=1m=1 (continuous line), m=2m=2 (dotted line) and m=3m=3 (dashed line).",
"In allthree cases we fix the work function W=1W=1.Figure: The shape of the threshold frequency ω tr (θ)\\omega _{tr}(\\theta ) for the three different values ofthe work function: W=1W=1 (continuous line), W=2W=2 (dotted line) and W=3W=3 (dashed line).",
"In allthree cases we fix the parameter m=1m=1."
]
] | 1605.05678 |
[
[
"RADON: Repairable Atomic Data Object in Networks"
],
[
"Abstract Erasure codes offer an efficient way to decrease storage and communication costs while implementing atomic memory service in asynchronous distributed storage systems.",
"In this paper, we provide erasure-code-based algorithms having the additional ability to perform background repair of crashed nodes.",
"A repair operation of a node in the crashed state is triggered externally, and is carried out by the concerned node via message exchanges with other active nodes in the system.",
"Upon completion of repair, the node re-enters active state, and resumes participation in ongoing and future read, write, and repair operations.",
"To guarantee liveness and atomicity simultaneously, existing works assume either the presence of nodes with stable storage, or presence of nodes that never crash during the execution.",
"We demand neither of these; instead we consider a natural, yet practical network stability condition $N1$ that only restricts the number of nodes in the crashed/repair state during broadcast of any message.",
"We present an erasure-code based algorithm $RADON_C$ that is always live, and guarantees atomicity as long as condition $N1$ holds.",
"In situations when the number of concurrent writes is limited, $RADON_C$ has significantly improved storage and communication cost over a replication-based algorithm $RADON_R$, which also works under $N1$.",
"We further show how a slightly stronger network stability condition $N2$ can be used to construct algorithms that never violate atomicity.",
"The guarantee of atomicity comes at the expense of having an additional phase during the read and write operations."
],
[
"Introduction",
"We consider the problem of designing algorithms for distributed storage systems (DSSs) that offer consistent access to stored data.",
"Large scale DSSs are widely used by several industries, and also widely studied by academia for a variety of applications ranging from e-commerce to sequencing genomic-data.",
"The most desirable form of consistency is atomicity, which in simple terms, gives the users of the data service the impression that the various concurrent read and write operations take place sequentially.",
"Implementations of atomicity on an asynchronous system under message passing framework, in the presence of failures, is often challenging.",
"Traditional implementations[1], [2] use replication of data as the mechanism of fault-tolerance; however they suffer from the problem of having high storage cost, and communication costs for read and write operations.",
"Erasure codes provide an efficient way to decrease storage and communication cost in atomicity implementations.",
"An $[n, k]$ erasure code splits the value $v$ , say of size 1 unit into $k$ elements, each of size $\\frac{1}{k}$ units, creates $n$ coded elements, and stores one coded element per server.",
"The size of each coded element is also $\\frac{1}{k}$ units.",
"A class of erasure codes known as Maximum Distance Separable (MDS) codes have the property that value $v$ can be reconstructed from any $k$ out of these $n$ coded elements.",
"While it is known that usage of erasure codes in asynchronous decentralized storage systems do not offer all the advantages as in synchronous centralized systems [3], erasure code based algorithms like in [4], [5], [6], or [7] for implementing consistent memory service offer significant storage and communication cost savings over replication based algorithms, in many regimes of operation.",
"For instance CASGC [6] improves the costs under the scenario when the number of writes concurrent with a read is known to be limited, whereas SODA [7] trades-off write cost in order to optimize storage cost, which is meaningful in systems with infrequent writes.",
"Both CASGC and SODA are based on MDS codes.",
"In this work, we consider the additional important issue of repairing crashed nodes without disrupting the storage service.",
"Failure of storage nodes is a norm rather than an exception in large scale DSSs today, primarily because of the usage of commodity hardware for affordability and scalability reasons.",
"Replication based algorithms in [1], [2] and erasure-code based algorithms in [4], [6], or [7] do not consider repair of crashed nodes; instead assume that a crashed node remains so for the rest of the execution.",
"Algorithms in [5], [8] consider background repair of crashed nodes; however they assume either the presence of nodes having stable storage, whose content is unaffected by crashes, or presence of a subset of nodes that never crash during the entire execution.",
"We relax both these assumptions in this work.",
"In our model, any one of the storage nodes can crash; further, we assume that a crashed node loses all its data, both volatile as well as stable storage.",
"A repair operation of a node in the crashed state is triggered externally, and is carried out by the concerned node via message exchanges with other active nodes in the system.",
"Upon completion of repair, the node (with the same id) re-enters active state, and resumes participation in ongoing and future read, write, and repair operations.",
"It is natural to expect a restriction on the number of crash and repair operations in relation to the read and write operations; the authors of [8] show an impossibility result in this direction, for guaranteeing liveness and atomicity, simultaneously.",
"We formulate network stability conditions $N1$ and $N2$ , which can be used to limit the number of crash and repairs operations overlapping with a client operation.",
"These conditions are algorithm independent, and most likely to be satisfied in any practical storage network.",
"At a high level, the condition $N1$ restricts the set of servers that can be in the crashed or repair state any time a process (client or server) pings all the $n$ servers with corresponding messages.",
"Condition $N2$ is slightly stronger than $N1$ , and restricts the set of servers that can be in the crashed or repair state if the process wants to ping-pong a fraction of the servers.",
"In a ping-pong, it is expected that the servers which receive a message also respond back to the sender of the message.",
"We first present an impossibility result for an asynchronous DSS allowing background repair of crashed nodes, where there is no restriction on the number of crash and repair operations that occur during a client operation.",
"We show that it is impossible to simultaneously achieve liveness and atomicity in such a system, even if all the crash and repair operations occur sequentially during the execution (i.e., at most one node remains in the crash or repair state at any point during the execution).",
"We then consider the problem of erasure-code based algorithm design under the network stability condition $N1$ .",
"We present the algorithm in two stages.",
"First we present a replication-based algorithm ${RADON_R}$ , which performs background-repair, and guarantees atomicity and liveness of operations under $N1$ , if more than ${3/4}^{\\text{th}}$ of all servers remain active during any ping operation.",
"The write and read phases are almost identical to those of the ABD algorithm [1], except that during a write we expect responses from more than ${3/4}^{\\text{th}}$ of all the servers, while in ABD responses are expected only from a majority of servers.",
"A repair operation in ${RADON_R}$ is simply a read operation initiated by the concerned server.",
"Thus the algorithm itself is simple; however, the proof of atomicity gets complicated because of the fact that a repair operation can potentially restore the contents of a node to a version that is older than what was present before the crash.",
"We show how the network stability condition can be used to prove atomicity, and this proof is the key takeaway from ${RADON_R}$ towards constructing the erasure-code based algorithm.",
"Our erasure-code based algorithm ${RADON_C}$ uses $[n, k]$ MDS codes, and is a natural adaptation of ${RADON_R}$ for the usage of codes.",
"A key challenge while using erasure codes is ensuring liveness of read operations, in the presence of concurrent write operations.",
"Various techniques are known in literature to handle this challenge; for instance, [5] assumes synchronous write phases, [6] limits the number of writes concurrent with a read, while [7] uses an $O(n^2)$ write protocol to guarantee liveness of reads.",
"In this work, like in [6], we make the assumption that the number of write operations concurrent with any read operation is limited by a parameter $\\delta $ , which is known a priori.",
"However, the usage of the concurrency bound differs from that of the CASGC algorithm in [6]; for instance, CASGC has three rounds for write operations, while ${RADON_C}$ uses only two rounds.",
"In ${RADON_C}$ , each server maintains a list of up to $\\delta +1$ coded elements, corresponding to the latest $\\delta +1$ versions received as a result of the various write operations.",
"In comparison with ${RADON_R}$ where a writer expects responses from more than ${3/4}^{\\text{th}}$ of all servers, a write operation in ${RADON_C}$ expects responses from more than $\\frac{3n+k}{4}$ servers.",
"During a read operation, the client reads the lists from more than $\\frac{n+k}{2}$ nodes before decoding the value $v$ .",
"Like in ${RADON_R}$ , a repair operation in ${RADON_C}$ is essentially a read operation by the concerned node; however this time the concerned node creates a list (instead of just one version) by decoding as many possibles versions that it can from the $\\left\\lceil \\frac{n+k}{2} \\right\\rceil $ responses.",
"Liveness and atomicity of operations are proved under network stability condition $N1$ , if more than $\\frac{3n+k}{4}$ servers remain active during any ping operation.",
"${RADON_C}$ has substantially improved storage and communication costs than ${RADON_R}$ , when the concurrency bound $\\delta $ is limited; see Table REF for a comparison.",
"In both ${RADON_R}$ and ${RADON_C}$ , violation of the network stability condition $N1$ can result in executions that are not atomic, which might not be preferable in certain applications.",
"The choice of consistency over liveness, or vice versa, is the subject matter of a wide range of discussions and perspectives among system designers and software engineers.",
"For example, BigTable, a DSS by Google, prefers safety over liveness [9], whereas, Amazon's Dynamo does not compromise liveness but settles for eventual consistency [10].",
"Our third algorithm ${RADON_R^{(S)}}$ , which is replication-based, is designed to guarantee atomicity during every execution.",
"Liveness is guaranteed under the slightly more stringent condition of $N2$ , with more than ${3/4}^{\\text{th}}$ of all servers remaining active during any ping-pong operation.",
"The guarantee of atomicity of every execution also needs extra phases for read and write operations, when compared to ${RADON_R}$ .",
"The design of an erasure-coded version of ${RADON_R^{(S)}}$ that never violates atomicity, is an interesting direction that we leave out for future work.",
"Table: Performance comparison of RADON R {{RADON_R}}, RADON C {{RADON_C}} and RADON R (S) {{RADON_R^{(S)}}}, where nn is the number of servers, and δ\\delta is the maximum number of writes concurrent with a read or a repair operation.",
"See Section for a justification of the costs."
],
[
"Other Related Work",
"Dynamic Reconfiguration: Our setting is closely related to the problem of implementing a consistent memory object in a dynamic setting, where nodes are allowed to voluntarily leave and join the network.",
"The problem involves dynamic reconfiguration of the set of nodes that take part in client operations, which is often implemented via a reconfig operation that is initiated by any of the participating processes, including the clients.",
"Any node that wants to leave/join the network makes an announcement, via a leave/join operation, before doing so.",
"The problem is extensively studied in the field of distributed algorithms [11], [12], [13], [14], [15]; review and tutorial articles appear in [16], [17], [18].",
"In our context, the problem of node repair could in fact be thought of as one of dynamic reconfiguration, wherein an involuntary crash is simulated by a voluntary leave operation without an explicit announcement.",
"In this case, a new node joins as a replacement node via the join operation, which can be considered as the analogue of a repair operation.",
"In the setting of dynamic reconfiguration, every node has a distinct identity; thus the replacement node joins the network with a new identity that is different from the identity of the crashed node [16].",
"This demands a reconfiguration of the set of participating nodes after every repair.",
"Such reconfigurations get in the way of client operations, and add to the latency of read and write operations [18], in practical implementations.",
"Clearly, a repair operation as considered in this work does not demand any reconfiguration, since a repaired node has the same identity as the crashed node.",
"Also, the current work shows that modeling repair via a static system, permits design of algorithms where clients remain oblivious to the presence of repair operations.",
"Furthermore, addressing storage and communication costs is not the focus of the works in dynamic reconfigurations; specifically, it is not known as to how erasure codes can be advantageously used in dynamic settings.",
"Our ${RADON_C}$ algorithm shows that when repair is carried out under a static model, it is indeed possible to advantageously use erasure to reduce costs, when the number of concurrent writes are limited.",
"We make additional comparisons between our model and results to those found in works on dynamic reconfiguration.",
"Several impossibility results exist in the context of implementing a dynamic atomic register and simultaneously guaranteeing liveness; the authors in [13] argue impossibility if there are infinitely many reconfigurations during an execution, while the authors in [14] argue an impossibility when there is no upper bound on message delay.",
"We see, not surprisingly, that even in the problem of repair, we need to suitably limit the number of crash and repair operations that occur in an execution, even if all crash and repairs are sequentially ordered.",
"In [15], the authors implement a dynamic atomic register under a model that has an (unknown) upper bound $D$ on any point-to-point message delay, and where the number of reconfigurations in any $D$ units of time is limited.",
"Our network condition $N1$ is similar, except that $1)$ we limit the number of crash and repairs during any broadcast messaging, instead of point-to-point messaging, and $2)$ we do not assume any bound on the message delay.",
"In practice, limiting number of repairs during broadcast instead of every point-to-point messaging offers resiliency against straggler nodes, which refer to the nodes having the worst delays among all nodes.",
"We would also like to note that the algorithm in [15] does not guarantee atomicity, if the number of reconfigurations in $D$ units of time is higher than a set number.",
"This appears similar to ${RADON_R}$ , where atomicity is not guaranteed if we do not satisfy stability condition $N1$ .",
"While we show how the slightly tighter model $N2$ can be used to always guarantee atomicity, it is an interesting question as to whether the model $N2$ can be adopted in the work of [15] so as to always guarantee atomicity.",
"Repair-Efficient Erasure Codes for Distributed Storage: Recently, a large class of new erasure/network codes for storage have been proposed (see [19] for a survey), and also tested in networks [20], [21], [22], where the focus is efficient storage of immutable data, such as, archival data.",
"These new codes are specifically designed to optimize performance metrics like repair-bandwidth and repair-time (of failed servers), and offer significant performance gains when compared to the traditional Reed-Solomon MDS codes [23].",
"It needs to be explored if these codes can be used in conjunction with the ${RADON_C}$ algorithm, to further improve the performance costs.",
"Other Works on using Erasure Codes: Applications of erasure codes to Byzantine fault tolerant DSSs are discussed in [24], [25], [26].",
"In [3], the authors consider algorithms that use erasure codes for emulating regular registers.",
"Regularity [27], [28] is a weaker consistency notion than atomicity.",
"The rest of the document is organized as follows.",
"Our system model appears in Section .",
"The impossibility result, and the network stability conditions appear in Section .",
"The three algorithms appear in Sections , and , respectively.",
"In Section , we discuss the storage and communication costs of the algorithms.",
"Section concludes the paper.",
"Proofs of various claims appear in the Appendix.",
"Processes and Asynchrony: We consider a distributed system consisting of asynchronous processes, each with a unique identifier (ID), of three types: a set of readers, ${\\mathcal {R}}$ ; a set of writers, ${\\mathcal {W}}$ ; and a set of $n$ servers, ${\\mathcal {S}}$ .",
"The readers and writers are together referred to as clients.",
"The set ${\\mathcal {R}}\\cup {\\mathcal {W}}\\cup {\\mathcal {S}}$ forms a totally ordered set under some defined relation ($>$ ).",
"The reader and writer processes initiate read and write operations respectively, and communicate with the servers using messages.",
"A reader or writer can invoke a new operation only after all previous operations invoked by it has completed.",
"The property is referred to as the well-formedness property of an execution.",
"We assume that every client/server is connected to every other server via a reliable communication link; thus as long as the destination process is non-faulty, any message sent on the link eventually reaches the destination process.",
"Crash and Recovery: A client may fail at any point during the execution.",
"At any point during the execution, a server can be in one (and only one) of the following three states: active, crashed or repair.",
"A crash event triggers a server to enter the crashed state from an active state.",
"The server remains in the crashed state for an arbitrary amount of time, but eventually is triggered by a repair event to enter the repair state.",
"Crash and repair events are assumed to be externally triggered.",
"A server in the repair state can experience another crash event, and go back to the crashed state.",
"A server in the crashed state does not perform any local computation.",
"The server also does not send or receive messages in the crashed state, i.e., any message reaching the server in a crashed state is lost.",
"A server which enters the repair state has all its local state variables set to default values, i.e., a crash event causes the server to lose all its state variables.",
"A server in the repair state can perform computations like in the active state.",
"Atomicity and Liveness: We aim to implement only one atomic read/write memory object, say $x$ , under the MWMR setting on a set of servers, because any shared atomic memory can be emulated by composing individual atomic objects.",
"The object value $v$ comes from some set $V$ ; initially $v$ is set to a distinguished value $v_0$ ($\\in V$ ).",
"Reader $r$ requests a read operation on object $x$ .",
"Similarly, a write operation is requested by a writer $w$ .",
"Each operation at a non-faulty client begins with an invocation step and terminates with a response step.",
"An operation is incomplete when its invocation step does not have the associated response step; otherwise it is complete.",
"By liveness of a read or a write operation, we mean that during any well-formed execution, any read or write operation respectively initiated by a non-faulty reader or writer completes, despite the crash failure of any other client.",
"By liveness of repair associated with a crashed server, we mean that the server which enters a crashed state eventually re-enters the active state, unless it experiences a crash event during every repair operation that the server attempts.",
"The liveness of repair holds despite the crash failure of any other client.",
"Background on Erasure coding: In ${RADON_C}$ , we use an $[n, k]$ linear MDS code [30] over a finite field $\\mathbb {F}_q$ to encode and store the value $v$ among the $n$ servers.",
"An $[n, k]$ MDS code has the property that any $k$ out of the $n$ coded elements can be used to recover (decode) the value $v$ .",
"For encoding, $v$ is dividedIn practice $v$ is a file, which is divided into many stripes based on the choice of the code, various stripes are individually encoded and stacked against each other.",
"We omit details of representability of $v$ by a sequence of symbols of $\\mathbb {F}_q$ , and the mechanism of data striping, since these are fairly standard in the coding theory literature.",
"into $k$ elements $v_1, v_2, \\ldots v_k$ with each element having size $\\frac{1}{k}$ (assuming size of $v$ is 1).",
"The encoder takes the $k$ elements as input and produces $n$ coded elements $c_1, c_2, \\ldots , c_n$ as output, i.e., $[c_1, \\ldots , c_n] = \\Phi ([v_1, \\ldots , v_k])$ , where $\\Phi $ denotes the encoder.",
"For ease of notation, we simply write $\\Phi (v)$ to mean $[c_1, \\ldots , c_n]$ .",
"The vector $[c_1, \\ldots , c_n]$ is referred to as the codeword corresponding to the value $v$ .",
"Each coded element $c_i$ also has size $\\frac{1}{k}$ .",
"In our scheme we store one coded element per server.",
"We use $\\Phi _i$ to denote the projection of $\\Phi $ on to the $i^{\\text{th}}$ output component, i.e., $c_i = \\Phi _i(v)$ .",
"Without loss of generality, we associate the coded element $c_i$ with server $i$ , $1 \\le i \\le n$ .",
"Storage and Communication Cost: We define the total storage cost as the size of the data stored across all servers, at any point during the execution of the algorithm.",
"The communication cost associated with a read or write operation is the size of the total data that gets transmitted in the messages sent as part of the operation.",
"We assume that metadata, such as version number, process ID, etc.",
"used by various operations is of negligible size, and is hence ignored in the calculation of storage and communication cost.",
"Further, we normalize both the costs with respect to the size of the value $v$ ; in other words, we compute the costs under the assumption that $v$ has size 1 unit."
],
[
"An Impossibility Result",
"The crash and recovery model described in Section does not impose any restriction on the rate of crash events, and repair operations that happen in the system.",
"In other words, the model described above does not limit in any manner the number of crash events/repair operations, which can overlap with any a client operation.",
"In [8], the authors showed that without such restrictions, it is impossible to implement a shared atomic memory service, which guarantees liveness of operations.",
"Below, we state an impossibility result which holds even if there is at most one server in the crashed/repair state at any point during the execution.",
"We then introduce network stability conditions that enable us impose restrictions on the number of crash/repair events that overlap with any operation.",
"It is impossible to implement an atomic memory service that guarantees liveness of reads and writes, under the system model described in Section , even if 1) there is at most one server in the crashed/repair state at any point during the execution, and 2) every repair operation completes, and takes the repaired server back to the active state."
],
[
"Network Stability Conditions $N1$ and {{formula:2aeeec28-18b6-42bb-b8f4-7a5e4ecb28ea}} ",
"We begin with the notions of a group-send operation, and effective consumption of a message.",
"group-send operation: The group-send operation is used to abstract the operation of a process sending a list of $n$ messages $\\lbrace m_1, \\cdots , m_n\\rbrace $ to the set of all $n$ servers $\\lbrace s_1, \\ldots , s_n\\rbrace = \\mathcal {S}$ , where message $m_i$ is send to server $s_i, 1 \\le i \\le n$ .",
"Note that this is a mere abstraction of the process sending out $n$ point-to-point messages sequentially to $n$ servers, without interleaving the “send\" operations with any significant local computations or waiting for any external inputs.",
"The operation is no more powerful then sending $n$ consecutive messages.",
"The operation is written as $group\\mathord {\\sf -}send([m_1, m_2, \\cdots , m_n])$ .",
"In the event $m_i = m, \\forall i$ , we simply write $group\\mathord {\\sf -}send(m)$ .",
"Our model allows the sender to fail while executing the $group\\mathord {\\sf -}send$ operation, in which case only a subset of the $n$ servers receive their corresponding messages.",
"Effective Consumption: We say a process effectively consumes a message $m$ , if it receives $m$ , and executes all steps of the algorithm that depend only on the local state of the process, and the message $m$ ; in other words, the process executes all the steps that do not require any further external messages.",
"[Network Stability Conditions] Consider a process $p$ executing a $group\\mathord {\\sf -}send$ $([m_1, m_2, \\cdots , m_n])$ operation, and consider the following statements: $(a)$ $(i)$ There exists a subset $\\mathcal {S}_{\\alpha } \\subseteq {\\mathcal {S}}$ of $|\\mathcal {S}_{\\alpha }| = \\left\\lceil \\alpha n\\right\\rceil $ servers, $0 < \\alpha < 1$ , all of which effectively consume their respective messages from the group-send operation, and $(ii)$ all the servers in $\\mathcal {S}_{\\alpha }$ remain in the active state during the interval $[T_1 \\ T_2]$ , where $T_1$ denotes the point of time of invocation of the $group\\mathord {\\sf -}send$ operation, and $T_2$ denotes the earliest point of time in the execution at which all of the servers in $\\mathcal {S}_{\\alpha }$ complete the effective consumption of their respective messages.",
"$(b)$ Further, if effective consumption of the message $m_i$ by server $s_i$ involves sending a response back to the process $p$ , for all $s_i \\in \\mathcal {S}_{\\alpha }$ , then all servers in $\\mathcal {S}_{\\alpha }$ remain in the active state during the interval $[T_1 \\ T_3]$ , where $T_3$ denotes the earliest point of time in the execution at which the process $p$ completes effective consumption of the responses from the all the servers in $\\mathcal {S}_{\\alpha }$ .",
"If the network satisfies Statement $(a)$ for every execution of a group-send operation by any process, we say that it satisfies network stability condition $N1$ with parameter $\\alpha $ .",
"If the network satisfies Statements $(a)$ and $(b)$ for every execution of a group-send operation by any process, we say that it satisfies network stability condition $N2$ with parameter $\\alpha $ .",
"Clearly, $N2$ implies $N1$ .",
"Note that the set $\\mathcal {S}_{\\alpha }$ which needs to satisfy the conditions need not be the same for various invocations of group-send operations by either the same or distinct processes.",
"Also, note that in condition $N2$ , the process $p$ might crash before completing the effective consumption of the responses from the servers in $\\mathcal {S}_{\\alpha }$ .",
"In this case we only expect Statement $(a)$ to be satisfied, and not Statement $(b)$ .",
"Furthermore, in both $N1$ and $N2$ , we do not expect any of these statements to be true, if process $p$ crashes after partial execution of the group-send operation."
],
[
"The ${RADON_R}$ Algorithm",
"In this section, we present the ${RADON_R}$ algorithm, and prove its liveness and atomicity properties for networks that satisfy the network condition $N1$ with $\\alpha > \\frac{3}{4}$ .",
"We begin with some useful notation.",
"Tags are used for version control of the object values.",
"A tag $t$ is defined as a pair $(z, w)$ , where $z \\in \\mathbb {N}$ and $w \\in \\mathcal {W}$ denotes the ID of a writer.",
"We use $\\mathcal {T}$ to denote the set of all the possible tags.",
"For any two tags $t_1, t_2 \\in \\mathcal {T}$ , we say $t_2 > t_1$ if $(i)$ $t_2.z > t_1.z$ or $(ii)$ $t_2.z = t_1.z$ and $t_2.w > t_1.w$ .",
"Note that $(\\mathcal {T}, >)$ is a totally ordered set.",
"The protocols for writer, reader, and servers are shown in Fig. .",
"Each server stores two state variables $(i)$ $(t_{loc}, v_{loc})$ - a tag and value pair, initially set to $(t_0, v_0)$ , $(ii)$ $status$ - a variable that can be in either active or repair state.",
"[!ht] 2write(v) get-tag $group\\mathord {\\sf -}send(\\text{\\sc {query-tag}})$ Await responses from majority Select the max tag $t^*$ put-data $t_w = (t^{*}.z + 1, w)$ $group\\mathord {\\sf -}send((\\text{\\sc {put-data}}, (t_w, v)))$ Terminate after $\\left\\lceil \\frac{3n + 1}{4} \\right\\rceil $ acks.",
"read get-data $group\\mathord {\\sf -}send(\\text{\\sc {query-tag-data}})$ Await responses from majority Select $(t_r, v_r)$ , with max tag.",
"put-data $group\\mathord {\\sf -}send((\\text{\\sc {put-data}}, (t_r, v_r)))$ Wait for $\\left\\lceil \\frac{3n + 1}{4}\\right\\rceil $ acks Return $v_r$ Server $s \\in \\mathcal {S}$$State~Variables$ $(t_{loc}, v_{loc}) \\in \\mathcal {T} \\times {\\mathcal {V}}$ , initially $(t_0, v_0)$ $status \\in \\lbrace active, repair\\rbrace $ , initially $active$ get-tag-resp, recv $\\text{\\sc {query-tag}}$ from writer $w$ $status = active$ Send $t_{loc}$ to $w$ get-data-resp, recv $\\text{\\sc {query-tag-data}}$ from reader $r$ $status = active$ Send $(t_{loc}, v_{loc})$ to $r$ put-data-resp, recv $\\text{\\sc {put-data}}, (t, v)$ from client $c$ $status = active$ $t > t_{loc}$ $(t_{loc}, v_{loc}) \\leftarrow (t, v)$ Send ack to $c$ .",
"init-repair $status \\leftarrow repair$ $(t_{loc}, v_{loc}) \\leftarrow (t_0, v_0)$ $group\\mathord {\\sf -}send(\\text{\\sc {repair-tag-data}})$ Await responses from majority Select $(t_{rep}, v_{rep})$ , for max tag $(t_{loc}, v_{loc}) \\leftarrow (t_{rep}, v_{rep})$ $status \\leftarrow active$ init-repair-resp, recv $\\text{\\sc {repair-tag-data}}$ from $s^{\\prime }$ $status = active$ Send $(t_{loc}, v_{loc})$ to $s^{\\prime }$ The protocols for writer, reader, and any server $s \\in {\\mathcal {S}}$ in ${RADON_R}$ .",
"The write and read operations are very similar to those in the ABD algorithm [1], and each consists of two phases.",
"In the first phase, get-tag, of a write operation $\\pi $ , the writer queries all servers for local tags, awaits responses from a majority of servers, and selects the maximum tag $t^*$ from among the responses.",
"Next, the writer executes the put-data phase, during which a new tag $t_w = tag(\\pi )$ is created by incrementing the integer part of $t^*$ , and by incorporating the writer's own ID.",
"The writer then sends pair $(t_w, v)$ to all servers, and awaits acknowledgments (acks) from $\\left\\lceil \\frac{3n+1}{4}\\right\\rceil $ servers before completing the operation.",
"The two phases are identical to those of the ABD algorithm [1], except for the fact that during the second phase, ABD expects acks from only a majority of servers, whereas here we need from $\\left\\lceil \\frac{3n+1}{4}\\right\\rceil $ servers.",
"During a read operation $\\rho $ , the reader in the get-data phase queries all the servers in $S$ for the respective local tag and value pairs.",
"Onces it receives responses from a majority of servers in $S$ , it picks the pair with the highest tag, which we designate as $t_r = tag(\\pi )$ .",
"In the subsequent put-data phase, the reader writes back the tag $t_r$ and the corresponding value $v_r$ to all servers, and terminates after receiving acknowledgments from $\\left\\lceil \\frac{3n+1}{4}\\right\\rceil $ servers.",
"Once again, we remark that both phases in the read are identical to those of the ABD algorithm, except for the difference in the number of the servers from which acks are expected in the second write-back phase.",
"Note that, during both the write and operations, a server responds to an incoming message only if it is in the active state.",
"A repair operation is initiated via the action init-repair, by an external trigger, at a server which is in the crashed state.",
"Note that we do not explicitly define a crashed state since a crash is not a part of the algorithm.",
"We assume that as soon as the repair operation starts, the variable status is set to the repair state, and also the local (tag, value) pair is set to the default sate $(t_0, v_0)$ .",
"The repair operation is essentially the first phase of the read operation, during which the server queries all the servers for the respective local tag and value pairs, and stores the tag and value pair corresponding to the highest tag after receiving responses from a majority of servers.",
"Finally, the repair operation is terminated setting variable status to active state.",
"A server in $S$ responds to a request generated from init-repair phase only if it is in the active state."
],
[
"Analysis of ${RADON_R}$",
"Liveness of read, write and repair operations in ${RADON_R}$ follows immediately if we assume condition $N1$ with $\\alpha > \\frac{3}{4}$ .",
"This is because liveness of any operation depends on sufficient number of responses from the servers during the various phases of the operation.",
"From Fig.",
", we know that the maximum number of responses that is expected in any phase is $\\left\\lceil \\frac{3n+1}{4}\\right\\rceil $ , which is guaranteed under $N1$ with $\\alpha > \\frac{3}{4}$ .",
"The tricky part is to prove atomicity of reads and writes.",
"The proof is based on Lemma $13.16$ of [31], a restatement of which can be found in [29].",
"Consider two completed write operations $\\pi _1$ and $\\pi _2$ , such that, $\\pi _2$ starts after the completion of $\\pi _1$ .",
"For any completed write operation $\\pi $ , we define $tag(\\pi ) = t_w$ , where $t_w$ is the tag which the writer uses in the put-data phase.",
"In this case, one of the requirements the algorithm needs to satisfy to ensure atomicity is $tag(\\pi _2) > tag(\\pi _1)$ .",
"While this fact is straightforward to prove for an algorithm like ABD, which does not have background repair, in ${RADON_R}$ , we need to consider the effect of those repair operations that overlap with $\\pi _1$ , and also those that occur in between $\\pi _1$ and $\\pi _2$ .",
"The point to note is that such repair operations can potentially restore the contents of the repaired node such that the restored tag is less than $tag(\\pi _1)$ .",
"We then need to show the absence of propagation of older tags (older than $tag(\\pi _1)$ ) into a majority of nodes, due to a sequence of repairs which happen before $\\pi _2$ decides its tag.",
"We do this via the following two observations: $1)$ In Lemma REF , we show that any successful repair operation, which begins after a point of time $T$ , always restores value to one, which corresponds to a tag which is at least as high as the minimum of the tags stored in any majority of active servers at time $T$ .",
"This fact is in turn used to prove a similar property for reads and writes, as well.",
"$2)$ We next show (as part of proof of Theorem REF ), under the assumption of $N1$ with $\\alpha > 3/4$ , the existence of a point of time $T$ before the completion of $\\pi _1$ such that a majority of nodes are active at $T$ , and all of whose tags are at least as high as $tag(\\pi _1)$ .",
"The two steps are together used to prove that $tag(\\pi _2) > tag(\\pi _1)$ .",
"A similar sequence of steps are used to show atomicity properties of read operations, as well.",
"For a completed read operation $\\pi $ , $tag(\\pi ) = t_r$ , where $t_r$ is the tag corresponding to the value $v_r$ returned by the reader.",
"For a completed repair $\\pi $ , $tag(\\pi ) = t_{rep}$ , where $t_{rep}$ is the tag corresponding to the value restored during the repair operation.",
"Let $\\beta $ denote a well-formed execution of ${RADON_R}$ .",
"Suppose $T$ denotes a point of time in $\\beta $ such that there exists a majority of servers $\\mathcal {S}_m$ , $\\mathcal {S}_m \\subset \\mathcal {S}$ all of which are in the active state at time $T$ .",
"Also, let $t_s$ denote the value of the local tag at server $s \\in \\mathcal {S}_m$ , at time $T$ .",
"Then, if $\\pi $ denotes any completed repair or read operation that is initiated after time $T$ , we have $tag(\\pi ) \\ge \\min _{s\\in \\mathcal {S}_m} t_s$ .",
"Also, if $\\pi $ denotes any completed write operation that is initiated after time $T$ , we have $tag(\\pi ) > \\min _{s\\in \\mathcal {S}_m} t_s$ .",
"[Liveness] Let $\\gamma $ denote a well-formed execution of ${RADON_R}$ , under the condition $N1$ with $\\alpha > \\frac{3}{4}$ .",
"Then every operation initiated by a non-faulty client completes.",
"[Atomcity] Every execution of the ${RADON_R}$ algorithm operating under the $N1$ network stability condition with $\\alpha > \\frac{3}{4}$ , is atomic.",
"We note that, though Lemma REF gives a result about completed operations, condition $N1$ is not a prerequisite for the result in Lemma REF .",
"In other words, the result in Lemma REF holds for any completed operation, even if condition $N1$ is violated.",
"As we will see, this is an important fact that we will use to establish atomicity of ${RADON_R^{(S)}}$ for any execution."
],
[
"Algorithm ${{RADON_C}}$ ",
" In this section, we present the erasure-code based ${RADON_C}$ algorithm for implementing atomic memory service, and performing repair of crashed nodes.",
"The algorithm uses $[n, k]$ MDS codes for storage.",
"Liveness and atomicity are guaranteed under the following assumptions: $1)$ the $N1$ network stability condition with $\\alpha \\ge \\frac{3n+k}{4n}$ , $2)$ the number of write operations concurrent with a read or repair operation is at most $\\delta $ .",
"The precise definition of concurrency depends on the algorithm itself, and appears later in this section.",
"The ${RADON_C}$ algorithm has significantly reduced storage and communication cost requirements than ${RADON_R}$ , when $\\delta $ is limited.",
"[!ht] 2write($v$ ) get-tag $group\\mathord {\\sf -}send(\\text{\\sc {query-tag}})$ Await responses from majority Select the max tag $t^*$ put-data $t_w = (t^{*}.z + 1, w)$ .",
"$code\\mathord {\\sf -}elems= [(t_w, c_1), \\ldots , (t_w, c_n)]$ , $c_i = \\Phi _i(v)$ $group\\mathord {\\sf -}send(\\text{\\sc {code-elements}}, code\\mathord {\\sf -}elems)$ Terminate after $\\left\\lceil \\frac{3n + k}{4}\\right\\rceil $ acks read get-data $group\\mathord {\\sf -}send(\\text{\\sc {query-list}})$ Wait for $\\left\\lceil \\frac{n+k}{2}\\right\\rceil $ $Lists$ Select the max tag, $t_r$ , whose corresponding value, $v_r$ , is decodable using the $Lists$ .",
"put-data $code\\mathord {\\sf -}elems= [(t_r, c_1), \\ldots , (t_r, c_n)]$ , $c_i = \\Phi _i(v_r)$ $group\\mathord {\\sf -}send(\\text{\\sc {code-elements}}, code\\mathord {\\sf -}elems)$ Wait for $\\left\\lceil \\frac{3n + k}{4}\\right\\rceil $ acks Return $v_r$ Server $s_i \\in \\mathcal {S}$$State~Variables$ $status \\in \\lbrace active, repair\\rbrace $ , initially $active$ $List \\subseteq \\mathcal {T} \\times \\mathcal {C}_s$ , initially $\\lbrace (t_0, \\Phi _i(v_0))\\rbrace $ get-tag-resp,recv $\\text{\\sc {query-tag}}$ from writer $w$ $status = active$ $t^* = \\max _{(t,c) \\in List}t$ Send $t^*$ to $w$ get-data-resp, recv $\\text{\\sc {query-list}}$ from reader $r$ $status = active$ Send $List$ to $r$ put-data-resp, recv $\\text{\\sc {code-elements}}, (t, c_i)$ from $p$ $status = active$ $List \\leftarrow List \\cup \\lbrace (t, c_i) \\rbrace $ $|List| > \\delta + 1$ Retain the (tag, coded-element) pairs for the $\\delta +1 $ highest tags in $List$ , and delete the rest.",
"Send ack to $p$ .",
"init-repair $status \\leftarrow repair$ $group\\mathord {\\sf -}send(\\text{\\sc {repair-list}})$ Wait for $\\left\\lceil \\frac{n+k}{2}\\right\\rceil $ $Lists$ Find (tag, value) pairs decodable from $Lists$ .",
"Restore local $List$ via re-encoding and retaining the (tag, coded-element) pairs corresponding to at most $\\delta +1 $ highest tags, from the above pairs $status \\leftarrow active$ init-repair-resp, recv $\\text{\\sc {repair-list}}$ from server $s^{\\prime }$ $status=active$ Send $List$ to $s^{\\prime }$ The protocols for write, reader, and any server $s_i \\in {\\mathcal {S}}$ in ${RADON_C}$ .",
"The algorithm (see Fig. )",
"is a natural generalization of the ${RADON_R}$ algorithm accounting for the fact that we use MDS codes.",
"The write operation has two phases, where the first phase finds the maximum tag in the system based on majority responses.",
"During the second phase, the writer computes the coded elements for each of the $n$ servers and uses the group-send operation to disperse them.",
"The $group\\mathord {\\sf -}send$ operation here uses a vector of length $n$ , where the $i^{\\text{th}}$ element denotes the message for the $i^{\\text{th}}$ server, $1 \\le i \\le n$ .",
"Each server keeps a $List$ of up to $(\\delta + 1)$ (tag, coded-element) pairs.",
"Every time a (tag, coded-element) message arrives from a writer, the pair gets added to the $List$ , which is then pruned to at most $(\\delta +1)$ pairs, corresponding to the highest tags.",
"The writer terminates after getting acks from $\\left\\lceil \\frac{3n+k}{4} \\right\\rceil $ servers.",
"During a read operation, the reader queries all servers for their entire local $Lists$ , and awaits responses from $\\left\\lceil \\frac{n+k}{2} \\right\\rceil $ servers.",
"Once the reader receives $Lists$ from $\\left\\lceil \\frac{n+k}{2} \\right\\rceil $ servers, it selects the highest tag $t_r$ whose corresponding value $v_r$ can be decoded using the using the coded elements in the lists.",
"The read operation completes following a write-back of $(t_r, v_r)$ using the put-data phase.",
"The repair operation is very similar to the first phase of the read operation, during which a server collects lists from $\\left\\lceil \\frac{n+k}{2} \\right\\rceil $ servers.",
"But this time, the server figures out the set of all the possible tags that can be decoded from among the $Lists$ , and prunes the set to the highest $(\\delta + 1)$ tags.",
"The repaired $List$ then consists of (tag, coded-element) pairs corresponding these (at most) $(\\delta + 1)$ tags.",
"Assuming repair of server $i$ , the creation of a coded-element corresponding to a value $v$ involves first decoding the value $v$ , and then computing $\\Phi _i(v)$ (referred to as re-encoding in Fig.",
")."
],
[
"Analysis of ${RADON_C}$",
"Throughout this section, we assume network stability condition $N1$ with $\\alpha \\ge \\frac{3n+k}{4n}$ .",
"Tags for completed read and write operations are defined in the same manner as we did for ${RADON_R}$ ; we avoid repeating them here.",
"We first discuss liveness properties of ${RADON_C}$ .",
"Let us first consider liveness of repair operations.",
"Towards this, note from the algorithm in Fig.",
"that a repair operation never gets stuck even if it does not find any set of $k$ $Lists$ among the responses, all of which have a common tag.",
"In such a case, the algorithm allows the possibility that the repaired $List$ is simply empty, at the point of execution when the server re-enters the active state.",
"In other words, liveness of a repair operation is trivially proved, i.e., a server in a repair state always eventually reenters the active state, as long as it does not experience a crash during the repair operation.",
"The triviality of liveness of repair operations, observed above, does not extend to read operations.",
"For a read operation to complete the get-data phase, it must be able to find a set of $k$ $Lists$ among the responses all of which contain coded-elements corresponding to a common tag; otherwise a read operation gets stuck.",
"The discussion above motivates the following definitions of valid read and valid repair operations.",
"[Valid Read and Repair Operations] A read operation will be called as a valid read if the associated reader remains alive at least until the reception of the $\\left\\lceil \\frac{n+k}{2} \\right\\rceil $ responses during the get-data phase.",
"Similarly, a repair operation will be called a valid repair if the associated server does not experience a further crash event during the repair operation.",
"[Writes Concurrent with a Valid Read (Repair)] Consider a valid read (repair) operation $\\pi $ .",
"Let $T_1$ denote the point of initiation of $\\pi $ .",
"For a valid read, let $T_2$ denote the earliest point of time during the execution when the associated reader receives all the $\\left\\lceil \\frac{n+k}{2} \\right\\rceil $ responses.",
"For a valid repair, let $T_2$ denote the point of time during the execution when the repair completes, and takes the associated server back to the active state.",
"Consider the set $\\Sigma = \\lbrace \\sigma : \\sigma $ is any write operation that completes before $\\pi \\text{ is initiated} \\rbrace $ , and let $\\sigma ^* = \\arg \\max _{\\sigma \\in \\Sigma }tag(\\sigma )$ .",
"Next, consider the set $\\Lambda = \\lbrace \\lambda : \\lambda $ is any write operation that starts before $T_2 \\text{ such that } tag(\\lambda ) > tag(\\sigma ^*)\\rbrace $ .",
"We define the number of writes concurrent with the valid read (repair) operation $\\pi $ to be the cardinality of the set $\\Lambda $ .",
"The above definition captures all the write operations that overlap with the read, until the time the reader has all data needed to attempt decoding a value.",
"However, we ignore those write operations that might have started in the past, and never completed yet, if their tags are less than that of any write that completed before the read started.",
"This allows us to ignore write operations due to failed writers, while counting concurrency, as long as the failed writes are followed by a successful write that completed before the read started.",
"The following lemma could be considered as the analogue of Lemma REF for ${RADON_C}$ .",
"The first part of the lemma shows that under $N1$ with $\\alpha \\ge \\frac{3n+k}{4n}$ , the repaired $List$ is never empty; there is always at least one (tag, coded-element) pair in the repaired $List$ .",
"Parts 2 and 3 are used to prove liveness and atomicity of client operations.",
"Consider any well-formed execution $\\beta $ of ${{RADON_C}}$ operating under the network stability condition $N1$ with $\\alpha \\ge \\frac{3n+k}{4n}$ .",
"Further assume that the number of writes concurrent with any valid read or repair operation is at most $\\delta $ .",
"For any operation $\\pi $ , consider the set $\\Sigma = \\lbrace \\sigma : \\sigma ~\\text{is a read }$ $ \\text{or a write in $\\beta $ }$ $\\text{that completes before}~\\pi $ $\\text{begins}\\rbrace $ , and also let $\\sigma ^{*} = \\arg \\max \\limits _{\\sigma \\in \\Sigma }{{tag(\\sigma )}}$ .",
"Then, the following statements hold: If $\\pi $ denotes a completed repair operation on a server $s \\in \\mathcal {S}$ , then the repaired $List$ of server $s$ due to $\\pi $ contains the pair $(tag(\\sigma ^*), c_s^*)$ .",
"If $\\pi $ denotes a read operation associated with a non-faulty reader $r$ , and further, if $\\mathcal {S}_1$ denotes the set of $\\left\\lceil \\frac{n+k}{2}\\right\\rceil $ servers whose responses, say $\\lbrace L_{\\pi }(s), s \\in \\mathcal {S}_1\\rbrace $ , are used by $r$ to attempt decoding of a value in the get-data phase, then there exists $\\mathcal {S}_2 \\subseteq \\mathcal {S}_{1}$ , $|\\mathcal {S}_{2}| = k $ , such that $\\forall s \\in \\mathcal {S}_2, (tag(\\sigma ^{*}), c_s^* ) \\in L_{\\pi }(s)$ .",
"If $\\pi $ denotes a write operation associated with a non-faulty writer $w$ , and further if $\\mathcal {S}_1$ denotes the set of majority servers whose responses are used by $w$ to compute max-tag in the get-tag phase, then there exists a server $s \\in \\mathcal {S}_1$ , whose response tag $t_s \\ge tag(\\sigma ^*)$ .",
"Here, $c_s^*$ denotes the coded-element of server $s$ for value $v^*$ , associated with $tag(\\sigma ^*)$ .",
"[Liveness] Let $\\beta $ denote a well-formed execution of ${RADON_C}$ , operating under the $N1$ network stability condition with $\\alpha \\ge \\frac{3n+k}{4n}$ and $\\delta $ be the maximum number of write operations concurrent with any valid read or repair operation.",
"Then every operation initiated by a non-faulty client completes.",
"[Atomicity] Any execution of ${RADON_C}$ , operating under condition $N1$ with $\\alpha \\ge \\frac{3n+k}{4n}$ , is atomic, if the maximum number of write operations concurrent with a valid read or repair operation is $\\delta $ ."
],
[
"The ${{RADON_R^{(S)}}}$ Algorithm",
"In this section, we present the ${RADON_R^{(S)}}$ algorithm having the property that every execution is atomic.",
"Liveness is guaranteed under the slightly stronger network stability condition $N2$ with $\\alpha > \\frac{3}{4}$ .",
"In comparison wtih ${RADON_R}$ , the algorithm has extra phases for both read and write operations, in order to guarantee safety of every execution.",
"[!ht] 2write(v) get-tag $group\\mathord {\\sf -}send(\\text{\\sc {query-tag}})$ Await responses from majority Select the max tag $t^*$ put-data $t_w = (t^{*}.z + 1, w)$ .",
"$group\\mathord {\\sf -}send( (\\text{\\sc {put-data}}, (t_w, v)))$ Wait for $\\left\\lceil \\frac{3n + 1}{4}\\right\\rceil $ acks (say from $\\mathcal {S}_{\\alpha }$ ) confirm-data $group\\mathord {\\sf -}send((\\text{\\sc {confirm-data}}, t_w))$ Terminate after acks from majority from among servers in $\\mathcal {S}_{\\alpha }$ read get-data $group\\mathord {\\sf -}send(\\text{\\sc {query-tag-data}})$ Await responses from majority Select $(t_r, v_r)$ , with max tag.",
"put-data $group\\mathord {\\sf -}send((\\text{\\sc {put-data}}, (t_r, v_r)))$ Wait for $\\left\\lceil \\frac{3n + 1}{4} \\right\\rceil $ acks (say from $\\mathcal {S}_{\\alpha }$ ) confirm-data $group\\mathord {\\sf -}send((\\text{\\sc {confirm-data}}, t_r))$ Await acks from a majority of servers in $\\mathcal {S}_{\\alpha }$ Return $v_r$ Server $s \\in \\mathcal {S}$$State~Variables$ $(t_{loc}, v_{loc}) \\in \\mathcal {T} \\times {\\mathcal {V}}$ , initially $(t_0, v_0)$ $status \\in \\lbrace active, repair\\rbrace $ , initially $active$ $Seen \\subseteq {\\mathcal {T}} \\times \\lbrace {\\mathcal {W}} \\cup {\\mathcal {R}}\\rbrace $ , initially empty get-tag-resp, recv $\\text{\\sc {query-tag}}$ from writer $w$ $status = active$ Send $t_{loc}$ to $w$ get-data-resp, recv $\\text{\\sc {query-tag-data}}$ from reader $r$ $status = active$ Send $(t_{loc}, v_{loc})$ to $r$ put-data-resp, recv $(\\text{\\sc {put-data}}, (t, v))$ from $c$ $status = active$ $t > t_{loc}$ $(t_{loc}, v_{loc}) \\leftarrow (t, v)$ $Seen \\leftarrow Seen \\cup \\lbrace (t, c) \\rbrace $ Send ack to $c$ .",
"confirm-data-resp, recv $(\\text{\\sc {confirm-data}}, t)$ from $c$ $status = active$ $(t, c) \\in Seen$ Remove $(t, c)$ from $Seen$ Send ack to client $c$ .",
"init-repair $status \\leftarrow repair$ $(t_{loc}, v_{loc}) \\leftarrow (t_0, v_0)$ $Seen \\leftarrow \\emptyset $ $group\\mathord {\\sf -}send(\\text{\\sc {repair-tag-data}})$ Await responses from majority.",
"Select $(t_{rep}, v_{rep})$ , with max tag $(t_{loc}, v_{loc}) \\leftarrow (t_{rep}, v_{rep})$ $status \\leftarrow active$ init-repair-resp, recv $\\text{\\sc {repair-tag-data}}$ from $s^{\\prime }$ $status=active$ Send $(t_{loc}, v_{loc})$ to $s^{\\prime }$ The protocols for writer, reader, and any server $s \\in {\\mathcal {S}}$ in ${RADON_R^{(S)}}$ .",
"The write operation has three phases (see Fig.",
").",
"The first two phases are identical to those of ${RADON_R}$ during which the writer queries for the local tags, and then sends out the new (tag, value) pair, respectively.",
"In the third phase, called confirm-data, the writer ensures the presence of at least a majority of servers, which the writer knows for sure that received its data during the second phase, put-data.",
"In order to facilitate the confirm-data phase, the servers maintain a $Seen$ variable.",
"Any time the server receives a value from a writer, the server adds the corresponding (tag, writer ID) pair to the $Seen$ list.",
"Next, during the confirm-data-resp phase, the server responds to the writer only if this (tag, writer ID) pair is part of the $Seen$ variable.",
"The idea is that if the server experiences a crash and a successful repair operation in between the put-data and confirm-data phases, the server no longer has the (tag, writer ID) pair in its $Seen$ variable, and hence does not respond to the confirm-data phase.",
"This is because, a crash removes all state variables, including $Seen$ , and the repair algorithm (see Fig. )",
"simply restores the $Seen$ variable to its default value, the empty set.",
"Further, by ensuring that the writer expects acks from among a majority of servers in confirm-data, from among the $\\frac{3n+1}{4}$ servers whose acks were obtained during put-data, we can guarantee that any execution is atomic.",
"The read operation also has three phases, first two of which are identical to those of ${RADON_R}$ , except for the use of the $Seen$ variable in the server during the put-data phase.",
"The third phase is the confirm-data phase as in the write operation.",
"The repair operation has one phase, and is nearly exactly identical to that of ${RADON_R}$ .",
"Note that the $Seen$ variable gets reset to its initial value during repair."
],
[
"Analysis of ${RADON_R^{(S)}}$",
" We overview the proofs of liveness and atomicity before formal claims.",
"For liveness of writes, we assume $N2$ with $\\alpha > \\frac{3}{4}$ , and argue the existence of a majority $\\mathcal {S}_m$ of servers all of which remain active from the point of time at which the $group\\mathord {\\sf -}send$ operation gets initiated in the put-data phase, till the point of time all the servers in $\\mathcal {S}_m$ effectively consume requests for confirm-data from the writer.",
"In this case, write operation completes after receiving acks from servers in $\\mathcal {S}_m$ during the confirm-data phase.",
"The set $\\mathcal {S}_m$ exists because, under $N2$ with $\\alpha > \\frac{3}{4}$ , a set $\\mathcal {S}_{\\alpha }$ of $\\left\\lceil \\frac{3n + 1}{4} \\right\\rceil $ servers remain alive from the start of the group-send, till the effective consumption of the acks by the writer in put-data phase.",
"Also, a second set $\\mathcal {S}^{\\prime }_{\\alpha }$ of $\\left\\lceil \\frac{3n + 1}{4} \\right\\rceil $ servers remain active from the start of the group-send in the confirm-data phase, till all servers in $\\mathcal {S}_{\\alpha }^{\\prime }$ complete the respective effective consumption from this group-send.",
"We note that $\\mathcal {S}_{\\alpha }^{\\prime } \\cap \\mathcal {S}_{\\alpha }$ is at least a majority.",
"We next use the observation that the $group\\mathord {\\sf -}send$ operation in the confirm-data phase forms part of the effective consumption of the last of the acks in the put-data phase.",
"Using this, we argue that the servers in $\\mathcal {S}_{\\alpha }^{\\prime } \\cap \\mathcal {S}_{\\alpha }$ remain active till they effectively consume message from $group\\mathord {\\sf -}send$ operation of the confirm-data phase, and thus $\\mathcal {S}_{\\alpha }^{\\prime } \\cap \\mathcal {S}_{\\alpha }$ is a candidate for $\\mathcal {S}_m$ .",
"The liveness of read is similar to that of write, while liveness of repair is straightforward under $N2$ with $\\alpha > \\frac{3}{4}$ .",
"Towards proving atomicity of reads and writes, we first define tags for completed reads, writes and repair operations exactly in the same manner as we did in ${RADON_R}$ .",
"Consider two completed write operations $\\pi _1$ and $\\pi _2$ such that $\\pi _2$ starts after the completion of $\\pi _1$ , and we need to show that $tag(\\pi _2) > tag(\\pi _1)$ .",
"As in ${RADON_R}$ , we do this in two parts: Lemma REF holds as it is for ${RADON_R^{(S)}}$ as well.",
"Recall that Lemma REF essentially shows that if a majority of active nodes is locked-on to any particular tag, say $t^{\\prime }$ , at a specific point of time $T$ during the execution of the algorithm, then any repair operation which begins after the time $T$ always restores the tag to one which is at least as high as $t^{\\prime }$ .",
"The challenge now is to show the existence of these favorable points of time instants $T$ as needed in the assumption of the lemma.",
"While in ${RADON_R}$ , we used the $N1$ to argue this, in ${RADON_R^{(S)}}$ , we do not use $N2$ ; instead we rely on the third confirm-data phase of the first write operation $\\pi _1$ .",
"[Liveness] Let $\\beta $ denote a well-formed execution of ${RADON_R^{(S)}}$ under condition $N2$ with $\\alpha > \\frac{3}{4}$ .",
"Then every operation initiated by a non-faulty client completes.",
"[Atomcity] Every execution of the ${RADON_R^{(S)}}$ algorithm is atomic."
],
[
"Storage and Communication Costs of Algorithms",
"We give a justification of storage and communication cost numbers of the three algorithms, appearing in Table REF .",
"Recall that the size of value $v$ is assumed to be 1 and also that we ignore the costs due to metadata.",
"It is clear that both ${RADON_R}$ and ${RADON_R^{(S)}}$ have storage cost $n$ , write cost $n$ , and read cost $2n$ (due to write back).",
"For ${RADON_C}$ , each server stores at most $\\delta +1 $ coded-elements, where each element has size $\\frac{1}{k}$ .",
"Thus storage cost of ${RADON_C}$ is $(\\delta + 1)\\frac{n}{k}$ .",
"The write cost of ${RADON_C}$ is simply $\\frac{n}{k}$ , and the contribution comes from the writer sending one coded-element to each of the $n$ servers.",
"For a read, getting the entire $Lists$ during the ${ \\it {get-data}}$ phase incurs a cost of $(\\delta + 1)\\frac{n}{k}$ .",
"The write-back phase incurs an additional cost of $\\frac{n}{k}$ .",
"Thus, the total read cost in ${RADON_C}$ is $(\\delta + 2)\\frac{n}{k}$ ."
],
[
"Conclusions",
"In this paper, we provided an erasure-code-based algorithm for implementing atomic memory, having the ability to perform repair of crashed nodes in the background, without affecting client operations.",
"We assumed a static model with a fixed, finite set of nodes, and also a practical network condition $N1$ to facilitate repair.",
"We showed how the usage of MDS codes significantly improve storage and communication costs over a replication based solution, when the number of writes concurrent with a read or repair is limited.",
"Liveness and atomicity are guaranteed as long as $N1$ is satisfied; however violation of $N1$ can lead to non-atomic executions.",
"We further showed how a slightly stringent network condition $N2$ can be used to construct a replication based algorithm that always guarantees atomicity.",
"Ongoing efforts include exploring possibility of using repair-efficient erasure codes [19] in ${RADON_C}$ , and testbed evaluations on cloud based infrastructure."
],
[
"Acknowledgments",
"The work is supported in part by AFOSR under grants FA9550-14-1-043, FA9550-14-1-0403, and in part by NSF under awards CCF-1217506, CCF-0939370."
],
[
"Proof of Theorem ",
"The theorem is restated for convenience.",
"(Theorem REF) It is impossible to implement an atomic memory service that guarantees liveness of reads and writes, under the system model described in Section , even if 1) there is at most one server in the crashed/repair state at any point during the execution, and 2) every repair operation completes, and takes the repaired server back to the active state.",
"We prove this result by contradiction, by assuming an algorithm $A_{alg}$ that guarantees liveness and atomicity, and also is such that every repair operation completes, and takes the repaired server back to the active state.",
"Let the initial value stored in the system be $v_o \\in V$ , where $V$ is the domain of all values.",
"Consider a non-faulty writer $w$ , and suppose $w$ initiates a write operation $\\pi ^{w}$ with the value $v_1$ , such that $v_1 \\ne v_0$ .",
"Let $\\mathcal {S}_w \\subseteq \\mathcal {S}$ be the set of all servers that writer $w$ sends messages before $w$ expects any response from any of the servers in $\\mathcal {S}$ .",
"Without loss of generalityClearly, the writer must send a message to at least one server, so we ignore the trivial case when $\\mathcal {S}_w$ is empty.",
"let $\\mathcal {S}_w = \\lbrace s_1, s_2, \\ldots , s_k\\rbrace $ , for some $k \\le n$ , and let $m_i$ denote the message sent by $w$ to server $s_i, 1 \\le i \\le k$ .",
"Note that if $w$ sends two or messages to a particular server, say $s_1$ , all these can be combined into $m_1$ , since all these messages are sent without expecting any response.",
"Consider an execution which starts with all the servers in the active state, the operation $\\pi ^{w}$ begins, messages get sent out to servers in $\\mathcal {S}_w$ .",
"Delay the messages such that message $m_1$ arrives at server $s_1$ before any other server in $\\mathcal {S}_w$ receives the respective message.",
"Assume that $s_1$ is in the crashed state when $m_1$ arrives, so $s_1$ does not receive $m_1$ .",
"Further assume that all the other servers are in the active state at this point of execution.",
"Let server $s_1$ undergo a successful repair operation, before any other server in $\\mathcal {S}_w$ receives its respective message.",
"Next, consider the case when server $s_2$ receives the message $m_2$ , and delay the messages to all other servers, and assume that $s_2$ is in the crashed state when $m_2$ arrives.",
"The sequence of crash and repair can be carried out in this manner one-by-one for every server in $\\mathcal {S}_w$ , where all these servers end up losing the writer message, though they get repaired.",
"Now if the algorithm is such that the writer expects a response from any of the servers in $\\mathcal {S}$ , clearly it will not happen, since no server in $\\mathcal {S}$ has received any message from $w$ while the server is in the active state.",
"Thus liveness of write is compromised.",
"We next consider the case when the writer decides to terminate without expecting any response from any server in $\\mathcal {S}$ , and show that such a method of guaranteeing liveness results in violation of atomicity.",
"Let us call this execution fragment (as discussed above) with such a write as $\\beta ^{w}(v_1)$ .",
"After the write $\\pi _w$ completes, a read $\\pi _r$ associated with a non-faulty reader, begins.",
"By liveness of read, and atomicity, the read must return $v_1$ .",
"Let the execution fragment associated with the read be denoted as $\\beta ^r$ , so that the overall execution fragment under consideration is $\\beta ^{w}(v_1) \\circ \\beta ^r$ .",
"Next, consider the execution fragment $\\beta ^{w}({v^{\\prime }}_1)$ obtained by replacing $v_1$ with $v_1^{\\prime }$ such that ${v_1}^{\\prime } \\ne v_1$ .",
"Since a crash causes a server to lose its entire state, it is clear that to the reader $r$ there is no distinction between the state of the system after $\\beta ^{w}(v_1)$ , and the state of the system after $\\beta ^{w}(v_1^{\\prime })$ .",
"In this case, if we consider the execution $\\beta ^{w}(v_1^{\\prime }) \\circ \\beta ^r$ , the read returns $v_1$ ($\\ne v_1^{\\prime }$ ), since in the execution $\\beta ^{w}(v_1) \\circ \\beta ^r$ also, $r$ returned $v_1$ .",
"However it violates atomicity of $\\beta ^{w}(v_1^{\\prime }) \\circ \\beta ^r$ , which completes the proof."
],
[
"Proof of Lemma ",
"The lemma is restated for convenience.",
"[Lemma REF ] Let $\\beta $ denote a well-formed execution of the ${RADON_R}$ algorithm.",
"Suppose $T$ denotes a point of time in the execution $\\beta $ such that there exists a majority of servers $\\mathcal {S}_m$ , $\\mathcal {S}_m \\subset \\mathcal {S}$ all of which are in the active state at the time $T$ .",
"Also, let $t_s$ denote the value of the local tag at server $s$ , at time $T$ .",
"Then, if $\\pi $ denotes any completed repair or read operation that is initiated after time $T$ , we have $tag(\\pi ) \\ge \\min _{s\\in \\mathcal {S}_m} t_s$ .",
"Also, if $\\pi $ denotes any completed write operation that is initiated after time $T$ , then we have $tag(\\pi ) > \\min _{s\\in \\mathcal {S}_m} t_s$ .",
"We use $\\rho $ to denote $\\min _{s\\in \\mathcal {S}_m} t_s$ .",
"Also, for any state variable $x(s)$ that is stored in server $s$ , we write $x(s)|_{T}$ to denote the value $x$ at time $T$ .",
"Below, we separately consider the cases when $\\pi $ denotes a successful repair, read and write operations, in this respective order.",
"(a) $\\pi $ is a successful repair operation: We prove the statement by contradiction, by starting with the assumption that $tag(\\pi ) < \\rho $ .",
"Let $T_{\\pi }$ denote the point of time in the execution $\\beta $ at which the operation $\\pi $ completes.",
"Let $\\Pi ^{\\prime }_R$ denote the set of all successful repair operations which start after the time $T$ , but start before $T_{\\pi }$ , and is such that $\\forall \\pi ^{\\prime } \\in \\Pi ^{\\prime }_R$ , we have $tag(\\pi ^{\\prime }) < \\rho $ .",
"Clearly, $\\pi \\in \\Pi ^{\\prime }_R$ .",
"Let $\\pi ^* \\in \\Pi ^{\\prime }_R$ denote the repair operation, which completes first.",
"Note that $\\pi ^*$ exists since the set $\\Pi ^{\\prime }_R$ is finite.",
"Now, let $\\hat{\\mathcal {S}}$ denote the set of majority servers based on whose responses the operation $\\pi ^*$ completed.",
"Clearly, $|\\hat{\\mathcal {S}} \\cap \\mathcal {S}_{m}| \\ge 1$ .",
"For any server $s \\in \\hat{\\mathcal {S}} \\cap \\mathcal {S}_{m}$ , let $T_s$ denote the point of time in the execution at which the server $s$ responds to $\\pi ^*$ with its local (tag, value) pair.",
"Clearly, the server $s$ must have remained in the active state during the entire interval $[T, T_s]$ .",
"This follows because $s$ is active at time $T$ , $\\pi ^*$ is the first completed repair operation that started after $T$ , and due to the fact that a server responds to a repair request only if it is in the active state.",
"In this case, we know thatAny read or write operation cannot decrease the local tag that is stored in an active server.",
"$t_{loc}(s)|_{T_s} \\ge t_{loc}(s)|_T \\ge \\rho $ for any $s$ in $\\hat{\\mathcal {S}} \\cap \\mathcal {S}_{m}$ .",
"Therefore, we have $tag(\\pi ^*) = \\max _{s \\in \\hat{\\mathcal {S}}} t_{loc}(s)|_{T_s} \\ge \\max _{s \\in \\hat{\\mathcal {S}} \\cap \\mathcal {S}_m} t_{loc}(s)|_{T_s} \\ge \\rho $ , which contradicts the existence of $\\pi ^* \\in \\Pi ^{\\prime }_R$ .",
"From, this we conclude that the set $\\Pi ^{\\prime }_R$ must be empty to avoid contradictions, and hence $tag(\\pi ) \\ge \\rho $ .",
"(b) $\\pi $ is a successful read operation: We prove this by contradiction by starting with the assumption that $tag(\\pi ) < \\rho $ .",
"Let $\\hat{\\mathcal {S}}$ denote the set of majority servers based on whose responses during the $get$ -$data$ phase (see Fig.",
"), the read operation completed.",
"As in Part $a)$ , we know that $|\\hat{\\mathcal {S}} \\cap \\mathcal {S}_{m}| \\ge 1$ .",
"In this case, let $T_s$ denote the point of time during the execution at which the server $s \\in \\hat{\\mathcal {S}} \\cap \\mathcal {S}_{m}$ responded to the reader.",
"Next, note that in the $get$ -$data$ phase, the reader picks the response with the highest tag.",
"Thus, since we assume that $tag(\\pi ) < \\rho $ , it must be true that $t_{loc}(s)|_{T_s} < \\rho , s \\in \\hat{\\mathcal {S}} \\cap \\mathcal {S}_{m}$ .",
"Since the server $s \\in \\hat{\\mathcal {S}} \\cap \\mathcal {S}_{m}$ is active at time $T$ such that $t_{loc}(s)|_T \\ge \\rho $ , this would imply that server $s$ experienced a crash event after time $T$ , and came back to the active state before the time $T_s$ via a successful repair operation $\\phi $ such that $tag(\\phi ) < \\rho $ .",
"But then, this contradicts Part a) of the theorem which we proved above, and hence we conclude that $tag(\\pi ) \\ge \\rho $ .",
"(c) $\\pi $ is a write operation: Once again we prove via contradiction, by starting with the assumption that $tag(\\pi ) \\le \\rho $ .",
"Let $\\hat{\\mathcal {S}}$ denote the set of majority servers based on whose responses during the $get$ -$tag$ phase, the writer determined $tag(\\pi )$ .",
"We know from the algorithm that $tag(\\pi )$ is strictly larger than all the tags among the responses from $\\hat{\\mathcal {S}}$ .",
"Since $|\\hat{\\mathcal {S}} \\cap \\mathcal {S}_{m}| \\ge 1$ , we argue like in Part $b)$ , and arrive at a contradiction to Part $a)$ ."
],
[
"Proof of Theorem ",
"The theorem is restated here for convenience.",
"[Theorem REF ] Every execution of the ${RADON_R}$ algorithm operating under the $N1$ network stability condition with $\\alpha > \\frac{3}{4}$ , is atomic."
],
[
"Partial Order on read and write operations",
"Consider any well-formed execution $\\beta $ of ${RADON_R}$ , all of whose invoked read or write operations complete.",
"Let $\\Pi _{RW}$ denote the set of all completed read and write operations in $\\beta $ .",
"We first define a partial order ($\\prec $ ) on $\\Pi _{RW}$ .",
"Towards this, recall that for any completed write operation $\\pi $ , we defined $tag(\\pi )$ as the tag created by the writer during the write-put phase.",
"Also, recall that for any completed read operation $\\pi $ , we define $tag(\\pi )$ as the tag corresponding to the value returned by the read.",
"The partial order ($\\prec $ ) in $\\Pi _{RW}$ is defined as follows: For any $\\pi , \\phi \\in \\Pi _{RW}$ , we say $\\pi \\prec \\phi $ if one of the following holds: $(i)$ $tag(\\pi ) < tag(\\phi )$ , or $(ii)$ $tag(\\pi ) = tag(\\phi )$ , and $\\pi $ and $\\phi $ are write and read operations, respectively.",
"The proof of atomicity is based on the following lemma, which is simply a restatement of the sufficiency condition for atomicity presented in [31].",
"Consider any well-formed execution $\\beta $ of the algorithm, such that all the invoked read and the write operations complete.",
"Now, suppose that all the invoked read and write operations in $\\beta $ can be partially ordered by an ordering $\\prec $ , so that the following properties are satisfied: P1.",
"The partial order ($\\prec )$ is consistent with the external order of invocation and responses, i.e., there are no operations $\\pi _1$ and $\\pi _2$ , such that $\\pi _1$ completes before $\\pi _2$ starts, yet $\\pi _2 \\prec \\pi _1$ .",
"P2.",
"All operations are totally ordered with respect to the write operations, i.e., if $\\pi _1$ is a write operation and $\\pi _2$ is any other operation then either $\\pi _1 \\prec \\pi _2$ or $\\pi _2 \\prec \\pi _1$ .",
"P3.",
"Every read operation returns the value of the last write preceding it (with respect to $\\prec $ ), and if no preceding write is ordered before it, then the read returns the initial value of the object.",
"Then, the execution $\\beta $ is atomic."
],
[
"Proof of Atomicity under $N1$ with {{formula:d8e34b6e-3991-41d4-b56f-372060db18d5}}",
"We need to prove the properties $P1$ , $P2$ and $P3$ of Lemma REF .",
"We do this under $N1$ with $\\alpha > 3/4$ , using Lemma REF .",
"Let $\\phi $ and $\\pi $ denote two operations in $\\Pi _{RW}$ such that $\\phi $ completed before $\\pi $ started.",
"Also, let $c_{\\phi }$ and $c_{\\pi }$ denote the clients that initiated the operations $\\phi $ and $\\pi $ , respectively."
],
[
"Property $P1$",
"We want to show that $\\pi \\lnot \\prec \\phi $ .",
"We show this in detail only for the case when $\\phi $ and $\\pi $ are both write operations.",
"The proofs of other three casesThese correspond to the case when $\\phi $ and $\\pi $ are both read operations, and the cases where one of them is a write and the other is a read.",
"are similar, and hence omitted.",
"By virtue of the definition of the partial order $\\prec $ , it is enough to prove that $tag(\\pi ) > tag(\\phi )$ .",
"Consider the $put$ -$data$ phase of $\\phi $ , where the writer sends the pair $(t_w, v)$ to all servers via the $group$ -$send$ operation.",
"Under the condition $N1$ with $\\alpha > 3/4$ , we know that there exists a set $\\mathcal {S}_{\\alpha } \\subseteq \\mathcal {S}$ of $\\lceil n\\alpha \\rceil \\ge \\left\\lceil \\frac{3n+1}{4} \\right\\rceil $ servers all of which remain in the active state during the interval $[T_1, T_2]$ where $T_1$ denotes the point of time of invocation of the group-send operation, and $T_2$ denotes the earliest point of time during the execution where all of the servers in $\\mathcal {S}_{\\alpha }$ complete effective consumption (including sending ack to the writer $c_{\\phi }$ ) of the message $(t_w, v)$ .",
"Also, let $\\mathcal {S}^{\\prime } \\subseteq \\mathcal {S}$ denote the set of $\\left\\lceil \\frac{3n+1}{4}\\right\\rceil $ servers whose acks are used by the writer to decide the completion of the write operation.",
"Clearly, $|\\mathcal {S}^{\\prime } \\cap \\mathcal {S}_{\\alpha }| > \\frac{n}{2}$ .",
"Let $T$ denote the earliest point of time during the execution when all servers in $\\mathcal {S}^{\\prime } \\cap \\mathcal {S}_{\\alpha }$ complete their respective effective consumption of the message $(t_w, v)$ .",
"In this case note that $a)$ $T$ occurs before the point of completion of the write operation, $b)$ all servers in $\\mathcal {S}^{\\prime } \\cap \\mathcal {S}_{\\alpha }$ are in the active state at $T$ , and $c)$ $t_{loc}(s)|_T \\ge tag(\\phi ), \\forall s \\in \\mathcal {S}^{\\prime } \\cap \\mathcal {S}_{\\alpha }$ .",
"We now apply Lemma REF to conclude that $tag(\\pi ) > tag(\\phi )$ .",
"This follows from the construction of tags, and the definition of the partial order ($\\prec $ ).",
"This follows from the definition of partial order ($\\prec $ ), and by noting that value returned by a read operation $\\pi $ is simply the value associated with $tag(\\pi )$ ."
],
[
"Proof of Lemma ",
"The lemma is restated below for easy reference.",
"[Lemma 2] Consider any well-formed execution $\\beta $ of ${{RADON_C}}$ operating under the network stability condition $N1$ with $\\alpha \\ge \\frac{3n+k}{4n}$ .",
"Further assume that the number of writes concurrent with any valid read or repair operation is at most $\\delta $ .",
"For any operation $\\pi $ , consider the set $\\Sigma = \\lbrace \\sigma : \\sigma ~\\text{is a read }$ $ \\text{or a write in $\\beta $ }$ $\\text{that completes before}~\\pi $ $\\text{begins}\\rbrace $ , and also let $\\sigma ^{*} = \\arg \\max \\limits _{\\sigma \\in \\Sigma }{{tag(\\sigma )}}$ .",
"Then, the following statements hold: If $\\pi $ denotes a completed repair operation on a server $s \\in \\mathcal {S}$ , then the repaired $List$ of server $s$ due to $\\pi $ contains the pair $(tag(\\sigma ^*), c_s^*)$ .",
"If $\\pi $ denotes a read operation associated with a non-faulty reader $r$ , and further, if $\\mathcal {S}_1$ denotes the set of $\\left\\lceil \\frac{n+k}{2}\\right\\rceil $ servers whose responses, say $\\lbrace L_{\\pi }(s), s \\in \\mathcal {S}_1\\rbrace $ , are used by $r$ to attempt decoding of a value in the get-data phase, then there exists $\\mathcal {S}_2 \\subseteq \\mathcal {S}_{1}$ , $|\\mathcal {S}_{2}| = k $ , such that $\\forall s \\in \\mathcal {S}_2, (tag(\\sigma ^{*}), c_s^* ) \\in L_{\\pi }(s)$ .",
"If $\\pi $ denotes a write operation associated with a non-faulty writer $w$ , and further if $\\mathcal {S}_1$ denotes the set of majority servers whose responses are used by $w$ to compute max-tag in the get-tag phase, then there exists a server $s \\in \\mathcal {S}_1$ , whose response tag $t_s \\ge tag(\\sigma ^*)$ .",
"Here, $c_s^*$ denotes the coded-element of server $s$ corresponding to the value $v^*$ , associated with $tag(\\sigma ^*)$ .",
"We prove the lemma separately for the cases of repair and read (Parts 1 and 2).",
"The proof for the third part for the case of write operations to similar to that of Part 2, and hence omitted."
],
[
"Proof of Part 1 of Lemma ",
"Consider the set $\\Sigma $ and the operation $\\sigma ^*$ as defined in the statement of Lemma .",
"Without loss of generality, let us assume that $\\sigma ^*$ is a write operation.",
"Since we assume condition $N1$ with $\\alpha \\ge \\frac{3n+k}{4n}$ , there exists a set $\\mathcal {S}_{\\alpha }$ of $\\left\\lceil \\frac{3n+k}{n} \\right\\rceil $ servers that respects $N1$ for the group-send operation (say gp*) in the put-data phase of $\\sigma ^*$ .",
"If $\\mathcal {S}_1$ denotes the set of $\\left\\lceil \\frac{3n+k}{n} \\right\\rceil $ servers, whose responses are used by the writer to decide termination, we then know that $1)$ $|\\mathcal {S}_{\\alpha } \\cap \\mathcal {S}_1| \\ge \\lceil \\frac{n+k}{2} \\rceil $ , and $2)$ if $T_{prop}$ denotes the earliest point of time during the execution when all the servers in $\\mathcal {S}_{prop} = \\mathcal {S}_{\\alpha } \\cap \\mathcal {S}_1$ complete effective consumption of their respective messages from the group-send operation gp*, then every server in $\\mathcal {S}_{prop}$ remains active at $T_{prop}$ , and has not experienced a crash after its effective consumption, until $T_{prop}$ .",
"Our goal is to show that the repair operation $\\pi $ always receives at least $k$ responses from among the servers in $\\mathcal {S}_{prop}$ , and must be able to decode (and then re-encode) the value corresponding to $tag(\\sigma ^*)$ .",
"Below we consider the effects of concurrent writes having higher tags, and repairs before $\\pi $ starts, both of which can potentially remove coded elements corresponding to $tag(\\sigma ^*)$ , from lists of various servers.",
"We show under the assumptions of the lemma, that neither of these cause a problem.",
"Let us first consider the effect of concurrent writes.",
"Towards this, consider the set $\\Lambda $ of writes concurrent with the valid repair operation $\\pi $ (see Definition REF ).",
"Recall that $\\Lambda = \\lbrace \\lambda : \\lambda \\text{ is a write operation that starts before completion of } $ $ \\pi , \\text{such that } tag(\\lambda ) > tag(\\sigma ^*)\\rbrace $ .",
"By assumption on the lemma, we know that $|\\Lambda | \\le \\delta $ .",
"In this case, it is clear that if a server $s \\in \\mathcal {S}_{prop}$ does not crash in the interval $[T_{prop} \\ \\ T]$ , the $List(s)|_T$ contains the pair corresponding to $tag(\\sigma ^*)$ , for any $T$ such that $T_{prop} \\le T \\le T_{end}(\\pi )$ .",
"Here $T_{end}(\\pi )$ denotes the point of completion of $\\pi $ .",
"Let us next consider the effect of repairs, let $\\widetilde{\\Pi } = \\lbrace \\widetilde{\\pi }:$ a repair which start after $T_{prop}$ , but also start before the completion of $\\pi \\rbrace $ .",
"Clearly, $\\pi \\in \\widetilde{\\Pi }$ .",
"Let $\\widetilde{\\pi }^* \\in \\widetilde{\\Pi }$ denote the repair operation that completes first.",
"Clearly, it must be true that $T_{prop} < T_{end}(\\widetilde{\\pi }^*) \\le T_{end}(\\pi )$ .",
"We prove Part 1 of the Lemma for $\\widetilde{\\pi }^*$ first.",
"Using this result, we prove the lemma for the repair operation in $\\widetilde{\\Pi }$ which completes second.",
"We continue in an inductive manner (on the finite set $\\widetilde{\\Pi }$ ), until we hit $\\pi $ .",
"Towards proving the lemma for $\\widetilde{\\pi }^*$ , consider the group-send operation, where $\\widetilde{\\pi }^*$ requests for local $Lists$ from all servers.",
"Let $\\mathcal {S}_{\\theta } \\subset \\mathcal {S}_{prop}$ denote the servers among $\\mathcal {S}_{prop}$ which are not in the active state when the repair request arrives.",
"Also, let $\\mathcal {S}_a \\subset \\mathcal {S}$ denote the set of all servers which are in the active state when the repair request arrives.",
"Clearly, $|\\mathcal {S}_a| \\le n - |\\mathcal {S}_{\\theta }|$ .",
"Next, let $\\mathcal {S}_{ack} \\subset \\mathcal {S}_a$ denote the set of $\\lceil \\frac{n+k}{2} \\rceil $ servers based on whose responses the repair operation $\\widetilde{\\pi }^*$ completes.",
"Now, since $\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta } \\subset \\mathcal {S}_a$ , we have $(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }) \\cup \\mathcal {S}_{ack} & \\subset & \\mathcal {S}_a \\\\\\Rightarrow |\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }| + |\\mathcal {S}_{ack}| - |(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }) \\cap \\mathcal {S}_{ack}| & \\le & \\mathcal {S}_a \\\\\\Rightarrow |\\mathcal {S}_{prop}| - |\\mathcal {S}_{\\theta }| + \\lceil \\frac{n+k}{2} \\rceil - |(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }) \\cap \\mathcal {S}_{ack}| & \\le & n - |\\mathcal {S}_{\\theta }| \\\\\\Rightarrow |(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }) \\cap \\mathcal {S}_{ack}| \\ge k, $ where the last inequality follows from our earlier observation that $|\\mathcal {S}_{prop}| \\ge \\lceil \\frac{n+k}{2} \\rceil $ .",
"Next, note that any server $s$ in $(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }) \\cap \\mathcal {S}_{ack}$ remains active from $T_{prop}$ until the point when $s$ responds to the repair request from $\\widetilde{\\pi }^*$ .",
"This follows because of the facts that $1)$ $s$ is active at $T_{prop}$ , $2)$ a server responds to a repair request only if it is in the active state, and $3)$ since $\\widetilde{\\pi }^*$ is the first repair operation that completes after $T_{prop}$ .",
"Also, recall our earlier observations that $1)$ if a server $s \\in \\mathcal {S}_{prop}$ does not crash in the interval $[T_{prop} \\ \\ T]$ , then $List(s)|_T$ contains the pair corresponding to $tag(\\sigma ^*)$ , for any $T$ such that $T_{prop} \\le T \\le T_{end}(\\pi )$ , and $2)$ $T_{prop} < T_{start}(\\widetilde{\\pi }^*) < T_{end}(\\widetilde{\\pi }^*) \\le T_{end}(\\pi )$ .",
"In this case, we know that the responses of all the servers in $(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }) \\cap \\mathcal {S}_{ack}$ to $\\widetilde{\\pi }^*$ , contain the pair corresponding to $tag(\\sigma ^*)$ .",
"From (), it follows that the repaired list for $\\widetilde{\\pi }^*$ , before pruning to $(\\delta + 1)$ entries, contains the pair corresponding to $tag(\\sigma ^*)$ .",
"Finally the fact that $tag(\\sigma ^*)$ is among the highest $\\delta +1$ tags, and hence part of the pruned list, follows from our earlier observationsNote that $\\Lambda $ need not be the set of writes concurrent with $\\widetilde{\\pi }^*$ .",
"The above argument where we say that the pruned list, after the repair $\\widetilde{\\pi }^*$ , is of size at most $\\delta +1$ can be argued entirely based on $\\Lambda $ itself.",
"that $1)$ $|\\Lambda | \\le \\delta $ , and $2)$ $T_{end}(\\widetilde{\\pi }^*) \\le T_{end}(\\pi )$ .",
"This completes our proof of Part 1 of Lemma for the repair operation $\\widetilde{\\pi }^*$ .",
"We next prove the lemma for the repair operation $\\pi _2 \\in \\widetilde{\\Pi }$ , which completes second.",
"The proof is mostly identical, and we will only highlight the place where we use the result on $\\widetilde{\\pi }^*$ .",
"Clearly, since we carry out the induction only until we hit $\\pi $ , it must be true that $T_{prop} < T_{end}({\\pi }_2) \\le T_{end}(\\pi )$ .",
"Consider the group-send operation, where $\\pi _2$ requests for local $Lists$ from all servers.",
"Let $\\mathcal {S}_{\\theta }^{(2)} \\subset \\mathcal {S}_{prop}$ denote the servers among $\\mathcal {S}_{prop}$ which are not in the active state when the repair request arrives.",
"Also, let $\\mathcal {S}_a^{(2)} \\subset \\mathcal {S}$ denote the set of all servers which are in the active state when the repair request arrives.",
"As before, $|\\mathcal {S}_a^{(2)}| \\le n - |\\mathcal {S}_{\\theta }^{(2)}|$ .",
"Next, let $\\mathcal {S}_{ack}^{(2)} \\subset \\mathcal {S}_a^{(2)}$ denote the set of $\\lceil \\frac{n+k}{2} \\rceil $ servers based on whose responses the repair operation $\\pi _2$ completes.",
"Along the lines of (REF )-(), one can show that $|(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }^{(2)}) \\cap \\mathcal {S}_{ack}^{(2)}| \\ge k$ .",
"Next, if we consider the set $(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }^{(2)}) \\cap \\mathcal {S}_{ack}^{(2)}$ , at most one of the servers in this set would have undergone a crash after the time $T_{prop}$ , and got repaired before the time the server responded to $\\pi _2$ .",
"Note that more than one repair operation on $(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }^{(2)}) \\cap \\mathcal {S}_{ack}^{(2)}$ cannot happen, since this will contradict the assumption that $\\pi _2$ is the second repair operation to complete after $T_{prop}$ .",
"Further, if one repair operation among a server in $(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }^{(2)}) \\cap \\mathcal {S}_{ack}^{(2)}$ has indeed occurred, this must be the operation $\\widetilde{\\pi }^*$ which we considered above.",
"Further, we know that the repaired $List$ due to $\\widetilde{\\pi }^*$ contains the pair corresponding to $tag(\\sigma ^*)$ .",
"In other words, irrespective of whether one repair operation occurred among the servers in $(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }^{(2)}) \\cap \\mathcal {S}_{ack}^{(2)}$ , or not, the responses of all the servers in $(\\mathcal {S}_{prop}\\backslash \\mathcal {S}_{\\theta }^{(2)}) \\cap \\mathcal {S}_{ack}^{(2)}$ contain the pair corresponding to $tag(\\sigma ^*)$ .",
"The rest of the proof is similar to that of $\\widetilde{\\pi }^*$ , where we argue that the pruned list after repair contains the pair corresponding to $tag(\\sigma ^*)$ .",
"The rest of the induction is similar, and this completes the proof of Part 1 of Lemma ."
],
[
"Proof of Part 2 of Lemma ",
"The proof follows mostly along the lines of proof of Part 1 of the lemma.",
"We will only highlight the main steps here.",
"Consider the set $\\Sigma $ and the operation $\\sigma ^*$ as defined in the statement of the lemma.",
"Without loss of generality, let us assume that $\\sigma ^*$ is a write operation.",
"Also, define the time $T_{prop}$ and the set $\\mathcal {S}_{prop}$ exactly in the same way as what we defined in the proof of Part 1 of the lemma.",
"Let $T_1$ denote the earliest point of time during the execution when the reader receives responses from all the servers in $\\mathcal {S}_1$ , where $\\mathcal {S}_1$ is as defined in the statement of this lemma.",
"Consider the set of writes concurrent with the valid read operation $\\pi $ .",
"Recall from Definition REF that $\\Lambda = \\lbrace \\lambda : \\lambda \\text{ is a write operation which starts before time } T_1 \\text{ such that } tag(\\lambda ) > tag(\\sigma ^*)\\rbrace $ .",
"From the assumption on concurrency in the lemma statement, we know that $|\\Lambda | \\le \\delta $ .",
"In this case, it is clear (like in the proof of Part 1 above) that if a server $s \\in \\mathcal {S}_{prop}$ does not crash in the interval $[T_{prop} \\ \\ T_1]$ , the $List(s)|_T$ contains the pair corresponding to $tag(\\sigma ^*)$ , for any $T$ such that $T_{prop} \\le T \\le T_1$ .",
"Now, if server $s \\in \\mathcal {S}_{prop}$ undergoes a crash and repair operation (say $\\rho $ ) during the interval $[T_{prop}\\ \\ T]$ (so that it is active again at $T$ ), we can argue exactly like in the proof of Part 1 above, and show that the repaired $List$ due to $\\rho $ contains the pair corresponding to $tag(\\sigma ^*)$ .",
"This can be done by considering the set $\\widetilde{\\Pi } = \\lbrace \\widetilde{\\pi }:$ a repair which start after $T_{prop}$ , but also start before $T_1 \\rbrace $ , and applying induction on $\\widetilde{\\Pi }$ based on the order of completion times of the repair operations.",
"The completes the proof of our claim about $List(s)|_T$ .",
"The rest of the proof follows simply by noting $|\\mathcal {S}_1 \\cap \\mathcal {S}_{prop}| \\ge k$ , and thus the value corresponding to $tag(\\sigma ^*)$ is surely a candidate for decoding, since we know that an $[n, k]$ linear MDS code can be uniquely decoded given any $k$ out of the $n$ coded-elements."
],
[
"Liveness: Proof of Theorem ",
"The theorem is restated below for easy reference: (Theorem REF ) Let $\\beta $ denote a well-formed execution of ${RADON_C}$ , operating under the $N1$ network stability condition with $\\alpha \\ge \\frac{3n+k}{4n}$ and $\\delta $ be the maximum number of write operations concurrent with any valid read or repair operation.",
"Then every operation initiated by a non-faulty client completes.",
"Liveness of writes depends only on sufficient number of responses in the two phases.",
"The maximum number of responses expected in either of the two phases is $\\frac{3n+k}{4n}$ , which we know is guaranteed under $N1$ with $\\alpha \\ge \\frac{3n+k}{4n}$ .",
"Liveness of reads follows by combining Lemma REF (for decodability of a value), and the liveness of write operations (for the write-back phase)."
],
[
"Atomicity: Proof of Theorem ",
"The theorem is restated first: (Theorem REF ) Any execution of ${RADON_C}$ , operating under condition $N1$ with $\\alpha \\ge \\frac{3n+k}{4n}$ , is atomic, if the maximum number of write operations concurrent with a valid read or repair operation is $\\delta $ .",
"The proof is based on Lemmas REF and REF .",
"In order to apply Lemma REF , consider any well-formed execution $\\beta $ of ${RADON_C}$ , all of whose invoked read and write operations, denoted by the set $\\Pi _{RW}$ , complete.",
"We define a partial order ($\\prec $ ) on $\\Pi _{RW}$ like in the proof of Theorem REF for case of ${RADON_R}$ .",
"To prove Property $P1$ of Lemma REF , consider two successful operations $\\phi $ and $\\pi $ such that $\\phi $ completes before $\\pi $ begins.",
"Firstly, consider the case $\\pi $ is a write, and $\\phi $ is either a read or write.",
"We need to show that $tag(\\pi ) > tag(\\phi )$ , which we note follows directly from Part 3 of Lemma REF .",
"Next, consider the case when consider the case $\\pi $ is a read, and $\\phi $ is either a read or write.",
"We need to show that $tag(\\pi ) \\ge tag(\\phi )$ , which we note follows directly from Part 2 of Lemma REF .",
"This completes the proof of Property $P1$ .",
"Proofs of Properties $P2$ and $P3$ are similar to those of the corresponding properties in Theorem REF , where we proved atomicity of ${RADON_R}$ ."
],
[
"Proof of Theorem ",
"The theorem is restated for convenience.",
"(Theorem REF ) Let $\\beta $ denote a well-formed execution of ${RADON_R^{(S)}}$ operating under condition $N2$ with $\\alpha > \\frac{3}{4}$ .",
"Then every operation initiated by a non-faulty client completes.",
"We will prove that a write operation associated with a non-faulty client always completes, the proof for a read is similar and hence is omitted.",
"The main step is to show the completion of the confirm-data phase.",
"Consider the put-data phase, and note that under $N2$ with $\\alpha > \\frac{3}{4}$ , we are guaranteed that there exists of set of $\\mathcal {S}_{\\alpha } \\subset \\mathcal {S}$ severs, such that $1)$ $|\\mathcal {S}_{\\alpha }| \\ge \\left\\lceil \\frac{3n + 1}{4} \\right\\rceil $ , and $2)$ every server in $ \\mathcal {S}_{\\alpha }$ remains active from the point of time $T_1$ of initiation of the group-send operation of put-data phase till the point of time $T_1^{\\prime }$ , when the writer effectively consumes all responses (acks) from the servers in $\\mathcal {S}_{\\alpha }$ .",
"Next, let $\\mathcal {S}_1 \\subset \\mathcal {S}$ denote the set of $\\left\\lceil \\frac{3n + 1}{4} \\right\\rceil $ whose acks are received by the writer before moving on to the confirm-data phase.",
"First of all note that the existence of the set $\\mathcal {S}_1$ is clearly guaranteed under $N2$ with $\\alpha > \\frac{3}{4}$ (since the set $\\mathcal {S}_{\\alpha }$ is a candidate for $\\mathcal {S}_1$ ).",
"Secondly, we note that the group-send operation in the confirm-data phase forms part of the effective consumption of the last ack that is received from the servers in $\\mathcal {S}_1$ .",
"This follows from the definition of effective-consumption, and by noting the execution of the group-send operation in the confirm-data phase does not depend on any more input after all the acks in the put-data phase are received.",
"Let $T_2$ denote the point of time at which the group-send operation in the confirm-data phase gets initiated.",
"Note that $T_1^{\\prime } \\ge T_2$ , in fact if $\\mathcal {S}_{1} \\ne \\mathcal {S}_{\\alpha }$ , we haveIn this case, some of the acks from the servers in $\\mathcal {S}_{\\alpha }$ get effectively consumed only after the required number $\\left\\lceil \\frac{3n + 1}{4} \\right\\rceil $ have already been consumed, the last of which includes execution of the group-send operation of the confirm-data phase.",
"We note that the effective consumption of these additional acks from servers in $\\mathcal {S}_{\\alpha }$ is the operation where server simply ignores these, which is not explicitly mentioned in the algorithm.",
"We also note that the notion of atomicity of any sequences of effective consumptions that are local to a server, is implicitly used when we argue that $T_1^{\\prime } > T_2$ .",
"By this we mean that if a server receives a message $m_1$ before $m_2$ , the effective consumption of message $m_1$ is assumed to be entirely completed before the effective consumption of the message $m_2$ starts.",
"$T_1^{\\prime } > T_2$ .",
"Next we apply the network condition to the group-send operation in the confirm-data phase.",
"From the $N1$ part of $N2$ , we know that there exists a $\\mathcal {S}^{\\prime }_{\\alpha }$ of $\\left\\lceil \\frac{3n + 1}{4} \\right\\rceil $ servers, all of which receive and effectively consume the message from the group-send operation, and remain active from $T_2$ till the point of time $T_2^{\\prime }$ when the last of the servers in $\\mathcal {S}^{\\prime }_{\\alpha }$ completes effective consumption.",
"Now if we let $\\mathcal {S}_{\\gamma } = \\mathcal {S}_{\\alpha } \\cap \\mathcal {S}^{\\prime }_{\\alpha }$ , observe that $1)$ $|\\mathcal {S}_{\\gamma }| > \\frac{n}{2}$ , and $2)$ all the servers in $\\mathcal {S}_{\\gamma }$ remain active from $T_1$ till $T_2^{\\prime }$ .",
"The second part follows from our earlier observation that $T_1^{\\prime } \\ge T_2$ .",
"In this case, we infer that all the servers in $\\mathcal {S}_{\\gamma }$ does indeed acknowledge back to writer as part of their effective consumption of the confirm-data message, and since $\\mathcal {S}_{\\gamma } \\subset \\mathcal {S}_{\\alpha }$ is at least a majority, we conclude that the write operation associated with the non faulty writer eventually completes."
],
[
"Proof of Theorem ",
"(Theorem REF ) Every execution of the ${RADON_R^{(S)}}$ algorithm is atomic."
],
[
"Some Preliminaries",
"The proof is based on Lemma REF , and the equivalent of Lemma REF for ${RADON_R^{(S)}}$ , which we state below for the sake of completion: Let $\\beta $ denote a well-formed execution of ${RADON_R^{(S)}}$ .",
"Suppose $T$ denotes a point of time in $\\beta $ such that there exists a majority of servers $\\mathcal {S}_m$ , $\\mathcal {S}_m \\subset \\mathcal {S}$ all of which are in the active state at time $T$ .",
"Also, let $t_s$ denote the value of the local tag at server $s$ , at time $T$ .",
"Then, if $\\pi $ denotes any completed repair or read operation that is initiated after time $T$ , we have $tag(\\pi ) \\ge \\min _{s\\in \\mathcal {S}_m} t_s$ .",
"Also, if $\\pi $ denotes any completed write operation that is initiated after time $T$ , we have $tag(\\pi ) > \\min _{s\\in \\mathcal {S}_m} t_s$ .",
"Similar to the proof of Lemma REF .",
"Next, in order to apply Lemma REF , consider any well-formed execution $\\beta $ of ${RADON_R^{(S)}}$ , all of whose invoked read and write operations, denoted by the set $\\Pi _{RW}$ , complete.",
"Recall the discussion in Section , where we noted that tags for completed operations in ${RADON_R^{(S)}}$ are defined exactly as we had done for ${RADON_R}$ .",
"Thus, for any completed write operation $\\pi $ , we define $tag(\\pi )$ as the tag created by the writer during the write-put phase.",
"For any completed read operation $\\pi $ , we define $tag(\\pi )$ as the tag corresponding to the value returned by the read.",
"Further, we define a partial order ($\\prec $ ) on $\\Pi _{RW}$ like in the proof of Theorem REF for case of ${RADON_R}$ .",
"These are restated for the sake of completion: For any $\\pi , \\phi \\in \\Pi _{RW}$ , we say $\\pi \\prec \\phi $ if one of the following holds: $(i)$ $tag(\\pi ) < tag(\\phi )$ , or $(ii)$ $tag(\\pi ) = tag(\\phi )$ , and $\\pi $ and $\\phi $ are write and read operations, respectively.",
"Consider two successful operations $\\phi $ and $\\pi $ such that $\\phi $ completes before $\\pi $ begins.",
"We want to prove that $\\pi \\lnot \\prec \\phi $ .",
"Consider the case when both $\\phi $ and $\\pi $ are write operations (the other cases are similar, so only one case is discussed).",
"By virtue of the definition of the partial order ($\\prec $ ), it is enough to prove that $tag(\\pi ) > tag(\\phi )$ .",
"Let $\\mathcal {S}_{\\alpha }$ and $\\mathcal {S}_1$ respectively denote the set of servers whose responses were used by the writer during the put-data and confirm-data phases of $\\phi $ .",
"Let $T$ denote the time of initiation of the confirm-data phase of $\\phi $ .",
"From the algorithm (see Fig.",
"), we know that $\\mathcal {S}_1 \\subset \\mathcal {S}_{\\alpha }$ .",
"Further, based on the algorithm, it is clear that all servers in $\\mathcal {S}_1$ (which is a majority) are active at time $T$ , such that $t_{loc}(s)|_T \\ge tag(\\phi )$ .",
"In this case, we apply Lemma REF to conclude that $tag(\\pi ) > tag(\\phi )$ ."
],
[
"Property $P2$",
"This follows from the construction of tags, and the definition of the partial order ($\\prec $ )."
],
[
"Property $P3$",
"This follows from the definition of partial order ($\\prec $ ), and by noting that value returned by a read operation $\\pi $ is simply the value associated with $tag(\\pi )$ ."
]
] | 1605.05717 |
[
[
"Bayesian Variable Selection for Globally Sparse Probabilistic PCA"
],
[
"Abstract Sparse versions of principal component analysis (PCA) have imposed themselves as simple, yet powerful ways of selecting relevant features of high-dimensional data in an unsupervised manner.",
"However, when several sparse principal components are computed, the interpretation of the selected variables is difficult since each axis has its own sparsity pattern and has to be interpreted separately.",
"To overcome this drawback, we propose a Bayesian procedure called globally sparse probabilistic PCA (GSPPCA) that allows to obtain several sparse components with the same sparsity pattern.",
"This allows the practitioner to identify the original variables which are relevant to describe the data.",
"To this end, using Roweis' probabilistic interpretation of PCA and a Gaussian prior on the loading matrix, we provide the first exact computation of the marginal likelihood of a Bayesian PCA model.",
"To avoid the drawbacks of discrete model selection, a simple relaxation of this framework is presented.",
"It allows to find a path of models using a variational expectation-maximization algorithm.",
"The exact marginal likelihood is then maximized over this path.",
"This approach is illustrated on real and synthetic data sets.",
"In particular, using unlabeled microarray data, GSPPCA infers much more relevant gene subsets than traditional sparse PCA algorithms."
],
[
"Introduction",
"From the children test results of the seminal paper of [21] to the challenging analysis of microarray data [55] and the recent successes of deep learning [12], principal component analysis (PCA) has become one of the most popular tools for data-preprocessing and dimension-reduction.",
"The original procedure consists in projecting the data onto a \"principal\" subspace spanned by the leading eigenvectors of the sample covariance matrix.",
"It was later shown that this subspace could also be retrieved from the maximum-likelihood estimator of a parameter, in a particular factor analysis model called probabilisitic PCA (PPCA) [58], [67].",
"This probabilistic framework led to diverse Bayesian analysis of PCA [6], [44], [49]."
],
[
"Local and global sparsity",
"A potential drawback of PCA is that the principal components are linear combinations of every single original variable, and can therefore be difficult to interpret.",
"To tackle this issue, several procedures have been designed to project the data onto subspaces generated by sparse vectors while retaining as much variance as possible.",
"Many of them were based on convex or partially convex relaxations of cardinality-constrained PCA problems – among these techniques are the popular $\\ell _1$ -based SPCA algorithm of [83] or the semidefinite relaxation of [13].",
"Another strategy is to use a sparsity-inducing prior distributions on the coefficients of the projection matrix [3], [18], [30].",
"However, when several principal components are computed, these various techniques do not enforce them to have the same sparsity pattern, and each component has to be interpreted individually.",
"While individual interpretation is particularly natural in several cases – when PCA serves visualization, for example –, it is not adapted to situations where the practitioner aims at globally selecting which features are relevant.",
"In these situations, a simple and popular approach has been to consider that the relevant variables correspond to the sparsity pattern of the first principal component [83], [82].",
"However, this procedure is limited, and several important aspects of the data may lie in the next principal components.",
"For example, in the colon cancer data set studied by [13], the most relevant genes were the ones selected not by the first but by the second principal component.",
"Another motivation for global sparsity is the fact that, in many real-life situations, the sparsity pattern of the axes computed by a sparse PCA algorithm are extremely close.",
"This is for example the case of the three axes of the template attacks application considered by [3].",
"In this setting, forcing these patterns to be equal will give the practitioner a precise idea of which variables are relevant.",
"Another interesting feature of global sparsity is the fact that, once the common sparsity pattern has been determined, performing PCA on the relevant variables yields orthogonal and uncorrelated principal components – conversely to most sparse PCA procedures."
],
[
"Related work",
"Since the seminal papers of [25], [26] and [57], several methods have been designed to discard features in PCA (see e.g.",
"[10] for a recent review).",
"However, these techniques were designed to eliminate redundant, rather that irrelevant variables, and are based on combinatorial algorithms that are not really suitable for high-dimensional problems.",
"A simple and scalable way of performing variable selection for PCA is to simply keep the features that have the largest marginal variance.",
"In certain cases, this technique is theoretically sound, and was applied for instance to the analysis of electrocardiogram (ECG) data [24].",
"[81] also proved that it could be used as an efficient preprocessing technique to reduce the dimensionality of ultra-high dimensional problems before applying a traditional sparse PCA algorithm.",
"However, this technique has two main drawbacks.",
"First, it is not robust to simple transformations of the data since simply multiplying a variable by a constant may wrongfully select (or discard) it.",
"An unfortunate consequence of this is the fact that this technique can not be applied to scaled data.",
"Moreover, since it ignores non-marginal information, this technique will behave badly in the case of correlated features.",
"A more refined approach to global sparsity is $\\ell _1$ -based regularization, which has imposed itself as one of the most versatile and efficient approaches to sparse statistical learning [20].",
"In a context of structured sparse PCA, [23] proposed to recast sparse PCA as a penalized matrix factorization problem and suggested that limiting the number of sparsity patterns allowed within the principal vectors could improve the feature extraction quality – particularly in face recognition problems.",
"Using the $\\ell _1-\\ell _2$ norm, they derived an algorithm (hereafter referred as SSPCA) that allows to compute $d$ sparse components with exactly $m\\le d$ sparsity patterns.",
"However, they only considered cases where $m$ is larger than 2 and therefore did not focus on global sparsity.",
"They were followed by [29] who, in a very close framework, argued that global sparsity (which they called joint sparsity) led to better representations of hyperspectral images.",
"Other similar approaches based on structured composite norms have been conducted by [41], [17] and [77].",
"[68], [69] used sparsity inducing penalties together with a PPCA model to enforce global sparsity.",
"They proposed an algorithm called sparse variable noisy PCA (hereafter refered as svnPCA) and fixed the amount of penalization using the Bayesian information criterion (BIC) of [61].",
"Eventually, it is worth mentioning that global sparsity has also been investigated in other contexts, such as partial least squares regression [35] or electroencephalography (EEG) imaging [74], [16]."
],
[
"Contributions and organization of the paper",
"We present in Section 2 a Bayesian approach that allows to project the data onto a globally sparse subspace (i.e a subspace spanned by vectors with the same sparsity pattern) while preserving a large part of the variance.",
"To this end, we use the noiseless PPCA model introduced by [58] together with an isotropic gaussian prior on the projection matrix and a binary vector that segregates relevant from irrelevant variables.",
"While past Bayesian PCA frameworks relied on variational [7], [3], [18] or Laplace [6], [44] methods to approximate the marginal likelihood, we derive here a closed-form expression for the evidence based on the multivariate Bessel distribution.",
"In order to avoid the drawbacks of discrete model selection and to treat high-dimensional data, we also present a relaxation of our model by replacing the binary vector with a continuous one.",
"Inference of this relaxed model can be performed using a variational expectation-maximization (VEM) algorithm.",
"Such a procedure allows to find a path of models.",
"The exact evidence is eventually maximized over this path, relying on Occam's razor [38], to select the relevant variables.",
"We illustrate the behaviour of our algorithm and compare it to other methods in Section 3.",
"In particular, we show that Bayesian model selection empirically outperforms $\\ell _1-\\ell _2$ -based regularization on a series of tasks.",
"Sections 4 and 5 are devoted to two applications showcasing the features of our method.",
"The first one concerns signal denoising with wavelets, and shows how global sparsity can surpass traditional sparse PCA algorithms within this context.",
"The second one treats about unsupervised gene selection.",
"Given an (unlabeled) microarray data matrix, we show how GSPPCA can select biologically relevant subsets of genes.",
"Interestingly, we exhibit an important correlation between our exact marginal likelihood expression and a criterion of biological relevance based on pathway enrichment.",
"Note that this paper is an extended version of previous work [42] published in the Proceedings of the $19^\\textup {th}$ Conference on Artificial Intelligence and Statistics."
],
[
"Bayesian variable selection for PCA",
"Let us assume that a centered i.i.d.",
"sample $\\mathbf {x}_1,...,\\mathbf {x}_n \\in \\mathbb {R}^p$ is observed which one wishes to project onto a $d$ -dimensional subspace while retaining as much variance as possible.",
"All the observations are stored in the $n\\times p$ matrix $\\mathbf {X}=(\\mathbf {x}_1,...,\\mathbf {x}_n)^T$ ."
],
[
"Probabilistic PCA",
"The PPCA model assumes that each observation is driven by the following generative model $ \\mathbf {x} = \\mathbf {W} \\mathbf {y} + \\varepsilon ,$ where $\\mathbf {y} \\sim \\mathcal {N}(0,\\mathbf {I}_d)$ is a low-dimensional Gaussian latent vector, $\\mathbf {W}$ is a $p \\times d$ parameter matrix called the loading matrix and $\\varepsilon \\sim \\mathcal {N}(0, \\sigma ^2 \\mathbf {I}_p)$ is a Gaussian noise term.",
"This model is a particular instance of factor analysis and was first introduced by [33].",
"Following [65], [67] confirmed that this generative model is equivalent to PCA in the sense that the principal components of $\\mathbf {X}$ can be retrieved using the maximum likelihood (ML) estimator $\\mathbf {W}_{\\textup {ML}}$ of $\\mathbf {W}$ .",
"Indeed, if $\\mathbf {A}$ is the $p \\times d$ matrix of ordered principal eigenvectors of $\\mathbf {X}^T\\mathbf {X}$ and if $\\Lambda $ is the $d \\times d$ diagonal matrix with corresponding eigenvalues, we have $ \\mathbf {W}_{\\textup {ML}} = \\mathbf {A}(\\Lambda -\\sigma ^2\\mathbf {I}_d)^{1/2}\\mathbf {R},$ where $\\mathbf {R}$ is an arbitrary orthogonal matrix.",
"Several Bayesian treatments of this model have been conducted by using different priors on the loading matrix.",
"However, the marginal likelihood of these models appeared to be untractable.",
"To tackle this issue, several computational techniques were considered.",
"The automatic relevance determination (ARD) prior was used together with Laplace [6] or variational [7], [3] approximations.",
"[44] introduced more complex conjugate priors to perform Bayesian model selection on the dimension $d$ of the latent space using the Laplace approximation.",
"Combined with variational inference, several sparsity inducing priors such as the Laplace [18], the generalized hyperbolic [3] or the spike-and-slab [34] prior were also chosen for $\\mathbf {W}$ .",
"In this work, we aim at avoiding these approximations.",
"Our approach is to investigate in which cases the marginal likelihood can be analytically computed.",
"To this end, we will use the fact that, within the PPCA model (REF ), the limit noiseless setting $\\sigma \\rightarrow 0$ also allows to recover the principal components.",
"This convenient framework was first studied by [58] and has proven to be useful in several situations.",
"The noiseless PPCA model was used for instance to facilitate inference in the presence of missing data [80], [22].",
"More importantly in our context, it was successfully used by [63] to enforce sparsity within an $\\ell _1$ penalized PPCA framework – which means that getting rid of the noise term is likely to be compatible with variable selection."
],
[
"A general framework for globally sparse PPCA",
"In a classical (locally) sparse PCA context, the loading matrix $\\mathbf {W}$ would be expected to contain few nonzero coefficients.",
"However, to reach global sparsity, several entire rows of $\\mathbf {W}$ have to be further constrained to be null.",
"In this work, we handle variable selection using a binary vector $\\mathbf {v} \\in \\lbrace 0, 1\\rbrace ^p$ whose nonzero entries correspond to relevant variables.",
"For technical purposes, we also denote $\\bar{\\mathbf {v}}$ the binary vector of $\\lbrace 0,1\\rbrace ^p$ whose support is exactly the complement of $\\textup {Supp}(\\mathbf {v})$ .",
"We denote $q=||\\mathbf {v}||_0$ the number of relevant variables.",
"In the PPCA framework, this leads to the following model for each observation $ \\mathbf {x} = \\mathbf {V} \\mathbf {W} \\mathbf {y} + \\varepsilon ,$ where $ \\mathbf {V} = \\textup {diag}( \\mathbf {v})$ .",
"Notice that the rows of $\\mathbf {V} \\mathbf {W}$ , corresponding to the zero entries of $\\mathbf {v}$ , are null.",
"Therefore, the principal subspace will be generated by a basis of vectors which shares the sparsity pattern of $\\mathbf {v}$ .",
"Such spaces spanned by a family of vectors sharing the same sparsity pattern will be called globally sparse subspaces.",
"This definition of global sparsity is closely related to the notion of row sparsity introduced by [71].",
"We further assume that the coefficients of the matrix $\\mathbf {W}$ are endowed with the Gaussian priors $w_{ij}\\sim \\mathcal {N}(0,1/\\alpha ^2)$ , for all $i,j$ .",
"Following the empirical Bayes framework leads to seeking the parameters $\\mathbf {v}$ , $\\alpha $ and $\\sigma $ that maximizes the marginal likelihood or evidence $p(\\mathbf {X}|\\mathbf {v},\\alpha ,\\sigma ) =\\prod _{i=1}^n p(\\mathbf {x}_i|\\mathbf {v},\\alpha ,\\sigma ) = \\prod _{i=1}^n\\int _{\\mathbb {R}^{p \\times d}} p(\\mathbf {x}_i|\\mathbf {W},\\mathbf {v},\\alpha ,\\sigma )p(\\mathbf {W})d\\mathbf {W}.$ In previous Bayesian PCA models, the marginal likelihood was never derived because it was too difficult to compute in practice or even intractable.",
"Here, conversely, the evidence of the model can be expressed analytically as a univariate integral using the isotropy of the prior on$\\mathbf {W}$ .",
"In the following, $\\mathbf {x}_\\mathbf {v}$ denotes the subvector of $\\mathbf {x}$ where only the columns corresponding to the nonzero indexes of $\\mathbf {v}$ have been kept.",
"Given a real order $\\nu $ , we denote by $J_\\nu $ and $K_\\nu $ the Bessel function of the first kind [1].",
"The density of $\\mathbf {x}$ is given by $ p(\\mathbf {x}|\\mathbf {v},\\alpha ,\\sigma )=e^{-\\frac{||\\mathbf {x}_\\mathbf {\\bar{v}}||_2^2}{2\\sigma ^2}} \\sigma ^{q-p} (2\\pi )^{-p/2} ||\\mathbf {x}_\\mathbf {v}||_2^{1-q/2} \\int _0^{\\infty }\\frac{u^{q/2}e^{-\\sigma ^2 u^2}}{(1+(u /\\alpha )^2)^{d/2}}J_{q/2-1}(u||\\mathbf {x}_\\mathbf {v}||_2)du.$ A proof of this theorem is given in Appendix A.",
"While reducing the dimension of the integration domain to one appears to be a valuable improvement, the integral of Equation (REF ), albeit univariate, falls within the category of Hankel-like integrals known to be particularly delicate to compute.",
"This is due to the fact that the integrand has singularities near the real axis [52].",
"To overcome this limitation, we investigate in the following subsection the use of the noiseless PPCA model to obtain a tractable expression."
],
[
"A closed-form evidence for globally sparse noiseless PPCA",
"To obtain a closed-form expression of the marginal likelihood, we consider the following modification of Model (REF ).",
"For the relevant variables, we use the noiseless PPCA model, and we assume that the irrelevant variables are generated by a Gaussian white noise.",
"More specifically, we write $\\mathbf {x} = \\mathbf {V} \\mathbf {W} \\mathbf {y} + \\mathbf {\\bar{V}} \\varepsilon _1+ \\mathbf { V} \\varepsilon _2,$ where $\\varepsilon _1 \\sim \\mathcal {N}(0, \\sigma _1^2 \\mathbf {I}_p)$ is the noise of the inactive variables and $\\varepsilon _2 \\sim \\mathcal {N}(0, \\sigma _2^2 \\mathbf {I}_p)$ is the noise of the active variables, having in mind that we aim at investigating the noiseless limit $\\sigma _2\\rightarrow 0$ .",
"We will see that, with this particular formulation of the problem, the evidence has a closed form expression which involves the multivariate Bessel distribution, introduced by [15].",
"A random vector is said to have a symmetric multivariate Bessel distribution with parameters $\\beta >0$ and $\\nu >-k/2$ if its density is $\\forall \\mathbf {z} \\in \\mathbb {R}^k, \\; \\textup {Bessel}(\\mathbf {z}|\\beta ,\\nu )=\\frac{2^{-k-\\nu +1}\\beta ^{-k-\\nu }}{\\Gamma (\\nu +k/2)\\pi ^{k/2}}||\\mathbf {z}||_2^\\nu K_\\nu (||\\mathbf {z}||_2/\\beta ).$ In the noiseless limit $\\sigma _2\\rightarrow 0$ , $\\mathbf {x}$ converges in probability to a random variable $\\tilde{\\mathbf {x}}$ whose density is $p(\\tilde{\\mathbf {x}}|\\mathbf {v},\\alpha ,\\sigma _1^2) = \\mathcal {N}(\\tilde{\\mathbf {x}}_\\mathbf {\\bar{v}}|0,\\sigma _1 \\mathbf {I}_{p-q}) \\textup {Bessel}(\\tilde{\\mathbf {x}}_\\mathbf {v}|1/\\alpha ,(d-q)/2).$ This theorem (proved in Appendix B) allows us to efficiently compute the noiseless marginal log-likelihood defined as $ \\mathcal {L}(\\mathbf {X},\\mathbf {v},\\alpha ,\\sigma _1)= \\sum _{i=1}^n \\log \\mathbb {P} (\\tilde{\\mathbf {x}}=\\mathbf {x}_i|\\mathbf {v},\\alpha ,\\sigma _1).$ Regarding hyper-parameter tuning, if we assume that $\\mathbf {v}$ is known, the regularization parameter $\\alpha $ can be optimized efficiently using univariate gradient ascent.",
"In fact, as stated by next proposition (proved in Appendix C), the marginal log-likelihood is even a strictly concave function of $\\alpha $ .",
"The function $ \\alpha \\mapsto \\mathcal {L}(\\mathbf {X},\\mathbf {v},\\alpha ,\\sigma _1)$ is strictly concave on $\\mathbb {R}^{*}_+$ .",
"The unique optimal value $\\hat{\\alpha }$ can therefore be found easily using univariate convex programming.",
"The noise variance $\\sigma _1$ can be estimated using (REF ) by computing the standard error of the variables which were not selected by $\\mathbf {v}$ .",
"However, since model (REF ) is a particular instance of PPCA, it is possible to use any regular PPCA noise variance estimator.",
"A discussion on which estimator to choose is provided in subsection REF"
],
[
"High-dimensional inference through a continuous relaxation",
"In spite of the results of the previous subsection, maximizing the evidence, even in the noiseless case, is particularly difficult (because of the discreteness of $\\mathbf {v}$ which can take $2^p$ possible values).",
"We therefore consider a simple continuous relaxation of the problem by replacing $\\mathbf {v}$ by a continuous vector $\\mathbf {u} \\in [0,1]^p$ .",
"This relaxation is close to the one considered by [32] in a sparse linear regression framework.",
"Denoting $\\mathbf {U}=\\textup {diag}(\\mathbf {u})$ , this relaxed model can be written as $ \\mathbf {x} = \\mathbf {U} \\mathbf {W} \\mathbf {y} + \\varepsilon .$ We denote $\\theta =(\\mathbf {u},\\alpha ,\\sigma )$ the vector of parameters.",
"In order to maximize the evidence $p(\\mathbf {X}|\\theta )$ , we adopt a variational approach [8].",
"We view $\\mathbf {y}_1,...\\mathbf {y}_n$ and$\\mathbf {W}$ as latent variables.",
"Given a (variational) distribution $q$ over the space of latent variables, the variational free energy is given by $\\mathcal {F}_q(\\mathbf {x_1},...\\mathbf {x_n}|\\theta )=-\\mathbb {E}_q[\\ln p(\\mathbf {X},\\mathbf {Y},\\mathbf {W}|\\theta )]-H(q),$ where $H$ denotes the differential entropy, and is an upper bound to the negative log-evidence $-\\ln p(\\mathbf {X}|\\theta )=\\mathcal {F}_q(\\mathbf {X}|\\theta )-\\textup {KL}(q||p(\\cdot |\\theta ))\\le \\mathcal {F}_q(\\mathbf {X}|\\theta ).$ To minimize $\\mathcal {F}_q(\\mathbf {X}|\\theta )$ , the following mean-field approximation is made on the variational distribution $ q(\\mathbf {Y},\\mathbf {W})=q(\\mathbf {Y})q(\\mathbf {W}).$ With this factorization, a variational expectation-maximization (VEM) algorithm can be derived.",
"For the E-step, the variational posterior distribution $q^*$ , which minimizes the free energy, is computed.",
"The variational posterior distribution of the latent variables which minimizes the free energy is given by $q^*(\\mathbf {Y})=\\prod _{i=1}^n\\mathcal {N}(\\mathbf {y}_i|\\mu _i,\\mathbf {\\Sigma }),$ and $q^*(\\mathbf {W})=\\prod _{k=1}^p\\mathcal {N}(\\mathbf {w}_k|\\mathbf {m}_k,\\mathbf {S}_k),$ where, for all $i \\in \\lbrace 1,...,n\\rbrace $ and $k \\in \\lbrace 1,...,p\\rbrace $ $ \\mu _i=\\frac{1}{\\sigma ^2}\\mathbf {\\Sigma } \\mathbf {M}^T\\mathbf {U} \\mathbf {x}_i\\textup {, } \\mathbf {m}_k=\\frac{u_k}{\\sigma ^2}\\mathbf {S}_k\\sum _{i=1}^nx_{i,k}\\mu _i,$ $\\mathbf {\\Sigma }^{-1}=\\mathbf {I}_d+\\frac{1}{\\sigma ^2}\\mathbf {M}^T\\mathbf {U}^2\\mathbf {M}+\\frac{1}{\\sigma ^2}\\sum _{k=1}^pu_k^2\\mathbf {S}_k,\\; \\mathbf {S}_k^{-1}=\\alpha ^2\\mathbf {I}_d + \\frac{nu_k^2}{\\sigma ^2}\\mathbf {\\Sigma }+\\frac{u_k^2}{\\sigma ^2}\\mathcal {M}^T\\mathcal {M},$ $\\mathbf {M}=(\\mathbf {m}_1,...\\mathbf {m}_p)^T \\; \\; { and } \\; \\; \\mathcal {M}=(\\mu _1,...\\mu _n)^T.$ It is worth noticing that two factorizations arise naturally.",
"The four equations of Proposition (REF ) (proved in Appendix D) will constitute the E-step of the VEM algorithm used to minimized the free energy.",
"We can now compute the negative free energy which will be maximized during the M-step.",
"Up to unnecessary additive constants, the negative free energy is given by $-\\mathcal {F}_q(\\mathbf {x_1},...\\mathbf {x_n}|\\theta )=-np\\ln \\sigma + dp \\ln \\alpha - \\frac{1}{2\\sigma ^2}\\textup {Tr}(\\mathbf {X}^T\\mathbf {X})- \\frac{1}{2\\sigma ^2}\\sum _{k=1}^pu_k^2\\textup {Tr}[(n \\Sigma +\\mathcal {M}^T\\mathcal {M})(S_k+\\mathbf {m}_k\\mathbf {m}_k^T)]\\\\+ \\frac{1}{\\sigma ^2}\\sum _{i=1}^n\\mathbf {x}_i^T\\mathbf {UM}\\mu _i + \\sum _{k=1}^p -\\frac{\\alpha ^2}{2}\\textup {Tr}(\\mathbf {S}_k+\\mathbf {m}_k\\mathbf {m}_k^T)-\\frac{1}{2}\\sum _{i=1}^n \\textup {Tr}(\\Sigma +\\mu _i\\mu _i^T) +\\frac{n}{2}\\ln |\\mathbf {\\Sigma }|+\\frac{1}{2}\\sum _{k=1}^p\\ln |\\mathbf {S}_k|.$ Minimizing the free energy leads to the following M-step updates $ \\alpha ^*=\\left(\\frac{1}{dp}\\sum _{k=1}^p\\textup {Tr}(\\mathbf {S}_k+\\mathbf {m}_k\\mathbf {m}_k^T)\\right)^{-1/2},$ $ \\sigma ^*=\\sqrt{\\frac{\\textup {Tr}(\\mathbf {X}\\mathbf {X}^T+ \\mathbf {X}\\mathbf {U}\\mathbf {M}\\mathcal {M})}{np} +\\frac{1}{np}\\sum _{i=1}^n\\sum _{k=1}^pu_k^2\\textup {Tr}[(\\Sigma +\\mu _i\\mu _i^T)(S_k+\\mathbf {m}_i\\mathbf {m}_i^T)]},$ and, for $k \\in \\lbrace 1,...,p\\rbrace $ , $ u_k^*=\\textup {argmin}_{u \\in [0,1]} \\frac{u^2}{2\\sigma ^2} \\sum _{i=1}^n\\textup {Tr}[(\\Sigma +\\mu _i\\mu _i^T)(S_k+\\mathbf {m}_i\\mathbf {m}_i^T)]-u\\sum _{i=1}^nx_{i,k}\\mathbf {m}_k^T\\mu _i.$ Note that the objective function of the optimization problem (REF ) is simply a univariate polynomial."
],
[
"The GSPPCA algorithm",
"Once the VEM algorithm has converged, the continuous vector $\\mathbf {u}$ still needs to be transformed into a binary one.",
"To do so, the following simple procedure, summarized in Algorithm REF , is considered: a family of $p$ nested models is built using the order of the coefficients of $\\mathbf {u}$ as a way of ranking the variables.",
"Specifically, for each $k\\le p$ , the $k$ -th element of this family is the binary vector $\\mathbf {v}^{(k)}$ such that the $k$ top coefficients of $\\mathbf {u}$ are set to 1 and the others to 0. the marginal likelihood $\\mathcal {L}$ of the non-relaxed model (computed using the formula of Theorem 3) is then maximized over this family of models.",
"the model $\\mathbf {v}$ with the largest marginal likelihood is kept.",
"Once the model is estimated, the globally sparse principal components of $\\mathbf {X}$ can be computed by simply performing PCA on $\\mathbf {X}_\\mathbf {v}$ .",
"This type of post-processing is similar to the variational renormalization introduced by [47].",
"In the case of local sparsity, variational renormalization can be achieved using an alternating maximization scheme [28].",
"However, the global sparsity structure greatly simplifies this procedure by reducing it to performing PCA on the relevant variables.",
"[t] GSPPCA algorithm for unsupervised variable selection data matrix $\\mathbf {X} \\in \\mathbb {R}^{n\\times p}$ , dimension of the latent space $d \\in \\mathbb {N}^*$ sparsity pattern $\\mathbf {v} \\in \\lbrace 0,1\\rbrace ^p$ // VEM algorithm to infer the path of models Initialize $\\mathbf {u},\\alpha ,\\sigma ,\\mu _1,...,\\mu _n,\\mathbf {m}_1,...,\\mathbf {m}_p,\\mathbf {S}_1,...,\\mathbf {S}_p$ and $\\Sigma $ convergence of the variational free energy E-step from Proposition REF M-step from equations (REF ),(REF ),(REF ) // Model selection using the exact marginal likelihood Compute $\\sigma _1$ k = 1..p Compute $\\mathbf {v}^{(k)}$ Find $ \\alpha _k = \\textup {argmax}_{\\alpha >0} \\lbrace \\alpha \\mapsto \\mathcal {L}(\\mathbf {X},\\mathbf {v}^{(k)},\\alpha ,\\sigma _1) \\rbrace $ using gradient ascent $ q=\\textup {argmax}_{1\\le k\\le p} \\mathcal {L}(\\mathbf {X},\\mathbf {v}^{(k)},\\alpha _k,\\sigma _1)$ $\\mathbf {v}=\\mathbf {v}^{(q)}$"
],
[
"Spike-and-slab models",
"Model (REF ) may be rewritten $\\mathbf {x}=\\tilde{\\mathbf {W}}\\mathbf {y}+\\varepsilon $ where $\\tilde{\\mathbf {W}}=\\mathbf {VW}$ .",
"The prior distribution for the parameter $\\tilde{\\mathbf {W}}$ is similar to the spike-and-slab prior introduced by [46] in a linear regression framework.",
"Indeed, each coefficient $\\tilde{w}_{ij}$ follows a priori either a Dirac distribution with mass at zero (if $v_i=0$ ) which is usually called the spike or a Gaussian distribution with variance $1/\\alpha ^2$ (if $v_i=1$ ) which is usually called the slab.",
"However, contrary to standard spike-and-slab models which would assume a product of Bernoulli prior distributions over $\\mathbf {v}$ , we see $\\mathbf {v}$ here as a deterministic parameter to be inferred from the data.",
"It is worth noticing that spike-and-slab priors have already been applied to locally sparse PCA by [34] and [48]."
],
[
"Automatic relevance determination",
"Introduced in the context of feedforward neural networks [37], [51], automatic relevance determination (ARD) is a popular empirical Bayes procedure to induce sparsity.",
"ARD was applied to Bayesian PCA models together with VEM algorithms in order to obtain automatic dimensionality selection [7] of local sparsity [3].",
"In order to obtain global sparsity, ARD may be built using Model (REF ) together with Gaussian priors $\\mathbf {w}_i\\sim \\mathcal {N}(0,a_i\\mathbf {I}_d)$ for $i \\in \\lbrace 1,...,p\\rbrace $ .",
"Similarly to [66], maximizing the marginal likelihood would discard irrelevant variables by leading several variance parameters $a_i$ to vanish.",
"Interestingly, this model is somehow related to the relaxed GSPPCA model.",
"Indeed the relaxed model (REF ) assumes that the $i$ -th line of the loading matrix $\\mathbf {UW}$ follows a priori a $\\mathcal {N}(0,u_i^2/\\alpha ^2\\mathbf {I}_d)$ distribution.",
"The relaxed model will consequently inherit the good properties of ARD – listed for example by [75].",
"However, similarly to [32], using the exact marginal likelihood to eventually obtain a sparse solution will avoid many classical drawbacks of ARD.",
"First, as pointed out by [73], convergences of EM algorithms are extremely slow in the case of the ARD models.",
"However, with our approach, since we only need the ordering of the coefficients of $\\mathbf {u}$ , we do not have to wait for the complete convergence of this parameter.",
"In practice, in all the experiments that we carried out, we only had to perform less than a few hundreds of iterations of the algorithm to obtain convergence of the free energy in order to perform variable selection.",
"It is worth mentioning that the fact that the objective function converges faster than the parameters of the model is a quite general property of EM algorithms [78].",
"Our procedure also avoids the lack of flexibility of ARD by computing posterior probabilities of models rather than simply giving an estimate of the best sparse model.",
"Combined with a greedy technique similar to Occam's window [39], this feature could allow for example to perform Bayesian model averaging, which is not possible with ARD.",
"Eventually, in the context of Bayesian PCA, ARD models such as the ones of [6], [7] or [3] have to rely on approximations of the marginal likelihood while we use an exact expression."
],
[
"Intrinsic dimension estimation",
"Since model (REF ) is a particular instance of PPCA, any intrinsic dimension estimator for PCA can be applied to estimate beforehand the intrinsic dimension $d$ .",
"Although the problem of finding $d$ is of critical importance, we assume in this work that a reasonable choice of dimension has already been made by the practitioner.",
"While it could be tempting to use the exact noiseless marginal likelihood to select $d$ , the close relationship existing between the noise level and $d$ in PPCA [67], [49] suggests that loosing the noise information is likely to be prejudicial for intrinsic dimension estimation."
],
[
"Initialization strategies for the VEM algorithm",
"Regarding the initialization of the relaxed model parameter $\\mathbf {u}$ , we chose to initialize all its coefficients to one.",
"This allows to avoid premature vanishing of these coefficients which is a common drawback of ARD-like techniques [73].",
"The noise standard error can be simply initialized using any classical PPCA noise estimator (cf.",
"subsection REF ).",
"Similarly to [32], the slab precision parameter $\\alpha $ controls the sparsity of the VEM solution and a too small initial value is likely to lead to a too sparse solution such as the useless local optimum $\\mathbf {u} = 0$ .",
"Following [5], we chose to perform short VEM runs (with less than 5 iterations) on a small grid (typically $\\alpha \\in \\lbrace 0.1,1,10\\rbrace $ ) and to select the value of $\\alpha $ that led to the lowest free energy.",
"The posterior means of the PCA loadings $\\mathbf {m}_1,...,\\mathbf {m}_p$ and of the corresponding scores $\\mu _1,...,\\mu _n$ can be initialized using the singular vectors of $\\mathbf {X}$ .",
"If the size of the data forbids to perform this SVD, using random standard Gaussian coefficients as starting points does not significantly alter the results.",
"Finally, the initial values chosen for the posterior covariance matrices are $\\Sigma = \\mathbf {I}_d$ and $\\mathbf {S}_1=...=\\mathbf {S}_p=\\alpha ^{-2}\\mathbf {I}_d$ ."
],
[
"Computational cost of VEM iterations",
"Thanks to the factorizations that arised naturally during variational inference, the cost of each VEM iteration is of order $O(pnd^3)$ which is linear both in sample size and dimensionality and therefore particularly suitable for high-dimensional inference."
],
[
"Estimation of the noise variance",
"As mentioned in seubsection REF , the standard error $\\sigma _1$ of irrelevant predictors can be estimated using any regular PPCA estimator.",
"Specifically, three important estimators are considered: the maximum likelihood estimator [67], its unbiased correction [53], or simply the median of the variances of all features [24].",
"Since the ML estimator is known to be biased in the high-dimensional regime, it is usually preferable to use its bias-corrected version.",
"Both of these estimators can also be computed using the singular value decomposition (SVD) of $\\mathbf {X}$ .",
"Note that since the median estimator does not need to perform this decomposition, it is therefore more suitable for large-scale inference."
],
[
"Large scale inference",
"In the GSPPCA algorithm, SVD is used twice.",
"Indeed, the top $d$ singular vectors can be used to initiate the VEM algorithm and the $p-d$ smallest singular values can be used to estimate the noise variance (both as a VEM starting point for $\\sigma $ and as an estimator for $\\sigma _1$ ).",
"This can be done efficiently using a truncated SVD algorithm.",
"We chose specifically the R interface [54] of the Spectrahttp://yixuan.cos.name/spectra/index.html C++ library.",
"However, for very large scale problems, even a fast truncated SVD algorithm appears computationally prohibitive.",
"To tackle this issue, we offer two alternatives.",
"First, the covariance matrices initialized using the eigenvectors can be initialized using random standard Gaussian coefficients.",
"Moreover, following [24], the noise variance can be estimated using the median of the variable variances.",
"This leads to a \"SVD-free\" version of the GSPPCA algorithm suitable for very large scale problems."
],
[
"Model selection speedup",
"The model selection step of the GSPPCA algorithm requires to perform $p$ univariate gradient ascents, which can be computationally expensive when $p$ is large.",
"A simple way to reduce the number of gradient ascents is to rely on the links between our relaxed model and ARD.",
"Specifically, we can discard before the model selection step all the variables corresponding to the subset $\\lbrace i \\in \\lbrace 1,...,p\\rbrace | u_i=0 \\rbrace $ where $\\mathbf {u}$ is the relaxed model parameter obtained after convergence of the VEM algorithm.",
"When $\\mathbf {u}$ is sparse, this will bring about a substantial speedup.",
"Notice that, since ARD is known to converge slowly, $\\mathbf {u}$ is unlikely to be sparse enough and the model selection step is still necessary."
],
[
"Evaluation of Bessel functions",
"The modified Bessel function of the second kind, which is used to compute the exact marginal likelihood and it gradient with respect to $\\alpha $ , can be delicate to compute as soon as its order or its argument is large.",
"In our experiments, we tackled this issue by using an asymptotic expansion based on Debye polynomials [1].",
"This is in particular implemented in the R package Bessel [40].",
"We found this approximation to be extremely accurate in all the experiments that we carried out."
],
[
"Numerical simulations",
"This section aims at highlighting the specific features and abilities of the proposed GSPPCA approach on simulated and real data sets."
],
[
"An introductory example",
"We consider here a simple introductory example to illustrate the proposed combination between a relaxed VEM algorithm and the closed-form expression of the marginal likelihood.",
"For this experiment, $n=50$ observations are simulated according to (REF ) with $p=30$ , $d=5$ and $q=10$ .",
"Each coefficient of $\\mathbf {W}$ is drawn at random according to a standard Gaussian distribution and the noise variance is equal to $0.1$ .",
"Figure 1 presents the results of GSPPCA on this toy data set.",
"The left panel presents in dark blue the coefficients of the estimated $\\mathbf {u}$ obtained after running the VEM algorithm (sorted in decreasing order) and the corresponding true values of $\\mathbf {v}$ (pale blue points) used in the simulations.",
"The right panel shows the values of evidence computed on the family of models inferred by the order of the coefficients of $\\mathbf {u}$ .",
"On this simple example, $\\mathbf {u}$ captures the true ranking of the variables and the model with the largest evidence is actually the true one.",
"Figure: Variable selection with GSPPCA on the introductory example."
],
[
"Range of the noiseless assumption",
"In all the experiments that we carried out, since the noiseless PPCA model is not a true generative $p$ -dimensional model (the random variable $\\tilde{\\mathbf {x}}$ belongs to a strict subspace of $\\mathbb {R}^p$ ), we chose not to use it to generate data in our experiments.",
"We rather chose the more realistic and natural Model (REF ).",
"Since this model includes a nonzero noise, it is important to know the limits of the noiseless assumption.",
"We therefore simulated two scenarios according to Model (REF ): a first one with $n=40$ observations and a second one with $n=200$ .",
"In both scenarios, $p=200$ , $d=10$ , $q=20$ , and each coefficient of $\\mathbf {W}$ is drawn according to a standard Gaussian distribution.",
"The sparsity pattern chosen is simply $ \\mathbf {v}=(\\overbrace{1,...,1}^{\\textup {20 times}},\\overbrace{0,...,0}^{\\textup {180 times}})^T.",
"$ In this simple simulation scheme, the signal-to-noise ratio (SNR) may be defined as $ \\textup {SNR}=\\frac{1}{p\\sigma ^2}\\mathbb {E}_\\mathbf {W}[(\\mathbf {VW})^T\\mathbf {VW}]p\\sigma ^2=\\frac{dq}{p\\sigma ^2}.$ We chose a linear grid of 20 SNR ranging from $0.1$ (most difficult scenario) to 3 (easiest scenario) and generated 100 datasets for each noise level.",
"To evaluate the quality of the variable selection, we computed the F-score between $\\hat{\\mathbf {v}}$ and $\\mathbf {v}$ on 100 runs.",
"We recall that the F-score is the harmonic mean of precision and recall, and is closer to 1 when the selection is faithful.",
"Unsurprisingly, when the SNR gets close to zero, the quality of the variable selection diminishes.",
"However, GSPPCA appears to be quite robust to noise, even though the data are not generated according to the underlying noiseless model.",
"Indeed, even in the case where $n=40$ , we observe an almost perfect recovery as long as SNR>0.5.",
"Figure: Median, first and third quartiles of the F-score for the experiment of subsection 3.2, based on 100 runs"
],
[
"Model selection",
"In this subsection, we compare the model selection accuracies of two global methods – GSPPCA, SSPCA [23] – and a local one – SPCA [83]."
],
[
"Simulation setup",
"While the simple simulation setup of Subsection REF conveniently allowed to compute the SNR in closed formed in order to assess the range of the noiseless assumption, we introduce here a more realistic scheme by considering a finer correlation structure as well as a non-Gaussian noise.",
"Specifically, first we generate $n$ i.i.d observations $(\\mathbf {z}_1,...,\\mathbf {z}_n)$ following multivariate normal distribution $\\mathcal {N}(0,\\mathbf {R})$ where $\\mathbf {R}=\\textup {diag}(\\mathbf {R}_1,...,\\mathbf {R}_4)$ is a 4-blocks diagonal matrix where $R_\\ell $ is such that $r_{\\ell ii} = 0.3$ and $r_{\\ell ij} = \\rho $ for $i,j=1,\\dots ,p/4$ and $i\\ne j$ .",
"Then, a globally sparse PCA model is obtained as followed.",
"First, PPCA is performed on the sample $(\\mathbf {z}_1,...,\\mathbf {z}_n)$ , which leads to a non-sparse ML estimate $\\mathbf {W}_{\\textup {ML}}$ for the loading matrix.",
"Then, given a sparsity pattern $\\mathbf {v} \\in \\lbrace 0,1\\rbrace ^p$ and denoting $\\mathbf {V}=\\textup {diag}(\\mathbf {v})$ as before, the loading matrix matrix is \"globally sparsified\" by considering $\\mathbf {VW}_{\\textup {ML}}$ .",
"The final observations are eventually generated according to the non-noiseless model $\\forall i \\le n, \\; \\; \\mathbf {x}_i = \\mathbf {V} \\mathbf {W}_{\\textup {ML}} \\mathbf {y}_i + \\varepsilon .$ The simple sparsity pattern (REF ) is kept and the vectors $\\mathbf {y_1},...,\\mathbf {y}_n$ are standard Gaussian as in regular PPCA.",
"Regarding the noise term $\\varepsilon $ , we consider two scenarios.",
"A first one with Gaussian noise and a second one with Laplacian noise, both centered with unit variance.",
"We choose $p=200$ , $d=10$ , $q=20$ and consider five cases for the sample size: $n=p/5$ , $p/4$ , $n=\\lfloor p/3\\rfloor $ , $n=p/2$ and $n=p$ .",
"More classical $n>p$ cases are not presented here since regular PCA is known to perform well in this context and variable selection thus may not be of great use [24].",
"Each experiment was repeated 50 times."
],
[
"Model selection criteria",
"Regarding SSPCA, we used the Matlab code available at the main author's webpage and chose the tuning parameter using 5-fold cross-validation on the reconstruction error.",
"We constrained the algorithm in order to obtain globally sparse solutions.",
"For SPCA, we used the elasticnet R package and an ad-hoc method by selecting enough variables to explain $99\\%$ of the total variance.",
"We also tried to apply another globally sparse algorithm, vsnPCA-$\\ell _0$ from [69].",
"However, their use of the Bayesian information criterion (BIC) led to selecting very few variables.",
"This is not very surprising: since BIC is an asymptotic sparsity criterion, it is thus likely to perform poorly when $p$ is larger than $n$ ."
],
[
"Results",
"Tables REF and REF reports the mean and standard error of the F-score for the experiments described is this subsection.",
"The two globally sparse methods vastly outperform SPCA, which is unable to identify the particular structure of the data.",
"When $p$ is larger than $n/2$ , both globally sparse algorithms perform very well, GSPPCA being slightly better in the Gaussian noise case.",
"It is not surprising to see SSPCA adapt efficiently to Laplacian noise because cross-validation is a model-free technique and is more likely to outperform model-based techniques when the data is not generated according to the model distribution.",
"However, when $n$ is smaller than $p/2$ , GSPPCA significantly outperforms SSPCA in both noise scenarios.",
"This reminds the fact that, is many $p\\gg n$ situations, Bayesian model selection empirically outperforms $\\ell _1$ -based methods [11], [32].",
"Table: F-score×100\\times 100 for the model selection experiment of subsection with Gaussian noiseTable: F-score×100\\times 100 for the model selection experiment of subsection with Laplacian noise"
],
[
"Global versus local",
"Here, we illustrate on real data sets how using GSPPCA instead of computing the leading sparse principal component for model selection can lead to selecting more relevant variables – i.e variables that retain more variance or are more interpretable."
],
[
"Explained variance",
"We consider the breast cancer data base from the breastCancerVDX R package [60], consisting in expression levels of $p=5391$ genes for $n=344$ breast cancer patients.",
"More details regarding this data set – including the preprocessing technique used – are given in Appendix F. Given a cardinality $q$ , we applied three methods to select relevant genes: we computed the first $q$ -sparse principal component using SPCA [83] we computed the support of the globally $q$ -sparse subspace of dimension $d=10$ using GSPPCA and SSPCA For each method, we projected the data onto a 10-dimensional globally $q$ -sparse subspace using the sparsity pattern found by the algorithm and computed the percentage of explained variance using the criterion introduced by [62] – for each method, we applied the post-processing technique of [47].",
"The results are plotted on Figure REF .",
"It is important to notice that both global methods explain much more variance than SPCA.",
"This fact is not surprising since the data is indeed projected onto a globally sparse subspace, but the significance of this variance gap highlights the fact that different dimensions lead to very different sparsity patterns.",
"This means that projecting the data onto a single sparse axis is likely to lead to an important information loss (this fact is confirmed in section 5).",
"The variables selected by GSPPCA retain significantly more variance than the ones selected by SSPCA, and may consequently be of superior interest.",
"Figure: Percentage of variance explained by the data projected onto a 10-dimensional globally sparse subspace"
],
[
"Interpretability",
"Inspired by [20], we consider the problem of learning which features are relevant on three data sets of handwritten digits.",
"We consider $n=500$ gray-scale images (with $p=758$ pixels) of handwritten sevens from three data sets introduced by [31]: mnist-basic which is simply a subsample of sevens from the original MNIST data set, mnist-back-rand in which random backgrounds were inserted in the images.",
"Each pixel value of the background was generated uniformly between 0 and 255, mnist-back-image in which random patches extracted from a set of 20 grey-scale natural images were used as backgrounds for the sevens.",
"On these three data sets, we apply SPCA (with $d=1$ ), SSPCA and GSPPCA (both with $d=100$ ) in order to select $q=200$ relevant pixels.",
"On mnist-basic, even if SPCA's result is a little bit more erratic than the two others, all selections are interpretable and we can easily recognize a seven.",
"On mnist-back-rand however, while the two globally sparse selections are still consistent, SPCA's pixels are more scattered and it is harder to recognize the shape of a seven.",
"Eventually, on mnist-back-image, GSPPCA's selection is less smooth but a seven can still be recognized, whereas SPCA appears to randomly select pixels almost everywhere but near the mean seven.",
"SSPCA seems to notice that the zone occupied by the upper bars of the sevens is of interest, but its selection does not appear interpretable.",
"Table: Variable selection of SPCA and GSPPCA for the three datasets of , selected variables are in white"
],
[
"Application to signal denoising",
"In this section, we focus on a first possible application of GSPPCA for signal denoising through the sparsification of a wavelet decomposition.",
"PCA is indeed a popular way to denoise multivariate signals [2], [24].",
"To illustrate the potential interest of GSPPCA in this context, we consider hereafter two simulation scenarios, each using a specific form of signal and wavelet.",
"The simulation scenarios are as follows: Scenario A: it consists in a square wave signal with 6 states of different lengths.",
"The observed signal is sampled with a time step of $5\\times 10^{-3}$ with an additional Gaussian noise with zero mean and $0.2$ standard deviation.",
"The Haar wavelet is used here for signal reconstruction.",
"Scenario B: the original signal is here a mixture of 4 Gaussian densities.",
"The observed signal is also sampled with a time step of $5\\times 10^{-3}$ with an additional Gaussian noise with zero mean and $0.2$ standard deviation.",
"The Daubechies D8 wavelet is used here for signal reconstruction.",
"Figure REF presents the original signals and observed signals for scenarios A and B.",
"In both cases, $n=100$ signals were sampled during the training phase and decomposed as $p=175$ wavelet coefficients.",
"For signal denoising, GSPPCA is applied on the $n\\times p$ wavelet coefficient matrix to extract $d=10$ globally sparse principal axes.",
"Then, a new sampled signal is projected on those extracted principal axes and back-projected in the original wavelet domain.",
"It is worth mentioning that the estimated value for $q=\\left\\Vert v\\right\\Vert _{0}$ is 17 on scenario A and 15 on scenario B.",
"As an illustration, we plotted on Figure REF the denoising results for newly sampled signals A and B with GSPPCA.",
"We used the same projection-reconstruction protocol for PCA, thresholded PCA (PCA loading smaller than $1\\times 10^{-3}$ are set to 0) and SPCA ($\\lambda $ is chosen such that $99\\%$ of the PCA projected variance is conserved).",
"Denoising results obtained with those methods are also supplied on Figure REF .",
"First, on both signal A and B, PCA achieves a very satisfying denoising and thus confirms his validity in this context.",
"One can also show that a simple thresholding of the PCA loadings allows a clear denoising improvement and turns out to be competitive with the one performed by SPCA.",
"The SPCA result is here somehow disappointing due to the fact that the sparsity is not global and most wavelet levels stay active in the final reconstruction.",
"Finally, the global sparsity of GSPPCA retains only a few wavelet levels and achieves here the best reconstruction in both scenarios.",
"Finally, Table REF presents the reconstruction error (sum of squared errors) averaged on 50 test signal reconstructions, on the two simulation scenarios.",
"The results confirms the observations made on Figure REF .",
"GSPPCA achieves particularly good performances on both scenarios and thus imposes itself as a competitive tool for signal denoising.",
"Moreover, the GSPPCA reconstruction uses fewer wavelet levels and is therefore visually smoother.",
"Figure: Denoising results for signals A (top) and B (bottom) with PCA, thresholded PCA,SPCA and GSPPCA.Table: Reconstruction error (sum of squared errors) for waveletsignal denoising on the two simulation scenarios (results are averagedon 50 signal reconstructions).",
"Standard deviations are also provided."
],
[
"Application to unsupervised gene selection",
"Considering again the breast cancer data set previously studied in Section 3, we address here the issue of the biological significance of the selected genes.",
"To this end, we will use the pathway enrichment index (PEI) introduced by [64] and used in a sparse PCA framework by [28]."
],
[
"Pathway enrichment as a measure of biological significance",
"In this subsection, we briefly review how the PEI can be computed in order to evaluate the quality of a given subset of genes.",
"For more details on the PEI, see [64] or [27], and on hypergeometric tests and enrichment, see [56].",
"Suppose that using a microarray data matrix $\\mathbf {X} \\in \\mathbb {R}^{n \\times p}$ where each variable corresponds to a gene, an algorithm infers a subset $\\mathbf {s} \\subset \\lbrace 1,...,p\\rbrace $ of genes.",
"A way to assess its biological significance is to compare $\\mathbf {s}$ to many other subsets which are known to be biologically relevant.",
"In this case, the biologically relevant subsets are defined by biological pathways, and are therefore groups of genes involved in series of biochemical reactions linked to a certain biological function.",
"Let us denote these known subsets $\\mathbf {b}_1,...,\\mathbf {b}_N \\subset \\lbrace 1,...,p\\rbrace $ .",
"For our breast cancer experiment, we use the $N=1116$ pathways from the Reactome database [14] included in the R package reactomePA [79].",
"For $k\\le N$ , the enrichment of $\\mathbf {s}$ in the $k$ -th pathway of this list is the statistical significance of its overlap with $\\mathbf {b}_k$ , evaluated using the hypergeometric test.",
"More specifically, for each $k\\le N$ , the null hypothesis of this test is that the genes in $\\mathbf {s}$ are chosen uniformly at random from the total gene population.",
"Under this hypothesis, the test statistic ${\\# (\\mathbf {s} \\cap \\mathbf {b}_k)}$ follows a hypergeometric distribution and a $p$ -value can be computed to assess the statistical significance of the overlap.",
"Because we are conducting one test for each pathway considered, these $p$ -value are then adjusted using the Benjamini-Hochberg procedure to control the false discovery rate [4].",
"The subset $\\mathbf {s}$ is eventually declared enriched for a certain pathway if the adjusted $p$ -value of the corresponding hypergeometric test is lower than $0.01$ .",
"The PEI is finally defined as the percentage of enriched pathways in the Reactome family."
],
[
"Results",
"We compare in Table REF the PEI obtained by GSPPCA with $d=10$ , SPCA and thresholded PCA for several fixed cardinalities.",
"Similarly to [83], the two local methods are computing a single sparse axis.",
"As in [28] SPCA appears to give slightly better results than thresholded PCA.",
"GSPPCA significantly outperforms the two other methods.",
"This means that the genes selected by GSPPCA are consistently more associated with the Reactome pathways, and are therefore more interpretable.",
"This highlights the fact that projecting the data onto a globally sparse subspace of dimension higher than one leads to significantly more interpretable and biologically plausible results.",
"Table: PEI for several fixed cardinalitiesRegarding the estimation of the sparsity level, choosing the one that explains $99\\%$ of the variance led SPCA to selecting 4810 genes, which is difficult to interpret.",
"For thresholded PCA, we selected the sparsity level using a criterion proposed by [64].",
"Even though it led to the sparsest solution, its PEI was very small.",
"Regarding GSPPCA, the noiseless marginal log-likelihood and the PEI of the corresponding models are plotted on Figure REF .",
"We can see that the marginal likelihood peak corresponds to highly interpretable genes: more than $5\\%$ of the biological pathways in the Reactome family have a significant overlap with the genes selected by GSPPCA.",
"Furthermore, models with a lower marginal likelihood have generally a lower PEI.",
"To a certain extend, this shows that our marginal likelihood expression can stand as an indicator of biological significance.",
"Figure: Marginal likelihood for the gene selection problem"
],
[
"Conclusion",
"Unsupervised feature selection is an hazy and exciting problem.",
"It becomes particularly difficult and ill-posed when no specific learning task (such as clustering) is driving it.",
"We have proposed in this paper a new method for unsupervised feature selection based on the idea that the data may lie close to a subspace of moderate dimension spanned by a basis with a shared sparsity pattern.",
"On several real data sets, this approach outperforms a popular method which consists in finding the sparsity pattern of the single leading principal vector of the data.",
"These results suggest that, on many real-life high-dimensional data sets, an important part of the information cannot be captured by one-dimensional subspace approximations.",
"While building our framework, we derived the first closed-form expression of the marginal likelihood of a Bayesian PCA model, using the noiseless model of [58].",
"Regarding future work, it would be interesting to see if more complex priors can be used and to what extend our expression can lead to a simultaneous estimation of the sparsity level and the dimension of the latent space.",
"Indeed, intrinsic dimension estimation, which was beyond the scope of this paper, has an enduring relationship with probabilistic versions of PCA [44], [9], [50] and would be an interesting direction."
],
[
"Acknowledgements",
"The authors would like to thank Magnus Ulfarsson for providing svnPCA software and Florentin Damiens for helpful discussions on Bessel functions.",
"Part of this work was done while PAM was visiting University College Dublin, funded by the Fondation Sciences Mathématiques de Paris (FSMP).",
"Let us first consider the case where all variables are active and assume that $\\mathbf {v}=(1,1,...,1)$ .",
"Therefore, $\\mathbf {V}=\\mathbf {I}_p$ and the considered model reduces to probabilistic PCA.",
"In this framework, we will derive the density of $\\mathbf {x}$ by computing the Fourier transform of its characteristic function.",
"In order to compute the characteristic function of $\\mathbf {x}$ , we first decompose the latent vector $\\mathbf {y}$ in the canonical base $\\mathbf {y}=y_1\\mathbf {e_1}+...+y_d\\mathbf {e_d},$ where $(\\mathbf {e_i})_{i\\ge d}$ is the canonical base of $\\mathbb {R}^d$ .",
"We can now write the vector $\\mathbf {W} \\mathbf {y}$ as a sum of of $d$ i.i.d variables $\\mathbf {W} \\mathbf {y} = y_1\\mathbf {W}\\mathbf {e_1} +...+y_d\\mathbf {W} \\mathbf {e_d}.$ Its characteristic function will consequently be $\\varphi _{\\mathbf {W} \\mathbf {y}}=(\\varphi _{y_1\\mathbf {W}\\mathbf {e_1}})^d.$ Now, for all $\\mathbf {u}\\in \\mathbb {R}^d$ , we have $\\varphi _{y_1\\mathbf {W}\\mathbf {e_1}}(\\mathbf {u}) &= \\mathbb {E}[\\exp ( i y_1\\mathbf {e_1}^T\\mathbf {W}^T\\mathbf {u})] \\\\ &=\\mathbb {E}\\left[\\exp \\left( i y_1 \\sum _{k=1}^p w_{k1}u_k\\right)\\right], $ but, since $w_{st}\\sim \\mathcal {N}(0,\\alpha )$ for all $s,t$ , we will have $\\frac{1}{\\sqrt{\\alpha }||\\mathbf {u}||_2}\\sum _{k=1}^p w_{k1}u_k \\sim \\mathcal {N}(0,1),$ thus, since $\\mathbf {y}$ and $\\mathbf {W}$ are independent, the law of $(\\sqrt{\\alpha }||\\mathbf {u}||_2)^{-1}y_1 \\sum _{k=1}^p w_{k1}u_k$ will be the one of a product of two standard Gaussian random variables, whose density is $1/\\pi K_{0}(| .",
"|)$ [76].",
"Therefore, we find that $\\varphi _{y_1\\mathbf {W}\\mathbf {e_1}}(\\mathbf {u}) &= \\frac{1}{\\pi }\\int _{-\\infty }^{+\\infty } K_0(|t|)e^{i \\sqrt{\\alpha }||\\mathbf {u}||_2 t}dt \\\\ &= \\frac{2}{\\pi } \\int _{0}^{+\\infty } K_0(t) \\cos (\\sqrt{\\alpha }||\\mathbf {u}||_2 t)dt,$ is simply the cosine Fourier transform of a univariate Bessel function.",
"Using a formula in [1], we eventually find that $\\varphi _{y_1\\mathbf {W}}(\\mathbf {u}) = \\frac{1}{\\sqrt{1+\\alpha ||\\mathbf {u}||_2^2}},$ which leads to $\\varphi _{\\mathbf {W} \\mathbf {y}} (\\mathbf {u})= \\frac{1}{(1+\\alpha ||\\mathbf {u}||_2^2)^{d/2}}.$ Finally, since the noise term and $\\mathbf {W} \\mathbf {y}$ are independent, the characteristic function of $\\mathbf {x}$ will be $\\varphi _\\mathbf {x}(\\mathbf {u})=\\varphi _{\\mathbf {W} \\mathbf {y}} (\\mathbf {u})\\varphi _{\\varepsilon }(\\mathbf {u})=\\frac{e^{-\\sigma ^2 ||\\mathbf {u}||_2^2}}{(1+\\alpha ||\\mathbf {u}||_2^2)^{d/2}}.$ The density of $\\mathbf {x}$ is then given by the Fourier transform of its characteristic function $p(\\mathbf {x})=\\frac{1}{(2\\pi )^p}\\int _{\\mathbb {R}^p}\\varphi _\\mathbf {x}(\\mathbf {u})e^{i \\mathbf {x}^T\\mathbf {u}}d\\mathbf {u},$ but, since $\\varphi _\\mathbf {x}(\\mathbf {u})$ is a radial function (i.e a function that only depends on the norm of its argument), its Fourier transform can be expressed as a univariate integral [59] and we can write $p(\\mathbf {x})=\\frac{||\\mathbf {x}||_2^{1-p/2}}{(2\\pi )^{p/2}}\\int _0^{+\\infty }\\frac{u^{p/2}e^{-\\sigma ^2 u^2}}{(1+\\alpha u^2)^{d/2}}J_{p/2-1}(u||\\mathbf {x}||_2)du, $ which is the desired form for the case with no inactive variable.",
"In the general case, $\\mathbf {v}$ is not necessarily equal to $(1,1,...,1)$ but we can notice that, since $\\mathbf {x}_\\mathbf {v}$ and $\\mathbf {x}_{\\mathbf {\\bar{v}}}$ are independent, we can write $p(\\mathbf {x})=p(\\mathbf {x}_{\\mathbf {\\bar{v}}})p(\\mathbf {x}_\\mathbf {v})$ .",
"Applying (REF ) to $\\mathbf {x}_\\mathbf {v}$ allows us to compute $p(\\mathbf {x}_\\mathbf {v})$ and to eventually obtain the expression of the density given by the theorem.",
"We begin by proving the following lemma, which links the distribution of the product between a Gaussian matrix and a Gaussian vector with the Bessel distribution.",
"This result may be of independent interest.",
"Let $\\mathbf {A}$ be a $q \\times d$ random matrix such that $a_{ij} \\sim \\mathcal {N}(0,s^2)$ with $s>0$ for all $i,j$ and let $\\mathbf {b} \\sim \\mathcal {N}0,\\mathbf {I}_d)$ .",
"Then $\\mathbf {Ab}$ follows a Bessel distribution with parameters $s$ and $(d-q)/2$ .",
"Using the decomposition arguments from the proof of Theorem 1, the characteristic function of $\\mathbf {Ab}$ is, for all $\\mathbf {u} \\in \\mathbb {R}^k$ , $\\varphi _{\\mathbf {Ab}}(\\mathbf {u})=\\frac{1}{(1+||\\mathbf {u}||^2_2/s)^{d/2}},$ which is exactly the characteristic function of the symmetric multivariate Bessel distribution [15].",
"We can now prove Theorem 2.",
"Let us first consider the case where all variables are active and assume that $\\mathbf {v}=(1,1,...,1)$ .",
"Using Lévy's continuity theorem, $\\varepsilon _2$ weakly converges to zero when $\\sigma _2$ vanishes.",
"Since zero is a constant, this convergence also happens to be in probability [70].",
"The variable $\\mathbf {x}$ therefore converges in probability to $\\mathbf {W}\\mathbf {y}$ , which follows a $\\textup {Bessel}(1/\\alpha ,(d-q)/2)$ distribution according to our lemma.",
"In the general case when $\\mathbf {v}$ is not necessarily equal to $(1,1,...,1)$ we can prove (REF ) by invoking the independence between $\\mathbf {x}_\\mathbf {v}$ and $\\mathbf {x}_\\mathbf {\\bar{v}}$ , similarly to the proof of Theorem 1.",
"Since a sum of concave functions is concave, it is sufficient to prove that the function $g: \\alpha \\mapsto p(\\tilde{\\mathbf {x}}|\\mathbf {v},\\alpha ,\\sigma _1) $ is strictly concave.",
"Up to unnecessary additive constants, we have for all $\\alpha >0$ , $g(\\alpha ) = d \\log \\alpha \\\\ + \\log \\left((\\alpha ||\\tilde{\\mathbf {x}}_\\mathbf {v}||_2)^{\\frac{q-d}{2}} K_{\\frac{q-d}{2}}\\left(||\\tilde{\\mathbf {x}}_\\mathbf {v}||_2\\alpha \\right) \\right).$ Using standard results about Bessel functions derivatives [1], it can be shown that $g^{\\prime }(u) = \\frac{d}{\\alpha } - ||\\tilde{\\mathbf {x}}_\\mathbf {v}||_2 h(u),$ where the $h$ is the ratio $h(\\alpha )=\\frac{K_{\\frac{q-d}{2}-1}\\left(||\\tilde{\\mathbf {x}}_\\mathbf {v}||_2\\alpha \\right)}{K_{\\frac{q-d}{2}}\\left(||\\tilde{\\mathbf {x}}_\\mathbf {v}||_2\\alpha \\right)}.$ As proven independently by [36] and [19], since $q-d\\ge 0$ , $h$ is a increasing function on $\\mathbb {R}^*_+$ .",
"Therefore $g^{\\prime }$ is stricly decreasing and $g$ is strictly concave.",
"Variational distribution of the latent vectors.",
"Using a standard result in variational mean-field approximations [8], we can write $\\ln q^*(\\mathbf {y})=\\mathbb {E}_{q(\\mathbf {W})}[\\ln p(\\mathbf {X},\\mathbf {Y},\\mathbf {W}|\\theta )]$ which leads to the factorization $q^*(\\mathbf {y})=\\prod _{i \\le n} q^*(\\mathbf {y}_i).$ Then, for each $i\\le n$ , we can write, up to unnecessary additive constants, $\\ln q^*(\\mathbf {y}_i)=\\mathbb {E}_{q(\\mathbf {W})}[\\ln p(\\mathbf {x}_i,\\mathbf {y}_i,\\mathbf {W}|\\theta )]=\\mathbb {E}_{q(\\mathbf {W})}\\left[\\frac{-1}{2\\sigma ^2}||\\mathbf {x}_i-\\mathbf {UWy}_i||_2^2\\right]-\\frac{1}{2}||\\mathbf {y}_i||_2^2,$ thus $\\ln q^*(\\mathbf {y}_i)= \\frac{-1}{2\\sigma ^2} \\mathbf {y_i}^T\\mathbb {E}_{q(\\mathbf {W})}[\\mathbf {W}^T\\mathbf {U}^2\\mathbf {W}]\\mathbf {y}_i+\\frac{1}{\\sigma ^2}\\mathbf {y}_i^T\\mathbb {E}_{q(\\mathbf {W})}[\\mathbf {W}]^T\\mathbf {U}\\mathbf {x}_i-\\frac{1}{2}||\\mathbf {y}_i||_2^2,$ which leads to the desired form.",
"Variational distribution of the loading matrix.",
"Similarly, up to unnecessary additive constants, $\\ln q^*(\\mathbf {W}) = \\frac{-1}{2\\sigma ^2}\\sum _{i=1}^n\\mathbb {E}_{q(\\mathbf {y}_i)}[||\\mathbf {x}_i-\\mathbf {UWy}_i||_2^2]-\\frac{\\alpha ^2}{2}\\sum _{i=1}^p||\\mathbf {w}_i||_2^2,$ $\\ln q^*(\\mathbf {W}) =\\sum _{i=1}^n\\left( \\frac{-1}{2\\sigma ^2}\\sum _{j=1}^p u_j^2\\mathbf {w}_j^T\\mathbb {E}_{q(\\mathbf {y}_i)}[\\mathbf {y}_i\\mathbf {y}_i^T]\\mathbf {w}_j + \\frac{1}{\\sigma ^2}\\sum _{j=1}^px_{i,j} u_j \\mathbf {w}_j^T\\mathbb {E}_{q(\\mathbf {y}_i)}[\\mathbf {y}_i] \\right)-\\frac{\\alpha ^2}{2}\\sum _{i=1}^p||\\mathbf {w}_i||_2^2,$ and $\\ln q^*(\\mathbf {W}) =\\sum _{i=1}^p\\left( \\frac{-1}{2\\sigma ^2}\\sum _{j=1}^p u_j^2\\mathbf {w}_j^T\\mathbb {E}_{q(\\mathbf {y}_i)}[\\mathbf {y}_i\\mathbf {y}_i^T]\\mathbf {w}_j + \\frac{1}{\\sigma ^2}\\sum _{j=1}^px_{i,j} u_j \\mathbf {w}_j^T\\mathbb {E}_{q(\\mathbf {y}_i)}[\\mathbf {y}_i] \\right)-\\frac{\\alpha ^2}{2}\\sum _{i=1}^p||\\mathbf {w}_i||_2^2,$ which leads to the factorization $q^*(\\mathbf {W})=\\prod _{j \\le p} q^*(\\mathbf {w}_i)$ and to the desired expression.",
"By definition, we have $-\\mathcal {F}_q(\\mathbf {x_1},...\\mathbf {x_n}|\\theta )=\\mathbb {E}_q[\\ln p(\\mathbf {X},\\mathbf {Y},\\mathbf {W}|\\theta )]+H(q),$ therefore $-\\mathcal {F}_q(\\mathbf {x_1},...\\mathbf {x_n}|\\theta )=-np\\ln \\sigma - \\frac{1}{2\\sigma ^2}\\textup {Tr}(\\mathbf {X}^T\\mathbf {X})- \\frac{1}{2\\sigma ^2}\\sum _{i=1}^n\\mathbb {E}_q[\\mathbf {y}_i\\mathbf {W}^T\\mathbf {U}^2\\mathbf {W}\\mathbf {y}_i]+ \\frac{1}{\\sigma ^2}\\sum _{i=1}^n\\mathbf {x}_i^T\\mathbf {UM}\\mu _i \\\\ + \\sum _{k=1}^p\\left(d \\ln \\alpha -\\frac{\\alpha ^2}{2}\\mathbb {E}_q[\\mathbf {w}_k^T\\mathbf {w}_k] \\right)- \\frac{1}{2}\\sum _{i=1}^n \\mathbb {E}_q[\\mathbf {y}_i^T\\mathbf {y}_i] +\\frac{n}{2}\\ln |\\mathbf {\\Sigma }|+\\frac{1}{2}\\sum _{k=1}^p\\ln |\\mathbf {S}_k|,$ and computing the expectations leads to $-\\mathcal {F}_q(\\mathbf {x_1},...\\mathbf {x_n}|\\theta )=-np\\ln \\sigma + dp \\ln \\alpha - \\frac{1}{2\\sigma ^2}\\textup {Tr}(\\mathbf {X}^T\\mathbf {X})- \\frac{1}{2\\sigma ^2}\\sum _{i=1}^n\\sum _{k=1}^pu_k^2\\textup {Tr}[(\\Sigma +\\mu _i\\mu _i^T)(S_k+\\mathbf {m}_k\\mathbf {m}_k^T)]\\\\+ \\frac{1}{\\sigma ^2}\\sum _{i=1}^n\\mathbf {x}_i^T\\mathbf {UM}\\mu _i + \\sum _{k=1}^p -\\frac{\\alpha ^2}{2}\\textup {Tr}(\\mathbf {S}_k+\\mathbf {m}_k\\mathbf {m}_k^T)-\\frac{1}{2}\\sum _{i=1}^n \\textup {Tr}(\\Sigma +\\mu _i\\mu _i^T) +\\frac{n}{2}\\ln |\\mathbf {\\Sigma }|+\\frac{1}{2}\\sum _{k=1}^p\\ln |\\mathbf {S}_k|,$ which allows us to conclude.",
"The microarray data set used in this paper is included in the breastCancerVDX R package [60] and contains the gene expression data published by [72] and [45].",
"It contains expression levels of 22283 probes for 344 patients.",
"In order to be able to provide an interpretation of feature selection, we reduced the data from probe-level to gene-level using the following procedure: first, the probes with no gene identifier were discarded then, the data was aggregated to gene-level using the collapseRows R function of [43], among the genes obtained, only the genes listed in the Reactome database [14] were kept in order to eventually perform pathway enrichment, finally, the data was centered but not standardized.",
"The resulting data matrix contains 5391 variables (genes) and 344 observations (patients)."
]
] | 1605.05918 |
[
[
"On two questions by Finch and Jones about perfect order subset groups"
],
[
"Abstract A finite group G is said to be a POS-group if the number of elements of every order occurring in G divides |G|.",
"We answer two questions by Finch and Jones by providing an infinite family of nonabelian POS-groups with orders not divisible by 3."
],
[
"verbose,tmargin=0.8in,bmargin=0.8in,lmargin=0.8in,rmargin=0.8in December 9, 2022 A finite group $G$ is said to be a POS-group if the number of elements of every order occurring in $G$ divides $|G|$ .",
"We answer two questions by Finch and Jones in [2] by providing an infinite family of nonabelian POS-groups with orders not divisible by 3.",
"Let $G$ be a finite group, and define the order subset of an element $x \\in G$ to be $\\lbrace g \\in G \\mid o(g)=o(x)\\rbrace $ , where $o(x)$ denotes the order of $x$ .",
"We say that $G$ has perfect order subsets if the number of elements in every order subset divides $|G|$ ; in this case, we say that $G$ is a POS-group.",
"It is easy to see that $\\mathbb {Z}_2$ , $\\mathbb {Z}_4$ , and the symmetric group $S_3$ are POS-groups, whereas $\\mathbb {Z}_3$ , $\\mathbb {Z}_5$ , and $S_4$ are not.",
"This definition is due to Finch and Jones, who worked with abelian groups [2] and direct products of abelian groups with $S_3$ [3].",
"They provided the following open questions at the end of [2].",
"Are there nonabelian POS-groups other than the symmetric group $S_3$ ?",
"If the order of a POS-group is not a power of 2, is the order necessarily divisible by 3?",
"This is also Conjecture 5.1 from [3], also by Finch and Jones.",
"Are there nonabelian POS-groups other than the symmetric group $S_3$ ?",
"If the order of a POS-group is not a power of 2, is the order necessarily divisible by 3?",
"This is also Conjecture 5.1 from [3], also by Finch and Jones.",
"The answers are “yes\" and “no,\" respectively.",
"The first question was answered by Finch and Jones in [3], although all groups were direct products of $S_3$ with an abelian group.",
"Das [1] answered both questions by proving that there exists an action $\\theta $ such that the semidirect product $\\mathbb {Z}_{p^{k}} \\rtimes _{\\theta } \\mathbb {Z}_{2^{l}}$ is a POS-group, where $p$ is a Fermat prime, $k \\ge 1$ , and $2^{l} \\ge p-1$ .",
"Feit also answered both questions by indicating that a Frobenius group of order $p(p-1)$ for a prime $p>3$ is a POS-group [3].",
"We now provide an infinite family of groups that simultaneously answers both questions.",
"Let $n \\ge 1$ , and consider the group $\\mathbb {Z}_4 \\rtimes \\mathbb {Z}_{2 \\cdot 5^{n}}$ with the inversion action.",
"The order of this group is $2^3 \\cdot 5^{n}$ , which is not divisible by 3.",
"Consequently, $S_3$ cannot appear as a subgroup or quotient of any of these groups, as 3 divides the order of $S_3$ .",
"Table REF summarizes the easy calculations required to find the size of each order subset of $\\mathbb {Z}_4 \\rtimes \\mathbb {Z}_{2 \\cdot 5^{n}}$ (one can use geometric sums to verify that all elements of the group are accounted for).",
"Note that the number of elements of each order divides the order of the group, thereby proving that the groups are POS-groups.",
"This confirms that $\\mathbb {Z}_4 \\rtimes \\mathbb {Z}_{2 \\cdot 5^{n}}$ is a POS-group.",
"Table: Here, nn is defined so that n≥1n \\ge 1 and mm is defined so that 1≤m≤n1 \\le m \\le n."
]
] | 1605.05994 |
[
[
"Multi-epoch Spectroscopy of Dwarf Galaxies with AGN Signatures:\n Identifying Sources with Persistent Broad H-alpha Emission"
],
[
"Abstract We use time-domain optical spectroscopy to distinguish between broad emission lines powered by accreting black holes (BHs) or stellar processes (i.e., supernovae) for 16 galaxies identified as AGN candidates by Reines \\etal (2013).",
"Our study is primarily focused on those objects with narrow emission-line ratios dominated by star formation.",
"Based on follow-up spectra taken with the Magellan Echellette Spectrograph (MagE), the Dual Imaging Spectrograph, and the Ohio State Multi-Object Spectrograph, we find that the broad H$\\alpha$ emission has faded or was ambiguous for all of the star-forming objects (14/16) over baselines ranging from 5 to 14 years.",
"For the two objects in our follow-up sample with narrow-line AGN signatures (RGG 9 and RGG 119), we find persistent broad H$\\alpha$ emission consistent with an AGN origin.",
"Additionally, we use our MagE observations to measure stellar velocity dispersions for 15 objects in the Reines et al.",
"(2013) sample, all with narrow-line ratios indicating the presence of an AGN.",
"Stellar masses range from $\\sim5\\times10^{8}$ to $3\\times10^{9}$~\\msun, and we measure $\\sigma_{\\ast}$ ranging from $28-71~{\\rm km~s^{-1}}$.",
"These $\\sigma_{\\ast}$ correspond to some of the lowest-mass galaxies with optical signatures of AGN activity.",
"We show that RGG 119, the one object which has both a measured $\\sigma_{\\ast}$ and persistent broad H$\\alpha$ emission, falls near the extrapolation of the $\\rm M_{BH}-\\sigma_{\\star}$ relation to the low-mass end."
],
[
"Introduction",
"While massive black holes (BHs; here defined as $\\rm M_{BH} \\gtrsim 10^{4-5}$ $M_{\\odot }$ ) are ubiquitous in the centers of galaxies of Milky Way mass and larger [48], little is known about the occurrence rate and properties of BHs in dwarf galaxies (i.e., galaxies with $\\rm M_{\\star } \\lesssim 10^{9.5}$ $M_{\\odot }$ ).",
"Studying the population of BHs in dwarf galaxies is useful for understanding BH formation and growth in the early universe.",
"Having had relatively quiet merger histories, their BHs may have masses similar to their “birth\" mass (e.g., [71], [6]).",
"Clues to the formation of BH seeds may reside in the well studied relation between BH mass and stellar velocity dispersion (see e.g., [16], [22], [30]).",
"Using semi-analytic modeling, [72] find that, while the relationship between BH mass and velocity dispersion (or the $\\rm M_{BH}-\\sigma $ relation) is in place by $z\\sim 4$ for massive halos, systems with lower mass BHs ($\\rm M_{BH} < 10^{6}~M_{\\odot }$ ) evolve onto it at later times.",
"If BH seeds form “light\" ($\\sim 10^{2}M_{\\odot }$ ; the remnants of Population III stars, see e.g., [47], [1], [46]), then BHs begin as undermassive with respect to the local $\\rm M_{BH}-\\sigma _{\\star }$ relation, and will evolve towards it from below.",
"Alternatively, if BH seeds form “heavy\" ($10^{4-5}M_{\\odot }$ ; through direct collapse of gas clouds, e.g., [5], [45]), then BHs are initially overmassive and systems evolve rightwards onto the relation (i.e., towards higher velocity dispersions) as the host galaxies grow through mergers.",
"The smallest $z~=~0$ halos ($\\sigma _{\\star } \\sim 20-50~\\,{\\rm {km\\, s^{-1}}}$ ) have not yet undergone major mergers, and thus may be present-day outliers on the local $\\rm M_{BH}-\\sigma _{\\star }$ relation.",
"Finding and weighing relatively low-mass BHs is difficult due to the small gravitational sphere of influence of the BH, which is typically $\\lesssim 1$ parsec for $\\rm M_{BH}\\sim 10^{5}$ $M_{\\odot }$ .",
"Thus, in order to identify BHs in dwarf galaxies outside the Local Group, it is necessary to look for signs of BH activity (i.e., active galactic nuclei, or AGN).",
"Large-scale surveys such as the Sloan Digital Sky Survey (SDSS) have made it possible to search for signatures of AGN in samples comprised of tens of thousands of galaxies.",
"Early work by [25] (see also [28]), identified galaxies in the SDSS with low-mass AGN by searching for broad H$\\alpha $ emission, which can be produced by virialized gas near the central BH.",
"These studies identified $\\sim 200$ low-mass AGN candidates with a median BH mass of $\\sim 10^{6}$ $M_{\\odot }$ .",
"More recent studies have pushed the search for low-mass AGN into the dwarf galaxy regime.",
"X-ray and radio observations led to the discovery of AGN in the dwarf starburst galaxy Henize 2-10 [60], [57], and in the dwarf galaxy pair Mrk 709 [59].",
"Optical spectroscopic observations revealed the well-studied AGN in NGC 4395 [19], [18] and POX 52 [4], and more recently led to the discovery of a BH with a mass in the range of $27,000-62,000$ $M_{\\odot }$ in the dwarf galaxy RGG 118 [2].",
"Additionally, multiwavelength studies using large samples of bona fide dwarf galaxies have produced a collective sample of hundreds of AGN candidates in dwarf galaxies ([58], [52], [41], [64], [54]).",
"[58] (hereafter R13) searched for dwarf galaxies with spectroscopic signatures of AGN activity based on their narrow and/or broad emission lines.",
"Out of $\\sim 25,000$ nearby ($z\\le 0.055$ ; D $\\lesssim 250$ Mpc) dwarf galaxies in the SDSS, they identified 136 galaxies with narrow-line ratios indicating photoionization at least partly due to an AGN based on the BPT (Baldwin, Phillips & Terlevich 1981) diagram, i.e., they were classified as either AGN or AGN/star-forming composites (see also [37]; [36]; [38]).",
"A fraction of these also had broad H$\\alpha $ emission.",
"Additionally, R13 identified 15 galaxies with broad H$\\alpha $ emission in their SDSS spectroscopy (FWHM ranging from $\\sim 600-3700~\\,{\\rm {km\\, s^{-1}}}$ ), but with narrow-line ratios that placed them in the star forming region of the BPT diagram.",
"We are particularly concerned with the origin of the broad emission in these systems.",
"Broad Balmer emission with FWHM up to several thousand kilometers per second can be produced by stellar processes such as Type II supernovae [17], [55], luminous blue variables [66], or Wolf-Rayet stars.",
"Follow-up spectroscopy, taken over a sufficiently long baseline, can help distinguish between an AGN and one of the aforementioned transient stellar phenomena.",
"Here, we present follow-up optical spectroscopic observations of 14 of the 15 BPT star forming galaxies with broad H$\\alpha $ presented in R13, as well as observations of ten BPT AGN (one with broad H$\\alpha $ ) and three BPT composites (one with broad H$\\alpha $ ).",
"Follow-up observations for RGG 118 – the current record holder for the lowest-mass BH in a galaxy nucleus – were presented separately in [2].",
"The goals of this work are two-fold.",
"First, for our targets which were identified by R13 to have broad H$\\alpha $ emission (broader than narrow lines such as [SII]$\\lambda \\lambda $ 6713,6731) with FWHM $>500\\,{\\rm {km\\, s^{-1}}}$ , we analyze their spectra and determine whether the broad H$\\alpha $ emission is still present and consistent with previous observations.",
"Second, for galaxies with sufficiently high-resolution follow-up spectroscopy, we measure stellar velocity dispersions.",
"In Section 2, we describe our observations and data reduction procedures.",
"In Section 3, we discuss our emission line fitting analysis and report stellar velocity dispersion measurements for objects with sufficiently high spectral resolution data.",
"In Section 4, we present results from the follow-up spectroscopy for the broad line objects."
],
[
"Observations and Data Reduction",
"The sample analyzed here is comprised of 27 total targets.",
"Of these, 16 comprise our “SDSS broad H$\\alpha $ sample\" or galaxies identified to have broad H$\\alpha $ emission in their SDSS spectroscopy.",
"In the SDSS broad H$\\alpha $ sample, 14 objects are BPT star forming, 1 is a BPT composite and 1 is a BPT AGN.",
"Additionally, we have follow-up observations for 11 objects (9 BPT AGN and 2 BPT composites) that did not have broad H$\\alpha $ emission in their SDSS spectroscopy.",
"In addition to the SDSS observations, each galaxy has one or two follow-up spectroscopic observations taken between 5 and 14 years after the SDSS spectrum.",
"Below, we outline our observations and data reduction procedures for each instrument used.",
"See Table REF for a summary of our sample and observations.",
"All spectra were corrected for Galactic extinction.",
"Clay/MagE.",
"We observed 16 targets (including RGG 118) with the Magellan Echellette Spectrograph (MagE; [49]).",
"MagE is a moderate-resolution (R=4100) spectrograph on the 6.5m Clay Telescope at Las Campanas Observatory.",
"Data were collected on the nights of 2013 April 18-19 using a 1 slit.",
"Spectral coverage spans roughly from 3200 $\\rm {Å}$ to 10000 $\\rm {Å}$ across 15 orders.",
"Seeing was measured to be 0.5$^{\\prime \\prime }$ -1.2 over the two nights.",
"Total exposure times per object ranged from 1800 to 4800s.",
"A thorium argon arc lamp was observed for wavelength calibration, and the flux standard $\\rm \\theta $ Crt was observed for flux calibration.",
"Flux calibrated reference spectra were obtained from the European Southern Observatory library of spectrophotometric standards.",
"Flat fielding, sky subtraction, extraction, wavelength calibration, and flux calibration were performed with the mage_reduce pipeline written by George Becker.The mage_reduce reduction pipeline is available for download at: ftp://ftp.ociw.edu/pub/gdb/mage_reduce/mage_reduce.tar.gz One-dimensional spectra were extracted with a 3 aperture (the larger signal within 3was useful for determining stellar velocity dispersions; see Section 3.2).",
"APO/DIS.",
"Eight targets were observed with the Dual Imaging Spectrograph (DIShttp://www.astro.princeton.edu/$\\sim $ rhl/dis.html) on the 3.5m Astrophysical Research Consortium telescope at Apache Point Observatory.",
"Observations were taken using a 1.5 slit.",
"DIS consists of a red channel and a blue channel, with a transition wavelength of $\\rm \\sim 5350Å$ .",
"We used the B1200 and R1200 gratings (R=1200), which give linear dispersions of 0.62 $\\rm Å$ /pix for the blue channel and 0.58 $\\rm Å$ /pix for the red channel.",
"DIS targets were observed on the nights of 2013 February 9, 2013 February 16, and 2013 March 15.",
"Each object had three exposures with individual exposure times of 1200s.",
"A Helium-Neon-Argon lamp was observed for wavelength calibration, and standard stars Feige 34 and Feige 66 were observed for flux calibration.",
"The flux calibrated reference spectra were obtained from the Hubble Space Telescope CALSPEC Calibration Database.",
"Reductions for the two-dimensional images, as well as extraction of the one-dimensional spectra and wavelength calibration were done using standard longslit reduction procedures in IRAFIRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Associate of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.",
"following Massey et al.",
"1992 and Massey 1997.",
"MDM/OSMOS.",
"We observed ten targets with the Ohio State Multi-Object Spectrograph (OSMOS; [50]) on the MDM Observatory 2.4 Hiltner Telescope on 2 February 2014 and 20-23 January 2015.",
"For each target, we used a $1.2^{\\prime \\prime }$ by 20$^{\\prime }$ slit, with a VPH grism ($R=1600$ , or 0.7 $\\rm Å$ / pixel), which covered an observed wavelength range of $3900 - 6900$ Å .",
"The seeing was measured to be 0.8$^{\\prime \\prime }$ -1.1$^{\\prime \\prime }$ for the February 2 observation, and between 0.8$^{\\prime \\prime }$ -1.8$^{\\prime \\prime }$ for the January 20-23 observations.",
"Each object was observed with multiple exposures of 1800s.",
"For wavelength calibration, we observed an Argon arc lamp, and for flux calibration, we observed the spectrophotometric standards Feige 34 and Feige 66.",
"The data were reduced following standard IRAF routines from the longslit package, including bias and dark subtraction, flat fielding, wavelength calibration, and telluric and background subtraction.",
"One-dimensional spectra were then extracted using a $\\sim 3^{\\prime \\prime }$ aperture along the slit, and corrected for heliocentric velocity.",
"SDSS.",
"For purposes of comparison, we also make use of the SDSS observations of the galaxies in our sample.",
"The objects were originally selected by R13 by analyzing SDSS spectroscopy of $\\sim 25,000$ dwarf emission line galaxies in the NASA-Sloan Atlas http://www.nsatlas.org."
],
[
"Emission line modeling",
"For each galaxy, we modeled the H$\\alpha $ -[NII] complex in order to ascertain whether broad H$\\alpha $ emission was present.",
"We summarize the general procedure here, though more thorough discussions can be found in R13 and [2].",
"In this work, we are mainly interested in whether a broad H$\\alpha $ emission line is present.",
"Thus, we simply fit the continuum as a line across the region surrounding the H$\\alpha $ -[NII] complex.",
"This also ensures that we do not artificially introduce a broad line by over-subtracting an absorption feature.",
"We then create a model for the narrow-line emission using an intrinsically narrow-line (e.g., [SII]$\\lambda \\lambda $ 6713,6731; forbidden lines are guaranteed not to be produced in the dense broad line region gas).",
"We use the narrow-line model to fit the narrow H$\\alpha $ emission and the [NII] lines simultaneously.",
"The width of the narrow H$\\alpha $ line is allowed to increase by up to 25%.",
"The relative amplitudes of the [NII] lines are fixed to their laboratory values.",
"We next add an additional Gaussian component to the model to represent broad H$\\alpha $ emission.",
"If the preferred FWHM for the component is $> 500\\,{\\rm {km\\, s^{-1}}}$ and the $\\chi ^{2}_{\\nu }$ is improved by at least 20%, then we classify the galaxy as having persistent broad H$\\alpha $ emission ([31]; R13).",
"As noted in R13, this threshold is somewhat arbitrary, but found to be empirically suitable for this work.",
"If the broad model is preferred, we attempt to add an additional broad component, effectively allowing the broad emission to be modeled with up to two Gaussians.",
"Once we have our best-fit model (i.e., adding additional components no longer improves the $\\chi ^{2}_{\\nu }$ by our threshold value), we measure the FWHM and luminosity of broad H$\\alpha $ , if present.",
"We model and measure emission line fluxes for H$\\beta $ and [OIII] $\\lambda 5007$ .",
"Below, we discuss the narrow-line models used for each instrument.",
"MagE: For the MagE spectra, we follow R13 and model the narrow-line emission by fitting Gaussian profiles to the [SII] $\\lambda \\lambda $ 6713,6731 doublet.",
"In our modeling, the two lines are required to have the same width.",
"OSMOS: The wavelength coverage of the OSMOS spectra does not extend to the [SII] doublet, so here we model the narrow emission with the [OIII] $\\lambda $ 5007 line.",
"The profile of [OIII] often has a core component and a wing component [32], [14], [75], [26], [53], [79].",
"We therefore fit the [OIII] line with two components if the addition of a second component improves the $\\chi ^{2}_{\\nu }$ by at least 20%.",
"If [OIII] is indeed best fit with both a core and wing component, then we use the width of the core component for our narrow-line model.",
"If it is instead best fit with a single component, we retain the width of the single component for our narrow-line model.",
"In order to determine whether the use of [OIII] as our narrow line would affect our fits to the H$\\alpha $ -[NII] complex, we compared the profiles of [SII] and [OIII] in the SDSS spectra for our OSMOS targets.",
"We find that, for a given object, the FWHMs of the [OIII] and [SII] lines differ by very little, with percent differences in FWHM ranging from 0.3 to 11%.",
"Since we allow the width of the H$\\alpha $ narrow line to increase by up to 25%, we do not expect this choice of narrow line to affect our results.",
"DIS: For the DIS spectra, we also use the [SII] lines to model the narrow-line emission.",
"However, the emission lines in the DIS spectra display some additional instrumental broadening, which we observe in the red channel arc lamp spectra (see Figure REF ).",
"Since this instrumental broadening is not visible in low S/N [SII] lines but can still affect the fit to H$\\alpha $ , we created a basic model for the narrow-line emission to use in fitting the [SII] lines.",
"Using the galaxy spectrum for which the [SII] lines had the most flux, we generated a narrow-line model consisting of two Gaussians with fixed relative amplitudes and widths.",
"This basic shape was then used to model [SII] lines in all other DIS galaxy spectra, and correspondingly, their narrow H$\\alpha $ and [NII] lines.",
"Figure REF shows the [SII] lines from a representative DIS spectrum.",
"Figure: Red channel He-Ne-Ar arc lamp spectrum taken as part of our observations with the Dual-Imaging Spectrograph on the ARC 3.5m telescope.",
"Note that the shape of the instrumental broadening has two components – a narrow core and a broad base.Figure: [SII]λλ6716,6731\\lambda \\lambda 6716,6731 lines from the DIS spectrum of RGG C (NSA 109990).",
"The black line shows the observed spectrum, the red line is our best fit to the two lines, and the solid blue lines show the various components corresponding to our best fit of the two [SII] lines.",
"We constrain the relative heights and widths of the “narrow\" and “broad\" components and use that model to fit the narrow-lines in the Hα\\alpha -[NII] complex for DIS observations.Given the instrumental broad component seen in the emission lines, we tested the degree to which the narrow-line model affects the presence/properties of broad H$\\alpha $ emission to ensure that our results were not influenced by this broadening.",
"To do this, we created two additional narrow-line models: one where the instrumental broadening is less prevalent (i.e., we reduced the relative width and relative amplitude of the “broad\" narrow-line component by 10%), and one where it is more prevalent (i.e., the relative width and amplitude are increased by 10%).",
"We then refit the spectra and assessed the security of our broad H$\\alpha $ detections.",
"We found this to have little effect on the FWHM and luminosity of broad H$\\alpha $ for secure detections (i.e., FWHM varied by less than 10%), but to greatly influence these measured quantities for more ambiguous detections (with the FWHM varying by as much as 90% for a slightly different narrow-line model).",
"We err on the side of caution in our classifications; objects which have ambiguous broad H$\\alpha $ detections in their DIS spectra are classified as ambiguous."
],
[
"Stellar velocity dispersions",
"We measured stellar velocity dispersions for 15 galaxies with MagE observations using pPXF (Penalized Pixel Fitting; [11]).",
"As a reminder, of these, 9 are classified as BPT AGN, 3 are BPT composites, and 3 are BPT star forming galaxies.",
"Four galaxies (1 composite and 3 star forming) overlap with our SDSS broad H$\\alpha $ sample.",
"PPXF uses a library of stellar templates to fit the stellar continuum and kinematics of a galaxy spectrum.",
"We used a library of 51 stars from the ELODIE spectral database [56].",
"Our library consists of stars covering spectral classes from O through M and luminosity classes from bright giants to dwarfs (see Appendix Table REF for a list of stars and spectral types).",
"The ELODIE spectra span from 3900Å to 6800Å in wavelength and have a spectral resolution of R=10,000.",
"We measured stellar velocity dispersions in region of the spectrum surrounding the Mg b triplet, spanning from 5100Å to 5250Å(see Figure REF ).",
"We fit each region using combinations of low-order multiplicative and additive polynomials (orders range from 1 to 4; see e.g., [77], [76]) , and report the mean value for all measurements.",
"Uncertainties are reported by PPXF for each individual fit (i.e.",
"each combination of multiplicative and additive polynomial).",
"We take the mean for all velocity dispersion measurements the Mg b band and add the errors in quadrature to obtain a velocity dispersion measurement and corresponding error estimate.",
"We measure $\\sigma _{\\star }$ values ranging from $28-71 \\,{\\rm {km\\, s^{-1}}}$ with a median $\\sigma _{\\star }$ of $41 \\,{\\rm {km\\, s^{-1}}}$ .",
"These values correspond to galaxies with NASA-Sloan Atlas stellar masses ranging from $\\sim 5\\times 10^{8}$ to $3\\times 10^{9}$ $M_{\\odot }$ .",
"All stellar velocity dispersion measurements are presented in Table REF .",
"Figure: Spectral region of the MagE spectrum of RGG 119 used to measure the stellar velocity dispersion in pPXF.",
"We use a region of the spectrum encompassing the Mg bb triplet.",
"The observed spectrum is shown in gray, while the shaded red region represents the range of outputs from pPXF for the chosen combinations of additive and multiplicative polynomials.Table: Stellar velocity dispersion measurements"
],
[
"Broad line AGN candidates: Results",
"We show the locations of all 16 SDSS broad H$\\alpha $ targets on the BPT diagram in Figure REF .",
"Based on our follow-up spectroscopy, we classify our targets as having SDSS broad H$\\alpha $ emission that is: transient (broad H$\\alpha $ is not present in follow-up spectra), persistent (broad H$\\alpha $ is present in follow-up spectroscopy and consistent with prior observations), or ambiguous (state of broad emission is unclear).",
"In the following subsections, we describe our classifications in more detail.",
"Table REF summarizes the results of this analysis.",
"Figures showing the fits to H$\\beta $ , [OIII], and H$\\alpha $ are given for each observation in the Appendix.",
"See Section 6 for a discussion on aperture effects and intrinsic variability as they relate to this work."
],
[
"Transient broad H$\\alpha $",
"Out of the 16 objects with SDSS broad H$\\alpha $ , we find 11 to have transient broad H$\\alpha $ emission, i.e., they lacked broad emission in their followup spectroscopic observations (see Figure REF for an example).",
"All 11 fall in the “HII\" region of the BPT diagram, which indicates the presence of recent star formation in the galaxy.",
"We note that several objects (e.g., RGG G, RGG F; see Appendix) had additional components in their H$\\alpha $ line models with FWHMs of a few hundred $\\,{\\rm {km\\, s^{-1}}}$ ; since these are all narrower than the broad emission in the SDSS spectra and fall below our FWHM cut of $500\\,{\\rm {km\\, s^{-1}}}$ , we do not classify those objects as having persistent broad emission.",
"Given the narrow-emission line ratios and host galaxy properties, we consider Type II supernovae (SNe) to be the most likely origin for the transient broad H$\\alpha $ (see Section 5 for more details).",
"We note that, as discussed in detail in R13, luminous blue variables [66] and Wolf-Rayet stars can also produce transient broad H$\\alpha $ emission.",
"However, Wolf-Rayet stars typically produce other notable spectral features, such as the Wolf-Rayet bump from $\\lambda $ 4650-5690$\\rm Å$ .",
"Moreover, their spectra don't typically have strong hydrogen lines [12], [13].",
"Finally, recent work has revealed the existence of “changing look\" quasars, i.e., quasars with broad Balmer emission lines that fade significantly over the course of 5-10 years (see e.g., [63], [62]).",
"In the case of changing look quasar SDSS J101152.98+544206.4, the broad H$\\alpha $ luminosity dropped by a factor of 55 on this timescale.",
"This behavior is suspected to be caused by a sudden drop in accretion rate.",
"Figure: The observed Hα\\alpha -[NII] complex and corresponding best-fit profile for the SDSS, DIS, and OSMOS spectra of RGG C (NSA 109990).",
"In each, the dark gray line represents the observed spectrum.",
"The narrow emission line fit is plotted in green, and the blue shows the fit to broad Hα\\alpha emission.",
"The overall best-fit model is given in red, and the light gray line below the observed spectrum shows the residual between the observed spectrum and the best fit, offset by an arbitrary amount.",
"This galaxy is classified as having transient broad Hα\\alpha emission."
],
[
"Persistent broad H$\\alpha $",
"We identified 2/16 objects with secure, persistent broad H$\\alpha $ emission.",
"RGG 9RGG denotes we are using the ID assigned by Reines, Greene, & Geha (2013).",
"(NSA 10779), is classified as an AGN on the BPT diagram based on its SDSS spectrum.",
"We observe broad H$\\alpha $ emission in both the SDSS spectrum and the DIS spectrum, taken 13 years apart (see Figure REF for the best fit to the DIS spectrum).",
"The second secure broad line object, RGG 119 (NSA 79874), is classified as a composite object on the BPT diagram.",
"This object has broad H$\\alpha $ emission in the SDSS, MagE, and DIS spectra, spanning five years (see Figure REF ).",
"We compute the mass of the central BH using standard virial techniques which employ only the FWHM and luminosity of the broad H$\\alpha $ emission [27], [7], [8].",
"The calculation also includes a scale factor $\\epsilon $ intended to account for the unknown geometry of the broad line region; we adopt $\\epsilon =1$ (see Equation 5 in R13).",
"This method makes use of multiple scaling relations between parameters (e.g., the relation between FWHM$_{\\rm H\\alpha }$ and FWHM$_{\\rm H\\beta }$ ), each of which has its own intrinsic scatter, giving a systematic uncertainty on the measured BH mass of 0.42 dex [2].",
"To ensure that our choice of continuum subtraction method would not affect our BH mass estimates, we measured BH masses for the SDSS spectra using both R13 and our continuum subtraction, and found the masses to be consistent with one another.",
"For RGG 9, we measure a BH mass from the SDSS observation of $\\rm 3.5(\\pm 0.7)\\times 10^{5}~M_{\\odot }$ .",
"From the DIS observation, we measure a BH mass of $\\rm 3.7(\\pm 0.3)\\times 10^{5}~M_{\\odot }$ .",
"Using these two measurements, we compute a mean mass of $\\rm M_{BH} = 3.6(\\pm 0.8)\\times 10^{5}M_{\\odot }$ .",
"Taking into account the systematic uncertainty of 0.42 dex, this gives us a final estimate of $\\rm M_{BH}=3.6^{+5.9}_{-2.3}\\times 10^{5}M_{\\odot }$ .",
"Using the stellar mass of RGG 9 from R13 ($\\rm M_{\\star }=2.3\\times 10^{9}~M_{\\odot }$ ), we compute a BH mass-to-stellar mass ratio of $\\rm M_{BH}/M_{\\star } = 1.6\\times 10^{-4}$ .",
"Figure: The observed Hα\\alpha -[NII] complex and corresponding best-fit profile for the SDSS and DIS spectra of RGG 9.",
"The dark gray line represents the observed spectrum.",
"The narrow emission line fit is plotted in green, and the blue shows the fit to broad Hα\\alpha emission.",
"The overall best-fit model is given in red, and the light gray line below the observed spectrum shows the residual between the observed spectrum and the best fit, offset by an arbitrary amount.",
"We classify this galaxy as having persistent broad Hα\\alpha .For RGG 119, we measure a BH mass of $\\rm 2.1(\\pm 0.5)\\times 10^{5}~M_{\\odot }$ from the SDSS spectroscopy.",
"The MagE observation yields a BH mass of $\\rm 2.9(\\pm 0.2)\\times 10^{5}~M_{\\odot }$ , and the DIS observation gives a BH mass of $\\rm 3.6(\\pm 0.3)\\times 10^{5}~M_{\\odot }$ .",
"We compute a mean BH mass of $\\rm M_{BH} = 2.9(\\pm 0.6)\\times 10^{5}M_{\\odot }$ using the three spectroscopic observations.",
"With the additional systematic uncertainty, our final BH mass estimate is $\\rm M_{BH} = 2.9^{+4.9}_{-1.8}\\times 10^{5}M_{\\odot }$ .",
"R13 reports a stellar mass of $\\rm M_{\\star } = 2.1\\times 10^{9}M_{\\odot }$ for RGG 119, giving a BH mass to stellar mass ratio of $\\rm M_{BH}/M_{\\star } = 1.3\\times 10^{-4}$ .",
"See Table REF for the measured broad H$\\alpha $ FWHM and luminosities, and corresponding BH masses for each observation.",
"We note that our measured BH masses are consistent with those measured in R13 ($2.5\\times 10^{5}M_{\\odot }$ and $5.0\\times 10^{5}M_{\\odot }$ for RGG 9 and RGG 119, respectively).",
"We also refer the reader to Section 6 for a discussion of the potential effect of instrumentation/aperture on the measured properties of the broad H$\\alpha $ emission.",
"Figure: The observed Hα\\alpha -[NII] complex and corresponding best-fit profile for the SDSS, MagE, and DIS spectra of RGG 119 (NSA 79874).",
"In each, the dark gray line represents the observed spectrum.",
"The narrow emission line fit is plotted in green, and the blue shows the fit to broad Hα\\alpha emission.",
"The overall best-fit model is given in red, and the light gray line below the observed spectrum shows the residual between the observed spectrum and the best fit, offset by an arbitrary amount.",
"We classify this galaxy as having persistent broad Hα\\alpha .Using the MagE spectrum of RGG 119 we measure a stellar velocity dispersion of $28 \\pm 6~\\,{\\rm {km\\, s^{-1}}}$ from the Mg b triplet, and place this galaxy on the M$_{BH}$ -$\\sigma _{\\star }$ relation (Figure REF ).",
"RGG 119 sits close to the extrapolation of M$_{BH}$ -$\\sigma _{\\star }$ to low BH masses, similar to well-studied low-mass AGN NGC 4395 [18] and Pox 52 [4].",
"Our BH-to-galaxy stellar mass ratios are also consistent with other low-mass AGNs [61].",
"Table: Broad Hα\\alpha parameters for galaxies with persistent broad Hα\\alpha .Figure: The relation between black hole mass and stellar velocity dispersion.",
"The M BH -σ ☆ \\rm M_{BH}-\\sigma _{\\star } relations determined in , , and are plotted.",
"We also show points for systems with dynamical BH mass measurements as compiled by (pseudobulges are shown as gray squares, while classical bulges are plotted as gray circles).",
"The black hole mass and stellar velocity dispersion for RGG 119 (NSA 79874; shown as a purple circle) were measured in this work.",
"Also plotted are the low-mass AGN RGG 118 , NGC 4395 , , and Pox 52 , and the dwarf elliptical M32 , ."
],
[
"Ambiguous galaxies",
"In three cases, we were unable to definitively state whether broad H$\\alpha $ emission was still present and thus classify these objects as ambiguous.",
"All three ambiguous galaxies fall in the HII region of the BPT diagram.",
"RGG B (NSA 15952) was observed to have broad H$\\alpha $ emission in its SDSS spectrum and OSMOS spectrum.",
"Moreover, if we fit the broad H$\\alpha $ emission from each spectrum with one Gaussian component, we get FWHM$_{\\rm H\\alpha , SDSS} = 1036 \\pm 69 \\,{\\rm {km\\, s^{-1}}}$ , and FWHM$_{\\rm H\\alpha , OSMOS} = 977 \\pm 32 \\,{\\rm {km\\, s^{-1}}}$ (i.e., the broad components are consistent with one another).",
"However, the DIS spectrum – taken in between the SDSS and OSMOS spectra – does not require a broad component according to our criteria.",
"We cannot discount that we do not detect a broad component in the DIS spectrum due to the atypical narrow-line shapes.",
"Nevertheless, additional follow-up spectroscopy of RGG B would be useful for determining the true nature of the SDSS broad H$\\alpha $ emission.",
"We note that RGG B is of particular interest since its position on the BPT diagram is consistent with a low-metallicity AGN [33].",
"Line ratios involving [NII] are affected by the metallicity of the AGN host galaxy [29] and AGN hosts with sub-solar metallicities can migrate left-ward into the star forming region of the BPT diagram.",
"RGG I (NSA 112250) has broad emission detected in the SDSS, DIS, and OSMOS spectra, but the FWHM measured for the DIS spectrum is highly dependent on the narrow-line model used, i.e., changing the parameters of the narrow-line model changes the measured FWHM by up to $\\sim 90\\%$ .",
"The FWHM is also not consistent between the three spectra ($1237 \\pm 69 \\,{\\rm {km\\, s^{-1}}}$ , $316 \\pm 27 \\,{\\rm {km\\, s^{-1}}}$ , and $1717 \\pm 32 \\,{\\rm {km\\, s^{-1}}}$ for SDSS, DIS, and OSMOS, respectively).",
"Finally, RGG K (NSA 91579) has a highly asymmetric broad line observed in the SDSS and OSMOS spectra.",
"However, the DIS spectrum, taken in between the SDSS and OSMOS observations, does not require a broad component.",
"We do observe asymmetric H$\\alpha $ emission by eye in the all three spectra.",
"Similarly to with RGG B, we cannot rule out that there is indeed persistent broad H$\\alpha $ emission, and that the DIS spectrum does not require it because of issues relating to the fitting of the unusually shaped narrow lines.",
"Additional observations of all three ambiguous broad H$\\alpha $ targets will be necessary to determine whether these objects do indeed host AGN.",
"See the Appendix for figures showing fits for all observations of the ambiguous objects."
],
[
"Type II supernovae in dwarf galaxies",
"We consider Type II supernovae the most-likely explanation for the fading broad H$\\alpha $ emission observed in 11/16 objects in our SDSS broad H$\\alpha $ sample.",
"In this section, we further examine our SNe II candidates, and discuss this hypothesis in more detail.",
"All 11 objects with transient broad H$\\alpha $ have narrow line ratios consistent with recent star formation.",
"Moreover, their broad emission properties and galaxy properties are offset from the remainder of the SDSS broad line objects in two important ways.",
"As seen in Figure REF , SDSS broad emission that we found to be transient tended to have larger FWHM and broad lines that were more offset from the H$\\alpha $ line center in velocity space.",
"All SDSS broad H$\\alpha $ emission lines with FWHM $\\gtrsim 2000\\,{\\rm {km\\, s^{-1}}}$ in our sample were found to be transient, as well as all those with absolute velocity offsets of more than $\\sim 200\\,{\\rm {km\\, s^{-1}}}$ .",
"Figure: Velocity offset from the center of the Hα\\alpha emission line in km s -1 \\,{\\rm {km\\, s^{-1}}} versus the FWHM of the broad Hα\\alpha emission from R13.",
"A red dot indicates the broad line is red-shifted in velocity space, blue indicates the broad line is blue-shifted.",
"The dots with circles around them are objects with transient broad Hα\\alpha (i.e., likely SNe II hosts).",
"Large FWHM and/or velocity shifts are more characteristic of galaxies with transient broad Hα\\alpha emission.Additionally, galaxies with transient broad H$\\alpha $ also tend to be bluer with respect to the sample of broad and narrow-line candidate AGN host galaxies from R13 (Figure REF ).",
"The narrow-line AGN candidates were found to have a median galaxy color of $g-r = 0.51$ (R13), while the transient broad H$\\alpha $ galaxies have a median host galaxy color of $g-r = 0.22$ (with all transient broad H$\\alpha $ galaxies having $g-r<0.4$ ).",
"This suggests the transient broad H$\\alpha $ galaxies have, in general, younger stellar populations more likely to produce SNe II.",
"Figure: Galaxy g-rg-r color versus host galaxy stellar mass for narrow and broad line AGN candidates from R13.",
"Stellar masses and g-rg-r colors are from the NASA-Sloan Atlas database.",
"Objects with transient broad Hα\\alpha are plotted as the larger blue circles.",
"These galaxies tend to have bluer colors with respect to the rest of the sample.",
"We also include SNe II hosts identified in the SDSS , in galaxies below our mass cut-off.",
"The masses and colors for these also come from the NASA-Sloan Atlas database (except for the lowest mass SDSS SN II host, for which the mass/color are derived from the MPA-JHU Galspec pipeline).Using time-domain spectroscopy, we have empirically identified targets most likely to be SNe based on the disappearance of broad emission lines over long ($\\sim 5-10$ year) time scales.",
"A complementary analysis was performed by Graur et al.",
"(2013, 2015).",
"They searched through most SDSS galaxy spectra for SN-like spectra, and used SN template matching, rather than broad emission line fitting, to identify potential SNe.",
"Here, we compare the results of the two methods.",
"Similar to Graur et al.",
"(2013, 2015), R13 did identify 9 SN candidates (in a sample of $\\sim 25,000$ objects) directly from the single-epoch SDSS spectra (Table 5 in R13; none of those have been followed up in this work).",
"Those objects were identified because they exhibited P Cygni profiles in H$\\alpha $ .",
"Of the 15 BPT star forming galaxies with broad H$\\alpha $ identified in R13 as AGN candidates (14 of which are considered in this paper), one was detected by Graur et al.",
"(2015) and classified as a SN II (RGG M, identified in Table 1 of Graur et al.",
"2015 as SDSS J131503.77+223522.7 or 2651-54507-488).",
"The remainder did not meet all of the detection criteria set by Graur et al.",
"(2013, 2015) and thus were not detected by Graur et al.",
"(2015).",
"The Graur et al.",
"(2013, 2015) method is superior at detecting bona fide SNe II near maximum light, when these objects display prominent P-Cygni profiles.",
"The nature of the objects considered in this paper is generally more ambiguous, as no P-Cygni profiles are seen and the emission line profiles are relatively symmetric (although generally broader and more asymmetric than the bona fide AGN; Figure REF ).",
"During their survey, Graur et al.",
"(2015) also identified five ambiguous broad-line objects in dwarf galaxies, which their pipeline classified as either SNe or AGNs.",
"Of these, three were not in the R13 parent sample due to the stellar mass cut in two cases and a glitch in the NSA catalog in the third.",
"Two are in R13; one was identified as a transient [40], and the other (RGG J) is included in this work.",
"To understand why the majority of the transient sources presented here were not identified by Graur et al.",
"(2013, 2015), and to try and learn something about their nature, we pass our continuum-subtracted SDSS spectra through the same supernova identification scheme used by Graur et al.",
"First, the stellar continuum and narrow emission line fits are removed.",
"We employ a different procedure from that used by Graur et al.",
"to fit the continuum, but the overall result should be similar.",
"Second, we de-redshift the spectra, remove their continua, and manually ran them through the Supernova Identification code (SNID;http://people.lam.fr/blondin.stephane/software/snid/ [10]).",
"Four objects were classified as old SNe II, i.e., when their spectra are dominated by a broad H$\\alpha $ feature, but with relatively low rlap values.",
"The SNID `rlap' parameter is used to assess the quality of the fit, and is equal to the height-to-noise ratio of the normalized correlation function times the overlap between the input and template spectra in $\\rm {ln}~\\lambda $ space (see [10] for details).",
"A value of rlap=5 is the default minimum and fits with rlap values close to 5 are regarded as suspect (see [10] for details).",
"Of the four objects identified as SNe II, two had rlap values $<5$ and two had rlap $\\sim 7$ .",
"One object was classified as an old SN Ia at z $\\sim $ 0.1.",
"As the spectra were de-redshifted, this classification is not trustworthy.",
"The other three objects failed to be classified by SNID.",
"See Figure REF for an example of the fit obtained by SNID to RGG C. We will discuss the special case of RGG J in some detail in Section 5.1.",
"Figure: The Supernova Identification (SNID) code fit to the SDSS spectrum of RGG C. This fit had an rlap value of 7.1.",
"The observed spectrum is shown in black, while the SNID fit is shown in red.",
"SNID finds that the spectrum is best fit by a post-plateau SN II-P.Since we consider SNe II the best explanation for the transient broad emission, in the following we compute the expected number of SNe II in the R13 parent sample, and check for consistency with our results.",
"R13 identified and removed 9 galaxies with SNe II from their sample based on P Cygni profiles in the broad H$\\alpha $ emission.",
"Considering the 11 objects with transient broad H$\\alpha $ and three ambiguous objects identified in this work, we have $20-23$ SNe II candidates.",
"The Lick Observatory Supernova Search [44] calculated the rate of SNe II in the local universe as a function of color and Hubble Type by observing more than 10,000 galaxies over the course of 12 years [42].",
"Using representative SDSS gri magnitudes and $(g-r)$ and $(r-i)$ colors for the star forming galaxies from R13, we use filter transformations [34], [9] to determine a typical $(B-K)$ color in order to select the correct SN rate from [42].",
"Choosing a galaxy stellar mass of $1.2\\times 10^{9}$ $M_{\\odot }$ and using the equation for $(B-K) < 2.3$ , we find the expected rate of SNe II in a given galaxy over the course of 6 months is $0.002^{+0.003}_{-0.001}$ .",
"We then convert this to a probability using Poisson statistics.",
"We choose to compute the rate over 6 months since SNe II typically begin to fade at $100-150$ days post explosion (see e.g., [35]).",
"Similarly, Graur et al.",
"(2015) measured a mass-normalized SN II rate of $5.5^{+3.7}_{-2.4}~{\\rm statistical}~^{+1.2}_{-0.7}~{\\rm systematic~\\times 10^{-12}~yr^{-1}~M_\\odot ^{-1}}$ in galaxies with stellar masses of $0.08^{+0.06}_{-0.05}~{\\times 10^{10}~\\rm M_\\odot }$ (see their table 2).",
"Thus, for a galaxy of $10^9~{\\rm M_\\odot }$ , we would expect $0.0026^{+0.0025}_{-0.0016}$ SNe II in a period of six months, consistent with the rate derived from Li et al.",
"(2011).",
"R13 analyzed spectroscopy of $\\sim 25,000$ dwarf galaxies.",
"For a sample of this size, the rate derived from [42] suggests we can expect $50^{+75}_{-25}$ SNe II.",
"Using the rate from Graur et al.",
"(2015), we expect $65^{+63}_{-40}$ SNe II in a sample of 25,000 galaxies.",
"Both rates are roughly consistent with the number we detect in our sample.",
"Using either rate, we compute that in a sample of 25,000 galaxies, we expect less than 1 to have an observed SN signature in the original SDSS observation and a signature from a new SN in the follow-up observation."
],
[
"A Possible SN IIn in RGG J",
"In the following, we discuss the interesting case of RGG J.",
"This particular source, classified in this work as having transient broad H$\\alpha $ emission, has fooled many of us.",
"It was identified as a broad line AGN in [28], [68], and [15].",
"Thus, it serves as an excellent cautionary tale that there may well be transient sources lurking in our broad-line AGN samples even at higher mass.",
"We also have more epochs for this target than others.",
"It was observed twice by the SDSS, once by [78] (spectrum taken in 2008), and once by us in 2013.",
"The SDSS spectra were taken on 11 March 2003 and 05 April 2003 (see Figure REF ).",
"Thus, our observations span a full decade.",
"The SDSS flagged the second epoch as the “primary” spectrum, and so the fits in the literature are most likely made to that later epoch.",
"During their survey, Graur et al.",
"(2015) classified this source as either a SN IIn or an AGN based on a fit to the April 2003 spectrum.",
"By 2008, when [78] observed this source with the Echellette Spectrograph and Imager on Keck II, there was still a (rather marginal) detection of broad Ha.",
"The broad line is undetectable in our DIS spectrum, so we conclude the broad line may be produced by a SN IIn (Figure REF ).",
"Unlike SNe II-P and II-L, which exhibit broad H emission lines with P-Cygni profiles, Type IIn supernovae (SNe IIn) are characterized by narrow lines (FWHM $$ < $$ 200$ km/s; hence the `n^{\\prime } in `IIn^{\\prime }) with intermediate-width bases (FWHM $ 1000-2000$ km/s) and bluer continua (see \\cite {1997ARA&A..35..309F} for a review).",
"SNe IIn comprise $ 5 2$\\% of all SNe and show a wide spread of peak luminosity \\cite {2011MNRAS.412.1441L}.",
"The narrow lines are thought to be the result of strong interaction between the SN ejecta and a dense circumstellar medium.",
"Though some SNe IIn have been associated with luminous blue variables (e.g., SN 2005gl; \\cite {2009Natur.458..865G}, SN 2010jl; \\cite {2011ApJ...732...63S}, and SN 2009ip; \\cite {2011ApJ...732...32F}), the exact nature of their progenitors is still debated (see \\cite {2014ARA&A..52..487S} for a review).$ Figure: The Hα\\alpha regime of the four observed spectra of RGG J.",
"Broad Hα\\alpha is clearly visibile in the two SDSS spectra taken in 2003.",
"An Keck II ESI spectrum was taken in 2008 (; spectrum provided by Aaron Barth, private communication).",
"This spectrum shows some broad Hα\\alpha emission as well, though it is distinctly blue-shifted with respect to the narrow Hα\\alpha line center.",
"The DIS spectrum, taken in 2013 and analyzed in this work, does not show evidence for broad Hα\\alpha emission."
],
[
"Discussion",
"We analyzed follow-up spectroscopy of 27 dwarf galaxies with AGN signatures identified in a sample of $\\sim 25,000$ dwarf galaxies in the SDSS (see R13).",
"Of our follow-up targets, 16 were found by R13 to have broad H$\\alpha $ emission in their SDSS spectra, and 14 had narrow emission line ratios indicative of AGN activity (there is some overlap between the broad and narrow-line objects).",
"Of the 16 SDSS broad H$\\alpha $ objects for which we have follow-up spectroscopy, we found that the broad H$\\alpha $ emission faded for eleven, all of which fall in the star forming region of the BPT diagram.",
"As stated above, we consider SNe II to be the most likely cause of the original broad emission for these objects, though luminous blue variables and changing look quasars also produce transient broad H$\\alpha $ (see R13 for a more complete discussion of potential contaminants).",
"We also find two of the SDSS broad H$\\alpha $ objects, both of which had narrow-line AGN signatures, to have persistent broad H$\\alpha $ emission, suggesting it is emission from virialized gas around a central BH.",
"This also suggests that the broad emission for the remainder of the R13 narrow-line AGN/composite objects is due to an AGN.",
"R13 calculates the type 1 fraction (or the fraction with detectable broad H$\\alpha $ emission) for BPT AGN to be $\\sim 17\\%$ (6/35 objects).",
"The type 1 fraction for BPT composites is lower; including RGG 118 [2], $\\sim 5\\%$ of the composites (5/101) are type 1.",
"Finally, three SDSS broad H$\\alpha $ objects (all BPT star forming) are ambiguous and will require further observations to determine the source of broad emission.",
"In the case of RGG J, (the expected Type II SN; see Section 5.1), the broad emission was persistent over at least 5 years, but the FWHM and velocity shift varied considerably.",
"Thus, in addition to determining whether broad emission is persistent, we also check for consistency between epochs.",
"We note that differences in aperture could slightly affect broad emission measurements since the measured luminosity and FWHM of broad H$\\alpha $ depend on the fit to the narrow emission lines.",
"Additionally, the instruments used all have different spectral resolutions, which can affect the measured FWHM.",
"Finally, broad line regions can vary intrinsically on these timescales [69], making it difficult to determine the degree to which aperture effects play a role.",
"Nevertheless, for RGG 9, we find the two measured FWHM to be consistent with one another within the errors.",
"For RGG 119, the measured FWHM of H$\\alpha $ increases by $\\sim 13\\%$ over the course of our three observations (though the first and last measurements are consistent within the errors).",
"In summary, we were able to securely detect broad H$\\alpha $ arising from an AGN in the follow-up observations of two SDSS broad H$\\alpha $ galaxies; both of these objects also have narrow-line ratios which support the presence of an AGN.",
"Both objects (RGG 9 and RGG 119) have BH masses of a few times $\\rm 10^{5}~M_{\\odot }$ .",
"Chandra X-ray Observatory observations (Cycle 16, PI: Reines) will provide further valuable information about the accretion properties of these AGN.",
"Three SDSS broad line objects (all BPT star forming) are classified as ambiguous and will require additional observations to determine whether an AGN is present.",
"For the remainder of the star forming SDSS broad H$\\alpha $ dwarf galaxies, we find that transient stellar phenomena are likely to be the source of detected broad H$\\alpha $ emission.",
"Given the relatively low FWHMs, faint flux levels, and the prevalence of young stars, we confirm that, in dwarf galaxies (and perhaps more massive galaxies) broad H$\\alpha $ alone should not be taken as evidence for an AGN – another piece of evidence (such as narrow line ratios) is required (see also discussions in R13 and [52]).",
"We also measure stellar velocity dispersions for 12 galaxies with narrow-line AGN signatures.",
"One of these – RGG 119 – also has persistent broad H$\\alpha $ emission.",
"In this case, we use the broad emission to measure the BH mass and find that RGG 119 lies near the extrapolation of the M$_{\\rm BH}$ –$\\sigma _{\\star }$ relation to low BH masses.",
"The measured stellar velocity dispersions for these targets range from $28-71\\,{\\rm {km\\, s^{-1}}}$ with a median value of $41\\,{\\rm {km\\, s^{-1}}}$ .",
"With galaxy stellar masses of just a few times $10^{9}$ , these stellar velocity dispersions correspond to some of the lowest-mass galaxies with accreting BHs.",
"If the remaining narrow-line AGN/composites similarly follow the M$_{\\rm BH}$ –$\\sigma _{\\star }$ relation, their BH masses would range from $\\sim 6\\times 10^{4}-2\\times 10^{6}~M_{\\odot }$ (using the relation reported in [30]).",
"Acknowledgments VFB is supported by the National Science Foundation Graduate Research Fellowship Program grant DGE 1256260.",
"Support for AER was provided by NASA through Hubble Fellowship grant HST-HF2-51347.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.",
"RCH acknowledges support from a Alfred P. Sloan Research Fellowship and a Dartmouth Class of 1962 Faculty Fellowship.",
"This work makes use of: observations obtained at the MDM Observatory, operated by Dartmouth College, Columbia University, Ohio State University, Ohio University, and the University of Michigan; observations obtained with the Apache Point Observatory 3.5-meter telescope, which is owned and operated by the Astrophysical Research Consortium; data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile.",
"We thank Aaron Barth for providing us with the Keck ESI spectrum of RGG J."
]
] | 1605.05731 |
[
[
"High Energy Neutrinos from Recent Blazar Flares"
],
[
"Abstract The energy density of cosmic neutrinos measured by IceCube matches the one observed by Fermi in extragalactic photons that predominantly originate in blazars.",
"This has inspired attempts to match Fermi sources with IceCube neutrinos.",
"A spatial association combined with a coincidence in time with a flaring source may represent a smoking gun for the origin of the IceCube flux.",
"In June 2015, the Fermi Large Area Telescope observed an intense flare from blazar 3C 279 that exceeded the steady flux of the source by a factor of forty for the duration of a day.",
"We show that IceCube is likely to observe neutrinos, if indeed hadronic in origin, in data that are still blinded at this time.",
"We also discuss other opportunities for coincident observations that include a recent flare from blazar 1ES 1959+650 that previously produced an intriguing coincidence with AMANDA observations."
],
[
"Introduction",
"Since the discovery of cosmic neutrinos [1], IceCube has been accumulating observations of these events [2], [3], [4].",
"The higher statistics data reinforce the observation that the flux is predominantly extragalactic and reveal a flux of neutrinos with a total energy density that matches the one observed by Fermi in extragalactic gamma rays.",
"This has bolstered the speculation that blazars, which are responsible for the majority of Fermi photons, are the sources of cosmic neutrinos.",
"Blazars are a subclass of active Galactic nuclei (AGN) with collimated jets aligned with the line of sight of the observer.",
"With gamma-ray bursts, they have been widely speculated to be the sources of the highest energy cosmic rays and of accompanying neutrinos and gamma rays of pionic origin.",
"Recent studies with the Fermi Large Area Telescope (Fermi LAT) have shown that blazars are responsible for more than 85% of the extragalactic isotropic gamma-ray background (IGB) [5].",
"From that along with the fact that a gamma-ray flux from neutral pions, accompanying the flux of the charged pions responsible for the IceCube neutrinos, matches the Fermi flux, blazars emerge as a plausible source of cosmic neutrinos [6].",
"Also, recent studies have argued for a correlation between cosmic neutrinos and blazar catalogs [7].",
"Because blazars are flaring sources coincident in time as well in direction, they provide a powerful opportunity to make the case for such a connection, possibly with a single observation.",
"The recent association of the second highest energy neutrino event of 2 PeV with the blazar PKS B1424-418 provides an interesting hint in this context [8].",
"The blazar spectral energy distribution generically has two components with two peaks in the IR/X-ray and the MeV/TeV photon energy ranges.",
"The two components can typically be described in leptonic and hadronic scenarios where the acceleration of electrons and protons, respectively, are the origin of the high-energy photons.",
"In the leptonic scenario, synchrotron radiation by electrons is responsible for the first peak, and Inverse Compton scattering on electrons produces the second [9].",
"In hadronic models [10], both protons and electrons are accelerated.",
"Synchrotron radiation still produces the low-energy peak in the spectrum, while the high-energy MeV/TeV photons are the decay products of pions produced in $pp$ or $p\\gamma $ interactions in the jet.",
"In one model, the protons interact with the synchrotron photons, for instance.",
"The charged pions inevitably produced with neutral pions will be the parents of neutrinos that provide incontrovertible evidence of cosmic-ray acceleration in the source.",
"Therefore, the detection of high-energy neutrinos accompanying photons represents direct evidence for the hadronic model of blazars.",
"The multiwavelength association of blazars and neutrinos is greatly facilitated by the fact that their emission is highly variable on different timescales, from flares that last minutes to days to several months in a high state of radiation.",
"Also, it is easier to identify a point source in a transient search because of the lower background accumulated over the relatively short duration of the burst [11].",
"It is noteworthy that AMANDA detected three neutrino events in temporal coincidence with a rare orphan flare of blazar 1ES 1959+650 [12].",
"No attempt was made to evaluate a posteriori statistics for this event although its significance by any account exceeds that of the coincidences presently under discussion.",
"In June 2015, blazar 3C 279 underwent an intense flare observed by Fermi LAT.",
"The gamma-ray flux increased up to forty times over the steady flux and developed a relatively hard spectrum during the flare [13].",
"An in-depth study of the data revealed an even higher photon flux and a harder spectral index [14], [15].",
"An increase in X-ray emission was observed by SWIFT [16].",
"The event represents an extraordinary opportunity to investigate the pionic origin of the gamma rays by identifying temporally coincident cosmic neutrinos in IceCube.",
"IceCube data is routinely subjected to a blind analysis.",
"The data covering this event, and others discussed in this paper, have not been unblinded, which follows a yearly process.",
"In this paper, we investigate the prospects of observing muon neutrinos in coincidence with this and other flares of blazar 3C 279 by calculating the number of neutrino events based on estimates previously developed in connection with the 2002 burst of blazar 1ES 1959+650.",
"We also comment on a recent flare of this blazar.",
"Note that we focus on muon neutrinos, whose directions can be reconstructed with a resolution of $0.3^{\\circ }$ , allowing for statistically compelling coincidences that are unlikely to emerge with electron and tau neutrinos, which at present are only reconstructed to about $10^{\\circ }$ ."
],
[
"Neutrino flux from a pionic gamma-ray source",
"In the hadronic scenario, MeV-TeV gamma rays are produced from protons colliding with radiation or gas surrounding the object.",
"These collisions generate charged and neutral pions which decay, producing high-energy gamma rays and neutrinos.",
"Here, we follow estimates [17] that relate the neutrino flux to the observed gamma-ray flux using energy conservation: $\\int ^{E_{\\gamma }^{\\rm {max}}}_{E_{\\gamma }^{\\rm {min}}} E_{\\gamma } \\frac{dN_{\\gamma }}{dE_{\\gamma }} dE_{\\gamma } = K \\int ^{E_{\\nu }^{\\rm {max}}}_{E_{\\nu }^{\\rm {min}}} E_{\\nu } \\frac{dN_{\\nu }}{dE_{\\nu }} dE_{\\nu },$ where the factor $K=1(4)$ for $pp(p\\gamma )$ interactions.",
"Considering multipion interaction channels, $K$ is changed to approximately 2 in the case of $p\\gamma $ interactions.",
"The proton spectrum, resulting from Fermi acceleration, as well as the accompanying photon and neutrino spectra are expected to follow a power law spectrum with $\\alpha \\approx 2$ .",
"However, the observed photon spectrum steepens because the gamma rays are absorbed by propagation in the EBL and, possibly, also in the source.",
"Therefore, the observed gamma-ray spectrum is assumed to follow $\\frac{dN_{\\gamma }}{dE_{\\gamma }} = A_{\\gamma } E_{\\gamma }^{- \\alpha },$ with $\\alpha > 2$ .",
"In contrast to the photons, the neutrino spectrum is not modified by absorption.",
"Although one may propagate the observed gamma rays in the EBL to find the de-absorbed spectrum, it is in general not possible to match the neutrino spectrum to the gamma rays because of gamma rays cascading inside the source.",
"However, energy is conserved in the process, and the total energy between neutrinos and photons can still be related.",
"Even this will result in a lower limit on the neutrino flux because photons absorbed in the source are not accounted for in the left hand side of Eq.",
"REF .",
"Using Eq.",
"REF and assuming $E_{\\gamma , \\rm {max}} \\gg E_{\\gamma , \\rm {min}}$ , we obtain the following neutrino spectrum: $\\frac{dN_{\\nu }}{dE_{\\nu }} \\approx A_{\\nu } \\, E_{\\nu }^{-2} \\approx \\frac{A_{\\gamma } E_{\\gamma , \\rm {min}}^{-\\alpha +2}}{(\\alpha -2) K \\ln {(E_{\\nu ,\\rm {max}}/E_{\\nu ,\\rm {min}})}} \\, E_{\\nu }^{-2}.$ Here, $E_{\\gamma , \\rm {min}}$ is the minimum energy of photons reflecting the threshold energy of pion production in $pp$ interactions and for the production of the delta resonance in $p\\gamma $ interactions.",
"For $pp$ collisions, the minimum energy required for pion production is: $E_{p}^{\\rm {min}} = \\Gamma \\, \\frac{(2 m_p + m_{\\pi })^2 -2 m_p^2 }{2 m_p} \\simeq \\Gamma \\times 1.23 \\, \\rm {GeV},$ where $\\Gamma $ is the Lorentz factor of the jet relative to the observer.",
"Given that three pions are produced and that, on average, each charged pion produces four leptons and each neutral pion two photons, the relation between proton energy and gamma-ray and neutrino energies is: $E_{\\gamma }^{\\rm {min}}=\\frac{E_p^{\\rm {min}}}{6}, \\,\\,\\, E_{\\nu }^{\\rm {min}}=\\frac{E_p^{\\rm {min}}}{12}.$ For $p\\gamma $ collisions, the energy threshold is set by the delta resonance $p \\gamma \\rightarrow \\Delta \\rightarrow \\pi N$ : $E_{p}^{\\rm {min}} = \\Gamma ^2 \\, \\frac{m_{\\Delta }^2 - m_p^2}{4 E_{\\gamma }} \\simeq \\Gamma ^2 \\, \\bigg (\\frac{1 \\,\\rm {MeV}}{E_{\\gamma }}\\bigg ) \\times 160 \\,\\rm {GeV}.$ In this case, the gamma-ray and neutrino energies are related to the proton energy by $E_{\\gamma }^{\\rm {min}}=\\frac{E_p^{\\rm {min}} <x_{p \\rightarrow \\pi }> }{2}, \\,\\,\\, E_{\\nu }^{\\rm {min}}=\\frac{E_p^{\\rm {min}} <x_{p \\rightarrow \\pi }>}{4},$ where $<x_{p \\rightarrow \\pi }> \\simeq 0.2$ is the average fraction of proton's energy transferred to the pion."
],
[
"Neutrinos in coincidence with 3C 279 Flares",
"Blazar 3C 279 is a flat spectrum radio quasar (FSRQ) located at declination $-5.8 ^{\\circ }$ and right ascension $194^{\\circ }$ with a redshift of 0.536.",
"It is one of the brightest sources in the EGRET catalogue [18] and was the first FSRQ discovered at TeV energy by MAGIC in 2006 [19].",
"The MAGIC collaboration has reported a gamma-ray flux from 3C 279 of $5.2 \\times 10^{-10} \\, TeV^{-1}cm^{-2}s^{-1}$ with a spectral index of 4.1.",
"The EBL corrected spectrum has a smaller spectral index of 2.94.",
"Blazar 3C 279 has consistently exhibited rapid variations in flux, and multiple flares have been observed.",
"On June 16, 2015, Fermi observed an intense flare of GeV gamma rays from 3C 279 reaching forty times the steady flux of this source.",
"The spectral study of this flux found a relatively hard spectral index.",
"Specifically, we will use in our estimates the average daily photon flux of $24.3\\times 10^{-6}\\, ph\\,cm^{-2}\\,s^{-1}$ with a spectral index of 2.1.",
"The expected number of well-reconstructed muon neutrinos in IceCube is calculated using Eq.",
"REF and $N_{\\nu _\\mu +\\overline{\\nu }_\\mu } = t \\int \\frac{dN_\\nu }{dE_\\nu }\\,A_{eff}(E,\\theta )\\,dE$ where the effective area $A_{eff}$ is taken from [20].",
"The neutrino flux calculated from the energy balance relation of Eq.",
"REF depends on the value of $E_{\\nu ,\\rm {max}}/E_{\\nu ,\\rm {min}}$ , which represents the energy interval over which proton interactions produce pionic gamma rays.",
"We will consider three possible values for this ratio: Case 1: We assume that neutrinos are exclusively produced in a specific energy range.",
"This is similar to the approach in reference [8] where the neutrino spectrum is assumed to peak at PeV energies.",
"The number of events for various values of $E_{\\nu ,\\rm {max}}/E_{\\nu ,\\rm {min}}$ is shown in Fig.",
"REF (REF ) for $pp(p\\gamma )$ interactions.",
"For $pp$ collisions, the number of neutrinos expected is within IceCube's sensitivity for the wide range of values for the Lorentz factor and the neutrino energy range considered.",
"For $p\\gamma $ collisions, large values of $E_\\gamma /\\Gamma ^2$ are required for observing the flare in neutrinos.",
"Note that for $p\\gamma $ interactions we label the energy in terms of the ratio $E_\\gamma /\\Gamma ^2$ .",
"The actual energy depends on the value of $\\Gamma $ which is often not directly measured and must be obtained from further modeling of the spectrum [15], [21].",
"Case 2: We assume that neutrinos are produced over the same energy range as the gamma rays, i.e., $E_{\\nu , min}$ is obtained from Eq.",
"REF (REF ) for $pp(p\\gamma )$ collisions.",
"The resulting number of neutrino events is shown in Figs.",
"REF and REF for different values of maximum neutrino energy.",
"Here, the estimated number of events strongly depends on the maximum energy achieved by the cosmic accelerator.",
"Case 3: Finally, we consider different threshold energies for neutrinos, assuming that the maximum neutrino energy is 10 PeV, the highest energy observed by IceCube so far.",
"The results are shown in Fig.",
"REF (REF ) for $pp$ ($p\\gamma $ ) collisions.",
"Notice that a higher minimum neutrino energy corresponds to a larger number of events observed.",
"Figure: Estimated number of events from blazar 3C 279 flare in June 2015 for different energy ranges of neutrino emission from pppp collision.",
"The events correspond to neutrino energies above 1 TeV.Figure: Estimated number of events from blazar 3C 279 flare in June 2015 for different energy ranges of neutrino emission from pγp\\gamma collision.",
"The events correspond to neutrino energies above 1 TeV.Figure: Estimated number of events from blazar 3C 279 flare in June 2015 for different values of maximum neutrino energy in pppp collision when neutrinos and gamma rays are produced in the same energy range.",
"The events correspond to neutrino energies above 1 TeV.Figure: Estimated number of events from blazar 3C 279 flare in June 2015 for different values of maximum neutrino energy in pγp\\gamma collision when neutrinos and gamma rays are produced in the same energy range.",
"The events correspond to neutrino energies above 1 TeV.Figure: Estimated number of events from blazar 3C 279 flare in June 2015 for different minimum energy ranges of neutrinos in pppp collision, assuming that the maximum neutrino energy is 10 PeV.",
"The events correspond to neutrino energies above 1 TeV.Figure: Estimated number of events from blazar 3C 279 flare in June 2015 for different minimum energy ranges of neutrinos in pγp\\gamma collision, assuming that the maximum neutrino energy is 10 PeV.",
"The events correspond to neutrino energies above 1 TeV."
],
[
"Discussion and Conclusion",
"The above calculations illustrate that blazar flares represent an extraordinary opportunity to identify the origin of IceCube neutrinos.",
"The short time window results in a lower number of events but also a suppressed background.",
"For the specific burst of 3C 279, we have shown that there is a clear opportunity for observing coincident neutrinos, especially in the case of $pp$ interaction.",
"In addition to the intense flare in June 2015, two previous flares were observed during December 2013 and April 2014 [22].",
"Although their photon flux is not as large as for the flare discussed above, stacking them will result in a higher likelihood of finding neutrinos.",
"The average daily photon flux of all three flares is listed in Table REF .",
"Assuming, for simplicity, that the neutrino spectrum would follow the gamma-ray spectrum, the total number of events for a flat spectrum of neutrinos will be about 4(2) for $pp$ ($p\\gamma $ ) collisions.",
"Table: The date and average daily photon flux of observed flares from 3C 279.",
"All fluxes are measured above 100 MeV , .We have also estimated the number of events for each flare using detailed information of fluxes and assuming same spectral behavior for neutrinos and gamma rays.",
"Detailed duration and flux are listed in Table REF .",
"The total number of events obtained is 4(2) for $pp$ ($p\\gamma $ ) collisions.",
"Statistics are straightforward with less than 0.001 background of atmospheric events per day within the resolution of $0.3^\\circ $ .",
"Table: The date, photon flux, spectral index, and duration of observed flares from 3C 279.",
"All fluxes are measured above 100 MeV .The FSRQ 3C 279 is included in the IceCube source list for both time-dependent [23] and time-independent point source searches [20].",
"The latest time-dependent search looked for a correlation of neutrinos with observed flares up until 2012.",
"This period did not include any flares from 3C 279.",
"Future time-dependent studies may reveal signals from these flares.",
"Recalling the temporal coincidence observed in AMANDA with flares of blazar 1ES 1959+650, it is noteworthy that a new very high energy flare from 1ES 1959+650 was observed by VERITAS during October 2015 [24].",
"According to the preliminary analysis of the data, the flux has reached $\\sim $ $50\\%$ of the Crab flux with a spectral index of 2.5.",
"Detailed analysis will provide more information about the duration and spectrum of this flare.",
"Based on the preliminary results, and provided that the gamma rays are hadronic in origin, IceCube expects to observe $\\sim $ 0.1 events per hour for this burst.",
"If the burst has lasted for more than a day, then it is very likely that accompanying neutrinos would be observed in IceCube.",
"Its location is obscured by the earth, and the highest energy events will therefore be absorbed."
],
[
"Acknowledgments",
"We would like to thank Markus Ahlers, Keith Bechtol, Markus Ackermann, and Marcos Santander for useful discussions and their informative comments during this study.",
"This research was supported in part by the U.S. National Science Foundation under Grants No.",
"ANT-0937462 and PHY- 1306958 and by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation."
]
] | 1605.06119 |
[
[
"Isotropic Schur roots"
],
[
"Abstract In this paper, we study the isotropic Schur roots of an acyclic quiver $Q$ with $n$ vertices.",
"We study the perpendicular category $\\mathcal{A}(d)$ of a dimension vector $d$ and give a complete description of it when $d$ is an isotropic Schur $\\delta$.",
"This is done by using exceptional sequences and by defining a subcategory $\\mathcal{R}(Q,\\delta)$ attached to the pair $(Q,\\delta)$.",
"The latter category is always equivalent to the category of representations of a connected acyclic quiver $Q_{\\mathcal{R}}$ of tame type, having a unique isotropic Schur root, say $\\delta_{\\mathcal{R}}$.",
"The understanding of the simple objects in $\\mathcal{A}(\\delta)$ allows us to get a finite set of generators for the ring of semi-invariants SI$(Q,\\delta)$ of $Q$ of dimension vector $\\delta$.",
"The relations among these generators come from the representation theory of the category $\\mathcal{R}(Q,\\delta)$ and from a beautiful description of the cone of dimension vectors of $\\mathcal{A}(\\delta)$.",
"Indeed, we show that SI$(Q,\\delta)$ is isomorphic to the ring of semi-invariants SI$(Q_{\\mathcal{R}},\\delta_{\\mathcal{R}})$ to which we adjoin variables.",
"In particular, using a result of Skowro\\'nski and Weyman, the ring SI$(Q,\\delta)$ is a polynomial ring or a hypersurface.",
"Finally, we provide an algorithm for finding all isotropic Schur roots of $Q$.",
"This is done by an action of the braid group $B_{n-1}$ on some exceptional sequences.",
"This action admits finitely many orbits, each such orbit corresponding to an isotropic Schur root of a tame full subquiver of $Q$."
],
[
"Introduction",
"Let $Q$ be an acyclic quiver with $n$ vertices and $k$ be an algebraically closed field.",
"One crucial tool in representation theory of acyclic quivers is the use of perpendicular categories.",
"If $V$ is a rigid representation of $Q$ , then the (left) perpendicular category $^\\perp V$ of $V$ is an exact extension-closed abelian subcategory of ${\\rm rep}(Q)$ , where ${\\rm rep}(Q)$ is the category of finite dimensional representations of $Q$ .",
"These perpendicular subcategories were first studied by Geigle and Lenzing in [9] and also by Schofield in [18].",
"One important fact is that such a subcategory has a projective generator, or equivalently, it is equivalent to the category of representations of some acyclic quiver.",
"There is a very natural way to generalize perpendicular categories of rigid representations, namely, by taking the perpendicular category $\\mathcal {A}(d)$ of a dimension vector $d$ ; see Section 3.",
"If $V$ is a rigid representation, then $^\\perp V = \\mathcal {A}(d_V)$ where $d_V$ is the dimension vector of $V$ .",
"If $d_V$ is not the dimension vector of a rigid representation, we show that the category $\\mathcal {A}(d)$ does not admit a projective generator.",
"However, this category plays a fundamental role for understanding the ring SI$(Q,d)$ of semi-invariants of $Q$ of dimension vector $d$ , the latter object being our second object of study in this paper.",
"Indeed, as a ring, SI$(Q,d)$ is generated by the Schofield semi-invariants $C^V$ (see [6]) where $V$ runs through the simple objects in $\\mathcal {A}(d)$ .",
"In general, $\\mathcal {A}(d)$ may have infinitely many non-isomorphic simple objects.",
"Since SI$(Q,d)$ is always finitely generated as a ring, we still need to decide how to pick a nice subset of those generators $C^V$ and find the relations among them.",
"There is no general method, so far, for doing this.",
"When we specialize to the case where $d=\\delta $ is an isotropic Schur root of $Q$ , then we can answer this problem.",
"It was proven in [20] that when $Q$ is of tame type and $d$ is arbitrary, the ring of semi-invariants ${\\rm SI}(Q, d)$ is either a polynomial ring or a hypersurface.",
"The smallest possible case of hypersurface occurs for the dimension vector $d=\\delta $ where $\\delta $ is the isotropic Schur root for $Q$ and where $Q$ is of type $\\widetilde{\\mathbb {E}_6}, \\widetilde{\\mathbb {E}_7},\\widetilde{\\mathbb {E}_8}$ or $\\widetilde{\\mathbb {D}_n}$ , for $n \\ge 4$ .",
"In such a case, the ring of semi-invariants is generated by the $C^V$ where $V$ runs through the quasi-simple exceptional regular representations of $Q$ , that is, the exceptional simple objects in $\\mathcal {A}(\\delta )$ .",
"The hypersurface equation in these cases comes from the fact that the subcategory of regular representations in ${\\rm rep}(Q)$ has exactly three non-homogeneous tubes.",
"If the regular exceptional quasi-simple representations in the first, second and third tubes are respectively denoted $E_1,\\ldots ,E_p$ , $E^{\\prime }_1,\\ldots , E^{\\prime }_q$ and $E^{\\prime \\prime }_1,\\ldots ,E^{\\prime \\prime }_r$ , then the hypersurface equation is given by $C^{E_1}\\cdots C^{E_p}+C^{E^{\\prime }_1}\\cdots C^{E^{\\prime }_q}+C^{E^{\\prime \\prime }_1}\\cdots C^{E^{\\prime \\prime }_r}=0.$ We investigate how much of this structure is preserved for the ring SI$(Q,\\delta )$ of an isotropic Schur root $\\delta $ of an arbitrary $Q$ .",
"We consider the cone of dimension vectors in $\\mathcal {A}(\\delta )$ .",
"Using convex geometry and Radon's theorem, we prove a certain decomposition property of the space of that cone (Proposition REF ).",
"Using this, we give a complete description of the simple objects in $\\mathcal {A}(\\delta )$ .",
"There exists an exact extension-closed abelian subcategory $\\mathcal {R}=\\mathcal {R}(Q,\\delta )$ of ${\\rm rep}(Q)$ which has a projective generator, connected and of tame type.",
"In particular, $\\mathcal {R}$ is equivalent to ${\\rm rep}(Q_{\\mathcal {R}})$ for a connected quiver of tame type $Q_{\\mathcal {R}}$ having a unique isotropic Schur root $\\delta _{\\mathcal {R}}$ , that can also be seen as an isotropic Schur root of $Q$ .",
"This subcategory $\\mathcal {R}$ is built from the data of an exceptional sequence $(M_{n-2}, \\ldots , M_1)$ of length $n-2$ of simple objects in $\\mathcal {A}(\\delta )$ .",
"Up to isomorphisms, the simple objects in $\\mathcal {A}(\\delta )$ are given by the objects $M_{n-2}, \\ldots , M_1$ together with the quasi-simple objects in $\\mathcal {R}$ .",
"In particular, there are finitely many, up to isomorphism, exceptional simple objects in $\\mathcal {A}(\\delta )$ .",
"Using this, we can get an explicit description of our ring of semi-invariants: SI$(Q,\\delta )$ is obtained by adjoining variables to the ring of semi-invariants ${\\rm SI}(Q_{\\mathcal {R}},\\delta _{\\mathcal {R}})$ .",
"In particular, ${\\rm SI}(Q, \\delta )$ is still a polynomial ring or a hypersurface.",
"The defining equation, in the hypersurface case, again comes from linear dependance of three products of semi-invariants $C^V$ where $V$ runs through the exceptional quasi-simple objects in $\\mathcal {R}(Q,\\delta )$ .",
"One difference between the general case and the case of quivers of tame type is that the isotropic Schur root $\\delta _{\\mathcal {R}}$ may differ from $\\delta $ .",
"Moreover, the root $\\delta _{\\mathcal {R}}$ needs not lie in the interior of the cone of dimension vectors of $\\mathcal {A}(\\delta )$ .",
"For the last part of this paper, we find an algorithm for finding all isotropic Schur roots of $Q$ .",
"We restrict to full exceptional sequences of ${\\rm rep}(Q)$ that are called exceptional of isotropic type: it is an exceptional sequence of the form $E=(X_1, \\ldots , X_{n-1}, X_n)$ where there is an integer $i$ such that the thick subcategory generated by $X_{i}, X_{i+1}$ contains an isotropic Schur root $\\delta _E$ , called the type of $E$ .",
"We explain how the braid group $B_{n-1}$ acts on these sequences to get all the isotropic Schur roots of $Q$ .",
"The action admits finitely many orbits, and each orbit contains an exceptional sequence whose type is an isotropic Schur root of a tame full subquiver of $Q$ ."
],
[
"Preliminaries",
"In this paper, $Q=(Q_0, Q_1)$ is always a connected acyclic quiver with $n$ vertices, unless otherwise indicated, and $k$ is an algebraically closed field.",
"We let ${\\rm rep}(Q)$ denote the category of ($k$ -linear) finite dimensional representations of $Q$ over $k$ ."
],
[
"Bilinear form, Schur roots",
"Given a representation $M$ , we denote by $d_M$ its dimension vector, which is an element in $(\\mathbb {Z}_{\\ge 0})^n$ .",
"We denote by $\\langle -,- \\rangle _Q$ , or simply by $\\langle -, - \\rangle $ when there is no risk of confusion, the Euler-Ringel form for $Q$ .",
"This is the unique $k$ -bilinear form in $\\mathbb {R}^n$ such that for $M,N \\in {\\rm rep}(Q)$ , we have $\\langle d_M, d_N \\rangle = {\\rm dim}_k {\\rm Hom}(M,N) - {\\rm dim}_k {\\rm Ext}^1(M,N).$ Recall that $M \\in {\\rm rep}(Q)$ is called a Schur representation if ${\\rm Hom}(M,M)$ is one dimensional.",
"It is well known from Kac [12] that in this case, $\\langle d_M, d_M \\rangle $ is at most one.",
"A Schur representation $M$ is called an exceptional representation if $\\langle d_M, d_M \\rangle = 1$ or, equivalently, ${\\rm Ext}^1(M,M)=0$ .",
"A dimension vector $d \\in (\\mathbb {Z}_{\\ge 0})^n$ is a Schur root if $d = d_M$ for some Schur representation $M$ .",
"We call such a $d$ real, isotropic or imaginary if $\\langle d,d\\rangle $ is one, zero or negative, respectively.",
"In this paper, $\\delta $ will always denote an isotropic Schur root of $Q$ ."
],
[
"Geometry and semi-invariants",
"For an arrow $\\alpha \\in Q_1$ , we denote by $t(\\alpha )$ its tail and by $h(\\alpha )$ its head.",
"We write ${\\rm Mat}_{u\\times v}(k)$ for the set of all $u\\times v$ matrices over $k$ .",
"For a dimension vector $d=(d_1, \\ldots , d_n)$ , we denote by ${\\rm rep}(Q,d)$ the space of representations of dimension vector $d$ with fixed vector spaces, that is, ${\\rm rep}(Q,d) = \\prod _{\\alpha \\in Q_1}{\\rm Mat}_{t(\\alpha )\\times h(\\alpha )}(k).$ This space is an affine space and the reductive group GL$_d(k):= \\prod _{i=1}^n{\\rm GL}_{d_i}(k)$ acts on it by simultaneous conjugation, so that for $M \\in {\\rm rep}(Q,d)$ , the GL$_d(k)$ -orbit of $M$ is the set of all representations in ${\\rm rep}(Q,d)$ that are isomorphic to $M$ .",
"Since $Q$ is acyclic, the ring of invariants $k[{\\rm rep}(Q,d)]^{{\\rm GL}_d(k)}$ is trivial.",
"Instead, one rather considers the ring of invariants $k[{\\rm rep}(Q,d)]^{{\\rm SL}_d(k)}$ , where ${\\rm SL}_d(k):= \\prod _{i=1}^n{\\rm SL}_{d_i}(k)\\subset {\\rm GL}_d(k).$ This ring, denoted SI$(Q,d)$ , is called the ring of semi-invariants of $Q$ of dimension vector $d$ .",
"It is always finitely generated, since ${\\rm SL}_{d}(k)$ is reductive.",
"Let $\\Gamma $ denote the group of all homomorphisms $\\mathbb {Z}^n \\rightarrow \\mathbb {Z}$ , which is identified to the set of multiplicative characters of GL$_d(k)$ .",
"A well known result states that SI$(Q,d)$ admits a weight space decomposition ${\\rm SI}(Q,d) = \\oplus _{\\tau \\in \\Gamma }{\\rm SI}(Q,d)_\\tau $ where ${\\rm SI}(Q,d)_\\tau $ is called the space of semi-invariants of weight $\\tau $ .",
"This makes ${\\rm SI}(Q,d)$ a $\\Gamma $ -graded ring.",
"If $V$ is a representation with $\\langle d_V, d \\rangle = 0$ and $f_V: P_1 \\rightarrow P_0$ denotes a fixed projective resolution of $V$ , then for $M \\in {\\rm rep}(Q,d)$ , we have a $k$ -linear map $d(V,M) = {\\rm Hom}(f_V, M)$ given by a square matrix.",
"We define $C^V(-)$ to be the polynomial function on ${\\rm rep}(Q,d)$ that takes a representation $M$ to the determinant of $d(V,M)$ .",
"If we change the projective resolution of $V$ , $C^V(-)$ only changes by a non-zero scalar.",
"This function $C^V(-)$ is a semi-invariant of weight $\\langle d_V, - \\rangle $ , and will be called determinantal semi-invariant; see [6], [18].",
"The following theorem was proven by Derksen and Weyman in [6] and also by Schofield and van den Bergh in [19] in the characteristic zero case.",
"Theorem 2.1 (Derksen-Weyman, Schofield-Van den Bergh) Let $d$ be a dimension vector.",
"Then the ring ${\\rm SI}(Q,d)$ is spanned over $k$ by the determinantal semi-invariants $C^V(-)$ where $\\langle d_V, d \\rangle = 0$ .",
"There is a dual way to define semi-invariants.",
"One can also take a representation $W$ with $\\langle d, d_W \\rangle = 0$ and $f_W: P_1^{\\prime } \\rightarrow P_0^{\\prime }$ a fixed projective resolution of $W$ .",
"For $M \\in {\\rm rep}(Q,d)$ , we have a $k$ -linear map $d(M,W) = {\\rm Hom}(M, f_W)$ given by a square matrix.",
"We define $C^-(W)$ the polynomial function on ${\\rm rep}(Q,d)$ that takes a representation $M$ to the determinant of $d(M,W)$ .",
"This function $C^-(W)$ is a semi-invariant of weight $-\\langle -, d_W\\rangle $ .",
"As in the above theorem, the ring ${\\rm SI}(Q,d)$ is spanned over $k$ by the semi-invariants $C^-(W)$ where $\\langle d, d_W \\rangle = 0$ ."
],
[
"Exceptional sequences, thick subcategories and braid groups",
"One of our main tools in this paper will be to make use of particular exceptional sequences.",
"Recall that a sequence $(X_1, \\ldots , X_r)$ of exceptional representations is an exceptional sequence if ${\\rm Hom}(X_i, X_j) = 0 = {\\rm Ext}^1(X_i, X_j)$ whenever $i < j$ .",
"It is full if $r = n$ .",
"For such an exceptional sequence $E$ , we denote by $E)$ the thick subcategory of ${\\rm rep}(Q)$ generated by the objects in $E$ .",
"Here thick means full, closed under extensions, under direct summands, under kernels of epimorphisms, and under cokernels of monomorphisms.",
"The following is well known.",
"We include a proof for the sake of completeness.",
"Proposition 2.2 A full subcategory $\\mathcal {A}$ of ${\\rm rep}(Q)$ is thick if and only if it is exact abelian and extension-closed.",
"The sufficiency is clear.",
"Suppose that $\\mathcal {A}$ is thick.",
"We only need to show that $\\mathcal {A}$ has kernels and cokernels.",
"We will prove this by showing that the kernel and cokernel in ${\\rm rep}(Q)$ of a morphism in $\\mathcal {A}$ lie in $\\mathcal {A}$ .",
"Let $f: M \\rightarrow N$ be a morphism in $\\mathcal {A}$ and let $u: K \\rightarrow M$ , $v: N \\rightarrow C$ and $g: M \\rightarrow E$ be the kernel, cokernel and coimage in ${\\rm rep}(Q)$ .",
"Since ${\\rm rep}(Q)$ is hereditary and since we have a monomorphism $g^{\\prime }: E \\cong {\\rm im}(f) \\rightarrow N$ , we have a surjective map ${\\rm Ext}^1(g^{\\prime },K): {\\rm Ext}^1(N,K) \\rightarrow {\\rm Ext}^1(E,K)$ .",
"The short exact sequence $0 \\rightarrow K \\rightarrow M \\rightarrow E \\rightarrow 0$ is an element in ${\\rm Ext}^1(E,K)$ and hence is the image of an element in ${\\rm Ext}^1(N,K)$ .",
"We have a pullback diagram ${0 [r] & K [r]^u @{=}[d] & M [r]^g [d] & E [r] [d]^{g^{\\prime }} & 0 \\\\0 [r] & K [r] & M^{\\prime } [r] & N [r] & 0}.$ This gives rise to a short exact sequence $0 \\rightarrow M \\rightarrow M^{\\prime } \\oplus E \\rightarrow N \\rightarrow 0.$ Since $M,N \\in \\mathcal {A}$ and $\\mathcal {A}$ is thick, we get $E \\in \\mathcal {A}$ .",
"Hence, $K,C \\in \\mathcal {A}$ .",
"Recall that if $E$ is an exceptional sequence of length $r \\le n$ , then $E)$ is equivalent to the category ${\\rm rep}(Q_E)$ of representations of an acyclic quiver $Q_E$ with $r$ vertices; see [18].",
"Denote by $S_1, \\ldots , S_r$ the non-isomorphic simple objects in $E)$ .",
"The Euler-Ringel form of ${\\rm rep}(Q)$ , restricted to the subgroup of $\\mathbb {Z}^n$ generated by the $d_{S_i}$ , is isometric to the Euler-Ringel form of ${\\rm rep}(Q_E)$ .",
"In other words, there is an equivalence $\\psi : {\\rm rep}(Q_E) \\rightarrow E)$ of categories such that for $X,Y \\in {\\rm rep}(Q_E)$ , we have $\\langle d_{\\psi (X)}, d_{\\psi (Y)} \\rangle _{Q} = \\langle d_X,d_Y \\rangle _{Q_E}.$ In this way, we will often identify a dimension vector for $Q_E$ to a dimension vector for $Q$ .",
"Moreover, using the above equivalence, a Schur root for $Q_E$ will be thought of as a Schur root for $Q$ .",
"Let $E = (X_1, \\ldots , X_r)$ be an exceptional sequence.",
"If $j > 1$ , we denote by $L_{X_{j-1}}(X_j)$ the reflection of $X_j$ to the left of $X_{j-1}$ .",
"This is the unique exceptional representation such that we have an exceptional sequence $(X_1, \\ldots , L_{X_{j-1}}(X_j), X_{j-1}, X_{j+1}, \\ldots , X_r).$ If $j<r$ , we denote by $R_{X_{j+1}}(X_j)$ the reflection of $X_j$ to the right of $X_{j+1}$ .",
"This is the unique exceptional representation such that we have an exceptional sequence $(X_1, \\ldots , X_{j-1}, X_{j+1}, R_{X_{j+1}}(X_j), X_{j+2}, \\ldots , X_r).$ These reflections actually have another interpretation in terms of the braid group.",
"Let $r = n$ .",
"For $1 \\le i \\le n-1$ , denote by $\\sigma _i$ the operation that takes an exceptional sequence $E$ and reflect the $(i+1)$ th object to the left of the $i$ th one.",
"Hence, $\\sigma _1, \\ldots \\sigma _{n-1}$ act on the set of all exceptional sequences.",
"It is not hard to check that for $1 \\le i \\le n-2$ and an exceptional sequence $E$ , we have $(\\sigma _i (\\sigma _{i+1} (\\sigma _iE))) = (\\sigma _{i+1} (\\sigma _{i} (\\sigma _{i+1}E)))$ and for $|i-j|\\ge 2$ , we have $(\\sigma _i(\\sigma _jE)) = (\\sigma _i(\\sigma _jE)).$ Therefore, the braid group $B_n := \\langle \\sigma _1, \\ldots , \\sigma _{n-1} \\mid \\sigma _i \\sigma _{i+1} \\sigma _i = \\sigma _{i+1} \\sigma _{i} \\sigma _{i+1}, \\sigma _i\\sigma _j = \\sigma _j\\sigma _i \\;\\; \\text{for}\\;|i-j|\\ge 2 \\rangle $ on $n$ strands acts on the set of all full exceptional sequences.",
"It is well known by a result of Crawley-Boevey [3] that this action is transitive, meaning that all exceptional sequences lie in a single orbit."
],
[
"Perpendicular subcategories, stable and semi-stable representations",
"In this section, we consider perpendicular subcategories of dimension vectors, study them and explain why these categories are related to semi-invariants of quivers.",
"Let $d$ be any dimension vector.",
"Consider $\\mathcal {A}(d):=\\lbrace X \\in {\\rm rep}(Q) \\mid {\\rm Hom}(X,M)=0={\\rm Ext}^1(X,M) \\; \\text{for some} \\; M \\in {\\rm rep}(Q,d)\\rbrace ,$ seen as a full subcategory.",
"This category is called the (left) perpendicular subcategory of $d$ .",
"Proposition 3.1 The subcategory $\\mathcal {A}(d)$ is a thick subcategory of ${\\rm rep}(Q)$ , hence exact extension-closed abelian.",
"It is clear that $\\mathcal {A}(d)$ is closed under direct summands.",
"Let $0 \\rightarrow X_1 \\rightarrow X_2 \\rightarrow X_3 \\rightarrow 0$ be a short exact sequence in ${\\rm rep}(Q)$ with exactly two terms $X_s, X_t$ in $\\mathcal {A}(d)$ .",
"There are $N_s, N_t \\in {\\rm rep}(Q,d)$ such that ${\\rm Hom}(X_i, N_i)=0 = {\\rm Ext}^1(X_i, N_i)$ whenever $i=s,t$ .",
"Now, for $i=s,t$ , there is an open set $\\mathcal {U}_i \\in {\\rm rep}(Q,d)$ such that for $Z_i \\in \\mathcal {U}_i$ , we have ${\\rm Hom}(X_i, Z_i)=0 = {\\rm Ext}^1(X_i, Z_i)$ .",
"Since $\\mathcal {U}_s \\cap \\mathcal {U}_t$ is non-empty, take $N$ a representation in $\\mathcal {U}_s \\cap \\mathcal {U}_t$ .",
"Applying ${\\rm Hom}(-,N)$ to the above exact sequence, we get that the third term lies in $\\mathcal {A}(d)$ .",
"Clearly, $\\mathcal {A}$ has a projective generator if and only if it is equivalent to a module category.",
"Since $\\mathcal {A}$ is a Hom-finite hereditary abelian category over an algebraically closed field, this means that $\\mathcal {A}$ has a projective generator if and only if it is equivalent to ${\\rm rep}(Q^{\\prime })$ for some finite acyclic quiver $Q^{\\prime }$ .",
"Lemma 3.2 Let $d$ be an imaginary or isotropic Schur root in ${\\rm rep}(Q)$ with $n=2$ .",
"Then $\\mathcal {A}(d)$ is not empty.",
"The result is trivial if $d$ is isotropic.",
"Therefore, assume that the quiver contains at least $m \\ge 3$ arrows and $d$ is imaginary.",
"Assume also that the vertices are $\\lbrace 1,2\\rbrace $ with 1 being the sink vertex.",
"It is sufficient to prove that the result holds for some dimension vector $f$ in the $\\tau $ -orbit of $d$ , where $\\tau $ denotes the Coxeter transformation.",
"Consider the cone $C = \\lbrace x \\in (\\mathbb {Z}_{\\ge 0})^2 \\mid \\langle x, x \\rangle < 0\\rbrace $ .",
"The Coxeter transformation is clearly $C$ -invariant.",
"Observe that $\\tau $ sends the vector $[m-1, 1]$ to $[1, m-1]$ .",
"Set $z = [m-1,1]$ and for $i \\in \\mathbb {Z}$ , set $z_i = \\tau ^iz$ .",
"We claim that for $i \\in \\mathbb {Z}$ , the cones $[z_i, z_{i+1}]$ , $[z_{i+1}, z_{i+2}]$ only intersect at the ray generated by $z_{i+1}$ .",
"Assume otherwise.",
"By continuity of $\\tau $ and since $\\tau z_i = z_{i+1}$ , there exists a ray in $[z_{i}, z_{i+1}]$ that is fixed by $\\tau $ .",
"Therefore, there is an eigenvector $v$ in $[z_{i}, z_{i+1}]$ .",
"Then $\\langle v, v \\rangle = - \\langle v, \\tau v \\rangle = -\\lambda \\langle v, v \\rangle $ where $\\lambda $ is the corresponding eigenvalue.",
"It is well known, see [16], that $\\tau $ has a real eigenvalue greater than 1 and a positive real eigenvalue less than one.",
"Therefore, $\\lambda \\ne -1$ meaning that $\\langle v, v \\rangle = 0$ so $v \\notin C$ , a contradiction.",
"This proves our claim.",
"This also proves that the two linearly independent eigenvectors of $\\tau $ lie on the two boundary rays of $C$ .",
"Now, the vectors $z_1, z_2, z_3, \\ldots $ converge to one such ray and $z_0, z_{-1}, z_{-2},\\ldots $ converge to the other ray.",
"Therefore, we see that the dimension vectors in the half open cone $[z_0, z_1)$ forms a fundamental domain in $C$ for the action of $\\tau $ .",
"Hence, we may assume that our dimension vector $d$ lies in $[z_0, z_1]$ .",
"Assume first that $d = [q,p]$ with $1 \\le p \\le q$ .",
"Since $d$ lies in $[z_0, z_1]$ , we have $1 \\le \\frac{q}{p} \\le m-1$ .",
"By [2], the Hilbert null-cone of ${\\rm rep}(Q,d)$ is not the entire space if $m > \\lceil \\frac{q}{p}\\rceil $ and $p > 1$ .",
"The first condition always holds since $d \\in [z_0, z_1]$ .",
"So if $p > 1$ , then there has to be a representation $M$ of dimension vector $d$ and a semi-invariant that does not vanish at $M$ .",
"By Theorem REF , this yields a representation $N$ with $C^N(M)\\ne 0$ , meaning that $N \\in \\mathcal {A}(d)$ .",
"Assume now that $p=1$ and take $M$ a general representation of dimension vector $d$ .",
"Consider a general representation $Z$ of dimension vector $[mq-1, q]$ .",
"By construction $\\langle d_Z, d_M \\rangle = 0$ .",
"Now, a proper subrepresentation of $M$ has dimension vector $d_i=[i,0]$ for $0 \\le i \\le q$ .",
"Notice that $\\langle d_Z, d_i \\rangle \\le 0$ .",
"Therefore, it follows from King's criterion [14] that $M$ is $\\langle d_Z, - \\rangle $ -semistable.",
"Therefore, there is a positive integer $r$ and a representation $Z^{\\prime }$ of dimension vector $rd_Z$ such that $C^{Z^{\\prime }}(M) \\ne 0$ , meaning that $Z^{\\prime } \\in \\mathcal {A}(d)$ .",
"Similarly, one can prove that $\\mathcal {A}(d)$ contains a non-zero representation if $d = [q,p]$ with $1 \\le q \\le p$ .",
"For two dimension vectors $d_1,d_2$ , let ${\\rm ext}(d_1,d_2)$ denote the minimal value of ${\\rm dim}_k {\\rm Ext}^1(M_1,M_2)$ where $(M_1,M_2) \\in {\\rm rep}(Q,d_1) \\times {\\rm rep}(Q,d_2)$ .",
"Similarly, we let $\\hom (d_1,d_2)$ denote the minimal value of ${\\rm dim}_k {\\rm Hom}(M_1,M_2)$ where $(M_1,M_2) \\in {\\rm rep}(Q,d_1) \\times {\\rm rep}(Q,d_2)$ .",
"There is an open subset $\\mathcal {U}_1$ of ${\\rm rep}(Q,d_1)$ and an open subset $\\mathcal {U}_2$ of ${\\rm rep}(Q,d_2)$ such that for $M_1 \\in \\mathcal {U}_1, M_2 \\in \\mathcal {U}_2$ , we have $\\langle d_1, d_2 \\rangle = {\\rm dim}_k {\\rm Hom}(M_1,M_2) -{\\rm dim}_k {\\rm Ext}^1(M_1,M_2) = \\hom (d_1,d_2) - {\\rm ext}(d_1,d_2).$ We write $d_1 \\bot d_2$ if $\\hom (d_1,d_2) = {\\rm ext}(d_1,d_2) = 0$ .",
"Observe that $d_1 \\bot d_2$ implies $\\langle d_1, d_2 \\rangle = 0$ .",
"A sequence $(d_1, \\ldots , d_r)$ of Schur roots with $d_i \\bot d_j$ whenever $i < j$ is called an orthogonal sequence of Schur roots.",
"Let $d$ be a dimension vector.",
"Due to results of Kac [13], there is a decomposition, denoted $d=\\alpha _1 \\oplus \\cdots \\oplus \\alpha _m$ , having the property that there exists an open (dense) subset $\\mathcal {U}_d$ of ${\\rm rep}(Q,d)$ such that for $M \\in \\mathcal {U}_d$ , we have a decomposition $M \\cong M_1 \\oplus \\cdots \\oplus M_m$ , where each $M_{i}$ is a Schur representation with $d_{M_{i}} = \\alpha _i$ .",
"Moreover, ${\\rm Ext}^1(M_i,M_j)=0$ when $i \\ne j$ .",
"The latter decomposition of $d$ is unique up to ordering, and is called the canonical decomposition of $d$ .",
"The dimension vectors $\\alpha _i$ are clearly Schur roots, however, they do not need be distinct.",
"Sometimes, it is more convenient to write the above decomposition as $(*) \\quad d=p_1\\beta _1 \\oplus \\cdots \\oplus p_r\\beta _r,$ where the $\\beta _i$ are pairwise distinct and $p_i$ is the number of $1 \\le j \\le m$ with $\\beta _i = \\alpha _j$ .",
"It follows from [17] that when $\\beta _i$ is imaginary, then $p_i=1$ .",
"When writing a canonical decomposition as in $(*)$ , we adopt the convention that when $\\alpha $ is an imaginary Schur root and $p$ is a positive integer, then $p\\alpha $ is just one root (not $p$ times the root $\\alpha $ as when $\\alpha $ is real or isotropic).",
"With this convention, Schofield has proven in [17] the following result.",
"Proposition 3.3 (Schofield) Let $d=p_1\\beta _1 \\oplus \\cdots \\oplus p_r\\beta _r$ be the canonical decomposition of $d$ .",
"If $p$ is a positive integer, then $pd=pp_1\\beta _1 \\oplus \\cdots \\oplus pp_r\\beta _r$ is the canonical decomposition of $pd$ , using the above convention for imaginary Schur roots.",
"The following generalizes Lemma REF .",
"Lemma 3.4 Let $d$ be be a dimension vector in ${\\rm rep}(Q)$ whose canonical decomposition involves an isotropic Schur root or an imaginary Schur root.",
"Then $\\mathcal {A}(d)$ is not empty.",
"We follow the algorithm in [4] to find the canonical decomposition of $d$ .",
"In particular, we apply this algorithm until the first step where an imaginary or isotropic Schur root is created.",
"In particular, there is an orthogonal sequence $(\\alpha _1, \\ldots , \\alpha _r)$ of real Schur roots with positive integers $p_1, \\ldots , p_r$ such that $d = \\sum _{i=1}^rp_i\\alpha _i$ .",
"Moreover, there is a pair $\\alpha _t, \\alpha _{t+1}$ with $\\langle \\alpha _{t+1}, \\alpha _t \\rangle < 0$ and $\\gamma = p_t\\alpha _t + p_{t+1}\\alpha _{t+1}$ is such that $\\langle \\gamma , \\gamma \\rangle \\le 0$ .",
"Assume first that $r<n$ .",
"Then we can find a real Schur root $\\alpha _0$ such that $\\alpha _0 \\bot \\alpha _j$ for $1 \\le j \\le r$ .",
"Then, an exceptional representation of dimension vector $\\alpha _0$ lies in $\\mathcal {A}(d)$ .",
"Therefore, we may assume that $r = n$ .",
"Let $\\gamma ^{\\prime } = \\gamma $ if $\\langle \\gamma , \\gamma \\rangle < 0$ and $\\gamma ^{\\prime }$ be the smallest indivisible dimension vector in the ray of $\\gamma $ , otherwise.",
"Set $p \\ge 1$ with $\\gamma = p \\gamma ^{\\prime }$ .",
"Observe that $\\gamma ^{\\prime }$ is a Schur root.",
"The next step of the algorithm replaces the sequence $(\\alpha _1, \\ldots , \\alpha _n)$ with positive integers $p_i$ by the orthogonal sequence $(\\alpha _1, \\ldots , \\alpha _{t-1}, \\gamma ^{\\prime }, \\alpha _{t+2},\\ldots ,\\alpha _n)$ of Schur roots with positive integers $p_1, \\ldots , p_{t-1},p,p_{t+2},\\ldots , p_n$ .",
"Now, $d = \\sum _{i=1}^{t-1}p_i\\alpha _i + p\\gamma ^{\\prime } + \\sum _{i=t+2}^np_i\\alpha _i.$ For $1 \\le i \\le n$ , let $M_i$ denote an exceptional representation of dimension vector $\\alpha _i$ .",
"The root $\\gamma ^{\\prime }$ is a root in $M_t, M_{t+1})$ .",
"Since $M_t, M_{t+1})$ is equivalent to the category of representations of a quiver with two vertices, it follows from Lemma REF that there is a dimension vector $\\nu $ in $M_t, M_{t+1})$ with $\\nu \\bot \\gamma ^{\\prime }$ .",
"Clearly, $\\langle \\nu , \\nu \\rangle < 0$ and hence $\\nu $ is an imaginary Schur root.",
"Let $Z$ be a general representation of dimension vector $\\nu $ .",
"Since $Z$ is in general position, we may assume that it lies in $M_j^\\perp $ for $1 \\le j \\le t-1$ and in $^\\perp M_j$ for $t+2 \\le j \\le n$ .",
"Therefore, we may assume it lies in $M_t, M_{t+1})$ .",
"Since $Z,M_{t-1}$ are in general position and ${\\rm Ext}^1(M_{t-1},Z)=0$ , either ${\\rm Hom}(Z,M_{t-1})=0$ or ${\\rm Ext}^1(Z,M_{t-1})=0$ by [17].",
"We construct a representation $Z^{\\prime }$ that lies in $M_j^\\perp $ for $1 \\le j \\le t-2$ , in $^\\perp M_j$ for $j = t-1$ and $t+2 \\le j \\le n$ and lies in $^\\perp N$ where $N$ is a general representation of dimension vector $\\gamma ^{\\prime }$ .",
"Assume first that ${\\rm Hom}(Z,M_{t-1})=0$ .",
"It follows from [4] that $\\nu ^{\\prime }=\\nu - \\langle \\nu , \\alpha _{t-1}\\rangle \\alpha _{t-1}$ is a Schur root with $\\hom (\\nu ^{\\prime }, \\alpha _{t-1})=0$ .",
"Moreover, since $\\langle \\nu ^{\\prime }, \\alpha _{t-1}\\rangle = 0$ , we also have ${\\rm ext}^1(\\nu ^{\\prime }, \\alpha _{t-1})=0$ .",
"Therefore, we have an orthogonal sequence $(\\alpha _1, \\ldots , \\alpha _{t-2}, \\nu ^{\\prime }, \\alpha _{t-1}, \\gamma ^{\\prime }, \\alpha _{t+2}, \\ldots , \\alpha _n)$ of Schur roots.",
"Consider $Z^{\\prime }$ a general representation of dimension vector $\\nu ^{\\prime }$ .",
"Then $Z^{\\prime }$ satisfies the above wanted conditions for $Z^{\\prime }$ .",
"Assume now that ${\\rm Ext}^1(Z,M_{t-1})=0$ .",
"It follows from Lemma 2.3 in [17] that the non-zero morphisms $Z \\rightarrow M_{t-1}$ are either all injective or all surjective.",
"Let $f = {\\rm dim}_k {\\rm Hom}(Z,M_{t-1})$ .",
"Assume first that all non-zero morphisms $Z \\rightarrow M_{t-1}$ are surjective.",
"We get an epimorphism $Z \\rightarrow M_{t-1}^f$ given by the basis elements of ${\\rm Hom}(Z,M_{t-1})$ .",
"It is not hard to check that the kernel $Z^{\\prime }$ lies in $^\\perp M_{t-1}$ .",
"Clearly, $Z^{\\prime }$ satisfies the above wanted conditions for $Z^{\\prime }$ .",
"Similarly, we can construct such a $Z^{\\prime }$ if the non-zero morphisms $Z \\rightarrow M_{t-1}$ are injective.",
"We can continue this process and construct a representation $X$ that lies in $^\\perp M_j$ for $1 \\le j \\le t-1$ and $t+2 \\le j \\le n$ and lies in $^\\perp N$ where $N$ is a general representation of dimension vector $\\gamma ^{\\prime }$ .",
"In particular, $X$ lies in $\\mathcal {A}(d)$ .",
"Observe that $X$ is not the zero representation since $\\alpha _1, \\ldots , \\alpha _{t-1}, \\nu , \\gamma ^{\\prime }$ are linearly independent.",
"Recall that a representation $V \\in {\\rm rep}(Q)$ is rigid if ${\\rm Ext}^1(V,V)=0$ .",
"Hence, the indecomposable rigid representations are the exceptional ones.",
"Observe that there is a one-to-one correspondence $\\lbrace \\text{real Schur roots}\\rbrace \\leftrightarrow \\lbrace \\text{iso.",
"classes of exceptional representations}\\rbrace .$ The dimension vector of a rigid representation is called prehomogeneous.",
"Observe that if $V$ is rigid, then the GL$(d_V)$ -orbit of $V$ is open in ${\\rm rep}(Q,d_V)$ .",
"In this case, the canonical decomposition of $d$ involves the dimension vectors of its indecomposable direct summands, so involves only real Schur roots.",
"Conversely, if the canonical decomposition of a dimension vector $d$ involves only real Schur roots, then ${\\rm rep}(Q,d)$ has an open orbit and hence, $d = d_V$ where $V$ is rigid.",
"Thus, the above correspondence extends to the following one-to-one correspondence: $\\lbrace d \\mid d \\; \\text{is prehomogeneous}\\rbrace \\leftrightarrow \\lbrace \\text{iso.",
"classes of rigid representations}\\rbrace .$ Proposition 3.5 The subcategory $\\mathcal {A}(d)$ has a projective generator if and only if $d$ is prehomogeneous.",
"In this case, $\\mathcal {A}(d) = ^\\perp V = \\lbrace X \\in {\\rm rep}(Q) \\mid {\\rm Hom}(X,V)=0={\\rm Ext}^1(X,V)\\rbrace $ where $V$ is rigid with $d = d_V$ .",
"Assume that $\\mathcal {A}(d)$ has a projective generator.",
"Therefore, it is equivalent to the category of representations of an acyclic quiver.",
"Let $M_1, \\ldots , M_r$ be the indecomposable simple objects in $\\mathcal {A}(d)$ , up to isomorphism.",
"We may assume that they are ordered so that $E:=(M_1, \\ldots , M_r)$ is an exceptional sequence in $\\mathcal {A}(d)$ and hence, also an exceptional sequence in ${\\rm rep}(Q)$ .",
"We have $\\mathcal {A}(d) = E)$ .",
"As argued in the proof of Proposition REF , for $1 \\le i \\le r$ , there are open sets $\\mathcal {U}_i$ in ${\\rm rep}(Q,d)$ such that for $N_i \\in \\mathcal {U}_i$ , we have ${\\rm Hom}(M_i,N_i)=0={\\rm Ext}^1(M_i,N_i)$ .",
"Now, $\\bigcap \\mathcal {U}_i$ is non-empty and we let $N$ lies in it.",
"Then, for $1 \\le i \\le r$ , we have ${\\rm Hom}(M_i,N)=0={\\rm Ext}^1(M_i,N)$ .",
"Since the $M_i$ are the simple objects in $, it follows that $ N E)$.",
"Thus, $ A(d) N$.",
"However, by definition of $ A(d)$, we have $ N A(d)$.",
"Therefore, $ A(d) = N$.",
"Observe that $ E)$ is equivalent to the category of representations of an acyclic quiver $ Q'$.",
"We can think of $ N$ as a representation in $ rep(Q')$ with dimension vector $ d$.",
"Assume that $ N$ is not rigid.",
"Then the canonical decomposition of $ d$ (as a dimension vector of $ Q'$) involves an isotropic or imaginary Schur root of $ Q'$.",
"It follows from Lemma \\ref {LemmaPerp} that there is a representation $ Z$ in the category $ A(d)$ for $ rep(Q')$.",
"This means that $ Z A(d)$ but $ Z E)$, a contradiction.",
"Therefore, $ N$ is rigid and this proves the necessity.",
"Assume now that $ d = dV$ where $ V$ is rigid.",
"Since $ V$ is in general position, any $ Z A(d)$ has to be in $ V$ and hence, $ A(d) = V.$$ Given a dimension vector $d$ , we fix $\\sigma _d:= -\\langle -, d \\rangle $ , which is called the weight associated to $d$ .",
"Recall that $M \\in {\\rm rep}(Q)$ is $\\sigma _d$ -semistable if there is a positive integer $m$ and a semi-invariant $f$ of weight $m\\sigma _d$ in SI$(Q, d_M)$ such that $f$ does not vanish at $M$ .",
"If $M$ is $\\sigma _d$ -semistable and has no proper (non-zero) $\\sigma _d$ -semistable subobject, then it is $\\sigma _d$ -stable.",
"It follows from King's criterion [14] that $M$ is $\\sigma _d$ -semistable (resp.",
"$\\sigma _d$ -stable) if and only if $\\sigma _d(d_M) = 0$ and $\\sigma _d(f) \\le 0$ (resp.",
"$\\sigma _d(f) < 0$ ) whenever $M$ has a proper non-zero subobject of dimension vector $f$ .",
"We will see that this notion of semistability is related to perpendicular subcategories.",
"Observe first that for any dimension vector $d$ , and a positive integer $m$ , we have $\\mathcal {A}(d) \\subseteq \\mathcal {A}(md)$ .",
"The other inclusion is not true, in general, if the canonical decomposition of $d$ involves an imaginary Schur root.",
"Proposition 3.6 Let $d$ be a dimension vector.",
"The (simple) objects in $\\cup _{m \\ge 1}\\mathcal {A}(md)$ are the $\\sigma _d$ -(semi)stable objects.",
"If the canonical decomposition of $d$ does not involve imaginary Schur roots, then $\\cup _{m \\ge 1}\\mathcal {A}(md) = \\mathcal {A}(d)$ .",
"Let $M \\in \\mathcal {A}(d)$ .",
"Then there exists $M(d) \\in {\\rm rep}(Q,d)$ such that ${\\rm Hom}(M,M(d)) = 0 = {\\rm Ext}^1(M,M(d)).$ Therefore, the semi-invariant $C^-(M(d))$ of weight $\\sigma _d$ in SI$(Q, d_M)$ does not vanish at $M$ .",
"Since there exists a semi-invariant in ${\\rm SI}(Q,d_M)_{\\sigma _d}$ that does not vanish on $M$ , $M$ is $\\sigma _d$ -semistable.",
"Conversely, assume that $M$ is $\\sigma _d$ -semistable.",
"Then there exists $m \\ge 1$ and a semi-invariant $f\\in {\\rm SI}(Q,d_M)_{m \\sigma _d}$ that does not vanish on $M$ .",
"Now, $f$ is given by a semi-invariant of the form $C^-(N)$ for some representation $N$ of dimension vector $m d$ .",
"We see that $M$ lies in $\\mathcal {A}(md) \\subseteq \\cup _{i \\ge 1}\\mathcal {A}(id)$ .",
"For $\\sigma _d$ -stable representations, one just needs to use king's criterion, as mentioned in the paragraph before this proposition.",
"Assume now that the canonical decomposition of $d$ does not involve imaginary Schur root.",
"It follows from Proposition REF that a general representation of dimension vector $md$ has a direct summand of dimension vector $d$ .",
"Let $M \\in \\mathcal {A}(md)$ .",
"Then there exists $N \\in {\\rm rep}(Q,md)$ such that $C^M(N) \\ne 0$ .",
"Also, $N$ can be taken to be in general position.",
"By the above observation, there is a summand $N^{\\prime }$ of $N$ of dimension vector $d$ .",
"Thus $C^M(N^{\\prime })\\ne 0$ , meaning that $M \\in \\mathcal {A}(d)$ .",
"Therefore, $\\mathcal {A}(md) = \\mathcal {A}(d)$ for all $m \\ge 1$ .",
"The following proposition explains why the study of perpendicular subcategories is directly related to the study of semi-invariants.",
"Proposition 3.7 The ring ${\\rm SI}(Q,d)$ is generated by the semi-invariants $C^V(-)$ where $V$ is simple in $\\mathcal {A}(d)$ .",
"We know from Theorem REF that the ring ${\\rm SI}(Q,d)$ is generated, over $k$ , by semi-invariants of the form $C^V(-)$ where $V$ is a representation with $\\langle d_V, d \\rangle = 0$ .",
"If ${\\rm Hom}(V,M) \\ne 0$ for all $M \\in {\\rm rep}(Q, d)$ , then $C^V(-)$ is the zero semi-invariant.",
"Otherwise, $V$ lies in $\\mathcal {A}(d)$ .",
"If $V$ is not a simple object in $\\mathcal {A}(d)$ , then there exists a simple subobject $V_1$ of $V$ in $\\mathcal {A}(d)$ .",
"Since $\\mathcal {A}(d)$ is thick, this yields a short exact sequence $0 \\rightarrow V_1 \\rightarrow V \\rightarrow V_2 \\rightarrow 0$ in $\\mathcal {A}(d)$ and it follows from [6] that $C^{V}(-)=aC^{V_1}(-)C^{V_2}(-)$ where $a \\in k$ is non-zero.",
"Repeating this reduction for the object $V_2$ yields that $C^V(-)$ is, up to a scalar, a product of semi-invariants as in the statement.",
"If $\\sigma $ is a weight, a dimension vector $d$ is called $\\sigma $ -semistable (resp.",
"$\\sigma $ -stable) if a general representation of dimension vector $d$ is $\\sigma $ -semistable (resp.",
"$\\sigma $ -stable).",
"Given a $\\sigma $ -semistable dimension vector $d$ , there exists a $\\sigma $ -stable decomposition $d = d_1 + \\cdots + d_r$ of $d$ where a general representation $M$ of dimension vector $d$ has a filtration $0 \\subset M_1 \\subset \\cdots \\subset M_r = M$ such that for $1 \\le i \\le r$ , $M_i/M_{i-1}$ has dimension vector $d_i$ and is $\\sigma $ -stable; see [5].",
"In this case, $d_i$ will be called a $\\sigma $ -stable factor of $d$ .",
"The $\\sigma $ -stable decomposition of a $\\sigma $ -semistable dimension vector is unique up to ordering.",
"Moreover, it was shown in [5] that the pairwise distinct $\\sigma $ -stable factors $f_1, \\ldots , f_s$ of $d$ can be ordered in such a way that $f_i \\bot f_j$ whenever $i < j$ .",
"So the sequence $(f_1, \\ldots , f_s)$ is an orthogonal sequence of Schur roots.",
"Lemma 3.8 Let $\\sigma $ be a weight and $d$ be a Schur root that is $\\sigma $ -semi-stable.",
"If $d$ is real, then all the $\\sigma $ -stable factors of $d$ are real Schur roots.",
"If $d$ is isotropic, then all the $\\sigma $ -stable factors of $d$ are real or isotropic Schur roots.",
"Let $d_1, \\ldots , d_r$ be the distinct $\\sigma $ -stable factors of $d$ .",
"We may assume that $(d_1, \\ldots , d_r)$ is an orthogonal sequence of Schur roots.",
"We can use the algorithm in [4] for finding the canonical decomposition of $d$ starting with the sequence $(d_1, \\ldots , d_r)$ .",
"If one of $d_i$ is imaginary (or isotropic), it follows from the algorithm that $d$ will be imaginary (resp.",
"isotropic or imaginary).",
"Therefore, if $d$ is real, then all $d_i$ are real.",
"If $d$ is isotropic, then no $d_i$ is imaginary.",
"Let $E=(X_1, \\ldots , X_r)$ be an exceptional sequence and consider the subcategory $E)=X_1, \\ldots , X_r)$ .",
"Assume that the dimension vector $d$ lies in $E)$ , that is, there is an object in $E)$ having $d$ as dimension vector.",
"The stability condition $\\sigma _d = -\\langle - , d \\rangle $ also gives rise to a stability condition, denoted $\\sigma _{E), d}$ , in $E)$ .",
"An object $X \\in E)$ is said to be relative $\\sigma _d$ -semistable (resp.",
"relative $\\sigma _d$ -stable) in $E)$ provided it is $\\sigma _{E), d}$ -semistable (resp.",
"$\\sigma _{E),d}$ -stable).",
"Proposition 3.9 Let $E=(X_1, \\ldots , X_r)$ be an exceptional sequence and let $d$ be a dimension vector lying in $E)$ .",
"Let $X \\in E)$ .",
"The object $X$ is relative $\\sigma _{d}$ -semistable in $E)$ if and only if it is $\\sigma _d$ -semistable.",
"If $X$ is $\\sigma _d$ -stable, then $X$ is relative $\\sigma _{d}$ -stable in $E)$ .",
"Observe that $X$ is $\\sigma _d$ -semistable if and only if $-\\langle d_X, d \\rangle = 0$ and for any subobject $X^{\\prime }$ of $X$ , we have $-\\langle d_{X^{\\prime }}, d \\rangle \\le 0$ .",
"Similarly, $X$ is $\\sigma _{E),d}$ -semistable if and only if $-\\langle d_X, d \\rangle = 0$ and for any subobject $X^{\\prime }$ of $X$ in $E)$ , we have $-\\langle d_{X^{\\prime }}, d \\rangle \\le 0$ .",
"Now, a subobject of $X$ in $E)$ is also a subobject of $X$ in ${\\rm rep}(Q)$ .",
"This shows the sufficiency of (1).",
"Suppose now that $X$ is $\\sigma _{E),d}$ -semistable.",
"Then there exists $M$ in $E)$ of dimension vector $md$ for some $m > 0$ such that $C^X(M)\\ne 0$ , that is, ${\\rm Hom}(X,M)={\\rm Ext}^1(X,M)=0$ .",
"Since $M$ has dimension vector $md$ in ${\\rm rep}(Q)$ , this gives that $X$ is $\\sigma $ -semistable.",
"Assume now that $X$ is $\\sigma $ -stable.",
"Then $X$ is $\\sigma _{d}$ -semistable.",
"If $X$ is not $\\sigma _{d}$ -stable, then there exists a proper subobject $X^{\\prime }$ of $X$ in $E)$ such that $-\\langle d_{X^{\\prime }}, d \\rangle = 0$ .",
"Now, $X^{\\prime }$ is also a subobject of $X$ in ${\\rm rep}(Q)$ with $-\\langle d_{X^{\\prime }}, d \\rangle = 0$ , and this contradicts the fact that $X$ is $\\sigma _d$ -stable.",
"The case of an isotropic Schur root In this section, $\\delta $ stands for an isotropic Schur root.",
"The weight $\\sigma _\\delta $ will simply be denoted $\\sigma $ , when there is no risk of confusion.",
"Our aim is to describe all simple objects in $\\mathcal {A}(\\delta )$ or, equivalently, all $\\sigma _\\delta $ -stable objects.",
"We start with the following proposition; see [15].",
"Proposition 4.1 There exists an exceptional sequence $(V,W)$ in ${\\rm rep}(Q)$ such that $V,W)$ is tame and $\\delta = d_V + d_W$ with $\\langle d_W, d_V \\rangle = -2$ .",
"In particular, $^\\perp V \\cap ^\\perp W \\subseteq \\mathcal {A}(\\delta )$ .",
"As shown in [15], there is an exceptional sequence $E=(V,W)$ of length two such that $\\delta $ is a root in $E)$ .",
"Since $E$ has length two, $E)$ is equivalent to the category of representations of an acyclic quiver $Q_E$ with two vertices.",
"But $Q_E$ has to have an isotropic Schur root.",
"Therefore, $Q_E$ is the Kronecker quiver.",
"With no loss of generality, we may assume that $V,W$ are the simple objects of $E)$ .",
"Therefore, $\\langle d_W, d_V \\rangle = -{\\rm dim}_k{\\rm Ext}^1(V,W)$ and ${\\rm dim}_k{\\rm Ext}^1(V,W)$ is the number of arrows in $Q_E$ .",
"The second part of the statement is trivial.",
"Lemma 4.2 Let $Q$ have at least three vertices.",
"Then there is at least one exceptional simple object in $\\mathcal {A}(\\delta )$ .",
"Suppose that all the $\\sigma $ -stable dimension vectors are isotropic or imaginary.",
"By Proposition REF , we have an exceptional sequence $(V,W)$ , such that $\\delta = d_V + d_W$ .",
"We can extend this to an exceptional sequence $(U,V,W)$ and hence, $U \\in \\mathcal {A}(\\delta )$ .",
"This means that $d_U$ is $\\sigma $ -semistable.",
"Now, we apply Lemma REF .",
"The subcategories of the form $A,B,C)$ where $(A,B,C)$ is an exceptional sequence will play a crucial role in our investigation.",
"The following lemma, which is easy to check, provides a description of the quivers with three vertices having an isotropic Schur root.",
"Lemma 4.3 Let $(A,B,C)$ be an exceptional sequence such that $A,B,C)$ contains $\\delta $ .",
"Then either the category $A,B,C)$ is wild and connected or the category $A,B,C)$ is equivalent to ${\\rm rep}(Q^{\\prime })$ , where $Q^{\\prime }$ is either of type $\\widetilde{\\mathbb {A}}_{2,1}$ or a union of the Kronecker quiver and a single vertex.",
"Let us denote by $\\tau $ the Auslander-Reiten translation in ${\\rm rep}(Q)$ .",
"An indecomposable representation that lies in the $\\tau $ -orbit of a projective (resp.",
"injective) representation is called preprojective (resp.",
"preinjective).",
"Baer and Strau${\\ss }$ have proven the following crucial result; see [1] or [21].",
"Lemma 4.4 (Baer, Strau${\\ss }$ ) Let $Q$ be of wild type and let $X$ be exceptional.",
"If $X^\\perp $ is of finite or tame type, then $X$ has to be preprojective or preinjective.",
"The following result describes a way to produce other isotropic Schur roots starting with an exceptional sequence $(U,V,W)$ where $\\delta = d_V + d_W$ .",
"Lemma 4.5 Let $E=(U,V,W)$ be an exceptional sequence such that $V,W)$ is tame with isotropic Schur root $\\delta = d_V + d_W$ .",
"Reflect both $V,W$ to the left of $U$ to get an exceptional sequence $(V^{\\prime }, W^{\\prime }, U)$ .",
"Let $\\delta ^{\\prime }$ be the unique isotropic Schur root in $V^{\\prime }, W^{\\prime })$ .",
"Then $\\delta ^{\\prime } = \\delta - \\langle \\delta , d_U \\rangle d_U$ .",
"If $U$ is preinjective in $E)$ , then $Q$ is wild connected and $\\langle \\delta , d_U \\rangle \\ge 0$ .",
"If $U$ is preprojective in $E)$ , then $Q$ is wild connected and $\\langle \\delta , d_U \\rangle \\le 0$ .",
"Otherwise, $E)$ is tame, $U$ is regular or is simple projective-injective in $E)$ , $\\langle \\delta , d_U \\rangle = 0$ and $\\delta ^{\\prime } = \\delta $ .",
"We may assume that ${\\rm rep}(Q) = E)$ .",
"If $Q$ is wild, then it follows from Lemma REF that $Q$ is connected.",
"In this case, by Lemma REF , $U$ cannot be regular.",
"Assume first that $Q$ is tame.",
"If $Q$ is tame connected, $Q$ is of type $\\widetilde{\\mathbb {A}}_{2,1}$ and $U$ has to be isomorphic to one of the two quasi-simple regular exceptional representations.",
"If $Q$ is tame disconnected, then we see that $U$ is the simple representation corresponding to the connected component of $Q$ of type $\\mathbb {A}_1$ .",
"So if $Q$ is tame, it is clear that $\\langle \\delta , d_U \\rangle = 0$ and $\\delta = \\delta ^{\\prime }$ .",
"So we may assume that $Q$ is wild connected.",
"We have an orthogonal sequence of Schur roots $(d_U, \\delta )$ .",
"Set $\\delta ^{\\prime \\prime } = \\delta - \\langle \\delta , d_U \\rangle d_U$ .",
"One easily checks that $\\langle \\delta ^{\\prime \\prime }, d_U \\rangle = 0$ and $\\langle \\delta ^{\\prime \\prime }, \\delta ^{\\prime \\prime } \\rangle = 0$ .",
"Since $^\\perp U = V^{\\prime }, W^{\\prime })$ is of tame type, the Euler-Ringel form $\\langle - , - \\rangle _{V^{\\prime }, W^{\\prime })}$ restricted to $V^{\\prime }, W^{\\prime })$ is positive semi-definite.",
"Therefore, $\\delta ^{\\prime \\prime }$ is an integral multiple of the isotropic root $\\delta ^{\\prime }$ in $V^{\\prime }, W^{\\prime })$ .",
"Since each of $\\delta ^{\\prime }, \\delta ^{\\prime \\prime }$ is a sum or difference of $d_{V^{\\prime }}, d_{W^{\\prime }}$ , we see that $\\delta ^{\\prime \\prime } = \\pm \\delta ^{\\prime }$ .",
"Assume first that $U$ is preinjective.",
"A general representation $M(\\delta )$ of dimension vector $\\delta $ is regular while a general representation of dimension vector $d_U$ is isomorphic to $U$ hence preinjective.",
"We have ${\\rm Ext}^1(M(\\delta ), U)=0$ and hence $\\langle \\delta , d_U \\rangle = {\\rm dim}_k{\\rm Hom}(M(\\delta ), U) \\ge 0$ .",
"If $U$ is preprojective, then ${\\rm Hom}(M(\\delta ), U) = 0$ .",
"This gives $\\langle \\delta , d_U \\rangle \\le 0$ .",
"Suppose that $\\delta ^{\\prime } = \\langle \\delta , d_U \\rangle d_U - \\delta $ .",
"This is only possible if $\\langle \\delta , d_U \\rangle > 0$ and hence, if $U$ is preinjective.",
"Then $\\langle \\delta , d_U \\rangle d_U = \\delta + \\delta ^{\\prime }$ .",
"The region ${R}$ given by $\\langle d, d \\rangle \\le 0$ is the cone over a two dimensional ellipse and hence, is a convex cone.",
"Since both $\\delta , \\delta ^{\\prime }$ lie on the boundary of ${R}$ , we see that $d_U$ lies in ${R}$ .",
"Hence, $\\langle d_U, d_U \\rangle \\le 0$ , a contradiction to $U$ being exceptional.",
"Therefore, $\\delta ^{\\prime \\prime } = \\delta ^{\\prime }$ is the wanted isotropic Schur root.",
"The root $\\delta ^{\\prime }$ defined above will be denoted $L_U(\\delta )$ , as it is the reflection of $\\delta $ to the left of the real Schur root $d_U$ .",
"Proposition 4.6 Let $Q$ be wild with three vertices, and let $(V,W)$ be an exceptional sequence of tame type containing $\\delta $ .",
"Complete this to a full exceptional sequence $(U,V,W)$ .",
"Then $U$ is preprojective or preinjective.",
"If $U$ is preprojective, then the $\\sigma $ -stable representations are, up to isomorphism, $U$ or $M(\\delta )$ where $M(\\delta )$ is a general representation of dimension vector $\\delta $ in $V,W)$ .",
"In particular, the $\\sigma $ -stable dimension vectors are $d_U, \\delta $ .",
"If $U$ is preinjective, then the $\\sigma $ -stable representations are, up to isomorphism, $U$ or $M(\\delta ^{\\prime })$ where $M(\\delta ^{\\prime })$ is a general representation of dimension vector $\\delta ^{\\prime } = L_U(\\delta )$ in $V^{\\prime },W^{\\prime })$ , where $(V^{\\prime }, W^{\\prime }, U)$ is exceptional.",
"In particular, the $\\sigma $ -stable dimension vectors are $d_U, \\delta ^{\\prime }$ .",
"In both cases, the $\\sigma $ -stable dimension vectors are the two extremal rays in the cone of $\\sigma $ -semistable dimension vectors.",
"Recall that the region $\\langle d, d \\rangle \\le 0$ is the cone over a two dimensional ellipse.",
"The ray $[\\delta ]$ lies on the quadric $\\langle d, d \\rangle =0$ .",
"Since $V, W)$ is tame, the Euler-Ringel form for $V, W)$ is positive semi-definite.",
"In particular, any linear combination of $d_V, d_W$ that lie on the region $\\langle d, d \\rangle \\le 0$ has to be a multiple of $\\delta $ .",
"Therefore, the cone $C_{V, W}$ generated by $d_V, d_W$ is the tangent plane of $\\langle d, d \\rangle =0$ at $\\delta $ .",
"If $U^{\\prime }$ is exceptional and $\\sigma $ -semistable, then we have an exceptional sequence $(U^{\\prime }, X, Y)$ such that $[\\delta ]$ lies in the cone $C_{X, Y}$ generated by $d_{X}, d_{Y}$ .",
"Therefore, as argued above, $C_{X, Y}$ is the tangent plane of $\\langle d, d \\rangle =0$ at $\\delta $ .",
"Hence, $C_{V, W} = C_{X, Y}$ and thus, since $\\langle - , - \\rangle $ is non-degenerate, the rays of $d_U$ , $d_{U^{\\prime }}$ coincide.",
"This gives $U \\cong U^{\\prime }$ since $U, U^{\\prime }$ are exceptional.",
"This proves that there is a unique $\\sigma $ -semistable real Schur root $d_U$ and a unique $\\sigma $ -semistable exceptional representation, up to isomorphism.",
"Hence, $d_U$ has to be $\\sigma $ -stable by Lemma REF .",
"Observe that $Q$ is connected by Lemma REF .",
"By Lemma REF , since $V,W)$ is tame, we know that $U$ is preinjective or preprojective.",
"Let $M$ be a $\\sigma $ -stable representation that is not isomorphic to $U$ .",
"By the previous argument, $M$ cannot be exceptional.",
"Suppose first that $U$ is preprojective.",
"If $M$ is not in $U^\\perp $ , then ${\\rm Ext}^1(U,M)\\ne 0$ as ${\\rm Hom}(U,M)=0$ .",
"Therefore, $M$ has to be preprojective, contradicting that $M$ is not exceptional.",
"Hence, $M$ is $\\sigma $ -stable and lies in $U^\\perp = V,W)$ .",
"By Proposition REF , $M$ is relative $\\sigma $ -stable in $U^\\perp $ .",
"Therefore, $M \\cong M(\\delta )$ is a general representation of dimension vector $\\delta $ in $V,W)$ .",
"Suppose finally that $U$ is preinjective.",
"If $M$ is not in $^\\perp U$ , then ${\\rm Ext}^1(M,U)\\ne 0$ and this means that $M$ is preinjective, thus exceptional, a contradiction.",
"Therefore, $M$ lies in $^\\perp U$ .",
"Now, $^\\perp U$ is also tame.",
"Consider the exceptional sequence $(V^{\\prime }, W^{\\prime }, U)$ where $V^{\\prime }, W^{\\prime }$ are obtained from $V,W$ by reflecting to the left of $U$ .",
"Let $\\delta ^{\\prime } = L_U(\\delta )$ be the unique isotropic Schur root in $V^{\\prime }, W^{\\prime })$ .",
"Since $M$ is Schur and not exceptional, $M$ has dimension vector $\\delta ^{\\prime }$ .",
"Let $E$ be an exceptional sequence of $\\sigma $ -stable representations.",
"Clearly, such a sequence has length at most $n-2$ .",
"The sequence $E$ is said to be full if $E$ has length $n-2$ .",
"The following result guarantees that such a full exceptional sequence of $\\sigma $ -stable representations always exists.",
"Lemma 4.7 Any exceptional sequence of $\\sigma $ -stable representations can be completed to a full exceptional sequence of $\\sigma $ -stable representations.",
"In particular, there exists an exceptional sequence $(M_{n-2}, \\ldots , M_2, M_{1})$ of $\\sigma $ -stable representations.",
"Let $(X_1, \\ldots , X_r)$ be an exceptional sequence with all $X_i$ $\\sigma $ -stable.",
"Assume that $r$ is not equal to $n-2$ .",
"Observe that $\\langle d_{X_i}, d_{X_j} \\rangle \\le 0$ whenever $i \\ne j$ .",
"By Proposition REF , and since the perpendicular category $X_1, \\ldots , X_r)^\\perp $ contains representations of dimension vector $\\delta $ , we can extend it to an exceptional sequence $(X_1, \\ldots , X_r, V,W)$ of length less than $n$ where $V,W)$ is tame and $\\delta = d_V + d_W$ .",
"Therefore, there exists an exceptional representation $Y$ such that $(X_1, \\ldots , X_r, Y, V, W)$ is exceptional.",
"Being left orthogonal to both $V,W$ , the representation $Y$ has to be $\\sigma $ -semistable.",
"It is well known and easy to see that for any acyclic quiver, there exists a sincere representation of it that is rigid.",
"Since the category $X_1, \\ldots , X_r,Y)$ is equivalent to the category of representations of an acyclic quiver with $r+1$ vertices, there exists a rigid object $Z \\in X_1, \\ldots , X_r, Y)$ such that its Jordan-Hölder composition factors in $X_1, \\ldots , X_r, Y)$ will consist of all the simple objects in $X_1, \\ldots , X_r, Y)$ .",
"Being $\\sigma $ -stable, the objects $X_1, \\ldots , X_r$ are non-isomorphic simple objects in $X_1, \\ldots , X_r, Y)$ .",
"Let $Y^{\\prime }$ be the other simple object.",
"Being rigid, $Z$ is a general representation.",
"It has some filtration where the subquotients are $\\lbrace X_1, \\ldots , X_r, Y^{\\prime }\\rbrace $ with possible multiplicities.",
"If $Y^{\\prime }$ is $\\sigma $ -stable, then we have a list $\\lbrace X_1, \\ldots , X_r, Y^{\\prime }\\rbrace $ of $r+1$ objects that are $\\sigma $ -stable.",
"Since these objects are the simple objects of $X_1, \\ldots , X_r, Y)$ , they could be ordered to form an exceptional sequence of length $r+1$ , which will be the wanted sequence.",
"If $Y^{\\prime }$ is not $\\sigma $ -stable, then the above filtration of $Z$ can be refined to a filtration in ${\\rm rep}(Q)$ where all subquotients are $\\sigma $ -stable.",
"In particular, there will be more than $r$ non-isomorphic subquotients.",
"Since $Z$ is rigid, we know that all subquotients have dimension vector a real Schur root, by Lemma REF .",
"Now, these real Schur roots can be ordered to form an orthogonal sequence of real Schur roots; see [5].",
"This sequence corresponds to an exceptional sequence of $\\sigma $ -stable representations of length at least $r+1$ .",
"Now, we treat the case where $n$ is arbitrary.",
"We first need some more notations.",
"From now on, let us fix $(M_{n-2}, \\ldots , M_{1})$ a full exceptional sequence of $\\sigma $ -stable representations.",
"Complete this sequence to get a full exceptional sequence $(M_{n-2}, \\ldots , M_{1}, V, W)$ .",
"Note that $\\delta $ is a root in $V,W)$ .",
"We construct a sequence of isotropic Schur roots $\\delta _1, \\ldots , \\delta _{n-2}$ and a sequence of rank three subcategories $i$ as follows.",
"Set $\\delta _1 = \\delta $ , $V_1 = V$ and $W_1=W$ and $1 = M_1, V_1, W_1)$ .",
"Observe that if 1 is wild, then it is connected and $M_1$ is preprojective or preinjective in 1.",
"If 1 is tame, then it is either disconnected (in which case $M_1$ is the unique simple object in the trivial component of 1) or else it is connected and $M_1$ is quasi-simple in 1.",
"For $1 \\le i \\le n-2$ , if $M_{i}$ is not preinjective in $i=M_{i}, V_i, W_i)$ , then we reduce to the case of the exceptional sequence $(M_{n-2}, \\ldots , M_{i+1}, V_{i+1}, W_{i+1})$ and we set $\\delta _{i+1} = \\delta _i, V_{i+1} = V_i, W_{i+1}=W_i$ .",
"If $M_{i}$ is preinjective in $M_{i}, V_i, W_i)$ , we reduce to the case of the exceptional sequence $(M_{n-2}, \\ldots , M_{i+1}, V_{i+1}, W_{i+1})$ where $V_{i+1}$ is the reflection of $V_i$ to the left of $M_{i}$ and $W_{i+1}$ is the reflection of $W_i$ to the left of $M_{i}$ .",
"We set $\\delta _{i+1} = \\delta _i - \\langle \\delta _i, d_{M_{i}} \\rangle d_{M_{i}}.$ In all cases, we set ${i+1}=M_{i+1}, V_{i+1}, W_{i+1})$ .",
"Note that for $1 \\le i \\le n-2$ , the root $\\delta _i$ is a root in $V_i, W_i)$ .",
"Finally, we set $\\bar{\\delta }= \\delta _{n-2}$ .",
"For two dimension vectors $d_1, d_2$ , we write $d_1 \\hookrightarrow d_2$ provided a general representation of dimension vector $d_2$ has a subrepresentation of dimension vector $d_1$ .",
"Lemma 4.8 We have $\\delta _{i+1} \\hookrightarrow \\delta _{i}$ and each $\\delta _{i}$ is an isotropic Schur root that is $\\sigma $ -semistable.",
"We proceed by induction on $i$ .",
"The case where $i=1$ is clear.",
"Let $M(\\delta _i)$ be a general representation of dimension vector $\\delta _i$ in $V_i, W_i)$ that is $\\sigma $ -semistable.",
"If $M_{i}$ is not preinjective in $i=M_{i}, V_i, W_i)$ , then $\\delta _{i+1} = \\delta _i$ and hence, it is clear that $\\delta _{i+1} \\hookrightarrow \\delta _{i}$ and that $\\delta _{i}$ is an isotropic Schur root that is $\\sigma $ -semistable.",
"So assume that $M_{i}$ is preinjective in $i$ .",
"Then ${\\rm Ext}^1(M(\\delta _i), M_{i})=0$ .",
"If we have a non-zero morphism $f_i: M(\\delta _i) \\rightarrow M_{i}$ that is not an epimorphism, then the cokernel $C_i$ of $f_i$ is $i$ -preinjective and $\\sigma $ -semistable in $i$ .",
"There is an epimorphism $C_i \\rightarrow N_i$ where $N_i$ is relative $\\sigma $ -stable in $i$ .",
"Since $M_i$ is $i$ -preinjective, $N_i$ (and $C_i$ ) has to be $i$ -preinjective and the only $i$ -preinjective relative $\\sigma $ -stable object in $i$ is $M_{i}$ .",
"This is a contradiction.",
"Therefore, any non-zero morphism $M(\\delta _i) \\rightarrow M_{i}$ is an epimorphism.",
"Now, take $e_{i} = \\langle \\delta _i, d_{M_{i}}\\rangle = {\\dim }{\\rm Hom}(M(\\delta _i), M_{i})$ .",
"We have a morphism $g_i: M(\\delta _i) \\rightarrow M_{i}^{e_{i}}$ given by a basis of ${\\rm Hom}(M(\\delta _i), M_{i})$ , and as argued above, $g_i$ is an epimorphism.",
"The kernel $K_i$ of $g_i$ is of dimension $\\delta _i - \\langle \\delta _i, d_{M_{i}}\\rangle d_{M_{i}}$ which is $\\delta _{i+1}$ .",
"Hence, $\\delta _{i+1}$ is relative $\\sigma $ -semistable and hence $\\sigma $ -semistable.",
"Consider the short exact sequence $0 \\rightarrow K_i \\rightarrow M(\\delta _i) \\rightarrow M_{i}^{e_{i}} \\rightarrow 0$ in $i$ .",
"Since $K_i \\in \\hspace{-2.0pt}^\\perp M_{i}$ by construction, and $K_i$ has dimension vector $\\delta _{i+1}$ , we have ${\\rm ext}(\\delta _{i+1}, e_i\\cdot d_{M_{i}})=0$ and by Schofield's result [17], $\\delta _{i+1} \\hookrightarrow \\delta _{i+1} + e_i\\cdot d_{M_{i}} = \\delta _i$ .",
"Call $\\delta $ of smaller type if the $\\tau $ -orbit of a general representation of dimension vector $\\delta $ contains a non-sincere representation.",
"Equivalently, if $\\tau $ denotes the Coxeter matrix of $Q$ , then there is an integer $r$ such that $\\tau ^r \\delta $ is not sincere.",
"Proposition 4.9 The following conditions are equivalent.",
"The root $\\delta $ is of smaller type.",
"There is a $\\sigma $ -stable representation that is preprojective or preinjective.",
"There is a $\\sigma $ -semistable representation that is preprojective or preinjective.",
"Let $Z$ be a $\\sigma $ -semistable representation that is preprojective or preinjective.",
"If $Z$ is preprojective, then there is some $i \\ge 0$ with $\\tau ^iZ=P$ projective.",
"Since $\\langle d_Z, \\delta \\rangle =0$ , we get $\\langle d_P, \\tau ^i\\delta \\rangle =0$ , showing that $\\tau ^i\\delta $ is not sincere, that is, $\\delta $ is of smaller type.",
"If $Z$ is preinjective, then there is some $j < 0$ with $\\tau ^jZ=Q[1]$ a shift of a projective representation $Q$ .",
"Since $\\langle d_Z, \\delta \\rangle =0$ , we get $\\langle -d_Q, \\tau ^j\\delta \\rangle =0$ , showing that $\\tau ^j\\delta $ is not sincere, that is, $\\delta $ is of smaller type.",
"This proves that $(3)$ implies $(1)$ .",
"Clearly, $(2)$ implies $(3)$ .",
"Assume that $\\delta $ is of smaller type.",
"Then there is some integer $i$ and some $x \\in Q_0$ such that ${\\rm Hom}(P_x, \\tau ^iM(\\delta ))=0$ whenever $M(\\delta )$ has dimension vector $\\delta $ .",
"Therefore, there exists a preprojective (if $i \\ge 0$ ) or preinjective (if $i < 0$ ) representation $Z$ such that $Z$ is left orthogonal to any representation $M(\\delta )$ of dimension vector $\\delta $ .",
"Therefore $Z$ is $\\sigma $ -semistable.",
"If $Z$ is not $\\sigma $ -stable and preprojective, then it has a $\\sigma $ -stable subrepresentation $Z^{\\prime }$ which has to be preprojective.",
"If $Z$ is not $\\sigma $ -stable and preinjective, then it has a $\\sigma $ -stable quotient $Z^{\\prime }$ which has to be preinjective.",
"We start with the following result, which describes the $\\sigma $ -stable objects when $n=4$ .",
"The core of it will be generalized to arbitrary $n$ later.",
"We think it is interesting to have a separate result for the $n=4$ case since more can be said for this small case.",
"Proposition 4.10 Let $Q$ be a connected wild quiver with 4 vertices and let $(M_2, M_1)$ be a full exceptional sequence of $\\sigma $ -stable representations.",
"Consider the exceptional sequence $(M_2, M_1, V, W)$ with $V,W$ the simple objects in $V,W)$ .",
"Let $M$ be $\\sigma $ -stable but not isomorphic to $M_1$ or $M_2$ .",
"Then The root $\\bar{\\delta }$ is the only $\\sigma $ -stable non-real Schur root.",
"The root $\\delta $ is of smaller type if and only if one of $M_1, M_2$ is preprojective or preinjective.",
"If $\\delta $ is not of smaller type, then the only exceptional $\\sigma $ -stable representations are $M_1, M_2$ .",
"The sufficiency of (2) follows from Proposition REF .",
"Assume now that both $M_1, M_2$ are regular.",
"We claim that $\\delta $ is not of smaller type and that the only exceptional $\\sigma $ -stable representations are $M_1, M_2$ .",
"In this case, by Lemma REF , since $1 =M_1, V, W)$ is wild and $V, W)$ is tame, we know that $M_1$ is either preprojective or preinjective in 1.",
"Observe that 1 is connected since it is wild and $\\delta $ is an isotropic root in 1.",
"If $M_1$ is preinjective in 1, then $2 = ^\\perp M_1$ is also wild, since $M_1$ is regular.",
"If $M_1$ is preprojective in 1, consider the exceptional sequence $(M_1^{\\prime }, M_2, V, W) = (M_1^{\\prime }, M_2, V_2, W_2)$ .",
"Note that $M_1, M_2$ are the relative simples in $M_2,M_1)$ .",
"Since any indecomposable object in $M_2,M_1)$ has a morphism from and to $M_1 \\oplus M_2$ , all objects of $M_2,M_1)$ , seen as objects in ${\\rm rep}(Q)$ , are regular.",
"Therefore, $M_1^{\\prime } \\in M_2,M_1)$ is regular in ${\\rm rep}(Q)$ and $M_1^{\\prime \\perp }= M_2, V_2, W_2)=2$ is wild (and connected).",
"Let $M$ be an arbitrary $\\sigma $ -stable representation not isomorphic to $M_1$ or $M_2$ .",
"Since ${\\rm Hom}(M_2, M)=0$ , there exists a non-negative integer $d_2$ and a short exact sequence $(*): \\quad 0 \\rightarrow M \\rightarrow E_2 \\stackrel{f_1}{\\rightarrow } M_2^{d_2} \\rightarrow 0$ with $E_2 \\in M_2^\\perp $ .",
"Observe that $d_2 = 0$ if and only if $M \\in M_2^\\perp $ .",
"Observe also that ${\\rm Hom}(E_2, M_1) = {\\rm Hom}(M_1, E_2)=0$ .",
"Since ${\\rm Hom}(M_1, E_2)=0$ , there exists a non-negative integer $d_1$ and a short exact sequence $(*): \\quad 0 \\rightarrow E_2 \\rightarrow E_1 \\stackrel{f_1}{\\rightarrow } M_1^{d_1} \\rightarrow 0$ with $E_1 \\in M_2^\\perp \\cap M_1^\\perp $ .",
"Observe that $d_1 = 0$ if and only if $E_2 \\in M_1^\\perp $ .",
"Since $E_1$ is relative $\\sigma $ -semistable in $M_2^\\perp \\cap M_1^\\perp =V,W)$ , there is a monomorphism $M(\\delta ) \\rightarrow E_1$ where $M(\\delta )$ is a Schur representation of dimension vector $\\delta $ in $V,W)$ .",
"If $M_1$ is preinjective in 1, then ${\\rm Ext}^1(M(\\delta ), M_1)=0$ .",
"Then $e_1:=\\langle \\delta , d_{M_1} \\rangle = {\\rm dim}{\\rm Hom}(M(\\delta ), M_1)>0$ .",
"We have a short exact sequence $0 \\rightarrow M(\\delta _1) \\rightarrow M(\\delta )\\rightarrow M_1^{e_1}\\rightarrow 0$ and this yields a monomorphism $M(\\delta _1) \\rightarrow E_2$ .",
"If $M_1$ is preprojective in 1, then ${\\rm Hom}(M(\\delta ), M_1)=0$ and hence, we get a monomorphism $M(\\delta _1) = M(\\delta ) \\rightarrow E_2$ .",
"If $M_2$ is preinjective in 2, then ${\\rm Ext}^1(M(\\delta _1), M_2)=0$ .",
"Then $e_2:=\\langle \\delta _1, d_{M_2} \\rangle = {\\rm dim}{\\rm Hom}(M(\\delta _1), M_2)>0$ .",
"We have a short exact sequence $0\\rightarrow M(\\delta _2) \\rightarrow M(\\delta _1)\\rightarrow M_2^{e_2}\\rightarrow 0$ and this yields a monomorphism $M(\\delta _2) \\rightarrow M$ .",
"If $M_2$ is preprojective in 2, then ${\\rm Hom}(M(\\delta _1), M_2)=0$ and hence, we get a monomorphism $M(\\delta _2) = M(\\delta _1) \\rightarrow M$ .",
"This shows that $M \\cong M(\\delta _2)=M(\\bar{\\delta })$ .",
"In particular, the exceptional sequence $(M_2,M_1)$ of $\\sigma $ -stable representations is unique.",
"It follows from Proposition REF that $\\delta $ is not of smaller type.",
"This prove our claim and hence $(2)$ and $(3)$ .",
"Statement $(1)$ in the case where $\\delta $ is of smaller type is a consequence of the next proposition, Proposition REF .",
"Given a subcategory $ of $ rep(Q)$, we denote by $ the Auslander-Reiten translation in $.",
"Fix a full exceptional sequence $ (Mn-2, ..., M2, M1)$ of $$-stable representations, and complete it to form an exceptional sequence $ (Mn-2, ..., M2, M1, V, W)$ where $ V,W$ are the non-isomorphic simple objects in $ V,W)$.",
"In particular, $ = dV + dW$ and $ dW, dV = -2$.",
"Recall that for $ 1 i n-2$, we denote by $ i$ the subcategory $ Mi, Vi, Wi)$, where $ Vi, Wi$ have been defined previously.",
"If $ i$ is tame and connected, then $ Mi$ is regular quasi-simple in $ i$ and lies in a tube of rank $ 2$ in $ i$.",
"We set $ Ni := iMi = i-1Mi$.",
"The exceptional sequence $ (Mn-2, ..., Mi+1)$ is denoted by $ Ei$.$ Proposition 4.11 Let $Q$ be a connected quiver with isotropic Schur root $\\delta $ .",
"Let $N$ be a $\\sigma $ -stable object not isomorphic to any $M_i$ .",
"The root $\\bar{\\delta }$ is the only $\\sigma $ -stable Schur root that is not real.",
"If $N$ is not exceptional, then $d_N = \\bar{\\delta }$ and $N \\in V_{n-2},W_{n-2})$ .",
"If $N$ is exceptional, then there exists some $i$ such that $i$ is tame connected with $M_i$ quasi-simple.",
"There exists a subsequence $\\mathcal {F}_i$ of $\\mathcal {E}_i$ such that $N$ is the reflection of $N_i$ to the left of $\\mathcal {F}_i$ .",
"We first need to check that there is at least one $\\sigma $ -stable root that is not real.",
"Assume otherwise.",
"As $\\delta $ is $\\sigma $ -stable, its $\\sigma $ -stable decomposition will involve only real Schur roots.",
"These roots may be ordered to form a Schur sequence.",
"Since all roots are real, this will correspond to an exceptional sequence $(F_1, \\ldots , F_r)$ .",
"Suppose that $r < n$ .",
"Complete the latter sequence to get a full exceptional sequence $(F_1, \\ldots , F_n)$ .",
"Since $\\delta $ is a root in each $F_i^\\perp $ for $1 \\le i \\le r$ , we see that $\\delta $ is a root in $F_{r+1}, \\ldots , F_n)$ .",
"Since $\\delta $ is also a root in $F_{1}, \\ldots , F_r)$ , we get that $\\delta $ is in the span of $d_{F_1}, \\ldots , d_{F_r}$ as well as in the span of $d_{F_{r+1}}, \\ldots , d_{F_n}$ .",
"Since $(F_1, \\ldots , F_n)$ is an exceptional sequence, the vectors $d_{F_1}, \\ldots , d_{F_n}$ are linearly independent, a contradiction.",
"Hence, $r=n$ .",
"But then, $F_{1}, \\ldots , F_n) = {\\rm rep}(Q)$ and all objects of ${\\rm rep}(Q)$ are $\\sigma $ -semistable, which is also a contradiction.",
"In the rest of the proof, we will see that the only possibility for a non-real $\\sigma $ -stable Schur root is $\\bar{\\delta }= \\delta _{n-2}$ .",
"Let $N$ be a $\\sigma $ -stable representation not isomorphic to any $M_i$ .",
"In particular, ${\\rm Hom}(N, M_i) = {\\rm Hom}(M_i,N)=0$ for all $i$ .",
"Let $E_{n-2}:=N$ .",
"For $i=n-2, n-3, \\ldots , 2, 1$ , we have a short exact sequence $0 \\rightarrow E_i \\rightarrow E_{i-1} \\rightarrow M_{i}^{d_{i}}\\rightarrow 0$ where $d_{i}$ is a non-negative integer with $d_{i} = -{\\rm dim}{\\rm Ext}^1(M_{i},E_i) = \\langle d_{M_{i}}, d_{E_i} \\rangle \\ge 0$ , as ${\\rm Hom}(M_i, E_i)=0$ by induction.",
"Observe that for $i < n-2$ , we have $E_i \\in M_{n-2}^\\perp \\cap \\cdots \\cap M_{i+1}^\\perp $ .",
"In particular, $E_0$ lies in $V,W)$ .",
"Observe also that all $E_i$ are $\\sigma $ -semistable.",
"Since $E_0$ is relative $\\sigma $ -semistable in $V,W)$ , there is a monomorphism $Z_0:=M(\\delta ) \\rightarrow E_0$ where $M(\\delta )$ is a Schur representation of dimension vector $\\delta $ in $V,W)$ .",
"If $M_1$ is preinjective in 1, then ${\\rm Ext}^1(M(\\delta ), M_1)=0$ .",
"Then $e_1:=\\langle \\delta , d_{M_1} \\rangle = {\\rm dim}{\\rm Hom}(M(\\delta ), M_1)$ .",
"We have a short exact sequence $0 \\rightarrow M(\\delta _2) \\rightarrow M(\\delta )\\rightarrow M_1^{e_1}\\rightarrow 0$ and this yields a monomorphism $M(\\delta _2) \\rightarrow E_1$ where $M(\\delta _2)$ lies in $V_2, W_2)$ is $\\sigma $ -semistable.",
"If $M_1$ is preprojective in 1, then ${\\rm Hom}(M(\\delta ), M_1)=0$ and hence, we get a monomorphism $M(\\delta _2) = M(\\delta ) \\rightarrow E_1$ .",
"If 1 is tame disconnected, then ${\\rm Hom}(M(\\delta ), M_1)=0$ and we get a monomorphism $M(\\delta _2) = M(\\delta ) \\rightarrow E_1$ as well.",
"Assume that 1 is tame connected.",
"If ${\\rm Hom}(M(\\delta ), M_1)=0$ , then we get a monomorphism $M(\\delta _2) = M(\\delta ) \\rightarrow E_1$ as previously.",
"Assume that ${\\rm Hom}(M(\\delta ), M_1)\\ne 0$ .",
"Then ${\\rm Hom}(N_1, M_1)=0$ and there is a monomorphism $N_1 \\rightarrow M(\\delta )$ .",
"Therefore, we get a monomorphism $N_1 \\rightarrow E_1$ .",
"Hence, in all cases, we have a monomorphism $Z_1 \\rightarrow E_1$ where $Z_1$ is either a Schur ($\\sigma $ -semistable) representation of dimension vector $\\delta _2$ in $V_2, W_2)$ or is $N_1$ .",
"In the first case, we consider the exceptional sequence $(M_{n-2}, \\ldots , M_2, V_2, W_2)$ and we proceed by induction starting with the monomorphism $Z_1 \\rightarrow E_1$ where $Z_1 = M(\\delta _2)$ .",
"For the second case, we consider the exceptional sequence $(M_{n-2}, \\ldots , M_2, N_1)$ .",
"If ${\\rm Hom}(N_1, M_2) \\ne 0$ , then ${\\rm Ext}^1(N_1, M_2) = 0$ as ${\\rm Ext}^1(M_2, N_1) = 0$ .",
"We have a short exact sequence $0 \\rightarrow Z_2 \\rightarrow N_1 \\rightarrow M_2^{f_2} \\rightarrow 0$ where $f_2 = {\\rm dim} {\\rm Hom}(N_1, M_2)$ .",
"Then, $Z_2 \\in ^\\perp M_2$ and hence, we get a monomorphism $Z_2 \\rightarrow E_2$ and $Z_2$ is the reflection of $N_1$ to the left of $M_2$ .",
"If ${\\rm Hom}(N_1, M_2) = 0$ , then we get a monomorphism $Z_2:=N_1 \\rightarrow E_2$ .",
"We proceed by induction until we get a monomorphism $Z_{n-2} \\rightarrow E_{n-2}=N$ .",
"This gives $Z_{n-2} \\cong N$ .",
"We see that $Z_{n-2}$ will be of the required form.",
"In particular, if $N$ is not exceptional, then $N$ has dimension vector $\\delta _{n-2}=\\bar{\\delta }$ and will be in $V_{n-2}, W_{n-2})$ .",
"The next two lemmas will help us in giving a better description of the simple objects in $\\mathcal {A}(\\delta )$ , that is, the $\\sigma $ -stable objects.",
"Lemma 4.12 Assume that both $M_{t-1}, \\ldots , M_1, V, W)$ and $t$ are tame connected with $\\delta $ the unique isotropic Schur root in both of these subcategories.",
"Then $M_t, M_{t-1}, \\ldots , M_1, V, W)$ is tame connected with only one isotropic Schur root $\\delta $ .",
"Assume that $=M_t, M_{t-1}, \\ldots , M_1, V, W)$ is wild.",
"Then $M_t$ lies in the preprojective or preinjective component of a wild connected component of $.",
"Observe that, in $ t$, the object $ Mt$ is left orthogonal to the unique isotropic Schur root (which is $ dVt + dWt$).",
"Since $ t$ is connected, this means that $ Mt$ is regular in $ t$.",
"In particular, there are infinitely many indecomposable objects $ Z$ of $ t$ (and hence of $ ) with ${\\rm Hom}(M_t, Z)\\ne 0$ .",
"Therefore, $M_t$ cannot lie in a preinjective component of $.",
"Similarly, $ Mt$ cannot lie in a preprojective component of $ .",
"This is a contradiction.",
"It remains to show that $ is connected (it is clear that then, $$ will be the unique isotropic Schur root in $ ).",
"Assume otherwise.",
"Then, we have $\\mathcal {B}_1 \\times \\mathcal {B}_2$ where each of $\\mathcal {B}_1, \\mathcal {B}_2$ is equivalent to a category of representations of a non-empty acyclic quiver.",
"Assume that $\\delta \\in \\mathcal {B}_2$ .",
"By assumption, we must have that both $M_{t-1}, \\ldots , M_1, V, W), t$ are subcategories of $\\mathcal {B}_2$ .",
"But then, $ \\mathcal {B}_2$ , a contradiction.",
"Therefore, $ is connected.$ Lemma 4.13 Assume that $i$ is tame connected for all $1 \\le i \\le s-1$ .",
"Fix $1 \\le t \\le s-1$ and consider the representation $N_t$ as defined above.",
"If $M_s$ is preprojective in the category $s$ , then ${\\rm Hom}(N_t,M_s)=0$ .",
"If $M_s$ is preinjective in the category $s$ , then $M_i \\in ^\\perp M_s$ for all $1 \\le i \\le s-1$ .",
"If $M_s$ is simple disconnected in $s$ , then all $\\sigma $ -stable representations, except possibly $M_{s-1}, \\ldots , M_1$ , lie in $^\\perp M_s$ .",
"Consider the categories $= M_s, \\ldots , M_1, V, W)$ and $:=M_{s-1}, \\ldots , M_1, V, W).$ We know that $$ is tame connected by Lemma REF .",
"Assume that $ is wild.",
"Therefore, $ Ms$ lies in a preprojective or preinjective component of $ .",
"For proving (1), assume that $M_s$ is preprojective in the category $s$ , which means that $M_s$ is preprojective in $.",
"Assume to the contrary that $ Hom(Nt,Ms)0$.",
"Since $ Nt , the object $N_t$ is preprojective in $.",
"Therefore, $ Nt$ cannot be regular in $ t$, a contradiction.",
"For proving (2), assume that $ Ms$ is preinjective in the category $ s$, which means that $ Ms$ is preinjective in $ .",
"Since $t$ is arbitrary and ${\\rm Hom}(M_t, M_i)=0$ , it is sufficient to prove that ${\\rm Ext}^1(M_t, M_s)=0$ .",
"If not, then the Auslander-Reiten formula in $ yields a nonzero morphism from $ Ms$ to $ M̏t$ and hence, $ M̏t$ (and thus $ Mt$) is preinjective in $ .",
"Thus, $M_t$ cannot be regular in $t$ , a contradiction.",
"Clearly, if $ is tame, then (1), (2) cannot occur, since a subcategory of a tame category cannot be wild.$ It remains to prove (3).",
"Let $N$ be a $\\sigma $ -stable object not isomorphic to any $M_i$ .",
"It follows from the proof of the last theorem that we have a short exact sequence $0 \\rightarrow E_s \\rightarrow E_{s-1} \\rightarrow M_s^{d_s}\\rightarrow 0$ and a monomorphism $Z \\rightarrow E_{s-1}$ where $Z$ is either indecomposable of dimension vector $\\delta $ in $V,W)$ or is exceptional and relative $\\sigma $ -semistable in $$ (a reflection of one of the $N_i$ for $1 \\le i \\le s-1$ ).",
"In the first case, since $s$ is tame disconnected, this yields a monomorphism $Z \\rightarrow E_s$ and the rest of the proof of the above theorem deals with representations in $^\\perp M_s$ .",
"In the second case, we get a monomorphism $M \\rightarrow Z$ where $M$ is relative $\\sigma $ -stable in $$ and exceptional, and hence a quasi-simple object of $$ .",
"Assume that $M$ lies in a tube $T$ of rank $r$ .",
"In particular, the other non-isomorphic $r-1$ quasi-simple objects of that tube are among the objects $M_{s-1}, \\ldots , M_1$ .",
"We claim that ${\\rm Ext}^1(M,M_s)=0$ .",
"There exists a short exact sequence $0 \\rightarrow M \\rightarrow Y \\rightarrow N \\rightarrow 0$ in $$ where $M,Y,N$ are indecomposable in $T$ with $Y$ of dimension vector $\\delta $ and $N$ of quasi-length $r-1$ which does not have $M$ as a quasi-simple composition factor.",
"Using that $(M_{s-1}, \\ldots , M_1, V,W)$ is an exceptional sequence and since, by construction, $M$ is the unique quasi-simple of $T$ with ${\\rm Hom}(M,Y)\\ne 0$ , we have $Y \\in V,W)$ .",
"The surjective map ${\\rm Ext}^1(Y,M_s) \\rightarrow {\\rm Ext}^1(M,M_s)$ together with the fact that $s = M_s, V, W)$ is disconnected gives that ${\\rm Ext}^1(M,M_s)=0$ .",
"This proves our claim.",
"Observe that $M$ lies in $M_s^\\perp $ .",
"We have a short exact sequence $0 \\rightarrow Z^{\\prime } \\rightarrow M \\rightarrow M_s^{f} \\rightarrow 0$ where $f = {\\rm dim} {\\rm Hom}(M, M_s) = \\langle d_M, d_{M_s} \\rangle $ .",
"Thus, $Z^{\\prime }$ is the reflection of $M$ to the left of $M_s$ and $Z^{\\prime } \\in ^\\perp M_s$ and we get a monomorphism $Z^{\\prime } \\rightarrow E_s$ .",
"The proof of the last theorem continues with $Z_s = Z^{\\prime }$ and the monomorphism $Z_s \\rightarrow E_s$ .",
"In all cases, we see that $M_s$ satisfies the required property.",
"Cone of $\\sigma $ -semi-stable dimension vectors Let $d$ be a dimension vector.",
"Let us denote by $C(\\sigma _d)$ the set of all $\\sigma _d$ -semistable dimension vectors.",
"We consider $C_\\mathbb {R}(\\sigma _d)$ the corresponding cone in $\\mathbb {R}^n$ , which lie in the positive orthant of $\\mathbb {R}^n$ .",
"Since the rays $x \\in C_\\mathbb {R}(\\sigma _d)$ satisfy $\\langle x, d \\rangle = 0$ , we rather consider $C_\\mathbb {R}(\\sigma _d)$ as a cone in $\\mathbb {R}^{n-1}$ .",
"The integral vectors in $C_\\mathbb {R}(\\sigma _d)$ correspond to the $\\sigma _d$ -semistable dimension vectors.",
"For $d=\\delta $ , since there exists a full exceptional sequence $(M_{n-2}, \\ldots , M_2, M_1)$ of $\\sigma _\\delta $ -stable representations and since the dimension vectors in an exceptional sequence are linearly independent, we see that $C_\\mathbb {R}(\\sigma _\\delta )$ is a cone of full dimension in $\\mathbb {R}^{n-1}$ .",
"In general, it is well known that $C_\\mathbb {R}(\\sigma _d)$ is a cone over a polyhedron where the indivisible dimension vectors in the extremal rays are $\\sigma _d$ -stable dimension vectors.",
"On the other hand, a $\\sigma _d$ -stable dimension vector needs not lie on an extremal ray.",
"Lemma 5.1 If $f$ is $\\sigma _d$ -stable and a real Schur root, then it lies in an extremal ray.",
"Let $f$ be a $\\sigma _d$ -stable real Schur root.",
"Assume that $f$ does not lie on an extremal ray.",
"According to [5], there exists dimension vectors $f_1, \\ldots , f_s$ , all lying on extremal rays, such that $f$ is a positive integral combination of $f_1, \\ldots , f_s$ and $f_1, \\ldots , f_s$ are linearly independent in $\\mathbb {R}^n$ .",
"Moreover, we have $\\langle f, f_i \\rangle \\le 0$ and $\\langle f_i, f \\rangle \\le 0$ for all $1 \\le i \\le s$ .",
"These conditions imply that $1=\\langle f, f \\rangle = \\langle f, a_1f_1 + \\cdots + a_sf_s \\rangle \\le 0,$ a contradiction.",
"There is a special case of interest, which is when there is some ray $[r]$ such that all $\\sigma _d$ -stable dimension vectors, except possibly the ones on $[s]$ , lie on extremal rays (this is the case when $d=\\delta $ where $\\delta $ is an isotropic Schur root).",
"In such a case, either all $\\sigma _d$ -stable dimension vectors lie on the boundary of the cone $C_\\mathbb {R}(\\sigma _d)$ or else $[s]$ lies in the interior of $C_\\mathbb {R}(\\sigma _d)$ .",
"Let $v_1, v_2, \\ldots , v_r$ be the extremal rays of $C_\\mathbb {R}(\\sigma _d)$ .",
"Take $f$ any dimension vector lying in $C_\\mathbb {R}(\\sigma _d)$ but that is neither in an extremal ray nor in the ray $[s]$ .",
"Then we know that $f$ has at least one $\\sigma _d$ -stable factor that lies on an extremal ray.",
"Since the $\\sigma _d$ -stable factors of $f$ can be ordered to form an orthogonal sequence of Schur roots, we see that there exists a Schur root $\\alpha $ (that corresponds to an extremal ray $v_i$ ) such that either $\\langle v_i, f \\rangle > 0$ or $\\langle f, v_i \\rangle > 0$ .",
"If $\\langle v_i, f \\rangle > 0$ , then $\\langle v_i, x \\rangle = 0$ defines an hyperplane cutting $C_\\mathbb {R}(\\sigma _d)$ such that $f, v_i$ lie on the same side while all other extremal rays $v_1, \\ldots , v_{i-1}, v_{i+1}, \\ldots , v_r$ of $C_\\mathbb {R}(\\sigma _d)$ lie on the other side or on the boundary of that hyperplane.",
"We get a similar situation if $\\langle f, v_i \\rangle > 0$ by considering the hyperplane $\\langle x, v_i \\rangle = 0$ .",
"For $1 \\le i \\le r$ , let $C_i(\\sigma _d)$ be the cone in $\\mathbb {R}^{n-1}$ generated by all the rays $v_1, \\ldots , v_r$ but $v_i$ .",
"By the above observation, we have that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)):=\\cap _{1 \\le i \\le r}C_i(\\sigma _d)$ is either empty or else contains only the ray $[s]$ .",
"We will see that this restriction yield a very beautiful description of $C_\\mathbb {R}(\\sigma _d)$ and, in particular, of $C_\\mathbb {R}(\\sigma _\\delta )$ .",
"Proposition 5.2 We have that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ if and only if $C_\\mathbb {R}(\\sigma _d)$ is the cone over a simplex.",
"The sufficiency is easy to see.",
"Assume that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ .",
"We may work in $\\mathbb {R}^{j}$ where $j \\le n-1$ and assume that $C_\\mathbb {R}(\\sigma _d)$ is of full dimension in $\\mathbb {R}^{j}$ .",
"By Radon's theorem, if the number of extremal rays $r$ of $C_\\mathbb {R}(\\sigma _d)$ is at least $(j-1)+2 = j+1$ , then we can partition the rays $v_1, \\ldots , v_r$ into two non-empty subsets $A,B$ such that the corresponding cones $C(A)$ and $C(B)$ generated by the rays in $A$ and by the rays in $B$ have a ray of intersection.",
"This ray of intersection will have to be in ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ , a contradiction.",
"Therefore, $r \\le j$ .",
"Since $C_\\mathbb {R}(\\sigma _d)$ is of full dimension in $\\mathbb {R}^{j}$ , then $r=j$ and $C_\\mathbb {R}(\\sigma _d)$ is the cone over an $(j-1)$ -simplex.",
"Now, we are interested in the case where ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d))$ is reduced to a single ray $[s]$ (which then has to be the ray of $\\bar{\\delta }$ if $d = \\delta $ ).",
"Let us take an affine slice $\\Delta $ of $C_\\mathbb {R}(\\sigma _d)$ .",
"The rays $v_1, \\ldots , v_r$ will correspond to points $u_1, \\ldots , u_r$ in $\\Delta $ and these points are the vertices of a polyhedron $\\mathcal {R}$ in $\\Delta $ defined as the convex hull of $u_1, \\ldots , u_r$ .",
"The ray $[s]$ corresponds to a point $s$ in $\\mathcal {R}$ .",
"In order to study the convex properties of $\\mathcal {R}$ , let us translate $\\mathcal {R}$ so that $s$ coincides with the origin.",
"In other words, set $w_i = u_i - s$ and consider the polyhedron $\\mathcal {P}$ which is the convex hull of $w_1, \\ldots , w_r$ .",
"Since $\\mathcal {R}$ lies on an affine slice, we see that $\\mathcal {P}$ lies in a subspace of dimension $n-2$ of $\\mathbb {R}^{n-1}$ .",
"Let $\\mathcal {P}_i$ be the convex hull of all points $w_1, \\ldots , w_r$ but $w_i$ .",
"We define ${\\rm Proper}(\\mathcal {P}):=\\cap _{1 \\le i \\le r}\\mathcal {P}_i$ and we will be interested in the case where ${\\rm Proper}(\\mathcal {P})$ only contains the origin.",
"The first two lemmas are easy to prove.",
"Lemma 5.3 Let $\\mathcal {P}^{\\prime }$ be the convex hull of a subset of $w_1, \\ldots , w_r$ .",
"Then ${\\rm Proper}(\\mathcal {P}^{\\prime })$ is either empty or reduced to the origin.",
"Lemma 5.4 Consider a non-trivial partition $\\lbrace w_{i_1}, \\ldots , w_{i_s}\\rbrace = A_1 \\cup A_2$ of a subset of $\\lbrace w_1, \\ldots , w_r\\rbrace $ .",
"Denote by $\\mathcal {P}_{A_i}$ the convex hull of the points in $A_i$ , for $i=1,2$ .",
"Then $\\mathcal {P}_{A_1} \\cap \\mathcal {P}_{A_2}$ is either empty or reduced to the origin.",
"Proposition 5.5 Suppose that ${\\rm Proper}(\\mathcal {P})$ is empty or reduced to the origin and is full dimensional in $\\mathbb {R}^{t}$ with $t \\le n-2$ .",
"Then there exists a vector space decomposition $\\mathbb {R}^{t} = V_1 \\oplus \\cdots \\oplus V_s$ of $\\mathbb {R}^{t}$ such that if $V_i$ has dimension $d_i$ , then it contains $d_i + 1$ points among $0, w_1, \\ldots , w_r$ that form a $d_i$ -simplex in $V_i$ containing the origin.",
"We may assume that $t = n-2$ so that $\\mathcal {P}$ is $(n-2)$ -dimensional.",
"If ${\\rm Proper}(\\mathcal {P})$ is empty, then $s=1$ and the result follows from Proposition REF .",
"Assume that ${\\rm Proper}(\\mathcal {P})$ is reduced to the origin.",
"Suppose first that the origin lies on a facet, say $F$ , of $\\mathcal {P}$ .",
"We claim that $F$ contains $r-1$ of the points $w_1, \\ldots , w_r$ .",
"Assume otherwise.",
"Consider an $(n-3)$ -simplex in $F$ generated by points $w_{i_1}, \\ldots , w_{i_{n-2}}$ .",
"Let $u, v \\in \\lbrace w_1, \\ldots , w_r\\rbrace $ be two distinct points not in $F$ .",
"By Radon's theorem, we can partition the points $\\lbrace w_{i_1}, \\ldots , w_{i_{n-2}}, u, v\\rbrace $ into two non-empty subsets $A_1, A_2$ such that $\\mathcal {P}_{A_1} \\cap \\mathcal {P}_{A_2} \\ne \\emptyset $ , where $\\mathcal {P}_{A_i}$ denotes the convex hull of the points in $A_i$ .",
"By Lemma REF , this intersection is the origin and hence lies on $F$ .",
"Since $F$ is a facet, $u,v$ lie on the same side of $F$ .",
"Therefore, for $i=1,2$ , the set $B_i:=A_i\\backslash \\lbrace u,v\\rbrace $ is not empty.",
"Now, $B_1$ and $B_2$ form a partition of $w_{i_1}, \\ldots , w_{i_{n-2}}$ such that $\\mathcal {P}_{B_1} \\cap \\mathcal {P}_{B_2} \\ne \\emptyset $ , where $\\mathcal {P}_{B_i}$ denotes the convex hull of the points in $B_i$ .",
"This contradicts Proposition REF .",
"Now, let us assume that the origin lies in the interior of $\\mathcal {P}$ .",
"By Radon's theorem, we can write $\\lbrace w_1, \\ldots , w_r\\rbrace = E_1 \\cup E_2$ where $E_1,E_2$ are disjoint and non-empty such that $\\mathcal {P}_{E_1} \\cap \\mathcal {P}_{E_2} = \\lbrace 0\\rbrace $ , where $\\mathcal {P}_{E_i}$ denotes the convex hull of the points in $E_i$ .",
"By Carathéodory's theorem, there is a simplex formed by some points $z_1, \\ldots , z_s$ in $E_1$ that contains the origin in its interior.",
"With no loss of generality, assume that $z_i = w_i$ and $s \\le r-1$ .",
"Let $V_1$ be the vector space spanned by the points $w_1, \\ldots , w_s$ and consider the vector space $V_2$ spanned by the points $w_{s+1}, \\ldots , w_r$ .",
"Let $C_1$ be the convex hull of the points $w_1, \\ldots , w_s$ and let $C_2$ be the convex hull of the points $w_{s+1}, \\ldots , w_r$ .",
"Since both $\\mathcal {P}_{E_1}, \\mathcal {P}_{E_2}$ contain the origin, we see that $C_1 \\cap C_2 = \\lbrace 0\\rbrace $ .",
"Since 0 lies in the interior of $C_1$ , we get also that $V_1 \\cap C_2=0$ .",
"We claim that $V_1 \\cap V_2 = 0$ .",
"Assume that $V_1 \\cap V_2$ is non-zero.",
"Observe that any element in $V_1$ can be written as a non-negative linear combination of $w_1, \\ldots , w_s$ .",
"There exists non-negative real numbers $a_1, \\ldots , a_s$ and real numbers $b_{s+1}, \\ldots , b_r$ such that $a_1w_1 + \\cdots + a_sw_s = b_{s+1}w_{s+1} + \\cdots + b_rw_r.$ Moreover, $a_1w_1 + \\cdots + a_sw_s$ is non-zero.",
"We may assume the $a_i$ small enough so that the left-hand side lies in $C_1$ .",
"Let us write $\\lbrace s+1, \\ldots , r\\rbrace = I_1 \\cup I_2$ where $I_1, I_2$ are disjoint and $i \\in I_1$ if and only if $b_i \\ge 0$ .",
"We may assume further that the $|b_i|$ are small enough so that both $\\sum _{i \\in I_1}b_iw_i, -\\sum _{j \\in I_2}b_jw_j$ lie in $C_2$ .",
"If $I_2 = \\emptyset $ , then $a_1w_1 + \\cdots + a_sw_s \\in C_1 \\cap C_2$ is non-zero, a contradiction.",
"If all $b_i$ are non-positive, then $-\\sum _{j \\in I_2}b_jw_j$ lies in $V_1\\cap C_2=\\lbrace 0\\rbrace $ , a contradiction.",
"If some $b_i$ are negative and some $b_i$ are positive, we can rewrite the sum as $a_1w_1 + \\cdots + a_sw_s + -\\sum _{j \\in I_2}b_jw_j= \\sum _{j \\in I_1}b_jw_j.$ Considering Lemma REF with the partition $(\\lbrace w_1, \\ldots , w_s\\rbrace \\cup \\lbrace w_{i}\\mid i \\in I_2\\rbrace ) \\cup (\\lbrace w_{j} \\mid j \\in I_1\\rbrace )$ of $\\lbrace w_1, \\ldots , w_r\\rbrace $ , we get that $a_1w_1 + \\cdots + a_sw_s + -\\sum _{j \\in I_2}b_jw_j$ is zero, which reduces to a case we have already considered.",
"Therefore, we have proven that $\\mathbb {R}^{n-2} = V_1 \\oplus V_2$ , where $V_1$ satisfies the property of the statement.",
"We proceed by induction on $V_2$ with the points $w_{s+1}, \\ldots , w_r$ and by using Lemma REF .",
"The ring of semi-invariants of an isotropic Schur root In this section, we denote by $\\delta $ an isotropic Schur root and by $\\sigma = \\sigma _\\delta $ the weight given by $-\\langle -, \\delta \\rangle $ .",
"Consider, as previously, a full exceptional sequence $(M_{n-2}, \\ldots , M_2, M_1, V,W)$ where $(M_{n-2}, \\ldots , M_2, M_1)$ is an exceptional sequence of simple objects in $\\mathcal {A}(\\delta )$ .",
"Take $I \\subseteq \\lbrace 1, \\ldots , n-2\\rbrace $ such that $i \\in I$ if and only if $i$ is tame connected.",
"Definition 6.1 The associated tame subcategory of $Q$ relative to $\\delta $, denoted $\\mathcal {R}(Q,\\delta )$ , is the thick subcategory of ${\\rm rep}(Q)$ generated by $(\\bigoplus _{i \\in I}M_i)\\oplus V_{n-2}\\oplus W_{n-2}$ .",
"Theorem 6.2 Let $Q$ be an acyclic connected quiver and $\\delta $ an isotropic Schur root.",
"Then The category $\\mathcal {R}(Q,\\delta )$ is tame connected with isotropic Schur root $\\bar{\\delta }$ and is uniquely determined by $\\delta $ .",
"The simple objects in $\\mathcal {A}(\\delta )$ , up to isomorphism, are given by the disjoint union $\\lbrace M_i \\mid i \\notin I\\rbrace \\cup \\lbrace \\text{quasi-simple objects in} \\; \\mathcal {R}(Q,\\delta )\\rbrace .$ We have ${\\rm SI}(Q, \\delta ) \\cong {\\rm SI}(\\mathcal {R}(Q,\\delta ), \\bar{\\delta })[x_{r+1}, \\ldots , x_n].$ Let $(M_{n-2}, \\ldots , M_1)$ be an exceptional sequence of $\\sigma _\\delta $ -stable representations with the corresponding full exceptional sequence $(M_{n-2}, \\ldots , M_1, V, W)$ in ${\\rm rep}(Q)$ , where $\\delta = d_V + d_W$ .",
"First, denote by $M_{l_r}, \\ldots , M_{l_1}$ with $l_r > \\cdots > l_1$ the $M_j$ such that $j$ is tame disconnected or such that $M_j$ is preprojective in $j$ .",
"We get an exceptional sequence $(*) \\qquad (N_{t}, \\ldots , N_2, N_1, V,W, M_{l_r}^{\\prime }, \\ldots , M_{l_1}^{\\prime })$ where all $M_{l_j}$ have been reflected, one by one, to the right of the exceptional sequence.",
"Observe that $\\lbrace M_i \\mid i \\in I\\rbrace \\subseteq \\lbrace N_1, \\ldots , N_t\\rbrace $ .",
"Let $ \\lbrace N_t, \\ldots , N_1\\rbrace \\backslash \\lbrace M_i \\mid i \\in I\\rbrace :=\\lbrace N_{j_s},\\ldots , N_{j_1}\\rbrace $ where $j_s > \\cdots > j_1$ .",
"Assume also that $I = \\lbrace m_1, \\ldots , m_q\\rbrace $ with $m_q > \\cdots > m_1$ .",
"We have $q+s=t$ and $t+r=n-2$ .",
"By Lemma REF (2), we may reflect all exceptional objects of $\\lbrace N_{j_s},\\ldots , N_{j_1}\\rbrace $ in $(*)$ so that we get an exceptional sequence $(M_{m_q}, \\ldots , M_{m_1}, N_{j_s},\\ldots ,N_{j_1}, V, W, M_{l_r}^{\\prime }, \\ldots , M_{l_1}^{\\prime }).$ Now, it follows from the definition of the $V_i, W_i$ that we get an exceptional sequence $(M_{m_q}, \\ldots , M_{m_1}, V_{n-2},W_{n-2}, N_{j_s},\\ldots ,N_{j_1}, M_{l_1}^{\\prime }, \\ldots , M_{l_r}^{\\prime }).$ We claim that for $1 \\le u \\le q$ , we have that ${m_u}=M_{m_u},V_{m_u},W_{m_u})$ is equivalent to $M_{m_u},V_{n-2},W_{n-2})$ .",
"Fix such a $u$ .",
"Note that there is an exceptional sequence of the form $(N_{j_s}, \\ldots , N_{j_p}, M_{m_u},V_{m_u},W_{m_u}).$ Now, it follows from Lemma REF (2) that $M_{m_u}$ lies in $^\\perp N_{j_i}$ for all $1 \\le i \\le p$ .",
"By reflecting, we get the exceptional sequence $(M_{m_u},V_{n-2},W_{n-2}, N_{j_s}, \\ldots , N_{j_p}).$ It follows that ${m_u}=M_{m_u},V_{m_u},W_{m_u})$ is equivalent to $M_{m_u},V_{n-2},W_{n-2})$ .",
"This proves our claim.",
"Let $E = (M_{m_q}, \\ldots , M_{m_1}, V_{n-2},W_{n-2})$ .",
"By Lemma REF and our claim, $\\mathcal {R}(Q,\\delta )=E)$ is tame connected.",
"Since $V_{n-2}, W_{n-2}$ lie in it, $\\bar{\\delta }$ is the (unique) isotropic Schur root of $\\mathcal {R}(Q,\\delta )$ .",
"It follows from Lemma REF that any $\\sigma _\\delta $ -stable representation not isomorphic to any $M_i$ for $1 \\le i \\le n-2$ will have to be (quasi-simple) in $E)$ .",
"Now, we need to show that all quasi-simple objects of $E)$ are $\\sigma _\\delta $ -stable.",
"Assume the contrary.",
"Let $f$ be the dimension vector of a quasi-simple object in $E)$ that is not $\\sigma _\\delta $ -stable, but $\\sigma _\\delta $ -semistable.",
"It follows from our previous observations that $f$ has to be a positive integral combination of the $\\sigma _\\delta $ -stable dimension vectors in $E)$ .",
"It follow from [11] that this is not possible to have such a decomposition.",
"Therefore, we have a complete list of the simple objects in $\\mathcal {A}(\\delta )$ .",
"These are given by the disjoint union $\\lbrace M_i \\mid i \\notin I\\rbrace \\cup \\lbrace \\text{quasi-simple objects in} \\; \\mathcal {R}(Q,\\delta )\\rbrace .$ Observe that, in $C_\\mathbb {R}(\\sigma _\\delta )$ , a dimension vector $d$ can be uniquely written as $d = d_1 + \\sum _{i \\notin I} \\lambda _i f_i$ where $d_1$ is a dimension vector in $E)$ and $f_i = d_{M_{m_i}}$ for $i \\notin I$ .",
"This decomposition is unique.",
"This implies the unicity of $\\mathcal {R}(Q,\\delta )$ and statement $(3)$ .",
"Corollary 6.3 Let $Q$ be an acyclic connected quiver and $\\delta $ an isotropic Schur root.",
"Then ${\\rm SI}(Q, \\delta )$ is a polynomial ring or a hypersurface.",
"More precisely, it is a hypersurface (and not a polynomial ring) if and only if $\\mathcal {R}(Q,\\delta )$ has quiver of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"In [20], it was proven that the ring of semi-invariant of an isotropic Schur root of a tame quiver is a polynomial ring or a hypersurface, where the second situation occurs precisely when the quiver is of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"Our result follows from this and Theorem REF .",
"Example 6.4 Consider the quiver $Q$ given by ${7pt}{& 2 [dl] & \\\\ 1 & & 4 [ul] [ll] [dl]\\\\ & 3 [ul] &}$ Consider the exceptional sequence $(P_2, S_1, I_3, S_3)$ where $P_2$ is the projective representation at vertex 2, $I_3$ is the injective representation at vertex 3 and $S_1, S_3$ are the simple representations at vertices $1,3$ , respectively.",
"Reflecting $S_1, I_3$ to the left of $P_2$ , we get an exceptional sequence whose dimension vectors are as follows.",
"$((0,1,0,0), (3, 3, 1, 1), (1,1,0,0), (0,0,1,0)).$ Then, using a sequence of reflections, we get the following exceptional sequences, where we put the corresponding dimension vectors.",
"$((0,1,0,0), (3, 3, 1, 1), (0,0,1,0), (1,1,1,0))$ $((0,1,0,0), (0,0,1,0), (3, 3, 3, 1), (1,1,1,0))$ $((0,0,1,0), (0,1,0,0), (3, 3, 3, 1), (1,1,1,0))$ $((0,0,1,0), (0,1,0,0), (8,8,8,3), (3, 3, 3, 1))$ $((0,0,1,0), (8,3,8,3), (0,1,0,0), (3, 3, 3, 1))$ $((8,3,3,3), (0,0,1,0), (0,1,0,0), (3, 3, 3, 1)).$ Observe that $\\langle (3,3,3,1),(0,1,0,0) \\rangle =2$ and $\\delta = (3,3,3,1) - (0,1,0,0) = (3,2,3,1)$ is an isotropic Schur root.",
"The Coxeter matrix $\\tau $ is $\\tau = \\left(\\begin{array}{cccc}-1 & 1 & 1 & 1 \\\\-1 & 0 & 1 & 2 \\\\-1 & 1 & 0 & 2 \\\\-3 & 2 & 2 & 4 \\\\\\end{array}\\right).$ This matrix has eigenvalues $\\lambda = 5/2 +\\sqrt{21}/2$ , $\\lambda ^{-1} = 5/2-\\sqrt{21}/2$ and $-1$ with (algebraic and geometric) multiplicity 2.",
"The eigenvector corresponding to $\\lambda $ is $v_1 = (10, 9+\\sqrt{21}, 9+\\sqrt{21}, 17 + \\sqrt{189})$ and the one corresponding to $\\lambda ^{-1}$ is $v_2 = (10, 9-\\sqrt{21}, 9-\\sqrt{21}, 17 - \\sqrt{189}).$ Now, $\\langle v_2, (8,3,3,3) \\rangle = -197 + 10\\sqrt{21} + 11\\sqrt{189} > 0$ and $\\langle v_2, (0,0,1,0) \\rangle = -8-\\sqrt{21}+\\sqrt{189} > 0$ .",
"Similarly, both $\\langle (8,3,3,3), v_1 \\rangle $ and $\\langle (0,0,1,0), v_1 \\rangle $ are positive.",
"Therefore, the exceptional objects with dimension vectors $(8,3,3,3), (0,0,1,0)$ are regular by the theorem at page 240 of [16].",
"It follows from Proposition REF that $\\delta $ is not of smaller type.",
"It also follows from the same proposition that there is a unique exceptional sequence $(M_2, M_1)$ of length 2 of $\\sigma $ -stable objects.",
"Let $M_1^{\\prime } = S_3$ and $M_2^{\\prime }$ be the exceptional representation with dimension vector $(8,3,3,3)$ .",
"Since $M_1^{\\prime }, M_2^{\\prime }$ lie in $M_2,M_1)$ by Lemma REF , we see that $M_2^{\\prime },M_1^{\\prime }) \\subseteq M_2,M_1)$ and thus, we have equality.",
"This means that $M_2^{\\prime } = M_2$ , $M_1^{\\prime } = M_1$ .",
"Since $\\langle \\delta , (0,0,1,0) \\rangle = 2 >0$ , we get $\\delta _1 = \\delta - 2(0,0,1,0) = (3,2,1,1)$ .",
"Now, $\\langle \\delta _1, (8,3,3,3) \\rangle = -2$ .",
"Therefore, $\\bar{\\delta }= \\delta _1 = (3,2,1,1)$ .",
"In this example, the cone of $\\sigma $ -semistable dimension vectors is as follows (where only an affine slice of that cone is shown).",
"Figure: The cone of σ\\sigma -semistable dimension vectors for δ=(3,2,3,1)\\delta = (3,2,3,1)The following is an easy observation.",
"The reader is referred to [8] for the notion of cluster algebra and to [7] for results in similar directions.",
"Corollary 6.5 If ${\\rm SI}(Q,\\delta )$ is not a polynomial ring, then it has a cluster algebra structure of type $\\mathbb {A}_1$ .",
"There are two cluster variables which are all $\\Gamma $ -homogeneous, and the coefficients are built from $n-1$ frozen variables, which are also $\\Gamma $ -homogeneous, where $\\Gamma $ is the set of all multiplicative characters of ${\\rm GL}_\\delta (k)$ .",
"From Theorem REF , it is enough to prove this for ${\\rm rep}(Q) = \\mathcal {R}(Q,\\delta )$ , that is, we may assume that $Q$ is tame connected.",
"Suppose that ${\\rm SI}(Q,\\delta )$ is not a polynomial ring.",
"Then $Q$ is of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"In particular, it is well known in these cases that there are exactly three non-homogeneous tubes $T_1, T_2, T_3$ in the Auslander-Reiten quiver of $\\mathcal {R}(Q,\\delta )$ .",
"One, say $T_1$ , has rank 2.",
"Let $M,N$ be the non-isomorphic exceptional quasi-simple objects in $T_1$ .",
"Then, let $E_1, \\ldots , E_r$ be the non-isomorphic quasi-simple objects of $T_2$ and let $E_1^{\\prime }, \\ldots , E_t^{\\prime }$ be the non-isomorphic quasi-simple objects of $T_3$ .",
"Now, the hypersurface equation can be written as $(*) \\qquad C^MC^N = C^{E_1}\\cdots C^{E_r} + C^{E_1^{\\prime }}\\cdots C^{E_t^{\\prime }}.$ Consider the indeterminates $x,y_1, \\ldots , y_r, z_1, \\ldots , z_t$ .",
"We define a cluster algebra $A$ as follows.",
"We start with the initial seed $\\lbrace x,y_1, \\ldots , y_r, z_1, \\ldots , z_t\\rbrace $ where $y_1, \\ldots , y_r$ and $z_1, \\ldots , z_t$ are declared to be frozen variables.",
"The exchange relation is $xx^{\\prime } = \\prod _{i=1}^ry_i + \\prod _{j=1}^tz_j$ which clearly produces exactly two cluster variables $x,x^{\\prime }$ .",
"The cluster algebra is the the $\\mathbb {Z}$ -subalgebra of $\\mathbb {Q}(x,y_1, \\ldots , y_r, z_1, \\ldots , z_t)$ generated by $x, x^{\\prime }$ and $y_1, \\ldots , y_r, z_1, \\ldots , z_t$ .",
"This algebra is clearly isomorphic to ${\\rm SI}(Q,\\delta )$ .",
"An interesting problem would be to find all acyclic quivers $Q$ and dimension vectors $d$ such that SI$(Q,d)$ has a cluster algebra structure whose variables (frozen or not) are all $\\Gamma $ -homogeneous.",
"Construction of all isotropic Schur roots In this section, we show that all of the isotropic Schur roots of ${\\rm rep}(Q)$ come from isotropic Schur roots of a tame full subquiver of $Q$ by applying special reflections.",
"We make this precise by defining an action of the braid group $B_{n-1}$ on $n-1$ strands on a special type of exceptional sequences that will encode all we need to study isotropic Schur roots.",
"We start with the definition of these sequences.",
"Definition 7.1 Let $E=(X_1, \\ldots , X_n)$ be a full exceptional sequence.",
"We say that $E$ is of isotropic type if there exists $1 \\le i \\le n-1$ such that $X_{i}, X_{i+1})$ is tame.",
"The integer $i$ is called the isotropic position of $E$ and the root type of $E$ , denoted $\\delta _E$ , is the isotropic Schur root in $X_{i}, X_{i+1})$ .",
"We denote by $\\mathcal {E}$ the set of all full exceptional sequences of isotropic type, up to isomorphism.",
"Not all elements of the braid group $B_{n}$ act on $\\mathcal {E}$ .",
"We rather consider the group $B_{n-1}$ and show that it acts on $\\mathcal {E}$ .",
"Let us denote the standard generators of $B_{n-1}$ by $\\gamma _1, \\ldots , \\gamma _{n-2}$ .",
"Let $E=(X_1, \\ldots , X_n) \\in \\mathcal {E}$ with isotropic position $r$ .",
"Let $1 \\le i \\le n-2$ .",
"If $i<r-1$ , then $\\gamma _iE := \\sigma _iE$ .",
"If $i>r$ , then $\\gamma _{i}E := \\sigma _{i+1}E$ .",
"Assume that $i=r$ with $r<n-1$ .",
"We can reflect $X_{r+2}$ to the left of $X_r, X_{r+1}$ to get the exceptional sequence: $E^{\\prime }=(X_1, \\ldots ,X_{r-1}, L_{X_{r}}(L_{X_{r+1}}(X_{r+2})), X_r, X_{r+1}, X_{r+3}, \\ldots , X_n).$ and this is an exceptional sequence of isotropic type with isotropic position $r+1$ .",
"We define $\\gamma _rE:=E^{\\prime }$ .",
"If $r > 1$ and $i=r-1$ , then we can reflect both $X_r, X_{r+1}$ to the left of $X_{r-1}$ as follows: $E^{\\prime \\prime } = (X_1, \\ldots ,X_{r-2}, L_{X_{r-1}}(X_r), L_{X_{r-1}}(X_{r+1}), X_{r-1}, X_{r+2}, \\ldots , X_n)$ and clearly, the subcategory $L_{X_{i-1}}(X_i), L_{X_{i-1}}(X_{i+1}))$ generates a tame subcategory of rank 2.",
"Therefore, $E^{\\prime \\prime }$ is an exceptional sequence of isotropic type with isotropic position $r-1$ and its root type is the unique isotropic Schur root in $L_{X_{i-1}}(X_i), L_{X_{i-1}}(X_{i+1}))$ , which is $\\delta _{E^{\\prime }} = \\delta _E - \\langle \\delta _E, d_{X_{r-1}}\\rangle d_{X_{r-1}}$ , by Lemma REF .",
"We define $\\gamma _{r-1}E=E^{\\prime \\prime }$ .",
"Similarly, we can define the action of $\\gamma _{i}^{-1}$ on $E$ for $1 \\le i \\le n-2$ .",
"The following is easy to check.",
"Proposition 7.2 The group $B_{n-1}$ acts on exceptional sequences of isotropic type, with the action defined above.",
"Definition 7.3 A sequence $E=(X_1, \\ldots , X_{n-1}, X_n)$ in $\\mathcal {E}$ is of tame type if it has isotropic position $n-1$ , and there is $0 \\le s \\le n-2$ such that $X_1, \\ldots , X_s$ are projective in ${\\rm rep}(Q)$ and $X_{s+1}, \\ldots , X_{n-2}, X_{n-1}, X_n)$ is tame connected.",
"By convention, $s=0$ means that ${\\rm rep}(Q)$ is already tame connected.",
"Observe that if $E \\in \\mathcal {E}$ is of tame type, then the isotropic Schur root $\\delta _E$ is the unique isotropic Schur root of the tame subcategory $X_{s+1}, \\ldots , X_{n-2}, X_{n-1}, X_n)$ and is an isotropic Schur root coming from a tame full subquiver of $Q$ .",
"In particular, there are finitely many roots $\\delta _E$ where $E \\in \\mathcal {E}$ is of tame type.",
"Example 7.4 Consider a quiver of rank $n=4$ and an exceptional sequence $E=(X,U,V,Y)$ of isotropic type with isotropic position 2.",
"The root type is the isotropic root $\\delta _E$ in $U,V)$ .",
"Figure: Correspondence between B 3 B_3 and some braids of B 4 B_4The first braid (A) in Figure REF corresponds to the element $g=\\gamma _2^{-1}\\gamma _1^{-1}\\gamma _2\\gamma _1^{-1}$ of $B_3$ while the second braid (B) corresponds to the element $h=\\sigma _2^{-1}\\sigma _1^{-1}\\sigma _3^{-1}\\sigma _2^{-1}\\sigma _3\\sigma _2^{-1}\\sigma _1^{-1}$ of $B_4$ .",
"Notice that $gE = hE$ .",
"Notice also that the braid in (A) is obtained from the braid in (B) by identifying the two strands starting at the positions of $U,V$ , that is, the second and third strands.",
"Our aim in this section is to prove that any $E \\in \\mathcal {E}$ lies in the $B_{n-1}$ -orbit of an exceptional sequence of tame type.",
"In the next lemmas, we will consider exceptional sequences in the bounded derived category $D^b({\\rm rep}(Q))$ of ${\\rm rep}(Q)$ .",
"Recall that an object $X$ in $D^b({\\rm rep}(Q))$ is exceptional if ${\\rm Hom}(X,X[i])=0$ for all non-zero $i$ (and then, ${\\rm Hom}(X,X)$ has to be one dimensional).",
"Equivalently, an exceptional object in $D^b({\\rm rep}(Q))$ is isomorphic to the shift of an exceptional representation.",
"A sequence $(X_1, \\ldots , X_r)$ of objects in $D^b({\\rm rep}(Q))$ is exceptional if every $X_i$ is exceptional and, for $i < j$ , we have ${\\rm Hom}_{D^b({\\rm rep}(Q))}(X_i, X_j[t])=0$ for all $t \\in \\mathbb {Z}$ .",
"For such a sequence, one can consider the smallest full additive subcategory $\\mathcal {D}(X_1, \\ldots , X_r)$ of $D^b({\\rm rep}(Q))$ containing $X_1, \\ldots , X_r$ and that is closed under direct sums, direct summands, taking the cone of a morphism and the shift of an object.",
"One can also consider the exceptional sequence $(X_1^{\\prime }, \\ldots , X_r^{\\prime })$ in ${\\rm rep}(Q)$ such that $X_i^{\\prime }$ is the unique shift of $X_i$ lying in ${\\rm rep}(Q)$ .",
"The indecomposable objects in $\\mathcal {D}(X_1, \\ldots , X_r)$ are just the shifts of the indecomposable objects in $X_1^{\\prime }, \\ldots , X_r^{\\prime })$ .",
"In what follows, the Auslander-Reiten translate in $D^b({\\rm rep}(Q))$ is denoted by $\\tau _{D}$ while the Auslander-Reiten translate in ${\\rm rep}(Q)$ is simply denoted $\\tau $ .",
"Recall that if $X$ is a non-projective indecomposable representation, then $\\tau _DX = \\tau X$ and, if $X = P_x$ with $x \\in Q_0$ , then $\\tau _D X = I_x[-1]$ .",
"When $d$ is a dimension vector, we denote by $\\tau d$ the product of the Coxeter matrix with $d$ .",
"In particular, if $X$ is a non-projective indecomposable representation, then $\\tau d_X = d_{\\tau X}$ and, if $X=P_x$ with $x \\in Q_0$ , then $\\tau d_{X} = -d_{I_x}$ .",
"We start our investigation with the following lemma that is crucial for the proof of the main result of this section.",
"Lemma 7.5 Let $(X_1, \\ldots , X_{n})$ be an exceptional sequence with $X_{r+1}, \\ldots , X_{n})$ tame and assume that $X_1, \\ldots , X_r$ are the simple objects in $X_1, \\ldots , X_r)$ .",
"Let $X \\in X_1, \\ldots , X_r)$ be the injective object with socle $X_1$ .",
"If $X$ is projective in $X,X_{r+1}, \\ldots , X_{n})$ , then $X_1$ is projective in ${\\rm rep}(Q)$ and in particular, an isotropic Schur root of $X_{r+1}, \\ldots , X_{n} )$ is not sincere.",
"Assume that $X$ is projective in $X,X_{r+1}, \\ldots , X_{n})$ .",
"Set $d_i = d_{X_i}$ for $1 \\le i \\le n$ .",
"Consider the linear form $f$ given by $f(x)=\\langle d_1, x \\rangle $ .",
"Then $f$ vanishes on $d_2, \\ldots , d_{n}$ and $f(d_1)>0$ .",
"Assume to the contrary that $X_1$ is not projective in ${\\rm rep}(Q)$ .",
"Observe that $f(x)=\\langle d_1, x \\rangle = -\\langle x, \\tau d_1 \\rangle $ .",
"Since $\\tau X_1$ is exceptional, $\\langle \\tau d_1, \\tau d_1 \\rangle = 1$ and hence $f(\\tau d_1) < 0$ .",
"Now, reflect $X_1$ to the right of $X_2, \\ldots X_r$ , so that we get an exceptional sequence $(X_2, \\ldots , X_r, Y)$ where $Y$ is in the cone spanned by $d_1, \\ldots , d_r$ .",
"Clearly, $X_1$ is simple projective in $X_1, \\ldots , X_r)$ and hence, $Y=X$ is the injective hull of $X_1$ in $X_1, \\ldots , X_r)$ .",
"Set $=X, X_{r+1}, \\ldots , X_{n})$ .",
"We know that $X$ is projective in $.",
"Reflecting $ X$ to the right of $ Xr+1, ..., Xn$ will give the exceptional representation $ X1$.",
"Therefore, $ d:=d1 = -d̏X$ where $ denotes the Coxeter transformation in $.",
"Take the linear form $ g$ in the Grothendieck group of $ given by $g(x)=\\langle d_{X}, x \\rangle $ .",
"Then $g$ vanishes on $d_{r+1}, \\ldots , d_{n}$ and $g(d_X)>0$ .",
"The form $f|_ has the same property since $ dX$ is a non-negative linear combination of $ d1, ..., dr$ with the coefficient of $ d1$ positive.",
"Thus, $ g = f| up to a positive scalar.",
"Therefore, $g(d)<0$ , which means that $X,\\tau X_1$ lie on opposite sides of the hyperplane $g(x)=0$ in $.",
"This contradicts that $ X$ is projective in $ .",
"Lemma 7.6 Let $(X_1, \\ldots , X_r)$ be an exceptional sequence and assume that $1:=X_2, \\ldots , X_r)$ is tame with an isotropic Schur root $\\gamma $ while $2:=X_1, X_2, \\ldots , X_r)$ is wild.",
"Then there is a unique minimal isotropic Schur root in the $\\tau $ -orbit of $\\gamma $ .",
"We may assume that $2 = {\\rm rep}(Q)$ for an acyclic quiver $Q$ .",
"Since 2 is wild and 1 is tame, we know that $X_1$ is preprojective or preinjective in 2.",
"Hence, there is some $r \\in \\mathbb {Z}$ such that $\\tau _D^r X_1$ is projective or the shift of a projective.",
"This means that $\\tau ^r\\gamma $ is not sincere.",
"Let $Y = \\tau _D^r X_1$ if $\\tau _D^r X_1$ is a representation or $Y = \\tau _D^r X_1[-1]$ if $\\tau _D^r X_1$ is the shift of a projective representation.",
"Observe that $Y^\\perp \\subseteq {\\rm rep}(Q)$ is also of tame representation type, where the quivers of $X_1^\\perp $ and $Y^\\perp $ only differ by a change of orientation; see for instance [10].",
"Therefore, $\\tau ^r\\gamma $ is an isotropic Schur root of a tame full subquiver of $Q$ .",
"Let $s \\in \\mathbb {Z}$ with $s \\ne r$ .",
"Consider $Z$ the unique shift of $\\tau _D^s X_1$ which is a representation.",
"Since the simples in $Y^\\perp $ are simples in ${\\rm rep}(Q)$ , and since there is a simple of $Z^\\perp \\subseteq {\\rm rep}(Q)$ that is not simple in ${\\rm rep}(Q)$ , we see that the isotropic Schur root $\\tau ^r\\gamma $ has smaller length than $\\tau ^s\\gamma $ .",
"This also proves unicity since only one object in the $\\tau $ -orbit of $X_1$ is projective or a shift of a projective.",
"Lemma 7.7 Let $E=(X_1, \\ldots , X_{n-2}, U, V)$ be in $\\mathcal {E}$ with isotropic position $n-1$ .",
"Let $E^{\\prime } = \\gamma _{1}\\cdots \\gamma _{n-3}\\gamma _{n-2}E = (U^{\\prime }, V^{\\prime }, X_1, \\ldots , X_{n-2})$ .",
"Then $\\tau ^{-1} \\delta _E = \\delta _{E^{\\prime }}$ .",
"If $V$ is not injective, then we have the exceptional sequence $(\\tau ^{-1}V, X_1, \\ldots , X_{n-2},U)$ in ${\\rm rep}(Q)$ .",
"Otherwise, we have the exceptional sequence $(\\tau ^{-1}V[-1], X_1, \\ldots , X_{n-2}, U)$ in ${\\rm rep}(Q)$ .",
"Let us write $\\tau ^{-1}V[0,-1]$ to indicate that we either take the shift $[0]$ or $[-1]$ for $\\tau ^{-1}V$ .",
"Then, we get an exceptional sequence $(\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1], X_1, \\ldots , X_{n-2}).$ The categories $U^{\\prime }, V^{\\prime })$ and $\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1])$ are equal in ${\\rm rep}(Q)$ .",
"Therefore, they have the same isotropic Schur root.",
"The isotropic Schur root of $\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1])$ is clearly $\\tau ^{-1}\\delta _E$ .",
"Of course, we have the dual version of the above lemma as follows.",
"Lemma 7.8 Let $E=(U,V, X_1, \\ldots , X_{n-2})$ be in $\\mathcal {E}$ with isotropic position 1.",
"Let $E^{\\prime } = \\gamma _{n-2}^{-1}\\gamma _{n-3}^{-1}\\cdots \\gamma _{1}^{-1}E = (X_1, \\ldots , X_{n-2}, U^{\\prime }, V^{\\prime })$ .",
"Then $\\tau \\delta _E = \\delta _{E^{\\prime }}$ .",
"We are now ready for the main result of this section.",
"Theorem 7.9 Let $\\delta $ be an isotropic Schur root.",
"Then there is $E \\in \\mathcal {E}$ of tame type and $g \\in B_{n-1}$ such that $gE$ has root type $\\delta $ .",
"It follows from Proposition REF that there is an exceptional sequence $F=(M_1, \\ldots , M_{n-2}, X,Y)$ in $\\mathcal {E}$ of isotropic position $n-1$ and of root type $\\delta $ .",
"Assume that $G\\in \\mathcal {E}$ is in the orbit of $E$ and the root type of $G$ is minimal, that is, has minimal length as a root in ${\\rm rep}(Q)$ .",
"We may assume that the isotropic position of $G$ is $n-1$ .",
"Therefore, we may assume that $G$ is of the form $(Y_1, \\ldots , Y_{n-2}, U, V).$ Assume first that there is an object $W$ in $Y_1, \\ldots , Y_{n-2})$ that is not projective in $W,U,V)$ .",
"We can apply a sequence of reflections to the subsequence $(Y_1, \\ldots , Y_{n-2})$ to get an exceptional sequence $H=(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W, U,V)$ in $\\mathcal {E}$ .",
"Now, applying $\\gamma _{n-2}^2$ to $H$ and using Lemma REF , we get the sequence $(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W^{\\prime }, U^{\\prime }, V^{\\prime })$ in $\\mathcal {E}$ whose root type is the inverse Auslander-Reiten translate of $\\delta $ in $W,U,V)$ .",
"Similarly, applying $(\\gamma _{n-2}^{-1})^2$ to $H$ , we get the sequence $(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W^{\\prime \\prime }, U^{\\prime \\prime }, V^{\\prime \\prime })$ in $\\mathcal {E}$ whose root type is the Auslander-Reiten translate of $\\delta $ in $W,U,V)$ .",
"We can iterate this to get a smaller root by Lemma REF , provided $W,U,V)$ is wild.",
"Therefore, whenever there is an object $W$ which is not projective in $W,U,V)$ , then $W,U,V)$ is of tame type (and hence connected).",
"Suppose, by induction, that we have an exceptional sequence $J=(W_{r+1},\\ldots ,W_{n-2},U,V)$ such that $J)$ is tame connected and $J$ has maximal length with respect to this property.",
"If $r=0$ , then $Q$ is a tame connected quiver and there is nothing to prove.",
"Complete this to get a full exceptional sequence $(Z_1, \\ldots , Z_r, W_{r+1}, \\ldots , W_{n-2}, U,V).$ If there is $W \\in ^\\perp J) = Z_1, \\ldots , Z_r)$ such that $W$ is not projective in $W,U,V)$ then, $W,U,V)$ is tame connected.",
"As in the proof of Lemma REF , we get that $W,W_{r+1}, \\ldots , W_{n-2},U,V)$ is tame connected, contradicting the maximality of $J$ .",
"Therefore, any object $Z$ in $Z_1, \\ldots , Z_r)$ is such that $Z$ is projective in $Z,W_{r+1}, \\ldots , W_{n-2},U,V)$ .",
"We may apply a sequence of reflections and assume that all of $Z_1, \\ldots , Z_r$ are simple in $Z_1, \\ldots , Z_r)$ .",
"It follows from Lemma REF that the injective hull of $Z_1$ in $Z_1, \\ldots , Z_r)$ is projective in ${\\rm rep}(Q)$ .",
"Then the proof goes by induction.",
"Here is another way to interpret this result.",
"Start with an isotropic Schur root $\\delta _0$ of a tame full subquiver $Q^{\\prime }$ of $Q$ and consider an exceptional sequence $(U_0,V_0)$ of length 2 in ${\\rm rep}(Q^{\\prime }) \\subset {\\rm rep}(Q)$ such that $\\delta _0 = d_{U_0} + d_{V_0}$ .",
"Consider an exceptional object $X_0$ such that $(X_0,U_0,V_0)$ is an exceptional sequence of length three (which generates a thick subcategory 0 of ${\\rm rep}(Q)$ ).",
"Then we can transform it into another exceptional sequence $(X_0^{\\prime }, U_1, V_1)$ with an isotropic Schur root $\\delta _1 = d_{U_1}+d_{V_1}$ such that $\\delta _1$ is a power $\\tau _{0}^{r_0} \\delta _0$ where $\\tau _{0}$ denotes the Coxeter matrix for 0.",
"Now, for $i \\ge 1$ , consider an exceptional object $X_i$ such that $(X_i,U_i, V_i)$ is an exceptional sequence.",
"Take a power $\\delta _{i+1}=\\tau _{i}^{r_i} \\delta _i$ where $i$ is the thick subcategory of ${\\rm rep}(Q)$ generated by $X_i, U_i, V_i$ and $\\tau _{i}$ denotes the Coxeter matrix for $i$ .",
"All the roots $\\delta _i$ constructed this way are isotropic Schur roots.",
"Moreover, all isotropic Schur roots of ${\\rm rep}(Q)$ can be obtained in this way.",
"There are clearly only finitely many starting roots $\\delta _0$ , but the choices of the $r_i$ and $X_i$ yield, in general, infinitely many possible isotropic Schur roots.",
"As observed in [15], when $Q$ is wild connected with more than 3 vertices, there are infinitely many $\\tau $ -orbit of isotropic Schur roots (provided there is at least one isotropic Schur root).",
"An interesting question would be to describe the minimal root types of the orbits of $\\mathcal {E}$ under $B_{n-1}$ .",
"It is not hard to check that when $n=3$ , these minimal root types correspond exactly to the tame full subquivers of $Q$ .",
"We do not know if this holds in general.",
"Conjecture 7.10 Let $E_1, E_2 \\in \\mathcal {E}$ .",
"Assume that there are $g_1, g_2 \\in B_{n-1}$ with $g_1E_1, g_2E_2$ of tame type but with different root types.",
"Then $E_1, E_2$ lie in distinct orbits under $B_{n-1}$ .",
"Acknowledgment.",
"The authors are thankful to Hugh Thomas for suggesting the decomposition in Proposition REF .",
"The second named author was supported by NSF grant DMS-1400740."
],
[
"The case of an isotropic Schur root",
"In this section, $\\delta $ stands for an isotropic Schur root.",
"The weight $\\sigma _\\delta $ will simply be denoted $\\sigma $ , when there is no risk of confusion.",
"Our aim is to describe all simple objects in $\\mathcal {A}(\\delta )$ or, equivalently, all $\\sigma _\\delta $ -stable objects.",
"We start with the following proposition; see [15].",
"Proposition 4.1 There exists an exceptional sequence $(V,W)$ in ${\\rm rep}(Q)$ such that $V,W)$ is tame and $\\delta = d_V + d_W$ with $\\langle d_W, d_V \\rangle = -2$ .",
"In particular, $^\\perp V \\cap ^\\perp W \\subseteq \\mathcal {A}(\\delta )$ .",
"As shown in [15], there is an exceptional sequence $E=(V,W)$ of length two such that $\\delta $ is a root in $E)$ .",
"Since $E$ has length two, $E)$ is equivalent to the category of representations of an acyclic quiver $Q_E$ with two vertices.",
"But $Q_E$ has to have an isotropic Schur root.",
"Therefore, $Q_E$ is the Kronecker quiver.",
"With no loss of generality, we may assume that $V,W$ are the simple objects of $E)$ .",
"Therefore, $\\langle d_W, d_V \\rangle = -{\\rm dim}_k{\\rm Ext}^1(V,W)$ and ${\\rm dim}_k{\\rm Ext}^1(V,W)$ is the number of arrows in $Q_E$ .",
"The second part of the statement is trivial.",
"Lemma 4.2 Let $Q$ have at least three vertices.",
"Then there is at least one exceptional simple object in $\\mathcal {A}(\\delta )$ .",
"Suppose that all the $\\sigma $ -stable dimension vectors are isotropic or imaginary.",
"By Proposition REF , we have an exceptional sequence $(V,W)$ , such that $\\delta = d_V + d_W$ .",
"We can extend this to an exceptional sequence $(U,V,W)$ and hence, $U \\in \\mathcal {A}(\\delta )$ .",
"This means that $d_U$ is $\\sigma $ -semistable.",
"Now, we apply Lemma REF .",
"The subcategories of the form $A,B,C)$ where $(A,B,C)$ is an exceptional sequence will play a crucial role in our investigation.",
"The following lemma, which is easy to check, provides a description of the quivers with three vertices having an isotropic Schur root.",
"Lemma 4.3 Let $(A,B,C)$ be an exceptional sequence such that $A,B,C)$ contains $\\delta $ .",
"Then either the category $A,B,C)$ is wild and connected or the category $A,B,C)$ is equivalent to ${\\rm rep}(Q^{\\prime })$ , where $Q^{\\prime }$ is either of type $\\widetilde{\\mathbb {A}}_{2,1}$ or a union of the Kronecker quiver and a single vertex.",
"Let us denote by $\\tau $ the Auslander-Reiten translation in ${\\rm rep}(Q)$ .",
"An indecomposable representation that lies in the $\\tau $ -orbit of a projective (resp.",
"injective) representation is called preprojective (resp.",
"preinjective).",
"Baer and Strau${\\ss }$ have proven the following crucial result; see [1] or [21].",
"Lemma 4.4 (Baer, Strau${\\ss }$ ) Let $Q$ be of wild type and let $X$ be exceptional.",
"If $X^\\perp $ is of finite or tame type, then $X$ has to be preprojective or preinjective.",
"The following result describes a way to produce other isotropic Schur roots starting with an exceptional sequence $(U,V,W)$ where $\\delta = d_V + d_W$ .",
"Lemma 4.5 Let $E=(U,V,W)$ be an exceptional sequence such that $V,W)$ is tame with isotropic Schur root $\\delta = d_V + d_W$ .",
"Reflect both $V,W$ to the left of $U$ to get an exceptional sequence $(V^{\\prime }, W^{\\prime }, U)$ .",
"Let $\\delta ^{\\prime }$ be the unique isotropic Schur root in $V^{\\prime }, W^{\\prime })$ .",
"Then $\\delta ^{\\prime } = \\delta - \\langle \\delta , d_U \\rangle d_U$ .",
"If $U$ is preinjective in $E)$ , then $Q$ is wild connected and $\\langle \\delta , d_U \\rangle \\ge 0$ .",
"If $U$ is preprojective in $E)$ , then $Q$ is wild connected and $\\langle \\delta , d_U \\rangle \\le 0$ .",
"Otherwise, $E)$ is tame, $U$ is regular or is simple projective-injective in $E)$ , $\\langle \\delta , d_U \\rangle = 0$ and $\\delta ^{\\prime } = \\delta $ .",
"We may assume that ${\\rm rep}(Q) = E)$ .",
"If $Q$ is wild, then it follows from Lemma REF that $Q$ is connected.",
"In this case, by Lemma REF , $U$ cannot be regular.",
"Assume first that $Q$ is tame.",
"If $Q$ is tame connected, $Q$ is of type $\\widetilde{\\mathbb {A}}_{2,1}$ and $U$ has to be isomorphic to one of the two quasi-simple regular exceptional representations.",
"If $Q$ is tame disconnected, then we see that $U$ is the simple representation corresponding to the connected component of $Q$ of type $\\mathbb {A}_1$ .",
"So if $Q$ is tame, it is clear that $\\langle \\delta , d_U \\rangle = 0$ and $\\delta = \\delta ^{\\prime }$ .",
"So we may assume that $Q$ is wild connected.",
"We have an orthogonal sequence of Schur roots $(d_U, \\delta )$ .",
"Set $\\delta ^{\\prime \\prime } = \\delta - \\langle \\delta , d_U \\rangle d_U$ .",
"One easily checks that $\\langle \\delta ^{\\prime \\prime }, d_U \\rangle = 0$ and $\\langle \\delta ^{\\prime \\prime }, \\delta ^{\\prime \\prime } \\rangle = 0$ .",
"Since $^\\perp U = V^{\\prime }, W^{\\prime })$ is of tame type, the Euler-Ringel form $\\langle - , - \\rangle _{V^{\\prime }, W^{\\prime })}$ restricted to $V^{\\prime }, W^{\\prime })$ is positive semi-definite.",
"Therefore, $\\delta ^{\\prime \\prime }$ is an integral multiple of the isotropic root $\\delta ^{\\prime }$ in $V^{\\prime }, W^{\\prime })$ .",
"Since each of $\\delta ^{\\prime }, \\delta ^{\\prime \\prime }$ is a sum or difference of $d_{V^{\\prime }}, d_{W^{\\prime }}$ , we see that $\\delta ^{\\prime \\prime } = \\pm \\delta ^{\\prime }$ .",
"Assume first that $U$ is preinjective.",
"A general representation $M(\\delta )$ of dimension vector $\\delta $ is regular while a general representation of dimension vector $d_U$ is isomorphic to $U$ hence preinjective.",
"We have ${\\rm Ext}^1(M(\\delta ), U)=0$ and hence $\\langle \\delta , d_U \\rangle = {\\rm dim}_k{\\rm Hom}(M(\\delta ), U) \\ge 0$ .",
"If $U$ is preprojective, then ${\\rm Hom}(M(\\delta ), U) = 0$ .",
"This gives $\\langle \\delta , d_U \\rangle \\le 0$ .",
"Suppose that $\\delta ^{\\prime } = \\langle \\delta , d_U \\rangle d_U - \\delta $ .",
"This is only possible if $\\langle \\delta , d_U \\rangle > 0$ and hence, if $U$ is preinjective.",
"Then $\\langle \\delta , d_U \\rangle d_U = \\delta + \\delta ^{\\prime }$ .",
"The region ${R}$ given by $\\langle d, d \\rangle \\le 0$ is the cone over a two dimensional ellipse and hence, is a convex cone.",
"Since both $\\delta , \\delta ^{\\prime }$ lie on the boundary of ${R}$ , we see that $d_U$ lies in ${R}$ .",
"Hence, $\\langle d_U, d_U \\rangle \\le 0$ , a contradiction to $U$ being exceptional.",
"Therefore, $\\delta ^{\\prime \\prime } = \\delta ^{\\prime }$ is the wanted isotropic Schur root.",
"The root $\\delta ^{\\prime }$ defined above will be denoted $L_U(\\delta )$ , as it is the reflection of $\\delta $ to the left of the real Schur root $d_U$ .",
"Proposition 4.6 Let $Q$ be wild with three vertices, and let $(V,W)$ be an exceptional sequence of tame type containing $\\delta $ .",
"Complete this to a full exceptional sequence $(U,V,W)$ .",
"Then $U$ is preprojective or preinjective.",
"If $U$ is preprojective, then the $\\sigma $ -stable representations are, up to isomorphism, $U$ or $M(\\delta )$ where $M(\\delta )$ is a general representation of dimension vector $\\delta $ in $V,W)$ .",
"In particular, the $\\sigma $ -stable dimension vectors are $d_U, \\delta $ .",
"If $U$ is preinjective, then the $\\sigma $ -stable representations are, up to isomorphism, $U$ or $M(\\delta ^{\\prime })$ where $M(\\delta ^{\\prime })$ is a general representation of dimension vector $\\delta ^{\\prime } = L_U(\\delta )$ in $V^{\\prime },W^{\\prime })$ , where $(V^{\\prime }, W^{\\prime }, U)$ is exceptional.",
"In particular, the $\\sigma $ -stable dimension vectors are $d_U, \\delta ^{\\prime }$ .",
"In both cases, the $\\sigma $ -stable dimension vectors are the two extremal rays in the cone of $\\sigma $ -semistable dimension vectors.",
"Recall that the region $\\langle d, d \\rangle \\le 0$ is the cone over a two dimensional ellipse.",
"The ray $[\\delta ]$ lies on the quadric $\\langle d, d \\rangle =0$ .",
"Since $V, W)$ is tame, the Euler-Ringel form for $V, W)$ is positive semi-definite.",
"In particular, any linear combination of $d_V, d_W$ that lie on the region $\\langle d, d \\rangle \\le 0$ has to be a multiple of $\\delta $ .",
"Therefore, the cone $C_{V, W}$ generated by $d_V, d_W$ is the tangent plane of $\\langle d, d \\rangle =0$ at $\\delta $ .",
"If $U^{\\prime }$ is exceptional and $\\sigma $ -semistable, then we have an exceptional sequence $(U^{\\prime }, X, Y)$ such that $[\\delta ]$ lies in the cone $C_{X, Y}$ generated by $d_{X}, d_{Y}$ .",
"Therefore, as argued above, $C_{X, Y}$ is the tangent plane of $\\langle d, d \\rangle =0$ at $\\delta $ .",
"Hence, $C_{V, W} = C_{X, Y}$ and thus, since $\\langle - , - \\rangle $ is non-degenerate, the rays of $d_U$ , $d_{U^{\\prime }}$ coincide.",
"This gives $U \\cong U^{\\prime }$ since $U, U^{\\prime }$ are exceptional.",
"This proves that there is a unique $\\sigma $ -semistable real Schur root $d_U$ and a unique $\\sigma $ -semistable exceptional representation, up to isomorphism.",
"Hence, $d_U$ has to be $\\sigma $ -stable by Lemma REF .",
"Observe that $Q$ is connected by Lemma REF .",
"By Lemma REF , since $V,W)$ is tame, we know that $U$ is preinjective or preprojective.",
"Let $M$ be a $\\sigma $ -stable representation that is not isomorphic to $U$ .",
"By the previous argument, $M$ cannot be exceptional.",
"Suppose first that $U$ is preprojective.",
"If $M$ is not in $U^\\perp $ , then ${\\rm Ext}^1(U,M)\\ne 0$ as ${\\rm Hom}(U,M)=0$ .",
"Therefore, $M$ has to be preprojective, contradicting that $M$ is not exceptional.",
"Hence, $M$ is $\\sigma $ -stable and lies in $U^\\perp = V,W)$ .",
"By Proposition REF , $M$ is relative $\\sigma $ -stable in $U^\\perp $ .",
"Therefore, $M \\cong M(\\delta )$ is a general representation of dimension vector $\\delta $ in $V,W)$ .",
"Suppose finally that $U$ is preinjective.",
"If $M$ is not in $^\\perp U$ , then ${\\rm Ext}^1(M,U)\\ne 0$ and this means that $M$ is preinjective, thus exceptional, a contradiction.",
"Therefore, $M$ lies in $^\\perp U$ .",
"Now, $^\\perp U$ is also tame.",
"Consider the exceptional sequence $(V^{\\prime }, W^{\\prime }, U)$ where $V^{\\prime }, W^{\\prime }$ are obtained from $V,W$ by reflecting to the left of $U$ .",
"Let $\\delta ^{\\prime } = L_U(\\delta )$ be the unique isotropic Schur root in $V^{\\prime }, W^{\\prime })$ .",
"Since $M$ is Schur and not exceptional, $M$ has dimension vector $\\delta ^{\\prime }$ .",
"Let $E$ be an exceptional sequence of $\\sigma $ -stable representations.",
"Clearly, such a sequence has length at most $n-2$ .",
"The sequence $E$ is said to be full if $E$ has length $n-2$ .",
"The following result guarantees that such a full exceptional sequence of $\\sigma $ -stable representations always exists.",
"Lemma 4.7 Any exceptional sequence of $\\sigma $ -stable representations can be completed to a full exceptional sequence of $\\sigma $ -stable representations.",
"In particular, there exists an exceptional sequence $(M_{n-2}, \\ldots , M_2, M_{1})$ of $\\sigma $ -stable representations.",
"Let $(X_1, \\ldots , X_r)$ be an exceptional sequence with all $X_i$ $\\sigma $ -stable.",
"Assume that $r$ is not equal to $n-2$ .",
"Observe that $\\langle d_{X_i}, d_{X_j} \\rangle \\le 0$ whenever $i \\ne j$ .",
"By Proposition REF , and since the perpendicular category $X_1, \\ldots , X_r)^\\perp $ contains representations of dimension vector $\\delta $ , we can extend it to an exceptional sequence $(X_1, \\ldots , X_r, V,W)$ of length less than $n$ where $V,W)$ is tame and $\\delta = d_V + d_W$ .",
"Therefore, there exists an exceptional representation $Y$ such that $(X_1, \\ldots , X_r, Y, V, W)$ is exceptional.",
"Being left orthogonal to both $V,W$ , the representation $Y$ has to be $\\sigma $ -semistable.",
"It is well known and easy to see that for any acyclic quiver, there exists a sincere representation of it that is rigid.",
"Since the category $X_1, \\ldots , X_r,Y)$ is equivalent to the category of representations of an acyclic quiver with $r+1$ vertices, there exists a rigid object $Z \\in X_1, \\ldots , X_r, Y)$ such that its Jordan-Hölder composition factors in $X_1, \\ldots , X_r, Y)$ will consist of all the simple objects in $X_1, \\ldots , X_r, Y)$ .",
"Being $\\sigma $ -stable, the objects $X_1, \\ldots , X_r$ are non-isomorphic simple objects in $X_1, \\ldots , X_r, Y)$ .",
"Let $Y^{\\prime }$ be the other simple object.",
"Being rigid, $Z$ is a general representation.",
"It has some filtration where the subquotients are $\\lbrace X_1, \\ldots , X_r, Y^{\\prime }\\rbrace $ with possible multiplicities.",
"If $Y^{\\prime }$ is $\\sigma $ -stable, then we have a list $\\lbrace X_1, \\ldots , X_r, Y^{\\prime }\\rbrace $ of $r+1$ objects that are $\\sigma $ -stable.",
"Since these objects are the simple objects of $X_1, \\ldots , X_r, Y)$ , they could be ordered to form an exceptional sequence of length $r+1$ , which will be the wanted sequence.",
"If $Y^{\\prime }$ is not $\\sigma $ -stable, then the above filtration of $Z$ can be refined to a filtration in ${\\rm rep}(Q)$ where all subquotients are $\\sigma $ -stable.",
"In particular, there will be more than $r$ non-isomorphic subquotients.",
"Since $Z$ is rigid, we know that all subquotients have dimension vector a real Schur root, by Lemma REF .",
"Now, these real Schur roots can be ordered to form an orthogonal sequence of real Schur roots; see [5].",
"This sequence corresponds to an exceptional sequence of $\\sigma $ -stable representations of length at least $r+1$ .",
"Now, we treat the case where $n$ is arbitrary.",
"We first need some more notations.",
"From now on, let us fix $(M_{n-2}, \\ldots , M_{1})$ a full exceptional sequence of $\\sigma $ -stable representations.",
"Complete this sequence to get a full exceptional sequence $(M_{n-2}, \\ldots , M_{1}, V, W)$ .",
"Note that $\\delta $ is a root in $V,W)$ .",
"We construct a sequence of isotropic Schur roots $\\delta _1, \\ldots , \\delta _{n-2}$ and a sequence of rank three subcategories $i$ as follows.",
"Set $\\delta _1 = \\delta $ , $V_1 = V$ and $W_1=W$ and $1 = M_1, V_1, W_1)$ .",
"Observe that if 1 is wild, then it is connected and $M_1$ is preprojective or preinjective in 1.",
"If 1 is tame, then it is either disconnected (in which case $M_1$ is the unique simple object in the trivial component of 1) or else it is connected and $M_1$ is quasi-simple in 1.",
"For $1 \\le i \\le n-2$ , if $M_{i}$ is not preinjective in $i=M_{i}, V_i, W_i)$ , then we reduce to the case of the exceptional sequence $(M_{n-2}, \\ldots , M_{i+1}, V_{i+1}, W_{i+1})$ and we set $\\delta _{i+1} = \\delta _i, V_{i+1} = V_i, W_{i+1}=W_i$ .",
"If $M_{i}$ is preinjective in $M_{i}, V_i, W_i)$ , we reduce to the case of the exceptional sequence $(M_{n-2}, \\ldots , M_{i+1}, V_{i+1}, W_{i+1})$ where $V_{i+1}$ is the reflection of $V_i$ to the left of $M_{i}$ and $W_{i+1}$ is the reflection of $W_i$ to the left of $M_{i}$ .",
"We set $\\delta _{i+1} = \\delta _i - \\langle \\delta _i, d_{M_{i}} \\rangle d_{M_{i}}.$ In all cases, we set ${i+1}=M_{i+1}, V_{i+1}, W_{i+1})$ .",
"Note that for $1 \\le i \\le n-2$ , the root $\\delta _i$ is a root in $V_i, W_i)$ .",
"Finally, we set $\\bar{\\delta }= \\delta _{n-2}$ .",
"For two dimension vectors $d_1, d_2$ , we write $d_1 \\hookrightarrow d_2$ provided a general representation of dimension vector $d_2$ has a subrepresentation of dimension vector $d_1$ .",
"Lemma 4.8 We have $\\delta _{i+1} \\hookrightarrow \\delta _{i}$ and each $\\delta _{i}$ is an isotropic Schur root that is $\\sigma $ -semistable.",
"We proceed by induction on $i$ .",
"The case where $i=1$ is clear.",
"Let $M(\\delta _i)$ be a general representation of dimension vector $\\delta _i$ in $V_i, W_i)$ that is $\\sigma $ -semistable.",
"If $M_{i}$ is not preinjective in $i=M_{i}, V_i, W_i)$ , then $\\delta _{i+1} = \\delta _i$ and hence, it is clear that $\\delta _{i+1} \\hookrightarrow \\delta _{i}$ and that $\\delta _{i}$ is an isotropic Schur root that is $\\sigma $ -semistable.",
"So assume that $M_{i}$ is preinjective in $i$ .",
"Then ${\\rm Ext}^1(M(\\delta _i), M_{i})=0$ .",
"If we have a non-zero morphism $f_i: M(\\delta _i) \\rightarrow M_{i}$ that is not an epimorphism, then the cokernel $C_i$ of $f_i$ is $i$ -preinjective and $\\sigma $ -semistable in $i$ .",
"There is an epimorphism $C_i \\rightarrow N_i$ where $N_i$ is relative $\\sigma $ -stable in $i$ .",
"Since $M_i$ is $i$ -preinjective, $N_i$ (and $C_i$ ) has to be $i$ -preinjective and the only $i$ -preinjective relative $\\sigma $ -stable object in $i$ is $M_{i}$ .",
"This is a contradiction.",
"Therefore, any non-zero morphism $M(\\delta _i) \\rightarrow M_{i}$ is an epimorphism.",
"Now, take $e_{i} = \\langle \\delta _i, d_{M_{i}}\\rangle = {\\dim }{\\rm Hom}(M(\\delta _i), M_{i})$ .",
"We have a morphism $g_i: M(\\delta _i) \\rightarrow M_{i}^{e_{i}}$ given by a basis of ${\\rm Hom}(M(\\delta _i), M_{i})$ , and as argued above, $g_i$ is an epimorphism.",
"The kernel $K_i$ of $g_i$ is of dimension $\\delta _i - \\langle \\delta _i, d_{M_{i}}\\rangle d_{M_{i}}$ which is $\\delta _{i+1}$ .",
"Hence, $\\delta _{i+1}$ is relative $\\sigma $ -semistable and hence $\\sigma $ -semistable.",
"Consider the short exact sequence $0 \\rightarrow K_i \\rightarrow M(\\delta _i) \\rightarrow M_{i}^{e_{i}} \\rightarrow 0$ in $i$ .",
"Since $K_i \\in \\hspace{-2.0pt}^\\perp M_{i}$ by construction, and $K_i$ has dimension vector $\\delta _{i+1}$ , we have ${\\rm ext}(\\delta _{i+1}, e_i\\cdot d_{M_{i}})=0$ and by Schofield's result [17], $\\delta _{i+1} \\hookrightarrow \\delta _{i+1} + e_i\\cdot d_{M_{i}} = \\delta _i$ .",
"Call $\\delta $ of smaller type if the $\\tau $ -orbit of a general representation of dimension vector $\\delta $ contains a non-sincere representation.",
"Equivalently, if $\\tau $ denotes the Coxeter matrix of $Q$ , then there is an integer $r$ such that $\\tau ^r \\delta $ is not sincere.",
"Proposition 4.9 The following conditions are equivalent.",
"The root $\\delta $ is of smaller type.",
"There is a $\\sigma $ -stable representation that is preprojective or preinjective.",
"There is a $\\sigma $ -semistable representation that is preprojective or preinjective.",
"Let $Z$ be a $\\sigma $ -semistable representation that is preprojective or preinjective.",
"If $Z$ is preprojective, then there is some $i \\ge 0$ with $\\tau ^iZ=P$ projective.",
"Since $\\langle d_Z, \\delta \\rangle =0$ , we get $\\langle d_P, \\tau ^i\\delta \\rangle =0$ , showing that $\\tau ^i\\delta $ is not sincere, that is, $\\delta $ is of smaller type.",
"If $Z$ is preinjective, then there is some $j < 0$ with $\\tau ^jZ=Q[1]$ a shift of a projective representation $Q$ .",
"Since $\\langle d_Z, \\delta \\rangle =0$ , we get $\\langle -d_Q, \\tau ^j\\delta \\rangle =0$ , showing that $\\tau ^j\\delta $ is not sincere, that is, $\\delta $ is of smaller type.",
"This proves that $(3)$ implies $(1)$ .",
"Clearly, $(2)$ implies $(3)$ .",
"Assume that $\\delta $ is of smaller type.",
"Then there is some integer $i$ and some $x \\in Q_0$ such that ${\\rm Hom}(P_x, \\tau ^iM(\\delta ))=0$ whenever $M(\\delta )$ has dimension vector $\\delta $ .",
"Therefore, there exists a preprojective (if $i \\ge 0$ ) or preinjective (if $i < 0$ ) representation $Z$ such that $Z$ is left orthogonal to any representation $M(\\delta )$ of dimension vector $\\delta $ .",
"Therefore $Z$ is $\\sigma $ -semistable.",
"If $Z$ is not $\\sigma $ -stable and preprojective, then it has a $\\sigma $ -stable subrepresentation $Z^{\\prime }$ which has to be preprojective.",
"If $Z$ is not $\\sigma $ -stable and preinjective, then it has a $\\sigma $ -stable quotient $Z^{\\prime }$ which has to be preinjective.",
"We start with the following result, which describes the $\\sigma $ -stable objects when $n=4$ .",
"The core of it will be generalized to arbitrary $n$ later.",
"We think it is interesting to have a separate result for the $n=4$ case since more can be said for this small case.",
"Proposition 4.10 Let $Q$ be a connected wild quiver with 4 vertices and let $(M_2, M_1)$ be a full exceptional sequence of $\\sigma $ -stable representations.",
"Consider the exceptional sequence $(M_2, M_1, V, W)$ with $V,W$ the simple objects in $V,W)$ .",
"Let $M$ be $\\sigma $ -stable but not isomorphic to $M_1$ or $M_2$ .",
"Then The root $\\bar{\\delta }$ is the only $\\sigma $ -stable non-real Schur root.",
"The root $\\delta $ is of smaller type if and only if one of $M_1, M_2$ is preprojective or preinjective.",
"If $\\delta $ is not of smaller type, then the only exceptional $\\sigma $ -stable representations are $M_1, M_2$ .",
"The sufficiency of (2) follows from Proposition REF .",
"Assume now that both $M_1, M_2$ are regular.",
"We claim that $\\delta $ is not of smaller type and that the only exceptional $\\sigma $ -stable representations are $M_1, M_2$ .",
"In this case, by Lemma REF , since $1 =M_1, V, W)$ is wild and $V, W)$ is tame, we know that $M_1$ is either preprojective or preinjective in 1.",
"Observe that 1 is connected since it is wild and $\\delta $ is an isotropic root in 1.",
"If $M_1$ is preinjective in 1, then $2 = ^\\perp M_1$ is also wild, since $M_1$ is regular.",
"If $M_1$ is preprojective in 1, consider the exceptional sequence $(M_1^{\\prime }, M_2, V, W) = (M_1^{\\prime }, M_2, V_2, W_2)$ .",
"Note that $M_1, M_2$ are the relative simples in $M_2,M_1)$ .",
"Since any indecomposable object in $M_2,M_1)$ has a morphism from and to $M_1 \\oplus M_2$ , all objects of $M_2,M_1)$ , seen as objects in ${\\rm rep}(Q)$ , are regular.",
"Therefore, $M_1^{\\prime } \\in M_2,M_1)$ is regular in ${\\rm rep}(Q)$ and $M_1^{\\prime \\perp }= M_2, V_2, W_2)=2$ is wild (and connected).",
"Let $M$ be an arbitrary $\\sigma $ -stable representation not isomorphic to $M_1$ or $M_2$ .",
"Since ${\\rm Hom}(M_2, M)=0$ , there exists a non-negative integer $d_2$ and a short exact sequence $(*): \\quad 0 \\rightarrow M \\rightarrow E_2 \\stackrel{f_1}{\\rightarrow } M_2^{d_2} \\rightarrow 0$ with $E_2 \\in M_2^\\perp $ .",
"Observe that $d_2 = 0$ if and only if $M \\in M_2^\\perp $ .",
"Observe also that ${\\rm Hom}(E_2, M_1) = {\\rm Hom}(M_1, E_2)=0$ .",
"Since ${\\rm Hom}(M_1, E_2)=0$ , there exists a non-negative integer $d_1$ and a short exact sequence $(*): \\quad 0 \\rightarrow E_2 \\rightarrow E_1 \\stackrel{f_1}{\\rightarrow } M_1^{d_1} \\rightarrow 0$ with $E_1 \\in M_2^\\perp \\cap M_1^\\perp $ .",
"Observe that $d_1 = 0$ if and only if $E_2 \\in M_1^\\perp $ .",
"Since $E_1$ is relative $\\sigma $ -semistable in $M_2^\\perp \\cap M_1^\\perp =V,W)$ , there is a monomorphism $M(\\delta ) \\rightarrow E_1$ where $M(\\delta )$ is a Schur representation of dimension vector $\\delta $ in $V,W)$ .",
"If $M_1$ is preinjective in 1, then ${\\rm Ext}^1(M(\\delta ), M_1)=0$ .",
"Then $e_1:=\\langle \\delta , d_{M_1} \\rangle = {\\rm dim}{\\rm Hom}(M(\\delta ), M_1)>0$ .",
"We have a short exact sequence $0 \\rightarrow M(\\delta _1) \\rightarrow M(\\delta )\\rightarrow M_1^{e_1}\\rightarrow 0$ and this yields a monomorphism $M(\\delta _1) \\rightarrow E_2$ .",
"If $M_1$ is preprojective in 1, then ${\\rm Hom}(M(\\delta ), M_1)=0$ and hence, we get a monomorphism $M(\\delta _1) = M(\\delta ) \\rightarrow E_2$ .",
"If $M_2$ is preinjective in 2, then ${\\rm Ext}^1(M(\\delta _1), M_2)=0$ .",
"Then $e_2:=\\langle \\delta _1, d_{M_2} \\rangle = {\\rm dim}{\\rm Hom}(M(\\delta _1), M_2)>0$ .",
"We have a short exact sequence $0\\rightarrow M(\\delta _2) \\rightarrow M(\\delta _1)\\rightarrow M_2^{e_2}\\rightarrow 0$ and this yields a monomorphism $M(\\delta _2) \\rightarrow M$ .",
"If $M_2$ is preprojective in 2, then ${\\rm Hom}(M(\\delta _1), M_2)=0$ and hence, we get a monomorphism $M(\\delta _2) = M(\\delta _1) \\rightarrow M$ .",
"This shows that $M \\cong M(\\delta _2)=M(\\bar{\\delta })$ .",
"In particular, the exceptional sequence $(M_2,M_1)$ of $\\sigma $ -stable representations is unique.",
"It follows from Proposition REF that $\\delta $ is not of smaller type.",
"This prove our claim and hence $(2)$ and $(3)$ .",
"Statement $(1)$ in the case where $\\delta $ is of smaller type is a consequence of the next proposition, Proposition REF .",
"Given a subcategory $ of $ rep(Q)$, we denote by $ the Auslander-Reiten translation in $.",
"Fix a full exceptional sequence $ (Mn-2, ..., M2, M1)$ of $$-stable representations, and complete it to form an exceptional sequence $ (Mn-2, ..., M2, M1, V, W)$ where $ V,W$ are the non-isomorphic simple objects in $ V,W)$.",
"In particular, $ = dV + dW$ and $ dW, dV = -2$.",
"Recall that for $ 1 i n-2$, we denote by $ i$ the subcategory $ Mi, Vi, Wi)$, where $ Vi, Wi$ have been defined previously.",
"If $ i$ is tame and connected, then $ Mi$ is regular quasi-simple in $ i$ and lies in a tube of rank $ 2$ in $ i$.",
"We set $ Ni := iMi = i-1Mi$.",
"The exceptional sequence $ (Mn-2, ..., Mi+1)$ is denoted by $ Ei$.$ Proposition 4.11 Let $Q$ be a connected quiver with isotropic Schur root $\\delta $ .",
"Let $N$ be a $\\sigma $ -stable object not isomorphic to any $M_i$ .",
"The root $\\bar{\\delta }$ is the only $\\sigma $ -stable Schur root that is not real.",
"If $N$ is not exceptional, then $d_N = \\bar{\\delta }$ and $N \\in V_{n-2},W_{n-2})$ .",
"If $N$ is exceptional, then there exists some $i$ such that $i$ is tame connected with $M_i$ quasi-simple.",
"There exists a subsequence $\\mathcal {F}_i$ of $\\mathcal {E}_i$ such that $N$ is the reflection of $N_i$ to the left of $\\mathcal {F}_i$ .",
"We first need to check that there is at least one $\\sigma $ -stable root that is not real.",
"Assume otherwise.",
"As $\\delta $ is $\\sigma $ -stable, its $\\sigma $ -stable decomposition will involve only real Schur roots.",
"These roots may be ordered to form a Schur sequence.",
"Since all roots are real, this will correspond to an exceptional sequence $(F_1, \\ldots , F_r)$ .",
"Suppose that $r < n$ .",
"Complete the latter sequence to get a full exceptional sequence $(F_1, \\ldots , F_n)$ .",
"Since $\\delta $ is a root in each $F_i^\\perp $ for $1 \\le i \\le r$ , we see that $\\delta $ is a root in $F_{r+1}, \\ldots , F_n)$ .",
"Since $\\delta $ is also a root in $F_{1}, \\ldots , F_r)$ , we get that $\\delta $ is in the span of $d_{F_1}, \\ldots , d_{F_r}$ as well as in the span of $d_{F_{r+1}}, \\ldots , d_{F_n}$ .",
"Since $(F_1, \\ldots , F_n)$ is an exceptional sequence, the vectors $d_{F_1}, \\ldots , d_{F_n}$ are linearly independent, a contradiction.",
"Hence, $r=n$ .",
"But then, $F_{1}, \\ldots , F_n) = {\\rm rep}(Q)$ and all objects of ${\\rm rep}(Q)$ are $\\sigma $ -semistable, which is also a contradiction.",
"In the rest of the proof, we will see that the only possibility for a non-real $\\sigma $ -stable Schur root is $\\bar{\\delta }= \\delta _{n-2}$ .",
"Let $N$ be a $\\sigma $ -stable representation not isomorphic to any $M_i$ .",
"In particular, ${\\rm Hom}(N, M_i) = {\\rm Hom}(M_i,N)=0$ for all $i$ .",
"Let $E_{n-2}:=N$ .",
"For $i=n-2, n-3, \\ldots , 2, 1$ , we have a short exact sequence $0 \\rightarrow E_i \\rightarrow E_{i-1} \\rightarrow M_{i}^{d_{i}}\\rightarrow 0$ where $d_{i}$ is a non-negative integer with $d_{i} = -{\\rm dim}{\\rm Ext}^1(M_{i},E_i) = \\langle d_{M_{i}}, d_{E_i} \\rangle \\ge 0$ , as ${\\rm Hom}(M_i, E_i)=0$ by induction.",
"Observe that for $i < n-2$ , we have $E_i \\in M_{n-2}^\\perp \\cap \\cdots \\cap M_{i+1}^\\perp $ .",
"In particular, $E_0$ lies in $V,W)$ .",
"Observe also that all $E_i$ are $\\sigma $ -semistable.",
"Since $E_0$ is relative $\\sigma $ -semistable in $V,W)$ , there is a monomorphism $Z_0:=M(\\delta ) \\rightarrow E_0$ where $M(\\delta )$ is a Schur representation of dimension vector $\\delta $ in $V,W)$ .",
"If $M_1$ is preinjective in 1, then ${\\rm Ext}^1(M(\\delta ), M_1)=0$ .",
"Then $e_1:=\\langle \\delta , d_{M_1} \\rangle = {\\rm dim}{\\rm Hom}(M(\\delta ), M_1)$ .",
"We have a short exact sequence $0 \\rightarrow M(\\delta _2) \\rightarrow M(\\delta )\\rightarrow M_1^{e_1}\\rightarrow 0$ and this yields a monomorphism $M(\\delta _2) \\rightarrow E_1$ where $M(\\delta _2)$ lies in $V_2, W_2)$ is $\\sigma $ -semistable.",
"If $M_1$ is preprojective in 1, then ${\\rm Hom}(M(\\delta ), M_1)=0$ and hence, we get a monomorphism $M(\\delta _2) = M(\\delta ) \\rightarrow E_1$ .",
"If 1 is tame disconnected, then ${\\rm Hom}(M(\\delta ), M_1)=0$ and we get a monomorphism $M(\\delta _2) = M(\\delta ) \\rightarrow E_1$ as well.",
"Assume that 1 is tame connected.",
"If ${\\rm Hom}(M(\\delta ), M_1)=0$ , then we get a monomorphism $M(\\delta _2) = M(\\delta ) \\rightarrow E_1$ as previously.",
"Assume that ${\\rm Hom}(M(\\delta ), M_1)\\ne 0$ .",
"Then ${\\rm Hom}(N_1, M_1)=0$ and there is a monomorphism $N_1 \\rightarrow M(\\delta )$ .",
"Therefore, we get a monomorphism $N_1 \\rightarrow E_1$ .",
"Hence, in all cases, we have a monomorphism $Z_1 \\rightarrow E_1$ where $Z_1$ is either a Schur ($\\sigma $ -semistable) representation of dimension vector $\\delta _2$ in $V_2, W_2)$ or is $N_1$ .",
"In the first case, we consider the exceptional sequence $(M_{n-2}, \\ldots , M_2, V_2, W_2)$ and we proceed by induction starting with the monomorphism $Z_1 \\rightarrow E_1$ where $Z_1 = M(\\delta _2)$ .",
"For the second case, we consider the exceptional sequence $(M_{n-2}, \\ldots , M_2, N_1)$ .",
"If ${\\rm Hom}(N_1, M_2) \\ne 0$ , then ${\\rm Ext}^1(N_1, M_2) = 0$ as ${\\rm Ext}^1(M_2, N_1) = 0$ .",
"We have a short exact sequence $0 \\rightarrow Z_2 \\rightarrow N_1 \\rightarrow M_2^{f_2} \\rightarrow 0$ where $f_2 = {\\rm dim} {\\rm Hom}(N_1, M_2)$ .",
"Then, $Z_2 \\in ^\\perp M_2$ and hence, we get a monomorphism $Z_2 \\rightarrow E_2$ and $Z_2$ is the reflection of $N_1$ to the left of $M_2$ .",
"If ${\\rm Hom}(N_1, M_2) = 0$ , then we get a monomorphism $Z_2:=N_1 \\rightarrow E_2$ .",
"We proceed by induction until we get a monomorphism $Z_{n-2} \\rightarrow E_{n-2}=N$ .",
"This gives $Z_{n-2} \\cong N$ .",
"We see that $Z_{n-2}$ will be of the required form.",
"In particular, if $N$ is not exceptional, then $N$ has dimension vector $\\delta _{n-2}=\\bar{\\delta }$ and will be in $V_{n-2}, W_{n-2})$ .",
"The next two lemmas will help us in giving a better description of the simple objects in $\\mathcal {A}(\\delta )$ , that is, the $\\sigma $ -stable objects.",
"Lemma 4.12 Assume that both $M_{t-1}, \\ldots , M_1, V, W)$ and $t$ are tame connected with $\\delta $ the unique isotropic Schur root in both of these subcategories.",
"Then $M_t, M_{t-1}, \\ldots , M_1, V, W)$ is tame connected with only one isotropic Schur root $\\delta $ .",
"Assume that $=M_t, M_{t-1}, \\ldots , M_1, V, W)$ is wild.",
"Then $M_t$ lies in the preprojective or preinjective component of a wild connected component of $.",
"Observe that, in $ t$, the object $ Mt$ is left orthogonal to the unique isotropic Schur root (which is $ dVt + dWt$).",
"Since $ t$ is connected, this means that $ Mt$ is regular in $ t$.",
"In particular, there are infinitely many indecomposable objects $ Z$ of $ t$ (and hence of $ ) with ${\\rm Hom}(M_t, Z)\\ne 0$ .",
"Therefore, $M_t$ cannot lie in a preinjective component of $.",
"Similarly, $ Mt$ cannot lie in a preprojective component of $ .",
"This is a contradiction.",
"It remains to show that $ is connected (it is clear that then, $$ will be the unique isotropic Schur root in $ ).",
"Assume otherwise.",
"Then, we have $\\mathcal {B}_1 \\times \\mathcal {B}_2$ where each of $\\mathcal {B}_1, \\mathcal {B}_2$ is equivalent to a category of representations of a non-empty acyclic quiver.",
"Assume that $\\delta \\in \\mathcal {B}_2$ .",
"By assumption, we must have that both $M_{t-1}, \\ldots , M_1, V, W), t$ are subcategories of $\\mathcal {B}_2$ .",
"But then, $ \\mathcal {B}_2$ , a contradiction.",
"Therefore, $ is connected.$ Lemma 4.13 Assume that $i$ is tame connected for all $1 \\le i \\le s-1$ .",
"Fix $1 \\le t \\le s-1$ and consider the representation $N_t$ as defined above.",
"If $M_s$ is preprojective in the category $s$ , then ${\\rm Hom}(N_t,M_s)=0$ .",
"If $M_s$ is preinjective in the category $s$ , then $M_i \\in ^\\perp M_s$ for all $1 \\le i \\le s-1$ .",
"If $M_s$ is simple disconnected in $s$ , then all $\\sigma $ -stable representations, except possibly $M_{s-1}, \\ldots , M_1$ , lie in $^\\perp M_s$ .",
"Consider the categories $= M_s, \\ldots , M_1, V, W)$ and $:=M_{s-1}, \\ldots , M_1, V, W).$ We know that $$ is tame connected by Lemma REF .",
"Assume that $ is wild.",
"Therefore, $ Ms$ lies in a preprojective or preinjective component of $ .",
"For proving (1), assume that $M_s$ is preprojective in the category $s$ , which means that $M_s$ is preprojective in $.",
"Assume to the contrary that $ Hom(Nt,Ms)0$.",
"Since $ Nt , the object $N_t$ is preprojective in $.",
"Therefore, $ Nt$ cannot be regular in $ t$, a contradiction.",
"For proving (2), assume that $ Ms$ is preinjective in the category $ s$, which means that $ Ms$ is preinjective in $ .",
"Since $t$ is arbitrary and ${\\rm Hom}(M_t, M_i)=0$ , it is sufficient to prove that ${\\rm Ext}^1(M_t, M_s)=0$ .",
"If not, then the Auslander-Reiten formula in $ yields a nonzero morphism from $ Ms$ to $ M̏t$ and hence, $ M̏t$ (and thus $ Mt$) is preinjective in $ .",
"Thus, $M_t$ cannot be regular in $t$ , a contradiction.",
"Clearly, if $ is tame, then (1), (2) cannot occur, since a subcategory of a tame category cannot be wild.$ It remains to prove (3).",
"Let $N$ be a $\\sigma $ -stable object not isomorphic to any $M_i$ .",
"It follows from the proof of the last theorem that we have a short exact sequence $0 \\rightarrow E_s \\rightarrow E_{s-1} \\rightarrow M_s^{d_s}\\rightarrow 0$ and a monomorphism $Z \\rightarrow E_{s-1}$ where $Z$ is either indecomposable of dimension vector $\\delta $ in $V,W)$ or is exceptional and relative $\\sigma $ -semistable in $$ (a reflection of one of the $N_i$ for $1 \\le i \\le s-1$ ).",
"In the first case, since $s$ is tame disconnected, this yields a monomorphism $Z \\rightarrow E_s$ and the rest of the proof of the above theorem deals with representations in $^\\perp M_s$ .",
"In the second case, we get a monomorphism $M \\rightarrow Z$ where $M$ is relative $\\sigma $ -stable in $$ and exceptional, and hence a quasi-simple object of $$ .",
"Assume that $M$ lies in a tube $T$ of rank $r$ .",
"In particular, the other non-isomorphic $r-1$ quasi-simple objects of that tube are among the objects $M_{s-1}, \\ldots , M_1$ .",
"We claim that ${\\rm Ext}^1(M,M_s)=0$ .",
"There exists a short exact sequence $0 \\rightarrow M \\rightarrow Y \\rightarrow N \\rightarrow 0$ in $$ where $M,Y,N$ are indecomposable in $T$ with $Y$ of dimension vector $\\delta $ and $N$ of quasi-length $r-1$ which does not have $M$ as a quasi-simple composition factor.",
"Using that $(M_{s-1}, \\ldots , M_1, V,W)$ is an exceptional sequence and since, by construction, $M$ is the unique quasi-simple of $T$ with ${\\rm Hom}(M,Y)\\ne 0$ , we have $Y \\in V,W)$ .",
"The surjective map ${\\rm Ext}^1(Y,M_s) \\rightarrow {\\rm Ext}^1(M,M_s)$ together with the fact that $s = M_s, V, W)$ is disconnected gives that ${\\rm Ext}^1(M,M_s)=0$ .",
"This proves our claim.",
"Observe that $M$ lies in $M_s^\\perp $ .",
"We have a short exact sequence $0 \\rightarrow Z^{\\prime } \\rightarrow M \\rightarrow M_s^{f} \\rightarrow 0$ where $f = {\\rm dim} {\\rm Hom}(M, M_s) = \\langle d_M, d_{M_s} \\rangle $ .",
"Thus, $Z^{\\prime }$ is the reflection of $M$ to the left of $M_s$ and $Z^{\\prime } \\in ^\\perp M_s$ and we get a monomorphism $Z^{\\prime } \\rightarrow E_s$ .",
"The proof of the last theorem continues with $Z_s = Z^{\\prime }$ and the monomorphism $Z_s \\rightarrow E_s$ .",
"In all cases, we see that $M_s$ satisfies the required property.",
"Cone of $\\sigma $ -semi-stable dimension vectors Let $d$ be a dimension vector.",
"Let us denote by $C(\\sigma _d)$ the set of all $\\sigma _d$ -semistable dimension vectors.",
"We consider $C_\\mathbb {R}(\\sigma _d)$ the corresponding cone in $\\mathbb {R}^n$ , which lie in the positive orthant of $\\mathbb {R}^n$ .",
"Since the rays $x \\in C_\\mathbb {R}(\\sigma _d)$ satisfy $\\langle x, d \\rangle = 0$ , we rather consider $C_\\mathbb {R}(\\sigma _d)$ as a cone in $\\mathbb {R}^{n-1}$ .",
"The integral vectors in $C_\\mathbb {R}(\\sigma _d)$ correspond to the $\\sigma _d$ -semistable dimension vectors.",
"For $d=\\delta $ , since there exists a full exceptional sequence $(M_{n-2}, \\ldots , M_2, M_1)$ of $\\sigma _\\delta $ -stable representations and since the dimension vectors in an exceptional sequence are linearly independent, we see that $C_\\mathbb {R}(\\sigma _\\delta )$ is a cone of full dimension in $\\mathbb {R}^{n-1}$ .",
"In general, it is well known that $C_\\mathbb {R}(\\sigma _d)$ is a cone over a polyhedron where the indivisible dimension vectors in the extremal rays are $\\sigma _d$ -stable dimension vectors.",
"On the other hand, a $\\sigma _d$ -stable dimension vector needs not lie on an extremal ray.",
"Lemma 5.1 If $f$ is $\\sigma _d$ -stable and a real Schur root, then it lies in an extremal ray.",
"Let $f$ be a $\\sigma _d$ -stable real Schur root.",
"Assume that $f$ does not lie on an extremal ray.",
"According to [5], there exists dimension vectors $f_1, \\ldots , f_s$ , all lying on extremal rays, such that $f$ is a positive integral combination of $f_1, \\ldots , f_s$ and $f_1, \\ldots , f_s$ are linearly independent in $\\mathbb {R}^n$ .",
"Moreover, we have $\\langle f, f_i \\rangle \\le 0$ and $\\langle f_i, f \\rangle \\le 0$ for all $1 \\le i \\le s$ .",
"These conditions imply that $1=\\langle f, f \\rangle = \\langle f, a_1f_1 + \\cdots + a_sf_s \\rangle \\le 0,$ a contradiction.",
"There is a special case of interest, which is when there is some ray $[r]$ such that all $\\sigma _d$ -stable dimension vectors, except possibly the ones on $[s]$ , lie on extremal rays (this is the case when $d=\\delta $ where $\\delta $ is an isotropic Schur root).",
"In such a case, either all $\\sigma _d$ -stable dimension vectors lie on the boundary of the cone $C_\\mathbb {R}(\\sigma _d)$ or else $[s]$ lies in the interior of $C_\\mathbb {R}(\\sigma _d)$ .",
"Let $v_1, v_2, \\ldots , v_r$ be the extremal rays of $C_\\mathbb {R}(\\sigma _d)$ .",
"Take $f$ any dimension vector lying in $C_\\mathbb {R}(\\sigma _d)$ but that is neither in an extremal ray nor in the ray $[s]$ .",
"Then we know that $f$ has at least one $\\sigma _d$ -stable factor that lies on an extremal ray.",
"Since the $\\sigma _d$ -stable factors of $f$ can be ordered to form an orthogonal sequence of Schur roots, we see that there exists a Schur root $\\alpha $ (that corresponds to an extremal ray $v_i$ ) such that either $\\langle v_i, f \\rangle > 0$ or $\\langle f, v_i \\rangle > 0$ .",
"If $\\langle v_i, f \\rangle > 0$ , then $\\langle v_i, x \\rangle = 0$ defines an hyperplane cutting $C_\\mathbb {R}(\\sigma _d)$ such that $f, v_i$ lie on the same side while all other extremal rays $v_1, \\ldots , v_{i-1}, v_{i+1}, \\ldots , v_r$ of $C_\\mathbb {R}(\\sigma _d)$ lie on the other side or on the boundary of that hyperplane.",
"We get a similar situation if $\\langle f, v_i \\rangle > 0$ by considering the hyperplane $\\langle x, v_i \\rangle = 0$ .",
"For $1 \\le i \\le r$ , let $C_i(\\sigma _d)$ be the cone in $\\mathbb {R}^{n-1}$ generated by all the rays $v_1, \\ldots , v_r$ but $v_i$ .",
"By the above observation, we have that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)):=\\cap _{1 \\le i \\le r}C_i(\\sigma _d)$ is either empty or else contains only the ray $[s]$ .",
"We will see that this restriction yield a very beautiful description of $C_\\mathbb {R}(\\sigma _d)$ and, in particular, of $C_\\mathbb {R}(\\sigma _\\delta )$ .",
"Proposition 5.2 We have that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ if and only if $C_\\mathbb {R}(\\sigma _d)$ is the cone over a simplex.",
"The sufficiency is easy to see.",
"Assume that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ .",
"We may work in $\\mathbb {R}^{j}$ where $j \\le n-1$ and assume that $C_\\mathbb {R}(\\sigma _d)$ is of full dimension in $\\mathbb {R}^{j}$ .",
"By Radon's theorem, if the number of extremal rays $r$ of $C_\\mathbb {R}(\\sigma _d)$ is at least $(j-1)+2 = j+1$ , then we can partition the rays $v_1, \\ldots , v_r$ into two non-empty subsets $A,B$ such that the corresponding cones $C(A)$ and $C(B)$ generated by the rays in $A$ and by the rays in $B$ have a ray of intersection.",
"This ray of intersection will have to be in ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ , a contradiction.",
"Therefore, $r \\le j$ .",
"Since $C_\\mathbb {R}(\\sigma _d)$ is of full dimension in $\\mathbb {R}^{j}$ , then $r=j$ and $C_\\mathbb {R}(\\sigma _d)$ is the cone over an $(j-1)$ -simplex.",
"Now, we are interested in the case where ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d))$ is reduced to a single ray $[s]$ (which then has to be the ray of $\\bar{\\delta }$ if $d = \\delta $ ).",
"Let us take an affine slice $\\Delta $ of $C_\\mathbb {R}(\\sigma _d)$ .",
"The rays $v_1, \\ldots , v_r$ will correspond to points $u_1, \\ldots , u_r$ in $\\Delta $ and these points are the vertices of a polyhedron $\\mathcal {R}$ in $\\Delta $ defined as the convex hull of $u_1, \\ldots , u_r$ .",
"The ray $[s]$ corresponds to a point $s$ in $\\mathcal {R}$ .",
"In order to study the convex properties of $\\mathcal {R}$ , let us translate $\\mathcal {R}$ so that $s$ coincides with the origin.",
"In other words, set $w_i = u_i - s$ and consider the polyhedron $\\mathcal {P}$ which is the convex hull of $w_1, \\ldots , w_r$ .",
"Since $\\mathcal {R}$ lies on an affine slice, we see that $\\mathcal {P}$ lies in a subspace of dimension $n-2$ of $\\mathbb {R}^{n-1}$ .",
"Let $\\mathcal {P}_i$ be the convex hull of all points $w_1, \\ldots , w_r$ but $w_i$ .",
"We define ${\\rm Proper}(\\mathcal {P}):=\\cap _{1 \\le i \\le r}\\mathcal {P}_i$ and we will be interested in the case where ${\\rm Proper}(\\mathcal {P})$ only contains the origin.",
"The first two lemmas are easy to prove.",
"Lemma 5.3 Let $\\mathcal {P}^{\\prime }$ be the convex hull of a subset of $w_1, \\ldots , w_r$ .",
"Then ${\\rm Proper}(\\mathcal {P}^{\\prime })$ is either empty or reduced to the origin.",
"Lemma 5.4 Consider a non-trivial partition $\\lbrace w_{i_1}, \\ldots , w_{i_s}\\rbrace = A_1 \\cup A_2$ of a subset of $\\lbrace w_1, \\ldots , w_r\\rbrace $ .",
"Denote by $\\mathcal {P}_{A_i}$ the convex hull of the points in $A_i$ , for $i=1,2$ .",
"Then $\\mathcal {P}_{A_1} \\cap \\mathcal {P}_{A_2}$ is either empty or reduced to the origin.",
"Proposition 5.5 Suppose that ${\\rm Proper}(\\mathcal {P})$ is empty or reduced to the origin and is full dimensional in $\\mathbb {R}^{t}$ with $t \\le n-2$ .",
"Then there exists a vector space decomposition $\\mathbb {R}^{t} = V_1 \\oplus \\cdots \\oplus V_s$ of $\\mathbb {R}^{t}$ such that if $V_i$ has dimension $d_i$ , then it contains $d_i + 1$ points among $0, w_1, \\ldots , w_r$ that form a $d_i$ -simplex in $V_i$ containing the origin.",
"We may assume that $t = n-2$ so that $\\mathcal {P}$ is $(n-2)$ -dimensional.",
"If ${\\rm Proper}(\\mathcal {P})$ is empty, then $s=1$ and the result follows from Proposition REF .",
"Assume that ${\\rm Proper}(\\mathcal {P})$ is reduced to the origin.",
"Suppose first that the origin lies on a facet, say $F$ , of $\\mathcal {P}$ .",
"We claim that $F$ contains $r-1$ of the points $w_1, \\ldots , w_r$ .",
"Assume otherwise.",
"Consider an $(n-3)$ -simplex in $F$ generated by points $w_{i_1}, \\ldots , w_{i_{n-2}}$ .",
"Let $u, v \\in \\lbrace w_1, \\ldots , w_r\\rbrace $ be two distinct points not in $F$ .",
"By Radon's theorem, we can partition the points $\\lbrace w_{i_1}, \\ldots , w_{i_{n-2}}, u, v\\rbrace $ into two non-empty subsets $A_1, A_2$ such that $\\mathcal {P}_{A_1} \\cap \\mathcal {P}_{A_2} \\ne \\emptyset $ , where $\\mathcal {P}_{A_i}$ denotes the convex hull of the points in $A_i$ .",
"By Lemma REF , this intersection is the origin and hence lies on $F$ .",
"Since $F$ is a facet, $u,v$ lie on the same side of $F$ .",
"Therefore, for $i=1,2$ , the set $B_i:=A_i\\backslash \\lbrace u,v\\rbrace $ is not empty.",
"Now, $B_1$ and $B_2$ form a partition of $w_{i_1}, \\ldots , w_{i_{n-2}}$ such that $\\mathcal {P}_{B_1} \\cap \\mathcal {P}_{B_2} \\ne \\emptyset $ , where $\\mathcal {P}_{B_i}$ denotes the convex hull of the points in $B_i$ .",
"This contradicts Proposition REF .",
"Now, let us assume that the origin lies in the interior of $\\mathcal {P}$ .",
"By Radon's theorem, we can write $\\lbrace w_1, \\ldots , w_r\\rbrace = E_1 \\cup E_2$ where $E_1,E_2$ are disjoint and non-empty such that $\\mathcal {P}_{E_1} \\cap \\mathcal {P}_{E_2} = \\lbrace 0\\rbrace $ , where $\\mathcal {P}_{E_i}$ denotes the convex hull of the points in $E_i$ .",
"By Carathéodory's theorem, there is a simplex formed by some points $z_1, \\ldots , z_s$ in $E_1$ that contains the origin in its interior.",
"With no loss of generality, assume that $z_i = w_i$ and $s \\le r-1$ .",
"Let $V_1$ be the vector space spanned by the points $w_1, \\ldots , w_s$ and consider the vector space $V_2$ spanned by the points $w_{s+1}, \\ldots , w_r$ .",
"Let $C_1$ be the convex hull of the points $w_1, \\ldots , w_s$ and let $C_2$ be the convex hull of the points $w_{s+1}, \\ldots , w_r$ .",
"Since both $\\mathcal {P}_{E_1}, \\mathcal {P}_{E_2}$ contain the origin, we see that $C_1 \\cap C_2 = \\lbrace 0\\rbrace $ .",
"Since 0 lies in the interior of $C_1$ , we get also that $V_1 \\cap C_2=0$ .",
"We claim that $V_1 \\cap V_2 = 0$ .",
"Assume that $V_1 \\cap V_2$ is non-zero.",
"Observe that any element in $V_1$ can be written as a non-negative linear combination of $w_1, \\ldots , w_s$ .",
"There exists non-negative real numbers $a_1, \\ldots , a_s$ and real numbers $b_{s+1}, \\ldots , b_r$ such that $a_1w_1 + \\cdots + a_sw_s = b_{s+1}w_{s+1} + \\cdots + b_rw_r.$ Moreover, $a_1w_1 + \\cdots + a_sw_s$ is non-zero.",
"We may assume the $a_i$ small enough so that the left-hand side lies in $C_1$ .",
"Let us write $\\lbrace s+1, \\ldots , r\\rbrace = I_1 \\cup I_2$ where $I_1, I_2$ are disjoint and $i \\in I_1$ if and only if $b_i \\ge 0$ .",
"We may assume further that the $|b_i|$ are small enough so that both $\\sum _{i \\in I_1}b_iw_i, -\\sum _{j \\in I_2}b_jw_j$ lie in $C_2$ .",
"If $I_2 = \\emptyset $ , then $a_1w_1 + \\cdots + a_sw_s \\in C_1 \\cap C_2$ is non-zero, a contradiction.",
"If all $b_i$ are non-positive, then $-\\sum _{j \\in I_2}b_jw_j$ lies in $V_1\\cap C_2=\\lbrace 0\\rbrace $ , a contradiction.",
"If some $b_i$ are negative and some $b_i$ are positive, we can rewrite the sum as $a_1w_1 + \\cdots + a_sw_s + -\\sum _{j \\in I_2}b_jw_j= \\sum _{j \\in I_1}b_jw_j.$ Considering Lemma REF with the partition $(\\lbrace w_1, \\ldots , w_s\\rbrace \\cup \\lbrace w_{i}\\mid i \\in I_2\\rbrace ) \\cup (\\lbrace w_{j} \\mid j \\in I_1\\rbrace )$ of $\\lbrace w_1, \\ldots , w_r\\rbrace $ , we get that $a_1w_1 + \\cdots + a_sw_s + -\\sum _{j \\in I_2}b_jw_j$ is zero, which reduces to a case we have already considered.",
"Therefore, we have proven that $\\mathbb {R}^{n-2} = V_1 \\oplus V_2$ , where $V_1$ satisfies the property of the statement.",
"We proceed by induction on $V_2$ with the points $w_{s+1}, \\ldots , w_r$ and by using Lemma REF .",
"The ring of semi-invariants of an isotropic Schur root In this section, we denote by $\\delta $ an isotropic Schur root and by $\\sigma = \\sigma _\\delta $ the weight given by $-\\langle -, \\delta \\rangle $ .",
"Consider, as previously, a full exceptional sequence $(M_{n-2}, \\ldots , M_2, M_1, V,W)$ where $(M_{n-2}, \\ldots , M_2, M_1)$ is an exceptional sequence of simple objects in $\\mathcal {A}(\\delta )$ .",
"Take $I \\subseteq \\lbrace 1, \\ldots , n-2\\rbrace $ such that $i \\in I$ if and only if $i$ is tame connected.",
"Definition 6.1 The associated tame subcategory of $Q$ relative to $\\delta $, denoted $\\mathcal {R}(Q,\\delta )$ , is the thick subcategory of ${\\rm rep}(Q)$ generated by $(\\bigoplus _{i \\in I}M_i)\\oplus V_{n-2}\\oplus W_{n-2}$ .",
"Theorem 6.2 Let $Q$ be an acyclic connected quiver and $\\delta $ an isotropic Schur root.",
"Then The category $\\mathcal {R}(Q,\\delta )$ is tame connected with isotropic Schur root $\\bar{\\delta }$ and is uniquely determined by $\\delta $ .",
"The simple objects in $\\mathcal {A}(\\delta )$ , up to isomorphism, are given by the disjoint union $\\lbrace M_i \\mid i \\notin I\\rbrace \\cup \\lbrace \\text{quasi-simple objects in} \\; \\mathcal {R}(Q,\\delta )\\rbrace .$ We have ${\\rm SI}(Q, \\delta ) \\cong {\\rm SI}(\\mathcal {R}(Q,\\delta ), \\bar{\\delta })[x_{r+1}, \\ldots , x_n].$ Let $(M_{n-2}, \\ldots , M_1)$ be an exceptional sequence of $\\sigma _\\delta $ -stable representations with the corresponding full exceptional sequence $(M_{n-2}, \\ldots , M_1, V, W)$ in ${\\rm rep}(Q)$ , where $\\delta = d_V + d_W$ .",
"First, denote by $M_{l_r}, \\ldots , M_{l_1}$ with $l_r > \\cdots > l_1$ the $M_j$ such that $j$ is tame disconnected or such that $M_j$ is preprojective in $j$ .",
"We get an exceptional sequence $(*) \\qquad (N_{t}, \\ldots , N_2, N_1, V,W, M_{l_r}^{\\prime }, \\ldots , M_{l_1}^{\\prime })$ where all $M_{l_j}$ have been reflected, one by one, to the right of the exceptional sequence.",
"Observe that $\\lbrace M_i \\mid i \\in I\\rbrace \\subseteq \\lbrace N_1, \\ldots , N_t\\rbrace $ .",
"Let $ \\lbrace N_t, \\ldots , N_1\\rbrace \\backslash \\lbrace M_i \\mid i \\in I\\rbrace :=\\lbrace N_{j_s},\\ldots , N_{j_1}\\rbrace $ where $j_s > \\cdots > j_1$ .",
"Assume also that $I = \\lbrace m_1, \\ldots , m_q\\rbrace $ with $m_q > \\cdots > m_1$ .",
"We have $q+s=t$ and $t+r=n-2$ .",
"By Lemma REF (2), we may reflect all exceptional objects of $\\lbrace N_{j_s},\\ldots , N_{j_1}\\rbrace $ in $(*)$ so that we get an exceptional sequence $(M_{m_q}, \\ldots , M_{m_1}, N_{j_s},\\ldots ,N_{j_1}, V, W, M_{l_r}^{\\prime }, \\ldots , M_{l_1}^{\\prime }).$ Now, it follows from the definition of the $V_i, W_i$ that we get an exceptional sequence $(M_{m_q}, \\ldots , M_{m_1}, V_{n-2},W_{n-2}, N_{j_s},\\ldots ,N_{j_1}, M_{l_1}^{\\prime }, \\ldots , M_{l_r}^{\\prime }).$ We claim that for $1 \\le u \\le q$ , we have that ${m_u}=M_{m_u},V_{m_u},W_{m_u})$ is equivalent to $M_{m_u},V_{n-2},W_{n-2})$ .",
"Fix such a $u$ .",
"Note that there is an exceptional sequence of the form $(N_{j_s}, \\ldots , N_{j_p}, M_{m_u},V_{m_u},W_{m_u}).$ Now, it follows from Lemma REF (2) that $M_{m_u}$ lies in $^\\perp N_{j_i}$ for all $1 \\le i \\le p$ .",
"By reflecting, we get the exceptional sequence $(M_{m_u},V_{n-2},W_{n-2}, N_{j_s}, \\ldots , N_{j_p}).$ It follows that ${m_u}=M_{m_u},V_{m_u},W_{m_u})$ is equivalent to $M_{m_u},V_{n-2},W_{n-2})$ .",
"This proves our claim.",
"Let $E = (M_{m_q}, \\ldots , M_{m_1}, V_{n-2},W_{n-2})$ .",
"By Lemma REF and our claim, $\\mathcal {R}(Q,\\delta )=E)$ is tame connected.",
"Since $V_{n-2}, W_{n-2}$ lie in it, $\\bar{\\delta }$ is the (unique) isotropic Schur root of $\\mathcal {R}(Q,\\delta )$ .",
"It follows from Lemma REF that any $\\sigma _\\delta $ -stable representation not isomorphic to any $M_i$ for $1 \\le i \\le n-2$ will have to be (quasi-simple) in $E)$ .",
"Now, we need to show that all quasi-simple objects of $E)$ are $\\sigma _\\delta $ -stable.",
"Assume the contrary.",
"Let $f$ be the dimension vector of a quasi-simple object in $E)$ that is not $\\sigma _\\delta $ -stable, but $\\sigma _\\delta $ -semistable.",
"It follows from our previous observations that $f$ has to be a positive integral combination of the $\\sigma _\\delta $ -stable dimension vectors in $E)$ .",
"It follow from [11] that this is not possible to have such a decomposition.",
"Therefore, we have a complete list of the simple objects in $\\mathcal {A}(\\delta )$ .",
"These are given by the disjoint union $\\lbrace M_i \\mid i \\notin I\\rbrace \\cup \\lbrace \\text{quasi-simple objects in} \\; \\mathcal {R}(Q,\\delta )\\rbrace .$ Observe that, in $C_\\mathbb {R}(\\sigma _\\delta )$ , a dimension vector $d$ can be uniquely written as $d = d_1 + \\sum _{i \\notin I} \\lambda _i f_i$ where $d_1$ is a dimension vector in $E)$ and $f_i = d_{M_{m_i}}$ for $i \\notin I$ .",
"This decomposition is unique.",
"This implies the unicity of $\\mathcal {R}(Q,\\delta )$ and statement $(3)$ .",
"Corollary 6.3 Let $Q$ be an acyclic connected quiver and $\\delta $ an isotropic Schur root.",
"Then ${\\rm SI}(Q, \\delta )$ is a polynomial ring or a hypersurface.",
"More precisely, it is a hypersurface (and not a polynomial ring) if and only if $\\mathcal {R}(Q,\\delta )$ has quiver of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"In [20], it was proven that the ring of semi-invariant of an isotropic Schur root of a tame quiver is a polynomial ring or a hypersurface, where the second situation occurs precisely when the quiver is of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"Our result follows from this and Theorem REF .",
"Example 6.4 Consider the quiver $Q$ given by ${7pt}{& 2 [dl] & \\\\ 1 & & 4 [ul] [ll] [dl]\\\\ & 3 [ul] &}$ Consider the exceptional sequence $(P_2, S_1, I_3, S_3)$ where $P_2$ is the projective representation at vertex 2, $I_3$ is the injective representation at vertex 3 and $S_1, S_3$ are the simple representations at vertices $1,3$ , respectively.",
"Reflecting $S_1, I_3$ to the left of $P_2$ , we get an exceptional sequence whose dimension vectors are as follows.",
"$((0,1,0,0), (3, 3, 1, 1), (1,1,0,0), (0,0,1,0)).$ Then, using a sequence of reflections, we get the following exceptional sequences, where we put the corresponding dimension vectors.",
"$((0,1,0,0), (3, 3, 1, 1), (0,0,1,0), (1,1,1,0))$ $((0,1,0,0), (0,0,1,0), (3, 3, 3, 1), (1,1,1,0))$ $((0,0,1,0), (0,1,0,0), (3, 3, 3, 1), (1,1,1,0))$ $((0,0,1,0), (0,1,0,0), (8,8,8,3), (3, 3, 3, 1))$ $((0,0,1,0), (8,3,8,3), (0,1,0,0), (3, 3, 3, 1))$ $((8,3,3,3), (0,0,1,0), (0,1,0,0), (3, 3, 3, 1)).$ Observe that $\\langle (3,3,3,1),(0,1,0,0) \\rangle =2$ and $\\delta = (3,3,3,1) - (0,1,0,0) = (3,2,3,1)$ is an isotropic Schur root.",
"The Coxeter matrix $\\tau $ is $\\tau = \\left(\\begin{array}{cccc}-1 & 1 & 1 & 1 \\\\-1 & 0 & 1 & 2 \\\\-1 & 1 & 0 & 2 \\\\-3 & 2 & 2 & 4 \\\\\\end{array}\\right).$ This matrix has eigenvalues $\\lambda = 5/2 +\\sqrt{21}/2$ , $\\lambda ^{-1} = 5/2-\\sqrt{21}/2$ and $-1$ with (algebraic and geometric) multiplicity 2.",
"The eigenvector corresponding to $\\lambda $ is $v_1 = (10, 9+\\sqrt{21}, 9+\\sqrt{21}, 17 + \\sqrt{189})$ and the one corresponding to $\\lambda ^{-1}$ is $v_2 = (10, 9-\\sqrt{21}, 9-\\sqrt{21}, 17 - \\sqrt{189}).$ Now, $\\langle v_2, (8,3,3,3) \\rangle = -197 + 10\\sqrt{21} + 11\\sqrt{189} > 0$ and $\\langle v_2, (0,0,1,0) \\rangle = -8-\\sqrt{21}+\\sqrt{189} > 0$ .",
"Similarly, both $\\langle (8,3,3,3), v_1 \\rangle $ and $\\langle (0,0,1,0), v_1 \\rangle $ are positive.",
"Therefore, the exceptional objects with dimension vectors $(8,3,3,3), (0,0,1,0)$ are regular by the theorem at page 240 of [16].",
"It follows from Proposition REF that $\\delta $ is not of smaller type.",
"It also follows from the same proposition that there is a unique exceptional sequence $(M_2, M_1)$ of length 2 of $\\sigma $ -stable objects.",
"Let $M_1^{\\prime } = S_3$ and $M_2^{\\prime }$ be the exceptional representation with dimension vector $(8,3,3,3)$ .",
"Since $M_1^{\\prime }, M_2^{\\prime }$ lie in $M_2,M_1)$ by Lemma REF , we see that $M_2^{\\prime },M_1^{\\prime }) \\subseteq M_2,M_1)$ and thus, we have equality.",
"This means that $M_2^{\\prime } = M_2$ , $M_1^{\\prime } = M_1$ .",
"Since $\\langle \\delta , (0,0,1,0) \\rangle = 2 >0$ , we get $\\delta _1 = \\delta - 2(0,0,1,0) = (3,2,1,1)$ .",
"Now, $\\langle \\delta _1, (8,3,3,3) \\rangle = -2$ .",
"Therefore, $\\bar{\\delta }= \\delta _1 = (3,2,1,1)$ .",
"In this example, the cone of $\\sigma $ -semistable dimension vectors is as follows (where only an affine slice of that cone is shown).",
"Figure: The cone of σ\\sigma -semistable dimension vectors for δ=(3,2,3,1)\\delta = (3,2,3,1)The following is an easy observation.",
"The reader is referred to [8] for the notion of cluster algebra and to [7] for results in similar directions.",
"Corollary 6.5 If ${\\rm SI}(Q,\\delta )$ is not a polynomial ring, then it has a cluster algebra structure of type $\\mathbb {A}_1$ .",
"There are two cluster variables which are all $\\Gamma $ -homogeneous, and the coefficients are built from $n-1$ frozen variables, which are also $\\Gamma $ -homogeneous, where $\\Gamma $ is the set of all multiplicative characters of ${\\rm GL}_\\delta (k)$ .",
"From Theorem REF , it is enough to prove this for ${\\rm rep}(Q) = \\mathcal {R}(Q,\\delta )$ , that is, we may assume that $Q$ is tame connected.",
"Suppose that ${\\rm SI}(Q,\\delta )$ is not a polynomial ring.",
"Then $Q$ is of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"In particular, it is well known in these cases that there are exactly three non-homogeneous tubes $T_1, T_2, T_3$ in the Auslander-Reiten quiver of $\\mathcal {R}(Q,\\delta )$ .",
"One, say $T_1$ , has rank 2.",
"Let $M,N$ be the non-isomorphic exceptional quasi-simple objects in $T_1$ .",
"Then, let $E_1, \\ldots , E_r$ be the non-isomorphic quasi-simple objects of $T_2$ and let $E_1^{\\prime }, \\ldots , E_t^{\\prime }$ be the non-isomorphic quasi-simple objects of $T_3$ .",
"Now, the hypersurface equation can be written as $(*) \\qquad C^MC^N = C^{E_1}\\cdots C^{E_r} + C^{E_1^{\\prime }}\\cdots C^{E_t^{\\prime }}.$ Consider the indeterminates $x,y_1, \\ldots , y_r, z_1, \\ldots , z_t$ .",
"We define a cluster algebra $A$ as follows.",
"We start with the initial seed $\\lbrace x,y_1, \\ldots , y_r, z_1, \\ldots , z_t\\rbrace $ where $y_1, \\ldots , y_r$ and $z_1, \\ldots , z_t$ are declared to be frozen variables.",
"The exchange relation is $xx^{\\prime } = \\prod _{i=1}^ry_i + \\prod _{j=1}^tz_j$ which clearly produces exactly two cluster variables $x,x^{\\prime }$ .",
"The cluster algebra is the the $\\mathbb {Z}$ -subalgebra of $\\mathbb {Q}(x,y_1, \\ldots , y_r, z_1, \\ldots , z_t)$ generated by $x, x^{\\prime }$ and $y_1, \\ldots , y_r, z_1, \\ldots , z_t$ .",
"This algebra is clearly isomorphic to ${\\rm SI}(Q,\\delta )$ .",
"An interesting problem would be to find all acyclic quivers $Q$ and dimension vectors $d$ such that SI$(Q,d)$ has a cluster algebra structure whose variables (frozen or not) are all $\\Gamma $ -homogeneous.",
"Construction of all isotropic Schur roots In this section, we show that all of the isotropic Schur roots of ${\\rm rep}(Q)$ come from isotropic Schur roots of a tame full subquiver of $Q$ by applying special reflections.",
"We make this precise by defining an action of the braid group $B_{n-1}$ on $n-1$ strands on a special type of exceptional sequences that will encode all we need to study isotropic Schur roots.",
"We start with the definition of these sequences.",
"Definition 7.1 Let $E=(X_1, \\ldots , X_n)$ be a full exceptional sequence.",
"We say that $E$ is of isotropic type if there exists $1 \\le i \\le n-1$ such that $X_{i}, X_{i+1})$ is tame.",
"The integer $i$ is called the isotropic position of $E$ and the root type of $E$ , denoted $\\delta _E$ , is the isotropic Schur root in $X_{i}, X_{i+1})$ .",
"We denote by $\\mathcal {E}$ the set of all full exceptional sequences of isotropic type, up to isomorphism.",
"Not all elements of the braid group $B_{n}$ act on $\\mathcal {E}$ .",
"We rather consider the group $B_{n-1}$ and show that it acts on $\\mathcal {E}$ .",
"Let us denote the standard generators of $B_{n-1}$ by $\\gamma _1, \\ldots , \\gamma _{n-2}$ .",
"Let $E=(X_1, \\ldots , X_n) \\in \\mathcal {E}$ with isotropic position $r$ .",
"Let $1 \\le i \\le n-2$ .",
"If $i<r-1$ , then $\\gamma _iE := \\sigma _iE$ .",
"If $i>r$ , then $\\gamma _{i}E := \\sigma _{i+1}E$ .",
"Assume that $i=r$ with $r<n-1$ .",
"We can reflect $X_{r+2}$ to the left of $X_r, X_{r+1}$ to get the exceptional sequence: $E^{\\prime }=(X_1, \\ldots ,X_{r-1}, L_{X_{r}}(L_{X_{r+1}}(X_{r+2})), X_r, X_{r+1}, X_{r+3}, \\ldots , X_n).$ and this is an exceptional sequence of isotropic type with isotropic position $r+1$ .",
"We define $\\gamma _rE:=E^{\\prime }$ .",
"If $r > 1$ and $i=r-1$ , then we can reflect both $X_r, X_{r+1}$ to the left of $X_{r-1}$ as follows: $E^{\\prime \\prime } = (X_1, \\ldots ,X_{r-2}, L_{X_{r-1}}(X_r), L_{X_{r-1}}(X_{r+1}), X_{r-1}, X_{r+2}, \\ldots , X_n)$ and clearly, the subcategory $L_{X_{i-1}}(X_i), L_{X_{i-1}}(X_{i+1}))$ generates a tame subcategory of rank 2.",
"Therefore, $E^{\\prime \\prime }$ is an exceptional sequence of isotropic type with isotropic position $r-1$ and its root type is the unique isotropic Schur root in $L_{X_{i-1}}(X_i), L_{X_{i-1}}(X_{i+1}))$ , which is $\\delta _{E^{\\prime }} = \\delta _E - \\langle \\delta _E, d_{X_{r-1}}\\rangle d_{X_{r-1}}$ , by Lemma REF .",
"We define $\\gamma _{r-1}E=E^{\\prime \\prime }$ .",
"Similarly, we can define the action of $\\gamma _{i}^{-1}$ on $E$ for $1 \\le i \\le n-2$ .",
"The following is easy to check.",
"Proposition 7.2 The group $B_{n-1}$ acts on exceptional sequences of isotropic type, with the action defined above.",
"Definition 7.3 A sequence $E=(X_1, \\ldots , X_{n-1}, X_n)$ in $\\mathcal {E}$ is of tame type if it has isotropic position $n-1$ , and there is $0 \\le s \\le n-2$ such that $X_1, \\ldots , X_s$ are projective in ${\\rm rep}(Q)$ and $X_{s+1}, \\ldots , X_{n-2}, X_{n-1}, X_n)$ is tame connected.",
"By convention, $s=0$ means that ${\\rm rep}(Q)$ is already tame connected.",
"Observe that if $E \\in \\mathcal {E}$ is of tame type, then the isotropic Schur root $\\delta _E$ is the unique isotropic Schur root of the tame subcategory $X_{s+1}, \\ldots , X_{n-2}, X_{n-1}, X_n)$ and is an isotropic Schur root coming from a tame full subquiver of $Q$ .",
"In particular, there are finitely many roots $\\delta _E$ where $E \\in \\mathcal {E}$ is of tame type.",
"Example 7.4 Consider a quiver of rank $n=4$ and an exceptional sequence $E=(X,U,V,Y)$ of isotropic type with isotropic position 2.",
"The root type is the isotropic root $\\delta _E$ in $U,V)$ .",
"Figure: Correspondence between B 3 B_3 and some braids of B 4 B_4The first braid (A) in Figure REF corresponds to the element $g=\\gamma _2^{-1}\\gamma _1^{-1}\\gamma _2\\gamma _1^{-1}$ of $B_3$ while the second braid (B) corresponds to the element $h=\\sigma _2^{-1}\\sigma _1^{-1}\\sigma _3^{-1}\\sigma _2^{-1}\\sigma _3\\sigma _2^{-1}\\sigma _1^{-1}$ of $B_4$ .",
"Notice that $gE = hE$ .",
"Notice also that the braid in (A) is obtained from the braid in (B) by identifying the two strands starting at the positions of $U,V$ , that is, the second and third strands.",
"Our aim in this section is to prove that any $E \\in \\mathcal {E}$ lies in the $B_{n-1}$ -orbit of an exceptional sequence of tame type.",
"In the next lemmas, we will consider exceptional sequences in the bounded derived category $D^b({\\rm rep}(Q))$ of ${\\rm rep}(Q)$ .",
"Recall that an object $X$ in $D^b({\\rm rep}(Q))$ is exceptional if ${\\rm Hom}(X,X[i])=0$ for all non-zero $i$ (and then, ${\\rm Hom}(X,X)$ has to be one dimensional).",
"Equivalently, an exceptional object in $D^b({\\rm rep}(Q))$ is isomorphic to the shift of an exceptional representation.",
"A sequence $(X_1, \\ldots , X_r)$ of objects in $D^b({\\rm rep}(Q))$ is exceptional if every $X_i$ is exceptional and, for $i < j$ , we have ${\\rm Hom}_{D^b({\\rm rep}(Q))}(X_i, X_j[t])=0$ for all $t \\in \\mathbb {Z}$ .",
"For such a sequence, one can consider the smallest full additive subcategory $\\mathcal {D}(X_1, \\ldots , X_r)$ of $D^b({\\rm rep}(Q))$ containing $X_1, \\ldots , X_r$ and that is closed under direct sums, direct summands, taking the cone of a morphism and the shift of an object.",
"One can also consider the exceptional sequence $(X_1^{\\prime }, \\ldots , X_r^{\\prime })$ in ${\\rm rep}(Q)$ such that $X_i^{\\prime }$ is the unique shift of $X_i$ lying in ${\\rm rep}(Q)$ .",
"The indecomposable objects in $\\mathcal {D}(X_1, \\ldots , X_r)$ are just the shifts of the indecomposable objects in $X_1^{\\prime }, \\ldots , X_r^{\\prime })$ .",
"In what follows, the Auslander-Reiten translate in $D^b({\\rm rep}(Q))$ is denoted by $\\tau _{D}$ while the Auslander-Reiten translate in ${\\rm rep}(Q)$ is simply denoted $\\tau $ .",
"Recall that if $X$ is a non-projective indecomposable representation, then $\\tau _DX = \\tau X$ and, if $X = P_x$ with $x \\in Q_0$ , then $\\tau _D X = I_x[-1]$ .",
"When $d$ is a dimension vector, we denote by $\\tau d$ the product of the Coxeter matrix with $d$ .",
"In particular, if $X$ is a non-projective indecomposable representation, then $\\tau d_X = d_{\\tau X}$ and, if $X=P_x$ with $x \\in Q_0$ , then $\\tau d_{X} = -d_{I_x}$ .",
"We start our investigation with the following lemma that is crucial for the proof of the main result of this section.",
"Lemma 7.5 Let $(X_1, \\ldots , X_{n})$ be an exceptional sequence with $X_{r+1}, \\ldots , X_{n})$ tame and assume that $X_1, \\ldots , X_r$ are the simple objects in $X_1, \\ldots , X_r)$ .",
"Let $X \\in X_1, \\ldots , X_r)$ be the injective object with socle $X_1$ .",
"If $X$ is projective in $X,X_{r+1}, \\ldots , X_{n})$ , then $X_1$ is projective in ${\\rm rep}(Q)$ and in particular, an isotropic Schur root of $X_{r+1}, \\ldots , X_{n} )$ is not sincere.",
"Assume that $X$ is projective in $X,X_{r+1}, \\ldots , X_{n})$ .",
"Set $d_i = d_{X_i}$ for $1 \\le i \\le n$ .",
"Consider the linear form $f$ given by $f(x)=\\langle d_1, x \\rangle $ .",
"Then $f$ vanishes on $d_2, \\ldots , d_{n}$ and $f(d_1)>0$ .",
"Assume to the contrary that $X_1$ is not projective in ${\\rm rep}(Q)$ .",
"Observe that $f(x)=\\langle d_1, x \\rangle = -\\langle x, \\tau d_1 \\rangle $ .",
"Since $\\tau X_1$ is exceptional, $\\langle \\tau d_1, \\tau d_1 \\rangle = 1$ and hence $f(\\tau d_1) < 0$ .",
"Now, reflect $X_1$ to the right of $X_2, \\ldots X_r$ , so that we get an exceptional sequence $(X_2, \\ldots , X_r, Y)$ where $Y$ is in the cone spanned by $d_1, \\ldots , d_r$ .",
"Clearly, $X_1$ is simple projective in $X_1, \\ldots , X_r)$ and hence, $Y=X$ is the injective hull of $X_1$ in $X_1, \\ldots , X_r)$ .",
"Set $=X, X_{r+1}, \\ldots , X_{n})$ .",
"We know that $X$ is projective in $.",
"Reflecting $ X$ to the right of $ Xr+1, ..., Xn$ will give the exceptional representation $ X1$.",
"Therefore, $ d:=d1 = -d̏X$ where $ denotes the Coxeter transformation in $.",
"Take the linear form $ g$ in the Grothendieck group of $ given by $g(x)=\\langle d_{X}, x \\rangle $ .",
"Then $g$ vanishes on $d_{r+1}, \\ldots , d_{n}$ and $g(d_X)>0$ .",
"The form $f|_ has the same property since $ dX$ is a non-negative linear combination of $ d1, ..., dr$ with the coefficient of $ d1$ positive.",
"Thus, $ g = f| up to a positive scalar.",
"Therefore, $g(d)<0$ , which means that $X,\\tau X_1$ lie on opposite sides of the hyperplane $g(x)=0$ in $.",
"This contradicts that $ X$ is projective in $ .",
"Lemma 7.6 Let $(X_1, \\ldots , X_r)$ be an exceptional sequence and assume that $1:=X_2, \\ldots , X_r)$ is tame with an isotropic Schur root $\\gamma $ while $2:=X_1, X_2, \\ldots , X_r)$ is wild.",
"Then there is a unique minimal isotropic Schur root in the $\\tau $ -orbit of $\\gamma $ .",
"We may assume that $2 = {\\rm rep}(Q)$ for an acyclic quiver $Q$ .",
"Since 2 is wild and 1 is tame, we know that $X_1$ is preprojective or preinjective in 2.",
"Hence, there is some $r \\in \\mathbb {Z}$ such that $\\tau _D^r X_1$ is projective or the shift of a projective.",
"This means that $\\tau ^r\\gamma $ is not sincere.",
"Let $Y = \\tau _D^r X_1$ if $\\tau _D^r X_1$ is a representation or $Y = \\tau _D^r X_1[-1]$ if $\\tau _D^r X_1$ is the shift of a projective representation.",
"Observe that $Y^\\perp \\subseteq {\\rm rep}(Q)$ is also of tame representation type, where the quivers of $X_1^\\perp $ and $Y^\\perp $ only differ by a change of orientation; see for instance [10].",
"Therefore, $\\tau ^r\\gamma $ is an isotropic Schur root of a tame full subquiver of $Q$ .",
"Let $s \\in \\mathbb {Z}$ with $s \\ne r$ .",
"Consider $Z$ the unique shift of $\\tau _D^s X_1$ which is a representation.",
"Since the simples in $Y^\\perp $ are simples in ${\\rm rep}(Q)$ , and since there is a simple of $Z^\\perp \\subseteq {\\rm rep}(Q)$ that is not simple in ${\\rm rep}(Q)$ , we see that the isotropic Schur root $\\tau ^r\\gamma $ has smaller length than $\\tau ^s\\gamma $ .",
"This also proves unicity since only one object in the $\\tau $ -orbit of $X_1$ is projective or a shift of a projective.",
"Lemma 7.7 Let $E=(X_1, \\ldots , X_{n-2}, U, V)$ be in $\\mathcal {E}$ with isotropic position $n-1$ .",
"Let $E^{\\prime } = \\gamma _{1}\\cdots \\gamma _{n-3}\\gamma _{n-2}E = (U^{\\prime }, V^{\\prime }, X_1, \\ldots , X_{n-2})$ .",
"Then $\\tau ^{-1} \\delta _E = \\delta _{E^{\\prime }}$ .",
"If $V$ is not injective, then we have the exceptional sequence $(\\tau ^{-1}V, X_1, \\ldots , X_{n-2},U)$ in ${\\rm rep}(Q)$ .",
"Otherwise, we have the exceptional sequence $(\\tau ^{-1}V[-1], X_1, \\ldots , X_{n-2}, U)$ in ${\\rm rep}(Q)$ .",
"Let us write $\\tau ^{-1}V[0,-1]$ to indicate that we either take the shift $[0]$ or $[-1]$ for $\\tau ^{-1}V$ .",
"Then, we get an exceptional sequence $(\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1], X_1, \\ldots , X_{n-2}).$ The categories $U^{\\prime }, V^{\\prime })$ and $\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1])$ are equal in ${\\rm rep}(Q)$ .",
"Therefore, they have the same isotropic Schur root.",
"The isotropic Schur root of $\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1])$ is clearly $\\tau ^{-1}\\delta _E$ .",
"Of course, we have the dual version of the above lemma as follows.",
"Lemma 7.8 Let $E=(U,V, X_1, \\ldots , X_{n-2})$ be in $\\mathcal {E}$ with isotropic position 1.",
"Let $E^{\\prime } = \\gamma _{n-2}^{-1}\\gamma _{n-3}^{-1}\\cdots \\gamma _{1}^{-1}E = (X_1, \\ldots , X_{n-2}, U^{\\prime }, V^{\\prime })$ .",
"Then $\\tau \\delta _E = \\delta _{E^{\\prime }}$ .",
"We are now ready for the main result of this section.",
"Theorem 7.9 Let $\\delta $ be an isotropic Schur root.",
"Then there is $E \\in \\mathcal {E}$ of tame type and $g \\in B_{n-1}$ such that $gE$ has root type $\\delta $ .",
"It follows from Proposition REF that there is an exceptional sequence $F=(M_1, \\ldots , M_{n-2}, X,Y)$ in $\\mathcal {E}$ of isotropic position $n-1$ and of root type $\\delta $ .",
"Assume that $G\\in \\mathcal {E}$ is in the orbit of $E$ and the root type of $G$ is minimal, that is, has minimal length as a root in ${\\rm rep}(Q)$ .",
"We may assume that the isotropic position of $G$ is $n-1$ .",
"Therefore, we may assume that $G$ is of the form $(Y_1, \\ldots , Y_{n-2}, U, V).$ Assume first that there is an object $W$ in $Y_1, \\ldots , Y_{n-2})$ that is not projective in $W,U,V)$ .",
"We can apply a sequence of reflections to the subsequence $(Y_1, \\ldots , Y_{n-2})$ to get an exceptional sequence $H=(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W, U,V)$ in $\\mathcal {E}$ .",
"Now, applying $\\gamma _{n-2}^2$ to $H$ and using Lemma REF , we get the sequence $(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W^{\\prime }, U^{\\prime }, V^{\\prime })$ in $\\mathcal {E}$ whose root type is the inverse Auslander-Reiten translate of $\\delta $ in $W,U,V)$ .",
"Similarly, applying $(\\gamma _{n-2}^{-1})^2$ to $H$ , we get the sequence $(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W^{\\prime \\prime }, U^{\\prime \\prime }, V^{\\prime \\prime })$ in $\\mathcal {E}$ whose root type is the Auslander-Reiten translate of $\\delta $ in $W,U,V)$ .",
"We can iterate this to get a smaller root by Lemma REF , provided $W,U,V)$ is wild.",
"Therefore, whenever there is an object $W$ which is not projective in $W,U,V)$ , then $W,U,V)$ is of tame type (and hence connected).",
"Suppose, by induction, that we have an exceptional sequence $J=(W_{r+1},\\ldots ,W_{n-2},U,V)$ such that $J)$ is tame connected and $J$ has maximal length with respect to this property.",
"If $r=0$ , then $Q$ is a tame connected quiver and there is nothing to prove.",
"Complete this to get a full exceptional sequence $(Z_1, \\ldots , Z_r, W_{r+1}, \\ldots , W_{n-2}, U,V).$ If there is $W \\in ^\\perp J) = Z_1, \\ldots , Z_r)$ such that $W$ is not projective in $W,U,V)$ then, $W,U,V)$ is tame connected.",
"As in the proof of Lemma REF , we get that $W,W_{r+1}, \\ldots , W_{n-2},U,V)$ is tame connected, contradicting the maximality of $J$ .",
"Therefore, any object $Z$ in $Z_1, \\ldots , Z_r)$ is such that $Z$ is projective in $Z,W_{r+1}, \\ldots , W_{n-2},U,V)$ .",
"We may apply a sequence of reflections and assume that all of $Z_1, \\ldots , Z_r$ are simple in $Z_1, \\ldots , Z_r)$ .",
"It follows from Lemma REF that the injective hull of $Z_1$ in $Z_1, \\ldots , Z_r)$ is projective in ${\\rm rep}(Q)$ .",
"Then the proof goes by induction.",
"Here is another way to interpret this result.",
"Start with an isotropic Schur root $\\delta _0$ of a tame full subquiver $Q^{\\prime }$ of $Q$ and consider an exceptional sequence $(U_0,V_0)$ of length 2 in ${\\rm rep}(Q^{\\prime }) \\subset {\\rm rep}(Q)$ such that $\\delta _0 = d_{U_0} + d_{V_0}$ .",
"Consider an exceptional object $X_0$ such that $(X_0,U_0,V_0)$ is an exceptional sequence of length three (which generates a thick subcategory 0 of ${\\rm rep}(Q)$ ).",
"Then we can transform it into another exceptional sequence $(X_0^{\\prime }, U_1, V_1)$ with an isotropic Schur root $\\delta _1 = d_{U_1}+d_{V_1}$ such that $\\delta _1$ is a power $\\tau _{0}^{r_0} \\delta _0$ where $\\tau _{0}$ denotes the Coxeter matrix for 0.",
"Now, for $i \\ge 1$ , consider an exceptional object $X_i$ such that $(X_i,U_i, V_i)$ is an exceptional sequence.",
"Take a power $\\delta _{i+1}=\\tau _{i}^{r_i} \\delta _i$ where $i$ is the thick subcategory of ${\\rm rep}(Q)$ generated by $X_i, U_i, V_i$ and $\\tau _{i}$ denotes the Coxeter matrix for $i$ .",
"All the roots $\\delta _i$ constructed this way are isotropic Schur roots.",
"Moreover, all isotropic Schur roots of ${\\rm rep}(Q)$ can be obtained in this way.",
"There are clearly only finitely many starting roots $\\delta _0$ , but the choices of the $r_i$ and $X_i$ yield, in general, infinitely many possible isotropic Schur roots.",
"As observed in [15], when $Q$ is wild connected with more than 3 vertices, there are infinitely many $\\tau $ -orbit of isotropic Schur roots (provided there is at least one isotropic Schur root).",
"An interesting question would be to describe the minimal root types of the orbits of $\\mathcal {E}$ under $B_{n-1}$ .",
"It is not hard to check that when $n=3$ , these minimal root types correspond exactly to the tame full subquivers of $Q$ .",
"We do not know if this holds in general.",
"Conjecture 7.10 Let $E_1, E_2 \\in \\mathcal {E}$ .",
"Assume that there are $g_1, g_2 \\in B_{n-1}$ with $g_1E_1, g_2E_2$ of tame type but with different root types.",
"Then $E_1, E_2$ lie in distinct orbits under $B_{n-1}$ .",
"Acknowledgment.",
"The authors are thankful to Hugh Thomas for suggesting the decomposition in Proposition REF .",
"The second named author was supported by NSF grant DMS-1400740."
],
[
"Cone of $\\sigma $ -semi-stable dimension vectors",
"Let $d$ be a dimension vector.",
"Let us denote by $C(\\sigma _d)$ the set of all $\\sigma _d$ -semistable dimension vectors.",
"We consider $C_\\mathbb {R}(\\sigma _d)$ the corresponding cone in $\\mathbb {R}^n$ , which lie in the positive orthant of $\\mathbb {R}^n$ .",
"Since the rays $x \\in C_\\mathbb {R}(\\sigma _d)$ satisfy $\\langle x, d \\rangle = 0$ , we rather consider $C_\\mathbb {R}(\\sigma _d)$ as a cone in $\\mathbb {R}^{n-1}$ .",
"The integral vectors in $C_\\mathbb {R}(\\sigma _d)$ correspond to the $\\sigma _d$ -semistable dimension vectors.",
"For $d=\\delta $ , since there exists a full exceptional sequence $(M_{n-2}, \\ldots , M_2, M_1)$ of $\\sigma _\\delta $ -stable representations and since the dimension vectors in an exceptional sequence are linearly independent, we see that $C_\\mathbb {R}(\\sigma _\\delta )$ is a cone of full dimension in $\\mathbb {R}^{n-1}$ .",
"In general, it is well known that $C_\\mathbb {R}(\\sigma _d)$ is a cone over a polyhedron where the indivisible dimension vectors in the extremal rays are $\\sigma _d$ -stable dimension vectors.",
"On the other hand, a $\\sigma _d$ -stable dimension vector needs not lie on an extremal ray.",
"Lemma 5.1 If $f$ is $\\sigma _d$ -stable and a real Schur root, then it lies in an extremal ray.",
"Let $f$ be a $\\sigma _d$ -stable real Schur root.",
"Assume that $f$ does not lie on an extremal ray.",
"According to [5], there exists dimension vectors $f_1, \\ldots , f_s$ , all lying on extremal rays, such that $f$ is a positive integral combination of $f_1, \\ldots , f_s$ and $f_1, \\ldots , f_s$ are linearly independent in $\\mathbb {R}^n$ .",
"Moreover, we have $\\langle f, f_i \\rangle \\le 0$ and $\\langle f_i, f \\rangle \\le 0$ for all $1 \\le i \\le s$ .",
"These conditions imply that $1=\\langle f, f \\rangle = \\langle f, a_1f_1 + \\cdots + a_sf_s \\rangle \\le 0,$ a contradiction.",
"There is a special case of interest, which is when there is some ray $[r]$ such that all $\\sigma _d$ -stable dimension vectors, except possibly the ones on $[s]$ , lie on extremal rays (this is the case when $d=\\delta $ where $\\delta $ is an isotropic Schur root).",
"In such a case, either all $\\sigma _d$ -stable dimension vectors lie on the boundary of the cone $C_\\mathbb {R}(\\sigma _d)$ or else $[s]$ lies in the interior of $C_\\mathbb {R}(\\sigma _d)$ .",
"Let $v_1, v_2, \\ldots , v_r$ be the extremal rays of $C_\\mathbb {R}(\\sigma _d)$ .",
"Take $f$ any dimension vector lying in $C_\\mathbb {R}(\\sigma _d)$ but that is neither in an extremal ray nor in the ray $[s]$ .",
"Then we know that $f$ has at least one $\\sigma _d$ -stable factor that lies on an extremal ray.",
"Since the $\\sigma _d$ -stable factors of $f$ can be ordered to form an orthogonal sequence of Schur roots, we see that there exists a Schur root $\\alpha $ (that corresponds to an extremal ray $v_i$ ) such that either $\\langle v_i, f \\rangle > 0$ or $\\langle f, v_i \\rangle > 0$ .",
"If $\\langle v_i, f \\rangle > 0$ , then $\\langle v_i, x \\rangle = 0$ defines an hyperplane cutting $C_\\mathbb {R}(\\sigma _d)$ such that $f, v_i$ lie on the same side while all other extremal rays $v_1, \\ldots , v_{i-1}, v_{i+1}, \\ldots , v_r$ of $C_\\mathbb {R}(\\sigma _d)$ lie on the other side or on the boundary of that hyperplane.",
"We get a similar situation if $\\langle f, v_i \\rangle > 0$ by considering the hyperplane $\\langle x, v_i \\rangle = 0$ .",
"For $1 \\le i \\le r$ , let $C_i(\\sigma _d)$ be the cone in $\\mathbb {R}^{n-1}$ generated by all the rays $v_1, \\ldots , v_r$ but $v_i$ .",
"By the above observation, we have that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)):=\\cap _{1 \\le i \\le r}C_i(\\sigma _d)$ is either empty or else contains only the ray $[s]$ .",
"We will see that this restriction yield a very beautiful description of $C_\\mathbb {R}(\\sigma _d)$ and, in particular, of $C_\\mathbb {R}(\\sigma _\\delta )$ .",
"Proposition 5.2 We have that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ if and only if $C_\\mathbb {R}(\\sigma _d)$ is the cone over a simplex.",
"The sufficiency is easy to see.",
"Assume that ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ .",
"We may work in $\\mathbb {R}^{j}$ where $j \\le n-1$ and assume that $C_\\mathbb {R}(\\sigma _d)$ is of full dimension in $\\mathbb {R}^{j}$ .",
"By Radon's theorem, if the number of extremal rays $r$ of $C_\\mathbb {R}(\\sigma _d)$ is at least $(j-1)+2 = j+1$ , then we can partition the rays $v_1, \\ldots , v_r$ into two non-empty subsets $A,B$ such that the corresponding cones $C(A)$ and $C(B)$ generated by the rays in $A$ and by the rays in $B$ have a ray of intersection.",
"This ray of intersection will have to be in ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d)) = \\emptyset $ , a contradiction.",
"Therefore, $r \\le j$ .",
"Since $C_\\mathbb {R}(\\sigma _d)$ is of full dimension in $\\mathbb {R}^{j}$ , then $r=j$ and $C_\\mathbb {R}(\\sigma _d)$ is the cone over an $(j-1)$ -simplex.",
"Now, we are interested in the case where ${\\rm Proper}(C_\\mathbb {R}(\\sigma _d))$ is reduced to a single ray $[s]$ (which then has to be the ray of $\\bar{\\delta }$ if $d = \\delta $ ).",
"Let us take an affine slice $\\Delta $ of $C_\\mathbb {R}(\\sigma _d)$ .",
"The rays $v_1, \\ldots , v_r$ will correspond to points $u_1, \\ldots , u_r$ in $\\Delta $ and these points are the vertices of a polyhedron $\\mathcal {R}$ in $\\Delta $ defined as the convex hull of $u_1, \\ldots , u_r$ .",
"The ray $[s]$ corresponds to a point $s$ in $\\mathcal {R}$ .",
"In order to study the convex properties of $\\mathcal {R}$ , let us translate $\\mathcal {R}$ so that $s$ coincides with the origin.",
"In other words, set $w_i = u_i - s$ and consider the polyhedron $\\mathcal {P}$ which is the convex hull of $w_1, \\ldots , w_r$ .",
"Since $\\mathcal {R}$ lies on an affine slice, we see that $\\mathcal {P}$ lies in a subspace of dimension $n-2$ of $\\mathbb {R}^{n-1}$ .",
"Let $\\mathcal {P}_i$ be the convex hull of all points $w_1, \\ldots , w_r$ but $w_i$ .",
"We define ${\\rm Proper}(\\mathcal {P}):=\\cap _{1 \\le i \\le r}\\mathcal {P}_i$ and we will be interested in the case where ${\\rm Proper}(\\mathcal {P})$ only contains the origin.",
"The first two lemmas are easy to prove.",
"Lemma 5.3 Let $\\mathcal {P}^{\\prime }$ be the convex hull of a subset of $w_1, \\ldots , w_r$ .",
"Then ${\\rm Proper}(\\mathcal {P}^{\\prime })$ is either empty or reduced to the origin.",
"Lemma 5.4 Consider a non-trivial partition $\\lbrace w_{i_1}, \\ldots , w_{i_s}\\rbrace = A_1 \\cup A_2$ of a subset of $\\lbrace w_1, \\ldots , w_r\\rbrace $ .",
"Denote by $\\mathcal {P}_{A_i}$ the convex hull of the points in $A_i$ , for $i=1,2$ .",
"Then $\\mathcal {P}_{A_1} \\cap \\mathcal {P}_{A_2}$ is either empty or reduced to the origin.",
"Proposition 5.5 Suppose that ${\\rm Proper}(\\mathcal {P})$ is empty or reduced to the origin and is full dimensional in $\\mathbb {R}^{t}$ with $t \\le n-2$ .",
"Then there exists a vector space decomposition $\\mathbb {R}^{t} = V_1 \\oplus \\cdots \\oplus V_s$ of $\\mathbb {R}^{t}$ such that if $V_i$ has dimension $d_i$ , then it contains $d_i + 1$ points among $0, w_1, \\ldots , w_r$ that form a $d_i$ -simplex in $V_i$ containing the origin.",
"We may assume that $t = n-2$ so that $\\mathcal {P}$ is $(n-2)$ -dimensional.",
"If ${\\rm Proper}(\\mathcal {P})$ is empty, then $s=1$ and the result follows from Proposition REF .",
"Assume that ${\\rm Proper}(\\mathcal {P})$ is reduced to the origin.",
"Suppose first that the origin lies on a facet, say $F$ , of $\\mathcal {P}$ .",
"We claim that $F$ contains $r-1$ of the points $w_1, \\ldots , w_r$ .",
"Assume otherwise.",
"Consider an $(n-3)$ -simplex in $F$ generated by points $w_{i_1}, \\ldots , w_{i_{n-2}}$ .",
"Let $u, v \\in \\lbrace w_1, \\ldots , w_r\\rbrace $ be two distinct points not in $F$ .",
"By Radon's theorem, we can partition the points $\\lbrace w_{i_1}, \\ldots , w_{i_{n-2}}, u, v\\rbrace $ into two non-empty subsets $A_1, A_2$ such that $\\mathcal {P}_{A_1} \\cap \\mathcal {P}_{A_2} \\ne \\emptyset $ , where $\\mathcal {P}_{A_i}$ denotes the convex hull of the points in $A_i$ .",
"By Lemma REF , this intersection is the origin and hence lies on $F$ .",
"Since $F$ is a facet, $u,v$ lie on the same side of $F$ .",
"Therefore, for $i=1,2$ , the set $B_i:=A_i\\backslash \\lbrace u,v\\rbrace $ is not empty.",
"Now, $B_1$ and $B_2$ form a partition of $w_{i_1}, \\ldots , w_{i_{n-2}}$ such that $\\mathcal {P}_{B_1} \\cap \\mathcal {P}_{B_2} \\ne \\emptyset $ , where $\\mathcal {P}_{B_i}$ denotes the convex hull of the points in $B_i$ .",
"This contradicts Proposition REF .",
"Now, let us assume that the origin lies in the interior of $\\mathcal {P}$ .",
"By Radon's theorem, we can write $\\lbrace w_1, \\ldots , w_r\\rbrace = E_1 \\cup E_2$ where $E_1,E_2$ are disjoint and non-empty such that $\\mathcal {P}_{E_1} \\cap \\mathcal {P}_{E_2} = \\lbrace 0\\rbrace $ , where $\\mathcal {P}_{E_i}$ denotes the convex hull of the points in $E_i$ .",
"By Carathéodory's theorem, there is a simplex formed by some points $z_1, \\ldots , z_s$ in $E_1$ that contains the origin in its interior.",
"With no loss of generality, assume that $z_i = w_i$ and $s \\le r-1$ .",
"Let $V_1$ be the vector space spanned by the points $w_1, \\ldots , w_s$ and consider the vector space $V_2$ spanned by the points $w_{s+1}, \\ldots , w_r$ .",
"Let $C_1$ be the convex hull of the points $w_1, \\ldots , w_s$ and let $C_2$ be the convex hull of the points $w_{s+1}, \\ldots , w_r$ .",
"Since both $\\mathcal {P}_{E_1}, \\mathcal {P}_{E_2}$ contain the origin, we see that $C_1 \\cap C_2 = \\lbrace 0\\rbrace $ .",
"Since 0 lies in the interior of $C_1$ , we get also that $V_1 \\cap C_2=0$ .",
"We claim that $V_1 \\cap V_2 = 0$ .",
"Assume that $V_1 \\cap V_2$ is non-zero.",
"Observe that any element in $V_1$ can be written as a non-negative linear combination of $w_1, \\ldots , w_s$ .",
"There exists non-negative real numbers $a_1, \\ldots , a_s$ and real numbers $b_{s+1}, \\ldots , b_r$ such that $a_1w_1 + \\cdots + a_sw_s = b_{s+1}w_{s+1} + \\cdots + b_rw_r.$ Moreover, $a_1w_1 + \\cdots + a_sw_s$ is non-zero.",
"We may assume the $a_i$ small enough so that the left-hand side lies in $C_1$ .",
"Let us write $\\lbrace s+1, \\ldots , r\\rbrace = I_1 \\cup I_2$ where $I_1, I_2$ are disjoint and $i \\in I_1$ if and only if $b_i \\ge 0$ .",
"We may assume further that the $|b_i|$ are small enough so that both $\\sum _{i \\in I_1}b_iw_i, -\\sum _{j \\in I_2}b_jw_j$ lie in $C_2$ .",
"If $I_2 = \\emptyset $ , then $a_1w_1 + \\cdots + a_sw_s \\in C_1 \\cap C_2$ is non-zero, a contradiction.",
"If all $b_i$ are non-positive, then $-\\sum _{j \\in I_2}b_jw_j$ lies in $V_1\\cap C_2=\\lbrace 0\\rbrace $ , a contradiction.",
"If some $b_i$ are negative and some $b_i$ are positive, we can rewrite the sum as $a_1w_1 + \\cdots + a_sw_s + -\\sum _{j \\in I_2}b_jw_j= \\sum _{j \\in I_1}b_jw_j.$ Considering Lemma REF with the partition $(\\lbrace w_1, \\ldots , w_s\\rbrace \\cup \\lbrace w_{i}\\mid i \\in I_2\\rbrace ) \\cup (\\lbrace w_{j} \\mid j \\in I_1\\rbrace )$ of $\\lbrace w_1, \\ldots , w_r\\rbrace $ , we get that $a_1w_1 + \\cdots + a_sw_s + -\\sum _{j \\in I_2}b_jw_j$ is zero, which reduces to a case we have already considered.",
"Therefore, we have proven that $\\mathbb {R}^{n-2} = V_1 \\oplus V_2$ , where $V_1$ satisfies the property of the statement.",
"We proceed by induction on $V_2$ with the points $w_{s+1}, \\ldots , w_r$ and by using Lemma REF ."
],
[
"The ring of semi-invariants of an isotropic Schur root",
"In this section, we denote by $\\delta $ an isotropic Schur root and by $\\sigma = \\sigma _\\delta $ the weight given by $-\\langle -, \\delta \\rangle $ .",
"Consider, as previously, a full exceptional sequence $(M_{n-2}, \\ldots , M_2, M_1, V,W)$ where $(M_{n-2}, \\ldots , M_2, M_1)$ is an exceptional sequence of simple objects in $\\mathcal {A}(\\delta )$ .",
"Take $I \\subseteq \\lbrace 1, \\ldots , n-2\\rbrace $ such that $i \\in I$ if and only if $i$ is tame connected.",
"Definition 6.1 The associated tame subcategory of $Q$ relative to $\\delta $, denoted $\\mathcal {R}(Q,\\delta )$ , is the thick subcategory of ${\\rm rep}(Q)$ generated by $(\\bigoplus _{i \\in I}M_i)\\oplus V_{n-2}\\oplus W_{n-2}$ .",
"Theorem 6.2 Let $Q$ be an acyclic connected quiver and $\\delta $ an isotropic Schur root.",
"Then The category $\\mathcal {R}(Q,\\delta )$ is tame connected with isotropic Schur root $\\bar{\\delta }$ and is uniquely determined by $\\delta $ .",
"The simple objects in $\\mathcal {A}(\\delta )$ , up to isomorphism, are given by the disjoint union $\\lbrace M_i \\mid i \\notin I\\rbrace \\cup \\lbrace \\text{quasi-simple objects in} \\; \\mathcal {R}(Q,\\delta )\\rbrace .$ We have ${\\rm SI}(Q, \\delta ) \\cong {\\rm SI}(\\mathcal {R}(Q,\\delta ), \\bar{\\delta })[x_{r+1}, \\ldots , x_n].$ Let $(M_{n-2}, \\ldots , M_1)$ be an exceptional sequence of $\\sigma _\\delta $ -stable representations with the corresponding full exceptional sequence $(M_{n-2}, \\ldots , M_1, V, W)$ in ${\\rm rep}(Q)$ , where $\\delta = d_V + d_W$ .",
"First, denote by $M_{l_r}, \\ldots , M_{l_1}$ with $l_r > \\cdots > l_1$ the $M_j$ such that $j$ is tame disconnected or such that $M_j$ is preprojective in $j$ .",
"We get an exceptional sequence $(*) \\qquad (N_{t}, \\ldots , N_2, N_1, V,W, M_{l_r}^{\\prime }, \\ldots , M_{l_1}^{\\prime })$ where all $M_{l_j}$ have been reflected, one by one, to the right of the exceptional sequence.",
"Observe that $\\lbrace M_i \\mid i \\in I\\rbrace \\subseteq \\lbrace N_1, \\ldots , N_t\\rbrace $ .",
"Let $ \\lbrace N_t, \\ldots , N_1\\rbrace \\backslash \\lbrace M_i \\mid i \\in I\\rbrace :=\\lbrace N_{j_s},\\ldots , N_{j_1}\\rbrace $ where $j_s > \\cdots > j_1$ .",
"Assume also that $I = \\lbrace m_1, \\ldots , m_q\\rbrace $ with $m_q > \\cdots > m_1$ .",
"We have $q+s=t$ and $t+r=n-2$ .",
"By Lemma REF (2), we may reflect all exceptional objects of $\\lbrace N_{j_s},\\ldots , N_{j_1}\\rbrace $ in $(*)$ so that we get an exceptional sequence $(M_{m_q}, \\ldots , M_{m_1}, N_{j_s},\\ldots ,N_{j_1}, V, W, M_{l_r}^{\\prime }, \\ldots , M_{l_1}^{\\prime }).$ Now, it follows from the definition of the $V_i, W_i$ that we get an exceptional sequence $(M_{m_q}, \\ldots , M_{m_1}, V_{n-2},W_{n-2}, N_{j_s},\\ldots ,N_{j_1}, M_{l_1}^{\\prime }, \\ldots , M_{l_r}^{\\prime }).$ We claim that for $1 \\le u \\le q$ , we have that ${m_u}=M_{m_u},V_{m_u},W_{m_u})$ is equivalent to $M_{m_u},V_{n-2},W_{n-2})$ .",
"Fix such a $u$ .",
"Note that there is an exceptional sequence of the form $(N_{j_s}, \\ldots , N_{j_p}, M_{m_u},V_{m_u},W_{m_u}).$ Now, it follows from Lemma REF (2) that $M_{m_u}$ lies in $^\\perp N_{j_i}$ for all $1 \\le i \\le p$ .",
"By reflecting, we get the exceptional sequence $(M_{m_u},V_{n-2},W_{n-2}, N_{j_s}, \\ldots , N_{j_p}).$ It follows that ${m_u}=M_{m_u},V_{m_u},W_{m_u})$ is equivalent to $M_{m_u},V_{n-2},W_{n-2})$ .",
"This proves our claim.",
"Let $E = (M_{m_q}, \\ldots , M_{m_1}, V_{n-2},W_{n-2})$ .",
"By Lemma REF and our claim, $\\mathcal {R}(Q,\\delta )=E)$ is tame connected.",
"Since $V_{n-2}, W_{n-2}$ lie in it, $\\bar{\\delta }$ is the (unique) isotropic Schur root of $\\mathcal {R}(Q,\\delta )$ .",
"It follows from Lemma REF that any $\\sigma _\\delta $ -stable representation not isomorphic to any $M_i$ for $1 \\le i \\le n-2$ will have to be (quasi-simple) in $E)$ .",
"Now, we need to show that all quasi-simple objects of $E)$ are $\\sigma _\\delta $ -stable.",
"Assume the contrary.",
"Let $f$ be the dimension vector of a quasi-simple object in $E)$ that is not $\\sigma _\\delta $ -stable, but $\\sigma _\\delta $ -semistable.",
"It follows from our previous observations that $f$ has to be a positive integral combination of the $\\sigma _\\delta $ -stable dimension vectors in $E)$ .",
"It follow from [11] that this is not possible to have such a decomposition.",
"Therefore, we have a complete list of the simple objects in $\\mathcal {A}(\\delta )$ .",
"These are given by the disjoint union $\\lbrace M_i \\mid i \\notin I\\rbrace \\cup \\lbrace \\text{quasi-simple objects in} \\; \\mathcal {R}(Q,\\delta )\\rbrace .$ Observe that, in $C_\\mathbb {R}(\\sigma _\\delta )$ , a dimension vector $d$ can be uniquely written as $d = d_1 + \\sum _{i \\notin I} \\lambda _i f_i$ where $d_1$ is a dimension vector in $E)$ and $f_i = d_{M_{m_i}}$ for $i \\notin I$ .",
"This decomposition is unique.",
"This implies the unicity of $\\mathcal {R}(Q,\\delta )$ and statement $(3)$ .",
"Corollary 6.3 Let $Q$ be an acyclic connected quiver and $\\delta $ an isotropic Schur root.",
"Then ${\\rm SI}(Q, \\delta )$ is a polynomial ring or a hypersurface.",
"More precisely, it is a hypersurface (and not a polynomial ring) if and only if $\\mathcal {R}(Q,\\delta )$ has quiver of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"In [20], it was proven that the ring of semi-invariant of an isotropic Schur root of a tame quiver is a polynomial ring or a hypersurface, where the second situation occurs precisely when the quiver is of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"Our result follows from this and Theorem REF .",
"Example 6.4 Consider the quiver $Q$ given by ${7pt}{& 2 [dl] & \\\\ 1 & & 4 [ul] [ll] [dl]\\\\ & 3 [ul] &}$ Consider the exceptional sequence $(P_2, S_1, I_3, S_3)$ where $P_2$ is the projective representation at vertex 2, $I_3$ is the injective representation at vertex 3 and $S_1, S_3$ are the simple representations at vertices $1,3$ , respectively.",
"Reflecting $S_1, I_3$ to the left of $P_2$ , we get an exceptional sequence whose dimension vectors are as follows.",
"$((0,1,0,0), (3, 3, 1, 1), (1,1,0,0), (0,0,1,0)).$ Then, using a sequence of reflections, we get the following exceptional sequences, where we put the corresponding dimension vectors.",
"$((0,1,0,0), (3, 3, 1, 1), (0,0,1,0), (1,1,1,0))$ $((0,1,0,0), (0,0,1,0), (3, 3, 3, 1), (1,1,1,0))$ $((0,0,1,0), (0,1,0,0), (3, 3, 3, 1), (1,1,1,0))$ $((0,0,1,0), (0,1,0,0), (8,8,8,3), (3, 3, 3, 1))$ $((0,0,1,0), (8,3,8,3), (0,1,0,0), (3, 3, 3, 1))$ $((8,3,3,3), (0,0,1,0), (0,1,0,0), (3, 3, 3, 1)).$ Observe that $\\langle (3,3,3,1),(0,1,0,0) \\rangle =2$ and $\\delta = (3,3,3,1) - (0,1,0,0) = (3,2,3,1)$ is an isotropic Schur root.",
"The Coxeter matrix $\\tau $ is $\\tau = \\left(\\begin{array}{cccc}-1 & 1 & 1 & 1 \\\\-1 & 0 & 1 & 2 \\\\-1 & 1 & 0 & 2 \\\\-3 & 2 & 2 & 4 \\\\\\end{array}\\right).$ This matrix has eigenvalues $\\lambda = 5/2 +\\sqrt{21}/2$ , $\\lambda ^{-1} = 5/2-\\sqrt{21}/2$ and $-1$ with (algebraic and geometric) multiplicity 2.",
"The eigenvector corresponding to $\\lambda $ is $v_1 = (10, 9+\\sqrt{21}, 9+\\sqrt{21}, 17 + \\sqrt{189})$ and the one corresponding to $\\lambda ^{-1}$ is $v_2 = (10, 9-\\sqrt{21}, 9-\\sqrt{21}, 17 - \\sqrt{189}).$ Now, $\\langle v_2, (8,3,3,3) \\rangle = -197 + 10\\sqrt{21} + 11\\sqrt{189} > 0$ and $\\langle v_2, (0,0,1,0) \\rangle = -8-\\sqrt{21}+\\sqrt{189} > 0$ .",
"Similarly, both $\\langle (8,3,3,3), v_1 \\rangle $ and $\\langle (0,0,1,0), v_1 \\rangle $ are positive.",
"Therefore, the exceptional objects with dimension vectors $(8,3,3,3), (0,0,1,0)$ are regular by the theorem at page 240 of [16].",
"It follows from Proposition REF that $\\delta $ is not of smaller type.",
"It also follows from the same proposition that there is a unique exceptional sequence $(M_2, M_1)$ of length 2 of $\\sigma $ -stable objects.",
"Let $M_1^{\\prime } = S_3$ and $M_2^{\\prime }$ be the exceptional representation with dimension vector $(8,3,3,3)$ .",
"Since $M_1^{\\prime }, M_2^{\\prime }$ lie in $M_2,M_1)$ by Lemma REF , we see that $M_2^{\\prime },M_1^{\\prime }) \\subseteq M_2,M_1)$ and thus, we have equality.",
"This means that $M_2^{\\prime } = M_2$ , $M_1^{\\prime } = M_1$ .",
"Since $\\langle \\delta , (0,0,1,0) \\rangle = 2 >0$ , we get $\\delta _1 = \\delta - 2(0,0,1,0) = (3,2,1,1)$ .",
"Now, $\\langle \\delta _1, (8,3,3,3) \\rangle = -2$ .",
"Therefore, $\\bar{\\delta }= \\delta _1 = (3,2,1,1)$ .",
"In this example, the cone of $\\sigma $ -semistable dimension vectors is as follows (where only an affine slice of that cone is shown).",
"Figure: The cone of σ\\sigma -semistable dimension vectors for δ=(3,2,3,1)\\delta = (3,2,3,1)The following is an easy observation.",
"The reader is referred to [8] for the notion of cluster algebra and to [7] for results in similar directions.",
"Corollary 6.5 If ${\\rm SI}(Q,\\delta )$ is not a polynomial ring, then it has a cluster algebra structure of type $\\mathbb {A}_1$ .",
"There are two cluster variables which are all $\\Gamma $ -homogeneous, and the coefficients are built from $n-1$ frozen variables, which are also $\\Gamma $ -homogeneous, where $\\Gamma $ is the set of all multiplicative characters of ${\\rm GL}_\\delta (k)$ .",
"From Theorem REF , it is enough to prove this for ${\\rm rep}(Q) = \\mathcal {R}(Q,\\delta )$ , that is, we may assume that $Q$ is tame connected.",
"Suppose that ${\\rm SI}(Q,\\delta )$ is not a polynomial ring.",
"Then $Q$ is of type $\\widetilde{\\mathbb {D}_n}$ with $n \\ge 4$ , $\\widetilde{\\mathbb {E}_6}$ , $\\widetilde{\\mathbb {E}_7}$ or $\\widetilde{\\mathbb {E}_8}$ .",
"In particular, it is well known in these cases that there are exactly three non-homogeneous tubes $T_1, T_2, T_3$ in the Auslander-Reiten quiver of $\\mathcal {R}(Q,\\delta )$ .",
"One, say $T_1$ , has rank 2.",
"Let $M,N$ be the non-isomorphic exceptional quasi-simple objects in $T_1$ .",
"Then, let $E_1, \\ldots , E_r$ be the non-isomorphic quasi-simple objects of $T_2$ and let $E_1^{\\prime }, \\ldots , E_t^{\\prime }$ be the non-isomorphic quasi-simple objects of $T_3$ .",
"Now, the hypersurface equation can be written as $(*) \\qquad C^MC^N = C^{E_1}\\cdots C^{E_r} + C^{E_1^{\\prime }}\\cdots C^{E_t^{\\prime }}.$ Consider the indeterminates $x,y_1, \\ldots , y_r, z_1, \\ldots , z_t$ .",
"We define a cluster algebra $A$ as follows.",
"We start with the initial seed $\\lbrace x,y_1, \\ldots , y_r, z_1, \\ldots , z_t\\rbrace $ where $y_1, \\ldots , y_r$ and $z_1, \\ldots , z_t$ are declared to be frozen variables.",
"The exchange relation is $xx^{\\prime } = \\prod _{i=1}^ry_i + \\prod _{j=1}^tz_j$ which clearly produces exactly two cluster variables $x,x^{\\prime }$ .",
"The cluster algebra is the the $\\mathbb {Z}$ -subalgebra of $\\mathbb {Q}(x,y_1, \\ldots , y_r, z_1, \\ldots , z_t)$ generated by $x, x^{\\prime }$ and $y_1, \\ldots , y_r, z_1, \\ldots , z_t$ .",
"This algebra is clearly isomorphic to ${\\rm SI}(Q,\\delta )$ .",
"An interesting problem would be to find all acyclic quivers $Q$ and dimension vectors $d$ such that SI$(Q,d)$ has a cluster algebra structure whose variables (frozen or not) are all $\\Gamma $ -homogeneous."
],
[
"Construction of all isotropic Schur roots",
"In this section, we show that all of the isotropic Schur roots of ${\\rm rep}(Q)$ come from isotropic Schur roots of a tame full subquiver of $Q$ by applying special reflections.",
"We make this precise by defining an action of the braid group $B_{n-1}$ on $n-1$ strands on a special type of exceptional sequences that will encode all we need to study isotropic Schur roots.",
"We start with the definition of these sequences.",
"Definition 7.1 Let $E=(X_1, \\ldots , X_n)$ be a full exceptional sequence.",
"We say that $E$ is of isotropic type if there exists $1 \\le i \\le n-1$ such that $X_{i}, X_{i+1})$ is tame.",
"The integer $i$ is called the isotropic position of $E$ and the root type of $E$ , denoted $\\delta _E$ , is the isotropic Schur root in $X_{i}, X_{i+1})$ .",
"We denote by $\\mathcal {E}$ the set of all full exceptional sequences of isotropic type, up to isomorphism.",
"Not all elements of the braid group $B_{n}$ act on $\\mathcal {E}$ .",
"We rather consider the group $B_{n-1}$ and show that it acts on $\\mathcal {E}$ .",
"Let us denote the standard generators of $B_{n-1}$ by $\\gamma _1, \\ldots , \\gamma _{n-2}$ .",
"Let $E=(X_1, \\ldots , X_n) \\in \\mathcal {E}$ with isotropic position $r$ .",
"Let $1 \\le i \\le n-2$ .",
"If $i<r-1$ , then $\\gamma _iE := \\sigma _iE$ .",
"If $i>r$ , then $\\gamma _{i}E := \\sigma _{i+1}E$ .",
"Assume that $i=r$ with $r<n-1$ .",
"We can reflect $X_{r+2}$ to the left of $X_r, X_{r+1}$ to get the exceptional sequence: $E^{\\prime }=(X_1, \\ldots ,X_{r-1}, L_{X_{r}}(L_{X_{r+1}}(X_{r+2})), X_r, X_{r+1}, X_{r+3}, \\ldots , X_n).$ and this is an exceptional sequence of isotropic type with isotropic position $r+1$ .",
"We define $\\gamma _rE:=E^{\\prime }$ .",
"If $r > 1$ and $i=r-1$ , then we can reflect both $X_r, X_{r+1}$ to the left of $X_{r-1}$ as follows: $E^{\\prime \\prime } = (X_1, \\ldots ,X_{r-2}, L_{X_{r-1}}(X_r), L_{X_{r-1}}(X_{r+1}), X_{r-1}, X_{r+2}, \\ldots , X_n)$ and clearly, the subcategory $L_{X_{i-1}}(X_i), L_{X_{i-1}}(X_{i+1}))$ generates a tame subcategory of rank 2.",
"Therefore, $E^{\\prime \\prime }$ is an exceptional sequence of isotropic type with isotropic position $r-1$ and its root type is the unique isotropic Schur root in $L_{X_{i-1}}(X_i), L_{X_{i-1}}(X_{i+1}))$ , which is $\\delta _{E^{\\prime }} = \\delta _E - \\langle \\delta _E, d_{X_{r-1}}\\rangle d_{X_{r-1}}$ , by Lemma REF .",
"We define $\\gamma _{r-1}E=E^{\\prime \\prime }$ .",
"Similarly, we can define the action of $\\gamma _{i}^{-1}$ on $E$ for $1 \\le i \\le n-2$ .",
"The following is easy to check.",
"Proposition 7.2 The group $B_{n-1}$ acts on exceptional sequences of isotropic type, with the action defined above.",
"Definition 7.3 A sequence $E=(X_1, \\ldots , X_{n-1}, X_n)$ in $\\mathcal {E}$ is of tame type if it has isotropic position $n-1$ , and there is $0 \\le s \\le n-2$ such that $X_1, \\ldots , X_s$ are projective in ${\\rm rep}(Q)$ and $X_{s+1}, \\ldots , X_{n-2}, X_{n-1}, X_n)$ is tame connected.",
"By convention, $s=0$ means that ${\\rm rep}(Q)$ is already tame connected.",
"Observe that if $E \\in \\mathcal {E}$ is of tame type, then the isotropic Schur root $\\delta _E$ is the unique isotropic Schur root of the tame subcategory $X_{s+1}, \\ldots , X_{n-2}, X_{n-1}, X_n)$ and is an isotropic Schur root coming from a tame full subquiver of $Q$ .",
"In particular, there are finitely many roots $\\delta _E$ where $E \\in \\mathcal {E}$ is of tame type.",
"Example 7.4 Consider a quiver of rank $n=4$ and an exceptional sequence $E=(X,U,V,Y)$ of isotropic type with isotropic position 2.",
"The root type is the isotropic root $\\delta _E$ in $U,V)$ .",
"Figure: Correspondence between B 3 B_3 and some braids of B 4 B_4The first braid (A) in Figure REF corresponds to the element $g=\\gamma _2^{-1}\\gamma _1^{-1}\\gamma _2\\gamma _1^{-1}$ of $B_3$ while the second braid (B) corresponds to the element $h=\\sigma _2^{-1}\\sigma _1^{-1}\\sigma _3^{-1}\\sigma _2^{-1}\\sigma _3\\sigma _2^{-1}\\sigma _1^{-1}$ of $B_4$ .",
"Notice that $gE = hE$ .",
"Notice also that the braid in (A) is obtained from the braid in (B) by identifying the two strands starting at the positions of $U,V$ , that is, the second and third strands.",
"Our aim in this section is to prove that any $E \\in \\mathcal {E}$ lies in the $B_{n-1}$ -orbit of an exceptional sequence of tame type.",
"In the next lemmas, we will consider exceptional sequences in the bounded derived category $D^b({\\rm rep}(Q))$ of ${\\rm rep}(Q)$ .",
"Recall that an object $X$ in $D^b({\\rm rep}(Q))$ is exceptional if ${\\rm Hom}(X,X[i])=0$ for all non-zero $i$ (and then, ${\\rm Hom}(X,X)$ has to be one dimensional).",
"Equivalently, an exceptional object in $D^b({\\rm rep}(Q))$ is isomorphic to the shift of an exceptional representation.",
"A sequence $(X_1, \\ldots , X_r)$ of objects in $D^b({\\rm rep}(Q))$ is exceptional if every $X_i$ is exceptional and, for $i < j$ , we have ${\\rm Hom}_{D^b({\\rm rep}(Q))}(X_i, X_j[t])=0$ for all $t \\in \\mathbb {Z}$ .",
"For such a sequence, one can consider the smallest full additive subcategory $\\mathcal {D}(X_1, \\ldots , X_r)$ of $D^b({\\rm rep}(Q))$ containing $X_1, \\ldots , X_r$ and that is closed under direct sums, direct summands, taking the cone of a morphism and the shift of an object.",
"One can also consider the exceptional sequence $(X_1^{\\prime }, \\ldots , X_r^{\\prime })$ in ${\\rm rep}(Q)$ such that $X_i^{\\prime }$ is the unique shift of $X_i$ lying in ${\\rm rep}(Q)$ .",
"The indecomposable objects in $\\mathcal {D}(X_1, \\ldots , X_r)$ are just the shifts of the indecomposable objects in $X_1^{\\prime }, \\ldots , X_r^{\\prime })$ .",
"In what follows, the Auslander-Reiten translate in $D^b({\\rm rep}(Q))$ is denoted by $\\tau _{D}$ while the Auslander-Reiten translate in ${\\rm rep}(Q)$ is simply denoted $\\tau $ .",
"Recall that if $X$ is a non-projective indecomposable representation, then $\\tau _DX = \\tau X$ and, if $X = P_x$ with $x \\in Q_0$ , then $\\tau _D X = I_x[-1]$ .",
"When $d$ is a dimension vector, we denote by $\\tau d$ the product of the Coxeter matrix with $d$ .",
"In particular, if $X$ is a non-projective indecomposable representation, then $\\tau d_X = d_{\\tau X}$ and, if $X=P_x$ with $x \\in Q_0$ , then $\\tau d_{X} = -d_{I_x}$ .",
"We start our investigation with the following lemma that is crucial for the proof of the main result of this section.",
"Lemma 7.5 Let $(X_1, \\ldots , X_{n})$ be an exceptional sequence with $X_{r+1}, \\ldots , X_{n})$ tame and assume that $X_1, \\ldots , X_r$ are the simple objects in $X_1, \\ldots , X_r)$ .",
"Let $X \\in X_1, \\ldots , X_r)$ be the injective object with socle $X_1$ .",
"If $X$ is projective in $X,X_{r+1}, \\ldots , X_{n})$ , then $X_1$ is projective in ${\\rm rep}(Q)$ and in particular, an isotropic Schur root of $X_{r+1}, \\ldots , X_{n} )$ is not sincere.",
"Assume that $X$ is projective in $X,X_{r+1}, \\ldots , X_{n})$ .",
"Set $d_i = d_{X_i}$ for $1 \\le i \\le n$ .",
"Consider the linear form $f$ given by $f(x)=\\langle d_1, x \\rangle $ .",
"Then $f$ vanishes on $d_2, \\ldots , d_{n}$ and $f(d_1)>0$ .",
"Assume to the contrary that $X_1$ is not projective in ${\\rm rep}(Q)$ .",
"Observe that $f(x)=\\langle d_1, x \\rangle = -\\langle x, \\tau d_1 \\rangle $ .",
"Since $\\tau X_1$ is exceptional, $\\langle \\tau d_1, \\tau d_1 \\rangle = 1$ and hence $f(\\tau d_1) < 0$ .",
"Now, reflect $X_1$ to the right of $X_2, \\ldots X_r$ , so that we get an exceptional sequence $(X_2, \\ldots , X_r, Y)$ where $Y$ is in the cone spanned by $d_1, \\ldots , d_r$ .",
"Clearly, $X_1$ is simple projective in $X_1, \\ldots , X_r)$ and hence, $Y=X$ is the injective hull of $X_1$ in $X_1, \\ldots , X_r)$ .",
"Set $=X, X_{r+1}, \\ldots , X_{n})$ .",
"We know that $X$ is projective in $.",
"Reflecting $ X$ to the right of $ Xr+1, ..., Xn$ will give the exceptional representation $ X1$.",
"Therefore, $ d:=d1 = -d̏X$ where $ denotes the Coxeter transformation in $.",
"Take the linear form $ g$ in the Grothendieck group of $ given by $g(x)=\\langle d_{X}, x \\rangle $ .",
"Then $g$ vanishes on $d_{r+1}, \\ldots , d_{n}$ and $g(d_X)>0$ .",
"The form $f|_ has the same property since $ dX$ is a non-negative linear combination of $ d1, ..., dr$ with the coefficient of $ d1$ positive.",
"Thus, $ g = f| up to a positive scalar.",
"Therefore, $g(d)<0$ , which means that $X,\\tau X_1$ lie on opposite sides of the hyperplane $g(x)=0$ in $.",
"This contradicts that $ X$ is projective in $ .",
"Lemma 7.6 Let $(X_1, \\ldots , X_r)$ be an exceptional sequence and assume that $1:=X_2, \\ldots , X_r)$ is tame with an isotropic Schur root $\\gamma $ while $2:=X_1, X_2, \\ldots , X_r)$ is wild.",
"Then there is a unique minimal isotropic Schur root in the $\\tau $ -orbit of $\\gamma $ .",
"We may assume that $2 = {\\rm rep}(Q)$ for an acyclic quiver $Q$ .",
"Since 2 is wild and 1 is tame, we know that $X_1$ is preprojective or preinjective in 2.",
"Hence, there is some $r \\in \\mathbb {Z}$ such that $\\tau _D^r X_1$ is projective or the shift of a projective.",
"This means that $\\tau ^r\\gamma $ is not sincere.",
"Let $Y = \\tau _D^r X_1$ if $\\tau _D^r X_1$ is a representation or $Y = \\tau _D^r X_1[-1]$ if $\\tau _D^r X_1$ is the shift of a projective representation.",
"Observe that $Y^\\perp \\subseteq {\\rm rep}(Q)$ is also of tame representation type, where the quivers of $X_1^\\perp $ and $Y^\\perp $ only differ by a change of orientation; see for instance [10].",
"Therefore, $\\tau ^r\\gamma $ is an isotropic Schur root of a tame full subquiver of $Q$ .",
"Let $s \\in \\mathbb {Z}$ with $s \\ne r$ .",
"Consider $Z$ the unique shift of $\\tau _D^s X_1$ which is a representation.",
"Since the simples in $Y^\\perp $ are simples in ${\\rm rep}(Q)$ , and since there is a simple of $Z^\\perp \\subseteq {\\rm rep}(Q)$ that is not simple in ${\\rm rep}(Q)$ , we see that the isotropic Schur root $\\tau ^r\\gamma $ has smaller length than $\\tau ^s\\gamma $ .",
"This also proves unicity since only one object in the $\\tau $ -orbit of $X_1$ is projective or a shift of a projective.",
"Lemma 7.7 Let $E=(X_1, \\ldots , X_{n-2}, U, V)$ be in $\\mathcal {E}$ with isotropic position $n-1$ .",
"Let $E^{\\prime } = \\gamma _{1}\\cdots \\gamma _{n-3}\\gamma _{n-2}E = (U^{\\prime }, V^{\\prime }, X_1, \\ldots , X_{n-2})$ .",
"Then $\\tau ^{-1} \\delta _E = \\delta _{E^{\\prime }}$ .",
"If $V$ is not injective, then we have the exceptional sequence $(\\tau ^{-1}V, X_1, \\ldots , X_{n-2},U)$ in ${\\rm rep}(Q)$ .",
"Otherwise, we have the exceptional sequence $(\\tau ^{-1}V[-1], X_1, \\ldots , X_{n-2}, U)$ in ${\\rm rep}(Q)$ .",
"Let us write $\\tau ^{-1}V[0,-1]$ to indicate that we either take the shift $[0]$ or $[-1]$ for $\\tau ^{-1}V$ .",
"Then, we get an exceptional sequence $(\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1], X_1, \\ldots , X_{n-2}).$ The categories $U^{\\prime }, V^{\\prime })$ and $\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1])$ are equal in ${\\rm rep}(Q)$ .",
"Therefore, they have the same isotropic Schur root.",
"The isotropic Schur root of $\\tau ^{-1}U[0,1], \\tau ^{-1}V[0,1])$ is clearly $\\tau ^{-1}\\delta _E$ .",
"Of course, we have the dual version of the above lemma as follows.",
"Lemma 7.8 Let $E=(U,V, X_1, \\ldots , X_{n-2})$ be in $\\mathcal {E}$ with isotropic position 1.",
"Let $E^{\\prime } = \\gamma _{n-2}^{-1}\\gamma _{n-3}^{-1}\\cdots \\gamma _{1}^{-1}E = (X_1, \\ldots , X_{n-2}, U^{\\prime }, V^{\\prime })$ .",
"Then $\\tau \\delta _E = \\delta _{E^{\\prime }}$ .",
"We are now ready for the main result of this section.",
"Theorem 7.9 Let $\\delta $ be an isotropic Schur root.",
"Then there is $E \\in \\mathcal {E}$ of tame type and $g \\in B_{n-1}$ such that $gE$ has root type $\\delta $ .",
"It follows from Proposition REF that there is an exceptional sequence $F=(M_1, \\ldots , M_{n-2}, X,Y)$ in $\\mathcal {E}$ of isotropic position $n-1$ and of root type $\\delta $ .",
"Assume that $G\\in \\mathcal {E}$ is in the orbit of $E$ and the root type of $G$ is minimal, that is, has minimal length as a root in ${\\rm rep}(Q)$ .",
"We may assume that the isotropic position of $G$ is $n-1$ .",
"Therefore, we may assume that $G$ is of the form $(Y_1, \\ldots , Y_{n-2}, U, V).$ Assume first that there is an object $W$ in $Y_1, \\ldots , Y_{n-2})$ that is not projective in $W,U,V)$ .",
"We can apply a sequence of reflections to the subsequence $(Y_1, \\ldots , Y_{n-2})$ to get an exceptional sequence $H=(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W, U,V)$ in $\\mathcal {E}$ .",
"Now, applying $\\gamma _{n-2}^2$ to $H$ and using Lemma REF , we get the sequence $(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W^{\\prime }, U^{\\prime }, V^{\\prime })$ in $\\mathcal {E}$ whose root type is the inverse Auslander-Reiten translate of $\\delta $ in $W,U,V)$ .",
"Similarly, applying $(\\gamma _{n-2}^{-1})^2$ to $H$ , we get the sequence $(Y_1^{\\prime }, \\ldots , Y_{n-3}^{\\prime }, W^{\\prime \\prime }, U^{\\prime \\prime }, V^{\\prime \\prime })$ in $\\mathcal {E}$ whose root type is the Auslander-Reiten translate of $\\delta $ in $W,U,V)$ .",
"We can iterate this to get a smaller root by Lemma REF , provided $W,U,V)$ is wild.",
"Therefore, whenever there is an object $W$ which is not projective in $W,U,V)$ , then $W,U,V)$ is of tame type (and hence connected).",
"Suppose, by induction, that we have an exceptional sequence $J=(W_{r+1},\\ldots ,W_{n-2},U,V)$ such that $J)$ is tame connected and $J$ has maximal length with respect to this property.",
"If $r=0$ , then $Q$ is a tame connected quiver and there is nothing to prove.",
"Complete this to get a full exceptional sequence $(Z_1, \\ldots , Z_r, W_{r+1}, \\ldots , W_{n-2}, U,V).$ If there is $W \\in ^\\perp J) = Z_1, \\ldots , Z_r)$ such that $W$ is not projective in $W,U,V)$ then, $W,U,V)$ is tame connected.",
"As in the proof of Lemma REF , we get that $W,W_{r+1}, \\ldots , W_{n-2},U,V)$ is tame connected, contradicting the maximality of $J$ .",
"Therefore, any object $Z$ in $Z_1, \\ldots , Z_r)$ is such that $Z$ is projective in $Z,W_{r+1}, \\ldots , W_{n-2},U,V)$ .",
"We may apply a sequence of reflections and assume that all of $Z_1, \\ldots , Z_r$ are simple in $Z_1, \\ldots , Z_r)$ .",
"It follows from Lemma REF that the injective hull of $Z_1$ in $Z_1, \\ldots , Z_r)$ is projective in ${\\rm rep}(Q)$ .",
"Then the proof goes by induction.",
"Here is another way to interpret this result.",
"Start with an isotropic Schur root $\\delta _0$ of a tame full subquiver $Q^{\\prime }$ of $Q$ and consider an exceptional sequence $(U_0,V_0)$ of length 2 in ${\\rm rep}(Q^{\\prime }) \\subset {\\rm rep}(Q)$ such that $\\delta _0 = d_{U_0} + d_{V_0}$ .",
"Consider an exceptional object $X_0$ such that $(X_0,U_0,V_0)$ is an exceptional sequence of length three (which generates a thick subcategory 0 of ${\\rm rep}(Q)$ ).",
"Then we can transform it into another exceptional sequence $(X_0^{\\prime }, U_1, V_1)$ with an isotropic Schur root $\\delta _1 = d_{U_1}+d_{V_1}$ such that $\\delta _1$ is a power $\\tau _{0}^{r_0} \\delta _0$ where $\\tau _{0}$ denotes the Coxeter matrix for 0.",
"Now, for $i \\ge 1$ , consider an exceptional object $X_i$ such that $(X_i,U_i, V_i)$ is an exceptional sequence.",
"Take a power $\\delta _{i+1}=\\tau _{i}^{r_i} \\delta _i$ where $i$ is the thick subcategory of ${\\rm rep}(Q)$ generated by $X_i, U_i, V_i$ and $\\tau _{i}$ denotes the Coxeter matrix for $i$ .",
"All the roots $\\delta _i$ constructed this way are isotropic Schur roots.",
"Moreover, all isotropic Schur roots of ${\\rm rep}(Q)$ can be obtained in this way.",
"There are clearly only finitely many starting roots $\\delta _0$ , but the choices of the $r_i$ and $X_i$ yield, in general, infinitely many possible isotropic Schur roots.",
"As observed in [15], when $Q$ is wild connected with more than 3 vertices, there are infinitely many $\\tau $ -orbit of isotropic Schur roots (provided there is at least one isotropic Schur root).",
"An interesting question would be to describe the minimal root types of the orbits of $\\mathcal {E}$ under $B_{n-1}$ .",
"It is not hard to check that when $n=3$ , these minimal root types correspond exactly to the tame full subquivers of $Q$ .",
"We do not know if this holds in general.",
"Conjecture 7.10 Let $E_1, E_2 \\in \\mathcal {E}$ .",
"Assume that there are $g_1, g_2 \\in B_{n-1}$ with $g_1E_1, g_2E_2$ of tame type but with different root types.",
"Then $E_1, E_2$ lie in distinct orbits under $B_{n-1}$ .",
"Acknowledgment.",
"The authors are thankful to Hugh Thomas for suggesting the decomposition in Proposition REF .",
"The second named author was supported by NSF grant DMS-1400740."
]
] | 1605.05719 |
[
[
"Robust Reserve Capacity Provision and Peak Load Reduction from Buildings\n in Smart Grids"
],
[
"Abstract This paper proposes a robust demand-side control algorithm in a smart grid environment for heating, ventilation and air conditioning (HVAC) systems.",
"A robust model predictive control (RMPC) scheme in a receding horizon fashion is deployed, which optimizes electricity cost and capacity market participation of the HVAC system, while satisfying comfort and operational constraints of the building and utility, respectively.",
"Thermal load uncertainties experienced by the HVAC system are included to perform a realistic assessment of the developed controller.",
"The National Electricity Market of Singapore (NEMS) is used as a case study and the developed RMPC scheme is tested for various price signals and scenarios.",
"Numerical simulation results show the effectiveness of the developed framework to be readily adopted by utilities -- interested in realizing a grid-friendly and economicaly eficient demand response (DR) strategy."
],
[
"Introduction",
"One of the key concepts in smart grids is the interaction of users and utilities.",
"To achieve it, bidirectional communication is promised among operators, retailers and consumers.",
"Under such conditions, one can realize adjusting local DR strategies, based on exegenous signals from utility.",
"This can help relieve the grid's need for a higher reserve requirements, due to the integration of highly variable renewable energy supply.",
"With a similar philosophy, various utilities have established demand response (DR) programs, aiming to improve the overall efficiency of the grid [1], [2], [3], [4].",
"Among all energy consumption sectors, buildings are considered as one of the major contributors of greenhouse gas emissions and electricity consumption.",
"Within a building, the HVAC system consumes the largest portion of the energy.",
"Space cooling/heating along with the thermal inertia of buildings provides an inherent flexibility in the consumption of electricity.",
"In principle, the usage of space cooling's flexibility can provide: $(1)$ reduced building operational cost, $(2)$ ancillary service provision to the grid and, $(3)$ grid secuirity.",
"Significant amount of work has been done for controlling energy consumption of buildings.",
"Recent contribution regarding price-based and direct load control was reported in [5].",
"The applicability of MPC to control building energy consumption was reported in [6], [7], [8].",
"The use of thermal electric loads for the provision of ancillary services, and reduction of the balancing groups' scheduled deviations were reported in [9] and [10], respectively.",
"Authors in [11] mentioned methods for dealing with uncertain thermal load of the HVAC system, by including them as bounded disturbances.",
"The aforementioned papers either deal with the market participation or the uncertainties of the HVAC system model.",
"Furthermore, the DR planning framework capable of analyzing the interaction of the market-oriented robust control scheme, subjected to various utility pricing structures and ancillary services provision has also not yet been presented.",
"The contribution of this paper is twofold.",
"First, it develops an RMPC scheme to co-optimize the energy schedule, with respect to both the energy and capacity market.",
"Second, it assesses the applicability of the developed control scheme for minimizing the total cost of the HVAC system under an exogenous peak load constraining utility signal.",
"The remainder of the paper is organized as follows.",
"Section explains the HVAC system and building model, along with the market and uncertainty settings used for developing the RMPC scheme.",
"The RMPC control scheme is designed in section .",
"In section , simulation results are presented.",
"Section concludes this paper with comments on adequacy of RMPC scheme for providing flexible demand, with corporation of utility peak load reduction signal.",
"The HVAC system considered in this paper has a variable air volume (VAV) mass flow rate.",
"This provides us the opportunity to change the variable frequency drive, to meet the energy demand of a building.",
"The cooling/heating demand is calculated based on a thermal dynamic model given in [12].",
"The validation of model is given in [13].",
"The external and internal loads are estimated in [14].",
"Essentially, the model provides a nonlinear relationship of the form $\\dot{x}_{t} = f\\left(x_t,u_t,\\hat{d_t}\\right)$ between the room temperature $x_{t} \\in \\mathbb {R}^{n_d}$ and the air flow input $u_{t} \\in \\mathbb {R}^{j}$ from the HVAC, experiencing disturbance $\\hat{d_t} \\in \\mathbb {R}^{n_d}$ .",
"The thermal dynamic model of a room is represented as a network of $n = i + j$ nodes.",
"Where $i$ and $j$ represent walls and room, respectively: $\\frac{dT_{wi}}{dt} &= \\frac{1}{C_{wi}}\\left[\\sum _{j{ \\in N_{wi}}} \\frac{T_j - T_{wi}}{R_{ij}} + r_i \\alpha _i A_i q^{^{\\prime \\prime }}_{radi}\\right], \\\\\\frac{dT_{ri}}{dt} &= \\frac{1}{C_{ri}}\\left[\\sum _{j_{\\exists N_{ri}}} \\frac{T_j - T_{ri}}{R_{ij}} + \\dot{m}_{ri} c_p \\left(T_{si} - T_{ri}\\right) \\right.",
"\\nonumber \\\\&\\qquad \\left.",
"{} + w_i \\tau _{wi} A_{wi} q^{^{\\prime \\prime }}_{radi} + \\dot{q}_{int}\\right],$ Where $T_{wi}$ , $C_{wi}$ , $\\alpha _{i}$ , $A_i$ and $c_p$ represent the temperature, thermal capacitance, absorptivity factor, area, and specific heat capacity of the room $i$ , respectively.",
"$N_{wi}$ shows the set of all neighboring nodes to $w_i$ .",
"The value of $r_i$ is equal to 0 for internal, and 1 for peripheral walls.",
"For the $i$ -th room, $T_{ri}$ , $C_{ri}$ and $\\dot{m}_{ri}$ show its temperature, thermal capacitance and air mass flow rate, respectively.",
"The transmittance and area of the $i$ -th window is given by $\\tau _{wi}$ and $A_i$ , respectively.",
"$q^{^{\\prime \\prime }}_{radi}$ is the solar irradiation experienced by room $i$ , and $\\dot{q}_{int}$ represents the internal heat generated due to equipments, furniture and occupancy.",
"$w_i$ shows if windows are present on the surrounding walls of the room.",
"It can be observed from (REF ), the system at hand is non-linear.",
"For the purpose of designing a controller, linear models are desirable.",
"In [12], author proposed a method based on Sequential Quadratic Programming (SQP), to obtain a linearized model.",
"After the linear system is obtained, Zero-order hold is performed to discretize it.",
"The resultant discrete time state system is represented as: ${\\begin{@align}{1}{-1}x_{k+1} &= A x_k + B u_k + E \\hat{d}_k\\end{@align}}$ Where $x_{k+1} \\in \\mathbb {R}^n$ is the temperature of all the states at step $k+1$ , due to control input $u_k \\in \\mathbb {R}^j$ and disturbance $\\hat{d}_k \\in \\mathbb {R}^n$ , at time step $k$ .",
"To calculate the cost of consumption as a function of fan power $u_k$ , $K\\left(u_k\\right)$ is represented in (REF ).",
"Where the electricity price at time step $k$ is represented as $c_k$ .",
"And sample time to convert power to energy is taken as $ \\Delta t $ : ${\\begin{@align}{1}{-1}K\\left(u_k\\right) &= \\Delta t \\ c_k \\left(P_{f,u_k} + P_{c,u_k} + P_{h,u_k}\\right)\\end{@align}}$ In (REF ), $P_{f,u_k}$ , $P_{c,u_k}$ and $P_{h,u_k}$ are power consumed by the fan, cooling and heating coil of the HVAC system, respectively (more details regarding units and dimensions are given in [12], [13], [14])."
],
[
"Model Extension",
"To align the model with our objectives, two modifications are performed to the original model presented above.",
"Firstly, to account for the uncertainties into the model, additive uncertainty $w_k$ $\\in $ $\\mathbb {R}^{n_w}$ is introduced.",
"Where $n_w$ are the number of uncertain variables.",
"The origin of disturbance is the in-ability to model part of the thermal behaviour of the room acurately.",
"We consider box-constrained disturbance uncertainty with uniform distribution i.e.",
"to be known bounded by some measure, other wise unknown.",
"${\\begin{@align}{1}{-1}x_{k+1}^{a} &= A x_{k} + B u_{k} + E (\\hat{d_{k}} + w_k), \\\\\\mathcal {W}_k &= \\lbrace w: \\left\\Vert w\\right\\Vert \\le \\sigma _k \\rbrace \\end{@align}}$ where $\\mathcal {W}_k$ is the set of all possible disturbance uncertainties $(\\forall k = 0, 1, \\ldots , N - 1)$ and $w_k \\in \\mathcal {W}_k$ .",
"$N$ is the prediction horizon to be used in developing the controller in section .",
"The dimensions of vectors and matrices in (REF ) follows directly from the original system.",
"Moreover, the introduction of box-constrained disturbance variable $w_k$ in (REF ) is done by adjusting the original disturbance vector $\\hat{d_{k}}$ .",
"The next extension to the model is performed based on the idea suggested by authors in [9].",
"To enable the HVAC system for offering its reserve capacity, we have extended the state space model of (REF ) as: ${\\begin{@align}{1}{-1}x_{k+1} &= A x_{k} + B u_{k} + E (\\hat{d_{k}} + w_k) + B_{r} r_{k}\\end{@align}}$ At each time step $k$ , the matrix $B_{r} \\in \\mathbb {R}^{n \\times j}$ translates the effect of the extra power in the form of reserve capacity $r_k \\in \\mathbb {R}^{j}$ on to the temperature of zones.",
"Matrix $B_{r}$ (by multiplying 0 or 1 with original input coefficients from $B$ ) indicates whether the HVAC system is considering to offer some reserve capacity or not.",
"Similar to (REF ), the cost $K\\left(r_k\\right)$ of allocating reserve capacity $r_k$ for the reserve price of $b_k$ at time step $k$ is calculated by: ${\\begin{@align}{1}{-1}K\\left(r_k\\right) &= \\Delta t \\ b_k \\left(P_{f,r_k} + P_{c,r_k} + P_{h,r_k}\\right)\\end{@align}}$ For the purpose of the development of a model-oriented control scheme, the modeled system presented above, with the given initial state $x_0$ , is used to predict the future states of the system as: ${\\begin{@align}{1}{-1}\\textbf {x}_k &= \\textbf {A} x_{0} + \\textbf {B} \\textbf {u}_k + \\textbf {E} \\mathbf {\\hat{d}}_{k} + \\textbf {w}_{k} + \\textbf {B}_{r} \\textbf {r}_{k}\\end{@align}}$ Where $\\textbf {x}_k = \\left[ x_{k|k}, x_{k|k+1} \\ldots ,x_{k|k+N} \\right]^{\\prime } \\in \\mathbb {R}^{n\\left(N+1\\right)}$ represents the predicted states at time step $k$ along a prediction horizon $N$ .",
"The subscript “$k|k+1$ \" is used to denote the prediction state at time $k$ for time $k+1$ .",
"Similar explanation is valid for other predicted state vectors $\\textbf {u}_k,\\textbf {r}_k \\in \\mathbb {R}^{j \\left(N\\right)}$ , $\\mathbf {\\hat{d}}_{k} \\in \\mathbb {R}^{n_{d} \\left(N\\right)}$ and $ \\textbf {w}_{k} \\in \\mathbb {R}^{nw \\left(N\\right)}$ .",
"The matrices $\\textbf {A}$ , $\\textbf {B}$ , $\\textbf {B}_{r}$ , and $\\textbf {E}$ are of appropriate dimensions."
],
[
"Market Environment",
"Note that in (REF ), the extension considers two scenarios i.e.",
"curtailment or not-curtailment of the HVAC's load, which means the offered capacity from the HVAC system is always kept positive.",
"This setting is used because in this paper the market framework of NEMS is used [15].",
"These two trajectories are implemented to replicate the interruptible load (IL) program, already in place in the NEMS [15].",
"In IL, the load operator can submit its bid for each 48 half-hourly period of the day.",
"In the case of load bid getting accepted, the load operator must then curtail its offered load [4].",
"Figure: Experimental set-up of the RMPC control scheme.As shown in (REF ), a perfect two way communication channel between the smart grid interface (SGI) and the Building Energy Management System (BEMS) is assumed.",
"The RMPC scheme for each time step $k$ is formulated as: LLr * u*k, -r*kmin K(uk) + K(-rk) + k + k subject to k' Pf,uk Pc,uk Ph,uk 1 1 1 k wk kmax xCk+1 = A xCk + B uk + E (dk + wk) wk kmax xNCk+1= A xNCk + B uk + E (dk + wk) + Br rk x-k - k xCk x+k + k x-k - k xNCk x+k + k u-k - rk uk u+k - rk rk, uk - rk, k 0 For the entire prediction horizon $N$ , the solution of the optimization problem formulated in () results in the cost optimal schedule $\\textbf {u}^{*}_{k}$ and reserve $\\textbf {r}^{*}_{k}$ capacity sequence.",
"The slack variable $\\mathbf {\\epsilon }_k$ – penalized by a scalar $\\rho $ in the objective function is implemented as a soft constraint on the upper $\\textbf {x}^{+}_{k}$ and lower limits $\\textbf {x}^{-}_{k}$ of both curtailed and not-curtailed scenarios.",
"A utility peak-power-penalty (PPP) $\\mathbf {\\phi }_k$ $\\in $ $\\mathbb {R}^N$ ($/kW) is communicated to the BEMS and the peak power term – defined as an epigraph $\\beta _k$ – is minimized in the objective function [16].",
"The benefit of restricting the peak load in the objective function provides: $(1)$ dynamic inclusion of an updated PPP signal from the utility signal at each time step $k$ , and $(2)$ keeps the economic objective function of the RMPC scheme generic and consistent, with and without the inclusion of PPP.",
"Constraints () and () restricts the not-curtailed $\\textbf {x}^{NC}_{k+1}$ and curtailed $\\textbf {x}^{C}_{k+1}$ trajectories within the feasibile region.",
"This procedure robustify the consumption schedule of both the curtailed $(\\textbf {u}_k)$ and not-curtailed $(\\textbf {u}_k - \\textbf {r}_{k})$ scenarios, to stay within their respective comfort zones of () and (), respectively.",
"() and () imposes the actuator limits of the HVAC system.",
"The maximization term presented in () and () can be manually made robust using standard procedures presented in the literature [17].",
"Essentially the procedure is to calculate the robust counterpart of the uncertain problem to yield a linear program.",
"For the curtailed case of (): LLr max xCk+1 = A xCk + B uk + E (dk + wk) s.t.",
"-k wk k Using Lagrangian multipliers $\\lambda _{w,1}$ and $\\lambda _{w,2}$ , dual of () can be expressed as: LLr min xCk+1 = A xCk + B uk + E dk + k(w,1 + w,2) s.t.",
"w,1 - w,2 = E w,1, w,2 0 When strong duality holds, then for any feasible $\\lambda _{w,1/2}$ in (), the maximization term of () becomes upper bounded.",
"Hence, the minimization term can be dropped.",
"The resulting robust counterparts of both cases (curtailed and not-curtailed) is jointly written as: LLr xCk+1 = A xCk + B uk + E dk + k(w,1 + w,2) xNCk+1= A xNCk + B uk + E dk + Br rk+ k(w,1 + w,2) w,1 - w,2 = E w,1, w,2 0 The RMPC scheme presented above, though deployed receding horizongly, still is an open-loop control strategy.",
"Because, while optimizing the schedule, RMPC scheme does not incorporate that the adjustment of input is also possible after the measurement of state is available for the next time step.",
"A better approach to deal with this is the closed-loop MPC, which incorporates affine policies of the uncertainty in the optimization problem.",
"Unfortunately, the robust counterpart of the closed-loop MPC results in larger number of variables than the open-loop MPC.",
"Various techniques are provided in the literature on how to overcome the problem of large number of variables [18].",
"But, for the case studies of this paper, the improvement in objective function from the closed-loop implementation was almost insignificant.",
"Hence for the purpose of saving the computational efforts, it was decided to deploy an open-loop RMPC.",
"In future studies.",
"Neverthless, in the future, the sensitivity analysis with respect to the computational tractability and accuracy of the closed-loop and the open-loop MPC strategies could serve as an interesting topic.",
"Figure: Real time energy and reserve price in Singaporean Dollars (SGD) taken from the NEMS."
],
[
"Simulation Results",
"The linear robust optimization problem of ()-() has been implemented using YALMIP [19] and CPLEX [20].",
"All simulations are performed with the real time energy and reserve price taken from the NEMS (REF ).",
"The prediction horizon of 1 day (48 periods) is chosen as a compromise between the stability of the MPC and computational efforts.",
"Since, in principle, a longer prediction horizon provides more stability to the MPC scheme.",
"But for our simulations, prediction horizon larger than 1 day showed very little improvement in the cost, but with great increase in computational expense.",
"3 Scenarios considered for evaluating the RMPC schemes are: a Nominal MPC scheme (NMPC) b RMPC scheme without PPP c RMPC scheme with PPP of 1.5 (SGD/kW) Figure: The curtailed and not-curtailed temperature evolution for the (a) NMPC, (b) RMPC without PPP and (c) RMPC with PPP scenarios.Figure: Power consumption and reserve capacity allocation for the (a) NMPC, (b) RMPC without PPP and (c) RMPC with PPP scenarios.For all scenarios, simulation results from 2 days of the year 2014 are presented in fig.",
"REF and REF .",
"Where XC* and XNC* from fig.",
"REF represent the curtailed and not-curtailed case for the given $*$ scenario.",
"To evaluate the performance of the RMPC scheme, we have assumed the disturbance prediction error of approximately 50%.",
"That means, maximum deviation from the actual disturbance is 50%.",
"The NMPC scheme doesn't consider uncertainty in the system.",
"Compared to the scenario in fig.",
"REF (a), both RMPC schemes – without PPP fig.",
"REF (b) and with PPP fig.",
"REF (c) – demonstrate successfully that they are capable of adhering to the comfort requirements in the presence of uncertainties.",
"One of the key observations from fig.",
"REF (b) and REF (b) is that the RMPC scheme takes care of the uncertainty in the model at the expense of extra consumption.",
"This is due to the fixed bounds on disturbances, which is seen by the controller as an extra thermal load to be cooled off in the room.",
"And as a result, the controller ends up consuming some energy also at high price periods.",
"This effect is even more pronounced in fig.",
"REF (c), due to the inclusion of PPP signal.",
"But nevertheless, the actual goal of the utility – minimizing the overall peak load – is achieved.",
"The simulation-setup of fig.",
"is repeated for the whole year of 2014.",
"Table REF shows monthly average cost of consumption and revenues from placing reserve capacity for the year 2014.",
"It can be observed that despite the increase in the cost of consumption, the increase in revenue has also occured for the scenario (b) and (c).",
"This is due to some of the load scheduled at high price periods, providing opportunity to also allocate reserves.",
"But due to low reserve prices, the magnitude of earnings from reserves are not comparable to the total cost of operation.",
"Fig.",
"REF shows the effect of increasing PPP on the cost and the peak load reduction.",
"Two new terms are introduced; $\\%$ Normalized Total Cost $= Cost_j/Max(Cost) \\ \\forall j =1,2, \\ldots z $ and $\\%$ Normalized Peak Load $= Peak_j / Max(Peak) \\ \\forall j =1,2, \\ldots z $ .",
"Where $z$ values are represented by the index $j$ , ranging from 0.5 SGD/kW to 30 SGD/kW.",
"Figure: Reduction of peak load and as increment of total cost due to the PPP.Table: Average cost per monthFig.",
"REF shows decrease in the load reduction after PPP signal of 5 SGD/kW.",
"Whereas, the opreational cost continues to rise.",
"Hence, the PPP beyond this value will only result in expensive operation of the HVAC system – without providing any significant improvement in peak load reduction for the utility."
],
[
"Conclusion and Future Work",
"The results have shown that the developed RMPC scheme provides a robust control framework for the HVAC system.",
"The developed controller optimizes energy consumption, reserve capacity provision and peak load reduction to achieve a cost effective and grid-friendly operation of buildings.",
"It can also be seen from the presented results that to improve the overall efficiency of distribution grid; utilities and buildings must co-optimize their underlying systems.",
"The simulation-based analysis presented in fig.",
"REF , can be use as a simplified control and planning framework, to design the incentive schemes for future load management schemes.",
"Future work regarding this paper is to incorporate the distribution grid constraints in the developed RMPC scheme.",
"It is also planned to extend building model presented in this paper to grid-oriented aggregated models."
],
[
"Acknowledgment",
"This work was financially supported by the Singapore National Research Foundation under its Campus for Research Excellence And Technological Enterprise (CREATE) programme.",
"This work was also sponsored by National Research Foundation, Prime Minister’s Office, Singapore under its Competitive Research Programme (CRP grant NRF2011NRF-CRP003-030, Power grid stability with an increasing share of intermittent renewables (such as solar PV) in Singapore)."
]
] | 1605.05795 |
[
[
"On circular flows: linear stability and damping"
],
[
"Abstract In this article we establish linear inviscid damping with optimal decay rates around 2D Taylor-Couette flow and similar monotone flows in an annular domain $B_{r_{2}}(0) \\setminus B_{r_{1}}(0) \\subset \\mathbb{R}^{2}$.",
"Following recent results by Wei, Zhang and Zhao, we establish stability in weighted norms, which allow for a singularity formation at the boundary, and additional provide a description of the blow-up behavior."
],
[
"Introduction",
"In this article we consider the linear stability and long-time asymptotic behavior of circular flows in an annular domain $(x,y) \\in B_{r_{2}}(0) \\setminus B_{r_{1}}(0)$ .",
"Such two-dimensional flows can for example be established experimentally in rotating cylinders, where the rotation is sufficiently slow as to not cause a (three-dimensional) Taylor-Couette instability.",
"In this setting, radial vorticities $\\begin{split}\\omega (x,y)&=\\omega (r), \\\\v(x,y)&= \\partial _r \\psi e_\\theta =\\begin{pmatrix}-y \\\\ x\\end{pmatrix}\\frac{\\psi ^{\\prime }(r)}{r}, \\\\\\psi ^{\\prime \\prime }(r)+\\frac{1}{r}\\psi ^{\\prime }(r)&=\\omega (r),\\end{split}$ are stationary solutions of the incompressible 2D Euler equations.",
"Considering a small perturbation to Taylor-Couette flow, $\\frac{\\phi ^{\\prime }(r)}{r}= A + \\frac{B}{r^{2}},$ we observe in Figure REF that for $B=0$ , i.e.",
"constant angular velocity, perturbations are rotated while keeping their shape.",
"However, in the general case when $B \\ne 0$ , $\\frac{\\phi ^{\\prime }(r)}{r}$ is strictly monotone and the perturbation is sheared in way reminiscent of plane Couette flow, as is depicted in Figure REF .",
"This mixing behavior underlies the phenomenon of (linear) inviscid damping.",
"Figure: Transport with constant angular velocity.We consider the Taylor-Couette flow rr in an annulus.The time 1 flow-lines are drawn as arrows.A perturbation initially concentrated on a line stays concentrated on aline.On the right this behavior is expressed in polar coordinates.Figure: Transport by a monotone flow.We consider the Taylor-Couette flow r+1 rr + \\frac{1}{r}, which we observe tobe mixing.",
"As time tends to infinity this mixing results in weak convergence to an averagedquantity.Considering polar coordinates, the linearized Euler equations around these stationary solutions are given by $\\begin{split}\\partial _tf + U(r) \\partial _{\\theta } f &= b(r) \\partial _{\\theta } \\phi , \\\\(\\partial _{r}^{2}+\\frac{1}{r}\\partial _{r}+\\frac{1}{r^{2}} \\partial _{\\theta }^{2})\\phi &= f, \\\\\\partial _{\\theta }\\phi |_{r=r_{1},r2_{2})}&=0, \\\\(t,\\theta ,r) & \\in \\mathbb {R}\\times [r_{1},r_{2}],\\end{split}$ where $U$ and $b$ are given by $U(r)&=\\frac{\\phi ^{\\prime }(r)}{r}, \\\\b(r)&=-\\frac{1}{r}\\partial _{r} (\\partial _{r}^{2}\\phi (r)+\\frac{1}{r}\\partial _{r}\\phi (r)),$ and $b(r)\\equiv 0$ if and only if one considers Taylor-Couette flow, $U(r)=A+\\frac{B}{r^2}$ .",
"As suggested by our notation, these equations share strong similarities with the linearized Euler equations around a shear flow $(U(y),0)$ in a plane finite periodic channel, $[0,1]$ : $\\begin{split}\\partial _t\\omega + U(y) \\partial _{x} \\omega - U^{\\prime \\prime }(y) \\partial _{x}\\phi &=0, \\\\(\\partial _{y}^{2}+\\partial _{x}^{2})\\phi &= \\omega , \\\\\\partial _{x}\\phi |_{y=0,1}&=0 , \\\\(t,x,y) &\\in \\mathbb {R}\\times [0,1].\\end{split}$ Here, various different approaches have been used to study this and related settings.",
"In [9], Stepin studies the asymptotic stability of monotone shear flows using spectral methods.",
"Under the assumption that the associated Rayleigh boundary value problem possesses no eigenvalues, he obtains an asymptotic description of the stream function and non-optimal decay rates.",
"In [6], Bouchet and Morita provide heuristic results which suggest that the algebraic decay rates of Couette flow should hold for general monotone flows as well.",
"However, their methods are not rigorous and do not provide sufficient error and stability estimates, especially in higher Sobolev regularity, in order to prove decay with optimal rates.",
"In [13] and [11], the author establishes linear inviscid damping and scattering for monotone shear flows in an infinite and finite periodic channel.",
"In the latter setting, we restrict to perturbations in $H^2 \\cap H^{1}_0$ in order to obtain the optimal decay rates.",
"Conversely, in the setting without vanishing Dirichlet boundary values, the sharp stability threshold is shown to be given by $H^{s},s=3/2$ due to asymptotic singularity formation at the boundary.",
"In [10], Wei, Zhang and Zhao follow similar methods as in [9] and establish linear inviscid damping with optimal decay rates for monotone shear flows under the condition of there being no embedded eigenvalues.",
"In particular, they remove the requirement of vanishing Dirichlet data and note that, using the boundary conditions of the velocity field and Hardy's inequality, one may allow for some blow-up at the boundary and still attain optimal decay rates.",
"In a seminal work [3], [4] Bedrossian and Masmoudi establish nonlinear inviscid damping for Couette flow in an infinite periodic channel.",
"There perturbations are required to be extremely regular, more precisely of Gevrey 2 class, in order to control nonlinear resonances.",
"In particular, due to the singularity formation at the boundary and the associated blow-up of relatively low Sobolev norms, the question of linear inviscid damping for settings with boundary remains open.",
"In addition to the inviscid setting, Bedrossian, Germain and Masmoudi also consider Couette flow as a solution of the Navier-Stokes equation in a two and three-dimensional infinite periodic channel.",
"There, in addition to inviscid damping, the interaction between the mixing and viscous behavior yields additional stabilization by enhanced dissipation.",
"Nonlinear inviscid damping is then established in Gevrey regularity [2] and more recently in Sobolev regularity [1], [5], where the threshold for stability results depends on $\\nu >0$ .",
"In the circular setting, research has focused on instability results, such as Taylor-Couette instability, bifurcation and turbulence.",
"For an introduction we refer to the book of Chossat and Iooss [7].",
"As the main results of this article we prove linear inviscid damping and scattering for a general class of circular flows, satisfying suitable monotonicity and smallness assumptions.",
"In comparison to our previous results, we note the following changes and improvements: We obtain optimal decay rates also for perturbations without vanishing Dirichlet data.",
"We show that $\\partial _yW$ splits into a bulk part $\\Gamma $ , which is stable also in unweighted higher Sobolev spaces, and a boundary correction $\\beta $ , which is stable in a suitably weighted $H^1$ space, but exhibits blow-up in $L^{\\infty }$ .",
"The smallness condition is strongly reduced for results in higher regularity.",
"In this circular setting, periodicity in $\\theta $ is a natural condition, unlike in the setting of a plane periodic channel.",
"We obtain a finer description of the boundary layer in terms of only the Dirichlet boundary values of the initial data."
],
[
"Main results",
"Our main results are summarized in the following theorem.",
"Theorem 1.1 (Linear inviscid damping with optimal decay rates) Let $0<r_1<r_2<\\infty $ and let $U:(r_1,r_2)\\rightarrow (a,b)$ be bilipschitz and suppose that $h(\\cdot )=b(U^{-1}(\\cdot )) \\in W^{3,\\infty }((a,b))$ and that $\\Vert h\\Vert _{W^{1,\\infty }}$ is sufficiently small.",
"Then, for any $f_{0}\\in H^{-1}_{\\theta }H^{2}_{r}$ there exists $v_{\\infty }(r)$ such that the solution $f$ of (REF ) satisfies $\\Vert v(t,\\theta ,r)- v_{\\infty }(r) e_{\\theta }\\Vert _{L^{2}} \\lesssim <t>^{-1}\\Vert f_{0}\\Vert _{H^{-1}_{\\theta }H^{1}_{r}}, \\\\\\Vert v(t,\\theta ,r)e_{r}\\Vert _{L^{2}} \\lesssim <t>^{-2} \\Vert f_{0}\\Vert _{H^{-1}_{\\theta }H^{2}_{r}},$ as $t \\rightarrow \\infty $ .",
"There exists $f_{\\infty } \\in L^{2}_{\\theta }H^{1}_{r}$ such that $f(t,\\theta -tU(r),r) \\rightarrow f_{\\infty } \\text{ in } L^{2},$ and $\\Vert f(t,\\theta -tU(r), r) - f_{\\infty }(\\theta , r)\\Vert _{L^{2}_{\\theta ,r}} \\lesssim <t>^{-1}\\Vert f_{0}\\Vert _{H^{-1}_{\\theta }H^{2}_{r}}.$ Furthermore, $f$ satisfies $\\Vert f(t,\\theta -rU(r),r)\\Vert _{H^{-1}H^{1}}+ \\Vert (r-r_{1})(r-r_{2})\\frac{d^{2}}{dr^{2}}f(t,\\theta -rU(r),r)\\Vert _{H^{-1}H^{1}} \\lesssim \\Vert f_{0}\\Vert _{H^{-1}H^{2}}.$ However, unless $bf|_{r=r_1,r_2}$ is constant, $\\sup _{t \\ge 0} \\Vert f(t,\\theta -rU(r),r)\\Vert _{H^{-1}H^{s}}= \\infty ,$ for any $s>3/2$ .",
"More precisely, there exists an (explicit) function $\\nu (t,\\theta ,r)$ determined solely by $f_0|_{r=r_1,r_2}$ and $U$ such that $\\Vert \\frac{d^2}{dr^2} f(t,\\theta -rU(r),r)- \\nu \\Vert _{L^2L^2} \\le \\Vert f_0\\Vert _{L^2H^2},$ and such that $\\Vert (r-r_1)(r-r_2)\\nu \\Vert _{L^2} \\le |f_0|_{r=r_1,r_2}|.$ Remark 1 While $h=b(U^{-1})$ is required to be regular, the smallness assumption is only imposed on the $W^{1,\\infty }$ norm.",
"This theorem summarizes the main results of Proposition REF and Theorems REF , REF and REF in terms of common norms in the variables $t,\\theta ,r$ .",
"In Section we introduce a scattering formulation, which is used throughout the article.",
"The function $\\nu $ is introduced in Section REF .",
"In [10] it has been observed that, by a use of Hardy's inequality, the second derivative of $W$ can be allowed to form a singularity as $t \\rightarrow \\infty $ while still attaining the optimal $t^{-2}$ decay rate.",
"Here, we stress that stability in $H^2$ indeed does not hold due to singularity formation at the boundary as $t\\rightarrow \\infty $ , as quantified in $\\nu $ and $\\beta $ (c.f.",
"Section ).",
"As we discuss in Section , our method of proof does not rely on cancellations or conserved quantities.",
"Hence, the results extend to complex-valued $b(r)$ and various modified equations in a straightforward manner.",
"In the case of the linearized Euler equations in a plane finite periodic channel, however, Wei, Zhang and Zhao [10] have shown, using different methods, that weaker assumptions suffice to obtain damping.",
"Similarly to [11] our strategy is to first establish the damping and scattering result, assuming stability in higher Sobolev norms.",
"We stress that the damping estimate necessarily loses regularity.",
"Hence, usual Duhamel fixed point iteration approaches or energy methods can not yield stability results.",
"Instead we employ a finer study of the damping mechanism, which allows us to construct a Lyapunov functional using the mode-wise decay to avoid the necessary loss of regularity of uniform damping estimates.",
"The remainder of the article is organized as follows: In Section , we show that regularity of the vorticity in coordinates moving with the flow can be exchanged for uniform damping estimates and that the problem of linear inviscid damping thus reduces to a stability problem.",
"As motivating examples, we discuss the specific cases of Taylor-Couette flow, a point vortex and of Couette flow in a plane channel, where explicit solutions are available and, in a sense, trivial.",
"In Section , we introduce several reductions and changes of variables to arrive at a scattering formulation of the linearized Euler equations.",
"Subsequently, we analyze the structure of the equation and establish $L^{2}$ stability.",
"Section considers higher regularity and singularity formation at the boundary.",
"Compared to [12], in addition to considering a circular setting, we introduce a splitting $\\partial _{y}W=\\Gamma +\\beta $ , where $\\Gamma $ is shown to be stable in higher regularity, regardless of Dirichlet boundary data.",
"On the other hand, $\\beta $ is determined solely by the underlying circular flow and the Dirichlet boundary data of the initial perturbation and provides an explicit characterization of the boundary layer.",
"Subsequently, we further split $\\partial _y\\beta $ to obtain an explicit characterization of the $H^2$ blow-up in the form of $\\nu $ and stability in weighted spaces.",
"Here, we rely on a new approach based on Duhamel's principle and an iterative estimate in order to control the evolution of the weighted quantities.",
"The Appendices and provide a description of boundary evaluations for elliptic ODEs and a variant of Duhamel's formula adapted to a time-dependent right-hand-side of the equation."
],
[
"Damping by mixing, the role of regularity and examples",
"As in the case of inviscid damping in a plane channel or Landau damping, decay of the velocity/force field and regularity of the solution in a coordinate system moving with the flow are closely linked.",
"More precisely, in this section we show that uniform damping estimates closely correspond to a control of the regularity of $W(t,\\theta ,r):=f(t,\\theta -tU(r),r)$ with respect to $r$ and that such a control is necessary.",
"The problem of linear inviscid damping with optimal decay rates thus turns out to be a stability problem, studied in Section , which is the main focus of this article.",
"We consider the linearized Euler equations $\\begin{split}\\partial _tf + U(r) \\partial _{\\theta } f &= b(r) \\partial _{\\theta } \\phi , \\\\(\\partial _{r}^{2}+\\frac{1}{r}\\partial _{r}+\\frac{1}{r^{2}} \\partial _{\\theta }^{2})\\phi &= f, \\\\\\partial _{\\theta }\\phi |_{r=r_{1},r_{2}}&=0, \\\\(t,\\theta ,r) & \\in \\mathbb {R}\\times [r_{1},r_{2}],\\end{split}$ as a perturbation around the transport problem $\\begin{split}\\partial _tg + U(r) \\partial _{\\theta } g &=0, \\\\(t,\\theta ,r) & \\in \\mathbb {R}\\times [r_{1},r_{2}].\\end{split}$ Based on this view, we measure the deviation of these equations by introducing the scattered vorticity $W(t,\\theta ,r):= f(t,\\theta -tU(r),r).$ Assuming regularity of $W$ uniformly in time, damping results for (REF ) then reduce to estimates for (REF ).",
"Here, we it has recently been observed by Wei, Zhang and Zhao [10] that quadratic decay rates only require control of a weighted $H^{2}$ norm $\\Vert W\\Vert _{H^{1}}+ \\Vert y (1-y)\\partial _{y}^{2}W\\Vert _{L^{2}}$ by using a Hardy inequality in the duality estimate.",
"The following two propositions provide damping estimates in terms of regularity of $W$ in the case of a plane channel and a circular domain, respectively.",
"Proposition 2.1 (Damping by regularity for plane channel [10], [12], [8]) Let $-\\infty \\le a<b \\le \\infty $ and let $U:(a,b) \\rightarrow \\mathbb {R}$ be locally $C^1$ and suppose that $U^{\\prime }(y)\\ne 0$ for almost every $y\\in (a,b)$ .",
"Let $W\\in H^{-1}_xH^{1}_y((a,b))$ with $\\int _{W dx=0 and let{\\begin{@align*}{1}{-1}\\omega (t,x,y)=W(t,x-tU(y),y).\\end{@align*}}Let further the associated velocity field v be defined by{\\begin{@align*}{1}{-1}v_1&=-\\partial _y\\phi ,\\\\v_2&=\\partial _x\\phi ,\\\\\\Delta \\phi &=\\omega , \\\\\\partial _x\\phi |_{y=a,b}&=0, \\\\\\phi &\\in \\dot{H}^1.\\end{@align*}}Then v satisfies{\\begin{@align}{1}{-1}\\Vert v(t)\\Vert _{L^2} &\\lesssim \\min \\Big ( \\left\\Vert W\\right\\Vert _{H^{-1}_{x}L^{2}_{y}}, t^{-1}\\left\\Vert \\frac{W}{U^{\\prime }}\\right\\Vert _{H^{-1}_{x}H^{1}_{y}}, \\\\& \\quad t^{-1} \\left(\\left\\Vert W|\\frac{1}{U^{\\prime }}| + W|\\partial _y\\frac{1}{U^{\\prime }}|\\right\\Vert _{H^{-1}_{x}L^{2}_{y}}+ \\left\\Vert (y-a)(y-b)\\frac{\\partial _yW}{U^{\\prime }}\\right\\Vert _{H^{-1}_{x}L^{2}_{y}}\\right)\\Big ).\\end{@align}}Furthermore, suppose that \\partial _{y}^2 W exists.",
"Then, v_2 additionally satisfies{\\begin{@align}{1}{-1}\\begin{split}\\Vert v_2(t)\\Vert _{L^2} &\\le t^{-2} \\Big (\\left\\Vert \\frac{W}{(U^{\\prime })^2}\\right\\Vert _{H^{-1}_{x}H^{1}_{y}} + \\left\\Vert W\\partial _y\\frac{1}{(U^{\\prime })^2}\\right\\Vert _{H^{-1}_{x}H^{1}_{y}}\\\\ & \\quad + \\min \\left(\\left\\Vert \\frac{(y-a)(y-b)}{(U^{\\prime })^2}\\partial _{y}^2W\\right\\Vert _{H^{-1}_{x}L^{2}_{y}},\\left\\Vert \\frac{\\partial _{y}^2W}{(U^{\\prime })^2}\\right\\Vert _{H^{-1}_{x}L^{2}_{y}}\\right)\\Big ).\\end{split}\\end{@align}}}$ We note that, by integration by parts, $\\Vert v\\Vert _{L^2}^2=\\Vert \\nabla \\phi \\Vert _{L^2}^2= -\\iint \\phi \\omega dx dy.$ Applying Plancherel's theorem with respect to $x$ and noting that, $\\mathcal {F}_x (\\omega (t,\\cdot ,y))(k)= e^{iktU(y)} \\hat{W}(t,k,y),$ this equals $\\sum _{k \\ne 0}\\int \\overline{\\hat{\\phi }}(t,k,y) e^{iktU(y)} \\hat{W}(t,k,y).$ Integrating $e^{iktU}= \\frac{1}{iktU^{\\prime }}\\partial _y e^{iktU(y)}$ by parts, we further obtain $\\sum _{k \\ne 0} \\frac{1}{t}\\int e^{iktU(y)}\\partial _y \\left(\\overline{\\hat{\\phi }} \\frac{\\hat{W}}{ikU^{\\prime }}\\right),$ which is controlled by $t^{-1} \\Vert \\phi \\Vert _{L^2H^1} \\left\\Vert \\frac{W}{U^{\\prime }}\\right\\Vert _{H^{-1}H^1}.$ The estimate () thus follows by noting that $\\Vert \\phi \\Vert _{L^2H^1} \\le \\Vert v\\Vert _{L^2}.$ In order to prove (), we note that $\\Delta v_2= \\partial _x \\omega $ and define $\\psi $ s.t.",
"$\\Delta \\psi &=v_2, \\\\\\partial _x\\psi |_{y=a,b}&=0, \\\\\\psi &\\in \\dot{H}^{1}.$ Then, using integration by parts, we obtain $\\Vert v\\Vert _{L^2}^2&= \\iint \\psi \\partial _x \\omega = \\sum _{k \\ne } \\overline{\\hat{\\psi }} e^{iktU(y)} ik\\hat{W} dy \\\\&= \\frac{1}{t^2} \\sum _{k \\ne 0} \\int e^{iktU(y)}\\partial _y \\left( \\frac{1}{U^{\\prime }} \\partial _y \\left(\\frac{1}{U^{\\prime }} \\overline{\\hat{\\psi }} \\frac{\\hat{W}}{k}\\right)\\right) dy \\\\& \\quad + \\frac{1}{t^2} \\sum _{k \\ne 0} e^{iktU(y)}\\frac{1}{U^{\\prime }} \\partial _y \\left(\\frac{1}{U^{\\prime }} \\overline{\\hat{\\psi }} \\frac{\\hat{W}}{k}\\right) \\Big |_{y=a}^b \\\\$ The result hence follows by the Cauchy-Schwarz inequality, the trace map and by using the estimates $\\Vert \\phi \\Vert _{H^1} &\\lesssim \\Vert v_2\\Vert _{L^2}, \\\\\\Vert \\frac{\\phi }{(y-a)(y-b)}\\Vert _{L^2} &\\lesssim \\Vert \\phi \\Vert _{H^1},$ The first estimate here follows by standard elliptic regularity theory, while the second one is given Hardy's inequality, as observed in [10].",
"The following proposition adapts these results to the setting of circular flows.",
"Proposition 2.2 (Damping for circular flows; [10], [12], [8]) Let $0<r_1<r_2<\\infty $ and let $U:(r_1,r_2)\\rightarrow \\mathbb {R}$ be locally $C^1$ with $U^{\\prime }(r)\\ne 0$ for almost every $r \\in (r_1,r_2)$ .",
"Let $\\frac{W(t,\\theta ,r)}{U^{\\prime }} \\in H^{-1}H^1((r_1,r_2), r drd\\theta )$ with $\\int _W̰ d\\theta =0$ and let $f(t,\\theta ,r)=W(t,\\theta -tU(r),r).$ Let further the associated velocity field be by defined by $\\begin{split}v_{r}(t,\\theta ,r)&= \\frac{1}{r}\\partial _{\\theta } \\phi (t,\\theta ,r), \\\\v_{\\theta }(t,\\theta ,r)&= \\partial _{r} \\phi (t,\\theta ,r), \\\\(\\partial _{r}^{2}+ \\frac{1}{r}\\partial _{r}+\\frac{1}{r^{2}}\\partial _{\\theta }^{2})\\phi &=f, \\\\\\partial _{\\theta }\\phi |_{r=r_{1},r_{2}}&=0, \\\\v &\\in L^2(r dr d\\theta ).\\end{split}$ Then $v$ satisfies $\\Vert v(t)\\Vert _{L^{2}(r dr d\\theta )} &\\lesssim \\min \\Big ( \\Vert W\\Vert _{L^{2}(rdr d\\theta )},t^{-1} \\Vert \\frac{W(t)}{U^{\\prime }}\\Vert _{H^{-1}_{\\theta }H^{1}_{r}( r dr d\\theta )}, \\\\& \\quad t^{-1} \\left(\\Vert W(t)|\\frac{1}{U^{\\prime }}|+ W r|\\partial _r\\frac{1}{U^{\\prime }}|\\Vert _{H^{-1}_{\\theta }L^{2}_{r}( r dr d\\theta )} + \\left\\Vert \\frac{(r-r_1)(r-r_2)\\partial _rW}{U^{\\prime }}\\right\\Vert _{H^{-1}L^2(r dr d\\theta )}\\right) \\Big ).$ Furthermore, suppose that $\\partial _{r}^2W$ exists.",
"Then, $v_r$ additionally satisfies $\\Vert v_{r}(t)\\Vert _{L^{2}(r dr \\theta )} &\\lesssim t^{-2} \\Big (\\left\\Vert \\frac{W}{(U^{\\prime })^2}\\right\\Vert _{H^{-1}H^1(r dr d\\theta )} + \\left\\Vert W \\partial _r \\frac{1}{(U^{\\prime })^2}\\right\\Vert _{H^{-1}H^1( rdr d\\theta )} \\\\& \\quad + \\min \\left( \\left\\Vert \\frac{\\partial _{r}^2 W}{(U^{\\prime })^2}\\right\\Vert _{H^{-1}L^2( r dr d\\theta )}, \\left\\Vert \\frac{(r-r_1)(r-r_2)\\partial _{r}^2 W}{(U^{\\prime })^2}\\right\\Vert _{H^{-1}L^2( rdr d\\theta )} \\right)\\Big ).$ Remark 2 We note that for any given $0<r_1<r_2<\\infty $ we could replace $r dr$ by just $dr$ in the above estimates at the cost of a constant $C(r_1,r_2)$ .",
"In this way the result and its proof can be made more similar to the setting of a finite channel.",
"However, the above formulation also allows us to pass to the limits $r_1\\downarrow 0$ and $r_2 \\uparrow \\infty $ .",
"In order to obtain a more tractable stream function formulation of Euler's equations, in this proof we consider conformal coordinates, i.e.",
"$(r,\\theta )&=(e^s,\\theta ),\\\\s \\in (\\log (r_1),\\log (r_2))&=:(s_1,s_2).$ With respect to these coordinates, the stream function $\\psi $ and the velocity field are given by $e^{-2s}(\\partial _{s}^2+ \\partial _{\\theta }^2)\\psi &= \\omega , \\\\v_r&=e^{-s}\\partial _\\theta \\psi , \\\\v_\\theta &= e^{-s}\\partial _s \\psi .$ Furthermore, the kinetic energy satisfies $\\int |v_r|^2 r dr d\\theta = \\int e^{-2s} |\\partial _\\theta \\psi |^2 e^{2s}ds d\\theta &= \\int |\\partial _\\theta \\psi |^2 ds d\\theta , \\\\\\int |v_\\theta |^2 r dr d\\theta &= \\int |\\partial _s \\psi |^2 ds d\\theta , \\\\\\int |v|^2 dr d\\theta &= - \\int \\psi \\omega e^{2s} ds d\\theta .$ Applying a Fourier transform in $\\theta $ and using the definition of $W$ , we hence obtain $\\int |\\partial _s \\psi |^2 + |\\partial _\\theta \\psi |^2 ds d\\theta = - \\sum _{k} \\int \\overline{\\hat{\\psi }} e^{2s} e^{iktU(e^s)} \\hat{W} ds.$ Integrating $e^{iktU(e^s)} = e^{-s} \\frac{1}{iktU^{\\prime }(e^s)}\\partial _s e^{iktU(e^s)}$ by parts, we further compute $\\int |\\partial _s \\psi |^2 + |\\partial _\\theta \\psi |^2 ds d\\theta =\\sum _{k \\ne 0} \\int e^{iktU(e^s)} \\partial _s \\left( \\frac{1}{iktU^{\\prime }(e^s)} \\overline{\\hat{\\psi }} e^{s} \\hat{W}\\right) ds.$ In order to estimate this integral, we use various different tools: If $\\partial _s$ does not fall on $\\psi $ , we control $\\sum _{k \\ne 0} \\int |\\hat{\\psi } \\frac{1}{k}X| ds \\le \\Vert \\psi \\Vert _{L^2(ds d\\theta )} \\Vert X\\Vert _{H^{-1}_{\\theta }L^{2}_s(ds d\\theta )}$ and use Poincaré's inequality to further estimate $\\Vert \\psi \\Vert _{L^2(ds d\\theta )} \\le C \\Vert \\partial _\\theta \\psi \\Vert _{L^2}.$ Alternatively, instead of Poincaré's inequality, duality yields an estimate by $\\Vert \\partial _\\theta \\psi \\Vert _{L^2(ds d\\theta )} \\Vert X\\Vert _{H^{-2}_{\\theta }L^{2}_s(ds d\\theta )}.$ Since $\\psi $ has zero boundary values, we can also use Hardy's inequality to control by $& \\quad \\Vert \\frac{\\psi }{(e^s-e^{s_1})(e^s-e^{s_2})}\\Vert _{L^2(ds d\\theta )} \\Vert (e^s-e^{s_1})(e^s-e^{s_2}) X\\Vert _{H^{-1}L^2(ds d\\theta )}\\\\ &\\le \\Vert \\partial _s \\psi \\Vert _{L^2(ds d\\theta )} \\Vert (e^s-e^{s_1})(e^s-e^{s_2}) X\\Vert _{H^{-1}L^2(ds d\\theta )}.$ In the case of a fixed annulus $(r_1,r_2),$ $ 0<r_1<r_2<\\infty ,$ the precise choice of estimate is not essential.",
"However, when considering a non-periodic setting, e.g.",
"$\\mathbb {R}\\times [a,b]$ , or a point vortex, i.e.",
"$r_1=0$ , or initial data with singularities at the boundary, all these estimates can yield improvements.",
"In order to obtain the quadratic decay estimate for $v_{r}=e^{-s}\\partial _\\theta \\psi $ , we note that $(\\partial _{s}^2+\\partial _{\\theta }^2)\\partial _\\theta \\psi = e^{2s}\\partial _{\\theta }\\omega .$ Thus, we define a potential $\\gamma $ by $(\\partial _{s}^2 + \\partial _{\\theta }^2)\\gamma &= \\partial _\\theta \\psi , \\\\\\gamma |_{s=s_1,s_2}&=0,\\\\\\nabla \\gamma &\\in L^2,$ and compute $\\int |v_r|^2 r dr d\\theta &= \\int |\\partial _\\theta \\psi |^2 ds d\\theta \\\\&= \\int \\gamma (\\partial _{s}^2+\\partial _{\\theta }^2)\\partial _\\theta \\psi \\\\&= \\int \\gamma e^{2s} \\partial _\\theta \\omega ds d\\theta = \\sum _{k\\ne 0} ik \\int \\overline{\\hat{\\gamma }} e^{2s}e^{iktU(e^s)} \\hat{W} ds.$ The result hence follows by integrating $e^{iktU(e^s)}$ by parts twice and using the Dirichlet data of $\\gamma $ and $\\partial _\\theta \\psi $ , the trace inequality and a variant of Hardy's inequality.",
"That is, since $\\gamma $ has zero Dirichlet boundary values, $\\Vert \\frac{\\gamma }{(e^s-e^{s_1})(e^s-e^{s_2})}\\Vert _{L^2(ds d\\theta )} &\\le \\Vert \\frac{\\gamma }{(s-s_1)(s-s_2)}\\Vert _{L^2(ds d\\theta )} \\\\ &\\lesssim \\Vert \\partial _s \\gamma \\Vert _{L^{2}(ds d\\theta )}= \\Vert v_r\\Vert _{L^2(rdr d\\theta )}.$ We stress that these uniform damping estimates necessarily lose regularity, since the associated change of coordinates is a unitary operator.",
"Thus, the operator norm of $f \\mapsto v$ considered as a mapping from $L^2$ to $L^2$ does not improve in time.",
"Hence, it is not possible to derive stability of (REF ) using a common Duhamel-type approach or a fixed point mapping.",
"Instead, in Sections REF and we have to make use of finer properties of the dynamics and the mode-wise decay of the principal symbol of the evolution operator.",
"Before that, in the following we discuss some examples for which explicit computations are possible."
],
[
"Taylor-Couette flow",
"As an application of the damping results, we discuss some exceptional cases for which $W$ can be trivially computed in terms of the initial datum.",
"Corollary 2.1 (Couette flow) Let $U(y)=y$ on $[a,b]$ with $a,b \\in [-\\infty ,\\infty ]$ , then the linearized Euler equations reduce to the free transport equations.",
"Furthermore, if $\\omega _0 \\in H^{-1}_xH^{2}_y$ , then the associated velocity field satisfies $\\Vert v(t)-\\langle v|_{t=0} \\rangle _x\\Vert _{L^2}&\\le t^{-1}\\Vert \\omega _0\\Vert _{H^{-1}H^1}, \\\\\\Vert v_2(t)\\Vert _{L^2} &\\le t^{-2}\\Vert \\omega _0\\Vert _{H^{-1}H^2}.$ Corollary 2.2 (Taylor-Couette flow; Point vortex) Let $A, B \\in \\mathbb {R}$ and let $0\\le r_1<r_2 \\le \\infty $ , then the linearized Euler equations around Taylor-Couette flow $U(r)= (Ar + \\frac{B}{r})e_{\\theta },$ are given by $\\partial _tf + (A+\\frac{B}{r^2})\\partial _\\theta f &=0, \\text{ on } (0,\\infty )\\times (r_1,r_2) \\\\f|_{t=0}&=f_0 \\text{ on } (r_1,r_2).$ Furthermore, the associated velocity field $v$ satisfies $\\Vert v\\Vert _{L^2(rdrd\\theta )} \\le C t^{-1}B^{-1} \\Vert f_0\\Vert _{H^{-1}H^1((r^7+r^5)drd\\theta )}, \\\\\\Vert v_r\\Vert _{L^2( rdr d\\theta )} \\le C t^{-2}B^{-2} \\Vert f_0\\Vert _{H^{-1}H^2((r^7+r^5)drd\\theta )}.$ Here, the case the case $r_1=0$ , $A=0, B \\ne 0$ corresponds to a point vortex.",
"We note that $(A+\\frac{B}{r^2})^{\\prime }=-B \\frac{2}{r^3}$ .",
"Hence, by direct computation $\\Vert \\frac{\\omega _0}{U^{\\prime }}\\Vert _{H^{-1}H^1(rdrd\\theta )} + \\Vert \\frac{\\omega _0}{U^{\\prime }}\\Vert _{L^2(r^{-1} dr \\theta )} \\le 2B^{-1} (\\Vert \\omega _0\\Vert _{H^{-1}L^2((r^7+r^5) dr \\theta )} + \\Vert \\partial _r\\omega _0\\Vert _{H^{-1}L^2(r^7 dr \\theta )}).$"
],
[
"Scattering formulation and $L^2$ stability",
"As established in Section , the core problem of (linear) inviscid damping consists of establishing a control of higher Sobolev norms of the vorticity moving with the flow: $W(t,\\theta ,r):= f(t,\\theta -tU(r),r).$ Here, we largely follow a similar approach as in the plane setting considered in [12].",
"As key improvements we obtain a less restrictive smallness condition and develop a splitting of $\\partial _{r}W$ into a well-behaved and more regular part $\\Gamma $ and a (relatively) explicit boundary layer $\\beta $ .",
"This then allows us to deduce damping with optimal decay rates and a detailed stability in suitable weighted Sobolev spaces, such as the ones considered in Proposition REF .",
"In order simplify our analysis, in this section we introduce several changes of variables as well as useful auxiliary functions."
],
[
"Scattering formulation",
"Expressing the linearized Euler equations $\\begin{split}\\partial _tf + U(r) \\partial _{\\theta } f &= b(r) \\partial _{\\theta } \\phi , \\\\(\\partial _{r}^{2}+\\frac{1}{r}\\partial _{r}+\\frac{1}{r^{2}} \\partial _{\\theta }^{2})\\phi &= f, \\\\\\partial _{\\theta }\\phi |_{r=r_{1},r_{2})}&=0, \\\\(t,\\theta ,r) & \\in \\mathbb {R}\\times [r_{1},r_{2}],\\end{split}$ in terms of the scattered quantities $\\begin{split}F(t,\\theta ,r)&= f(t,\\theta -tU(r),r), \\\\\\Upsilon (t,\\theta ,r)&= \\phi (t,\\theta -tU(r),r),\\end{split}$ we obtain $\\begin{split}\\partial _tF &= b(r) \\partial _{\\theta } \\Upsilon , \\\\((\\partial _{r}-tU^{\\prime }(r)\\partial _{\\theta })^{2}+\\frac{1}{r}(\\partial _{r}-tU^{\\prime }(r)\\partial _{\\theta })+\\frac{1}{r^{2}} \\partial _{\\theta }^{2})\\Upsilon &= F, \\\\\\partial _{\\theta }\\Upsilon |_{r=r_{1},r2_{2})}&=0, \\\\(t,\\theta ,r) & \\in \\mathbb {R}\\times [r_{1},r_{2}],\\end{split}$ As none of the coefficient functions depend on $\\theta $ , our system decouples with respect to Fourier modes $k$ in $\\theta $ .",
"$\\begin{split}\\partial _t\\hat{F} &= b(r) ik \\hat{\\Upsilon }, \\\\((\\partial _{r}-iktU^{\\prime }(r))^{2}+\\frac{1}{r}(\\partial _{r}-iktU^{\\prime }(r))-\\frac{k^{2}}{r^{2}})\\hat{\\Upsilon }&= \\hat{F}, \\\\ik\\hat{\\Upsilon }|_{r=r_{1},r2_{2})}&=0, \\\\(t,k,r) & \\in \\mathbb {R}\\times 2\\pi \\mathbb {Z}\\times [r_{1},r_{2}],\\end{split}$ We in particular note that the mode $k=0$ , which corresponds to a purely circular flow, is conserved in time.",
"Using the linearity of our equations, in the following we hence without loss of regularity consider $k \\in 2\\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace )$ as a given parameter.",
"In view of the structure of the differential equation for $\\Phi $ , it is further advantageous to use that $U$ , as a strictly monotone function, is invertible.",
"Introducing a change of coordinates $r \\mapsto y=U(r).$ as well a denoting $\\begin{split}h(y) &= \\frac{(\\omega _{0})^{\\prime }}{r}|_{r=U^{-1}(y)}, \\\\g(y)&= U^{\\prime }(r)|_{r=U^{-1}(y)}, \\\\W(t,y,k) &= \\hat{F}(t,r,k)|_{r=U^{-1}(y)},\\\\\\Phi (t,y,k) &= \\frac{1}{k^{2}}\\hat{\\Upsilon }(t,r,k)|_{r=U^{-1}(y)},\\end{split}$ our system is then given by the following definition.",
"Definition 3.1 (Euler's equations in scattering formulation) Let $U:[r_1,r_2]\\rightarrow \\mathbb {R}$ be strictly monotone and let $h(y)=b|_{r=U^{-1}(y)}$ and $g=U^{\\prime }(U^{-1}(y))$ .",
"Then Euler's equations in scattering formulation are given by $\\begin{split}\\partial _tW = \\frac{ih(y)}{k}\\Phi &=: \\frac{ih(y)}{k}L_{t}W, \\\\\\mathcal {E}_{t}\\Phi := \\left(\\left(g(y)(\\frac{\\partial _{y}}{k}-it)\\right)^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)}\\right)\\Phi &=W, \\\\\\Phi |_{y=a,b}&=0, \\\\(t,k,y) &\\in \\mathbb {R}\\times 2\\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace ) \\times [a,b],\\end{split}$ where $a=\\min (U^{-1}(r_{1}), U^{-1}(r_{2}))$ , $b=\\max (U^{-1}(r_{1}),U^{-1}(r_{2}))$ and $k \\in 2 \\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace )$ .",
"Remark 3 Our methods do not rely on the specific form of $h$ or $g$ in terms of $U$ .",
"For example, we can allow for $h$ to be an arbitrary complex valued $W^{1,\\infty }$ function.",
"Here the notation $L_{t}W$ is used to stress that the mapping $W \\mapsto \\Phi $ is a linear operator in $W$ .",
"As this system decouples with respect to $k$ , we will often treat $k\\ne 0$ as a fixed given external parameter and with slight abuse of notation use $W(t,y)$ to refer to $W(t,k,y)$ for the given $k$ ."
],
[
"Shifted elliptic regularity and modified spaces",
"We note that in this scattering formulation $\\mathcal {E}_{t}$ is obtained from an elliptic operator by conjugation with $e^{ikty}$ and hence define suitable replacements of the $H^1$ and $H^{-1}$ energies: Definition 3.2 ( $\\tilde{H}^{1}_{t}$ and $\\tilde{H}^{-1}_{t}$ energies) Let $u \\in H^{1}([a,b])$ and let $k \\in 2\\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace )$ be given, then for every $t \\in \\mathbb {R}$ , we define $\\Vert u\\Vert _{\\tilde{H}^{1}_t}^2:= \\Vert e^{ikty}u\\Vert _{H^1}^2= \\Vert u\\Vert _{L^2}^2 + \\Vert (\\frac{\\partial _y}{k}-it)u\\Vert _{L^2}^2.$ Furthermore, we define a dual quantity in the following way.",
"Let $v \\in L^2$ and let $\\Psi [v]$ be the unique solution of $(-1+(\\frac{\\partial _y}{k}-it)^2) \\Psi [v]&=v, \\\\\\Psi [v]|_{y=a,b}&=0.$ Then we define $\\Vert v\\Vert _{\\tilde{H}^{-1}_t}:= \\Vert \\Psi [v]\\Vert _{\\tilde{H}^{1}_t}.$ Lemma 3.1 (Duality) Let $W \\in L^{2}$ and let $k \\in (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace )$ be given.",
"Then $\\Vert W\\Vert _{\\tilde{H}^{-1}_{t}}=\\sup \\lbrace \\langle W, \\alpha \\rangle _{L^{2}}: \\alpha \\in H^{1}_{0}, \\Vert \\alpha \\Vert _{\\tilde{H}^{1}}\\le 1 \\rbrace ,$ i.e.",
"$\\tilde{H}^{-1}_{t}$ is dual to $\\tilde{H}^{1}_{t}$ .",
"Since multiplication by $e^{ikty}$ is a unitary operation and preserves zero Dirichlet boundary values and $\\Psi _t[v]= e^{-ikty}\\Psi _0[e^{ikty}v],$ it suffices to consider the case $t=0$ , which is given by the usual $H^1$ and $H^{-1}$ norms (where we use $\\frac{\\partial _y}{k}$ instead of $\\partial _y$ ).",
"The result then follows using integration by parts: $-\\langle W, \\alpha \\rangle = \\langle (1-\\frac{\\partial _{y}}{k}^{2}) \\Psi [W], \\alpha \\rangle \\\\= \\langle \\Psi [W], \\alpha \\rangle + \\langle \\frac{\\partial _{y}}{k}\\Psi [W], \\frac{\\partial _{y}}{k}\\alpha \\rangle \\le \\Vert W\\Vert _{H^{-1}}\\Vert \\alpha \\Vert _{H^{1}},$ with equality if $\\alpha = -\\frac{1}{\\Vert \\Psi [W]\\Vert _{H^1}}\\Psi [W]$ .",
"Taking the supremum over all $\\alpha $ with $\\Vert \\alpha \\Vert _{H^{1}}$ we hence obtain the result."
],
[
"Heuristics and obstructions",
"On a heuristic level, in order to establish stability in $L^2$ , we use that $\\frac{d}{dt}\\Vert W(t)\\Vert _{L^2}^2 =2\\Re \\langle W, \\frac{ih}{k}L_t W\\rangle \\lesssim C(h,k)\\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2,$ and that for fixed functions $u \\in L^2$ , which do not depend on time, $\\int _{0}^{\\infty } \\Vert u\\Vert _{\\tilde{H}^{-1}_t}^2 dt \\le C \\Vert u\\Vert _{L^2}^2,$ as can be computed from a Fourier characterization.",
"Hence, it seems reasonable to expect that solutions $W(t)$ of (REF ) satisfy an estimate of the form $\\Vert W(t)\\Vert _{L^2} \\le \\exp (C\\Vert h\\Vert _{L^\\infty }|k|^{-1}) \\Vert f_0\\Vert _{L^2},$ also for complex valued $h$ , which is the case for some explicit model problems (c.f. [11]).",
"However, we stress that this heuristic is very rough and does not account for several obstructions: We note that integrability in time in general fails for time-dependent $u \\in L^{\\infty }_t(L^2)$ .",
"For example, choosing $u(t,k,y)=e^{ikty}u_0(k,y),$ we observe that $\\int _{0}^{T} \\Vert u\\Vert _{\\tilde{H}^{-1}_t}^2 dt = T \\Vert u_0\\Vert _{H^{-1}}^2,$ which diverges as $T \\rightarrow \\infty $ despite $\\Vert u(t,y)\\Vert _{L^2}= \\Vert u_0\\Vert _{L^2}$ being uniformly bounded.",
"Since the first estimate does not account for antisymmetric operators in $\\frac{d}{dt}W$ it is not sufficient to establish $L^2$ stability.",
"For example, this estimate is satisfied by solutions $u(t,y)$ to $\\partial _tu +iy u &= \\Phi , \\\\(-1+(\\partial _y-it)^2)\\Phi &=u, \\\\\\Phi |_{y=a,b}&=0.$ Considering $v(t,y)=e^{ity}u(t,y)$ , we observe that $v$ solves $\\partial _tv &= \\phi , \\\\(1-\\partial _{y}^2)\\phi &= v, \\\\\\phi |_{y=a,b}&=0.$ Hence, choosing $u|_{t=0}$ to be an eigenfunction of $(1-\\partial _{y}^2)$ , we obtain an exponentially growing solution."
],
[
"$L^2$ stability",
"As the main result of this section, we adapt the Lyapunov functional approach of [13] to this circular setting and prove stability of (REF ).",
"In the following we formulate the main ingredients of our approach as a series of Lemmata, which are then used to prove $L^2$ stability in Theorem REF .",
"Subsequently, we elaborate on the theorem's statement and assumptions in comparison to existing results and prove the lemmata.",
"Here, the lemmata are formulated in a general way in order to facilitate their use for higher regularity estimates in later sections.",
"Lemma 3.2 Let $L_{t}$ be given by (REF ) and let $\\kappa \\in W^{1,\\infty }$ .",
"Then, for any $u,v \\in L^{2}$ $| \\langle u, \\kappa L_{t}v \\rangle | \\le (\\Vert \\kappa \\Vert _{L^\\infty } + \\frac{1}{|k|} \\Vert \\partial _y\\kappa \\Vert _{L^\\infty }) \\Vert u\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert L_{t} v\\Vert _{\\tilde{H}^{1}_t}$ Lemma 3.3 Let $L_{t}$ be as in (REF ).",
"Then there exists a constant $C=C(a,b,g)$ such that for any $u \\in L^{2}$ and any $t\\ge 0$ $\\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}} \\le C \\Vert u\\Vert _{\\tilde{H}^{-1}_{t}}.$ Lemma 3.4 ([12]) Let $u \\in L^2([a,b])$ and let $\\sum _{n \\in (b-a)\\mathbb {N}} u_{n} \\sin (ny)$ be its series expansion.",
"Define the symmetric, positive definite, non-increasing operator $A$ by $\\langle u, A u \\rangle := \\sum _{n} \\exp (\\arctan (\\frac{n}{k}-t)) |u_{n}|^{2}.$ Then $A$ is symmetric, positive definite, non-increasing, $C^{1}$ in time and comparable to the identity, i.e.",
"$e^{-\\pi } \\Vert u\\Vert _{L^{2}} \\le \\langle u, A u \\rangle \\le e^\\pi \\Vert u\\Vert _{L^{2}},$ for all $u \\in L^{2}$ .",
"Furthermore, there exists a constant $e^{-\\pi }\\le C_{2} \\le e^{\\pi }$ and $\\delta >0$ such that $\\Vert u\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert Au\\Vert _{\\tilde{H}^{-1}_{t}} \\le -C_{2} \\langle u, \\dot{A} u \\rangle $ Using the preceding lemmata, we can establish $L^{2}$ stability.",
"Theorem 3.1 ($L^{2}$ stability) Let $A$ and $C_{2}$ be given by Lemma REF and let $C$ be as in Lemma REF .",
"Further suppose that there exists $\\delta >0$ such that $|k|^{-1}(\\Vert h\\Vert _{L^{\\infty }}+|k|^{-1}\\Vert \\partial _y h\\Vert _{L^\\infty }) \\le \\frac{1}{C}(\\frac{1}{C_2}-\\delta ).$ Then for any solution $W$ to the Euler equations in scattering formulation (REF ) the functional $I(t):= \\langle W, A(t) W \\rangle .$ is non-increasing and satisfies $\\partial _tI(t) \\le -\\delta \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}\\Vert A(t)W(t)\\Vert _{\\tilde{H}^{-1}_t} \\le 0.$ In particular, this implies $e^{-\\pi }\\Vert W(t)\\Vert _{L^{2}}^{2} \\le I(t) \\le I(0) \\le e^\\pi \\Vert \\omega _{0}\\Vert _{L^{2}}^{2}.$ Using Lemma REF , we estimate $\\partial _tI(t) \\le \\langle W, \\dot{A} W \\rangle + 2 |k|^{-1}(\\Vert h\\Vert _{L^{\\infty }}+|k|^{-1}\\Vert \\partial _y h\\Vert _{L^\\infty }) \\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert L_tW\\Vert _{\\tilde{H}^{1}_{t}}.$ Applying Lemma REF and Young's inequality, we further control $\\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert L_tW\\Vert _{\\tilde{H}^{-1}_{t}} \\le \\frac{C}{2} \\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}} \\Vert W\\Vert _{\\tilde{H}^{-1}_{t}}.$ The result then follows by an application of Lemma REF and noting that, by our smallness assumption, $\\partial _tI(t) \\le \\langle W, \\dot{A} W \\rangle + |k|^{-1}(\\Vert h\\Vert _{L^{\\infty }}+|k|^{-1}\\Vert \\partial _y h\\Vert _{L^\\infty })C\\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert W\\Vert _{\\tilde{H}^{-1}_{t}} \\\\\\le \\langle W, \\dot{A} W \\rangle + (\\frac{1}{C_2}-\\delta )\\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}} \\Vert W\\Vert _{\\tilde{H}^{-1}_{t}} \\\\\\le -\\delta \\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert W\\Vert _{\\tilde{H}^{-1}_{t}} \\le 0$ Let us briefly remark on this result and its assumptions: We require a smallness condition on $\\frac{ih}{k}L_t$ in order to rule out the obstacles mentioned in Section REF .",
"Since $h$ is allowed to be complex-valued, we do not rely on conserved quantities or classical stability results such as the ones of Rayleigh, Fjortoft or Arnold.",
"In the setting of a plane finite periodic channel, in [10] Wei, Zhang and Zhao use a spectral approach to establish linear stability and decay with optimal rates for monotone shear flows under the assumption that the strictly monotone shear flow $U(y)$ possesses no embedding eigenvalues.",
"In comparison, our smallness assumption is more restrictive, but extends to related problems such as stability in fractional Sobolev spaces, complex valued functions $h$ and fractional operators $L_t$ in a straightforward way.",
"In Section , we show that $\\partial _{y}^2W$ can be split into a very regular, stable part $\\Gamma $ and a boundary layer part $\\beta $ which develops a singularity at the boundary.",
"Here, $\\beta $ is determined solely by the Dirichlet boundary data of the initial datum, $\\omega _0$ , and allows for a detailed study of the stability properties of the evolution.",
"It remains to prove Lemmata REF and REF .",
"Let $u,v \\in L^2$ and let $\\Psi [u]$ be the unique solution of $(-1+(\\partial _{y}-ikt)^{2})\\Psi [u]&=u, \\\\\\Psi [u]|_{y=a,b}&=0.$ Then we directly compute $| \\langle u, \\kappa L_{t}v \\rangle | = | \\langle (1-(\\frac{\\partial _{y}}{k}-ikt)^{2})\\Psi [u], \\kappa L_{t}v \\rangle | \\\\\\le \\Vert \\Psi [u]\\Vert _{L^{2}} \\Vert \\kappa L_{t}\\Vert _{L^{2}} + \\Vert (\\frac{\\partial _{y}}{k}-it)\\Psi [u]\\Vert _{L^{2}}\\Vert (\\frac{\\partial _{y}}{k}-it) \\kappa L_{t}v\\Vert _{L^{2}} \\\\\\le \\Vert \\Psi [u]\\Vert _{\\tilde{H}^{1}_{t}}(\\Vert \\kappa \\Vert _{L^\\infty } + \\frac{1}{|k|} \\Vert \\partial _y\\kappa \\Vert _{L^\\infty }) \\Vert L_{t}v\\Vert _{\\tilde{H}^{1}_{t}}.$ Here, we used that $L_tv$ by definition satisfies zero Dirichlet boundary conditions and hence no boundary contributions appear when integrating by parts.",
"We recall that $L_{t}u$ is the solution of $\\mathcal {E}_{t}L_{t} u= \\left(\\left(g(y)(\\frac{\\partial _{y}}{k}-it)\\right)^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)}\\right)L_{t}u&=0 , \\\\L_{t}u|_{y=a,b}&=0,$ and that $g(y)$ and $\\frac{g(y)}{r(y)}$ are bounded from below (and above).",
"Hence $\\mathcal {E}_{t}$ is a shifted elliptic operator and testing by $L_{t}u$ (or $\\frac{1}{g}L_{t}u$ ) we obtain that $\\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}}^{2} \\le -C \\langle L_{t}u, u \\rangle ,$ for some $C>0$ .",
"Applying Lemma REF , we thus obtain $\\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}}^{2} &\\le C \\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}}\\Vert \\Psi [u]\\Vert _{\\tilde{H}^{1}_{t}},\\\\\\Leftrightarrow \\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}} &\\le C \\Vert u\\Vert _{\\tilde{H}^{-1}_{t}}.$ Having introduced the basic tools of our approach, in the following section we consider higher stability of $W$ , i.e.",
"control of $\\partial _{y}W$ .",
"Here, boundary effects qualitatively change the dynamics and necessitate a modification of the weight $A(t)$ ."
],
[
"Higher stability and boundary layers",
"In this section we show that the $L^{2}$ stability result can be extended to higher Sobolev regularity.",
"However, unlike in the setting of an infinite periodic channel, boundary effects can not be neglected and result in the formation of singularities.",
"As the main improvements over our previous work for the plane channel in [12], we provide an explicit splitting into a more regular good parts and a boundary layer exhibiting blow-up as well as an improved smallness condition.",
"This splitting then also allows to provide a more detailed description of the blow-up also in weighted Sobolev spaces.",
"For this purpose we also introduce a different method of proof.",
"Let thus $W$ be a solution to (REF ) $\\partial _tW &= \\frac{ih}{k}L_{t}W, \\\\\\mathcal {E}_{t}L_t W &=W, \\\\L_t|_{y=a,b}&=0, \\\\(t,k,y) &\\in \\mathbb {R}\\times 2\\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace ) \\times [a,b].$ We begin by studying $\\partial _yW$ , which satisfies $\\partial _t\\partial _yW &= \\frac{ih}{k}L_t \\partial _y W + \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW + \\frac{ih}{k}H^{(1)}, \\\\\\mathcal {E}_{t}H^{(1)}&=0 ,\\\\H^{(1)}_{y=a,b}&=\\partial _yL_tW|_{y=a,b}.$ In contrast to the $L^2$ setting (or a setting without boundary such as $\\mathbb {R}$ ) we hence obtain a correction $H^{(1)}$ due to $\\partial _y L_tW$ not satisfying zero Dirichlet boundary conditions.",
"As a main result of Appendix , we study the boundary behavior of $\\partial _yL_t$ (also confer [12]) and obtain the following description of $H^{(1)}$ : Lemma 4.1 Let $W$ be a solution of (REF ) and let $H^{(1)}$ be the unique solution of $\\mathcal {E}_{t}H^{(1)}&=0 ,\\\\H^{(1)}_{y=a,b}&=\\partial _yL_tW|_{y=a,b}.$ Then there exist functions $u_1,u_2,\\tilde{u}_1,\\tilde{u}_2 \\in H^{2}$ (depending on $a,b,k$ and $g$ but not on $t$ ) and constants $c_1,c_2$ such that $H^{(1)}(t,y)&= c_1\\langle W, e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)}u_1 \\\\& \\quad + c_2\\langle W, e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)}u_2.$ Furthermore, for instance for $u_1$ for any $t>0$ $\\langle W, e^{ikt(y-a)}\\tilde{u}_1 \\rangle = \\frac{\\omega _0(a)}{ikt} - \\frac{1}{ikt} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle - \\frac{1}{ikt} \\langle \\partial _yW, e^{ikt(y-a)}\\tilde{u}_1 \\rangle .$ Based on this characterization of $H^{(1)}$ , we introduce a splitting of $\\partial _yW$ into a function $\\beta $ depending only on $\\omega _0|_{y=a,b}$ and $\\Gamma =\\partial _yW-\\beta $ .",
"As we show in Theorem REF , $\\Gamma $ is stable also in higher regularity.",
"In contrast, unless $\\omega _0|_{y=a,b}$ is trivial, $\\beta $ asymptotically develops singularities at the boundary and exhibits blow-up in $H^{s},s>1/2$ .",
"If one however considers weighted spaces, it is possible to compensate for these singularities by vanishing weights and hence establish sufficient control for damping with optimal decay rates.",
"Lemma 4.2 Let $W$ be a solution of (REF ) and let $\\Gamma $ be the solution of $\\begin{split}\\partial _t\\Gamma &= \\frac{ih}{k}L_t \\Gamma - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad + \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW -c_1 \\frac{h}{k^2t} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 \\\\ & \\quad -c_2 \\frac{h}{k^2t} \\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2, \\\\\\Gamma |_{t=0}&=\\partial _y \\omega _0,\\end{split}$ and let $\\beta $ be the solution of $\\begin{split}\\partial _t\\beta &= \\frac{ih}{k}L_t \\beta - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad -\\frac{c_1h\\omega _0(a)}{k^2t} e^{ikt(y-a)}u_1 -\\frac{c_2h\\omega _0(b)}{k^2t} e^{ikt(y-b)}u_2, \\\\\\beta |_{t=0}&=0.\\end{split}$ Then $\\partial _yW=\\Gamma +\\beta $ .",
"The function $\\beta $ is called the boundary layer.",
"Theorem 4.1 ($H^2$ regularity of $\\Gamma $ ) Suppose that $g,h$ satisfy the assumptions of Theorem REF .",
"Suppose that additionally $g \\in W^{2,\\infty }$ and $h \\in W^{2,\\infty }$ .",
"Then there exists a constant $C_1$ such that for all $\\omega _0 \\in H^1$ and any $t\\ge 0$ , the solution $\\Gamma $ of (REF ) satisfies $\\Vert \\Gamma (t)\\Vert _{L^2} \\le C \\Vert \\omega _0\\Vert _{H^1}.$ Suppose that additionally $g \\in W^{3,\\infty }$ and $h \\in W^{3,\\infty }$ , then there exists a second constant $C_2$ such that for any $\\omega _0 \\in H^2$ and for any $t \\ge 0$ , $\\Vert \\Gamma (t)\\Vert _{H^1} \\le C_2 \\Vert \\omega _0\\Vert _{H^2}.$ Theorem 4.2 ($H^2$ regularity of $\\beta $ ) Suppose $g,h$ satisfy the assumptions of Theorem REF .",
"Then there exists a constant $C_1$ such that for all $t\\ge 0$ , the solution $\\beta $ of (REF ) satisfies $\\Vert \\beta (t)\\Vert _{L^2} \\le C_1 (|\\omega _0(a)|+|\\omega _0(b)|).$ Suppose that additionally $g,h \\in W^{2,\\infty }$ , then there exists a second constant $C_2$ such that $\\Vert (y-a)(y-b)\\partial _y\\beta (t)\\Vert _{L^2} \\le C_2 (|\\omega _0(a)|+|\\omega _0(b)|).$ However, if for instance $|\\omega _0(a)|>0$ , then $|\\beta (t,a)| \\gtrsim \\log (t)$ as $t \\rightarrow \\infty $ (similarly for $b$ ).",
"In particular, by the Sobolev embedding, we obtain blow-up in $H^{s},s>1/2$ .",
"Remark 4 Combining Theorems REF , REF and REF and Proposition REF , we obtain Theorem REF .",
"It is possible to further split $\\Gamma $ into functions controlled solely in terms of $\\Vert \\omega _0\\Vert _{L^2}$ , $\\Vert \\partial _y \\omega _0\\Vert _{L^2}$ and $\\Vert \\partial _{y}^2\\omega _0\\Vert _{L^2}$ , if finer control is desired.",
"Like Theorem REF , in addition to these stability results we obtain Lyapunov functionals.",
"As a key difference, these functionals are however in general only decreasing for times $t\\ge T>0$ .",
"Control up time $T$ is hence provided by a Gronwall-type argument, which determines the constants $C_1,C_2$ .",
"We stress that we do not require higher norms of $g,h$ to be small but only finite, so that derivatives of the equation are well-defined as mappings in $L^2$ .",
"When considering a setting without boundary contributions such as $\\mathbb {R}$ or $, no boundary correction $$ is needed.Thus (a suitable modification of) this result alreadyyields the desired stability for decay with optimal rates.Furthermore, this result generalizes to higher derivatives in astraightforward way, where again only finiteness of higher norms has to be required.$ Stability of $\\Gamma $ As the main result of this subsection we provide a proof of Theorem REF .",
"Here, the $L^2$ stability result is self-contained, while the $H^1$ estimate presupposes the $L^2$ stability of $\\beta $ , which is established in the following subsection.",
"Furthermore, we briefly discuss the implications of Theorem REF for settings without boundary and provide an improved stability result for the setting of an infinite plane periodic channel, $L\\times \\mathbb {R}$ .",
"We recall that $\\Gamma $ is the solution of $\\partial _t\\Gamma &= \\frac{ih}{k}L_t \\Gamma - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad + \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW -c_1 \\frac{h}{k^2t} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 \\\\ & \\quad -c_2 \\frac{h}{k^2t} \\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2, \\\\\\Gamma |_{t=0}&=\\partial _y \\omega _0,$ In addition to the estimates for $L_t$ derived in Section REF , we hence need to control contributions of the form $\\frac{1}{ikt} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle \\langle A \\Gamma , e^{ikt(y-b)} u_2 \\rangle ,$ which can not be controlled by the previous choice of $A(t)$ .",
"Instead, we construct a modified weight $A_1(t)$ , which is introduced in the following Lemmata (cf.",
"[13] for a similar construction adapted to fractional Sobolev spaces).",
"Lemma 4.3 Let $u \\in H^1$ , then for $0<\\mu <1/2$ and for every $v=\\sum _n v_n e^{iny} \\in L^2$ $|\\langle v, e^{ikt(y-a)} u \\rangle |^{2} \\le C_\\mu \\Vert u\\Vert _{H^1}^2 \\sum _n <n-kt>^{-2\\mu }|v|_{n}^2.$ By expanding the $L^2$ inner product in a basis, we obtain that $\\langle v, e^{ikt(y-a)} u\\rangle = \\sum _n v_n \\langle e^{iny},e^{ikt(y-a)} u \\rangle .$ Integrating by parts and using the trace inequality, we further estimate $|\\langle e^{iny} ,e^{ikt(y-a)} u \\rangle | \\le <n-kt>^{-1} \\Vert u\\Vert _{H^1}.$ The result hence follows by an application of the Cauchy-Schwarz inequality: $|\\langle v, e^{ikt(y-a)} u\\rangle | &\\lesssim \\sum _n v_n <n-kt>^{-\\mu } <n-kt>^{1-\\mu } \\\\&\\le \\Vert v_n <n-kt>^{-\\mu }\\Vert _{l^2} \\Vert <n-kt>^{1-\\mu }\\Vert _{l^2} \\\\&\\le 2 \\Vert v_n <n-kt>^{-\\mu }\\Vert _{l^2} \\Vert <n>^{-(1-\\mu )}\\Vert _{l^2} \\\\&=: C_\\mu \\Vert v_n <n-kt>^{-\\mu }\\Vert _{l^2},$ where we used that $<n>^{-(1-\\mu )} \\in l^2$ if $\\mu <1/2$ .",
"Lemma 4.4 Let $0<\\lambda ,\\mu <1$ with $\\lambda +2\\mu >1$ and let $\\epsilon >0$ and define the symmetric operator $A_1(t)$ by its action on the basis: $A_1(t): e^{iny} \\mapsto \\exp \\left(\\arctan \\left(\\frac{\\eta }{k}-t\\right)- \\epsilon \\int ^{t} <\\tau >^{-\\lambda } <n-k\\tau >^{-2\\mu } d\\tau \\right).$ Then for every $u \\in L^2$ and every $t \\in \\mathbb {R}$ $C \\Vert u\\Vert _{L^2}^2\\le \\langle u, A_1(t) u \\rangle &\\le C^{-1} \\Vert u\\Vert _{L^2}^2, \\\\\\langle u, \\dot{A}_1(t)u \\rangle \\le -C_1\\Vert u\\Vert _{\\tilde{H}^{-1}_t}^2 - C\\epsilon \\sum _n <t>^{-\\lambda } <n-kt>^{-2\\mu }|u_n|^2 &\\le 0.$ We note that $<t>^{-\\lambda } <n-kt>^{-2\\mu } \\in L^1(\\mathbb {R})$ and that $- \\epsilon \\int ^{t} <\\tau >^{-\\lambda } <n-k\\tau >^{-2\\mu } d\\tau $ is monotonically decreasing.",
"The properties of $A_1(t)$ hence follow by direct computation, where $C=\\exp (-\\pi - \\epsilon \\Vert <\\cdot >^{-\\lambda } <n-k\\cdot >^{-2\\mu }\\Vert _{L^1(\\mathbb {R})}).$ and $C_1$ is determined by $C$ and Lemma REF .",
"Lemma 4.5 Let $g \\in W^{2,\\infty }$ , $g \\ge c>0$ , then for every $u \\in L^2$ and for every $t \\ge 0$ , $\\Vert L_t[\\mathcal {E}_{t}, \\partial _y]L_tu\\Vert _{\\tilde{H}^{1}_{t}} \\lesssim \\Vert u\\Vert _{\\tilde{H}^{-1}_t}.$ By Lemma REF , we obtain that $\\Vert L_t[\\mathcal {E}_{t}, \\partial _y]L_tu\\Vert _{\\tilde{H}^{1}_{t}} \\lesssim \\Vert [\\mathcal {E}_{t}, \\partial _y]L_tu\\Vert _{\\tilde{H}^{-1}_{t}}.$ We further note that $[\\mathcal {E}_{t}, \\partial _y]= e^{-ikty}[\\mathcal {E}_0, \\partial _y-ikt]e^{ikty}= e^{-ikty}[\\mathcal {E}_0, \\partial _y]e^{ikty},$ and that, by direct computation, $[\\mathcal {E}_0, \\partial _y]$ is a second-order operator.",
"Hence, using integration by parts, we further estimate $\\Vert [\\mathcal {E}_{t}, \\partial _y]L_tu\\Vert _{\\tilde{H}^{-1}_{t}} \\lesssim \\Vert L_tu\\Vert _{\\tilde{H}^{1}_t} \\lesssim \\Vert u\\Vert _{\\tilde{H}^{-1}_t}.$ Using these results, we can now provide a proof of Theorem REF and thus establish $L^2$ stability.",
"Fix $0<\\lambda ,\\mu <1$ with $2\\mu +\\lambda >1$ and let $A_1$ be given by Lemma REF , where $0<\\epsilon < \\frac{1}{100} \\Vert <n-k\\cdot >^{-2\\mu }<\\cdot >^{-\\lambda }\\Vert _{L^1(\\mathbb {R})}^{-1}.$ Then we define $I(t):= \\langle \\Gamma , A_1(t)\\Gamma \\rangle + C_1 \\langle W, A(t)W \\rangle ,$ where $C_1 \\gg 0$ is to be chosen later.",
"We then claim that there exists $T>0$ such that for all initial data and for all $t \\ge 0$ , $I(t)$ satisfies $\\frac{d}{dt}I(t) \\le C t^{-2(1-\\mu /2)} \\Vert \\omega _0\\Vert _{L^2}^2 \\in L^1(\\mathbb {R}).$ Using Gronwall's inequality, we further obtain that $I(T)\\le \\exp (CT)I(0),$ which concludes the proof.",
"It remains to prove the claim.",
"Using Theorem REF and Lemma REF , we directly compute $\\frac{d}{dt}I(t) &\\le -C\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t}^2 - C\\epsilon \\sum _n <t>^{-\\lambda } <n-kt>^{-2\\mu }|\\Gamma _n|^2 \\\\ & \\quad - C_1\\delta \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2 + 2 \\Re \\langle \\frac{d}{dt}\\Gamma , A_{1}(t)\\Gamma \\rangle .$ Using Lemma REF and Lemma REF and recalling (REF ), we further estimate $2 \\Re \\langle \\frac{d}{dt}\\Gamma , A_{1}(t)\\Gamma \\rangle &\\le C(h,k)\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t} \\Vert A_1\\Gamma \\Vert _{\\tilde{H}^{-1}_t} + C(h,k,\\mu ) \\frac{1}{t} \\left( \\sum _n <n-kt>^{-2\\mu }|\\Gamma _n|^2 \\right) \\\\& \\quad + C(h,h^{\\prime },k) \\Vert A_{1}\\Gamma \\Vert _{\\tilde{H}^{-1}_t} (\\Vert L_tW\\Vert _{\\tilde{H}^{1}_t} + \\Vert L_t [\\mathcal {E}_{t},\\partial _y]L_t\\Vert _{\\tilde{H}^{1}_t}) \\\\& \\quad + C(h,k,g) t^{-1} \\Vert \\omega _0\\Vert _{L^2} \\sqrt{\\sum _n <n-kt>^{-2\\mu }|\\Gamma _n|^2}.$ Splitting $t=t^{-(1-\\mu )} t^{-\\mu }$ and using Young's inequality and Lemmata REF and REF , we further control $\\frac{1}{t} \\left( \\sum _n <n-kt>^{-2\\mu }|\\Gamma _n|^2 \\right)= t^{-(1-\\lambda )}\\sum _n t^{-\\lambda } <n-kt>^{-2\\mu }|\\Gamma _n|^2 , \\\\\\Vert A_{1}\\Gamma \\Vert _{\\tilde{H}^{-1}_t} (\\Vert L_tW\\Vert _{\\tilde{H}^{1}_t} + \\Vert L_t [\\mathcal {E}_{t},\\partial _y]L_t\\Vert _{\\tilde{H}^{1}_t}) \\le \\sigma \\Vert A_{1}\\Gamma \\Vert _{\\tilde{H}^{-1}_t}^{2} + \\sigma ^{-1} \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2 , \\\\t^{-1} \\Vert \\omega _0\\Vert _{L^2} \\sqrt{\\sum _n <n-kt>^{-2\\mu }|\\Gamma _n|^2} \\le \\sigma \\langle \\Gamma , \\dot{A}_1(t) \\Gamma \\rangle | + \\sigma ^{-1} t^{-2(1-\\mu /2)}\\Vert \\omega _0\\Vert _{L^2}^2.$ Choosing $\\sigma $ sufficiently small and letting $T>0$ be sufficiently large and using the smallness assumption of Theorem REF , we observe that $-C\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t}^2 - C\\epsilon \\sum _n <t>^{-\\lambda } <n-kt>^{-2\\mu }|\\Gamma _n|^2+ C(h,k)\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t} \\Vert A_1\\Gamma \\Vert _{\\tilde{H}^{-1}_t} \\\\+ (C(h,k,\\mu )t^{-(1-\\mu )}) \\sum _n <t>^{-\\lambda } <n-kt>^{-2\\mu }|\\Gamma _n|^2 \\\\+ \\sigma \\Vert A_{1}\\Gamma \\Vert _{\\tilde{H}^{-1}_t}^{2} + \\sigma \\langle \\Gamma , \\dot{A}_1(t) \\Gamma \\rangle | \\le 0.$ Similarly, choosing $C_1$ sufficiently large, we observe that $- C_1\\delta \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2 + \\sigma ^{-1} \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2 \\le 0.$ Hence, we conclude that for $t \\ge T>0$ , $I(t)$ satisfies $\\frac{d}{dt}I(t) \\le \\sigma ^{-1} t^{-2(1-\\mu /2)}\\Vert \\omega _0\\Vert _{L^2}^2.$ which finishes the proof of the claim and hence of the $L^2$ stability result, $(1)$ .",
"Next, we consider the evolution of $\\partial _y\\Gamma $ : $\\partial _t\\partial _y\\Gamma &= \\frac{ih}{k}L_t \\partial _y \\Gamma + \\frac{ih^{\\prime }}{k}L_t \\Gamma + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t\\Gamma - (\\partial _yL_t\\Gamma )(a) e^{ikt(y-a)}u_1 - (\\partial _yL_t\\Gamma )(b) e^{ikt(y-b)}u_1\\\\& \\quad + \\partial _y\\Big ( \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad -c_1 \\frac{h}{k^2t} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 -c_2 \\frac{h}{k^2t} \\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\Big ) \\\\& \\quad + \\partial _y \\left( \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW \\right), \\\\\\partial _y \\Gamma |_{t=0}&=\\partial _{y}^2 \\omega _0$ Since we here also have to compute $\\partial _yW=\\Gamma +\\beta $ in order to control $\\Vert \\partial _y\\Gamma \\Vert _{L^2}$ , we require $L^2$ estimates on $\\beta $ .",
"Before continuing with the proof of Theorem REF , we hence prove the first part of Theorem REF as well as some further properties of the evolution of $\\beta $ , which are formulated in the following proposition.",
"Proposition 4.1 Suppose $g,h$ satisfy the assumptions of Theorem REF .",
"Let $\\beta $ be the solution of (REF ) and let $A_1(t)$ be given by Lemma REF .",
"Then there exists $T>0$ such that for all $t\\ge 0$ $I_2(t)=\\langle \\beta , A_1(t)\\beta \\rangle $ satisfies $\\frac{d}{dt}I_2(t) \\le \\delta \\langle \\beta , \\dot{A}_1(t)\\beta \\rangle + C t^{-2(1-\\mu /2)}|\\omega _0|_{y=a,b}|^2.$ Using the same weight $A_{1}$ , we observe that $\\Re \\langle A_{1}(t)\\beta , \\frac{ih}{k}L_t \\beta -\\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\rangle \\\\\\le (C(h,k)+ C(h,k,g)t^{-(1-\\mu )})|\\langle \\beta , \\dot{A}_1(t)\\beta \\rangle |.$ Using the smallness assumption and restricting to $t\\ge T>0$ , this contribution can thus be absorbed by $\\langle \\beta , \\dot{A}_1(t)\\beta \\rangle \\le 0.$ Hence, we focus on $\\Re \\langle A_{1}\\beta , \\omega _{0}(a)\\frac{1}{ikt}e^{ikty}u \\rangle \\lesssim C_{\\lambda } \\Vert \\beta _{n}<n-kt>^{-\\lambda }\\Vert _{l^{2}} |\\omega _{0}|_{y=a,b}| \\frac{1}{|kt|}.$ Using Young's inequality and choosing $\\sigma $ sufficiently small, we thus obtain that $\\frac{d}{dt}\\langle \\beta , A_1 \\beta \\rangle \\le \\delta \\langle \\beta , \\dot{A}_1 \\beta \\rangle + C\\sigma ^{-1}t^{-2(1-\\mu /2)} |\\omega _0|_{y=a,b}|^2.$ The first part of Theorem REF then follows by integrating this inequality and using a Gronwall-type estimate to control the growth up to time $T$ .",
"Additionally, we make use of the following estimates for boundary evaluations of $L_t\\Gamma , W$ and $\\Gamma $ , which are obtained as an application of the results of Appendix .",
"Lemma 4.6 Let $g,h,k$ satisfy the assumptions of the second part of Theorem REF .",
"Then, $(\\partial _yL_t\\Gamma )(a)=c_1 \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle , \\\\(\\partial _yL_t\\Gamma )(b)=c_2 \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle ,$ and the following estimates hold: $|\\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle | &\\lesssim \\frac{C_\\mu }{kt}\\sqrt{\\sum _n |(\\partial _y\\Gamma )|_{n}^2 <n-kt>^{-2\\lambda }} + \\frac{C}{kt}|\\Gamma (a,t)|, \\\\|\\langle W, e^{ikt(y-a)}\\tilde{u}_1 \\rangle | &\\lesssim \\frac{C_\\mu }{kt}(\\Vert \\Gamma \\Vert _{L^2}+\\Vert \\beta (t)\\Vert _{L^2})+ \\frac{C}{kt}|\\omega _0(a,t)|, \\\\|\\Gamma (a,t)| &\\le \\log (t)(|\\omega _0(a)|+\\Vert \\omega _0\\Vert _{L^2}).$ The evaluations of $\\partial _yL_t\\Gamma $ at the boundary are obtained as an application of Lemma REF .",
"The first two estimates follow by integration by parts.",
"In order to show the last estimate, we restrict (REF ) to the boundary and obtain that $|\\partial _t\\Gamma (a,t)|\\lesssim \\frac{1}{kt}(\\Vert \\Gamma (t)\\Vert _{L^2}+\\Vert W(t)\\Vert _{L^2}),$ where we used that $L_t$ enforces zero Dirichlet data.",
"The result hence follows by using Theorem REF and the first part of Theorem REF to control $\\Vert \\Gamma (t)\\Vert _{L^2}+\\Vert W(t)\\Vert _{L^2}\\lesssim |\\omega _0(a)|+\\Vert \\omega _0\\Vert _{L^2},$ and then integrating the inequality.",
"Lemma 4.7 Let $W$ be the solution of (REF ) with initial datum $\\omega _0 \\in H^1$ and let $\\Gamma $ and $\\beta $ be as in Lemma REF .",
"Then, for any $\\sigma >0$ , $\\Re \\langle A_1 \\partial _y\\Gamma , \\left[\\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right), \\partial _y \\right] W \\rangle \\le \\sigma |\\langle \\partial _y\\Gamma , \\dot{A}_1 \\partial _y \\Gamma \\rangle | + C\\sigma ^{-1} \\Vert W\\Vert _{\\tilde{H}^{-1}_t}^2.$ The contribution due to $\\frac{ih^{\\prime }}{k}L_t$ can be estimated as in Lemma REF .",
"In the following we thus focus on the commutator and decompose the commutator into the cases where $\\partial _y$ falls on $h$ , $\\frac{ih^{\\prime \\prime }}{k}L_tW + \\frac{ih^{\\prime }}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW,$ the terms solving an elliptic equation with vanishing Dirichlet data, $\\frac{ih^{\\prime }}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t W + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t[\\mathcal {E}_{t},\\partial _y]L_tW,$ and the homogeneous corrections, $\\frac{ih^{\\prime }}{k} ((\\partial _yL_tW)(a,t) e^{ikt(y-a)}u_1+(\\partial _yL_tW)(b,t) e^{ikt(y-b)}u_2), \\\\\\frac{ih}{k} ((\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(a,t) e^{ikt(y-a)}u_1+(\\partial _yL_t[\\mathcal {E}_{t},\\partial _y]L_tW)(b,t) e^{ikt(y-b)}u_2),$ In the first and second case, we use Lemmata REF and REF to estimate by $\\Vert \\partial _y\\Gamma \\Vert _{\\tilde{H}^{-1}_t} \\Vert W\\Vert _{\\tilde{H}^{-1}_t},$ which is of the desired form by Young's inequality.",
"It hence only remains to consider the homogeneous corrections.",
"Here, we estimate $\\Re \\Big \\langle A_1(t)\\partial _y \\Gamma &, \\frac{ih^{\\prime }}{k} ((\\partial _yL_tW)(a,t) e^{ikt(y-a)}u_1+(\\partial _yL_tW)(b,t) e^{ikt(y-b)}u_2) \\\\ &+ \\frac{ih}{k} ((\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(a,t) e^{ikt(y-a)}u_1+(\\partial _yL_t[\\mathcal {E}_{t},\\partial _y]L_tW)(b,t) e^{ikt(y-b)}u_2) \\\\&\\le C_\\mu \\sqrt{\\sum _n |\\partial _y\\Gamma _n|^{2}<n-kt>^{-2\\mu }} \\Big (|\\partial _yL_tW)(a,t)|+|\\partial _yL_tW)(b,t)| \\\\& \\quad +|\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(a,t)| + |\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(b,t)|\\Big ).$ We further recall from Section that boundary evaluations can be obtained by testing with suitable homogeneous solution to the adjoint problem.",
"Hence, $|\\partial _yL_tW)(a,t)|+|\\partial _yL_tW)(b,t) \\lesssim t^{-1} \\Vert \\partial _yW\\Vert _{L^2}\\le t^{-1}\\Vert \\omega _0\\Vert _{H^1}, \\\\|\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(a,t)| + |\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(b,t)| \\lesssim t^{-1} \\Vert [\\mathcal {E}_{t},\\partial _y]L_tW\\Vert _{H^1}.$ We can thus conclude the proof, if we can show that $\\Vert [\\mathcal {E}_{t},\\partial _y]L_tW\\Vert _{H^1} \\lesssim \\Vert W\\Vert _{H^1}.$ Expressing $\\partial _y [\\mathcal {E}_{t},\\partial _y]L_tW= [\\mathcal {E}_{t},\\partial _y]L_t \\partial _yW +[[\\mathcal {E}_{t},\\partial _y]L_t,\\partial _y]W$ , this estimate follows from elliptic regularity theory for $[\\mathcal {E}_{t},\\partial _y]L_t|_{t=0}$ and using that multiplication by $e^{ikty}$ is an isometry.",
"Building on these results, we can now complete the proof of Theorem REF .",
"Following a similar strategy as in the previous part, we consider $I_{2}(t):= \\langle \\partial _y \\Gamma , A_1(t)\\partial _y \\Gamma \\rangle + C_1 \\langle \\Gamma , A_1(t)\\Gamma \\rangle + C_2 \\langle \\beta , A_1(t)\\beta \\rangle + C_3 \\langle W, A(t)W\\rangle ,$ where $C_1,C_2,C_3>0$ are to be chosen later.",
"Using the preceding results and strategy, it suffices to study $\\Re \\langle \\partial _t \\partial _y \\Gamma , A_1 \\partial _y \\Gamma \\rangle .$ Following the same strategy as in the previous part of the proof and using Lemma REF , we estimate $& \\quad \\Re \\langle A_1 \\partial _y\\Gamma , \\frac{ih}{k}L_t \\partial _y \\Gamma + \\frac{ih^{\\prime }}{k}L_t \\Gamma + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t\\Gamma \\\\ & \\quad - (\\partial _yL_t\\Gamma )(a) e^{ikt(y-a)}u_1 - (\\partial _yL_t\\Gamma )(b) e^{ikt(y-b)}u_1 \\rangle \\\\&\\le (C+\\sigma +ct^{-(1-\\mu )}\\log (t)) \\Vert \\partial _y\\Gamma \\Vert _{\\tilde{H}^{-1}_t}^2+ \\sigma ^{-1}|\\langle \\Gamma , \\dot{A} \\Gamma \\rangle |,$ which can be absorbed.",
"Furthermore, applying Lemma REF , we can control $& \\quad \\Re \\Big \\langle A_1 \\partial _y\\Gamma ,\\partial _y\\Big ( \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad -c_1 \\frac{h}{k^2t} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 -c_2 \\frac{h}{k^2t} \\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\Big ) \\Big \\rangle \\\\&\\le C(\\mu ,g,h,k) \\left( \\sum _n |\\Gamma _n|^2<n-kt>^{-2\\mu } \\right)^{1/2}\\Big ( \\left|\\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle \\right| \\\\& \\quad +\\left|\\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle \\right| +\\left|\\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle \\right|+\\left|\\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle \\right| \\Big ).$ Applying the estimates of Lemma REF and using Young's inequality, these contributions can hence again be partially absorbed provided $\\sigma $ is sufficiently small and $T>0$ is sufficiently large.",
"The remaining non-absorbed terms can be estimated by $t^{-2(1-\\mu /2)} (|\\Gamma (a,t)| + |\\Gamma (b,t)| + \\Vert \\partial _y W(t)\\Vert _{L^2} + |\\omega _0(a)| + |\\omega _0(b)|) \\lesssim t^{-2(1-\\mu /2)} \\Vert \\omega _0\\Vert _{H^1},$ where we used Theorem REF , the first part of Theorem REF and the Sobolev embedding.",
"It remains to estimate $\\Re \\left\\langle A_1(t)\\partial _y\\Gamma , \\partial _y \\left( \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW \\right) \\right\\rangle .$ Recalling the definition of $\\Gamma $ and $\\beta $ , we express the right function as $\\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right) (\\Gamma +\\beta ) \\\\+ \\left[\\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right), \\partial _y \\right] W.$ We then estimate $\\Re \\left\\langle A_1(t)\\partial _y\\Gamma , \\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right) (\\Gamma +\\beta ) \\right\\rangle \\lesssim \\Vert \\partial _y\\Gamma \\Vert _{\\tilde{H}^{-1}_t} (\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t} + \\Vert \\beta \\Vert _{\\tilde{H}^{-1}_t}).$ Using Young's inequality, the respective terms can then again be controlled, given a suitable choice of $\\sigma $ .",
"Finally, using Lemma REF , $\\Re \\langle A_1 \\partial _y\\Gamma , \\left[\\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right), \\partial _y \\right] W \\rangle \\\\ \\le \\sigma |\\langle \\partial _y\\Gamma , \\dot{A}_1 \\partial _y \\Gamma \\rangle | + C\\sigma ^{-1} \\Vert W\\Vert _{\\tilde{H}^{-1}_t}^2,$ which can again be absorbed and hence concludes the proof.",
"Weighted stability of $\\partial _y\\beta $ and boundary blow-up In this section we consider the evolution of $\\partial _y\\beta $ .",
"Since the behavior at both boundary points is similar and separates, we for simplicity of notation consider the case $\\omega _0(a)\\ne 0$ , $\\omega _0(b)=0$ .",
"The general case can then be obtained by switching $a$ and $b$ and using the linearity of the equation.",
"The function $\\beta $ then satisfies (REF ): $\\begin{split}\\partial _t\\beta -\\frac{ih}{k}L_{t}\\beta - \\frac{h}{k^2t}\\langle \\beta ,e^{ikt(y-a)}u \\rangle e^{ikt(y-a)}u &= \\omega _{0}(a)\\frac{h}{k^2t}e^{ikt(y-a)}u, \\\\\\beta |_{t=0}&=0.\\end{split}$ We note that, if $\\omega _{0}|_{y=a,b}=0$ , then $\\beta $ identically vanishes.",
"We recall that by Proposition REF under suitable assumptions on $h,g$ and $k$ , $\\beta $ is stable in $L^2$ .",
"However, stability in $H^{1}$ or, indeed in $H^{s},s>1/2$ , does not hold due to the asymptotic formation of singularities at the boundary.",
"Lemma 4.8 (Boundary blow-up) Suppose that for some $s>0$ , $\\sup _{t>0} \\Vert \\beta (t)\\Vert _{H^{s}} = C <\\infty .$ Then $\\beta (a,t)$ satisfies $|\\beta (a,t)-h(a)\\omega _{0}(a)k^{-2} \\log (t)| \\le C_{s}C,$ as $t \\rightarrow \\infty $ In particular, if $\\omega _{0}(a)\\ne 0$ , then $\\sup _{t}\\Vert \\beta (t)\\Vert _{C^{0}}=\\infty .$ Hence, by the Sobolev embedding, in that case, $\\sup _{t} \\Vert \\beta (t)\\Vert _{H^{s}} \\ge \\sup _{t} \\log (t) = \\infty ,$ for any $s>\\frac{1}{2}$ .",
"Restricting the evolution by (REF ) to the boundary, we obtain $\\partial _t\\beta (a) + \\frac{h(a)}{k^2t} \\langle \\beta , e^{ikt(y-a)}u \\rangle &= \\frac{h(a)\\omega _{0}(a)}{k^2t}.$ Let $s>0$ and without loss of generality $s<1/2$ , then by direct computation $|\\langle \\beta , e^{ikt(y-a)}u \\rangle | \\lesssim C t^{-s}\\Vert \\beta \\Vert _{H^s}.$ Hence, $\\beta (a,t)$ satisfies $\\partial _t\\beta (a) - \\frac{\\omega _{0}(a)h(a)}{k^2} \\partial _t\\log (t) = t^{-1} \\mathcal {O}(t^{-s}) \\in L^{1}_{t}.$ The result hence follows by integrating in time.",
"Letting $s=1$ in the preceding Lemma, we in particular note that in general $H^1$ stability of $\\beta $ fails.",
"Following a similar approach as in [13], one can further show that $s=1/2$ is indeed critical in the sense that stability holds for $H^{s},s<1/2$ .",
"As this is however not sufficient for optimal decay rates in the damping estimate of Section , in the following we prove weighted $H^1$ stability as formulated in Theorem REF .",
"Here, we use a different method of proof based Duhamel's formula, the details of which can be found in Appendix .",
"Splitting $\\partial _y\\beta $ We recall that $\\beta $ solves $\\begin{split}\\partial _t\\beta - \\frac{ih}{k}L_{t}\\beta - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}u \\rangle e^{ikt(y-a)} u &= \\omega _{0}(a) \\frac{h}{k^2t}e^{ikt(y-a)}u, \\\\\\beta |_{t=0}&=0.\\end{split}$ Applying one $y$ derivative to this equation, we obtain $\\begin{split}& \\quad \\partial _t\\partial _{y}\\beta - \\frac{ih}{k}L_{t}\\partial _{y}\\beta + \\frac{h}{k^2t} \\langle \\partial _{y} \\beta , e^{ikt(y-a)}u \\rangle e^{ikt(y-a)} u\\\\ &= [\\frac{ih}{k}L_{t},\\partial _{y}]\\beta + \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)} \\partial _{y} u \\rangle e^{ikt(y-a)} u + \\frac{h}{k^2t} \\beta (a,t) e^{ikt(y-a)} u \\\\ &\\quad - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}u \\rangle e^{ikt(y-a)} u- \\omega _{0}(a) \\frac{h^{\\prime }}{k^2t}e^{ikt(y-a)}u \\\\ & \\quad +\\omega _{0}(a) \\frac{h}{k^2t}e^{ikt(y-a)}\\partial _{y}u + \\frac{i\\omega _{0}(a)h}{k} e^{ikt(y-a)}u,\\end{split}$ where we used that $\\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}u \\rangle \\partial _y(e^{ikt(y-a)})u= \\frac{h}{k^2t} (\\langle \\partial _y(\\beta u), e^{ikt(y-a)} \\rangle - \\beta u e^{ikt(y-a)}|_{y=a}^b) e^{ikt(y-a)}u.$ We note that most terms in (REF ) are very similar to ones in equation (REF ) satisfied by $\\partial _{y}\\Gamma $ , with the exception of $\\frac{i\\omega _{0}(a)h}{k} e^{ikt(y-a)}u,$ which is hence identified as the term driving the blow-up.",
"Based on this reasoning the following lemma introduces a splitting of $\\partial _y \\beta $ .",
"Lemma 4.9 Let $\\beta _{I}$ be the solution of $\\begin{split}\\partial _t\\beta _{I} - \\frac{ih}{k}L_{t}\\beta _{I} + \\frac{1}{ikt} \\langle \\beta _{I}, e^{ikty}u \\rangle e^{ikty} u&= [\\frac{ih}{k}L_{t},\\partial _{y}]\\beta + \\frac{h}{k^2t} \\langle \\beta , e^{ikty} \\partial _{y} u \\rangle e^{ikty} u \\\\ & \\quad + \\frac{h}{k^2t} \\beta (a,t) e^{ikty} u - \\frac{h}{k^2t} \\langle \\beta , e^{ikty}u \\rangle e^{ikty} u \\\\& \\quad - \\omega _{0}(a) \\frac{h}{k^2t}e^{ikty}\\partial _{y}u +\\omega _{0}(a) \\frac{h}{k^2t}e^{ikty}\\partial _{y}u, \\\\\\beta _{I}|_{t=0}&=0,\\end{split}$ and let $\\beta _{II}$ be the solution of $\\partial _t\\beta _{II} - \\frac{ih}{k}L_{t}\\beta _{II} + \\frac{h}{k^2t} \\langle \\beta _{II}, e^{ikty}u \\rangle e^{ikty} u &= \\omega _{0}(a) e^{ikty}u, \\\\\\beta _{V}|_{t=0}&=0.$ Then $\\partial _y\\beta =\\beta _{I}+\\beta _{II}$ .",
"Following the same strategy as in Section REF , we obtain $L^{2}$ stability of $\\beta _{I}$ .",
"Proposition 4.2 Suppose the assumptions of Theorem REF are satisfied, then $\\Vert \\beta _{I}(t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}|_{y=a,b}|.$ Following the same strategy as in the proof of Theorem REF , we show that, $\\frac{d}{dt}\\langle \\beta _{I}, A_1(t) \\beta _{I} \\rangle \\le \\langle \\beta _{I}, \\dot{A}_1(t) \\beta _{I} \\rangle + C \\Vert \\beta _{I}\\Vert _{\\tilde{H}^{-1}_t}^2 \\\\+ (Ct^{-(1-\\mu )} + \\sigma ) \\sum _n |(\\beta _{I})_n|^2<n-kt>^{-2\\lambda }t^{-\\mu } \\\\+ C\\sigma ^{-1} t^{-2(1-\\mu /2)} (|\\beta (a,t)|^2 + \\Vert \\beta \\Vert _{L^2}^2 + |\\omega _0(a)|^2).$ Hence, restricting to $t \\ge T>0$ and choosing $\\sigma $ sufficiently small, $\\frac{d}{dt}\\langle \\beta _{I}, A_1(t) \\beta _{I} \\rangle \\le C\\sigma ^{-1} t^{-2(1-\\mu /2)} (|\\beta (a,t)|^2 + \\Vert \\beta \\Vert _{L^2}^2 + |\\omega _0(a)|^2) \\\\ \\le C\\sigma ^{-1} t^{-2(1-\\mu /2)}\\log (t)^2 |\\omega _0(a)|^2,$ where we used Proposition REF and that, by equation (REF ), $|\\beta (a,t)|\\lesssim \\int ^t \\tau ^{-1}\\Vert \\beta (\\tau )\\Vert _{L^2} d\\tau \\lesssim \\log (t) |\\omega _0(a)|.$ For later reference, we note that we have thus also proven the following proposition.",
"Proposition 4.3 Suppose that $g,h,k$ satisfy the assumptions of the second part of Theorem REF .",
"Then, for any $\\omega _0 \\in L^2$ , the solution $W$ of (REF ) satisfies $\\Vert W(t)\\Vert _{H^1} + \\Vert \\partial _y^2W(t) - \\beta _{II}(t)\\Vert _{L^2} \\lesssim \\Vert \\omega _0\\Vert _{H^2},$ where $\\beta _{II}$ is given by Lemma REF .",
"This result combines Theorems REF and REF and Propositions REF and REF .",
"Weighted stability of $\\beta _{II}$ In order to complete the proof of Theorem REF , it only remains to study the stability of $\\partial _t\\beta _{II} - \\frac{ih}{k}L_{t}\\beta _{II} + \\frac{h}{k^2t} \\langle \\beta _{II}, e^{ikty}u \\rangle e^{ikty} u &= \\frac{ih}{k}\\omega _{0}(a) e^{ikty}u, \\\\\\beta _{II}|_{t=0}&=0.$ While it would be possible to study this equation directly, we instead build on our previous analysis of $\\partial _t- \\frac{ih}{k}L_{t}$ and introduce an additional boundary layer $\\nu $ (c.f.",
"Theorem REF ) solving $(\\partial _t- \\frac{ih}{k}L_{t})\\nu &= \\frac{h}{k}\\omega _{0}(a)e^{ikty}, \\\\\\nu |_{t=0} &=0,$ and also define $\\beta _{V}=\\beta _{II}-\\nu $ .",
"Then $\\beta _{V}$ solves $\\begin{split}\\partial _t\\beta _{V} - \\frac{ih}{k}L_{t}\\beta _{V} + \\frac{\\langle \\beta _{V}, e^{ikty}\\tilde{u} \\rangle }{ikt} e^{ikty}u &= \\frac{\\langle \\nu , e^{ikty}\\tilde{u} \\rangle }{ikt} e^{ikty}u.",
"\\\\\\beta _{V}|_{t=0}&=0.\\end{split}$ Remark 5 Instead of $\\nu $ one might attempt to choose the explicit function $\\int ^t \\frac{ih}{k}\\omega _0(a)e^{ik\\tau y} d\\tau = \\frac{ih}{k}\\omega _0(a)\\frac{e^{ikty}-1}{iky}=:\\chi .$ However, we note that part of this function oscillates like $e^{ikty}$ and that $L_t\\chi = e^{ikty}L_0 \\frac{h}{k^2y}\\omega _0(a) + L_t \\frac{h}{k^2y},$ where $L_0 \\frac{h}{k^2y}\\omega _0(a)$ is independent of $t$ .",
"Hence, even for a constant function $u$ $\\langle u, L_t \\chi \\rangle $ would not decay or oscillate rapidly enough to be an integrable perturbation.",
"As the main result of this section we establish the following proposition, which concludes the proof of Theorem REF .",
"Proposition 4.4 Suppose the assumptions of Theorem REF are satisfied.",
"Then the functions $\\beta _V$ and $\\nu $ satisfy $\\Vert \\beta _{V}(t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}(a)|, \\\\\\Vert (y-a)(y-b)\\nu (t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}(a)|.$ As the evolution of $\\beta _{V}$ depends on $\\nu $ via $\\langle \\nu , e^{ikt(y-a)}\\tilde{u} \\rangle $ and as our estimates of $\\nu $ rely on properties of the solution operator of (REF ) (and hence $W$ ), we follow a multi-step approach: Using Propositions REF and REF , we show that (REF ) grows at most like $\\sqrt{t}$ .",
"By direct computation, we show that $\\Vert (y-a)(y-b) \\nu \\Vert _{L^{2}}$ grows at most like $\\log (t)$ .",
"This yields a weaker form of Proposition REF with an estimate by $\\sqrt{t} |\\omega _{0}(a)|$ .",
"Combining this estimate with the damping result of Section , the estimate of (REF ) improves to $\\log (t)$ and we obtain a uniform bound of $\\Vert (y-a)(y-b) \\nu \\Vert _{L^{2}}$ .",
"Finally, we establish $L^{2}$ stability of $\\beta _{V}$ and thus conclude the proof of Proposition REF .",
"Lemma 4.10 Assume that the assumptions of Theorem REF are satisfied.",
"Then (REF ) satisfies $|\\langle e^{ikty} \\tilde{u} , \\nu (t)\\rangle | \\lesssim \\sqrt{t}|\\omega _{0}(a)|$ as $t \\rightarrow \\infty $ .",
"Lemma 4.11 Assume that the assumptions of Theorem REF are satisfied.",
"Then $\\nu (t)$ satisfies $\\Vert (y-a)(y-b)\\nu (t)\\Vert _{L^{2}} \\lesssim \\log (t)|\\omega _{0}(a)|$ as $t\\rightarrow \\infty $ .",
"Lemma 4.12 Assume that the assumptions of Theorem REF are satisfied.",
"Then, as $t \\rightarrow \\infty $ , $\\beta _{V}$ and $\\nu $ satisfy $\\Vert \\beta _{V}(t)\\Vert _{L^{2}} &\\lesssim \\sqrt{t}|\\omega _{0}(a)|, \\\\\\Vert (y-a)(y-b)\\nu (t)\\Vert _{L^{2}} &\\lesssim \\log (t)|\\omega _{0}(a)|.$ In particular, we conclude that the solution opertor $S(t,0): H^{2}(dy) &\\rightarrow H^{2}\\left( (y-a)(y-b) dy\\right), \\\\\\omega _0 &\\mapsto W(t),$ satisfies $||| S(t,0)||| \\lesssim \\sqrt{t}.$ Lemma 4.13 Assume that the assumptions of Theorem REF are satisfied.",
"Then $\\nu $ satisfies $\\Vert (y-a)(y-b)\\nu (t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}(a)|$ as $t\\rightarrow \\infty $ .",
"Lemma 4.14 Assume that the assumptions of Proposition REF are satisfied.",
"Then $\\beta _{V}$ satisfies $\\Vert \\beta _{V}(t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}(a)|$ as $t\\rightarrow \\infty $ .",
"In our proof of Lemmata REF to REF , we rely on more detailed, (semi-explicit) characterization of $\\nu (t)$ via Duhamel's formula, which is established in Appendix .",
"We directly compute $\\langle e^{ikty}\\tilde{u}, \\int _{0}^{t} e^{ikty}S(t,\\tau ) e^{ik\\tau y} u d\\tau \\rangle \\\\= \\langle \\tilde{u}, \\int _{0}^{t} e^{ik(t-\\tau ) y}S(t-\\tau ,0) u d\\tau \\rangle .$ Next, we integrate $e^{ik(t-\\tau ) y} = \\partial _{\\tau } \\frac{e^{ik(t-\\tau )y}-1}{iky}$ by parts in $\\tau $ .",
"Here, we obtain a boundary term $\\langle \\tilde{u},\\frac{e^{ikty}-1}{iky} S(t,0) u \\rangle $ and an integral term $\\langle \\tilde{u},\\int _{0}^{t}\\frac{e^{ik(t-\\tau ) y}-1}{iky} \\partial _{\\tau } S(t-\\tau ,0) u d\\tau \\rangle .$ For (REF ) we apply Hölder's inequality and control by $\\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert \\frac{e^{ikty}-1}{iky}\\Vert _{L^{1}_{y}} \\Vert S(t,0) u\\Vert _{L^{\\infty }} \\lesssim \\log (t) \\Vert u\\Vert _{H^{1}}.$ In the integral term we use the damping estimate, Proposition REF , to control by $\\int _{0}^{t} \\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert \\frac{e^{ik(t-\\tau )y}-1}{iky}\\Vert _{L^{2}} \\Vert \\partial _{\\tau } S(t-\\tau ,0) u\\Vert _{L^{2}} d\\tau \\\\\\lesssim \\int _{0}^{t}\\sqrt{|t-\\tau |} <t-\\tau >^{-1} \\Vert S(t-\\tau ,0) u\\Vert _{H^{1}} d\\tau \\\\\\lesssim \\int <t-\\tau >^{-1/2} d\\tau \\lesssim \\sqrt{t}.$ Using Lemmata REF and REF , we obtain that $\\nu (t)= \\int _{0}^{t} e^{ikt(t-\\tau )(y-a)}S(t-\\tau ,0)u d\\tau $ Multiplying with $(y-a)$ , we use that $-\\partial _{\\tau } \\frac{e^{ik(t-\\tau )(y-a)}-1}{ik}= (y-a)e^{ik(t-\\tau )(y-a)}$ and hence control $\\Vert (y-a)\\nu (t)\\Vert _{L^{2}} & \\le \\Vert \\frac{e^{ik(t-\\tau )(y-a)}-1}{ik} S(t-\\tau ,0)u|_{\\tau =0}^{t}\\Vert _{L^{2}} \\\\&\\quad + \\int _{0}^{t} \\Vert \\frac{e^{ik(t-\\tau )(y-a)}-1}{ik} \\partial _{\\tau }S(t-\\tau ,0)u \\Vert _{L^{2}} d\\tau \\\\&\\lesssim |k|^{-1} \\Vert u\\Vert _{L^{2}} + |k|^{-2} \\Vert h\\Vert _{L^{\\infty }} \\int _{0}^{t} \\Vert L_{t-\\tau } S(t-\\tau ,0)u\\Vert _{L^{2}} \\\\& \\lesssim |k|^{-1} \\Vert u\\Vert _{L^{2}} + |k|^{-2} \\Vert h\\Vert _{L^{\\infty }} \\int _{0}^{t} <t-\\tau >^{-1} \\Vert S(t-\\tau ,0)u\\Vert _{H^{1}} d\\tau \\\\& \\lesssim |k|^{-1} \\Vert u\\Vert _{L^{2}} + |k|^{-2} \\Vert h\\Vert _{L^{\\infty }} \\Vert u\\Vert _{H^{1}}\\log (t),$ where we used Proposition REF and Theorem REF .",
"Using our Lyapunov functional approach on $\\beta _{V}$ , we need to estimate $\\langle A_{1}\\beta _{V}, e^{ikty}u\\rangle \\frac{\\langle \\nu , e^{ikty}\\tilde{u} \\rangle }{ikt}.$ By Lemma REF , we control $|\\frac{\\langle \\nu , e^{ikty}\\tilde{u} \\rangle }{ikt}|\\lesssim t^{-1/2},$ and using Lemma REF , we estimate.",
"$|\\langle A_{1}\\beta _{V}, e^{ikty}u\\rangle | \\le C_{\\lambda } \\Vert (\\beta _{V})<n-kt>^{-\\lambda }\\Vert _{l^{2}},$ where $0<\\lambda <\\frac{1}{2}$ .",
"Hence, using Young's inequality, we can control (REF ) by $\\epsilon \\Vert (\\beta _{V})_n<n-kt>^{-\\lambda }\\Vert _{l^{2}}t^{-1/2} + C(\\epsilon ,\\lambda ) t^{-1/2}.$ Here, for $\\epsilon $ sufficiently small, the first term can be absorbed by $\\langle \\beta _{V}, \\dot{A}_{1} \\beta _{V} \\rangle $ and in summary we obtain $\\partial _t\\langle \\beta _{V}, A_{1} \\beta _{V} \\rangle \\le C(\\epsilon ,\\lambda ) t^{-1/2}.$ Integrating this inequality then yields the result.",
"We remark that already in step 3 we could obtain a better growth bound by optimizing in $\\lambda $ and the splitting of $t^{-1/2}$ in Young's inequality.",
"However, since $t^{-1/2}\\notin L^{2}$ this would only yield a non-uniform bound and our multi-step proof only requires a better than linear growth bound.",
"Following the proof of Lemma REF it suffices to show that $\\int _{0}^{t}\\Vert \\partial _{\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} d\\tau \\lesssim 1,$ uniformly in $t$ .",
"Using Hölder's inequality and Proposition REF , we estimate $\\Vert \\partial _{\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} &= \\Vert \\frac{ih}{k}L_{t-\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} \\le \\Vert h\\Vert _{L^{\\infty }}|k|^{-1} \\Vert L_{t-\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} \\\\&\\le \\Vert h\\Vert _{L^{\\infty }}|k|^{-1}<t-\\tau >^{-2}(\\Vert (y-a)(y-b)\\partial _{y}^{2}S(t-\\tau ,0)u\\Vert _{L^{2}} \\\\ & \\quad +\\Vert S(t-\\tau ,0)u\\Vert _{H^{1}}) \\le \\Vert h\\Vert _{L^{\\infty }}|k|^{-1}<t-\\tau >^{-2} |||S(t-\\tau ,0)||| \\Vert u\\Vert _{H^{2}},$ the operator norm of $S(t-\\tau ,0)$ is given by Lemma REF .",
"Hence, we obtain that $\\Vert \\partial _{\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} \\lesssim \\Vert h\\Vert _{L^{\\infty }}|k|^{-1}\\Vert u\\Vert _{H^{2}}<t-\\tau >^{-2} \\sqrt{t-\\tau },$ which is integrable in $\\tau $ and thus concludes the proof.",
"We claim that $|\\langle e^{ikty} \\tilde{u} , \\nu (t)\\rangle | \\lesssim \\log (t)|\\omega _{0}(a)|.$ Following the proof of Lemma REF , this implies that $|\\langle A_{1}\\beta _{V}, e^{ikty}u\\rangle \\frac{\\langle \\nu , e^{ikty}\\tilde{u} \\rangle }{ikt} | \\le C \\Vert (\\beta _{V})<n-kt>^{-\\lambda }\\Vert _{l^{2}} \\frac{\\log (t)}{t} \\\\\\le \\epsilon \\Vert (\\beta _{V})<n-kt>^{-\\lambda }\\Vert _{l^{2}}^{2} t^{-2\\mu } + C(\\epsilon ) \\log (t)^{2}t^{-2(1-\\mu )},$ where $C(\\epsilon )$ is given by Young's inequality and $0<\\mu <1$ is chosen such that $2\\lambda +2\\mu >1$ and $2(1-\\mu )>1$ .",
"Choosing $\\epsilon $ sufficiently small, we thus obtain $\\partial _t\\langle \\beta _{V}, A_{1}\\beta _{V} \\rangle &\\le \\langle \\beta _{V}, \\dot{A}_{1}\\beta _{V} \\rangle + \\epsilon \\Vert (\\beta _{V})<n-kt>^{-\\lambda }\\Vert _{l^{2}}^{2} t^{-2\\mu } + C(\\epsilon ) \\log (t)^{2}t^{-2(1-\\mu )} \\\\ &\\le C(\\epsilon ) \\log (t)^{2}t^{-2(1-\\mu )} \\in L^{1}_{t}([1,\\infty )).$ Integrating this inequality then yields the desired result.",
"It remains to prove the claim (REF ).",
"Here, we estimate $|\\langle e^{ikty} \\tilde{u}, \\nu (t) \\rangle | \\lesssim \\log (t) \\Vert u\\Vert _{H^{1}} + \\int _{0}^{t}\\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert \\frac{e^{ik(t-\\tau )y}-1}{iky}\\Vert _{L^{2}} \\Vert \\partial _{\\tau } S(t-\\tau ,0) u\\Vert _{L^{2}} d\\tau .$ Using Lemma REF and Proposition REF we control $\\Vert \\partial _{\\tau } S(t-\\tau ,0) u\\Vert _{L^{2}} \\le <t-\\tau >^{-2}|||S(t-\\tau ,0)||| \\Vert u\\Vert _{H^{2}} \\lesssim <t-\\tau >^{-3/2} \\Vert u\\Vert _{H^{2}},$ and we directly compute that $\\Vert \\frac{e^{ik(t-\\tau )y}-1}{iky}\\Vert _{L^{2}} \\lesssim \\sqrt{t-\\tau }.$ Hence, we control $\\int _{0}^{t}\\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert \\frac{e^{ik(t-\\tau )y}-1}{iky}\\Vert _{L^{2}} \\Vert \\partial _{\\tau } S(t-\\tau ,0) u\\Vert _{L^{2}} d\\tau \\\\\\lesssim \\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert u\\Vert _{H^{2}} \\int _{0}^{t} <t-\\tau >^{-1} d\\tau \\le \\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert u\\Vert _{H^{2}} \\log (t),$ which proves the claim.",
"appendix Auxiliary functions and boundary evaluations In this section we introduce several auxiliary functions, which can be used to compute boundary evaluations of of derivatives of $L_tW$ and related quantities.",
"Lemma 1.1 Let $u_{1},u_{2}$ be solutions of $\\begin{split}\\mathcal {E}_{t}u&=0, \\\\z &\\in [a,b],\\end{split}$ with boundary values $\\begin{pmatrix}u_{1}(a) & u_{2}(a) \\\\ u_{1}(b) & u_{2}(b)\\end{pmatrix}=\\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}.$ Let further $\\tilde{u}_{1},\\tilde{u}_{2}$ be solutions to the adjoint problem $\\begin{split}\\mathcal {E}_{t}^{*}\\tilde{u}:= \\left(\\left((\\frac{\\partial _{y}}{k}-it)g(y)\\right)^{2}- (\\frac{\\partial _{y}}{k}-it)\\frac{g(y)}{kr(y)}-\\frac{1}{r^{2}(y)}\\right)\\tilde{u}&=0, \\\\y &\\in [a,b],\\end{split}$ with boundary values $\\begin{pmatrix}\\tilde{u}_{1}(a) & \\tilde{u}_{2}(a) \\\\\\tilde{u}_{1}(b) & \\tilde{u}_{2}(b)\\end{pmatrix}=\\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}.$ Then $u_{1}, u_{2}, \\tilde{u}_{1}, \\tilde{u}_{2}$ satisfy $\\begin{split}u_{1}(t,r,k)&=e^{ikt(y-a)}u_{1}(0,r,k), \\\\u_{2}(t,r,k)&=e^{ikt(y-b)}u_{2}(0,r,k), \\\\\\tilde{u}_{1}(t,r,k)&=e^{ikt(y-a)}\\tilde{u}_{1}(0,r,k), \\\\\\tilde{u}_{2}(t,r,k)&=e^{ikt(y-b)}\\tilde{u}_{2}(0,r,k).\\end{split}$ We note that the operators in equations (REF ) and (REF ) are obtained by conjugating by $e^{iktz}$ and are complex linear.",
"The result hence follows by noting that multiplication by $e^{ikt(y-a)}$ or $e^{ikt(y-b)}$ is compatible with the boundary conditions (REF ) and (REF ).",
"Lemma 1.2 Let $W$ be a given function and let $\\Phi $ be a solution of $\\begin{split}\\mathcal {E}_{t}\\Phi &=W, \\\\\\Phi |_{y=a,b}&=0,\\end{split}$ and let $u_{1},u_{2},\\tilde{u}_{1}, \\tilde{u}_{2}$ be as in Lemma REF .",
"Define $\\begin{split}H^{(1)}&= \\frac{k^{2}}{g^{2}(a)}\\langle \\Phi , \\tilde{u}_{1} \\rangle _{L^{2}}u_{1} +\\frac{k^{2}}{g^{2}(b)}\\langle \\Phi , \\tilde{u}_{2} \\rangle _{L^{2}}u_{2}, \\\\\\Phi ^{(1)}&= \\partial _{r}\\Phi - H^{(1)}\\end{split}$ Then $\\Phi $ satisfies $\\begin{split}\\langle W, \\tilde{u}_{1} \\rangle _{L^{2}}&= \\frac{g^{2}(a)}{k^{2}}\\partial _{r}\\Phi (t,k,a) \\\\\\langle W, \\tilde{u}_{2} \\rangle _{L^{2}}&=\\frac{g^{2}(b)}{k^{2}}\\partial _{r}\\Phi (t,k,b),\\end{split}$ and $\\Phi ^{(1)}$ solves $\\begin{split}\\mathcal {E}_{t}\\Phi ^{(1)} &=\\partial _{y}W + \\left[\\mathcal {E}_{t}, \\partial _{y} \\right]\\Phi , \\\\\\Phi ^{(1)}|_{y=a,b}&=0.\\end{split}$ The function $H^{(1)}$ is a solution of (REF ) and is called the (first) homogeneous correction.",
"Testing the equation (REF ) with the homogeneous solutions of Lemma REF , the results follow by integration by parts and direct calculations.",
"Lemma 1.3 Let $\\Phi ,W$ as in Lemma REF and let $u_{1},u_{2},\\tilde{u}_{1}, \\tilde{u}_{2}$ as in Lemma REF .",
"Then $\\Phi $ satisfies $\\begin{split}\\frac{g^{2}(a)}{k^{2}}\\partial _{r}^{2}\\Phi (t,k,a)&= -\\frac{g(a)g^{\\prime }(a)}{k^{2}} \\partial _{r}\\Phi (t,k,a)- \\frac{g(y)}{k^{2}r(y)}\\partial _{y}\\Phi (t,k,a) + W(t,k,a), \\\\\\frac{g^{2}(a)}{k^{2}}\\partial _{r}^{2}\\Phi (t,k,b)&= -\\frac{g(b)g^{\\prime }(b)}{k^{2}} \\partial _{r}\\Phi (t,k,b)- \\frac{g(y)}{k^{2}r(y)}\\partial _{y}\\Phi (t,k,b) + W(t,k,b).\\end{split}$ Define $\\begin{split}H^{(2)}&= \\partial _{y}^{2}\\Phi (t,k,a)u_{1} + \\partial _{y}^{2}\\Phi (t,k,b)u_{2}, \\\\\\Phi ^{(2)}&= \\partial _{y}^{2}\\Phi - H^{(2)},\\end{split}$ then $\\Phi ^{(2)}$ satisfies $\\begin{split}((g(y)(\\frac{\\partial _{y}}{k}-it))^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)})\\Phi ^{(2)}\\\\=\\partial _{y}^{2}W + \\left[((g(y)(\\frac{\\partial _{y}}{k}-it))^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)}), \\partial _{y}^{2} \\right]\\Phi , \\\\\\Phi ^{(2)}|_{y=a,b}=0.\\end{split}$ The function $H^{(2)}$ is a solution of (REF ) and is called the (second) homogeneous correction.",
"Direct computation.",
"Duhamel's formula and shearing Lemma 2.1 (Time dependent Duhamel) Let $(L(t))_{t \\in \\mathbb {R}}$ be a given family of linear operators and denote by $S(t,t^{\\prime })$ the solution operator of $(\\partial _t + \\frac{ih}{k}L_t) a=0,$ mapping a prescribed $a(t^{\\prime })$ to $a(t)$ .",
"Then for any given function $F$ the unique solution of $(\\partial _t + \\frac{ih}{k}L_t) u&=F,\\\\u(0)&=u_0,$ is given by $u(t)= S(t,0)u_0 + \\int _{0}^{t} S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime }.$ Since $S(0,0)=Id$ , we observe that the such defined $u(t)$ satisfies $u(0)=u_0$ .",
"It remains to show that $u$ satisfies the equation.",
"We directly compute $(\\partial _t+ \\frac{ih}{k}L_t) u(t)&= (\\partial _t+\\frac{ih}{k}L_t)S(t,0)u_0 + \\int _{0}^{t} (\\partial _t +\\frac{ih}{k}L_t) S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime } + S(t,t)F(t)\\\\&= 0 + \\int _{0}^t 0 S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime } +\\text{Id} F(t) = F(t).$ Here we used that for any $t^{\\prime }$ $(\\partial _t +\\frac{ih}{k}L_t) S(t,t^{\\prime })=0.$ We stress that $(\\partial _t +\\frac{ih}{k}L_{\\tilde{t}}) S(t,t^{\\prime })$ does not vanish in general for any $\\tilde{t} \\ne t$ .",
"Applying Lemma REF to (REF ), we obtain that $\\nu (t)= \\omega _{0}(a) \\int _{0}^{t}S(t,\\tau ) e^{ik\\tau y}u d\\tau ,$ where $S(t,\\tau )$ is the solution operator corresponding to (REF ).",
"Since $L_{t}$ was defined by a conjugation of $L_{0}$ with $e^{ikty}$ , we can also conjugate $S(t,\\tau )$ .",
"Lemma 2.2 Let $\\sigma >0$ , then for any $0\\le s \\le \\tau \\le t$ the solution operator $S$ satisfies $S(t,\\tau )e^{ik \\sigma y} f = e^{ik \\sigma y} S(t-\\sigma ,\\tau -\\sigma ) f$ for any $f \\in L^{2}$ .",
"We note that for any $t$ $e^{-ik\\sigma y} \\mathcal {E}_{t}e^{ik\\sigma y} = \\mathcal {E}_{t-\\sigma }$ and that also $e^{ik\\sigma y} \\langle e^{ik \\sigma y}f, e^{ikty}u \\rangle e^{ikty}u= \\langle f, e^{ik(t-\\sigma )y}u \\rangle e^{ik(t-\\sigma )y}u.$ Hence, conjugating the equation by $e^{ikt\\sigma y}$ is equivalent to a shift in time, which yields the desired result."
],
[
"Scattering formulation and $L^2$ stability",
"As established in Section , the core problem of (linear) inviscid damping consists of establishing a control of higher Sobolev norms of the vorticity moving with the flow: $W(t,\\theta ,r):= f(t,\\theta -tU(r),r).$ Here, we largely follow a similar approach as in the plane setting considered in [12].",
"As key improvements we obtain a less restrictive smallness condition and develop a splitting of $\\partial _{r}W$ into a well-behaved and more regular part $\\Gamma $ and a (relatively) explicit boundary layer $\\beta $ .",
"This then allows us to deduce damping with optimal decay rates and a detailed stability in suitable weighted Sobolev spaces, such as the ones considered in Proposition REF .",
"In order simplify our analysis, in this section we introduce several changes of variables as well as useful auxiliary functions."
],
[
"Scattering formulation",
"Expressing the linearized Euler equations $\\begin{split}\\partial _tf + U(r) \\partial _{\\theta } f &= b(r) \\partial _{\\theta } \\phi , \\\\(\\partial _{r}^{2}+\\frac{1}{r}\\partial _{r}+\\frac{1}{r^{2}} \\partial _{\\theta }^{2})\\phi &= f, \\\\\\partial _{\\theta }\\phi |_{r=r_{1},r_{2})}&=0, \\\\(t,\\theta ,r) & \\in \\mathbb {R}\\times [r_{1},r_{2}],\\end{split}$ in terms of the scattered quantities $\\begin{split}F(t,\\theta ,r)&= f(t,\\theta -tU(r),r), \\\\\\Upsilon (t,\\theta ,r)&= \\phi (t,\\theta -tU(r),r),\\end{split}$ we obtain $\\begin{split}\\partial _tF &= b(r) \\partial _{\\theta } \\Upsilon , \\\\((\\partial _{r}-tU^{\\prime }(r)\\partial _{\\theta })^{2}+\\frac{1}{r}(\\partial _{r}-tU^{\\prime }(r)\\partial _{\\theta })+\\frac{1}{r^{2}} \\partial _{\\theta }^{2})\\Upsilon &= F, \\\\\\partial _{\\theta }\\Upsilon |_{r=r_{1},r2_{2})}&=0, \\\\(t,\\theta ,r) & \\in \\mathbb {R}\\times [r_{1},r_{2}],\\end{split}$ As none of the coefficient functions depend on $\\theta $ , our system decouples with respect to Fourier modes $k$ in $\\theta $ .",
"$\\begin{split}\\partial _t\\hat{F} &= b(r) ik \\hat{\\Upsilon }, \\\\((\\partial _{r}-iktU^{\\prime }(r))^{2}+\\frac{1}{r}(\\partial _{r}-iktU^{\\prime }(r))-\\frac{k^{2}}{r^{2}})\\hat{\\Upsilon }&= \\hat{F}, \\\\ik\\hat{\\Upsilon }|_{r=r_{1},r2_{2})}&=0, \\\\(t,k,r) & \\in \\mathbb {R}\\times 2\\pi \\mathbb {Z}\\times [r_{1},r_{2}],\\end{split}$ We in particular note that the mode $k=0$ , which corresponds to a purely circular flow, is conserved in time.",
"Using the linearity of our equations, in the following we hence without loss of regularity consider $k \\in 2\\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace )$ as a given parameter.",
"In view of the structure of the differential equation for $\\Phi $ , it is further advantageous to use that $U$ , as a strictly monotone function, is invertible.",
"Introducing a change of coordinates $r \\mapsto y=U(r).$ as well a denoting $\\begin{split}h(y) &= \\frac{(\\omega _{0})^{\\prime }}{r}|_{r=U^{-1}(y)}, \\\\g(y)&= U^{\\prime }(r)|_{r=U^{-1}(y)}, \\\\W(t,y,k) &= \\hat{F}(t,r,k)|_{r=U^{-1}(y)},\\\\\\Phi (t,y,k) &= \\frac{1}{k^{2}}\\hat{\\Upsilon }(t,r,k)|_{r=U^{-1}(y)},\\end{split}$ our system is then given by the following definition.",
"Definition 3.1 (Euler's equations in scattering formulation) Let $U:[r_1,r_2]\\rightarrow \\mathbb {R}$ be strictly monotone and let $h(y)=b|_{r=U^{-1}(y)}$ and $g=U^{\\prime }(U^{-1}(y))$ .",
"Then Euler's equations in scattering formulation are given by $\\begin{split}\\partial _tW = \\frac{ih(y)}{k}\\Phi &=: \\frac{ih(y)}{k}L_{t}W, \\\\\\mathcal {E}_{t}\\Phi := \\left(\\left(g(y)(\\frac{\\partial _{y}}{k}-it)\\right)^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)}\\right)\\Phi &=W, \\\\\\Phi |_{y=a,b}&=0, \\\\(t,k,y) &\\in \\mathbb {R}\\times 2\\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace ) \\times [a,b],\\end{split}$ where $a=\\min (U^{-1}(r_{1}), U^{-1}(r_{2}))$ , $b=\\max (U^{-1}(r_{1}),U^{-1}(r_{2}))$ and $k \\in 2 \\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace )$ .",
"Remark 3 Our methods do not rely on the specific form of $h$ or $g$ in terms of $U$ .",
"For example, we can allow for $h$ to be an arbitrary complex valued $W^{1,\\infty }$ function.",
"Here the notation $L_{t}W$ is used to stress that the mapping $W \\mapsto \\Phi $ is a linear operator in $W$ .",
"As this system decouples with respect to $k$ , we will often treat $k\\ne 0$ as a fixed given external parameter and with slight abuse of notation use $W(t,y)$ to refer to $W(t,k,y)$ for the given $k$ ."
],
[
"Shifted elliptic regularity and modified spaces",
"We note that in this scattering formulation $\\mathcal {E}_{t}$ is obtained from an elliptic operator by conjugation with $e^{ikty}$ and hence define suitable replacements of the $H^1$ and $H^{-1}$ energies: Definition 3.2 ( $\\tilde{H}^{1}_{t}$ and $\\tilde{H}^{-1}_{t}$ energies) Let $u \\in H^{1}([a,b])$ and let $k \\in 2\\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace )$ be given, then for every $t \\in \\mathbb {R}$ , we define $\\Vert u\\Vert _{\\tilde{H}^{1}_t}^2:= \\Vert e^{ikty}u\\Vert _{H^1}^2= \\Vert u\\Vert _{L^2}^2 + \\Vert (\\frac{\\partial _y}{k}-it)u\\Vert _{L^2}^2.$ Furthermore, we define a dual quantity in the following way.",
"Let $v \\in L^2$ and let $\\Psi [v]$ be the unique solution of $(-1+(\\frac{\\partial _y}{k}-it)^2) \\Psi [v]&=v, \\\\\\Psi [v]|_{y=a,b}&=0.$ Then we define $\\Vert v\\Vert _{\\tilde{H}^{-1}_t}:= \\Vert \\Psi [v]\\Vert _{\\tilde{H}^{1}_t}.$ Lemma 3.1 (Duality) Let $W \\in L^{2}$ and let $k \\in (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace )$ be given.",
"Then $\\Vert W\\Vert _{\\tilde{H}^{-1}_{t}}=\\sup \\lbrace \\langle W, \\alpha \\rangle _{L^{2}}: \\alpha \\in H^{1}_{0}, \\Vert \\alpha \\Vert _{\\tilde{H}^{1}}\\le 1 \\rbrace ,$ i.e.",
"$\\tilde{H}^{-1}_{t}$ is dual to $\\tilde{H}^{1}_{t}$ .",
"Since multiplication by $e^{ikty}$ is a unitary operation and preserves zero Dirichlet boundary values and $\\Psi _t[v]= e^{-ikty}\\Psi _0[e^{ikty}v],$ it suffices to consider the case $t=0$ , which is given by the usual $H^1$ and $H^{-1}$ norms (where we use $\\frac{\\partial _y}{k}$ instead of $\\partial _y$ ).",
"The result then follows using integration by parts: $-\\langle W, \\alpha \\rangle = \\langle (1-\\frac{\\partial _{y}}{k}^{2}) \\Psi [W], \\alpha \\rangle \\\\= \\langle \\Psi [W], \\alpha \\rangle + \\langle \\frac{\\partial _{y}}{k}\\Psi [W], \\frac{\\partial _{y}}{k}\\alpha \\rangle \\le \\Vert W\\Vert _{H^{-1}}\\Vert \\alpha \\Vert _{H^{1}},$ with equality if $\\alpha = -\\frac{1}{\\Vert \\Psi [W]\\Vert _{H^1}}\\Psi [W]$ .",
"Taking the supremum over all $\\alpha $ with $\\Vert \\alpha \\Vert _{H^{1}}$ we hence obtain the result."
],
[
"Heuristics and obstructions",
"On a heuristic level, in order to establish stability in $L^2$ , we use that $\\frac{d}{dt}\\Vert W(t)\\Vert _{L^2}^2 =2\\Re \\langle W, \\frac{ih}{k}L_t W\\rangle \\lesssim C(h,k)\\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2,$ and that for fixed functions $u \\in L^2$ , which do not depend on time, $\\int _{0}^{\\infty } \\Vert u\\Vert _{\\tilde{H}^{-1}_t}^2 dt \\le C \\Vert u\\Vert _{L^2}^2,$ as can be computed from a Fourier characterization.",
"Hence, it seems reasonable to expect that solutions $W(t)$ of (REF ) satisfy an estimate of the form $\\Vert W(t)\\Vert _{L^2} \\le \\exp (C\\Vert h\\Vert _{L^\\infty }|k|^{-1}) \\Vert f_0\\Vert _{L^2},$ also for complex valued $h$ , which is the case for some explicit model problems (c.f. [11]).",
"However, we stress that this heuristic is very rough and does not account for several obstructions: We note that integrability in time in general fails for time-dependent $u \\in L^{\\infty }_t(L^2)$ .",
"For example, choosing $u(t,k,y)=e^{ikty}u_0(k,y),$ we observe that $\\int _{0}^{T} \\Vert u\\Vert _{\\tilde{H}^{-1}_t}^2 dt = T \\Vert u_0\\Vert _{H^{-1}}^2,$ which diverges as $T \\rightarrow \\infty $ despite $\\Vert u(t,y)\\Vert _{L^2}= \\Vert u_0\\Vert _{L^2}$ being uniformly bounded.",
"Since the first estimate does not account for antisymmetric operators in $\\frac{d}{dt}W$ it is not sufficient to establish $L^2$ stability.",
"For example, this estimate is satisfied by solutions $u(t,y)$ to $\\partial _tu +iy u &= \\Phi , \\\\(-1+(\\partial _y-it)^2)\\Phi &=u, \\\\\\Phi |_{y=a,b}&=0.$ Considering $v(t,y)=e^{ity}u(t,y)$ , we observe that $v$ solves $\\partial _tv &= \\phi , \\\\(1-\\partial _{y}^2)\\phi &= v, \\\\\\phi |_{y=a,b}&=0.$ Hence, choosing $u|_{t=0}$ to be an eigenfunction of $(1-\\partial _{y}^2)$ , we obtain an exponentially growing solution."
],
[
"$L^2$ stability",
"As the main result of this section, we adapt the Lyapunov functional approach of [13] to this circular setting and prove stability of (REF ).",
"In the following we formulate the main ingredients of our approach as a series of Lemmata, which are then used to prove $L^2$ stability in Theorem REF .",
"Subsequently, we elaborate on the theorem's statement and assumptions in comparison to existing results and prove the lemmata.",
"Here, the lemmata are formulated in a general way in order to facilitate their use for higher regularity estimates in later sections.",
"Lemma 3.2 Let $L_{t}$ be given by (REF ) and let $\\kappa \\in W^{1,\\infty }$ .",
"Then, for any $u,v \\in L^{2}$ $| \\langle u, \\kappa L_{t}v \\rangle | \\le (\\Vert \\kappa \\Vert _{L^\\infty } + \\frac{1}{|k|} \\Vert \\partial _y\\kappa \\Vert _{L^\\infty }) \\Vert u\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert L_{t} v\\Vert _{\\tilde{H}^{1}_t}$ Lemma 3.3 Let $L_{t}$ be as in (REF ).",
"Then there exists a constant $C=C(a,b,g)$ such that for any $u \\in L^{2}$ and any $t\\ge 0$ $\\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}} \\le C \\Vert u\\Vert _{\\tilde{H}^{-1}_{t}}.$ Lemma 3.4 ([12]) Let $u \\in L^2([a,b])$ and let $\\sum _{n \\in (b-a)\\mathbb {N}} u_{n} \\sin (ny)$ be its series expansion.",
"Define the symmetric, positive definite, non-increasing operator $A$ by $\\langle u, A u \\rangle := \\sum _{n} \\exp (\\arctan (\\frac{n}{k}-t)) |u_{n}|^{2}.$ Then $A$ is symmetric, positive definite, non-increasing, $C^{1}$ in time and comparable to the identity, i.e.",
"$e^{-\\pi } \\Vert u\\Vert _{L^{2}} \\le \\langle u, A u \\rangle \\le e^\\pi \\Vert u\\Vert _{L^{2}},$ for all $u \\in L^{2}$ .",
"Furthermore, there exists a constant $e^{-\\pi }\\le C_{2} \\le e^{\\pi }$ and $\\delta >0$ such that $\\Vert u\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert Au\\Vert _{\\tilde{H}^{-1}_{t}} \\le -C_{2} \\langle u, \\dot{A} u \\rangle $ Using the preceding lemmata, we can establish $L^{2}$ stability.",
"Theorem 3.1 ($L^{2}$ stability) Let $A$ and $C_{2}$ be given by Lemma REF and let $C$ be as in Lemma REF .",
"Further suppose that there exists $\\delta >0$ such that $|k|^{-1}(\\Vert h\\Vert _{L^{\\infty }}+|k|^{-1}\\Vert \\partial _y h\\Vert _{L^\\infty }) \\le \\frac{1}{C}(\\frac{1}{C_2}-\\delta ).$ Then for any solution $W$ to the Euler equations in scattering formulation (REF ) the functional $I(t):= \\langle W, A(t) W \\rangle .$ is non-increasing and satisfies $\\partial _tI(t) \\le -\\delta \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}\\Vert A(t)W(t)\\Vert _{\\tilde{H}^{-1}_t} \\le 0.$ In particular, this implies $e^{-\\pi }\\Vert W(t)\\Vert _{L^{2}}^{2} \\le I(t) \\le I(0) \\le e^\\pi \\Vert \\omega _{0}\\Vert _{L^{2}}^{2}.$ Using Lemma REF , we estimate $\\partial _tI(t) \\le \\langle W, \\dot{A} W \\rangle + 2 |k|^{-1}(\\Vert h\\Vert _{L^{\\infty }}+|k|^{-1}\\Vert \\partial _y h\\Vert _{L^\\infty }) \\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert L_tW\\Vert _{\\tilde{H}^{1}_{t}}.$ Applying Lemma REF and Young's inequality, we further control $\\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert L_tW\\Vert _{\\tilde{H}^{-1}_{t}} \\le \\frac{C}{2} \\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}} \\Vert W\\Vert _{\\tilde{H}^{-1}_{t}}.$ The result then follows by an application of Lemma REF and noting that, by our smallness assumption, $\\partial _tI(t) \\le \\langle W, \\dot{A} W \\rangle + |k|^{-1}(\\Vert h\\Vert _{L^{\\infty }}+|k|^{-1}\\Vert \\partial _y h\\Vert _{L^\\infty })C\\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert W\\Vert _{\\tilde{H}^{-1}_{t}} \\\\\\le \\langle W, \\dot{A} W \\rangle + (\\frac{1}{C_2}-\\delta )\\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}} \\Vert W\\Vert _{\\tilde{H}^{-1}_{t}} \\\\\\le -\\delta \\Vert AW\\Vert _{\\tilde{H}^{-1}_{t}}\\Vert W\\Vert _{\\tilde{H}^{-1}_{t}} \\le 0$ Let us briefly remark on this result and its assumptions: We require a smallness condition on $\\frac{ih}{k}L_t$ in order to rule out the obstacles mentioned in Section REF .",
"Since $h$ is allowed to be complex-valued, we do not rely on conserved quantities or classical stability results such as the ones of Rayleigh, Fjortoft or Arnold.",
"In the setting of a plane finite periodic channel, in [10] Wei, Zhang and Zhao use a spectral approach to establish linear stability and decay with optimal rates for monotone shear flows under the assumption that the strictly monotone shear flow $U(y)$ possesses no embedding eigenvalues.",
"In comparison, our smallness assumption is more restrictive, but extends to related problems such as stability in fractional Sobolev spaces, complex valued functions $h$ and fractional operators $L_t$ in a straightforward way.",
"In Section , we show that $\\partial _{y}^2W$ can be split into a very regular, stable part $\\Gamma $ and a boundary layer part $\\beta $ which develops a singularity at the boundary.",
"Here, $\\beta $ is determined solely by the Dirichlet boundary data of the initial datum, $\\omega _0$ , and allows for a detailed study of the stability properties of the evolution.",
"It remains to prove Lemmata REF and REF .",
"Let $u,v \\in L^2$ and let $\\Psi [u]$ be the unique solution of $(-1+(\\partial _{y}-ikt)^{2})\\Psi [u]&=u, \\\\\\Psi [u]|_{y=a,b}&=0.$ Then we directly compute $| \\langle u, \\kappa L_{t}v \\rangle | = | \\langle (1-(\\frac{\\partial _{y}}{k}-ikt)^{2})\\Psi [u], \\kappa L_{t}v \\rangle | \\\\\\le \\Vert \\Psi [u]\\Vert _{L^{2}} \\Vert \\kappa L_{t}\\Vert _{L^{2}} + \\Vert (\\frac{\\partial _{y}}{k}-it)\\Psi [u]\\Vert _{L^{2}}\\Vert (\\frac{\\partial _{y}}{k}-it) \\kappa L_{t}v\\Vert _{L^{2}} \\\\\\le \\Vert \\Psi [u]\\Vert _{\\tilde{H}^{1}_{t}}(\\Vert \\kappa \\Vert _{L^\\infty } + \\frac{1}{|k|} \\Vert \\partial _y\\kappa \\Vert _{L^\\infty }) \\Vert L_{t}v\\Vert _{\\tilde{H}^{1}_{t}}.$ Here, we used that $L_tv$ by definition satisfies zero Dirichlet boundary conditions and hence no boundary contributions appear when integrating by parts.",
"We recall that $L_{t}u$ is the solution of $\\mathcal {E}_{t}L_{t} u= \\left(\\left(g(y)(\\frac{\\partial _{y}}{k}-it)\\right)^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)}\\right)L_{t}u&=0 , \\\\L_{t}u|_{y=a,b}&=0,$ and that $g(y)$ and $\\frac{g(y)}{r(y)}$ are bounded from below (and above).",
"Hence $\\mathcal {E}_{t}$ is a shifted elliptic operator and testing by $L_{t}u$ (or $\\frac{1}{g}L_{t}u$ ) we obtain that $\\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}}^{2} \\le -C \\langle L_{t}u, u \\rangle ,$ for some $C>0$ .",
"Applying Lemma REF , we thus obtain $\\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}}^{2} &\\le C \\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}}\\Vert \\Psi [u]\\Vert _{\\tilde{H}^{1}_{t}},\\\\\\Leftrightarrow \\Vert L_{t}u\\Vert _{\\tilde{H}^{1}_{t}} &\\le C \\Vert u\\Vert _{\\tilde{H}^{-1}_{t}}.$ Having introduced the basic tools of our approach, in the following section we consider higher stability of $W$ , i.e.",
"control of $\\partial _{y}W$ .",
"Here, boundary effects qualitatively change the dynamics and necessitate a modification of the weight $A(t)$ ."
],
[
"Higher stability and boundary layers",
"In this section we show that the $L^{2}$ stability result can be extended to higher Sobolev regularity.",
"However, unlike in the setting of an infinite periodic channel, boundary effects can not be neglected and result in the formation of singularities.",
"As the main improvements over our previous work for the plane channel in [12], we provide an explicit splitting into a more regular good parts and a boundary layer exhibiting blow-up as well as an improved smallness condition.",
"This splitting then also allows to provide a more detailed description of the blow-up also in weighted Sobolev spaces.",
"For this purpose we also introduce a different method of proof.",
"Let thus $W$ be a solution to (REF ) $\\partial _tW &= \\frac{ih}{k}L_{t}W, \\\\\\mathcal {E}_{t}L_t W &=W, \\\\L_t|_{y=a,b}&=0, \\\\(t,k,y) &\\in \\mathbb {R}\\times 2\\pi (\\mathbb {Z}\\setminus \\lbrace 0\\rbrace ) \\times [a,b].$ We begin by studying $\\partial _yW$ , which satisfies $\\partial _t\\partial _yW &= \\frac{ih}{k}L_t \\partial _y W + \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW + \\frac{ih}{k}H^{(1)}, \\\\\\mathcal {E}_{t}H^{(1)}&=0 ,\\\\H^{(1)}_{y=a,b}&=\\partial _yL_tW|_{y=a,b}.$ In contrast to the $L^2$ setting (or a setting without boundary such as $\\mathbb {R}$ ) we hence obtain a correction $H^{(1)}$ due to $\\partial _y L_tW$ not satisfying zero Dirichlet boundary conditions.",
"As a main result of Appendix , we study the boundary behavior of $\\partial _yL_t$ (also confer [12]) and obtain the following description of $H^{(1)}$ : Lemma 4.1 Let $W$ be a solution of (REF ) and let $H^{(1)}$ be the unique solution of $\\mathcal {E}_{t}H^{(1)}&=0 ,\\\\H^{(1)}_{y=a,b}&=\\partial _yL_tW|_{y=a,b}.$ Then there exist functions $u_1,u_2,\\tilde{u}_1,\\tilde{u}_2 \\in H^{2}$ (depending on $a,b,k$ and $g$ but not on $t$ ) and constants $c_1,c_2$ such that $H^{(1)}(t,y)&= c_1\\langle W, e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)}u_1 \\\\& \\quad + c_2\\langle W, e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)}u_2.$ Furthermore, for instance for $u_1$ for any $t>0$ $\\langle W, e^{ikt(y-a)}\\tilde{u}_1 \\rangle = \\frac{\\omega _0(a)}{ikt} - \\frac{1}{ikt} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle - \\frac{1}{ikt} \\langle \\partial _yW, e^{ikt(y-a)}\\tilde{u}_1 \\rangle .$ Based on this characterization of $H^{(1)}$ , we introduce a splitting of $\\partial _yW$ into a function $\\beta $ depending only on $\\omega _0|_{y=a,b}$ and $\\Gamma =\\partial _yW-\\beta $ .",
"As we show in Theorem REF , $\\Gamma $ is stable also in higher regularity.",
"In contrast, unless $\\omega _0|_{y=a,b}$ is trivial, $\\beta $ asymptotically develops singularities at the boundary and exhibits blow-up in $H^{s},s>1/2$ .",
"If one however considers weighted spaces, it is possible to compensate for these singularities by vanishing weights and hence establish sufficient control for damping with optimal decay rates.",
"Lemma 4.2 Let $W$ be a solution of (REF ) and let $\\Gamma $ be the solution of $\\begin{split}\\partial _t\\Gamma &= \\frac{ih}{k}L_t \\Gamma - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad + \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW -c_1 \\frac{h}{k^2t} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 \\\\ & \\quad -c_2 \\frac{h}{k^2t} \\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2, \\\\\\Gamma |_{t=0}&=\\partial _y \\omega _0,\\end{split}$ and let $\\beta $ be the solution of $\\begin{split}\\partial _t\\beta &= \\frac{ih}{k}L_t \\beta - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad -\\frac{c_1h\\omega _0(a)}{k^2t} e^{ikt(y-a)}u_1 -\\frac{c_2h\\omega _0(b)}{k^2t} e^{ikt(y-b)}u_2, \\\\\\beta |_{t=0}&=0.\\end{split}$ Then $\\partial _yW=\\Gamma +\\beta $ .",
"The function $\\beta $ is called the boundary layer.",
"Theorem 4.1 ($H^2$ regularity of $\\Gamma $ ) Suppose that $g,h$ satisfy the assumptions of Theorem REF .",
"Suppose that additionally $g \\in W^{2,\\infty }$ and $h \\in W^{2,\\infty }$ .",
"Then there exists a constant $C_1$ such that for all $\\omega _0 \\in H^1$ and any $t\\ge 0$ , the solution $\\Gamma $ of (REF ) satisfies $\\Vert \\Gamma (t)\\Vert _{L^2} \\le C \\Vert \\omega _0\\Vert _{H^1}.$ Suppose that additionally $g \\in W^{3,\\infty }$ and $h \\in W^{3,\\infty }$ , then there exists a second constant $C_2$ such that for any $\\omega _0 \\in H^2$ and for any $t \\ge 0$ , $\\Vert \\Gamma (t)\\Vert _{H^1} \\le C_2 \\Vert \\omega _0\\Vert _{H^2}.$ Theorem 4.2 ($H^2$ regularity of $\\beta $ ) Suppose $g,h$ satisfy the assumptions of Theorem REF .",
"Then there exists a constant $C_1$ such that for all $t\\ge 0$ , the solution $\\beta $ of (REF ) satisfies $\\Vert \\beta (t)\\Vert _{L^2} \\le C_1 (|\\omega _0(a)|+|\\omega _0(b)|).$ Suppose that additionally $g,h \\in W^{2,\\infty }$ , then there exists a second constant $C_2$ such that $\\Vert (y-a)(y-b)\\partial _y\\beta (t)\\Vert _{L^2} \\le C_2 (|\\omega _0(a)|+|\\omega _0(b)|).$ However, if for instance $|\\omega _0(a)|>0$ , then $|\\beta (t,a)| \\gtrsim \\log (t)$ as $t \\rightarrow \\infty $ (similarly for $b$ ).",
"In particular, by the Sobolev embedding, we obtain blow-up in $H^{s},s>1/2$ .",
"Remark 4 Combining Theorems REF , REF and REF and Proposition REF , we obtain Theorem REF .",
"It is possible to further split $\\Gamma $ into functions controlled solely in terms of $\\Vert \\omega _0\\Vert _{L^2}$ , $\\Vert \\partial _y \\omega _0\\Vert _{L^2}$ and $\\Vert \\partial _{y}^2\\omega _0\\Vert _{L^2}$ , if finer control is desired.",
"Like Theorem REF , in addition to these stability results we obtain Lyapunov functionals.",
"As a key difference, these functionals are however in general only decreasing for times $t\\ge T>0$ .",
"Control up time $T$ is hence provided by a Gronwall-type argument, which determines the constants $C_1,C_2$ .",
"We stress that we do not require higher norms of $g,h$ to be small but only finite, so that derivatives of the equation are well-defined as mappings in $L^2$ .",
"When considering a setting without boundary contributions such as $\\mathbb {R}$ or $, no boundary correction $$ is needed.Thus (a suitable modification of) this result alreadyyields the desired stability for decay with optimal rates.Furthermore, this result generalizes to higher derivatives in astraightforward way, where again only finiteness of higher norms has to be required.$ Stability of $\\Gamma $ As the main result of this subsection we provide a proof of Theorem REF .",
"Here, the $L^2$ stability result is self-contained, while the $H^1$ estimate presupposes the $L^2$ stability of $\\beta $ , which is established in the following subsection.",
"Furthermore, we briefly discuss the implications of Theorem REF for settings without boundary and provide an improved stability result for the setting of an infinite plane periodic channel, $L\\times \\mathbb {R}$ .",
"We recall that $\\Gamma $ is the solution of $\\partial _t\\Gamma &= \\frac{ih}{k}L_t \\Gamma - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad + \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW -c_1 \\frac{h}{k^2t} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 \\\\ & \\quad -c_2 \\frac{h}{k^2t} \\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2, \\\\\\Gamma |_{t=0}&=\\partial _y \\omega _0,$ In addition to the estimates for $L_t$ derived in Section REF , we hence need to control contributions of the form $\\frac{1}{ikt} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle \\langle A \\Gamma , e^{ikt(y-b)} u_2 \\rangle ,$ which can not be controlled by the previous choice of $A(t)$ .",
"Instead, we construct a modified weight $A_1(t)$ , which is introduced in the following Lemmata (cf.",
"[13] for a similar construction adapted to fractional Sobolev spaces).",
"Lemma 4.3 Let $u \\in H^1$ , then for $0<\\mu <1/2$ and for every $v=\\sum _n v_n e^{iny} \\in L^2$ $|\\langle v, e^{ikt(y-a)} u \\rangle |^{2} \\le C_\\mu \\Vert u\\Vert _{H^1}^2 \\sum _n <n-kt>^{-2\\mu }|v|_{n}^2.$ By expanding the $L^2$ inner product in a basis, we obtain that $\\langle v, e^{ikt(y-a)} u\\rangle = \\sum _n v_n \\langle e^{iny},e^{ikt(y-a)} u \\rangle .$ Integrating by parts and using the trace inequality, we further estimate $|\\langle e^{iny} ,e^{ikt(y-a)} u \\rangle | \\le <n-kt>^{-1} \\Vert u\\Vert _{H^1}.$ The result hence follows by an application of the Cauchy-Schwarz inequality: $|\\langle v, e^{ikt(y-a)} u\\rangle | &\\lesssim \\sum _n v_n <n-kt>^{-\\mu } <n-kt>^{1-\\mu } \\\\&\\le \\Vert v_n <n-kt>^{-\\mu }\\Vert _{l^2} \\Vert <n-kt>^{1-\\mu }\\Vert _{l^2} \\\\&\\le 2 \\Vert v_n <n-kt>^{-\\mu }\\Vert _{l^2} \\Vert <n>^{-(1-\\mu )}\\Vert _{l^2} \\\\&=: C_\\mu \\Vert v_n <n-kt>^{-\\mu }\\Vert _{l^2},$ where we used that $<n>^{-(1-\\mu )} \\in l^2$ if $\\mu <1/2$ .",
"Lemma 4.4 Let $0<\\lambda ,\\mu <1$ with $\\lambda +2\\mu >1$ and let $\\epsilon >0$ and define the symmetric operator $A_1(t)$ by its action on the basis: $A_1(t): e^{iny} \\mapsto \\exp \\left(\\arctan \\left(\\frac{\\eta }{k}-t\\right)- \\epsilon \\int ^{t} <\\tau >^{-\\lambda } <n-k\\tau >^{-2\\mu } d\\tau \\right).$ Then for every $u \\in L^2$ and every $t \\in \\mathbb {R}$ $C \\Vert u\\Vert _{L^2}^2\\le \\langle u, A_1(t) u \\rangle &\\le C^{-1} \\Vert u\\Vert _{L^2}^2, \\\\\\langle u, \\dot{A}_1(t)u \\rangle \\le -C_1\\Vert u\\Vert _{\\tilde{H}^{-1}_t}^2 - C\\epsilon \\sum _n <t>^{-\\lambda } <n-kt>^{-2\\mu }|u_n|^2 &\\le 0.$ We note that $<t>^{-\\lambda } <n-kt>^{-2\\mu } \\in L^1(\\mathbb {R})$ and that $- \\epsilon \\int ^{t} <\\tau >^{-\\lambda } <n-k\\tau >^{-2\\mu } d\\tau $ is monotonically decreasing.",
"The properties of $A_1(t)$ hence follow by direct computation, where $C=\\exp (-\\pi - \\epsilon \\Vert <\\cdot >^{-\\lambda } <n-k\\cdot >^{-2\\mu }\\Vert _{L^1(\\mathbb {R})}).$ and $C_1$ is determined by $C$ and Lemma REF .",
"Lemma 4.5 Let $g \\in W^{2,\\infty }$ , $g \\ge c>0$ , then for every $u \\in L^2$ and for every $t \\ge 0$ , $\\Vert L_t[\\mathcal {E}_{t}, \\partial _y]L_tu\\Vert _{\\tilde{H}^{1}_{t}} \\lesssim \\Vert u\\Vert _{\\tilde{H}^{-1}_t}.$ By Lemma REF , we obtain that $\\Vert L_t[\\mathcal {E}_{t}, \\partial _y]L_tu\\Vert _{\\tilde{H}^{1}_{t}} \\lesssim \\Vert [\\mathcal {E}_{t}, \\partial _y]L_tu\\Vert _{\\tilde{H}^{-1}_{t}}.$ We further note that $[\\mathcal {E}_{t}, \\partial _y]= e^{-ikty}[\\mathcal {E}_0, \\partial _y-ikt]e^{ikty}= e^{-ikty}[\\mathcal {E}_0, \\partial _y]e^{ikty},$ and that, by direct computation, $[\\mathcal {E}_0, \\partial _y]$ is a second-order operator.",
"Hence, using integration by parts, we further estimate $\\Vert [\\mathcal {E}_{t}, \\partial _y]L_tu\\Vert _{\\tilde{H}^{-1}_{t}} \\lesssim \\Vert L_tu\\Vert _{\\tilde{H}^{1}_t} \\lesssim \\Vert u\\Vert _{\\tilde{H}^{-1}_t}.$ Using these results, we can now provide a proof of Theorem REF and thus establish $L^2$ stability.",
"Fix $0<\\lambda ,\\mu <1$ with $2\\mu +\\lambda >1$ and let $A_1$ be given by Lemma REF , where $0<\\epsilon < \\frac{1}{100} \\Vert <n-k\\cdot >^{-2\\mu }<\\cdot >^{-\\lambda }\\Vert _{L^1(\\mathbb {R})}^{-1}.$ Then we define $I(t):= \\langle \\Gamma , A_1(t)\\Gamma \\rangle + C_1 \\langle W, A(t)W \\rangle ,$ where $C_1 \\gg 0$ is to be chosen later.",
"We then claim that there exists $T>0$ such that for all initial data and for all $t \\ge 0$ , $I(t)$ satisfies $\\frac{d}{dt}I(t) \\le C t^{-2(1-\\mu /2)} \\Vert \\omega _0\\Vert _{L^2}^2 \\in L^1(\\mathbb {R}).$ Using Gronwall's inequality, we further obtain that $I(T)\\le \\exp (CT)I(0),$ which concludes the proof.",
"It remains to prove the claim.",
"Using Theorem REF and Lemma REF , we directly compute $\\frac{d}{dt}I(t) &\\le -C\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t}^2 - C\\epsilon \\sum _n <t>^{-\\lambda } <n-kt>^{-2\\mu }|\\Gamma _n|^2 \\\\ & \\quad - C_1\\delta \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2 + 2 \\Re \\langle \\frac{d}{dt}\\Gamma , A_{1}(t)\\Gamma \\rangle .$ Using Lemma REF and Lemma REF and recalling (REF ), we further estimate $2 \\Re \\langle \\frac{d}{dt}\\Gamma , A_{1}(t)\\Gamma \\rangle &\\le C(h,k)\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t} \\Vert A_1\\Gamma \\Vert _{\\tilde{H}^{-1}_t} + C(h,k,\\mu ) \\frac{1}{t} \\left( \\sum _n <n-kt>^{-2\\mu }|\\Gamma _n|^2 \\right) \\\\& \\quad + C(h,h^{\\prime },k) \\Vert A_{1}\\Gamma \\Vert _{\\tilde{H}^{-1}_t} (\\Vert L_tW\\Vert _{\\tilde{H}^{1}_t} + \\Vert L_t [\\mathcal {E}_{t},\\partial _y]L_t\\Vert _{\\tilde{H}^{1}_t}) \\\\& \\quad + C(h,k,g) t^{-1} \\Vert \\omega _0\\Vert _{L^2} \\sqrt{\\sum _n <n-kt>^{-2\\mu }|\\Gamma _n|^2}.$ Splitting $t=t^{-(1-\\mu )} t^{-\\mu }$ and using Young's inequality and Lemmata REF and REF , we further control $\\frac{1}{t} \\left( \\sum _n <n-kt>^{-2\\mu }|\\Gamma _n|^2 \\right)= t^{-(1-\\lambda )}\\sum _n t^{-\\lambda } <n-kt>^{-2\\mu }|\\Gamma _n|^2 , \\\\\\Vert A_{1}\\Gamma \\Vert _{\\tilde{H}^{-1}_t} (\\Vert L_tW\\Vert _{\\tilde{H}^{1}_t} + \\Vert L_t [\\mathcal {E}_{t},\\partial _y]L_t\\Vert _{\\tilde{H}^{1}_t}) \\le \\sigma \\Vert A_{1}\\Gamma \\Vert _{\\tilde{H}^{-1}_t}^{2} + \\sigma ^{-1} \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2 , \\\\t^{-1} \\Vert \\omega _0\\Vert _{L^2} \\sqrt{\\sum _n <n-kt>^{-2\\mu }|\\Gamma _n|^2} \\le \\sigma \\langle \\Gamma , \\dot{A}_1(t) \\Gamma \\rangle | + \\sigma ^{-1} t^{-2(1-\\mu /2)}\\Vert \\omega _0\\Vert _{L^2}^2.$ Choosing $\\sigma $ sufficiently small and letting $T>0$ be sufficiently large and using the smallness assumption of Theorem REF , we observe that $-C\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t}^2 - C\\epsilon \\sum _n <t>^{-\\lambda } <n-kt>^{-2\\mu }|\\Gamma _n|^2+ C(h,k)\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t} \\Vert A_1\\Gamma \\Vert _{\\tilde{H}^{-1}_t} \\\\+ (C(h,k,\\mu )t^{-(1-\\mu )}) \\sum _n <t>^{-\\lambda } <n-kt>^{-2\\mu }|\\Gamma _n|^2 \\\\+ \\sigma \\Vert A_{1}\\Gamma \\Vert _{\\tilde{H}^{-1}_t}^{2} + \\sigma \\langle \\Gamma , \\dot{A}_1(t) \\Gamma \\rangle | \\le 0.$ Similarly, choosing $C_1$ sufficiently large, we observe that $- C_1\\delta \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2 + \\sigma ^{-1} \\Vert W(t)\\Vert _{\\tilde{H}^{-1}_t}^2 \\le 0.$ Hence, we conclude that for $t \\ge T>0$ , $I(t)$ satisfies $\\frac{d}{dt}I(t) \\le \\sigma ^{-1} t^{-2(1-\\mu /2)}\\Vert \\omega _0\\Vert _{L^2}^2.$ which finishes the proof of the claim and hence of the $L^2$ stability result, $(1)$ .",
"Next, we consider the evolution of $\\partial _y\\Gamma $ : $\\partial _t\\partial _y\\Gamma &= \\frac{ih}{k}L_t \\partial _y \\Gamma + \\frac{ih^{\\prime }}{k}L_t \\Gamma + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t\\Gamma - (\\partial _yL_t\\Gamma )(a) e^{ikt(y-a)}u_1 - (\\partial _yL_t\\Gamma )(b) e^{ikt(y-b)}u_1\\\\& \\quad + \\partial _y\\Big ( \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad -c_1 \\frac{h}{k^2t} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 -c_2 \\frac{h}{k^2t} \\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\Big ) \\\\& \\quad + \\partial _y \\left( \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW \\right), \\\\\\partial _y \\Gamma |_{t=0}&=\\partial _{y}^2 \\omega _0$ Since we here also have to compute $\\partial _yW=\\Gamma +\\beta $ in order to control $\\Vert \\partial _y\\Gamma \\Vert _{L^2}$ , we require $L^2$ estimates on $\\beta $ .",
"Before continuing with the proof of Theorem REF , we hence prove the first part of Theorem REF as well as some further properties of the evolution of $\\beta $ , which are formulated in the following proposition.",
"Proposition 4.1 Suppose $g,h$ satisfy the assumptions of Theorem REF .",
"Let $\\beta $ be the solution of (REF ) and let $A_1(t)$ be given by Lemma REF .",
"Then there exists $T>0$ such that for all $t\\ge 0$ $I_2(t)=\\langle \\beta , A_1(t)\\beta \\rangle $ satisfies $\\frac{d}{dt}I_2(t) \\le \\delta \\langle \\beta , \\dot{A}_1(t)\\beta \\rangle + C t^{-2(1-\\mu /2)}|\\omega _0|_{y=a,b}|^2.$ Using the same weight $A_{1}$ , we observe that $\\Re \\langle A_{1}(t)\\beta , \\frac{ih}{k}L_t \\beta -\\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\rangle \\\\\\le (C(h,k)+ C(h,k,g)t^{-(1-\\mu )})|\\langle \\beta , \\dot{A}_1(t)\\beta \\rangle |.$ Using the smallness assumption and restricting to $t\\ge T>0$ , this contribution can thus be absorbed by $\\langle \\beta , \\dot{A}_1(t)\\beta \\rangle \\le 0.$ Hence, we focus on $\\Re \\langle A_{1}\\beta , \\omega _{0}(a)\\frac{1}{ikt}e^{ikty}u \\rangle \\lesssim C_{\\lambda } \\Vert \\beta _{n}<n-kt>^{-\\lambda }\\Vert _{l^{2}} |\\omega _{0}|_{y=a,b}| \\frac{1}{|kt|}.$ Using Young's inequality and choosing $\\sigma $ sufficiently small, we thus obtain that $\\frac{d}{dt}\\langle \\beta , A_1 \\beta \\rangle \\le \\delta \\langle \\beta , \\dot{A}_1 \\beta \\rangle + C\\sigma ^{-1}t^{-2(1-\\mu /2)} |\\omega _0|_{y=a,b}|^2.$ The first part of Theorem REF then follows by integrating this inequality and using a Gronwall-type estimate to control the growth up to time $T$ .",
"Additionally, we make use of the following estimates for boundary evaluations of $L_t\\Gamma , W$ and $\\Gamma $ , which are obtained as an application of the results of Appendix .",
"Lemma 4.6 Let $g,h,k$ satisfy the assumptions of the second part of Theorem REF .",
"Then, $(\\partial _yL_t\\Gamma )(a)=c_1 \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle , \\\\(\\partial _yL_t\\Gamma )(b)=c_2 \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle ,$ and the following estimates hold: $|\\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle | &\\lesssim \\frac{C_\\mu }{kt}\\sqrt{\\sum _n |(\\partial _y\\Gamma )|_{n}^2 <n-kt>^{-2\\lambda }} + \\frac{C}{kt}|\\Gamma (a,t)|, \\\\|\\langle W, e^{ikt(y-a)}\\tilde{u}_1 \\rangle | &\\lesssim \\frac{C_\\mu }{kt}(\\Vert \\Gamma \\Vert _{L^2}+\\Vert \\beta (t)\\Vert _{L^2})+ \\frac{C}{kt}|\\omega _0(a,t)|, \\\\|\\Gamma (a,t)| &\\le \\log (t)(|\\omega _0(a)|+\\Vert \\omega _0\\Vert _{L^2}).$ The evaluations of $\\partial _yL_t\\Gamma $ at the boundary are obtained as an application of Lemma REF .",
"The first two estimates follow by integration by parts.",
"In order to show the last estimate, we restrict (REF ) to the boundary and obtain that $|\\partial _t\\Gamma (a,t)|\\lesssim \\frac{1}{kt}(\\Vert \\Gamma (t)\\Vert _{L^2}+\\Vert W(t)\\Vert _{L^2}),$ where we used that $L_t$ enforces zero Dirichlet data.",
"The result hence follows by using Theorem REF and the first part of Theorem REF to control $\\Vert \\Gamma (t)\\Vert _{L^2}+\\Vert W(t)\\Vert _{L^2}\\lesssim |\\omega _0(a)|+\\Vert \\omega _0\\Vert _{L^2},$ and then integrating the inequality.",
"Lemma 4.7 Let $W$ be the solution of (REF ) with initial datum $\\omega _0 \\in H^1$ and let $\\Gamma $ and $\\beta $ be as in Lemma REF .",
"Then, for any $\\sigma >0$ , $\\Re \\langle A_1 \\partial _y\\Gamma , \\left[\\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right), \\partial _y \\right] W \\rangle \\le \\sigma |\\langle \\partial _y\\Gamma , \\dot{A}_1 \\partial _y \\Gamma \\rangle | + C\\sigma ^{-1} \\Vert W\\Vert _{\\tilde{H}^{-1}_t}^2.$ The contribution due to $\\frac{ih^{\\prime }}{k}L_t$ can be estimated as in Lemma REF .",
"In the following we thus focus on the commutator and decompose the commutator into the cases where $\\partial _y$ falls on $h$ , $\\frac{ih^{\\prime \\prime }}{k}L_tW + \\frac{ih^{\\prime }}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW,$ the terms solving an elliptic equation with vanishing Dirichlet data, $\\frac{ih^{\\prime }}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t W + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t[\\mathcal {E}_{t},\\partial _y]L_tW,$ and the homogeneous corrections, $\\frac{ih^{\\prime }}{k} ((\\partial _yL_tW)(a,t) e^{ikt(y-a)}u_1+(\\partial _yL_tW)(b,t) e^{ikt(y-b)}u_2), \\\\\\frac{ih}{k} ((\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(a,t) e^{ikt(y-a)}u_1+(\\partial _yL_t[\\mathcal {E}_{t},\\partial _y]L_tW)(b,t) e^{ikt(y-b)}u_2),$ In the first and second case, we use Lemmata REF and REF to estimate by $\\Vert \\partial _y\\Gamma \\Vert _{\\tilde{H}^{-1}_t} \\Vert W\\Vert _{\\tilde{H}^{-1}_t},$ which is of the desired form by Young's inequality.",
"It hence only remains to consider the homogeneous corrections.",
"Here, we estimate $\\Re \\Big \\langle A_1(t)\\partial _y \\Gamma &, \\frac{ih^{\\prime }}{k} ((\\partial _yL_tW)(a,t) e^{ikt(y-a)}u_1+(\\partial _yL_tW)(b,t) e^{ikt(y-b)}u_2) \\\\ &+ \\frac{ih}{k} ((\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(a,t) e^{ikt(y-a)}u_1+(\\partial _yL_t[\\mathcal {E}_{t},\\partial _y]L_tW)(b,t) e^{ikt(y-b)}u_2) \\\\&\\le C_\\mu \\sqrt{\\sum _n |\\partial _y\\Gamma _n|^{2}<n-kt>^{-2\\mu }} \\Big (|\\partial _yL_tW)(a,t)|+|\\partial _yL_tW)(b,t)| \\\\& \\quad +|\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(a,t)| + |\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(b,t)|\\Big ).$ We further recall from Section that boundary evaluations can be obtained by testing with suitable homogeneous solution to the adjoint problem.",
"Hence, $|\\partial _yL_tW)(a,t)|+|\\partial _yL_tW)(b,t) \\lesssim t^{-1} \\Vert \\partial _yW\\Vert _{L^2}\\le t^{-1}\\Vert \\omega _0\\Vert _{H^1}, \\\\|\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(a,t)| + |\\partial _yL_t[\\mathcal {E}_{t},p_y]L_tW)(b,t)| \\lesssim t^{-1} \\Vert [\\mathcal {E}_{t},\\partial _y]L_tW\\Vert _{H^1}.$ We can thus conclude the proof, if we can show that $\\Vert [\\mathcal {E}_{t},\\partial _y]L_tW\\Vert _{H^1} \\lesssim \\Vert W\\Vert _{H^1}.$ Expressing $\\partial _y [\\mathcal {E}_{t},\\partial _y]L_tW= [\\mathcal {E}_{t},\\partial _y]L_t \\partial _yW +[[\\mathcal {E}_{t},\\partial _y]L_t,\\partial _y]W$ , this estimate follows from elliptic regularity theory for $[\\mathcal {E}_{t},\\partial _y]L_t|_{t=0}$ and using that multiplication by $e^{ikty}$ is an isometry.",
"Building on these results, we can now complete the proof of Theorem REF .",
"Following a similar strategy as in the previous part, we consider $I_{2}(t):= \\langle \\partial _y \\Gamma , A_1(t)\\partial _y \\Gamma \\rangle + C_1 \\langle \\Gamma , A_1(t)\\Gamma \\rangle + C_2 \\langle \\beta , A_1(t)\\beta \\rangle + C_3 \\langle W, A(t)W\\rangle ,$ where $C_1,C_2,C_3>0$ are to be chosen later.",
"Using the preceding results and strategy, it suffices to study $\\Re \\langle \\partial _t \\partial _y \\Gamma , A_1 \\partial _y \\Gamma \\rangle .$ Following the same strategy as in the previous part of the proof and using Lemma REF , we estimate $& \\quad \\Re \\langle A_1 \\partial _y\\Gamma , \\frac{ih}{k}L_t \\partial _y \\Gamma + \\frac{ih^{\\prime }}{k}L_t \\Gamma + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t\\Gamma \\\\ & \\quad - (\\partial _yL_t\\Gamma )(a) e^{ikt(y-a)}u_1 - (\\partial _yL_t\\Gamma )(b) e^{ikt(y-b)}u_1 \\rangle \\\\&\\le (C+\\sigma +ct^{-(1-\\mu )}\\log (t)) \\Vert \\partial _y\\Gamma \\Vert _{\\tilde{H}^{-1}_t}^2+ \\sigma ^{-1}|\\langle \\Gamma , \\dot{A} \\Gamma \\rangle |,$ which can be absorbed.",
"Furthermore, applying Lemma REF , we can control $& \\quad \\Re \\Big \\langle A_1 \\partial _y\\Gamma ,\\partial _y\\Big ( \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 - \\frac{h}{k^2t} \\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\\\& \\quad -c_1 \\frac{h}{k^2t} \\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle e^{ikt(y-a)} u_1 -c_2 \\frac{h}{k^2t} \\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle e^{ikt(y-b)} u_2 \\Big ) \\Big \\rangle \\\\&\\le C(\\mu ,g,h,k) \\left( \\sum _n |\\Gamma _n|^2<n-kt>^{-2\\mu } \\right)^{1/2}\\Big ( \\left|\\langle \\Gamma , e^{ikt(y-a)}\\tilde{u}_1 \\rangle \\right| \\\\& \\quad +\\left|\\langle \\Gamma , e^{ikt(y-b)}\\tilde{u}_2 \\rangle \\right| +\\left|\\langle W, e^{ikt(y-a)}\\partial _y\\tilde{u}_1 \\rangle \\right|+\\left|\\langle W, e^{ikt(y-b)}\\partial _y\\tilde{u}_2 \\rangle \\right| \\Big ).$ Applying the estimates of Lemma REF and using Young's inequality, these contributions can hence again be partially absorbed provided $\\sigma $ is sufficiently small and $T>0$ is sufficiently large.",
"The remaining non-absorbed terms can be estimated by $t^{-2(1-\\mu /2)} (|\\Gamma (a,t)| + |\\Gamma (b,t)| + \\Vert \\partial _y W(t)\\Vert _{L^2} + |\\omega _0(a)| + |\\omega _0(b)|) \\lesssim t^{-2(1-\\mu /2)} \\Vert \\omega _0\\Vert _{H^1},$ where we used Theorem REF , the first part of Theorem REF and the Sobolev embedding.",
"It remains to estimate $\\Re \\left\\langle A_1(t)\\partial _y\\Gamma , \\partial _y \\left( \\frac{ih^{\\prime }}{k}L_tW + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_tW \\right) \\right\\rangle .$ Recalling the definition of $\\Gamma $ and $\\beta $ , we express the right function as $\\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right) (\\Gamma +\\beta ) \\\\+ \\left[\\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right), \\partial _y \\right] W.$ We then estimate $\\Re \\left\\langle A_1(t)\\partial _y\\Gamma , \\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right) (\\Gamma +\\beta ) \\right\\rangle \\lesssim \\Vert \\partial _y\\Gamma \\Vert _{\\tilde{H}^{-1}_t} (\\Vert \\Gamma \\Vert _{\\tilde{H}^{-1}_t} + \\Vert \\beta \\Vert _{\\tilde{H}^{-1}_t}).$ Using Young's inequality, the respective terms can then again be controlled, given a suitable choice of $\\sigma $ .",
"Finally, using Lemma REF , $\\Re \\langle A_1 \\partial _y\\Gamma , \\left[\\left(\\frac{ih^{\\prime }}{k}L_t\\cdot + \\frac{ih}{k}L_t[\\mathcal {E}_{t},\\partial _y]L_t \\cdot \\right), \\partial _y \\right] W \\rangle \\\\ \\le \\sigma |\\langle \\partial _y\\Gamma , \\dot{A}_1 \\partial _y \\Gamma \\rangle | + C\\sigma ^{-1} \\Vert W\\Vert _{\\tilde{H}^{-1}_t}^2,$ which can again be absorbed and hence concludes the proof.",
"Weighted stability of $\\partial _y\\beta $ and boundary blow-up In this section we consider the evolution of $\\partial _y\\beta $ .",
"Since the behavior at both boundary points is similar and separates, we for simplicity of notation consider the case $\\omega _0(a)\\ne 0$ , $\\omega _0(b)=0$ .",
"The general case can then be obtained by switching $a$ and $b$ and using the linearity of the equation.",
"The function $\\beta $ then satisfies (REF ): $\\begin{split}\\partial _t\\beta -\\frac{ih}{k}L_{t}\\beta - \\frac{h}{k^2t}\\langle \\beta ,e^{ikt(y-a)}u \\rangle e^{ikt(y-a)}u &= \\omega _{0}(a)\\frac{h}{k^2t}e^{ikt(y-a)}u, \\\\\\beta |_{t=0}&=0.\\end{split}$ We note that, if $\\omega _{0}|_{y=a,b}=0$ , then $\\beta $ identically vanishes.",
"We recall that by Proposition REF under suitable assumptions on $h,g$ and $k$ , $\\beta $ is stable in $L^2$ .",
"However, stability in $H^{1}$ or, indeed in $H^{s},s>1/2$ , does not hold due to the asymptotic formation of singularities at the boundary.",
"Lemma 4.8 (Boundary blow-up) Suppose that for some $s>0$ , $\\sup _{t>0} \\Vert \\beta (t)\\Vert _{H^{s}} = C <\\infty .$ Then $\\beta (a,t)$ satisfies $|\\beta (a,t)-h(a)\\omega _{0}(a)k^{-2} \\log (t)| \\le C_{s}C,$ as $t \\rightarrow \\infty $ In particular, if $\\omega _{0}(a)\\ne 0$ , then $\\sup _{t}\\Vert \\beta (t)\\Vert _{C^{0}}=\\infty .$ Hence, by the Sobolev embedding, in that case, $\\sup _{t} \\Vert \\beta (t)\\Vert _{H^{s}} \\ge \\sup _{t} \\log (t) = \\infty ,$ for any $s>\\frac{1}{2}$ .",
"Restricting the evolution by (REF ) to the boundary, we obtain $\\partial _t\\beta (a) + \\frac{h(a)}{k^2t} \\langle \\beta , e^{ikt(y-a)}u \\rangle &= \\frac{h(a)\\omega _{0}(a)}{k^2t}.$ Let $s>0$ and without loss of generality $s<1/2$ , then by direct computation $|\\langle \\beta , e^{ikt(y-a)}u \\rangle | \\lesssim C t^{-s}\\Vert \\beta \\Vert _{H^s}.$ Hence, $\\beta (a,t)$ satisfies $\\partial _t\\beta (a) - \\frac{\\omega _{0}(a)h(a)}{k^2} \\partial _t\\log (t) = t^{-1} \\mathcal {O}(t^{-s}) \\in L^{1}_{t}.$ The result hence follows by integrating in time.",
"Letting $s=1$ in the preceding Lemma, we in particular note that in general $H^1$ stability of $\\beta $ fails.",
"Following a similar approach as in [13], one can further show that $s=1/2$ is indeed critical in the sense that stability holds for $H^{s},s<1/2$ .",
"As this is however not sufficient for optimal decay rates in the damping estimate of Section , in the following we prove weighted $H^1$ stability as formulated in Theorem REF .",
"Here, we use a different method of proof based Duhamel's formula, the details of which can be found in Appendix .",
"Splitting $\\partial _y\\beta $ We recall that $\\beta $ solves $\\begin{split}\\partial _t\\beta - \\frac{ih}{k}L_{t}\\beta - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}u \\rangle e^{ikt(y-a)} u &= \\omega _{0}(a) \\frac{h}{k^2t}e^{ikt(y-a)}u, \\\\\\beta |_{t=0}&=0.\\end{split}$ Applying one $y$ derivative to this equation, we obtain $\\begin{split}& \\quad \\partial _t\\partial _{y}\\beta - \\frac{ih}{k}L_{t}\\partial _{y}\\beta + \\frac{h}{k^2t} \\langle \\partial _{y} \\beta , e^{ikt(y-a)}u \\rangle e^{ikt(y-a)} u\\\\ &= [\\frac{ih}{k}L_{t},\\partial _{y}]\\beta + \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)} \\partial _{y} u \\rangle e^{ikt(y-a)} u + \\frac{h}{k^2t} \\beta (a,t) e^{ikt(y-a)} u \\\\ &\\quad - \\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}u \\rangle e^{ikt(y-a)} u- \\omega _{0}(a) \\frac{h^{\\prime }}{k^2t}e^{ikt(y-a)}u \\\\ & \\quad +\\omega _{0}(a) \\frac{h}{k^2t}e^{ikt(y-a)}\\partial _{y}u + \\frac{i\\omega _{0}(a)h}{k} e^{ikt(y-a)}u,\\end{split}$ where we used that $\\frac{h}{k^2t} \\langle \\beta , e^{ikt(y-a)}u \\rangle \\partial _y(e^{ikt(y-a)})u= \\frac{h}{k^2t} (\\langle \\partial _y(\\beta u), e^{ikt(y-a)} \\rangle - \\beta u e^{ikt(y-a)}|_{y=a}^b) e^{ikt(y-a)}u.$ We note that most terms in (REF ) are very similar to ones in equation (REF ) satisfied by $\\partial _{y}\\Gamma $ , with the exception of $\\frac{i\\omega _{0}(a)h}{k} e^{ikt(y-a)}u,$ which is hence identified as the term driving the blow-up.",
"Based on this reasoning the following lemma introduces a splitting of $\\partial _y \\beta $ .",
"Lemma 4.9 Let $\\beta _{I}$ be the solution of $\\begin{split}\\partial _t\\beta _{I} - \\frac{ih}{k}L_{t}\\beta _{I} + \\frac{1}{ikt} \\langle \\beta _{I}, e^{ikty}u \\rangle e^{ikty} u&= [\\frac{ih}{k}L_{t},\\partial _{y}]\\beta + \\frac{h}{k^2t} \\langle \\beta , e^{ikty} \\partial _{y} u \\rangle e^{ikty} u \\\\ & \\quad + \\frac{h}{k^2t} \\beta (a,t) e^{ikty} u - \\frac{h}{k^2t} \\langle \\beta , e^{ikty}u \\rangle e^{ikty} u \\\\& \\quad - \\omega _{0}(a) \\frac{h}{k^2t}e^{ikty}\\partial _{y}u +\\omega _{0}(a) \\frac{h}{k^2t}e^{ikty}\\partial _{y}u, \\\\\\beta _{I}|_{t=0}&=0,\\end{split}$ and let $\\beta _{II}$ be the solution of $\\partial _t\\beta _{II} - \\frac{ih}{k}L_{t}\\beta _{II} + \\frac{h}{k^2t} \\langle \\beta _{II}, e^{ikty}u \\rangle e^{ikty} u &= \\omega _{0}(a) e^{ikty}u, \\\\\\beta _{V}|_{t=0}&=0.$ Then $\\partial _y\\beta =\\beta _{I}+\\beta _{II}$ .",
"Following the same strategy as in Section REF , we obtain $L^{2}$ stability of $\\beta _{I}$ .",
"Proposition 4.2 Suppose the assumptions of Theorem REF are satisfied, then $\\Vert \\beta _{I}(t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}|_{y=a,b}|.$ Following the same strategy as in the proof of Theorem REF , we show that, $\\frac{d}{dt}\\langle \\beta _{I}, A_1(t) \\beta _{I} \\rangle \\le \\langle \\beta _{I}, \\dot{A}_1(t) \\beta _{I} \\rangle + C \\Vert \\beta _{I}\\Vert _{\\tilde{H}^{-1}_t}^2 \\\\+ (Ct^{-(1-\\mu )} + \\sigma ) \\sum _n |(\\beta _{I})_n|^2<n-kt>^{-2\\lambda }t^{-\\mu } \\\\+ C\\sigma ^{-1} t^{-2(1-\\mu /2)} (|\\beta (a,t)|^2 + \\Vert \\beta \\Vert _{L^2}^2 + |\\omega _0(a)|^2).$ Hence, restricting to $t \\ge T>0$ and choosing $\\sigma $ sufficiently small, $\\frac{d}{dt}\\langle \\beta _{I}, A_1(t) \\beta _{I} \\rangle \\le C\\sigma ^{-1} t^{-2(1-\\mu /2)} (|\\beta (a,t)|^2 + \\Vert \\beta \\Vert _{L^2}^2 + |\\omega _0(a)|^2) \\\\ \\le C\\sigma ^{-1} t^{-2(1-\\mu /2)}\\log (t)^2 |\\omega _0(a)|^2,$ where we used Proposition REF and that, by equation (REF ), $|\\beta (a,t)|\\lesssim \\int ^t \\tau ^{-1}\\Vert \\beta (\\tau )\\Vert _{L^2} d\\tau \\lesssim \\log (t) |\\omega _0(a)|.$ For later reference, we note that we have thus also proven the following proposition.",
"Proposition 4.3 Suppose that $g,h,k$ satisfy the assumptions of the second part of Theorem REF .",
"Then, for any $\\omega _0 \\in L^2$ , the solution $W$ of (REF ) satisfies $\\Vert W(t)\\Vert _{H^1} + \\Vert \\partial _y^2W(t) - \\beta _{II}(t)\\Vert _{L^2} \\lesssim \\Vert \\omega _0\\Vert _{H^2},$ where $\\beta _{II}$ is given by Lemma REF .",
"This result combines Theorems REF and REF and Propositions REF and REF .",
"Weighted stability of $\\beta _{II}$ In order to complete the proof of Theorem REF , it only remains to study the stability of $\\partial _t\\beta _{II} - \\frac{ih}{k}L_{t}\\beta _{II} + \\frac{h}{k^2t} \\langle \\beta _{II}, e^{ikty}u \\rangle e^{ikty} u &= \\frac{ih}{k}\\omega _{0}(a) e^{ikty}u, \\\\\\beta _{II}|_{t=0}&=0.$ While it would be possible to study this equation directly, we instead build on our previous analysis of $\\partial _t- \\frac{ih}{k}L_{t}$ and introduce an additional boundary layer $\\nu $ (c.f.",
"Theorem REF ) solving $(\\partial _t- \\frac{ih}{k}L_{t})\\nu &= \\frac{h}{k}\\omega _{0}(a)e^{ikty}, \\\\\\nu |_{t=0} &=0,$ and also define $\\beta _{V}=\\beta _{II}-\\nu $ .",
"Then $\\beta _{V}$ solves $\\begin{split}\\partial _t\\beta _{V} - \\frac{ih}{k}L_{t}\\beta _{V} + \\frac{\\langle \\beta _{V}, e^{ikty}\\tilde{u} \\rangle }{ikt} e^{ikty}u &= \\frac{\\langle \\nu , e^{ikty}\\tilde{u} \\rangle }{ikt} e^{ikty}u.",
"\\\\\\beta _{V}|_{t=0}&=0.\\end{split}$ Remark 5 Instead of $\\nu $ one might attempt to choose the explicit function $\\int ^t \\frac{ih}{k}\\omega _0(a)e^{ik\\tau y} d\\tau = \\frac{ih}{k}\\omega _0(a)\\frac{e^{ikty}-1}{iky}=:\\chi .$ However, we note that part of this function oscillates like $e^{ikty}$ and that $L_t\\chi = e^{ikty}L_0 \\frac{h}{k^2y}\\omega _0(a) + L_t \\frac{h}{k^2y},$ where $L_0 \\frac{h}{k^2y}\\omega _0(a)$ is independent of $t$ .",
"Hence, even for a constant function $u$ $\\langle u, L_t \\chi \\rangle $ would not decay or oscillate rapidly enough to be an integrable perturbation.",
"As the main result of this section we establish the following proposition, which concludes the proof of Theorem REF .",
"Proposition 4.4 Suppose the assumptions of Theorem REF are satisfied.",
"Then the functions $\\beta _V$ and $\\nu $ satisfy $\\Vert \\beta _{V}(t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}(a)|, \\\\\\Vert (y-a)(y-b)\\nu (t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}(a)|.$ As the evolution of $\\beta _{V}$ depends on $\\nu $ via $\\langle \\nu , e^{ikt(y-a)}\\tilde{u} \\rangle $ and as our estimates of $\\nu $ rely on properties of the solution operator of (REF ) (and hence $W$ ), we follow a multi-step approach: Using Propositions REF and REF , we show that (REF ) grows at most like $\\sqrt{t}$ .",
"By direct computation, we show that $\\Vert (y-a)(y-b) \\nu \\Vert _{L^{2}}$ grows at most like $\\log (t)$ .",
"This yields a weaker form of Proposition REF with an estimate by $\\sqrt{t} |\\omega _{0}(a)|$ .",
"Combining this estimate with the damping result of Section , the estimate of (REF ) improves to $\\log (t)$ and we obtain a uniform bound of $\\Vert (y-a)(y-b) \\nu \\Vert _{L^{2}}$ .",
"Finally, we establish $L^{2}$ stability of $\\beta _{V}$ and thus conclude the proof of Proposition REF .",
"Lemma 4.10 Assume that the assumptions of Theorem REF are satisfied.",
"Then (REF ) satisfies $|\\langle e^{ikty} \\tilde{u} , \\nu (t)\\rangle | \\lesssim \\sqrt{t}|\\omega _{0}(a)|$ as $t \\rightarrow \\infty $ .",
"Lemma 4.11 Assume that the assumptions of Theorem REF are satisfied.",
"Then $\\nu (t)$ satisfies $\\Vert (y-a)(y-b)\\nu (t)\\Vert _{L^{2}} \\lesssim \\log (t)|\\omega _{0}(a)|$ as $t\\rightarrow \\infty $ .",
"Lemma 4.12 Assume that the assumptions of Theorem REF are satisfied.",
"Then, as $t \\rightarrow \\infty $ , $\\beta _{V}$ and $\\nu $ satisfy $\\Vert \\beta _{V}(t)\\Vert _{L^{2}} &\\lesssim \\sqrt{t}|\\omega _{0}(a)|, \\\\\\Vert (y-a)(y-b)\\nu (t)\\Vert _{L^{2}} &\\lesssim \\log (t)|\\omega _{0}(a)|.$ In particular, we conclude that the solution opertor $S(t,0): H^{2}(dy) &\\rightarrow H^{2}\\left( (y-a)(y-b) dy\\right), \\\\\\omega _0 &\\mapsto W(t),$ satisfies $||| S(t,0)||| \\lesssim \\sqrt{t}.$ Lemma 4.13 Assume that the assumptions of Theorem REF are satisfied.",
"Then $\\nu $ satisfies $\\Vert (y-a)(y-b)\\nu (t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}(a)|$ as $t\\rightarrow \\infty $ .",
"Lemma 4.14 Assume that the assumptions of Proposition REF are satisfied.",
"Then $\\beta _{V}$ satisfies $\\Vert \\beta _{V}(t)\\Vert _{L^{2}} \\lesssim |\\omega _{0}(a)|$ as $t\\rightarrow \\infty $ .",
"In our proof of Lemmata REF to REF , we rely on more detailed, (semi-explicit) characterization of $\\nu (t)$ via Duhamel's formula, which is established in Appendix .",
"We directly compute $\\langle e^{ikty}\\tilde{u}, \\int _{0}^{t} e^{ikty}S(t,\\tau ) e^{ik\\tau y} u d\\tau \\rangle \\\\= \\langle \\tilde{u}, \\int _{0}^{t} e^{ik(t-\\tau ) y}S(t-\\tau ,0) u d\\tau \\rangle .$ Next, we integrate $e^{ik(t-\\tau ) y} = \\partial _{\\tau } \\frac{e^{ik(t-\\tau )y}-1}{iky}$ by parts in $\\tau $ .",
"Here, we obtain a boundary term $\\langle \\tilde{u},\\frac{e^{ikty}-1}{iky} S(t,0) u \\rangle $ and an integral term $\\langle \\tilde{u},\\int _{0}^{t}\\frac{e^{ik(t-\\tau ) y}-1}{iky} \\partial _{\\tau } S(t-\\tau ,0) u d\\tau \\rangle .$ For (REF ) we apply Hölder's inequality and control by $\\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert \\frac{e^{ikty}-1}{iky}\\Vert _{L^{1}_{y}} \\Vert S(t,0) u\\Vert _{L^{\\infty }} \\lesssim \\log (t) \\Vert u\\Vert _{H^{1}}.$ In the integral term we use the damping estimate, Proposition REF , to control by $\\int _{0}^{t} \\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert \\frac{e^{ik(t-\\tau )y}-1}{iky}\\Vert _{L^{2}} \\Vert \\partial _{\\tau } S(t-\\tau ,0) u\\Vert _{L^{2}} d\\tau \\\\\\lesssim \\int _{0}^{t}\\sqrt{|t-\\tau |} <t-\\tau >^{-1} \\Vert S(t-\\tau ,0) u\\Vert _{H^{1}} d\\tau \\\\\\lesssim \\int <t-\\tau >^{-1/2} d\\tau \\lesssim \\sqrt{t}.$ Using Lemmata REF and REF , we obtain that $\\nu (t)= \\int _{0}^{t} e^{ikt(t-\\tau )(y-a)}S(t-\\tau ,0)u d\\tau $ Multiplying with $(y-a)$ , we use that $-\\partial _{\\tau } \\frac{e^{ik(t-\\tau )(y-a)}-1}{ik}= (y-a)e^{ik(t-\\tau )(y-a)}$ and hence control $\\Vert (y-a)\\nu (t)\\Vert _{L^{2}} & \\le \\Vert \\frac{e^{ik(t-\\tau )(y-a)}-1}{ik} S(t-\\tau ,0)u|_{\\tau =0}^{t}\\Vert _{L^{2}} \\\\&\\quad + \\int _{0}^{t} \\Vert \\frac{e^{ik(t-\\tau )(y-a)}-1}{ik} \\partial _{\\tau }S(t-\\tau ,0)u \\Vert _{L^{2}} d\\tau \\\\&\\lesssim |k|^{-1} \\Vert u\\Vert _{L^{2}} + |k|^{-2} \\Vert h\\Vert _{L^{\\infty }} \\int _{0}^{t} \\Vert L_{t-\\tau } S(t-\\tau ,0)u\\Vert _{L^{2}} \\\\& \\lesssim |k|^{-1} \\Vert u\\Vert _{L^{2}} + |k|^{-2} \\Vert h\\Vert _{L^{\\infty }} \\int _{0}^{t} <t-\\tau >^{-1} \\Vert S(t-\\tau ,0)u\\Vert _{H^{1}} d\\tau \\\\& \\lesssim |k|^{-1} \\Vert u\\Vert _{L^{2}} + |k|^{-2} \\Vert h\\Vert _{L^{\\infty }} \\Vert u\\Vert _{H^{1}}\\log (t),$ where we used Proposition REF and Theorem REF .",
"Using our Lyapunov functional approach on $\\beta _{V}$ , we need to estimate $\\langle A_{1}\\beta _{V}, e^{ikty}u\\rangle \\frac{\\langle \\nu , e^{ikty}\\tilde{u} \\rangle }{ikt}.$ By Lemma REF , we control $|\\frac{\\langle \\nu , e^{ikty}\\tilde{u} \\rangle }{ikt}|\\lesssim t^{-1/2},$ and using Lemma REF , we estimate.",
"$|\\langle A_{1}\\beta _{V}, e^{ikty}u\\rangle | \\le C_{\\lambda } \\Vert (\\beta _{V})<n-kt>^{-\\lambda }\\Vert _{l^{2}},$ where $0<\\lambda <\\frac{1}{2}$ .",
"Hence, using Young's inequality, we can control (REF ) by $\\epsilon \\Vert (\\beta _{V})_n<n-kt>^{-\\lambda }\\Vert _{l^{2}}t^{-1/2} + C(\\epsilon ,\\lambda ) t^{-1/2}.$ Here, for $\\epsilon $ sufficiently small, the first term can be absorbed by $\\langle \\beta _{V}, \\dot{A}_{1} \\beta _{V} \\rangle $ and in summary we obtain $\\partial _t\\langle \\beta _{V}, A_{1} \\beta _{V} \\rangle \\le C(\\epsilon ,\\lambda ) t^{-1/2}.$ Integrating this inequality then yields the result.",
"We remark that already in step 3 we could obtain a better growth bound by optimizing in $\\lambda $ and the splitting of $t^{-1/2}$ in Young's inequality.",
"However, since $t^{-1/2}\\notin L^{2}$ this would only yield a non-uniform bound and our multi-step proof only requires a better than linear growth bound.",
"Following the proof of Lemma REF it suffices to show that $\\int _{0}^{t}\\Vert \\partial _{\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} d\\tau \\lesssim 1,$ uniformly in $t$ .",
"Using Hölder's inequality and Proposition REF , we estimate $\\Vert \\partial _{\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} &= \\Vert \\frac{ih}{k}L_{t-\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} \\le \\Vert h\\Vert _{L^{\\infty }}|k|^{-1} \\Vert L_{t-\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} \\\\&\\le \\Vert h\\Vert _{L^{\\infty }}|k|^{-1}<t-\\tau >^{-2}(\\Vert (y-a)(y-b)\\partial _{y}^{2}S(t-\\tau ,0)u\\Vert _{L^{2}} \\\\ & \\quad +\\Vert S(t-\\tau ,0)u\\Vert _{H^{1}}) \\le \\Vert h\\Vert _{L^{\\infty }}|k|^{-1}<t-\\tau >^{-2} |||S(t-\\tau ,0)||| \\Vert u\\Vert _{H^{2}},$ the operator norm of $S(t-\\tau ,0)$ is given by Lemma REF .",
"Hence, we obtain that $\\Vert \\partial _{\\tau }S(t-\\tau ,0)u\\Vert _{L^{2}} \\lesssim \\Vert h\\Vert _{L^{\\infty }}|k|^{-1}\\Vert u\\Vert _{H^{2}}<t-\\tau >^{-2} \\sqrt{t-\\tau },$ which is integrable in $\\tau $ and thus concludes the proof.",
"We claim that $|\\langle e^{ikty} \\tilde{u} , \\nu (t)\\rangle | \\lesssim \\log (t)|\\omega _{0}(a)|.$ Following the proof of Lemma REF , this implies that $|\\langle A_{1}\\beta _{V}, e^{ikty}u\\rangle \\frac{\\langle \\nu , e^{ikty}\\tilde{u} \\rangle }{ikt} | \\le C \\Vert (\\beta _{V})<n-kt>^{-\\lambda }\\Vert _{l^{2}} \\frac{\\log (t)}{t} \\\\\\le \\epsilon \\Vert (\\beta _{V})<n-kt>^{-\\lambda }\\Vert _{l^{2}}^{2} t^{-2\\mu } + C(\\epsilon ) \\log (t)^{2}t^{-2(1-\\mu )},$ where $C(\\epsilon )$ is given by Young's inequality and $0<\\mu <1$ is chosen such that $2\\lambda +2\\mu >1$ and $2(1-\\mu )>1$ .",
"Choosing $\\epsilon $ sufficiently small, we thus obtain $\\partial _t\\langle \\beta _{V}, A_{1}\\beta _{V} \\rangle &\\le \\langle \\beta _{V}, \\dot{A}_{1}\\beta _{V} \\rangle + \\epsilon \\Vert (\\beta _{V})<n-kt>^{-\\lambda }\\Vert _{l^{2}}^{2} t^{-2\\mu } + C(\\epsilon ) \\log (t)^{2}t^{-2(1-\\mu )} \\\\ &\\le C(\\epsilon ) \\log (t)^{2}t^{-2(1-\\mu )} \\in L^{1}_{t}([1,\\infty )).$ Integrating this inequality then yields the desired result.",
"It remains to prove the claim (REF ).",
"Here, we estimate $|\\langle e^{ikty} \\tilde{u}, \\nu (t) \\rangle | \\lesssim \\log (t) \\Vert u\\Vert _{H^{1}} + \\int _{0}^{t}\\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert \\frac{e^{ik(t-\\tau )y}-1}{iky}\\Vert _{L^{2}} \\Vert \\partial _{\\tau } S(t-\\tau ,0) u\\Vert _{L^{2}} d\\tau .$ Using Lemma REF and Proposition REF we control $\\Vert \\partial _{\\tau } S(t-\\tau ,0) u\\Vert _{L^{2}} \\le <t-\\tau >^{-2}|||S(t-\\tau ,0)||| \\Vert u\\Vert _{H^{2}} \\lesssim <t-\\tau >^{-3/2} \\Vert u\\Vert _{H^{2}},$ and we directly compute that $\\Vert \\frac{e^{ik(t-\\tau )y}-1}{iky}\\Vert _{L^{2}} \\lesssim \\sqrt{t-\\tau }.$ Hence, we control $\\int _{0}^{t}\\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert \\frac{e^{ik(t-\\tau )y}-1}{iky}\\Vert _{L^{2}} \\Vert \\partial _{\\tau } S(t-\\tau ,0) u\\Vert _{L^{2}} d\\tau \\\\\\lesssim \\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert u\\Vert _{H^{2}} \\int _{0}^{t} <t-\\tau >^{-1} d\\tau \\le \\Vert \\tilde{u}\\Vert _{L^{\\infty }} \\Vert u\\Vert _{H^{2}} \\log (t),$ which proves the claim.",
"appendix Auxiliary functions and boundary evaluations In this section we introduce several auxiliary functions, which can be used to compute boundary evaluations of of derivatives of $L_tW$ and related quantities.",
"Lemma 1.1 Let $u_{1},u_{2}$ be solutions of $\\begin{split}\\mathcal {E}_{t}u&=0, \\\\z &\\in [a,b],\\end{split}$ with boundary values $\\begin{pmatrix}u_{1}(a) & u_{2}(a) \\\\ u_{1}(b) & u_{2}(b)\\end{pmatrix}=\\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}.$ Let further $\\tilde{u}_{1},\\tilde{u}_{2}$ be solutions to the adjoint problem $\\begin{split}\\mathcal {E}_{t}^{*}\\tilde{u}:= \\left(\\left((\\frac{\\partial _{y}}{k}-it)g(y)\\right)^{2}- (\\frac{\\partial _{y}}{k}-it)\\frac{g(y)}{kr(y)}-\\frac{1}{r^{2}(y)}\\right)\\tilde{u}&=0, \\\\y &\\in [a,b],\\end{split}$ with boundary values $\\begin{pmatrix}\\tilde{u}_{1}(a) & \\tilde{u}_{2}(a) \\\\\\tilde{u}_{1}(b) & \\tilde{u}_{2}(b)\\end{pmatrix}=\\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}.$ Then $u_{1}, u_{2}, \\tilde{u}_{1}, \\tilde{u}_{2}$ satisfy $\\begin{split}u_{1}(t,r,k)&=e^{ikt(y-a)}u_{1}(0,r,k), \\\\u_{2}(t,r,k)&=e^{ikt(y-b)}u_{2}(0,r,k), \\\\\\tilde{u}_{1}(t,r,k)&=e^{ikt(y-a)}\\tilde{u}_{1}(0,r,k), \\\\\\tilde{u}_{2}(t,r,k)&=e^{ikt(y-b)}\\tilde{u}_{2}(0,r,k).\\end{split}$ We note that the operators in equations (REF ) and (REF ) are obtained by conjugating by $e^{iktz}$ and are complex linear.",
"The result hence follows by noting that multiplication by $e^{ikt(y-a)}$ or $e^{ikt(y-b)}$ is compatible with the boundary conditions (REF ) and (REF ).",
"Lemma 1.2 Let $W$ be a given function and let $\\Phi $ be a solution of $\\begin{split}\\mathcal {E}_{t}\\Phi &=W, \\\\\\Phi |_{y=a,b}&=0,\\end{split}$ and let $u_{1},u_{2},\\tilde{u}_{1}, \\tilde{u}_{2}$ be as in Lemma REF .",
"Define $\\begin{split}H^{(1)}&= \\frac{k^{2}}{g^{2}(a)}\\langle \\Phi , \\tilde{u}_{1} \\rangle _{L^{2}}u_{1} +\\frac{k^{2}}{g^{2}(b)}\\langle \\Phi , \\tilde{u}_{2} \\rangle _{L^{2}}u_{2}, \\\\\\Phi ^{(1)}&= \\partial _{r}\\Phi - H^{(1)}\\end{split}$ Then $\\Phi $ satisfies $\\begin{split}\\langle W, \\tilde{u}_{1} \\rangle _{L^{2}}&= \\frac{g^{2}(a)}{k^{2}}\\partial _{r}\\Phi (t,k,a) \\\\\\langle W, \\tilde{u}_{2} \\rangle _{L^{2}}&=\\frac{g^{2}(b)}{k^{2}}\\partial _{r}\\Phi (t,k,b),\\end{split}$ and $\\Phi ^{(1)}$ solves $\\begin{split}\\mathcal {E}_{t}\\Phi ^{(1)} &=\\partial _{y}W + \\left[\\mathcal {E}_{t}, \\partial _{y} \\right]\\Phi , \\\\\\Phi ^{(1)}|_{y=a,b}&=0.\\end{split}$ The function $H^{(1)}$ is a solution of (REF ) and is called the (first) homogeneous correction.",
"Testing the equation (REF ) with the homogeneous solutions of Lemma REF , the results follow by integration by parts and direct calculations.",
"Lemma 1.3 Let $\\Phi ,W$ as in Lemma REF and let $u_{1},u_{2},\\tilde{u}_{1}, \\tilde{u}_{2}$ as in Lemma REF .",
"Then $\\Phi $ satisfies $\\begin{split}\\frac{g^{2}(a)}{k^{2}}\\partial _{r}^{2}\\Phi (t,k,a)&= -\\frac{g(a)g^{\\prime }(a)}{k^{2}} \\partial _{r}\\Phi (t,k,a)- \\frac{g(y)}{k^{2}r(y)}\\partial _{y}\\Phi (t,k,a) + W(t,k,a), \\\\\\frac{g^{2}(a)}{k^{2}}\\partial _{r}^{2}\\Phi (t,k,b)&= -\\frac{g(b)g^{\\prime }(b)}{k^{2}} \\partial _{r}\\Phi (t,k,b)- \\frac{g(y)}{k^{2}r(y)}\\partial _{y}\\Phi (t,k,b) + W(t,k,b).\\end{split}$ Define $\\begin{split}H^{(2)}&= \\partial _{y}^{2}\\Phi (t,k,a)u_{1} + \\partial _{y}^{2}\\Phi (t,k,b)u_{2}, \\\\\\Phi ^{(2)}&= \\partial _{y}^{2}\\Phi - H^{(2)},\\end{split}$ then $\\Phi ^{(2)}$ satisfies $\\begin{split}((g(y)(\\frac{\\partial _{y}}{k}-it))^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)})\\Phi ^{(2)}\\\\=\\partial _{y}^{2}W + \\left[((g(y)(\\frac{\\partial _{y}}{k}-it))^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)}), \\partial _{y}^{2} \\right]\\Phi , \\\\\\Phi ^{(2)}|_{y=a,b}=0.\\end{split}$ The function $H^{(2)}$ is a solution of (REF ) and is called the (second) homogeneous correction.",
"Direct computation.",
"Duhamel's formula and shearing Lemma 2.1 (Time dependent Duhamel) Let $(L(t))_{t \\in \\mathbb {R}}$ be a given family of linear operators and denote by $S(t,t^{\\prime })$ the solution operator of $(\\partial _t + \\frac{ih}{k}L_t) a=0,$ mapping a prescribed $a(t^{\\prime })$ to $a(t)$ .",
"Then for any given function $F$ the unique solution of $(\\partial _t + \\frac{ih}{k}L_t) u&=F,\\\\u(0)&=u_0,$ is given by $u(t)= S(t,0)u_0 + \\int _{0}^{t} S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime }.$ Since $S(0,0)=Id$ , we observe that the such defined $u(t)$ satisfies $u(0)=u_0$ .",
"It remains to show that $u$ satisfies the equation.",
"We directly compute $(\\partial _t+ \\frac{ih}{k}L_t) u(t)&= (\\partial _t+\\frac{ih}{k}L_t)S(t,0)u_0 + \\int _{0}^{t} (\\partial _t +\\frac{ih}{k}L_t) S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime } + S(t,t)F(t)\\\\&= 0 + \\int _{0}^t 0 S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime } +\\text{Id} F(t) = F(t).$ Here we used that for any $t^{\\prime }$ $(\\partial _t +\\frac{ih}{k}L_t) S(t,t^{\\prime })=0.$ We stress that $(\\partial _t +\\frac{ih}{k}L_{\\tilde{t}}) S(t,t^{\\prime })$ does not vanish in general for any $\\tilde{t} \\ne t$ .",
"Applying Lemma REF to (REF ), we obtain that $\\nu (t)= \\omega _{0}(a) \\int _{0}^{t}S(t,\\tau ) e^{ik\\tau y}u d\\tau ,$ where $S(t,\\tau )$ is the solution operator corresponding to (REF ).",
"Since $L_{t}$ was defined by a conjugation of $L_{0}$ with $e^{ikty}$ , we can also conjugate $S(t,\\tau )$ .",
"Lemma 2.2 Let $\\sigma >0$ , then for any $0\\le s \\le \\tau \\le t$ the solution operator $S$ satisfies $S(t,\\tau )e^{ik \\sigma y} f = e^{ik \\sigma y} S(t-\\sigma ,\\tau -\\sigma ) f$ for any $f \\in L^{2}$ .",
"We note that for any $t$ $e^{-ik\\sigma y} \\mathcal {E}_{t}e^{ik\\sigma y} = \\mathcal {E}_{t-\\sigma }$ and that also $e^{ik\\sigma y} \\langle e^{ik \\sigma y}f, e^{ikty}u \\rangle e^{ikty}u= \\langle f, e^{ik(t-\\sigma )y}u \\rangle e^{ik(t-\\sigma )y}u.$ Hence, conjugating the equation by $e^{ikt\\sigma y}$ is equivalent to a shift in time, which yields the desired result."
],
[
"Auxiliary functions and boundary evaluations",
"In this section we introduce several auxiliary functions, which can be used to compute boundary evaluations of of derivatives of $L_tW$ and related quantities.",
"Lemma 1.1 Let $u_{1},u_{2}$ be solutions of $\\begin{split}\\mathcal {E}_{t}u&=0, \\\\z &\\in [a,b],\\end{split}$ with boundary values $\\begin{pmatrix}u_{1}(a) & u_{2}(a) \\\\ u_{1}(b) & u_{2}(b)\\end{pmatrix}=\\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}.$ Let further $\\tilde{u}_{1},\\tilde{u}_{2}$ be solutions to the adjoint problem $\\begin{split}\\mathcal {E}_{t}^{*}\\tilde{u}:= \\left(\\left((\\frac{\\partial _{y}}{k}-it)g(y)\\right)^{2}- (\\frac{\\partial _{y}}{k}-it)\\frac{g(y)}{kr(y)}-\\frac{1}{r^{2}(y)}\\right)\\tilde{u}&=0, \\\\y &\\in [a,b],\\end{split}$ with boundary values $\\begin{pmatrix}\\tilde{u}_{1}(a) & \\tilde{u}_{2}(a) \\\\\\tilde{u}_{1}(b) & \\tilde{u}_{2}(b)\\end{pmatrix}=\\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}.$ Then $u_{1}, u_{2}, \\tilde{u}_{1}, \\tilde{u}_{2}$ satisfy $\\begin{split}u_{1}(t,r,k)&=e^{ikt(y-a)}u_{1}(0,r,k), \\\\u_{2}(t,r,k)&=e^{ikt(y-b)}u_{2}(0,r,k), \\\\\\tilde{u}_{1}(t,r,k)&=e^{ikt(y-a)}\\tilde{u}_{1}(0,r,k), \\\\\\tilde{u}_{2}(t,r,k)&=e^{ikt(y-b)}\\tilde{u}_{2}(0,r,k).\\end{split}$ We note that the operators in equations (REF ) and (REF ) are obtained by conjugating by $e^{iktz}$ and are complex linear.",
"The result hence follows by noting that multiplication by $e^{ikt(y-a)}$ or $e^{ikt(y-b)}$ is compatible with the boundary conditions (REF ) and (REF ).",
"Lemma 1.2 Let $W$ be a given function and let $\\Phi $ be a solution of $\\begin{split}\\mathcal {E}_{t}\\Phi &=W, \\\\\\Phi |_{y=a,b}&=0,\\end{split}$ and let $u_{1},u_{2},\\tilde{u}_{1}, \\tilde{u}_{2}$ be as in Lemma REF .",
"Define $\\begin{split}H^{(1)}&= \\frac{k^{2}}{g^{2}(a)}\\langle \\Phi , \\tilde{u}_{1} \\rangle _{L^{2}}u_{1} +\\frac{k^{2}}{g^{2}(b)}\\langle \\Phi , \\tilde{u}_{2} \\rangle _{L^{2}}u_{2}, \\\\\\Phi ^{(1)}&= \\partial _{r}\\Phi - H^{(1)}\\end{split}$ Then $\\Phi $ satisfies $\\begin{split}\\langle W, \\tilde{u}_{1} \\rangle _{L^{2}}&= \\frac{g^{2}(a)}{k^{2}}\\partial _{r}\\Phi (t,k,a) \\\\\\langle W, \\tilde{u}_{2} \\rangle _{L^{2}}&=\\frac{g^{2}(b)}{k^{2}}\\partial _{r}\\Phi (t,k,b),\\end{split}$ and $\\Phi ^{(1)}$ solves $\\begin{split}\\mathcal {E}_{t}\\Phi ^{(1)} &=\\partial _{y}W + \\left[\\mathcal {E}_{t}, \\partial _{y} \\right]\\Phi , \\\\\\Phi ^{(1)}|_{y=a,b}&=0.\\end{split}$ The function $H^{(1)}$ is a solution of (REF ) and is called the (first) homogeneous correction.",
"Testing the equation (REF ) with the homogeneous solutions of Lemma REF , the results follow by integration by parts and direct calculations.",
"Lemma 1.3 Let $\\Phi ,W$ as in Lemma REF and let $u_{1},u_{2},\\tilde{u}_{1}, \\tilde{u}_{2}$ as in Lemma REF .",
"Then $\\Phi $ satisfies $\\begin{split}\\frac{g^{2}(a)}{k^{2}}\\partial _{r}^{2}\\Phi (t,k,a)&= -\\frac{g(a)g^{\\prime }(a)}{k^{2}} \\partial _{r}\\Phi (t,k,a)- \\frac{g(y)}{k^{2}r(y)}\\partial _{y}\\Phi (t,k,a) + W(t,k,a), \\\\\\frac{g^{2}(a)}{k^{2}}\\partial _{r}^{2}\\Phi (t,k,b)&= -\\frac{g(b)g^{\\prime }(b)}{k^{2}} \\partial _{r}\\Phi (t,k,b)- \\frac{g(y)}{k^{2}r(y)}\\partial _{y}\\Phi (t,k,b) + W(t,k,b).\\end{split}$ Define $\\begin{split}H^{(2)}&= \\partial _{y}^{2}\\Phi (t,k,a)u_{1} + \\partial _{y}^{2}\\Phi (t,k,b)u_{2}, \\\\\\Phi ^{(2)}&= \\partial _{y}^{2}\\Phi - H^{(2)},\\end{split}$ then $\\Phi ^{(2)}$ satisfies $\\begin{split}((g(y)(\\frac{\\partial _{y}}{k}-it))^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)})\\Phi ^{(2)}\\\\=\\partial _{y}^{2}W + \\left[((g(y)(\\frac{\\partial _{y}}{k}-it))^{2}+ \\frac{g(y)}{kr(y)}(\\frac{\\partial _{y}}{k}-it)-\\frac{1}{r^{2}(y)}), \\partial _{y}^{2} \\right]\\Phi , \\\\\\Phi ^{(2)}|_{y=a,b}=0.\\end{split}$ The function $H^{(2)}$ is a solution of (REF ) and is called the (second) homogeneous correction.",
"Direct computation."
],
[
"Duhamel's formula and shearing",
"Lemma 2.1 (Time dependent Duhamel) Let $(L(t))_{t \\in \\mathbb {R}}$ be a given family of linear operators and denote by $S(t,t^{\\prime })$ the solution operator of $(\\partial _t + \\frac{ih}{k}L_t) a=0,$ mapping a prescribed $a(t^{\\prime })$ to $a(t)$ .",
"Then for any given function $F$ the unique solution of $(\\partial _t + \\frac{ih}{k}L_t) u&=F,\\\\u(0)&=u_0,$ is given by $u(t)= S(t,0)u_0 + \\int _{0}^{t} S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime }.$ Since $S(0,0)=Id$ , we observe that the such defined $u(t)$ satisfies $u(0)=u_0$ .",
"It remains to show that $u$ satisfies the equation.",
"We directly compute $(\\partial _t+ \\frac{ih}{k}L_t) u(t)&= (\\partial _t+\\frac{ih}{k}L_t)S(t,0)u_0 + \\int _{0}^{t} (\\partial _t +\\frac{ih}{k}L_t) S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime } + S(t,t)F(t)\\\\&= 0 + \\int _{0}^t 0 S(t,t^{\\prime }) F(t^{\\prime }) dt^{\\prime } +\\text{Id} F(t) = F(t).$ Here we used that for any $t^{\\prime }$ $(\\partial _t +\\frac{ih}{k}L_t) S(t,t^{\\prime })=0.$ We stress that $(\\partial _t +\\frac{ih}{k}L_{\\tilde{t}}) S(t,t^{\\prime })$ does not vanish in general for any $\\tilde{t} \\ne t$ .",
"Applying Lemma REF to (REF ), we obtain that $\\nu (t)= \\omega _{0}(a) \\int _{0}^{t}S(t,\\tau ) e^{ik\\tau y}u d\\tau ,$ where $S(t,\\tau )$ is the solution operator corresponding to (REF ).",
"Since $L_{t}$ was defined by a conjugation of $L_{0}$ with $e^{ikty}$ , we can also conjugate $S(t,\\tau )$ .",
"Lemma 2.2 Let $\\sigma >0$ , then for any $0\\le s \\le \\tau \\le t$ the solution operator $S$ satisfies $S(t,\\tau )e^{ik \\sigma y} f = e^{ik \\sigma y} S(t-\\sigma ,\\tau -\\sigma ) f$ for any $f \\in L^{2}$ .",
"We note that for any $t$ $e^{-ik\\sigma y} \\mathcal {E}_{t}e^{ik\\sigma y} = \\mathcal {E}_{t-\\sigma }$ and that also $e^{ik\\sigma y} \\langle e^{ik \\sigma y}f, e^{ikty}u \\rangle e^{ikty}u= \\langle f, e^{ik(t-\\sigma )y}u \\rangle e^{ik(t-\\sigma )y}u.$ Hence, conjugating the equation by $e^{ikt\\sigma y}$ is equivalent to a shift in time, which yields the desired result."
]
] | 1605.05959 |
[
[
"A Microwave Josephson Refrigerator"
],
[
"Abstract We present a microwave quantum refrigeration principle based on the Josephson effect.",
"When a superconducting quantum interference device (SQUID) is pierced by a time-dependent magnetic flux, it induces changes in the macroscopic quantum phase and an effective finite bias voltage appears across the SQUID.",
"This voltage can be used to actively cool well below the lattice temperature one of the superconducting electrodes forming the interferometer.",
"The achievable cooling performance combined with the simplicity and scalability intrinsic to the structure pave the way to a number of applications in quantum technology."
],
[
" Introduction",
"One of the key lessons we learn from themodynamics is that in order to extract heat from a system we must spend energy in the form of work.",
"We can thereby use this effect to cool that system.",
"However, as soon we generate a thermal gradient between the system and its surrounding, a heat flow opposite to the thermal gradient tends to restore thermodynamic equilibrium.",
"For these reasons, to be of practical use, we must be able to sustain over time the thermal gradient by performing continuously work on the system.",
"The simplest way to do this is to cyclicly drive the system out-of-equilibrium.",
"These principles are at the basis of any (macroscopic or microscopic) thermal machine and refrigerator.",
"In the push towards device miniaturization and quantum technologies, it has become of pivotal importance the realization of high-performance nanoscale electronic coolers [1], [2].",
"There are several successful solid-state quantum refrigeration schemes exploiting superconductors most of which are based either on normal metal-insulator-superconductor (NIS) [3], [4], [5], [6], [7], [8], [9], [10], [11] or superconductor-insulator-superconductor (SIS) [12], [13], [14], [15] tunnel junctions, even in combination with magnetic elements [16], [17], [18], [19].",
"In such systems, electronic refrigeration occurs thanks to the presence of the energy gap in the superconducting density of states.",
"The latter provides an effective energy-filtering mechanism yielding a substantial electron cooling upon voltage biasing the tunnel junction near the gap edge [1], [2].",
"Here, we propose and analyze the concept for a microwave Josephson refrigerator (MJR) [see Fig.",
"REF a)].",
"The structure we envision is a superconducting quantum interference device (SQUID) which allows us to control the dynamics of the macroscopic quantum phase through an externally applied time-dependent microwave magnetic field.",
"The operating principle of this refrigeration method is based on the recent discovery that a driven SQUID can generate intense voltage pulses [20], [21], [22].",
"These voltage pulses can be used to actively transfer heat from one superconductor to the other and, therefore, to cool one of them.",
"The whole process is fully phase-coherent since it critically depends on the induced dynamics of the superconducting phase.",
"Figure: a) A superconducting quantum interference device (SQUID) pierced by a time-dependent magnetic flux Φ(t)\\Phi (t).S i S_i, Δ i \\Delta _i, T i T_i, ϕ i \\varphi _i denote the superconductor, the energy gap, the temperature and the superconducting phase difference, respectively.I bias I_{bias} is the dissipationless supercurrent used to bias the interferometer.b) Equivalent electric circuital description of the SQUID.",
"The parameter CC, R T R_T, L J L_J are the total capacitance, resistance and Josephson inductance of the SQUID, respectively; I bias I_{bias} is the biasing current and V(t)V(t) is the effective voltage generated by the external drive.c) Critical current I C I_C versus magnetic flux Φ\\Phi for an asymmetric SQUID (black line).",
"I + I_+ is the maximum critical current of the SQUID, and Φ 0 \\Phi _0 is the flux quantum.The red curve represents the modulation of the magnetic flux centered around one of the interference nodes.d) Heat current flowing through a Josephson junction vs bias voltage VV for Δ 2 /Δ 1 =3.3\\Delta _2/\\Delta _1=3.3: P qp P_{qp}, P cos P_{\\cos } and P sin P_{\\sin } are represented in solid blue, red dashed and purple dotted curves, respectively.The yellow shaded region in the figure denotes the working voltage interval in the discussed example.Here, ℰ=Δ 2 2 /(e 2 R T i )\\mathcal {E}=\\Delta _2^2/(e^2 R_{T_i}), Δ 2 =200μ\\Delta _2 = 200~\\mu eV, and R T i R_{T_i} is the normal-state resistance of each Josephson junction.For a realistic structure, we obtain sizeable cooling performance.",
"The superconducting electronic temperature can be reduced from $70\\%$ to $40\\%$ depending on the fabrication parameters, and the temperature working regime.",
"In particular, the MJR behavior depends on the resistance and capacitance of the SQUID junctions as well as on the gap engineering of the two superconductors forming the interferometer.",
"Depending on the final purpose these can be, in principle, fine-tuned to optimize this refrigeration scheme.",
"Important advantages of this cooling structure stem from the simplicity of its design, from its scalability, and from the fact it can be operated at a distance.",
"As a matter of fact, the only requirement is an external microwave magnetic field.",
"These facts open the way to a number of possible applications.",
"Connected to other quantum devices it can be used to remotely cool down them.",
"Furthermore, arrays of parallel MJRs can be engineered to extract heat from large devices, and to quickly and efficiently cool down them.",
"The paper is organized as follows.",
"In Sec.",
"and we discuss the device heat transport properties and its dynamics, respectively.",
"In Sec.",
", by using a power balance equation, we estimate the device cooling performances.",
"Section contains our conclusions."
],
[
" Heat current",
"We consider a SQUID [as shown in Fig.",
"REF a)] composed by two different superconductors $S_1$ and $S_2$ .",
"Its electric and thermal state is characterized by the two superconducting phases $\\varphi _1$ and $\\varphi _2$ across the Josephson junctions (JJs).",
"We neglect the inductance of the superconducting loop so that the phases are related by the flux quantization condition, $\\varphi _1 - \\varphi _2 + 2 \\pi \\Phi /\\Phi _0 = 2 \\pi n$ , where $n$ is an integer, $\\Phi $ is the applied magnetic flux through the SQUID and $\\Phi _0\\simeq 2\\times 10^{-15}$ Wb is the flux quantum.",
"The SQUID is connected to a generator that supplies a small non-dissipative bias current $I_{bias}$ [see Fig.",
"REF a) and -b)].",
"Its only purpose is to give a preferred direction for the dynamics of the phase [20], [21], [22].",
"A part from this, there is no need for additional connections, and the device can therefore be isolated.",
"The coherent thermal transport properties of Josephson tunnel junctions have been studied both theoretically [23], [24], [25], [26], [27], [28] and experimentally [29], [30], [31], [32].",
"We denote with $V$ the voltage drop across the JJs, with $\\Delta _i$ and $T_i$ the energy gap and the temperature of the $i$ -th superconductor, respectively.",
"The heat current $P_i(t)$ flowing between two tunnel-coupled superconductors $S_1$ and $S_2$ through the $i$ -th JJ consists of three contributions [23], [24], [25], [26], [27], [28], $P_i(t)= P_{qp,i} (V) + P_{\\cos ,i} (V) \\cos \\varphi _i(t)+P_{\\sin ,i} (V) \\sin \\varphi _i(t).$ The powers $P_{qp,i} (V)$ , $P_{\\cos ,i} (V)$ and $P_{\\sin ,i} (V)$ are the quasi-particle, and the anomalous heat currents, respectively.",
"They read $P_{qp,i} (V) &=& \\frac{1}{e^2 R_{T_i}} \\int dE~ N_1(E-e V) N_2(E) (E- e V) [f_1(E- e V) - f_2(E)] \\nonumber \\\\P_{\\cos ,i} (V) &=& - \\frac{1}{e^2 R_{T_i}} \\int dE ~N_1(E-e V) N_2(E) \\frac{\\Delta _1 \\Delta _2}{E} [f_1(E- e V) - f_2(E)] \\nonumber \\\\P_{\\sin ,i} (V) &=& \\frac{e V}{2 \\pi e^2 R_{T_i}} \\int d\\epsilon _1 \\int d\\epsilon _2 \\frac{\\Delta _1 \\Delta _2}{E_2}\\Big [ \\frac{1-f_1(E_1) - f_2(E_2)}{(E_1+E_2)^2-e^2 V^2 } +\\frac{f_1(E_1) - f_2(E_2)}{(E_1-E_2)^2-e^2 V^2 } \\Big ].$ Here, $f_j(E) = 1/(1+ e^{E/k_BT_j})$ is the Fermi distribution function, $ N_j(E) = \\left|\\Re {\\rm e} \\left[ \\frac{E + i \\gamma }{\\sqrt{[E+i \\gamma ]^2 -\\Delta _j^2}} \\right]\\right|$ is the smeared BCS density of states, $\\gamma =10^{-4}\\Delta _2$ is the Dynes broadening parameter [33], [5], and $R_{T_i}$ is the normal-state resistance of each junction composing the SQUID.",
"By choosing superconductors with different energy gaps allows us to create a thermal asymmetry in the structure [34], [35], [36].",
"Its effect is captured by the asymmetry parameter $r=\\Delta _2/\\Delta _1$ which, in the MJR, has the purpose to improve and optimize the device performance.",
"An example of the heat current contributions (REF ) vs bias voltage across the SQUID is shown in Fig.",
"REF d) for $r=3.3$ and $T_2=T_1$ .",
"To optimize the heat transport we will focus in the region around $V=(\\Delta _2-\\Delta _1)/e$ where the quasi-particle and cosine are maximal.",
"Equations (REF ) remain valid even in presence of a time-dependent voltage if the quasi-particles can be considered in a Fermi distribution, i.e., not out of equilibrium.",
"This assumption is at the basis of any calculation and experiment based on the Josephson effect and it is usually satisfied even in presence of fast oscillating voltage [37], [38].",
"Since the physical process underlying the heat and the charge transport is the same [38], the above equations should have similar vast range of validity."
],
[
"SQUID dynamics",
"The dynamics of a driven SQUID can be complex and must be solved numerically.",
"We rely on the driven resistively and capacitively shunted Josephson junction (RCSJ) equation [20], [21], [22].",
"By introducing the phase $\\varphi = (\\varphi _1+\\varphi _2)/2$ , the Josephson current flowing through the SQUID $I_J= I_{c_1} \\sin \\varphi _1 + I_{c_2} \\sin \\varphi _2$ can be written as $I_J[\\varphi ;\\phi (\\tau )] = I_+ [\\cos \\phi \\sin \\varphi + \\mathcal {R}~ \\sin \\phi \\cos \\varphi ]$ .",
"Here, $\\phi = \\pi \\Phi /\\Phi _0$ is the normalized applied magnetic flux, $I_+ = I_{c_1} + I_{c_2}$ , $\\mathcal {R}= (I_{c_1} - I_{c_2})/ (I_{c_1}+I_{c_2})=(R_{T_2}-R_{T_1})/ (R_{T_1}+R_{T_2})$ (assuming that $I_{c_i} \\propto 1/R_{T_i}$ ) and $I_{ci}$ is the critical current of the $i$ -th junction.",
"We assume junctions with the same capacitance $C_i$ but different critical current and resistance.",
"We can write the RCSJ equation as $\\frac{\\hbar C}{2 e } \\ddot{{\\varphi }} + \\frac{\\hbar }{2 e R_T } \\dot{\\varphi } + \\frac{\\hbar \\mathcal {R}}{2 e R_T } \\dot{\\phi } + I_J[\\varphi ;\\phi (\\tau )] = I_{bias}$ where $C=C_1+ C_2=2 C_1$ and $R_T = R_{T_1} R_{T_2}/(R_{T_1}+R_{T_2})$ are the total SQUID capacitance and resistance, respectively (see Appendix ).",
"The solution of Eq.",
"(REF ) combined with the flux quantization condition gives immediately the dynamics of $\\varphi _i$ and, through the Josephson relation, the voltage generated across the junctions ($V$ ).",
"From these, we can directly calculate the time-dependent heat current transferred across the $i$ -th junction $P_i$ , and the total heat current flowing through the SQUID as $P = \\sum _{i=1,2} P_i$ [39].",
"Since the system is driven and has, in general, a complex dynamics, the relevant quantity is the average power transferred within a time interval $t_0$ .",
"This is obtained by calculating the heat transferred as $P_{av} (t)= (1/ t_0) \\int _t^{t+t_0} dt P$ .",
"In the following, we consider a simple monochromatic drive of the magnetic field, $ \\Phi (t) = \\Phi _M \\cos \\left( 2 \\pi \\nu t \\right) + \\Phi _m $ , where $\\nu $ is the drive frequency, while $\\Phi _M$ and $\\Phi _m$ are the maximum and minimum magnetic flux, respectively.",
"Figure: Time dependence of the voltage VV a) and of the heat current PP b) trough the SQUID for different tunneling resistances R T =7,6,5R_T=7, 6, 5~Ohm.The parameters are C=100C=100~pF, normalized bias current I bias /I + =2×10 -2 I_{bias}/I_+=2\\times 10^{-2}, and drive frequency ν=1\\nu =1~GHz for a SQUID with junction asymmetry of ℛ=0.05\\mathcal {R}=0.05.",
"The voltage dynamics is averaged over the heat transport time τ heat =10\\tau _{heat}=10~ns.The inset in panel b) shows the oscillating behaviour of the heat current for R T =7R_T=7~Ohm.The resistances are expressed in Ohm.The dynamics of the phase $\\varphi $ has very different behaviors depending on the parameters of the SQUID and the external drive.",
"For our purpose, they can be distinghuised by the presence or the absence of phase jumps [20], [21], [22].",
"We can select a priori which dynamics to induce by driving the magnetic flux across or avoiding an interference node of the critical current at $n \\Phi _0/2$ , as shown in Fig.",
"REF c) [37].",
"In the absence of crossing, the phase dynamics follows the drive modulation.",
"In such a case, from the Josephson relation, the effective voltage appearing across the SQUID is quite small (i.e., of the order of a fraction of $\\mu V\\sim 10^{-3}\\Delta _2/e$ for $1~$ GHz frequency drive).",
"For a realistic drive source, it therefore leads to a somewhat limited heat transfer across the interferometer preventing an efficient electron cooling [see Fig.",
"REF d)].",
"The alternative choice is to let the magnetic flux to cross an interference node, as displayed in Fig.",
"REF c).",
"Recently, it has been shown [20], [21], [22] that when this occurs the superconducting phase undergoes a sequence of fast $\\pi $ jumps.",
"This corresponds to sharp voltage pulses developed across the SQUID junctions.",
"Under this condition, a moderate frequency drive can generate from some hundreds to thousands of higher harmonics [20], [21], [22].",
"This frequency up-conversion allows us to reach sufficiently high voltages and, thus, the peaks in Fig.",
"REF d) where the heat current and the cooling power are maximized.",
"The other important ingredient is to include a finite SQUID capacitance $C$ of the order of $\\sim 50-100~$ pF.",
"The latter introduces a time scale $\\tau = R_TC$ in the RCSJ dynamics, and allows to sustain a large effective applied voltage across the interferometer over long times.",
"In other words, if $\\tau $ is large enough, near voltage pulses are broadened until they merge together leading to a constant effective voltage applied across the SQUID.",
"The dynamics of the interferometer depends on the combination of the SQUID intrinsic parameters, i.e., $C$ , $R_T$ , $\\mathcal {R}$ and $r$ , and on the external ones, i.e., $\\nu $ .",
"Even if the details may change, we always observe that the system, after an initial transient, reaches a stationary state characterized by constant voltage with fast oscillations superimposed [see Fig.",
"REF ].",
"These oscillations are usually fast with respect to the heat transport time $\\tau _{heat}$ between superconductors.",
"This can be estimated of the order of $\\sim 10-10^3~$ ns (see Ref.",
"[40] and Appendix ).",
"Therefore the effective voltage relevant for thermal transport is averaged over this time-scale.",
"Examples of the voltage dynamics average over $\\tau _{heat}=10~$ ns are presented in Fig.",
"REF a) for different values of the tunneling resistance.",
"Notably, the stationary voltage seems to depends on $R_T$ and $C$ but not on the driving frequency as soon as $\\nu >100~$ MHz.",
"In our calculation we set a standard drive frequency of $\\nu =1~$ GHz.",
"Such a frequency allows us to safely neglect the heating due to the photon-assisted tunneling induced by the drive, as this usually becomes relevant above frequencies of the order of $10~$ GHz [41].",
"The SQUID fabrication parameters can be tailored so that $V_{stat}$ is close to the matching peak at $(\\Delta _2-\\Delta _1)/e$ thereby maximizing the cooling power [see Fig.",
"REF d)] An example of the instantaneous transferred power $P$ is plotted in Fig.",
"REF b).",
"From this we can obtain the time averaged power $P_{av}$ in the stationary regime.",
"To calculate it we have taken different $t_0$ so to have $P_{av}$ independent of its specific value.",
"It is worthwhile to emphasize that in the stationary regime the phase grows linearly in time, i.e., $\\varphi \\propto V_{stat} t$ .",
"Therefore, the average cosine and sine heat current terms vanish, and thermal transport essentially occurs thanks to the quasi-particle contribution.",
"Yet, we stress that this effect is completely phase coherent since phase dynamics is the essential ingredient to enable heat transfer across the structure."
],
[
"Cooling performance",
"Let us now analyze the cooling performance achievable in a realistic structure.",
"We consider the electrode $S_2$ to be large enough so that it can be treated as having infinite heat capacitance, and to be well thermalized with the lattice phonons residing at bath temperature, $T_2=T_{bath}$ .",
"Differently, the superconducting lead $S_1$ is taken to be small so that any heat current flowing through it may easily change its temperature and cooled.",
"It is useful to first discuss the cooling process from the point of view of the SQUID alone.",
"If the two superconductors reside initially at the same temperature equal to the bath temperature, i.e., $T_1^{initial}=T_2=T_{bath}$ , the driving magnetic field leads to quasiparticle cooling in $S_1$ .",
"As soon as we establish a thermal gradient across the SQUID, the heat flows in the thermodynamical direction from the hotter to the colder superconductor.",
"The final stationary temperature and temperature gradient in the system is reached when these two competing effects balance each other.",
"In the above picture we have neglected the energy exchanged by quasiparticles in $S_1$ with the phononic bath.",
"The heat current flowing between the electrons at temperature $T_1$ and lattice phonons at temperature $T_{bath}$ is given by [42], [43] $P_{qp-ph} (T_1,T_{bath}) = -\\frac{ \\Sigma \\mathcal {V}}{ 96 \\zeta (5) k_B^2} \\int dE E \\int d\\epsilon \\epsilon ^2 {\\rm sgn} (\\epsilon ) L_{E,E+\\epsilon } \\nonumber \\\\\\times \\Big [ \\coth \\left( \\frac{\\epsilon }{2 k_B T_{bath}} \\right) \\left(f_E^{(1)} - f_{E+\\epsilon }^{(1)}\\right) - f_E^{(1)} f_{E+\\epsilon }^{(1)} +1 \\Big ], \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,$ where $f_E^{(1)} = f_1(-E) -f_1(E)$ , $L_{E,E^{\\prime }}= N(E) N(E^{\\prime }) (1-\\frac{\\Delta _1^2}{E E^{\\prime }})$ , $\\Sigma =2\\times 10^{-8}~{\\rm W}/({\\rm m}^3~{\\rm K}^5)$ is the electron-phonon coupling constant of Al [1], and $\\mathcal {V}=10^{-17}~{\\rm m}^3$ is the $S_1$ electrode volume.",
"Furthermore, additional heating can come through the radiative electron-photon heat exchange occurring between the two superconductors, $P_{\\gamma } (T_1,T_2,\\Phi )$ [44].",
"The latter can be written as $P_{\\gamma } (T_1,T_2,\\Phi )=\\int _0^{\\infty }\\frac{\\text{d}\\omega }{2\\pi }\\hbar \\omega \\mathcal {T}(\\omega ,T_1,T_2,\\Phi )[n(\\omega ,T_2)-n(\\omega ,T_1)],$ where $n(\\varepsilon ,T)=[\\text{exp}(\\varepsilon /k_BT)-1]^{-1}$ is the Bose-Einstein photons distribution at temperature $T$ , and $\\mathcal {T}(\\omega ,T_1,T_2,\\Phi )$ is the effective SQUID photonic transmission coefficient [44].",
"The final steady-state thermal balance equation which must solved in order to obtain $T_1$ is therefore $P_{av}(T_1,T_2)+ P_{qp-ph} (T_1,T_{bath})+P_{\\gamma }^{av} (T_1,T_2)=0$ [1].",
"Here, $P_{\\gamma }^{av} (T_1,T_2)=(1/t_0)\\int _t^{t+t_0}dt P_{\\gamma }(T_1,T_2,\\Phi )$ is the average transferred radiative power within a time interval $t_0$ .",
"Its time-dependency comes from the modulation of the magnetic flux $\\Phi $ .",
"Since $\\Phi $ is periodically driven close to $n \\Phi _0/2$ , $P_{\\gamma }\\approx P_{\\gamma }^{av} $ .",
"The performances of the MJR are shown in Fig.",
"REF .",
"In particular, Fig.",
"REF a) displays the final temperature $T_1^{min}$ as a function of $T_2=T_{bath}=T_1^i$ for different $\\Delta _2/\\Delta _1$ ratios.",
"As a prototypical refrigerator we choose the one with $\\Delta _2/\\Delta _1=3.3$ .",
"In this case, the cooling performance ranges from $\\Delta T_1=T_1^i-T_1^{min} =186~$ mk at $T_2=300~$ mK to $\\Delta T_1 =40~$ mk at $T_2=100~$ mK.",
"In general, the cooling process is more efficient at higher $T_2$ , and for strongly asymmetric superconductors, i.e., for large $\\Delta _2/\\Delta _1$ ratios.",
"However, it seems that for $\\Delta _2/\\Delta _1> 5$ the achievable minimum temperature tends to saturate and no substantial improvement in cooling is obtained [see the inset of Fig.",
"REF a)].",
"The final achievable minimum temperature $T_1^{min}$ as a function of $T_2=T_{bath}=T_1^i$ and for different $R_T$ values is shown in Fig.",
"REF b).",
"Here we notice a non-monotonic behaviour as a function of $R_T$ [see also the inset in Fig.",
"REF b)].",
"This can be explained on the basis of the heat current [see Fig.",
"REF d)] and the voltage dynamics [see Fig.",
"REF a)].",
"Keeping fixed all the other parameters, the stationary voltage increases with the resistance $R_T$ [see Fig.",
"REF a)].",
"Therefore, an increase in junction resistance allows us to reach the maximum cooling power [at $(\\Delta _2-\\Delta _1)/e$ ] for $R_T=7~$ Ohm [as shown in Fig.",
"REF b)].",
"From this value any further resistance enhancement would lead to a decrease of the heat current, and to a worsening of the cooling performance.",
"This feature is well-captured by the plot shown in the inset of Fig.",
"REF b).",
"The performance of the present cooling principle can be compared to that of other time-dependent refrigeration methods.",
"For instance, in Ref.",
"[45] with a Coulombic single-electron refrigerator (SER), $\\Delta T_1 \\sim 130~$ mk at $T_2=300~$ mK and $\\Delta T_1 \\sim 30~$ mk at $T_2=100~$ mK were in principle achievable.",
"Therefore, an optimized MJR may outperform the SER in all the considered temperature ranges.",
"Figure: a) Minimum achievable electron temperature T 1 min T_1^{min} vs T 2 =T bath T_2=T_{bath}.The curves refer to different superconducting gap ratios: Δ 2 /Δ 1 =2,2.5,3.3\\Delta _2/\\Delta _1=2, 2.5, 3.3 and 5 from top to bottom.The red dot in the Δ 2 /Δ 1 =5\\Delta _2/\\Delta _1=5 curve represents the temperature at which S 1 S_1 becomes a normal metal.Inset: minimum temperature T 1 min T_1^{min} reached as a function of Δ 2 /Δ 1 \\Delta _2/\\Delta _1 at T 2 =200T_2=200~mK.b) Minimum achievable electron temperature T 1 min T_1^{min} as a function of T 2 T_2 for different junction resistance R T R_T values (expressed in Ohm).Inset: minimum temperature T 1 min T_1^{min} reached as a function of R T R_T at T 2 =200T_2=200~mK.For these calculations we set Δ 2 /Δ 1 =3.3\\Delta _2/\\Delta _1=3.3.In addition, the MJR has other practical advantages.",
"First, the structure stands out for the simplicity of fabrication and control.",
"Second, the superconductor can be cooled at a distance.",
"Third, due to its scalability and flexibility, it can be assembled to respond to different needs.",
"For instance, one can envision a network of parallel MJRs yielding a large cooling power.",
"Yet, the exploitation of the MJR depends on the temperature one intends to achieve.",
"It can be used as direct electron cooler if we are planning to work at temperatures around 100 or $50~$ mK.",
"Alternatively, it can be used as an efficient intermediate-stage electron refrigerator."
],
[
"Conclusions",
"In summary, we have proposed a principle of coherent electron cooling based on the Josephson effect.",
"The microwave Josephson refrigerator is build from a SQUID made of superconductors with different gaps, and exploits the work performed by a microwave magnetic field to efficiently cool the superconductor with smaller energy gap.",
"The working principle stems from the dynamics induced in the macroscopic quantum phase by an external time-dependent magnetic drive.",
"The latter yields a finite effective voltage drop appearing across the SQUID which enables electron cooling of one of the superconductors.",
"Finally, the MJR can be tuned by fabrication to reach optimal cooling performance.",
"Fruitful discussions with C. Altimiras, S. Gasparinetti and A. Fornieri are gratefully acknowledged.",
"P.S.",
"and R.B.",
"have received funding from the European Union FP7/2007-2013 under REA grant agreement no 630925 – COHEAT and from MIUR-FIRB2013 – Project Coca (Grant No. RBFR1379UX).",
"F.G. acknowledges the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013)/ERC Grant agreement No.",
"615187-COMANCHE for partial financial support."
],
[
"Dynamics of the asymmetric SQUID",
"To describe the dynamics of the driven SQUID we rely on the resistively and capacitively shunted Josephson junction (RCSJ) equation.",
"The current $I_i$ flowing through the $i-$ th junction is [37] $I_i=\\frac{\\hbar C_i}{2 e } \\ddot{{\\varphi }}_i + \\frac{\\hbar }{2 e R_{T_i}} \\dot{\\varphi }_i + I_{J_i}[\\varphi _i;\\phi _i(\\tau )],$ where $C_i$ , $ R_{T_i}$ , $I_{J_i} = I_{c_i} \\sin \\varphi _i$ and $\\varphi _i$ are the capacitance, resistance, Josephson current and superconducting phase across the junction, respectively.",
"The parameter $I_{c_i}$ is the critical current of the $i$ -th junction.",
"The phases $\\varphi _i$ are related through the flux quantization condition $\\varphi _1 - \\varphi _2 + 2 \\pi \\Phi /\\Phi _0 = 2 \\pi n$ , where $n$ is an integer, $\\Phi $ is the applied magnetic flux through the SQUID and $\\Phi _0\\simeq 2\\times 10^{-15}$ Wb is the flux quantum.",
"By introducing the phase $\\varphi = (\\varphi _1+\\varphi _2)/2$ and the normalized applied magnetic flux $\\phi = \\pi \\Phi /\\Phi _0$ , we have that $\\varphi _1 = \\varphi + \\phi $ and $\\varphi _2 = \\varphi - \\phi $ .",
"If the SQUID is biased with a current $I_{bias}$ , the total current passing though the interferometer, i.e., $I_1+I_2=I_{bias}$ , can be written as $\\frac{\\hbar (C_1+C_2)}{2 e } \\ddot{{\\varphi }} + \\frac{\\hbar (C_1-C_2)}{2 e } \\ddot{{\\phi }} + \\frac{\\hbar (R_{T_1}+R_{T_2})}{2 e R_{T_1} R_{T_2} } \\dot{\\varphi } - \\frac{\\hbar (R_{T_1}-R_{T_2})}{2 e R_{T_1} R_{T_2} } \\dot{\\phi } + (I_{c_1} + I_{c_2}) \\cos \\phi \\sin \\varphi + (I_{c_1} - I_{c_2}) \\sin \\phi \\cos \\varphi = I_{bias}$ The asymmetry in the SQUID is captured by the factor $\\mathcal {R}= (I_{c_1} - I_{c_2})/ (I_{c_1}+I_{c_2}) =(R_{T_2}-R_{T_1})/ (R_{T_1}+R_{T_2})$ (assuming that $I_{c_i} \\propto 1/R_{T_i}$ ).",
"Assuming that $C_1=C_2$ and introducing the parameter $I_+ = I_{c_1} + I_{c_2}$ , $C=C_1+C_2= 2 C_1$ , $R_T = R_{T_1} R_{T_2}/(R_{T_1}+R_{T_2})$ , we have $\\frac{\\hbar C}{2 e } \\ddot{{\\varphi }} + \\frac{\\hbar }{2 e R_T } \\dot{\\varphi } + \\frac{\\hbar \\mathcal {R}}{2 e R_T } \\dot{\\phi } + I_+ \\left( \\cos \\phi \\sin \\varphi + \\mathcal {R} \\sin \\phi \\cos \\varphi \\right) = I_{bias}.$"
],
[
"Heat transfer time-scale",
"For a driven system the voltage applied to the device can be obtained by Eq.",
"(REF ).",
"We consider a simple monochromatic drive of the magnetic field, $ \\Phi (t) = \\Phi _M \\cos \\left( 2 \\pi \\nu t \\right) + \\Phi _m $ , where $\\nu =1~$ GHz is the drive frequency, while $\\Phi _M$ and $\\Phi _m$ are the maximum and minimum magnetic flux, respectively.",
"The drive is taken to cross the critical current interference node at $\\Phi _0/2$ .",
"Under these conditions, the phase dynamics can be complex as discussed in Refs.",
"[20], [21], [22] and higher harmonics of the fundamental frequency can be generated.",
"For this reason, the voltage across the SQUID can show modulation over short time-scales (below $1~$ ns).",
"However, the interferometer cannot react and transport heat over such time scale and, thus, the effective voltage for thermal transport in Eqs.",
"(REF ) in the main text is averaged over a heat transport time-scale $\\tau _{heat}$ .",
"This thermal time-scale can be estimated in the following way.",
"The electronic entropy of the small superconductor $S_1$ is given by [40] $S &=& -4 k_B N_F \\mathcal {V} \\int _0^{\\infty } d \\epsilon N(E) \\Big [ (1- f(\\epsilon , T_1)) \\log (1- f(\\epsilon , T_1)) \\nonumber \\\\&&+(f(\\epsilon , T_1) \\log (f(\\epsilon , T_1))\\Big ],$ where $k_B$ is the Boltzmann constant, $ N_F$ is the density of states at the Fermi energy, $\\mathcal {V}$ and $ \\Delta _1$ are the volume and the energy gap of the superconducting electrode $S_1$ .",
"The function $f(\\epsilon , T_1)$ is the Fermi-Dirac energy distribution, $N(E) = \\left|\\Re {\\rm e} \\left[ \\frac{E + i \\gamma }{\\sqrt{[E+i \\gamma ]^2 -\\Delta _j^2}} \\right]\\right|$ is the smeared BCS density of states, and $\\gamma $ is the Dynes broadening parameter.",
"At temperature $T_1$ , the heat transferred is $Q= S T_1$ .",
"By supposing a constant quasi-particle heat current $P_{qp}$ , we have that $Q= P_{qp} \\tau _{heat}$ .",
"To provide an estimate for $\\tau _{heat}$ we can take $S \\propto 4 k_B N_F \\mathcal {V} \\Delta _1$ and $P_{qp} \\propto \\Delta _1^2/(e^2 R_T)$ , so that we obtain $\\tau _{heat} = \\frac{4 k_B N_F \\mathcal {V} e^2 R_T}{\\Delta _1} T_1.$ By taking $N_F = 10^{47} J^{-1}m^{-3}$ , $\\mathcal {V}= 10^{-17}$ m$^3$ , $R_T=1-10~$ Ohm, $\\Delta _1\\approx 3.4-0.6 ~\\times 10^{-23}~J$ and $T_1=100~$ K, we get $\\tau _{heat} \\approx 10-10^3~$ ns.",
"For the numerical analysis we have chosen the lowest value of $\\tau _{heat}=10~$ ns."
]
] | 1605.05884 |
[
[
"Nonlinear trend removal should be carefully performed in heart rate\n variability analysis"
],
[
"Abstract $\\bullet$ Background : In Heart rate variability analysis, the rate-rate time series suffer often from aperiodic non-stationarity, presence of ectopic beats etc.",
"It would be hard to extract helpful information from the original signals.",
"10 $\\bullet$ Problem : Trend removal methods are commonly practiced to reduce the influence of the low frequency and aperiodic non-stationary in RR data.",
"This can unfortunately affect the signal and make the analysis on detrended data less appropriate.",
"$\\bullet$ Objective : Investigate the detrending effect (linear \\& nonlinear) in temporal / nonliear analysis of heart rate variability of long-term RR data (in normal sinus rhythm, atrial fibrillation, 15 congestive heart failure and ventricular premature arrhythmia conditions).",
"$\\bullet$ Methods : Temporal method : standard measure SDNN; Nonlinear methods : multi-scale Fractal Dimension (FD), Detrended Fluctuation Analysis (DFA) \\& Sample Entropy (Sam-pEn) analysis.",
"$\\bullet$ Results : The linear detrending affects little the global characteristics of the RR data, either 20 in temporal analysis or in nonlinear complexity analysis.",
"After linear detrending, the SDNNs are just slightly shifted and all distributions are well preserved.",
"The cross-scale complexity remained almost the same as the ones for original RR data or correlated.",
"Nonlinear detrending changed not only the SDNNs distribution, but also the order among different types of RR data.",
"After this processing, the SDNN became indistinguishable be-25 tween SDNN for normal sinus rhythm and ventricular premature beats.",
"Different RR data has different complexity signature.",
"Nonlinear detrending made the all RR data to be similar , in terms of complexity.",
"It is thus impossible to distinguish them.",
"The FD showed that nonlinearly detrended RR data has a dimension close to 2, the exponent from DFA is close to zero and SampEn is larger than 1.5 -- these complexity values are very close to those for 30 random signal.",
"$\\bullet$ Conclusions : Pre-processing by linear detrending can be performed on RR data, which has little influence on the corresponding analysis.",
"Nonlinear detrending could be harmful and it is not advisable to use this type of pre-processing.",
"Exceptions do exist, but only combined with other appropriate techniques to avoid complete change of the signal's intrinsic dynamics.",
"35 Keywords $\\bullet$ heart rate variability $\\bullet$ linear / nonlinear detrending $\\bullet$ complexity analysis $\\bullet$ mul-tiscale analysis $\\bullet$ detrended fluctuation analysis $\\bullet$ fractal dimension $\\bullet$ sample entropy;"
],
[
"Introduction",
"Heart rate variability (HRV) has been since long time a standard method to evaluate the heart's performance.",
"In 1996, the Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology standardized the HRV measurement, its physiological interpretation and clinical use [1].",
"The fluctuation of intervals between normal heartbeats are mediated by autonomic inputs to the sinus node [2].",
"This means that by analyzing these fluctuations, information about the cardiac autonomic modulation and its changes can be obtained.",
"HRV reflects these fluctuations, more precisely, the phasic modulation of heart rate.",
"In pathological cases, the heart loses (part of) its central modulation capability of heart rate or there would be a lack of response of the sinus node.",
"So, the HRV values will be lower than in normal case.",
"There exists several ways to analyze the variability of heart rate.",
"The basic ones are the traditional statistics in Time domain, like SDNN (the standard deviation of all N–N intervals; SDANN (the standard deviation of the average of N–N intervals).",
"These measures have been clinically proven useful.",
"Analysis can also be performed in Frequency-domain: high–frequency power; low–frequency power; very–low–frequency band; ultra–low–frequency band; total power [2], [3], [4], [5].",
"Due to the strong nonlinear properties of this type of signals, methods of the third class based on nonlinear dynamics have been shown to be more robust [6].",
"They can be divided into several families [7], [8] : symbolic dynamics [9], [10], entropy [11], [12], fractality-multifractality [13], [14], predictability [15], empirical mode decomposition [16], [17], and Poincaré plots [18].",
"The physiological signals are often noised by perturbations which could arise from electrode changes due to perspiration, movement and respiration, from the electronic data acquisition systems themselves or be interferences from other organs.",
"It is thus necessary to reduce these noises, especially in case of electrocardiological signals analysis where the baseline interferences are an unavoidable preprocessing step.",
"Many techniques exist to remove these trends [19], [20], [21], [22].",
"In HRV analysis, the RR time series suffer often from aperiodic non-stationarity, presence of ectopic beats.",
"It would be hard to extract helpful information from the original signals.",
"It is also necessary to perform the above-mentioned preprocessing – trend removal which is commonly practiced [23], [24], [25], [26], [27], [28], [29].",
"The typical detrending in HRV analysis can be divided into two classes : linear (first order or higher polynomial model [26], [30], moving polynomial model [24], [27]) and nonlinear detrending (smoothness priors method [28], wavelet [29], wavelet packets [31], nonparametric regression [19], [23]).",
"The objective remained the same : removing the low frequency and aperiodic non-stationary components and rejecting low periodic non-sinusoidal activity which may have higher frequency harmonics [24].",
"However, this pre-processing step by detrending RR time series can affect the following analysis.",
"In many cases, the influence can be dramatic.",
"Comparison studies of the detrending influence in HRV analysis can be found in literature [32], [29], [31], [33], [34].The results are all interesting, but the conclusions often varied from on study to another or are sometimes contradictory.",
"For example, detrending (linear or nonlinear) is suggested in study but not in another one; nonlinear analysis of detrended RR time series is not recommended in some studies, but is considered as a good practice to discriminate the different heart rate data in others'.",
"Though in reality, all these results are true in their own dataset conditions (dependence of data samples, pathology) / methodology (methods choices, combination of other pre-processing steps), it is still a little confusing.",
"From a point-view of pure signal processing, the linear detrending is tolerable in HRV analysis.",
"Since linear detrending would not technically change too much intrinsic dynamics of the signal (especially in case of electrocardiological signals), it should not affect largely the analysis either in linear or nonlinear studies.",
"But for nonlinear detrending, it should be performed carefully in both linear/nonlinear studies.",
"In biomedical signal, it is the rich fluctuations which consists the true value of the signal.",
"The nonlinear trend removal is in fact the same process of signal denoising.",
"Removing the nonlinear trend, is somehow to remove the fundamental dynamics from the signal, what's left can be considered as “noise”.",
"Analyzing these “nonlinearly detrended signal” could lead to some wrong conclusions.",
"If in some cases where the nonlinear detrending is required, the detrending should be appropriately determined.",
"So, the objective of this work is to study the linear / nonlinear detrending effects on linear / nonlinear HRV analysis, with long-term & large datasets for different pathologies to quantify the effects and to explain qualitatively the effects."
],
[
"Data",
"The data used in HRV analysis can be on either short-term (like 500 seconds, 10 minutes, 1024 R-R intervals) or long-term data (24 hours).",
"The former is most used because it is affordable in terms of time-cost and its results are relatively reasonable.",
"Though it is supposed that the data acquisition would be performed in standard conditions, this is not often respected.",
"What's more, since the human heart rate / ECG could be influenced by many factors, such as emotion, exercise, daylight, sleep $\\ldots $ .",
"the robustness became thus the main concern for short-term data analysis.",
"Considering also the circadian rhythms, the long-term measure can provide more reliable data.",
"So in this study, we used long-term data analysis ($1 \\sim 1.2 \\times 10^5$ samples).",
"Four type of RR data are analyzed : normal sinus rhythm as baseline; three cardiac diseases : congestive heart failure, atrial fibrillation and ventricular premature arrhythmia.",
"Normal Sinus Rhythm (NSR) [35] : beat-to-beat data for 54 long-term (24 hours) ECG recordings of subjects in normal sinus rhythm.",
"The original ECG recordings were digitized at 128 samples per second.",
"The beat annotations were obtained by automated analysis with manual review and correction.",
"http://physionet.org/physiobank/database/nsr2db/.",
"Atrial Fibrillation (AF) [36] : The Long-Term AF Database includes 84 long-term ECG recordings of subjects with paroxysmal or sustained atrial fibrillation.",
"Same data samples per second as for NSR and CHF database.",
"http://physionet.org/physiobank/database/ltafdb/.",
"Congestive Heart Failure (CHF) [37] : beat-to-beat data for 29 long-term ECG recordings of subjects with congestive heart failure (NYHA classes I, II, and III).",
"Digitized at 128 samples per second, these data have been manually reviewed and corrected.",
"http://physionet.org/physiobank/database/chf2db/.",
"Ventricular Premature Arrhythmia (VPA) : from PhysioNet Cardiac Arrhythmia Suppression Trial (CAST) database comprising long-term ECG recordings of more than 800 patients [38] (some data is missed, around 600 data are used).",
"http://physionet.org/physiobank/database/crisdb/."
],
[
"Detrending Methods",
"As above-mentioned, there exists many detrending methods, but the typical methods are linear and nonlinear detrending.",
"For each category, the outputs will change little – never dramatically.",
"This is also the basic principle of every method, violating this rule will make the detrending be failed.",
"So, in this work, we used to typical linear and nonlinear detrending methods : linear detrending : by computing the least-squares fit of a straight line to the data and subtracts the resulting function from the data which has been implemented in Matlab® as a standard function.",
"nonlinear detrending : the trend is often taken as the reconstructed signal from the wavelet decomposition [39], [29].",
"For comparative reason, we use the same method as in [29] : the trend is the reconstruction of the sixth level from wavelet db3 decomposition."
],
[
"Analysis Methods",
"In single scale analysis (directly on the original RR data), the standard HRV measure – SDNN is used.",
"For nonlinear analysis, we are interested in complexity analysis methods.",
"It's known that special patterns or shifts can be often found in electrophysiological signals reflecting the system's dynamics.",
"So, analyzing the complexity of these patterns can help to explore the searched physiological mechanisms.",
"Three typical complexity analysis methods are used here : Fractal Dimension (FD), Detrended Fluctuation Analysis (DFA) and Sample Entropy (SampEn).",
"We give only the principles of each method, the detailed algorithms can be found in original works.",
"Fractal Dimension.",
"By quantifying their graph complexity as a ratio of the change in detail to the change in scale, Fractal dimension helps to measure the roughness or smoothness for time series or spatial data.",
"Many FD estimators have been proposed : from the basic box-counting to variogram or by spectrum.",
"[40], [41].",
"But the principle remained the same: [(i)] measure the quantities of the object using various step sizes; use least-squares regression to fit the graph (generally the log-log plot, measures quantities vs. step sizes); estimate $m_{\\mathrm {FD}}$ as the slope of the regression line.",
"Detrended Fluctuation Analysis.",
"DFA is one of the most used methods to determine/quantify signal's statistical self-similarity, so complexity as well.",
"The principle is that, if the subset of an object can be rescaled to resemble statistically the original object, this object can be considered as self-similar.",
"This implies that the self-similarity can be defined by the rescaling process [42].",
"It is basically a modified root-mean-square analysis [43].",
"Firstly, the time series is converted into unbounded series.",
"Then, it is filtered by a linear-local-trend-removal to eliminate any inferences.Once this is done, root-mean-square fluctuations analysis is performed over different scales so that the scaling exponent can be characterized by the slope of the log-log plot (fluctuations vs. scales) [44].",
"Sample Entropy (SampEn).",
"Sample Entropy is designed to examine the regularity or fluctuations of a time series.",
"It detects the changes in underlying episodic behavior not reflected in peak occurrences or amplitudes [45].",
"If in a time series, there exist repetitive patterns of fluctuation, it will be more predictable than a time series in which such patterns are absent.",
"So the basic idea of SampEn is to determine if similar patterns in current observation exists in the following observations.",
"SampEn $\\ldots $ “is precisely the negative natural logarithm of the conditional probability that a dataset of length $N$ , having repeated itself within a tolerance $r$ for $m$ points, will also repeat itself for $m+1$ points, without allowing self-matches”$\\ldots $ [46].",
"So, by quantifying this probability of repeatability, the regularity of the time series is examined.",
"The human physiological systems are very complex systems consisted of multiple organs, each of them has their own mechanical / electrophysiological properties.",
"The interactions / interferences of these sub-systems make the output of the whole system extremely complex.",
"So, single scale analysis of this output – acquired signals could give global information but fail to provide more comprehensive understandings.",
"In fact, the physiological systems exhibit nonlinear dynamics with highly irregularity or even randomness which the single scale analysis of the system output often failed to reveal the true dynamics.",
"What's more, the cardiac arrhythmia are often associated with highly erratic fluctuations which have statistically uncorrelated noise [47] which brings yet more challenges to single-scale analysis.",
"Multi-scale analysis methods can overcome those shortcomings and reveal the spatial-temporal structures at multiple scales that provide more information about the system.",
"It is thus more robust.",
"The scaling is performed with coarse-graining method.",
"Its principle is to smooth the original signal, in such a way that the intrinsic dynamics could be revealed by eliminating the local fluctuation.",
"The lager the scale is, the smoother the obtained signal is.",
"Given a one-dimensional discrete time series $x(t)$ , ${x_1, \\ldots , x_i, \\ldots , x_N}$ , the consecutive coarse-grained time series ${y^{(\\tau )}}$ , determined by the scale factor $\\tau $ : $\\displaystyle y^{(\\tau )}_{j} = \\frac{1}{\\tau } \\sum _{i=(j-1) \\tau +1}^{j \\tau } x_i, \\; 1 \\le j \\le \\frac{N}{\\tau }$ .",
"The length of each coarse-grained time series is equal to the length of the original time series divided by the scale factor $\\tau $ [47].",
"The coarse-graining method is generally used in multiscale sample entropy (MSE) analysis.",
"To compare with MSE, we performed the same scaling procedure for detrended fluctuation analysis and fractal dimension."
],
[
"Temporal Analysis : SDNN",
"The detrending effects on SDNN are shown in figsdnn.",
"For better description, the results are presented in both boxplot and probability density distribution.",
"Their probability densities are normalized for comparison reason – these densities are from different groups, comparison of absolute densities is meaningless in this case.",
"For the four RR data types, the global conclusion is the same : linear trend removal changes very little the SDNN (figsdnn).",
"Their values are still in the same range and distribution.",
"Comparing the SDNN values of the original RR time series and the detrended ones by Kruskal-Wallis tests, we can see that the related $p$ -values are all larger than 0.01 (tablesdnnpv) and showed the strong correlation.",
"So, they can be considered from the same distribution.",
"However, nonlinear detrending changed everything.",
"For SDNN of NSR and VPB, their distributions are separated (fig1a,fig1d).",
"The Kruskal-Wallis test on SDNN for groups of NSR and VPB showed their $p$ -values are close to zero ($p<10^{-19}$ , tablesdnnpv), which rejected completely the correlation hypothesis of the two groups.",
"The same observation happened for AF and CHF groups.",
"The only difference is that the $p$ -values are relatively larger, but they are still small enough ($p < 10^{-5}$ , tablesdnnpv) to confirm the correlation rejection.",
"So, at this step, the results showed that, linear detrending does not indeed change the dynamics of the original data, only small shifting of the distribution of SDNN.",
"The larger $p$ -values showed that the SDNN for detrended data and the original data have strong correlation.",
"Due to this correlation, the SDNN for detrended data can be regarded as an alternative to the original SDNN or improved ones in case of data classification.",
"After the nonlinear detrending, the SDNN changed in two ways : [(i)] the values are completely different from the original SDNN; distribution changed as well so that this is no more correlation between the two groups.",
"It is thus certain that nonlinear detrending should not be used in SDNN analysis.",
"Figure: SDNN of HRV in original, linear trend removal and nonlinear trend removal conditions.",
"Four types RR data : NSR, AF, CHF & VPB.We know now that the linear or nonlinear detrending effect on the SDNN.",
"The comparisons are performed on themselves : original data vs. linearly detrended data vs. nonlinearly detrended data.",
"How about the comparison the three pathological RR data (AF, CHF, VPB) to the baseline RR data in normal conditions ?",
"What kind of effect by linear or nonlinear detrending on SDNN ?",
"Comparing the SDNN of the original data (1st row, figsdnnhrv), it is clear that SDNN of NSR is larger than any other cases (AF, CHF, VPB).",
"This confirmed that in case of cardiac diseases, the heart losses part of its central modulation capability.",
"So the SDNN is is then reasonably smaller.",
"When applying linear detrending, the SDNN distributions are slightly shifted, but their order remained the same as in the first row ($\\textrm {NSR} > \\textrm {VPB} > \\textrm {AF} > \\textrm {CHF}$ ).",
"Nonlinear detrending changed unfortunately this order (3rd row, figsdnnhrv), the SDNN distribution of NSE and VPB are almost overlapped.",
"Figure: SDNN and their normalized probability density of four types RR data : NSR, AF, CHF & VPB in three conditions : original data, after linear detrending and after nonlinear detrending.Table: pp-values from Kruskal-Wallis tests for original SDNN vs. SDNN after linear detreding and original SDNN vs. SDNN after nonlinear detredingThis section suggests that pre-processing by linear detrending RR time series has little negative effect on typical analysis in time domain.",
"The nonlinear detrending changed a lot the analysis, especially for RR data in normal sinus rhythm and in ventricular premature beats conditions.",
"Complexity Analysis The SDNN in time domain gives the basic statistics / quantification of the RR time series.",
"The analysis of nonlinear dynamics will provide yet another dimension of quantification and appropriate qualification.",
"These nonlinear indexes are shown in fig3,fig4,fig5,fig6,fig7,fig8.",
"The very first values of each curve are the basic measures, while the curves presented the multiscale analysis.",
"One important note : all the curves in the figures shown in this section are the median values of that group at each scale.",
"They are not value for only one patient.",
"The normal distribution tests have been performed for all values at each scale.",
"The results showed that all these values at each scale have normal distribution.",
"So the median value use is justified.",
"Multiscale Fractal Dimension analysis The fractal dimension $m_{\\mathrm {FD}}$ of an irregular time series is always $m_{\\mathrm {FD}} \\in [1,2]$ .",
"The lower limit associated with a smooth curve, and the upper limit, $m_{\\mathrm {FD}} =2$ , corresponding to a space-filling exceedingly rough graph.",
"The original data and the linearly detrended data have almost the identical $m_{\\mathrm {FD}}$ for the four types of RR data – the curves are overlapped.",
"The values of $m_{\\mathrm {FD}}$ are also in the typical range as in literature.",
"After nonlinear detrending, $m_{\\mathrm {FD}}$ is changed a lot.",
"At smaller scales, the values' change is significant for NSR, CHF and VPB data, but varied little for AF data.",
"When increasing the scale, all is changed : the $m_{\\mathrm {FD}}$ are close or equal to 2 – typical value for random noise, this indicates that the nonlinearly detrended data changed the nature of the data.",
"Figure: Multiscale Fractal Dimension analysis of detrending effect on normal RR time series and pathological ones.",
"(each curve represents median values for each group)In fig3, we compared the detrending effect for each type RR data.",
"The comparison can also be done among the $m_{\\mathrm {FD}}$ for the four types of RR data in three conditions : original, linear/nonlinear detrending (fig4).",
"An interesting point about the multi-scale $m_{\\mathrm {FD}}$ of these data is that they have different signatures.",
"For normal heart rate data, the $m_{\\mathrm {FD}}$ is linear to scales.",
"It is no more the case for pathological heart rate time series.",
"The $m_{\\mathrm {FD}}$ are asymptotic to certain values for each pathology.",
"This suggested that multiscale FD could be used a marker of these diseases.",
"So, these signatures are well preserved after linear detrending.",
"Unfortunately, the nonlinear detrending made all these signatures almost indistinguishable.",
"Figure: Multiscale Fractal Dimension comparison for normal RR time series and pathological ones in three situations : normal, linear detrending and nonlinear detrending.",
"(each curve represents median values for each group) Multiscale Detrended Fluctuation Analysis Similar observations happened in multi-scale DFA (fig5).",
"The exponent $\\alpha _{\\mathrm {DFA}}$ gives another quantification of the complexity.",
"When the linearity in the data is dominant, $\\alpha _{\\mathrm {DFA}}$ is larger than $0.5$ (around $0.5$ for short-term correlation, $ 0.5 < \\alpha \\le 1$ for persistent long-range power-law correlations).",
"If the data is completely uncorrelated, $\\alpha _{\\mathrm {DFA}} = 0.5$ .",
"Outside the range of $[0.5, 1]$ , $0 < \\alpha < 0.5$ signals anti-persistent power-low correlations; $\\alpha > 1$ means that the data is non-stationary, unbounded.",
"The multi-scale DFA in this section confirmed the conclusion in fractal dimension analysis.",
"The linearly detrended data has the same dynamics as the original data, the exponents $\\alpha _{\\mathrm {DFA}}$ are larger than 1, indicating that these data are non-stationary.",
"However, the $\\alpha _{\\mathrm {DFA}}$ for nonlinearly detrended data switched completely to other side of the spectrum – smaller than $0.5$ and even converged to zero.",
"This suggested strong random and anti-persistent component in the data.",
"Since this is very likely impossible for heart rate time series, the nonlinear detrending could not be considered thus as a good choice.",
"Figure: Multiscale Detrended Fluctuation Analysis of detrending effect on normal RR time series and pathological ones.",
"(each curve represents median values for each group)Figure: Multiscale Detrended Fluctuation Analysis comparison for normal RR time series and pathological ones in three situations : normal, linear detrending and nonlinear detrending.",
"(each curve represents median values for each group)If we put the $\\alpha _{\\mathrm {DFA}}$ for the four RR data in the same figure, and in three conditions (normal, linnear & nonlinear detrending), the changes are more visible (fig6).",
"It showed that in original and linear detrending conditions, the difference of $\\alpha _{\\mathrm {DFA}}$ for each type of RR data can be still viewed.",
"Once nonlinearly detrending the data, the $\\alpha _{\\mathrm {DFA}}$ for the four RR data are almost identical – it suggested that these nonlinearly detrended data can be in fact put into the same category – noise.",
"It proved once again that nonlinear detrending would not be an appropriate practice.",
"Multiscale Sample Entropy Analysis As above-mentioned, Sample Entropy examines the probability of similar patterns presence in the signal.",
"If a time series contains more similar patterns, the SampEn values will be smaller; otherwise, they would be higher.",
"It is then also a predictability term.",
"The linear detrending changed little the MSE analysis (fig7), the curves (original and linear detrending) are close enough each other or just slightly shifted.",
"Their values are smaller than those for nonlinearly detrended RR data.",
"After nonlinear detrending, the SampEn values jumped up to 1.5 or larger and have a increasing trend.",
"These values correspond to SampEn for random signals.",
"Figure: Multiscale Sample Entropy of detrending effect on normal RR time series and pathological ones.",
"(each curve represents median values for each group)Figure: Multiscale Sample Entropy comparison for normal RR time series and pathological ones in three situations : normal, linear detrending and nonlinear detrending.",
"(each curve represents median values for each group)Another observation of MSE for NSR, CHF and VPB is that the MSE for nonlinearly detrended data are completely separated from the original and linearly detrended data, including the first scales.",
"However, in case of atrial fibrillation, the signature is different.",
"Firstly, MSE at the very first scales are close in the three conditions (original, linear & nonlinear detrending).",
"Secondly, in original and linear detrending conditions, the MSE for NSR, CHF and VPB are relatively increasing and asymptotic to some thresholds.",
"For AF, these curves are decreasing instead of increasing.",
"The fig8 showed that RR data in normal condition has smallest Sample Entropy, indicating that this type of data contains more similar patterns and is thus more predictable.",
"In case of arrhythmia, the rhythm / harmony is broken, so the RR variation becomes less regular and less predictable, so the Sample Entropy values are larger.",
"After the nonlinear detrending, the data are all normalized so that the original intrinsic differences are disappeared.",
"That's why they all looked the same, as MSE values suggested (fig8).",
"Figure: Complexity Space based on FD, DFA & SampEn, for RR data in NSR, AF, CHF & VPB conditions.Constructing a space with the three complexity indexes, as shown in fig9, the complexity of the nonlinearly detrended RR data situated in a position where is far from the original (and linearly detrended) data.",
"Considering all previous results, in terms of either temporal analysis or nonlinear complexity analysis, it is now clear that, nonlinear detrending will change the intrinsic dynamics of the RR data.",
"It is certain that nonlinear detrending in HRV analysis should not be advised, at least for RR data in NSR, AF, CHF & VPB conditions or without other combined processing.",
"Conclusion The pre-processing in biomedical signal processing played an essential but often underestimated role.",
"The reason is that, we know very little about the biomedical system in too many cases.",
"In fact, the acquired signals come from a black box.",
"The related processing is, based on very limited information, to study a high-dimensional system.",
"In consequence, any inappropriate pre-processing would affect the fundamental dynamics.",
"In case of heart rate variability analysis, the pre-processing of detrending could modify a lot the basic characteristics of the signal leading to an important change of the system.",
"Analysis based on these modifications could be deviated.",
"So, careful attention should be paid on this delicate procedure.",
"The linear detrending affects little the global characteristics of the RR data.",
"After linear detrending, the SDNNs are just slightly shifted and all distributions are well preserved.",
"The cross-scale complexity (the Fractal Dimensions and exponents from Detrended Fluctuation Analysis) is almost the same as the ones for original RR data.",
"Though there are still some differences revealed with multi-scale Sample Entropy analysis, their changes are correlated.",
"We can conclude that pre-processing by linear detrending can be performed on RR data which does not modify the intrinsic dynamics of the data.",
"The same analysis with nonlinear detrending showed that this type of pre-processing could be harmful.",
"It changed not only the SDNNs distribution, but also the order among different types of RR data.",
"After nonlinear detrending, the SDNN for normal sinus rhythm became indistinguishable from the SDNN for RR with ventricular premature beats.",
"This is dangerous and cannot be accepted for clinical applications.",
"This problem can be explained by multi-scale complexity analysis.",
"The different RR data has different complexity signature.",
"Nonlinear detrending made the all RR data to be similar.",
"It is thus impossible to distinguish them.",
"In fact, the Fractal Dimension showed that nonlinearly detrended RR data has a dimension close to 2, the exponent from DFA is close to zero and SampEn is larger than 1.5 – these complexity values are very close to those for random signal.",
"So, when one processing completely change the nature of the data, how to draw useful / exploitable conclusion ?",
"This work investigated the detrending effect on the complexity analysis of heart rate.",
"Though in HRV analysis the data used could have some influences on the conclusion, and the authors could argue that the combination with other pre-precessing techniques can avoid or bypass the detrending effect.",
"The results are indeed not appropriate.",
"The problem does exist.",
"So for any clinical application, the pre-processing by detrending, if needed, should be careful conducted in order to find an optimal way which would not affect the fundamental dynamics of the original data.",
"Unfortunately, it is hard to propose a general solution for this compensation.",
"Because all depends on the used data."
],
[
"Conclusion",
"The pre-processing in biomedical signal processing played an essential but often underestimated role.",
"The reason is that, we know very little about the biomedical system in too many cases.",
"In fact, the acquired signals come from a black box.",
"The related processing is, based on very limited information, to study a high-dimensional system.",
"In consequence, any inappropriate pre-processing would affect the fundamental dynamics.",
"In case of heart rate variability analysis, the pre-processing of detrending could modify a lot the basic characteristics of the signal leading to an important change of the system.",
"Analysis based on these modifications could be deviated.",
"So, careful attention should be paid on this delicate procedure.",
"The linear detrending affects little the global characteristics of the RR data.",
"After linear detrending, the SDNNs are just slightly shifted and all distributions are well preserved.",
"The cross-scale complexity (the Fractal Dimensions and exponents from Detrended Fluctuation Analysis) is almost the same as the ones for original RR data.",
"Though there are still some differences revealed with multi-scale Sample Entropy analysis, their changes are correlated.",
"We can conclude that pre-processing by linear detrending can be performed on RR data which does not modify the intrinsic dynamics of the data.",
"The same analysis with nonlinear detrending showed that this type of pre-processing could be harmful.",
"It changed not only the SDNNs distribution, but also the order among different types of RR data.",
"After nonlinear detrending, the SDNN for normal sinus rhythm became indistinguishable from the SDNN for RR with ventricular premature beats.",
"This is dangerous and cannot be accepted for clinical applications.",
"This problem can be explained by multi-scale complexity analysis.",
"The different RR data has different complexity signature.",
"Nonlinear detrending made the all RR data to be similar.",
"It is thus impossible to distinguish them.",
"In fact, the Fractal Dimension showed that nonlinearly detrended RR data has a dimension close to 2, the exponent from DFA is close to zero and SampEn is larger than 1.5 – these complexity values are very close to those for random signal.",
"So, when one processing completely change the nature of the data, how to draw useful / exploitable conclusion ?",
"This work investigated the detrending effect on the complexity analysis of heart rate.",
"Though in HRV analysis the data used could have some influences on the conclusion, and the authors could argue that the combination with other pre-precessing techniques can avoid or bypass the detrending effect.",
"The results are indeed not appropriate.",
"The problem does exist.",
"So for any clinical application, the pre-processing by detrending, if needed, should be careful conducted in order to find an optimal way which would not affect the fundamental dynamics of the original data.",
"Unfortunately, it is hard to propose a general solution for this compensation.",
"Because all depends on the used data."
]
] | 1605.05891 |
[
[
"Quantifying Model Form Uncertainty in RANS Simulation of Wing-Body\n Junction Flow"
],
[
"Abstract Wing-body junction flows occur when a boundary layer encounters an airfoil mounted on the surface.",
"The corner flow near the trailing edge is challenging for the linear eddy viscosity Reynolds Averaged Navier-Stokes (RANS) models, due to the interaction of two perpendicular boundary layers which leads to highly anisotropic Reynolds stress at the near wall region.",
"Recently, Xiao et al.",
"proposed a physics-informed Bayesian framework to quantify and reduce the model-form uncertainties in RANS simulations by utilizing sparse observation data.",
"In this work, we extend this framework to incorporate the use of wall function in RANS simulations, and apply the extended framework to the RANS simulation of wing-body junction flow.",
"Standard RANS simulations are performed on a 3:2 elliptic nose and NACA0020 tail cylinder joined at their maximum thickness location.",
"Current results show that both the posterior mean velocity and the Reynolds stress anisotropy show better agreement with the experimental data at the corner region near the trailing edge.",
"On the other hand, the prior velocity profiles at the leading edge indicate the restriction of uncertainty space and the performance of the framework at this region is less effective.",
"By perturbing the orientation of Reynolds stress, the uncertainty range of prior velocity profiles at the leading edge covers the experimental data.",
"It indicates that the uncertainty of RANS predicted velocity field is more related to the uncertainty in the orientation of Reynolds stress at the region with rapid change of mean strain rate.",
"The present work not only demonstrates the capability of Bayesian framework in improving the RANS simulation of wing-body junction flow, but also reveals the major source of model-form uncertainty for this flow, which can be useful in assisting RANS modeling."
],
[
"Introduction",
"Wing–body junction flows are common in many engineering applications, such as the airfoil/fuselage junction of an aircraft, the sail/hull junction of a submarine and the blade/hub assembly of a wind turbine.",
"Due to the flow stagnation at the leading edge of the wing, strong streamwise adverse pressure gradient occurs at upstream of the stagnation point.",
"Such adverse pressure gradient leads to the separation of the incoming boundary layer, which forms spanwise vortex upstream of the leading edge.",
"This vortex will move around the wing and stretch along the streamwise direction, which is known as the horseshoe vortex system.",
"The formation of this vortex system has been well studied for decades [1], [2].",
"On the other hand, the flow near the trailing edge of the junction is much more complex, due to the combination effect of vortex stretching, adverse pressure gradient and the interaction between the boundary layer on the wing and that on the body.",
"This interaction of two perpendicular boundary layers leads to the stress-induced secondary flow, which cannot be predicted by the linear eddy viscosity RANS models, including $k$ –$\\varepsilon $ model, $k$ –$\\omega $ model and S–A model.",
"This is because these linear viscosity RANS models have difficulty in predicting the anisotropy state of Reynolds stresses at the near wall region.",
"Although the nonlinear eddy viscosity models are able to predict the stress-induced secondary flow, these models are still restricted by the assumption that the Reynolds stress is determined by the local mean flow quantities.",
"Compared to RANS simulations, high fidelity simulations such as Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) are more accurate for the prediction of such stress-induced secondary flow.",
"However, the computational cost of these high fidelity simulations is currently prohibitive for most real engineering applications of junction flow, since the wall bounded flow at high Reynolds number demands fine resolution of the near wall region.",
"Many previous studies of junction flow have focused on the horseshoe vortex system.",
"Rodi et al.",
"[3] compared the algebraic Reynolds stress model and the non-linear $k$ –$\\varepsilon $ models with the linear eddy viscosity models, and showed that the more complex RANS models were no better than linear eddy viscosity models for the practical mean quantities such as mean velocity.",
"Coombs at al.",
"[4] tested eight turbulence models including RNG $k$ –$\\varepsilon $ model and LRR model and found that all models significantly under-predicted the turbulent kinetic energy (TKE).",
"Aspley and Leschziner [5] compared twelve RANS models including non-linear eddy viscosity models and Reynolds stress transport models (RSTM), and suggested that the second-moment closure models offered better prediction of Reynolds stress over other models, although no model achieved close agreement with the experimental data.",
"Similarly, Chen [6] showed that RSTM had an overall better prediction than the isotropic eddy viscosity models.",
"However, the convergence could be difficult to achieve for RSTM [7].",
"All these previous studies indicate that further increasing the complexity of RANS model barely improve the prediction of mean quantities that is of the most interest in engineering applications.",
"The hybrid RANS/LES simulations, such as Detached Eddy Simulations (DES), have been used in several previous studies and provide better prediction for the horseshoe vortex system.",
"However, the predicted location of the horseshoe vortex system is still not satisfactory.",
"Alin and Fureby [8] showed that the DES typically predicts the horseshoe vortex located too further away from the body than the experimental measurements.",
"Paik et al.",
"[9] also reported the discrepancy in predicting the location of horseshoe vortex system.",
"In addition, Paik et al.",
"[9] pointed out that the DES simulation result depends closely on the flow-specific adjustment of the DES length scale.",
"Compared to the horseshoe vortex system at the leading edge, the flow separation at the trailing edge of junction is much less investigated.",
"Huser and Biringen [10] showed that the normal stress imbalance was important for the generation of stress-induced secondary flow within the junction of two flat plates.",
"For a more realistic configuration with NACA0012 airfoil, Gand et al.",
"[11] found that the prediction of flow separation at the trailing edge varied based on different linear eddy viscosity RANS models, and none of the prediction had a close agreement with the experimental measurement.",
"In addition, Gand et al.",
"[11], [12] also pointed out that the linear eddy viscosity models were not able to accurately predict the corner separation of junction flow.",
"Bordji et al.",
"[13] showed that the quadratic constitutive relation (QCR) closure can provide better prediction of corner separation, and they suggested that the effect of corner flow separation may also be associated to the local increase of turbulent kinetic energy.",
"Rumsey et al.",
"[14] confirmed that the QCR closure improved the prediction of corner separation compared to the linear model.",
"However, he also pointed out that the existing comparisons were not all consistent and more efforts were still required in understanding the corner separation of wing–body junction flow.",
"Recently, Xiao et al.",
"[15] proposed a physics-informed, data-driven Bayesian framework for calibrating the RANS simulations by quantifying and reducing the model-form uncertainties in RANS simulations with a small amount of velocity observation data.",
"In their framework, uncertainties were introduced to the Reynolds stresses and a Bayesian inference procedure based on an iterative ensemble Kalman method [16] was used to quantify and reduce the uncertainties by incorporating observation data.",
"By applying their framework to the RANS simulation of the flow in a square duct, they had demonstrated that the prediction of normal stress imbalance can be improved and the secondary flow was therefore captured.",
"However, to avoid the possible complexity caused by the wall functions, wall resolved RANS simulations were used in the framework by Xiao [15], which demand large computational cost for wall bounded flow at high Reynolds number.",
"In practical scenarios, many RANS simulations are performed with wall functions to make the computational cost affordable.",
"Therefore, in the present work we explore the compatibility of this Bayesian framework and the wall function approach commonly used in RANS simulations.",
"Specifically, we discuss how the wall function is incorporated into the RANS simulation, and demonstrate how the wall function is taken into consideration in our current framework.",
"To illustrate this concept, we use the implementation of OpenFOAM as an example.",
"Based on this extension, we are able to apply the Bayesian framework to quantify and reduce the model-form uncertainty in RANS simulations with complex geometry at higher Reynolds numbers.",
"In addition to the compatibility with wall function, the arrangement of observation is also discussed in this work.",
"In the original framework proposed by Xiao et al.",
"[15], the arrangement of observation is determined based on the physical understanding of the particular flow.",
"Specifically, it is determined based on an empirical estimation of the mean flow correlation.",
"Since such correlation is important for the performance of Bayesian inference with sparse observation data, it is more rigorous to arrange the observations with an objective criteria rather than the subjective judgement by the user, especially for the complex flow problem in which the mean flow correlation is less obvious.",
"In this work, we apply the correlation analysis to the mean velocity field to estimate the mean flow correlation, and determine the arrangement of observation accordingly.",
"The objective of the present work is to use the Bayesian framework proposed by Xiao et al.",
"[15] to improve the prediction of the RANS simulation of the wing–body junction flow.",
"This is a much more complicated flow problem compared to the flow problem that studied in the work by Xiao et al. [15].",
"We first extend the original Bayesian framework by Xiao et al.",
"[15] for complex wall-bounded flows at high Reynolds numbers.",
"Based on the extended framework, we further calibrate the RANS simulation of wing–body junction flow, and demonstrate that the predicted mean flow field by the linear eddy viscosity RANS model can be improved.",
"The remaining of the paper is organized as follows.",
"Section 2 outlines the Bayesian framework proposed by Xiao et al.",
"[15] and presents the extension in the current work.",
"Section 3 presents the numerical simulation results of wing–body junction flow and compares the results with the experimental data [1].",
"Section 4 discusses the limitation of the current framework and the possible extension.",
"Finally, Section 5 concludes the paper.",
"We first summarize the Bayesian framework, which is proposed by Xiao et al.",
"[15] for quantifying and reducing model-form uncertainties in RANS simulations.",
"In RANS simulations, the modeled Reynolds stress term is considered as the main source of model-form uncertainty [17].",
"To quantify this model-form uncertainty, perturbations are directly injected to the RANS-modeled Reynolds stress.",
"Specifically, the Reynolds stress $\\tau (x)$ term is modeled as a random field whose prior mean is the RANS-predicted Reynolds stress $\\tilde{\\tau }^{rans}(x)$ , in which $x$ represents the spatial coordinates.",
"It should be noted that Reynolds stress tensor is a positive semi-definite matrix, which leads to the realizability requirement of Reynolds stress [18].",
"This realizability requirement may be violated if arbitrary perturbation is injected into each element of the Reynolds stress tensor.",
"To guarantee the realizability of Reynolds stress, Iaccarino and co-workers [19], [20], [21], [22], [23] first proposed an uncertainty quantification approach to perform the perturbation within the Barycentric triangle, which guarantees the realizability of Reynolds stress.",
"In details, the Reynolds stress tensor is decomposed into physical meaningful components and transformed into the coordinates of Barycentric triangle [24] as follows: $\\tau = 2 k \\left( \\frac{1}{3} \\mathbf {I} + \\mathbf {a} \\right)= 2 k \\left( \\frac{1}{3} \\mathbf {I} + \\mathbf {V} \\Lambda \\mathbf {V}^T \\right)where k is the turbulent kinetic energy, which indicates the magnitude of \\tau ; \\mathbf {I}is the second order identity tensor; \\mathbf {a} is the anisotropy tensor;\\mathbf {V} = [\\mathbf {v}_1, \\mathbf {v}_2, \\mathbf {v}_3] and\\Lambda = \\textrm {diag}[\\lambda _1, \\lambda _2, \\lambda _3] with\\lambda _1+\\lambda _2+\\lambda _3=0 are the eigenvectors and eigenvalues of \\mathbf {a},respectively.",
"The eigenvalues \\lambda _1,\\lambda _2, and \\lambda _3 are mapped to a Barycentric coordinate (C_1, C_2, C_3) with C_1 +C_2 + C_3 = 1.",
"Consequently, the Barycentrictriangle shown in Fig.~\\ref {fig:bary}a encloses all physically realizable states of Reynolds stress.",
"To facilitate the parameterization, the Barycentriccoordinate is further transformed to the natural coordinate (\\xi , \\eta ) as shown in Fig.~\\ref {fig:bary}b.",
"Finally, uncertainties are introduced to the mappedquantities k, \\xi , and \\eta by adding discrepancy terms to the corresponding RANS predictions,i.e.,\\begin{subequations}{\\begin{@align}{3}{2}\\log k(x) & = &\\ \\log \\tilde{k}^{rans}(x) & + \\delta ^k(x) \\\\\\xi (x) & = &\\ \\tilde{\\xi }^{rans}(x) & + \\delta ^\\xi (x) \\\\\\eta (x) & = &\\ \\tilde{\\eta }^{rans}(x) & + \\delta ^\\eta (x)\\end{@align}}\\end{subequations}Currently, uncertainties are not introduced to the orientation (\\mathbf {v}_1, \\mathbf {v}_2, \\mathbf {v}_3) ofthe Reynolds stress in the Bayesian inference framework.",
"This is due to the consideration of numerical stability.",
"More details has been discussed in~\\cite {wu2015bayesian}.$ Figure: Mapping between the Barycentric coordinate to the natural coordinate, transforming theBarycentric triangle enclosing all physically realizablestates , to a square via standard finiteelement shape functions.",
"Corresponding edges in the twocoordinates are indicated with matching colors.The prior discrepancies are chosen as nonstationary zero-mean Gaussian random fields $\\mathcal {GP}(0, K)$ (also known as Gaussian processes), and the basis set is chosen as the eigenfunctions of the kernel $K$ [26].",
"Therefore, the discrepancies can be reconstructed as follows: $\\delta (x) = \\sum _{i=1}^\\infty \\omega _{i} \\; \\phi _i (x) ,where \\phi _i (x) are the eigenfunctions of the kernel K, and the coefficients \\omega _{i} are independent standard Gaussian random variables.",
"In practice, theinfinite series is truncated to m terms with m depending on the smoothness of the kernel K.$ With the projections above, the discrepancies are parameterized by the coefficients $\\omega ^k_{i},\\, \\omega ^\\xi _{i}, \\, \\omega ^\\eta _{i}$ with $i = 1, 2, \\cdots , m$ .",
"These coefficients are then inferred by an iterative ensemble Kalman method [16], [27].",
"Specifically, this iterative ensemble Kalman method incorporates sparse observation data of mean velocity from experiment to infer the coefficients $\\omega _{i}$ of the Reynolds stress discrepancies as shown in Eq.",
"REF .",
"More details of the Bayesian inference procedure can be found in ref.",
"[15]."
],
[
"Compatibility of the Uncertainty Quantification Framework with Wall Function",
"The original Bayesian framework proposed by Xiao et al.",
"[15] is developed for wall-resolved RANS simulations and is restricted for many engineering applications, since resolving the near wall region is costly for wall-bounded flow at high Reynolds number.",
"In this work, we discuss the compatibility of this Bayesian framework for RANS simulations with wall functions.",
"For the wall resolved RANS simulation, the wall shear stress can be estimated as follow: $\\tau _w = \\nu \\frac{U_{p}}{y_p}where U_{p} represents the mean velocity tangential to the wall at the first cell near the wall, and y_p is the distance between the wall and the center of that cell.However, Eq.~\\ref {eq:tauw_resolved} is only a valid approximation for wall shear stress if the viscous sublayer is resolved, which demands much higher computational cost than RANS simulation with wall functions.",
"In the practice of RANS simulation with wall functions, the center of first cell is arranged in the logarithmic layer.",
"Therefore, Eq.~\\ref {eq:tauw_resolved} is no longer appropriate to estimate the wall shear stress.",
"Instead, the wall shear stress is estimated at the first layer of cell near wall as follow:{\\equation \\tau _w = (\\nu _t+\\nu ) \\frac{U_{p}}{y_p}\\endequation }where \\nu _t is calculated as follow:{\\equation \\nu _t = \\nu \\left(\\frac{y^+ \\kappa }{\\log {(E y^+)}} -1\\right)\\endequation }Essentially, the estimation of wall shear stress based on Eq.~\\ref {eq:tauw} and Eq.~\\ref {eq:nut} makes use of the law of the wall.",
"By substituting Eq.~\\ref {eq:nut} into Eq.~\\ref {eq:tauw} and recalling that u_{\\tau }=\\sqrt{\\tau _w / \\rho }, y^+=y \\sqrt{\\tau _w} / \\nu and u^+=U / u_{\\tau }, we can obtain the log law of the wall:{\\equation u^+=\\frac{1}{\\kappa }\\log {y^+} + \\frac{1}{\\kappa } \\log {E}\\endequation }where the constant E is 9.8~\\cite {popebook} and the constant term C^+=\\frac{1}{\\kappa } \\log {E} \\approx 5.",
"It should be noted that the true value of y^+ is unknown before the wall shear stress \\tau _w is solved.",
"In practice, y^* as calculated by Eq.~\\ref {eq:ystar} is used as an approximation of y^+~\\cite {launder1974numerical}.",
"{\\equation y^*=\\frac{\\left( C_{\\mu }^{1/2} k_p \\right)^{1/2}}{\\nu }\\endequation }where k_p represents the turbulent kinetic energy at the first cell near the wall.$ The above derivation shows that the wall shear stress calculated by Eq.",
"is compatible with the law of the wall, by specifying the boundary condition of eddy viscosity $\\nu _t$ at wall based on Eq. .",
"To illustrate how we incorporate the wall functions in our current uncertainty quantification framework, we use the implementation in OpenFOAM as an illustration.",
"Compared to the built-in solvers of OpenFOAM, the CFD solver tauFOAM used in the Bayesian framework directly takes the Reynolds stress field as an input.",
"Theoretically, each component of Reynolds stress at wall should be zero, due to the non-slip condition.",
"However, if we specify all the components of Reynolds stress as zero at wall, the wall shear stress turns to Eq.",
"REF , which underestimates the wall shear stress if the near wall mesh is coarse and the viscous sublayer is not resolved.",
"In such case, to estimate the wall shear stress based on the law of wall, we first calculate the value of $\\nu _t$ at wall based on Eq. .",
"In our uncertainty quantification framework, the Reynolds stress boundary condition at wall is specified according to Eq.",
"REF if the viscous sublayer is not resolved: $\\widetilde{\\tau }_w = \\nu _t \\frac{U_{p}}{y_p}The wall shear stress is calculated as follow:{\\equation \\tau _w = \\widetilde{\\tau }_w+\\nu \\frac{U_{p}}{y_p}\\endequation }which is essentially the same as Eq.~\\ref {eq:tauw}.",
"Therefore, the wall shear stress is still compatible with the law of the wall as shown in Eq.~\\ref {eq:law_wall_2} due to the specification of boundary condition of Reynolds stress.$"
],
[
"Case Setup",
"The configuration of the computational domain and the coordinate system are shown in Fig.",
"REF .",
"The airfoil shown in Fig.",
"REF is a `Rood' wing, which has a 3:2 elliptic nose and NACA 0020 tail cylinder joined at the maximum thickness position.",
"This configuration follows that used in the experiment performed by Devenport et al. [1].",
"The Reynolds number based on the airfoil thickness $T$ and the free stream velocity $U_{\\inf }$ is $Re_T = 115000$ .",
"Free stream boundary conditions are applied at the far fields, zero gradient boundary condition is applied at the outlet, and non-slip boundary conditions are applied at the walls.",
"The mean flow is symmetrical about $x$ -$z$ plane, and thus only half of the physical domain is simulated in the RANS simulation.",
"Figure: Domain shape of the wing–body junction flow.",
"The xx-, yy- and zz-coordinates arealigned with streamwise, spanwise of the plate and spanwise of the airfoil, respectively.",
"The locations where velocitiesare observed are indicated as crosses (×\\times ).Velocity observations are marked as crosses ($\\times $ ) in Fig.",
"REF , which are generated by adding Gaussian random noises with standard deviation $\\sigma _{obs}$ to the experimental data.",
"According to the experimental results [1], the uncertainties of streamwise velocity $U$ and secondary velocity $W$ are 5% and 7.2%, respectively.",
"However, it was also mentioned that some bias error not included in this uncertainty estimation.",
"Therefore, the standard deviation $\\sigma _{obs}$ is set as 10% of the true mean value in this work, which is a total estimation of the uncertainty in the experimental data.",
"The arrangement of observations are based on the analysis of mean flow correlation as shown in Fig.",
"REF .",
"Plane A is the symmetrical plane upstream of leading edge, where the horseshoe vortex system develops.",
"Plane C is the secondary flow plane downstream of the corner region of the trailing edge.",
"Plane B is another secondary flow plane within the junction.",
"It can be seen that the velocity correlation between plane A and plane C is weak, which indicates that the observation information from plane A has little influence upon the mean flow field around the corner separation region.",
"Such weak correlation is also confirmed by a recent NASA experiment [14].",
"In contrast, the correlation between plane B and plane C is much stronger as shown in Fig.",
"REF b.",
"Therefore, the observation data from plane B would have more influence upon the inference performance in plane C. It should be noted that the Ensemble Kalman Filter used in this framework relies on the mean flow correlation between observation locations and the regions without observations.",
"Since both the horseshoe vortex system and the possible corner separation are of interest in this work, the observations are arranged at both plane A and plane B as shown in Fig.",
"REF to ensure the correlation between the observation locations and the regions of interest.",
"Figure: Mean flow velocity correlation between different planes.",
"Plane A is the symmetrical plane at the upstream of leading edge.",
"Plane B is the secondary flow plane at x/T=3.187x/T=3.187 within the junction.",
"Plane C is the secondary plane at x/T=4.4618x/T=4.4618, which is at the downstream of trailing edge.",
"The locations of these three planes are chosen based on the available experimental data .The perturbations $\\delta ^\\xi $ , $\\delta ^\\eta $ and $\\delta ^k$ are modeled as non-stationary Gaussian process.",
"The variance field $\\sigma (x)$ of the Gaussian process is based on physical prior knowledge, i.e., the RANS prediction is more unreliable at the near wall region of the airfoil.",
"To achieve a more compact representation of the Gaussian process, Karhunen-Loève (KL) expansion is used in the framework to reconstruct the Gaussian process.",
"Specifically, 6 modes are used in this work.",
"All the modes are illustrated in Fig.",
"REF .",
"It should be noted that these modes do not have any variation along the $y$ direction.",
"This is a simplification in this framework to reduce the number of modes.",
"Although the true Reynolds stress discrepancy field is more likely to be 3D in this flow, the number of modes will grow rapidly if 3D mode is used.",
"Since the amount of observation points and the computational cost are both related to the number of modes, it is impractical to introduce a large amount of modes in real applications.",
"Therefore, we make an assumption that the variation of the discrepancy field around the airfoil is much more complex than that within the boundary layer along the $y$ direction.",
"Based on this assumption, the choice of 2D modes as shown in Fig.",
"REF can be justified.",
"Figure: Illutstration of KL expansion modes of the perturbation field.",
"All the modes have beenshifted and scaled into the range between 0 (lightest) and 1 (darkest) to facilitatepresentation, and the legend is thus omitted.",
"Panels (a) to (f) represent modes 1 to 6,respectively.",
"Lower modes are more important."
],
[
"Results at Corner Region",
"The posterior secondary velocity profiles at the plane B ($x/T=3.187$ ) within the junction are shown in Fig.",
"REF .",
"It can be seen that the posterior velocity profile $U_y$ has a much better agreement with the experimental data at the location $-y/T=0.38$ , where observation data is available.",
"In addition to the $U_y$ profiles at $-y/T=0.38$ where observation data are available, the posterior velocity profiles $U_y$ at $-y/T=0.48$ and $-y/T=0.58$ are also improved, while over-correction is noticeable.",
"It indicates that the length scale we used to construct the gaussian random field is larger than the true length scale of the mean flow.",
"As a consequence, the mean flow correlation is over-estimated and it leads to some over-correction in the region where observation data are not available.",
"Compared to the posterior velocity $U_y$ , the posterior velocity $U_z$ profiles have less improvement as shown in Fig.",
"REF b.",
"A possible reason is that the discrepancy field is 2D according to the KL modes as shown in Fig.",
"REF .",
"The complexity of 2D discrepancy field would not exactly satisfy the true discrepancy of Reynolds stress.",
"Therefore, the posterior velocity profiles may not exactly agree with the observation data, which explains the mismatch of $U_z$ at $-y/T=0.38$ where observation data is available.",
"Figure: Posterior ensemble of secondary velocity at plane B (x/T=3.187x/T=3.187) along three locations -y/T=0.38-y/T=0.38, 0.480.48 and 0.580.58 with increasing distance from the airfoil surface.",
"A smaller value of -y/T-y/T represents a location closer to the airfoil.",
"The velocity profiles of U y U_y is scaled by a factor of 0.5 for clarity.",
"Black crosses (×\\times ) denote locations where velocity observations are available.Figure REF shows the posterior secondary velocity profiles at plane C in the wake downstream of the trailing edge.",
"Compared to the baseline RANS prediction, the posterior velocity profiles show significant improvement, even though no observation data is available at this plane.",
"Such improvement is due to the correlation between this plane and plane B where observation data is available.",
"Specifically, the prediction of posterior velocity profiles at upstream plane B shows better agreement with experimental data in Fig.",
"REF , especially for the $U_y$ profiles at the region near airfoil ($-y/T=0.38$ ).",
"It demonstrates that the Bayesian framework can provide better prediction for this complex flow problem, even though there is no local observation data available.",
"Figure: Posterior ensemble of secondary velocity at plane C (x/T=4.4618x/T=4.4618) along three locations -y/T=0.38-y/T=0.38, 0.480.48 and 0.580.58, with increasing distance from the airfoil surface.",
"A smaller value of -y/T-y/T represents a location closer to the airfoil.",
"The velocity profiles are scaled by a factor of 0.5 for clarity.",
"No velocity observation is applied at this plane.The comparison of prior and posterior Reynolds stress is shown in Fig.",
"REF in Barycentric triangle.",
"The Reynolds stress are sampled from the line along $-y/T=0.38$ at plane B ($x/T=3.1817$ ).",
"The length along $z$ direction of the sample region is about 0.2 thickness of the airfoil.",
"It can be seen from Fig.",
"REF that the baseline RANS Reynolds stress is quite different from the experimental data.",
"By injecting uncertainties into the baseline RANS Reynolds stresses, the prior ensemble of Reynolds stress explores most part of Barycentric triangle as shown in Fig.",
"REF a.",
"Compared to the prior Reynolds stresses, the posterior ones have a better agreement with the experimental data, as shown in Fig.",
"REF b.",
"However, it should be noted that the posterior Reynolds stresses still do not cover the experimental data.",
"A possible reason is that the baseline RANS Reynolds stresses have a more clustered distribution than the experimental data in Barycentric triangle.",
"Such clustered distribution is largely preserved due to the choice of length scale in constructing the Gaussian random field for the perturbation.",
"Specifically, the length scale of Gaussian random field is chosen as the thickness of the airfoil, which accounts for the estimation of mean flow correlation when KL expansion is performed.",
"On the other hand, it can be seen in Fig.",
"REF that the discrepancy of Reynolds stress anisotropy still has large variation within the sampled region.",
"Such variation is not considered in this work for two reasons.",
"First, the variation at such small length scale indicates much more modes of KL expansion, which increase the degree of freedom of the unknown parameters and require much more observation data to constrain those unknown parameters.",
"However, a large amount of observation data is usually impractical in most engineering applications.",
"Second, the variation at this small region is difficult to estimate beforehand without a comprehensive experimental database.",
"Therefore, it is not feasible to take such variation into account for most engineering applications of interest.",
"Due to these two reasons, the variation of Reynolds stress discrepancy at such small scale is not considered in this work when the uncertainty is injected into the Reynolds stress.",
"Consequently, the relative locations of Reynolds stresses is largely preserved in the Barycentric triangle if the physical distance is small.",
"Such limitation is referred to as the preservation of relative locations of Reynolds stress in the rest part of this work.",
"It indicates that the perturbed Reynolds stress will not explore the true uncertainty space as shown in Fig.",
"REF , and it is expected that the inferred Reynolds stress would not exactly match with the experimental data in the Barycentric triangle.",
"Although it is a limitation of this framework, it should be noted that the posterior Reynolds stress anisotropy indeed demonstrates some improvement as indicated by the locations in Barycentric triangle in Fig.",
"REF b.",
"In addition, the mean velocity profiles are also improved as shown in Fig.",
"REF and Fig.",
"REF , which is of more interest in many engineering applications.",
"Figure: Prior and posterior ensemble of Reynolds stress in Barycentric triangle.",
"The Reynolds stresses are sampled at plane B (x/T=3.187x/T=3.187) along -y/T=0.38-y/T=0.38.",
"The Reynolds stresses from the same sample are linked by line to illustrate the distribution of different samples.To examine the inference of Reynolds stress components, we use $\\tau _{yy}$ and $\\tau _{zz}$ as illustration.",
"Figure REF shows these two components of Reynolds stress at plane B ($x/T=3.187$ ) along $-y/T=0.38$ .",
"It can be seen from Fig.",
"REF that both components show little improvement.",
"It is due to the fact that the mapping from velocity field to Reynolds stress field is not unique based on the RANS equations.",
"Therefore, different sets of Reynolds stress field may satisfy the same velocity field, and the information incorporated by the velocity observation is not sufficient to infer each Reynolds stress component individually.",
"Figure: Posterior ensemble of Reynolds stress components at plane B (x/T=3.187x/T=3.187) along three locations -y/T=0.38-y/T=0.38, 0.480.48 and 0.580.58, with increasing distance from the airfoil surface.",
"A smaller value of -y/T-y/T represents a location closer to the airfoil.",
"Two components τ yy \\tau _{yy} and τ zz \\tau _{zz} are shown.",
"The results of other components have the same trend and are omitted for simplicity.Unlike each component of Reynolds stress tensor, the normal stress imbalance $\\tau _{yy}-\\tau _{zz}$ is known to have direct impact upon the secondary flow at the corner region [10].",
"The posterior ensemble of the normal stress imbalance $\\tau _{yy}-\\tau _{zz}$ is shown in Fig.",
"REF .",
"It can be seen that the baseline RANS prediction is close to zero due to the linear eddy viscosity assumption.",
"According to the eddy viscosity assumption, the normal stress imbalance predicted by RANS simulation is related to the mean strain rates $\\frac{\\partial V}{\\partial y}$ and $\\frac{\\partial W}{\\partial z}$ .",
"These mean strain rates are close to zero based on the prediction of RANS models with linear eddy viscosity assumption.",
"Consequently, the normal stress imbalance is close to zero and the stress-induced secondary flow is not captured by the baseline RANS simulation.",
"Compared to the baseline RANS prediction, the posterior normal stress imbalance $\\tau _{yy}-\\tau _{zz}$ shows much better agreement with the experimental data, which is the main reason that the secondary flow prediction are improved by the Bayesian framework as shown in Fig.",
"REF and Fig.",
"REF .",
"Figure: Posterior ensemble of normal stress imbalance τ yy -τ zz \\tau _{yy}-\\tau _{zz} in plane B (x/T=3.187x/T=3.187) along three locations -y/T=0.38-y/T=0.38, 0.480.48 and 0.580.58, with increasing distance from the airfoil surface.",
"A smaller value of -y/T-y/T represents a location closer to the airfoil..",
"The normal stress imbalance is normalized with square of free stream velocity U ref 2 U_{ref}^2."
],
[
"Results at Upstream of Leading Edge",
"Figure REF a shows the prior ensemble of Reynolds stresses upstream of the leading edge near $x/T=-0.2$ , where the flow is approaching the stagnation point ($x/T$ = 0) at the leading edge.",
"Near this region, the horseshoe vortex develops when the incoming boundary layer experiences the strong adverse pressure gradient caused by the stagnation.",
"According to the experimental measurement, the center of the horseshoe vortex system approximately locates at $x/T=-0.2$ .",
"Since the RANS models have difficulty in predicting the separation of the boundary layer, the RANS predicted Reynolds stresses are unreliable at this region and thus large perturbation is required.",
"Based on this physics-informed knowledge, larger variance $\\sigma $ is specified around this region.",
"Consequently, the prior ensemble of Reynolds stresses shows a large scattering in the Barycentric triangle in Fig.",
"REF a.",
"Figure: Prior and posterior ensemble of Reynolds stresses in Barycentric triangle at x/Tx/T=-0.2.",
"Only the Reynolds stresses at first 9 mesh points from the flat plate are shown for clarity.",
"The experimental results fall on the bottom edge of the Barycentric triangle, which represents the two-component limit of turbulence.The posterior ensemble of Reynolds stress is shown in Fig.",
"REF b.",
"It can be seen that the inferred Reynolds stresses show no better agreement with the experimental data.",
"Specifically, the experimental data shows that the Reynolds stress anisotropy starts from the one component limit state, since the Reynolds stress is restricted in two directions at the point that close to both the wall and the leading edge.",
"For the sampled point further away from the plate, the Reynolds stress anisotropy becomes closer to the two component limit state, indicating that the restriction along the $z$ direction due to the wall is gradually reduced.",
"We also examined the inferred velocity at this region.",
"The prior and posterior velocity profiles at the plane upstream of the leading edge are shown in Figure REF .",
"It can be seen that the posterior velocity profiles shown in Fig.",
"REF b barely show better agreement with the experimental data compared to the RANS baseline prediction.",
"Therefore, it indicates that the inference of Reynolds stress is not satisfactory and thus the propagated velocity field is not improved.",
"In addition, it should be noted that the uncertainty as shown in the prior velocity profiles in Fig.",
"REF a is not able to cover the experimental data, which indicates that the uncertainty space of Reynolds stress is restricted and may not cover the true Reynolds stress.",
"Such restriction of uncertainty space of Reynolds stress can explain the unsatisfactory performance of the Bayesian inference at this region.",
"There are two possible reasons for the restriction of uncertainty space.",
"First, the Reynolds stress anisotropy is not the dominant factor of the discrepancy of velocity field at this region.",
"Since the turbulence time scale is known to be comparable with the time scale of the strain rate, the Reynolds stress is not able to adjust to the rapid change of strain rate [28].",
"Therefore, there is a misalignment of principal axis between the Reynolds stress tensor and strain rate tensor.",
"However, the RANS models with linear eddy viscosity assumption predicts the Reynolds stress tensor that have the same principal axis as the strain rate tensor.",
"Therefore, the principal axis of the Reynolds stress tensor predicted by RANS models has a large misalignment with the experimental data near the stagnation point, and the uncertainty space do not cover the true Reynolds stress since the orientation of the RANS predicted Reynolds stress is not perturbed in this framework.",
"Consequently, the propagated velocity profiles barely show improvement compared to the RANS baseline prediction.",
"Another reason is that the preservation of relative location of Reynolds stresses in Barycentric triangle as shown in Fig.",
"REF a.",
"Such preservation of relative location has been discussed in Sec.",
"REF .",
"Figure: The (a) prior and (b) posterior ensemble of streamwise velocity at the plane A upstream of the leading edge.",
"The profiles are shown along three locations at x/T=-0.25x/T=-0.25, -0.2-0.2 and -0.15-0.15, with x/T=-0.25x/T=-0.25 being furtherest from the leading edge and x/T=-0.15x/T=-0.15 being closest.To further examine the dominant reason for the unsatisfactory inference performance at upstream of the leading edge, we first constructed another perturbation field via Gaussian process with smaller length scale at this region.",
"The objective is to reduce the preservation of relative locations of Reynolds stresses and thus to reduce the restriction of the uncertainty space.",
"By propagating the perturbed Reynolds stresses field to the velocity field, we find that both the prior and posterior velocity profiles are still similar to the ones as shown in Fig.",
"REF and is omitted here for simplicity.",
"It shows that the preservation of relative location of Reynolds stress in Barycentric triangle is not the dominant reason for the restriction of uncertainty space of velocity field near the stagnation point.",
"Therefore, it is more likely that the misalignment of the principal axis of Reynolds stress is the main reason that accounts for the unsatisfactory inference performance at this region.",
"Figure REF shows the prior velocity profiles near the stagnation point by enabling the perturbation of the orientation of Reynolds stresses.",
"Compared to the velocity profiles as shown in Fig.",
"REF a, the velocity profiles as shown in Fig.",
"REF demonstrates that the velocity field at this region can cover the experimental data if the orientation of RANS predicted Reynolds stresses is perturbed.",
"Figure: Streamwise velocity profiles at the plane upstream of the leading edge obtained from the prior Reynolds stresses, where both the Reynolds stress anisotropy and their orientations are perturbed.",
"The profiles are shown along three locations at x/T=-0.25x/T=-0.25, -0.2-0.2 and -0.15-0.15 with x/T=-0.25x/T=-0.25 being furtherest away from the leading edge and x/T=-0.15x/T=-0.15 being closest.To illustrate the misalignment of principal axis, the RANS predicted orientation of Reynolds stress ($\\mathbf {v}_1$ , $\\mathbf {v}_2$ and $\\mathbf {v}_3$ ) and the experimental data are shown in Fig.",
"REF .",
"The perturbed orientations of Reynolds stress, which correspond to the prior velocity profiles as shown in Fig.",
"REF , are also shown in Fig.",
"REF .",
"The Reynolds stresses are sampled at three locations, one of which is located at the first cell near the flat plate ($z/T=2.5\\times 10^{-3}$ ), and the other two are further away from the plate at $z/T=0.015$ and $z/T=0.035$ , respectively.The unit vectors as shown in Fig.",
"REF represents the direction of principal axis.",
"It can be seen from Fig.",
"REF that the misalignment of principal axis between RANS predicted Reynolds stress and the experimental data is more severe at the region near the flat plate.",
"With the increase of the distance away from the plate, the misalignment of principal axis is reduced, especially for the orientation of $\\mathbf {v}_1$ .",
"By injecting uncertainties into the orientation of Reynolds stress, it can be seen in Fig.",
"REF that the samples of Reynolds stress orientation show a scattering around the baseline RANS result and cover the experimental data.",
"It explains the better coverage of velocity profiles as shown in Fig.",
"REF .",
"In addition, it demonstrates that enabling the perturbation of the orientation $\\mathbf {v}_1$ , $\\mathbf {v}_2$ , $\\mathbf {v}_3$ can better explore the uncertainty space of Reynolds stress and indicates that the inference performance at the leading edge can be improved.",
"However, it should be noted that the perturbation of the orientation $\\mathbf {v}_1$ , $\\mathbf {v}_2$ , $\\mathbf {v}_3$ can lead to the momentum flux from low momentum cell to high momentum one, which can cause numerical instability.",
"Moreover, given the same amount of observation data, introducing more unknowns into the problem increases the dimensionality of the uncertainty space and thus inevitably increases the difficulty for the inference.",
"Extensions of our current Bayesian inference framework to include perturbed orientations and applications to the wing-body junction flow problem are under way.",
"Figure: The comparison of orientations (as indicated by the normal eigenvectors 𝐯 1 \\mathbf {v}_1and 𝐯 2 \\mathbf {v}_2) of the Reynolds stresses.",
"The Reynolds stresses are sampled at threelocations with vertical coordinates z/T=0.035z/T=0.035 (panels a and d), 0.0150.015 (panels b and e) and2.5×10 -3 2.5 \\times 10^{-3} (panels c and f).",
"The third point z/T=2.5×10 -3 z/T=2.5 \\times 10^{-3} is located atthe center of the first cell next to the bottom flat plate.",
"All three points are located alongthe line x/T=-0.2x/T=-0.2 upstream the leading edge (indicated below the legend).",
"The orientationvector 𝐯 3 \\mathbf {v}_3 is omitted for clarity since it can be uniquely determined from𝐯 1 \\mathbf {v}_1 and 𝐯 2 \\mathbf {v}_2, i.e., 𝐯 3 =𝐯 1 ×𝐯 2 \\mathbf {v}_3 = \\mathbf {v}_1 \\times \\mathbf {v}_2."
],
[
"Conclusion",
"In this work we apply the Bayesian framework proposed by Xiao et al.",
"[15] to the wing–body junction flow, which is featured by the horseshoe vortex system and the possible corner separation.",
"Both features are studied in this work to evaluate the performance of the Bayesian framework for the complex flow problems.",
"To reduce the computational cost, we extend the original Bayesian framework to account for the RANS simulations with wall function.",
"Simulation results suggest that, at the corner region, both the posterior mean velocities and the Reynolds stress anisotropy show better agreement with the experimental data, even though the posterior Reynolds stress components do not demonstrate noticeable improvement.",
"The improvement in the prediction of mean flow field and Reynolds stress anisotropy demonstrate the capability of this Bayesian framework in predicting the complex flow problem.",
"In addition, the analysis of posterior Reynolds stress components indicates that the secondary flow at corner region is largely governed by the Reynolds stress anisotropy, and it is not necessary to accurately model each component of Reynolds stress to improve the mean flow field prediction at this region.",
"On the other hand, at the center region of horseshoe vortex, the prior mean velocities are still close to the baseline RANS prediction, even though the prior Reynolds stress anisotropy shows a significant difference from the baseline predicted Reynolds stress in Barycentric triangle.",
"In addition, the posterior mean velocities at leading edge show little improvement compared with the experimental data.",
"These results indicate that the mean flow field at this region is not as sensitive to the Reynolds stress anisotropy as the flow at the corner region.",
"This is attributed to the rapid change of strain rate near the stagnation point, which leads to a misalignment of principal axis between RANS predicted Reynolds stresses and the experimental data.",
"In this work, the orientations of RANS predicted Reynolds stress are not perturbed, and thus the uncertainty space would not cover the experimental data.",
"It explains the unsatisfactory inference performance near the leading edge where the strain rate changes rapidly as the flow is approaching the stagnation point.",
"In order to account for the flow problem with such rapid change of mean strain rate, consideration of the uncertainties in the orientations of Reynolds stress is necessary to extend the capability of the current framework."
]
] | 1605.05962 |
[
[
"Extrinsic electromagnetic chirality in all-photodesigned one-dimensional\n THz metamaterials"
],
[
"Abstract We suggest that all-photodesigned metamaterials, sub-wavelength custom patterns of photo-excited carriers on a semiconductor, can display an exotic extrinsic electromagnetic chirality in terahertz (THz) frequency range.",
"We consider a photo-induced pattern exhibiting 1D geometrical chirality, i.e.",
"its mirror image can not be superposed onto itself by translations without rotations and, in the long wavelength limit, we evaluate its bianisotropic response.",
"The photo-induced extrinsic chirality turns out to be fully reconfigurable by recasting the optical illumination which supports the photo-excited carriers.",
"The all-photodesigning technique represents a feasible, easy and powerful method for achieving effective matter functionalization and, combined with the chiral asymmetry, it could be the platform for a new generation of reconfigurable devices for THz wave polarization manipulation."
],
[
"Introduction",
"In the last two decades, metamaterial science has provided novel tools for manipulating electromagnetic radiation [1].",
"In the terahertz (THz) frequency domain, many researchers exploited the potential of metamaterials to fill the so-called THz gap and to conceive efficient optical devices, with particular effort on THz polarization manipulators [2], [3].",
"Semiconductor based reconfigurable metamaterials show electromagnetic response that can be rapidly modified through photo-doping [4], [5], application of a bias voltage [6] and thermal excitation [7].",
"In addition, a spatially modulated optical beam can induce a pattern of photo-carriers which mimics a standard metal-dielectric structure [8], [9], [10], [11].",
"Considering the homogenized regime (i.e.",
"the light modulation scale generating the photo-carriers pattern is much smaller than the wavelength), we suggested the first example of tunable hyperbolic all-photodesigned THz metamaterial [12].",
"In Ref.",
"[13], using femtosecond laser technology, authors investigated all-photodesigned transient metamaterials allowing the manipulation of THz waveforms polarization with subcycle switch-on times.",
"Recently, Mezzapesa et al.",
"[14] devised an all-photodesigned metamaterial reflector and they exploited it to control the emission properties of a THz QCL.",
"All-photodesigning technique provides ultra-fast reconfigurability and it is a type of grey-scale lithography [13].",
"Furthermore, in the quasi-homogenized regime, by tailoring the geometry of the photoinduced carrier profile, one can achieve a strong electromagnetic chiral response.",
"In Ref.",
"[15], the authors experimentally realized all-photodesigned gammadion-type metamaterials showing a highly tunable optical activity.",
"Artificial electromagnetic chirality and bi-anisotropy (due to the magneto-electric coupling) have attracted much attention in metamaterial research and they support several relevant effects such as, for example, negative refraction [16], giant optical activity, asymmetric transmission [17].",
"It is worth noting that the reciprocal bianisotropy response can also be achieved in a metamaterial whose basic meta-atoms do not show the standard 2D-3D geometrical chiral asymmetry.",
"This peculiar electromagnetic response is termed as extrinsic electromagnetic chirality [18] and it can yield giant linear [19], [20] and nonlinear [21] circular dichroism.",
"Moreover, in Ref.",
"[22], the authors introduced the concept of 1D geometrical chirality (namely the mirror image of the considered pattern can not be superposed onto itself by translations without rotations) and they proved that a 1D multi-layered structure, without intrinsic chiral inclusions, can support strong extrinsic electromagnetic chirality.",
"In this paper, we introduce all-photodesigned THz metamaterials exhibiting 1D chirality.",
"From full-wave simulations, we numerically retrieve the dielectric and chiral tensor of the considered metamaterial.",
"By fully exploiting the all-photodesign technique, we show that the extrinsic chirality is highly reconfigurable by varying the intensity and the chiral degree of the light pattern generating photo-excited carriers.",
"We support the numerical evidences through a simplified approach providing an analytical description of the non-local effective medium."
],
[
"THz metamaterial photodesigning",
"In Fig.1, we report the sketch of an all-photodesigned 1D chiral THz metamaterial and its corresponding waves scattering geometry.",
"An infrared (IR) plane wave, modulated along the $x$ -axis by a spatial light modulator (SLM), normally impinges onto the vacuum-semiconductor interface.",
"The light beam produces a photo-generated charge carrier pattern which spatially modulates the local plasma frequency thus yielding a spatially modulated dielectric permittivity profile at THz frequencies [12].",
"Figure: (Color online) Sketch of the all-photodesigned 1D metamaterial and waves scattering.",
"An infrared (IR) plane wave is modulated by a spatial light modulator (SLM) and it generates a photo-induced 1D chiral structure (whose strength and orientation is identified by 1D chiral vector κ\\kappa ) within the bulk of a semiconductor slab of thickness L\\mathrm {L}.",
"THz waves (dashed arrows) normally impinge onto the slab.Here, we consider the situation where the modulated beam intensity and the corresponding \"microscopic\" dielectric permittivity $\\varepsilon ^{(THz)}$ are 1D periodic functions showing 1D chiral asymmetry [22] and whose period $\\Lambda $ is much smaller than the THz wavelength $\\lambda ^{(THz)}$ .",
"In the long wavelength limit ($\\eta = \\Lambda / \\lambda ^{(THz)} \\ll 1$ ), the effective constitutive relations are ${ D}_x &=& \\varepsilon _0 \\left[ \\varepsilon _{\\parallel } { E}_x + \\frac{\\kappa }{k_0^{(THz)}} \\left( \\partial _y { E}_y +\\partial _z { E}_z \\right) \\right], \\nonumber \\\\{ D}_y &=& \\varepsilon _0 \\left( \\varepsilon _{\\perp } { E}_y - \\frac{\\kappa }{k_0^{(THz)}} \\partial _y { E}_x \\right), \\nonumber \\\\{ D}_z &=& \\varepsilon _0\\left( \\varepsilon _{\\perp } {\\ E}_z - \\frac{\\kappa }{k_0^{(THz)}} \\partial _z { E}_x \\right),$ where $k_0^{(THz)}=2\\pi /\\lambda ^{(THz)} = \\omega ^{(THz)}/c$ and $\\varepsilon _{\\parallel }$ ,$\\varepsilon _{\\perp }$ and $\\kappa $ are the three parameters describing the effective electromagnetic response.",
"The parameter $\\kappa $ measures the effects of the first order spatial dispersion and its amplitude is proportional to $\\eta $ [22].",
"As it is well-known, first order spatial dispersion is physically equivalent to reciprocal bianisotropy [23] and such equivalence is expressed by the Serdyukov-Fedorov transformation [23], i.e.",
"${\\bf { D}}^{\\prime }={\\bf { D}} - \\nabla \\times {\\bf Q}$ , ${\\bf { H}}^{\\prime }={\\bf { H}} + i \\omega ^{(THz)} {\\bf Q}$ with ${\\bf Q}=-\\varepsilon _0 \\kappa / k_0^{(THz)} \\left( { E}_z {\\hat{ \\bf e}_y}-{ E}_y {\\hat{ \\bf e}_z} \\right)$ .",
"Here, we use such transformation to investigate the chiral properties of the photoinduced metamaterial and we get ${\\bf D}^{\\prime } = \\varepsilon _0 \\varepsilon ^{(eff)} {\\bf E} - \\frac{i}{c} \\kappa \\times {\\bf H}^{\\prime }, \\quad {\\bf B} = -\\frac{i}{c} \\kappa \\times {\\bf E} + \\mu _0 {\\bf H}^{\\prime },$ where $\\varepsilon ^{(eff)}= \\textrm {diag}(\\varepsilon _{\\parallel },\\varepsilon _{\\perp }^{\\prime },\\varepsilon _{\\perp }^{\\prime })$ ($\\varepsilon _{\\perp } ^{\\prime } = \\varepsilon _{\\perp } + \\kappa ^2$ ).",
"In Eqs.",
"(REF ), we have introduced the 1D chiral vector $\\kappa = \\kappa \\hat{\\bf e}_x$ which does not vanish only if the photo-induced pattern has 1D chiral asymmetry along the $x$ -axis [24].",
"In order to investigate the chiral electromagnetic response of 1D photoinduced metamaterials, we have performed 2D full-wave simulations [25] where Maxwell's equations for both the 1+1D monochromatic IR beam and the THz field are coupled with the rate equation for the carriers dynamics.",
"We have focused on a GaAs slab and, by using the microscopic approach reported in Ref.",
"[26], we have evaluated the density-dependent optical semiconductor dielectric constant $\\varepsilon ^{(IR)}(N)$ (see Supplementary Material).",
"In the steady state, the 2D spatial dynamics of the carrier density $N(x,z)$ is described by the rate equation $D (\\partial _x^2+\\partial _z^2) N - \\frac{N}{\\tau (N)} + \\frac{\\varepsilon _0 }{2 \\hbar } Im \\left[ \\varepsilon ^{(IR)}(N) \\right] |{\\bf E}^{(IR)}|^2=0,$ where ${\\bf E}^{(IR)}(x,z)$ is the electric field at the considered optical frequency inside the semiconductor slab, $\\tau (N)$ is the density dependent electron-hole recombination time and $D$ is the ambipolar diffusion coefficient ($\\hbar $ is the reduced Planck's constant) [27].",
"Here, the density dependent electron-hole recombination time is given by $1/\\tau =A N + B N^2 + C N^3$ , where $A$ and $B$ are the non-radiative and radiative recombination rates, respectively, while the term proportional to $N^3$ describes Auger recombinations.",
"The THz dielectric response is described by the Drude model $\\varepsilon ^{(THz)} &= &\\varepsilon _b + g N,$ where $g = - e^2/ \\left(\\varepsilon _0 m^* \\right) \\left[\\omega ^{(THz)} \\left( \\omega ^{(THz)}+i \\gamma _D \\right) \\right]^{-1}$ , $\\varepsilon _b$ is the THz background dielectric constant in the absence of the optical beam, $-e$ is the electron charge unit, $m^*$ is the electron effective mass and $\\gamma _D$ is the free electron relaxation rate."
],
[
"Photoinduced extrinsic chirality",
"As a theoretical benchmark to discuss the properties of photo-induce chiral phenomena, we focus on the case where the incident IR illumination is given by $ I_{in}(x)= I_0 \\left\\lbrace 1+ \\frac{1}{2} \\left[ \\cos {\\left(\\frac{2 \\pi }{\\Lambda } x \\right)} +\\chi \\sin \\left(\\frac{4 \\pi }{\\Lambda } x\\right) \\right] \\right\\rbrace ,$ where $I_0$ is the average intensity and $\\chi $ is a parameter controlling the illumination chiral symmetry and, consequently, the photo-induced grating one (the allowed values of $\\chi $ are those for which $I_{in}$ is positive).",
"It is worth noting that, for $\\chi \\ne 0$ , the considered intensity profile $I_{in}(x)$ exhibits geometrical 1D chirality, namely its mirror image ($I_{in}(-x)$ ) can not be superposed onto itself by 1D translations ($I_{in}(x+x_0) \\ne I_{in}(-x)$ for all $x_0$ ); whereas, for $\\chi =0$ , the intensity profile is an even function and it is achiral.",
"In the considered numerical examples, we have set $\\lambda ^{(IR)}=870$ nm for the IR wavelength, $\\Lambda =10$ $\\mu $ m for the grating period, $L=2.5$ $\\mu $ m for the semiconductor thickness and we have used typical values for GaAs paramaters: $A=10^8$ s$^{-1}$ [26], $B=7.2 \\cdot 10^{-16}$ m$^3$ /s, $C=10^{-42}$ m$^6$ /s [28], $D=A L_D^2$ where the diffusion length is $L_D=2$ $\\mu $ m, $m^*=0.067 \\quad m_0$ ($m_0$ is the electron mass) and $\\gamma _D=3.09$ THz [29].",
"In addition, we have modeled the background dielectric constant $\\varepsilon _b$ (appearing in Eq.",
"(REF )) with the expression $\\varepsilon _b=\\varepsilon _1 - \\left[\\varepsilon _1 (\\omega _L^2-\\omega _T^2)\\right]/ \\left[\\omega ^{(THz)} \\left(\\omega ^{(THz)} + i \\gamma \\right) -\\omega _T^2 \\right]$ , where $\\varepsilon _1=11$ , $\\omega _L=55.06$ THz, $\\omega _T=50.65$ THz and $\\gamma =0.45$ THz [30].",
"The IR beam is a transverse electric (TE) wave, i.e.",
"${\\bf E}_{in}^{(IR)}= E_{in}^{(IR)}(x,z) \\exp {(-i\\omega ^{(IR)} t)} \\hat{\\bf e}_y$ (being $\\omega ^{(IR)}$ the angular IR frequency) whose incident intensity is $I_{in}(x)$ of Eq.",
"(REF ).",
"The incident THz fields are monochromatic plane waves normally impinging onto the GaAs slab (linearly polarized in the $x$ -$y$ plane).",
"The semiconductor slab has been placed between two vacuum layers.",
"Along the $x$ -axis, we have considered periodic conditions for IR, THz and carrier density; whereas, along the $z$ -axis, we have set scattering boundary and matched boundary conditions for IR and THz waves, respectively.",
"We have required the charge current to vanish (i.e.",
"$\\partial _z N=0$ ) at the facets of the semiconductor slab.",
"Figure: THz frequency dependence of the real (a) and the imaginary (b) parts of the dielectric permittivities (ε ∥ \\varepsilon _{\\parallel } , ε ⊥ \\varepsilon _{\\perp }).",
"(c) THz frequency dependence of the real and imaginary parts of the chiral electromagnetic parameter κ\\kappa .",
"Here I 0 =1.85I_0=1.85 kW/cm 2 ^2 and χ=1\\chi =1.We have focused on THz frequencies $\\nu ^{(THz)}$ in the range $1.5 - 4$ THz and accordingly, since the period to wavelength ratio $\\eta $ is smaller than $0.13$ , we have assumed the THz electromagnetic response of the photo-induced grating to be quasi-homogenized and to be given by the effective medium description of Eqs.",
"(REF ).",
"Strictly, Eqs.",
"(REF ) are valid only for a one-dimensional configuration where the microscopic permittivity only depends on $x$ .",
"In the considered situations, the microscopic dielectric profile retains a dependence from $z$ which is inherited from the spatial pattern of the IR illumination within the slab.",
"However, the chosen diffusion length $L_D$ (characterizing the spatial spreading of the photo-carrier distribution) is much larger than the IR wavelength so that the profile of the photo-induced carrier density $N$ is largely smoothed by carrier diffusion to the point that $N$ , and hence $\\varepsilon ^{(THz)}$ of Eq.",
"(REF ), can be regarded as uniform along the $z$ -axis within the slab (see Supplementary Material) [31].",
"In order to numerically retrieve the effective parameters appearing in Eqs.",
"(REF ) from the full-wave simulations we have averaged the fields along the $x$ -axis and subsequently integrated such averages along the slab thickness.",
"Therefore, after integrating both sides of Eqs.",
"(REF ) along the $z$ -axis inside the slab, we have retrieved $\\varepsilon _{\\parallel }$ , $\\varepsilon _{\\perp }$ and $\\kappa $ by solving the three algebraic equations thus obtained.",
"The general features of the retrieved THz effective parameters are reported in Fig.2 where we plot their characteristic spectral profiles for an IR illumination with $I_0=1.85$ kW/cm$^2$ and $\\chi =1$ .",
"In Fig.2(a) and 2(b) we plot the real and imaginary parts of the effective dielectric permittivities $\\varepsilon _{\\parallel }$ , $\\varepsilon _{\\perp }$ ; whereas, in panel (c), we report the real and the imaginary part of the chiral electromagnetic parameter $\\kappa $ .",
"The homogenized response is slightly birefringent ($ \\varepsilon _{\\parallel } \\simeq \\varepsilon _{\\perp } $ ) and it shows extrinsic electromagnetic chirality ($ \\kappa \\ne 0$ ).",
"In the considered situation, we have chosen the IR intensity profile in such a way that the effective permittivity has a zero crossing-point within the considered frequency range (i.e.",
"$Re(\\varepsilon _{\\perp })=0$ for $\\nu ^{(THz)}_0=2.7$ THz).",
"Note that the profile of $\\kappa $ shows a distinct and broad resonance-like shape in a spectral region surrounding the crossing point $\\nu ^{(THz)}_0$ which is a consequence of the competition between semiconductor dispersion (Drude model of Eq.",
"(REF )) and averaging features yielding electromagnetic chirality, as will become clearer in the following.",
"Bearing in mind that standard SRR-based THz metamaterials are characterized by a narrow-band frequency response, the considered broadband magneto-electric response can be a relevant feature for different applications [32].",
"Figure: |ε ∥ ||\\varepsilon _{\\parallel }| (a), |ε ⊥ ||\\varepsilon _{\\perp }| (b) and |κ||\\kappa | (c) as functions of the THz frequency for I 0 =1.85I_0=1.85 kW/cm 2 ^2 (solid lines) and I 0 =3.7I_0=3.7 kW/cm 2 ^2 (dashed lines).",
"Here χ=1\\chi = 1.Compared with active composite metamaterials (see e.g.",
"[33]) where the geometry design is structural and thus rigid, all-photodesigned metamaterials offer improved flexibility.",
"The high tunability of all-photodesigned THz metamaterials is due to two significant reasons: (i) the incident IR profile can be easily and rapidly modified at will (the dielectric permittivity is written, erased and rewritten even on ultrafast time scales [13]), (ii) the possibility to achieve a continuous spatial variation of the microscopic permittivity (grey-lithography) greatly increases the possibilities for the effective electromagnetic behavior.",
"In our examples, the freedom to independently set the IR intensity $I_0$ and the chirality degree $\\chi $ of the considered intensity profile allow to investigate the tunability features of all-photodesigned THz metamaterials.",
"In Fig.3, we plot the absolute value of $\\varepsilon _{\\parallel }$ , $\\varepsilon _{\\perp }$ and $\\kappa $ (panels (a), (b) and (c), respectively) for two different optical intensities, namely $I_0=1.85$ kW/cm$^2$ , $I_0=3.7$ kW/cm$^2$ and $\\chi =1$ .",
"For both intensities, the permittivity absolute value has a minimum at the zero-crossing point.",
"Accordingly, the profile of $|\\kappa |$ shows a broad hump, due to the broad resonance of $Re(\\kappa )$ and $Im(\\kappa )$ , which for both intensities is located around the corresponding permittivity crossing-point.",
"Note that, remarkably, the higher IR intensity provides the stronger electromagnetic chirality.",
"In Fig.4, we plot the absolute value of $\\varepsilon _{\\parallel }$ , $\\varepsilon _{\\perp }$ and $\\kappa $ (panels (a), (b) and (c), respectively) for three different chirality degrees, namely $\\chi =0, 0.5, 1$ for the IR intensity $I_0=1.85$ kW/cm$^2$ .",
"While $|\\varepsilon _{\\parallel }|$ and $|\\varepsilon _{\\perp }|$ are almost independent of $\\chi $ , $|\\kappa |$ is effectively proportional to $\\chi $ and it vanishes for $\\chi =0$ , as it must be, since in this case the IR intensity profile is achiral.",
"In panel (d) of Fig.4, for the same values of $\\chi $ and $I_0$ as above, we plot $|\\langle E_z \\rangle |$ , where $\\langle f(z) \\rangle =\\frac{1}{L} \\int _0^L dz f(z)$ labels the spatial average over the slab thickness and $E_z$ is the $z$ -component of the THz electric field.",
"Note that the fact that $|\\langle E_z \\rangle |$ is not vanishing is a key signature of the photoinduced chiral response.",
"In fact the launched THz plane-wave normally impinging at the entrance slab facet has vanishing longitudinal component so that the matching conditions across such facet requires the vanishing of $D_z$ within the slab and the third of Eqs.",
"(REF ) correspondingly implies that $E_z \\ne 0$ if $\\kappa \\ne 0$ .",
"Figure: THz frequency dependence of |ε ∥ ||\\varepsilon _{\\parallel }| (a), |ε ⊥ ||\\varepsilon _{\\perp }| (b), |κ||\\kappa | (c) and |〈E z 〉||\\langle E_z \\rangle | (d), for χ=0\\chi =0 (squares and solid lines), χ=0.5\\chi =0.5 (stars and dashed lines), χ=1\\chi =1 (solid line) for I 0 =1.85I_0=1.85 kW/cm 2 ^2."
],
[
"Discussion",
"In order to physically grasp the basic phenomenological features of the all-photoinduced metamaterials we exploit the approach of Ref.",
"[34] which, for a metamaterial with low dielectric contrast, provides an analytical description of the effective medium bianisotropy.",
"Such approach is suitable to describe the examples considered in this paper since the two permittivities $\\varepsilon _{\\parallel }$ and $\\varepsilon _{\\perp }$ can be slightly different (see Fig.2) only if the absolute value of the photoinduced dielectric modulation $|\\Delta \\varepsilon ^{(THz)}| =| \\varepsilon ^{(THz)} - \\bar{\\varepsilon }|$ is much smaller than $|\\bar{\\varepsilon }|$ ($\\bar{\\varepsilon }$ is the spatial average of the THz permittivity).",
"Since we are interested in understanding the main features of the obtained numerical results, to simplify the treatment we do not consider IR propagation within the thin slab ($|E^{(IR)}|^2=2 I_{in}/(\\varepsilon _0 c)$ ) and we neglect both nonlinear contributions and the $z$ -dependence of $N$ in Eq.",
"(REF ), thus obtaining for the photoinduced carried density $ N = \\beta I_0 \\left\\lbrace {1 + d_1 \\cos \\left( {\\frac{{2\\pi }}{\\Lambda }x} \\right) + \\chi d_2 \\sin \\left( {\\frac{{4\\pi }}{\\Lambda }x} \\right)} \\right\\rbrace ,$ where $\\beta = \\frac{{{\\mathop {\\rm Im}\\nolimits } \\left( {\\varepsilon ^{(IR)}(0) } \\right)}}{{A \\hbar c}}$ , $d_1 = \\frac{1}{2}\\left[ {1 + \\left( {2\\pi \\frac{{L_D }}{\\Lambda }} \\right)^2 } \\right]^{ - 1}$ , $d_2 = \\frac{1}{2}\\left[ {1 + \\left( {4\\pi \\frac{{L_D }}{\\Lambda }} \\right)^2 } \\right]^{ - 1}$ and $Im(\\varepsilon ^{(IR)}(0))=0.14$ .",
"After substituting Eq.",
"(REF ) into Eq.",
"(REF ) and following the approach of Ref.",
"[34] (see Supplementary Material), we obtain $ \\varepsilon _\\parallel &=&\\bar{\\varepsilon }- \\frac{1}{4} \\left[ d_1^2 + \\left( d_2 \\chi \\right)^2 \\right] \\frac{\\left(g \\beta I_0 \\right)^2 }{\\bar{\\varepsilon }},\\quad \\quad \\varepsilon _ \\perp = \\bar{\\varepsilon }, \\nonumber \\\\\\kappa &=& \\eta \\chi \\frac{3 d_1^2 d_2}{8} \\frac{\\left({ {g\\beta I_0 } }\\right)^3}{{\\bar{\\varepsilon }}^2},$ where $\\bar{\\varepsilon }= \\varepsilon _b + g \\beta I_0$ .",
"From Eqs.",
"(REF ) it is evident that the birefringence $|\\varepsilon _\\parallel - \\varepsilon _{\\perp }|$ and the chiral parameter $\\kappa $ are proportional to $(\\bar{\\varepsilon })^{-1}$ and to $(\\bar{\\varepsilon })^{-2}$ , respectively.",
"Accordingly, in the spectral region around the permittivity crossing point, this proves both the slight discrepancies between $\\varepsilon _{\\parallel }$ and $\\varepsilon _{\\perp }$ in panel (a) and (b) of Fig.2 and the resonant-like shape of $\\kappa $ in panel (c) of Fig.2.",
"Therefore in the low-contrast regime, the smaller values of the average permittivity entail a larger electromagnetic chirality.",
"Physically this is due to the fact that, when the dielectric contrast is very low, the difference between the dielectric profile and its mirror image is magnified for lower values of the average $\\bar{\\varepsilon }$ .",
"Note that the dependence of $\\bar{\\varepsilon }$ on $I_0$ agrees with the metamaterial tunability features reported in Fig.3.",
"Finally, the expression of $\\kappa $ in the third of Eqs.",
"(REF ) suggests different ways for increasing the value of such chiral parameter.",
"This can be done, for example, by shaping the IR illumination pattern with higher chiral degree $\\chi $ (as shown in panel (c) of Fig.4) and with higher average intensity $I_0$ or by designing the semiconductor bulk with smaller diffusion length $L_d$ and higher IR absorption coefficient $\\beta $ ."
],
[
"Conclusions",
"In conclusion, we have introduced all-photodesigned THz metamaterials displaying 1D chirality.",
"We have investigated the tunability of the extrinsic electromagnetic chiral response of a 1D photo-induced structure, where the magneto-electric coupling is due the 1D chiral asymmetry of the photo-carrier density profile.",
"Such 1D photo-induced structures show the same kind of electromagnetic chiral response exhibited by chiral layered media which support optical activity (see Ref.",
"[22]) and therefore they can be exploited to manipulate THz radiation.",
"The strength of electromagnetic chirality can in principle be increased by choosing highly asymmetric infrared illumination profiles, by extending our approach to photo-induced 2D structures or by resorting to femto-second laser technology to enhance the photo-induced dielectric modulation depth [13].",
"In view of the high spatial and temporal reconfigurability shown by this kind of THz metamaterials, all-photodesigned chirality holds the potential to conceive a novel class of active optical devices such as, for examples, circular polarization beam splitters and polarization spectral filters.",
"A. C. and C. R. thank the U.S. Army International Technology Center Atlantic for financial support (Grant No.",
"W911NF-14-1-0315)."
],
[
"Light-matter coupling in a bulk semiconductor",
"Following the approach in [26], we report here the First-Principles calculation of the semiconductor optical susceptibility $\\chi (N)$ used to evaluate at the frequency $\\omega ^{(IR)}=2168$ THz ($\\lambda ^{(IR)}=870$ nm) the density-dependent dielectric constant $\\varepsilon ^{(IR)}(N)$ in Eq.",
"(3) of the main manuscript.",
"We will focus on a GaAs bulk semiconductor medium.",
"In the free carriers, quasi-equilibrium and the two-bands approximations the expression of the complex susceptibility which describes the radiation-matter interaction reads [35], [36] $\\chi (N)=\\frac{-i}{\\hbar \\varepsilon _{0}V}\\sum _{\\mathbf {k}}\\frac{\\mu _{\\mathbf {k}}^{2}(f_{e\\mathbf {k}}(N)+f_{h\\mathbf {k}}(N)-1)}{\\gamma _{p}+i(\\omega _{\\mathbf {k}}-\\omega )}, $ where $\\mathbf {k}$ is the carrier momentum, $\\mu _{\\mathbf {k}}$ is the dipole matrix element associated with an optical transition, $V$ is the active volume, $\\gamma _{p}$ is the polarization decay rate ($\\sim 3.4 \\times 10^{13} \\, {\\rm s}^{-1}$ for GaAs), $N$ is the total carrier density, $\\omega $ is the injected field frequency and $\\hbar \\omega _{\\mathbf {k}}=E_g+\\epsilon _{e}+\\epsilon _{h}=E_g+ \\frac{\\hbar ^{2}k^{2}}{2m_{e}}+ \\frac{\\hbar ^{2}k^{2}}{2m_{h}}$ is the transition energy at the carrier momentum $\\mathbf {k}$ , being $E_g$ the energy gap ($E_g=1.424$ eV for GaAs) and $m_{e,h}$ the effective mass of the electrons in the conduction band and of the holes in the valence band ($m_{e}=0.067m_{0}$ , $m_{h}=0.57m_{0}$ for GaAs [37] where $m_{0}$ is the free electron mass).",
"We set the zeros of $\\epsilon _{e}$ and $\\epsilon _{h}$ equal to the bottom of the conduction band and the top of the valence band respectively.",
"The Fermi-Dirac distributions in Eq.",
"(REF ) are given by $f_{e\\mathbf {k}} (N)=\\frac{1}{\\exp (\\frac{\\epsilon _{e}-\\mu _{e}}{k_{b}T})+1}, \\quad f_{h\\mathbf {k}} (N)=\\frac{1}{\\exp (\\frac{\\epsilon _{h}-\\mu _{h}}{k_{b}T})+1},$ where $\\mu _{e}(N)$ and $\\mu _{h}(N)$ are the electron and holes quasi-chemical potential, $k_{b}$ is the Boltzmann constant and $T$ is the device temperature.",
"Using a series representation for the Fermi-Dirac functions and the Padè approximation, it is possible to derive the following analytical approximations for the chemical potential valid in the bulk case [35], [36] $\\mu _{e} &=&(\\ln N_{e}+k_{1}\\ln (k_{2}N_{e}+1)+k_{3}N_{e})k_{b}T, \\nonumber \\\\\\mu _{h} &=&(\\ln N_{h}+k_{1}\\ln (k_{2}N_{h}+1)+k_{3}N_{h})k_{b}T,$ where $k_{1}=4.8966851$ , $k_{2}=0.04496457$ , $k_{3}=0.1333760$ and $N_{e}$ and $N_{h}$ are the normalized total carrier densities given by $N_{e} =4N\\left( \\frac{\\pi \\hbar ^{2}}{2k_{b}Tm_{e}}\\right) ^{3/2}, \\quad N_{h} =4N\\left( \\frac{\\pi \\hbar ^{2}}{2k_{b}Tm_{h}}\\right) ^{3/2}.$ Moreover, for the square modulus of the electric dipole element $\\mu _{\\mathbf {k}}$ we assume the expression [38], [39], [40]: $\\mu _{\\mathbf {k}}^{2}=\\frac{E_g(E_g+\\Delta _{s})}{4\\left(E_g+\\frac{2}{3}\\Delta _{s}\\right)}\\left( \\frac{1}{m_{e}}-\\frac{1}{m_{0}}\\right)\\left( \\frac{e\\hbar }{\\hbar \\omega _{\\mathbf {k}}}\\right)^{2}$ where $\\Delta _{s}$ is the spin-orbit energy splitting ($\\Delta _{s}=0.34$ eV for GaAs).",
"In order to phenomenologically correct the overestimation of the homogeneous broadening effects of the Lorentzian slowly decaying tails (leading to an excessive absorption at photon frequencies below the band gap) we have considered an exponential dependence on $k$ in the polarization decay rate $\\gamma _{k}$ (Urbach tail correction) [35], [38] $\\gamma _{p} \\longrightarrow \\gamma _{k}=\\frac{2 \\gamma _{p}}{\\exp (\\frac{\\hbar \\omega _{\\mathbf {k}}-\\hbar \\omega }{E_{0}})+1},$ where $E_{0}$ is an empirical parameter linked to the LO phonon energy ( $E_{0} \\simeq 0.032$ eV for GaAs).",
"The value of the energy gap $E_g$ is corrected to account for many-body effects by extrapolating the band gap reduction $\\Delta _{g}$ as a function of the carrier density from a semianalytical curve well known in literature [36] $\\Delta _{g}=E_{R}\\left( -4.29115 e^{-(r_s+0.09274)/1.3888}-27.14073e^{-(r_s+0.09274)/0.27683} \\right)$ where $r_s=\\left(\\frac{3}{4\\pi N a_{0}^{3}}\\right)^{1/3}$ is the scaled interparticle distance, being $a_{0}=\\frac{4\\pi \\epsilon _{0}n_{0}^{2}\\hbar ^{2}}{e^{2}m_{r}}$ the exciton Bohr radius and $E_{R}=\\frac{e^{2}}{8\\pi \\epsilon _{0}n_{0}^{2}a_{0}}$ the $3D$ exciton Rydberg energy.",
"In the expression of $a_{0}$ the constant $n_{0}$ represents the background refractive index ($n_{0}=3.6$ for GaAs) and $m_{r}$ is the reduced electron-hole mass defined as $\\frac{1}{m_{r}}=\\frac{1}{m_{e}}+\\frac{1}{m_{h}}$ .",
"We further observe that the semi-analytical curve giving $\\Delta _{g}/E_{R}$ as a function of the interparticle distance $r_s$ is almost the same for several semiconductor materials [41].",
"Finally for a bulk medium, assuming an essentially continuous distribution of the states, we can replace the summation $\\frac{1}{V}\\sum \\limits _{\\mathbf {k}}$ with the integral $\\frac{2}{\\left( 2\\pi \\right) ^{3}} \\int \\limits _{0}^{\\infty }4\\pi k^{2}dk$ .",
"Figure: Real and imaginary parts of the susceptibility χ\\chi for bulk GaAs versus the carrier density NN for ω=ω (IR) =2168\\omega =\\omega ^{(IR)}=2168 THz.In Fig.REF we plot the real and imaginary part of $\\chi $ versus $N$ for $\\omega =\\omega ^{(IR)}=2168$ THz.",
"A polynomial fitting gives for the carriers density-dependence of the real and imaginary part of $\\chi $ respectively $Re \\left[\\chi (N) \\right] &=& 0.004(N/N_{0})^{3}-0.017(N/N_{0})^{2}-0.093(N/N_{0})+0.024, \\nonumber \\\\Im\\left[\\chi (N)\\right]&=&0.03(N/N_{0})^{7}-0.25(N/N_{0})^{6}+0.83(N/N_{0})^{5}-1.5(N/N_{0})^{4} \\nonumber \\\\&+&1.6(N/N_{0})^{3}-0.96(N/N_{0})^{2}+0.22(N/N_{0})+0.14$ where $N_{0}=10^{18} cm^{-3}$ , which are used in turn to evaluate the dielectric constant $\\varepsilon ^{(IR)}$ according to the formula $\\varepsilon ^{(IR)}(N)=n_{0}^{2}+{\\chi }(N).$"
],
[
"Effect of carrier diffusion on the photoinduced THz dielectric profile",
"The basic assumptions on the photoinduced THz dielectric profile $\\varepsilon ^{(THz)}$ in the manuscript are that 1) it does not depend on $z$ and 2) it reproduces the modulation of the infrared illumination along the $x$ -axis.",
"Such assumptions can be made consistent by choosing the diffusion length $L_D$ satisfying the constraint $\\lambda ^{(IR)} \\ll L_D \\ll \\Lambda $ where $\\lambda ^{(IR)}$ and $\\Lambda $ are the radiation wavelength and the infrared pattern period.",
"For the numerical parameters used in the main manuscript, and for the IR illumination of Eq.",
"(5) of the main manuscript with $\\chi =1$ and $I_0=1.85$ kW/cm$^2$ we have evaluated the photoinduced carrier distribution and in Fig.REF , we report both $|E^{(IR)}(x,z)|$ (a) and $N(x,z)$ (b).",
"Note that longitudinal dynamics of the IR field is dominated by the interference between the forward and backward components, whereas the photo-carrier density $N$ is almost independent on $z$ .",
"On the other hand the transverse modulation of the IR field is encoded in the transverse profile of $N$ .",
"Figure: (Color on-line) (a) Infrared electric field absolute value |E (IR) ||E^{(IR)}| and (b) photo-carrier density NN for χ=1\\chi =1 and I 0 =1.85I_0=1.85 kW/cm 2 ^2."
],
[
"A First-Principles Homogenization approach for one-dimensional photo-induced THz metamaterials",
"In this section we report the derivation and the main results of a homogenization approach suitable for describing the effective bianisotropic response of a one-dimensional photo-induced THz metamaterial."
],
[
"Two-scale approach to the homogenization of a periodic metamaterial",
"We here summarize the results of the two-scale homogenization approach developed in Ref.",
"[24].",
"Let us consider propagation of a monochromatic electromagnetic field through a periodic media $\\varepsilon _r (\\textbf {r})=\\varepsilon _r \\left( \\mathbf {r}+\\mathbf {\\Lambda } \\right)$ , where $\\mathbf {\\Lambda }$ is any arbitrary lattice vector.",
"The electric $\\textbf {E}$ and magnetic $\\textbf {H}$ field amplitudes satisfy Maxwell's equations $\\nabla \\times \\textbf {E} = i \\omega \\mu _0 \\textbf {H}$ , $\\nabla \\times \\textbf {H} = -i \\omega \\varepsilon _0 \\varepsilon _r \\textbf {E}$ , where time dependence $e^{-i\\omega t}$ has been assumed.",
"Here, the main assumption is that dielectric permittivity is characterized by a spatial subwavelength modulation and we introduce the parameter $\\eta = d/\\lambda $ where $d$ is the largest of the lattice basis vector lengths.",
"Considering the long-wavelength limit $\\eta \\ll 1$ , we can develop an asymptotic analysis of electromagnetic propagation.",
"Any field $\\mathbf {A}$ ($\\mathbf {A}=\\mathbf {E},\\mathbf {H}$ ) separately depends on the slow and fast coordinates ($\\mathbf {r}$ , $\\mathbf {R} = \\mathbf {r} / \\eta $ , respectively) and $A(\\mathbf {r},\\mathbf {R})$ can be decomposed as $\\mathbf {A} (\\mathbf {r},\\mathbf {R}) = \\overline{\\mathbf {A}}(\\mathbf {r}) + \\tilde{\\mathbf {A}}(\\mathbf {r},\\mathbf {R})$ where the overline denotes the spatial average over the metamaterial unit cell, namely $ \\overline{\\mathbf {A}}(\\mathbf {r}) = \\frac{1}{V} \\int _C d^3 R \\mathbf {A}(\\mathbf {r},\\mathbf {R})$ ($C$ is the unit cell and $V$ is its volume scaled by $\\eta ^3$ ), and the tilde denotes the rapidly varying zero mean residual, i.e.",
"$\\tilde{\\mathbf {A}} = \\mathbf {A} - \\overline{\\mathbf {A}} $ .",
"In our approach, the relative dielectric permittivity only depends on the fast coordinates ($\\varepsilon (\\mathbf {R}) = \\varepsilon _r(\\eta \\mathbf {R})$ ) and it can be decomposed as $\\varepsilon (\\mathbf {R}) = \\overline{\\varepsilon } + \\tilde{\\varepsilon }(\\mathbf {R})$ .",
"Representing each field $\\mathbf {A}=\\overline{\\mathbf {A}}+\\tilde{\\mathbf {A}}$ as a Taylor expansion up to the first order in $\\eta $ , we get $\\overline{\\mathbf {A}} = \\overline{\\mathbf {A}}_0 (\\mathbf {r}) + \\overline{\\mathbf {A}}_1 (\\mathbf {r}) \\eta $ , $\\tilde{\\mathbf {A}} = \\tilde{\\mathbf {A}}_0 (\\mathbf {r},\\mathbf {R}) + \\tilde{\\mathbf {A}}_1 (\\mathbf {r},\\mathbf {R}) \\eta $ .",
"From Maxwell's equations, we obtain the averaged equations $\\nabla \\times \\overline{\\mathbf {E}} = i \\omega \\mu _0 \\overline{\\mathbf {H}}$ , $\\nabla \\times \\overline{\\mathbf {H}} = -i \\omega \\overline{\\mathbf {D}}$ , where $\\overline{\\mathbf {D}} = \\varepsilon _0 \\Big [ \\overline{\\varepsilon } \\overline{\\mathbf {E}} + \\overline{ \\varepsilon (\\tilde{\\mathbf {E}}_0 + \\tilde{\\mathbf {E}}_1 \\eta ) } \\Big ]$ .",
"After some calculations, we obtain $\\tilde{\\mathbf {E}}_{0} = \\hat{\\mathbf {e}}_i \\left(\\partial _i f_j\\right) \\overline{E}_{0j}$ , $\\tilde{\\mathbf {E}}_1 = \\hat{\\mathbf {e}}_i \\left[ \\left( \\partial _i f_j \\right) \\overline{E}_{1j} + \\left( \\delta _{ir} \\tilde{f}_j+ \\partial _i W_{rj} \\right) \\frac{\\partial \\overline{E}_{0j}}{\\partial x_r} \\right]$ , where the sum is hereafter understood over repeated indices, $\\hat{\\textbf {e}}_i$ is the unit vector along the $i$ -th direction, $\\partial _i$ is the partial derivative along $X_i = \\hat{\\textbf {e}}_i \\cdot \\mathbf {R}$ , $\\overline{E}_{0j}= \\hat{\\textbf {e}}_j \\cdot \\overline{\\mathbf {E}}_{0}$ .",
"Here, we have introduced the functions $f_j$ ($\\tilde{f}_j$ is the zero mean residual of $f_j$ ) and $W_{rj}$ satisfying the equations $ \\nabla _\\mathbf {R} \\cdot \\left( \\varepsilon \\nabla _\\mathbf {R} f_j \\right) = -\\partial _j \\varepsilon , \\quad \\nabla _\\mathbf {R} \\cdot \\left( \\varepsilon \\nabla _\\mathbf {R} W_{rj} \\right) = -\\partial _r \\left( \\varepsilon \\tilde{f}_j\\right) - \\left( Q_{rj} - \\overline{Q_{rj}} \\right),$ respectively, where $Q_{rj}=\\varepsilon (\\delta _{rj}+\\partial _r f_j)$ ($\\delta _{rj}$ is the Kronecker's delta).",
"Next, after long calculations [24], the effective constitutive relations can be written as $\\overline{D}_i =\\varepsilon _0 \\left(\\varepsilon ^{(eff)}_{ij} \\overline{E}_j +\\alpha ^{(eff)}_{ijr} \\frac{\\partial \\overline{E}_j}{\\partial x_r} \\right)$ , $\\overline{B}_i = \\mu _0 \\overline{H}_i$ , where $\\varepsilon ^{(eff)}_{ij} = \\frac{1}{2} \\overline{ \\left( Q_{ij} + Q_{ji} \\right)}$ and $\\alpha ^{(eff)}_{ijr} = \\eta \\overline{ \\left( Q_{ri} \\tilde{f}_j - Q_{rj} \\tilde{f}_i \\right) }$ .",
"Exploiting the Serdyukov-Fedorov transformation, the constitutive relations can be transformed to a symmetric form [23], i.e.",
"$ \\overline{\\textbf {D}^{\\prime }} = \\varepsilon _0 \\varepsilon ^{\\prime (eff)} \\overline{\\textbf {E}^{\\prime }} - \\frac{i}{c} \\kappa ^{(eff)T} \\overline{\\textbf {H}^{\\prime }}, \\quad \\overline{\\textbf {B}^{\\prime }} = \\frac{i}{c} \\kappa ^{(eff)} \\overline{\\textbf {E}^{\\prime }} + \\mu _0 \\overline{\\textbf {H}^{\\prime }},$ where $ \\varepsilon ^{\\prime (eff)}_{ij} = \\varepsilon ^{(eff)}_{ij} + \\kappa ^{(eff)}_{ir} \\kappa ^{(eff)}_{rj}, \\quad \\kappa ^{(eff)}_{ij} = \\eta k_0 \\left[ \\epsilon _{imj} \\overline{\\varepsilon f_m} +\\left(\\epsilon _{imn} \\delta _{jq} + \\frac{1}{2} \\epsilon _{mqn} \\delta _{ij} \\right) \\overline{\\varepsilon f_m \\partial _q f_n}\\right],$ where $\\epsilon _{imn}$ is the Levi-Civita symbol.",
"$\\kappa _{ij}^{(eff)}$ is the effective chiral medium tensor and it is provided by the first order spatial dispersion."
],
[
"Extended Landau-Lifshitz-Looyenga Effective-Medium Approach",
"Here, we summarize the derivation on a homogenization approach, developed in Ref[34], which can be considered the extension of Landau-Lifshitz-Looyenga (LLL) effective-medium approach in the context of periodic metamaterials.",
"The LLL approach is an effective medium theory for evaluating the electrostatic effective dielectric permittivity of an isotropic mixture in the case where the dielectric contrast is low [42].",
"Specifically we use the approach summarized in the above section in the situation where $ \\varepsilon = \\overline{\\varepsilon }+ \\tilde{\\varepsilon }({\\bf R}) \\equiv \\overline{\\varepsilon }+ \\tau { \\Delta \\varepsilon }({\\bf R}),$ where $\\tau \\ll 1$ .",
"Accordingly, we expand the potential field ${\\bf f} = f_j \\hat{\\bf e}_j$ as a power series in the small parameter $\\tau $ up to the second order $ {\\bf f}= {\\bf f}^{(0)}+\\tau {\\bf f}^{(1)}+ \\tau ^2 {\\bf f}^{(2)}.$ Substituting Eq.",
"(REF ) and Eq.",
"(REF ) into the first of Eq.",
"(REF ) and extracting equations for each order in $\\tau $ , we get $ \\nabla _{\\bf R}^2 f_j^{(0)}&=&0, \\nonumber \\\\\\overline{\\varepsilon } \\nabla _{\\bf R}^2 f_j^{(1)}&=& -\\partial _j \\Delta \\varepsilon - \\nabla _{\\bf R} \\cdot ({ \\Delta \\varepsilon } \\nabla _{\\bf R} f^{(0)}_j ) , \\nonumber \\\\\\overline{\\varepsilon } \\nabla _{\\bf R}^2 f_j^{(2)}&=&- \\nabla _{\\bf R} \\cdot ({ \\Delta \\varepsilon } \\nabla _{\\bf R} f^{(1)}_j ).$ In order to solve such differential equations, we consider the Fourier series for ${ \\Delta \\varepsilon }$ and for the fields $f_j^{(m)}$ ($m=0,1,2$ ) which are given by, respectively, $ {\\bf \\Delta \\varepsilon }&=&\\sum _{{\\bf G} \\ne {\\bf 0}} { \\Delta \\varepsilon }_{\\bf G} e^{i {\\bf G} \\cdot {\\bf R}}, \\nonumber \\\\{\\bf f}^{(m)}&=&\\sum _{\\bf G} {\\bf f}^{(m)}_{\\bf G} e^{i {\\bf G} \\cdot {\\bf R}}.$ Using the Fourier series of Eqs.",
"(REF ), we obtain an explicit solution of the set of Eqs.",
"(REF ) which reads $ f_j= i \\tau \\sum _{{\\bf G} \\ne {\\bf 0}} \\left( \\frac{{ \\Delta \\varepsilon }_{\\bf G}}{\\overline{\\varepsilon }} \\frac{G_j}{|{\\bf G}|^2}- \\tau \\sum _{{ {\\bf G}^{\\prime }}\\ne {\\bf 0}} \\frac{{ \\Delta \\varepsilon }_{{\\bf G}^{\\prime }} { \\Delta \\varepsilon }_{{\\bf G}-{ {\\bf G}^{\\prime }}} }{\\overline{\\varepsilon }^2} \\frac{({\\bf G} \\cdot { {\\bf G}^{\\prime }}) {G_j^{\\prime }} }{|{\\bf G}|^2 |{ {\\bf G}^{\\prime }}|^2} \\right) e^{i {\\bf G} \\cdot {\\bf R}}.$ Substituting Eq.",
"(REF ) into Eqs.",
"(REF ), the analytical expression for the electromagnetic dielectric and chiral tensors are given by, respectively, $ \\varepsilon ^{\\prime (eff)}_{ij} & = & \\overline{\\varepsilon } \\delta _{ij}-\\frac{\\tau ^2}{\\overline{\\varepsilon }}\\sum _{{\\bf G} \\ne {\\bf 0}} \\frac{{ \\Delta \\varepsilon }_{- \\bf G} G_i}{|{\\bf G}|^2}\\Bigg [ { \\Delta \\varepsilon }_{\\bf G} G_j- \\tau \\sum _{{ {\\bf G}^{\\prime }} \\ne {\\bf 0}} \\frac{ { \\Delta \\varepsilon }_{\\bf K} { \\Delta \\varepsilon }_{{\\bf G}-{ {\\bf G}^{\\prime }}}}{\\overline{\\varepsilon }} \\frac{({\\bf G} \\cdot { {\\bf G}^{\\prime }}) {G_j^{\\prime }}}{|{ {\\bf G}^{\\prime }}|^2} \\Bigg ],\\nonumber \\\\\\kappa _{ij}^{(eff)}& = & i \\eta k_0 \\frac{\\tau ^3}{\\overline{\\varepsilon }^2} \\sum _{{\\bf G} \\ne {\\bf 0}, { {\\bf G}^{\\prime }}\\ne {\\bf 0}}\\frac{{ \\Delta \\varepsilon }_{-{\\bf G}-{ {\\bf G}^{\\prime }}} {\\Delta \\varepsilon }_{{\\bf G}} { \\Delta \\varepsilon }_{{ {\\bf G}^{\\prime }}} }{|{\\bf G}|^2 |{ {\\bf G}^{\\prime }}|^2} \\times \\nonumber \\\\&& \\Bigg [ ({\\bf G} \\cdot { {\\bf G}^{\\prime }}) \\epsilon _{imj} { {G}^{\\prime }_m} + \\left( 1+ 2 \\frac{{\\bf G} \\cdot { {\\bf G}^{\\prime }} }{|{\\bf G}|^2} \\right) \\left( \\epsilon _{imn} \\delta _{jq}+\\frac{1}{2} \\epsilon _{mqn} \\delta _{ij} \\right)G_q G_m { {G}^{\\prime }_n} \\Bigg ],$ where we have neglected the fourth and higher order terms in $\\tau $ ."
],
[
"Analytical effective medium theory for all-photodesigned one-dimensional THz metamaterials",
"Here, by using Eqs.",
"(REF ), we obtain analytical expressions for the dielectric and chiral tensors of an all-photodesigned 1D THz metamaterial.",
"In the simplified situation considered in the Sec.IV of the main manuscript (where we neglect the IR propagation within the thin slab together with the nonlinear contributions to the carrier dynamics and the $z$ -dependence of the carrier density $N$ ), we get the THz microscopic dielectric contrast $\\varepsilon ^{(THz)} &= &\\varepsilon _b + g N,$ where $ N = \\beta I_0 \\left\\lbrace {1 + d_1 \\cos \\left( {\\frac{{2\\pi }}{\\Lambda }x} \\right) + \\chi d_2 \\sin \\left( {\\frac{{4\\pi }}{\\Lambda }x} \\right)} \\right\\rbrace .$ In the Eq.",
"(REF ) and Eq.",
"(REF ), $g = - e^2/ \\left(\\varepsilon _0 m^* \\right) \\left[\\omega ^{(THz)} \\left( \\omega ^{(THz)}+i \\gamma _D \\right) \\right]^{-1}$ , $\\varepsilon _b$ is the THz background dielectric constant in the absence of the optical beam, $-e$ is the electron charge unit, $m^*$ is the electron effective mass and $\\gamma _D$ is the free electron relaxation rate; $\\beta = \\frac{{{\\mathop {\\rm Im}\\nolimits } \\left( {\\varepsilon ^{(IR)}(0) } \\right)}}{{A \\hbar c}}$ , $d_1 = \\frac{1}{2}\\left[ {1 + \\left( {2\\pi \\frac{{L_D }}{\\Lambda }} \\right)^2 } \\right]^{ - 1}$ , $d_2 = \\frac{1}{2}\\left[ {1 + \\left( {4\\pi \\frac{{L_D }}{\\Lambda }} \\right)^2 } \\right]^{ - 1}$ , $Im \\left( \\varepsilon ^{(IR)}(0) \\right) = 0.14$ .",
"Comparing Eq.",
"(REF ) with Eq.",
"(REF ) we get $\\bar{\\varepsilon }= \\varepsilon _b + g \\beta I_0$ and $\\Delta \\varepsilon ^{(THz)} = \\tau \\Delta \\varepsilon = g \\left( N - \\beta I_0 \\right)$ so that Eqs.",
"(REF ) yield $\\varepsilon ^{\\prime (eff)} = \\textrm {diag}(\\varepsilon _\\parallel ,\\varepsilon _ \\perp ,\\varepsilon _ \\perp )$ and $\\kappa ^{(eff)}_{ij}=\\varepsilon _{ij1} \\kappa $ , where $\\varepsilon _\\parallel &=&\\bar{\\varepsilon }- \\frac{1}{4} \\left[ d_1^2 + \\left( d_2 \\chi \\right)^2 \\right] \\frac{\\left(g \\beta I_0 \\right)^2 }{\\bar{\\varepsilon }},\\quad \\quad \\varepsilon _ \\perp = \\bar{\\varepsilon }, \\nonumber \\\\\\kappa &=& \\eta \\chi \\frac{3 d_1^2 d_2}{8} \\frac{\\left({ {g\\beta I_0 } }\\right)^3}{{\\bar{\\varepsilon }}^2},$"
]
] | 1605.05897 |
[
[
"RadioAstron gravitational redshift experiment: status update"
],
[
"Abstract A test of a cornerstone of general relativity, the gravitational redshift effect, is currently being conducted with the RadioAstron spacecraft, which is on a highly eccentric orbit around Earth.",
"Using ground radio telescopes to record the spacecraft signal, synchronized to its ultra-stable on-board H-maser, we can probe the varying flow of time on board with unprecedented accuracy.",
"The observations performed so far, currently being analyzed, have already allowed us to measure the effect with a relative accuracy of $4\\times10^{-4}$.",
"We expect to reach $2.5\\times10^{-5}$ with additional observations in 2016, an improvement of almost a magnitude over the 40-year old result of the GP-A mission."
],
[
"Introduction",
"Quantum theory (QT) and general relativity (GR) are the two pillars of modern physics.",
"However, they are incompatible.",
"Theoretical attempts to unify QT and GR lead to violations of GR and, in particular, the equivalence principle (EP).",
"It is hard to estimate the level at which EP may be violated.",
"Therefore equivalence principle tests are considered to “stand out as our deepest possible probe of new physics” .",
"We intend to test the EP with RadioAstron.",
"According to the Einstein equivalence principle, an electromagnetic wave of frequency $f$ , propagating in a region of space where the gravitational potential is not constant, experiences a gravitational frequency shift: ${\\Delta f_\\mathrm {grav} \\over f} = { \\Delta U \\over c^{2} },$ where $\\Delta U$ is the gravitational potential difference between the measurement points and $c$ is the speed of light .",
"Any violation of Eq.",
"(REF ) in an experiment with two identical atomic frequency standards may be parameterized as ${\\Delta f_\\mathrm {grav} \\over f} = {\\Delta U \\over c^2} (1+\\varepsilon ),$ where the violation parameter, $\\varepsilon $ , may depend on element composition of the gravitational field sources and on the specific kind of quantum transition exploited by the frequency standards.",
"It is generally agreed that the best test of Eq.",
"(REF ) to date was performed with the NASA-SAO Gravity Probe A (GP-A) mission 40 years ago which measured $\\varepsilon =(0.05\\pm 1.4)\\times 10^{-4}$ , giving the accuracy $\\delta \\varepsilon =1.4\\times 10^{-4}$ .",
"The gravitational potential modulation experienced by RadioAstron is comparable to that of GP-A: $\\Delta U/c^2 \\sim 3\\times 10^{-10}$ .",
"The better stability of the RadioAstron on-board H-maser and the possibility of repeating observations promise a factor of $\\sim $ 10 improvement on the GP-A result.",
"Testing the gravitational redshift effect has recently become an active field of experimental research.",
"The experiment with Galileo 5 & 6 navigational satellites is expected to probe the effect with (3–4)$\\times 10^{-5}$ accuracy .",
"The specialized ACES mission , to be launched in 2017, is expected to achieve $2\\times 10^{-6}$ ."
],
[
"Outline of the Experiment",
"In the gravitational redshift experiment with RadioAstron we detect the frequency change of the RadioAstron's on-board H-maser due to gravitation by comparing it, by means of radio links, with an H-maser at a ground station.",
"Either one of the mission's dedicated tracking stations (TS), Pushchino or Green Bank, or a ground radio telescope (GRT) equipped with a 8.4 or 15 GHz receiver and appropriate data acquisition instrument may be used for receiving the spacecraft signal.",
"The frequency variation due to the small gravitational frequency shift ($\\Delta f/f\\sim 10^{-10}$ ) needs to be separated from a number of other effects influencing the signal sent from the spacecraft to the ground station : ${\\Delta f_{1\\mathrm {w}}} &=f \\left(- {\\dot{D} \\over c}- {v_\\mathrm {s}^2 - v_\\mathrm {e}^2 \\over 2 c^2}+ { (\\mathbf {v}_\\mathrm {s} \\cdot \\mathbf {n})^2- (\\mathbf {v}_\\mathrm {e} \\cdot \\mathbf {n}) \\cdot (\\mathbf {v}_\\mathrm {s} \\cdot \\mathbf {n})\\over c^2 }\\right) \\nonumber \\\\&+ {\\Delta f_\\mathrm {grav} }+ {\\Delta f_\\mathrm {ion} }+ {\\Delta f_\\mathrm {trop} }+ {\\Delta f_0 }+ O\\left({v\\over c}\\right)^3,$ where “$\\mathrm {1w}$ ” stands for “1-way” (space to ground link), $\\mathbf {v}_\\mathrm {s}$ and $\\mathbf {v}_\\mathrm {e}$ are the velocities of the spacecraft and the ground station (in an Earth-centered inertial reference frame), $\\dot{D}$ is the radial velocity of the spacecraft relative to the ground station, $\\Delta f_\\mathrm {grav}$ is the gravitational redshift, $\\mathbf {n}$ is a unit vector in the direction opposite to that of signal propagation, $\\Delta f_\\mathrm {ion}$ and $\\Delta f_\\mathrm {trop}$ are the ionospheric and tropospheric shifts, and $\\Delta f_0$ is the frequency bias between the ground and space H-masers.",
"There are two major problems in using Eq.",
"(REF ) to determine $\\Delta f_\\mathrm {grav}$ directly, at least for RadioAstron.",
"The first is caused by the unknown frequency bias, $\\Delta f_0$ , which cannot be determined after launch without making use of Eq.",
"(REF ).",
"We solve this problem by measuring only the variation of the gravitational effect and taking into account the bias drift.",
"The second problem is that the nonrelativistic Doppler shift, $-\\dot{D}/c$ , cannot be calculated accurately enough from the available spacecraft state vector data.",
"We solve this problem with the help of the frequency measurements of the 2-way link, which let us cancel the nonrelativistic Doppler term: $\\Delta f_{1\\mathrm {w}} - {1\\over 2}\\Delta f_{2\\mathrm {w}}&=\\Delta f_\\mathrm {grav} + f \\left(- {|\\mathbf {v}_\\mathrm {s}^2 - \\mathbf {v}_\\mathrm {e}^2| \\over 2 c^2}+ {\\mathbf {a}_\\mathrm {e} \\cdot \\mathbf {n}_{} \\over c} \\Delta t\\right) + O(v/c)^3,$ where $\\mathbf {a}_e$ is the ground station acceleration and $\\Delta t$ is the signal light time.",
"(Eq.",
"(REF ) is relevant for a TS, similar but more complex equation holds for the case of the 2-way link signal received by a nearby GRT.)",
"The idea of the compensation scheme based on Eq.",
"(REF ) was first implemented in the GP-A experiment.",
"For RadioAstron, however, it is not directly applicable due to impossibility of the 1- and 2-way links to be using different reference signals simultaneously (Fig. 1).",
"Nevertheless, two options for realizing the compensation scheme (REF ) with RadioAstron are available.",
"The first option requires interleaving the 1-way (Fig.",
"1a) and 2-way (Fig.",
"1b) modes .",
"The data recorded by GRTs (and the TS) contain only one kind of signal at any given time.",
"However, if the switching cycle is short enough ($\\sim $ 4 min at 8.4 GHz) we are able to interpolate into the gaps with an error of $\\Delta f/f\\sim 5\\times 10^{-15}$ .",
"Thus we obtain quasi-simultaneous frequency measurements of both kinds and can apply the compensation scheme (REF ) to them directly.",
"Figure: NO_CAPTION"
]
] | 1605.05832 |
[
[
"Collapsing objects with the same gravitational trajectory can radiate\n away different amount of energy"
],
[
"Abstract We study radiation emitted during the gravitational collapse from two different types of shells.",
"We assume that one shell is made of dark matter and is completely transparent to the test scalar (for simplicity) field which belongs to the standard model, while the other shell is made of the standard model particles and is totally reflecting to the scalar field.",
"These two shells have exactly the same mass, charge and angular momentum (though we set the charge and angular momentum to zero), and therefore follow the same geodesic trajectory.",
"However, we demonstrate that they radiate away different amount of energy during the collapse.",
"This difference can in principle be used by an asymptotic observer to reconstruct the physical properties of the initial collapsing object other than mass, charge and angular momentum.",
"This result has implications for the information paradox and expands the list of the type of information which can be released from a collapsing object."
],
[
"Introduction",
"In Einstein-Maxwell theory a stationary black hole solution is generally characterized by its mass, electric charge and angular momentum.",
"In more general theories, some scalar field hairs have also been found [1], [2], [3], and they can be considered as generalized (or Noether) charges.",
"All additional information about the initial state of matter that formed the black hole is lost during the collapse.",
"This includes the global charges (e.g.",
"lepton number, baryon number, flavor [4]), angular momentum, charge and energy distributions (as opposed to their total values which are conserved) etc.",
"To recover this information after the black hole is formed seems to be impossible without invoking some exotic physics.",
"Instead of looking at the $t \\rightarrow \\infty $ , i.e.",
"an exact Schwarzschild solution in an asymptotically flat space-time, we can take a look at the near horizon region.",
"Information about the initial state might be released during the collapse, since once the collapse is over there is no much one can do.",
"It is well known that during the collapse an object radiates away its higher multipoles and other irregularities in the so-called balding phase before a perfect spherically symmetric horizon is formed.",
"The problem is that these are all gravitational degrees of freedom, and cannot account for other non-gravitational information content.",
"In [5], [6], it was shown that gravitational collapse is followed by the so-called pre-Hawking radiation from the very beginning of the collapse, simply because the metric is time dependent.",
"This radiation becomes completely thermal Hawking radiation only in $t \\rightarrow \\infty $ limit when the event horizon is formed.",
"Since the collapsing object has only finite amount of mass, an asymptotic observer would never witness the formation of the horizon at $t \\rightarrow \\infty $ .",
"For him, the collapsing shell will slowly get converted into not-quite-thermal radiation before it reaches its own Schwarzschild radius.",
"It was demonstrated in [7] that the evolution is completely unitary in such a setup.",
"In this paper, we also concentrate on the pre-Hawking radiation, but we are using the standard analysis as defined in [8], [9].",
"We explicitly construct an example in which two shells have exactly the same mass, charge and angular momentum (though we set the charge and angular momentum to zero for simplicity).",
"By construct, they follow the same gravitational trajectory, however they emit different radiation during the collapse.",
"We achieve this by giving different physical properties to the collapsing shells, other than mass, charge and angular momentum.",
"In particular, one of the shells is completely transparent to radiation, and the other is totally reflecting.",
"This is for example the situation where one of the shells is made of dark matter and the other of the standard model particles.",
"If one studies emission of the standard model particles from these shells, then the dark matter shell will be completely transparent to radiation, and the standard model shell will be totally or partially reflecting.",
"Of course, there is a whole continuum of cases between the totally reflecting and totally transparent shells, but for the purpose of illustration, these two extremes will suffice.",
"For simplicity, we use a spherically symmetric falling shell.",
"In this case only s-wave scalar field is relevant, and therefore the radiation field is chosen to be a scalar field.",
"In the realistic standard model, one could use any other field.",
"We show that the flux of energy and power spectra of radiation emitted from these two shells is notably different, though in the limit of $t \\rightarrow \\infty $ the fluxes become identical.",
"Thus, an observer studying the flux of the standard model particles from a collapsing shell could in principle tell if the shell is made of the dark or ordinary matter."
],
[
"The trajectory of the collapsing shell",
"For our purpose, we consider a freely falling massive spherical shell.",
"The time dependent radius of the shell is $R(\\tau )$ , where $\\tau $ is the proper time of the observer located on the shell.",
"The geometry outside the shell is Schwarzschild $&&ds^2=-\\left(1-\\frac{2M}{r}\\right)dt^2+\\left(1-\\frac{2M}{r}\\right)^{-1} dr^2+r^2d\\Omega \\\\&&d\\Omega =d\\theta ^2+\\sin ^2\\theta d\\phi ^2 .$ The geometry inside the shell is by the Birkhoff theorem flat Minkowski space $ds^2=-dT^2+dr^2+r^2d\\Omega $ The motion of the shell can be found by matching the geometry inside and outside the shell [10].",
"The equation of motion is given in terms of the conserved quantity $\\mu $ , which is just the rest mass of the shell.",
"$ \\mu =-R\\left[ (1-\\frac{2M}{R}+\\dot{R}^2)^{\\frac{1}{2}}-(1+\\dot{R}^2)^{\\frac{1}{2}}\\right] .$ Here, $\\dot{R}=\\frac{dR}{d\\tau }$ .",
"From Eq.",
"(REF ) we have $ \\dot{R}=\\Big ( \\frac{M^2}{\\mu ^2}-1+\\frac{M}{R}+\\frac{\\mu ^2}{4R^2}\\Big )^{\\frac{1}{2}}$ Then, the proper time on the shell is given by $\\tau = \\int d\\tau =\\int \\frac{dR}{\\dot{R}}$ The time coordinate of an asymptotic observer on the shell is $t = \\int dt =\\int \\frac{\\Big (1+\\frac{\\dot{R}^2}{1-\\frac{2M}{R}}\\Big )^\\frac{1}{2}}{\\Big (1-\\frac{2M}{R}\\Big )^\\frac{1}{2}}d\\tau $ The time coordinate of an observer on the shell shell is $T = \\int dT =\\int \\Big (1+\\dot{R}^2\\Big )^\\frac{1}{2}d\\tau $"
],
[
"Reflecting and transparent shells",
"We are set to study whether two massive shells with the same gravitational trajectory can have different pre-Hawking radiation.",
"To achieve this we consider two shells of equal mass, but one is completely transparent to a scalar field that propagates in this background, while the other one reflects the scalar field totally.",
"The evolution of the scalar field in a curved background outside the shell is described by $\\Box \\phi =0$ where the $\\Box $ operator is covariant.",
"Inside the shell, the $\\Box $ operator is Minkowski.",
"Because of the spherical symmetry, as usual, we simplify the discussion and focus on a $1+1$ dimensional scalar field, $\\phi (t,r)$ , which satisfies the wave equation $&& \\partial _t^2 \\phi - \\partial _{r^*}^2\\phi =0 \\mbox{, for $r>R$}\\\\&& \\partial _T^2 \\phi - \\partial _r^2 \\phi =0 \\mbox{, for $r<R$}$ Here $r^*=\\int \\frac{dr}{1-\\frac{2M}{r}}$ is the usual tortoise coordinate.",
"The trajectory of the spherical shell is given by Eq.",
"(REF ), and it is the same for both shells since they have the same mass (and carry no charge nor angular momentum).",
"There are two types of solutions to the wave equation for $r> R$ , i.e.",
"$f(t\\pm r^*)$ .",
"The function $f(t-r^*)$ represents a wave moving to the right, while $f(t+r^*)$ represents a wave moving to the left.",
"When a plane wave is propagating inward toward the origin, it is considered as an ingoing mode and can be written as $\\phi _{\\rm in} \\sim \\exp (-i\\omega v)$ where we defined the ingoing and outgoing null coordinates $v=t+r^*$ and $u=t-r^*$ .",
"When the ingoing mode passes through the center, it starts propagating outward (away from the center), and it becomes an outgoing mode.",
"The form of wave function is the same as before, but its argument must be a function of an outgoing coordinate $f(u)$ , i.e.",
"$\\phi _{\\rm out} \\sim \\exp (-i\\omega p(u)) ,$ where $p(u)$ is a function of the coordinate $u$ .",
"The shells in our discussion here are massive, which is different from the massless shells discussed in usual cases.",
"The massless scalar field is moving faster than the shell and will pass through or be reflected by the matter on the shell.",
"We now consider the transparent shell first.",
"While the shell is collapsing, the incoming scalar field mode passes through the shell and reaches the center of the shell.",
"Once it passes through the center, it becomes an outgoing mode.",
"As shown in Fig.",
"REF , the mode crosses the shell at some initial time $\\tau _i$ , passes through the center, and crosses again at some final time $\\tau _f$ .",
"Since the field moves at the speed of light, $\\tau _i$ and $\\tau _f$ must satisfy the condition $R(\\tau _f)+R(\\tau _i)=T(\\tau _f)-T(\\tau _i) .$ A scalar field coming from $R(\\tau _i)$ passing through the center and arriving to $R(\\tau _f)$ travels the distance $R(\\tau _i)+R(\\tau _f)$ .",
"An inside observer measures the time of this process $T(\\tau _f)-T(\\tau _i)$ .",
"Since the massless field travels at the speed of light, we arrive to Eq.",
"(REF ).",
"Figure: Penrose diagram for the transparent collapsing shell.",
"The mode crosses the shell at some initial time τ i \\tau _i, passes through the center, and crosses again at some final time τ f \\tau _f.The shell's radius at the moments of these two crossings is shown in Fig.",
"REF .",
"Figure: In this figure we set u=t(τ f )-r * (τ f )u=t(\\tau _f)-r^*(\\tau _f ) and M=1M =1.",
"R i R_i is the radius of the shell when the scalar field mode crosses it for the first time at τ i \\tau _i on the way inside the shell, and R f R_f is the radius when the mode crosses it for the second time on the way out τ f \\tau _f.When the wave mode comes out of the shell we have $u=t(\\tau _f)-r^*(\\tau _f)$ , but $p=v(\\tau _i)=t(\\tau _i)+r^*(\\tau _i)$ .",
"The function $p$ can be written in terms of the variable $u$ with the help of Eq.",
"(REF ).",
"The totally reflecting case is shown in Fig.",
"REF .",
"An ingoing mode is reflected by the shell immediately at $\\tau _f$ .",
"Therefore in this case $p=v(\\tau _f)=t(\\tau _f)+r^* (\\tau _f)$ and $u=t(\\tau _f)-r^* (\\tau _f)$ .",
"The function $p$ can now be written in terms of the variable $u$ by replacing $\\tau _f$ with $u$ .",
"We will now study the energy flux coming from the shell in these two cases.",
"Figure: Penrose diagram for the totally reflecting collapsing shell.",
"The scalar field ingoing mode is reflected by the shell at time τ f \\tau _f."
],
[
"Energy flux and power spectrum",
"The renormalized stressed-energy tensor for a massless scalar field was computed in terms of $p(u)$ by Fulling and Davies[8], [9].",
"In a $1+1$ dimensional spacetime, it can be written as $T_{uu}&=&\\frac{1}{24\\pi }\\Bigg ( -\\frac{M}{r^3}+\\frac{3}{2}\\frac{M^2}{r^2}+\\frac{3}{2}\\Big (\\frac{p^{\\prime \\prime }}{p^{\\prime }}\\Big )^2-\\frac{p^{\\prime \\prime \\prime }}{p^{\\prime }}\\Bigg )\\\\T_{uv}&=&-\\frac{1}{24\\pi }\\Big ( 1-\\frac{2M}{r}\\Big ) \\frac{M}{r^3}\\\\T_{vv}&=&\\frac{1}{24\\pi }\\Big ( -\\frac{M}{r^3}+\\frac{3}{2}\\frac{M^2}{r^2}\\Big )$ Primes denote derivatives with respect to the coordinate $u$ .",
"As $r\\rightarrow \\infty $ , only $T_{uu}$ survives, which is the radiated energy flux we are looking for.",
"To get concrete numerical results, for simplicity we set $\\mu =M =1$ .",
"We can find the coordinates $t$ and $T$ from Eqs.",
"(REF ) and (REF ).",
"There will be an arbitrary integration constant in these two integrals, but their values will not affect the result, so we set them to zero.",
"We plot the term $T_{uu}$ as a function of $r$ in Fig.",
"REF .",
"It is obvious that these two shells emits different fluxes as seen at infinity.",
"The reflecting shell has stronger pre-Hawking radiation and emits more energy than the transparent shell.",
"The difference is maximal around the value of the time parameter $u\\approx -8.3$ (see Fig.",
"REF ).",
"Obviously, the totally reflecting shell emits more energy than a transparent shell (Fig.",
"REF ).",
"This difference is coming from the interaction between the shell and the $\\phi $ field in vacuum.",
"Since the reflecting shell can interact with $\\phi $ field, it affects the vacuum stronger than the transparent shell does.",
"This interaction that does not exist in the transparent case may be interpreted as that the fact that reflecting shell contains more information than the transparent one.",
"This information then needs to be released before a static featureless black hole is formed.",
"At late time, in the $t\\rightarrow \\infty $ limit, radiation from both shells is indistinguishable and matches the radiation from a static black hole.",
"In this limit, all the black holes with the same mass emit the same radiation, no matter what they are made of (e.g.",
"dark vs. ordinary matter).",
"As the main result of our analysis, we state that pre-Hawking radiation from two shells is different, even though the gravitational trajectories are the same.",
"Figure: The black curve represents the energy flux T uu T_{uu} at r→∞r\\rightarrow \\infty for the totally reflecting shell.",
"The dashed curve represents the energy flux T uu T_{uu} at r→∞r\\rightarrow \\infty for the totally transparent shell.",
"The doted curve is the Hawking radiation for a static black hole, which is just 1 768π\\frac{1}{768\\pi }.",
"At late times, the two shells match the Hawking radiation from a pre-existing black hole, but the fluxes are substantially different during the collapse.Figure: The black curve represents the energy flux difference between totally reflecting shell and transparent shell, ΔT uu \\Delta T_{uu}, at r→∞r\\rightarrow \\infty .",
"At first, at u→-∞u\\rightarrow -\\infty , there is no radiation and therefore no difference, so ΔT uu →0\\Delta T_{uu}\\rightarrow 0.",
"ΔT uu \\Delta T_{uu} achieves its maximal at u≈-8.3u\\approx -8.3.",
"After that, radiation from both shells becomes very close to Hawking radiation, and ΔT uu \\Delta T_{uu} goes back to zero."
],
[
"Conclusion",
"In Einstein-Maxwell theory, physical properties of a black hole are completely determined by its mass, electric charge and angular momentum.",
"There is no additional information left after a black hole is formed.",
"Therefore black holes made of matter with different properties (other than mass, charge and angular momentum) emit exactly the same Hawking radiation.",
"Then one cannot recover any additional information with conventional physics if only Hawking radiation from a static black hole is considered.",
"It is well known that a collapsing object can shed its higher multipole moments in the so-called balding phase before reaching a perfect spherically symmetric form.",
"However it is important to extend this balding phase to other non-gravitational degrees of freedom.",
"In this paper, we considered a specific problem whether a collapsing object made of dark matter radiates away energy in the same way as an object made of the standard model particles.",
"If an asymptotic observer registers the flux of the standard model particles (for a simplicity a standard model scalar field), then the collapsing shell made of dark matter will be almost completely transparent to the scalar field due to the very weak interactions between the dark matter and the standard model.",
"In contrast, the shell made of the standard model particles will be almost completely reflecting.",
"These two shells have the same mass, charge and angular momentum, therefore, their trajectories dictated by gravity will be exactly the same.",
"Both of these shells will give rise to pre-Hawking radiation, so the question is whether this radiation will be identical for both shells or not.",
"We calculate the trajectory for the massive shell, and then the components of energy momentum tensor in a $(1+1)$ -dim spherically symmetric space-time defined by $(t,r)$ .",
"The scalar field ingoing mode passes through the transparent shell, reaches the center and then reappears on the other side of the shell as an outgoing mode.",
"In contrast, scalar field ingoing mode gets reflected back from the reflecting shell and becomes an outgoing mode immediately.",
"This difference is sufficient to give different amount and power spectrum of radiation.",
"The main results are shown in Fig.",
"REF .",
"The shells indeed radiate in a different way before becoming a black hole.",
"The reflecting shell emits more energy than the transparent shell.",
"This difference is caused by the interaction between the shell and the vacuum.",
"At late time, in the $t\\rightarrow \\infty $ limit, radiation from both shells is indistinguishable and matches the radiation from a static black hole.",
"We therefore demonstrated that pre-Hawking radiation can be used in principle to recover some physical quantities of the collapsing matter.",
"In this concrete example, an observer can tell if an object was made of ordinary or dark matter.",
"This information is released out during the collapse and it should be part of the balding phase in the black hole formation.",
"One should not confuse this result with the usual balding phase, which only includes quantities that affect gravity, i.e.",
"mass, charge, angular momentum and position.",
"This is an extra balding effect which reveals other physical quantities of the collapsing object.",
"D.C Dai was supported by the National Science Foundation of China (Grant No.",
"11433001 and 11447601), National Basic Research Program of China (973 Program 2015CB857001), No.14ZR1423200 from the Office of Science and Technology in Shanghai Municipal Government, the key laboratory grant from the Office of Science and Technology in Shanghai Municipal Government (No.",
"11DZ2260700) and the Program of Shanghai Academic/Technology Research Leader under Grant No.",
"16XD1401600.",
"DS partially supported by the US National Science Foundation, under Grant No.",
"PHY-1417317."
]
] | 1605.06026 |
[
[
"False Discovery Rate Control and Statistical Quality Assessment of\n Annotators in Crowdsourced Ranking"
],
[
"Abstract With the rapid growth of crowdsourcing platforms it has become easy and relatively inexpensive to collect a dataset labeled by multiple annotators in a short time.",
"However due to the lack of control over the quality of the annotators, some abnormal annotators may be affected by position bias which can potentially degrade the quality of the final consensus labels.",
"In this paper we introduce a statistical framework to model and detect annotator's position bias in order to control the false discovery rate (FDR) without a prior knowledge on the amount of biased annotators - the expected fraction of false discoveries among all discoveries being not too high, in order to assure that most of the discoveries are indeed true and replicable.",
"The key technical development relies on some new knockoff filters adapted to our problem and new algorithms based on the Inverse Scale Space dynamics whose discretization is potentially suitable for large scale crowdsourcing data analysis.",
"Our studies are supported by experiments with both simulated examples and real-world data.",
"The proposed framework provides us a useful tool for quantitatively studying annotator's abnormal behavior in crowdsourcing data arising from machine learning, sociology, computer vision, multimedia, etc."
],
[
"Introduction",
"In applications, building good predictive models is challenging primarily due to the difficulties in obtaining annotated training data.",
"A traditional way for data labeling is to hire a small group of experts to provide labels for the entire set of data.",
"However, such an approach can be expensive and time consuming for large scale data.",
"Thanks to the wide spread of crowdsourcing platforms (e.g., MTurk, Innocentive, CrowdFlower, CrowdRank, and Allourideas), a much more efficient way is to post unlabeled data to a crowdsourcing marketplace, where a big crowd of low-paid workers can be hired instantaneously to perform labeling tasks [27], [28], [16], [22], [8].",
"Despite of its high efficiency and immediate availability, crowd labeling raises many new challenges.",
"Since typical crowdsourced tasks are tedious and annotators usually come from a diverse pool including genuine experts, novices, biased workers, and malicious annotators, labels generated by the crowd suffer from low quality.",
"Thus, all crowdsourcers need strategies to ensure the reliability of answers.",
"In other words, outlier detection is a critical task in order to achieve a robust labeling results.",
"Various methods have been developed in literature for outlier detection, of which majority voting strategy [14], [18] is the most typical one.",
"In this setting, each pair is allocated to multiple annotators and their opinions are averaged over so as to identify and discard noisy data provided by unreliable raters.",
"They thus require large amount of pairwise labels to be collected.",
"More importantly as a local outlier detection method, majority voting is ineffective in identifying outliers that can cause global ranking inconsistencies [12], [13].",
"The work in [30] attacks this problem and formulates the outlier detection as a LASSO problem based on sparse approximations of the cyclic ranking projection of paired comparison data in Hodge decomposition.",
"Regularization paths of the LASSO problem could provide an order on samples tending to be outliers.",
"However, these work all treat pairwise comparison judgements as independent random outliers, which are typically defined to be data samples that have unusual deviations from the remaining data.",
"In this paper, instead of modeling the random effect of sample-wise outliers, we are primarily interested in the fixed effect where the annotators are influenced by positions when labeling in pairwise comparison setting.",
"Such an annotator's position bias [10] is ubiquitous in uncontrolled crowdsourced ranking experiments.",
"In our studies, annotator's position bias typically arises from: i) the ugly: one typically clicks one side more often than another.",
"As some pairs are highly confusing or annotators get too tired, in these cases, some annotators tend to click one side hoping to simply raise their record to receive more payment; while for pairs with substantial differences, they click as usual.",
"ii) the bad: some extremely careless annotators, or robots pretending to be human annotators, actually do not look at the instances and click one side all the time to quickly receive pay for work.",
"Such kinds of annotators may significantly deteriorate the quality of crowdsourcing data and increase the cost of acquiring annotations (since each raw feedback comes with a cost: the task requestor has to pay workers a pre-specified monetary reward for each labeling they provide, usually, regardless of the feedback correctness).",
"Although it might be relatively easy to identify the bad annotators above by inspecting their inputs, it is impossible for eye inspection to pick up those ugly annotators with mixed behaviors.",
"Therefore it is desired to design a statistical framework to quantitatively detect and eliminate annotator's position bias for crowdsourcing platforms in market.",
"Such a systematic study, up to the author's knowledge, however has not been seen in literature.",
"In this paper, we propose a linear model with annotator's position bias and new algorithms to find good estimates with an automatic control on the false discovery rate (FDR) – the expected fraction of false discoveries among all discoveries.",
"To understand FDR, imagine that we have a detection method that has just made 100 discoveries.",
"Then, if our method is known to control the FDR at the 10% level, this means that with high probability, we can expect at most 10 of these discoveries to be false and, therefore, at least 90 to be true and replicable.",
"Such a FDR control is desired when we don't have a prior knowledge about the amount of bad or ugly annotators and typical statistical estimates will lead to an over-identification of them.",
"Specifically, our contributions in this work are highlighted as follows: (A) A linear model with annotator's position bias as fixed effects; (B) New algorithms to find good estimates of such position bias, etc., based on Inverse Scale Space dynamics and its discretization Linearized Bregman Iteration; (C) New knockoff filters for FDR control adapted to our setting, which aims to mimic the correlation structure found within the original features for position bias; (D) Extensive experimental validation based on one simulated and four real-world crowdsourced datasets."
],
[
"Methodology",
"In this section, we systematically introduce the methodology for annotator's position bias estimation.",
"Specifically, we first start from a basic linear model with different types of noise models, which have been successfully used widely in literature.",
"Then we introduce a new dynamic approach with unbiased estimator called Inverse Scale Space (ISS).",
"Based on this, we present the modified knockoff filter for FDR control in details."
],
[
"Basic Linear Model",
"Let $V = \\lbrace 1,2,\\dots ,n\\rbrace $ be the set of nodes and $E = \\lbrace (\\alpha ,i,j): i,j\\in V, \\alpha \\in U\\rbrace $ be the set of edges, where $U$ is the set of all annotators.",
"Suppose the pairwise ranking data is given as $Y: E\\rightarrow R$ .",
"$Y_{ij}^\\alpha >0$ means $\\alpha $ prefers $i$ to $j$ and $Y_{ij}^{\\alpha }\\le 0$ otherwise.",
"The magnitude of $Y_{ij}^\\alpha $ can represent the degree of preference and it varies in applications.",
"It can be dichotomous choice $\\lbrace \\pm 1\\rbrace $ , $k$ -point Likert scale (e.g.",
"$k=3,4,5$ ), or even real values.",
"In this paper, consider the following linear model: $ Y_{ij}^\\alpha = \\theta _i - \\theta _j + z_{ij}^\\alpha $ where $\\theta : V\\rightarrow \\mathbb {R}$ is some common score on $V$ and the residue $z_{ij}^\\alpha $ may have interesting structures in crowdsourcing settings.",
"The annotators might have different effects on the residues.",
"While for most annotators, the deviations from the common score are due to random noise; occasionally the annotators deviate from the common behavior regularly – some careless ones always choose the left or the right candidate in comparisons, but others only do this when they get too confused to decide.",
"Such behaviors can be modeled in the following way, $ z_{ij}^\\alpha =\\gamma ^\\alpha + \\varepsilon _{ij}^\\alpha ,$ where $\\gamma ^\\alpha $ measures an annotator's position bias in a fixed effect, and the remainder $\\varepsilon _{ij}^\\alpha $ measures the random effect in sampling which is assumed to be sub-gaussian noise.",
"For example, a positive value of $\\gamma ^\\alpha $ means the annotator $\\alpha $ is more likely to prefer the left choice.",
"Under the random design of pairwise comparison experiments, a candidate should be placed on the left or the right randomly, so the position should not affect the choice of a careful (good) annotator.",
"Therefore $\\gamma ^\\alpha $ is assumed to be sparse, i.e., zero for most of annotators, and a nonzero position bias $\\gamma ^\\alpha $ means the annotator $\\alpha $ is either always choosing one position over the other (bad) or occasionally incurring this when they get too confused or tired (ugly).",
"We note that (REF ) should not be confused with recent studies in [12], [30] on outlier detection problem, $z_{ij}^\\alpha =\\gamma _{ij}^\\alpha + \\varepsilon _{ij}^\\alpha $ , where $\\gamma _{ij}^\\alpha $ models sparse outliers for each sample $(\\alpha ,i,j)$ , which only measures the random effect of samples rather than the fixed effect of annotators.",
"By modeling the annotator's fixed effect on position bias, one can systematically classify the annotators into the good, the ugly, and the bad according to their behaviors."
],
[
"ISS/LBI",
"Define the gradient operator by $\\delta _0:\\mathbb {R}^{|V|}\\rightarrow \\mathbb {R}^{|E|}$ such that $(\\delta _0 \\theta )(i,j,\\alpha ) = \\theta _i - \\theta _j$ , and the annotator operator $A:\\mathbb {R}^{|\\mathcal {A}|} \\rightarrow \\mathbb {R}^{|E|}$ by $(A\\gamma )(i,j,\\alpha ) = \\gamma ^\\alpha $ , then the model above can be rewritten as: $Y = \\delta _0 \\theta + A \\gamma + \\varepsilon ,$ In this case, detecting the annotators affected by position bias can be reformulated as: learning a sparse vector $\\gamma $ from given data $(\\delta _0,A,Y)$ .",
"To solve such a problem, in this paper, we choose a new approach based on the following dynamics, $\\frac{dp}{dt} &= A^T(Y-\\delta _0 \\theta -A\\gamma ) \\\\0 &= \\delta _0^T (Y-\\delta _0 \\theta -A\\gamma ) \\\\p &\\in \\partial \\Vert \\gamma \\Vert _1.",
"$ Its solution path can be easily solved by a sequence of nonnegative least squares, see [24] and references therein.",
"In this paper we use the free R-package [29].",
"In [24], it has been shown that the dynamics above has several advantages over the traditional LASSO approach, which can be formulated as follows in our setting $\\min _{\\theta ,\\gamma } \\frac{1}{2}\\Vert Y - \\delta _0 \\theta - A \\gamma \\Vert _2^2 + \\lambda \\Vert \\gamma \\Vert _1.$ First of all, the dynamics above is statistically equivalent to LASSO in terms of model selection consistency but may render oracle estimator which is bias-free, while the LASSO estimator is well-known biased.",
"In this sense the ISS path can be better than the LASSO path.",
"Here the solution path $\\hat{\\gamma }(t)_{t:0\\rightarrow \\infty }$ plays the same role of the regularization path of LASSO $\\hat{\\gamma }(\\lambda )_{\\lambda :\\infty \\rightarrow 0}$ with roughly $t=1/\\lambda $ , where the important features (variables) are selected before the noisy ones.",
"Following the tradition in image processing, such a dynamics is called Inverse Scale Space (ISS).",
"Beyond the charming statistical properties, ISS also admits an extremely simple discrete approximation, i.e., the Linearized Bregman Iteration (LBI), which has been widely used in image reconstruction with TV-regularization.",
"Adapted to our setting, the discretized algorithm is illustrated in Algorithm REF , which is scalable, easy for parallelization, and particularly suitable for large scale crowdsourced ranking data analysis.",
"LBI in correspondence to (REF ) Initialization: Given parameter $\\kappa $ and $\\triangle t$ , define $k=0, w^0=0, \\theta ^0 = (\\delta _0^T\\delta _0)^{\\dag }\\delta _0^TY,\\gamma ^0=0$ .",
"Iteration: $w^{k+1} &= w^k + A^T(Y - \\delta _0\\theta ^k - A\\gamma ^k)\\triangle t. \\\\\\gamma ^{k+1}&=\\kappa \\,\\mathrm {shrink}(w^{k+1}).",
"\\\\\\theta ^{k+1} &= \\theta ^k + \\kappa \\delta _0^T (Y-\\delta _0 \\theta ^k-A\\gamma ^k)\\triangle t. $ Stopping: exit when $k\\triangle t > t $ .",
"where $\\mathrm {shrink}(x) := \\mathrm {sign}(x)\\max \\lbrace |x|-1,0\\rbrace $ ."
],
[
"FDR Control and New Knockoff Filter",
"A crucial question for LASSO and ISS is how to choose the regularization parameter $\\lambda $ and $t$ in real-world data.",
"After all, different parameters can give different bad or ugly annotator sets.",
"Traditional methods either require a prior knowledge on the amount of such annotators which is often unknown in practice, or some statistically optimal choice of such regularization parameters.",
"Extensive studies in statistics have shown that such parameter tuning typically lead to an over estimation of the sparse signal, therefore False Discovery Rate (FDR) control is necessary [3] which is adopted in this paper.",
"FDR is defined as the expected proportion of false discoveries among the discoveries.",
"Putting in a mathematical way, here we consider $FDR = \\mathbb {E}\\left[\\frac{\\#\\lbrace \\alpha :\\gamma ^\\alpha =0,\\hat{\\gamma }^\\alpha \\ne 0\\rbrace }{\\#\\lbrace \\alpha :\\hat{\\gamma }^\\alpha \\ne 0\\rbrace \\wedge 1}\\right].$ To control the FDR means to control the accuracy of the bad/ugly annotators we detected to see if they are reasonable ones.",
"Recently, a new method called knockoff filter [3] is proposed to automatically control FDR in standard linear regression without a prior knowledge on the sparsity.",
"In this paper, such an approach will be extended to our linear model (REF ) and the algorithms, where both non-sparse $\\theta $ and sparse $\\gamma $ co-exist in the model.",
"The extended method consists of the same three steps as in [3], where the key difference lies in the knockoff feature construction adapted to our setting.",
"Construct knockoff features: let $\\tilde{A}$ be knockoff features that satisfy $\\tilde{A}^T\\tilde{A} = A^TA,~ A^T\\tilde{A} = A^TA- \\mathrm {diag}(s),~\\delta _0^T\\tilde{A} = \\delta _0^TA$ where $s$ is positive and can be solved by SDP: $\\max _s & &\\sum _j s_j\\\\s.t.",
"& &0 \\le s_j \\le 1\\\\& &\\mathrm {diag}(s) \\preceq 2A^T(I - H)A,$ with $H:=\\delta _0(\\delta _0^T\\delta _0)^{\\dag }\\delta _0^T$ .",
"Let $Q\\in \\mathbb {R}^{|E|\\times |\\mathcal {A}|}$ be an orthonormal matrix such that $\\delta _0^TQ=0, A^TQ=0$ , which requires $|E|\\ge 2|\\mathcal {A}|+|V|$ easily met in crowdsourcing.",
"Then (REF ) can be satisfied by defining $\\tilde{A} := A - (I - H)A(A^T(I - H)A)^{-1}\\mathrm {diag}(s) +QC$ where $C\\in \\mathbb {R}^{|\\mathcal {A}|\\times |\\mathcal {A}|}$ satisfies $C^TC = 2\\mathrm {diag}(s) - \\mathrm {diag}(s)(A^T(I - H)A)^{-1}\\mathrm {diag}(s)$ .",
"Now define the extended design matrix $A_{ko} = [A,\\tilde{A}]$ and $\\gamma _{ko} = [\\gamma ,\\tilde{\\gamma }]^T$ , then replace $A$ with $A_{ko}$ and $\\gamma $ with $\\gamma _{ko}$ in (REF ) , (REF ) or Alg.",
"REF , we can get solution path $\\hat{\\gamma }_{ko}(\\lambda )$ (or $\\hat{\\gamma }_{ko}(t)$ ).",
"Generate knockoff statistics for every original feature: define $Z_{j}$ to be the first entering time for $A_j$ , i.e., $Z_{j} = \\sup \\lbrace \\lambda : \\hat{\\gamma }_j(\\lambda ) \\ne 0\\rbrace $ for LASSO (or $\\sup \\lbrace 1/t: \\hat{\\gamma }_j(t) \\ne 0\\rbrace $ for ISS/LBI) and $\\tilde{Z}_j$ can be defined similarly.",
"Then the knockoff statistics becomes $W_j = \\max (Z_j,\\tilde{Z}_j) \\mathrm {sign}(Z_j - \\tilde{Z}_j)$ Choose variables based on the knockoff statistics: define the selected variable set $\\hat{S} = \\lbrace j: W_j\\ge T_{0/1}\\rbrace $ , where $T_{0/1} = \\min \\lbrace t: \\frac{0/1+\\#\\lbrace j:W_j\\le -t\\rbrace }{\\#\\lbrace j:W_j\\ge t\\rbrace }\\le q\\rbrace .$ $T_0$ is knockoff cut and $T_1$ is knockoff+ cut.",
"It can be shown that the new knockoff filter above indeed controls FDR in the following sense, whose proof is similar to that of [3] (collected in Supplementary Materials for completeness).",
"Theorem 1 If $\\epsilon $ is i.i.d $N(0,\\sigma ^2)$ and $|E|\\ge 2|\\mathcal {A}|+|V|$ , then for any $q\\in [0,1]$ , the knockoff filter with ISS/LBI (or LASSO) satisfies $\\mathbb {E}\\left[\\frac{\\#\\lbrace j:\\gamma _j =0~and~j\\in \\hat{S}\\rbrace }{\\#\\lbrace j:j\\in \\hat{S}\\rbrace +q^{-1}}\\right] \\le q$ and the knockoff+ method satisfies $\\mathbb {E}\\left[\\frac{\\#\\lbrace j:\\gamma _j =0~and~j\\in \\hat{S}\\rbrace }{\\#\\lbrace j:j\\in \\hat{S}\\rbrace }\\right] \\le q$ Remark 1 There is an equivalent reformulation of (REF ) to eliminate the non-sparse structure variable $\\theta $ and convert it to a standard LASSO.",
"Let $\\delta _0$ have a full SVD decomposition $\\delta _0 = U \\Sigma V^T$ and $U=[U_1,U_2]$ , where $U_1$ is an orthonormal basis of the column space $\\rm {col}(\\delta _0)$ and $U_2$ becomes an orthonormal basis for $\\rm {ker} (\\delta _0^T)$ .",
"Then $U_2^TY = U_2^TA \\gamma + U_2^T\\varepsilon .$ Let $y = U_2^TY, X = U_2^TA, e = U_2^T\\varepsilon $ , then $e$ is i.i.d $N(0,\\sigma ^2)$ $y = X \\gamma + e.$ Based on this, we can use the original knockoff filter $\\tilde{X}$ in [3] to select the position-biased annotators.",
"A shortcoming of this approach lies in the full SVD decomposition which might be too expensive for large scale problem.",
"The former approach will not suffer from this.",
"However, one can see in the following theorem both approaches are in fact equivalent.",
"Therefore such a reformulation provides us a conceptual insight in understanding the construction of knockoff filters.",
"Theorem 2 The approach in Remark 1 is equivalent to what we proposed above in the following sense: The knockoff features of (REF ) satisfies $\\tilde{X} = U_2^T \\tilde{A}$ and $\\tilde{A} = U_2\\tilde{X} + U_1U_1^TA$ ; The knockoff statistics constructed by ISS (or LASSO) for both procedures are exactly the same.",
"Both knockoff filters above can choose variables with FDR control but the estimator $(\\hat{\\theta },\\hat{\\gamma }_{ko})$ consists of knockoff features, so we need to reestimate $\\hat{\\theta },\\hat{\\gamma }$ after bad annotator detection by passing to a least square while only keeping those nonzero parameters and features.",
"Suppose that $\\hat{S}$ is the set of bad or ugly annotators given by knockoff filters, then one can find the final estimators by $(\\hat{\\theta },\\hat{\\gamma }) = \\arg \\min _{\\theta ,\\gamma _{\\hat{S}}} \\Vert Y - \\delta _0 \\theta - A_{\\hat{S}}\\gamma _{\\hat{S}}\\Vert _2^2.$ 0pt Table: Knockoff with q=10%q = 10\\% via ISS."
],
[
"EXPERIMENTS",
"In this section, five examples are exhibited with both simulated and real-world data to illustrate the validity of the analysis above and applications of the methodology proposed.",
"The first example is with simulated data while the latter four exploit real-world data collected by crowdsourcing."
],
[
"Simulated Study",
"Settings We first validate the proposed algorithm on simulated binary data labeled by 150 annotators.",
"Of the 150 annotators we have 100 good annotators (annotators 1 to 100 without position bias) and 50 bad/ugly annotators (annotators 101 to 150 with position bias).",
"We note that for good annotators, it does not mean that each worker always present the correct labels.",
"Instead, it means that they also have the probability to make incorrect judgements due to certain reasons, rather than position effect.",
"Specifically, we first create a random total order on $n$ candidates $V$ as the ground-truth and add paired comparison edges $(i,j)\\in E$ to graph $G=(V,E)$ until a complete graph, with the preference direction following the ground-truth order.",
"Here we choose $n=|V|=16$ , which is consistent with the third real-world dataset with smallest node size.",
"Then, for good annotators, they make judgements with an incorrect probability $p_1$ (i.e., $p_1 \\%$ of $E$ is reversed in preference direction), while for bad/ugly annotators, they are with a probability of $p_2$ disturbed by position effect.",
"Evaluation metrics Two metrics are employed to evaluate the performance of the proposed algorithms.",
"The first one is Control of Actual FDR, the second is Number of True Discoveries.",
"Experimental results With different choices of $p_1$ and $p_2$ , the mean Control of Actual FDR and Number of True Discoveries with $q = 10\\%$ over 100 runs are shown in Table REF to measure the performance of knockoff filter via ISS in position biased annotator detection.",
"It can be seen that via knockoff filter, ISS can provide an accurate detection of position biased annotators (indicated by control of actual FDR around $10\\%$ and Number of True Discoveries around 50).",
"Comparable results of LASSO with $q = 10\\%$ can be found in Table REF .",
"It can be seen that via knockoff filter, both LASSO and ISS can provide an accurate detection of position biased annotators.",
"This result is consistent with the theoretical comparison between LASSO and ISS discussed in [24], where ISS/LBI has similar theoretical guarantees as LASSO, but with bias-free and simpler implementation (the 3 line algorithm in Sec.",
"REF ) properties.",
"0pt Table: Knockoff with q=10%q = 10\\% via LASSO."
],
[
"Real-world Datasets",
"As there is no ground-truth for position biased annotators in real-world data, one can not compute control of actual FDR and Number of True Discoveries as in simulated data to evaluate the detection performance here.",
"In this subsection, we inspect the annotators returned by knockoff filter via ISS/LASSO under $q = 10\\%$ to see if they are reasonably good position biased workers.",
"3pt Table: Position biased annotators detected in Human age dataset, together with the click counts of each side (i.e., Left and Right).Figure: ISS regularization path of four real-world datasets (Green: the good; Red: the bad; Blue: the ugly)."
],
[
"Human Age",
"In this dataset, 30 images from human age dataset FG-NET http://www.fgnet.rsunit.com/ are annotated by a group of volunteer users on ChinaCrowds platform.",
"The groundtruth age ranking is known to us.",
"The annotator is presented with two images and given a binary choice of which one is older.",
"Totally, we obtain 14,011 pairwise comparisons from 94 annotators.",
"By adopting the knockoff-based algorithm we proposed, LASSO and ISS identify exactly the same set of abnormal annotators (i.e., 17 users) at q=10%, as is shown in Table REF .",
"It is easy to see that these annotators can be divided into two types: (1) the bad: click one side all the time (with ID in red); (2) the ugly: click one side with high probability (with ID in blue).",
"Besides, the regularization paths of ISS can be found in Figure REF (a), where the position biased annotators detected mostly lie outside the majority of the paths.",
"Note that since we allow a small percentage of false positives, some ugly annotators might be good in reality as well.",
"To see the effect of position biased annotators on global ranking scores, Table REF shows the outcomes of two ranking algorithms, namely original and corrected.",
"The original is calculated by least squares problems on all of the pairwise comparisons, while the corrected is obtained by the correction step via knockoff illustrated in Section REF .",
"It is easy to see that the removal of position biased annotators often changes the orders of some competitive images, such as ID=11 and ID=21, ID=30 and ID=8, etc.",
"To see which ranking is more reasonable, Table REF shows the groundtruth ranking of these competitive images.",
"We can find from this table that, compared with the original ranking, the corrected one is in more agreement with the groundtruth ranking, which further shows that: i) position biased annotators may disturb the ranking to a departure from the real ranking.",
"ii) pairs with little differences are more likely to lead to position biased annotations.",
"From this viewpoint, we can see that the knockoff-based FDR-controlling method indeed effectively selects the position biased annotators.",
"0pt Table: Comparison of original vs. corrected rankings on Human age dataset.",
"The integer represents the ranking position and the number in parenthesis represents the global ranking score returned by the corresponding algorithm.0pt Table: Groundtruth ranking of the competitive images highlighted with red color in Table ."
],
[
"Reading Level",
"The second dataset is a subset of reading level dataset [9], which contains 490 documents.",
"8,000 pairwise comparisons are collected from 346 annotators using CrowdFlower crowdsourcing platform.",
"More specifically, each annotator is asked to provide his/her opinion on which text is more challenging to read and understand.",
"Table REF shows the position biased annotators detected from this dataset, together with the ISS regularization path shown in Figure REF (b).",
"It is easy to see that LASSO and ISS picked out the same 6 annotators as position biased ones.",
"In terms of the small number of bad annotators detected, we can say that the overall quality of annotators on this task is relatively high.",
"0pt Table: Position biased annotators detected in Reading level."
],
[
"Image Quality Assessment",
"The third dataset is a pairwise comparison dataset for subjective image quality assessment (IQA), which contains 15 reference images and 15 distorted versions of each reference, for a total of 240 images which come from two publicly available datasets LIVE, [2] and IVC [1].",
"Totally, 342 observers, each of whom performs a varied number of comparisons via Internet, provide $52,043$ paired comparisons for crowdsourced subjective image quality assessment.",
"Note that the number of responses each reference image received is different in this dataset.",
"To validate whether the annotators we detected are good position biased annotators or not, we randomly take reference image 1 as an illustrative example while other reference images exhibit similar results.",
"Table REF shows the annotators with position bias picked by knockoff filter and the ISS regularization path is shown in Figure REF (c).",
"In this dataset, the abnormal annotators picked out by LASSO and ISS are also exactly the same.",
"It is easy to see that annotators picked out are mainly those clicking on one side almost all the time.",
"Besides, it is interesting to see that all these bad annotators highlighted with red color in Table REF click the left side all the time.",
"We then go back to the crowdsourcing platform and find out that the reason behind this is a default choice on the left button thus induces some lazy annotators cheat for the annotation task.",
"0pt Table: Position biased annotators detected in reference image 1."
],
[
"WorldCollege Ranking",
"We now apply the knockoff filter to the WorldCollege dataset, which is composed of 261 colleges.",
"Using the Allourideas crowdsourcing platform, a total of 340 distinct annotators from various countries (e.g., USA, Canada, Spain, France, Japan) are shown randomly with pairs of these colleges, and asked to decide which of the two universities is more attractive to attend.",
"Finally, we obtain a total of 8,823 pairwise comparisons.",
"We then apply knockoff filter to the resulting dataset and find out that both LASSO and ISS selected 36 annotators as position biased ones, as is shown in Table REF and Figure REF (d).",
"It is easy to see that similar to the human age dataset, the annotators picked out are either clicking one side all the time, or clicking one side with high probability.",
"0pt Table: Position biased annotators detected in WorldCollege."
],
[
"Discussion",
"Someone may argue that setting a threshold on the ratio of left/right answers can be an easy way to detect position biased annotators.",
"To illustrate why simply setting a threshold does not work, Figure REF shows the click counts of each side (i.e., X-axis: number of left clicks; Y-axis: number of right clicks), where each color $\\circ /\\times $ represents one annotator.",
"It is easy to see that there are indeed some overlaps between abnormal and normal annotators.",
"For example, in reading level dataset, annotators with ID=69 and ID=57 both provide 6:0 on the ratio of left/right clicks.",
"However, ID=69 is detected as abnormal annotator, while ID=57 as normal one.",
"To figure out the reason behind this, we further compute the Match Ratio (MR) of these two annotators with the global ranking scores obtained by all pairwise comparisons and find that $MR_{ID=69}=3/6$ and $MR_{ID=57}=5/6$ .",
"This indicates that the position biased annotator (i.e., ID=69) we picked out is the one not only with one-side click but also with a large deviation with the majority.",
"Similar results can be easily found in other three datasets."
],
[
"Outlier Detection",
"Outliers are often referred to as abnormalities, discordants, deviants, or anomalies in data.",
"Generally speaking, there can be two types of outliers: (1) samples as outliers; (2) subjects as outliers.",
"Hawkins formally defined in [15] the concept of an outlier as follows: “An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism.\"",
"Outliers are rare events, but once they have occurred, they may lead to a large instability of models estimated from the noisy data.",
"For type (1), many methods have been developed for outlier detection, such as distribution-based [4], depth-based [19], distance-based [20], [21], density-based [6], and clustering-based [17] methods.",
"For subject-based outlier detection, some sophisticated methods have been proposed to model annotators' judgements.",
"Recently, [9] propose a Crowd-BT algorithm to detect spammers and malicious annotators: spammers assign random labels while malicious annotators assign the wrong label most of the time.",
"Besides, [25] defines a score to rank the annotators for crowdsourced labeling tasks.",
"Furthermore, [26] presents an empirical Bayesian algorithm called SpEM to eliminate the spammers and estimate the consensus labels based only on the good annotators.",
"However, a phenomenon that has annoyed researchers who have used paired comparison tests is position bias or testing order effects.",
"Until now, little work have been found for such kind of position biased annotator detection, which is our main focus in this paper."
],
[
"FDR Control and Knockoff Method",
"Most variable selection techniques in statistics such as LASSO suffer from over-selection as picking up too many false positives by leaving out few true positives.",
"In order to offer guarantees on the accuracy of the selection, it is desired to control the false discovery rate (FDR) among all the selected variables.",
"The Benjamini-Hochberg (BH) procedure [5] is a typical method known to control FDR under independence scenarios.",
"Recently, [3] developed a new knockoff filter filtermethod for FDR control for general dependent features as long as the sample size is larger than that of parameters.",
"In this paper, we extend this method to our setting with mixed parameters of both nonsparse and sparse ones to achieve the same FDR control."
],
[
"Inverse Scale Space and Linearized Bregman Iteration",
"Linearized Bregman Iteration (LBI) has been widely used in image processing and compressed sensing [23], [31] even before its limit form as Inverse Scale Space (ISS) dynamics [7].",
"ISS/LBI at least have two advantages over the popular LASSO in variable selection: (1) ISS may give unbiased estimator [24], under nearly the same condition for model selection consistency as LASSO whose estimators are however always biased [11].",
"(2) LBI, regarded as a discretization of ISS dynamics, is an extremely simple algorithm which combines an iterative gradient descent algorithm together with a soft thresholding.",
"It only runs in a single path and regularization is achieved by early stopping like boosting algorithms [24], which may save the computational cost greatly and thus suitable for large scale implementation [32]."
],
[
"Conclusion",
"Annotator's position bias is ubiquitous in crowdsourced ranking data, which, up to our knowledge, has not been systematically addressed in literature.",
"In this paper, we propose a statistical model for annotator's position bias with pairwise comparison data on graphs, together with new algorithms to reach statistically good estimates with a FDR control based on some new design of knockoff filters.",
"FDR control here does not need a prior knowledge on the sparsity of position bias, i.e., the amount of bad or ugly annotators.",
"Such a framework is valid for both traditional LASSO estimator and the new dynamic approach based on ISS/LBI with debiased estimator and scalable implementations which is desired for crowdsourcing experiments.",
"Experimental studies are conducted with both simulated examples and real-world datasets.",
"Our results suggest that the proposed methodology is an effective tool to investigate annotator's abnormal behavior in modern crowdsourcing data."
],
[
"Acknowledgements",
"The research of Qianqian Xu was supported in part by National Natural Science Foundation of China (No.",
"61422213, 61402019, 61390514, 61572042), China Postdoctoral Science Foundation (2015T80025), “Strategic Priority Research Program\" of the Chinese Academy of Sciences (XDA06010701), and National Program for Support of Top-notch Young Professionals.",
"The research of Jiechao Xiong and Yuan Yao was supported in part by National Basic Research Program of China under grant 2015CB85600, 2012CB825501, and NSFC grant 61370004, 11421110001 (A3 project), as well as grants from Baidu and Microsoft Research-Asia.",
"Xiaochun Cao and Yuan Yao are the corresponding authors.",
"We would like to thank Yongyi Guo for helpful discussions and anonymous reviewers who gave valuable suggestions to help improve the manuscript.",
"Sketchy Proof of Theorem REF .",
"Similar to the treatment in [3], we only need to prove that the knockoff statistics $W_j$ satisfy the following two properties: sufficiency property: $W = f([\\delta _0,A_{ko}]^T[\\delta _0,A_{ko}], [\\delta _0,A_{ko}]^TY)$ , which indicates $W$ depends only on $[\\delta _0,A_{ko}]^T[\\delta _0,A_{ko}]$ and $[\\delta _0,A_{ko}]^TY$ .",
"antisymmetry property: Swapping $A_j$ and $\\tilde{A}_j$ has the effect of switching the sign of $W_j$ .",
"The second property is obvious because $W_j$ is constructed using entering time difference.",
"Now we go to prove the first property.",
"For ISS and LBI, the whole path is only determined by $A_{ko}^T(Y-\\delta _0 \\theta -A_{ko}\\gamma _{ko}) &= A_{ko}^TY - A_{ko}^T[\\delta _0 ,A_{ko}][\\theta ^T,\\gamma _{ko}^T]^T), \\\\\\delta _0^T (Y-\\delta _0 \\theta -A_{ko}\\gamma _{ko}) &= \\delta _0^TY - \\delta _0^T[\\delta _0 ,A_{ko}][\\theta ^T,\\gamma _{ko}^T]^T),$ which is only based on $[\\delta _0,A_{ko}]^T[\\delta _0,A_{ko}]$ and $[\\delta _0,A_{ko}]^TY$ , so is the entering time $Z_{j}$ The same reasoning holds for LASSO since $\\min _{\\theta ,\\gamma } \\frac{1}{2}\\Vert Y - [\\delta _0,A_{ko}] [\\theta ^T,\\gamma _{ko}^T]^T\\Vert _2^2 + \\lambda \\Vert \\gamma _{ko}\\Vert _1$ is equivalent to $\\min _{\\theta ,\\gamma } &\\frac{1}{2}(\\Vert Y\\Vert _2^2 + [\\theta ^T,\\gamma _{ko}^T] [\\delta _0,A_{ko}] ^T[\\delta _0,A_{ko}] [\\theta ^T,\\gamma _{ko}^T]^T\\\\&-2 [\\theta ^T,\\gamma _{ko}^T] [\\delta _0,A_{ko}] ^TY )+ \\lambda \\Vert \\gamma _{ko}\\Vert _1$ So the entire path is determined by $[\\delta _0,A_{ko}]^T[\\delta _0,A_{ko}]$ and $[\\delta _0,A_{ko}]^TY$ .",
"Proof of Theorem REF .",
"Suppose $\\tilde{X}$ is the knockoff statistics for (REF ), then it satisfies $\\tilde{X}^T\\tilde{X} = X^TX, X^T\\tilde{X} = X^TX - \\mathrm {diag}(s).$ Let $B = A + U_2(\\tilde{X} - X)$ , then $\\tilde{X} = U_2^TB$ and it can be verified $B^TB= A^TA, A^TB = A^TA-\\mathrm {diag}(s), \\delta _0^TB=\\delta _0^TA$ which means $B$ is a valid knockoff feature matrix for (REF ).",
"On the reverse, let $\\tilde{A}$ be knockoff features for (REF ), it is also easy to verify $\\tilde{X} = U_2^T\\tilde{A}$ satisfies condition (REF ).",
"This establishes an injection between $\\tilde{X}$ and $\\tilde{A}$ .",
"The equivalence of knockoff statistics comes from the equivalence of solution paths in both approaches.",
"To see this, () actually means $\\hat{\\theta } = (\\delta _0^T\\delta _0)^{\\dag }\\delta _0^T(Y - A_{ko}\\gamma _{ko})$ , plugging $\\hat{\\theta }$ in (REF ), we get $\\frac{dp}{dt} &=&A_{ko}^T (Y-\\delta _0 \\hat{\\theta }-A_{ko}\\gamma _{ko}) \\\\&=& A_{ko}^T(U_2U_2^T(Y-A_{ko}\\gamma _{ko}))\\\\&= & (U_2^TA_{ko})^T(U_2^TY-U_2^TA_{ko}\\gamma _{ko})$ This is equivalent to the ISS for the second procedure model (REF ) in Remark 1.",
"So in both approaches, the two ISS solution paths are identical.",
"The same reasoning holds for LASSO, the derivative of (REF ) w.r.t.",
"$\\theta $ is zero at the optimal estimator which means $0 = \\delta _0^T (Y-\\delta _0 \\hat{\\theta }-A_{ko}\\gamma _{ko})$ this is actually ().",
"So plugging $\\hat{\\theta }$ in (REF ), we get $&&\\Vert Y-\\delta _0 \\theta -A_{ko}\\gamma _{ko}\\Vert _2^2 \\\\&=& \\Vert (I - \\delta _0(\\delta _0^T\\delta _0)^{\\dag }\\delta _0^T)^T(Y-A_{ko}\\gamma _{ko})\\Vert _2^2\\\\&= & \\Vert U_2U_2^T(Y-A_{ko}\\gamma _{ko})\\Vert _2^2\\\\&= & \\Vert U_2^TY-U_2^TA_{ko}\\gamma _{ko})\\Vert _2^2.$ This is in fact the $l_2$ loss for the second procedure in Remark 1.",
"Finally identical paths lead to the same knockoff statistics which ends the proof."
]
] | 1605.05860 |
[
[
"Magnetoresistivity in a Tilted Magnetic Field in p-Si/SiGe/Si\n Heterostructures with an Anisotropic g-Factor: Part II"
],
[
"Abstract The magnetoresistance components $\\rho_{xx}$ and $\\rho_{xy}$ were measured in two p-Si/SiGe/Si quantum wells that have an anisotropic g-factor in a tilted magnetic field as a function of temperature, field and tilt angle.",
"Activation energy measurements demonstrate the existence of a ferromagnetic-paramagnetic (F-P) transition for a sample with a hole density of $p$=2$\\times10^{11}$\\,cm$^{-2}$.",
"This transition is due to crossing of the 0$\\uparrow$ and 1$\\downarrow$ Landau levels.",
"However, in another sample, with $p$=7.2$\\times10^{10}$\\,cm$^{-2}$, the 0$\\uparrow$ and 1$\\downarrow$ Landau levels coincide for angles $\\Theta$=0-70$^{\\text{o}}$.",
"Only for $\\Theta$ > 70$^{\\text{o}}$ do the levels start to diverge which, in turn, results in the energy gap opening."
],
[
"Introduction",
"Magnetotransport measurements on dilute p-Si/SiGe/Si structures, with two-dimensional hole gas (2DHG) densities of about 10$^{11}$ cm$^{-2}$ , have revealed an unusual phenomenon at filling factor $\\nu $ =3/2, the so-called \"re-entrant\" metal-insulator transition.",
"[1], [2], [3], [4], [5], [6] This phenomenon manifests itself as an additional peak of the magnetoresistance $\\rho _{xx}(T, \\Theta )$ at $\\nu $ =3/2.",
"The peak demonstrates an insulator type behavior, i.e.",
"its magnitude increases with decreasing sample temperature.",
"[3], [5] The authors of Ref.",
"2 explained this appearance by the presence of smooth long-range potential fluctuations having a magnitude comparable to the Fermi energy.",
"However, in Refs.",
"3,4,5 the magnetoresistance anomaly was attributed to a crossing of Landau levels (LLs) with different spin directions 0$\\uparrow $ and 1$\\downarrow $ as the magnetic field increased.",
"It appears that some p-Si/SiGe/Si systems show a magnetoresistance anomaly at $\\nu =3/2$ that depends on the tilt angle between the magnetic field and sample normal, [6] whereas in other p-Si/SiGe/Si systems this anomaly is not manifested at all.",
"[4] A third set of p-Si/SiGe/Si systems have such anomaly in $\\rho _{xx}$ at $\\nu =3/2$ , but it does not depend on the tilt angle.",
"[3] In our earlier article [7], we analyzed the conductivity at $\\nu $ =2 in tilted magnetic fields in a sample with $p$ =2$\\times $ 10$^{11}$ cm$^{-2}$ and demonstrated the presence of a ferromagnetic-paramagnetic (F-P) transition at a tilt angle of about 60$^{\\text{o}}$ .",
"It should be noted that at $\\nu $ =3/2 we did not observe any significant variation of the conductivity, instead a resistivity peak of the re-entrant-transition-type occurred in this region of filling factor.",
"We therefore focused our research on the $\\nu $ =2 region, i.e.",
"in the vicinity of the ferromagnetic-paramagnetic transition.",
"The magnetoresistance components $\\rho _{xx}$ and $\\rho _{xy}$ for the p-Si/SiGe/Si structure were measured in a tilted magnetic field, from which the conductivity $\\sigma _{xx}$ was calculated together with its dependence on temperature $T$ , magnetic field, and the tilt angle $\\Theta $ .",
"Such an approach allowed us to approximately calculate values of the Landau level energies, rather than just providing a qualitative description of the phenomenon, as was presented in Refs. 1,2,3,4,5,6.",
"The F-P phase transition seen at $\\nu \\cong $ 2, $T$ =0.3 K, and $\\Theta \\approx $ 60$^{\\text{o}}$ , is the result of crossing of the 0$\\uparrow $ and 1$\\downarrow $ LLs.",
"This transition is characterized by a jump in the filling factor and by a coexistence of both phases in the transition region.",
"A F-P transition has previously been reported in p-Si/SiGe/Si at $\\nu $ =4 and 6 in a tilted magnetic field by the authors of Ref. 8.",
"The present paper is an continuation of our previous article [7] and has three aims: (i) to study the dependence of the energy gap between LLs 0$\\uparrow $ and 1$\\downarrow $ on the magnetic field tilt angle $\\Theta $ to provide further confirmation of the crossing of these levels, in the p-Si/SiGe/Si sample with $p$ =2$\\times $ 10$^{11}$ cm$^{-2}$ ; (ii) to investigate the conductivity anisotropy in this sample, by measuring the conductivity at different orientations of the magnetic field component in the sample plane with respect to the current: $B_{\\parallel }\\parallel I$ and $B_{\\parallel }\\perp I$ , and comparing this with the theoretical model proposed in [9]; (iii) to measure the magnetoresistance in a tilted magnetic field for another p-Si/SiGe/Si sample with a lower density of $p$ =7.2$\\times $ 10$^{10}$ cm$^{-2}$ and compare it with the experimental data obtained by other groups on similar samples [3], [4], [6], with the hope of clearing up the inconsistency of the previous results mentioned above."
],
[
"Experiment and Discussion",
"In this research we studied two p-Si/SiGe/Si systems grown on a Si (100) substrate that consisted of a 300 nm Si buffer layer followed by a 30 nm Si$_{(1-x)}$ Ge$_{x}$ layer, 20 nm undoped Si spacer, and 50 nm layer of B-doped Si with a doping concentration of 2.5$\\times $ 10$^{18}$ cm$^{-3}$ .",
"One sample had x=0.08, yielding $p$ =7.2$\\times $ 10$^{10} $ cm$^{-2}$ , and the second had x=0.13, with $p$ =2$\\times $ 10$^{11}$ cm$^{-2}$ .",
"Both samples had a hole mobility of about 1$\\times $ 10$^4$ cm$^2$ /Vs at liquid-helium temperatures.",
"Figure: Dependence of the activation energy on tilt angleΘ\\Theta .",
"Inset: Dependence of the conductivity σ xx \\sigma _{xx} onΘ\\Theta at ν≈\\nu \\approx 2; TT=0.3 K.In the sample with $p$ =2$\\times $ 10$^{11}$ cm$^{-2}$ we measured the temperature dependence of the conductivity at different tilt angles $\\Theta $ over the temperature range 20 mK to 1 K, from which we were able to determine the activation energy $\\Delta E$ at various angles via the slope of the Arrhenius curves: $\\ln \\sigma _{xx} \\propto 1 /T$ .",
"The dependence of the activation energy on the tilt angle $\\Theta $ is shown in Figure REF , where it can clearly be seen that the activation energy achieves a minimum at $\\Theta \\approx $ 60$^{\\text{o}}$ .",
"The conductivity $\\sigma _{xx}$ ($\\Theta $ ) at the minima of oscillations at $\\nu \\cong $ 2, also shows a maximum as a function of tilt angle at $\\Theta \\approx $ 60$^{\\text{o}}$ , as shown in the inset to Figure REF .",
"It is worth noting that when the measurements are performed with the magnetic field normal to the sample plane the energy gap related to $\\nu \\cong $ 2 is about 3.2 K (0.28 meV).",
"Thus, we are justified in extracting the energy gap value from the temperature range of 200 mK - 1 K. When the tilt angle approaches 60$^{\\text{o}}$ the size of the energy gap is very small, due to the LLs crossing.",
"So, whilst the actual gap value obtained here is subject to considerable uncertainty, the observation of a minimum of the energy gap value at about 60$^{\\text{o}}$ qualitatively supports our model.",
"Figure: Energies of the LLs 0↑\\uparrow and 1↓\\downarrow vs.angle Θ\\Theta for the sample withpp=2×\\times 10 11 ^{11} cm -2 ^{-2}.These facts confirm that the observed F-P transition is indeed associated with the crossing of the LLs 0$\\uparrow $ and 1$\\downarrow $ at 60$^{\\textrm {o}}$ .",
"Now, knowing the activation energy dependence on $\\Theta $ and using the value $\\Delta E$ =0.28 meV found in Ref.",
"7 for $\\Theta $ =0, we can get a more accurate angle dependence of the energies of the levels 0$\\uparrow $ and 1$\\downarrow $ .",
"It is presented in Figure REF .",
"The F-P transition is expected to be accompanied by the formation of ferromagnetic domains.",
"According to Ref.",
"9, the domain formation should be manifested in an anisotropy of the magnetoresistance, i.e.",
"in a tilted field the value of the magnetoresistance should depend on the orientation of $B_{\\parallel }$ , the in-plane projection of the magnetic field, with respect to the current.",
"For example, an anisotropy in the region where LLs cross has been reported in several papers for GaAs/AlGaAs [10] and n-Si/SiGe [11], [12] heterostructures.",
"We tilted the sample in the two possible orientations, keeping the field projection ($B_{\\parallel } \\parallel I$ ) parallel and ($B_{\\parallel } \\perp I$ ) perpendicular to the current, but did not observe any anisotropy of the magneto-resistance in the vicinity of the transition.",
"Figure REF illustrates the dependence of the conductivity on the normal component of the applied magnetic field $B_{\\perp }$ at different angles and for both orientations of the in-plane projection of $B$ relative to the current.",
"Figure: Dependences of the σ xx \\sigma _{xx} on the normal componentof the magnetic field for different tilt angles shown for twoorientations of the magnetic field B ∥ ∥IB_{\\parallel } \\parallel I andB ∥ ⊥IB_{\\parallel } \\perp I at T=0.3T=0.3 K. The curves for each angle areshifted by 5×\\times 10 -6 ^{-6} Ω -1 \\Omega ^{-1} for clarity.As seen in Figure REF , the curves for the different directions of the in-plane projection of the magnetic field ($B_{\\parallel }\\parallel I$ and $B_{\\parallel } \\perp I$ ) virtually coincide, i. e. in our case the anisotropy of the conductivity is absent with a high degree of accuracy.",
"We also carried out similar studies at $T$ =(18 - 200) mK for the lower density p-Si/SiGe/Si sample with $p=7.2\\times 10^{10}$ cm$^{-2}$ .",
"The dependence of the resistivity $\\rho _{xx}$ on the magnetic field for different tilt angles are shown in Figure REF .",
"We particularly notice that, at tilt angles $\\Theta >$ from $0^{\\text{o}}$ to 70$^{\\text{o}}$ , the oscillations corresponding to $\\nu $ =2 are extremely weak.",
"They only start manifesting themselves for $\\Theta >$ 70$^{\\text{o}}$ .",
"At $\\nu $ =3/2, a maximum of resistance appears similar to the one we observed in the other sample, with a magnitude that depends strongly on the tilt angle.",
"Figure: Dependences of the ρ xx \\rho _{xx} on the normal component ofthe magnetic field for different tilt angles.",
"TT=0.2 K.Yet the oscillations at $\\nu $ =2 are clearly visible in another way of measuring the magnetoresistance: when the sample is rotated in a fixed total magnetic field, the perpendicular field component $B_{\\perp }$ causes oscillations at the angles determined by the concentration of charge carriers in the sample.",
"Figure REF shows such an angle dependence of the magnetoresistance measured at several fixed magnetic fields, where the oscillation related to $\\nu $ =2 can be seen to move from a tilt of about 9$^{\\text{o}}$ at 10 T to 5$^{\\text{o}}$ at 18 T. This corresponds to $B_{\\perp }$ = 1.7 T in each case, as shown in the Figure REF inset.",
"[11] The field value for $\\nu $ =2 $B$ =1.7 T is slightly different from data shown above.",
"This is probably a result of an ageing of the sample as the experiments of Ref.",
"11 were done much earlier.",
"Figure: Resistance ρ xx \\rho _{xx} as a function of the field tiltangle with respect to the plane of the 2D layer at different valuesof the total magnetic field, T≈T \\approx 0.4 K. Inset: ρ xx \\rho _{xx}as a function of the normal component of the magnetic field BB.The dependence of the conductivity $\\sigma _{xx}$ on the normal component of the magnetic field $B_{\\perp }$ is shown in Figure REF at different tilt angles, with $B_{\\parallel } \\parallel I$ .",
"Since the oscillations of $\\rho _{xx}$ at high tilt angles are observed against a background of high resistance with $\\rho _{xx} \\gg \\rho _{xy}$ , it turns out that $\\sigma _{xx} \\sim 1/\\rho _{xx}$ , so minima in $\\rho _{xx}$ correspond to maxima in $\\sigma _{xx}$ , as observed at $B_{\\perp }\\approx $ 1.5 T in Figure REF .",
"Figure: Dependences of the σ xx \\sigma _{xx} on the normal componentof the magnetic field for different tilt angles for the samplep=7.2×10 10 p=7.2 \\times 10^{10} cm -2 ^{-2}; T=0.2T=0.2 K.The absence of oscillations at magnetic fields corresponding to $\\nu $ =2 in the range of angles (0-70)$^{\\text{o}}$ indicates that the 0$\\uparrow $ and 1$\\downarrow $ LLs coincide.",
"The appearance of these oscillations for $\\Theta > $ 70$^{\\text{o}}$ is due, in our opinion, to the fact that the levels begin to diverge, resulting in the energy gap opening up.",
"Apparently, the gap opening in the sample with $p=7.2 \\times 10^{10}$ cm$^{-2}$ is associated with the angle dependence of the g-factor.",
"The g-factor in this material is anisotropic [1] and depends on the magnetic field tilt angle relative to the sample surface normal.",
"If the g-factor had an axial symmetry we could write $g^*=\\sqrt{g_{\\perp }^2\\cos ^2(\\Theta )+g_{\\parallel }^2 \\sin ^2(\\Theta )}$ , where $g_{\\perp }$ is the g-factor with the magnetic field perpendicular to the 2DHG, and $g_{\\parallel }$ is with the magnetic field parallel to the 2DHG.",
"For strong anisotropy, when $g_{\\parallel }$ =0 (as it should be in our structure) this reduces to $g^* = g_{\\perp } \\cos \\Theta $ .",
"However, if such a dependence of the g-factor were to occur, then the F-P transition should not be observed.",
"Unfortunately, we are unable to make reliable calculations and determine the width of the gap appearing in the sample with $p=7.2\\times 10^{10}$ cm$^{-2}$ due to the large magnetoresistance produced by the parallel magnetic field in this sample.",
"[13] It should be noted that the values of $\\rho _{xx}(B)$ and $\\sigma _{xx}(B)$ , on which background the oscillations develop, strongly depend on the magnetic field, and the greater the angle the stronger is this dependence.",
"So, it does not seem to be possible to reliably separate the small oscillations at $\\Theta > $ 70$^{\\text{o}}$ from the smooth background of $\\rho _{xx}(B)$ , which is about 10$^6$ ohms.",
"(Such problem for the sample with $p=2 \\times 10^{11}$ cm$^{-2}$ did not arise because the overall change $\\rho _{xx}(B)$ /$\\rho _{xx}$ (0) in a parallel magnetic field of 18 T did not exceed a factor of 4, and the in-plane resistance was only about 10$^4$ ohms).",
"Thus, the complete F-P transition in the sample with $p=7.2 \\times 10^{10}$ cm$^{-2}$ is not observed in tilted fields.",
"In a wide range of angles $\\Theta $ =(0-70)$^{\\text{o}}$ the 0$\\uparrow $ and 1$\\downarrow $ LLs are still coinciding, and only for $\\Theta >$ 70$^{\\text{o}}$ is there a gap in the hole energy spectrum arising as a result of a divergence of the LLs."
],
[
"Conclusion",
"The ferromagnetic-paramagnetic transition is observed in a p-Si/GeSi/Si sample with $p=2 \\times 10^{11}$ cm$^{-2}$ at a magnetic field corresponding to filling factor $\\nu \\approx $ 2.",
"It appears as a result of a change in the relative position of the 0$\\uparrow $ and 1$\\downarrow $ LLs as a function of the tilt angle $\\Theta $ .",
"This fact was first demonstrated in Ref.",
"7 and is confirmed in this paper by measurements of the energy gap dependence on the angle $\\Theta $ .",
"For this sample we also demonstrate an absence of anisotropy of xx with respect to the magnetic field projection on to the sample plane, despite such an anisotropy having been proposed in Ref. 9.",
"At the same time, in the sample with $p=7.2\\times 10^{10}$ cm$^{-2}$ the ferromagnetic-paramagnetic transition is not observed.",
"In a wide range of angles $\\Theta =$ 0-70$^{\\text{o}}$ the LLs 0$\\uparrow $ and 1$\\downarrow $ coincide, and only for $\\Theta > $ 70$^{\\text{o}}$ does a gap open in the hole spectrum as a result of the LLs diverging.",
"Ambiguity in the results observed by various authors [1], [2], [3], [4], [5], [6], as well as ourselves, on different p-Si/GeSi/Si samples is due, in our opinion, to dissimilar dependences of the g-factors on the magnetic field tilt angle.",
"This is caused by different levels of disorder in all these samples, since disorder can lead to breaking of the axial symmetry."
],
[
"Acknowledgments",
"The authors are grateful to E. Palm, T. Murphy, J.H.",
"Park, and G. Jones for their help with the experiments.",
"This work was supported by grant of RFBR 11-02-00223, grant of the Presidium of the Russian Academy of Science, the Program \"Spintronika\" of Branch of Physical Sciences of RAS.",
"The NHMFL is supported by the NSF through Cooperative Agreement No.",
"DMR-0654118, the State of Florida, and the US Department of Energy."
]
] | 1204.1144 |
[
[
"A stratification on the moduli spaces of symplectic and orthogonal\n bundles over a curve"
],
[
"Abstract A symplectic or orthogonal bundle $V$ of rank $2n$ over a curve has an invariant $t(V)$ which measures the maximal degree of its isotropic subbundles of rank $n$.",
"This invariant $t$ defines stratifications on moduli spaces of symplectic and orthogonal bundles.",
"We study this stratification by relating it to another one given by secant varieties in certain extension spaces.",
"We give a sharp upper bound on $t(V)$, which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces.",
"In particular, we compute the dimension of each stratum.",
"We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite.",
"We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case."
],
[
"Introduction",
"Let $X$ be a smooth algebraic curve over $\\mathbb {C}$ .",
"A vector bundle $V$ of rank two over $X$ determines a ruled surface $\\mathbb {P}V$ .",
"Such surfaces have been studied since the 19th century.",
"A line subbundle $L$ of $V$ gives a section $\\sigma _L$ of $\\mathbb {P}V$ , called a directrix curve.",
"The self-intersection number of $\\sigma _L$ is given by $\\sigma _{L} \\cdot \\sigma _{L} \\ = \\ \\deg (V/L) - \\deg L \\ = \\ \\deg V - 2 \\deg L.$ The Segre invariant of $\\mathbb {P}V$ is defined as the minimal value of $\\sigma _{L} \\cdot \\sigma _L$ over all $L \\subset V$ .",
"Via the above formula, this invariant also provides a measure of the difference of the slopes of $V$ and $L$ .",
"The Segre invariant yields a natural stratification on the moduli space of vector bundles of rank two over $X$ , which was studied by Lange and Narasimhan [13].",
"A generalization of this stratification to the moduli of vector bundles over $X$ of arbitrary rank was considered by Lange [11] and the details were settled in Brambila-Paz–Lange [2], and Russo–Teixidor i Bigas [19].",
"The aim of the present article is to establish parallel results for symplectic and orthogonal bundles over $X$ .",
"Let us define a few notions and fix notations.",
"A vector bundle $V$ over $X$ of rank $2n$ is called symplectic (resp., orthogonal) if there is a nondegenerate alternating (resp., symmetric) bilinear form $\\omega \\colon V \\otimes V \\rightarrow \\mathcal {O}_{X}$ .",
"A subbundle $E$ of $V$ is called isotropic if $\\omega |_{E \\otimes E} = 0$ .",
"By linear algebra, an isotropic subbundle of $V$ has rank $\\le n$ .",
"When $V$ is symplectic, a rank $n$ isotropic subbundle of $V$ is often called a Lagrangian subbundle.",
"We say that a symplectic or orthogonal bundle $V$ is stable (resp., semistable) if for every isotropic subbundle $E$ of $V$ , we have $\\mu (V) = \\frac{ \\deg V}{\\mathrm {rk}V} \\ > \\ \\frac{ \\deg E}{\\mathrm {rk}E} = \\mu (E) \\ \\ \\ (\\text{resp., } \\mu (V) \\ge \\mu (E)).$ Note that this is a priori weaker than the stability condition for $V$ as a vector bundle; compare with Ramanathan [18].",
"However, Ramanan [17] proved that semistability as an orthogonal bundle is equivalent to semistability of the underlying vector bundle, and moreover that a general stable orthogonal bundle is a stable vector bundle.",
"The same argument (worked through in [5]) shows that the analogous statement is true for symplectic bundles.",
"We denote by $SU_X(2n,\\mathcal {O}_{X})$ the moduli space of semistable vector bundles of rank $2n$ and trivial determinant, and write $\\mathcal {M}S_X(2n)$ (resp., $\\mathcal {M}O_X(2n)$ ) for the sublocus in $SU_X(2n,\\mathcal {O}_{X})$ of bundles admitting a symplectic (resp., orthogonal) structure.",
"In the symplectic case it has been proven by Serman [20] that the forgetful map $\\mathcal {M}_X(\\mathrm {Sp}_{2n}\\mathbb {C}) \\rightarrow SU_X(2n, \\mathcal {O}_{X})$ associated to the extension of the structure group $\\mathrm {Sp}_{2n}\\mathbb {C}\\subset \\mathrm {SL}_{2n} \\mathbb {C}$ , is an embedding, where $\\mathcal {M}_X(\\mathrm {Sp}_{2n}\\mathbb {C})$ is the moduli space of semistable principal $\\mathrm {Sp}_{2n}\\mathbb {C}$ -bundles over $X$ .",
"So $\\mathcal {M}S_X(2n)$ coincides with the embedded image of $\\mathcal {M}_X(\\mathrm {Sp}_{2n}\\mathbb {C})$ .",
"The orthogonal case is more delicate.",
"By [20], the forgetful map $\\mathcal {M}_X(\\mathrm {SO}_{2n}\\mathbb {C}) \\rightarrow SU_X(2n, \\mathcal {O}_{X})$ is generically two-to-one, amounting to forgetting the data of an orientation on a principal $\\mathrm {SO}_{2n}\\mathbb {C}$ -bundle.",
"On the other hand, the map $\\mathcal {M}_X(\\mathrm {O}_{2n}\\mathbb {C}) \\rightarrow \\mathcal {M}_X(\\mathrm {GL}_{2n}\\mathbb {C})$ is an embedding.",
"The moduli space $\\mathcal {M}_X(\\mathrm {O}_{2n}\\mathbb {C})$ of semistable principal $\\mathrm {O}_{2n}\\mathbb {C}$ -bundles over $X$ has several components, which are indexed by the first and second Stiefel–Whitney classes $(w_{1}, w_{2}) \\in H^{1}(X, \\mathbb {Z}_{2}) \\times H^{2}(X, \\mathbb {Z}_{2})$ .",
"The class $w_1$ corresponds to the determinant, and there are two components of $\\mathcal {M}_X(\\mathrm {O}_{2n}\\mathbb {C})$ with $w_1$ trivial.",
"We write $\\mathcal {M}O_{X}(2n)^\\pm $ for the embedded images of these components in $SU_X(2n, \\mathcal {O}_{X})$ .",
"Let $V$ be a symplectic or orthogonal bundle of rank $2n$ .",
"Generalizing the Segre invariant for bundles of rank two, we define $t(V) := -2\\max \\lbrace \\deg E : E \\hbox{ a rank $n$ isotropic subbundle of } V \\rbrace .$ In particular, if $V$ is stable (resp., semistable), then $t(V) > 0$ (resp., $t(V) \\ge 0$ ).",
"(Note that when $V$ is a symplectic bundle of rank $>2$ , this differs from the invariant $s_{\\mathrm {Lag}}$ defined in [3] by a constant: $t(V) = \\frac{2}{n+1} s_{\\mathrm {Lag}}(V) $ .)",
"For a symplectic or orthogonal bundle $V$ , we denote by $M(V)$ the space of rank $n$ isotropic subbundles $E \\subset V$ such that $t(V) = -2 \\deg E$ .",
"It can be viewed as a closed subscheme of a Quot scheme, so it has a natural structure of projective variety.",
"The invariant $t(V)$ induces stratifications on moduli spaces of semistable symplectic and orthogonal bundles over $X$ .",
"For each positive even integer $t$ , we define $\\mathcal {M}S_X(2n;t) := \\lbrace V \\in \\mathcal {M}S_X(2n) : t(V) = t \\rbrace $ and $\\mathcal {M}O_X(2n;t) := \\lbrace V \\in \\mathcal {M}O_X(2n) : t(V) = t \\rbrace .$ By semi-continuity of the invariant $t(V)$ , these subloci are constructible sets.",
"For the symplectic case, we show: Theorem 1.1 Let $X$ be a smooth algebraic curve of genus $g \\ge 2$ .",
"For any symplectic bundle $V$ of rank $2n$ , we have $t(V) \\le n(g-1) +1$ .",
"This is sharp in the sense that for a general $V \\in \\mathcal {M}S_X(2n)$ , $n(g-1) \\le t(V) \\le n(g-1 ) +1.$ For each even integer $t$ with $ 2 \\le t \\le n (g-1)$ , the stratum $\\mathcal {M}S_X(2n;t)$ is nonempty and irreducible, and $\\dim \\mathcal {M}S_X(2n;t) \\ = \\ \\frac{1}{2} \\left( n(3n+1)(g-1) + (n+1)t \\right).$ Furthermore, if $t < n(g-1)$ , then $\\mathcal {M}S_X(2n;t)$ is contained in the closure of $\\mathcal {M}S_X(2n;t+2)$ in $\\mathcal {M}S_X(2n)$ .",
"For each positive even integer $t < n(g-1)$ , the space $M(V)$ is a single point for a general $V \\in \\mathcal {M}S_X(2n;t)$ .",
"On the other hand, for a general $V \\in \\mathcal {M}S_X(2n)$ , $\\dim M(V) \\ = \\ {\\left\\lbrace \\begin{array}{ll}0& \\text{when} \\ n(g-1) \\text{ is even, } \\\\\\frac{n+1}{2}& \\text{when} \\ n(g-1) \\text{ is odd}.\\end{array}\\right.",
"}$ Let us give some historical remarks.",
"For vector bundles, the Segre stratifications are well understood; see Lange–Narasimhan [13], Hirschowitz [4], Brambila-Paz–Lange [2] and Russo–Teixidor i Bigas [19].",
"Holla and Narasimhan [8] defined a generalized Segre invariant for principal $G$ -bundles for an arbitrary reductive group $G$ , and obtained a bound on the invariant in general.",
"This bound was sharpened for symplectic bundles ($G = \\mathrm {Sp}_{2n}\\mathbb {C}$ ) by the present authors [3], where Theorem REF (1) was proven using the Terracini lemma in projective geometry.",
"Also Theorem REF was proven for symplectic bundles of rank four in [3].",
"The special case when the curve has genus two has earlier been studied in detail by the second named author [7].",
"In this paper, we provide another simpler proof of (1), and prove (2) and (3) for arbitrary rank $n$ and genus $g$ .",
"In the orthogonal case, the moduli space $\\mathcal {M}O_X(2n)$ has two connected components, as was discussed before.",
"We denote by $\\mathcal {M}O_{X}(2n)^+$ (resp.",
"$\\mathcal {M}O_X(2n)^-$ ) the component consisting of bundles of trivial (resp., nontrivial) second Stiefel–Whitney class.",
"We first prove the following.",
"Theorem 1.2 Let $E_1$ and $E_2$ be isotropic rank $n$ subbundles of an orthogonal bundle of rank $2n$ .",
"Then $\\deg E_1$ and $\\deg E_2$ have the same parity.",
"A semistable orthogonal bundle $V$ belongs to $\\mathcal {M}O_{X}(2n)^+$ (resp., $\\mathcal {M}O_{X}(2n)^-$ ) if and only if its isotropic rank $n$ subbundles have even degree (resp., odd degree).",
"We then show the following on the stratification on each component.",
"Theorem 1.3 Let $X$ be a smooth algebraic curve of genus $g \\ge 2$ .",
"For any orthogonal bundle $V$ of rank $2n$ , we have $t(V) \\le n(g-1) + 3$ .",
"This is sharp in the sense that two even numbers $t$ with $n(g-1) \\le t \\le n(g-1)+3$ correspond to the values of $t(V)$ for general $V$ in $\\mathcal {M}O_X(2n)^+$ and $\\mathcal {M}O_X(2n)^-$ .",
"For each even integer $t$ with $2 \\le t \\le n (g-1) $ , the stratum $\\mathcal {M}O_X(2n;t)$ is nonempty and irreducible, and $\\dim \\mathcal {M}O_X(2n;t) = \\frac{1}{2} \\left( n(3n-1)(g-1) + (n-1)t \\right).$ Furthermore, if $t < n(g-1)$ then $\\mathcal {M}O_X(2n;t)$ is contained in the closure of $\\mathcal {M}O_X(2n;t+4)$ in the relevant component $\\mathcal {M}O_X(2n)^{\\pm }$ .",
"For each positive even integer $t < n(g-1)$ , the space $M(V)$ is a single point for a general $V \\in \\mathcal {M}S_X(2n;t)$ .",
"We also compute the dimension of $M(V)$ for general bundles in each component $\\mathcal {M}O_{X}(2n)^{\\pm }$ .",
"It turns out to depend on the class of $n(g-1)$ modulo 4; the precise statement is set out in §5.3.",
"A more detailed description of the top strata of $\\mathcal {M}O_X(2n)$ is given in §5.4.",
"To prove these statements, we consider families of symplectic and orthogonal extensions.",
"The main strategy is to relate the invariant $t(V)$ to the geometry of certain higher secant varieties in the projectivized extension spaces.",
"This idea goes back to the work of Lange and Narasimhan [13], where Theorem REF is proven for rank two bundles using higher secant varieties of the curve $X$ embedded in the extension spaces.",
"In §2, we generalize the geometric framework in [13] to our situation in Criterion REF .",
"The embedded curve $X$ in the extension space is replaced by the quadric bundle $\\mathbb {P}E$ and the Grassmannian bundle $\\mathrm {Gr}(2, E)$ for symplectic and orthogonal cases respectively.",
"To work with bundles of rank $2n \\ge 4$ , one has also to understand the situation when two rank $n$ isotropic subbundles intersect non-transversely.",
"It turns out that this case can also be understood geometrically (Criterion REF ).",
"This is a key advance upon the methods in [3], which enables us to argue for symplectic and orthogonal bundles of arbitrary rank.",
"In §3, we construct “universal extension spaces” parameterizing all the extensions of fixed type, and show that the rational classifying maps to $\\mathcal {M}S_X(2n)$ or $\\mathcal {M}O_X(2n)$ are defined on dense subsets.",
"In §4 and §5, we prove the main results for symplectic and orthogonal bundles respectively, using dimension counts based on the geometric information from the preceding sections.",
"In §5, we also observe some interesting properties of certain families of orthogonal bundles, stemming from the richer topological structure of the moduli space.",
"Acknowledgements: The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.",
"2011-A423-0004).",
"Also he would like to thank Høgskolen i Vestfold, Bakkenteigen for the invitation in June 2011.",
"The second author gratefully acknowledges the hospitality and generous financial support of Konkuk University, Seoul.",
"Both authors thank Christian Pauly and Olivier Serman for helpful discussions."
],
[
"Symplectic and orthogonal extensions and lifting criteria",
"In this section we discuss symplectic and orthogonal extensions.",
"Most of the results in this section were obtained for symplectic extensions in [3].",
"Those results will be restated here for the reader's convenience, and the modified versions for orthogonal extensions will be proven in detail.",
"Let $V \\rightarrow X$ be a vector bundle of rank $2n$ equipped with a nondegenerate bilinear form $\\omega \\colon V \\otimes V \\rightarrow \\mathcal {O}_{X}$ , and let $E \\subset V$ be a subbundle.",
"Then there is an exact sequence $ 0 \\rightarrow E^{\\perp } \\rightarrow V \\rightarrow E^{*} \\rightarrow 0, $ where $E^\\perp $ is the orthogonal complement of $E$ .",
"If $E$ is isotropic of rank $n$ , then $E = E^\\perp $ and $V$ defines a class $\\delta (V) \\in H^1(X, \\mathrm {Hom}(E^{*}, E)) \\cong H^1(X, E \\otimes E)$ .",
"Criterion 2.1 Suppose $E$ is simple.",
"An extension $0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0$ is induced by a symplectic (resp., orthogonal) structure on $V$ with respect to which $E$ is isotropic if and only if the extension class $\\delta (V)$ belongs to the subspace $ H^{1}(X, \\mathrm {Sym}^{2}E) \\quad \\left( \\hbox{resp.,} \\quad H^{1}(X, \\wedge ^{2}E ) \\right) $ of $H^{1}(X, E \\otimes E)$ .",
"This is due to S. Ramanan.",
"A detailed proof for the symplectic case is given in [6], and the proof for the orthogonal case is practically identical."
],
[
"Cohomological criterion for lifting",
"Here we recall the notion of a bundle-valued principal part (see Kempf [10] for corresponding results on line bundles).",
"A locally free sheaf $W$ on $X$ has the flasque resolution $ 0 \\rightarrow W \\rightarrow \\underline{\\mathrm {Rat}}(W) \\rightarrow \\mathrm {\\underline{Prin}}(W) \\rightarrow 0, $ where $\\underline{\\mathrm {Rat}}(W)$ is the sheaf of rational sections of $W$ and $\\mathrm {\\underline{Prin}}(W)$ the sheaf of $W$ -valued principal parts.",
"We denote their groups of global sections by $\\mathrm {Rat}(W)$ and $\\mathrm {Prin}(W)$ respectively.",
"Taking global sections, we obtain $ 0 \\rightarrow H^0(X, W) \\rightarrow \\mathrm {Rat}(W) \\rightarrow \\mathrm {Prin}(W) \\rightarrow H^{1}(X, W) \\rightarrow 0.$ For a principal part $p \\in \\mathrm {Prin}(W)$ , we write its class in $H^{1}(X, W)$ as $[p]$ .",
"Now consider an extension $0 \\rightarrow E \\rightarrow V \\rightarrow E^* \\rightarrow 0$ , and an elementary transformation $F$ of $E^*$ defined by the sequence $ 0 \\rightarrow F \\stackrel{\\mu }{\\rightarrow } E^* \\rightarrow \\tau \\rightarrow 0 $ for some torsion sheaf $\\tau $ .",
"We say that $F$ lifts to $V$ if there is a sheaf injection $F \\rightarrow V$ such that the composition $F \\rightarrow V \\rightarrow E^*$ coincides with $\\mu $ .",
"The following statements are proven in [6]: Lemma 2.2 Suppose $h^{0}(X, \\mathrm {Hom}(E^{*}, E)) = 0$ , and consider an extension $0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0$ with class $\\delta (V) \\in H^{1}(X, E \\otimes E)$ .",
"There is a bijection between principal parts $p \\in \\mathrm {Prin}(E \\otimes E)$ such that $\\delta (V) = [p]$ , and elementary transformations of $E^*$ lifting to subbundles of $V$ .",
"The bijection is given by $ p \\ \\leftrightarrow \\ \\mathrm {Ker}\\left(p \\colon E^{*} \\rightarrow \\mathrm {\\underline{Prin}}(E) \\right).",
"$ Suppose that $\\delta (V) = [p] \\in H^{1}(X, \\mathrm {Sym}^{2}E)$ , corresponding to a symplectic extension.",
"The subbundle corresponding to $\\mathrm {Ker}(p)$ is isotropic in $V$ if and only if $p$ is a symmetric principal part; that is, $^tp = p$ .",
"Suppose that $\\delta (V) = [p] \\in H^{1}(X, \\wedge ^{2}E)$ , corresponding to an orthogonal extension.",
"The subbundle corresponding to $\\mathrm {Ker}(p)$ is isotropic in $V$ if and only if $p$ is an antisymmetric principal part; that is, $^tp = -p$ .",
"$\\Box $"
],
[
"Subvarieties of the extension spaces",
"Given any vector bundle $W$ , consider the projectivization $\\pi \\colon \\mathbb {P}W \\rightarrow X$ .",
"Then we have a natural rational map $ \\mathbb {P}W \\dashrightarrow \\mathbb {P}H^{1}(X, W) $ defined as follows (a slightly different description was given in [3]): Consider the evaluation map $ X \\times H^{0}(X, K_X \\otimes W^{*}) \\rightarrow K_{X}\\otimes W^{*} $ .",
"Via Serre duality, the dual of this map is identified with $ W \\otimes T_{X} \\rightarrow X \\times H^{1}(X, W).",
"$ Projectivizing this map and then composing with the projection $X \\times \\mathbb {P}H^{1}(X, W) \\rightarrow \\mathbb {P}H^{1}(X, W)$ , we get a map $ \\phi \\colon \\mathbb {P}W \\dashrightarrow \\mathbb {P}H^{1}(X, W).",
"$ On a fibre $W|_x$ , this map is identified with the projectivized coboundary map in the sequence $ 0 \\rightarrow H^0 (X, W) \\rightarrow H^{0}(X, W(x)) \\rightarrow \\frac{W(x)}{W}|_{x} \\rightarrow H^{1}(X, W) \\rightarrow H^1(X, W(x)) \\rightarrow 0 .",
"$ Thus for $w \\in \\mathbb {P}W$ , the image $\\phi (w)$ may be realized as the cohomology class of a $W$ -valued principal part supported at $x$ with a simple pole along $w$ .",
"As discussed in [3], the rational map $\\phi $ is induced by the complete linear system of the line bundle $\\pi ^{*}K_{X}\\otimes \\mathcal {O}_{\\mathbb {P}W}(1)$ over $\\mathbb {P}W$ .",
"For $W = \\mathrm {Sym}^{2} E$ and $ W= \\wedge ^2 E$ respectively, we have the rational maps $ \\mathbb {P}(\\mathrm {Sym}^{2} E) \\dashrightarrow \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E) \\quad \\hbox{and} \\quad \\mathbb {P}(\\wedge ^2 E )\\dashrightarrow \\mathbb {P}H^{1}(X, \\wedge ^{2} E).",
"$ Note that both of these are restrictions of the map $\\mathbb {P}(E \\otimes E) \\dashrightarrow \\mathbb {P}H^{1}(X, E \\otimes E)$ .",
"In the symplectic case, we consider the chain of maps $\\mathbb {P}E \\ \\hookrightarrow \\ \\mathbb {P}(\\mathrm {Sym}^{2} E) \\dashrightarrow \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E) ,$ where the first inclusion is given by the Segre embedding $[v] \\mapsto [v \\otimes v]$ .",
"In the orthogonal case, we consider the chain of maps $\\mathrm {Gr}(2,E) \\ \\hookrightarrow \\ \\mathbb {P}(\\wedge ^{2} E) \\dashrightarrow \\mathbb {P}H^{1}(X, \\wedge ^{2} E),$ where the Grassmannian bundle $\\mathrm {Gr}(2,E)$ is embedded in $\\mathbb {P}(\\wedge ^{2} E)$ via the Plücker embedding.",
"These rational maps are denoted as $\\phi _s \\colon \\mathbb {P}E \\dashrightarrow \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E) \\quad \\hbox{and} \\quad \\phi _a: \\mathrm {Gr}(2,E) \\dashrightarrow \\mathbb {P}H^{1}(X, \\wedge ^{2} E).",
"$ Lemma 2.3 Let $W = E \\otimes E$ for a stable bundle $E$ over a curve of genus $g \\ge 2$ .",
"The rational map $\\phi \\colon \\mathbb {P}(W) \\dashrightarrow \\mathbb {P}H^{1}(X, W)$ is base point free if $\\mu (E) < - \\frac{1}{2}$ , and an embedding if $\\mu (E) < -1$ .",
"In particular, if $\\mu (E) < -1$ , the rational maps $\\phi _s$ and $\\phi _a$ in (REF ) are embeddings.",
"If $E$ is general of negative degree, $\\phi $ is injective on a general fiber of $\\mathbb {P}(W)$ .",
"If we further assume that $g \\ge 3$ then $\\phi $ separates two general fibers of $\\mathbb {P}(W)$ .",
"(1) (A similar version was stated in [3], unfortunately with a flawed proof.)",
"One can check that $\\phi $ is base point free (resp., an embedding) if $h^{0}(X, W(D)) = 0$ for all effective divisors $D$ of degree one (resp., of degree two) on $X$ .",
"Since $W = E \\otimes E$ is semistable, these vanishing results follow from the assumption that $\\mu (E) < - \\frac{1}{2}$ (resp., $\\mu (E) < -1$ ).",
"(2) Let $L$ and $M$ be line bundles with $\\deg L = \\deg E$ and $\\deg M = 0$ , and put $E_0 = L \\oplus M^{\\oplus (n-1)}$ , where $n = \\mathrm {rk}(E)$ .",
"One can check that for a general choice of $L$ and $M$ , the bundle $E_0 \\otimes E_0 (x) $ has no sections.",
"Deforming $E_0$ to a general stable bundle $E$ , we see that $h^{0}(X, W(x)) = 0$ for a general $x \\in X$ .",
"A similar argument shows that if $g \\ge 3$ , then $h^{0}(X, W(x+y)) = 0$ for general $ x,y \\in X$ ."
],
[
"Irreducibility of the space of principal parts",
"In this subsection, we prove some technical facts on principal parts.",
"Let $p$ be a principal part with values in $E \\otimes E$ .",
"Definition 2.4 The degree $\\deg (p)$ of $p$ is defined as the length of the torsion sheaf $ \\mathrm {Im}\\left( p \\colon E^{*} \\rightarrow \\mathrm {\\underline{Prin}}(E) \\right).",
"$ The support of $p$ is defined as the support of $\\mathrm {Im}(p)$ .",
"A symmetric (resp., antisymmetric) principal part $p$ of degree $k$ (resp., $2k$ ) is called general if it is supported at $k$ distinct points.",
"Obviously, the general symmetric (resp., antisymmetric) principal parts of degree $k$ (resp., $2k$ ) are parameterized by a quasi-projective irreducible variety.",
"We want to confirm that an arbitrary symmetric or antisymmetric principal part can be obtained as a limit of a continuous family of general ones.",
"Lemma 2.5 ([3]) Let $p $ be a $\\mathrm {Sym}^2 E$ -valued principal part of degree $k$ , supported at a single point $x$ .",
"Then there exists a local frame $e_{1}, \\ldots , e_n$ for $E$ in a neighborhood of $x$ , in terms of which $p$ is expressed as $ p = \\sum _{i=1}^{n} \\frac{e_{i} \\otimes e_{i}}{z^{k_i}} $ where $z$ is a uniformizer at $x$ , and $ k_{1} \\ge k_{2} \\ge \\cdots \\ge k_n \\ge 0$ , and $k = \\sum _{i=1}^{n} k_{i} $ .",
"$\\Box $ For antisymmetric principal parts, we have the following analogue: Lemma 2.6 Let $p$ be a $\\wedge ^{2}E$ -valued principal part, supported at a single point $x$ .",
"Then there exists a local frame $e_{1}, \\ldots , e_n$ for $E$ in a neighborhood of $x$ , in terms of which $p$ is expressed as $ p = \\sum _{i=1}^{s} \\frac{e_{2i-1} \\wedge e_{2i}}{z^{k_i}} $ where $z$ is a uniformizer at $x$ , and $ k_{1} \\ge k_{2} \\ge \\cdots \\ge k_s \\ge 0$ , and $\\deg (p) = 2 \\left( \\sum _{i=1}^{s} k_{i} \\right)$ .",
"In particular, any antisymmetric principal part has even degree.",
"This argument is adapted from the proof of [3].",
"Locally, $p$ can be expressed as $ p = \\frac{1}{z^{k_1}} A $ for some $n \\times n$ antisymmetric matrix $A$ with entries in the ring $R = \\mathbb {C}[z]/ (z^{k_1})$ , since we are concerned with the principal parts at $x$ only.",
"In this context, it suffices to show that there exists a matrix $P \\in M_{n} (R)$ , such that $\\det P$ is a unit in $R$ and $^t P A P = \\text{diag}(z^{d_1}J, z^{d_2}J, \\ldots , z^{d_s}J),$ where $0 = d_1 \\le d_2 \\le \\cdots \\le d_s$ and $J = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}$ .",
"This can be shown by mimicking the standard process to get the normal form of an antisymmetric matrix over $\\mathbb {C}$ .",
"The only difference lies in that $R$ has non-units contained in the principal ideal $(z)$ , and this can be taken care of by allowing the terms $z^{d_1}, \\ldots , z^{d_s}$ on the diagonal.",
"Alternatively, one may work instead over the formal power series ring $\\mathbb {C}[[z]]$ , which is a PID, and truncate the terms in the ideal $(z^{k_1})$ at the final step.",
"For the process over a PID, see Adkins–Weintraub [1].",
"Hence we have a frame for $E$ on this neighborhood in terms of which $p$ appears as $ \\sum _{i=1}^{s} \\frac{e_{2i-1} \\wedge e_{2i}}{z^{k_i}}, $ where $k_{i} = k_1 - d_i$ .",
"Since the terms in the sum (REF ) impose independent conditions on sections of $E$ , we have $ \\deg ( p ) = \\sum _{i=1}^{s} \\deg \\left( \\frac{e_{2i-1} \\wedge e_{2i}}{z^{k_i}} \\right) = 2 \\sum _{i=1}^{s} k_{i} , $ as required.",
"Corollary 2.7 For a fixed vector bundle $E$ and for each $k >0$ , the spaces of symmetric principal parts of degree $k$ and antisymmetric principal parts of degree $2k$ are irreducible.",
"We consider the antisymmetric case; the symmetric case can be proven similarly.",
"It suffices to show the irreducibility of the space of antisymmetric principal parts supported at a single point $x$ .",
"By Lemma REF , we may choose a trivialization of $E$ near $x$ with respect to which $p$ is expressed as $\\sum _{i=1}^{s} p_i$ , where $ p_{i} = \\frac{e_{2i-1} \\wedge e_{2i}}{z^{k_i}} $ and $2 \\sum _{i=1}^{s} k_{i} = k$ .",
"For each $i$ , choose distinct $\\lambda ^{i}_{1}, \\ldots , \\lambda ^{i}_{k_i} \\in \\mathbb {C}$ , and define a family of principal parts $ p_{i}(t) = \\frac{e_{2i-1} \\wedge e_{2i}}{(z - \\lambda ^{i}_{1}t) \\cdots (z - \\lambda ^{i}_{k_i}t)} $ where $t$ is a complex parameter.",
"We then put $p(t) = \\sum _{i=1}^{s}p_{i}(t)$ .",
"By construction, $p(0) = p$ , while for small $t \\ne 0$ we can rewrite $p(t)$ as a sum of $k$ antisymmetric principal parts of degree 2, supported at $k$ distinct points.",
"Hence we have shown that any antisymmetric principal part of degree $2k$ is a limit of a continuous family of general antisymmetric principal parts of degree $2k$ .",
"But we know that the general antisymmetric principal parts are parameterized by an irreducible variety.",
"This proves the claim."
],
[
"Geometric criteria for lifting",
"In this subsection, we find a geometric interpretation of the cohomological criterion on isotropic liftings in Lemma REF .",
"Throughout this subsection, $E$ is a general stable bundle of negative degree.",
"In particular, $E$ is simple and $h^0(X, \\mathrm {Hom}(E^*, E)) = 0$ .",
"Consider the rational maps $\\phi _s: \\mathbb {P}E \\dashrightarrow \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E) \\quad \\hbox{and} \\quad \\phi _a: \\mathrm {Gr}(2, E) \\dashrightarrow \\mathbb {P}H^1(X, \\wedge ^2 E).$ As discussed in Lemma REF , these maps are defined on dense subsets, and furthermore are embeddings if $\\mu (E)<-1$ .",
"In general, abusing notation, we denote by $\\mathbb {P}E$ and $\\mathrm {Gr}(2, E)$ the closures of the images $\\phi _s (\\mathbb {P}E)$ in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ and $\\phi _a (\\mathrm {Gr}(2, E))$ in $\\mathbb {P}H^1(X, \\wedge ^2 E)$ respectively.",
"Next, for a quasi-projective variety $Z \\subset \\mathbb {P}^N$ , we write $\\mathrm {Sec}^k Z$ for the $k$ -th secant variety of $\\bar{Z}$ , which is the closure of the union of linear subspaces spanned by $k$ general points of $Z$ .",
"In particular, $\\mathrm {Sec}^1 Z = \\bar{Z}$ .",
"Criterion 2.8 Consider an extension given by $ \\delta (V) : \\quad 0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0.",
"$ When $\\delta (V) \\in \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ , there is an elementary transformation $F$ of $E^*$ with $\\deg (E^*/F) \\le k$ lifting to a Lagrangian subbundle of $V$ if and only if $\\delta (V) \\in \\mathrm {Sec}^{k} \\mathbb {P}E$ .",
"When $\\delta (V) \\in \\mathbb {P}H^1(X, \\wedge ^2 E)$ , there is an elementary transformation $F$ of $E^*$ with $\\deg (E^*/F) \\le 2k$ lifting to a rank $n$ isotropic subbundle of $V$ if and only if $ \\delta (V) \\in \\mathrm {Sec}^{k} \\mathrm {Gr}(2,E)$ .",
"In this case, $\\deg E $ and $\\deg F$ have the same parity.",
"In the symplectic case, this is the content of [3].",
"The same idea works for antisymmetric case as follows: By Lemma REF (3), an elementary transformation $F \\subseteq E^*$ with $\\deg (E^*/F) \\le 2k$ lifts to an isotropic subbundle of $V$ if and only if the extension class $\\delta (V)$ is of the form $[p]$ where $p$ is an antisymmetric principal part of degree $2l \\le k$ .",
"If $p$ is general, it is of the form $ \\sum _{i=1}^{l} \\frac{e_{i} \\wedge f_i}{z_i}, $ where $z_1, z_2, \\ldots , z_l$ are uniformizers at $l$ distinct points $x_1, x_2,\\ldots , x_l \\in X$ .",
"According to the description of the map $\\phi _a$ in §REF , the class $[p] \\in \\mathbb {P}H^1(X, \\wedge ^2 E)$ lies on the secant plane spanned by $l$ distinct points of $\\mathrm {Gr}(2,E)$ .",
"Hence $[p] \\in \\mathrm {Sec}^l \\mathrm {Gr}(2, E)$ .",
"Conversely, a general point of $\\mathrm {Sec}^l \\mathrm {Gr}(2, E)$ corresponds to a class $\\delta (V) =[p]$ where $p$ is of the form (REF ).",
"From this correspondence on the general points, we get the desired statement by Corollary REF .",
"In this case, $F = \\mathrm {Ker}\\left(p \\colon E^{*} \\rightarrow \\mathrm {\\underline{Prin}}(E) \\right)$ with $\\deg p = 2l$ , so $\\deg F$ has the same parity as $\\deg E$ .",
"Corollary 2.9 Consider an extension given by $ \\delta (V) : \\quad 0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0.",
"$ Assume $\\delta (V) \\in H^1(X, \\mathrm {Sym}^2 E)$ .",
"If $\\delta (V) \\le \\mathrm {Sec}^k \\mathbb {P}E$ , then $t(V) \\le 2(k + \\deg E)$ .",
"Assume $\\delta (V) \\in H^1(X, \\wedge ^2 E)$ .",
"If $\\delta (V) \\le \\mathrm {Sec}^k \\mathrm {Gr}(2, E)$ , then $t(V) \\le 2(2k + \\deg E)$ .",
"By definition of the invariant, $t(V) \\le - 2 \\deg F$ for any rank $n$ isotropic subbundle $F \\subset V$ .",
"The above bounds follow as a direct consequence of Criterion REF .",
"The converse of the above Corollary is not true in general.",
"For instance, assume $\\delta (V) \\in H^1(X, \\mathrm {Sym}^2 E)$ .",
"The bound $t(V) \\le 2(k + \\deg E)$ tells us that there is a Lagrangian subbundle $F$ of degree $\\ge -\\deg E -k$ , but this does not imply that $F$ lifts from an elementary transformation of $E^*$ .",
"So in general, we are led to consider a diagram of the form ${ 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\0 [r] & H [r] [u] & F [r] [u] & G [r] [u] & 0 } $ where $H \\subset E$ is a subbundle of degree $-h \\le 0$ and rank $r \\ge 0$ , and $G$ is a locally free subsheaf of $E^*$ of rank $n-r$ .",
"When $r = 0$ so that $E|_x$ and $F|_x$ meet transversely for general $x \\in X$ , this reduces to the situation of Criterion REF of the lifting of elementary transformations.",
"The remaining part of this subsection will be devoted to finding a criterion for the existence of such a diagram with $r>0$ .",
"For $H \\subseteq E$ , let $q \\colon E \\rightarrow E/H$ be the quotient map.",
"Since $H$ is isotropic, $H^{\\perp }$ fits into the diagram ${ 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\0 [r] & E [r] @{=}[u] & H^{\\perp } [r] [u] & \\left( E/H \\right)^{*} [r] [u]^{^tq} & 0.",
"}$ For $\\delta (V) \\in H^1(X, \\mathrm {Hom}(E^*, E))$ , we have $\\delta \\left( H^{\\perp } \\right) = \\, ^tq^{*} (\\delta (V)) \\ \\in \\ H^{1}(X, \\mathrm {Hom}((E/H)^{*}, E)).$ Furthermore, $H^{\\perp }$ inherits a (degenerate) bilinear form from $V$ .",
"Since $(H^{\\perp })^{\\perp } = H$ , the quotient $H^{\\perp }/H$ is equipped with a nondegenerate bilinear form coming from $V$ .",
"In fact, $H^{\\perp }/H$ is a symplectic (resp., orthogonal) extension in the upper exact sequence of ${ 0 [r] & E/H [r] & H^{\\perp } /H [r] & \\left( E/H \\right)^{*} [r] & 0 \\\\0 [r] & E [r] [u]^q & H^{\\perp } [r] [u] & \\left( E/H \\right)^{*} [r] @{=}[u] & 0 }$ corresponding to the class $\\delta (H^{\\perp }/H) = q_{*}(^tq)^{*} (\\delta (V))$ in $ H^{1} \\left( X, \\mathrm {Sym}^{2} (E/H) \\right)$ (resp., $H^{1} \\left( X, \\wedge ^{2} (E/H) \\right)$ ).",
"Lemma 2.10 The induced maps $ q_{*}(^tq)^{*} \\colon H^{1}(X, \\mathrm {Sym}^{2} E ) \\rightarrow H^{1} \\left( X, \\mathrm {Sym}^{2} (E/H) \\right) $ and $ q_{*}(^tq)^{*} \\colon H^{1}(X, \\wedge ^{2} E ) \\rightarrow H^{1} \\left( X, \\wedge ^{2} (E/H) \\right) $ are surjective.",
"These maps are induced from $ (^tq)^{*} : E \\otimes E \\rightarrow (E/H) \\otimes E \\quad \\hbox{and} \\quad q_{*} \\colon (E/H) \\otimes E \\rightarrow (E/H) \\otimes (E/H) $ respectively.",
"By local computation, it can be seen that the images of $q_{*}(^tq)^{*}$ in $(E/H) \\otimes (E/H)$ are precisely $\\mathrm {Sym}^{2} \\left( E/H \\right)$ and $\\wedge ^{2} \\left( E/H \\right)$ respectively.",
"Hence on the first cohomology level, the induced maps are surjective.",
"We obtain the following geometric criterion on liftings: Criterion 2.11 Let $V$ be an extension of $E^*$ by $E$ with class $\\delta (V)$ .",
"Fix a subbundle $0 \\ne H \\subset E$ and write $\\deg (E) = -e, \\deg (H) = -h$ .",
"Let $f > 2h - e$ .",
"Assume $\\delta (V) \\in H^1(X, \\mathrm {Sym}^2 E)$ .",
"Then $V$ admits a Lagrangian subbundle $F$ of degree $\\ge -f$ inducing a diagram of the form (REF ) if and only if $q_{*}(^tq)^{*} (\\delta (V)) \\ \\in \\ \\mathrm {Sec}^{(e+f -2h)} \\mathbb {P}(E/H) .$ Suppose $\\delta (V) \\in H^1(X, \\wedge ^2 E)$ and $e \\equiv f \\mod {2}$ .",
"Then $V$ admits a rank $n$ isotropic subbundle $F$ of degree $\\ge -f$ inducing a diagram of the form (REF ) if and only if $q_{*}(^tq)^{*} (\\delta (V)) \\ \\in \\ \\mathrm {Sec}^{\\frac{1}{2}(e+f - 2h)} \\mathrm {Gr}(2, E/H) .$ Since the arguments for (1) and (2) are parallel, we prove (2) only.",
"The orthogonal bundle $V$ admits an isotropic subbundle $F$ inducing the diagram (REF ) if and only if $H^{\\perp }$ admits a subbundle $F$ of degree $\\ge -f$ which is isotropic with respect to the antisymmetric form inherited from $V$ , yielding the diagram ${ 0 [r] & E [r] & H^{\\perp } [r] & \\left( E/H \\right)^{*} [r] & 0 \\\\0 [r] & H [r] [u] & F [r] [u] & G [r] [u] & 0 }$ Since $G = F/H$ has the same rank as $(E/H)^*$ , the map $G \\rightarrow (E/H)^*$ is an elementary transformation whose quotient is a torsion sheaf of degree $\\le e+f - 2h$ .",
"Factorizing by $H$ , we get ${ 0 [r] & E/H [r] & H^{\\perp }/H [r] & \\left( E/H \\right)^{*} [r] & 0 \\\\& & F/H [r]^= [u] & G [r] [u] & 0.",
"}$ We are in this situation precisely when the orthogonal extension $H^{\\perp }/H$ with class $ q_{*}(^tq)^{*} \\delta (V) \\in H^1 (X, \\wedge ^2 (E/H)) $ admits an isotropic lifting of some elementary transformation of $(E/H)^*$ of degree $\\le e+f - 2h$ .",
"By Criterion REF (2), this is equivalent to the class $\\delta \\left( H^{\\perp }/H \\right)$ belonging to $\\mathrm {Sec}^{\\frac{1}{2}(e+f -2h)} \\mathrm {Gr}(2,E/H)$ in $\\mathbb {P}H^{1}(X, \\wedge ^{2}(E/H))$ ."
],
[
"Parameter spaces of extensions",
"Here we construct “universal extension spaces”, following Lange [12] (see also [3]), and investigate stability of the corresponding symplectic and orthogonal bundles."
],
[
"Construction of the families",
"For a positive integer $e$ , let $U_X(n,-e)^s$ denote the moduli space of stable vector bundles of rank $n$ and degree $-e < 0$ over $X$ .",
"By Narasimhan–Ramanan [16], there exist a finite étale cover $ \\pi _{e} \\colon \\tilde{U}_e \\rightarrow U_X(n, -e)^s $ and a bundle $\\mathcal {E}_e \\rightarrow \\tilde{U}_e \\times X$ with the property that $\\mathcal {E}_e|_{\\lbrace E \\rbrace \\times X} \\cong \\pi _e(E)$ for all $E \\in \\tilde{U}_e$ .",
"When $\\gcd (n, e)=1$ , it is well known that $\\pi _e$ reduces to the identity map.",
"By Riemann-Roch and semistability, for each $E \\in U_X(n, -e)^s$ , we have $ h^1(X, \\mathrm {Sym}^{2}E) = (n+1)e + \\frac{n(n+1)}{2}(g - 1).",
"$ Therefore, the sheaf $R^{1} {p}_{*} (\\mathrm {Sym}^2 (\\mathcal {E}_e))$ is locally free of rank $(n+1)(e + \\frac{1}{2}n(g - 1))$ on $\\tilde{U}_e$ .",
"We denote its projectivization by $\\mu \\colon \\mathbb {S}_{e} \\rightarrow \\tilde{U}_e$ .",
"We have a diagram $ {\\mathbb {S}_{e} [dd]_{\\mu } & \\mathbb {S}_{e} \\times X [d]^{\\mu \\times \\mathrm {Id}_X} [l]_{r} & \\\\& \\tilde{U}_{e} \\times X [dl]_{p} [dr]^{q} & \\\\\\tilde{U}_{e} & & X}$ We write $r \\colon \\mathbb {S}_e \\times X \\rightarrow \\mathbb {S}_e$ for the projection.",
"By Lange [12], there is an exact sequence of vector bundles $ 0 \\rightarrow (\\mu \\times \\mathrm {Id}_{X})^{*}\\mathcal {E}_e \\otimes r^{*}\\mathcal {O}_{\\mathbb {S}_e}(1) \\rightarrow \\mathcal {W}_{e} \\rightarrow (\\mu \\times \\mathrm {Id}_{X})^{*}\\mathcal {E}_e^{*} \\rightarrow 0 $ over $\\mathbb {S}_{e} \\times X$ , with the property that for $\\delta \\in \\mathbb {S}_e$ with $\\mu (\\delta ) = E$ , the restriction of $\\mathcal {V}_{e}$ to $\\lbrace \\delta \\rbrace \\times X$ is isomorphic to the extension of $E^*$ by $E$ defined by $\\delta \\in \\mathbb {P}H^1(X, \\mathrm {Sym}^{2}E)$ .",
"By Lemma REF , the space $\\mathbb {S}_{e} $ classifies all symplectic extensions of $E^*$ by $E$ for all $E \\in U(n, -e)^s$ , up to homothety.",
"In the same way, we define a bundle $\\mathbb {A}_{e} \\rightarrow \\tilde{U}_e$ whose fibre at $E$ is $\\mathbb {P}H^{1}(X, \\wedge ^{2}E)$ , a projective space of dimension $(n-1)(e + \\frac{1}{2}n(g - 1)) - 1$ .",
"There is a sequence of vector bundles $ 0 \\rightarrow (\\mu \\times \\mathrm {Id}_{X})^{*}\\mathcal {E}_e \\otimes r^{*}\\mathcal {O}_{\\mathbb {A}_e}(1) \\rightarrow \\mathcal {V}_{e} \\rightarrow (\\mu \\times \\mathrm {Id}_{X})^{*}\\mathcal {E}_e^{*} \\rightarrow 0 $ over $\\mathbb {A}_{e} \\times X$ , with the property that for $\\delta \\in \\mathbb {A}_e$ with $\\mu (\\delta ) = E$ , the restriction of $\\mathcal {W}_{e}$ to $\\lbrace \\delta \\rbrace \\times X$ is isomorphic to the extension of $E^*$ by $E$ defined by $\\delta \\in \\mathbb {P}H^1(X, \\wedge ^{2}E)$ .",
"Again by Lemma REF , the space $\\mathbb {A}_{e} $ classifies all the orthogonal extensions of $E^*$ by $E$ for all $E \\in U(n, -e)^s$ , up to homothety.",
"For each $e>0$ , the universal bundles $\\mathcal {W}_e \\rightarrow \\mathbb {S}_e \\times X$ and $\\mathcal {V}_e \\rightarrow \\mathbb {A}_e \\times X$ induce classifying maps $\\sigma _e: \\mathbb {S}_e \\dashrightarrow \\mathcal {M}S_X(2n)\\ \\ \\text{and} \\ \\ \\alpha _e: \\mathbb {A}_e \\dashrightarrow \\mathcal {M}O_X(2n)$ respectively.",
"The indeterminacy loci of these maps consist of precisely the points whose associated symplectic/orthogonal bundles are not semistable."
],
[
"Stability of extensions",
"The universal extension spaces $\\mathbb {S}_e$ and $\\mathbb {A}_e$ provide a natural way to study the Segre stratification on the moduli space of symplectic/orthogonal bundles, via the classifying maps $\\sigma _e$ and $\\alpha _e$ .",
"In order to proceed in this direction, we must verify that a general bundle with extension class represented in $\\mathbb {S}_e$ (resp., $\\mathbb {A}_e$ ) is a stable symplectic (resp., stable orthogonal) bundle.",
"The same question for vector bundles was formulated by Lange [11], and solved in Brambila-Paz–Lange [2] and Russo–Teixidor i Bigas [19].",
"In both papers, elementary transformations were used to construct stable bundles with the prescribed Segre invariant.",
"In this subsection, we prove the existence of a stable orthogonal/symplectic bundle in $\\mathbb {S}_e$ and $\\mathbb {A}_e$ for each $e >0$ .",
"Elementary transformations are used, but in a somewhat different context.",
"We begin by establishing the statement for $e = 1$ and $e = 2$ .",
"We will need the following bound on the classical Segre invariants of vector bundles, due to Hirschowitz [4]: Lemma 3.1 Let $E$ be a general stable vector bundle of rank $n$ and degree $-e$ .",
"Let $H \\subset E$ be a subbundle of rank $r$ and degree $-h$ .",
"Then $r(n-r) (g-1) \\ \\le \\ nh -re \\ < \\ r(n-r) (g-1) +n .$ Also we need the following result called the Hirschowitz lemma and its variant.",
"Lemma 3.2 Let $H_1$ and $H_2$ be general stable bundles, $\\mathrm {rk}(H_i) = r_i$ and $\\deg (H_i) = d_i$ for $i = 1,2$ .",
"If $r_1 d_2 + r_2 d_1 \\ge r_1 r_2(g-1)$ so that $\\mu (H_1 \\otimes H_2) \\ge g-1$ , then $h^1(X, H_1 \\otimes H_2) = 0$ .",
"Let $F$ be a general stable bundle of rank $n$ .",
"If $\\deg F \\ge \\frac{1}{2}n(g-1)$ so that $\\mu (F \\otimes F) \\ge g-1$ , then $h^1(X, F \\otimes F) = 0$ .",
"Part (1) was proven by Hirschowitz [4]; see also Russo–Teixidor i Bigas [19].",
"The variant (2) is [3].",
"Proposition 3.3 Let $E \\in U(n, -e)^s$ be general, $e = 1,2$ .",
"For $e=1$ , every point in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ outside a sublocus $Y$ with $\\dim Y \\le n = \\dim \\mathbb {P}E$ corresponds to a stable symplectic bundle.",
"Hence a general point of $\\mathbb {S}_1$ represents a stable symplectic bundle.",
"For $e=1$ , every point in $\\mathbb {P}H^1(X, \\wedge ^2 E)$ corresponds to a stable orthogonal bundle.",
"For $e=2$ , every point of $\\mathbb {P}H^1(X, \\wedge ^2 E)$ outside a sublocus $Z$ with $\\dim Z \\le 2(n-2) +1 = \\dim Gr(2,E)$ corresponds to a stable orthogonal bundle.",
"Hence a general point of $\\mathbb {A}_1$ or $\\mathbb {A}_2$ represents a stable orthogonal bundle.",
"Consider a symplectic or orthogonal extension $0 \\rightarrow E \\rightarrow V \\rightarrow E^{*} \\rightarrow 0$ where $E$ is a general stable bundle of rank $n$ and degree $-e \\in \\lbrace -1,-2 \\rbrace $ .",
"Assume that $V$ is not stable, so there is an isotropic subbundle $F$ of $V$ of rank $r (\\le n)$ and degree $\\ge 0$ .",
"The intersection of $E$ and $F$ contains a subbundle $F_1$ of rank $r_1$ (possibly zero), and the image of $F$ in $E^*$ is a locally free subsheaf $F_2$ of rank $r_2$ , yielding a diagram $ { 0 [r] & E [r] & V [r] & E^{*} [r] & 0 \\\\0 [r] & F_{1} [r] [u] & F [r] [u] & F_{2} [r] [u] & 0. }",
"$ We will bound the dimension of the locus of the extensions $V$ admitting this kind of diagram.",
"First assume $r_{1} \\ne 0$ and so $r_2 < n$ .",
"Since $E$ is general, by Lemma REF we have $ \\deg F_{1} \\le - \\frac{r_1}{n} \\left( e + (n-r_{1})(g-1) \\right) \\ <0 .$ In the same way (if $r_2 \\ne 0$ ), $ \\deg F_{2} \\le \\frac{r_2}{n} \\left( e - (n-r_{2})(g-1) \\right),$ which implies $\\deg F_2 \\le 0$ since $r_{2} < n$ .",
"Therefore, $\\deg F < 0$ .",
"Next, assume $r_1 = 0$ and $ r = r_{2} = n$ .",
"In this case, $F$ is an elementary transformation of $E^*$ .",
"If $\\deg F \\ge 0$ , then the torsion sheaf $E/F$ has degree $\\le e$ .",
"In the symplectic case, we need only consider the case when $e = 1$ .",
"By Criterion REF (1), if $F$ lifts to $V$ as a Lagrangian subbundle, then $\\delta (V) \\in \\phi _s(\\mathbb {P}E ) $ in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ .",
"In the orthogonal case, $e \\le 2$ .",
"By Criterion REF (2), if $F$ lifts to $V$ as an isotropic subbundle, then in fact $e=2$ and $\\delta (V) \\in { \\mathrm {Gr}(2, E) }$ in $\\mathbb {P}H^1(X, \\wedge ^2 E)$ .",
"Finally assume $r_1 = 0$ and $r=r_2 < n$ .",
"From the inequality (REF ), $\\deg F = \\deg F_2 \\le 0$ .",
"The possibility $\\deg F = 0$ appears only for the following special cases: $e = 1$ ; $r = n - 1$ ; $g = 2$ $e = 2$ ; $r = n - 1$ ; $g = 3$ $e = 2$ ; $r = n - 1$ ; $g = 2$ $e = 2$ ; $r = n - 2$ ; $g = 2$ .",
"From now on, we show that in each of these cases, the dimension of the locus of extensions admitting a lifting of a subsheaf $F$ of $E^*$ as an isotropic subbundle of $V$ of degree zero and rank $r<n$ is bounded by $n= \\dim \\mathbb {P}E$ in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ and $2(n-2) +1 = \\dim Gr(2,E)$ in $\\mathbb {P}H^1 (X, \\wedge ^2 E)$ respectively.",
"Note that in each case (i)–(iv), $F$ is a maximal subbundle of $E^*$ , which is easily seen from the inequalities in Lemma REF .",
"Hence the quotient $Q := E^{*}/F$ is torsion-free of degree $e$ and we obtain the diagrams ${ 0 [r] & E [r] & \\left( F^{\\perp } \\right)^{*} [r] & Q [r] & 0 \\\\0 [r] & E [r] [u]^{=} & V [r] [u] & E^{*} [r] [u] & 0 \\\\& & F [r]^{=} [u] & F [u] & } $ and ${ 0 [r] & Q^{*} [r] & G [r] & Q [r] & 0 \\\\0 [r] & Q^{*} [r] [u]^{=} & F^{\\perp } [r] [u] & E^{*} [r] [u] & 0 \\\\& & F [r]^{=} [u] & F. [u] & } $ Since $(F^{\\perp })^{\\perp } = F$ , the bundle $G = F^{\\perp }/F$ inherits the nondegenerate bilinear form from $V$ .",
"Since $F^{\\perp } \\cap E = Q^*$ is contained in the isotropic subbundle $E \\subset V$ , it is an isotropic subbundle of $G$ .",
"Thus the class of the extension $G$ belongs to either $H^1(X, \\mathrm {Sym}^2 Q^*)$ or $H^{1}(X, \\wedge ^{2} Q^{*})$ by Criterion REF .",
"For a given general $E$ , there are finitely many choices for $F$ in cases (i), (ii), and (iv), while in case (iii), the subbundles of degree zero and rank $n - 1$ in $E$ vary in a Quot scheme of dimension $n - 1$ .",
"Once $F$ is chosen, the quotient $Q = E^{*} / F$ is fixed.",
"After we choose a symplectic or orthogonal extension $G$ of $Q$ by $Q^*$ , the bundles $F^\\perp $ and $V$ are determined from the above diagrams by $(F^{\\perp })^* = \\left( E \\oplus G \\right) / Q^*$ and $V = \\left( E \\oplus F^{\\perp } \\right) / Q^{*}.$ Therefore, for a fixed general $E$ , the dimension of the locus of $V$ appearing in the above class of diagram is bounded by that of the deformations of $F$ and $G$ .",
"Let us consider the orthogonal case first.",
"In cases (i)–(iii), the bundle $Q$ has rank 1, so there is no nontrivial orthogonal extension of $Q$ by $Q^*$ .",
"Hence the only possibility for $V$ is the direct sum $E \\oplus E^*$ , which is excluded.",
"In case (iv), we have $h^1(X, \\wedge ^2 Q^*) = 3$ , and so $ \\dim \\lbrace F : F \\subset E^* \\rbrace + h^{1}(X, \\wedge ^{2}Q^{*}) - 1 = 2.",
"$ Since $0 <r = n-2$ , we have $2 \\le 2(n-2)+1$ as was claimed.",
"For the symplectic case, we assumed $e = 1$ , so (i) is the only case to be considered.",
"In this case, there are finitely many choices of $F$ , and $h^{1}(X, \\mathrm {Sym}^{2}Q^{*}) - 1 = 1 \\le n$ , as was claimed.",
"This confirms that a general point of $\\mathbb {S}_1$ , $\\mathbb {A}_1$ or $\\mathbb {A}_2$ represents a stable orthogonal bundle.",
"Theorem 3.4 For each $e > 0$ , a general point of $\\mathbb {S}_e$ (resp., $\\mathbb {A}_e$ ) represents a stable symplectic (resp., orthogonal) bundle.",
"We consider the orthogonal case first: For each value of $e>0$ , we will exhibit a stable orthogonal bundle represented in $\\mathbb {A}_e$ .",
"The statement will then follow from the openness of the stable objects in families.",
"Let $E \\in U_{X}(n,-1)$ be general.",
"For any $k \\ge 2$ , choose a general antisymmetric principal part $p$ of degree $2k$ , which defines an element of $\\mathrm {Sec}^k \\mathrm {Gr}(2, E)$ .",
"Since $\\mathrm {Gr}(2,E)$ is nondegenerate and properly contained in $\\mathrm {Sec}^{k} \\mathrm {Gr}(2,E)$ , we may assume that $[p]$ does not lie on the image of $\\mathrm {Gr}(2, E)$ .",
"By Lemma REF , the sheaf $F := \\mathrm {Ker}\\left( p \\colon E^{*} \\rightarrow \\mathrm {\\underline{Prin}}(E) \\right)$ lifts to a rank $n$ isotropic subbundle of the extension $V$ with $\\delta (V) = [p]$ .",
"Deforming $p$ if necessary, we can assume that $F$ is stable.",
"By Proposition REF (2), moreover, $V$ is a stable orthogonal bundle.",
"Since $V$ fits into an orthogonal extension $ 0 \\rightarrow F \\rightarrow V \\rightarrow F^{*} \\rightarrow 0, $ it is associated to a point in $\\mathbb {A}_{2k-1}$ .",
"In the same way, consider a general bundle $E \\in U_X(n, -2)$ and choose a general antisymmetric principal part of degree $2k+2$ for each $k \\ge 2$ , defining a point of $\\mathrm {Sec}^{k+1} \\mathrm {Gr}(2,E)$ .",
"By Proposition REF (2), we may assume that the extension $V$ associated to the point $[p]$ is a stable orthogonal bundle, if we avoid a subvariety $Z$ with $\\dim Z \\le \\dim \\mathrm {Gr}(2,E)$ ; and this is possible since $\\mathrm {Gr}(2,E)$ is nondegenerate and properly contained in $\\mathrm {Sec}^{k+1} \\mathrm {Gr}(2,E)$ .",
"As in the previous case, $V$ fits into an orthogonal extension (REF ) where this time $\\deg (F) = -2k$ , so $V$ is represented in $\\mathbb {A}_{2k}$ .",
"In the symplectic case, we argue similarly.",
"For each $k \\ge 2$ , we can choose a general symmetric principal part $q$ of degree $k+1$ which defines a point of $\\mathrm {Sec}^{k+1} \\mathbb {P}E $ .",
"By the above argument, the extension $V$ determined by $q$ is a stable symplectic bundle which is represented in $\\mathbb {S}_{k}$ ."
],
[
"Description of the Segre strata for symplectic bundles",
"In this section, we use the map $\\sigma _e: \\mathbb {S}_e \\dashrightarrow \\mathcal {M}S_X(2n)$ discussed in the previous section to prove Theorem REF .",
"Recall that for each positive even integer $t$ , we denote by $\\mathcal {M}S_X(2n;t)$ the sublocus of $\\mathcal {M}S_X(2n)$ consisting of symplectic bundles $V$ with $t(V) = t$ .",
"Also, for $V \\in \\mathcal {M}S_X(2n)$ , we write $M(V)$ for the space of Lagrangian subbundles of $V$ of (maximal) degree $-\\frac{1}{2}t(V)$ .",
"Theorem 4.1 Consider a positive even integer $ t=2e \\le n (g-1)+1$ .",
"A general point of $\\mathbb {S}_e$ corresponds to a bundle $V$ with $t(V) = t$ ; in particular, $\\mathcal {M}S_X(2n;t)$ is nonempty.",
"Furthermore, $\\mathcal {M}S_X(2n;t)$ is irreducible.",
"If $t < n(g-1)$ , then $M(V)$ is a single point for a general $V \\in \\mathcal {M}S_X(2n;t)$ .",
"If $t \\in \\lbrace n(g-1), \\: n(g-1) +1 \\rbrace $ and $V$ is general in $\\mathcal {M}S_X(2n;t)$ then any pair of Lagrangian subbundles in $M(V)$ intersect transversely in $V$ in a general fiber.",
"Remark 4.2 Statements (2) and (3) can be viewed as a symplectic analogue of Lange–Newstead [14].",
"Let $E$ be a general bundle in $U(n, -e)^s$ for $e \\ge 1$ .",
"By Theorem REF , the moduli map $\\mathbb {S}_{e}|_{E} \\cong \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E) \\dashrightarrow \\mathcal {M}S_X(2n)$ is defined on a dense subset.",
"We want to compute a bound on the dimension of the locus in $H^{1}(X, \\mathrm {Sym}^{2}E)$ of extensions $V$ which admit a Lagrangian subbundle $F$ of degree $\\ge -e$ other than $E$ .",
"There are two possibilities: either $F$ is an elementary transformation of $E^*$ lifting to $V$ , or $F$ fits into a diagram of the form (REF ).",
"Step 1.",
"We show first that the latter situation does not arise for a general $V$ in $H^1(X, \\mathrm {Sym}^2 E)$ for $t = 2e \\le n(g-1) +1$ .",
"As before, we write $\\deg (H) = -h$ .",
"Recall from the diagram (REF ) that $G$ is an elementary transformation of $(E/H)^*$ whose quotient is a torsion sheaf of degree $\\deg (E/H)^* - \\deg G = (e-h) - (\\deg F + h) \\le 2(e-h).$ In particular, $e-h \\ge 0.$ Now for each fixed $H \\subset E$ , Criterion REF says that the locus of extensions in $H^1(X, \\mathrm {Sym}^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\ge -e$ is bounded by $ \\dim \\left( \\mathrm {Sec}^{2(e-h)}\\mathbb {P}(E/H) \\right) + 1 + \\dim \\mathrm {Ker}\\left( q_{*} \\, ^tq^{*} \\right).",
"$ The secant variety has dimension bounded by $ 2(e-h)(n-r) + 2(e-h) - 1 = 2(e-h)(n-r+1) - 1.",
"$ Also by Proposition REF , we have $ \\dim \\mathrm {Ker}\\left(q_{*} \\, ^tq^{*} \\right) &= h^{1}(X, \\mathrm {Sym}^{2}E) - h^{1} \\left(X, \\mathrm {Sym}^{2}(E/H) \\right) \\\\&\\le h^{1}(X, \\mathrm {Sym}^{2}E) - (n-r+1)(e-h) - \\frac{1}{2} (n-r)(n-r+1)(g-1).",
"$ Finally we take account of the deformations of $H$ by computing the dimension of the appropriate Quot scheme of $E$ .",
"Since $E$ is general, by Lemma REF we have $nh - re \\ \\ge \\ r(n-r)(g-1).$ We may furthermore assume that $H$ and $E/H$ are general, and so $ h^1(X, \\mathrm {Hom}(H, E/H)) = 0 $ by Lemma REF (1).",
"Therefore $[H]$ is a smooth point of the Quot scheme (cf.",
"Le Potier [15]), and the dimension of the Quot scheme is given by $ h^{0}(X, \\mathrm {Hom}(H, E/H)) = nh - re - r(n-r)(g-1).",
"$ Adding up these three terms, we see that the dimension of the locus of extensions in $H^1(X, \\mathrm {Sym}^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\ge -e$ is bounded by $\\dim \\left( \\mathrm {Sec}^{2(e-h)} \\mathbb {P}(E/H) \\right) + 1 + \\dim \\mathrm {Ker}(q_* \\, ^t q^*) + h^0(X, \\mathrm {Hom}(H, E/H)) \\\\\\le \\ (e-h)(n-r+1) - \\frac{1}{2} (n-r)(n+r+1) (g-1) + nh-re + h^1(X, \\mathrm {Sym}^2 E) \\\\= \\ e(n-r) - (e-h)(r-1) - \\frac{1}{2} (n-r)(n+r+1) (g-1) + h^1(X, \\mathrm {Sym}^2 E).$ By the assumption $e \\le \\frac{1}{2} (n (g-1)+1)$ and the inequality (REF ), this is smaller than the dimension of the whole extension space $h^1(X, \\mathrm {Sym}^2 E)$ .",
"This shows that for $t = 2e \\le n (g-1) +1$ , a general $V \\in H^1(X, \\mathrm {Sym}^2 E)$ does not admit a Lagrangian subbundle $F$ of degree $\\ge -e$ which fits into a diagram of the form (REF ).",
"Step 2.",
"Next we consider the case $r = 0$ , so $F$ is an elementary transformation of $E^*$ lifting to $V$ isotropically.",
"Let $\\deg F = -f \\ge -e$ .",
"By Criterion REF , the dimension of the locus of extensions in $H^1(X, \\mathrm {Sym}^2 E)$ admitting such an $F$ is bounded by $\\dim \\left( \\mathrm {Sec}^{e+f} \\mathbb {P}E \\right) +1 \\ \\le \\ (e+f)(n+1).$ If $f < \\frac{1}{2} n (g-1)$ , this bound is smaller than $h^1(X, \\mathrm {Sym}^2 E) = (n+1)e + \\frac{1}{2}n(n+1)(g-1).$ Step 3.",
"Combining the dimension counts for two possibilities in the above, we conclude: (i) if $t=2e < n (g-1)$ , a general $V \\in H^1(X, \\mathrm {Sym}^2 E)$ does not have a Lagrangian subbundle of degree $\\ge -e$ other than $E$ itself.",
"This shows (2).",
"(ii) if $t=2e \\in \\lbrace n (g-1), \\: n(g-1) + 1 \\rbrace $ , a general $V \\in H^1(X, \\mathrm {Sym}^2 E)$ does not have a Lagrangian subbundle of degree $> -e$ .",
"Also, any Lagrangian subbundle of degree $-e$ different from $E$ intersects $E$ transversely at a general fiber.",
"This shows (3).",
"Hence for a general $V \\in H^1(X, \\mathrm {Sym}^2 E)$ with $t = 2e \\le n(g-1) +1$ , we have $t(V) = t$ .",
"It follows that the rational map $\\sigma _e : \\mathbb {S}_e \\dashrightarrow \\mathcal {M}S_X(2n)$ sends a general point of $\\mathbb {S}_e$ to $\\mathcal {M}S_X(2n;t)$ , so $\\mathcal {M}S_X(2n;t)$ is non-empty for each $t \\le n (g-1) +1$ .",
"Since $\\mathbb {S}_e$ is irreducible, so is its image.",
"To see that $\\mathcal {M}S_X(2n;t)$ is irreducible, it suffices to show that every point of $\\mathcal {M}S_X(2n;t)$ is in the closure of the image of $\\mathbb {S}_e$ .",
"Any $V$ in $\\mathcal {M}S_X(2n;t)$ admits a Lagrangian subbundle $E$ of degree $-e$ which might be unstable.",
"But every such $E$ is contained in an irreducible family of bundles whose general member is a stable bundle in $U_X(n, -e)$ .",
"This shows (1).",
"To finish the proof of Theorem REF , we need the following description of the tangent space of $M(V)$ .",
"Lemma 4.3 For a symplectic bundle $V \\in \\mathcal {M}S_X(2n)$ and a point $[\\eta : E \\subset V]$ of $M(V)$ , the Zariski tangent space of $M(V)$ at $\\eta $ is identified with $H^0(X, \\mathrm {Sym}^2 E^*)$ .",
"It is well known that the Zariski tangent space to the Quot scheme of a vector bundle $V$ at a point $[E\\subset V]$ is given by $H^0(X, \\mathrm {Hom}(E, V/E))$ .",
"Intuitively, this can be explained as follows: A rank $n$ subbundle of $V$ is equivalent to a global section of the Grassmannian bundle $\\mathrm {Gr}(n, V)$ over $X$ .",
"A tangent vector to the Quot scheme of $V$ at $[E \\subset V]$ corresponds, at each fiber $x \\in X$ , to a tangent vector to the Grassmannian $\\mathrm {Gr}(n, V_x)$ at $[E_x \\subset V_x]$ .",
"Therefore, the tangent vector is a global section of the bundle $\\mathrm {Hom}(E, V/E)$ .",
"When $E$ is a Lagrangian subbundle of a symplectic bundle $V$ , we have $V/E \\cong E^*$ and so $\\mathrm {Hom}(E, V/E) \\cong E^* \\otimes E^*$ .",
"In this case, a similar argument shows that a tangent vector to $M(V)$ at $[E \\subset V]$ corresponds to a global section of $\\mathrm {Sym}^2 E^*$ , since for each $x \\in X$ the tangent space of the Lagrangian Grassmannian at $[E_x \\subset V_x]$ is identified with $\\mathrm {Sym}^{2}E^{*}|_x$ .",
"Proof of Theorem REF We begin with part (3) of the statement.",
"The first part of (3) is Theorem REF (2).",
"For the latter part, we invoke Lemma REF : $\\dim M(V) = h^0(X, \\mathrm {Sym}^2 E^*)$ for a smooth point $[E \\subset V]$ of $M(V)$ .",
"By Lemma REF (2), for a general $E \\in U_X(n,-e)$ , we have $\\dim H^0(X, \\mathrm {Sym}^2 E^*) \\ = \\ {\\left\\lbrace \\begin{array}{ll}0& \\text{if} \\ 2e = n(g-1), \\\\\\frac{n+1}{2}& \\text{if} \\ 2e=n(g-1)+1.\\end{array}\\right.",
"}$ (1) A straightforward computation shows that when $ n(g-1) \\le 2e \\le n(g-1)+1 $ , $ \\dim M(V) = \\dim \\mathbb {S}_e - \\dim \\mathcal {M}S_X(2n).",
"$ Since $ \\dim M(V) \\ge \\dim \\sigma _{e}^{-1}(V)$ , this equality implies that $\\sigma _e$ is dominant.",
"This shows that for a general $V \\in \\mathcal {M}S_X(2n)$ , we have $n(g-1) \\le t(V) \\le n(g-1) +1.$ By semicontinuity, $t(V) \\le n(g-1) +1$ for any symplectic bundle $V$ .",
"(2) If $t = 2e \\ge n(g-1) $ , then $\\dim \\mathcal {M}S_X(2n;t) = \\dim \\mathcal {M}S_X(2n)$ by (1).",
"Assume $t = 2e < n(g-1)$ .",
"By Theorem REF (2), the map $\\sigma _e \\colon \\mathbb {S}_{e} \\dashrightarrow \\mathcal {M}S_X(2n;t)$ is generically finite (of degree $\\deg \\pi _e$ ), and so $\\dim \\mathcal {M}S_X(2n;t) \\ = \\ \\dim \\mathbb {S}_e \\ = \\ \\frac{1}{2}n(3n+1)(g-1) + (n+1)e.$ For the last part of (2), we can use Criterion REF to show that the Segre stratification matches the stratification given by the higher secant varieties: For a general $V \\in \\mathcal {M}S_X(2n;2e)$ , let $E \\in M(V)$ so that $E$ is general in $U_{X}(n, -e)^s$ .",
"Let $k$ be the smallest integer satisfying $ \\mathrm {Sec}^{k} \\mathbb {P}E = \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}E)$ .",
"Then by Criterion REF , there is some elementary transformation $F$ of $E^*$ with $\\deg E/F = k$ lifting to $V$ as a Lagrangian subbundle.",
"By deforming $V$ and $E$ inside $\\mathbb {S}_e$ , we may assume that $F$ is general in $U_X(n,e-k)^s$ .",
"Now consider the symplectic extension $ 0 \\rightarrow F \\rightarrow V \\rightarrow F^* \\rightarrow 0.",
"$ In the same way, $\\delta (V) \\in \\mathbb {P}H^{1}(X, \\mathrm {Sym}^{2}F)$ belongs to $\\mathrm {Sec}^{k}\\mathbb {P}F$ , since the elementary transformation $E \\rightarrow F^*$ lifts to $V$ .",
"Note that $\\dim \\mathrm {Sec}^k \\mathbb {P}F \\le k(n+1) -1 \\ < \\ (n+1)(k-e) + \\frac{1}{2}n(n+1)(g-1) -1 = \\dim \\mathbb {P}H^1 (X, \\mathrm {Sym}^2 F),$ since we assumed $2e < n(g-1)$ .",
"Therefore, $\\mathrm {Sec}^{k}\\mathbb {P}F$ is properly contained in $\\mathbb {P}H^1 (X, \\mathrm {Sym}^2 F)$ , and certainly it is inside the closure of $\\mathrm {Sec}^{k+1} \\mathbb {P}F \\setminus \\mathrm {Sec}^k \\mathbb {P}F$ .",
"Thus, again by Criterion REF , the class $\\delta (V)$ belongs to the closure of a family of bundles admitting liftings of elementary transformations of degree $(k-e)-(k+1) = -(e+1)$ .",
"In particular, $V$ belongs to the closure of $\\mathcal {M}S_{X}(2n;2e+2)$ in $\\mathcal {M}S_{X}(2n)$ .",
"By the irreducibility of $\\mathcal {M}S_{X}(2n;2e)$ , the same holds for arbitrary $V \\in \\mathcal {M}S_X(2n;2e)$ .",
"$\\Box $ Remark 4.4 Suppose $t=2e < n(g-1)$ .",
"If $\\gcd (n,e) = 1$ , then $\\tilde{U}_e = U_X(n,-e)^s$ and $\\mathcal {M}S_X(2n;t)$ is birational to the fibration $\\mathbb {S}_e$ over $U_X(n,-e)^s$ whose fiber at $ E$ is $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ .",
"Finally we give a geometric interpretation of the cardinality of $M(V)$ , when it is finite.",
"Assume $t =2e = n (g-1) $ .",
"Let $V$ be general in $\\mathcal {M}S_X(2n;t)$ and choose $E \\in M(V)$ with $\\deg E = -e = -\\frac{1}{2} n(g-1)$ .",
"Consider the subvariety $\\mathbb {P}E \\subset \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ .",
"When $g \\ge 4$ , we have $\\mu (E) < -1$ , and by Lemma REF , the map $\\phi _s : \\mathbb {P}E \\rightarrow \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ is an embedding.",
"Furthermore, it was proven in [3] that the secant variety $\\mathbb {P}E$ is not defective in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ in the sense that $\\dim \\mathrm {Sec}^{k} \\mathbb {P}E \\ = \\ \\min \\lbrace (n+1)k -1, \\ \\dim \\mathbb {P}H^1 (X, \\mathrm {Sym}^2 E) \\rbrace .$ In particular, for $k = n(g-1)$ we have $\\dim \\mathrm {Sec}^{k} \\mathbb {P}E \\ = \\ (n+1)k -1 \\ = \\ \\dim \\mathbb {P}H^1 (X, \\mathrm {Sym}^2 E).",
"$ By a $k$ -secant space of $\\mathbb {P}E$ , we mean a linear subspace in $\\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ spanned by some $k$ points of $\\mathbb {P}E$ .",
"By (REF ), there are finite number of $n(g-1)$ -secant spaces of $\\mathbb {P}E$ which pass through a general point of $\\mathbb {P}H^1 (X, \\mathrm {Sym}^2 E)$ .",
"The following result generalizes Lange-Narasimhan [13] for rank 2 bundles.",
"Theorem 4.5 Assume $g \\ge 4$ and $n(g-1)$ is even.",
"Let $V \\in \\mathcal {M}S_X(2n)$ be a general symplectic bundle with $t(V) = n(g-1)$ , and let $E \\in M(V)$ .",
"There is a one-to-one correspondence between $M(V) \\setminus \\lbrace E\\rbrace $ and $n(g-1)$ -secant spaces of $\\mathbb {P}E$ passing through the point $[E \\subset V] \\in \\mathbb {P}H^1(X, \\mathrm {Sym}^2 E)$ .",
"By Criterion REF , a general class $\\delta (V)$ lies on a $n(g-1)$ -secant space of $\\mathbb {P}E$ if and only if there is an associated elementary transformation $F$ of $E^*$ with $\\deg (E^*/ F) = n(g-1)$ , lifting to a Lagrangian subbundle of $V$ .",
"Since $t(V) = n(g-1)$ , the Lagrangian subbundle $F$ is an element of $M(V)$ .",
"Conversely, by Theorem REF (3) every element of $M(V)$ other than $E$ appears in this way, since $V$ is general.",
"Remark 4.6 It is an interesting problem to compute the cardinality of $M(V)$ explicitly.",
"Let $N_{n,g}$ denote the cardinality of $M(V)$ for a general symplectic bundle $V$ of rank $2n$ over a curve of genus $g$ so that $n(g-1)$ is even.",
"It is well known that $N_{1,g} = 2^g$ .",
"The same problem for maximal subbundles of vector bundles was solved by Holla [9] (see also Lange–Newstead [13]).",
"As far as we are aware, the number $N_{n,g}$ is not known in general."
],
[
"Description of Segre strata for orthogonal bundles",
"In this section, we investigate the geometry of the Segre stratification on the moduli space of orthogonal bundles $\\mathcal {M}O_{X}(2n)$ .",
"In contrast to $\\mathcal {M}S_{X}(2n)$ , this has two connected components $\\mathcal {M}O_{X}(2n)^{\\pm }$ , corresponding to bundles of trivial and nontrivial second Stiefel–Whitney class (see Serman [20] for more details).",
"We begin by determining the component to which each stratum $\\mathcal {M}O_{X}(2n;t)$ belongs."
],
[
"Topological classification",
"Theorem 5.1 (1) Let $E_1$ and $E_2$ be isotropic rank $n$ subbundles of an orthogonal bundle $V$ of rank $2n$ .",
"Then $\\deg E_1$ and $\\deg E_2$ have the same parity.",
"(2) A semistable orthogonal bundle $V$ belongs to $\\mathcal {M}O_{X}(2n)^+$ (resp., $\\mathcal {M}O_{X}(2n)^-$ ) if and only if its isotropic rank $n$ subbundles have even degree (resp., odd degree).",
"By Theorem REF , the moduli maps $\\alpha _{e} \\colon \\mathbb {A}_{e} \\dashrightarrow \\mathcal {M}O_{X}(2n)$ are defined on dense subsets for each $e>0$ .",
"Since each family $\\mathbb {A}_e$ is connected, the image of each $\\mathbb {A}_e$ is entirely contained in either $\\mathcal {M}O_{X}(2n)^+$ or $\\mathcal {M}O_{X}(2n)^-$ .",
"As was already noted in the proof of Theorem REF , whenever $e_1$ and $e_2$ have the same parity, the images of $\\mathbb {A}_{e_1}$ and $\\mathbb {A}_{e_2}$ have nonempty intersection.",
"Hence they lie on the same component of $\\mathcal {M}O_X(2n)$ .",
"This is essentially due to the fact that any antisymmetric principal part has an even degree (see Lemma REF ).",
"For each $k>0$ , the image of $\\mathbb {A}_{2k}$ is contained in $\\mathcal {M}O_{X}(2n)^+$ since the trivial bundle has trivial second Stiefel–Whitney class.",
"Thus the image of $\\mathbb {A}_{2k-1}$ is contained in $\\mathcal {M}O_{X}(2n)^-$ .",
"This shows the above statements (1) and (2) for the case when $V$ is a stable orthogonal bundle and the isotropic subbundles are stable bundles.",
"By deforming an arbitrary bundle to a stable one, we see that (1) and (2) hold in general."
],
[
"An extra stratum",
"The local deformations of an orthogonal bundle $V$ are given by $H^1(X, \\wedge ^2 V)$ .",
"Since $V$ is self-dual, $\\wedge ^2 V \\subset \\mathrm {End}(V)$ .",
"Hence both components $\\mathcal {M}O_{X}(2n)^{\\pm }$ have dimension $ h^{1}(X, \\wedge ^{2}V) = n(2n-1)(g-1), $ since we may assume that $V$ is simple.",
"On the other hand, $\\dim \\mathbb {A}_e & = & \\dim U_X(n,-e) + \\dim \\mathbb {P}H^1 (X, \\wedge ^2 E) \\\\& = & (n-1)e + \\frac{1}{2} n (3n-1) (g-1)$ for a general $E \\in U_X(n,-e)$ .",
"Comparing dimensions, we see that in order for an $\\mathbb {A}_e$ to cover either component of $\\mathcal {M}O_{X}(2n)$ , we need $e \\ge \\left\\lceil \\frac{1}{2}n(g-1) \\right\\rceil $ .",
"However, a connected set $\\mathbb {A}_e$ can cover at most one component.",
"Hence we are led to consider at least two top strata, corresponding to distinct values of $e \\ge \\left\\lceil \\frac{1}{2}n(g-1) \\right\\rceil $ .",
"This is one important difference between the stratifications in the symplectic and orthogonal cases."
],
[
"Geometry of the strata",
"The arguments in this subsection are essentially the same as those appearing in §4, except for the complication coming from the topological property described in Theorem REF .",
"Theorem 5.2 Consider a positive even integer $t = 2e \\le n(g-1)+3$ .",
"A general point of $\\mathbb {A}_e$ corresponds to a bundle $V$ with $t(V) = t$ ; in particular, $\\mathcal {M}O_{X}(2n;t)$ is nonempty and irreducible.",
"If $t < n(g-1)$ then $M(V)$ is a single point for general $V \\in \\mathcal {M}O_{X}(2n;t)$ .",
"Let $E$ be a general bundle in $U(n, -e)^s$ for $e \\ge 1$ .",
"As before, we will bound the dimension of the locus in $H^{1}(X, \\wedge ^{2}E)$ of extensions $V$ which admit a rank $n$ isotropic subbundle $F$ of degree $\\ge -e$ other than $E$ .",
"Again, either $F$ is an elementary transformation of $E^*$ lifting to $V$ , or $F$ fits into a diagram of the form (REF ).",
"Step 1.",
"We first show that the latter situation does not arise for a general $V$ in $H^1(X, \\wedge ^2 E)$ for $t \\le n(g-1) - 1$ (not $t \\le n(g-1) + 1$ as in the symplectic case; see Remark REF ).",
"For each fixed $H \\subset E$ , by Criterion REF (2), the locus of extensions fitting into a diagram of the form (REF ) for some $F$ of degree $\\ge -e$ is bounded by $ \\dim \\left( \\mathrm {Sec}^{e-h}\\mathrm {Gr}(2, E/H) \\right) + 1 + \\dim \\mathrm {Ker}\\left( q_{*} \\, ^tq^{*} \\right).",
"$ As in the symplectic case, we obtain $e-h \\ge 0$ .",
"Since $ \\dim \\mathrm {Gr}(2,E/H) = 2(n-r-2) + 1 = 2(n-r) - 3, $ the secant variety has dimension bounded by $ (e-h)(2(n-r) - 3) + (e-h) - 1 = 2(e-h)(n-r-1) - 1.",
"$ By Proposition REF , we have $ \\dim \\mathrm {Ker}\\left(q_{*} \\, ^tq^{*} \\right) &= h^{1}(X, \\wedge ^{2}E) - h^{1} \\left(X, \\wedge ^{2}(E/H) \\right) \\\\&\\le h^{1}(X, \\wedge ^{2}E) - (n-r-1)(e-h) - \\frac{1}{2} (n-r)(n-r-1)(g-1).",
"$ As before, the subbundle $H$ of $E$ varies in a Quot scheme of dimension $ h^{0}(X, \\mathrm {Hom}(H, E/H)) = nh - re - r(n-r)(g-1).",
"$ Thus the dimension of the locus of extensions in $H^1(X, \\wedge ^2 E)$ admitting a diagram of the form (REF ) for some $F$ of degree $\\ge -e$ is bounded by $\\dim \\left( \\mathrm {Sec}^{e-h} \\mathrm {Gr}(2, E/H) \\right) + 1 + \\dim \\mathrm {Ker}(q_* \\, ^t q^*) + h^0(X, \\mathrm {Hom}(H, E/H)) \\\\\\le \\ (e-h)(n-r-1) - \\frac{1}{2} (n-r)(n-r-1) (g-1) + nh-re + h^1(X, \\wedge ^2 E) \\\\= \\ e(n-r) - (e-h)(r+1) - \\frac{1}{2} (n-r)(n-r-1) (g-1) + h^1(X, \\wedge ^2 E).$ Since $e-h \\ge 0$ and we have assumed $e \\le \\frac{1}{2} (n (g-1)-1)$ , this is smaller than $h^1(X, \\wedge ^2 E)$ .",
"Therefore, for $t = 2e \\le n (g-1) -1$ , a general $V \\in H^1(X, \\wedge ^2 E)$ does not admit a rank $n$ isotropic subbundle $F$ of degree $\\ge -e$ which fits into a diagram of the form (REF ).",
"Step 2.",
"Next we consider the case $r = 0$ , so $F$ is an elementary transformation of $E^*$ lifting to $V$ isotropically.",
"Write $\\deg F = -f \\ge -e$ .",
"By Criterion REF (3) and Lemma REF , we have $e+f \\equiv 0 \\mod {2}$ .",
"By Criterion REF (2), the dimension of the locus of extensions in $H^1(X, \\wedge ^2 E)$ admitting such an $F$ is bounded by $\\dim \\left( \\mathrm {Sec}^{\\frac{1}{2}(e+f)} \\mathrm {Gr}(2, E) \\right) + 1 \\ \\le \\ \\frac{1}{2}(e+f) (2(n-2)+2) = (e+f)(n-1).$ If $f < \\frac{1}{2} n (g-1)$ , this bound is smaller than $h^1(X, \\wedge ^2 E) = (n-1)e + \\frac{1}{2}n(n-1)(g-1)$ so a general extension in $H^{1}(X, \\wedge ^{2}E)$ does not admit any such lifting.",
"Step 3 of the proof of Theorem REF carries over, mutatis mutandis, to the orthogonal case, to prove statement (2), and (1) for $t \\le n(g-1)-1$ .",
"To prove (1) for $n(g-1) \\le t \\le n(g-1)+3$ , we just repeat the arguments in Steps 1 and 2 under the assumption $-f \\ge -e + 2$ , to show that for general $E \\in U_{X}(n, -e)$ , a general $V \\in H^{1}(X, \\wedge ^{2}E)$ does not admit a rank $n$ isotropic subbundle $F$ of degree $\\ge -e + 2$ .",
"Remark 5.3 Note that we do not prove an analogue of Theorem REF (3) for the orthogonal case.",
"A dimension count analogous to that in the proof of Theorem REF (3) does not exclude the possibility that two maximal isotropic rank $n$ subbundles of a general bundle in $\\mathcal {M}O_{X}(2n)$ intersect in a line bundle, or in a rank 2 subbundle if $g = 2$ .",
"It is unclear to us at this stage whether or not this in fact happens.",
"Next, we compute the dimension of $M(V)$ for a general $V \\in \\mathcal {M}O_X(2n)^{\\pm }$ .",
"We will use the following analogue of Lemma REF , which is proven in the same way as in the symplectic case: Lemma 5.4 For an orthogonal bundle $V \\in \\mathcal {M}O_X(2n)$ and a point $[\\eta : E \\subset V]$ of $M(V)$ , the Zariski tangent space of $M(V)$ at $\\eta $ is identified with $H^0(X, \\wedge ^2 E^*)$ .",
"$\\Box $ Combining with Lemma REF (2), we obtain: Corollary 5.5 Let $t = 2e = n(g-1) + \\varepsilon $ with $\\varepsilon \\in \\lbrace 0,1,2,3 \\rbrace $ .",
"Then for a general $V \\in \\mathcal {M}O_X(2n;t)$ , we have $ \\dim M(V) = \\frac{\\varepsilon (n-1)}{2}.",
"$ $\\Box $ Proof of Theorem REF .",
"(1) Consider the rational map $\\alpha _e: \\mathbb {A}_e \\rightarrow \\mathcal {M}O_X(2n)$ .",
"For each $t = 2e = n(g-1) + \\varepsilon $ where $\\varepsilon \\in \\lbrace 0,1,2,3 \\rbrace $ , a direct computation shows that $\\dim M(V) = \\dim \\mathbb {A}_e - \\dim \\mathcal {M}O_X(2n).$ Since $\\dim M(V) \\ge \\dim \\alpha _e^{-1} (V)$ for any $V \\in \\mathcal {M}O_X(2n;2e)$ , this equality implies that $\\alpha _e$ is dominant to a component of $\\mathcal {M}O_X(2n)$ .",
"This shows that for general $V$ in some one of the components $ \\mathcal {M}O_X(2n)^{\\pm }$ , we have $n(g-1) \\le t(V) \\le n(g-1) +3.$ By semicontinuity, $t(V) \\le n(g-1) +3$ for any orthogonal bundle $V$ .",
"(2) The nonemptiness and irreducibility were proven in Theorem REF (1).",
"The remaining part can be proven in the same way as Theorem REF (2).",
"(3) This was proven in Theorem REF (2).",
"$\\Box $"
],
[
"Configuration of the dense strata",
"For each $2e = n(g-1 ) + \\varepsilon $ with $0 \\le \\varepsilon \\le 3$ , the locations of the images of the $\\alpha _{e}$ depend on the congruence class of $n(g-1)$ modulo 4: the trivial bundle of rank $2n$ is contained in $\\mathcal {M}O_X(2n)^+$ , hence for each $k$ , we have $\\mathcal {M}O_X(2n;4k) \\subset \\mathcal {M}O_X(2n)^+$ and $\\mathcal {M}O_X(2n;4k+2) \\subset \\mathcal {M}O_X(2n)^-$ .",
"We may summarize the situation for the dense strata as follows: $ n(g-1) \\equiv 0 \\mod {4}: \\quad \\quad \\begin{array}{c|c|c} t & \\hbox{Component} & \\dim M(V) \\\\ \\hline n(g-1) & \\mathcal {M}O_{X}(2n)^{+} & 0 \\\\n(g-1) + 2 & \\mathcal {M}O_{X}(2n)^{-} & \\hbox{$\\frac{1}{2}(n-1)$} \\end{array} $ $ n(g-1) \\equiv 1 \\mod {4}: \\quad \\quad \\begin{array}{c|c|c} t & \\hbox{Component} & \\dim M(V) \\\\ \\hline n(g-1) + 1 & \\mathcal {M}O_{X}(2n)^{-} & \\hbox{$\\frac{1}{2}(n-1)$} \\\\n(g-1) + 3 & \\mathcal {M}O_{X}(2n)^{+} & \\hbox{$n-1$} \\end{array} $ $ n(g-1) \\equiv 2 \\mod {4}: \\quad \\quad \\begin{array}{c|c|c} t & \\hbox{Component} & \\dim M(V) \\\\ \\hline n(g-1) & \\mathcal {M}O_{X}(2n)^{-} & \\hbox{0} \\\\n(g-1) + 2 & \\mathcal {M}O_{X}(2n)^{+} & \\hbox{$\\frac{1}{2}(n-1)$} \\end{array} $ $ n(g-1) \\equiv 3 \\mod {4}: \\quad \\quad \\begin{array}{c|c|c} t & \\hbox{Component} & \\dim M(V) \\\\ \\hline n(g-1) + 1 & \\mathcal {M}O_{X}(2n)^{+} & \\hbox{$\\frac{1}{2}(n-1)$} \\\\n(g-1) + 3 & \\mathcal {M}O_{X}(2n)^{-} & \\hbox{$n-1$} \\end{array} $ Remark 5.6 In [3], symplectic bundles $W$ of rank four were studied which satisfy $s_{2}(W) < t(W)$ ; that is, whose maximal vector subbundles of half rank are all nonisotropic.",
"Orthogonal bundles $V$ belonging to the top stratum $\\mathcal {M}O_{X}(2n;n(g-1)+2)$ provide a different example of this phenomenon.",
"Due to the Hirschowitz bound [4], all such $V$ have a vector subbundle of degree at least $- \\left\\lceil \\frac{1}{2}n(g-1) \\right\\rceil $ , but no isotropic rank $n$ subbundle of this degree or greater.",
"When $n(g-1)$ is even, such an orthogonal $V$ is nongeneral as a vector bundle (compare with Lange–Newstead [14]): Since any maximal rank $n$ subbundle $F \\subset V$ is nonisotropic, we must have $h^{0}(X, \\mathrm {Sym}^{2}F^{*}) > 0$ .",
"But in this case $\\mu (\\mathrm {Sym}^{2}F^{*}) \\le g-1$ .",
"By Lemma REF (2), this is a condition of positive codimension on $F$ , so none of the maximal vector subbundles of $V$ are general.",
"Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-Gu, Seoul 143-701, Korea.",
"E-mail: [email protected] Høgskolen i Oslo og Akershus, Postboks 4, St. Olavs plass, 0130 Oslo, Norway.",
"E-mail: [email protected]"
]
] | 1204.0834 |
[
[
"Entanglement Detection Using Majorization Uncertainty Bounds"
],
[
"Abstract Entanglement detection criteria are developed within the framework of the majorization formulation of uncertainty.",
"The primary results are two theorems asserting linear and nonlinear separability criteria based on majorization relations, the violation of which would imply entanglement.",
"Corollaries to these theorems yield infinite sets of scalar entanglement detection criteria based on quasi-entropic measures of disorder.",
"Examples are analyzed to probe the efficacy of the derived criteria in detecting the entanglement of bipartite Werner states.",
"Characteristics of the majorization relation as a comparator of disorder uniquely suited to information-theoretical applications are emphasized throughout."
],
[
"Introduction",
"Quantum measurements in general have indeterminate outcomes, with the results commonly expressed as a vector of probabilities corresponding to the set of outcomes.",
"In case of noncommuting observables, the joint indeterminacy of their measurement outcomes has an inviolable lower bound, in stark contrast to classical expectations.",
"This was discovered by Heisenberg in his desire to advance the physical understanding of the newly discovered matrix mechanics by relating the unavoidable disturbances caused by the act of measurement to the fundamental commutation relations of quantum dynamics [1].",
"Heisenberg's arguments relied on the statistical spread of the measured values of the observables to quantify uncertainty.",
"This gave rise to the variance formulation of the uncertainty principle which remains a powerful source of intuition on the structure and spectral properties of microscopic systems.",
"With the prospect of quantum computing and the development of quantum information theory in recent decades, on the other hand, the need for a measure of uncertainty that can better capture its information theoretical aspects, especially in dealing with noncanonical observables, has inspired new formulations.",
"Among these are the entropic measure developed in the eighties, and the majorization formulation proposed recently.",
"Uncertainty relations resulting from these formulations have found application to quantum cryptography, information locking, and entanglement detection, in addition to providing uncertainty limits [2], [3].",
"In this paper we develop applications of the majorization formulation of uncertainty introduced in Ref.",
"[3] to the problem of entanglement detection.",
"Deciding whether a given quantum state is entangled is a central problem of quantum information theory and known to be computationally intractable in general [4].",
"As a result, computationally tractable necessary conditions for separability, which provide a partial solution to this problem, have been the subject of active research in recent years.",
"Among these, the Peres-Horodecki positive partial transpose criterion actually provides necessary and sufficient separability conditions for $2\\otimes 2$ and $2\\otimes 3$ dimensional systems, and necessary conditions otherwise.",
"Other notable results are the reduction and global versus local disorder criteria, both necessary conditions in general [5].",
"An observable that has non-negative expectation values for all separable states and negative ones for a subset of entangled states provides an operational method of entanglement detection and is known as an entanglement witness [6].",
"It has also long been known that uncertainty relations can serve a similar purpose by providing inequalities that must be satisfied by separable states and if violated signal entanglement [7], [8].",
"In this paper we extend the majorization formulation of uncertainty developed in Ref.",
"[3] to the problem of entanglement detection.",
"As discussed in that paper (hereafter referred to as Paper I) and in the following, majorization as a comparator of uncertainty is qualitatively different from, and stronger than, scalar measures of uncertainty.",
"Consequently, the entanglement detection results developed here represent a qualitative strengthening of the existing variance and entropic results.",
"This will be evident, among other things, by the fact that they yield, as corollaries, an infinite class of entanglement detectors based on scalar measures.",
"As mentioned above, results of quantum measurements are in general probability vectors deduced from counter statistics, and an information theoretical formulation of uncertainty is normally based on an order of uncertainty defined on such vectors (see Paper I).",
"Thus a scalar measure of uncertainty is commonly a real, non-negative function defined on probability vectors whose value serves to define the uncertainty in question.",
"For example, the Shannon entropy function is the measure of uncertainty for the standard entropic formulation of uncertainty [9].",
"By contrast, majorization provides a partial order of uncertainty on probability vectors that is in general more stringent, and fundamentally stronger, than a scalar measure [10].",
"Note that, unlike scalar measures, majorization does not assign a quantitative measure of uncertainty to probability vectors, and as a partial order may find a pair of vectors to be incomparable.",
"A characterization of majorization that clarifies the foregoing statements can be attained by considering the quasi-entropic set of measures.",
"These were defined in Paper I as the set of concave, symmetric functions defined on probability vectors, and include the Shannon, Tsallis, and (a subfamily of) Rényi entropies as special cases [11].",
"It is important to realize that the uncertainty order determined by one member of the quasi-entropic set for a given pair of probability vectors may contradict that given by another.",
"While additional considerations may justify the use of, e.g., Shannon entropy in preference to the others, the foregoing observation clearly indicates the relative nature of the uncertainty order given by a specific measure, and immediately raises the following question: are there pairs of vectors for which all quasi-entropic measures determine the same uncertainty order?",
"The answer is yes, and the common determination of the quasi-entropic set in such cases defines the uncertainty order given by majorization.",
"What about the cases where there are conflicting determinations by the members of the quasi-entropic set?",
"The majorization relation defines such pairs as incomparable, whence the “partial” nature of the order defined by majorization.",
"We may therefore consider the majorization order to be equivalent to the collective order determination of the entire quasi-entropic set (see §IV).",
"This characterization of the majorization relation clearly shows its standing vis-à-vis the scalar measures of uncertainty.",
"More practical definitions of the majorization relation will be considered in §II.",
"The rest of this paper is organized as follows.",
"In §II we review elements of the majorization formulation of uncertainty needed for our work and establish the notation.",
"In §III we present the central results of this paper on entanglement detection, and in §IV we derive entire classes of entanglement detectors for quasi-entropic measures as corollaries to the theorems of §III.",
"Concluding remarks are presented in §V.",
"Details of certain mathematical proofs are given in the Appendix."
],
[
"Majorization formulation of uncertainty",
"This section presents a review of the basic elements of majorization theory and the formulation of the uncertainty principle based on it.",
"It follows the treatment given in Paper I."
],
[
"Majorization",
"The basic element of our formulation is the majorization protocol for comparing the degree of uncertainty, or disorder, among probability vectors, i.e., sequences of non-negative numbers summing to unity [10].",
"By definition, ${\\lambda }^{1}$ is no less uncertain than ${\\lambda }^{2}$ if ${\\lambda }^{1}$ equals a mixture of the permutations of ${\\lambda }^{2}$ .",
"Then ${\\lambda }^{1}$ is said to be majorized by ${\\lambda }^{2}$ and written ${\\lambda }^{1}\\prec {\\lambda }^{2}$ .",
"An equivalent definition that flushes out the details of the foregoing is based on the vector ${\\lambda }^{\\downarrow }$ which is obtained from $\\lambda $ by arranging the components of the latter in a nonincreasing order.",
"Then, ${\\lambda }^{1}\\prec {\\lambda }^{2}$ if ${\\sum }_{i}^{j} {\\lambda }^{1\\downarrow }_{i} \\le {\\sum }_{i}^{j} {\\lambda }^{2\\downarrow }_{i} $ for $j=1,2, \\ldots d-1$ , where $d$ is the larger of the two dimensions and trailing zeros are added where needed.",
"As stated earlier, the majorization relation is a partial order, i.e., not every two vectors are comparable under majorization.",
"As suggested in §I, two vectors are found to be incomparable when the difference in their degrees of uncertainty does not rise to the level required by majorization.",
"As evident from the second definition given above, majorization requires the satisfaction of $N-1$ inequalities if the number of non-zero components of the majorizing vector is $N$ .",
"This accounts for the strength of the majorization relation as compared to scalar measures of disorder.",
"Indeed as alluded to in §I, for any quasi-entropic function $F(\\lambda )$ , ${\\lambda }^{1} \\prec {\\lambda }^{2}$ implies $F({\\lambda }^{1}) \\ge F({\\lambda }^{2})$ , but not conversely.",
"On the other hand, if for every quasi-entropic function $F(\\lambda )$ we have $F({\\lambda }^{1}) \\ge F({\\lambda }^{2})$ , then ${\\lambda }^{1} \\prec {\\lambda }^{2}$ [10].",
"To establish uncertainty bounds, we need to characterize the greatest lower bound, or infimum, and the least upper bound, or supremum, of a set of probability vectors [12].",
"The infimum is defined as the vector that is majorized by every element of the set and in turn majorizes any vector with that property .",
"The supremum is similarly defined.",
"We will briefly outline the construction of the infimum here and refer the reader to Paper I for further details.",
"Given a set of probability vectors $\\lbrace {\\lambda }^{a} {\\rbrace }_{a=1}^{N}$ , consider the vector ${\\mu }^{inf}$ defined by ${\\mu }_{0}^{inf}=0&,\\,\\,\\,{\\mu }_{j}^{inf}=\\min \\big ( {\\sum }_{i=1}^{j} {\\lambda }^{1\\downarrow }_{i},{\\sum }_{i=1}^{j} {\\lambda }^{2\\downarrow }_{i},\\ldots , \\nonumber \\\\&{\\sum }_{i=1}^{j}{\\lambda }^{N\\downarrow }_{i} \\big ), \\,\\,\\, 1 \\le j \\le {d}_{max}, $ where ${d}_{max}$ is the largest dimension found in the set.",
"The infimum is then given by ${\\lambda }^{inf}_{i}={[\\inf ({\\lambda }^{1},{\\lambda }^{2}, \\ldots , {\\lambda }^{N})]}_{i}={\\mu }_{i}^{inf}-{\\mu }_{i-1}^{inf}, $ where $1 \\le i \\le {d}_{max}$ .",
"The construction of the supremum starts with ${\\mu }^{sup}$ in parallel with Eq.",
"(REF ), but with “max” replacing “min,” and may require further steps detailed in Paper I.",
"It is worth noting here that the infimum (supremum) of a pair of probability vectors will in general be more (less) disordered than either.",
"Important special cases are (i) one of the two majorizes the other, in which case the latter is the infimum and the former the supremum of the two, and (ii) the two are equal, in which case either is both the infimum and the supremum.",
"Furthermore, a useful qualitative rule is that the more “different” are two probability vectors, the further will the infimum or supremum of the two be from at least one of them.",
"Finally, we note that while the infimum or supremum of a set of probability vectors always exists, it need not be a member of the set."
],
[
"Measurement and uncertainty",
"A generalized measurement may be defined by a set of positive operators $\\lbrace {\\hat{\\mathrm {E}}}_{\\alpha } \\rbrace $ called measurement elements and subject to the completeness condition ${\\sum }_{\\alpha }{\\hat{\\mathrm {E}}}_{\\alpha } = \\hat{\\mathbb {1}}$ .",
"The probability that outcome $\\alpha $ turns up in a measurement of the state $\\hat{\\rho }$ is given by the Born rule ${P}_{\\alpha }(\\hat{\\rho })=\\textrm {tr} [\\hat{\\mathrm {E}}_{\\alpha } \\hat{\\rho } ]$ .",
"A generalized measurement can always be considered to be the restriction of a more basic type, namely a projective measurement, performed on an enlarged system to the system under generalized measurement [13].",
"A projective measurement is usually associated with an observable of the system represented by a self-adjoint operator $\\hat{M}$ , and entails a partitioning of the spectrum of $\\hat{M}$ into a collection of subsets $\\lbrace {b}_{\\alpha }^{M} \\rbrace $ called measurement bins.",
"We call a projective measurement maximal if each bin consists of a single point of the spectrum of the measured observable.",
"Having assembled the necessary concepts, we can now characterize uncertainty by means of majorization relations in a natural manner.",
"To start, we define the probability vector ${P}^{X}(\\hat{\\rho })$ resulting from a measurement $\\mathrm {X}$ on a state $\\hat{\\rho }$ to be uncertain if it is majorized by $\\mathcal {I}=(1,0,\\ldots ,0)$ but not equal to it.",
"As such, ${P}^{X}(\\hat{\\rho })$ is said to be strictly majorized by $\\mathcal {I}$ and written ${P}^{X}(\\hat{\\rho })\\prec \\prec \\mathcal {I}$ .",
"Similarly, given a pair of measurements $\\mathrm {X}$ and $\\mathrm {Y}$ on a state $\\hat{\\rho }$ , we say ${P}^{\\mathrm {X}}(\\hat{\\rho })$ is more uncertain, equivalently more disordered, than ${P}^{\\mathrm {Y}}(\\hat{\\rho })$ if ${P}^{\\mathrm {X}}(\\hat{\\rho }) \\prec {P}^{\\mathrm {Y}}(\\hat{\\rho })$ .",
"Further, we define the joint uncertainty of a pair of measurements $\\mathrm {X}$ and $\\mathrm {Y}$ to be the outer product ${P}^{\\mathrm {X}}\\otimes {P}^{\\mathrm {Y}}$ , i.e., ${P}_{\\alpha \\beta }^{\\mathrm {X} \\oplus \\mathrm {Y}}={P}^{\\mathrm {X}}_{\\alpha } {P}^{\\mathrm {Y}}_{\\beta }$ .",
"Since $H({P}^{\\mathrm {X}}\\otimes {P}^{\\mathrm {Y}})=H({P}^{\\mathrm {X}})+H({P}^{\\mathrm {Y}})$ , where $H(\\cdot )$ is the Shannon entropy function, this definition is seen to be consistent with its entropic counterpart.",
"As stated earlier, ${P}^{\\mathrm {X}} \\prec {P}^{\\mathrm {Y}}$ implies $H({P}^{\\mathrm {X}}) \\ge H({P}^{\\mathrm {Y}})$ but not conversely.",
"These definitions naturally extend to an arbitrary number of states and measurements.",
"We are now in a position to state the majorization statement of the uncertainty principle established in Paper I: “The joint results of a set of generalized measurements of a given state are no less uncertain than a probability vector that depends on the measurement set but not the state, and is itself uncertain unless the measurement elements have a common eigenstate.\"",
"In symbols, ${P}^{\\mathrm {X}}(\\hat{\\rho })\\otimes {P}^{\\mathrm {Y}}(\\hat{\\rho }) \\otimes \\ldots \\otimes {P}^{\\mathrm {Z}}(\\hat{\\rho }) \\prec {{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}} \\prec \\prec \\mathcal {I}, $ where ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}={\\sup }_{\\hat{\\rho }} [{P}^{\\mathrm {X}}(\\hat{\\rho })\\otimes {P}^{\\mathrm {Y}}(\\hat{\\rho }) \\otimes \\ldots \\otimes {P}^{\\mathrm {Z}}(\\hat{\\rho })] , $ unless the measurement elements $\\lbrace \\hat{\\mathrm {E}}^{\\mathrm {X}}, \\hat{\\mathrm {E}}^{\\mathrm {Y}}, \\ldots , \\hat{\\mathrm {E}}^{\\mathrm {Z}} \\rbrace $ have a common eigenstate in which case ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}} = \\mathcal {I}$ .",
"Note that ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ is the majorization uncertainty bound for the measurement set considered."
],
[
"majorization uncertainty bounds",
"As stated above, the uncertainty bound ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ given by Eq.",
"(REF ) depends on the measurement set but not the state of the system.",
"As the supremum of all possible measurement outcomes, it is the probability vector that sets the irreducible lower bound to uncertainty for the set.",
"As such, it is the counterpart of the variance product or entropic lower bound in the traditional formulations of the uncertainty principle.",
"Unlike the latter, however, ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ is in general not realizable or even approachable by any state of the system.",
"Therefore, there is in general no such thing as a “minimum uncertainty state” within the majorization framework.",
"This is a consequence of the fact, mentioned earlier, that the infimum or supremum of a set of vectors need not be a member of the set.",
"An obvious special case is the trivial example of zero uncertainty for which ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}= \\mathcal {I}$ , signaling the existence of a common eigenstate for the measurement elements.",
"Intuitively, we expect that mixing states can only increase their uncertainty, as is known to be the case for scalar measures of uncertainty.",
"In case of majorization, this expectation is manifested in the property that the uncertainty bound ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ is in general realized on the class of pure states.",
"Indeed for two and three mutually unbiased observables on a two-dimensional Hilbert space considered, we found in Paper I that the required maxima for the components of ${\\mu }^{sup}$ that serve to define the uncertainty bound are reached on pure states.",
"More specifically, as outlined in §IIA above and detailed in §IIB, IVA and IVB of Paper I, the calculation of ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ involves a component-wise determination of ${\\mu }^{sup}$ by a series of maximizations over all possible density matrices $\\hat{\\rho }$ .",
"The property in question then guarantees that the maximization process can be limited to density matrices representing pure states only.",
"We will prove this assertion in general, as well as for the class of separable states, in the Appendix.",
"As discussed earlier, an entanglement detector can be effective in deciding whether a given density matrix is separable by providing a condition that is satisfied by all separable states and if violated signals entanglement.",
"A large body of entanglement detection strategies have been developed in recent years which are primarily based on scalar conditions, including those based on variance and entropic type uncertainty relations, which can be found in Refs.",
"[5], [6], [7], [8].",
"Here we shall develop majorization conditions for entanglement detection based on the formulation of measurement uncertainty given in Paper I and outlined in §II above.",
"In particular, the entanglement condition given in Theorem 1 below is linear and susceptible to experimental implementation, so it can be formulated as an entanglement witness.",
"Nonlinear detectors developed below, on the other hand, rely on majorization-based uncertainty relations.",
"We will also introduce the notion of subsystem disorder and a sharpened version of the Nielsen-Kempe [14] separability condition as a nonlinear entanglement detector.",
"As majorization relations, our results in general entail multiple inequalities whose number will grow with the uncertainty levels involved.",
"As discussed in §IIA, this is an important feature of the majorization relation as comparator of disorder, one that sets it apart from scalar conditions and provides for the refinement needed in comparing highly disordered vectors.",
"As will be seen in §IV, this property has the consequence that each majorization condition yields a scalar condition for the entire quasi-entropic class of uncertainty measures."
],
[
"Linear detectors",
"Our detection strategy is based on the intuitive expectation that the measurement uncertainty bound for the class of separable states of a multipartite system must be majorized by the corresponding bound for all states, and that this hierarchy can be exploited for entanglement detection.",
"This is the majorization rendition of the strategy often used to derive separability conditions [7], [8].",
"We will first consider the case of one generalized measurement resulting in a linear detection condition.",
"Theorem 1.",
"Let the results of the generalized measurement ($\\mathrm {X}, \\lbrace \\hat{\\mathrm {E}}_{\\alpha }^{X} \\rbrace $ ) on a multipartite state ${\\hat{\\rho }}^{ABC \\dots F}$ of parties $(A, B, C, \\dots , F)$ defined on a finite-dimensional Hilbert space be bounded by ${{P}}_{sup}^{\\mathrm {X}}={\\sup }_{{\\hat{\\rho }}^{ABC \\dots F}}{{P}}^{\\mathrm {X}}({\\hat{\\rho }}^{ABC \\dots F}), $ and in the case of a separable state ${\\hat{\\rho }}^{ABC \\dots F}_{sep}$ by ${{P}}_{sep;sup}^{\\mathrm {X}}={\\sup }_{{\\hat{\\rho }}^{ABC \\dots F}_{sep}}{{P}}^{\\mathrm {X}}({\\hat{\\rho }}^{ABC \\dots F}_{sep}).",
"$ Then, (i) the suprema in the foregoing pair of equations may be taken over the class of pure and pure, product states, respectively, and (ii) given an arbitrary state ${\\hat{\\sigma }}^{ABC \\dots F}$ , the condition ${{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F}) \\nprec {{P}}_{sep;sup}^{\\mathrm {X}}$ implies that ${\\hat{\\sigma }}^{ABC \\dots F}$ is entangled [15].",
"Part (i) of Theorem 1 is the majorization version of the familiar result that uncertainty bounds are realized on pure states and is proved in the Appendix [16].",
"Part (ii) is the statement that the bound in Eq.",
"(REF ) is a necessary condition for separability and its violation signals the existence of entanglement.",
"As noted above, Theorem 1 provides an operational method of entanglement detection and can be reformulated as a set of entanglement witnesses.",
"Clearly, the efficacy of Theorem 1 in detecting entanglement depends on the choice of the measurement ${\\mathrm {X}}$ for a given quantum state.",
"This suggests looking for the optimum measurement, in parallel with the optimization problem for entanglement witnesses [17].",
"Note that here the choice of the generalized measurement ${\\mathrm {X}}$ affects both ${{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F})$ and ${{P}}_{sep;sup}^{\\mathrm {X}}$ , thus making the task of finding a fully optimized solution rather difficult.",
"It is therefore fortunate that we can achieve a partial optimization on the basis of Theorem 3 of Paper I, the relevant content of which we can state as follows: Lemma Under the conditions of Theorem 1 and with ${\\mathrm {X}}$ restricted to rank 1 measurements, we have ${{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F}) \\prec {{P}}^{{\\mathrm {X}}^{\\star }}({\\hat{\\sigma }}^{ABC \\dots F})={\\lambda }({\\hat{\\sigma }}^{ABC \\dots F}), $ where ${\\mathrm {X}}^{\\star }$ is a maximal projective measurement whose elements are the set of rank 1 orthogonal projection operators onto the eigenvectors of ${\\hat{\\sigma }}^{ABC \\dots F}$ and ${\\lambda }({\\hat{\\sigma }}^{ABC \\dots F})$ is the corresponding spectrum.",
"It should be noted here that while the choice of ${{\\mathrm {X}}^{\\star }}$ succeeds in optimizing ${{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F})$ which appears in part (ii) of Theorem 1, its effect on ${{P}}_{sep;sup}^{\\mathrm {X}}$ , the other object appearing therein, is unknown and may result in a poor overall choice.",
"Moreover, any degeneracy in the spectrum of ${\\hat{\\sigma }}^{ABC \\dots F}$ renders ${{\\mathrm {X}}^{\\star }}$ nonunique and subject to further optimization.",
"These features will be at play in the example considered below where, the foregoing caveats notwithstanding, the resulting detector turns out to be fully optimal.",
"The example in question is the Werner state ${\\hat{\\rho }}^{wer}_{d}(q)$ for a bipartite system of two $d$ -level parties $A$ and $B$ defined on a $d^2$ -dimensional Hilbert space [18], [19] ${\\hat{\\rho }}^{wer}_{d}(q)=\\frac{1}{d^2}({1-q})\\hat{{\\mathbb {1}}} + q \\mid {\\mathfrak {B}}_{1} \\rangle \\langle {\\mathfrak {B}}_{1}\\mid , $ where $\\mid {\\mathfrak {B}}_{1} \\rangle \\ =\\frac{1}{\\sqrt{d}} {\\sum }_{j=0}^{d-1} \\mid A,j \\rangle \\otimes \\mid B,j \\rangle .",
"$ Here $\\lbrace \\mid A,j \\rangle \\rbrace $ and $\\lbrace \\mid B,j \\rangle \\rbrace $ ) are orthonormal bases for subsystem $A$ and $B$ respectively.",
"Also, here and elsewhere we use the symbol $\\hat{{\\mathbb {1}}}$ to denote the identity operator of dimension appropriate to the context.",
"Note that $\\mid {\\mathfrak {B}}_{1} \\rangle $ is the generalization of the two-qubit Bell state $(\\mid 00 \\rangle + \\mid 11 \\rangle )/\\sqrt{2}$ .",
"As such, it is totally symmetric, as well as maximally entangled in the sense that its marginal states are maximally disordered and equal to $\\hat{{\\mathbb {1}}}/d$ .",
"An important fact to be used in the following is that ${\\hat{\\rho }}^{wer}_{d}(q)$ is separable if and only if $q \\le (1+d)^{-1}$ [19].",
"In order to probe the separability of ${\\hat{\\rho }}^{wer}_{d}(q)$ by means of Theorem 1, we must first choose a measurement.",
"We will use the above Lemma to guide our choice of the putative optimum measurement ${\\mathrm {X}}^{\\star }$ .",
"The Lemma requires that the measurement elements be equal to the orthogonal projections onto the eigenvectors of ${\\hat{\\rho }}^{wer}_{d}(q)$ .",
"However, a simple calculation shows that the spectrum of ${\\hat{\\rho }}^{wer}_{d}(q)$ is degenerate, consisting of a simple eigenvalue equal to $q+{d}^{-2}(1-q)$ and a $({d}^{2}-1)$ -fold degenerate eigenvalue equal to ${d}^{-2}(1-q)$ .",
"A natural choice of basis for the degenerate subspace is a set of $d^2-1$ generalized Bell states which, together with $\\mid {\\mathfrak {B}}_{1} \\rangle $ of Eq.",
"(REF ), constitute the orthonormal eigenbasis ${\\lbrace \\mid {\\mathfrak {B}}_{\\alpha } \\rangle \\rbrace }_{\\alpha =1}^{d^2}$ for ${\\hat{\\rho }}^{wer}_{d}(q)$ .",
"This generalized Bell basis is thus characterized by the fact that each $\\mid {\\mathfrak {B}}_{\\alpha } \\rangle $ is maximally entangled and possesses equal Schmidt coefficients.",
"The measurement elements of the optimal measurement ${\\mathrm {X}}^{\\star }$ are thus given by ${\\hat{\\mathrm {E}}}_{\\alpha }^{{X}^{\\star }}= \\mid {\\mathfrak {B}}_{\\alpha } \\rangle \\langle {\\mathfrak {B}}_{\\alpha } \\mid $ , $\\alpha =1, 2, \\dots , d^2$ .",
"With ${\\mathrm {X}}^{\\star }$ so defined, we readily find ${{P}}^{{\\mathrm {X}}^{\\star }}[{\\hat{\\rho }}^{wer}_{d}(q)]=&[q+{d}^{-2}(1-q), {d}^{-2}(1-q), \\nonumber \\\\&{d}^{-2}(1-q), \\ldots , {d}^{-2}(1-q)], $ which corresponds to the spectrum of ${\\hat{\\rho }}^{wer}_{d}(q)$ as expected.",
"Note that the Lemma guarantees that any other rank 1 measurement in place of ${\\mathrm {X}}^{\\star }$ would produce a more disordered probability vector on the right-hand side of Eq.",
"(REF ).",
"The next step is the calculation of ${{P}}_{sep;sup}^{{\\mathrm {X}}^{\\star }}$ of Theorem 1, which is the least disordered probability vector that can result from the measurement of ${\\mathrm {X}}^{\\star }$ on a pure, product state of two $d$ -level subsystems.",
"As can be seen in Eq.",
"(REF ) of the Appendix, this calculation requires finding the maximum overlap of every measurement element $\\mid {\\mathfrak {B}}_{\\alpha } \\rangle \\langle {\\mathfrak {B}}_{\\alpha } \\mid $ with the class of pure, product states.",
"This overlap is given by the square of the largest Schmidt coefficient of the respective Bell state $\\mid {\\mathfrak {B}}_{\\alpha }\\rangle $ [20].",
"Since all Schmidt coefficients are equal to $1/\\sqrt{d}$ for every Bell state $\\mid {\\mathfrak {B}}_{\\alpha }\\rangle $ , we can conclude that ${{P}}_{sep;sup}^{{\\mathrm {X}}^{\\star }}=(1/d, 1/d, \\ldots , 1/d,0,0, \\ldots ,0).",
"$ We are now in a position to apply the entanglement criterion of Theorem 1, which reads ${{P}}^{\\mathrm {{X}^{\\star }}}[{\\hat{\\rho }}^{wer}_{d}(q)] \\nprec {{P}}_{sep;sup}^{\\mathrm {{X}^{\\star }}}$ in this instance [21].",
"Using the information in Eqs.",
"(REF ) and (REF ), we readily see that this criterion translates to the single inequality $q+{d}^{-2}(1-q) > 1/d$ , or $q > 1/(1+d)$ , which is the necessary and sufficient condition for the inseparability of the Werner state ${\\hat{\\rho }}^{wer}_{d}(q)$ .",
"Thus every entangled Werner state of two $d$ -level systems is detected by the measurement ${\\mathrm {X}}^{\\star }$ defined above.",
"While the efficacy of the generalized Bell states for detecting entanglement in Werner states is well established [5], [6], [7], [8], we note its natural emergence from the general majorization results of this subsection.",
"It should also be noted that Theorem 1 and the foregoing analysis can be extended to multipartite states."
],
[
"Nonlinear detectors",
"As an example of nonlinear entanglement detectors, we will derive majorization conditions for separability based on uncertainty relations.",
"The method we will follow relies on the degeneracy properties of projective measurements on a single system versus the products of such measurements on a multipartite system [7], [8].",
"For example, with measurement ${\\mathrm {X}}^{A}$ on ${\\hat{\\rho }}^{A}$ and ${\\mathrm {X}}^{B}$ on ${\\hat{\\rho }}^{B}$ and each having a simple spectrum, the product projective measurement ${\\mathrm {X}}^{A} \\otimes {\\mathrm {X}}^{B}$ on the bipartite system ${\\hat{\\rho }}^{AB}$ may be degenerate.",
"A specific case is ${{\\hat{\\sigma }}_{x}}^{A}\\otimes {{\\hat{\\sigma }}_{x}}^{B}$ , whose spectrum $(+1/4,+1/4,-1/4,-1/4)$ is doubly degenerate, versus ${{\\hat{\\sigma }}_{x}}^{A}$ or ${{\\hat{\\sigma }}_{x}}^{B}$ each of which has the simple spectrum $(+1/2,-1/2)$ .",
"Note the fact that here we are following common practice by defining ${{\\hat{\\sigma }}_{x}}^{A}\\otimes {{\\hat{\\sigma }}_{x}}^{B}$ to be the projective measurement whose two elements project into the subspaces corresponding to the eigenvalues $+1/4$ and $-1/4$ (and not the rank 1 projective measurement that resolves the degeneracies by virtue of using all four product elements).",
"Consequently, it may happen that ${\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}$ and ${\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}$ have a common eigenstate while ${\\mathrm {X}}^{A}$ and ${\\mathrm {Y}}^{A}$ do not, and that the said common eigenstate is an entangled pure state.",
"In such cases, the corresponding probability vectors will reflect the stated differences and may be capable of detecting entanglement as in the case of linear detectors.",
"Consider two product projective measurements ${\\mathrm {X}}^{A} \\otimes {\\mathrm {X}}^{B}$ and ${\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}$ performed on the bipartite state ${\\hat{\\rho }}^{AB}$ .",
"The measurement results are then given by ${{P}}^{{\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}}({\\hat{\\rho }}^{AB})= \\textrm {tr}&[{\\hat{\\Pi }}^{{\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}} {\\hat{\\rho }}^{AB} ], \\nonumber \\\\ {{P}}^{{\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}}({\\hat{\\rho }}^{AB})= \\textrm {tr}&[{\\hat{\\Pi }}^{{\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}} {\\hat{\\rho }}^{AB} ], \\nonumber \\\\{P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\rho }}^{AB}) &={P}^{ {\\mathrm {X}}^{A} \\otimes {\\mathrm {X}}^{B} }({\\hat{\\rho }}^{AB})\\nonumber \\\\\\otimes {P}^{{\\mathrm {Y}}^{A} \\otimes {\\mathrm {Y}}^{B} }({\\hat{\\rho }}^{AB}), $ where ${\\hat{\\Pi }}^{{\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}}$ and ${\\hat{\\Pi }}^{{\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}}$ , both projection operators but not necessarily rank 1, represent the measurement elements of the two product measurements.",
"Note the majorization definition of joint uncertainty, given earlier, at work on the last line of Eq.",
"(REF ).",
"Under these conditions, we have the following general result.",
"Theorem 2.",
"The results of measurements ${\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}$ and ${\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B}$ on a separable state ${\\hat{\\rho }}^{AB}_{sep}$ satisfy ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\rho }}^{AB}_{sep}) \\prec {P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}, $ where ${P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}$ is defined in Eq.",
"(REF ), hence the condition ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\sigma }}^{AB}) \\nprec {P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}$ implies that the state ${\\hat{\\sigma }}^{AB}$ is entangled.",
"To establish Theorem 2, consider product states of the form ${\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}$ .",
"Then Lemma 1 of Ref.",
"[8] asserts that ${P}^{ {\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B} }({\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}) &\\prec {P}^{ {\\mathrm {X}}^{A}}({\\hat{\\rho }}^{A}), \\nonumber \\\\{P}^{ {\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B} }({\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}) &\\prec {P}^{ {\\mathrm {Y}}^{A}}({\\hat{\\rho }}^{A}).",
"$ Using these relations and Eq.",
"(REF ), we find [22] ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}) \\prec {P}^{ {\\mathrm {X}}^{A}}({\\hat{\\rho }}^{A})\\otimes {P}^{ {\\mathrm {Y}}^{A}}({\\hat{\\rho }}^{A}).",
"$ Since the right-hand side of Eq.",
"(REF ) is by definition majorized by ${P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}$ , we conclude that ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}({\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}) \\prec {P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}.",
"$ At this point we note that ${P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B}) \\oplus ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})}_{sep;sup}$ , which by definition majorizes all probability vectors that can appear on the left-hand side of Eq.",
"(REF ), can be found among pure, product states [16].",
"This assertion is established in Eq.",
"(REF ) et seq.",
"of the Appendix.",
"Consequently, the product state ${\\hat{\\rho }}^{A} \\otimes {\\hat{\\rho }}^{B}$ in Eq.",
"(REF ) may be replaced by any separable state ${\\hat{\\rho }}^{AB}_{sep}$ , thereby establishing Eq.",
"(REF ) and Theorem 2.",
"We note in passing that, because the measurements considered in Theorem 2 are in general not rank 1, the Lemma which we used earlier for detector optimization cannot be applied here.",
"To illustrate Theorem 2, we will consider the case of three mutually unbiased observables measured on bipartite states of two-level systems, as in §IVB of Paper I.",
"In effect, this amounts to measuring products of the three spin components of a pair of spin-1/2 systems (or qubits).",
"For this case, Eq.",
"(REF ) reads ${P}^{ ({{\\sigma }_{x}}^{A}\\otimes {{\\sigma }_{x}}^{B}) \\oplus ({{\\sigma }_{y}}^{A}\\otimes {{\\sigma }_{y}}^{B})\\oplus ({{\\sigma }_{z}}^{A}\\otimes {{\\sigma }_{z}}^{B})}({\\hat{\\rho }}^{AB}_{sep})\\prec {P}_{sup}^{{\\sigma }_{x}\\oplus {\\sigma }_{y}\\oplus {\\sigma }_{z}}, $ where ${\\hat{\\rho }}^{AB}_{sep}$ is any separable state of two qubits.",
"Thus a violation of this relation by a two-qubit state implies that it is entangled.",
"For the set of states ${\\hat{\\rho }}^{AB}$ to be probed by Theorem 2, we will consider the Werner family of two-qubit states ${\\hat{\\rho }}^{wer}(q)=\\frac{1}{4}({1-q}){\\mathbb {1}}+ q \\mid {\\mathfrak {B}}_{1} \\rangle \\langle {\\mathfrak {B}}_{1} \\mid , $ which is known to be entangled if and only if $q >1/3$ .",
"Here $0 \\le q \\le 1$ and $\\mid {\\mathfrak {B}}_{1} \\rangle =(\\mid 00 \\rangle + \\mid 11 \\rangle )/\\sqrt{2}$ is the totally symmetric Bell state of Eq.",
"(REF ) for the present case.",
"A calculation of the probability vector for this measurement gives $(1 \\pm q)(1 \\pm q)(1 \\pm q)/8$ for the 8 components of ${P}^{({{\\sigma }_{x}}^{A}\\otimes {{\\sigma }_{x}}^{B}) \\oplus ({{\\sigma }_{y}}^{A}\\otimes {{\\sigma }_{y}}^{B})\\oplus ({{\\sigma }_{z}}^{A}\\otimes {{\\sigma }_{z}}^{B})}[{\\hat{\\rho }}^{wer}(q)]$ .",
"Arranged in a non-ascending order, these are ${(1+q)}^{3}/8$ , ${(1+q)}^{2}{(1-q)}/8$ and ${(1+q)}{(1-q)}^{2}/8$ both three-fold degenerate, and ${(1-q)}^{3}/8$ .",
"The supremum ${P}_{sup}^{{\\sigma }_{x}\\oplus {\\sigma }_{y}\\oplus {\\sigma }_{z}}$ , on the other hand, was calculated in §IVB of Paper I, where it was found as ${P}_{sup}^{{\\sigma }_{x}\\oplus {\\sigma }_{y}\\oplus {\\sigma }_{z}}=\\frac{1}{8}\\big [{(1+1/\\sqrt{3})}^{3},2{(1+1/\\sqrt{2})}^{2} \\nonumber \\\\-{(1+1/\\sqrt{3})}^{3}, 4-{(1+1/\\sqrt{2})}^{2}, 4-{(1+1/\\sqrt{2})}^{2}\\nonumber \\\\,0,0,0,0 \\big ].",
"$ According to Theorem 2, ${\\hat{\\rho }}^{wer}(q)$ violates the separability condition of Eq.",
"(REF ) if ${P}_{sup}^{{\\sigma }_{x}\\oplus {\\sigma }_{y}\\oplus {\\sigma }_{z}}$ fails to majorize ${P}^{({{\\sigma }_{x}}^{A}\\otimes {{\\sigma }_{x}}^{B}) \\oplus ({{\\sigma }_{y}}^{A}\\otimes {{\\sigma }_{y}}^{B})\\oplus ({{\\sigma }_{z}}^{A}\\otimes {{\\sigma }_{z}}^{B})}[{\\hat{\\rho }}^{wer}(q)]$ , which occurs for $q > 1/\\sqrt{3}=0.577$ in this example.",
"The entanglement in ${\\hat{\\rho }}^{wer}(q)$ is thus detected by Theorem 2 for $q > 0.577$ , and missed for the range $0.333 < q \\le 0.577$ .",
"By comparison, a similar method based on the Shannon entropy in Ref.",
"[7] detects entanglement in ${\\hat{\\rho }}^{wer}(q)$ for $q >0.65$ .",
"Reference [8], using the same method but relying on the family of Tsallis entropies, matches our condition $q > 0.577$ by means of a numerical calculation that searches over the Tsallis family for optimum performance.",
"We end this subsection by a brief description of a sharpened version of the Nielsen-Kempe theorem as a nonlinear entanglement detector [12].",
"This celebrated theorem is an elegant separability condition based on majorization relations.",
"It asserts that the spectrum of a separable bipartite state is majorized by each of its marginal spectra.",
"By extension, this theorem guarantees that the spectra of all possible subsystems of a separable multipartite system must majorize its global spectrum [14].",
"Our version of this theorem employs the notion of the infimum of a set of probability vectors discussed in §IIA, and is based on the observation that if a probability vector is majorized by each member of a set of probability vectors, then it must also be majorized by the infimum of that set.",
"Consider the multipartite state ${\\hat{\\rho }}^{ABC \\dots F}$ together with all its subsystem states ${\\lbrace {\\hat{\\rho }}^{{X}_{a}} \\rbrace }_{a=1}^{f}$ and subsystem spectra ${\\lbrace {\\lambda }^{{X}_{a}} \\rbrace }_{a=1}^{f}$ , where ${\\lbrace {{X}_{a}} \\rbrace }_{a=1}^{f}$ represent all proper subsets of the set of parties $(A, B, C, \\dots , F)$ .",
"Then the infimum of the subsystem spectra, ${\\Lambda }^{ABC \\dots F}={\\inf }[{\\lambda }^{{X}_{1}},{\\lambda }^{{X}_{2}}, \\ldots , {\\lambda }^{{X}_{f}}]$ , embodies the subsystem disorder of the state ${\\hat{\\rho }}^{ABC \\dots F}$ , as expressed in the following theorem.",
"Theorem 3 (Nielsen-Kempe).",
"A multipartite state is entangled if its system disorder fails to exceed its subsystem disorder, i.e., ${\\lambda }^{ABC \\dots F}$ is entangled if ${\\lambda }^{ABC \\dots F}\\nprec {\\Lambda }^{ABC \\dots F}$ ."
],
[
"Quasi-Entropic Detectors",
"The majorization based theorems of the previous section have direct corollaries that yield scalar entanglement detectors for the entire class of Schur-concave measures.",
"A Schur-concave function $G$ is defined by the property that ${\\lambda }^{1} \\prec {\\lambda }^{2}$ implies $G({\\lambda }^{1}) \\ge G({\\lambda }^{2})$ [23].",
"Equivalently, Schur-concave functions are characterized by being monotonic with respect to the majorization relation.",
"They include all functions $G(\\cdot )$ that are concave and symmetric with respect to the arguments, a subset which we have defined as quasi-entropic.",
"An important subclass of quasi-entropic functions is obtained if we restrict $G$ to have the following trace structure: $G(\\lambda )=\\mathrm {tr}[g(\\lambda )]= {\\sum }_{\\alpha } g({\\lambda }_{\\alpha }), $ where $g(\\cdot )$ is a concave function of a single variable and $\\lambda $ is treated as a diagonal matrix for the purpose of calculating the trace.",
"Note that $G(\\cdot )$ as constructed in Eq.",
"(REF ) is manifestly symmetric and, as a sum of concave functions, it is also concave.",
"We shall refer to this class of functions, which include the Shannon and Tsallis entropies, as trace type quasi-entropic.",
"An important fact regarding this class of measures is that if, for a given pair of probability vectors $\\lambda $ and $\\mu $ , we have $G({\\lambda })>G({\\mu })$ for every trace type quasi-entropic measure $G$ , then $\\lambda \\prec \\mu $ .",
"The standard entropic measure of uncertainty [9], which is based on the Shannon entropy function, is trace type quasi-entropic and corresponds to the choice $g(x)=H(x)=-x \\ln (x)$ in Eq.",
"(REF ).",
"An example of an information theoretically relevant measure that is quasi-entropic but not trace type is the Rényi subfamily of entropies of order less than one [24].",
"Accordingly, although the scalar entanglement detectors that follow from the theorems of §III actually hold for the entire class of Schur-concave measures, we will continue to focus on the quasi-entropic class as the most suitable for information theoretical applications.",
"Scalar entanglement detectors based on Shannon, Tsallis, and Rényi entropies are already well known, albeit with detection bounds that may differ from those reported here [7], [8], [5].",
"Our primary purpose here is to emphasize the natural and categorical manner in which majorization based entanglement detectors yield entire classes of scalar detectors.",
"With $G(\\cdot )$ a quasi-entropic function, the aforementioned monotonicity property implies the following corollaries to Theorems 1-3.",
"Corollary 1.",
"Under the conditions of Theorem 1, the multipartite state ${\\hat{\\sigma }}^{ABC \\dots F}$ is entangled if $G[{{P}}^{\\mathrm {X}}({\\hat{\\sigma }}^{ABC \\dots F})] < G[{{P}}_{sep;sup}^{\\mathrm {X}}].",
"$ Corollary 2.",
"Under the conditions of Theorem 2, the bipartite state ${\\hat{\\sigma }}^{AB}$ is entangled if $G[{P}^{ ({\\mathrm {X}}^{A}\\otimes {\\mathrm {X}}^{B})}({\\hat{\\sigma }}^{AB})]+ G[{P}^{ ({\\mathrm {Y}}^{A}\\otimes {\\mathrm {Y}}^{B})} ({\\hat{\\sigma }}^{AB})] < G[{P}_{sup}^{ {\\mathrm {X}}\\oplus {\\mathrm {Y}}}].",
"$ Corollary 3.",
"Under the conditions of Theorem 3, the multipartite state ${\\hat{\\sigma }}^{ABC \\dots F}$ is entangled if $G[{\\lambda }^{ABC \\dots F}] < G[{\\Lambda }^{ABC \\dots F}].",
"$ As an application of the above corollaries, we will consider the entanglement detection threshold for the bipartite Werner state considered in Eq.",
"(REF ) et seq.",
"Using Corollary 1 together with the Tsallis entropy function, we find that ${\\hat{\\rho }}^{wer}_{d}(q)$ is entangled if ${S}^{tsa}_{r}\\big [ {{P}}^{{\\mathrm {X}}^{\\star }}[{\\hat{\\rho }}^{wer}_{d}(q)] \\big ] < {S}^{tsa}_{r}\\big [ {{P}}_{sep;sup}^{{\\mathrm {X}}^{\\star }} \\big ], $ where ${S}^{tsa}_{r}(\\cdot )$ is the Tsallis entropy of order $r$ , and the two probability vectors appearing in Eq.",
"(REF ) are those in Eqs.",
"(REF ) and (REF ).",
"The Tsallis entropy function ${S}^{tsa}_{r}(\\cdot )$ corresponds to the choice $g(\\lambda )= (\\lambda -{\\lambda }^{r})/(r-1)$ , $1<r<\\infty $ , in Eq.",
"(REF ).",
"The Tsallis entropy of order 1 is defined by continuity and equals the Shannon entropy [25].",
"A straightforward calculation turns Eq.",
"(REF ) into $&\\frac{1-{[q+(1-q)/{d}^{2})]}^{r}-({d}^{2}-1){[(1-q)/{d}^{2})]}^{r}}{r-1} \\nonumber \\\\< &\\frac{1-{d}^{1-r}}{r-1}.",
"$ For each value of $r$ , Eq.",
"(REF ) yields an entanglement detection threshold value for $q$ which depends on $d$ , and for a fixed value of the latter, decreases with increasing $r$ .",
"This corresponds to improving detection performance with increasing $r$ , a behavior which is known for $d$ equal to 2 and 3 [8].",
"Figure REF shows a plot of the threshold value $q$ versus the dimension $d$ for four different values of the order $r$ , respectively increasing from top to bottom.",
"The improvement in entanglement detection for fixed $d$ and increasing $r$ , as well as fixed $r$ and increasing $d$ , is clearly in evidence in Fig.",
"REF .",
"Figure: Entanglement detection threshold qq for a bipartite Werner state versus dimension dd of each party, using Corollary 1 and the Tsallis entropy.",
"The order of the Tsallis entropy equals, from top to bottom respectively, 1(red), 2 (magenta), 5 (turquoise), and ∞\\infty (blue).",
"The top graph corresponds to the Shannon entropy and the bottom one (d) reproduces the known entanglement threshold for the state.The weakest performance obtains for $r=1$ , top graph in Fig.",
"1, and can be found by taking the limit of Eq.",
"(REF ) as $r \\rightarrow 1$ .",
"The result is $H[q+(1-q)/{d}^{2}])+ ({d}^{2}-1)H[q+(1-q)/{d}^{2}] < \\ln (d)$ , which corresponds to the choice of Shannon entropy function $H(\\cdot )$ together with the set of generalized Bell states for the measurement of $d \\otimes d$ Werner states [26].",
"The best performance obtains for $r \\rightarrow \\infty $ in Eq.",
"(REF ), which simplifies to $q > 1/(1+d)$ , the known entanglement threshold for a the two-qudit Werner state.",
"This limiting case corresponds to the bottom graph in Fig.",
"REF .",
"The foregoing discussion of how well Corollary 1 does in detecting entanglement notwithstanding, the aim of this section is not to promote quasi-entropic detectors given in Corollaries 1-3 per se, but rather to emphasize the reach of the majorization results of §III whence they are inherited.",
"In addition, the above example highlights the fact that, while the entanglement detection bounds given in Corollaries 1-3 are optimal for Schur-concave functions as a class, their performance need not be optimal for individual members of the class, a fact that was emphasized in paper I as well."
],
[
"Concluding Remarks",
"In this paper we have developed entanglement detection criteria within the majorization formulation of uncertainty presented in paper I.",
"These are inherently stronger than similar scalar conditions in the sense that they are equivalent to and imply infinite classes of such scalar criteria.",
"Majorization relations in effect envelope the set of scalar measures of disorder based on Schur-concave functions, a huge set that includes the information-theoretically relevant class of quasi-entropic measures as a subset.",
"This enveloping property elucidates the exceptional effectiveness of majorization relations in dealing with problems of quantum information theory.",
"Entanglement detection criteria that can be experimentally implemented are especially useful in studies involving entangled microsystems, hence the importance of linear detectors and entanglement witnesses.",
"As already mentioned, Theorem 1 can be formulated as an entanglement witness inasmuch as it is a linear majorization condition on measurement results.",
"What may not be so obvious is that in principle the spectrum of a state is also experimentally accessible on the basis of the Lemma of §IIIA.",
"That Lemma states that the spectrum of a quantum state is the supremum of probability vectors that may be deduced from the counter statistics of all possible rank 1 generalized measurements performed on the state.",
"In practice, this provides an experimental method for estimating the spectrum, since a high precision determination may require a search over a large number of possible measurements.",
"There is already an extensive literature on entanglement detection with many important results.",
"With the notable exception of the Nielsen-Kempe theorem, these results are not formulated within the majorization framework.",
"The present contribution in part serves to demonstrate the effectiveness of the majorization formulation of some of the existing methods.",
"One can reasonably expect a sharpening of the results of some of the other entanglement detection strategies when reformulated within the majorization framework.",
"It is also not unreasonable to expect useful, albeit computationally intractable, necessary and sufficient separability criteria to emerge from such formulations.",
"This work was in part supported by a grant from California State University, Sacramento.",
"*"
],
[
"majorization bounds are found on pure states",
"Our task here is to establish that majorization uncertainty bounds can be reached on pure states.",
"We will prove this for two important cases, first for the uncertainty bound on all states of the system, and second on the class of separable states.",
"To avoid unnecessary clutter and technical distractions, we will limit the number of generalized measurements to three and the underlying Hilbert spaces to finite dimensions in the following treatment.",
"Suppose generalized measurements ($\\mathrm {X}, \\lbrace \\hat{\\mathrm {E}}_{\\alpha }^{X} \\rbrace $ ), ($\\mathrm {Y}, \\lbrace \\hat{\\mathrm {E}}_{\\beta }^{Y} \\rbrace $ ), and ($\\mathrm {Z}, \\lbrace \\hat{\\mathrm {E}}_{\\beta }^{Z} \\rbrace $ ) are performed on a state $\\hat{\\rho }$ that is supported on a finite-dimensional Hilbert space.",
"What we will show below is that the search for the components of ${\\mu }^{sup}$ which serve to define ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}$ can be limited to pure states.",
"By definition, ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}={\\sup }_{\\hat{\\rho }}[{{P}}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })], $ where ${{P}}_{\\alpha \\beta \\gamma }^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })=\\textrm {tr}({\\hat{\\mathrm {E}}}_{\\alpha }^{X} \\hat{\\rho }) \\textrm {tr}({\\hat{\\mathrm {E}}}_{\\beta }^{Y} \\hat{\\rho }) \\textrm {tr}({\\hat{\\mathrm {E}}}_{\\gamma }^{Z} \\hat{\\rho }).",
"$ We recall from §IIA above and §IIB, IVA and IVB of Paper I that the supremum defined by Eqs.",
"(REF ) and (REF ) is found by calculating ${\\mu }^{sup}_{i}$ for $i=1,2,\\ldots $ (${\\mu }^{sup}_{0}=0$ by definition).",
"Furthermore, ${\\mu }^{sup}_{1}$ is the maximum value of a single component of ${{P}}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })$ as $\\hat{\\rho }$ is varied, ${\\mu }^{sup}_{2}$ is the maximum value of the sum of two (different) components of ${{P}}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })$ as $\\hat{\\rho }$ is varied, and ${\\mu }^{sup}_{i}$ is the maximum value of the sum of $i$ (different) components of ${{P}}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\mathrm {Z}}(\\hat{\\rho })$ as $\\hat{\\rho }$ is varied.",
"It is thus sufficient to prove that the said maximum for ${\\mu }^{sup}_{i}$ is in fact realized on a pure state.",
"Let $\\hat{\\rho }={\\sum }_{a}\\, {\\lambda }_{a}(\\hat{\\rho }) \\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{a} \\mid $ be the principal ensemble representation for $\\hat{\\rho }$ .",
"Then the above maximization process defines ${\\mu }^{sup}_{i}$ as the maximum of ${\\sum }_{a,b,c}{\\lambda }_{a}(\\hat{\\rho }){\\lambda }_{b}(\\hat{\\rho }){\\lambda }_{c}(\\hat{\\rho }) \\big [ \\langle {\\psi }_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{1}}^{X}\\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{1}}^{Y}\\mid {\\psi }_{b} \\rangle \\nonumber \\\\ \\times \\langle {\\psi }_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{1}}^{Z}\\mid {\\psi }_{c} \\rangle + \\langle {\\psi }_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{2}}^{X}\\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{2}}^{Y}\\mid {\\psi }_{b} \\rangle \\nonumber \\\\ \\times \\langle {\\psi }_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{2}}^{Z}\\mid {\\psi }_{c} \\rangle + \\ldots \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\nonumber \\\\+ \\langle {\\psi }_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{i}}^{X}\\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{i}}^{Y}\\mid {\\psi }_{b} \\rangle \\langle {\\psi }_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{i}}^{Z}\\mid {\\psi }_{c} \\rangle \\big ] \\,\\,\\,\\,\\,\\, \\nonumber \\\\+\\,\\, {\\xi }_{i} \\big (1-{\\sum }_{a} {\\lambda }^{\\rho }_{a}\\big )+{\\sum }_{a,b} {\\eta }_{i,ab} \\big ({\\delta }_{ab}-\\langle {\\psi }_{b}\\mid {\\psi }_{a} \\rangle \\big ), $ as the state vectors $\\lbrace \\mid {\\psi }_{a}\\rangle \\rbrace $ which are the eigenstates of $\\hat{\\rho }$ , the probabilities $\\lbrace {\\lambda }_{a}(\\hat{\\rho }) \\rbrace $ which constitute the spectrum of $\\hat{\\rho }$ , and the Lagrange multipliers ${\\xi }_{i}$ and ${\\eta }_{i,ba}$ which serve to enforce the structure of ${\\sum }_{a}\\, {\\lambda }^{\\rho }_{a} \\mid {\\psi }_{a} \\rangle \\langle {\\psi }_{a} \\mid $ as an orthogonal ensemble are varied.",
"The symbol ${\\delta }_{ab}$ in the above expression is the Kronecker delta, and the three sets of indices $\\lbrace {({\\alpha }, {\\beta }, {\\gamma })}_{1}, {({\\alpha }, {\\beta }, {\\gamma })}_{2}, \\dots , {({\\alpha }, {\\beta }, {\\gamma })}_{i} \\rbrace $ are understood to be distinct, i.e., ${\\alpha }_{1} \\ne {\\alpha }_{2} \\ne \\ldots \\ne {\\alpha }_{i}$ , and similarly for the other two sets.",
"Note also that since $\\langle {\\psi }_{a}\\mid {\\psi }_{b} \\rangle $ is in general a Hermitian matrix (in the indices $a$ and $b$ ), the Lagrange multipliers ${\\eta }_{i,ba}$ may also be taken to constitute a Hermitian matrix.",
"A variation with respect to $\\langle {\\psi }_{a} \\mid $ gives ${\\lambda }_{a}(\\hat{\\rho }) \\big [ {\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z} \\big ]\\mid {\\psi }_{a} \\rangle ={\\sum }_{b} {\\eta }_{i,ba} \\mid {\\psi }_{b} \\rangle , $ where ${\\hat{\\mathcal {E}}}_{i}^{X}={\\sum }_{k=1}^{i} {P}^{\\mathrm {Y}}_{{\\beta }_{k}}(\\hat{\\rho }){P}^{\\mathrm {Z}}_{{\\gamma }_{k}}(\\hat{\\rho }){\\hat{\\mathrm {E}}}_{{\\alpha }_{k}}^{X}, $ and ${\\hat{\\mathcal {E}}}_{i}^{Z}$ and ${\\hat{\\mathcal {E}}}_{i}^{Y}$ are defined similarly.",
"Note that the three operators introduced in Eqs.",
"(REF ) and (REF ) are Hermitian and positive.",
"Let ${\\hat{\\mathcal {E}}}_{i}={\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z}$ and ${\\mathcal {E}}_{i,aa}=\\langle {\\psi }_{a} \\mid {\\hat{\\mathcal {E}}}_{i} \\mid {\\psi }_{a} \\rangle $ .",
"Then the maximum sought in (REF ), namely ${\\mu }^{sup}_{i}$ , is given by ${\\sum }_{a}{\\lambda }_{a}(\\hat{\\rho }){\\mathcal {E}}_{i,aa}/3=\\textrm {tr}[{\\hat{\\mathcal {E}}}_{i}\\hat{\\rho }]/3$ .",
"A key fact at this juncture is that the diagonal elements ${\\mathcal {E}}_{i,aa}$ do not depend on $a$ and are all equal.",
"In other words, each pure state $\\mid {\\psi }_{a} \\rangle $ in the ensemble contributes equally to ${\\mu }^{sup}_{i}$ .",
"This property follows from a variation of (REF ) with respect to ${\\lambda }_{a}(\\hat{\\rho })$ , which leads to ${\\xi }_{i}=\\langle {\\psi }_{a} \\mid {\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z} \\mid {\\psi }_{a} \\rangle ={\\mathcal {E}}_{i,aa}.",
"$ Clearly then, each pure state $\\mid {\\psi }_{a} \\rangle $ in the ensemble must realize the same maximum ${\\mu }^{sup}_{i}$ as the entire ensemble $\\hat{\\rho }$ .",
"We conclude therefore that the sought maximum can be found among the pure states of the system.",
"What if we are looking for ${\\mu }^{sup}_{i}$ but with $\\hat{\\rho }$ limited to separable states?",
"It turns out that here too the search can be limited to pure states, which would be pure product states in this instance.",
"To establish this result, we will modify the foregoing analysis by stipulating that $\\hat{\\rho }={\\sum }_{a}\\, {q}_{a} \\mid {\\phi }^{A}_{a} \\rangle \\langle {\\phi }^{A}_{a} \\mid \\otimes \\mid {\\phi }^{B}_{a} \\rangle \\langle {\\phi }^{B}_{a} \\mid $ representing a separable, bipartite state of parties $A$ and $B$ .",
"Then the corresponding ${\\mu }^{sup}_{sep,i}$ is the maximum of ${\\sum }_{a,b,c}{q}_{a}{q}_{b}{q}_{c} \\big [\\langle {\\phi }^{B}_{a}\\mid \\otimes \\langle {\\phi }^{A}_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{1}}^{X}\\mid {\\phi }^{A}_{a} \\rangle \\otimes \\mid {\\phi }^{B}_{a} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{b}\\mid \\otimes \\langle {\\phi }^{A}_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{1}}^{Y}\\mid {\\phi }^{A}_{b} \\rangle \\otimes \\mid {\\phi }^{B}_{b} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{c}\\mid \\otimes \\langle {\\phi }^{A}_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{1}}^{Z}\\mid {\\phi }^{A}_{c} \\rangle \\otimes \\mid {\\phi }^{B}_{c} \\rangle \\nonumber \\\\+ \\langle {\\phi }^{B}_{a}\\mid \\otimes \\langle {\\phi }^{A}_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{2}}^{X}\\mid {\\phi }^{A}_{a} \\rangle \\otimes \\mid {\\phi }^{B}_{a} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{b}\\mid \\otimes \\langle {\\phi }^{A}_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{2}}^{Y}\\mid {\\phi }^{A}_{b} \\rangle \\otimes \\mid {\\phi }^{B}_{b} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{c}\\mid \\otimes \\langle {\\phi }^{A}_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{2}}^{Z}\\mid {\\phi }^{A}_{c} \\rangle \\otimes \\mid {\\phi }^{B}_{c} \\rangle \\nonumber \\\\+ \\ldots \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\, \\nonumber \\\\+ \\langle {\\phi }^{B}_{a}\\mid \\otimes \\langle {\\phi }^{A}_{a} \\mid {\\hat{\\mathrm {E}}}_{{\\alpha }_{i}}^{X}\\mid {\\phi }^{A}_{a} \\rangle \\otimes \\mid {\\phi }^{B}_{a} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{b}\\mid \\otimes \\langle {\\phi }^{A}_{b} \\mid {\\hat{\\mathrm {E}}}_{{\\beta }_{i}}^{Y}\\mid {\\phi }^{A}_{b} \\rangle \\otimes \\mid {\\phi }^{B}_{b} \\rangle \\nonumber \\\\\\times \\langle {\\phi }^{B}_{c}\\mid \\otimes \\langle {\\phi }^{A}_{c} \\mid {\\hat{\\mathrm {E}}}_{{\\gamma }_{i}}^{Z}\\mid {\\phi }^{A}_{c} \\rangle \\otimes \\mid {\\phi }^{B}_{c} \\rangle \\nonumber \\\\+ {\\sum }_{a} {\\eta }_{i,a}^{A} \\big [ 1-\\langle {\\phi }^{A}_{a}\\mid {\\phi }^{A}_{a} \\rangle ]+{\\sum }_{a} {\\eta }_{i,a}^{B} \\big [ 1-\\langle {\\phi }^{B}_{a}\\mid {\\phi }^{B}_{a} \\rangle ] \\nonumber \\\\+ {\\xi }_{i} \\big (1-{\\sum }_{a} {q}_{a}\\big ), \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, $ as the state vectors $ \\lbrace \\mid {\\phi }^{A}_{a} \\rangle \\rbrace $ , $\\lbrace \\mid {\\phi }^{B}_{a} \\rangle \\rbrace $ , the probabilities $\\lbrace {q}_{a} \\rbrace $ , and the Lagrange multipliers ${\\eta }_{i,a}^{A}$ , ${\\eta }_{i,a}^{B}$ , and ${\\xi }_{i}$ which serve to enforce normalization conditions and probability conservation for the ensemble are varied.",
"Variations with respect to $ \\langle {\\phi }^{A}_{a} \\mid $ and $ \\langle {\\phi }^{B}_{a} \\mid $ now give ${q}_{a} \\big [ \\langle {\\phi }^{B}_{a}\\mid ({\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z}) \\mid {\\phi }^{B}_{a} \\rangle \\big ] \\mid {\\phi }^{A}_{a} \\rangle &={\\eta }_{i,a}^{A} \\mid {\\phi }^{A}_{a} \\rangle \\nonumber \\\\{q}_{a} \\big [ \\langle {\\phi }^{A}_{a}\\mid ({\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z}) \\mid {\\phi }^{A}_{a} \\rangle \\big ] \\mid {\\phi }^{B}_{a} \\rangle &={\\eta }_{i,a}^{B} \\mid {\\phi }^{B}_{a} \\rangle , $ where $({\\hat{\\mathcal {E}}}_{i}^{X},{\\hat{\\mathcal {E}}}_{i}^{Y},{\\hat{\\mathcal {E}}}_{i}^{Z})$ are defined as in Eq.",
"(REF ) et seq.",
"Equations (REF ) directly imply that ${\\eta }_{i,a}^{A}={\\eta }_{i,a}^{B}$ , and as in the general case above, the maximum sought in (REF ), namely ${\\mu }^{sup}_{sep,i}$ , is found to equal ${\\sum }_{a}{q}_{a} {\\mathcal {E}}_{i,aa}/3=\\textrm {tr}[{\\hat{\\mathcal {E}}}_{i}\\hat{\\rho }]/3$ , where ${\\hat{\\mathcal {E}}}_{i}={\\hat{\\mathcal {E}}}_{i}^{X}+ {\\hat{\\mathcal {E}}}_{i}^{Y}+{\\hat{\\mathcal {E}}}_{i}^{Z}$ .",
"Furthermore, a variation with respect to the probabilities $\\lbrace {q}_{a} \\rbrace $ shows the equality of the matrix elements ${\\mathcal {E}}_{i,aa}$ just as in the general case above, whereupon we learn that each pure product state in the ensemble makes the same contribution to ${\\mu }^{sup}_{sep,i}$ .",
"In other words, the separable density matrix which maximizes (REF ) may be taken to be a pure product state [16].",
"It should be clear from the above arguments that the results hold for any number of generalized measurements as well as any number of parties in the multipartite state.",
"In summary, then, we have found that the joint majorization uncertainty bound, ${{P}}_{sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ , for a set of generalized measurements performed on an arbitrary quantum state supported on a finite-dimensional Hilbert space can be found on the class of pure states of the system, and in the case of a separable states, ${{P}}_{sep;sup}^{\\mathrm {X} \\oplus \\mathrm {Y} \\oplus \\ldots \\oplus \\mathrm {Z}}$ can be found on the class of pure product states of the system."
]
] | 1204.1289 |
[
[
"New Lower Bounds for Matching Vector Codes"
],
[
"Abstract A Matching Vector (MV) family modulo $m$ is a pair of ordered lists $U=(u_1,...,u_t)$ and $V=(v_1,...,v_t)$ where $u_i,v_j \\in \\mathbb{Z}_m^n$ with the following inner product pattern: for any $i$, $< u_i,v_i>=0$, and for any $i \\ne j$, $< u_i,v_j> \\ne 0$.",
"A MV family is called $q$-restricted if inner products $< u_i,v_j>$ take at most $q$ different values.",
"Our interest in MV families stems from their recent application in the construction of sub-exponential locally decodable codes (LDCs).",
"There, $q$-restricted MV families are used to construct LDCs with $q$ queries, and there is special interest in the regime where $q$ is constant.",
"When $m$ is a prime it is known that such constructions yield codes with exponential block length.",
"However, for composite $m$ the behaviour is dramatically different.",
"A recent work by Efremenko [STOC 2009] (based on an approach initiated by Yekhanin [JACM 2008]) gives the first sub-exponential LDC with constant queries.",
"It is based on a construction of a MV family of super-polynomial size by Grolmusz [Combinatorica 2000] modulo composite $m$.",
"In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families.",
"When $q$ is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length.",
"When the modulus $m$ is constant (as it is in the construction of Efremenko) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over $\\mathbb{Z}_m$."
],
[
"Introduction",
"A Matching Vector Family (MV Family) is a combinatorial object that arises in several contexts including Ramsey graphs, weak representation of OR polynomials and recently in constant query locally decodable codes (LDCs).",
"It is defined by two ordered lists $U=\\left(u_1, \\cdots u_t\\right)$ and $V=\\left(v_1, \\cdots v_t\\right)$ where $u_i, v_j \\in \\mathbb {Z}_m^n$ and $m$ and $n$ are integers greater than 1.",
"The property that the two lists have to satisfy is the following: for all $i\\in [t]$ , $\\langle u_i, v_i \\rangle = 0\\ (mod\\ m)$ whereas for all $i \\ne j\\in [t]$ , $\\langle u_i, v_j \\rangle \\ne 0\\ (mod\\ m)$ .",
"By $\\langle \\cdot , \\cdot \\rangle $ we denote the standard inner product.",
"Let us call this the standard definition of a MV family.",
"If in addition, all the inner products $\\langle u_i, v_j \\rangle \\ (mod\\ m)$ lie in a set of size $q$ , then it is called a $q-restricted$ MV family.",
"Note that $q=m$ corresponds to the standard MV family.",
"The size of the MV family is $t$ , the number of vectors in the list.",
"In this paper, we shall prove upper bounds on $q-restricted$ MV families in the first part and on standard MV families in the later part.",
"Let $\\mathbf {MV}(m,n)$ denote the largest $t$ such that there exists a MV family of size $t$ in $\\mathbb {Z}_m^n$ .",
"Analogously, let $\\mathbf {MV}(m,n,q)$ denote the largest $t$ such that there exists a $q-restricted$ MV family of size $t$ in $\\mathbb {Z}_m^n$ .",
"The question of bounding $\\mathbf {MV}(m,n)$ (or $\\mathbf {MV}(m,n,q)$ ) is closely related to the well-known combinatorial problem of set systems with restricted modular intersections [2], [22], [12], [13] (in this setting the vectors $u_i,v_i$ are required to have entries that are either 0 or 1).",
"The systematic study of this more general problem, in the context of MV codes, was initiated in [8].",
"The setting of prime $m$ is well understood.",
"For large prime $m=p$ , it is known that $\\mathbf {MV}(p,n)=O\\left(p^{n/2}\\right)$ [8].",
"Infact, this is almost tight.",
"When $m$ is a small prime, again we have a tight upper bound of $O\\left(n^{p-1}\\right)$ [2].",
"Surprisingly, the setting of small composite $m$ leads to very useful constructions of Ramsey graphs and constant query LDCs.",
"This is due to a construction of MV family over $Z_6$ by Grolmusz [12] of superpolynomial size in contrast to a polynomial upper bound when $m$ is a small prime.",
"Thus, it is interesting to study the behavior of MV families for small composite $m$ , and more generally arbitrary general composites.",
"We will see later the connection between upper bounds on $\\mathbf {MV}(m,n,q)$ and lower bounds on the encoding lengths of MV Codes (a family of LDCs).",
"For general $m$ , the best upper bound known [8] is $\\mathbf {MV}(m,n) \\le m^{n-1 + o_m(1)},$ with $o_m(1)$ denoting a function that goes to zero when $m$ grows.",
"It was conjectured in [8] that an upper bound of $ \\sim m^{n/2}$ should hold for any $m$ (not just prime).",
"This would be tight for large $m$ as there are constructions of MV families almost meeting this bound [28].",
"However, the proof method used in [8] to prove the $O\\left(p^{n/2}\\right)$ bound does not extend to non primes.",
"In this work, we prove the conjecture for $q-restricted$ MV families in $\\mathbb {Z}_m^n$ , for any $m$ as long as $q= \\frac{o(n)\\log m}{\\log \\left(o(n)\\log m\\right)}$ (See Theorem REF ).",
"When $m=p$ is a fixed small prime, it follows from [2] that $\\mathbf {MV}(p,n) = O\\left(n^{p-1}\\right)$ .",
"On the other hand, when $m$ is a fixed composite, say $m=6$ , there exists a MV family of superpolynomial size $\\Omega \\left(\\exp \\left(\\log ^2n /\\log \\log n\\right)\\right)$ ) [12].",
"We prove a stronger upper bound on $\\mathbf {MV}(m,n)$ , compared to Theorem REF in such a case assuming a well known conjecture in additive combinatorics (see Theorem REF ).",
"Table REF lists the known and new upper bounds on MV families.",
"Table: List of upper bounds on 𝐌𝐕(m,n)\\mathbf {MV}(m,n), 𝐌𝐕(m,n,q)\\mathbf {MV}(m,n,q)Theorem 1 For all $m\\ge 2, n \\ge 1$ we have $ \\mathbf {MV}(m,n,q) \\le q^{O(q\\log q)}m^{n/2} $ Hence, Theorem REF resolves the conjecture of [8] for any $m$ and for $q = \\frac{o(n)\\log m}{\\log \\left(o(n)\\log m\\right)}$ .",
"When $m >> n$ , our bound is quite close to the best known construction of MV families which gives $\\mathbf {MV}(m,n) \\ge \\left( \\frac{m+1}{n-2} \\right)^{n/2 - 1}$ [28].",
"Our second result assumes the polynomial Freiman-Ruzsa conjecture (PFR) conjecture (discussed below) and gives a stronger upper bound on the size of MV families when $m$ is a constant and $n$ grows.",
"Before we state the conjecture, we need to define what a difference set is.",
"For an abelian group $G$ let $A \\subseteq G$ .",
"Then the difference set $A-A=\\lbrace a_1-a_2:a_1,a_2 \\in A\\rbrace $ Conjecture 1 (PFR Conjecture in $\\mathbb {Z}_m^n$ ) Suppose $A \\subseteq \\mathbb {Z}_m^n$ and $|A-A|\\le \\lambda \\cdot |A|$ .",
"Then one can find a subgroup $H$ of size at most $|A|$ such that $A$ can be covered by $\\lambda ^{\\prime } = \\lambda ^{c_m}$ many translates of $H$ , where $c_m$ depends only on $m$ .",
"We note that the PFR conjecture has already found several applications in computer science.",
"Ben-Sasson and Zewi [6] used it to construct two-source extractors from affine extractors; and Ben-Sasson, Lovett and Zewi [5] used it to bound the deterministic communication complexity of functions whose corresponding matrix has low rank.",
"Our work provides another application for the PFR and demonstrates its wide-reaching applicability.",
"We further note that a quasi-polynomial version of the PFR conjecture was recently proved by Sanders [21] (see also the exposition in [18]).",
"Unfortunately, all the applications discussed above require the truly polynomial version of the conjecture, and so cannot apply to Sanders' result.",
"We now state the second theorem.",
"Theorem 2 Assuming the PFR conjecture over $\\mathbb {Z}_m^n$ (Conjecture REF ) we have $\\mathbf {MV}(m,n) \\le \\exp \\left( c(m)\\frac{n}{\\log n} \\right), $ with $c(m)$ an explicit function of $m$ .",
"From a technical point of view, one of the ingredients in this work builds on the recent work of Ben-Sasson, Lovett and Zewi [5] who used the PFR conjecture to show that matrices over $\\mathbb {Z}_2$ with large bias (say, with many more ones than zeros) and small rank must contain a large monochromatic sub-matrix.",
"An important ingredient in our proof is a generalization of their results from $\\mathbb {Z}_2$ to $\\mathbb {Z}_m$ for all $m$ , not necessarily prime.",
"We note however that this is just one ingredient in our overall proof."
],
[
"Lower Bounds on LDCs: Motivation for MV Family",
"Locally Decodable Codes (LDCs) are a special kind of Error Correcting Codes (ECCs) that allow the receiver to decode a single symbol of the message by querying a small number of positions in a corrupted encoding.",
"More formally, an $(q,\\delta ,\\epsilon )$ -LDC encodes $K$ -symbol messages $x$ to $N$ -symbol codewords $C(x),$ such that for every $i\\in [K],$ the symbol $x_i$ can be recovered with probability $1-\\epsilon ,$ by a randomized decoding procedure that makes only $q$ queries, even if the codeword $C(x)$ is corrupted in up to $\\delta N$ locations.",
"Since the early 90's, LDC's have found exciting applications in various areas ranging from data transmission to complexity theory to cryptography/privacy.",
"We refer the reader to [23], [27] for more background.",
"A central research question, which is far from being solved, has to do with understanding the best possible `stretch' of an LDC with a constant number of queries.",
"That is, how large $N$ has to be as a function of $K$ for constant $q$ and with constant $\\delta ,{\\varepsilon }$ (these two last parameters are not our focus here and we will generally assume them to be small fixed constants).",
"For $q=1,2$ this question is completely answered.",
"There are no LDC's for $r=1$ [16] and the best LDC's with $q=2$ have exponential encoding length [10], [15].",
"For $q > 2$ there are huge gaps in our understanding.",
"Katz and Trevisan were the first to study this problem [16] and, today, the best general lower bounds on $N$ are slightly super-linear bounds of the form $\\tilde{\\Omega }\\left(K^{1+1/\\left(\\lceil r/2\\rceil -1\\right)}\\right)$ [24].",
"Notice that, when the number of queries is 3 or 4, these bounds are quadratic (see also [15], [25] for the $q=3,4$ case).",
"The upper bounds were, until recently, those coming from polynomial codes and were of the order of $N \\le \\exp \\left(K^{\\frac{1}{q-1}}\\right)$ .",
"Improved upper bounds, breaking this barrier slightly, were given in [3].",
"This state of affairs changed dramatically when, in a breakthrough paper, Yekhanin [26] developed a new approach for constructing LDCs, called MV codes, that have much shorter codeword length than polynomial codes.",
"Efremenko [9] was the first to show that this approach could yield codes with subexponential encoding length (Yekhanin's paper showed this under a number theoretic assumption).",
"More refinements and improvements to this new framework were obtained [20], [17], [14], [19], [8], [1] to give LDC's with $q$ queries and with encoding length that grows, when $q$ is a constant, roughly like $N \\sim \\exp \\exp \\left((\\log K)^{O(1/\\log q)}(\\log \\log K)^{1-1/\\log q}\\right).$ While significantly smaller than the length of polynomial codes, the codeword length of these new codes is still super polynomial in $K$ .",
"The most general setting of parameters was addressed in [8] where the authors had given a black box construction of $q$ query MV codes using $q-restricted$ MV families in $\\mathbb {Z}_m^n$ .",
"Using the standard definition of MV families, this implied $m$ query MV codes using MV families in $Z_m^n$ .",
"In this basic, yet general reduction, it was shown that upper bounds on MV families would lead to lower bounds on the encoding length of MV codes.",
"With this motivation in mind, the authors in [8] made a conjecture on the upper bound on the size of MV families which would lead to lower bounds on the encoding length of MV codes under the basic framework.",
"We note that Yekhanin in [26] used restricted MV families in $\\mathbb {Z}_p^n$ where $p$ is a very large Mersenne prime and used a specialized technique to reduce the number of queries from $p$ to 3.",
"Another instance of reduction in the number of queries from what the standard construction gives, was given by Efremenko [9] where he again used restricted MV families.",
"A certain gadget was discovered using computer search whereby the author worked in $\\mathbb {Z}_{511}$ but got down the number of queries to 3 from the basic bound of 511.",
"The following is a corollary of Theorem REF .",
"Corollary 3 For an arbitrary positive integer $m$ , consider an infinite family of $q$ -query Matching Vector code $C_n:{\\mathbb {F}}^k \\rightarrow {\\mathbb {F}}^N$ for $n \\in \\mathbb {N}$ , where $k(n)$ and $N(n)$ are growing functions of $n$ , constructed using the black box reduction from a $q$ -restricted Matching Vector Family in $\\mathbb {Z}_m^n$ ([8]).",
"For large enough $n$ , if $q=\\frac{o(n)\\log m}{\\log \\left(o(n)\\log m\\right)}$ , then $ N \\ge k^{2-o(1)}$ Specifically, if $q=O(1)$ , then $N = \\Omega \\left(k^{2}\\right)$ .",
"Next we have the following corollary from Theorem REF .",
"Corollary 4 For some arbitrary positive integer $m$ , assume the PFR conjecture over $\\mathbb {Z}_m^n$ (Conjecture REF ).",
"Consider an infinite family of $m$ -query Matching Vector code $C_n:{\\mathbb {F}}_q^k \\rightarrow {\\mathbb {F}}_q^N$ for $n \\in \\mathbb {N}$ , where $k(n)$ and $N(n)$ are growing functions of $n$ , constructed using the black box reduction from a standard Matching Vector Family in $\\mathbb {Z}_m^n$ ([8]).",
"For large enough $n$ , if $m=O(1)$ , then $ N = \\exp \\left(\\Omega _m\\left(\\log k \\ \\log \\log k\\right)\\right)$ Thus Corollary REF states that, assuming Conjecture REF , MV codes with constant number of queries must have super polynomial encoding length in the basic framework.",
"Note that we get the same bound in Efremenko's framework for 3 queries.",
"This is because the form of the superpolynomial bound is assuming a constant $m$ and applying our bound to Efremenko's work again leads to a superpolynomial bound as $m=511$ in his setting (another constant).",
"(He uses $\\mathbb {Z}_{511}$ to construct the MV family and further reduces the number of queries to 3.)",
"This essentially means that in order to construct polynomial length codes, one needs to construct MV families in $\\mathbb {Z}_m^n$ for non-constant $m$ and use some specialized gadget to reduce the number of queries.",
"One way is to ensure it is a $q-restricted$ (constant $q$ ) MV family.",
"This automatically ensures $q$ query decoding.",
"However, the quadratic lower bound continues to hold even in this scenario for constant $q$ .",
"To beat the quadratic lower bound for constant query MV codes, one needs to construct $q-restricted$ MV families for growing $m$ and $q=\\frac{o(n)\\log m}{\\log \\left(o(n)\\log m\\right)}$ and then develop some special gadget to get the number of queries down further from $q$ to some constant."
],
[
"Proof Overview",
"The proof of Theorem REF relies on intuitions coming from the theory of two-source extractors [7], which are functions of two variables $F(X,Y)$ such that the output of $F$ is distributed in a close-to-uniform fashion whenever the two inputs are drawn, independently, from two distributions of sufficiently high entropy.",
"Since our proof does not use two-source extractors explicitly we do not define them formally and just use them to explain the high level idea behind the proof.",
"It is a well known fact [7] that the inner product function $F(X,Y) = \\langle X,Y \\rangle $ , say over $\\mathbb {Z}_2^n \\times \\mathbb {Z}_2^n$ is a good two source extractor when the two inputs $X$ and $Y$ are both drawn uniformly from sets $S_X,S_Y \\subseteq \\mathbb {Z}_2^n$ of size larger $2^{n/2}$ .",
"This immediately suggests a connection to MV families, since, if we take $S_X = U$ and $S_Y = V$ for a MV family $U,V$ in $\\mathbb {Z}_2^n$ , we would get a completely non-uniform output (it will be zero with exponentially small probability).",
"This means that the size of $U,V$ is bounded from above by approximately $2^{n/2}$ .",
"If we try to use a similar argument over $\\mathbb {Z}_m$ we run into trouble since the inner product function modulo $m$ is not a good two source extractors for sources of size $m^{n/2}$ .",
"Take, for example, $S_X = S_Y=\\lbrace 0,2,4\\rbrace ^n \\subseteq \\mathbb {Z}_6^n$ and observe that $\\langle X,Y \\rangle $ is always divisible by 2 and so is far from being uniformly distributed over $\\mathbb {Z}_6$ .",
"It is, however, possible to show that this example is, in some sense, the only example and that, in general, we can always find a certain number of elements of either $S_X$ or $S_Y$ that `agree' modulo some factor of $m$ .",
"This observation suggests proving Theorem REF by induction on the number of factors of $m$ , which is the way we proceed.",
"The proof of Theorem REF uses a slightly different view of MV families as matrices with certain zero/non-zero pattern and small rank.",
"Specifically, for a MV family $U,V$ of size $t$ in $\\mathbb {Z}_m^n$ consider the $t \\times t$ matrix $P$ whose $(i,j)$ 'th entry is $\\langle u_i,v_j \\rangle \\mod {m}$ .",
"The definition of a MV family implies that $P$ has zeros on the diagonal and non-zeros everywhere else.",
"If $m$ was a prime, we could think of $\\mathbb {Z}_m$ as a field ${\\mathbb {F}}$ and say that, since $P$ is the inner product matrix of vectors of length $n$ over a field, it must have rank at most $n$ .",
"Conversely, every $t \\times t$ matrix over a field ${\\mathbb {F}}$ with these properties (zero on the diagonal and non-zero off the diagonal) and with rank $n$ gives a MV family of size $t$ in ${\\mathbb {F}}^n$ .",
"We can call a matrix with this pattern of zeros/non-zeros an MV matrix.",
"Thus, when $m$ is prime, the question of bounding the size of a MV family is the same as lower bounding the rank of a MV-matrixFor technical reasons, the actual proof will not be entirely using matrices and will keep the MV family in the background.",
"This is because we need to keep certain invariants throughout the proof and these are easier to define for families of vectors than for matrices..",
"When $m$ is composite, this whole approach should be re-examined since $\\mathbb {Z}_m$ is no longer a field and our familiar understanding of matrices and linear algebra over a field are no longer valid.",
"We do, however, manage to carry over this correspondence between the two problems by defining the notion of rank in a careful way (more on this issue below).",
"Assume for the purpose of this overview that the usual notion of rank and other intuitions from linear algebra are valid over $\\mathbb {Z}_m$ and let us proceed with sketching the proof of Theorem REF using the equivalent formulation as bounding (from below) the rank of a MV matrix $P$ .",
"The starting point is a generalization of a result of [5], mentioned above, from $\\mathbb {Z}_2$ to $\\mathbb {Z}_m$ .",
"We show that every matrix $P$ over $\\mathbb {Z}_m$ that is biased (i.e., its values are not distributed close to uniformly) and has low rank, contains a large monochromatic sub-matrix modulo some factor $m^{\\prime }$ of $m$.",
"The size of the sub-matrix is bounded from below by $\\sim |P|\\exp (-r^{\\prime }/\\log (r^{\\prime }))$ , where $r^{\\prime }$ is the rank of $P$ modulo $m^{\\prime }$ (this factor depends on the specific way the matrix is biased).",
"This generalizes the result of [5] which assumes $m=2$ and finds a large monochromatic sub-matrix (modulo 2).",
"We note that the sub-matrix lemma is the only component in the proof that relies on the PFR conjecture.",
"Let us refer to this result from now on as the sub-matrix lemma.",
"We can apply the sub-matrix lemma to a MV matrix $P$ since its values are far from uniform (the probability of zero is much less than $1/m$ ) and since its rank is assumed (towards a contradiction) to be low.",
"Suppose for the sake of simplicity that $m=p \\cdot q$ , with $p,q$ distinct primes (the proof for general $m$ is significantly more technical but relies on the same basic intuitions).",
"Applying the sub-matrix lemma we obtain a sub-matrix $P_1$ of $P$ that is constant modulo some factor $m_1$ of $m$ (so $m_1$ is either $p$ , $q$ or $m$ ) of size at least $|P|\\exp (-r_1/\\log (r_1))$ , where $r_1\\le n$ is the rank of $P \\mod {m}_1$ .",
"Using some matrix manipulations, and subtracting a rank one matrix, we can get a large sub-matrix $P_1^{\\prime }$ that does not intersect the diagonal of $P$ and s.t all of the entries of $P_1^{\\prime }$ are zero modulo $m_1$ .",
"Suppose $|P_1^{\\prime }| = t_1$ and consider the $2t_1 \\times 2t_1$ sub-matrix $P_1^{\\prime \\prime }$ of $P$ that has $P_1^{\\prime }$ as its top-right (or bottom-left) block and s.t the top-left and bottom-right blocks are taken to have zero diagonal elements.",
"Formally, if $P_1^{\\prime }$ is indexed by rows in $R$ and columns in $T$ with $R \\cap T = \\emptyset $ then the rows/columns of $P_1^{\\prime \\prime }$ will be indexed by $R \\cup T$ .",
"If we consider the matrix $P_1^{\\prime \\prime }$ modulo $m_1$ then it has top-right block which is all zero and so its rank (modulo $m_1$ ) will be the sum of the ranks of the top-left and bottom right blocks.",
"Thus, one of these blocks, w.l.o.g the top-left one, must have rank at most $n/2$ (over $\\mathbb {Z}_{m_1}$ ).",
"Notice also that both of these blocks are themselves MV matrices modulo $m$ since they are sub-matrices of $P$ with the same row and column sets.",
"Let $\\tilde{P}_1$ be the top-left block of $P_1^{\\prime \\prime }$ .",
"We can now apply, again, the monochromatic sub-matrix lemma to find a large sub-matrix $P_2$ of $\\tilde{P}_1$ which is constant modulo some other factor $m_2$ of $m$ .",
"The size of $P_2$ will be $t_1 \\cdot \\exp (-r_2/\\log (r_2)) = |P|\\cdot \\exp (-r_1/\\log (r_1) - r_2/\\log (r_2) ).$ The factor $m_2$ is also either $p$ or $q$ .",
"If it happens to be that $m_1=m_2$ then $r_2 \\le n/2$ and so we gain in the size of $P_2$ in this second step (the expression $r_2/\\log (r_2)$ is smaller than $n/2\\log (n/2)$ which is smaller by roughly a factor of two than our bound on $r_1/\\log (r_1)$ .",
"Suppose we continue with this iterative process of finding constant sub-matrices for $\\ell $ steps and that, by luck, all the factors $m_1,m_2, \\ldots $ are equal to the same factor of $m$ (say $p$ ).",
"Then, after roughly $\\log (n)$ iteration, we will reduce the rank modulo $p$ to one and still have at least $|P|\\cdot \\exp \\left(-\\sum _{i=1}^\\ell \\frac{n}{2^i\\log (n/2^i)}\\right)$ rows, which is close to the original size of $P$ if we assume (in contradiction) that $|P| >> \\exp ( n/\\log n)$ (this calculation is given in Claim REF ).",
"In this case we obtain a new large MV family $U^{\\prime },V^{\\prime }$ modulo $m$ such that all inner products $\\langle u_i^{\\prime },v_j^{\\prime } \\rangle $ of elements $u_i^{\\prime }\\in U^{\\prime }, v_j^{\\prime } \\in V^{\\prime }$ are fixed modulo $p$ .",
"From this we can easily construct a MV family of roughly the same size in $\\mathbb {Z}_q^n$ and then use the bounds on $\\mathbf {MV}(q,n)$ for primes to get a contradiction.",
"In the `unlucky' case we will have different factors $m_1,m_2,\\ldots $ in each stage, but we can adapt the analysis to consider the decrease in rank simultaneously for all factors of $m$ .",
"The full proof is by induction on the number of factors of $m$ and uses the iterative sub-matrix argument above to go from a MV family modulo $m$ to a MV family of roughly the same size modulo some proper factor of $m$ (and then uses the inductive hypothesis on this new MV family)."
],
[
"Matrix rank over $\\mathbb {Z}_m$",
"An important technical issue, which was already hinted at above, is in the definition of the rank of a matrix with entries in a ring $\\mathbb {Z}_m$ .",
"There are two main properties of matrix rank over a field that we relied on in the proof sketch above.",
"The first is that a rank $r$ matrix is always the inner product matrix of vectors in $r$ dimensions.",
"Equivalently, a $t \\times t$ matrix of rank $r$ can be written as a product of a $t \\times r$ matrix and an $r \\times t$ matrix.",
"This is important if we are to go back and forth between matrices and MV families.",
"Another property we used is that, if we have a $2t \\times 2t$ matrix composed of 4 blocks of size $t \\times t$ and the top-right block is zero, then the rank of the matrix is the sum of the ranks of the top-left block and the bottom right block.",
"Ideally, we would like to define rank over $\\mathbb {Z}_m$ so that both properties are satisfied.",
"This is, however, impossible as the following example shows: Consider the $2 \\times 2$ matrix with the two rows $(4,0)$ and $(0,3)$ over $\\mathbb {Z}_6$ .",
"This matrix can be written as the product of the two vectors $(2,3)^T$ and $(2,3)$ and so should have rank one, if we are to satisfy the first property.",
"However, if we are to satisfy the second property, its rank should be the sum of the ranks of the two $1 \\times 1$ matrices $(4)$ and $(3)$ , which clearly cannot have rank zero!",
"Our solution to this problem is to give two different definitions of rank, each satisfying one of the two properties.",
"We then show that the two definitions of rank can differ from each other by a multiplicative factor of $\\log m$ , which our proof can handle.",
"The first definition of rank is as the smallest $r$ such that our $t \\times t$ matrix can be written as a product of a $t \\times r$ matrix and an $r \\times t$ matrix.",
"Clearly this would satisfy the first property (but not the second).",
"The second definition of rank is termed column-rank and is defined as the logarithm to the base $m$ of the size of the additive subgroup of $\\mathbb {Z}_m^t$ generated by the columns of the matrix.",
"Notice that this definition of rank can result in the rank being non-integer.",
"For example, the rank of the matrix with a single column $(2,0)$ over $\\mathbb {Z}_6$ would be equal to $\\log _6(3)$ since the subgroup generated by this column is composed of the three vectors $(2,0),(4,0),(0,0)$ .",
"It is not hard to show (see Claim REF ) that this definition satisfies the second property described above regarding block matrices.",
"Clearly, the two definitions agree for matrices over a field.",
"We show (see Claim REF ) that the two notions of rank can differ by a multiplicative factor of at most $\\log m$ .",
"This allows us to use both definitions in different parts of the proof without losing too much in the transition.",
"We finish this discussion by noting that in no part of the proof do we use the characterization of rank using determinants, which is often very useful when working over a field."
],
[
"Organization",
"We begin with some preliminaries in Section .",
"We prove Theorem REF in Section .",
"Section contains some claims about matrices over $\\mathbb {Z}_m$ .",
"Section introduces collision free MV families.",
"Both Section and Section will be used in the proof of Theorem REF in Section .",
"The proof of Theorem REF also requires the sub-matrix lemma, whose proof appears in Section ."
],
[
"Notations:",
"Throughout the paper we will be handling ordered lists of elements.",
"A list $A$ of size $t$ over a finite set $\\Omega $ is an ordered $t$ -tuple $A=\\left(a_1,a_2,\\cdots ,a_t\\right)$ where each $a_i \\in \\Omega $ .",
"A list can have repetitions.",
"If it doesn't we say it is twin free.",
"When discussing sublists $A \\subseteq B$ with $B = (b_1,\\ldots ,b_t)$ we will use the convention that, unless specified otherwise, $A$ maintains the ordering induced by $B$ .",
"For a positive integer $t$ , we let $[t]$ denote the list $\\left(1,\\cdots t\\right)$ .",
"So, for example, when we say that $T \\subseteq [t]$ we mean that $T$ is a list of integers in increasing order belonging to $[t]$ .",
"We say that a list $A = (a_1,\\ldots ,a_t)$ over $\\Omega $ is constant if $a_i=a_j$ for all $i,j \\in [t]$ .",
"We assume all logarithms are in base 2 unless otherwise specified."
],
[
"MV Families: Basic Facts and Definitions",
"We now start with some basic definition and claims regarding MV families.",
"Definition 2.1 (Matching Vector Family) Let $U=\\left(u_1, u_2, \\cdots u_t\\right)$ and $V=\\left(v_1, v_2, \\cdots v_t\\right)$ be lists over $\\mathbb {Z}_m^n$ .",
"Then $\\left(U,V\\right)$ is called a matching vector family of size $t$ in $\\mathbb {Z}_m^n$ if $\\langle u_i,v_i \\rangle =0\\ \\left(mod\\ m\\right), \\quad \\forall i$ .",
"$\\langle u_i,v_j \\rangle \\ne 0\\ \\left(mod\\ m\\right), \\quad \\forall i \\ne j$ .",
"If in addition, we $|\\lbrace \\langle u,v \\rangle : u \\in U, v \\in V\\rbrace |=q$ , we call such a MV family an $q-restricted$ MV family.",
"We denote the size of $\\left(U,V\\right)$ by $\\left|\\left(U,V\\right)\\right|$ .",
"For instance, $\\left|\\left(U,V\\right)\\right|=t$ above.",
"Definition 2.2 (Subset of Matching Vector Family) Let $U=\\left(u_1, u_2, \\cdots u_t\\right), V=\\left(v_1, v_2, \\cdots v_t\\right)$ form a matching vector family in $\\mathbb {Z}_m^n$ of size $t$ .",
"By $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ , we mean there exists a sublist $T \\subseteq [t]$ such that $U^{\\prime }=\\left(u_i:i \\in T\\right), V^{\\prime }=\\left(v_i:i \\in T\\right)$ .",
"Observe that $\\left(U^{\\prime },V^{\\prime }\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ .",
"Definition 2.3 ($\\mathbf {MV}\\left(m,n\\right)$ ) We denote by $\\mathbf {MV}\\left(m,n\\right)$ the maximum size of a matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ .",
"Similarly, we denote by $\\mathbf {MV}\\left(m,n,q\\right)$ the maximum size of an $q-restricted$ matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ .",
"We shall use the following simple facts implicitly throughout the paper.",
"Fact 2.4 $\\mathbf {MV}\\left(m,n\\right)$ is an increasing function of $n$ .",
"For $n_1<n_2$ , we show $\\mathbf {MV}\\left(m,n_1\\right)\\le \\mathbf {MV}\\left(m,n_2\\right)$ .",
"Given $\\left(U,V\\right)$ , a matching vector family in $\\mathbb {Z}_m^{n_1}$ , we can pad each element in $U$ and $V$ by $n_2-n_1$ zeros and obtain a matching vector family in $\\mathbb {Z}_m^{n_2}$ of the same size.",
"Fact 2.5 If $\\left(U,V\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ , then $U$ and $V$ are twin free.",
"Let $U=\\left(u_1,u_2,\\cdots u_t\\right), V=\\left(v_1,v_2,\\cdots v_t\\right)$ .",
"We prove $U$ is twin free.",
"By symmetry $V$ is also twin free.",
"Suppose $u_i=u_j$ for some $i \\ne j$ .",
"Now, $\\langle u_i,v_j\\rangle =\\langle u_j,v_j\\rangle =0$ which is a contradiction.",
"To facilitate writing in the proofs to follow we introduce the following notation for taking lists, matrices, etc.",
"modulo an integer $r$ .",
"Definition 2.6 (Modulo $r$ notation) Let $2 \\le r \\le m$ be such that $r$ divides $m$ .",
"Given $a=\\left(a_1, \\cdots , a_n\\right) \\in \\mathbb {Z}_m^n$ , we denote by $a^{\\left(r\\right)}=\\left(a_1 \\pmod {r},\\cdots , a_n \\pmod {r}\\right) \\in \\mathbb {Z}_r^n$ .",
"For a list $U=\\left(u_1,u_2,\\cdots u_t\\right)$ over $\\mathbb {Z}_m^n$ , let $U^{\\left(r\\right)}=\\left(u_1^{\\left(r\\right)},u_2^{\\left(r\\right)},\\cdots u_t^{\\left(r\\right)}\\right)$ .",
"Also, if $u^{\\left(r\\right)}$ is constant for all $u \\in U$ , we say $U^{\\left(r\\right)}$ is constant.",
"Similarly, for a $t \\times t$ matrix $M$ over $\\mathbb {Z}_m$ , define $M^{\\left(r\\right)}$ to be the $t \\times t$ matrix over $\\mathbb {Z}_r$ such that $M^{\\left(r\\right)}\\left(j,k\\right)=M\\left(j,k\\right)\\pmod {r}$ for all $1 \\le j,k \\le t$ .",
"We will also need the following definitions.",
"Definition 2.7 (Bucket $B_r\\left(w,A\\right)$ ) Let $A \\subseteq \\mathbb {Z}_m^n$ be a list.",
"For any $w \\in \\mathbb {Z}_r^n$ , we denote by $B_r\\left(w,A\\right)=\\left(a \\in A:a^{\\left(r\\right)}=w\\right)$ the sub-list of elements of $A$ which are equal to $w$ modulo $r$ .",
"Definition 2.8 (Matrix $P_{U,V}$ ) Let $U=\\left(u_1, u_2, \\cdots u_t\\right)$ and $V=\\left(v_1, v_2, \\cdots v_t\\right)$ be lists over $\\mathbb {Z}_m^n$ .",
"We let $P_{U,V}$ be the $t \\times t$ matrix over $\\mathbb {Z}_m$ defined by $P_{U,V}\\left(i,j\\right)=\\langle u_i, v_j \\rangle $ for $1 \\le i,j \\le t$ .",
"We will use the following lemma from [8] mentioned informally in the introduction.",
"Lemma 2.9 [8] For any positive integer $n$ and prime $p$ , $\\mathbf {MV}\\left(p,n\\right) \\le 1+{n+p-2 \\atopwithdelims ()p-1}$ ."
],
[
"Probability Distributions",
"Definition 2.10 For a distribution $\\mu $ over a finite set $\\Omega $ , we write $X \\sim \\mu $ to denote a random variable $X$ drawn according to $\\mu $ .",
"We will also treat $\\mu $ as a function $\\mu : \\Omega \\mapsto [0,1]$ such that $\\mu (x) = \\mathbf {Pr}[ X = x]$ .",
"For a list $A$ over $\\Omega $ , $x \\sim A$ denotes a point sampled as per the uniform distribution on $A$ (taking repetitions into account).",
"Definition 2.11 (Statistical distance between distributions) Let $\\mu _1$ and $\\mu _2$ be two distributions over a finite set $\\Omega $ .",
"The statistical distance (or simply distance) between $\\mu _1$ and $\\mu _2$ , denoted $\\Delta \\left(\\mu _1,\\mu _2\\right)$ , is defined as $\\Delta \\left(\\mu _1, \\mu _2\\right)=\\frac{1}{2}\\sum _{x \\in \\Omega }\\left|\\mu _1\\left(x\\right)-\\mu _2\\left(x\\right)\\right|.",
"$ Definition 2.12 (Collision probability) Given a distribution $\\mu $ over a finite set $\\Omega $ the collision probability of $\\mu $ , denoted $\\textnormal {cp}(\\mu )$ , is defined as $\\textnormal {cp}\\left(\\mu \\right)=\\mathbf {Pr}_{x,y \\sim \\mu }[x=y]=\\sum _{x\\in \\Omega }\\mu \\left(x\\right)^2.$ The following two lemmas are standard and their proofs are included, for completeness, in Appendix .",
"Lemma 2.13 Let $\\mu $ be a distribution over $\\mathbb {Z}_m$ and let $\\mathcal {U}_m$ denote the uniform distriution over $\\mathbb {Z}_m$ .",
"If $\\Delta \\left(\\mu ,\\mathcal {U}_m\\right)\\ge \\epsilon $ then for some $1 \\le j \\le m-1$ , $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^j\\right)^{x} \\right]\\right|\\ge \\frac{2\\epsilon }{\\sqrt{m}},$ where $\\omega =exp\\left(2\\pi i/m\\right)$ is a primitive root of unity of order $m$ .",
"Lemma 2.14 Let $\\omega $ be a primitive root of unity of order $m$ .",
"Let $\\mu _1$ and $\\mu _2$ be two probability distributions over $\\mathbb {Z}_m^n$ .",
"If $ \\left|\\mathbb {E}_{x \\sim \\mu _1,y\\sim \\mu _2} \\left[\\omega ^{\\langle x,y \\rangle }\\right]\\right|\\ge \\epsilon $ , then $\\textnormal {cp}\\left(\\mu _1\\right)\\textnormal {cp}\\left(\\mu _2\\right) \\ge \\epsilon ^2/m^n$ ."
],
[
"Proof of Theorem ",
"In this section we prove Theorem REF , restated here with explicit constants.",
"Theorem 3.1 Let $m \\ge 2$ ,$2 \\le q\\le m$ and $n$ be arbitrary positive integers.",
"Then $\\mathbf {MV}\\left(m,n,q\\right)\\le 12q \\cdot q^{24\\left(1+\\log q^{10q} \\right)}m^{n/2}.$ For the purpose of the proof, we introduce the following notation that will be used only in this section.",
"Definition 3.2 ($\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right)$ ) Let $r_1,r_2$ be integers such that $r_1r_2|m$ .",
"We denote by $\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right)$ the maximum size of a $q-restricted$ MV family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ satisfying $U^{\\left(r_1\\right)}$ and $V^{\\left(r_2\\right)}$ are constants.",
"$\\langle u,v \\rangle =0 \\ \\left(mod\\ r_1r_2\\right)$ for all $u \\in U, v \\in V$ .",
"Note that $\\mathbf {MV}_{1,1}\\left(m,n,q\\right)=\\mathbf {MV}\\left(m,n,q\\right)$ (with the convention that $x \\pmod {1} =0$ for any integer $x$ ).",
"Before we go to the proof of Theorem REF , we have the following claims.",
"Claim 3.3 Let $(U,V)$ be a q-restricted matching vector in $\\mathbb {Z}_m^n$ .",
"Then, without loss of generality, $m$ has at most $q$ prime factors.",
"Assume $m=\\prod _{i=1}^r p_i^{e_i}$ with possible $r>q$ .",
"Let $v_1,\\cdots ,v_q \\in Z_m$ be the $q$ possible nonzero values that the inner products $\\langle u,v \\rangle $ attain.",
"For each $v_j$ there is some prime $p_{i_j}$ where $v_j \\ne 0 \\left( \\mod {p}_{i_j}^{e_{i_j}}\\right)$ .",
"So, we can replace $m$ with just $\\prod _{j=1}^q p_{i_j}^{e_{i_j}}$ and discard all primes other than $p_{i_1}, \\cdots ,p_{i_q}$ .",
"Claim 3.4 If $N$ has $r$ prime factors, then $|\\lbrace x \\in Z_N: order(x)<N/S\\rbrace | \\le N/S \\cdot \\left(\\log S\\right)^r$ .",
"Assume $N=\\prod _{i=1}^r p_i^{e_i}$ .",
"An element $x$ with $order(x) \\le N/S$ is divisible by some $\\prod _{i=1}^r p_i^{f_i} \\ge S$ .",
"Let $T= \\lbrace (f_1, \\cdots ,f_r): \\prod p_i^{f_i} \\ge S\\rbrace $ .",
"Define a partial order on $T$ by $(f_1,\\cdots ,f_r) \\le (f^{\\prime }_1,\\cdots ,f^{\\prime }_r)$ if $f_i \\le f^{\\prime }_i$ .",
"Let $T^{\\prime }$ be a subset of $T$ such that for any $t \\in T$ there is $t^{\\prime } \\in T^{\\prime }$ such that $t^{\\prime }\\le t$ .",
"Note that if $x$ has order $\\le N/S$ then $x$ must be divisible by $\\prod _i p_i^{f_i}$ for some $(f_1,\\cdots ,f_r)$ in $T^{\\prime }$ .",
"So, the number of elements of order $< N/S$ is at most $N|T^{\\prime }|/S$ .",
"We can bound the size of $T^{\\prime }$ as follows: any element $f_i$ is between 0 and $\\log _{p_i} S$ , since clearly if $f_i$ is larger we can reduce $f_i$ by one.",
"So, $|T^{\\prime }| \\le \\prod _{i=1}^r (\\log S / \\log p_i) <= (log S)^r$ .",
"The proof of Theorem REF will follow immediately from the following two lemmas, which will be proved below.",
"Lemma 3.5 Let $m=r_1r_2r_3$ where $r_1,r_2,r_3$ are arbitrary positive integers such that $r_3\\ge 2$ .",
"Let $q \\ge 2, t \\ge 12q$ and $n$ be arbitrary positive integers.",
"Let $\\left(U,V\\right)$ be a $q-restricted$ matching vector family in $\\mathbb {Z}_m^n$ with $\\left|\\left(U,V\\right)\\right|=t$ such that $U^{\\left(r_1\\right)}$ and $V^{\\left(r_2\\right)}$ are constants.",
"$\\langle u,v \\rangle =0 \\ \\left(mod\\ r_1r_2\\right)$ for all $u \\in U, v \\in V$ .",
"Then, there exists $s|r_3$ with $s \\ge \\max \\lbrace 2, r_3/q^{10q}\\rbrace $ and a $q-restricted$ matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ such that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right| \\ge s^{-n/2}q^{-24}t$ where $\\langle u^{\\prime },v^{\\prime } \\rangle =0 \\ \\left(mod\\ r_1r_2s\\right)$ for all $u^{\\prime } \\in U^{\\prime }, v^{\\prime } \\in V^{\\prime }$ .",
"Either $U^{\\prime \\left(r_1s\\right)}$ is constant or $V^{\\prime \\left(r_2s\\right)}$ is constant.",
"Applying Lemma REF iteratively we can prove the following bound.",
"Lemma 3.6 $\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right) \\le 12q \\cdot q^{24\\log \\frac{m}{r_1 r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}$ .",
"Given Lemma REF and Lemma REF , we now show how to deduce Theorem REF .",
"[Proof of Theorem REF ] Observe that for any matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ , $U^{\\left(1\\right)}$ and $V^{\\left(1\\right)}$ are constants and $\\langle u,v \\rangle =0 \\ \\left(mod\\ 1\\right)$ for all $u \\in U, v \\in V$ .",
"Thus, $\\mathbf {MV}\\left(m,n,q\\right)=\\mathbf {MV}_{1,1}\\left(m,n,q\\right)$ .",
"Case 1: $m \\le q^{10q}$.",
"Applying Lemma REF , we get $\\mathbf {MV}\\left(m,n,q\\right)=\\mathbf {MV}_{1,1}\\left(m,n,q\\right) \\le 12q \\cdot q^{24\\log q^{10q}}\\left(m\\right)^{n/2} \\le 12q \\cdot q^{24(1+\\log q^{10q})}\\left(m\\right)^{n/2}$ .",
"Case 2: $m > q^{10q}$.",
"By Lemma REF , we know that for $s\\ge m/q^{10q}$ , $\\mathbf {MV}_{1,s}\\left(m,n,q\\right)\\le 12q \\cdot q^{24\\log \\frac{m}{s}}\\left(\\frac{m}{s}\\right)^{n/2}\\le 12q \\cdot q^{24\\log q^{10q}}\\left(\\frac{m}{s}\\right)^{n/2}$ .",
"Similarly, we have for $s\\ge m/4q$ , $\\mathbf {MV}_{s,1}\\left(m,n,q\\right)\\le 12q \\cdot q^{24\\log q^{10q}}\\left(\\frac{m}{s}\\right)^{n/2}$ .",
"Now, suppose there is a $q-restricted$ MV family $(U,V)$ in $\\mathbb {Z}_m^n$ of size $t>12q \\cdot q^{24\\left(1+\\log q^{10q} \\right)}m^{n/2}$ .",
"Applying Lemma REF with $r_1=r_2=1$ , we get a $q-restricted$ MV family $(U^{\\prime },V^{\\prime })\\subseteq (U,V)$ of size $t^{\\prime } \\ge s^{-n/2}q^{-24}t >q^{24\\log q^{10q}}\\left(\\frac{m}{s}\\right)^{n/2}$ where $s\\ge m/q^{10q}$ such that $\\langle u^{\\prime },v^{\\prime } \\rangle =0 \\ \\left(mod\\ s\\right)$ for all $u^{\\prime } \\in U^{\\prime }, v^{\\prime } \\in V^{\\prime }$ .",
"Either $U^{\\prime \\left(s\\right)}$ is constant or $V^{\\prime \\left(s\\right)}$ is constant.",
"But, by the previous paragraph, we have for $s\\ge m/q^{10q}$ , $\\mathbf {MV}_{s,1}\\left(m,n,q\\right)$ and $\\mathbf {MV}_{1,s}\\left(m,n,q\\right)$ are at most $12q \\cdot q^{24\\log q^{10q}}\\left(\\frac{m}{s}\\right)^{n/2}$ .",
"This leads to a contadiction."
],
[
"Proof of Lemma ",
"By assumption we have that $\\langle u,v \\rangle =0 \\ \\left(mod\\ r_1r_2\\right)$ for all $u \\in U, v \\in V$ .",
"So, we can consider $\\frac{\\langle u,v \\rangle }{r_1r_2} \\in \\mathbb {Z}_{r_3}$ .",
"Also, by hypothesis, the inner products $\\frac{\\langle u,v \\rangle }{r_1r_2}$ occupy $q^{\\prime }\\le q$ residues in $\\mathbb {Z}_{r_3}$ .",
"We have that For $1 \\le i \\le t$ , $\\frac{\\langle u_i,v_i \\rangle }{r_1r_2}=0 \\ \\left(mod\\ r_3\\right)$ since $\\langle u_i,v_i \\rangle =0 \\pmod {m}$ .",
"For $1 \\le i,j \\le t$ , $i \\ne j$ , $\\frac{\\langle u_i,v_j \\rangle }{r_1r_2} \\ne 0 \\ \\left(mod\\ r_3\\right)$ since $\\langle u_i,v_j \\rangle \\ne 0 \\pmod {m}$ .",
"Let $\\mu $ denote the distribution over $\\mathbb {Z}_{r_3}$ defined by $\\frac{\\langle u_i,v_j \\rangle }{r_1r_2} \\mod {r}_3$ where $u_i, v_j$ are drawn independently and uniformly from $U,V$ respectively.",
"Case 1: $4q^{\\prime } \\ge r_3$.",
"Observe that $\\mu $ outputs 0 only when $i=j$ .",
"Therefore, $\\mathbf {Pr}[\\mu =0]=1/t \\le 1/12q^{\\prime } \\le 1/3r_3$ .",
"On the other hand, $\\mathbf {Pr}[\\mathcal {U}_{r_3}=0]=1/r_3$ .",
"This implies that $\\Delta \\left(\\mu ,\\mathcal {U}_{r_3}\\right) \\ge 1/3r_3$ .",
"Thus, applying Lemma REF with $\\omega =exp\\left(2\\pi i/r_3\\right)$ , we get that for some $1 \\le j \\le r_3-1$ , $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^j\\right)^{x} \\right]\\right|\\ge \\frac{2}{3r_3\\sqrt{r_3}} \\ge \\frac{1}{12q^{\\prime 3/2}}.$ Let $\\omega ^{\\prime }=\\omega ^j$ and $ord(\\omega ^{\\prime })$ (the order of $\\omega ^{\\prime }$ ) be $s=r_3/gcd\\left(r_3,j\\right)$ .",
"Also, note that as $j\\ge 1$ , we have $s \\ge 2$ .",
"Also, trivially, $s \\ge r_3/q^{\\prime 10q^{\\prime }} \\ge r_3/q^{10q}$ .",
"Case 2: $4q^{\\prime } <r_3$.",
"Let $X$ be the random variable that picks a random $0\\le j \\le r_3-1$ and outputs $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^j\\right)^{x} \\right]\\right|$ .",
"We will now show that with significant probability $X^2 \\ge 1/2q^{\\prime }$ .",
"First observe that $X\\le 1$ .",
"On the other hand, we will show that $E\\left[X^2\\right]$ is large.",
"To see this, let $Z=\\lbrace z_1, \\cdots z_{q^{\\prime }}\\rbrace $ be the $q^{\\prime }$ residues forming the support of $\\mu $ .",
"Also, for $1 \\le i \\le q^{\\prime }$ , let $\\alpha _i\\stackrel{def}{=}\\mu (z_i)$ .",
"Then, $\\mathbb {E}_j\\left[X^2\\right]&=&\\mathbb {E}_j\\left[\\sum _{1 \\le i,i^{\\prime } \\le q^{\\prime }}\\alpha _i \\alpha _{i^{\\prime }} \\omega ^{j\\left(z_i-z_{i^{\\prime }}\\right)}\\right]\\\\&=&\\sum _{1 \\le i \\le q^{\\prime }}\\alpha _i^2\\\\&\\ge &1/q^{\\prime }$ Therefore, we claim that $\\mathbf {Pr}[X^2\\ge 1/2q^{\\prime }]\\ge 1/2q^{\\prime } \\ge 1/2q$ .",
"If not, then $E_j\\left[X^2\\right]&=&\\mathbf {Pr}[X^2\\ge 1/2q^{\\prime }]E_j\\left[X^2|X^2 \\ge 1/2q^{\\prime }\\right]+\\mathbf {Pr}[X^2 <1/2q^{\\prime }]E_j\\left[X^2|X^2 <1/2q^{\\prime }\\right]\\\\&<& 1/2q^{\\prime } + 1/2q^{\\prime }\\\\&=&1/q^{\\prime }$ which is a contradiction.",
"By the above, we already have that there exists some $\\omega ^{\\prime }$ such that $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^{\\prime }\\right)^{x} \\right]\\right| \\ge 1/\\sqrt{2q^{\\prime }}$ and $ord\\left(\\omega ^{\\prime }\\right)\\ge 2$ since $r_3/2q^{\\prime }>1$ and thus $\\omega ^{\\prime }$ is not trivial.",
"Now, we shall show the existence of $\\omega ^{\\prime }$ of much higher order provided $r_3>q^{\\prime 10q^{\\prime }}$ .",
"By Claim REF , for $S=q^{\\prime 10q^{\\prime }}$ and $N=r_3$ , and noting that $r_3$ has atmost $q$ prime factors by Claim REF , we have $Pr_j[ord\\left(\\omega ^j\\right)\\le r_3/S]\\le 1/4q^{\\prime }$ Thus, with probabilty at least $1/2q^{\\prime }-1/4q^{\\prime }=1/4q^{\\prime }$ , a random $j$ satisfies $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^j\\right)^{x} \\right]\\right| \\ge 1/\\sqrt{2q^{\\prime }} \\ge \\frac{1}{12q^{3/2}}$ $s=ord\\left(\\omega ^j\\right) \\ge r_3/S$ Also, as $r_3/4q^{\\prime }>1$ the above two conditions are true for some $j \\ne 0$ .",
"Now, we combine the above two cases as follows.",
"Let $\\omega ^{\\prime }=\\omega ^j$ and ${\\varepsilon }= \\frac{1}{12q^{3/2}}$ .",
"We have shown by the above case-by-case analysis that $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^{\\prime }\\right)^{x} \\right]\\right|\\ge \\epsilon $ $s=ord(\\omega ^{\\prime })$ is such that $s \\ge \\max \\lbrace 2,r_3/q^{10q}\\rbrace $ Using the Cauchy-Schwartz inequality twice we get $&&\\left|\\mathbb {E}_{u \\sim U, v \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle u,v \\rangle /r_1r_2}\\right]\\right|\\ge \\epsilon \\\\&\\Rightarrow & \\left|\\mathbb {E}_{u,\\tilde{u} \\sim U, v \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle u-\\tilde{u},v \\rangle /r_1r_2}\\right]\\right|\\ge \\epsilon ^2\\\\&\\Rightarrow & \\left|\\mathbb {E}_{u,\\tilde{u} \\sim U, v,\\tilde{v} \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle u-\\tilde{u},v-\\tilde{v} \\rangle /r_1r_2}\\right]\\right|\\ge \\epsilon ^4\\\\&\\Rightarrow & \\left|\\mathbb {E}_{u,\\tilde{u} \\sim U, v,\\tilde{v} \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle \\left(u-\\tilde{u}\\right)/r_1,\\left(v-\\tilde{v}\\right)/r_2 \\rangle }\\right]\\right|\\ge \\epsilon ^4.$ We need to explain the last expression.",
"Since by assumption $U^{\\left(r_1\\right)}$ and $V^{\\left(r_2\\right)}$ are constants, $\\left(u-\\tilde{u}\\right)/r_1 \\in \\mathbb {Z}_{m}^n$ and $\\left(v-\\tilde{v}\\right)/r_2 \\in \\mathbb {Z}_{m}^n$ are well defined.",
"Thus, we can fix $\\tilde{u}$ and $\\tilde{v}$ by an averaging argument such that $\\left|\\mathbb {E}_{u \\sim U, v \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle \\left(u-\\tilde{u}\\right)/r_1,\\left(v-\\tilde{v}\\right)/r_2 \\rangle }\\right]\\right|\\ge \\epsilon ^4.$ Let $U^{\\prime }=\\left(u_1^{\\prime },u_2^{\\prime }, \\cdots u_t^{\\prime }\\right), V^{\\prime }=\\left(v_1^{\\prime },v_2^{\\prime }, \\cdots v_t^{\\prime }\\right)$ where $u_i^{\\prime }= \\left(u_i -\\tilde{u}\\right)/r_1$ and $v_i^{\\prime }= \\left(v_i -\\tilde{v}\\right)/r_2$ .",
"Notice that $U^{\\prime }$ and $V^{\\prime }$ are not assumed to be a MV family (later we will derive from them a MV family).",
"We now define two probability distributions $\\mu ^{U^{\\prime }}$ and $\\mu ^{V^{\\prime }}$ over $\\mathbb {Z}_s^n$ .",
"For each $w \\in \\mathbb {Z}_s^n$ , let $\\mu ^{U^{\\prime }}\\left(w\\right)=\\left|B_s\\left(w,U^{\\prime }\\right)\\right|/\\left|U^{\\prime }\\right|$ and $\\mu ^{V^{\\prime }}\\left(w\\right)=\\left|B_s\\left(w,V^{\\prime }\\right)\\right|/\\left|V^{\\prime }\\right|$ .",
"That is, $\\mu ^{U^{\\prime }}\\left(w\\right)$ is the probability that $u^{\\prime (s)}=w$ where $u^{\\prime }$ is chosen uniformly in $U^{\\prime }$ , and similarly for $\\mu ^{V^{\\prime }}\\left(w\\right)$ .",
"Therefore, since the order of $w^{\\prime }$ is $s$ , we have that $ \\left|\\mathbb {E}_{w_1 \\sim \\mu ^{U^{\\prime }},w_2 \\sim \\mu ^{V^{\\prime }}}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle w_1,w_2 \\rangle }\\right]\\right|\\ge \\epsilon ^4.$ Recalling that $s$ is the order of $\\omega ^{\\prime }$ and applying Lemma REF , we get $\\textnormal {cp}\\left(\\mu ^{U^{\\prime }}\\right)\\textnormal {cp}\\left(\\mu ^{V^{\\prime }}\\right) \\ge \\epsilon ^8/s^n$ .",
"Therefore, one of $\\textnormal {cp}\\left(\\mu ^{U^{\\prime }}\\right)$ , $\\textnormal {cp}\\left(\\mu ^{V^{\\prime }}\\right)$ , say $\\textnormal {cp}\\left(\\mu ^{U^{\\prime }}\\right)$ , is at least $\\epsilon ^4/s^{n/2}$ .",
"Let $w^*$ be the point of maximum probability mass given by $\\mu ^{U^{\\prime }}$ .",
"Then, $\\mu ^{U^{\\prime }}\\left(w^*\\right)=\\mu ^{U^{\\prime }}\\left(w^*\\right)\\sum _{w \\in \\mathbb {Z}_s^n}\\mu ^{U^{\\prime }}\\left(w\\right)\\ge \\sum _{w \\in \\mathbb {Z}_s^n}\\mu ^{U^{\\prime }}\\left(w\\right)^2=\\textnormal {cp}\\left(\\mu ^{U^{\\prime }}\\right)\\ge \\epsilon ^4/s^{n/2}.$ Now, $\\mu ^{U^{\\prime }}\\left(w^*\\right) \\ge \\epsilon ^4/s^{n/2}$ means that $\\left|\\lbrace u \\in U: \\frac{u-\\tilde{u}}{r_1}=w^*\\ \\left(mod\\ s\\right)\\rbrace \\right|\\ge t \\epsilon ^4/s^{n/2} $ .",
"Equivalently, $\\bigg |\\big \\lbrace u \\in U: u-\\tilde{u}=r_1w^*\\ \\left(mod\\ r_1s\\right)\\big \\rbrace \\bigg |\\ge t \\epsilon ^4/s^{n/2}.$ Let $T^{\\prime }=\\left(i:u_i=\\tilde{u}+r_1w^*\\ \\left(mod\\ r_1s\\right)\\right)$ .",
"Now, define $U^{\\prime \\prime }=\\left(u_i:i \\in T^{\\prime }\\right)$ and $V^{\\prime \\prime }=\\left(v_i:i \\in T^{\\prime }\\right)$ .",
"Observe that $\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ such that $U^{\\prime \\prime \\left(r_1s\\right)}$ and $V^{\\prime \\prime \\left(r_2\\right)}$ are constants.",
"$\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right| \\ge t\\left(\\epsilon ^4/s^{n/2}\\right)$ .",
"The only thing left is to show that $\\langle u,v\\rangle =0\\ \\left(mod\\ r_1r_2s\\right)$ for all $u \\in U^{\\prime \\prime },v \\in V^{\\prime \\prime }$ .",
"This may not be true in general.",
"However, we can take a large subset of the matching vector family so that the resulting matching vector family satisfies this condition.",
"To see this, let $u \\in U^{\\prime \\prime }, v \\in V^{\\prime \\prime }$ be arbitrary.",
"Now, $u=r_1s \\cdot u^{\\prime }+u_0$ and $v=r_2 \\cdot v^{\\prime }+v_0$ where $u^{\\prime }, v^{\\prime }$ depend on $u,v$ respectively and $u_0,v_0$ are independent of $u,v$ .",
"Then, $\\langle u,v \\rangle =r_1r_2s \\langle u^{\\prime },v^{\\prime }\\rangle +r_1s \\langle u^{\\prime },v_0\\rangle +r_2 \\langle u_0,v^{\\prime }\\rangle + \\langle u_0,v_0\\rangle .$ As $u$ varies over $U^{\\prime \\prime }$ , $\\langle u^{\\prime },v_0\\rangle $ takes at most $q$ values modulo $r_2$ .",
"Hence, $r_1s\\langle u^{\\prime },v_0\\rangle $ takes at most $q$ values modulo $r_1r_2s$ .",
"Therefore, there exist at least $\\left(1/q\\right)\\left|U^{\\prime \\prime }\\right|$ elements of $U^{\\prime \\prime }$ such that $r_1s\\langle u^{\\prime },v_0\\rangle $ is a constant modulo $r_1r_2s$ .",
"We take the corresponding elements from $V^{\\prime \\prime }$ to form a matching vector family $\\left(U^{\\prime \\prime \\prime },V^{\\prime \\prime \\prime }\\right)\\subseteq \\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)$ .",
"We apply another round using the same idea on $U^{\\prime \\prime \\prime },V^{\\prime \\prime \\prime }$ , this time ensuring that $r_2\\langle u_0,v^{\\prime }\\rangle $ is constant modulo $r_1r_2s$ as $v$ varies over a large fraction of $V^{\\prime \\prime \\prime }$ .",
"Thus, we end up with $\\tilde{V}$ of size at least $\\left(1/q\\right)\\left|V^{\\prime \\prime \\prime }\\right|$ such that $r_2\\langle u_0,v_i\\rangle $ is a constant modulo $r_1r_2s$ .",
"We take the corresponding subset $\\tilde{U}$ from $U^{\\prime \\prime \\prime }$ so that $(\\tilde{U}, \\tilde{V})\\subseteq \\left(U^{\\prime \\prime \\prime },V^{\\prime \\prime \\prime }\\right)$ is a matching vector family.",
"Denote the size of $(\\tilde{U}, \\tilde{V})$ by $\\tilde{t}$ .",
"Note that $\\tilde{U}=\\left(\\tilde{u}_1,\\cdots ,\\tilde{u}_{\\tilde{t}}\\right), \\tilde{V}=\\left(\\tilde{v}_1,\\cdots ,\\tilde{v}_{\\tilde{t}}\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ of size at least $\\left(1/q^2\\right)t\\left(\\epsilon ^4/s^{n/2}\\right)=s^{-n/2}q^{-\\left(8+4\\log _q \\left(12\\right)\\right)}t\\ge s^{-n/2}q^{-\\left(8+4\\log _2 \\left(12\\right)\\right)}t \\ge s^{-n/2}q^{-24}t$ .",
"Also, as $\\langle u,v\\rangle $ is a constant modulo $r_1r_2s$ , for $u \\in \\tilde{U}, v \\in \\tilde{V}$ , and $\\langle \\tilde{u}_i,\\tilde{v}_i \\rangle =0\\ \\left(mod \\ r_1r_2s\\right)$ , we get that $\\langle u,v\\rangle =0 \\ \\left(mod\\ r_1r_2s\\right)$ , for $u \\in \\tilde{U}, v \\in \\tilde{V}$ .",
"This concludes the proof.",
"$\\Box $"
],
[
"Proof of Lemma ",
"We prove the lemma by backward induction on $r_1r_2|m$ .",
"That is, to prove the claim about $\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right)$ , we assume the inductive hypothesis for $\\mathbf {MV}_{r_1^{\\prime },r_2^{\\prime }}\\left(m,n,q\\right)$ where $r_1^{\\prime }r_2^{\\prime }>r_1r_2$ and $r_1^{\\prime }r_2^{\\prime }|m$ .",
"Base Case.",
"The base case of $r_1r_2=m$ is trivial.",
"To see this, observe that if $\\langle u,v \\rangle =0 \\ \\left(mod\\ m\\right)$ for all $u \\in U, v \\in V$ , then by the definition of a matching vector family in $\\mathbb {Z}_m^n$ , the size of such a family cannot exceed 1.",
"Hence, for $r_1r_2=m$ , $\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right)=1 \\le 12q\\cdot q^{24\\log \\frac{m}{r_1r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}$ .",
"Inductive Step.",
"Let $m=r_1r_2r_3$ with $r_1r_2<m$ (that is, $r_3 \\ge 2$ ).",
"By the inductive hypothesis we have $\\mathbf {MV}_{r_1^{\\prime },r_2^{\\prime }}\\left(m,n,q\\right) \\le 12q \\cdot q^{24\\log \\frac{m}{r_1^{\\prime }r_2^{\\prime }}}\\left(\\frac{m}{r_1^{\\prime }r_2^{\\prime }}\\right)^{n/2}$ for all $r_1^{\\prime },r_2^{\\prime }$ such that $r_1^{\\prime }r_2^{\\prime }>r_1r_2$ and $r_1^{\\prime }r_2^{\\prime }|m$ .",
"We need to show that $\\mathbf {MV}_{r_1,r_2}\\left(m,n,q\\right)\\le 12q \\cdot q^{24\\log \\frac{m}{r_1r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}$ .",
"Suppose this is false, so that there exists a $q-restricted$ matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ with $U=\\left(u_1, \\cdots u_t\\right),V=\\left(v_1,\\cdots v_t\\right)$ where $t >12q \\cdot q^{24\\log \\frac{m}{r_1r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}$ such that $U^{\\left(r_1\\right)}$ and $V^{\\left(r_2\\right)}$ are constants.",
"$\\langle u,v \\rangle =0 \\ \\left(mod\\ r_1r_2\\right)$ for all $u \\in U, v \\in V$ .",
"Note that $t \\ge 12q$ .",
"Therefore, applying Lemma REF , there exists $s|r_3$ with $s \\ge 2$ and matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ such that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right| \\ge s^{-n/2}q^{-24}t$ where $\\langle u^{\\prime },v^{\\prime } \\rangle =0 \\ \\left(mod\\ r_1r_2s\\right)$ for all $u^{\\prime } \\in U^{\\prime }, v^{\\prime } \\in V^{\\prime }$ .",
"either $U^{\\prime \\left(r_1s\\right)}$ is constant or $V^{\\prime \\left(r_2s\\right)}$ is constant.",
"Without loss of generality, we assume that $U^{\\prime \\left(r_1s\\right)}$ is a constant.",
"Therefore, $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|& > & s^{-n/2}q^{-24}\\cdot 12q\\cdot q^{24\\log \\frac{m}{r_1r_2}}\\left(\\frac{m}{r_1r_2}\\right)^{n/2}\\\\&&=12q \\cdot q^{24\\left(\\log \\frac{m}{r_1r_2}-1\\right)}\\left(\\frac{m}{r_1r_2s}\\right)^{n/2}\\\\&&\\ge 12q \\cdot q^{24\\log \\frac{m}{r_1r_2s}}\\left(\\frac{m}{r_1r_2s}\\right)^{n/2},$ where the last inequality used the fact that $s \\ge 2$ .",
"This however contradicts the inductive hypothesis.",
"$\\Box $"
],
[
"Notations:",
"For a $t \\times s$ matrix $M$ over $\\mathbb {Z}_m$ and for lists $T \\subseteq [t], S \\subseteq [s]$ the $T \\times S$ submatrix of $M$ is the matrix with rows in $T$ and columns in $S$ .",
"For $i\\in [s]$ and $j \\in [t]$ we denote the $i$ 'th row of $M$ by $M(i:)$ and the $j$ 'th column by $M(:j)$ .",
"Definition 4.1 (Span of a set) For $A \\subseteq \\mathbb {Z}_m^n$ let $\\textnormal {span}\\left(A\\right)$ denote the additive subgroup generated by $A$ .",
"We say that a set $A$ spans $u \\in \\mathbb {Z}_m^n$ if $u \\in \\textnormal {span}(A)$ .",
"Definition 4.2 (Rank of a matrix over $\\mathbb {Z}_m$ ) Let $M$ be a $t \\times t$ matrix over $\\mathbb {Z}_m$ .",
"Then $\\textnormal {rank}\\left(M\\right)$ is the smallest $r$ such that $M=AB$ where $A$ is an $t \\times r$ martrix over $\\mathbb {Z}_m$ and $B$ is an $r \\times t$ matrix over $\\mathbb {Z}_m$ .",
"Definition 4.3 (Column rank of a matrix over $\\mathbb {Z}_m$ ) Let $M$ be a $t \\times t$ matrix over $\\mathbb {Z}_m$ .",
"Let $\\textnormal {colspan}\\left(M\\right)$ denote the subgroup of $\\mathbb {Z}_m^t$ generated by the columns of $M$ .",
"The column rank of $M$ over $\\mathbb {Z}_m$ is defined as $\\textnormal {colrank}\\left(M\\right)=\\log _m \\left|\\textnormal {colspan}\\left(M\\right)\\right|.$ The column rank is, in general, a real number in the range $[0,t]$ .",
"Since the rank can behave in unexpected ways over $\\mathbb {Z}_m$ , we make sure to prove some of the basic facts that we will be using later on.",
"Fact 4.4 Let $M$ be a $t \\times t$ matrix over $\\mathbb {Z}_m$ and let $M^{\\prime }$ be any submatrix of $M$ .",
"Then $\\textnormal {colrank}\\left(M^{\\prime }\\right)\\le \\textnormal {colrank}\\left(M\\right)$ .",
"Suppose $M^{\\prime }$ is given by the first $t^{\\prime }$ rows and the first $t^{\\prime \\prime }$ columns of $M$ .",
"We will define an injective map $f:\\textnormal {colspan}\\left(M^{\\prime }\\right) \\rightarrow \\textnormal {colspan}\\left(M\\right)$ .",
"Given any $x \\in \\textnormal {colspan}\\left(M^{\\prime }\\right)$ we can write $x = \\sum _{j=1}^{t^{\\prime \\prime }} \\alpha _j \\cdot M^{\\prime }(:j)$ in some fixed way (there might be several choices of $\\alpha _j$ ).",
"Define $f(x) = \\sum _{j=1}^{t^{\\prime \\prime }} \\alpha _j \\cdot M(:j)$ .",
"Then, $x$ is clearly the restriction of $f(x)$ to the first $t^{\\prime }$ indices and so the map is injective.",
"Fact 4.5 Let $M$ be a $t \\times t$ matrix over $\\mathbb {Z}_m$ and let $s|m$ .",
"Then $\\textnormal {rank}\\left(M^{\\left(s\\right)}\\right)\\le \\textnormal {rank}\\left(M\\right)$ .",
"Suppose there exist an $t \\times r$ matrix $A$ and an $r \\times t$ matrix $B$ over $\\mathbb {Z}_m$ such that $M=AB$ .",
"Then $M^{\\left(s\\right)}=A^{\\left(s\\right)}B^{\\left(s\\right)}$ and so the rank of $M^{(s)}$ is at most $r$ .",
"We will need the following claims relating the rank and the column rank of matrices over $\\mathbb {Z}_m$ .",
"Claim 4.6 Let $M$ be an $t \\times t$ matrix over $\\mathbb {Z}_m$ .",
"Then, $\\frac{\\textnormal {rank}\\left(M\\right)}{\\log m} \\le \\textnormal {colrank}\\left(M\\right) \\le \\textnormal {rank}\\left(M\\right).$ Let $r=\\textnormal {rank}\\left(M\\right)$ and $r^{\\prime }=\\textnormal {colrank}\\left(M\\right)$ .",
"We first prove that $r^{\\prime } \\le r$ .",
"This is equivalent to proving that $\\left|\\textnormal {colspan}\\left(M\\right)\\right| \\le m^r$ .",
"Let $M=AB$ where $A$ is an $t \\times r$ martrix over $\\mathbb {Z}_m$ and $B$ is an $r \\times t$ matrix over $\\mathbb {Z}_m$ .",
"Since the columns of $M$ are all in the span of the columns of $A$ we have that the column span of $M$ can contain at most $m^r$ elements.",
"We now prove that $r^{\\prime } \\ge r/\\left(\\log m\\right)$ or, equivalently, $\\left|\\textnormal {colspan}\\left(M\\right)\\right| \\ge 2^r$ .",
"Suppose in contradiction that $\\left|\\textnormal {colspan}\\left(M\\right)\\right| < 2^r$ .",
"Take a minimal spanning set $S$ of $\\textnormal {colspan}\\left(M\\right)$ (that is, a set that spans $\\textnormal {colspan}\\left(M\\right)$ and such that no proper subset of it does).",
"Suppose $|S| \\ge r$ and consider all linear combinations (over $\\mathbb {Z}_m$ ) of elements of $S$ with coefficients in $\\lbrace 0,1\\rbrace \\subseteq \\mathbb {Z}_m$ .",
"Since $\\left|\\textnormal {colspan}\\left(M\\right)\\right| < 2^r$ there are two distinct $0-1$ linear combinations that map to the same element.",
"This means that there is a linear combination with coefficients in $\\lbrace 1,-1\\rbrace $ of the elements of $S$ that is equal to zero.",
"Since both 1 and $-1$ are invertible modulo $m$ we can write one of the elements of $S$ as a linear combination of the other elements.",
"This contradicts the minimality of $S$ and so, we must have $|S| < r$ .",
"This implies that $\\textnormal {rank}(M) < r$ , a contradiction, since we can write $M$ as the product of the matrix with columns in $S$ with the matrix of coefficients giving the columns of $M$ .",
"Claim 4.7 Let $M$ be an $t \\times t$ matrix over $\\mathbb {Z}_m$ , let $r=\\textnormal {rank}\\left(M\\right)$ .",
"There exists $r^{\\prime }$ columns of $M$ that span the rest of $M^{\\prime }$ s columns such that $r^{\\prime } \\le r\\log m$ .",
"Take a minimal spanning set $S$ of the columns of $M$ (that is, a set that spans all other columns and such that no proper subset of it spans all columns).",
"If $2^{\\left|S\\right|} > m^r$ , then $2^{\\left|S\\right|} > \\left|\\textnormal {colspan}(M)\\right|$ (by Claim REF ) and we proceed as in the proof from Claim REF above.",
"If we look at all the $0-1$ combinations of the columns of $S$ , then there are two distinct $0-1$ linear combinations of the columns that map to the same element of $\\textnormal {colspan}\\left(M\\right)$ .",
"Thus, let $\\sum _i\\alpha _iS\\left(:i\\right)=\\sum _i\\beta _iS\\left(:i\\right)$ where $\\alpha _i \\ne \\beta _i$ for at least one $i$ , say $i_0$ .",
"Therefore, we have $\\sum _i\\left(\\alpha _i-\\beta _i\\right)S\\left(:i\\right)=0$ .",
"Note that $\\left(\\alpha _{i_0}-\\beta _{i_0}\\right)=\\pm 1$ and hence is invertible.",
"This lets us write $S\\left(:i_0\\right)$ as a linear combinations of the remaining columns contradicting the minimality of $S$ .",
"Thus, $r^{\\prime }=\\left|S\\right| \\le r\\log m$ .",
"The following claim shows that the column rank behaves similar to rank in terms of subadditivity.",
"Claim 4.8 Let $A, B$ be $t \\times t$ matrices over $\\mathbb {Z}_m$ .",
"Then, $\\textnormal {colrank}\\left(A+B\\right)\\le \\textnormal {colrank}\\left(A\\right)+\\textnormal {colrank}\\left(B\\right)$ .",
"We show that $\\left|\\textnormal {colspan}\\left(A+B\\right)\\right|\\le \\left|\\textnormal {colspan}\\left(A\\right)\\right|\\left|\\textnormal {colspan}\\left(B\\right)\\right|$ .",
"Note that $\\textnormal {colspan}\\left(A+B\\right) \\subseteq \\textnormal {colspan}\\left(A\\right)+ \\textnormal {colspan}\\left(B\\right) \\stackrel{def}{=} \\lbrace a+b|a\\in \\textnormal {colspan}\\left(A\\right), b \\in \\textnormal {colspan}\\left(B\\right)\\rbrace $ .",
"Therefore, $\\left|\\textnormal {colspan}\\left(A+B\\right)\\right| \\le \\left|\\textnormal {colspan}\\left(A\\right)+ \\textnormal {colspan}\\left(B\\right)\\right| \\le \\left|\\textnormal {colspan}\\left(A\\right)\\right|\\left|\\textnormal {colspan}\\left(B\\right)\\right|$ .",
"Claim 4.9 Let $M$ be a $2t \\times 2t$ matrix over $\\mathbb {Z}_m$ , such that $M=\\left(\\begin{array}{cc}A&0\\\\\\star &B\\end{array}\\right)$ where $A, B $ and $\\star $ are $t \\times t$ matrices.",
"Then, $\\textnormal {colrank}\\left(A\\right)+\\textnormal {colrank}\\left(B\\right) \\le \\textnormal {colrank}\\left(M\\right)$ .",
"We show that $\\left|\\textnormal {colspan}\\left(A\\right)\\right|\\left|\\textnormal {colspan}\\left(B\\right)\\right|\\le \\left|\\textnormal {colspan}\\left(M\\right)\\right|$ .",
"Let $\\textnormal {colspan}\\left(A\\right)=R_1$ , $\\textnormal {colspan}\\left(B\\right)=R_2$ , $\\textnormal {colspan}\\left(M\\right)=R$ .",
"We define $f:R_1 \\times R_2 \\rightarrow R$ and show that $f$ is injective.",
"Given $r_1 \\in R_1$ and $r_2 \\in R_2$ , let $\\alpha _1, \\cdots \\alpha _t$ and $\\beta _1,\\cdots \\beta _t$ denote coefficients for linear combinations of the columns of $A$ and $B$ respectively that give $r_1$ and $r_2$ .",
"There might be many such linear combinations but we fix one for each $r_i$ .",
"Then, $f\\left(r_1,r_2\\right)=\\sum _{i=1}^{t}\\alpha _iM\\left(:i\\right)+\\sum _{i=t+1}^{2t}\\beta _{i-t}M\\left(:i\\right)$ .",
"Now, given a column vector $f\\left(r_1,r_2\\right) \\in R$ , we uniquely identify $r_1$ and $r_2$ as follows.",
"We look at the first $t$ rows and call it $s_1$ .",
"Now $s_1=r_1$ and let $\\alpha _1, \\cdots \\alpha _{t}$ be the linear combination fixed for $r_1$ while defining $f$ .",
"Now, consider $f\\left(r_1,r_2\\right)-\\sum _{i=1}^{t}\\alpha _iM\\left(:i\\right)$ and call the last $t$ rows $s_2$ .",
"Note that $s_2=r_2$ .",
"Claim 4.10 Let $M$ be a $t \\times t$ square matrix over $\\mathbb {Z}_m$ with zero diagonal entries.",
"If for some $s |m$ , $\\textnormal {colrank}\\left(M^{\\left(s\\right)}\\right)\\le 2$ , then there exists at least $t^{\\prime }=t/m^2$ indices such that $M$ restricted to those indices as rows and columns is the all zero matrix modulo $s$ .",
"As $\\textnormal {colrank}\\left(M^{\\left(s\\right)}\\right)\\le 2$ , it follows that $\\left|\\textnormal {colspan}\\left(M^{\\left(s\\right)}\\right)\\right|\\le s^2 \\le m^2$ .",
"Hence, $M^{\\left(s\\right)}$ has at most $m^2$ distinct columns.",
"Therefore, there exists a set of indices $S$ of size $t^{\\prime }\\ge t/m^2$ with $S=\\lbrace r_1, r_2, \\cdots r_{t^{\\prime }}\\rbrace $ such that all the columns $M^{\\left(s\\right)}\\left(:r_i\\right)$ are identical.",
"Also, as the diagonal elements are zero modulo $m$ , they are zero modulo $s$ .",
"Thus, the $S \\times S$ submatrix is the all zero matrix modulo $s$ ."
],
[
"Collision-Free MV families",
"In the proof of Theorem REF it will be useful to assume that the elements of the MV family do not `collide' when reduced modulo an integer $s$ dividing $m$ .",
"In this section we develop the necessary machinery to allow for this assumption.",
"We start by defining a collision free matching vector family.",
"Definition 5.1 (Collision free MV family) A collision free matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ is a matching vector family such that for all $s|m, s \\ge 2$ , all elements of $U$ are distinct modulo $s$ , and all elements of $V$ are distinct modulo $s$ .",
"Note that if $\\left(U,V\\right)$ is a collision free matching vector family, then so is any $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ .",
"Lemma 5.2 Let $m\\ge 2$ be an arbitrary integer.",
"Let $s$ be a divisor of $m$ , such that $1<s<m$ .",
"Let $\\left(U,V\\right)$ be a matching vector family in $\\mathbb {Z}_m^n$ such that $\\langle u,v \\rangle =0\\ \\left(mod\\ s\\right)$ for all $u \\in U, v \\in V$ .",
"Then, $|(U,V)| \\le \\mathbf {MV}\\left(m/s,n\\log m\\right)$ .",
"Let $U=\\left(u_1, u_2, \\cdots u_t\\right)$ and $V=\\left(v_1, v_2, \\cdots v_t\\right) $ .",
"Recall that $P_{U,V}$ is the inner product matrix.",
"We shall write $P_{U,V}$ as $P$ in the rest of the proof for brevity.",
"Let $r=\\textnormal {rank}\\left(P\\right) \\le n$ .",
"Hence, by Claim REF , there exists $r^{\\prime } \\le r \\cdot \\log m$ columns of $P$ which span all the columns of $P$ .",
"As each entry of $P$ is a multiple of $s$ we can define a matrix $P^{\\prime }$ over $\\mathbb {Z}_{m/s}$ by $P^{\\prime }=\\left(1/s\\right)P$ .",
"We have $P^{\\prime }_{i,i}=0 \\quad \\forall i$ .",
"$P^{\\prime }_{i,j} \\ne 0 \\quad \\forall i \\ne j$ .",
"We next show that the $r^{\\prime }$ columns that span the columns of $P$ also span the columns in $P^{\\prime }$ .",
"Without loss of generality, let the first $r^{\\prime }$ columns of $P$ span the remaining columns of $P$ .",
"For any column $j$ , let $P\\left(:j\\right)=\\sum _{i=1}^{r^{\\prime }}c_iP\\left(:i\\right) \\pmod {m}$ .",
"Since all entries of $P$ are divisible by $s$ , we can divide the expression by $s$ and obtain that $P^{\\prime }\\left(:j\\right)=\\sum _{i=1}^{r^{\\prime }}c_i P^{\\prime }\\left(:i\\right) \\pmod {m/s}$ .",
"Hence, we deduce that $r_{P^{\\prime }}=\\textnormal {rank}\\left(P^{\\prime }\\right) \\le r^{\\prime } \\le r\\log m \\le n\\log m$ .",
"This implies that $P^{\\prime }=AB$ for some $t \\times r_{P^{\\prime }}$ matrix $A$ and some $r_{P^{\\prime }} \\times t$ matrix $B$ over $\\mathbb {Z}_{m/s}$ .",
"Thus, the rows of $A$ and the columns of $B$ form a matching vector family in $\\mathbb {Z}_{m/s}^{r_{P^{\\prime }}}$ .",
"Therefore, $t \\le \\mathbf {MV}\\left(m/s,n\\log m \\right)$ as claimed.",
"Lemma 5.3 (Bucket Lemma) For any $m$ , let $\\left(U,V\\right)$ be a matching vector family in $\\mathbb {Z}_m^n$ .",
"Let $1<s<m$ be any divisor of $m$ .",
"Then, for any $w \\in \\mathbb {Z}_s^n$ , $\\left|B_s\\left(w,U\\right)\\right| \\le \\mathbf {MV}\\left(m/s,n\\log m\\right)$ .",
"By symmetry, $\\left|B_s\\left(w,V\\right)\\right| \\le \\mathbf {MV}\\left(m/s,n\\log m\\right)$ .",
"We prove that $\\left|B_s\\left(w,U\\right)\\right| \\le \\mathbf {MV}\\left(m/s,n\\right)$ .",
"For $U=\\left(u_1, u_2, \\cdots u_t\\right) $ , consider any bucket $B_s\\left(w,U\\right)=U^{\\prime } \\ \\left(say\\right)$ .",
"Let $U^{\\prime }=\\left(u_{j_1}, u_{j_2}, \\cdots u_{j_{t^{\\prime }}}\\right)$ where $1 \\le j_1 < j_2 < \\cdots j_{t^{\\prime }} \\le t$ .",
"Let $V^{\\prime }=\\left(v_{j_1}, v_{j_2}, \\cdots v_{j_{t^{\\prime }}}\\right)$ .",
"Now, for any $l,m \\in [t^{\\prime }]$ , $\\langle u_{j_l},v_{j_l} \\rangle =0 \\left(mod \\ m\\right)$ .",
"Therefore, $\\langle u_{j_m},v_{j_l} \\rangle =0 \\left(mod \\ s\\right)$ .",
"By Lemma REF on $\\left(U^{\\prime },V^{\\prime }\\right)$ , $t^{\\prime } \\le \\mathbf {MV}\\left(m/s,n\\log m\\right)$ .",
"We use the above lemma repeatedly to obtain a collision free matching vector family.",
"Lemma 5.4 Let $m\\ge 2$ be any positive integer.",
"Suppose there is a matching vector family $\\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ .",
"Then, there exists a collision free matching vector family $\\left(U^{\\prime },V^{\\prime }\\right) \\subseteq \\left(U,V\\right)$ such that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right| \\ge \\frac{\\left|\\left(U,V\\right)\\right|}{\\left(\\prod _{s|m, 1<s<m} \\mathbf {MV}\\left(s,n\\log m\\right)\\right)^2}.$ We will get rid of collisions iteratively by repeatedly applying Lemma REF .",
"Let us write the divisors of $m$ in ascending order as $2 \\le s_1 < s_2 <\\cdots < s_l\\le m/2$ .",
"Perform the following operation for each $s|m$ starting from the smallest divisor greater than 1.",
"For $0 \\le i \\le l$ , let $U_i,V_i$ be the matching vector after stage $i$ with $U_0=U$ and $V_0=V$ .",
"Now suppose that we have $U_i,V_i$ after the $i$ 'th stage such that there is no collision modulo $s_j$ in $U_i$ for $1 \\le j \\le i$ .",
"The $\\left(i+1\\right)$ 'th stage is performed as follows.",
"Let us construct $U_{i+1},V_{i+1}$ from $U_i,V_i$ to ensure no collision among the elements of $U_{i+1}$ modulo $s_{i+1}$ as well.",
"For each $w \\in \\mathbb {Z}_{s_{i+1}}^n$ , by Lemma REF , $\\left|B_{s_{i+1}}\\left(w,U_i\\right)\\right|\\le \\mathbf {MV}\\left(m/s_{i+1},n\\log m\\right)$ .",
"Pick one element from each bucket in $U_i$ and the corresponding matching vector from $V_i$ to form $\\left(U_{i+1},V_{i+1}\\right)$ .",
"Thus, $\\left|\\left(U_{i+1},V_{i+1}\\right)\\right| \\ge \\left|U_i\\right|/\\mathbf {MV}\\left(m/s_{i+1},n\\log m\\right)$ .",
"We end up with matching vector family $U_l,V_l$ such that $\\left|\\left(U_l,V_l\\right)\\right| \\ge \\frac{\\left|\\left(U,V\\right)\\right|}{\\prod _{s|m, 1<s<m} \\mathbf {MV}\\left(m/s,n\\log m\\right)}$ and $U_l$ is collision free.",
"We repeat the same process this time pruning $V_l$ in order to make it collision free as well.",
"Thus, eventually we end up with a collision free matching vector family $\\left(U_l^{\\prime },V_l^{\\prime }\\right) \\subseteq \\left(U,V\\right)$ such that $\\left|\\left(U_l^{\\prime },V_l^{\\prime }\\right)\\right| \\ge \\frac{\\left|\\left(U,V\\right)\\right|}{\\left(\\prod _{s|m, 1<s<m} \\mathbf {MV}\\left(m/s,n\\log m\\right)\\right)^2}=\\frac{\\left|\\left(U,V\\right)\\right|}{\\left(\\prod _{s|m, 1<s<m} \\mathbf {MV}\\left(s,n\\log m\\right)\\right)^2}.$"
],
[
"Proof of Theorem ",
"Before proceeding with the proof we give yet another definition.",
"Definition 6.1 Let $A, B \\subseteq \\mathbb {Z}_m^n$ be twin-free lists (or sets).",
"Let $\\omega $ be a primitive root of unity of order $m$ .",
"The duality measure of $A,B$ with respect to $\\omega $ is defined as $D_{\\omega }\\left(A,B\\right)= \\left|\\mathbb {E}_{a \\sim A, b \\sim B}\\left[\\omega ^{\\langle a,b \\rangle }\\right]\\right|.$ Notice that, if $\\omega \\ne 1$ , $D_{\\omega }(A,B)=1$ implies that there is some $c \\in \\mathbb {Z}_m$ such that all the entries of the inner product matrix $P_{A,B}$ equal $c$ .",
"We often refer to such submatrices as monochromatic rectangles.",
"The following is an easy consequence of Lemma REF .",
"Lemma 6.2 Let $\\left(U,V\\right)$ be a MV family in $\\mathbb {Z}_m^n$ of size $t \\ge 3m$ and let $\\omega =exp\\left(2\\pi i/m\\right)$ be a primitive root of unity of order $m$ .",
"Then there exists some $1 \\le j \\le m-1$ such that $D_{\\omega ^j}\\left(U,V\\right)\\ge \\frac{2}{3m^{3/2}}.$ Let $\\mu $ be the random variable which chooses $u \\in U$ and $v \\in V$ randomly and outputs $\\langle u,v \\rangle $ and let $\\mathcal {U}_m$ be the uniform distribution over $\\mathbb {Z}_m$ .",
"Now, $\\Delta \\left(\\mu ,\\mathcal {U}_m\\right)\\ge \\left(1/2\\right)\\left(\\mathbf {Pr}[\\mathcal {U}_m=0]-\\mathbf {Pr}[\\mu =0]\\right)= \\left(1/2\\right)\\left(1/m-1/t\\right)\\ge 1/3m$ as $t \\ge 3m$ .",
"By Lemma REF , for some $1 \\le j \\le m-1$ , $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^j\\right)^{x} \\right]\\right|\\ge \\frac{2}{3m^{3/2}}.$ Thus, we have $\\left|\\mathbb {E}_{u \\sim U, v \\sim V}\\left[\\left(\\omega ^j\\right)^{\\langle u,v \\rangle } \\right]\\right|\\ge \\frac{2}{3m^{3/2}}$ as claimed.",
"An important ingredient in the proof of Theorem REF is the following lemma, referred to in the introduction as the `sub-matrix lemma' which is a generalization of a result of [5].",
"Lemma 6.3 (Sub-Matrix Lemma) Let $s,m,n \\ge 2$ where $s$ divides $m$ , and let $\\omega $ be a primitive root of unity of order $s$ .",
"Let $A,B \\subset \\mathbb {Z}_s^n$ be two twin-free lists satisfying $D_{\\omega }\\left(A,B\\right) \\ge \\frac{2}{3m^{3/2}}$ .",
"Let $\\textnormal {rank}\\left(P_{A,B}\\right)=r \\ge 2$ .",
"Then assuming Conjecture REF (PFR conjecture), there exist lists $A^{\\prime }\\subseteq A, B^{\\prime } \\subseteq B$ such that $D_{\\omega }\\left(A^{\\prime }, B^{\\prime }\\right) = 1$ , where $|A^{\\prime }| \\ge 2^{-c\\left(m\\right)r/\\log r}\\left|A\\right|$ , $|B^{\\prime }|\\ge 2^{-c\\left(m\\right)r/\\log r} \\left|B\\right|$ for some constant $c\\left(m\\right)$ which depends only on $m$ .",
"Without loss of generality, we can assume $c(m) \\ge 1$ above (it will be convenient to assume it in the proof of Theorem REF ).",
"In other words, we can replace the $c(m)$ above by $\\max \\lbrace c(m),1\\rbrace $ .",
"We postpone the proof of Lemma REF to Section and proceed now with the proof of Theorem REF .",
"We restate Theorem REF here for convenience and with the explicit function $d(m)$ .",
"Theorem 6.4 Let $n,m \\ge 2$ be arbitrary positive integers.",
"Then, assuming Conjecture REF (PFR conjecture), we have $ \\mathbf {MV}\\left(m,n\\right) < 2^{d\\left(m\\right)n/\\log n},$ where $d\\left(m\\right)=1200c\\left(m\\right)m^{6\\log m}$ and $c\\left(m\\right)$ is as in Lemma REF .",
"We prove the theorem by induction on the number of (not necessarily distinct) prime factors of $m$ ."
],
[
"Choice of $d\\left(m\\right)$ .",
"Let $d,d_1,d_2,d_3:\\mathbb {Z}^+ \\rightarrow \\mathbb {R}$ be functions and $d_4$ be a constant.",
"We want the following conditions to be satisfied for all $m,n \\ge 2$ .",
"$d\\left(m\\right),d_1\\left(m\\right),d_2\\left(m\\right),d_3\\left(m\\right)$ are monotonically increasing in $m$ $\\left(2n\\right)^m \\le 2^{d\\left(m\\right)n/\\log n}$ $\\left(2m\\right)^m \\le 2^{d\\left(m\\right)n/\\log n}$ $d\\left(m\\right) \\ge d\\left(m/2\\right) \\cdot 4 m \\log m$ $-d_2\\left(m\\right)+\\left(1/2\\right)d\\left(m\\right)>d\\left(m/2\\right)\\log m$ $2^{\\left(1/2\\right)d\\left(m\\right)n/\\log n} \\ge 3m2^{d_2\\left(m\\right)n/\\log n}$ $d_2\\left(m\\right)n/\\log n \\ge 2\\log m+d_3\\left(m\\right)n/\\log n$ $d_3\\left(m\\right) \\ge d_1\\left(m\\right) \\cdot d_4 \\cdot m \\log m$ $d_4 \\ge 300$ $d_1\\left(m\\right)\\ge 2c\\left(m\\right)$ $d_2 \\ge d_3+1$ It can be verified that the following choice for the functions meets the above conditions.",
"$d\\left(m\\right)=1200 \\cdot c\\left(m\\right) \\cdot m^{6\\log m}$ $d_1\\left(m\\right)=2 \\cdot c\\left(m\\right)$ $d_2\\left(m\\right)=602 \\cdot c\\left(m\\right) \\cdot m \\log m$ $d_3\\left(m\\right)=600 \\cdot c\\left(m\\right)\\cdot m\\log m$ $d_4=300$ We shall explicitly mention which conditions of the above functions are being used in different parts of the proof."
],
[
"Base Case.",
"The base case is where $m=p$ is prime.",
"Lemma REF implies that $\\mathbf {MV}\\left(p,n\\right) \\le 1+{n+p-2 \\atopwithdelims ()p-1}<\\left(2\\max \\lbrace n,p\\rbrace \\right)^p$ .",
"If we show $\\left(2n\\right)^p \\le 2^{d\\left(p\\right)n/\\log n}$ and $\\left(2p\\right)^p \\le 2^{d\\left(p\\right)n/\\log n}$ we will be done.",
"Indeed, by the choice of $d\\left(m\\right)$ (Condition 2 and 3) both of the above will hold."
],
[
"Inductive Case.",
"Let $n \\ge 2, m \\ge 2$ be arbitrary positive integers.",
"Suppose, by induction, that $\\mathbf {MV}\\left(s,n\\right) < 2^{d\\left(s\\right)n/\\log n}$ for all $s|m, s<m$ .",
"We need to show that, assuming Conjecture REF , $ \\mathbf {MV}\\left(m,n\\right) < 2^{d\\left(m\\right)n/\\log n}$ Suppose not.",
"That is, there exists a matching vector family $\\left(U,V\\right)$ of size $t \\ge 2^{d\\left(m\\right)n/\\log n}$ .",
"First, we shall apply Lemma REF to $\\left(U,V\\right)$ to obtain a large enough collision free matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)$ ."
],
[
" A large collision free matching vector family.",
"We show that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge 2^{\\left(1/2\\right)d\\left(m\\right)n/\\log n}$ .",
"Let $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|=t^{\\prime }$ .",
"Observe that by Lemma REF , the inductive hypothesis and the monotonicity of $d\\left(m\\right)$ (Condition 1), $t^{\\prime } \\ge 2^{d\\left(m\\right)n/\\log n-2m \\cdot d\\left(m/2\\right) \\cdot n \\log m/\\log n}$ where we have used a loose upper bound of $m$ for the number of factors of $m$ .",
"Now, $&&t^{\\prime }\\ge 2^{\\left(1/2\\right)d\\left(m\\right)n/\\log n}\\\\&if &d\\left(m\\right)n/\\log n-2m \\cdot d\\left(m/2\\right) \\cdot n \\log m/\\log n \\ge \\left(1/2\\right)d\\left(m\\right)n/\\log n\\\\&\\Leftrightarrow &d\\left(m\\right) \\ge d\\left(m/2\\right)\\cdot 4m \\log m$ which is satisfied by the choice of $d\\left(m\\right)$ (Condition 4)."
],
[
" Two key claims.",
"We will need two claims from which the inductive claim follows easily.",
"We shall provide proofs to these claims after the proof of the inductive claim.",
"Claim 6.5 Let $\\left(U,V\\right)$ be a collision free matching vector family in $\\mathbb {Z}_m^n$ with $\\left|\\left(U,V\\right)\\right| \\ge 3m$ and $\\textnormal {colrank}\\left(P_{U,V}^{\\left(s^{\\prime }\\right)}\\right) > 2$ for all $s^{\\prime }|m, s^{\\prime }\\ge 2$ .",
"Then, for some $s|m, s\\ge 2$ , there exists a collision free matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ satisfying $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge 2^{-d_1\\left(m\\right)r_s/\\log r_s}\\left|\\left(U,V\\right)\\right|$ where $r_s=\\textnormal {rank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ .",
"Either $\\textnormal {colrank}\\left(P_{U^{\\prime },V^{\\prime }}^{\\left(s\\right)}\\right) \\le \\left(3/4\\right)\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ or $\\textnormal {colrank}\\left(P_{U^{\\prime },V^{\\prime }}^{\\left(s\\right)}\\right) \\le 2$ .",
"Claim 6.6 Let $\\left(U,V\\right)$ be a collision free matching vector family in $\\mathbb {Z}_m^n$ such that $\\left|\\left(U,V\\right)\\right|\\ge 3m \\cdot 2^{d_2\\left(m\\right)n/\\log n}$ .",
"Then, there exists a collision free matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ satisfying $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge 2^{-d_2\\left(m\\right)n/\\log n}\\left|\\left(U,V\\right)\\right|$ .",
"$P_{U,V}^{\\left(s\\right)}$ is the all zero matrix for some $s|m, s \\ge 2$ .",
"Let us proceed with the proof of the inductive claim assuming these two claims.",
"We have a collision free matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)$ with $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge 2^{\\left(1/2\\right)d\\left(m\\right)n/\\log n} \\ge 3m \\cdot 2^{d_2\\left(m\\right)n/\\log n}$ .",
"(Condition 6 satisfied by the choice of $d\\left(m\\right), d_2\\left(m\\right)$ ) Applying Claim REF , there exists a collision free matching vector family $\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\subseteq \\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U,V\\right)$ in $\\mathbb {Z}_m^n$ satisfying $\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right|\\ge 2^{-d_2\\left(m\\right)n/\\log n}2^{\\left(1/2\\right)d\\left(m\\right)n/\\log n}$ .",
"$P_{U^{\\prime \\prime },V^{\\prime \\prime }}^{\\left(s\\right)}$ is the all zero matrix for some $s|m, s \\ge 2$ .",
"By the choice of $d\\left(m\\right)$ , it can be verified that $-d_2\\left(m\\right)+\\left(1/2\\right)d\\left(m\\right)>d\\left(m/2\\right)\\log m$ (Condition 5).",
"Thus, $\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right| > 2^{d\\left(m/2\\right)n\\log m/\\log n}$ .",
"We now show that this is enough to get a contradiction.",
"If $s=m$ , we have $\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right|\\le 1$ as $\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)$ is a matching vector family in $\\mathbb {Z}_m^n$ .",
"If $s<m$ , by Lemma REF and the inductive hypothesis, we have $\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right|\\le 2^{d\\left(m/s\\right)n\\log m/\\log \\left(n \\log m\\right)} \\le 2^{d\\left(m/2\\right)n\\log m/\\log n}$ by monotonicity of $d\\left(m\\right)$ (Condition 1).",
"Thus, irrespective of $s$ , $\\left|\\left(U^{\\prime \\prime },V^{\\prime \\prime }\\right)\\right|\\le 2^{d\\left(m/2\\right)n\\log m/\\log n}$ which is a contradiction.",
"This completes the proof."
],
[
"Proof of Claim ",
"Let $\\left|\\left(U,V\\right)\\right|=t \\ge 3m$ .",
"Let $\\omega $ be a root of unity of order $m$ .",
"By Lemma REF , for some $1\\le j \\le m-1$ , $D_{\\omega ^j}\\left(U,V\\right)\\ge \\frac{2}{3m^{3/2}}$ .",
"Note that $s=m/gcd\\left(m,j\\right)$ is the order of $\\omega ^{\\prime }=\\omega ^j$ .",
"Observe that $s|m, s\\ge 2$ as $1\\le j \\le m-1$ .",
"Recall from the statement of the claim that $r_s=rank\\left(P_{U,V}^{\\left(s\\right)}\\right)$ .",
"Thus, by the collision free property of $\\left(U,V\\right)$ , $D_{\\omega ^{\\prime }}\\left(U^{\\left(s\\right)},V^{\\left(s\\right)}\\right)=\\left|\\mathbb {E}_{u \\sim U^{\\left(s\\right)}, v \\sim V^{\\left(s\\right)}}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle u,v \\rangle }\\right]\\right|=\\left|\\mathbb {E}_{u \\sim U, v \\sim V}\\left[\\left(\\omega ^{\\prime }\\right)^{\\langle u,v \\rangle }\\right]\\right|=D_{\\omega ^{\\prime }}\\left(U,V\\right) \\ge \\frac{2}{3m^{3/2}}.$ Applying Lemma REF on $U^{\\left(s\\right)},V^{\\left(s\\right)}$ with $\\omega ^{\\prime }$ a primitive root of unity of order $s$ , we can get an $\\left(R \\times S\\right)$ submatrix of $P_{U,V}$ with $|R|=|S| \\ge 2^{-c\\left(m\\right)r_s/\\log r_s}t$ .",
"(we can make $|R|=|S|$ as throwing away rows and columns from a monochromatic rectangle still keeps it monochromatic) Let $T=R \\cap S$ .",
"We divide our analysis to two cases: either $|T|>|R|/2$ or $|T| \\le |R|/2$ .",
"In both cases, we shall exhibit a matching vector family as required in the statement of the claim.",
"Case 1: $|T|>|R|/2$ .",
"For $U=\\left(u_1, u_2, \\cdots u_t\\right)$ , $V=\\left(v_1, v_2, \\cdots v_t\\right)$ , let $U^{\\prime }=\\left(u_j|j\\in T\\right)$ and $V^{\\prime }=\\left(v_j|j\\in T\\right)$ , and $P^{\\prime }=P_{U^{\\prime },V^{\\prime }}$ .",
"Now, as $P^{\\prime \\left(s\\right)}$ is monochromatic, and $\\langle u_j,v_j\\rangle =0\\ \\left(mod\\ s\\right)$ for $j \\in T$ , we have $\\langle u^{\\prime },v^{\\prime } \\rangle =0 \\left(mod\\ s\\right)$ for all $u^{\\prime } \\in U^{\\prime }, v^{\\prime } \\in V^{\\prime }$ .",
"Observe that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right| \\ge 2^{-1-c\\left(m\\right)r_s/\\log r_s}t \\ge 2^{-2c\\left(m\\right)r_s/\\log r_s}t\\ge 2^{-d_1\\left(m\\right)r_s/\\log r_s}t$ (by the choice of $d_1\\left(m\\right)$ , Condition 10) $\\textnormal {colrank}\\left(P_{U^{\\prime },V^{\\prime }}^{\\left(s\\right)}\\right) =0 \\le 2$ This finishes Case 1.",
"Case 2: $|T| \\le |R|/2$ .",
"Let $R^{\\prime }=R \\setminus T$ and $S^{\\prime }=S \\setminus T$ .",
"Note that $R^{\\prime } \\cap S^{\\prime }=\\emptyset $ and $|R^{\\prime }|=|S^{\\prime }|$ .",
"Consider the $R^{\\prime }\\cup S^{\\prime } \\times R^{\\prime }\\cup S^{\\prime }$ submatrix of $P_{U,V}$ .",
"Call it $P^{\\prime }$ .",
"Note that $P^{\\prime \\left(s\\right)}=\\left(\\begin{array}{cc}P_1^{\\prime }&C\\\\\\star &P_2^{\\prime }\\end{array}\\right)$ where $P_1^{\\prime }$ and $P_2^{\\prime }$ are the $R^{\\prime } \\times R^{\\prime }$ and the $S^{\\prime } \\times S^{\\prime }$ submatrices of $P_{U,V}^{\\left(s\\right)}$ respectively and $C$ is monochromatic.",
"We add a matrix of column rank at most 1 to $P^{\\prime \\left(s\\right)}$ to yield $P^{\\prime \\prime \\left(s\\right)}$ which is the same as $P^{\\prime \\left(s\\right)}$ except that $C$ is replaced by the all zero block matrix.",
"Thus, $P^{\\prime \\prime \\left(s\\right)}=\\left(\\begin{array}{cc}P_1^{\\prime }&0\\\\\\star &P_2^{\\prime }\\end{array}\\right)$ Note that by Claim REF , $\\textnormal {colrank}\\left(P^{\\prime \\prime \\left(s\\right)}\\right) \\le \\textnormal {colrank}\\left(P^{\\prime \\left(s\\right)}\\right)+1$ .",
"Now, using Claim REF , $\\textnormal {colrank}\\left(P_1^{\\prime }\\right)+\\textnormal {colrank}\\left(P_2^{\\prime }\\right)\\le \\textnormal {colrank}\\left(P^{\\prime \\left(s\\right)}\\right)+1 \\le \\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)+1 \\le \\left(3/2\\right)\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ as $\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right) >2$ .",
"Therefore, one of $P_1^{\\prime }, P_2^{\\prime }$ , say $P_1^{\\prime }$ satisfies $\\textnormal {colrank}\\left(P_1^{\\prime }\\right)\\le \\left(3/4\\right)\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ .",
"Construct the matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)$ as follows.",
"Let $U^{\\prime }=\\left(u_j|j\\in R^{\\prime }\\right)$ and $V^{\\prime }=\\left(v_j|j\\in R^{\\prime }\\right)$ .",
"Again, observe that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right| \\ge 2^{-1-c\\left(m\\right)r_s/\\log r_s}t \\ge 2^{-2c\\left(m\\right)r_s/\\log r_s}t\\ge 2^{-d_1\\left(m\\right)r_s/\\log r_s}t$ (by the choice of $d_1\\left(m\\right)$ , Condition 10).",
"$\\textnormal {colrank}\\left(P_{U^{\\prime },V^{\\prime }}^{\\left(s\\right)}\\right) \\le (3/4)\\textnormal {colrank}\\left(P_{U,V}^{\\left(s\\right)}\\right)$ .",
"This completes the proof of Case 2.",
"$\\Box $"
],
[
"Proof of Claim ",
"We will use Claim REF iteratively.",
"For this, we first set up some notations."
],
[
"The setup.",
"Define a sequence of collision free matching vector families for $i=0,\\ldots ,z$ .",
"$\\left(U,V\\right)=\\left(U_0,V_0\\right), \\left(U_1,V_1\\right) \\cdots $ Let $t_i=\\left|\\left(U_i,V_i\\right)\\right|$ .",
"Each step $i$ has label $s_i|m$ (this label will be given by Claim REF ).",
"Let $cr_i:\\mathbb {Z}^+ \\rightarrow \\mathbb {R}$ be defined by $cr_i\\left(s\\right)=\\textnormal {colrank}\\left(P_{U_i,V_i}^{\\left(s\\right)}\\right).$ Let $r_i:\\mathbb {Z}^+ \\rightarrow \\mathbb {Z}$ be defined by $r_i\\left(s\\right)=\\textnormal {rank}\\left(P_{U_i,V_i}^{\\left(s\\right)}\\right).$"
],
[
"Invariants.",
"We will show how to go from step $i$ to step $i+1$ .",
"We stop after stage $z$ when $cr_z\\left(s\\right) \\le 2$ for some $s|m, s\\ge 2$ .",
"We shall maintain the following invariants for $0 \\le i \\le z-1$ .",
"$\\left(U_{i+1},V_{i+1}\\right)\\subseteq \\left(U_{i},V_{i}\\right)$ and hence is a collision free matching vector family in $\\mathbb {Z}_m^n$ .",
"$t_{i+1} \\ge 2^{-d_1\\left(m\\right)r_i\\left(s_i\\right)/\\log r_i\\left(s_i\\right)}t_i$ .",
"$cr_{i+1}\\left(s_i\\right)\\le \\left(3/4\\right)cr_i\\left(s_i\\right)$ or $cr_{i+1}\\left(s_i\\right)\\le 2$ .",
"$cr_{i+1}\\left(s^{\\prime }\\right)\\le cr_i\\left(s^{\\prime }\\right)$ for all $s^{\\prime }|m$ ."
],
[
"Step $i$ {{formula:ef557492-5072-4f74-b6af-082231e35300}} Step {{formula:d641f54b-beb7-403b-a4b6-d7eea3906b1e}} .",
"We state a claim that we will prove below.",
"Claim 6.7 $\\sum _{i=0}^{z-1}d_1\\left(m\\right)r_i\\left(s_i\\right)/\\log r_i\\left(s_i\\right)\\le d_3\\left(m\\right)n/log n$ .",
"In order to apply Claim REF , we need to satisfy $t_i \\ge 3m$ .",
"Observe that by Claim REF , $t_i\\ge t_z &\\ge & t_0\\prod _{j=0}^{z-1}2^{-d_1\\left(m\\right)r_j\\left(s_j\\right)/\\log r_j\\left(s_j\\right)}\\\\ &\\ge & 2^{-d_3\\left(m\\right)n/\\log n}t_0 \\\\&\\ge & 3m \\cdot 2^{-d_3\\left(m\\right)n/\\log n+d_2\\left(m\\right)n/\\log n}\\ge 3m,$ (by the choice of $d_2\\left(m\\right), d_3\\left(m\\right)$ in Condition 11).",
"Apply Claim REF to $\\left(U_{i},V_{i}\\right)$ to get label $s_i$ for step $i$ and $\\left(U_{i+1},V_{i+1}\\right)\\subseteq \\left(U_{i},V_{i}\\right)$ .",
"The first three invariants are maintained by the statement of Claim REF .",
"The last invariant follows from Fact REF .",
"Note that by the inequality we just established, $t_z\\ge 2^{-d_3\\left(m\\right)n/\\log n}t_0$ .",
"Also, by the stopping condition, $cr_z\\left(s^{\\prime }\\right) \\le 2$ for some $s^{\\prime }|m, s^{\\prime }\\ge 2$ .",
"Thus, applying Claim REF , we get another matching vector family $\\left(U^{\\prime },V^{\\prime }\\right)\\subseteq \\left(U_z,V_z\\right)\\subseteq \\left(U,V\\right)$ such that $\\left|\\left(U^{\\prime },V^{\\prime }\\right)\\right|\\ge t_z/m^2 \\ge 2^{-2\\log m-d_3\\left(m\\right)n/\\log n}\\left|\\left(U,V\\right)\\right|\\ge 2^{-d_2\\left(m\\right)n/\\log n}\\left|\\left(U,V\\right)\\right|$ (Condition 7 satisfied by the choice of $d_2\\left(m\\right)$ and $d_3\\left(m\\right)$ ).",
"$P_{U^{\\prime },V^{\\prime }}^{\\left(s^{\\prime }\\right)}$ is the all zero matrix.",
"This finishes the Proof of Claim REF .",
"Proof of Claim REF : Let $t_s$ be the number of steps with label $s$ .",
"Note that as the column rank modulo $s$ goes down by a factor of at least $3/4$ each time we are in a step labeled $s$ , it is easy to see that $t_s \\le \\log _{4/3}cr_0\\left(s\\right) \\le \\log _{4/3}n$ .",
"We shall rely on the monotonic increasing nature of $x/\\log x$ when $x \\ge e$ .",
"As $cr_i\\left(s\\right)>2$ , by Claim REF , $r_i\\left(s\\right)\\ge cr_i\\left(s\\right)> 2$ which means $r_i(s) \\ge 3 > e$ as the rank is always an integer.",
"We thus have $&&\\sum _{i=0}^{z-1}d_1\\left(m\\right)\\frac{r_i\\left(s_i\\right)}{\\log r_i\\left(s_i\\right)}\\\\&\\le & d_1\\left(m\\right)\\log m\\sum _{i=0}^{z-1}\\frac{cr_i\\left(s_i\\right)}{\\log cr_i\\left(s_i\\right)}\\ \\ \\text{(by Claim \\ref {clm-rankcolrank}) and monotonicity of $x/\\log x$ as discussed above}\\\\&\\le & d_1\\left(m\\right)\\log m\\sum _{s|m, s\\ge 2}\\sum _{j=1}^{\\lfloor \\log _{4/3}n\\left(s\\right)\\rfloor }\\left(\\frac{cr_0\\left(s\\right)}{\\left(4/3\\right)^{j-1}\\log \\left(cr_0\\left(s\\right)/\\left(4/3\\right)^{j-1}\\right)}\\right)\\\\&\\le & d_1\\left(m\\right)\\log m\\sum _{s|m, s\\ge 2}d_4cr_0\\left(s\\right)/\\log cr_0\\left(s\\right)\\ \\ \\text{(by Claim \\ref {clm-sum} and Condition $9$ satisfied by $d_4$)}\\\\&\\le & d_1\\left(m\\right)\\log m\\sum _{s|m, s\\ge 2}d_4n/\\log n\\ \\ \\text{(as $cr_0\\left(s\\right)\\le r_0\\left(s\\right)\\le r_0\\left(m\\right)\\le n$, by Claim \\ref {clm-rankcolrank} and Fact \\ref {rankmon})}\\\\&\\le & d_4d_1\\left(m\\right)m\\left(\\log m\\right)n/\\log n\\\\&\\le & d_3\\left(m\\right)n/\\log n\\ \\ \\text{(by the choice of $d_3\\left(m\\right)$, Condition $8$)}$ This completes the proof.",
"$\\Box $"
],
[
"Monochromatic rectangles from low rank matrices",
"In this section we prove Lemma REF (the Sub-Matrix Lemma).",
"We begin with some preliminary definitions.",
"The following is a standard result in algebra and can be find in any introductory text.",
"Theorem 7.1 (Fundamental Theorem of finitely generated abelian groups) Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of prime power order and an infinite cyclic group.",
"More precisely, $G \\cong \\mathbb {Z}^n \\times \\mathbb {Z}_{q_1} \\times \\mathbb {Z}_{q_2} \\cdots \\times \\mathbb {Z}_{q_r}$ where $q_i$ 's are prime powers with $q_1 \\le q_2 \\cdots \\le q_r$ .",
"The decomposition is unique after applying this ordering on $q_i$ 's.",
"If the group $G$ is finite, then $n=0$ .",
"We will use the following two definitions regarding sumsets.",
"Definition 7.2 (Difference Set) For $A \\subseteq \\mathbb {Z}_m^n$ define its difference set as $A-A{=}\\lbrace a-a^{\\prime }|a,a^{\\prime }\\in A\\rbrace $ .",
"Definition 7.3 ($rep_S\\left(x\\right)$ ) For any $S \\subseteq \\mathbb {Z}_m^n$ and $x \\in \\mathbb {Z}_m^n$ , $rep_S\\left(x\\right)$ is the number of different representations of $x$ as an expression of the form $s-s^{\\prime }$ where $s,s^{\\prime } \\in S$ .",
"Next, we define the $\\epsilon $ -spectrum of $B$ with respect to a primitive root of unity of order $m$ .",
"Definition 7.4 (Spectrum) For $B \\subseteq \\mathbb {Z}_m^n$ , and $\\epsilon \\in [0,1]$ , the $\\epsilon $ -spectrum of $B$ with respect to $\\omega $ , a primitive root of unity of order $m$ , is the set $\\textnormal {Spec}_{\\epsilon }\\left(B\\right)=\\left\\lbrace x \\in \\mathbb {Z}_m^n:\\left|\\mathbb {E}_{b \\sim B}\\left[\\omega ^{\\langle x,b \\rangle }\\right]\\right| \\ge \\epsilon \\right\\rbrace .$ When $\\omega $ is implicit in the context, we will drop the phrase \"with respect to $\\omega $ \".",
"We start by proving the following lemma which is a generalization of a lemma from [5].",
"Lemma 7.5 Let $A,B \\subseteq \\mathbb {Z}_m^n$ be sets.",
"Let $\\omega $ be a primitive root of unity of order $m$ .",
"If $A \\subseteq \\textnormal {Spec}_{\\epsilon }\\left(B\\right)$ , then there exist sets $A^{\\prime }\\subseteq A, B^{\\prime }\\subseteq B$ , such that $|A^{\\prime }| \\ge |A|/m$ and $|B^{\\prime }| \\ge \\epsilon ^2\\frac{|A|}{|span\\left(A\\right)|}|B|$ such that $D_{\\omega }\\left(A^{\\prime },B^{\\prime }\\right)=1$ .",
"We start by setting up some notations.",
"Let $W=\\textnormal {span}\\left(A\\right)$ be the subgroup of $\\mathbb {Z}_m^n$ spanned by $A$ .",
"By Theorem REF , there exists an isomorphism $\\tau :\\prod _{i=1}^r\\mathbb {Z}_{q_i}\\rightarrow W$ .",
"Let $\\prod _{i=1}^r\\mathbb {Z}_{q_i}$ and note that we can think of elements of $ as vectors with integer coordinates where the $ i$^{\\prime }th coordinate is in $ Zqi$.",
"Let $ e1,e2, er where $e_i$ is the vector that has 1 in the $i$ 'th coordinate and 0 everywhere else.",
"Given $x \\in , $ 1, r$, with $ i Zqi$ such that $$x=\\sum _{i=1}^r\\alpha _ie_i.$$Then $ (x)=i=1ri(ei)$.",
"Let $ vi=(ei)$ for $ 1 i r$.",
"We can think of the $ vi$^{\\prime }s as a basis of $ W$.",
"Therefore, for $ =(1, 2, r) we have $\\tau \\left(\\alpha \\right)=\\sum _{i=1}^r\\alpha _iv_i$ .",
"Let $\\Theta =\\lbrace \\left(\\beta _1, \\cdots \\beta _r\\right) \\in \\mathbb {Z}_m^r|\\exists u \\in \\mathbb {Z}_m^n \\text{ such that } \\forall i, \\beta _i=\\langle v_i,u\\rangle \\rbrace .$ Claim 7.6 For $1 \\le i \\le r$ , $q_iv_i=0^n\\ \\left(mod \\ m\\right)$ .",
"Let $x =0^r \\in .",
"Now $ (x)=0n (mod m)$.",
"Note that $ x$ can also be written as $ x=qiei$.",
"Applying $$ on both sides, we get $ (x)=qivi$.",
"Thus, $ qivi=0n (mod m)$.$ Claim 7.7 For $\\beta \\in \\Theta $ , $1 \\le i \\le r$ , $q_i\\beta _i=0\\ \\left(mod \\ m\\right)$ .",
"As $\\beta \\in \\Theta $ , there is a $u \\in \\mathbb {Z}_m^n$ such that $\\forall i$ , $\\beta _i=\\langle v_i,u\\rangle $ .",
"Then, $q_i\\beta _i=q_i\\langle v_i,u\\rangle =0\\ \\left(mod\\ m\\right)$ by Claim REF .",
"For $\\alpha \\in \\beta \\in \\Theta $ we define their inner product $\\langle \\alpha , \\beta \\rangle \\in \\mathbb {Z}_m$ by considering $\\alpha _i \\in \\lbrace 0,\\ldots ,q_i-1\\rbrace , \\beta _i \\in \\lbrace 0,\\ldots ,m-1\\rbrace $ , taking the inner product over the integers and then reducing the result modulo $m$ .",
"This is indeed an inner product by Claim REF .",
"Claim 7.8 Given $\\beta \\in \\Theta \\setminus \\lbrace 0\\rbrace $ , $\\sum _{a \\in W}\\omega ^{\\langle \\tau ^{-1}\\left(a\\right),\\beta \\rangle }=\\sum _{\\alpha \\in \\omega ^{\\langle \\alpha ,\\beta \\rangle }=0.",
"}\\begin{proof}Let \\beta _i \\ne 0.",
"Then \\sum _{\\alpha \\in C}\\omega ^{\\langle \\alpha ,\\beta \\rangle }=0 whenever \\sum _{j=0}^{q_i-1}\\omega ^{j\\beta _i}=0.",
"Now, \\sum _{j=0}^{q_i-1}\\omega ^{j\\beta _i}=\\frac{\\omega ^{q_i\\beta _i}-1}{\\omega ^{\\beta _i}-1}.",
"This is well defined because \\omega is of order m and \\beta _i\\ne 0.",
"The claim now follows from Claim \\ref {clm-beta} which makes the expression zero.\\end{proof}$ With the above setup in place, we can now proceed with the proof of Lemma REF .",
"For $\\beta \\in \\Theta $ , define $S_{\\beta }=\\lbrace x \\in \\mathbb {Z}_m^n|\\langle v_i,x\\rangle =\\beta _i, 1\\le i\\le r\\rbrace .$ Denoting $\\mu \\left(\\beta \\right)=\\mathbf {Pr}_{b \\in B}[b \\in S_{\\beta }]$ , we observe that $\\cup _{\\beta \\in \\Theta }\\left(B\\cap S_{\\beta }\\right)=B$ .",
"Hence, $\\sum _{\\beta \\in \\Theta }\\mu \\left(\\beta \\right)=1$ .",
"For $a \\in W$ , define $h\\left(a\\right)=\\mathbb {E}_{b \\in B}\\left[\\omega ^{\\langle a,b \\rangle }\\right]$ .",
"If $a=\\sum _{i=1}^r\\alpha _iv_i$ then $h\\left(a\\right)&=&\\mathbb {E}_{b \\in B}\\left[\\omega ^{\\langle a,b \\rangle }\\right]\\\\&=&\\mathbb {E}_{b \\in B}\\left[\\omega ^{\\langle \\sum _{i=1}^r\\alpha _iv_i,b \\rangle }\\right]\\\\&=&\\sum _{\\beta \\in \\Theta }\\mu \\left(\\beta \\right)\\omega ^{\\langle \\alpha ,\\beta \\rangle }\\\\&=&\\sum _{\\beta \\in \\Theta }\\mu \\left(\\beta \\right)\\omega ^{\\langle \\tau ^{-1}\\left(a\\right) ,\\beta \\rangle }.$ We will prove upper and lower bounds for the sum $\\sum _{a \\in A}\\left|h\\left(a\\right)\\right|^2$ .",
"On the one hand, $\\sum _{a \\in A}\\left|h\\left(a\\right)\\right|^2&\\ge &\\frac{1}{|A|}\\left(\\sum _{a \\in A}\\left|h\\left(a\\right)\\right|\\right)^2 \\ \\text{(Cauchy Scwartz inequality)}\\\\&\\ge &\\frac{1}{|A|}\\left(\\sum _{a \\in A}\\epsilon \\right)^2 \\ \\text{($A \\subseteq \\textnormal {Spec}_{\\epsilon }\\left(B\\right)$ implies $\\left|h\\left(a\\right)\\right|\\ge \\epsilon $ )}\\\\&\\ge & |A|\\epsilon ^2.$ On the other hand, $\\sum _{a \\in A}\\left|h\\left(a\\right)\\right|^2&\\le & \\sum _{a \\in W}\\left|h\\left(a\\right)\\right|^2\\\\&=&\\sum _{a \\in W}\\sum _{\\beta \\in \\Theta , \\beta ^{\\prime } \\in \\Theta ^{\\prime }}\\mu \\left(\\beta \\right)\\mu \\left(\\beta ^{\\prime }\\right)\\omega ^{\\langle \\tau ^{-1}\\left(a\\right),\\beta - \\beta ^{\\prime } \\rangle }\\\\&=&\\sum _{\\beta ,\\beta ^{\\prime } \\in \\Theta ,}\\mu \\left(\\beta \\right)\\mu \\left(\\beta ^{\\prime }\\right)\\sum _{a \\in W}\\omega ^{\\langle \\tau ^{-1}\\left(a\\right),\\beta - \\beta ^{\\prime } \\rangle }\\\\&=&\\sum _{\\beta \\in \\Theta }\\mu \\left(\\beta \\right)^2|W| \\qquad \\text{(Claim \\ref {clm-zero})}\\\\&\\le & |W| \\max _{\\beta \\in \\Theta }\\lbrace \\mu \\left(\\beta \\right)\\rbrace .$ Now, combining the upper and lower bounds, $\\max _{\\beta \\in \\Theta }\\lbrace \\mu \\left(\\beta \\right)\\rbrace \\ge \\epsilon ^2\\frac{|A|}{|W|}$ .",
"Thus, there exists a $\\beta \\in \\Theta $ such that $\\mu \\left(\\beta \\right)\\ge \\epsilon ^2\\frac{|A|}{|W|}$ .",
"This means that the subset $B^{\\prime }=B \\cap S_{\\beta }$ is of size at least $\\epsilon ^2\\frac{|A|}{|W|}|B|$ .",
"Now, any $a \\in A$ can be written as $a=\\sum _{i=1}^r \\alpha _i v_i$ , and for $b \\in B^{\\prime }$ , the inner product $\\langle a,b \\rangle =\\langle \\alpha ,\\beta \\rangle $ is independent of $b$ .",
"Now, for $i \\in [m]$ , let $A_i \\subseteq A$ be such that for $a \\in A_i$ , for all $b \\in B^{\\prime }$ , $\\langle a,b \\rangle =\\langle \\alpha ,\\beta \\rangle =i$ .",
"Now there exists some $A_i$ , call it $A^{\\prime }$ , of size at least $|A|/m$ such that $\\langle a^{\\prime },b^{\\prime } \\rangle =i$ for all $a^{\\prime }\\in A^{\\prime },b^{\\prime } \\in B^{\\prime }$ , that is, $D_{\\omega }\\left(A^{\\prime },B^{\\prime }\\right)=1$ and this proves the lemma.",
"We continue along the lines of [5] and prove the following lemma.",
"Lemma 7.9 Suppose the twin free lists $U, V \\subseteq \\mathbb {Z}_m^n$ satisfy $D_{\\omega }\\left(U, V\\right) \\ge \\epsilon $ ̨ where $\\omega $ is a primitive root of unity of order $m$ .",
"Also, let $\\textnormal {rank}\\left(P_{U,V}\\right)=r$ .",
"Then assuming Conjecture REF , for every $K > 1$ , letting $\\ell = r/\\log _m K$ , there exist lists $U^{\\prime }\\subseteq U, V^{\\prime }\\subseteq V$ such that $D_{\\omega }\\left(U^{\\prime }, V^{\\prime }\\right) = 1$ , and $|U^{\\prime }| \\ge poly_m\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell }}}{rK}\\right)\\left(2mr\\right)^{-\\ell }|U|$ , $|V^{\\prime }| \\ge poly_m\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell }}}{rK}\\right)m^{-\\ell }|V|$ .",
"Let $U=\\left(u_1, \\cdots u_t\\right)$ and $V=\\left(v_1, \\cdots v_t\\right)$ .",
"Since $P_{U,V}$ has rank $r$ there exists a $t\\times r$ matrix $U_M$ and $r \\times t$ matrix $V_M$ so that $U_MV_M=P_{U,V}$ .",
"Thus if we let $A$ denote the rows of $U_M$ and $B$ denote the columns of $V_M$ , then $A,B \\subseteq \\mathbb {Z}_m^r$ .",
"The proof does not care about the order of elements and hence we now consider $A,B$ which are sets.",
"Note that $|A|=|B|=t$ and if $A=\\left(a_1, \\cdots a_t\\right)$ and $B=\\left(b_1, \\cdots b_t\\right)$ then $\\langle a_i,b_j \\rangle =\\langle u_i, v_j \\rangle $ for $1\\le i,j \\le t$ .",
"Thus, $D_{\\omega }\\left(U, V\\right) \\ge \\epsilon $ implies $D_{\\omega }\\left(A, B\\right) \\ge \\epsilon $ .",
"Following [5] consider a sequence of constants $\\epsilon _1=\\epsilon /2$ , $\\epsilon _2=\\epsilon _1^2/2$ , $\\epsilon _3=\\epsilon _2^2/2$ , $\\cdots $ and a sequence of sets $A_1=A \\cap \\textnormal {Spec}_{\\epsilon _1}\\left(B\\right)$ and $A_{i} \\subseteq \\left(A_{i-1}-A_{i-1}\\right) \\cap \\textnormal {Spec}_{\\epsilon _i}\\left(B\\right)$ .",
"The way the subsets are chosen for $A_i$ 's will be made precise shortly.",
"Now by the pigeonhole principle, there exists a minimal index $\\ell \\le r/\\log _m K$ such that $|A_{\\ell +1}| \\le K|A_{\\ell }|$ .",
"To give a precise definition of the $A_i$ 's , we have the following.",
"Let $A_1=A \\cap \\textnormal {Spec}_{\\epsilon /2}\\left(B\\right)$ .",
"For $i\\ge 2$ , assuming $\\epsilon _{i-1}$ and $A_{i-1}$ , let $j_i$ be the the integer index which maximizes the size of $\\lbrace \\left(a,a^{\\prime }\\right) \\in A_{i-1}\\times A_{i-1}|a-a^{\\prime } \\in \\textnormal {Spec}_{\\epsilon _i}\\left(B\\right) \\ and \\ m^{j_i}\\le rep_{A_{i-1}}\\left(a-a^{\\prime }\\right) \\le m^{j_i+1}\\rbrace ,$ and let $A_i=\\lbrace a-a^{\\prime } | a,a^{\\prime } \\in A_{i-1}, a-a^{\\prime } \\in \\textnormal {Spec}_{\\epsilon _i}\\left(B\\right) \\ and \\ m^{j_i}\\le rep_{A_{i-1}}\\left(a-a^{\\prime }\\right) \\le m^{j_i+1}\\rbrace .$ Claim 7.10 For $i=1$ we have $|A_1| \\ge \\left(\\epsilon ̨/2\\right)|A|$ .",
"For $i>1$ we have $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{i-1}}[a-a^{\\prime } \\in A_i] \\ge \\epsilon _i/r$ and additionally $|A_i| \\ge \\frac{\\epsilon _i}{m^{j_i+1}r}|A_{i-1}|^2$ .",
"The case of $i=1$ follows from Markov inequality.",
"For larger $i$ , we show that $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{i-1}}[a-a^{\\prime } \\in \\textnormal {Spec}_{\\epsilon _i}\\left(B\\right)] \\ge \\epsilon _i.$ This follows from the fact that $\\epsilon _{i-1}^2 \\le \\left|\\mathbb {E}_{b \\in B, a \\in A_{i-1}}\\left[\\omega ^{\\langle a,b \\rangle }\\right]\\right|^2 \\le \\mathbb {E}_{b \\in B}\\left|\\mathbb {E}_{a \\in A_{i-1}}\\left[\\omega ^{\\langle a,b \\rangle }\\right]\\right|^2 = \\mathbb {E}_{a,a^{\\prime }\\in A_{i-1}}\\mathbb {E}_{b \\in B}\\left[\\omega ^{\\langle a-a^{\\prime },b \\rangle }\\right].$ Now applying Markov inequality we get that $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{i-1}}[a-a^{\\prime } \\in \\textnormal {Spec}_{\\epsilon _i}(B)] \\ge \\epsilon _i=\\epsilon _{i-1}^2/2$ .",
"Now selecting $j_i$ as in the construction gives that $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{i-1}}[a-a^{\\prime } \\in A_i] \\ge \\epsilon _i/r$ .",
"To prove the second part of the lemma, observe that by the above, we have shown that $\\left|\\lbrace \\left(a,a^{\\prime }\\right) \\in A_{i-1}\\times A_{i-1}|a-a^{\\prime } \\in A_i\\rbrace \\right| \\ge \\frac{\\epsilon _i}{r}|A_{i-1}|^2.$ Also, by construction of $A_i$ , since every $x \\in A_i$ can be represented as $x=a-a^{\\prime }$ with $a,a^{\\prime } \\in A_{i-1}$ in at most $m^{j_i+1}$ ways, we have that $|A_i| \\ge \\frac{\\epsilon _i}{m^{j_i+1}r}\\left|A_{i-1}\\right|^2$ .",
"This completes the proof.",
"Below we will use the following additive-combinatorics lemma.",
"Theorem 7.11 ([4], [11]) There exists an absolute constant $c>0$ such that the following holds.",
"Let $A$ be any arbitrary subset of an abelian group $G$ .",
"Let $S \\subseteq G$ be such that $|S|\\le C|A|$ .",
"If $\\mathbf {Pr}_{a,a^{\\prime } \\in A}[a-a^{\\prime } \\in S] \\ge 1/C$ , then there exists a subset $A^{\\prime } \\subseteq A$ such that $|A^{\\prime }|\\ge \\frac{|A|}{C^c}$ and $|A^{\\prime }-A^{\\prime }| \\le C^c|A|$ .",
"Now we come to the main claim.",
"Claim 7.12 For $i=\\ell ,\\ell -1, \\cdots 1$ there exist subsets $A_i^{\\prime } \\subseteq A_i$ , $B_i^{\\prime } \\subseteq B$ such that $D_{\\omega }\\left(A_i^{\\prime },B_i^{\\prime }\\right)=1$ and $|A_i^{\\prime }|\\ge \\alpha _i|A_i|$ and $|B_i^{\\prime }|\\ge \\beta _i|B|$ where $\\alpha _i=poly_m\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)\\left(2mr\\right)^{-\\left(\\ell -i\\right)}\\left(\\prod _{j=i}^{\\ell }\\epsilon _{j+1}\\right), \\beta _i=poly_m\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)m^{-\\left(\\ell -i\\right)}$ Base Case.",
"The base case of $i=\\ell $ is proved by an application of the Balog-Szemeredi-Gowers theorem followed by Conjecture REF followed by Lemma REF .",
"To see this, we know that $|A_{\\ell +1}| \\le K|A_{\\ell }|$ and $\\mathbf {Pr}_{a,a^{\\prime } \\in A_{\\ell }}[a-a^{\\prime } \\in A_{\\ell +1}] \\ge \\epsilon _{\\ell +1}/r$ .",
"Hence by Theorem REF (with $C=\\frac{rK}{\\epsilon _{\\ell +1}}$ ), there exists a set $A_{\\ell }^{\\prime \\prime }\\subseteq A_{\\ell }$ such that $|A_{\\ell }^{\\prime \\prime }| \\ge poly\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)\\left|A_{\\ell }\\right|$ and $|A_{\\ell }^{\\prime \\prime } - A_{\\ell }^{\\prime \\prime }|\\le poly\\left(\\frac{rK}{\\epsilon _{\\ell +1}}\\right)\\left|A_{\\ell }^{\\prime \\prime }\\right|$ .",
"Now by Conjecture REF applied to $A_{\\ell }^{\\prime \\prime }$ , there exists a set $A_{\\ell }^{\\prime \\prime \\prime }\\subseteq A_{\\ell }^{\\prime \\prime }$ such that $\\left|A_{\\ell }^{\\prime \\prime \\prime }\\right| \\ge poly_m\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)\\left|A_{\\ell }^{\\prime \\prime }\\right|$ and $\\left|\\textnormal {span}\\left(A_{\\ell }^{\\prime \\prime \\prime }\\right)\\right|\\le m|A_{\\ell }^{\\prime \\prime }| = poly_m\\left(\\frac{rK}{\\epsilon _{\\ell +1}}\\right)\\left|A_{\\ell }^{\\prime \\prime \\prime }\\right|$ .",
"(Note the extra factor of $m$ in front of $|A_{\\ell }^{\\prime \\prime }|$ as we get a coset of size $|A_{\\ell }^{\\prime \\prime }|$ and its span incurs an additional factor of $m$ ) Also, as $A_{\\ell }^{\\prime \\prime \\prime } \\in \\textnormal {Spec}_{\\epsilon _{\\ell }}\\left(B\\right)$ , applying Lemma REF to $A_{\\ell }^{\\prime \\prime \\prime }$ and $B$ , we get $A_{\\ell }^{\\prime } \\subseteq A_{\\ell }^{\\prime \\prime \\prime }$ and $B_{\\ell }^{\\prime } \\subseteq B$ such that $D_{\\omega }\\left(A_{\\ell }^{\\prime },B_{\\ell }^{\\prime }\\right)=1$ , $|A_{\\ell }^{\\prime }|\\ge poly_m\\left(\\frac{\\epsilon _{{\\ell }+1}}{rK}\\right)\\left|A_{\\ell }\\right|$ and $|B_{\\ell }^{\\prime }|\\ge poly_m\\left(\\frac{\\epsilon _{\\ell +1}}{rK}\\right)\\left|B\\right|$ .",
"This completes the base case.",
"Let us come to the inductive case.",
"$\\Box $ Inductive Case.",
"Suppose the statement is true for $i$ and let us argue for $i-1$ .",
"Let $G=\\left(A_{i-1},E\\right)$ be the graph whose vertices are the elements in $A_{i-1}$ and $\\left(a,a^{\\prime }\\right)$ is an edge if $a-a^{\\prime } \\in A_i^{\\prime }$ .",
"Now, $|E|&\\ge &m^{j_i}|A_i^{\\prime }|\\\\&\\ge & m^{j_i}\\alpha _i|A_i| \\ \\ \\text{(inductive hypothesis)}\\\\&\\ge & m^{j_i}\\alpha _i\\frac{\\epsilon _i}{m^{j_i+1}r}|A_{i-1}|^2 \\ \\ \\text{(Claim \\ref {clm-claim24})}\\\\&=&2\\alpha _{i-1}|A_{i-1}|^2$ Now the graph has at least $2\\alpha _{i-1}|A_{i-1}|^2$ edges and $|A_{i-1}|$ vertices and therefore has a connected component of size at least $2\\alpha _{i-1}|A_{i-1}|$ vertices.",
"Let us call these vertices $A_{i-1}^{\\prime \\prime }$ .",
"Let $\\tilde{a}$ be any element of $A_{i-1}^{\\prime \\prime }$ .",
"Partition $B_i^{\\prime }$ into $B_{i,j}^{\\prime }$ for $0 \\le j \\le m-1$ such that all elements of $B_{i,j}^{\\prime }$ have inner product $j$ with $\\tilde{a}$ .",
"Let $B_{i-1}^{\\prime }=B_{i,j_1}$ be the largest of them.",
"Note that $|B_{i-1}^{\\prime }| \\ge |B_i^{\\prime }|/m$ .",
"By assumption $D_{\\omega }\\left(A_i^{\\prime },B_i^{\\prime }\\right)=1$ .",
"Hence, $D_{\\omega }\\left(A_i^{\\prime },B_{i-1}^{\\prime }\\right)=1$ .",
"Therefore, for some $j_2$ , $\\langle a,b \\rangle = j_2$ for all $a \\in A_i^{\\prime }$ and $b \\in B_{i-1}^{\\prime }$ .",
"Now, in the connected component obtained above, whenever $a,a^{\\prime } \\in A_{i-1}^{\\prime \\prime }$ are neighbours, $\\langle a-a^{\\prime },b \\rangle =j_2$ for $b \\in B_{i-1}^{\\prime }$ .",
"Thus, starting with $\\tilde{a}$ as the anchor and propagating throughout the connected component, we can classify the vertices in $A_{i-1}^{\\prime \\prime }$ based on the inner product it has with all elements in $B_{i-1}^{\\prime }$ , which is either $j_1$ or $j_2-j_1$ .",
"Pick the larger set and call it $A_{i-1}^{\\prime }$ .",
"Hence, $D_{\\omega }\\left(A_{i-1}^{\\prime },B_{i-1}^{\\prime }\\right)=1$ .",
"Thus, $|A_{i-1}^{\\prime }|\\ge |A_{i-1}^{\\prime \\prime }|/2 \\ge \\alpha _{i-1}|A_{i-1}|$ and $B_{i-1}^{\\prime }\\ge |B_i^{\\prime }|/m \\ge \\frac{\\beta _i}{m}|B|= \\beta _{i-1}|B|$ .",
"This completes the inductive case.",
"$\\Box $ Put $i=1$ in the above claim.",
"Also observe that as $\\epsilon _{j+1}=\\epsilon ^{2^j}/2^{2^j-1}\\ge \\left(\\epsilon /2\\right)^{2^j}$ .",
"Thus, $\\epsilon _{\\ell +1}\\ge \\left(\\epsilon /2\\right)^{2^{\\ell }}$ and $\\prod _{j=1}^{\\ell }\\epsilon _{j+1}\\ge \\left(\\epsilon /2\\right)^{2^{\\ell +1}}$ there exist $A^{\\prime } \\subseteq A_1 \\subseteq A$ , $B^{\\prime } \\subseteq B$ , such that $|A^{\\prime }| \\ge poly\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell }}}{rK}\\right)\\left(2mr\\right)^{-\\ell }\\left|A\\right|$ and $ |B^{\\prime }| \\ge poly\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell }}}{rK}\\right)m^{-\\ell }\\left|B\\right|$ .",
"Observing that the lower bounds grow weaker with increasing $\\ell $ ,and that $\\ell \\le \\ell ^{\\prime }=r/\\log _m K$ we get $|A^{\\prime }| \\ge poly\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell ^{\\prime }}}}{rK}\\right)\\left(2mr\\right)^{-\\ell ^{\\prime }}\\left|A\\right|$ and $|B^{\\prime }| \\ge poly\\left(\\frac{\\left(\\epsilon /2\\right)^{2^{\\ell ^{\\prime }}}}{rK}\\right)m^{-\\ell ^{\\prime }}\\left|B\\right|$ where $\\ell ^{\\prime }=r/\\log _m K$ .",
"Therefore, if we take the list $U^{\\prime }\\subseteq U$ (corresponding to $A^{\\prime }\\subseteq A$ ) and $V^{\\prime }\\subseteq V$ (corresponding to $B^{\\prime }\\subseteq B$ ) then as $\\langle a_i,b_j \\rangle =\\langle u_i, v_j \\rangle $ the statement of the lemma follows.",
"This completes the proof of Lemma REF We can now prove the Sub-Matrix Lemma, Lemma REF ."
],
[
"Proof of Lemma ",
"Set $K=s^{4r/\\log r}, \\ell =\\frac{\\log r}{4}, \\epsilon =1/2m^{3/2}$ while applying Lemma REF over $\\mathbb {Z}_s$ .",
"We get $|A^{\\prime }|\\ge \\delta _s |A|$ , $|B^{\\prime }|\\ge \\delta _s |B|$ where $\\delta _s&=&poly_s\\left(\\frac{1}{m^{r^{1/4}}}\\right)2^{-c_1\\left(s\\right)r/\\log r}\\ \\ \\text{(for some constant $c_1\\left(s\\right)$ depending only on $s$)}\\\\&\\ge & poly_m\\left(\\frac{1}{m^{r^{1/4}}}\\right)2^{-c_1\\left(s\\right)r/\\log r}$ Now let $c_2\\left(m\\right)=\\max _{s|m, s\\ge 2}\\lbrace c_1\\left(s\\right)\\rbrace $ .",
"Thus, $\\delta _s \\ge poly_m\\left(\\frac{1}{m^{r^{1/4}}}\\right)2^{-c_2\\left(m\\right)r/\\log r}\\ge 2^{-c\\left(m\\right)r/\\log r}$ for some constant $c$ that depends only on $m$ .",
"$\\Box $"
],
[
"A Calculation",
"Claim A.1 Let $b>1, n \\ge 2$ be arbitrary integers.",
"Then $\\sum _{i=1}^{\\lfloor \\log _b n\\rfloor }\\frac{1}{b^{i-1}\\log \\left(n/b^{i-1}\\right)}\\le f(b)/\\log n$ where $f(b)=\\frac{10b}{b-1}+\\frac{10}{\\log b}+\\frac{16e}{\\log ^2b}$ .",
"When $b=4/3$ , $f(b)<300$ .",
"We divide the summation into two parts.",
"The first part consists of the first $\\lfloor \\log _b \\log n \\rfloor $ terms and the second part consists of the remaining terms.",
"In the first part, $\\frac{1}{b^{i-1}\\log n/b^{i-1}}\\le \\frac{1}{b^{i-1}0.1\\log n}$ whenever $n \\ge 2$ and hence the first part summation is bounded from above by $\\frac{10b}{(b-1)\\log n}$ .",
"In the second part of the summation, we use the monotonicity of $x \\log \\left(n/x\\right)$ .",
"The function increases with $x$ as long as $x\\le n/e$ .",
"Therefore, for terms with $b^{i-1}\\le n/e$ , the maximum value of each summand is given by substituting $i=\\log _b\\log n$ which gives an upper bound of $\\frac{1}{0.1\\log ^2 n}$ .",
"The remaining terms corresponding to $n/b \\ge b^{i-1} > n/e$ (note that these extra terms arise only if $b<e$ ) can be analysed as follows.",
"Observe that each summand in that range can be upperbounded by $\\frac{e}{n\\log b}$ .",
"Therefore, we have at most $\\log _b n$ terms each at most $\\frac{10}{\\log ^2 n}+\\frac{e}{n\\log b}$ .",
"Thus, the second part of the summation is bounded from above by $\\log _b n\\left(\\frac{10}{\\log ^2 n}+\\frac{e}{n\\log b}\\right)$ .",
"$\\log _b n\\left(\\frac{10}{\\log ^2 n}+\\frac{e}{n\\log b}\\right)&=&\\frac{10}{\\log b}\\frac{1}{\\log n}+\\frac{e}{\\log ^2b}\\frac{\\log n}{n}\\\\&\\le &\\frac{10}{\\log b}\\frac{1}{\\log n}+\\frac{e}{\\log ^2b}\\frac{16}{\\log n}\\ \\ \\text{(as $16n\\ge \\log ^2 n$)}\\\\&= &\\left(\\frac{10}{\\log b}+\\frac{16e}{\\log ^2b}\\right)\\frac{1}{\\log n}$ This completes the proof."
],
[
"Proof of Lemma ",
"Let $f:\\mathbb {Z}_m \\rightarrow \\mathbb {C}$ be any function.",
"Recall that, for $0 \\le j \\le m-1$ , the Fourier coefficients of $f$ are given by $\\hat{f}\\left(j\\right)=\\frac{1}{m}\\sum _{x \\in \\mathbb {Z}_m} f\\left(x\\right)\\exp \\left(-2\\pi i jx/m\\right).$ It is well known that the set of functions $\\lbrace \\exp \\left(2\\pi i jx/m\\right)\\rbrace _{0 \\le j \\le m-1}$ is an orthonormal basis for all functions of the above form, and that $f$ can be expressed as $f\\left(x\\right)=\\sum _{j=0}^{m-1}\\hat{f}\\left(j\\right)\\exp \\left(2\\pi i jx/m\\right).$ Let us consider $f:\\mathbb {Z}_m \\rightarrow [0,1]$ .",
"Thus, Parseval's identity states that $\\sum _{j=0}^{m-1}\\left|\\hat{f}\\left(j\\right)\\right|^2 =\\frac{1}{m}\\sum _{x \\in \\mathbb {Z}_m}f\\left(x\\right)^2 \\le 1.$ Observe that as $\\mathcal {U}_m\\left(x\\right)=1/m$ is the constant function, $\\hat{\\mathcal {U}_m}\\left(j\\right)=0$ for $j \\ne 0$ .",
"Also, for any distribution $\\mu $ , $\\hat{\\mu }\\left(0\\right)=1/m$ .",
"Now $2\\epsilon &\\le & \\sum _{x \\in \\mathbb {Z}_m}\\left|\\mu \\left(x\\right)-\\mathcal {U}_m\\left(x\\right)\\right|\\\\&\\le & \\sqrt{m}\\sqrt{\\sum _{x \\in \\mathbb {Z}_m}\\left|\\mu \\left(x\\right)-\\mathcal {U}_m\\left(x\\right)\\right|^2} \\qquad \\text{(Cauchy Schwartz Inequality)}\\\\&= & m\\sqrt{\\sum _{i=0}^{m-1}\\left|\\left(\\hat{\\mu }\\left(i\\right)-\\hat{\\mathcal {U}_m}\\left(i\\right)\\right)\\right|^2} \\\\&= & m\\sqrt{\\sum _{i=1}^{m-1}\\left|\\hat{\\mu }\\left(i\\right)\\right|^2} \\qquad \\text{($\\hat{\\mathcal {U}_m}\\left(j\\right)=0$ for $j \\ne 0$, and $\\hat{\\mu }\\left(0\\right)=\\hat{\\mathcal {U}_m}\\left(0\\right)=1/m$)}\\\\&\\le & m^{3/2}\\max _{i \\ne 0}\\left\\lbrace \\left|\\hat{\\mu }\\left(i\\right)\\right|\\right\\rbrace .$ Thus, for some $j \\ne 0$ , we have $\\left|\\hat{\\mu }\\left(j\\right)\\right| \\ge \\frac{2\\epsilon }{m^{3/2}}.$ Observe that $\\hat{\\mu }\\left(j\\right)&=&\\frac{1}{m}\\sum _{x \\in \\mathbb {Z}_m} \\mu \\left(x\\right)\\exp \\left(-2\\pi i jx/m\\right) \\\\&=&\\frac{1}{m}\\mathbb {E}_{x \\sim \\mu }\\left[\\exp \\left(-2\\pi i jx/m\\right)\\right]\\\\&=&\\frac{1}{m}\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^{m-j}\\right)^{x}\\right].$ Let $j^{\\prime }=m-j$ .",
"Thus, $\\left|\\hat{\\mu }\\left(j\\right)\\right| \\ge \\frac{2\\epsilon }{m^{3/2}}$ implies that $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^{j^{\\prime }}\\right)^{x}\\right]\\right| \\ge \\frac{2\\epsilon }{\\sqrt{m}}.$ This concludes the proof.",
"$\\Box $"
],
[
"Proof of Lemma ",
"$\\Box $"
],
[
"A Calculation",
"Claim A.1 Let $b>1, n \\ge 2$ be arbitrary integers.",
"Then $\\sum _{i=1}^{\\lfloor \\log _b n\\rfloor }\\frac{1}{b^{i-1}\\log \\left(n/b^{i-1}\\right)}\\le f(b)/\\log n$ where $f(b)=\\frac{10b}{b-1}+\\frac{10}{\\log b}+\\frac{16e}{\\log ^2b}$ .",
"When $b=4/3$ , $f(b)<300$ .",
"We divide the summation into two parts.",
"The first part consists of the first $\\lfloor \\log _b \\log n \\rfloor $ terms and the second part consists of the remaining terms.",
"In the first part, $\\frac{1}{b^{i-1}\\log n/b^{i-1}}\\le \\frac{1}{b^{i-1}0.1\\log n}$ whenever $n \\ge 2$ and hence the first part summation is bounded from above by $\\frac{10b}{(b-1)\\log n}$ .",
"In the second part of the summation, we use the monotonicity of $x \\log \\left(n/x\\right)$ .",
"The function increases with $x$ as long as $x\\le n/e$ .",
"Therefore, for terms with $b^{i-1}\\le n/e$ , the maximum value of each summand is given by substituting $i=\\log _b\\log n$ which gives an upper bound of $\\frac{1}{0.1\\log ^2 n}$ .",
"The remaining terms corresponding to $n/b \\ge b^{i-1} > n/e$ (note that these extra terms arise only if $b<e$ ) can be analysed as follows.",
"Observe that each summand in that range can be upperbounded by $\\frac{e}{n\\log b}$ .",
"Therefore, we have at most $\\log _b n$ terms each at most $\\frac{10}{\\log ^2 n}+\\frac{e}{n\\log b}$ .",
"Thus, the second part of the summation is bounded from above by $\\log _b n\\left(\\frac{10}{\\log ^2 n}+\\frac{e}{n\\log b}\\right)$ .",
"$\\log _b n\\left(\\frac{10}{\\log ^2 n}+\\frac{e}{n\\log b}\\right)&=&\\frac{10}{\\log b}\\frac{1}{\\log n}+\\frac{e}{\\log ^2b}\\frac{\\log n}{n}\\\\&\\le &\\frac{10}{\\log b}\\frac{1}{\\log n}+\\frac{e}{\\log ^2b}\\frac{16}{\\log n}\\ \\ \\text{(as $16n\\ge \\log ^2 n$)}\\\\&= &\\left(\\frac{10}{\\log b}+\\frac{16e}{\\log ^2b}\\right)\\frac{1}{\\log n}$ This completes the proof."
],
[
"Proof of Lemma ",
"Let $f:\\mathbb {Z}_m \\rightarrow \\mathbb {C}$ be any function.",
"Recall that, for $0 \\le j \\le m-1$ , the Fourier coefficients of $f$ are given by $\\hat{f}\\left(j\\right)=\\frac{1}{m}\\sum _{x \\in \\mathbb {Z}_m} f\\left(x\\right)\\exp \\left(-2\\pi i jx/m\\right).$ It is well known that the set of functions $\\lbrace \\exp \\left(2\\pi i jx/m\\right)\\rbrace _{0 \\le j \\le m-1}$ is an orthonormal basis for all functions of the above form, and that $f$ can be expressed as $f\\left(x\\right)=\\sum _{j=0}^{m-1}\\hat{f}\\left(j\\right)\\exp \\left(2\\pi i jx/m\\right).$ Let us consider $f:\\mathbb {Z}_m \\rightarrow [0,1]$ .",
"Thus, Parseval's identity states that $\\sum _{j=0}^{m-1}\\left|\\hat{f}\\left(j\\right)\\right|^2 =\\frac{1}{m}\\sum _{x \\in \\mathbb {Z}_m}f\\left(x\\right)^2 \\le 1.$ Observe that as $\\mathcal {U}_m\\left(x\\right)=1/m$ is the constant function, $\\hat{\\mathcal {U}_m}\\left(j\\right)=0$ for $j \\ne 0$ .",
"Also, for any distribution $\\mu $ , $\\hat{\\mu }\\left(0\\right)=1/m$ .",
"Now $2\\epsilon &\\le & \\sum _{x \\in \\mathbb {Z}_m}\\left|\\mu \\left(x\\right)-\\mathcal {U}_m\\left(x\\right)\\right|\\\\&\\le & \\sqrt{m}\\sqrt{\\sum _{x \\in \\mathbb {Z}_m}\\left|\\mu \\left(x\\right)-\\mathcal {U}_m\\left(x\\right)\\right|^2} \\qquad \\text{(Cauchy Schwartz Inequality)}\\\\&= & m\\sqrt{\\sum _{i=0}^{m-1}\\left|\\left(\\hat{\\mu }\\left(i\\right)-\\hat{\\mathcal {U}_m}\\left(i\\right)\\right)\\right|^2} \\\\&= & m\\sqrt{\\sum _{i=1}^{m-1}\\left|\\hat{\\mu }\\left(i\\right)\\right|^2} \\qquad \\text{($\\hat{\\mathcal {U}_m}\\left(j\\right)=0$ for $j \\ne 0$, and $\\hat{\\mu }\\left(0\\right)=\\hat{\\mathcal {U}_m}\\left(0\\right)=1/m$)}\\\\&\\le & m^{3/2}\\max _{i \\ne 0}\\left\\lbrace \\left|\\hat{\\mu }\\left(i\\right)\\right|\\right\\rbrace .$ Thus, for some $j \\ne 0$ , we have $\\left|\\hat{\\mu }\\left(j\\right)\\right| \\ge \\frac{2\\epsilon }{m^{3/2}}.$ Observe that $\\hat{\\mu }\\left(j\\right)&=&\\frac{1}{m}\\sum _{x \\in \\mathbb {Z}_m} \\mu \\left(x\\right)\\exp \\left(-2\\pi i jx/m\\right) \\\\&=&\\frac{1}{m}\\mathbb {E}_{x \\sim \\mu }\\left[\\exp \\left(-2\\pi i jx/m\\right)\\right]\\\\&=&\\frac{1}{m}\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^{m-j}\\right)^{x}\\right].$ Let $j^{\\prime }=m-j$ .",
"Thus, $\\left|\\hat{\\mu }\\left(j\\right)\\right| \\ge \\frac{2\\epsilon }{m^{3/2}}$ implies that $\\left|\\mathbb {E}_{x \\sim \\mu }\\left[\\left(\\omega ^{j^{\\prime }}\\right)^{x}\\right]\\right| \\ge \\frac{2\\epsilon }{\\sqrt{m}}.$ This concludes the proof.",
"$\\Box $"
],
[
"Proof of Lemma ",
"$\\Box $"
]
] | 1204.1367 |
[
[
"The Galaxy-Dark Matter Connection: A Cosmological Perspective"
],
[
"Abstract We present a method that uses observations of galaxies to simultaneously constrain cosmological parameters and the galaxy-dark matter connection (aka halo occupation statistics).",
"The latter describes how galaxies are distributed over dark matter haloes, and is an imprint of the poorly understood physics of galaxy formation.",
"A generic problem of using galaxies to constrain cosmology is that galaxies are a biased tracer of the mass distribution, and this bias is generally unknown.",
"The great advantage of simultaneously constraining cosmology and halo occupation statistics is that this effectively allows cosmological constraints marginalized over the uncertainties regarding galaxy bias.",
"Not only that, it also yields constraints on the galaxy-dark matter connection, this time properly marginalized over cosmology, which is of great value to inform theoretical models of galaxy formation.",
"We use a combination of the analytical halo model and the conditional luminosity function to describe the galaxy-dark matter connection, which we use to model the abundance, clustering and galaxy-galaxy lensing properties of the galaxy population.",
"We use a Fisher matrix analysis to gauge the complementarity of these different observables, and present some preliminary results from an analysis based on data from the Sloan Digital Sky Survey.",
"Our results are complementary to and perfectly consistent with the results from the 7 year data release of the WMAP mission, strengthening the case for a true 'concordance' cosmology."
],
[
"Introduction",
"Cosmology has reached an important cross-road in the last couple of decades, transitioning from a data-craved to a data-driven field of science.",
"The concordance cosmological picture of a Universe dominated in energy density by dark energy and dark matter has emerged from a vast number of cosmological investigations (see Figure 2 in the contribution by P. J. E. Peebles in the current volume).",
"Ordinary matter forms a fairly small component ($\\sim 4.5\\%$ ) of the energy density of the Universe, with most of it present in the form of intergalactic gas.",
"The energy density of stars in galaxies has an extremely negligible contribution to the energy budget.",
"However, unlike dark matter or dark energy, we can observe the star-light from galaxies directly, and use galaxies as tracers of the underlying matter density field to investigate the properties of the Universe.",
"Unfortunately, this connection between galaxies and (dark) matter is complicated by the fact that galaxies are biased tracers of the mass distribution.",
"Although this `galaxy bias' is generally considered a nuisance when trying to use galaxies to constrain cosmology, it also contains a wealth of information regarding galaxy formation.",
"After all, it is the physics of galaxy formation that determines where, how and with what efficiency galaxies form within the dark matter density field.",
"Therefore, ideally one would like to simultaneously solve for cosmology and galaxy bias.",
"In this paper, we present a method that can do this, and show some preliminary results.",
"The overdensity of galaxies, $\\delta _{\\rm g}$ , at a given position $\\vec{x}$ , is related to the overdensity of matter, $\\delta _{\\rm m}$ , at that position, by a multiplicative term called the galaxy bias, $\\delta _{\\rm g}(\\vec{x}) = b_{\\rm g}\\,\\delta _{\\rm m}(\\vec{x})\\,,$ and this implies that the power spectrum of the galaxy overdensity field on a particular scale is related to the matter overdensity power spectrum by $P_{\\rm gg}(k) = b_{\\rm g}^2(k)\\,P_{\\rm mm}(k)\\,.$ In general, the galaxy bias defined in the above manner is expected to be scale dependent [1].",
"However, on large scales, gravitation is the only relevant physics and galaxy bias is expected to be scale free and equal to a constant.",
"The shape of the galaxy power spectrum on large scales, therefore, mimics the shape of the matter power spectrum.",
"The investigations of cosmological parameters, in particular, the shape parameter $\\Gamma =\\Omega _m h$ , have primarily focussed on large scale precisely for this reason [2], [3], [4].",
"However, it is also clear from the above equations that on large scales, the amplitude of the power spectrum, quantified by $\\sigma _8$ , is perfectly degenerate with the galaxy bias, and large scale observations can only constrain the product $b\\sigma _8$ very well [2].",
"The problem is further complicated by the well known result that brighter galaxies cluster more strongly than fainter galaxies, thus implying that the galaxy bias is also luminosity dependent [5].",
"It is crucial to understand why and how galaxies are biased with respect to the matter distribution in order to break the degeneracy between galaxy bias and the amplitude of the matter power spectrum.",
"The matter distribution in the Universe collapses to form bound clumps of matter called halos.",
"Since halos form preferentially at the peaks of matter density field, halos themselves are biased tracers of the underlying matter density field [6].",
"Galaxies form and reside within these halos of dark matter, and therefore they inherit the bias of their parent halos.",
"Observations of the abundance of galaxies [7], [8], the clustering of galaxies on small scales [9], [10], the gravitational lensing signal due to the dark matter around galaxies [11], [12], and the kinematics of satellite galaxies around halos [13], [14], [15] can all provide important clues regarding this “galaxy-dark matter connection” (i.e., what galaxies resides in what halo).",
"Using this information one can predict the galaxy bias, both as function of scale and as function of galaxy properties (e.g., luminosity).",
"This allows one to break (some of) the degeneracies between galaxy bias and cosmology, and thus to use the observed distribution of galaxies to constrain cosmology.",
"Unfortunately, the clustering of galaxies is not sufficient to fully break degeneracies.",
"This is easy to understand.",
"A generic prediction of hierarchical formation scenarios is that more massive haloes are more strongly clustered.",
"Hence, the observed clustering strength of a particular subset of galaxies (i.e., galaxies in a narrow luminosity bin) on large scales, is a direct measure for the characteristic mass of their dark matter haloes.",
"However, different cosmologies predict different clustering properties of the dark matter haloes.",
"Consequently, the galaxy-dark matter connection inferred from measurements of galaxy clustering are strongly cosmology dependent [26].",
"This degeneracy can be broken using additional, independent constraints on the galaxy-dark matter connection, such as those provided by satellite kinematics or galaxy-galaxy lensing.",
"In this paper, we demonstrate the strength and complementarity of a variety of galaxy observations to constrain cosmological parameters.",
"In particular, we show how observations of galaxy abundances, galaxy clustering and galaxy-galaxy lensing, can be used to constrain cosmological parameters such as the the matter density in units of the critical density, $\\Omega _{\\rm m}$ , and the amplitude of the power spectrum of matter fluctuations, as characterized by $\\sigma _8$ .",
"We rely on the framework of the halo model to analytically predict these observations.",
"The halo model assumes that all the dark matter in the Universe is partitioned over dark matter halos of different sizes and masses [16].",
"The abundance and clustering of these halos of dark matter is set by the underlying cosmological parameters, and this dependence has been well calibrated with the use of numerical simulations [17], [18].",
"A parametric form of how galaxies populate halos, called the halo occupation distribution function, can then be used to predict the abundance and clustering of galaxies using the abundance and clustering of halos [19].",
"In this paper, we use a Fisher matrix analysis to highlight the complementarity of using these different data sets, and we present some preliminary results from an analysis based on existing data."
],
[
"Data",
"We use data from the main galaxy sample of the Sloan Digital Sky Survey (SDSS) [20], [21].",
"In particular, we use the galaxy luminosity function, $\\Phi (L)$ , of [22], the projected two-point correlation functions, $w_{\\rm p}(r_{\\rm p})$ , for six different luminosity bins and $0.2 h^{-1}\\:{\\rm Mpc}$ < $$ rp$\\; < \\over \\sim \\;$ 40 h-1 Mpc$, obtained by \\cite {Zehavi2011}, and the excess surfacedensities $ (rp)$, for the same six luminosity binsbut for $ 0.05 h-1Mpc$\\; < \\over \\sim \\;$ rp$\\; < \\over \\sim \\;$ 2 h-1 Mpc$ from\\cite {Mandelbaum2006}.",
"The latter is proportional to the tangentialshear induced by the mass distribution associated with the galaxies inquestion, and can be measured in the form of weak distortions ofbackground galaxies due to weak gravitational lensing (galaxy-galaxylensing).",
"Our goal is to use a unified model that can describe allthese observables in terms of a simple parametric model, and to usethe existing data to simultaneously constrain cosmological parametersand halo occupation statistics.$"
],
[
"Analytical framework",
"We use the conditional luminosity function (CLF) to specify the halo occupation distribution of galaxies [23].",
"The CLF, $\\Phi (L|M)dL$ , describes the average number of galaxies with luminosity $L\\pm dL/2$ that reside in a halo of mass $M$ , and consists of two components; one for central galaxies and the other for satellites.",
"Motivated by results obtained from a large SDSS galaxy group catalogue [24], we assume that the CLF for central galaxies is described by a log-normal distribution with a logarithmic mean luminosity that depends on mass and a scatter which we assume to be mass-independent.",
"The dependence of the logarithmic mean luminosity, $\\tilde{L_{\\rm c}}$ on halo mass is parameterized using four central CLF parameters, $L_0,\\,M_1,\\,\\gamma _1$ and $\\gamma _2$ , and is given by $\\tilde{L_{\\rm c}}(M) = L_0 \\frac{\\left( M/M_1\\right)^{\\gamma _1}}{\\left(1+M/M_1\\right)^{\\gamma _1-\\gamma _2}} \\,.$ For the satellite component, we assume that it is well described by a Schechter-like function $\\Phi _{\\rm s}(L|M) = \\Phi _* \\left( \\frac{L}{L_{\\rm s}}\\right)^{\\alpha _{\\rm s}}\\exp \\left[-\\left(\\frac{L}{L_{\\rm s}}\\right)^2\\right]\\,,$ where the parameters, $L_{\\rm s}$ , $\\Phi _*$ and $\\alpha _{\\rm s}$ are, in general, functions of halo mass.",
"Guided by the results of [24] based on a galaxy group catalog, we assume that $L_{\\rm s}(M) = 0.562 \\tilde{L}_{\\rm c}(M)$ and that $\\alpha _{\\rm s}$ is a constant, independent of halo mass.",
"The function $\\log \\Phi _{\\rm s}$ is assumed to have a quadratic dependence on $\\log M$ , which is described by three free parameters, $b_0$ , $b_1$ and $b_2$ .",
"Hence, the CLF, which describes the halo occupation distribution as function of galaxy luminosity, is described by a total of 9 free parameters.",
"In addition to specifying the luminosity dependence of the halo occupation distribution, we also need to specify the spatial distribution of galaxies in dark matter halos.",
"Throughout we assume that central galaxies reside at the centers of their halos and that the satellite galaxies follow the density distribution of dark matter without any spatial bias.",
"We have verified that this assumption has a negliglible impact on our cosmological constraints.",
"Given the parameters of the central and satellite CLF, and the cosmological parameters which set the abundance and clustering of halos, we can predict all the observables that we wish to model.",
"For example, the luminosity function simply follows from multiplying the average number of galaxies in a halo of mass $M$ by the number density of halos of that mass, $n(M)$ , and simply integrating this product over all halo masses [25], $\\Phi (L)=\\int \\Phi (L|M) \\, n(M) \\, {\\rm d}M\\,.$ Similarly, the large scale bias of galaxies of luminosity $L$ can be obtained by the following weighted average of the large scale bias of dark matter halos, $b_{\\rm h}(M)$ , according to $b_{\\rm g}(L) = \\frac{ \\int \\Phi (L|M) \\, b_{\\rm h}(M) \\, n(M) \\, {\\rm d}M }{\\int \\Phi (L|M) \\, n(M) \\, {\\rm d}M}\\,,$ Because of the page-limits of these proceedings, we cannot provide the detailed, analytical expresions that we use to calculate the observables $\\Phi (L)$ , $w_{\\rm p}(r_{\\rm p})$ , and $\\Delta \\Sigma (r_{\\rm p})$ for a given model (i.e., cosmology plus CLF).",
"These will be presented in van den Bosch et al.",
"(2012, in preparation).",
"We emphasize, though, that our implementation of the `halo model' [16] properly accounts for (i) the scale dependence of halo bias, (ii) halo exclusion and (iii) residual redshift space distortions that can affect the determinations of galaxy bias [27].",
"Detailed tests using realistic mock galaxy catalogs indicate that our analytical model is accurate to better than 5 percent over the entire range of scales covered by the data.",
"Throughout we adopt a `standard' flat $\\Lambda $ CDM cosmology (i.e., gravity is described by standard General Relativity, neutrino mass is neglected, initial power spectrum is a single power-law, and dark energy is modeled as Einstein's cosmological constant), which is described by 5 cosmological parameters: the matter density parameter $\\Omega _{\\rm m}$ , the baryon density parameter $\\Omega _{\\rm b}$ , the hubble parameter $h$ , the power law index $n_{\\rm s}$ and the parameter $\\sigma _8$ .",
"Our goal is to constrain (subsets) of these cosmological parameters, fully marginalizing over the galaxy-dark matter connection as parameterized by our 9-parameter CLF model."
],
[
"Fisher forecasts",
"In this section we use the Fisher information matrix in order to gauge the accuracy with which constraints on the cosmological parameters $\\Omega _{\\rm m}$ and $\\sigma _8$ can be obtained given the current accuracy of the observables that we wish to model.",
"Since we have three different observables, the luminosity function, galaxy-galaxy clustering and galaxy-galaxy lensing, we start by investigating how each of these different data sets contribute to our constraining power.",
"The different panels of Figure REF show the 68, 95 and 99 percent confidence intervals that can be placed on the cosmological parameters, $\\Omega _{\\rm m}$ and $\\sigma _8$ under varying assumptions of prior information from the 7 year analysis of the WMAP mission [28].",
"The left-hand panel assumes uninformative priors on all of the cosmological parameters in our model.",
"The dashed contours are used to indicate the confidence levels when we perform a joint analysis of the abundance of galaxies and the galaxy-galaxy lensing signal around them.",
"The constraints are fairly weak, in particular, because the galaxy-galaxy lensing signal has only been measured on fairly small scales ($r_{\\rm p}$ < $$ 2h-1 Mpc$).",
"Thisresults in a number of degeneracies between the cosmologicalparameters and the CLF parameters such that $ m$ and$ 8$ are only weakly constrained.",
"The dotted contours show theconfidence contours obtained by combining the luminosity function withthe galaxy-galaxy clustering data.",
"The constraints are significantlytighter, and the improvement is largely due to the addition ofinformation on intermediate scales ($ 2 h-1Mpc$\\; < \\over \\sim \\;$ rp$\\; < \\over \\sim \\;$ 40 h-1Mpc$).",
"Finally the solid contours show the result of a jointanalysis of all three observables.",
"Even in the absence of priorinformation, this joint analysis breaks a number of degeneracies thatare present in our model.",
"The resulting cosmological constraints arecompetitive with the existing constraints on these parameters,demonstrating the potential power of this method.$ The middle panel of Figure REF shows the effect of adding prior information on the secondary cosmological parameters, $\\Omega _b$ , $n_s$ and $h$ from the 7 year WMAP results (hereafter WMAP7).",
"Notice how the addition of prior information can flip the directions of degeneracies (compare the dashed contours in the left-hand and the middle panels).",
"The degeneracy between $\\Omega _{\\rm m}$ and $\\sigma _8$ from our analysis is such that it runs perpendicular to the WMAP7 constraints (shown by the dot-dashed contours).",
"Adding these WMAP7 constraints as additional priors on $\\Omega _{\\rm m}$ and $\\sigma _8$ , further improves the constraints, as shown in the right-hand panel.",
"The cosmological constraints presented above are also competitive with those obtained from studies of the abundance of galaxy clusters as a function of redshift.",
"These galaxy clusters are detected either via their X-ray emission [29], or as overdensities in optical galaxy catalog [30], or with the Sunyaev-Zel'dovich effect [31], [32] and require extensive followup to calibrate the cluster mass-observable relationship.",
"Measurements of cosmic shear have been used to obtain cosmological constraints, albeit weaker, on $\\Omega _m$ and $\\sigma _8$ [33], [34], [35], [36].",
"Cosmological constraints have also been recently obtained by analysing galaxy clustering and the mass-to-number ratio on cluster scales by [37].",
"It is important to note that all of these methods have very different systematics and are complementary to each other and our method: they are all part of a network of tests designed to validate the $\\Lambda $ CDM paradigm."
],
[
"Cosmological constraints",
"We have carried out a joint analysis of all three observables, the luminosity function, the projected galaxy clustering and the galaxy-galaxy lensing signal, and obtained the posterior distribution of our model parameters given these data.",
"We use a Monte-Carlo Markov chain to sample from posterior probability distribution of the parameters.",
"For our fiducial analysis, we impose priors on the secondary cosmological parameters $\\Omega _{\\rm b}$ , $n_{\\rm s}$ and $h$ and completely uninformative priors on the parameters $\\Omega _{\\rm m}$ and $\\sigma _8$ .",
"Our model is able to fit the data sufficiently well with $\\chi ^2$ per degree of freedom of the order of 2.",
"The fits to the data will be presented in Cacciato et al.",
"(2012, in preparation).",
"Preliminary results of our analysis, in the form of 68, 95 and 99 percent confidence contours, are shown in Fig.REF and compared to the WMAP7 results.",
"Figure: 68, 95 and 99 percentconfidence limits on the cosmological parameters Ω m \\Omega _m andσ 8 \\sigma _8 from our analysis (shown in chrome yellow) comparedwith the confidence limits obtained by the analysis of theseven year data from the cosmic microwave background experimentWMAP (shown in green).",
"For a more definitive version, see Cacciatoet al.",
"(2012, in preparation).There are two points worth making.",
"First of all, the constraints obtained from our analysis are in remarkably good agreement with the WMAP7 results, even though we have used no prior information on $\\Omega _{\\rm m}$ and $\\sigma _8$ .",
"The WMAP7 results are based on observations of the microwave background at a very early time in the Universe ($z\\sim 1080$ ) and primarily rest on the physics of perturbations that can be treated with the help of linear perturbation theory.",
"The results from our analysis derive from galaxy observations at redshift $z\\sim 0.1$ and are obtained by modelling extremely non-linear scales, properly marginalizing over the uncertainties related to galaxy bias (i.e., the galaxy-dark matter connection).",
"The agreement in cosmological constraints obtained from these two completely disjunct analyses is extremely striking and provides strong support for the notion of a true `concordance' cosmology: clearly $\\Lambda $ CDM provides an excellent description of data over a large range of scales and cosmic epochs.",
"Secondly, the constraints obtained from our analysis are both competitive with and complementary to those obtained by the WMAP analysis.",
"This is also in agreement with the complementarity expected from the Fisher forecast presented in the previous subsection."
],
[
"Summary",
"To summarize, observations of galaxies are an excellent way of probing the underlying matter distribution in the Universe and thereby obtaining precise constraints on the cosmological model.",
"We have shown that a joint analysis of the abundance of galaxies (characterized by the galaxy luminosity function), the clustering of galaxies (characterized by the projected two-point correlation functions), and the clustering of dark matter around galaxies (characterized by galaxy-galaxy lensing) can be a useful way to constrain the cosmological parameters.",
"We have modelled each of these observations in the analytical framework of the halo model.",
"The halo occupation distribution of galaxies in our model was specified by the parametric CLF model.",
"Using a Fisher matrix analysis, we have shown that the cosmological information contained in the three observables described above is complementary to each other and a joint analysis of these datasets is able to break a number of degeneracies between the CLF parameters and the cosmological parameters.",
"We followed up our Fisher forecast results, by constraining our model parameters using the actual data.",
"We have shown that the resulting constraints on the cosmological parameters $\\Omega _{\\rm m}$ and $\\sigma _8$ are in remarkable agreement with constraints from the analysis of the WMAP data.",
"This is yet another jewel in the crown of the $\\Lambda $ CDM model, which continues to reign king.",
"We are currently exploring the use of our method to constrain extensions of the $\\Lambda $ CDM model that include cosmological parameters such as the neutrino density parameter, the dark energy equation of state, non-gaussianity in the initial density fluctuations and modifications to gravity.",
"However, a large amount of work is still required in order to calibrate the predictions of the abundance and clustering of dark matter haloes in these alternative cosmologies.",
"Although many such calibrations for the extended $\\Lambda $ CDM already exist in the literature, the halo mass definitions used in these calibrations are often not suitable for use in the halo model [38].",
"We expect to address these issues and extensions of our methods in future work.",
"Finally, an interesting by-product of our analysis is a detailed, statistical description of the galaxy-dark matter connection, as parameterized by the CLF, fully marginalized over cosmological uncertainties.",
"This galaxy-dark matter connection is the outcome of a large number of poorly understood astrophysical processes, such as the formation of stars and the regulation of galaxy growth by feedback, that shape the formation and evolution of galaxies in our Universe.",
"By constraining the CLF we are therefore constraining the integral effect of these (and other) processes.",
"Hence, the constraints on the CLF parameters obtained from our analysis will be of great value to inform theoretical models of galaxy formation (both semi-analytic models and direct numerical simulations)."
],
[
"Acknowledgements",
"This research has spanned more than three years during which the affiliations of the different authors have changed.",
"SM, FvdB and MC would like to acknowledge support from the Max Planck Institute for Astronomy and the University of Utah during the partial conduct of this research.",
"We also thank Jeremy Tinker, Alexie Leauthaud, Martin White, Andrey Kravtsov, Eduardo Rozo, Matt Becker, Yin Li and Wayne Hu for many interesting discussions and possible extensions of this research work."
]
] | 1204.0786 |
[
[
"An Introduction to Quantum Bayesian Networks for Mixed States"
],
[
"Abstract This paper is intended to be a pedagogical introduction to quantum Bayesian networks (QB nets), as I personally use them to represent mixed states (i.e., density matrices, and open quantum systems).",
"A special effort is made to make contact with notions used in textbooks on quantum Shannon Information Theory (quantum SIT), such as the one by Mark Wilde (arXiv:1106.1445)"
],
[
"Abstract",
"This paper is intended to be a pedagogical introduction to quantum Bayesian networks (QB nets), as I personally use them to represent mixed states (i.e., density matrices, and open quantum systems).",
"A special effort is made to make contact with notions used in textbooks on quantum Shannon Information Theory (quantum SIT), such as the one by Mark Wilde (arXiv:1106.1445)"
],
[
"Introduction",
"This paper is intended to be a pedagogical introduction to quantum Bayesian networks (QB nets), as I personally use them to represent mixed states (i.e., density matrices, and open quantum systems).",
"A special effort is made to make contact with notions used in textbooks on quantum Shannon Information Theory (quantum SIT), such as the one by Mark Wilde[1].",
"QB nets are a generalization of classical Bayesian networks (CB nets) to quantum mechanics.",
"CB nets have been hot topic in AI circles since the seminal work of Judea Pearl and collaborators that started in the 1980's.",
"A very complete book on CB nets is the one by Koller and Friedman[2].",
"Just like mankind has devised many names for the idea of God, there are other names for CB nets and variations of the idea.",
"Others have called them causal probabilistic diagrams, factor graphs, probabilistic system diagrams, etc.",
"To be sure, there are some differences between some of these diagrams and CB nets, but all seem to be striving to conjure up the same divine concept.",
"Some of the close siblings of CB nets are discussed in Ref.",
"[3], an IEEE magazine article by Loeliger.",
"One variation on the CB net idea involves using graphs in which the arrows (a.k.a.",
"directed edges) represent tensor indices and the boxes (a.k.a.",
"nodes, vertices) represent transition matrices.",
"In this approach, call it the tensor-graphs approach, each arrow coming out of a fixed node carries different stuff.",
"In the CB nets approach, the nodes again represent transition matrices, but the arrows perform a very different job.",
"Each arrow coming out of a fixed node carries the same stuff, namely the name of the node the arrow originates at.",
"This difference might appear subtle or even insignificant to the untrained eye, but Bayesian network believers (like me) swear by it, claiming that it is clearer and more powerful than the tensor-graphs approach when dealing with probabilities.",
"Bayesian network believers think that using tensor graphs to describe probability networks is like trying to fill round holes with square pegs.",
"Even when the pegs fit, they don't do a very good job.",
"Classical information theorists have been using tensor-graphs in their field for a long time.",
"See, for example, the chapter on “network information theory\" in the book by Cover and Thomas, Ref.[4].",
"Or look at the book by El Gamal and Kim, Ref.",
"[5], which is devoted exclusively to the subject of network information theory.",
"Quantum information theorists have been using tensor-graphs in their field for a long time too, at least since the seminal work by Schumacher and collaborators.",
"For an early example of a paper by Schumacher that uses tensor-graphs, see, for example, Ref.",
"[6], published in 1996.",
"Some early quantum information papers, for example the seminal paper Ref.",
"[7] by Bennett et al.",
"also use a version of tensor-graphs, but they use them in a very loose, ambiguous, imprecise way.",
"Ref.",
"[7] even has some diagrams that sound like sacrilege to the ears of this Bayesian nets believer, such as diagrams that have nodes intended to represent buckets of sewage (Fig.13 in Ref.[7]).",
"Not only do adherents to “the QB net way\" espouse being good parents by treating all arrows (= children) coming out of a fixed node (= parent) the same.",
"We are also very strict about our nodes.",
"Each node has a numerical “value\", and the whole graph also has a value which equals the product of the node values.",
"Nodes aren't there just for decoration, or as mere labels with no numerical value assigned to them, or only to convey an abstract notion like “do this operation now\".",
"Many quantum information papers don't use diagrams at all.",
"They specify their quantum “protocols\" or algorithms in terms of “pseudo code\".",
"In my opinion, those papers would be much clearer if they described their algorithms using both, pseudo-code and QB nets, whenever this is possible.",
"The recent book on quantum information theory by Mark Wilde[1] earns high marks in this regard, as it uses a lot of diagrams.",
"Wilde's diagrams also have a fairly precise meaning.",
"However, they are tensor-graphs, not the wonderful QB nets.",
"My own work on QB nets started about 15 years ago with Ref.[8].",
"In that early paper, I dealt only with QB nets for pure quantum states.",
"I've been using QB nets for mixed states since at least Ref.[9].",
"This paper repeats some of the ideas of Ref.",
"[9] for QB nets of mixed states, with (hopefully) some small improvements.",
"I've also written a Mac application that does QB nets called Quantum Fog.I stopped Quantum Fog development in 2006.",
"The application is still available for free at Ref.[10].",
"Quantum Fog's last version (Version 2.0) is known to work with Mac OS X $\\le $ 10.4.",
"It probably works with some higher versions of Mac OS X too.",
"Quantum Fog only does QB nets for pure states.",
"That's not because QB nets can't deal with mixed states as some people think.",
"It's only because I stopped developing Quantum Fog before I had a chance to add to it the capability to do mixed state calculations.",
"I also have a blog called Quantum Bayesian Networks (Ref.",
"[11]) in which I regularly post articles about Bayesian networks and quantum computing.",
"Subsequent to Refs.",
"[8], [9], other workers have devised their own types of diagrams for doing quantum information theory.",
"Their diagrams are very different from the QB nets in this paper.",
"Leifer and Poulin in Ref.",
"[12], and later Leifer and Spekkens in Ref.",
"[13], postulate some directed acyclic graphs, but they assign a whole density matrix to each node of the graph.",
"Furthermore, their node density matrices would be quite hard to calculate in practice, especially for complicated graphs.",
"In comparison, a whole QB net is used to describe one density matrix.",
"And the transition matrix that a QB net assigns to each node comes from the definition of a probability amplitude, a really basic thing that requires almost no calculation— certainly less calculation than the node density matrices of Leifer and coworkers.",
"I like to think of QB nets as: A light container of data, useful as a data structure in computer programming.",
"A vehicle rather than a destination.",
"A transparent, tidy way of organizing a lot of data in a pictorial way, prior to intensive calculation, not as being itself the outcome of a major calculation.",
"Coecke (Ref.",
"[14]) and collaborators use category theory to define their diagrams.",
"In comparison, QB nets are much less abstract.",
"Defining them requires no category theory, just standard, run-of-the-mill quantum mechanics.",
"The QB nets in this paper are much simpler than the diagrams of (a) and (b).",
"Simplicity can be a virtue in mathematics (consider, for example, abstract algebra's definition of a group, which is simplicity itself).",
"QB nets are, however, complicated enough to be very expressive and useful; that is, they allow one to express numerous quantum mechanical concepts in a useful, practical and enlightening way.",
"QB nets are a very parsimonious extension of CB nets to quantum mechanics.",
"That is, the definition of QB nets is the smallest possible modification of the definition of CB nets that I can come up with, but enough of a modification so that one can do proper quantum mechanics with them.",
"Keeping QB nets close to CB nets can be very fruitful, because much is already known about CB nets.",
"And CB nets use classical probability so we can sharpen our classical understanding of a problem with them and then try to extrapolate that understanding to QB nets.",
"QB nets retain the same structure as CB nets and can be reduced to them very easily, simply by applying the dephasing operator “cl\" (defined below) to each node.",
"Thus, QB nets make the connection to the classical case very direct and explicit."
],
[
"Basic Notation",
"As usual, $,, $ will denote the integers, real numbers, and complex numbers, respectively.",
"For $a,b\\in $ such that $a\\le b$ , let $Z_{a,b}=\\lbrace a,a+1,a+2,\\ldots , b\\rbrace $ .",
"Let $\\delta ^x_{y}=\\delta (x,y)$ denote the Kronecker delta function; it equals 1 if $x=y$ and 0 if $x\\ne y$ .",
"For any matrix $M\\in ^{p\\times q}$ , $M^*$ will denote its complex conjugate, $M^T$ its transpose, and $M^\\dagger = M^{*T}$ its Hermitian conjugate.",
"Random variablesWe will use the term “random variables\" in both classical and quantum physics.",
"Normally, random variables are defined only in classical physics, where they are defined to be functions from an outcome space to a range of values.",
"For technical simplicity, here we define a random variable $$ , in both classical and quantum physics, to be merely the label of a node in a graph, or an n-tuple $_K$ of such labels.",
"Each node or random variable of a CB or QB net is akin to a spacetime event or a collection of them.",
"will be denoted by underlined letters; e.g., $$ .",
"The (finite) set of values (states) that $$ can assume will be denoted by $S_$ .",
"Let $N_=|S_|$ .",
"The probability that $=a$ will be denoted by $P(=a)$ or $P_(a)$ , or simply by $P(a)$ if the latter will not lead to confusion in the context it is being used.",
"We will use $pd(S_)$ to denote the set of all probability distributions with domain $S_$ .",
"In quantum physics, $$ has a fixed, orthonormal basis $\\lbrace {a}_:a\\in S_\\rbrace $ associated with it.",
"The vector space spanned by this basis will be denoted by $_$ .",
"Other spans of $_$ that are not necessarily orthonormal will be denoted by Greek letters with subscripts as in $\\lbrace {\\psi _j}_\\rbrace _{\\forall j}$ .",
"In quantum physics, instead of probabilities $P(=a)$ , we use “probability amplitudes\" (or just “amplitudes\" for short) $A(=a)$ (also denoted by $A_(a)$ or $A(a)$ ).",
"In place of $P(a)\\ge 0$ and $\\sum _a P(a)=1$ , one has $\\sum _a |A(a)|^2=1$ .",
"Besides probability amplitudes, we also use density matrices.",
"A density matrix $\\rho _$ is a Hermitian, non-negative, unit trace, square matrix (or the associated linear operator) acting on $_$ .",
"We will use $dm(_)$ to denote the set of all density matrices acting on $_$ .",
"If $\\rho _{,}\\in dm(_{,})$ , and $\\rho _= _(\\rho _{,})= \\sum _a {a}_\\rho _{,}{a}_\\in dm(_)$ , we will say that $\\rho _$ is a partial trace of $\\rho _{,}$ .",
"Given a density matrix $\\rho _{_1,_2,_3,\\ldots }\\in dm(_{_1,_2,_3,\\ldots })$ , its partial traces will be denoted by omitting its subscripts for the random variables that have been traced over.",
"For example, $\\rho _{_2} = tr_{_1,_3}\\rho _{_1,_2,_3}$ .",
"Sometimes, when two random variables 1 and 2 satisfy $S_{{1}}=S_{{2}}$ , we will omit the indices 1 and 2 and refer to both random variables as $$ .",
"We shall do this sometimes even if the random variables 1 and 2 are not identically distributed!",
"This notation, if used with caution, does not lead to confusion and does avoid a lot of index clutter.",
"When we want to make explicit that an operator $\\Omega $ maps states in $_$ to states in $_$ , we will indicate this with a subscript (or superscript) as $\\Omega _{\\leftarrow }$ or as $\\Omega _{|}$ .",
"In cases where $S_=S_$ , we will sometimes write $\\Omega _{}$ instead of the clearer but longer $\\Omega _{\\leftarrow }$ or $\\Omega _{|}$ .",
"The tensor product symbol $\\otimes $ will often be omitted.",
"Sometimes, when two vectors are being tensored, we will list the two vectors vertically instead of horizontally (the latter is more common in the literature).",
"For example, we might write = = c .",
"This doesn't lead to confusion as long as we indicate what vector space each vector lives in.",
"(In the above example, ${\\phi }_$ clearly lives in $_$ and ${\\psi }_$ in $_$ ).",
"In this paper, we consider networks (graphs) with $N$ nodes.",
"Each node is labeled by a random variable $_j$ , where $j\\in $ .",
"For any $J\\subset $ , the ordered set of random variables $_j$ $\\forall j\\in J$ (ordered so that the integer indices $j$ increase from left to right) will be denoted by $_{J}$ .",
"For example, $_{\\lbrace 2,4\\rbrace }=(_2,_4)$ .",
"We will often call the values that $_{J}$ can assume $x_{J}$ .",
"For example, $x_{\\lbrace 2,4\\rbrace }=(x_2,x_4)$ .",
"We will often abbreviate $_{}$ by just $$ .",
"We will often call the values that $$ can assume $x.$ ."
],
[
"The Sandbox and its Dual",
"For any expressions $\\Omega (x)$ and $\\rho $ for which this makes sense, we will use the shorthand notation: (x) c xx' = (x) (x') .",
"Here “h.c.\"",
"is an abbreviation of “hermitian conjugate\".",
"We will usually use this notation with $\\rho =1$ .",
"This notation is especially useful when $\\Omega (x)$ is a long expression and we want to avoid writing it twice.",
"We will refer to the space inside the set of square brackets to the left (resp., right) of $\\rho $ as the sandbox (resp., its dual or mirror sandbox)."
],
[
"The Meta State",
"QB nets for pure quantum states were first defined in Ref.[8].",
"A QB net for a pure state consists of a directed acyclic graph (DAG) and a transition matrix (a complex matrix) assigned to each node of the graph.",
"The transition matrices must satisfy certain requirements.",
"An example of such a pure state QB net is: =++[o][F-] *[dl]* *[ul][ll] If $a\\in S_, b\\in S_, c\\in S_$ , $A_{|,}(a|b,c)$ is the transition matrix associated with node $$ , $A_{|}(b|c)$ is the transition matrix for node $$ , and $A_(c)$ is the transition matrix for node $$ .",
"We must have a |A|,(a|b,c)|2=1 , b |A|(b|c)|2=1 , c |A(c)|2 = 1 .",
"Define the total probability amplitude $A_{,,}(a,b,c)$ by A,,(a,b,c) = A|,(a|b,c) A|(b|c) A(c) .",
"Note that Eqs.",
"() imply a,b,c |A,,(a,b,c)|2 = 1 .",
"Henceforth, we will sometimes omit the node subscripts from the probability amplitudes.",
"For example, we might use $A(a|b,c)$ instead of $A_{|,}(a|b,c)$ , if no confusion will arise.",
"This is analogous to probability theory, where we often use $P(a|b,c)$ instead of $P_{|,}(a|b,c)$ or $P(=a|=b,=c)$ for a probability.",
"More generally, suppose the graph has $N$ nodes $_1,_2, \\ldots ,_N$ .",
"For $j\\in $ , a node $_j$ with possible states $x_j\\in {_j}$ and with parent nodes $_{pa(_j)}$ where $pa(_j)\\subset $ , has a transition matrix $A(x_j|x_{pa(_j)})$ which satisfies xjj |A(xj|xpa(j))|2=1 for all $x_{pa(_j)}\\in {_{pa(_j)}}$ .",
"Let $x_.=(x_1,x_2, \\ldots ,x_N)$ If the total amplitude $A(x.",
")$ is defined by A(x.)",
"= j A(xj|xpa(j)) , then Eqs.",
"() imply x.|A(x.",
")|2=1 .",
"Given any transition matrix of the form $A(r|\\vec{c})$ , call $r$ the row index and $\\vec{c}$ the column indices.",
"Call all $A(r|\\vec{c})$ entries with any $r$ but fixed $\\vec{c}$ , a column vector of the transition matrix.",
"Eqs.",
"() say that each column vector of a transition matrix is normalized.",
"The column vectors may also be mutually orthogonal, in which case we say that the column vectors are orthonormal.",
"For example, the transition matrix for node $$ in the QB net above might also satisfy: a A|,(a|b,c) c b,cb',c' = bb'cc' .",
"The isometry nodes defined below are another example of a case where the column vectors of the transition matrix are orthonormal.",
"In general, it is not necessary that the column vectors be orthonormal.",
"For example, for the marginalizer nodes defined below, they aren't.",
"What is always necessary is that the total amplitude be normalized, as in Eq.",
"(), so as to enforce the “unitarity\" of quantum mechanics.",
"The meta state of a QB net was first defined in Ref.[9].",
"A meta ket state is a pure quantum state represented as a ket or as a QB net.",
"For example: meta,,= =++[o][F-] *[dl]* *[ul][ll] = a,b,c A,,(a,b,c) c a b c , where $A(a,b,c)$ is defined by Eq.().",
"We assume the states $\\lbrace {a}_\\rbrace _{\\forall a}$ are orthonormal, and likewise for $\\lbrace {b}_\\rbrace _{\\forall b}$ and $\\lbrace {c}_\\rbrace _{\\forall c}$ .",
"Note that each node of the QB net has its own ket and its own index that is summed over (i.e., bound).",
"For example, node $$ has ket ${b}_$ and bound index $b$ .",
"The projection operator of a meta ket defines a density matrix which we will call the meta density matrix of the protocol under consideration.",
"For example, the meta density matrix of the above meta ket is given by: (meta),, = meta,, = =++[o][F-] *[dl]* *[ul][ll] ."
],
[
"Generic Nodes",
"We find it convenient to define certain special, generic types of nodes.",
"Marginalizer nodes were first defined in Ref.[8].",
"In the current version of Quantum Fog, marginalizer nodes are usually denoted by black bullets, whereas non-marginalizer nodes are denoted by larger colored circles.",
"In this paper, we will represent marginalizer nodes by writing a small delta near them.",
"This node “decoration\" or subscript is easy to draw by hand and also easy for the eye to spot.",
"For example: = =+++++[o][F-] 2** *(1,1) [ull]> [dll]> 2** , where $_{{1}}$ and $_{{2}}$ have the same state space, call it $S_$ .",
"Likewise, $_{{1}}$ and $_{{2}}$ have the same state space, call it $S_$ .",
"Note that the subscripts 1 and 2 are acting like a “time\" index along a sort of timeline.",
"For all $a,a^{\\prime }\\in S_$ and $b,b^{\\prime }\\in S_$ , l A2|1,1(a|a',b')= aa' A2|1,1(b|a',b')= bb' .",
"Thus some column vectors of a marginalizer node are equal to each other.",
"To avoid index clutter, we will sometimes omit the indices 1 and 2 from the graph of Eq.",
"(), and draw instead the following graph: = =++[o][F-] ** *(,) [ull]> [dll]> ** Grounded nodes are root nodes (i.e., nodes with no incoming arrows, only outgoing ones) which have a deterministic probability amplitude (i.e., an amplitude that equals 1 for just one of the possible states of the node and zero for all the other states).",
"Grounded nodes will be indicated by writing a zero near them.",
"Here is an example of a QB net with a grounded node: =++[o][F-] **[dll] (,)** *[ull]<0 , whereWe are assuming that $0\\in S_$ .",
"It doesn't matter if the amplitude $A(b)$ of node $$ equals $\\delta _b^0$ , or $\\delta _b^{b_0}$ where $b_0\\in S_$ .",
"Either way, it's still a grounded node.",
"An alternative to writing a zero next to a node to indicate that it's grounded might be writing instead the letters “grd\" or the electrical symbol for a ground.",
"A(b)= b0 , for all $b\\in S_$ ."
],
[
"Isometries",
"Consider the following QB net =++[o][F-] [l] , where $A(b|a)$ satisfies bSA(b|a) c aa' = aa' , for all $a,a^{\\prime }\\in S_$ .",
"The node $$ in the above QB net is called an isometry node, or just an isometry for simplicity.",
"Eq.",
"() is saying that the column vectors of the transition matrix $A(b|a)$ are orthonormal.",
"This is only possible if $N_\\ge N_$ .",
"If $N_= N_$ (i.e., transition matrix is square), then the transition matrix is unitary.",
"If $N_> N_$ (i.e., transition matrix is rectangular with more rows than columns), then we can use the well-known, so called Gram-Schmidt procedure to add more columns to the transition matrix (“extend it\") to produce a unitary matrix.",
"Since $N_\\ge N_$ and the sets $S_$ , $S_$ are finite, we may assume without loss of generality that $S_\\supset S_$ .",
"For every $b,A\\in S_$ , letWe are using the symbol $A$ both for an element $A$ of $S_$ , and for amplitudes $A(\\cdot )$ .",
"It's easy to tell which usage is intended in each instance, so this should cause no confusion.",
"A(b|A) ={ lA(b|a) ifA=aS given by Gram-Schmidt procedureifAS-S. .",
"Then bS A(b|A) c AA' =AA' for all $A,A^{\\prime }\\in S_$ .",
"Thus, A(b|A)= bUA , where $U_{}$ is unitary.",
"Here is a pictorial representation, in terms of QB nets, of the procedure just outlined for extending an isometry to a unitary matrix: =++[o][F-] [l] =++[o][F-] A[l] , where $S_=S_{A}\\supset S_$ .",
"Consider the following QB net =++[o][F-] (,)[l] , where $A(a,b|a^{\\prime })$ satisfies aS bSA(a,b|a') c a'a” = a'a” , for all $a^{\\prime },a^{\\prime \\prime }\\in S_$ .",
"The node $(, )$ in the above QB net is a special case of the general isometry node presented previously.",
"Just as in the case of a general isometry, the transition matrix $A(a,b |a^{\\prime })$ can be extended to a unitary matrix.",
"Assume $0\\in S_$ .",
"If we define A(a, b|a', b'=0)= A(a,b|a') , then we can find a unitary matrix $U_{,}$ such that, for all $a,a^{\\prime }\\in S_$ and $b,b^{\\prime }\\in S_$ , A(a, b|a', b')= ccc aa' U, bb' .",
"Here is a pictorial representation, in terms of QB nets, of the procedure just outlined for extending an isometry to a unitary matrix: =++[o][F-] (,)[l] = =++[o][F-] **[dll] (,)** *[ull]<0 =++[o][F-] **[dll] (,)** *[ull]"
],
[
"Freeing a Bound Index",
"We've defined the QB net corresponding to the meta ket state as having for each node $$ : (1) an index $b$ that is summed over (bound), and (2) a ket ${b}_$ .",
"One can free an index $b$ of a node $$ of a meta ket state by multiplying that node by ${b}$ or ${b}{b}$ .",
"One can use QB nets to represent these two operations.",
"For example, if the meta state is, =++[o][F-] [l] = a,b A(b|a)A(a)ba, then =++[o][F-] * @[l]<<b[l] = a A(b|a)A(a)a, and =++[o][F-] * @[l]<<<<bb[l] = ba A(b|a)A(a)a."
],
[
"Classical Communication",
"Quantum information theorists call “classical communication\" the act of measuring an observable at one event and then using the result of that measurement to start a new event.",
"Classical communication can be represented using QB nets.",
"For example, if $S_= S_$ , then =++[o][F-] [l]<<b@[l]<<b[l] = d,a A|(d|b)A(b) A|(b|a)A(a) da."
],
[
"(Coherent or Incoherent)-(Scalar or Vector) Sums",
"For any $\\rho _{,}\\in dm(_{,})$ , define (,) = = b b , , cl(,) = cl, = b bb , , sl(,) = = bb , .",
"Note that (1) = N, cl(1) = 1, sl(1) = N. Note also that the product of any two operators in the set $F=\\lbrace 1, _, {\\rm cl}_,{\\rm sl}_\\rbrace $ can be expressed in terms of a single one of them.",
"For example, cl(,)= (,) , (,)= N(,) , etc..",
"Hence, a product of any number of the operators in $F$ can be expressed in terms of a single one of them.More formally, if we define $c= {\\rm cl}_,$ $\\sigma = {\\rm sl}_/N_,$ $\\tau = _/N_$ , and $F= \\lbrace 1, c,\\sigma ,\\tau \\rbrace $ , then it is easy to check that for all $f\\in F$ , $f\\tau = \\tau $ , $f\\sigma =\\sigma $ , and $f c=\\left\\lbrace \\begin{array}{l}c\\;\\;{\\rm if}\\;\\;f\\in \\lbrace 1,c\\rbrace \\\\\\tau \\;\\;{\\rm otherwise}\\end{array}\\right.$ .",
"Although $F$ is closed under composition, it is not a group.",
"This is not surprising since $c,\\rho ,\\sigma $ are irreversible transformations.",
"For each node $$ of a meta density matrix, there is an index $b$ that is summed over and a ket ${b}_$ .",
"Furthermore, the $\\sum _b$ is inside the sandbox, so we say that it's a coherent sum.",
"Because the term being summed (i.e., the summand) includes the ket ${b}_$ , we say it's a vector sum.",
"If the $\\sum _b$ were outside the sandbox (and index $b$ appeared in both the sandbox and its dual), we would call it an incoherent sum.",
"If the summand did not include ${b}_$ , we would call it a scalar sum.",
"The operators $_(\\cdot )$ , ${\\rm cl}_(\\cdot )$ , ${\\rm sl}_(\\cdot )$ act on the meta density matrix to change the coherent-vector sum over a node $$ to an incoherent-scalar, or an incoherent-vector, or a coherent-scalar sum.",
"Let's illustrate this with an example.",
"Consider the following meta density matrix as an example.",
"Note that in this meta density matrix, for the random variable $$ , there is a coherent-vector sum over the index $b$ .",
", = a,b A(b|a)A(a)ba = =++[o][F-] [l] “Tracing\" (i.e., taking a partial trace of) the random variable $$ means doing an incoherent-scalar sum over the index $b$ .",
"(,) = = ba A(b|a)A(a)a = =++[o][F-] [l]>>tr =b =++[o][F-] * @[l]<<b[l] “Classicizing\", or “Making classical\" the random variable $$ means doing an incoherent-vector sum over the index $b$ .",
"(This operation is also sometimes described as “dephasing\" because we are throwing away some off-diagonal terms).",
"cl(, )= cl, = ba A(b|a)A(a)ba = =++[o][F-] [l]>>cl =b =++[o][F-] *@[l]<<<<bb[l] “Slashing\" the random variable $$ means doing a coherent-scalar sum over the index $b$ .",
"sl(, )= , = a,b A(b|a)A(a)a = =++[o][F-] [l] = b =++[o][F-] *@[l]<<b[l] Note that the operators in $F$ are all irreversible transformations (except for the 1).",
"The meta state is truly “at the top of the food chain\": Once the operators in $F$ take the meta state to something else, no operator or combination of operators in $F$ can bring back the same meta state."
],
[
"Ensembles, Purification",
"An ensemble is a set $\\lbrace \\sqrt{w_j}{\\psi _j}_\\rbrace _{\\forall j}$ where the weights $w_j$ are non-negative numbers that sum to 1, and for all $j$ , the states ${\\psi _j}_\\in _$ are normalized but they are not necessarily mutually orthogonal.",
"The density matrix for this ensemble is = j wj j .",
"Define two ensembles as being equivalent if they have the same density matrix.",
"This defines an equivalence relation.",
"Elements of the same equivalence class are physically indistinguishable.",
"The density matrix Eq.",
"() can be purified, meaning that it can be expressed as a partial trace of a pure state.",
"One way of doing this is as follows.",
"Clearly, $\\rho _$ also equals = r x xx|j j wjj .",
"Thus =, , where , = x,jA(x,j) c x j = =++[o][F-] [l] , where A(x,j) = A(x|j)A(j) , and A(x|j) = x|j , A(j) = wj ."
],
[
"Measurement Superoperators",
"A superoperator is a linear operator that maps $dm(_)$ into $dm(_)$ .",
"A measurement is defined as a set $\\lbrace K_\\mu |\\mu \\in S_{{\\mu }}\\rbrace $ of operators $K_\\mu $ called Krauss operators that map states in $_$ to states in $_$ .",
"We assume $N_\\le N_$ .",
"($N_= |S_| = dim(_)$ and the same for $$ ).",
"The Krauss operators must also satisfy: KK= 1 .",
"Each Krauss operator $K_\\mu $ can be used to define a superoperator $\\$_\\mu (\\cdot )$ as follows.",
"Let $\\rho _\\in dm(_)$ , and, for each $\\mu $ , let $\\sigma _{|\\mu }\\in dm(_)$ .",
"Then define the measurement superoperator $\\$_\\mu (\\cdot )$ by $()= | = KK P() , where P() = ( KK)= (KK) .",
"Note that the $P(\\mu )$ are non-negative and P()=1 .",
"Note also that (|)=1 for all $\\mu $ .",
"A von Neumann measurement (for instance, $K_\\mu = {\\mu }{\\mu }$ ) is a measurement $\\lbrace K_\\mu \\rbrace _{\\forall \\mu }$ that satisfies: K= K, KK' = ', K= 1 .",
"Here are some other examples of measurements (you can check that $\\sum _a K_a^\\dagger K_a =1$ for each example) Tracing: $K_a = {a}_$ Making a node classical (i.e., dephasing it): $K_a = {a}_{a}_$ Classical (incoherent) communication: $K_a = {a}_{}{a}_{}$ , where $S_=S_$ .",
"Coherent communication (only one Krauss operator): $K = \\sum _a{a}_{}{a}_{}$ , where $S_=S_$ .",
"A measurement $\\lbrace K_\\mu \\rbrace _{\\forall \\mu }$ can be extended to a unitary operator as follows.",
"For every $b\\in S_$ , $\\mu \\in S_{{\\mu }}$ , $a\\in S_$ , define A(b,|a)=b|K|a .",
"Since for all $a,a^{\\prime }\\in S_$ , b, A(b,|a) c aa' = b, b|K|a a'|K|b =aa' , $A(b,\\mu |a)$ defines an isometry.",
"Assume $0\\in S_{{\\mu }}$ .",
"Let A(b,|a,'=0)=A(b,|a) .",
"Since $N_\\le N_$ , we can use Gram Schmidt to find a unitary operator $U_{,{\\mu }}$ such that A(b,|A, ')= lcr b AA U, ' for all $b,A\\in S_=S_{{A}}$ and $\\mu ,\\mu ^{\\prime }\\in S_{{\\mu }}$ ."
],
[
"RINNO (POVM)",
"A POVM, which I prefer to call a RINNO, is a Resolution of the Identity by Non Negative Operators.",
"Thus, a RINNO $\\lbrace R_\\mu \\rbrace _{\\forall \\mu }$ satisfies R= 1, R0 .",
"(Each $R_\\mu $ is a square matrix.",
"A square matrix $M$ is said to be non-negative, or said to satisfy $M\\ge 0$ , if $v^\\dagger M v\\ge 0$ for all complex column vectors $v$ .)",
"Suppose $\\rho _\\in dm(_)$ , and for each $\\mu $ , $R_\\mu $ maps $_$ into itself.",
"For each $\\mu $ , define P() = (R) .",
"By Eqs.",
"(), the $P(\\mu )$ are non-negative and satisfy $\\sum _\\mu P(\\mu )=1$ .",
"A RINNO $\\lbrace R_\\mu \\rbrace _{\\forall \\mu }$ can be constructed from a measurement $\\lbrace K_\\mu \\rbrace _{\\forall \\mu }$ by setting R= KK for each $\\mu $ .",
"The definition of a measurement $\\lbrace K_\\mu \\rbrace _{\\forall \\mu }$ and Eqs.",
"() imply Eqs.",
"()."
],
[
"Channel Superoperators",
"Suppose $\\lbrace K_\\mu |\\mu \\in S_{{\\mu }}\\rbrace $ is a measurement with Krauss operators $K_\\mu :__$ .",
"Let $\\rho _\\in dm(_)$ and $\\sigma _{}\\in dm(_)$ .",
"Then define the channel superoperator $\\$(\\cdot )$ byKrauss showed that for any superoperator $\\$(\\cdot )$ , $\\$(\\cdot )$ is a channel superoperator iff $\\$(\\cdot )$ is “completely positive\".",
"$()= = KK.",
"Note that a channel superoperator is a weighted sum of measurement superoperators (i.e., $\\$ = \\sum _\\mu P(\\mu ) \\$_\\mu $ ).",
"$\\rho _$ can always be expressed as = j wj j , where the weights $\\lbrace w_j\\rbrace _{\\forall j}$ are non-negative numbers that sum to one, and the states $\\lbrace {\\psi _j}_\\rbrace _{\\forall j}$ are all normalized but not necessarily mutually orthogonal.",
"Note that for all $b,b^{\\prime }\\in S_$ , b||b' = ,j b|K|jwj c bb' =,j ccc bjA U, 0 wj c bb' where, as discussed in Section , $U_{,{\\mu }}$ is a unitary matrix that extends the measurement $\\lbrace K_\\mu |\\mu \\in S_{{\\mu }}\\rbrace $ .",
"Eq.",
"() for $\\rho _$ and Eq.",
"() for $\\sigma _$ can be represented as follows in terms of QB nets: = =++[o][F-] [l] , = , =+++[o][F-] *A[dl][l] (,) [ul]>> [dl]>>** * [ul]<0* , where A() = 0 , A(j) = wj , A(A|j) = {l A|jif AS 0if AS-S. , A(b,|A,')= ccc bAA U, ' , A(b|b',')=bb' , A(|b',')=' ."
],
[
"Complementary Channel",
"The channel superoperator $\\$(\\cdot )$ given by Eq.",
"() can be used to define a complementary channel superoperator $\\$^{\\prime }(\\cdot )$ .",
"If $\\$(\\cdot )$ is generated using a measurement $\\lbrace K_\\mu \\rbrace _{\\forall \\mu }$ , then we can find a unitary operator $U_{,{\\mu }}$ such that KA = lcr U, 0 .",
"Now define a measurement $\\lbrace L_b\\rbrace _{\\forall b}$ using the same unitary operator $U_{,{\\mu }}$ : Lb = lcr b 0A U, .",
"Then = $()= KK and = $'()= b Lb Lb .",
"We've already shown how density matrices $\\rho _$ and $\\sigma _=\\$(\\rho _)$ can be represented by QB nets (see Eqs.",
"() and ()).",
"Likewise, density matrices $\\rho _\\mu $ and $\\sigma _\\mu =\\$^{\\prime }(\\rho _\\mu )$ can be represented by QB nets as follows: = =++[o][F-] [l] = , =+++[o][F-] *A[dl]<0* (,) [ul]>> [dl]>> ** *[ul][l]"
]
] | 1204.1550 |
[
[
"Electron Distribution in the Galactic Disk - Results From a\n Non-Equilibrium Ionization Model of the ISM"
],
[
"Abstract Using three-dimensional non-equilibrium ionization (NEI) hydrodynamical simulation of the interstellar medium (ISM), we study the electron density, $n_{e}$, in the Galactic disk and compare it with the values derived from dispersion measures towards pulsars with known distances located up to 200 pc on either side of the Galactic midplane.",
"The simulation results, consistent with observations, can be summarized as follows: (i) the DMs in the simulated disk lie between the maximum and minimum observed values, (ii) the log <n_e> derived from lines of sight crossing the simulated disk follows a Gaussian distribution centered at \\mu=-1.4 with a dispersion \\sigma=0.21, thus, the Galactic midplane <n_e>=0.04\\pm 0.01$ cm$^{-3}$, (iii) the highest electron concentration by mass (up to 80%) is in the thermally unstable regime (200<T<10^{3.9} K), (iv) the volume occupation fraction of the warm ionized medium is 4.9-6%, and (v) the electrons have a clumpy distribution along the lines of sight."
],
[
"Introduction",
"Dispersion measures (DMs) towards pulsars with known distance $d$ can be used to derive the mean electron density in the Galaxy through the relation $\\langle n_{e}\\rangle =DM/d$ .",
"For pulsars located at $\\left|b\\right|<5$ the derived mean electron density in the Galactic plane is $\\langle n_{e}\\rangle \\sim 0.02-0.1$ cm$^{-3}$ , and $0.01-0.017$ cm$^{-3}$ in the spiral arms and inter-arm regions, respectively (Ferrière 2001; Gaensler et al.",
"2008).",
"Berkhuijsen & Fletcher (2008), using the DMs towards 34 pulsars, mostly outside the galactic plane, showed that the $\\log \\langle n_{e}\\rangle $ follows a Gaussian distribution, relating it to the nature of the interstellar turbulence (see e.g.",
"reviews by Elmegreen & Scalo 2004).",
"So far, numerical (magnetized and unmagnetized) 3D simulations have assumed collisional ionization equilibrium (CIE) conditions for the ISM.",
"In fact the thermal evolution of the ISM is determined by heating and cooling processes, which in general are not synchronized with ionization and recombination processes, respectively (e.g., Kafatos 1973; Shapiro & Moore 1976).",
"Hence, below $10^6$ K, deviations from CIE conditions occur, thereby affecting the ionization structure of the interstellar gas, and thus the local electron density.",
"In particular, if delayed recombination plays a role, the number of free electrons may be severely underestimated (Breitschwerdt & Schmutzler 1994).",
"Here we study the electron density, $n_{e}$ , and it's mean, $\\langle n_e \\rangle $ , in the Galactic disk (up to 200 pc on either side of the midplane) using the first to date global hydrodynamical simulation of the interstellar gas, evolving under NEI conditions.",
"Furthermore, we compare simulation results with estimates of $\\langle n_e \\rangle $ obtained from available pulsar DMs.",
"In forthcoming papers we explore the $n_{e}$ distribution and it's topology in the thick disk and halo of the Milky Way and other galaxies.",
"This paper is organized as follows: Section 2 deals with the model setup; Sections 3 and 4 present the observational data, and simulation results, respectively.",
"A discussion and final remarks in Section 5 close the paper."
],
[
"Model and Numerical Setup",
"We simulate hydrodynamically the supernova-driven ISM in a patch of the Galaxy centered at the Solar circle with an area of 1 kpc$^{2}$ and extending to 15 kpc on either side of the Galactic midplane following de Avillez & Breitschwerdt (2005, 2007; AB0507).",
"The simulations are carried out with the EAF-parallel adaptive mesh refinement code coupled to the newly developed E(A+M)PEC codewww.lca.uevora.pt/research.html presents a description of the code, ionization fractions, cooling, and emission spectra.",
"; Avillez & Breitschwerdt 2012) featuring the time-dependent calculation on the spot (at each grid cell) of the ionization structure of H, He, C, N, O, Ne, Mg, Si, S and Fe and emissivities.",
"The physical model includes supernovae (SNe) types Ia, Ib+c, and II, a gravitational field provided by the stars in the disk, local self-gravity (excluding the contribution from the newly formed stars), heat conduction (Dalton & Balbus 1993), uniform heating due to a UV radiation field normalized to the Galactic value and varying with $z$ , and photoelectric heating of grains and polycyclic aromatic hydrocarbons.",
"E(A+M)PEC uses the recommended abundances of Asplund et al.",
"(2009), and calculates electron impact ionization, inner-shell excitation auto-ionization, radiative and dielectronic recombination, charge-exchange reactions (recombination with Hi and Hei , and ionization with Hii and Heii ), continuum (bremsstrahlung, free-bound, two-photon) and line (permitted, semi-forbidden and forbidden) emission in the range 1Å -610$\\mu $ .",
"The code also includes ionization of Hi by Lyman continuum photons emitted during the recombination of helium.",
"The internal energy of the plasma includes the contributions due to the thermal translational energy plus the energy stored in (or delivered) from high ionization stages.",
"Electron impact ionization rates are taken from Dere (2007), while radiative and dielectronic recombination rates are based on AUTOSTRUCTURE calculations (Badnell et al.",
"2003, Badnell 2006aamdpp.phys.strath.ac.uk/tamoc/DATA/) including the latest corrections to Fe ions by Badnell (2006b), Nikolic et al.",
"(2010), and Schmidt et al.",
"(2008).",
"Radiative and dielectronic recombination rates for Sii and Fevii are from Mazzotta el al.",
"(1998).",
"For the remaining ions we adopt the total recombination rates derived with the unified electron-ion recombination method (Nahar & Pradhan 1994) and available at NORAD-Atomic-Datawww.astronomy.ohio-state.edu/$\\sim $ nahar and references therein.. A coarse grid resolution of 8 pc is used, while the finest AMR resolution is $0.5\\,$ pc (4 levels of refinement) for $\\left|z\\right|\\le 2$ kpc, 4 pc for $\\left|z\\right|>4$ kpc, and 1 pc elsewhere.",
"Periodic and outflow boundary conditions are set along the vertical faces and top/bottom ($z=\\pm 15$ kpc) of the grid, respectively.",
"Figure: Histogram (solid black line) and Gaussian fit (dashed black line) of log〈n e 〉\\log \\langle n_{e} \\rangle distribution as obtained with a series of 24 pulsars located at 200<distance<8000200 < \\mbox{distance} <8000 pc from the Sun, with z≤200\\left|z\\right|\\le 200 pc.",
"The mean value is log〈n e 〉=-1.47±0.02\\log \\langle n_{e}\\rangle = -1.47\\pm 0.02 and σ=0.17±0.02\\sigma =0.17\\pm 0.02.",
"Red solid and dashed lines refer, respectively, tothe histogram and Gaussian fit (centered at μ=-1.44\\mu =-1.44 cm -3 ^{-3} and having a dispersionσ=0.21\\sigma =0.21) of the 〈n e 〉 350-400Myr \\langle n_{e}\\rangle _{350-400\\mbox{ Myr}} (§4)."
],
[
"The Electron Density in the Disk",
"To make a detailed comparison between simulation results (discussed below) and the electron density distribution derived from DMs towards pulsars, we reassessed the existing data to select pulsars with best possible distance estimates.",
"Density PDFs were created using pulsars located up to 8 kpc away from the Sun and with $|z|<200$ pc from the Galaxy's midplane.",
"Figure: Total (left) and electron (right) density distributions (in log scale) in the Galactic midplane at400 Myr of evolution.",
"Red regions in the left panel represent high density material, with molecular clouds beingrepresented by black.",
"Electron densities smaller than logn e =-4\\log n_{e}=-4 (white regions;right panel) are located in both high (atomic and molecular clouds) and lower (bubbles) density regions.For our analysis we chose pulsars from the ATNF catalog [21] with independent distance estimates, i.e., estimates without using NE2001 model (Cordes & Lazio 2003; CL03).",
"These estimates resulted from parallax measurements, absorption-line (Hi 21-cm or OH line) studies, physical associations or timing towards pulsars in the galactic disk ($\\left|z\\right|< 200$ pc).",
"Most parallax estimates were obtained from [30], which are updated by Chatterjee, S.\"www.astro.cornell.edu/ shami/psrvlb/parallax.html\".",
"For our study we augmented the ATNF catalog by including distance estimates from absorption studies wherever available from the literature, including two measurements from associations (Frail et al, 1996; Saravanan et al.",
"1996; Johnston et al.",
"2003; Minter et al 2008; Weisberg et al 2008; see references therein).",
"To quantify errors in distance estimates, we compute the fractional error involved in parallax and absorption studies from the difference between the upper and lower estimates.",
"Firm error estimates are not available for several pulsars in the ATNF catalog, whereby the fractional error was computed from the difference between the distance estimate and the best estimate available from NE2001 model (parameter DIST1 in ATNF catalog).",
"Errors in distance estimates using parallax measurements range between $2-35\\%$ .",
"The median error in distance estimate in our overall sample is $18\\%$ , a significant improvement over estimates using NE2001 model (CL03).",
"A total of 122 pulsars with reliable distance estimates were thus selected, 33 of which have $\\left|z\\right|< 200$ pc.",
"Nine pulsars were removed from this sample as they are located within the vicinity of the Sun, and measurements of electron density towards those sightlines are significantly affected by local structures, such as the Local Bubble, Gum nebula and Loop I (e.g., CL03).",
"The histogram of the computed $\\log \\langle n_{e} \\rangle $ derived from the sample DMs and the best-fit gaussian curve are shown in Figure REF (black lines).",
"The fit is centered at $\\log \\langle n_{e}\\rangle =-1.47\\pm 0.02$ and has a dispersion of $\\sigma =0.17\\pm 0.02$ , with a $\\chi ^2/\\mbox{dof}=1.62$ .",
"These values compare well with those derived directly from our pulsar sample data: $\\log \\langle n_{e}\\rangle \\sim 1.42$ and $\\sigma \\sim 0.23$ ."
],
[
"Simulation Results",
"The simulated supernova-driven interstellar medium is characterized by several evolutionary phases that have already been described in previous works: (i) domination of initial conditions being only wiped out after some 80 Myr of evolution, (ii) the full establishment of the continuous disk-halo-disk circulation (also known as the Galactic fountain), and (iii) the dynamical equilibrium (occurring at 200-220 Myr of evolution) in a statistical sense that determines the dynamics of the interstellar medium in the Galactic disk and its interaction with the halo.",
"As a result of this evolutionary path driven mainly by supervovae, the ISM becomes frothy and turbulent with a mean Mach number of 3 (AB0507).",
"Low temperature gas (high density) is concentrated into filamentary structures and molecular clouds (black regions in the left panel of Figure REF , showing the Galactic midplane density at 400 Myr of evolution), while hot gas (low density regions in the same panel) is concentrated into bubbles and superbubbles that dominate the landscape.",
"Figure: Top panel: Time evolution between 300 and 400 Myr of the electrons volume (solid lines) and mass (dashedlines) occupation fractions in the simulated disk for thermally unstable regime (orange), gas in shells andfilamentary structures (red), and warm ionized medium (WIM; cyan).",
"Bottom panel: Time averaged histogram of thetotal (black ), and the two gaussian fits (dashed black lines) of the electron density in the simulated disk over theevolution time 300-400 Myr.",
"The electron density in different temperature regimes are displayed.The turbulent nature of the simulated ISM, the ongoing physical processes and driving mechanisms (e.g., SNe and stellar winds) affect the electron distribution, which follows the topology of the medium.",
"Most of the electrons are distributed into the thermally unstable regime ($200<\\mbox{T}<10^{3.9}$ K) with a volume ($f_v$ ) and mass ($f_M$ ) filling fractions of $56-60\\%$ and $77-80\\%$ , respectively (Figure REF : top panel); $32-40\\%$ of the electron mass is locked in filamentary structures and shells, having $f_{v}=6-7\\%$ .",
"The electron distribution in warm ionized medium has $f_v\\simeq 4.9-6\\%$ and $f_M=7-10\\%$ .",
"These are time-dependent variations whose time average (over a period of 100 Myr using 1001 disk snapshots taken at every 0.1 Myr) histogram of $\\log n_{e}$ (Figure REF : bottom panel), is fitted with a composition of two gaussians centered at $\\log n_{e}=-1.437$ (for T$<10^{4.2}$ K gas) and $\\log n_{e}=-2.64$ (for T$>10^{4.2}$ K gas), respectively, with the latter representing mostly photoionized regions.",
"Further insight into the simulated electron distribution can be obtained directly from DMs along LOS, with length $d$ , crossing the simulated ISM, allowing the determination of the mean electron density.",
"The LOS have increasing lengths (with a step length of 10 pc) up to 1 kpc from a vantage point located at $(x,y,z)=(0,0,0)$ (left bottom corner of the electron density maps in Figure REF ) for $\\left|z\\right|\\le 200$ pc and spanning 90 (with a 1 separation) when projected onto the Galactic midplane.",
"Figure: Angular distribution of the DMs along LOS with lengths varying between 100 and 1000pc at 350 and 400 Myr (top two panels) and the time averaged 〈DM〉 t \\langle DM\\rangle _{t} (bottom panel) over theevolution time 350-400 Myr).Figure REF shows the DM along the LOS with lengths varying between 100 and 1000 pc, with a step length of 100 pc at 350 and 400 Myr (top two panels) and its time average (bottom panel) over 501 disk snapshots taken every 0.1 Myr for $350-400$ Myr of evolution.",
"Due to the ongoing turbulent processes in the Galactic disk the DM has variability along and between the lines of sight (Figure REF : top two panels).",
"The longitudinal variation is due to the inhomogeineity of the ISM (e.g., bubbles and superbubbles) as well as due to the turbulent nature of the regions crossed by the LOS.",
"These variations with distance and among LOS are still present in the time averaged DM, although the longitudinal variations are smoothed in the averaging process.",
"However, a record of any event that prevailed over a large period of time (e.g., a superbubble whose contribution to the DM is small as in the case of the observed hump in the DM between 48and 65) is kept.",
"Figure: Minimum (dotted line), mean (solid line) and maximum (dashed line) of the averaged DM (top panel) andn e n_{e} (bottom panel) for all LOS crossing the disk up to 4 kpc from the Sun at 310 Myr (〈X〉 310Myr \\langle X\\rangle _{310\\mbox{ Myr}}; red lines), 360 Myr (〈X〉 360Myr \\langle X\\rangle _{360\\mbox{ Myr}}; green lines), and overthe period 350-400 Myr (〈X〉 350-400Myr \\langle X\\rangle _{350-400\\mbox{ Myr}}; black lines).",
"The blue circles representthe DMs (top) and 〈n e 〉\\langle n_{e}\\rangle (bottom) derived from pulsars observations.We further compare the DMs and $\\langle n_{e}\\rangle $ derived from pulsars observations (§3), with those resulting from averaging over all the LOS crossing the simulated disk ($|z|<200$ pc) and up to 4 kpc from the Sun at specific times (e.g., 310 and 360 Myr) and over the period 350-400 Myr.",
"Top panel of Figure REF displays the observationally derived DMs and the averaged DMs at 310 Myr ($\\langle DM\\rangle _{310\\mbox{Myr}}$ ; red lines), 360 Myr ($\\langle DM\\rangle _{360\\mbox{ Myr}}$ ; green lines), and over the period 350-400 Myr ($\\langle DM\\rangle _{350-400\\mbox{ Myr}}$ ; black lines).",
"Due to the limitation of the box size in our simulation, we calculated the DMs for distance $d>1$ kpc taking advantage of the periodic boundary conditions used in the simulations.",
"The DMs were calculated from different vantage points in the midplane (including $(x=0,y=0,x=0)$ and between 500 and 1400 pc from it) using LOS crossing the disk volume at different angles, up to 4 kpc from the vantage point and up to $z=\\pm 200 $ pc.",
"This procedure, although inevitable here, may introduce artefacts in the simulation results, but we do not expect that our results are strongly affected though, because the simulation shows a certain pattern repetition in the ISM density and temperature distributions on scales of the order of a few correlation lengths (that is, a few times 75 pc, according to AB0507).",
"The averaged $\\langle DM\\rangle _{350-400\\mbox{ Myr}}$ , after a steep growth in the first 100 pc, reaches a smooth increase for $d>500$ pc, with the deviations of the minimum and maximum regarding the mean becoming constant over distance - consistent with the DMs variations in $d$ observed in Figure REF .",
"The averaged DMs at 310 and 360 Myr vary widely, but with values within the pulsar derived DMs.",
"Therefore, the terms minimum and maximum of the $\\langle DM\\rangle _{350-400\\mbox{ Myr}}$ should not be taken as strict upper and lower values for all times, as they only represent an average over a specific time window of the simulation.",
"After the large variation in the first few hundred parsecs, the mean of the simulated $\\langle n_{e}\\rangle $ , for the cases shown in the bottom panel of Figure REF , reach a constant value ($0.04\\pm 0.002$ cm$^{-3}$ ) and dispersion over large distances.",
"This reflects the clumpy nature of the electrons in the turbulent ISM as would be expected if they are predominantly found in filaments and shells (Berkhuijsen & Müller 2008).",
"The histogram of the $\\log \\langle n_{e}\\rangle _{350-400\\mbox{ Myr}}$ and it's best Gaussian fit are displayed in Figure REF by solid and dashed red lines, respectively.",
"The fit is centered at $\\log \\langle n_{e}\\rangle =-1.4\\pm 0.01$ with $\\sigma =0.21\\pm 0.01$ .",
"In the simulated disk, the clumpy material dominates the LOS and volume averaged PDFs which explains the similar means between the electron density of the T$<10^{4.2}$ K regime (Figure REF ) and the time averaged $\\langle n_{e}\\rangle _t$ (Figure REF ; red lines).",
"These fit parameters are similar to those displayed in Figure REF (black lines) for observationally derived $\\langle n_{e}\\rangle $ .",
"This similarity is indicative of the smoothing out of low and high density regions along the lines of sight, thus, not appearing in the PDFs, and leading to reduced Gaussian dispersion."
],
[
"Discussion and Final Remarks",
"In this letter we discuss the electron density distribution in the Galactic disk using the first to date 3D high resolution NEI simulations of the ISM.",
"The simulations trace the dynamical and thermal evolution of the interstellar gas, calculating on-the-spot the ionization structure, electron distribution and cooling function of the gas at each cell of the grid into which the computational domain is discretized.",
"We did not take into account stellar ionizing photons and their transport, but an averaged diffuse photon field, which should be a reasonable approximation, when describing the mesoscale ISM.",
"The NEI structure modifies the cooling function, which in turn enhances the number of free electrons.",
"Both the simulated DMs and electron density are consistent with the observations: (i)nmost DMs lie within the maximum and minimum observed values, (ii) $\\log \\langle n_{e} \\rangle $ is consistent with a gaussian distribution, (iii) the mean electron density is $0.04\\pm 0.01$ cm$^{-3}$ , and (iv) the volume filling fraction of the WIM is bracketed between 4.9 and 6%.",
"These results lye in between the estimates by Gaensler et al.",
"(2008) and Berkuijsen & Müller (2008) for the electron density in the disk and WIM.",
"Furthermore, the observed Gaussian distribution is present, irrespective of whether the ISM is isothermal (see discussion in Berkuijsen & Fletcher (2008) and refereces therein) or not.",
"This can be explained in two complementary ways: a fully developed turbulent system can be considered as a large number of independent random variables, with the logarithm of each having a certain distribution approaching a Gaussian as the number of variables goes to infinity (Central Limit Theorem), being the case of supernova-driven ISM simulations reaching a statistical equilibrium (see, e.g., Vázquez-Semadeni & Garcia 2001).",
"Or, alternatively, we can use the principle of maximum entropy.",
"The information entropy of a continuous random variable $X$ with probability density function $p(x)$ is defined as $H(X)=\\int _{-\\infty }^{\\infty } p(x) \\log p(x) dx$ .",
"The lognormal distribution of $X$ maximizes $H(X)$ , implying the least prior knowledge of the system (Sveshnikov, 1968).",
"This is exactly what is expected in homogeneous and isotropic turbulence with a large number of independent random variables.",
"We study deviations from lognormal distributions in a forthcoming paper."
],
[
"Acknowledgements",
"We thank the anonymous referee for the detailed report and valuable suggestions that allowed us to improved this letter.",
"The Milipeia Supercomputer (Univ.",
"of Coimbra) and the ISM-cluster (Univ.",
"of Évora) were heavily used for the calculations.",
"This research is supported by the FCT project PTDC/CTE-AST/70877/2006."
]
] | 1204.1511 |
[
[
"Milliarcsecond structure of water maser emission in two young high-mass\n stellar objects associated with methanol masers"
],
[
"Abstract The 22.2 GHz water masers are often associated with the 6.7 GHz methanol masers but owing to the different excitation conditions they likely probe independent spatial and kinematic regions around the powering young massive star.",
"We compared the emission of these two maser species on milliarcsecond scales to determine in which structures the masers arise and to test a disc-outflow scenario where the methanol emission arises in a circumstellar disc while the water emission comes from an outflow.",
"We obtained high-angular and spectral resolution 22.2 GHz water maser observations of the two sources G31.581+00.077 and G33.641-00.228 using the EVN.",
"In both objects the water maser spots form complex and filamentary structures of sizes 18-160 AU.",
"The emission towards the source G31.581+00.077 comes from two distinct regions of which one is related to the methanol maser source of ring-like shape.",
"In both targets the main axis of methanol distribution is orthogonal to the water maser distribution.",
"Most of water masers appear to trace shocks on a working surface between an outflow/jet and a dense envelope.",
"Some spots are possibly related to the disc-wind interface which is as close as 100-150 AU to the regions of methanol emission."
],
[
"Introduction",
"High-angular resolution observations of high-mass star forming sites in the 6.7 GHz methanol maser emission have revealed a diversity of morphological structures from very simple composed of a few milliarcsecond (mas) wide spots to complex and extended clouds of arcsecond sizes (Phillips et al.",
"[28]; Walsh et al.",
"[51]; Minier et al.",
"[21]; Dodson et al.",
"[13]; Bartkiewicz et al.",
"[3]; Pandian et al. [27]).",
"Linear structures with a velocity gradient are thought to be the signature of an edge-on disc or torus around young massive protostars (Norris et al.",
"[26]; Minier et al.",
"[21]) but there is evidence that some of them are associated with outflows (De Buizer [12]) or even a shock propagating through a rotating dense molecular clump (Dodson et al. [13]).",
"Arc-like or ring-like morphologies of methanol emission seem to prove a disc scenario (Bartkiewicz et al.",
"[3]), although most of those structures do not show signs of rotation but rather inflow or/and outflow dominate (van Langevelde et al.",
"[50]), it is then postulated that methanol masers arise in the shock interface between the large scale accretion and a circumstellar disc.",
"While the 6.7 GHz methanol maser line is radiatively pumped in warm ($T\\sim 150$ K) and dense ($n\\le 10^8$ cm$^{-3}$ ) regions (Cragg et al.",
"[8]; Sobolev et al.",
"[32]), the 22 GHz water maser emission is collisionally pumped and probes denser ($n\\ge 10^8$ cm$^{-3}$ ) and hotter ($T\\sim 400$ K) gas (Elitzur et al.",
"[14]).",
"The water maser is often excited in strong shocks, driven by young low-mass and high-mass (proto)stellar objects, on a working surface between a outflow and a dense envelope.",
"A number of morphologies of this emission is observed from collimated jet, wide-angle flows, expanding shells to equatorial flows (Goddi et al.",
"[18]; Moscadelli et al.",
"[23], [24]; Sanna et al.",
"[30], [31]; Torrelles et al. [42]).",
"One or more types of those maser structures occur simultaneously in star forming regions on scale sizes of a few mas to several arcsec and is possibly related to the geometry of the envelopes or the mechanisms driving the outflows.",
"Our recent VLA survey of 22 GHz water masers in a sample of 31 methanol sources has yielded 22 detections (Bartkiewicz et al.",
"[4]) where both maser species are excited by the same central objects.",
"We noted that a distinct group of methanol sources with ring-like structure show either no associated water masers at all or water masers that are distributed orthogonally to the major axis of the ring.",
"It is argued that the methanol maser structure traces a circumstellar disc/torus around a high-mass young stellar object while the water masers originate in outflows.",
"As the VLA data are of moderate angular and spectral resolutions of $\\sim $ 1 and 0.65 km s$^{-1}$ , respectively, we undertook the VLBI observations to examine a disc-outflow scenario in the two brightest water maser targets, which show the arc-like or ring-like morphologies that are characteristic of methanol masers.",
"The source G31.581$+$ 00.077The names of two targets follow the Galactic coordinates of the brightest methanol maser spots derived by Bartkiewicz et al. ([3]).",
"has been recognized as a massive young stellar object based on 6.7 GHz methanol maser observations (Szymczak et al. [36]).",
"The methanol maser spots are distributed along an arc or ring of 217 mas size (Bartkiewicz et al.",
"[3]), which corresponds to $\\sim $ 1200 AU for the assumed near kinematic distance of 5.5 kpc (Reid et al. [29]).",
"The detection of an infrared source of bolometric luminosity of 3$\\times $ 10$^4$ L$_{}$ (Urquhart et al.",
"[48]), 22 GHz H$_2$ O maser emission (Bartkiewicz et al.",
"[4]), 1665 and 1667 MHz weak OH masers (Szymczak & Gérard [38]), millimeter molecular thermal lines (Szymczak et al.",
"[40]; Urquhart et al.",
"[45]), and 5 and 8.4 GHz continuum emission at location of 31.582$+$ 00.075 (i.e., 9 apart from the maser arc/ring) (Urquhart et al.",
"[46]; Bartkiewicz et al.",
"[3]) indicates that this is a cluster of recent star formation.",
"The source G33.641$-$ 00.228 detected in the 6.7 GHz methanol line (Szymczak et al.",
"[36]) has an arc spot distribution of length 630 AU (Bartkiewicz et al.",
"[3]) for the assumed near kinematic distance of 3.8 kpc (Reid et al.",
"[29]), which seems to be more likely than the far kinematic distance (Zhang priv.comm.).",
"The site also contains water and OH masers (Bartkiewicz et al.",
"[4]; Szymczak & Gérard [38]) but no 8.4 GHz continuum emission was detected with an upper limit of 0.15 Jy (Bartkiewicz et al. [3]).",
"The detection of radio recombination lines at about 103 km s$^{-1}$ (Anderson et al.",
"[1]) and a multi-feature $^{13}$ CO line spectrum (Urquhart et al.",
"[45]) illustrates the complexity of this molecular cloud and its clustered star formation."
],
[
"Observations and data reduction",
"The EVNThe European VLBI Network is a joint facility of European, Chinese, South African and other radio astronomy institutes funded by their national research councils.",
"observations of G31.581$+$ 00.077 and G33.641$-$ 00.228 with the antennas at Jodrell Bank, Effelsberg, Medicina, Metsähovi, Onsala, and Yebes, were carried out at 22.23508 GHz on 2010 October 30 for 8 h (the project EB047).",
"The tracking phase centres were estimated from the VLA survey for the water maser spots that were located nearest to the methanol emission in each target (Bartkiewicz et al.",
"[4]) at $\\alpha $ =18$^{\\rm h}$ 48$^{\\rm m}$ 41951, $\\delta $ =$-$ 0110'02578 and $\\alpha $ =18$^{\\rm h}$ 53$^{\\rm m}$ 32563, $\\delta $ =$+$ 0031'39130 (J2000), respectively.",
"A phase-referencing scheme was applied with a reference source J1851$+$ 0035 (from the VLA calibrator catalogue), using a cycle time between the maser and phase-calibrator of 60 s$+$ 90 s. This strategy yielded 2 h on-source times for each target.",
"The bandwidth was set to 8 MHz and data were correlated with the Mk IV Data Processor operated by JIVE with 1024 spectral channels.",
"The resulting spectral resolution was 0.1 km s$^{-1}$ .",
"The velocity was measured with respect to the local standard of rest (LSR).",
"The data calibration and reduction were carried out with NRAO's Astronomical Image Processing System (AIPS) using standard procedures for spectral line observations.",
"We used the Effelsberg antenna as a reference.",
"The amplitude was calibrated by performing measurements of the system temperature at each telescope and applying the antenna gain curves.",
"The parallactic angle corrections were subsequently added to the data.",
"The source 3C454.3 was used as a delay, rate, and bandpass calibrator.",
"The phase-calibrator was imaged and a flux density of 190 mJy was obtained.",
"The maser data were corrected for all Doppler effects and self-calibrated using the brightest and most compact maser spot.",
"Finally, naturally weighted maps of spectral channels were created in the velocity range where the emission was seen in the scalar-averaged spectrum.",
"For imaging, the resulting synthesized beam was 1.0$\\times $ 2.4 mas$^2$ with a position angle (PA) of $-$ 38, while the pixel separation was 0.2 mas in both coordinates.",
"The rms noise levels (1$\\sigma _{\\rm rms}$ ) in line-free channels was typically 10 mJy beam$^{-1}$ .",
"The weakest detected maser spot had a brightness of 94 mJy beam$^{-1}$ that is more than 9$\\sigma _{\\rm rms}$ .",
"The positions of water maser spots in all channel maps were determined by fitting two-dimensional Gaussian models.",
"The formal fitting errors resulting from the beamsize/signal-to-noise ratio were smaller than 0.1 mas.",
"To determine the position accuracy of registered maser spots, we need to consider the following factors (Diamond et al.",
"[10]): i) the uncertainty in the phase-reference source position of 2 masThe GSFC ICRF2 VLBI Source Position Catalog.",
"; ii) the antenna positions, where the claimed accuracy of $\\sim $ 1 cm corresponds to 1 mas in RA and 2 mas in Dec; iii) the phase transfer over the separation between targets and the phase-calibrator: 192 (between G31.581$+$ 00.077 and J1851$+$ 0035) and 045 (between G33.641$-$ 00.228 and J1851$+$ 0035).",
"These caused potential phase-solution transfer errors corresponding to 1 mas in RA and 2 mas in Dec for G31.581$+$ 00.077 and to 0.2 mas in RA and 0.4 mas in Dec for G33.641$-$ 00.228, respectively.",
"In total, the absolute position accuracy (1$\\sigma _{\\rm pos}$ ) is 2.5 mas in RA and 3.5 mas in Dec for both targets."
],
[
"Results",
"Water maser emission towards both targets was mapped, after successful phase-referencing, in the areas of 5$\\times $ 5 and the velocity range explored by Bartkiewicz et al.",
"([4]) using the VLA.",
"The EVN maps of water maser spots (Figs.",
"REF and REF ) are overlayed on the VLA maps and the 6.7 GHz methanol maser distributions obtained with the EVN (Bartkiewicz et al. [3]).",
"The Spitzer GLIMPSE mapshttp://irsa.ipac.caltech.edu/data/SPITZER/GLIMPSE/ of the 4.5 $\\mu $ m$-$ 3.6 $\\mu $ m emission excess are added.",
"Following the procedure described in Bartkiewicz et al.",
"([3]), the water maser spots are analysed to identify individual velocity-coherent maser clouds.",
"Their basic parameters, such as positions, $\\Delta $ RA, $\\Delta $ Dec, LSR velocities, V$_{\\rm LSR}$ , and intensities of the brightest spot, S$_{\\rm p}$ , of each cloud are listed in Tables REF and REF .",
"The water and methanol maser spectra of single clouds are combined in Figures REF and REF with overlays of individual Gaussians, if emission was seen in at least three consecutive channels.",
"A Gaussian analysis of individual clouds is carried out and the fitted flux amplitude, S$_{\\rm fit}$ , the full width at half-maximum, FWHM, the projected extent between the most separated single spot centres within a cloud, L$_{\\rm proj}$ , and the velocity gradient, V$_{\\rm grad}$ are listed in Table REF .",
"The lower limit to the brightness temperature, T$_{\\rm b}$ , of each cloud is also calculated according to Eq.",
"9$-$ 27 of Wrobel & Walker ([49]).",
"For comparison purposes, the same analysis is done for the 6.7 GHz methanol maser data (Table REF ) obtained in 2007 June (G31.581$+$ 00.077) or 2003 June (G33.641$-$ 00.228) with the EVN.",
"Below we comment in more detail on each source."
],
[
"G31.581$+$ 00.077",
"A total of 91 water maser spots were detected.",
"They are concentrated in two distinct regions (Fig.",
"REF ).",
"The south-east (SE) region of size 50$\\times $ 30 mas$^2$ containing weak ($<$ 1.35 Jy) emission in the velocity ranges from 90.1 to 93.9 km s$^{-1}$ and from 100.0 to 103.3 km s$^{-1}$ is located close to the phase centre.",
"The second region (NW) of middle velocity emission (96.8$-$ 100.7 km s$^{-1}$ ) and flux density of 2.6$-$ 16.5 Jy is located offset by $\\sim $ 5 to the north-west (PA$=-60$ ) of the phase centre.",
"The SE region is divided into the cluster of clouds 1, 2, and 3 separated by 50 mas from the cluster of clouds 6, 7, and 8 (Fig.",
"REF ).",
"The clouds in the SE region show Gaussian velocity profiles with FWHM linewidths of 0.54$-$ 0.94 km s$^{-1}$ .",
"Their projected sizes are 0.07$-$ 0.47 mas, which correspond to 0.9$-$ 2.6 AU for the adopted distance of 5.5 kpc.",
"The brightness temperature is 0.24$-$ 1.38$\\times $ 10$^9$ K. We note that clouds 3 and 6 show the velocity gradient of 1.3$-$ 1.8 km s$^{-1}$ mas$^{-1}$ oriented along PAs of 1 and $-$ 23, respectively.",
"It is remarkable that the two clusters of clouds (1, 2, 3 and 6, 7, 8) appear very close in space (Table REF ) forming arc-like filaments of projected length of 4 and 3.2 mas corresponding to 22 and 18 AU, respectively, and elongated at PA=$-$ 58.",
"The closest clouds 3 and 6 are separated by 285 AU and a velocity difference of 6.7 km s$^{-1}$ .",
"The mean linear separation between the two clusters along PA=65 is 310$\\pm $ 8 AU, while the velocity spread is 13.2 km s$^{-1}$ .",
"The velocities of the two filaments are approximately symmetric with the regard of the systemic velocity of 96.0 km s$^{-1}$ (Szymczak et al. [40]).",
"The two SE filaments are likely signatures of flattened shock surfaces (Torrelles et al. [41]).",
"We found that the water cloud 7 coincides within $\\pm $ 0.1 km s$^{-1}$ with the methanol maser cloud k, while the spatial separation is 24.4 mas which corresponds to the projected distance of 134 AU.",
"Although these two lines were observed within a time span of three years their separation seems to be real.",
"This confirms that the water and methanol masers probe different parts of the environment of young massive stars (e.g., Beuther et al.",
"[5]; Sanna et al.",
"[30], [31]) because of the different pumping mechanisms affecting both species (Elitzur et al.",
"[14]; Cragg et al. [8]).",
"The NW region of elongation of 26 mas is composed of two clouds 4 and 5 of brightness temperatures of 2.7$-$ 17.1$\\times $ 10$^9$ K (Table REF ).",
"Their velocity profiles are obviously asymmetric and nicely fitted by the sums of three and two Gaussian components (Fig.",
"REF ) of FWHM linewidth of 0.5$-$ 0.9 km s$^{-1}$ .",
"The velocities of the NW water maser emission largely coincide with those of the methanol masers but their linear separation is about 27000 AU.",
"The map of 4.5 $\\mu $ m$-$ 3.6 $\\mu $ m emission excess (Fig.",
"REF ), which possibly traces shocked molecular gas in outflows from massive stars (e.g., Davies et al.",
"[11]), shows that the SE and NW maser clusters lie in a very complex region.",
"Both maser clusters are likely associated with different powering sources.",
"The comparison of our maps with those obtained about 14 months earlier with the VLA in a CnB configuration (Bartkiewicz et al.",
"[4]) implies that the cloud 1 coincides to within 13 mas, the cloud 4 to within 56 mas, and the cloud 7 to within 64 mas with the matching VLA spots.",
"The emission seen with the VLA from three components located $\\sim $ 1$-$ 2 eastward of the clouds 4 and 5 is not detected.",
"We also note that the water maser components detected have very similar amplitudes in both EVN and VLA observations.",
"Figure: G31.581++00.077.",
"a) Spitzer GLIMPSE 4.5 μ\\mu m--3.6 μ\\mu m excess image overlaidwith the water maser positions (yellow asterisks) from the EVN observation and H II region(yellow cross) detected by Bartkiewicz et al.",
"().b) Shows the distribution of water masers observed with the EVN (filled squares)from this paper and the VLA(open squares) from Bartkiewicz et al. ().",
"The colours ofthe symbols relate to theLSR velocities as indicated on the right-hand side of the plot.",
"The origin ofthe map isthe position of the brightest 6.7 GHz methanol maser spot (Bartkiewicz et al.",
")(see also Table ).",
"c) Shows an enlargement of the north-western (NW) water masers.d) Shows an enlargement of the south-eastern (SE) water masers together with the distributionof the 6.7 GHz methanol masers, marked by open circles, from Bartkiewicz et al. ().",
"In (c) and(d), thearrows represent the velocity gradients (from blue- to red-shifted LSR velocities) detectedwithin individual cloud.",
"The black numbers and letters correspond to the clouds withinternal velocity gradients, while the grey ones correspond to the clouds without internalvelocity gradients (Table ).Figure: Individual component spectra with Gaussian velocity profiles of 22.2 GHz water masers(red squares and lines) and 6.7 GHz methanol masers (blue circles and lines) towardsG31.581++00.077 detected using the EVN in 2010 and 2007, respectively.",
"The numbers andletters correspond to the cloud labels as given in Table and Fig.",
".The grey lines show individual Gaussian profiles fitted to the blended features.The clouds with only two single spots are not marked to improvethe clarity of the figure (i.e., clouds b, e, g)."
],
[
"G33.641$-$ 00.228",
"Twenty-one water maser spots were detected in the velocity range of 54.4$-$ 57.6 km s$^{-1}$ over the north-south elongated area 50$\\times $ 280 mas$^2$ close to the phase centre (Fig.",
"REF ).",
"Using the above-mentioned procedure, we found four maser clouds where the emission is seen in at least three contiguous spectral channels.",
"There are also two other clouds (5 and 6) that do not obey this criterion but seem to be real (Table REF ).",
"All but one of the cloud are in the southward cluster at a distance $\\sim $ 270 mas from the phase centre.",
"This filament cluster of size 41.6 mas is aligned along a PA of 78.",
"For the adopted distance of 3.8 kpc, the corresponding linear scales are 1020 AU and 157 AU.",
"The clouds 1$-$4 in the southern cluster have Gaussian profiles with FWHM linewidths of 0.25$-$ 0.88 km s$^{-1}$ .",
"We note that none of the water clouds have an internal velocity gradient.",
"The linear size of the individual clouds is 0.3$-$ 1.0 AU and the brightness temperature is 0.12$-$ 0.27$\\times $ 10$^9$ K (Table REF ).",
"The strongest emission (T$_{\\rm b}=0.63\\times 10^9$ K) comes from cloud 2, which is a blend of two Gaussian components (Fig.",
"REF ).",
"Cloud 2 is $\\sim $ 20 mas (75 AU) away from the brightest methanol maser clouds a and r and differ in terms of velocity by 1.5 km s$^{-1}$ .",
"This is probably the first such tight association of methanol and water masers.",
"We note that the emission of cloud 2 is blue-shifted by 4.2 km s$^{-1}$ and that of cloud a is red-shifted by 1.2 km s$^{-1}$ relative to the systemic velocity of 61.5 km s$^{-1}$ (Szymczak et al. [40]).",
"It is therefore possible that these clouds signpost a shock front in which the water emission originates behind the front, while the methanol emission (cloud a) appears outside of the shock interface in the infalling gas.",
"The image of the 4.5 $\\mu $ m$-$ 3.6 $\\mu $ m emission excess (Fig.",
"REF ) shows that the water maser emission is located 12 to the south-east of single mid-infrared object.",
"This is likely the source powering the outflow along a PA of 165 traced by the water masers.",
"The positions of the clouds 2, 3, 4 and 6 differ by about 92 mas from the VLA positions of the matching clouds (Bartkiewicz et al.",
"[4]), which is well within the $\\sim $ 150 mas accuracy of the CnB configuration VLA data.",
"A weak ($<$ 0.9 Jy) red-shifted emission in the velocity range of 83.8$-$ 85.1 km s$^{-1}$ seen with the VLA towards cloud 2 (Fig.",
"REF ) was not recovered in the VLBI observation.",
"The brightness of the emission from cloud 2 determined in the EVN observation is only slightly lower than that measured with the VLA.",
"In contrast, the maser spots from the southern region (clouds 1, 3$-$6) are about ten times weaker in the EVN maps that may suggest the presence of diffuse and/or highly variable emission.",
"Figure: Same as Fig.",
"but for G33.641--00.228.Figure: Same as Fig.",
"but for G33.641--00.228.",
"Thescale on the left side describes the water maser intensity, while theright side corresponds to the methanol maser intensity.",
"The clouds with onlytwo single spots or without Gaussian characteristic are not marked for theclarity of the figure (i.e., clouds 5, 6, e, g, i, n, p).Table: List of 22.2 GHz water maser clouds as found in EVN observationstowards G31.581++00.077.",
"The(0,0) point corresponds to the position of the brightest methanol maser in this source(Bartkiewicz et al.",
"): RA=18 h ^{\\rm h}48 m ^{\\rm m}4194108,Dec=--0110025281 (J2000).Table: List of 22.2 GHz water maser clouds as observed using EVN towards G33.641--00.228.The (0,0) point correspond to the brightest methanol maser in this source(Bartkiewicz et al.",
"): RA=18 h ^{\\rm h}53 m ^{\\rm m}32563,Dec=++003139180 (J2000).Table: Parameters of 22.2 GHz water and 6.7 GHz methanol maser clouds."
],
[
"Maser environments",
"The Spitzer GLIMPSE map (Fig.",
"REF ) clearly indicates that the source G31.581$+$ 00.077 lies in a large (35$\\times $ 25) complex area with an 4.5 $\\mu $ m emission excess in the form of clumped and diffuse structures.",
"This extended emission is proposed to be a tracer of shocked gas from regions where outflowing gas interacts with the surrounding medium (Davis et al.",
"[11]; Cyganowski et al. [9]).",
"The $^{13}$ CO(1$-$ 0) spectrum obtained with a 46 beam (Urquhart et al.",
"[45]) towards the position 31.5808+0.0757 has two Gaussian components centred at 96.2 and 109.8 km s$^{-1}$ .",
"A weak emission line is clearly detected in the range of 85 to 105 km s$^{-1}$ , i.e., in the wings of the first component suggests the presence of outflows.",
"A signature of ordered motions also appears in the HCO$^+$ (1$-$ 0) spectrum (Szymczak et al.",
"[40]), where in this slightly asymmetric profile the strongest emission is blueward of the source systemic velocity and can be interpreted as the result of infall (e.g., Fuller et al. [16]).",
"We then find 92 east of the SE water maser an H II region of flux density 10.3 and 15 mJy at 5 GHz (White et al.",
"[53]) and 8.4 GHz (Bartkiewicz et al.",
"[3]), respectively.",
"This continuum source coincides to within 06 with a clump of 3.6 $\\mu $ m$-$ 4.5 $\\mu $ m emission excess (Fig.",
"REF ).",
"The NW water masers are located at the borders of two clumps of excited gas, and their powering source is unclear although it seems unlikely that the SE and NW masers are associated with the same central object.",
"A comparison of the present EVN maps with those obtained with the VLA 14 months earlier (Bartkiewicz et al.",
"[4]) implies that there is a lack of water masers eastward of the NW clouds (Fig.",
"REF ).",
"It is likely that these are diffuse and low intensity masers resolved out with the EVN beam.",
"The spectrum obtained with a 40 beam sometime between November 2009 and December 2010 (Urquhart et al.",
"[48]) indicates that the peak flux density of 125 Jy at 99.7 km s$^{-1}$ is about one order of magnitude higher than that measured with the EVN.",
"We suggest that the water masers of G31.581$+$ 00.077 are produced by different young stellar objects in a complex region composed of at least a few high-mass stars well-signposted by the radio continuum and the methanol and water masers.",
"The water maser in G33.641$-$ 00.228 lies only 12 south-eastward of the maximum of the 3.6 $\\mu $ m$-$ 4.5 $\\mu $ m emission excess of cometary-like morphology (Fig.",
"REF ).",
"The HCO$^+$ (1$-$ 0) spectrum obtained towards the position 33.648$-$ 0.224 (Szymczak et al.",
"[40]) shows a small dip near 61.5 km s$^{-1}$ and a slight redward asymmetry that is indicative of outflow motions.",
"The less optically thin H$^{13}$ CO$^+$ (1$-$ 0) transition is detected as marginal red-shifted emission, which is consistent with an outflow.",
"In the region where the OH 1665, 1667, and 1720 MHz masers were detected (Szymczak & Gérard [38]) and all of them peak at 60.2 km s$^{-1}$ .",
"The strongest 1720 MHz emission and a broad (7.4 km s$^{-1}$ ) 1667 MHz absorption profile near 56.2 km s$^{-1}$ are indicative of shock fronts and a continuum background source, respectively.",
"No continuum emission at 8.4 GHz was detected with a 3$\\sigma _{\\rm rms}$ limit of 0.15 mJy beam$^{-1}$ (Bartkiewicz et al. [3]).",
"A weak ($\\sim $ 20 mJy) 8.6 GHz hydrogen radio recombination-line near 102.9 km s$^{-1}$ detected at position G033.645$-$ 0.227 (Anderson et al.",
"[1]) differs so greatly in terms of velocity from the water masers that this may be a chance projection that is unassociated with the methanol and water masers.",
"We argue that the water masers of G33.641$-$ 00.228 as well as the associated methanol and hydroxyl masers are excited by individual high-mass stars."
],
[
"Physical parameters of circumstellar medium",
"Lower limits to the brightness temperature of the SE water maser components associated with G31.581$+$ 00.077 are always lower than 1.4$\\times $ 10$^9$ K, while the measured linewidths range from 0.54 to 0.94 km s$^{-1}$ are quite commonly in the star-forming sources (e.g., Goldreich & Kwan [19]; Surcis et al.",
"[34], [35]).",
"Since the kinetic temperature of the masing gas might be expected to be about 400 K (Elitzur et al.",
"[14]), the intrinsic thermal linewidth given by $\\Delta v_{\\rm FWHM}=2.35482\\times \\sqrt{k T_k/m}$ , where $T_k$ is a kinetic temperature, $k$ the Boltzmann constant, and $m$ is a molecular mass should be $\\sim $ 1 km s$^{-1}$ .",
"This value is larger than the observed linewidths and suggests that the masers are unsaturated.",
"In the same volume of gas where the water maser cloud 7 and the methanol maser cloud k appear to coincide (Fig.",
"REF ), the methanol thermal linewidth would be 0.75 km s$^{-1}$ , whereas the measured value is 0.30 km s$^{-1}$ .",
"The narrowing of the line profile is expected when the maser is unsaturated.",
"Detailed calculations have shown that in one of the strongest known methanol sources NGC7538 about 92% of the components are unsaturated (Surcis et al. [35]).",
"In the source G33.641$-$ 00.228, the observed linewidths of methanol components are generally narrower than in G31.581$+$ 00.077, which suggests that the maser is also unsaturated in this region.",
"The mas-scale velocity gradients in both lines are observed in G31.581$+$ 00.077.",
"We note that the velocity gradient at 22 GHz is about one order of magnitude higher than at 6.7 GHz (Table REF ).",
"Extensive discussion of the velocity gradients for methanol masers by Moscadelli et al.",
"([25]) suggests that there is a kinematical interpretation of their origin.",
"We note that the methanol maser gradients in both targets can reflect the ordered motions on scales of 10$-$ 60 AU.",
"Large velocity gradients for water masers in G31.581$+$ 00.077 suggest that they are related to the outflow motions with velocities larger than 50 km s$^{-1}$ (see Sect.",
"4.3).",
"Table: Parameters derived by fitting the kinematics of the rotatingand expanding disc model.",
"The signs ++ and -- of the rotation andexpansion velocities refer to the clockwise or anti-clockwise rotation andoutflow or inflow for positive i.",
"Both rotation and floware reversed in the case of negative i.For each source, both signs together with the sign of i could be reversedsince our model does not give the directions unambiguously.",
"In the case ofG33.641--00.228, two model fits are presented as described in Sect.",
"4.3.Table: Parameters derived from fitting the biconical outflow modelof Moscadelli et al. ().",
"In the case ofG31.581++00.077, two model fits are presented as described in Sect.",
"4.3."
],
[
"Kinematic models",
"The two sources investigated in this study possibly belong to a group of objects where a ring-like or arc-like methanol maser distribution traces a circumstellar disc/torus around a high-mass young stellar object, whereas the water maser distribution is orthogonal to the major axis of the methanol structure (Bartkiewicz et al. [4]).",
"Furthermore, these objects are not associated with detectable continuum emission at cm wavelengths (Bartkiewicz et al.",
"[3]) and may represent an early stage of evolution.",
"The new EVN observations of both sources have shown that all of the individual water maser spots detected previously with the VLA (Bartkiewicz et al.",
"[4]) unfold into complex and filament structures of sizes 18$-$ 160 AU.",
"To examine a disc-outflow scenario in the two sources, we used a model of a rotating and expanding thin disc (Uscanga et al.",
"[44]) for the methanol masers and a model of the outflow (Moscadelli et al.",
"[22]) for the water masers.",
"Detailed descriptions of the modelling are given in the above cited works, and elsewhere for methanol masers in discs (Bartkiewicz et al.",
"[3]) and outflows (Bartkiewicz et al. [2]).",
"We note that in the following we analyse the new water maser data with the methanol maser data obtained 3$-$ 7 years earlier (Bartkiewicz et al. [3]).",
"However, our inspection of a few sources published so far and systematic EVN monitoring demonstrate that the overall methanol maser morphologies are stable on a scale time of 6-8 years (Bartkiewicz et al., in prep.).",
"In Tables 4 and 5, we summarize the best-fit values of both models.",
"For the rotating and expanding thin disc, the rotation (V$_{\\rm rot}$ ), expansion (V$_{\\rm exp}$ ), and systemic (V$_{\\rm sys}$ ) velocities as well as the inclination angle, i, i.e., the angle between the line-of-sight and the normal to the ring plane defined to be i$={\\rm acos} (\\frac{\\rm b}{\\rm a})$ is given for each source.",
"The solutions were based on the minimization of the $\\chi _{\\rm V}^2$ function given by eq.",
"8 in Uscanga et al.",
"([44]).",
"The outflow model for water masers is characterized by the vertex of the cone (X$_{\\rm o}$ , Y$_{\\rm o}$ ), the x–axis coinciding with the projection of the outflow on the plane of the sky at the position angle PA, the inclination angle between the outflow axis and the line-of-sight (i.e., the z–axis) $\\Psi $ , and the opening angle of the outflow/cone 2$\\theta $ .",
"The systemic LSR velocities, V$_{\\rm c}$ , were assumed to be the same as given in Sect. 3.",
"The $\\chi ^2$ function was assumed to be expressed as in Eq.",
"3 of Moscadelli et al.",
"([22]).",
"The methanol emission in G31.581$+$ 00.077 is most accurately reproduced by the model where the powering source is at the centre of the best-fit ellipse and the rotation velocity of 0.9 km s$^{-1}$ is a factor of two lower than the expansion velocity (Tab.",
"REF , Fig.",
"REF ).",
"These values are typical of the class of ring-like methanol masers (Bartkiewicz et al.",
"[3]) and suggest that the methanol structure of $\\sim $ 1000 AU diameter cannot be interpreted as a Keplerian disc.",
"It is instead proposed that the methanol masers arise on the interface region between a large-scale accretion flow and a stellar disc (van Langevelde et al.",
"[50]; Torstensson et al. [43]).",
"We also note that the best-fit systemic velocity of 98.54 km s$^{-1}$ corresponds well to that estimated from molecular lines (Sect.3.1).",
"The SE water emission is reasonably well-fitted by two different models.",
"In the first model, O1, the vertex of the cone with 2$\\theta $ =30 coincides with the centre of the methanol ellipse within 25 mas and the velocity of masers is 77 km s$^{-1}$ (Tab.",
"REF , Fig.",
"REF ).",
"We argue that this fit is consistent with the outflow scenario.",
"As we failed to identify any mid-infrared (MIR) counterpart to the centre of the methanol ellipse as a powering source of the methanol structure and perpendicular to the water maser outflow, we propose an alternative model.",
"This second model, O2, assumes that the position of the cone vertex is just between the red-shifted and blue-shifted voids of SE water masers.",
"The best-fit velocity of the outflow is $\\sim $ 8 km s$^{-1}$ , while the projection of the axis outflow onto the plane of the sky is roughly parallel to the main axis of the methanol structure (Tab.",
"REF , Fig.",
"REF ).",
"In star forming regions, different centres of activity are reported on scale sizes of a few to several hundred AUs (e.g., Torrelles et al.",
"[41], [42]).",
"The proper motion studies of G31.581$+$ 00.077 can clearly provide conclusive evidence for or against this hypothesis.",
"In the case of methanol masers in G33.641$-$ 00.228, we present the two kinematical models for a powering source located at the centre of the best-fit ellipse ($\\Delta $ RA, $\\Delta $ Dec)=(59 mas,17 mas) and that located at the MIR position ($-$ 495 mas, 380 mas).",
"In the first model, D1, the infall velocity of about 1 km s$^{-1}$ is slightly higher than the rotation velocity (Tab.",
"REF ).",
"The structure of water masers is fitted by a narrow ouflow of 2$\\theta $ =18 and velocity of 15 km s$^{-1}$ (Tab.",
"REF , Fig.",
"REF ).",
"It is consistent with a jet-like outflow of water masers roughly perpendicular to the main axis of methanol structure.",
"In the second model, D2, the visible methanol structure is only a small part of the disc radius of $\\sim $ 1900 AU and the rotation velocity of 3.7 km s$^{-1}$ (Tab.",
"REF ), which implies that the enclosed mass is about 30M$_{}$ .",
"In this case, the water masers may trace the wind gas from a disc.",
"We note that the kinematic models are unable to accurately reproduce the observed structures of the methanol and water masers in both targets, and that only measurements of the proper motions of maser spots of the two species can unambiguously constrain the proposed scenarios.",
"Figure: Velocity of the methanol maser spots (open circles) in G31.581++00.077and G33.641++00.228 versus azimuth angle measured from the major axis.The sinusoidal line represents the best-fit kinematical model of a rotatingand expanding disc of infinitesimal thickness to the methanolspots with the parameters listed inTable 4.",
"For completeness, the water maser spots are shown as squares.In a case of G33.641++00.228 two kinematical models arepresented for a powering source located as follow: D1 – in the centre of the best-fittedellipse, D2 – at the MIR position.Figure: Outflow models fitted to water (squares)in G31.581++00.077 and G33.641++00.228 according to themodel of Moscadelli et al. ().",
"The relevant parameters are listed inTable 5.",
"The right panel presents a comparison of obtained data (squares) vs. model (crosses).V c _{\\rm c} is the systemic LSR velocity as given in Sect.",
"3."
],
[
"Conclusions",
"We have carried out high-angular resolution studies of the 22.2 GHz of water maser line towards two methanol maser sources G31.581$+$ 00.077 and G33.641$-$ 00.228 using the EVN.",
"Although their morphologies did not differ significantly from the previous VLA results, the astrometry at the mas level and the properties of the maser clusters could be estimated owing to the high-angular and spectral resolution.",
"In total, we detected eight and six water maser clouds towards G31.581$+$ 00.077 and G33.641$-$ 00.228, respectively.",
"In the first target, the water maser components are associated with different centres of star-forming activity, and the components associated with the methanol ring-like structure possibly trace the outflow.",
"In the source G33.641$-$ 00.228, southern water masers possibly trace a wind from a disc.",
"The kinematic models containing ring-like or arc-like methanol maser structures are able to trace a circumstellar disc/torus around a high-mass young stellar object, whereas the water maser distribution is orthogonal to the major axis of the applied methanol structure and poorly constrained by the present data.",
"The present studies show that the two sources are good targets for proper motion studies in order to understand more clearly the kinematics of gas in the environments of high-mass stellar objects.",
"They also encourage us to extend the multi-epoch EVN observations for a whole sample.",
"AB and MS acknowledge support by the Polish Ministry of Science and Higher Education through grant N N203 386937.",
"This work has also been supported by the European Community Framework Programme 7, Advanced Radio Astronomy in Europe, grant agreement nl.",
": 227290."
]
] | 1204.0806 |
[
[
"Bounds on the non-real spectrum of differential operators with\n indefinite weights"
],
[
"Abstract Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces.",
"Under the assumption that 0 and $\\infty$ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set.",
"The main results are quantitative estimates for this set, which are applied to Sturm-Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains."
],
[
"Introduction",
"We consider linear operators associated with an indefinite differential expression $\\mathcal {L}=\\frac{1}{w}\\,\\ell ,$ where $w\\ne 0$ is a real-valued, locally integrable weight function which changes its sign and $\\ell $ is an ordinary or partial differential expression of the form $\\ell =-\\frac{d}{dx} p\\frac{d}{dx}+q\\qquad \\text{or}\\qquad \\ell =-\\sum _{j,k=1}^n \\frac{\\partial }{\\partial x_j} \\,a_{jk} \\,\\frac{\\partial }{\\partial x_k} +a$ acting on a real interval or domain $\\Omega \\subset \\mathbb {R}^n$ , respectively.",
"In the first case $p^{-1}$ and $q$ are assumed to be real and (locally) integrable.",
"In the second case $a\\in L^\\infty (\\Omega )$ is real and $a_{jk}$ are $C^{\\infty }$ -coefficients such that $\\ell $ is formally symmetric and uniformly elliptic on $\\Omega \\subset \\mathbb {R}^n$ , and, additionally, the weight function $w$ and its inverse are essentially bounded.",
"Together with appropriate boundary conditions (if necessary) the differential expression $\\ell $ gives rise to a selfadjoint operator $T$ in an $L^2$ Hilbert space.",
"Multiplication with $1/w$ leads to the corresponding indefinite differential operator $A=\\frac{1}{w} T$ associated with $\\mathcal {L}$ in (REF ), which is selfadjoint in a weighted $L^2$ Krein space.",
"Most of the existing literature for differential operators with an indefinite weight focuses on regular or left-definite problems.",
"The spectral properties of the operators associated to $\\mathcal {L}$ in the case of a regular Sturm-Liouville expression $\\ell $ were investigated in detail, we refer to [20], the monograph [56] and the detailed references therein.",
"Also singular left-definite Sturm-Liouville problems are well studied.",
"Here the selfadjoint operator $T$ associated with $\\ell $ is uniformly positive and, hence, the corresponding indefinite differential operator $A$ associated with $\\mathcal {L}$ in (REF ) has real spectrum with a gap around zero, cf.",
"e.g., [15], [16], [17], [41], [42] and [56].",
"In the case $T\\ge 0$ it is of particular interest whether the operator $A$ is similar to a selfadjoint operator; necessary and sufficient similarity criteria can be found in, e.g., [37], [38].",
"The slightly more general situation where the indefinite Sturm-Liouville operator $A$ has finitely many negative squares or is quasi-uniformly positive is discussed in, e.g., [12], [14], [22].",
"The general non-left-definite case is more difficult, especially the situation where the essential spectrum of the selfadjoint operator $T$ associated with $\\ell $ is no longer contained in $\\mathbb {R}^+$ .",
"In this case subtle problems appear, as, e.g., accumulation of non-real eigenvalues to the real axis, see [8], [11], [13], [39].",
"The spectral properties of indefinite elliptic partial differential operators have been investigated in, e.g., [25], [26], [27], [49], [50], [51] on bounded domains and in [9], [21] on unbounded domains.",
"In the non-left-definite situation the indefinite differential operator $A$ typically possesses non-real spectrum.",
"General perturbation results for selfadjoint operators in Krein spaces from [4], [7], [30] ensure that the non-real spectrum is contained in a compact set.",
"To the best of our knowledge, explicit bounds on the size of this set do not exist in the literature.",
"For singular indefinite Sturm-Liouville operators it is conjectured in [56] that the lower bound of the spectrum of $T$ is related to a bound for the non-real eigenvalues; the numerical examples contained in [11], [12] support this conjecture.",
"It is one of our main objectives to confirm this conjecture and to provide explicit bounds on the non-real eigenvalues of indefinite Sturm-Liouville operators and indefinite elliptic partial differential operators.",
"For this, we develop here an abstract Krein space perturbation approach which is designed for applications to differential operators with indefinite weights.",
"Besides bounds on the non-real spectrum, our general perturbation results Theorem REF and Theorem REF in Section provide also quantitative estimates for intervals containing only spectrum of positive/negative type in terms of the norm of the perturbation and some resolvent integral, as well as information on the critical point $\\infty $ of the perturbed operator.",
"The basic idea of our approach is simple: If $T$ is a semibounded selfadjoint operator in a Hilbert space with some negative lower bound $\\gamma $ , then the operator $T+\\gamma $ is non-negative or uniformly positive.",
"Hence $A_0:=A+\\frac{\\gamma }{w}=\\frac{1}{w} (T+\\gamma )$ (or, more generally, $A_0=G^{-1}(T+\\gamma )$ with some Gram operator $G$ connecting the Hilbert and Krein space inner product) is a non-negative operator in a Krein space and therefore the spectrum of $A_0$ is real.",
"Moreover, the difference of $A_0$ and $A$ is a bounded operator.",
"In general, a bounded perturbation of $A_0$ may lead to unbounded non-real spectrum, but under the additional assumption that 0 and $\\infty $ are not singular critical points of $A_0$ , the influence of the perturbation on the non-real spectrum can be controlled.",
"The proofs of Theorem REF and REF are based on general Krein space perturbation techniques, norm estimates and local spectral theory; they are partly inspired by methods developed in [9], [48].",
"The abstract perturbation results from Section are applied to ordinary and partial differential operators with indefinite weights in Section .",
"First we investigate the singular indefinite Sturm-Liouville operator $A$ with $p=1$ , a real potential $q\\in L^\\infty (\\mathbb {R})$ and the particularly simple weight function $w(x) =\\operatorname{sgn}(x)$ .",
"It turns out in Theorem REF that the nonreal spectrum of $A$ is contained in the set $\\bigl \\lbrace \\lambda \\in \\operatorname{dist}\\,\\bigl (\\lambda , (-d,d)\\bigr )\\le 5 \\Vert q\\Vert _\\infty ,\\, |\\operatorname{Im}\\lambda |\\le 2 \\Vert q\\Vert _\\infty \\bigr \\rbrace ,$ where $-d= \\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x)$ is assumed to be negative, and estimates on real spectral points of positive and negative type are obtained as well.",
"Our second example is a second order uniformly elliptic operator defined on an unbounded domain $\\Omega \\subset \\mathbb {R}^n$ with bounded coefficients and an essentially bounded weight function $w$ having an essentially bounded inverse.",
"We emphasize that the estimates for the non-real spectrum of indefinite differential operators seem to be the first ones in the mathematical literature.",
"Notation: As usual, $\\mathbb {C}^+$ ($\\mathbb {C}^-$ ) denotes the open upper half-plane (the open lower half-plane, respectively), $\\mathbb {R}^+:=(0,\\infty )$ and $\\mathbb {R}^-:=(-\\infty , 0)$ .",
"The compactification $\\mathbb {R} \\cup \\lbrace \\infty \\rbrace $ of $\\mathbb {R}$ is denoted by $\\overline{\\mathbb {R}}$ , the compactification $\\mathbb {C} \\cup \\lbrace \\infty \\rbrace $ by $\\overline{\\mathbb {C}}$ .",
"By $L(\\mathcal {H},\\mathcal {K})$ we denote the set of all bounded and everywhere defined linear operators from a Hilbert space $\\mathcal {H}$ to a Hilbert space $\\mathcal {K}$ .",
"We write $L(\\mathcal {H})$ for $L(\\mathcal {H},\\mathcal {H})$ .",
"For a closed, densely defined linear operator $T$ in $\\mathcal {H}$ we denote the spectrum and the resolvent set by $\\sigma (T)$ and $\\rho (T)$ , respectively.",
"A point $\\lambda \\in belongs to the {\\it approximate point spectrum} $ ap(T)$ of $ T$ if there exists a sequence $ (fn)$ in $ domT$ with $ fn = 1$ for $ n$\\mathbb {N}$$ and $ (T - )fn0$ as $ n$.",
"The essentialspectrum of $ T$ is the set $ {C : A- is not Fredholm}$.$"
],
[
"Selfadjoint operators in Krein spaces",
"Let $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ be a Krein space, let $J$ be a fixed fundamental symmetry in $\\mathcal {H}$ and denote by $(\\cdot \\,,\\cdot )$ the Hilbert space scalar product induced by $J$ , i.e.",
"$(\\cdot \\,,\\cdot )= [J\\,\\cdot \\,,\\cdot ]$ .",
"The induced norm is denoted by $\\Vert \\cdot \\Vert $ .",
"For a detailed treatment of Krein spaces and operators therein we refer to the monographs [5] and [18].",
"For a densely defined linear operator $A$ in $\\mathcal {H}$ the adjoint with respect to the Krein space inner product $[\\cdot ,\\cdot ]$ is denoted by $A^+$ .",
"We mention that $A^+ = JA^*J$ , where $A^*$ denotes the adjoint of $A$ with respect to the scalar product $(\\cdot \\,,\\cdot )$ .",
"The operator $A$ is called selfadjoint in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ (or $[\\cdot \\,,\\cdot ]$ -selfadjoint) if $A = A^+$ .",
"This is equivalent to the selfadjointness of the operator $JA$ in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ .",
"The spectrum of a selfadjoint operator $A$ in a Krein space is in general not contained in $\\mathbb {R}$ , but its real spectral points belong to the approximate point spectrum (see, e.g., [18]), $\\sigma (A)\\cap \\mathbb {R}\\subset \\sigma _{{ap}}(A).$ A selfadjoint operator $A$ in $(\\mathcal {H}, [\\cdot \\,,\\cdot ])$ is said to be non-negative if $\\rho (A)\\ne \\varnothing $ and if $[Af,f]\\ge 0$ holds for all $f\\in \\operatorname{dom}A$ .",
"If, for some $\\gamma > 0$ , $[Af,f]\\ge \\gamma \\Vert f\\Vert ^2$ holds for all $f\\in \\operatorname{dom}A$ , then $A$ is called uniformly positive.",
"A selfadjoint operator $A$ in $(\\mathcal {H}, [\\cdot \\,,\\cdot ])$ is uniformly positive if and only if $A$ is non-negative and $0\\in \\rho (A)$ .",
"It is well-known that the spectrum of a non-negative operator $A$ in a Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ is real.",
"Moreover, $A$ possesses a spectral function $E_A$ on $\\overline{\\mathbb {R}}$ which is defined for all Borel sets $\\Delta \\subset \\overline{\\mathbb {R}}$ whose boundary does not contain the points 0 and $\\infty $ .",
"The corresponding spectral projection $E_A(\\Delta )$ is bounded and $[\\cdot \\,,\\cdot ]$ -selfadjoint; see [3], [31], [43], [45], [46] for further details.",
"The point 0 ($\\infty $ ) is called a critical point of $A$ if for each Borel set $\\Delta \\subset \\overline{\\mathbb {R}}$ with $0\\in \\Delta $ ($\\infty \\in \\Delta $ , respectively) such that $E_A(\\Delta )$ is defined the inner product $[\\cdot \\,,\\cdot ]$ is indefinite on the subspace $E_A(\\Delta )\\mathcal {H}$ .",
"Moreover, if the point 0 ($\\infty $ ) is a critical point of $A$ , it is called regular if there exists $C > 0$ such that $\\Vert E_A([-\\varepsilon ,\\varepsilon ])\\Vert \\le C$ ($\\Vert E_A([-\\varepsilon ^{-1},\\varepsilon ^{-1}])\\Vert \\le C$ , respectively) holds for each $\\varepsilon \\in (0,1)$ .",
"Otherwise, the critical point 0 or $\\infty $ is called singular.",
"In this paper we will investigate operators which are non-negative outside of some compact set $K$ with $0\\in K$ , see Definition REF belowDefinition REF is slightly more general than the definition used in [10].",
"Contrary to the definition in [10], a spectral projector for an operator non-negative over $\\overline{\\setminus K in the sense of Definition \\ref {d:lnn}corresponding to the set \\overline{\\setminus K does, in general, not exist.}}$.",
"This notion is a generalization of the above concept of non-negative operators in Krein spaces and will be used in the study of additive bounded perturbations of non-negative operators in Section and .",
"For a set $\\Delta \\subset we denote by $ *$ the set which is obtained by reflecting $$ in the real axis, that is, $ * = {: }$.$ Definition 2.1 Let $K=K^*\\subset be a compact set such that $ 0K$ and $ +K$ is simply connected.",
"A selfadjoint operator$ A$ in the Krein space $ (H,[,])$ is said to be {\\em non-negative over} $K$ if for any bounded openneighborhood $ U$ of $ K$ in $ there exists a bounded $[\\cdot \\,,\\cdot ]$ -selfadjoint projection $E_\\infty $ such that with respect to the decomposition $\\mathcal {H}= (I-E_\\infty )\\mathcal {H}\\,[\\dotplus ]\\,E_\\infty \\mathcal {H}$ the operator $A$ can be written as a diagonal operator matrix $A = \\begin{pmatrix}A_0 & 0\\\\0 & A_\\infty \\end{pmatrix},$ where $A_0$ is a bounded selfadjoint operator in the Krein space $((I-E_\\infty )\\mathcal {H},[\\cdot \\,,\\cdot ])$ whose spectrum is contained in $\\overline{\\mathcal {U}}$ and $A_\\infty $ is a non-negative operator in $(E_\\infty \\mathcal {H},[\\cdot \\,,\\cdot ])$ with $\\mathcal {U}\\subset \\rho (A_\\infty )$ .",
"Relations (REF ) and (REF ) encode the non-negativity outside of the set $K$ , which can also be seen in the following example.",
"Example 2.2 Let $A$ be the direct sum of a bounded selfadjoint operator $A_1$ in the Krein space $(\\mathcal {H}_1, [\\cdot \\,,\\cdot ]_1)$ and of a non-negative operator $A_2$ in the Krein space $(\\mathcal {H}_2, [\\cdot \\,,\\cdot ]_2)$ , $A := A_1\\times A_2.$ Let $K_{r_1}$ be the closed disc around zero whose radius is the spectral radius $r_1$ of $A_1$ , then for a bounded open neighborhood $\\mathcal {U}$ of $K_{r_1}$ in $ we set$ E:= E2(RU) $, where $ E2$ is the spectral function of $ A2$.",
"We obtain a decomposition of the form (\\ref {e:dec_lnn})and the operator $ A$ can be written as in (\\ref {e:deco}),where $ A0$ and $ A$ have the properties stated inDefinition \\ref {d:lnn}.",
"Therefore, the operator $ A$ isnon-negative over $Kr1$.$ Let $A$ be non-negative over $\\overline{\\setminus K. Then from the representation (\\ref {e:deco}) it is seen thatfor each bounded open neighborhood \\mathcal {U} of K the non-real spectrum of A is contained in \\overline{\\mathcal {U}}.",
"Therefore,\\begin{equation}\\sigma (A)\\setminus \\mathbb {R}\\subset K.\\end{equation}By (\\ref {e:deco})the spectral properties of A and A_\\infty in \\overline{\\mathbb {R}}\\setminus K are the same.",
"Therefore, we say that \\infty is a{\\em singular (regular) critical point} of A if \\infty is a singular (regular, respectively) critical point of A_\\infty .The following proposition is a direct consequence of \\cite [Theorem~3.2]{cu} andthe representation (\\ref {e:deco}).",
"}\\begin{proposition}Let A be a selfadjoint operator in the Krein space (\\mathcal {H},[\\cdot \\,,\\cdot ]) and assume that A is non-negative over \\overline{\\setminus K.Then \\infty is not a singular critical point of A if and only if there exists a uniformly positive operator W in the Kreinspace (\\mathcal {H},[\\cdot \\,,\\cdot ]) such that W\\operatorname{dom}A\\subset \\operatorname{dom}A.",
"}\\end{proposition}$ In order to characterize operators which are non-negative over $\\overline{\\setminus K in Theorem \\ref {t:eqthm} below we recallthe notions of the spectral points of positive and negative type of selfadjoint operators in Krein spaces.",
"The following definitioncan be found in, e.g., \\cite {j03,LcMM,lmm}.",
"}\\begin{definition}Let A be a selfadjoint operator in the Krein space (\\mathcal {H},[\\cdot \\,,\\cdot ]).",
"A point \\lambda \\in \\sigma _{{ap}}(A) is called a {\\em spectral point ofpositive {\\rm (}negative{\\rm )} type} of A if for every sequence (f_n) in \\operatorname{dom}A with \\Vert f_n\\Vert = 1 and (A - \\lambda )f_n\\rightarrow 0as n\\rightarrow \\infty we have\\liminf _{n\\rightarrow \\infty }\\,[f_n,f_n] > 0\\quad \\Big (\\limsup _{n\\rightarrow \\infty }\\,[f_n,f_n] < 0,\\;\\text{respectively}\\Big ).The set of all spectral points of positive {\\rm (}negative{\\rm )} type of A will be denoted by \\sigma _+(A) {\\rm (}\\sigma _-(A),respectively{\\rm )}.",
"A set \\Delta \\subset is said to be of {\\em positive {\\rm (}negative{\\rm )} type} with respect to Aif each spectral point of A in \\Delta is of positive type {\\rm (}negative type, respectively{\\rm )}.\\end{definition}$ The sets $\\sigma _+(A)$ and $\\sigma _-(A)$ are contained in $\\mathbb {R}$ , open in $\\sigma (A)$ and the non-real spectrum of $A$ cannot accumulate to $\\sigma _+(A)\\cup \\sigma _-(A)$ .",
"Moreover, at a spectral point $\\lambda _0$ of positive or negative type the growth of the resolvent is of order one in the sense of the following definition, see also [6], [36], [48].",
"Definition 2.3 Let $A$ be a selfadjoint operator in the Krein space $\\mathcal {H}$ and assume that $\\lambda _0\\in \\overline{\\mathbb {R}}$ is not an accumulation point of $\\sigma (A)\\setminus \\mathbb {R}$ .",
"We say that the growth of the resolvent of $A$ at $\\lambda _0$ is of order $m\\ge 1$ if there exist an open neighborhood $\\mathcal {U}$ of $\\lambda _0$ in $\\overline{ and M > 0 such that \\mathcal {U}\\setminus \\overline{\\mathbb {R}}\\subset \\rho (A) and\\begin{equation}\\Vert (A - \\lambda )^{-1}\\Vert \\le M\\,\\frac{(1+|\\lambda |)^{2m-2}}{|\\operatorname{Im}\\lambda |^{m}}\\end{equation}holds for all \\lambda \\in \\mathcal {U}\\setminus \\overline{\\mathbb {R}}.",
"}$ Clearly, if $\\lambda _0\\ne \\infty $ , Definition REF can be formulated equivalently by replacing the enumerator in () by 1.",
"Moreover, if the growth of the resolvent of $A$ at $\\lambda _0$ is of order $m$ , then it is of order $n$ for each $n > m$ .",
"The following theorem gives an equivalent characterization for non-negative operators in a neighbourhood of $\\infty $ .",
"Theorem 2.4 Let $A$ be a selfadjoint operator in the Krein space $\\mathcal {H}$ and let $K=K^*\\subset be a compact set such that $ 0K$ and$ +K$ is simply connected.",
"Then $ A$ is non-negative over $K$ if and only if the following conditions are satisfied:\\begin{enumerate}\\item [{\\rm (i)}] \\sigma (A)\\setminus \\mathbb {R}\\subset K and (\\sigma (A)\\setminus K)\\cap \\mathbb {R}^\\pm \\subset \\sigma _\\pm (A).\\item [{\\rm (ii)}] The growth of the resolvent of A at \\infty is of finite order.\\end{enumerate}$ Let $A$ be non-negative over $\\overline{\\setminus K.It is well-known that the growth of the resolvent of a non-negative operator in a Krein space is of finite orderfor all \\lambda _0 \\in \\overline{\\mathbb {R}} and that spectral points in \\mathbb {R}^+ (\\mathbb {R}^-) are ofpositive (resp.\\ negative) type, see, e.g., \\cite [Proposition II.2.1 and Proposition II.3.1]{L82} and \\cite [Section 2.1]{J88}.Therefore, the statements (i) and (ii) are consequences ofthe representation (\\ref {e:deco}); cf.",
"(\\ref {nrspec}).",
"}Conversely, assume that (i) and (ii) are satisfied.",
"Then the operator $ A$ is definitizable over $K$, see \\cite {j03}.In particular $ A$ possesses a local spectral function $ E$ which is defined on all Borel sets $$\\mathbb {R}$K$ forwhich neither $$ nor points of $ K$\\mathbb {R}$$ are boundary points.",
"For such aset $$ the following holds:\\begin{enumerate}\\item [{\\rm (a)}] E(\\Delta ) is a bounded [\\cdot \\,,\\cdot ]-selfadjoint projection and commuteswith every bounded operator which commutes with the resolvent of A;\\item [{\\rm (b)}] \\sigma (A|E(\\Delta )\\mathcal {H})\\,\\subset \\,\\sigma (A)\\cap \\overline{\\Delta };\\item [{\\rm (c)}] \\sigma (A|(I - E(\\Delta ))\\mathcal {H})\\,\\subset \\,\\sigma (A)\\setminus \\operatorname{int}(\\Delta ), where\\operatorname{int}(\\Delta ) is the interior of \\Delta with respect to the topology of\\overline{\\mathbb {R}};\\item [{\\rm (d)}] If, in addition, \\Delta is a neighbourhood of \\infty (with respect to the topology of\\overline{\\mathbb {R}}), then A|(I-E(\\Delta )) \\mathcal {H} is a bounded operator;\\end{enumerate}cf.",
"\\cite [Definition 3.13, Theorems 3.15 and 4.8, Remark 4.9]{j03} and note that (d) follows from the definition of the local spectral function\\cite [Definition 3.13]{j03} and the definition of the extended spectrum $ (A)$in \\cite {j03}.$ Let $\\mathcal {U}$ be a bounded open neighborhood of $K$ in $ and set $ E:= E($\\mathbb {R}$U)$.",
"By (a), with respect to the decomposition$$\\mathcal {H}= (I - E_\\infty )\\mathcal {H}\\,[\\dotplus ]\\,E_\\infty \\mathcal {H},$$the operator $ A$ is represented by$$A = \\begin{pmatrix}A_0 & 0\\\\0 & A_\\infty \\end{pmatrix},$$and by (b) the spectrum of the operator $ A$ is real, $ (A) R U$.",
"According to (i) and (ii),the operator $ A$ is definitizable over $$ and thus definitizable, see \\cite [Theorem 4.7]{j03}.Therefore, \\cite [Corollary 3 of Proposition II.5.2]{L82} implies that $ A-1$ (and thus $ A$) is non-negative in $ EH$.By (d), the operator $ A0= A|(I-E) H$ is bounded and, by (c),$ (A0) (A) int($\\mathbb {R}$U)U$.\\Box $"
],
[
"Bounded selfadjoint perturbations of non-negative operators",
"In this section we prove two abstract results on additive bounded perturbations of non-negative (and some closely connected class of) operators in Krein spaces that lead to perturbed operators which are non-negative over some neighborhood of infinity.",
"In both cases the neighborhood is given in quantitative terms.",
"The results will be applied in Section to singular indefinite Sturm-Liouville operators and to second order elliptic operators with indefinite weights."
],
[
"Two perturbation results",
"The following notation will be useful when formulating our main results below: For a set $\\Delta \\subset \\mathbb {R}$ and $r > 0$ we define $K_r(\\Delta ) := \\lbrace z\\in \\operatorname{dist}(z,\\Delta )\\,\\le \\,r\\rbrace .$ Our first main theorem concerns bounded selfadjoint perturbations of non-negative operators in Krein spaces.",
"Theorem 3.1 Let $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ be a Krein space with fundamental symmetry $J$ and norm $\\Vert \\cdot \\Vert = [J\\cdot ,\\cdot ]^{1/2}$ .",
"Let $A_0$ be a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ such that 0 and $\\infty $ are not singular critical points of $A_0$ , and $0\\notin \\sigma _p(A_0)$ .",
"Furthermore, let $V\\in L(\\mathcal {H})$ be a selfadjoint operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"Then the following holds: (i) If $V$ is non-negative, then $A_0+V$ is a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"(ii) If $V$ is not non-negative, then $A_0+V$ is non-negative over $\\overline{\\setminus K_r((-d,d)),where\\begin{equation}r = \\frac{1+\\tau _0}{2}\\Vert V\\Vert ,\\qquad \\quad d = -\\frac{1+\\tau _0}{2}\\,\\min \\sigma (JV),\\end{equation}and\\begin{equation}\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert <\\infty .\\end{equation}}Moreover, in both cases $$ is not a singular critical point of $ A0 + V$.$ The following simple example shows that without further assumptions a bounded selfadjoint perturbation $V$ of a non-negative or uniformly positive $A_0$ in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ may lead to an operator $A_0+V$ with unbounded non-real spectrum.",
"Example 3.2 Let $(\\cal K,(\\cdot ,\\cdot ))$ be a Hilbert space and let $H$ be an unbounded selfadjoint operator in $\\cal K$ such that $\\sigma (H)\\subset (0,\\infty )$ .",
"Equip $\\cal H:=\\cal K\\oplus \\cal K$ with the Krein space inner product $\\left[\\begin{pmatrix}k_1 \\\\ k_2\\end{pmatrix},\\begin{pmatrix}l_1 \\\\ l_2\\end{pmatrix}\\right] :=(k_1,l_2)+(k_2,l_1),\\qquad \\begin{pmatrix}k_1 \\\\ k_2\\end{pmatrix},\\begin{pmatrix}l_1 \\\\ l_2\\end{pmatrix}\\in \\cal H,$ and consider the operators $A_0=\\begin{pmatrix} 0 & I \\\\ H & 0\\end{pmatrix},\\quad V=\\begin{pmatrix} 0 & -I \\\\ 0 & 0\\end{pmatrix},\\quad \\text{and}\\quad A_0+V =\\begin{pmatrix} 0 & 0 \\\\ H & 0\\end{pmatrix}.$ It is easy to see that $A_0$ is a non-negative operator and $V$ is a bounded selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"Moreover, as $\\operatorname{dom}(A_0+V)=\\operatorname{dom}H\\oplus {\\cal H}$ we conclude $\\operatorname{ran}(A_0+V-\\lambda )\\ne \\cal H$ for every $\\lambda \\in , that is, $ (A0+V)=.",
"Our second main result applies to operators of the form $A = G^{-1}T$ , where $T$ is a semibounded selfadjoint operator in a Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ and $G$ is a selfadjoint bounded and boundedly invertible operator in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ .",
"In other words, $A$ is selfadjoint in the Krein space $(\\mathcal {H},(G\\cdot ,\\cdot ))$ and $A + \\eta G^{-1}$ is uniformly positive for suitable $\\eta $ .",
"Such a situation arises, e.g., when considering elliptic differential operators with an indefinite weight function; cf.",
"Section 4.2.",
"Theorem 3.3 Let $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ be a Hilbert space, $\\Vert \\cdot \\Vert = (\\cdot \\,,\\cdot )^{1/2}$ and $[\\cdot \\,,\\cdot ]=(G\\cdot ,\\cdot )$ be as above.",
"Let $A$ be a selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ such that for some $\\gamma > 0$ we have $[Af,f]\\ge -\\gamma \\Vert f\\Vert ^2\\quad \\text{for all }f\\in \\operatorname{dom}A.$ Assume furthermore that for some $\\eta > \\gamma $ $\\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A + \\eta G^{-1} - it)^{-1}\\,dt\\,\\right\\Vert < \\infty ,$ holds.",
"Then $A$ is non-negative over $\\overline{\\setminus K_r((-r,r)), where\\begin{equation*}r = \\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert G^{-1}\\Vert .\\end{equation*}Moreover, \\infty is not a singular critical point of A.",
"}$ Remark 3.4 Setting $A_0 := A + \\eta G^{-1}$ in Theorem REF and $V:=-\\eta G^{-1}$ we have $A = A_0 + V$ and hence Theorem REF can also be seen as a variant of Theorem REF .",
"Here $A_0=A + \\eta G^{-1}$ is uniformly positive so that $0\\in \\rho (A_0)$ is not a critical point of $A_0$ and the entire imaginary axis belongs to $\\rho (A + \\eta G^{-1})$ .",
"It follows from [55] and [32] that $\\tau _\\eta < \\infty $ is equivalent to $\\infty $ not being a singular critical point of $A_0$ .",
"In what follows we describe the structure of the proofs of Theorem REF (ii) and Theorem REF .",
"Let $E$ denote the spectral function of the non-negative operator $A_0$ .",
"As 0 and $\\infty $ are not singular critical points of $A_0$ , the spectral projections $E_+ := E((0,\\infty ))$ and $E_- := E((-\\infty ,0))$ , and the corresponding spectral subspaces $\\mathcal {H}_\\pm := E_\\pm \\mathcal {H}$ of $A_0$ exist.",
"From $0\\notin \\sigma _p(A_0)$ it follows that $\\mathcal {H}= \\mathcal {H}_+\\,[\\dotplus ]\\,\\mathcal {H}_-$ and $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ are Hilbert spaces, see, e.g., [46].",
"Therefore (REF ) is a fundamental decomposition of the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"With respect to this decomposition the operator $A_0$ and the fundamental symmetry $\\widetilde{J}$ corresponding to (REF ) can be written as operator matrices: $A_0 = \\begin{pmatrix}A_{0,+} & 0\\\\0 & A_{0,-}\\end{pmatrix}\\quad \\text{and}\\quad \\widetilde{J} = \\begin{pmatrix}I & 0\\\\0 & -I\\end{pmatrix}.$ Note that the operator $\\pm A_{0,\\pm }$ is a selfadjoint non-negative operator in the Hilbert space $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ .",
"Hence, $A_0$ is selfadjoint in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim )$ , where $(f,g)_\\sim := [\\widetilde{J}f,g] = [E_+f,g] - [E_-f,g],\\qquad f,g\\in \\mathcal {H}.$ Now, let $V\\in L(\\mathcal {H})$ be a bounded selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"Then, with respect to the decomposition (REF ) it admits an operator matrix representation $V = \\begin{pmatrix}V_+ & B\\\\C & V_-\\end{pmatrix}.$ From the selfadjointness of $V$ in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ one concludes that $V_\\pm $ is selfadjoint in the Hilbert space $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ]) = (\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim )$ and $C = -B^{\\widetilde{*}},$ where $B^{\\widetilde{*}}$ denotes the adjoint of $B$ with respect to the scalar product $(\\cdot \\,,\\cdot )_\\sim $ .",
"Hence, the perturbed operator $A := A_0 + V$ is represented by $A = \\begin{pmatrix}A_{0,+} + V_+ & B\\\\-B^{\\widetilde{*}} & A_{0,-} + V_-\\end{pmatrix}.$ For operators as in (REF ) we show in Theorem REF below that $\\mathbb {R}\\setminus K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,-} + V_-))$ is of positive type, $\\mathbb {R}\\setminus K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,+} + V_+))$ is of negative type, the non-real spectrum of $A$ is contained in $K:=K_{\\Vert B\\Vert _\\sim }\\bigl (\\sigma (A_{0,+} + V_+)\\bigr )\\cap K_{\\Vert B\\Vert _\\sim } \\bigl (\\sigma (A_{0,-} + V_-)\\bigr ),$ and the growth of the resolvent of $A$ at $\\infty $ is of order one.",
"Here $\\Vert \\cdot \\Vert _\\sim $ stands for the operator norm induced by (REF ).",
"Then, by Theorem REF , the operator $A$ is non-negative over $\\overline{\\setminus K, where K is as in (\\ref {BadHersfeld}).In a final step it remains to bound the quantities\\Vert B\\Vert _\\sim \\quad \\mbox{and} \\quad K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,\\pm } + V_\\pm ))by the quantitiesr and \\tau _0 as in the statement of Theorem \\ref {t:main1}.",
"}$ Some auxiliary statements In this section we formulate and prove some auxiliary statements which will be used in the proofs Theorems REF and REF .",
"For the special case of bounded operators the first part of the following theorem was already proved in [48], see also [47] and [53], [54] for similar results with unbounded entries on the diagonal.",
"Theorem 3.5 Let $S_+$ and $S_-$ be selfadjoint operators in some Hilbert spaces $\\mathfrak {H}_+$ and $\\mathfrak {H}_-$ , respectively, let $M\\in L(\\mathfrak {H}_-,\\mathfrak {H}_+)$ and define the operators $S := \\begin{pmatrix}S_+ & M\\\\-M^* & S_-\\end{pmatrix}\\quad \\text{and}\\quad \\mathfrak {J}:= \\begin{pmatrix}I & 0\\\\0 & -I\\end{pmatrix}$ in the Hilbert space $\\mathfrak {H}:= \\mathfrak {H}_+\\oplus \\mathfrak {H}_-$ .",
"Then the operator $S$ is selfadjoint in the Krein space $(\\mathfrak {H},(\\mathfrak {J}\\cdot ,\\cdot ))$ and with $\\nu := \\Vert M\\Vert $ the following statements hold: $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_+))\\cap K_\\nu (\\sigma (S_-))$ .",
"$\\mathbb {R}\\setminus K_\\nu (\\sigma (S_-))$ is of positive type with respect to $S$ .",
"$\\mathbb {R}\\setminus K_\\nu (\\sigma (S_+))$ is of negative type with respect to $S$ .",
"Moreover, if $S_+$ is bounded from below and $S_-$ is bounded from above (or vice versa), the non-real spectrum of $S$ is bounded and the growth of the resolvent of $S$ at $\\infty $ is of order one.",
"We set $[\\cdot \\,,\\cdot ]:= (\\mathfrak {J}\\cdot ,\\cdot )$ .",
"Let $\\lambda \\in \\mathbb {C} \\setminus K_\\nu (\\sigma (S_-))$ and $\\alpha := \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_-)) < 1$ .",
"We claim that for some $r(\\alpha )>0$ and all $f\\in \\operatorname{dom}S$ the following implication $\\Vert (S - \\lambda )f\\Vert \\le \\frac{1-\\alpha ^2}{4\\alpha }\\nu \\Vert f\\Vert \\quad \\Longrightarrow \\quad [f,f]\\,\\ge \\,r(\\alpha )\\Vert f\\Vert ^2$ holds.",
"In fact, set $\\varepsilon := \\frac{1-\\alpha ^2}{4\\alpha }$ and let $f\\in \\operatorname{dom}S$ with $\\Vert (S - \\lambda )f\\Vert \\le \\varepsilon \\nu \\Vert f\\Vert .$ With respect to the decomposition $\\mathfrak {H}= \\mathfrak {H}_+\\oplus \\mathfrak {H}_-$ we write $f$ and $g := (S - \\lambda )f$ as column vectors $f = \\begin{pmatrix}f_+\\\\f_-\\end{pmatrix}\\in \\operatorname{dom}S_+\\oplus \\operatorname{dom}S_-, \\quad g = \\begin{pmatrix}g_+\\\\g_-\\end{pmatrix}.$ Then $g_- = -M^*f_+ + (S_- - \\lambda )f_-$ , or, equivalently, $f_- = (S_- - \\lambda )^{-1}g_- + (S_- - \\lambda )^{-1}M^*f_+.$ As $\\Vert g_-\\Vert \\le \\Vert g\\Vert = \\Vert (S - \\lambda )f\\Vert \\le \\varepsilon \\nu \\Vert f\\Vert $ and $\\Vert (S_- - \\lambda )^{-1}\\Vert = \\alpha /\\nu $ this yields $\\Vert f_-\\Vert \\,\\le \\,\\alpha \\varepsilon \\Vert f\\Vert + \\alpha \\Vert f_+\\Vert $ and hence $\\Vert f_-\\Vert ^2\\,\\le \\,\\alpha ^2\\big (\\varepsilon ^2\\Vert f\\Vert ^2 + 2\\varepsilon \\Vert f\\Vert ^2 + \\Vert f_+\\Vert ^2\\big ).$ Using $\\Vert f_+\\Vert ^2 = \\Vert f\\Vert ^2 - \\Vert f_-\\Vert ^2$ we conclude $\\Vert f_-\\Vert ^2\\,\\le \\,\\frac{\\alpha ^2}{1+\\alpha ^2}(1 + \\varepsilon )^2\\Vert f\\Vert ^2.$ Hence, as $[f,f] = \\Vert f_+\\Vert ^2 - \\Vert f_-\\Vert ^2 = \\Vert f\\Vert ^2 - 2\\Vert f_-\\Vert ^2$ , we obtain $[f,f]\\,\\ge \\,r(\\alpha )\\Vert f\\Vert ^2$ , where $r(\\alpha ) := 1 - \\frac{2\\alpha ^2}{1+\\alpha ^2}(1 + \\varepsilon )^2 = \\frac{(1 - \\alpha )^2(1 + \\alpha )(7 - \\alpha )}{8(1+\\alpha ^2)} > 0,$ i.e., the implication (REF ) holds.",
"It follows that $(K_\\nu (\\sigma (S_-)))\\cap \\sigma _{{ap}}(S)\\subset \\sigma _{+}(S)$ .",
"Hence (ii) is proved and, as $\\sigma _{+}(S)$ is real, the non-real spectrum of $S$ satisfies $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_-)).$ Similarly, as in the proof of (REF ), one proves that for $\\lambda \\in \\mathbb {C} \\setminus K_\\nu (\\sigma (S_+))$ , $f\\in \\operatorname{dom}S$ , and for $\\beta := \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_+))<1$ the implication $\\Vert (S - \\lambda )f\\Vert \\le \\frac{1-\\beta ^2}{4\\beta }\\nu \\Vert f\\Vert \\quad \\Longrightarrow \\quad [f,f]\\,\\le \\,-r(\\beta )\\Vert f\\Vert ^2$ holds with $r(\\beta )>0$ as in (REF ).",
"This shows (iii) and $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_+))$ .",
"Together with (REF ) we obtain (i).",
"Assume now that $S_+$ is bounded from below and $S_-$ is bounded from above.",
"Then the non-real spectrum of $S$ is bounded by (i) and it remains to show that the growth of the resolvent of $S$ is of order one at $\\infty $ .",
"Note first that for $S_0=\\begin{pmatrix}S_+ & 0 \\\\ 0 & S_-\\end{pmatrix}\\qquad \\text{and}\\qquad V=\\begin{pmatrix}0 & M \\\\ -M^* & 0\\end{pmatrix},$ and $\\lambda \\in such that $ Im2V= 2$ we have $ (S0-)-1V12$ and\\begin{equation}\\Vert (S-\\lambda )^{-1}\\Vert = \\bigl \\Vert \\bigl ((S_0-\\lambda )(I+(S_0-\\lambda )^{-1}V)\\bigr )^{-1}\\bigr \\Vert \\le \\frac{2}{\\vert \\operatorname{Im}\\lambda \\vert }.\\end{equation}For $ with $\\vert \\operatorname{Im}\\lambda \\vert < 2\\nu $ and $\\operatorname{dist}(\\lambda ,\\sigma (S_-)) > \\nu + \\delta $ with some fixed $\\delta > 0$ again define the value $\\alpha = \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_-))$ .",
"Then $\\alpha \\in (0,\\delta ^{\\prime })$ , where $\\delta ^{\\prime } = \\nu /(\\nu + \\delta )\\in (0,1)$ , and for $f\\in \\operatorname{dom}S$ we either have $\\Vert (S - \\lambda )f\\Vert \\ge \\frac{1-\\alpha ^2}{4\\alpha }\\nu \\Vert f\\Vert > \\frac{1-\\alpha ^2}{8\\alpha }\\vert \\operatorname{Im}\\lambda \\vert \\,\\Vert f\\Vert ,$ or, by (REF ), $r(\\alpha )|\\operatorname{Im}\\lambda |\\Vert f\\Vert ^2\\le |\\operatorname{Im}[\\lambda f,f]| = |\\operatorname{Im}[(S - \\lambda )f,f]|\\le \\Vert (S - \\lambda )f\\Vert \\Vert f\\Vert ,$ and hence $\\Vert (S - \\lambda )f\\Vert \\ge r(\\alpha )|\\operatorname{Im}\\lambda |\\,\\Vert f\\Vert .$ For $\\lambda \\in with $ Im<2$ and $ dist(,(S+)) > + $ the estimates (\\ref {abs2}) and (\\ref {abs3}) hold in a similar form.",
"Therefore, (\\ref {abs1})-(\\ref {abs3}) imply that for all non-real $ with $\\operatorname{dist}(\\lambda ,\\sigma (S_-)) > \\nu + \\delta $ or $\\operatorname{dist}(\\lambda ,\\sigma (S_+)) > \\nu + \\delta $ we have $\\Vert (S-\\lambda )^{-1}\\Vert \\le \\frac{C}{\\vert \\operatorname{Im}\\lambda \\vert }$ with some $C > 0$ which does not depend on $\\lambda $ .",
"This shows that the growth of the resolvent $(S-\\lambda )^{-1}$ is of order one at $\\infty $ .$\\Box $ Remark 3.6 If $\\operatorname{dist}(\\sigma (S_+),\\sigma (S_-))\\ge 2\\Vert M\\Vert $ , then the spectrum of $S$ in Theorem REF is real.",
"This result can be improved in certain special cases, cf.",
"[1] and also [2], where sharp norm bounds on the operator angles between reducing subspaces of $S$ and $S_+\\oplus S_-$ are given.",
"We mention here also [34], [35] for comparable results in the study of operators of Klein-Gordon type, where bounded perturbations of operators with unbounded off-diagonal entries are investigated.",
"The following simple example shows that the bounds on the non-real spectrum of $S$ in Theorem REF are sharp.",
"Example 3.7 Let $\\mathfrak {H}_+ = \\mathfrak {H}_- = and $ S= 1$, $ M = z and hence $S = \\begin{pmatrix}1 & z\\\\-\\overline{z} & -1\\end{pmatrix}.$ Then $\\operatorname{dist}(\\sigma (S_+),\\sigma (S_-)) = 2$ .",
"If $|z|\\le 1$ , then $\\sigma (S) = \\lbrace \\pm \\sqrt{1 - |z|^2}\\rbrace $ is real; cf.",
"Remark REF .",
"If $|z| > 1$ , then $\\sigma (S) = \\lbrace \\pm i\\sqrt{|z|^2 - 1}\\rbrace $ and in this case the eigenvalues of $S$ lie on the boundary of $ K_{|z|}(\\lbrace 1\\rbrace )\\cap K_{|z|}(\\lbrace -1\\rbrace )$ .",
"Lemma 3.8 Let $T$ be a selfadjoint operator in the Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ with $0\\notin \\sigma _p(T)$ and let $E$ be its spectral function.",
"If $(a_n)$ and $(b_n)$ are sequences in $[0,\\infty )$ such that $a_n\\downarrow 0$ and $b_n\\uparrow \\infty $ as $n\\rightarrow \\infty $ , and $0\\in \\rho (T)$ if $a_k=0$ for some $k\\in \\mathbb {N}$ , then $E(\\mathbb {R}^+) - E(\\mathbb {R}^-) = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{a_n}^{b_n}\\,\\left((T + it)^{-1} + (T - it)^{-1}\\right)\\,dt\\,.$ First of all we observe that $(T + it)^{-1} + (T - it)^{-1} = 2T(T^2 + t^2)^{-1}.$ For all $f,g\\in \\mathfrak {H}$ we have $\\int _{a_n}^{b_n}\\,\\big (T(T^2 + t^2)^{-1}f,g\\big )\\,dt&= \\int _{a_n}^{b_n}\\,\\int _{\\mathbb {R}}\\,\\frac{s}{s^2 + t^2}\\,d(E_sf,g)\\,dt\\\\&= \\int _{\\mathbb {R}}\\,\\int _{a_n}^{b_n}\\,\\frac{s}{s^2 + t^2}\\,dt\\,d(E_sf,g)\\\\&= \\int _{\\mathbb {R}}\\,\\big (\\arctan (b_n/s) - \\arctan (a_n/s)\\big )\\,d(E_sf,g).\\\\&= \\big (\\arctan (b_nT^{-1})f - \\arctan (a_nT^{-1})f,g\\big ).$ Fubini's theorem can be applied here since on $\\mathbb {R}\\times [a_n,b_n]$ the integrand is bounded and the measure $d(E_sf,g)\\otimes dt$ considered as a measure in $\\mathbb {R} \\times [a_n,b_n]$ is finite.",
"Hence, $\\int _{a_n}^{b_n}\\,2T(T^2 + t^2)^{-1}f\\,dt = 2\\arctan (b_nT^{-1})f - 2\\arctan (a_nT^{-1})f.$ Now, we apply [52] and observe that $2\\arctan (a_nT^{-1})f$ tends to zero and $2\\arctan (b_nT^{-1})f$ tends to $\\pi \\operatorname{sgn}(T^{-1})f = \\pi \\operatorname{sgn}(T)f = \\pi (E(\\mathbb {R}^+) - E(\\mathbb {R}^-))f$ as $n\\rightarrow \\infty $ .$\\Box $ A subspace $\\mathfrak {L}$ of a Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ is called uniformly positive if there exists $\\delta > 0$ such that $[f,f]\\,\\ge \\,\\delta \\Vert f\\Vert ^2 \\quad \\mbox{for all } f\\in \\mathfrak {L}.$ Here $\\Vert \\cdot \\Vert $ is a Hilbert space norm on $\\mathfrak {H}$ with respect to which $[\\cdot \\,,\\cdot ]$ is continuous.",
"All such norms are equivalent, see [46].",
"Lemma 3.9 Let $G$ be a bounded and boundedly invertible selfadjoint operator in a Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ and set $[\\cdot \\,,\\cdot ]:= (G\\cdot ,\\cdot )$ .",
"Let $\\mathfrak {L}$ be a closed uniformly positive subspace in the Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ .",
"If $P$ ($E$ ) denotes the orthogonal projection in the Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ (in the Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ , respectively) onto $\\mathfrak {L}$ , then $P(G|\\mathfrak {L})$ is a uniformly positive selfadjoint operator in $(\\mathfrak {L},(\\cdot \\,,\\cdot ))$ , and its inverse is given by $\\bigl (P(G|\\mathfrak {L})\\bigr )^{-1} = E\\bigl (G^{-1}|\\mathfrak {L}\\bigr ).$ As $\\mathfrak {L}$ is uniformly positive, for all $f\\in \\mathfrak {L}$ we have $\\big (P(G|\\mathfrak {L})f,f\\big ) = (Gf,f) = [f,f]\\,\\ge \\,\\delta \\Vert f\\Vert ^2$ with some $\\delta > 0$ .",
"Thus, $P(G|\\mathfrak {L})$ is a uniformly positive operator in $(\\mathfrak {L},(\\cdot \\,,\\cdot ))$ and, in particular, $0\\in \\rho (P(G|\\mathfrak {L}))$ .",
"Let $g\\in \\mathfrak {L}$ and set $f := E(G^{-1}|\\mathfrak {L})g\\in \\mathfrak {L}$ .",
"Since for all $h\\in \\mathfrak {L}$ $(g - Gf,h) = \\bigl (G(I - E)G^{-1}g,h\\bigr ) = \\bigl [(I - E)G^{-1}g,h\\bigr ] = 0,$ we conclude $g - Gf\\in \\mathfrak {L}^\\perp $ and hence $g = PGf = P(G|\\mathfrak {L})f$ .",
"Therefore, $\\bigl (P(G|\\mathfrak {L})\\bigr )^{-1}g = f = E\\bigl (G^{-1}|\\mathfrak {L}\\bigr )g,$ which proves the lemma.$\\Box $ Proof of Theorem REF Assertion (i) in Theorem REF is simple: As both $A_0$ and $V$ are non-negative in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ we have $[(A_0+V)f,f]\\ge 0\\quad \\text{for all}\\quad f\\in \\operatorname{dom}(A_0+V)=\\operatorname{dom}A_0.$ The operator $A_0$ is selfadjoint in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim )$ (cf.",
"(REF ) and (REF )) and, as $V$ is bounded, $\\rho (A_0+V)$ is nonempty.",
"Thus $A_0+V$ is a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ and (i) in Theorem REF is proved.",
"Let $E$ be the spectral function of the non-negative operator $A_0$ , define the projections $E_+ := E((0,\\infty ))$ , $E_-:=E((-\\infty ,0))$ and the subspaces $\\mathcal {H}_\\pm := E_\\pm \\mathcal {H}$ .",
"Then with respect to the fundamental decomposition (REF ) the operators $A_0$ and $V$ have the form in (REF ) and (REF ), respectively.",
"The fundamental symmetry $\\widetilde{J}$ associated with the decomposition (REF ) is also given in (REF ).",
"The inner product $(\\cdot \\,,\\cdot )_\\sim $ is defined as in (REF ).",
"Moreover, $A_{0,\\pm }$ and $V_\\pm $ from (REF ) are selfadjoint in $(\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim ) = (\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ , $\\sigma (A_{0,\\pm })\\subset \\mathbb {R}^\\pm \\cup \\lbrace 0\\rbrace $ and (REF ) holds.",
"We apply Lemma REF to the non-negative operators $\\pm A_{0,\\pm }$ and obtain $\\pm I_{\\mathcal {H}_\\pm } = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{1/n}^n\\,\\left((A_{0,\\pm } + it)^{-1} + (A_{0,\\pm } - it)^{-1}\\right)\\,dt.$ Therefore, $\\widetilde{J} = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{1/n}^n\\,\\left((A_{0} + it)^{-1} + (A_{0} - it)^{-1}\\right)\\,dt,$ where the strong limit is with respect to the norm $\\Vert \\cdot \\Vert _\\sim := (\\cdot \\,,\\cdot )_\\sim ^{1/2}$ .",
"But as $\\Vert \\cdot \\Vert _\\sim $ and $\\Vert \\cdot \\Vert $ are equivalent norms, the limit on the right hand side of (REF ) also exists with respect to the norm $\\Vert \\cdot \\Vert $ , and coincides with $\\widetilde{J}$ .",
"The uniform boundedness principle now yields that $\\tau _0$ in () is finite and that $\\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _0<\\infty .$ Define the constants $\\nu := \\Vert B\\Vert _\\sim ,\\quad a := \\min \\sigma (V_+)\\quad \\text{and}\\quad b := \\max \\sigma (V_-).$ Then $A_{0,+} + V_+$ is bounded from below by $a$ and $\\sigma (A_{0,+} + V_+)\\subset [a,\\infty )$ , and $A_{0,-} + V_-$ is bounded from above by $b$ and $\\sigma (A_{0,-} + V_-)\\subset (-\\infty ,b]$ .",
"Making use of the representation (REF ) of $A = A_0 + V$ and Theorem REF we conclude: $\\sigma (A)\\backslash \\mathbb {R}\\subset K_\\nu ((-\\infty ,b])\\cap K_\\nu ([a,\\infty ))$ .",
"$(b + \\nu ,\\infty )$ is of positive type with respect to $A$ .",
"$(-\\infty ,a - \\nu )$ is of negative type with respect to $A$ .",
"The growth of the resolvent of $A$ at $\\infty $ of order one, and hence of finite order.",
"The statement in Theorem REF (ii) follows from Theorem REF if we show that the constants $\\nu $ , $a$ and $b$ in (REF ) satisfy $\\nu \\le r, \\quad a\\ge -d, \\quad b\\le d,$ where $r$ and $d$ are as in (), because in that case the properties (a)-(d) above hold with $\\nu $ , $a$ and $b$ replaced by $r$ , $-d$ and $d$ , respectively, and $K_\\nu ((-\\infty ,d])\\cap K_\\nu ([-d,\\infty ))= K_\\nu ((-d,d)).$ First we check $\\nu \\le r$ .",
"Denote by $(\\cdot \\,,\\cdot )$ the scalar product corresponding to the Hilbert space norm $\\Vert \\cdot \\Vert $ .",
"Using the well-known fact that the spectrum of a bounded operator is always a subset of the closure of its numerical range we obtain $\\begin{split}\\nu ^2&= \\Vert B\\Vert _\\sim ^2 = \\Vert B^{\\tilde{*}}B\\Vert _\\sim = \\sup \\lbrace |\\lambda | : \\lambda \\in \\sigma (B^{\\tilde{*}}B)\\rbrace \\\\&\\le \\sup \\lbrace |(B^{\\tilde{*}}Bf,f)| : \\Vert f\\Vert =1\\rbrace \\,\\le \\,\\Vert B^{\\tilde{*}}\\Vert \\,\\Vert B\\Vert \\\\&= \\Vert E_-(V|\\mathcal {H}_+)\\Vert \\,\\Vert E_+(V|\\mathcal {H}_-)\\Vert \\,\\le \\,\\Vert E_+\\Vert \\,\\Vert E_-\\Vert \\,\\Vert V\\Vert ^2.\\end{split}$ As $E_\\pm = \\frac{1}{2}(I\\pm \\widetilde{J})$ , it follows from (REF ) that $\\Vert E_\\pm \\Vert \\le \\frac{1+\\tau _0}{2}$ which shows $\\nu \\le r$ .",
"Next we verify $a\\ge -d$ .",
"Denote by $P_\\pm $ the orthogonal projection onto $\\mathcal {H}_\\pm $ with respect to $(\\cdot \\,,\\cdot )$ .",
"Then, according to Lemma REF , the operator $P_+(J|\\mathcal {H}_+)$ is selfadjoint and uniformly positive in the Hilbert space $(\\mathcal {H}_+,(\\cdot \\,,\\cdot ))$ .",
"Hence, the same holds for its inverse $E_+(J|\\mathcal {H}_+)$ ; cf.",
"Lemma REF .",
"This implies $\\min \\sigma (P_+(J|\\mathcal {H}_+))&= \\big (\\max \\sigma (E_+(J|\\mathcal {H}_+))\\big )^{-1} = \\Vert E_+(J|\\mathcal {H}_+)\\Vert ^{-1}\\\\&\\ge \\Vert E_+J\\Vert ^{-1} \\ge \\Vert E_+\\Vert ^{-1}\\,\\ge \\,\\frac{2}{1+\\tau _0}\\,.$ Hence, for $f_+\\in \\mathcal {H}_+$ we have $[f_+,f_+] = (Jf_+,f_+) = (P_+(J|\\mathcal {H}_+)f_+,f_+)\\,\\ge \\,\\frac{2}{1+\\tau _0}\\,\\Vert f_+\\Vert ^2.$ We also use $(V_+f_+,f_+)_\\sim = [V_+f_+,f_+] = [V_+f_+ - B^{\\tilde{*}}f_+,f_+] = [Vf_+,f_+] = (JVf_+,f_+)$ for $f\\in \\mathcal {H}_+$ to conclude $a=\\min \\sigma (V_+)&=\\inf \\lbrace (V_+f_+,f_+)_\\sim : \\Vert f_+\\Vert _\\sim = 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&\\ge \\inf \\lbrace (V_+f_+,f_+)_\\sim : \\Vert f_+\\Vert _\\sim \\le 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&= \\inf \\lbrace (JVf_+,f_+) : [f_+,f_+]\\le 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&\\ge \\inf \\bigl \\lbrace (JVf_+,f_+) : \\Vert f_+\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f_+\\in \\mathcal {H}_+\\bigr \\rbrace \\\\&\\ge \\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace .$ By assumption $V$ is not non-negative in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ and hence $0>\\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace =\\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2=\\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace .$ Therefore we finally obtain $a=\\min \\sigma (V_+)&\\ge \\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2=\\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace \\\\&= \\frac{1+\\tau _0}{2}\\,\\inf \\lbrace (JVf,f) : \\Vert f\\Vert = 1,\\,f\\in \\mathcal {H}\\rbrace \\\\&= \\frac{1+\\tau _0}{2}\\,\\min \\sigma (JV)=-d.$ The proof of the inequality $b\\le d$ is analogous and is left to the reader.",
"This completes the proof of (ii).",
"As $\\infty $ is not a singular critical point of $A_0$ , the last statement in Theorem REF on the critical point $\\infty $ of $A_0+V$ follows in both cases (i) and (ii) from Proposition and $\\operatorname{dom}A=\\operatorname{dom}(A_0+V)$ .$\\Box $ Proof of Theorem REF Let $\\gamma $ and $\\eta $ be as in Theorem REF .",
"Then the operator $A_0 := A + \\eta G^{-1}$ is uniformly positive in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ , $[\\cdot \\,,\\cdot ]= (G\\cdot ,\\cdot )$ , since for $f\\in \\operatorname{dom}A$ we have $[A_0 f,f] = [Af,f] + \\eta [G^{-1}f,f] \\ge -\\gamma \\Vert f\\Vert ^2 + \\eta \\Vert f\\Vert ^2 = (\\eta - \\gamma )\\Vert f\\Vert ^2.$ We will show that $A$ is non-negative over $\\overline{\\setminus K_r((-r,r)).In particular, 0\\in \\rho (A_0) is not a singular critical point of A_0and, as \\tau _\\eta < \\infty , the point \\infty is not aa singular critical point of A_0, see Remark \\ref {Mai2012}.With V := -\\eta G^{-1} we haveA = A_0 + V\\qquad \\text{and}\\qquad \\Vert V\\Vert =\\eta \\Vert G^{-1}\\Vert .As in the proof of Theorem~\\ref {t:main1} let E be the spectral function of the operator A_0.",
"Then the fundamentaldecomposition \\mathcal {H}= E_+\\mathcal {H}\\,[\\dotplus ]\\,E_-\\mathcal {H} in (\\ref {e:dec_expl}) generates matrix representationsof A_0, V and the corresponding fundamental symmetry \\widetilde{J} as in (\\ref {e:AJ}) and (\\ref {e:V}).The operator A_0 is selfadjoint in the Hilbert space(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim ), where (\\cdot \\,,\\cdot )_\\sim is defined in (\\ref {e:sim}).",
"With the help of Lemma \\ref {l:veselic} we obtain{\\begin{@align*}{1}{-1}\\widetilde{J}&= \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _0^n\\,\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\big )\\,dt\\\\&= \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{-n}^n\\,(A + \\eta G^{-1} - it)^{-1}\\,dt\\,.\\end{@align*}}This yields\\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _\\eta .Following the lines of the proof of Theorem \\ref {t:main1}, it suffices to show the inequalities\\nu \\le \\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert ,\\quad \\min \\sigma (V_+)\\ge -\\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert \\quad \\text{and}\\quad \\max \\sigma (V_-)\\le \\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert ,where \\nu =\\Vert E_\\mp (V\\vert \\cal H_\\pm ) \\Vert _\\sim .The first relation is proved as in (\\ref {e:rel_proof}) and the remaining two inequalities follow from the selfadjointness of V_\\pm in (\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim ) and{\\begin{@align*}{1}{-1}\\max \\lbrace |\\min \\sigma (V_\\pm )|,|\\max \\sigma (V_\\pm )|\\rbrace &= \\Vert V_\\pm \\Vert _\\sim = \\sup \\lbrace |\\lambda | : \\lambda \\in \\sigma (V_\\pm )\\rbrace \\\\&\\le \\sup \\lbrace |(V_\\pm f_\\pm ,f_\\pm )| : \\Vert f_\\pm \\Vert =1,\\,f\\in \\mathcal {H}_\\pm \\rbrace \\\\&\\le \\Vert V_\\pm \\Vert = \\Vert E_\\pm (V|\\mathcal {H}_\\pm )\\Vert \\,\\le \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert .\\end{@align*}}Hence, A is non-negative over \\overline{\\setminus K_r((-r,r)).",
"As\\operatorname{dom}A= \\operatorname{dom}A_0, we conclude from Proposition \\ref {p:curgus} that \\infty is also not a critical point of A.This completes the proof of Theorem~\\ref {t:main2}.\\Box }\\begin{remark}If, in addition to the assumptions in Theorem \\ref {t:main2}, the point 0 is neither an eigenvalue nor a singular critical pointof the non-negative operator A_0 := A + \\gamma G^{-1}, then the operator A is non-negative over the set \\overline{\\setminus K_r((-r,r)), where\\begin{equation}r = \\gamma \\,\\frac{1+\\tau _\\gamma }{2}\\,\\Vert G^{-1}\\Vert \\end{equation}and\\tau _\\gamma := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\big )\\,dt\\right\\Vert .Indeed, under the assumptions of Theorem \\ref {t:main2} for any \\varepsilon > 0 the operator A + (\\gamma + \\varepsilon )G^{-1}is selfadjoint in some Hilbert space, and thus the resolvent set ofA_0 is non-empty as A + (\\gamma + \\varepsilon )G^{-1} and A_0 differ just by a bounded operator.Therefore, A_0 is a non-negative operator in the Krein space(\\cal H,[\\cdot \\,,\\cdot ]).",
"Moreover,the point \\infty is not a singular critical point of theoperator A + (\\gamma + \\varepsilon )G^{-1}, see Remark \\ref {Mai2012},and by Proposition \\ref {p:curgus} \\infty is not a singular critical point of A_0.Therefore, the operator A_0 admits a matrix representation as in (\\ref {e:AJ}) and the corresponding fundamental symmetry \\widetilde{J}satisfies \\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _\\gamma (see the proof of Theorem \\ref {t:main2}).Then by a similar reasoning as in the proof ofTheorem \\ref {t:main2} we see that A is non-negativeover \\overline{\\setminus K_r((-r,r)), where r is as in (\\ref {Juni2012}).",
"}}\\end{remark}}$ Differential operators with indefinite weights In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights.",
"Indefinite Sturm-Liouville operators We consider Sturm-Liouville differential expressions of the form $\\mathcal {L}(f)(x) = \\operatorname{sgn}(x)\\big (-f^{\\prime \\prime }(x) + q(x)f(x)\\big ),\\quad x\\in \\mathbb {R}\\,,$ with a real-valued potential $q\\in L^\\infty (\\mathbb {R})$ and the indefinite weight function $\\operatorname{sgn}(\\cdot )$ .",
"The corresponding differential operator in $L^2(\\mathbb {R})$ is defined by $Af := \\mathcal {L}(f), \\qquad \\operatorname{dom}A := H^2(\\mathbb {R}).$ Here $H^2(\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.",
"By $(\\cdot \\,,\\cdot )$ and $\\Vert \\cdot \\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\mathbb {R})$ .",
"Let $J$ be the operator of multiplication with the function $\\operatorname{sgn}(x)$ .",
"This operator is obviously selfadjoint and unitary in $L^2(\\mathbb {R})$ .",
"Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\prime \\prime } + qf,\\quad \\operatorname{dom}T := H^2(\\mathbb {R}),$ is selfadjoint in $L^2(\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\mathbb {R}),[\\cdot \\,,\\cdot ])$ , where $[f,g] := (Jf,g) = \\int _\\mathbb {R}\\,f(x)\\overline{g(x)}\\operatorname{sgn}(x)\\,dx,\\quad f,g\\in L^2(\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.",
"We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].",
"For this, denote by $m_\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\pm f_\\pm := - f_\\pm ^{\\prime \\prime } + q_\\pm f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ defined on $\\operatorname{dom}T_\\pm := \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ) : f_\\pm (0) = 0\\rbrace $ , where $q_\\pm $ is the restriction of the function $q$ to $\\mathbb {R}^\\pm $ .",
"The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\pm \\infty $ , then $m_\\pm =\\lim _{x\\rightarrow \\pm \\infty }\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.",
"Furthermore the following holds.",
"If $m_+ > -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ is finite; If $m_+\\le -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .",
"To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.",
"One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].",
"The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .",
"Here only the nontrivial case $\\operatornamewithlimits{ess\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).",
"Theorem 4.2 Assume that $\\operatornamewithlimits{ess\\,inf}q<0$ holds.",
"Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\overline{\\setminus K_r((-d,d)), wherer := 5\\Vert q\\Vert _\\infty ,\\quad \\text{and}\\quad d := -5\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\cap \\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .",
"}\\begin{proof}The proof of Theorem~\\ref {t:SL2} is split into 4 parts.",
"The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\ref {t:main1}.",
"In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\end{proof}\\ \\\\{\\it 1.\\ Preparation: } The operator $ A0$,\\begin{equation*}A_0 f:=-\\operatorname{sgn}(\\cdot )f^{\\prime \\prime },\\qquad \\operatorname{dom}A_0=H^2(\\mathbb {R}),\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\mathbb {R}$ ),[,])$.",
"Furthermore, we have $ (A0) = $\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\cite {cn}.",
"Hence $ A0$ satisfies theassumptions in Theorem~\\ref {t:main1}.",
"Define $ V$ as\\begin{equation*}Vf:=\\operatorname{sgn}(\\cdot ) qf,\\qquad \\operatorname{dom}V=L^2(\\mathbb {R}),\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\mathbb {R}$ ),[,])$ with$$\\min \\sigma (JV) = \\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}\\,q(x)\\quad \\text{and}\\quad \\Vert V\\Vert = \\Vert q\\Vert _\\infty .$$By Theorem \\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\frac{1+\\tau _0}{2}\\,\\Vert q\\Vert _\\infty ,\\qquad d = -\\frac{1+\\tau _0}{2}\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x),$$and$$\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert .$$Therefore, we have to show\\begin{equation}\\tau _0\\le 9\\end{equation}and\\begin{equation}\\sigma (A)\\setminus \\mathbb {R}\\,\\subset \\,\\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .\\end{equation}$ 2.",
"Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\begin{pmatrix} B_+ & 0 \\\\ 0 & B_- \\end{pmatrix}\\quad \\text{in}\\quad L^2(\\mathbb {R})=L^2(\\mathbb {R}^+)\\times L^2(\\mathbb {R}^-),$ where $B_\\pm f_\\pm :=\\mp f_\\pm ^{\\prime \\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\mathbb {R}^\\pm )$ which are defined on $\\operatorname{dom}B_\\pm = \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ):f_\\pm (0)=0\\rbrace $ .",
"Here and in the following we will denote the restrictions of a function $f$ defined on $\\mathbb {R}$ to $\\mathbb {R}^\\pm $ by $f_\\pm $ .",
"For $\\lambda = re^{it}$ , $r > 0$ , $t\\in [0,2\\pi )$ , we set $\\sqrt{\\lambda } := \\sqrt{r}e^{it/2}$ .",
"For $\\lambda \\in \\mathbb {R}$ define the function $f_\\lambda (x) :={\\left\\lbrace \\begin{array}{ll}e^{i\\sqrt{\\lambda }x}, &x> 0,\\\\e^{-i\\sqrt{-\\lambda }x}, &x < 0.\\end{array}\\right.",
"}$ The function $f_\\lambda $ is a solution of the equations $\\mp f_\\pm ^{\\prime \\prime } = \\lambda f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ .",
"From $\\overline{\\sqrt{\\lambda }} = -\\sqrt{\\overline{\\lambda }}$ it follows that $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ holds for all $\\lambda \\in \\mathbb {R}$ .",
"Let us now prove that for all $f\\in L^2(\\mathbb {R})$ and $\\lambda \\in \\mathbb {R}$ we have $(A_0 - \\lambda )^{-1}f = (B_0 - \\lambda )^{-1}f - \\frac{[f,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda .$ Since $\\sigma (A_0)=\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\lambda \\in \\mathbb {R}$ .",
"In particular, for $\\lambda \\in \\mathbb {R}$ there exists $g\\in \\operatorname{dom}B_0$ such that $f=(B_0-\\lambda )g$ holds.",
"The right hand side in (REF ) has the form $h:=g-\\frac{[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda $ and we will show that the function $h$ belongs to $\\operatorname{dom}A_0=H^2(\\mathbb {R})$ .",
"As $g_\\pm $ and $f_{\\lambda ,\\pm }$ are elements of $H^2(\\mathbb {R}^\\pm )$ the same is true for $h_\\pm $ .",
"Moreover, $g$ and $f_\\lambda $ are continuous at 0 and so is $h$ .",
"Hence it remains to check that $h^\\prime $ is continuous at 0.",
"For this note first that $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=\\int _0^\\infty (-g_+^{\\prime \\prime }-\\lambda g_+)\\overline{f_{\\overline{\\lambda },+}}\\, dx- \\int _{-\\infty }^0 (g_-^{\\prime \\prime }-\\lambda g_-)\\overline{f_{\\overline{\\lambda },-}}\\, dx$ and since $f_{\\overline{\\lambda },\\pm }$ are solutions of $\\mp f_\\pm ^{\\prime \\prime }=\\overline{\\lambda }f_\\pm $ , integration by parts yields $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=g_+^\\prime (0)\\overline{f_{\\overline{\\lambda },+}(0)}-g_+(0) \\overline{f^\\prime _{\\overline{\\lambda },+}(0)}-g_-^\\prime (0)\\overline{f_{\\overline{\\lambda },-}(0)}+g_-(0) \\overline{f^\\prime _{\\overline{\\lambda },-}(0)}.$ As $g\\in \\operatorname{dom}B_0$ we have $g_\\pm (0)=0$ and together with $f_{\\overline{\\lambda },\\pm }(0)=1$ we find $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]= g_+^\\prime (0)-g^\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\pm ^\\prime $ on $\\mathbb {R}^\\pm $ of the function $h$ from (REF ): $h^\\prime _\\pm =g^\\prime _\\pm -\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_{\\lambda ,\\pm }^\\prime $ and as $f^\\prime _{\\lambda ,+}(0)=i\\sqrt{\\lambda }$ and $f^\\prime _{\\lambda ,-}(0)=-i\\sqrt{-\\lambda }$ we conclude $h^\\prime _+(0)-h^\\prime _-(0)= \\bigl (g_+^\\prime (0)-g^\\prime _-(0)\\bigr )-\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}\\bigl (f^\\prime _{\\lambda ,+}(0)-f^\\prime _{\\lambda ,-}(0)\\bigr )=0,$ that is, $h^\\prime $ is continuous at 0 and therefore $h\\in \\operatorname{dom}A_0$ .",
"Now a straightforward computation shows that $(A_0-\\lambda )h=(B_0-\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).",
"3.",
"Proof of (): Let $f\\in L^2(\\mathbb {R})$ and $t > 0$ .",
"Then (REF ) and $\\sqrt{it} + \\sqrt{-it} = i\\sqrt{2t}$ yield $\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\& + \\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\in \\mathbb {R}^+$ , we have $\\big |[f,f_{it}][f_{-it},f]\\big |\\,\\le \\,\\left(\\int _0^\\infty \\,e^{-x\\sqrt{t/2}}g(x)\\,dx\\right)^2$ and thus for $n\\in \\mathbb {N}$ $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt&\\le \\int _{1/n}^n\\,\\int _0^\\infty \\,\\int _0^\\infty \\,g(x)g(y)\\,\\frac{e^{-(x+y)\\sqrt{t/2}}}{\\sqrt{2t}}\\,dy\\,dx\\,dt\\\\&= 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\left(e^{-\\frac{x+y}{\\sqrt{2n}}} - e^{-\\frac{\\sqrt{n}(x+y)}{\\sqrt{2}}}\\right)\\,dy\\,dx\\\\&\\le 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\,dy\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt\\,\\le \\,2\\pi \\Vert g\\Vert _{L^2(\\mathbb {R}^+)}^2\\,\\le \\,4\\pi \\Vert f\\Vert ^2.$ and therefore we have $\\frac{1}{\\pi }\\int _{1/n}^n\\,\\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\,dt\\,\\le \\,8\\Vert f\\Vert ^2.$ Denote by $E_0$ ($E_\\pm $ ) the spectral function of the operator $A_0$ ($B_\\pm $ , respectively).",
"Moreover define $J_{n,\\pm } := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}\\right)\\,dt,$ and, in addition, $J_{n,0} := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt.$ As $\\pm B_\\pm $ is non-negative, it follows that $J_{n,\\pm }$ converges strongly to $\\pm I_{L^2(\\mathbb {R}^\\pm )}$ when $n\\rightarrow \\infty $ , cf.",
"Lemma REF .",
"Moreover, by $(B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}= 2B_\\pm (B_\\pm + it)^{-1} (B_\\pm - it)^{-1}$ the sequence $\\pm J_{n,\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\mathbb {R}^\\pm )$ .",
"This implies $\\frac{1}{\\pi }\\,\\int _{1/n}^n\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\&\\le \\Vert f_+\\Vert _{L^2(\\mathbb {R}^+)}^2 + \\Vert f_-\\Vert _{L^2(\\mathbb {R}^-)}^2 = \\Vert f\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\,\\le \\,\\Vert f\\Vert ^2 + 8\\Vert f\\Vert ^2 = 9\\Vert f\\Vert ^2,\\quad f\\in L^2(\\mathbb {R}).$ Hence, as $J = \\operatorname{sgn}(\\cdot )$ is unitary in $L^2(\\mathbb {R})$ , we finally obtain $\\Vert J_{n,0}\\Vert = \\Vert J J_{n,0}\\Vert \\le 9$ and thus $\\tau _0 = \\limsup _{n\\rightarrow \\infty }\\,\\Vert J_{n,0}\\Vert \\le 9$ .",
"That is, () holds.",
"4.",
"Proof of (): Let $\\lambda \\in +$ .",
"Using (REF ), for $f\\in L^2(\\mathbb {R})$ we obtain $\\Vert (A_0 - \\lambda )^{-1}f\\Vert \\,\\le \\,\\Vert (B_0 - \\lambda )^{-1}f\\Vert + \\frac{|[f,f_{\\overline{\\lambda }}]|\\,\\Vert f_\\lambda \\Vert }{|\\sqrt{\\lambda } + \\sqrt{-\\lambda }|}\\,\\le \\,\\frac{\\Vert f\\Vert }{|\\operatorname{Im}\\lambda |} + \\frac{\\Vert f_\\lambda \\Vert ^2\\Vert f\\Vert }{\\sqrt{2|\\lambda |}}\\,.$ Here we have used the identity $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ .",
"Let $r > 0$ and $t\\in (0,\\pi )$ such that $\\lambda = re^{it}$ .",
"Then $\\Vert f_\\lambda \\Vert ^2&= \\int _0^\\infty \\,\\left|e^{i\\sqrt{r}e^{it/2}x}\\right|^2\\,dx + \\int _{-\\infty }^0\\,\\left|e^{-i\\sqrt{r}e^{i(t+\\pi )/2}x}\\right|^2\\,dx\\\\&= \\int _0^\\infty \\,e^{-2\\sqrt{r}\\sin (t/2)x}\\,dx + \\int _{-\\infty }^0\\,e^{2\\sqrt{r}\\cos (t/2)x}\\,dx\\\\&= \\frac{\\cos (t/2) + \\sin (t/2)}{\\sqrt{r}\\,\\sin (t)}\\,\\le \\,\\frac{\\sqrt{2}}{\\sqrt{r}\\,\\operatorname{Im}(\\lambda /r)} = \\frac{\\sqrt{2|\\lambda |}}{|\\operatorname{Im}\\lambda |}.$ Therefore, $\\Vert (A_0 - \\lambda )^{-1}\\Vert \\,\\le \\,\\frac{2}{|\\operatorname{Im}\\lambda |}\\,.$ The same estimate holds for $\\lambda \\in -$ .",
"Now, assume that $|\\operatorname{Im}\\lambda | > 2\\Vert q\\Vert _\\infty =2\\Vert V\\Vert $ .",
"Then $\\Vert V(A_0 - \\lambda )^{-1}\\Vert < 1$ , and hence $A - \\lambda = A_0 - \\lambda + V = \\left(I + V(A_0 - \\lambda )^{-1}\\right)(A_0 - \\lambda )$ is boundedly invertible, which proves ().$\\Box $ Second order elliptic operators Let $\\Omega \\subset \\mathbb {R}^n$ be a domain and let $\\ell $ be the \"formally symmetric\" uniformly elliptic second order differential expression $\\ell (f)(x):=-\\sum _{j,k=1}^n \\left( \\frac{\\partial }{\\partial x_j} a_{jk}\\frac{\\partial f}{\\partial x_k}\\right)(x)+a(x)f(x),\\quad x\\in \\Omega ,$ with bounded coefficients $a_{jk}\\in C^\\infty (\\Omega )$ satisfying $a_{jk}(x)=\\overline{a_{kj}(x)}$ for all $x\\in \\Omega $ and $j,k=1,\\dots ,n$ , the function $a\\in L^\\infty (\\Omega )$ is real valued and $\\sum _{j,k=1}^n a_{jk}(x)\\xi _j\\xi _k\\ge C\\sum _{k=1}^n\\xi _k^2$ holds for some $C>0$ , all $\\xi =(\\xi _1,\\dots ,\\xi _n)^\\top \\in \\mathbb {R}^n$ and $x\\in \\Omega $ .",
"With the differential expression $\\ell $ we associate the elliptic differential operator $Tf:=\\ell (f),\\qquad \\operatorname{dom}T=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\ell (f) \\in L^2(\\Omega )\\bigr \\rbrace ,$ where $H^1_0(\\Omega )$ stands for the closure of $C_0^\\infty (\\Omega )$ in the Sobolev space $H^1(\\Omega )$ .",
"It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\Omega ),(\\cdot ,\\cdot ))$ with spectrum semibounded from below by $\\operatornamewithlimits{ess\\,inf}\\,a$ ; cf.",
"[24].",
"Let $w$ be a real valued function such that $w,w^{-1}\\in L^\\infty (\\Omega )$ and each of the sets $\\Omega _+:=\\bigl \\lbrace x\\in \\Omega : w(x)>0\\bigr \\rbrace \\quad \\text{and}\\quad \\Omega _-:=\\bigl \\lbrace x\\in \\Omega : w(x)<0\\bigr \\rbrace $ has positive Lebesgue measure.",
"We define a second order elliptic differential expression $\\mathcal {L}$ with the indefinite weight $w$ by $\\mathcal {L}(f)(x):=\\frac{1}{w(x)}\\,\\ell (f)(x),\\qquad x\\in \\Omega .$ The multiplication operator $G_w f=w f$ , $f\\in L^2(\\Omega )$ , is an isomorphism in $L^2(\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\in L^2(\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\int _\\Omega f(x)\\overline{g(x)}\\,w(x)\\,dx,\\qquad f,g\\in L^2(\\Omega ).$ The differential operator associated with $\\mathcal {L}$ is defined as $Af=\\mathcal {L}(f),\\qquad \\operatorname{dom}A=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\mathcal {L}(f) \\in L^2(\\Omega )\\bigr \\rbrace .$ Since for $f\\in H^1_0(\\Omega )$ we have $\\ell (f)\\in L^2(\\Omega )$ if and only if $\\mathcal {L}(f)\\in L^2(\\Omega )$ it follows that $\\operatorname{dom}A=\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.",
"Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ .",
"In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\min \\sigma _{\\rm ess}(T)\\le 0$ holds.",
"This also implies that the domain $\\Omega $ is unbounded as otherwise $\\sigma _{\\rm ess}(T)=\\varnothing $ .",
"A discussion of the cases $\\sigma _{\\rm ess}(T)=\\varnothing $ and $\\min \\sigma _{\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].",
"Fix some $\\eta >0$ such that $-\\eta <\\min \\sigma (T)$ and define the spaces $\\mathcal {H}_s$ , $s\\in [0,2]$ , as the domains of the $\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\eta $ in $L^2(\\Omega )$ , $\\mathcal {H}_s:=\\operatorname{dom}\\bigl ((T+\\eta )^\\frac{s}{2}\\bigr ),\\qquad s\\in [0,2].$ Note that $\\mathcal {H}=\\mathcal {H}_0$ , $\\operatorname{dom}T=\\mathcal {H}_2$ and the form domain of $T$ is $\\mathcal {H}_1$ .",
"The spaces $\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\eta $ ; cf.",
"[40].",
"The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.",
"[9].",
"Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ such that $W\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds for some $s\\in (0,2]$ .",
"Then $A$ is non-negative over $ \\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r=\\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert w^{-1}\\Vert _\\infty \\quad \\text{and}\\quad \\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A+\\eta G_w^{-1} - it)^{-1} \\,dt\\,\\right\\Vert \\,<\\,\\infty .$ Moreover, $\\infty $ is not a singular critical point of $A$ .",
"In the next corollary the special case $\\Omega =\\mathbb {R}^n$ is treated; cf.",
"[9].",
"From now on we assume that $\\Omega =\\mathbb {R}^n$ and $\\Omega _\\pm =\\lbrace x\\in \\mathbb {R}^n:\\pm w(x)>0\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\in C^\\infty (\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.",
"Note that either $\\Omega _+$ or $\\Omega _-$ is bounded.",
"Corollary 4.4 Suppose that for some $s\\in (0,\\frac{1}{2})$ the Sobolev spaces $H^s(\\Omega _\\pm )$ are invariant under multiplication with $w\\!\\upharpoonright _{\\Omega _\\pm }$ .",
"Then the indefinite elliptic operator $A$ is non-negative over $\\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .",
"Moreover, $\\infty $ is not a singular critical point of $A$ .",
"A sufficient criterion for the invariance of $H^s(\\Omega _\\pm )$ in the above corollary can be deduced from [28].",
"Corollary 4.5 Suppose that the weight function $w\\!\\upharpoonright _{\\Omega _\\pm }$ belongs to some Hölder space $C^{0,\\alpha }(\\Omega _\\pm )$ , $\\alpha \\in (0,\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.",
"Then the claim in Corollary REF holds.",
"of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\operatorname{dom}A=\\operatorname{dom}T=H^2(\\mathbb {R}^n)\\qquad \\text{and}\\qquad \\mathcal {H}_s=H^s(\\mathbb {R}^n),\\qquad s\\in [0,2],$ by elliptic regularity and interpolation.",
"Since the Sobolev spaces $H^s(\\Omega _\\pm )$ are assumed to be invariant under multiplication with the functions $w\\!\\upharpoonright _{\\Omega _\\pm }$ for some $s\\in (0,\\frac{1}{2})$ it follows from [9] that also $H^s(\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds.",
"Furthermore, $G_w$ is uniformly positive in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\ge \\operatornamewithlimits{ess\\,inf}w^2 \\Vert f\\Vert ^2$ and $w^{-1}\\in L^\\infty (\\mathbb {R}^n)$ .",
"Now the assertion follows from Theorem REF .$\\Box $ Acknowledgements The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.",
"Contact information Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, [email protected] Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.",
"Juni 136, 10623 Berlin, Germany, [email protected] Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, [email protected]"
],
[
"Bounded selfadjoint perturbations of non-negative operators",
"In this section we prove two abstract results on additive bounded perturbations of non-negative (and some closely connected class of) operators in Krein spaces that lead to perturbed operators which are non-negative over some neighborhood of infinity.",
"In both cases the neighborhood is given in quantitative terms.",
"The results will be applied in Section to singular indefinite Sturm-Liouville operators and to second order elliptic operators with indefinite weights."
],
[
"Two perturbation results",
"The following notation will be useful when formulating our main results below: For a set $\\Delta \\subset \\mathbb {R}$ and $r > 0$ we define $K_r(\\Delta ) := \\lbrace z\\in \\operatorname{dist}(z,\\Delta )\\,\\le \\,r\\rbrace .$ Our first main theorem concerns bounded selfadjoint perturbations of non-negative operators in Krein spaces.",
"Theorem 3.1 Let $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ be a Krein space with fundamental symmetry $J$ and norm $\\Vert \\cdot \\Vert = [J\\cdot ,\\cdot ]^{1/2}$ .",
"Let $A_0$ be a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ such that 0 and $\\infty $ are not singular critical points of $A_0$ , and $0\\notin \\sigma _p(A_0)$ .",
"Furthermore, let $V\\in L(\\mathcal {H})$ be a selfadjoint operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"Then the following holds: (i) If $V$ is non-negative, then $A_0+V$ is a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"(ii) If $V$ is not non-negative, then $A_0+V$ is non-negative over $\\overline{\\setminus K_r((-d,d)),where\\begin{equation}r = \\frac{1+\\tau _0}{2}\\Vert V\\Vert ,\\qquad \\quad d = -\\frac{1+\\tau _0}{2}\\,\\min \\sigma (JV),\\end{equation}and\\begin{equation}\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert <\\infty .\\end{equation}}Moreover, in both cases $$ is not a singular critical point of $ A0 + V$.$ The following simple example shows that without further assumptions a bounded selfadjoint perturbation $V$ of a non-negative or uniformly positive $A_0$ in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ may lead to an operator $A_0+V$ with unbounded non-real spectrum.",
"Example 3.2 Let $(\\cal K,(\\cdot ,\\cdot ))$ be a Hilbert space and let $H$ be an unbounded selfadjoint operator in $\\cal K$ such that $\\sigma (H)\\subset (0,\\infty )$ .",
"Equip $\\cal H:=\\cal K\\oplus \\cal K$ with the Krein space inner product $\\left[\\begin{pmatrix}k_1 \\\\ k_2\\end{pmatrix},\\begin{pmatrix}l_1 \\\\ l_2\\end{pmatrix}\\right] :=(k_1,l_2)+(k_2,l_1),\\qquad \\begin{pmatrix}k_1 \\\\ k_2\\end{pmatrix},\\begin{pmatrix}l_1 \\\\ l_2\\end{pmatrix}\\in \\cal H,$ and consider the operators $A_0=\\begin{pmatrix} 0 & I \\\\ H & 0\\end{pmatrix},\\quad V=\\begin{pmatrix} 0 & -I \\\\ 0 & 0\\end{pmatrix},\\quad \\text{and}\\quad A_0+V =\\begin{pmatrix} 0 & 0 \\\\ H & 0\\end{pmatrix}.$ It is easy to see that $A_0$ is a non-negative operator and $V$ is a bounded selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"Moreover, as $\\operatorname{dom}(A_0+V)=\\operatorname{dom}H\\oplus {\\cal H}$ we conclude $\\operatorname{ran}(A_0+V-\\lambda )\\ne \\cal H$ for every $\\lambda \\in , that is, $ (A0+V)=.",
"Our second main result applies to operators of the form $A = G^{-1}T$ , where $T$ is a semibounded selfadjoint operator in a Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ and $G$ is a selfadjoint bounded and boundedly invertible operator in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ .",
"In other words, $A$ is selfadjoint in the Krein space $(\\mathcal {H},(G\\cdot ,\\cdot ))$ and $A + \\eta G^{-1}$ is uniformly positive for suitable $\\eta $ .",
"Such a situation arises, e.g., when considering elliptic differential operators with an indefinite weight function; cf.",
"Section 4.2.",
"Theorem 3.3 Let $(\\mathcal {H},(\\cdot \\,,\\cdot ))$ be a Hilbert space, $\\Vert \\cdot \\Vert = (\\cdot \\,,\\cdot )^{1/2}$ and $[\\cdot \\,,\\cdot ]=(G\\cdot ,\\cdot )$ be as above.",
"Let $A$ be a selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ such that for some $\\gamma > 0$ we have $[Af,f]\\ge -\\gamma \\Vert f\\Vert ^2\\quad \\text{for all }f\\in \\operatorname{dom}A.$ Assume furthermore that for some $\\eta > \\gamma $ $\\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A + \\eta G^{-1} - it)^{-1}\\,dt\\,\\right\\Vert < \\infty ,$ holds.",
"Then $A$ is non-negative over $\\overline{\\setminus K_r((-r,r)), where\\begin{equation*}r = \\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert G^{-1}\\Vert .\\end{equation*}Moreover, \\infty is not a singular critical point of A.",
"}$ Remark 3.4 Setting $A_0 := A + \\eta G^{-1}$ in Theorem REF and $V:=-\\eta G^{-1}$ we have $A = A_0 + V$ and hence Theorem REF can also be seen as a variant of Theorem REF .",
"Here $A_0=A + \\eta G^{-1}$ is uniformly positive so that $0\\in \\rho (A_0)$ is not a critical point of $A_0$ and the entire imaginary axis belongs to $\\rho (A + \\eta G^{-1})$ .",
"It follows from [55] and [32] that $\\tau _\\eta < \\infty $ is equivalent to $\\infty $ not being a singular critical point of $A_0$ .",
"In what follows we describe the structure of the proofs of Theorem REF (ii) and Theorem REF .",
"Let $E$ denote the spectral function of the non-negative operator $A_0$ .",
"As 0 and $\\infty $ are not singular critical points of $A_0$ , the spectral projections $E_+ := E((0,\\infty ))$ and $E_- := E((-\\infty ,0))$ , and the corresponding spectral subspaces $\\mathcal {H}_\\pm := E_\\pm \\mathcal {H}$ of $A_0$ exist.",
"From $0\\notin \\sigma _p(A_0)$ it follows that $\\mathcal {H}= \\mathcal {H}_+\\,[\\dotplus ]\\,\\mathcal {H}_-$ and $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ are Hilbert spaces, see, e.g., [46].",
"Therefore (REF ) is a fundamental decomposition of the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"With respect to this decomposition the operator $A_0$ and the fundamental symmetry $\\widetilde{J}$ corresponding to (REF ) can be written as operator matrices: $A_0 = \\begin{pmatrix}A_{0,+} & 0\\\\0 & A_{0,-}\\end{pmatrix}\\quad \\text{and}\\quad \\widetilde{J} = \\begin{pmatrix}I & 0\\\\0 & -I\\end{pmatrix}.$ Note that the operator $\\pm A_{0,\\pm }$ is a selfadjoint non-negative operator in the Hilbert space $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ .",
"Hence, $A_0$ is selfadjoint in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim )$ , where $(f,g)_\\sim := [\\widetilde{J}f,g] = [E_+f,g] - [E_-f,g],\\qquad f,g\\in \\mathcal {H}.$ Now, let $V\\in L(\\mathcal {H})$ be a bounded selfadjoint operator in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ .",
"Then, with respect to the decomposition (REF ) it admits an operator matrix representation $V = \\begin{pmatrix}V_+ & B\\\\C & V_-\\end{pmatrix}.$ From the selfadjointness of $V$ in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ one concludes that $V_\\pm $ is selfadjoint in the Hilbert space $(\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ]) = (\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim )$ and $C = -B^{\\widetilde{*}},$ where $B^{\\widetilde{*}}$ denotes the adjoint of $B$ with respect to the scalar product $(\\cdot \\,,\\cdot )_\\sim $ .",
"Hence, the perturbed operator $A := A_0 + V$ is represented by $A = \\begin{pmatrix}A_{0,+} + V_+ & B\\\\-B^{\\widetilde{*}} & A_{0,-} + V_-\\end{pmatrix}.$ For operators as in (REF ) we show in Theorem REF below that $\\mathbb {R}\\setminus K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,-} + V_-))$ is of positive type, $\\mathbb {R}\\setminus K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,+} + V_+))$ is of negative type, the non-real spectrum of $A$ is contained in $K:=K_{\\Vert B\\Vert _\\sim }\\bigl (\\sigma (A_{0,+} + V_+)\\bigr )\\cap K_{\\Vert B\\Vert _\\sim } \\bigl (\\sigma (A_{0,-} + V_-)\\bigr ),$ and the growth of the resolvent of $A$ at $\\infty $ is of order one.",
"Here $\\Vert \\cdot \\Vert _\\sim $ stands for the operator norm induced by (REF ).",
"Then, by Theorem REF , the operator $A$ is non-negative over $\\overline{\\setminus K, where K is as in (\\ref {BadHersfeld}).In a final step it remains to bound the quantities\\Vert B\\Vert _\\sim \\quad \\mbox{and} \\quad K_{\\Vert B\\Vert _\\sim } (\\sigma (A_{0,\\pm } + V_\\pm ))by the quantitiesr and \\tau _0 as in the statement of Theorem \\ref {t:main1}.",
"}$ Some auxiliary statements In this section we formulate and prove some auxiliary statements which will be used in the proofs Theorems REF and REF .",
"For the special case of bounded operators the first part of the following theorem was already proved in [48], see also [47] and [53], [54] for similar results with unbounded entries on the diagonal.",
"Theorem 3.5 Let $S_+$ and $S_-$ be selfadjoint operators in some Hilbert spaces $\\mathfrak {H}_+$ and $\\mathfrak {H}_-$ , respectively, let $M\\in L(\\mathfrak {H}_-,\\mathfrak {H}_+)$ and define the operators $S := \\begin{pmatrix}S_+ & M\\\\-M^* & S_-\\end{pmatrix}\\quad \\text{and}\\quad \\mathfrak {J}:= \\begin{pmatrix}I & 0\\\\0 & -I\\end{pmatrix}$ in the Hilbert space $\\mathfrak {H}:= \\mathfrak {H}_+\\oplus \\mathfrak {H}_-$ .",
"Then the operator $S$ is selfadjoint in the Krein space $(\\mathfrak {H},(\\mathfrak {J}\\cdot ,\\cdot ))$ and with $\\nu := \\Vert M\\Vert $ the following statements hold: $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_+))\\cap K_\\nu (\\sigma (S_-))$ .",
"$\\mathbb {R}\\setminus K_\\nu (\\sigma (S_-))$ is of positive type with respect to $S$ .",
"$\\mathbb {R}\\setminus K_\\nu (\\sigma (S_+))$ is of negative type with respect to $S$ .",
"Moreover, if $S_+$ is bounded from below and $S_-$ is bounded from above (or vice versa), the non-real spectrum of $S$ is bounded and the growth of the resolvent of $S$ at $\\infty $ is of order one.",
"We set $[\\cdot \\,,\\cdot ]:= (\\mathfrak {J}\\cdot ,\\cdot )$ .",
"Let $\\lambda \\in \\mathbb {C} \\setminus K_\\nu (\\sigma (S_-))$ and $\\alpha := \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_-)) < 1$ .",
"We claim that for some $r(\\alpha )>0$ and all $f\\in \\operatorname{dom}S$ the following implication $\\Vert (S - \\lambda )f\\Vert \\le \\frac{1-\\alpha ^2}{4\\alpha }\\nu \\Vert f\\Vert \\quad \\Longrightarrow \\quad [f,f]\\,\\ge \\,r(\\alpha )\\Vert f\\Vert ^2$ holds.",
"In fact, set $\\varepsilon := \\frac{1-\\alpha ^2}{4\\alpha }$ and let $f\\in \\operatorname{dom}S$ with $\\Vert (S - \\lambda )f\\Vert \\le \\varepsilon \\nu \\Vert f\\Vert .$ With respect to the decomposition $\\mathfrak {H}= \\mathfrak {H}_+\\oplus \\mathfrak {H}_-$ we write $f$ and $g := (S - \\lambda )f$ as column vectors $f = \\begin{pmatrix}f_+\\\\f_-\\end{pmatrix}\\in \\operatorname{dom}S_+\\oplus \\operatorname{dom}S_-, \\quad g = \\begin{pmatrix}g_+\\\\g_-\\end{pmatrix}.$ Then $g_- = -M^*f_+ + (S_- - \\lambda )f_-$ , or, equivalently, $f_- = (S_- - \\lambda )^{-1}g_- + (S_- - \\lambda )^{-1}M^*f_+.$ As $\\Vert g_-\\Vert \\le \\Vert g\\Vert = \\Vert (S - \\lambda )f\\Vert \\le \\varepsilon \\nu \\Vert f\\Vert $ and $\\Vert (S_- - \\lambda )^{-1}\\Vert = \\alpha /\\nu $ this yields $\\Vert f_-\\Vert \\,\\le \\,\\alpha \\varepsilon \\Vert f\\Vert + \\alpha \\Vert f_+\\Vert $ and hence $\\Vert f_-\\Vert ^2\\,\\le \\,\\alpha ^2\\big (\\varepsilon ^2\\Vert f\\Vert ^2 + 2\\varepsilon \\Vert f\\Vert ^2 + \\Vert f_+\\Vert ^2\\big ).$ Using $\\Vert f_+\\Vert ^2 = \\Vert f\\Vert ^2 - \\Vert f_-\\Vert ^2$ we conclude $\\Vert f_-\\Vert ^2\\,\\le \\,\\frac{\\alpha ^2}{1+\\alpha ^2}(1 + \\varepsilon )^2\\Vert f\\Vert ^2.$ Hence, as $[f,f] = \\Vert f_+\\Vert ^2 - \\Vert f_-\\Vert ^2 = \\Vert f\\Vert ^2 - 2\\Vert f_-\\Vert ^2$ , we obtain $[f,f]\\,\\ge \\,r(\\alpha )\\Vert f\\Vert ^2$ , where $r(\\alpha ) := 1 - \\frac{2\\alpha ^2}{1+\\alpha ^2}(1 + \\varepsilon )^2 = \\frac{(1 - \\alpha )^2(1 + \\alpha )(7 - \\alpha )}{8(1+\\alpha ^2)} > 0,$ i.e., the implication (REF ) holds.",
"It follows that $(K_\\nu (\\sigma (S_-)))\\cap \\sigma _{{ap}}(S)\\subset \\sigma _{+}(S)$ .",
"Hence (ii) is proved and, as $\\sigma _{+}(S)$ is real, the non-real spectrum of $S$ satisfies $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_-)).$ Similarly, as in the proof of (REF ), one proves that for $\\lambda \\in \\mathbb {C} \\setminus K_\\nu (\\sigma (S_+))$ , $f\\in \\operatorname{dom}S$ , and for $\\beta := \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_+))<1$ the implication $\\Vert (S - \\lambda )f\\Vert \\le \\frac{1-\\beta ^2}{4\\beta }\\nu \\Vert f\\Vert \\quad \\Longrightarrow \\quad [f,f]\\,\\le \\,-r(\\beta )\\Vert f\\Vert ^2$ holds with $r(\\beta )>0$ as in (REF ).",
"This shows (iii) and $\\sigma (S)\\setminus \\mathbb {R}\\,\\subset \\, K_\\nu (\\sigma (S_+))$ .",
"Together with (REF ) we obtain (i).",
"Assume now that $S_+$ is bounded from below and $S_-$ is bounded from above.",
"Then the non-real spectrum of $S$ is bounded by (i) and it remains to show that the growth of the resolvent of $S$ is of order one at $\\infty $ .",
"Note first that for $S_0=\\begin{pmatrix}S_+ & 0 \\\\ 0 & S_-\\end{pmatrix}\\qquad \\text{and}\\qquad V=\\begin{pmatrix}0 & M \\\\ -M^* & 0\\end{pmatrix},$ and $\\lambda \\in such that $ Im2V= 2$ we have $ (S0-)-1V12$ and\\begin{equation}\\Vert (S-\\lambda )^{-1}\\Vert = \\bigl \\Vert \\bigl ((S_0-\\lambda )(I+(S_0-\\lambda )^{-1}V)\\bigr )^{-1}\\bigr \\Vert \\le \\frac{2}{\\vert \\operatorname{Im}\\lambda \\vert }.\\end{equation}For $ with $\\vert \\operatorname{Im}\\lambda \\vert < 2\\nu $ and $\\operatorname{dist}(\\lambda ,\\sigma (S_-)) > \\nu + \\delta $ with some fixed $\\delta > 0$ again define the value $\\alpha = \\nu /\\operatorname{dist}(\\lambda ,\\sigma (S_-))$ .",
"Then $\\alpha \\in (0,\\delta ^{\\prime })$ , where $\\delta ^{\\prime } = \\nu /(\\nu + \\delta )\\in (0,1)$ , and for $f\\in \\operatorname{dom}S$ we either have $\\Vert (S - \\lambda )f\\Vert \\ge \\frac{1-\\alpha ^2}{4\\alpha }\\nu \\Vert f\\Vert > \\frac{1-\\alpha ^2}{8\\alpha }\\vert \\operatorname{Im}\\lambda \\vert \\,\\Vert f\\Vert ,$ or, by (REF ), $r(\\alpha )|\\operatorname{Im}\\lambda |\\Vert f\\Vert ^2\\le |\\operatorname{Im}[\\lambda f,f]| = |\\operatorname{Im}[(S - \\lambda )f,f]|\\le \\Vert (S - \\lambda )f\\Vert \\Vert f\\Vert ,$ and hence $\\Vert (S - \\lambda )f\\Vert \\ge r(\\alpha )|\\operatorname{Im}\\lambda |\\,\\Vert f\\Vert .$ For $\\lambda \\in with $ Im<2$ and $ dist(,(S+)) > + $ the estimates (\\ref {abs2}) and (\\ref {abs3}) hold in a similar form.",
"Therefore, (\\ref {abs1})-(\\ref {abs3}) imply that for all non-real $ with $\\operatorname{dist}(\\lambda ,\\sigma (S_-)) > \\nu + \\delta $ or $\\operatorname{dist}(\\lambda ,\\sigma (S_+)) > \\nu + \\delta $ we have $\\Vert (S-\\lambda )^{-1}\\Vert \\le \\frac{C}{\\vert \\operatorname{Im}\\lambda \\vert }$ with some $C > 0$ which does not depend on $\\lambda $ .",
"This shows that the growth of the resolvent $(S-\\lambda )^{-1}$ is of order one at $\\infty $ .$\\Box $ Remark 3.6 If $\\operatorname{dist}(\\sigma (S_+),\\sigma (S_-))\\ge 2\\Vert M\\Vert $ , then the spectrum of $S$ in Theorem REF is real.",
"This result can be improved in certain special cases, cf.",
"[1] and also [2], where sharp norm bounds on the operator angles between reducing subspaces of $S$ and $S_+\\oplus S_-$ are given.",
"We mention here also [34], [35] for comparable results in the study of operators of Klein-Gordon type, where bounded perturbations of operators with unbounded off-diagonal entries are investigated.",
"The following simple example shows that the bounds on the non-real spectrum of $S$ in Theorem REF are sharp.",
"Example 3.7 Let $\\mathfrak {H}_+ = \\mathfrak {H}_- = and $ S= 1$, $ M = z and hence $S = \\begin{pmatrix}1 & z\\\\-\\overline{z} & -1\\end{pmatrix}.$ Then $\\operatorname{dist}(\\sigma (S_+),\\sigma (S_-)) = 2$ .",
"If $|z|\\le 1$ , then $\\sigma (S) = \\lbrace \\pm \\sqrt{1 - |z|^2}\\rbrace $ is real; cf.",
"Remark REF .",
"If $|z| > 1$ , then $\\sigma (S) = \\lbrace \\pm i\\sqrt{|z|^2 - 1}\\rbrace $ and in this case the eigenvalues of $S$ lie on the boundary of $ K_{|z|}(\\lbrace 1\\rbrace )\\cap K_{|z|}(\\lbrace -1\\rbrace )$ .",
"Lemma 3.8 Let $T$ be a selfadjoint operator in the Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ with $0\\notin \\sigma _p(T)$ and let $E$ be its spectral function.",
"If $(a_n)$ and $(b_n)$ are sequences in $[0,\\infty )$ such that $a_n\\downarrow 0$ and $b_n\\uparrow \\infty $ as $n\\rightarrow \\infty $ , and $0\\in \\rho (T)$ if $a_k=0$ for some $k\\in \\mathbb {N}$ , then $E(\\mathbb {R}^+) - E(\\mathbb {R}^-) = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{a_n}^{b_n}\\,\\left((T + it)^{-1} + (T - it)^{-1}\\right)\\,dt\\,.$ First of all we observe that $(T + it)^{-1} + (T - it)^{-1} = 2T(T^2 + t^2)^{-1}.$ For all $f,g\\in \\mathfrak {H}$ we have $\\int _{a_n}^{b_n}\\,\\big (T(T^2 + t^2)^{-1}f,g\\big )\\,dt&= \\int _{a_n}^{b_n}\\,\\int _{\\mathbb {R}}\\,\\frac{s}{s^2 + t^2}\\,d(E_sf,g)\\,dt\\\\&= \\int _{\\mathbb {R}}\\,\\int _{a_n}^{b_n}\\,\\frac{s}{s^2 + t^2}\\,dt\\,d(E_sf,g)\\\\&= \\int _{\\mathbb {R}}\\,\\big (\\arctan (b_n/s) - \\arctan (a_n/s)\\big )\\,d(E_sf,g).\\\\&= \\big (\\arctan (b_nT^{-1})f - \\arctan (a_nT^{-1})f,g\\big ).$ Fubini's theorem can be applied here since on $\\mathbb {R}\\times [a_n,b_n]$ the integrand is bounded and the measure $d(E_sf,g)\\otimes dt$ considered as a measure in $\\mathbb {R} \\times [a_n,b_n]$ is finite.",
"Hence, $\\int _{a_n}^{b_n}\\,2T(T^2 + t^2)^{-1}f\\,dt = 2\\arctan (b_nT^{-1})f - 2\\arctan (a_nT^{-1})f.$ Now, we apply [52] and observe that $2\\arctan (a_nT^{-1})f$ tends to zero and $2\\arctan (b_nT^{-1})f$ tends to $\\pi \\operatorname{sgn}(T^{-1})f = \\pi \\operatorname{sgn}(T)f = \\pi (E(\\mathbb {R}^+) - E(\\mathbb {R}^-))f$ as $n\\rightarrow \\infty $ .$\\Box $ A subspace $\\mathfrak {L}$ of a Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ is called uniformly positive if there exists $\\delta > 0$ such that $[f,f]\\,\\ge \\,\\delta \\Vert f\\Vert ^2 \\quad \\mbox{for all } f\\in \\mathfrak {L}.$ Here $\\Vert \\cdot \\Vert $ is a Hilbert space norm on $\\mathfrak {H}$ with respect to which $[\\cdot \\,,\\cdot ]$ is continuous.",
"All such norms are equivalent, see [46].",
"Lemma 3.9 Let $G$ be a bounded and boundedly invertible selfadjoint operator in a Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ and set $[\\cdot \\,,\\cdot ]:= (G\\cdot ,\\cdot )$ .",
"Let $\\mathfrak {L}$ be a closed uniformly positive subspace in the Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ .",
"If $P$ ($E$ ) denotes the orthogonal projection in the Hilbert space $(\\mathfrak {H},(\\cdot \\,,\\cdot ))$ (in the Krein space $(\\mathfrak {H},[\\cdot \\,,\\cdot ])$ , respectively) onto $\\mathfrak {L}$ , then $P(G|\\mathfrak {L})$ is a uniformly positive selfadjoint operator in $(\\mathfrak {L},(\\cdot \\,,\\cdot ))$ , and its inverse is given by $\\bigl (P(G|\\mathfrak {L})\\bigr )^{-1} = E\\bigl (G^{-1}|\\mathfrak {L}\\bigr ).$ As $\\mathfrak {L}$ is uniformly positive, for all $f\\in \\mathfrak {L}$ we have $\\big (P(G|\\mathfrak {L})f,f\\big ) = (Gf,f) = [f,f]\\,\\ge \\,\\delta \\Vert f\\Vert ^2$ with some $\\delta > 0$ .",
"Thus, $P(G|\\mathfrak {L})$ is a uniformly positive operator in $(\\mathfrak {L},(\\cdot \\,,\\cdot ))$ and, in particular, $0\\in \\rho (P(G|\\mathfrak {L}))$ .",
"Let $g\\in \\mathfrak {L}$ and set $f := E(G^{-1}|\\mathfrak {L})g\\in \\mathfrak {L}$ .",
"Since for all $h\\in \\mathfrak {L}$ $(g - Gf,h) = \\bigl (G(I - E)G^{-1}g,h\\bigr ) = \\bigl [(I - E)G^{-1}g,h\\bigr ] = 0,$ we conclude $g - Gf\\in \\mathfrak {L}^\\perp $ and hence $g = PGf = P(G|\\mathfrak {L})f$ .",
"Therefore, $\\bigl (P(G|\\mathfrak {L})\\bigr )^{-1}g = f = E\\bigl (G^{-1}|\\mathfrak {L}\\bigr )g,$ which proves the lemma.$\\Box $ Proof of Theorem REF Assertion (i) in Theorem REF is simple: As both $A_0$ and $V$ are non-negative in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ we have $[(A_0+V)f,f]\\ge 0\\quad \\text{for all}\\quad f\\in \\operatorname{dom}(A_0+V)=\\operatorname{dom}A_0.$ The operator $A_0$ is selfadjoint in the Hilbert space $(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim )$ (cf.",
"(REF ) and (REF )) and, as $V$ is bounded, $\\rho (A_0+V)$ is nonempty.",
"Thus $A_0+V$ is a non-negative operator in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ and (i) in Theorem REF is proved.",
"Let $E$ be the spectral function of the non-negative operator $A_0$ , define the projections $E_+ := E((0,\\infty ))$ , $E_-:=E((-\\infty ,0))$ and the subspaces $\\mathcal {H}_\\pm := E_\\pm \\mathcal {H}$ .",
"Then with respect to the fundamental decomposition (REF ) the operators $A_0$ and $V$ have the form in (REF ) and (REF ), respectively.",
"The fundamental symmetry $\\widetilde{J}$ associated with the decomposition (REF ) is also given in (REF ).",
"The inner product $(\\cdot \\,,\\cdot )_\\sim $ is defined as in (REF ).",
"Moreover, $A_{0,\\pm }$ and $V_\\pm $ from (REF ) are selfadjoint in $(\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim ) = (\\mathcal {H}_\\pm ,\\pm [\\cdot \\,,\\cdot ])$ , $\\sigma (A_{0,\\pm })\\subset \\mathbb {R}^\\pm \\cup \\lbrace 0\\rbrace $ and (REF ) holds.",
"We apply Lemma REF to the non-negative operators $\\pm A_{0,\\pm }$ and obtain $\\pm I_{\\mathcal {H}_\\pm } = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{1/n}^n\\,\\left((A_{0,\\pm } + it)^{-1} + (A_{0,\\pm } - it)^{-1}\\right)\\,dt.$ Therefore, $\\widetilde{J} = \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{1/n}^n\\,\\left((A_{0} + it)^{-1} + (A_{0} - it)^{-1}\\right)\\,dt,$ where the strong limit is with respect to the norm $\\Vert \\cdot \\Vert _\\sim := (\\cdot \\,,\\cdot )_\\sim ^{1/2}$ .",
"But as $\\Vert \\cdot \\Vert _\\sim $ and $\\Vert \\cdot \\Vert $ are equivalent norms, the limit on the right hand side of (REF ) also exists with respect to the norm $\\Vert \\cdot \\Vert $ , and coincides with $\\widetilde{J}$ .",
"The uniform boundedness principle now yields that $\\tau _0$ in () is finite and that $\\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _0<\\infty .$ Define the constants $\\nu := \\Vert B\\Vert _\\sim ,\\quad a := \\min \\sigma (V_+)\\quad \\text{and}\\quad b := \\max \\sigma (V_-).$ Then $A_{0,+} + V_+$ is bounded from below by $a$ and $\\sigma (A_{0,+} + V_+)\\subset [a,\\infty )$ , and $A_{0,-} + V_-$ is bounded from above by $b$ and $\\sigma (A_{0,-} + V_-)\\subset (-\\infty ,b]$ .",
"Making use of the representation (REF ) of $A = A_0 + V$ and Theorem REF we conclude: $\\sigma (A)\\backslash \\mathbb {R}\\subset K_\\nu ((-\\infty ,b])\\cap K_\\nu ([a,\\infty ))$ .",
"$(b + \\nu ,\\infty )$ is of positive type with respect to $A$ .",
"$(-\\infty ,a - \\nu )$ is of negative type with respect to $A$ .",
"The growth of the resolvent of $A$ at $\\infty $ of order one, and hence of finite order.",
"The statement in Theorem REF (ii) follows from Theorem REF if we show that the constants $\\nu $ , $a$ and $b$ in (REF ) satisfy $\\nu \\le r, \\quad a\\ge -d, \\quad b\\le d,$ where $r$ and $d$ are as in (), because in that case the properties (a)-(d) above hold with $\\nu $ , $a$ and $b$ replaced by $r$ , $-d$ and $d$ , respectively, and $K_\\nu ((-\\infty ,d])\\cap K_\\nu ([-d,\\infty ))= K_\\nu ((-d,d)).$ First we check $\\nu \\le r$ .",
"Denote by $(\\cdot \\,,\\cdot )$ the scalar product corresponding to the Hilbert space norm $\\Vert \\cdot \\Vert $ .",
"Using the well-known fact that the spectrum of a bounded operator is always a subset of the closure of its numerical range we obtain $\\begin{split}\\nu ^2&= \\Vert B\\Vert _\\sim ^2 = \\Vert B^{\\tilde{*}}B\\Vert _\\sim = \\sup \\lbrace |\\lambda | : \\lambda \\in \\sigma (B^{\\tilde{*}}B)\\rbrace \\\\&\\le \\sup \\lbrace |(B^{\\tilde{*}}Bf,f)| : \\Vert f\\Vert =1\\rbrace \\,\\le \\,\\Vert B^{\\tilde{*}}\\Vert \\,\\Vert B\\Vert \\\\&= \\Vert E_-(V|\\mathcal {H}_+)\\Vert \\,\\Vert E_+(V|\\mathcal {H}_-)\\Vert \\,\\le \\,\\Vert E_+\\Vert \\,\\Vert E_-\\Vert \\,\\Vert V\\Vert ^2.\\end{split}$ As $E_\\pm = \\frac{1}{2}(I\\pm \\widetilde{J})$ , it follows from (REF ) that $\\Vert E_\\pm \\Vert \\le \\frac{1+\\tau _0}{2}$ which shows $\\nu \\le r$ .",
"Next we verify $a\\ge -d$ .",
"Denote by $P_\\pm $ the orthogonal projection onto $\\mathcal {H}_\\pm $ with respect to $(\\cdot \\,,\\cdot )$ .",
"Then, according to Lemma REF , the operator $P_+(J|\\mathcal {H}_+)$ is selfadjoint and uniformly positive in the Hilbert space $(\\mathcal {H}_+,(\\cdot \\,,\\cdot ))$ .",
"Hence, the same holds for its inverse $E_+(J|\\mathcal {H}_+)$ ; cf.",
"Lemma REF .",
"This implies $\\min \\sigma (P_+(J|\\mathcal {H}_+))&= \\big (\\max \\sigma (E_+(J|\\mathcal {H}_+))\\big )^{-1} = \\Vert E_+(J|\\mathcal {H}_+)\\Vert ^{-1}\\\\&\\ge \\Vert E_+J\\Vert ^{-1} \\ge \\Vert E_+\\Vert ^{-1}\\,\\ge \\,\\frac{2}{1+\\tau _0}\\,.$ Hence, for $f_+\\in \\mathcal {H}_+$ we have $[f_+,f_+] = (Jf_+,f_+) = (P_+(J|\\mathcal {H}_+)f_+,f_+)\\,\\ge \\,\\frac{2}{1+\\tau _0}\\,\\Vert f_+\\Vert ^2.$ We also use $(V_+f_+,f_+)_\\sim = [V_+f_+,f_+] = [V_+f_+ - B^{\\tilde{*}}f_+,f_+] = [Vf_+,f_+] = (JVf_+,f_+)$ for $f\\in \\mathcal {H}_+$ to conclude $a=\\min \\sigma (V_+)&=\\inf \\lbrace (V_+f_+,f_+)_\\sim : \\Vert f_+\\Vert _\\sim = 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&\\ge \\inf \\lbrace (V_+f_+,f_+)_\\sim : \\Vert f_+\\Vert _\\sim \\le 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&= \\inf \\lbrace (JVf_+,f_+) : [f_+,f_+]\\le 1,\\,f_+\\in \\mathcal {H}_+\\rbrace \\\\&\\ge \\inf \\bigl \\lbrace (JVf_+,f_+) : \\Vert f_+\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f_+\\in \\mathcal {H}_+\\bigr \\rbrace \\\\&\\ge \\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace .$ By assumption $V$ is not non-negative in $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ and hence $0>\\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2\\le \\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace =\\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2=\\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace .$ Therefore we finally obtain $a=\\min \\sigma (V_+)&\\ge \\inf \\bigl \\lbrace (JVf,f) : \\Vert f\\Vert ^2=\\tfrac{1+\\tau _0}{2},\\,f\\in \\mathcal {H}\\bigr \\rbrace \\\\&= \\frac{1+\\tau _0}{2}\\,\\inf \\lbrace (JVf,f) : \\Vert f\\Vert = 1,\\,f\\in \\mathcal {H}\\rbrace \\\\&= \\frac{1+\\tau _0}{2}\\,\\min \\sigma (JV)=-d.$ The proof of the inequality $b\\le d$ is analogous and is left to the reader.",
"This completes the proof of (ii).",
"As $\\infty $ is not a singular critical point of $A_0$ , the last statement in Theorem REF on the critical point $\\infty $ of $A_0+V$ follows in both cases (i) and (ii) from Proposition and $\\operatorname{dom}A=\\operatorname{dom}(A_0+V)$ .$\\Box $ Proof of Theorem REF Let $\\gamma $ and $\\eta $ be as in Theorem REF .",
"Then the operator $A_0 := A + \\eta G^{-1}$ is uniformly positive in the Krein space $(\\mathcal {H},[\\cdot \\,,\\cdot ])$ , $[\\cdot \\,,\\cdot ]= (G\\cdot ,\\cdot )$ , since for $f\\in \\operatorname{dom}A$ we have $[A_0 f,f] = [Af,f] + \\eta [G^{-1}f,f] \\ge -\\gamma \\Vert f\\Vert ^2 + \\eta \\Vert f\\Vert ^2 = (\\eta - \\gamma )\\Vert f\\Vert ^2.$ We will show that $A$ is non-negative over $\\overline{\\setminus K_r((-r,r)).In particular, 0\\in \\rho (A_0) is not a singular critical point of A_0and, as \\tau _\\eta < \\infty , the point \\infty is not aa singular critical point of A_0, see Remark \\ref {Mai2012}.With V := -\\eta G^{-1} we haveA = A_0 + V\\qquad \\text{and}\\qquad \\Vert V\\Vert =\\eta \\Vert G^{-1}\\Vert .As in the proof of Theorem~\\ref {t:main1} let E be the spectral function of the operator A_0.",
"Then the fundamentaldecomposition \\mathcal {H}= E_+\\mathcal {H}\\,[\\dotplus ]\\,E_-\\mathcal {H} in (\\ref {e:dec_expl}) generates matrix representationsof A_0, V and the corresponding fundamental symmetry \\widetilde{J} as in (\\ref {e:AJ}) and (\\ref {e:V}).The operator A_0 is selfadjoint in the Hilbert space(\\mathcal {H},(\\cdot \\,,\\cdot )_\\sim ), where (\\cdot \\,,\\cdot )_\\sim is defined in (\\ref {e:sim}).",
"With the help of Lemma \\ref {l:veselic} we obtain{\\begin{@align*}{1}{-1}\\widetilde{J}&= \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _0^n\\,\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\big )\\,dt\\\\&= \\frac{1}{\\pi }\\;\\,\\underset{n\\rightarrow \\infty }{{\\rm s}{\\text{-}}\\!\\lim }\\,\\!\\int _{-n}^n\\,(A + \\eta G^{-1} - it)^{-1}\\,dt\\,.\\end{@align*}}This yields\\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _\\eta .Following the lines of the proof of Theorem \\ref {t:main1}, it suffices to show the inequalities\\nu \\le \\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert ,\\quad \\min \\sigma (V_+)\\ge -\\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert \\quad \\text{and}\\quad \\max \\sigma (V_-)\\le \\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert ,where \\nu =\\Vert E_\\mp (V\\vert \\cal H_\\pm ) \\Vert _\\sim .The first relation is proved as in (\\ref {e:rel_proof}) and the remaining two inequalities follow from the selfadjointness of V_\\pm in (\\mathcal {H}_\\pm ,(\\cdot \\,,\\cdot )_\\sim ) and{\\begin{@align*}{1}{-1}\\max \\lbrace |\\min \\sigma (V_\\pm )|,|\\max \\sigma (V_\\pm )|\\rbrace &= \\Vert V_\\pm \\Vert _\\sim = \\sup \\lbrace |\\lambda | : \\lambda \\in \\sigma (V_\\pm )\\rbrace \\\\&\\le \\sup \\lbrace |(V_\\pm f_\\pm ,f_\\pm )| : \\Vert f_\\pm \\Vert =1,\\,f\\in \\mathcal {H}_\\pm \\rbrace \\\\&\\le \\Vert V_\\pm \\Vert = \\Vert E_\\pm (V|\\mathcal {H}_\\pm )\\Vert \\,\\le \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert V\\Vert .\\end{@align*}}Hence, A is non-negative over \\overline{\\setminus K_r((-r,r)).",
"As\\operatorname{dom}A= \\operatorname{dom}A_0, we conclude from Proposition \\ref {p:curgus} that \\infty is also not a critical point of A.This completes the proof of Theorem~\\ref {t:main2}.\\Box }\\begin{remark}If, in addition to the assumptions in Theorem \\ref {t:main2}, the point 0 is neither an eigenvalue nor a singular critical pointof the non-negative operator A_0 := A + \\gamma G^{-1}, then the operator A is non-negative over the set \\overline{\\setminus K_r((-r,r)), where\\begin{equation}r = \\gamma \\,\\frac{1+\\tau _\\gamma }{2}\\,\\Vert G^{-1}\\Vert \\end{equation}and\\tau _\\gamma := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\big ((A_0 + it)^{-1} + (A_0 - it)^{-1}\\big )\\,dt\\right\\Vert .Indeed, under the assumptions of Theorem \\ref {t:main2} for any \\varepsilon > 0 the operator A + (\\gamma + \\varepsilon )G^{-1}is selfadjoint in some Hilbert space, and thus the resolvent set ofA_0 is non-empty as A + (\\gamma + \\varepsilon )G^{-1} and A_0 differ just by a bounded operator.Therefore, A_0 is a non-negative operator in the Krein space(\\cal H,[\\cdot \\,,\\cdot ]).",
"Moreover,the point \\infty is not a singular critical point of theoperator A + (\\gamma + \\varepsilon )G^{-1}, see Remark \\ref {Mai2012},and by Proposition \\ref {p:curgus} \\infty is not a singular critical point of A_0.Therefore, the operator A_0 admits a matrix representation as in (\\ref {e:AJ}) and the corresponding fundamental symmetry \\widetilde{J}satisfies \\Vert \\widetilde{J}\\Vert \\,\\le \\,\\tau _\\gamma (see the proof of Theorem \\ref {t:main2}).Then by a similar reasoning as in the proof ofTheorem \\ref {t:main2} we see that A is non-negativeover \\overline{\\setminus K_r((-r,r)), where r is as in (\\ref {Juni2012}).",
"}}\\end{remark}}$ Differential operators with indefinite weights In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights.",
"Indefinite Sturm-Liouville operators We consider Sturm-Liouville differential expressions of the form $\\mathcal {L}(f)(x) = \\operatorname{sgn}(x)\\big (-f^{\\prime \\prime }(x) + q(x)f(x)\\big ),\\quad x\\in \\mathbb {R}\\,,$ with a real-valued potential $q\\in L^\\infty (\\mathbb {R})$ and the indefinite weight function $\\operatorname{sgn}(\\cdot )$ .",
"The corresponding differential operator in $L^2(\\mathbb {R})$ is defined by $Af := \\mathcal {L}(f), \\qquad \\operatorname{dom}A := H^2(\\mathbb {R}).$ Here $H^2(\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.",
"By $(\\cdot \\,,\\cdot )$ and $\\Vert \\cdot \\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\mathbb {R})$ .",
"Let $J$ be the operator of multiplication with the function $\\operatorname{sgn}(x)$ .",
"This operator is obviously selfadjoint and unitary in $L^2(\\mathbb {R})$ .",
"Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\prime \\prime } + qf,\\quad \\operatorname{dom}T := H^2(\\mathbb {R}),$ is selfadjoint in $L^2(\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\mathbb {R}),[\\cdot \\,,\\cdot ])$ , where $[f,g] := (Jf,g) = \\int _\\mathbb {R}\\,f(x)\\overline{g(x)}\\operatorname{sgn}(x)\\,dx,\\quad f,g\\in L^2(\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.",
"We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].",
"For this, denote by $m_\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\pm f_\\pm := - f_\\pm ^{\\prime \\prime } + q_\\pm f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ defined on $\\operatorname{dom}T_\\pm := \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ) : f_\\pm (0) = 0\\rbrace $ , where $q_\\pm $ is the restriction of the function $q$ to $\\mathbb {R}^\\pm $ .",
"The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\pm \\infty $ , then $m_\\pm =\\lim _{x\\rightarrow \\pm \\infty }\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.",
"Furthermore the following holds.",
"If $m_+ > -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ is finite; If $m_+\\le -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .",
"To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.",
"One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].",
"The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .",
"Here only the nontrivial case $\\operatornamewithlimits{ess\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).",
"Theorem 4.2 Assume that $\\operatornamewithlimits{ess\\,inf}q<0$ holds.",
"Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\overline{\\setminus K_r((-d,d)), wherer := 5\\Vert q\\Vert _\\infty ,\\quad \\text{and}\\quad d := -5\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\cap \\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .",
"}\\begin{proof}The proof of Theorem~\\ref {t:SL2} is split into 4 parts.",
"The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\ref {t:main1}.",
"In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\end{proof}\\ \\\\{\\it 1.\\ Preparation: } The operator $ A0$,\\begin{equation*}A_0 f:=-\\operatorname{sgn}(\\cdot )f^{\\prime \\prime },\\qquad \\operatorname{dom}A_0=H^2(\\mathbb {R}),\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\mathbb {R}$ ),[,])$.",
"Furthermore, we have $ (A0) = $\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\cite {cn}.",
"Hence $ A0$ satisfies theassumptions in Theorem~\\ref {t:main1}.",
"Define $ V$ as\\begin{equation*}Vf:=\\operatorname{sgn}(\\cdot ) qf,\\qquad \\operatorname{dom}V=L^2(\\mathbb {R}),\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\mathbb {R}$ ),[,])$ with$$\\min \\sigma (JV) = \\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}\\,q(x)\\quad \\text{and}\\quad \\Vert V\\Vert = \\Vert q\\Vert _\\infty .$$By Theorem \\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\frac{1+\\tau _0}{2}\\,\\Vert q\\Vert _\\infty ,\\qquad d = -\\frac{1+\\tau _0}{2}\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x),$$and$$\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert .$$Therefore, we have to show\\begin{equation}\\tau _0\\le 9\\end{equation}and\\begin{equation}\\sigma (A)\\setminus \\mathbb {R}\\,\\subset \\,\\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .\\end{equation}$ 2.",
"Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\begin{pmatrix} B_+ & 0 \\\\ 0 & B_- \\end{pmatrix}\\quad \\text{in}\\quad L^2(\\mathbb {R})=L^2(\\mathbb {R}^+)\\times L^2(\\mathbb {R}^-),$ where $B_\\pm f_\\pm :=\\mp f_\\pm ^{\\prime \\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\mathbb {R}^\\pm )$ which are defined on $\\operatorname{dom}B_\\pm = \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ):f_\\pm (0)=0\\rbrace $ .",
"Here and in the following we will denote the restrictions of a function $f$ defined on $\\mathbb {R}$ to $\\mathbb {R}^\\pm $ by $f_\\pm $ .",
"For $\\lambda = re^{it}$ , $r > 0$ , $t\\in [0,2\\pi )$ , we set $\\sqrt{\\lambda } := \\sqrt{r}e^{it/2}$ .",
"For $\\lambda \\in \\mathbb {R}$ define the function $f_\\lambda (x) :={\\left\\lbrace \\begin{array}{ll}e^{i\\sqrt{\\lambda }x}, &x> 0,\\\\e^{-i\\sqrt{-\\lambda }x}, &x < 0.\\end{array}\\right.",
"}$ The function $f_\\lambda $ is a solution of the equations $\\mp f_\\pm ^{\\prime \\prime } = \\lambda f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ .",
"From $\\overline{\\sqrt{\\lambda }} = -\\sqrt{\\overline{\\lambda }}$ it follows that $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ holds for all $\\lambda \\in \\mathbb {R}$ .",
"Let us now prove that for all $f\\in L^2(\\mathbb {R})$ and $\\lambda \\in \\mathbb {R}$ we have $(A_0 - \\lambda )^{-1}f = (B_0 - \\lambda )^{-1}f - \\frac{[f,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda .$ Since $\\sigma (A_0)=\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\lambda \\in \\mathbb {R}$ .",
"In particular, for $\\lambda \\in \\mathbb {R}$ there exists $g\\in \\operatorname{dom}B_0$ such that $f=(B_0-\\lambda )g$ holds.",
"The right hand side in (REF ) has the form $h:=g-\\frac{[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda $ and we will show that the function $h$ belongs to $\\operatorname{dom}A_0=H^2(\\mathbb {R})$ .",
"As $g_\\pm $ and $f_{\\lambda ,\\pm }$ are elements of $H^2(\\mathbb {R}^\\pm )$ the same is true for $h_\\pm $ .",
"Moreover, $g$ and $f_\\lambda $ are continuous at 0 and so is $h$ .",
"Hence it remains to check that $h^\\prime $ is continuous at 0.",
"For this note first that $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=\\int _0^\\infty (-g_+^{\\prime \\prime }-\\lambda g_+)\\overline{f_{\\overline{\\lambda },+}}\\, dx- \\int _{-\\infty }^0 (g_-^{\\prime \\prime }-\\lambda g_-)\\overline{f_{\\overline{\\lambda },-}}\\, dx$ and since $f_{\\overline{\\lambda },\\pm }$ are solutions of $\\mp f_\\pm ^{\\prime \\prime }=\\overline{\\lambda }f_\\pm $ , integration by parts yields $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=g_+^\\prime (0)\\overline{f_{\\overline{\\lambda },+}(0)}-g_+(0) \\overline{f^\\prime _{\\overline{\\lambda },+}(0)}-g_-^\\prime (0)\\overline{f_{\\overline{\\lambda },-}(0)}+g_-(0) \\overline{f^\\prime _{\\overline{\\lambda },-}(0)}.$ As $g\\in \\operatorname{dom}B_0$ we have $g_\\pm (0)=0$ and together with $f_{\\overline{\\lambda },\\pm }(0)=1$ we find $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]= g_+^\\prime (0)-g^\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\pm ^\\prime $ on $\\mathbb {R}^\\pm $ of the function $h$ from (REF ): $h^\\prime _\\pm =g^\\prime _\\pm -\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_{\\lambda ,\\pm }^\\prime $ and as $f^\\prime _{\\lambda ,+}(0)=i\\sqrt{\\lambda }$ and $f^\\prime _{\\lambda ,-}(0)=-i\\sqrt{-\\lambda }$ we conclude $h^\\prime _+(0)-h^\\prime _-(0)= \\bigl (g_+^\\prime (0)-g^\\prime _-(0)\\bigr )-\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}\\bigl (f^\\prime _{\\lambda ,+}(0)-f^\\prime _{\\lambda ,-}(0)\\bigr )=0,$ that is, $h^\\prime $ is continuous at 0 and therefore $h\\in \\operatorname{dom}A_0$ .",
"Now a straightforward computation shows that $(A_0-\\lambda )h=(B_0-\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).",
"3.",
"Proof of (): Let $f\\in L^2(\\mathbb {R})$ and $t > 0$ .",
"Then (REF ) and $\\sqrt{it} + \\sqrt{-it} = i\\sqrt{2t}$ yield $\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\& + \\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\in \\mathbb {R}^+$ , we have $\\big |[f,f_{it}][f_{-it},f]\\big |\\,\\le \\,\\left(\\int _0^\\infty \\,e^{-x\\sqrt{t/2}}g(x)\\,dx\\right)^2$ and thus for $n\\in \\mathbb {N}$ $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt&\\le \\int _{1/n}^n\\,\\int _0^\\infty \\,\\int _0^\\infty \\,g(x)g(y)\\,\\frac{e^{-(x+y)\\sqrt{t/2}}}{\\sqrt{2t}}\\,dy\\,dx\\,dt\\\\&= 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\left(e^{-\\frac{x+y}{\\sqrt{2n}}} - e^{-\\frac{\\sqrt{n}(x+y)}{\\sqrt{2}}}\\right)\\,dy\\,dx\\\\&\\le 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\,dy\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt\\,\\le \\,2\\pi \\Vert g\\Vert _{L^2(\\mathbb {R}^+)}^2\\,\\le \\,4\\pi \\Vert f\\Vert ^2.$ and therefore we have $\\frac{1}{\\pi }\\int _{1/n}^n\\,\\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\,dt\\,\\le \\,8\\Vert f\\Vert ^2.$ Denote by $E_0$ ($E_\\pm $ ) the spectral function of the operator $A_0$ ($B_\\pm $ , respectively).",
"Moreover define $J_{n,\\pm } := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}\\right)\\,dt,$ and, in addition, $J_{n,0} := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt.$ As $\\pm B_\\pm $ is non-negative, it follows that $J_{n,\\pm }$ converges strongly to $\\pm I_{L^2(\\mathbb {R}^\\pm )}$ when $n\\rightarrow \\infty $ , cf.",
"Lemma REF .",
"Moreover, by $(B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}= 2B_\\pm (B_\\pm + it)^{-1} (B_\\pm - it)^{-1}$ the sequence $\\pm J_{n,\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\mathbb {R}^\\pm )$ .",
"This implies $\\frac{1}{\\pi }\\,\\int _{1/n}^n\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\&\\le \\Vert f_+\\Vert _{L^2(\\mathbb {R}^+)}^2 + \\Vert f_-\\Vert _{L^2(\\mathbb {R}^-)}^2 = \\Vert f\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\,\\le \\,\\Vert f\\Vert ^2 + 8\\Vert f\\Vert ^2 = 9\\Vert f\\Vert ^2,\\quad f\\in L^2(\\mathbb {R}).$ Hence, as $J = \\operatorname{sgn}(\\cdot )$ is unitary in $L^2(\\mathbb {R})$ , we finally obtain $\\Vert J_{n,0}\\Vert = \\Vert J J_{n,0}\\Vert \\le 9$ and thus $\\tau _0 = \\limsup _{n\\rightarrow \\infty }\\,\\Vert J_{n,0}\\Vert \\le 9$ .",
"That is, () holds.",
"4.",
"Proof of (): Let $\\lambda \\in +$ .",
"Using (REF ), for $f\\in L^2(\\mathbb {R})$ we obtain $\\Vert (A_0 - \\lambda )^{-1}f\\Vert \\,\\le \\,\\Vert (B_0 - \\lambda )^{-1}f\\Vert + \\frac{|[f,f_{\\overline{\\lambda }}]|\\,\\Vert f_\\lambda \\Vert }{|\\sqrt{\\lambda } + \\sqrt{-\\lambda }|}\\,\\le \\,\\frac{\\Vert f\\Vert }{|\\operatorname{Im}\\lambda |} + \\frac{\\Vert f_\\lambda \\Vert ^2\\Vert f\\Vert }{\\sqrt{2|\\lambda |}}\\,.$ Here we have used the identity $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ .",
"Let $r > 0$ and $t\\in (0,\\pi )$ such that $\\lambda = re^{it}$ .",
"Then $\\Vert f_\\lambda \\Vert ^2&= \\int _0^\\infty \\,\\left|e^{i\\sqrt{r}e^{it/2}x}\\right|^2\\,dx + \\int _{-\\infty }^0\\,\\left|e^{-i\\sqrt{r}e^{i(t+\\pi )/2}x}\\right|^2\\,dx\\\\&= \\int _0^\\infty \\,e^{-2\\sqrt{r}\\sin (t/2)x}\\,dx + \\int _{-\\infty }^0\\,e^{2\\sqrt{r}\\cos (t/2)x}\\,dx\\\\&= \\frac{\\cos (t/2) + \\sin (t/2)}{\\sqrt{r}\\,\\sin (t)}\\,\\le \\,\\frac{\\sqrt{2}}{\\sqrt{r}\\,\\operatorname{Im}(\\lambda /r)} = \\frac{\\sqrt{2|\\lambda |}}{|\\operatorname{Im}\\lambda |}.$ Therefore, $\\Vert (A_0 - \\lambda )^{-1}\\Vert \\,\\le \\,\\frac{2}{|\\operatorname{Im}\\lambda |}\\,.$ The same estimate holds for $\\lambda \\in -$ .",
"Now, assume that $|\\operatorname{Im}\\lambda | > 2\\Vert q\\Vert _\\infty =2\\Vert V\\Vert $ .",
"Then $\\Vert V(A_0 - \\lambda )^{-1}\\Vert < 1$ , and hence $A - \\lambda = A_0 - \\lambda + V = \\left(I + V(A_0 - \\lambda )^{-1}\\right)(A_0 - \\lambda )$ is boundedly invertible, which proves ().$\\Box $ Second order elliptic operators Let $\\Omega \\subset \\mathbb {R}^n$ be a domain and let $\\ell $ be the \"formally symmetric\" uniformly elliptic second order differential expression $\\ell (f)(x):=-\\sum _{j,k=1}^n \\left( \\frac{\\partial }{\\partial x_j} a_{jk}\\frac{\\partial f}{\\partial x_k}\\right)(x)+a(x)f(x),\\quad x\\in \\Omega ,$ with bounded coefficients $a_{jk}\\in C^\\infty (\\Omega )$ satisfying $a_{jk}(x)=\\overline{a_{kj}(x)}$ for all $x\\in \\Omega $ and $j,k=1,\\dots ,n$ , the function $a\\in L^\\infty (\\Omega )$ is real valued and $\\sum _{j,k=1}^n a_{jk}(x)\\xi _j\\xi _k\\ge C\\sum _{k=1}^n\\xi _k^2$ holds for some $C>0$ , all $\\xi =(\\xi _1,\\dots ,\\xi _n)^\\top \\in \\mathbb {R}^n$ and $x\\in \\Omega $ .",
"With the differential expression $\\ell $ we associate the elliptic differential operator $Tf:=\\ell (f),\\qquad \\operatorname{dom}T=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\ell (f) \\in L^2(\\Omega )\\bigr \\rbrace ,$ where $H^1_0(\\Omega )$ stands for the closure of $C_0^\\infty (\\Omega )$ in the Sobolev space $H^1(\\Omega )$ .",
"It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\Omega ),(\\cdot ,\\cdot ))$ with spectrum semibounded from below by $\\operatornamewithlimits{ess\\,inf}\\,a$ ; cf.",
"[24].",
"Let $w$ be a real valued function such that $w,w^{-1}\\in L^\\infty (\\Omega )$ and each of the sets $\\Omega _+:=\\bigl \\lbrace x\\in \\Omega : w(x)>0\\bigr \\rbrace \\quad \\text{and}\\quad \\Omega _-:=\\bigl \\lbrace x\\in \\Omega : w(x)<0\\bigr \\rbrace $ has positive Lebesgue measure.",
"We define a second order elliptic differential expression $\\mathcal {L}$ with the indefinite weight $w$ by $\\mathcal {L}(f)(x):=\\frac{1}{w(x)}\\,\\ell (f)(x),\\qquad x\\in \\Omega .$ The multiplication operator $G_w f=w f$ , $f\\in L^2(\\Omega )$ , is an isomorphism in $L^2(\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\in L^2(\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\int _\\Omega f(x)\\overline{g(x)}\\,w(x)\\,dx,\\qquad f,g\\in L^2(\\Omega ).$ The differential operator associated with $\\mathcal {L}$ is defined as $Af=\\mathcal {L}(f),\\qquad \\operatorname{dom}A=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\mathcal {L}(f) \\in L^2(\\Omega )\\bigr \\rbrace .$ Since for $f\\in H^1_0(\\Omega )$ we have $\\ell (f)\\in L^2(\\Omega )$ if and only if $\\mathcal {L}(f)\\in L^2(\\Omega )$ it follows that $\\operatorname{dom}A=\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.",
"Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ .",
"In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\min \\sigma _{\\rm ess}(T)\\le 0$ holds.",
"This also implies that the domain $\\Omega $ is unbounded as otherwise $\\sigma _{\\rm ess}(T)=\\varnothing $ .",
"A discussion of the cases $\\sigma _{\\rm ess}(T)=\\varnothing $ and $\\min \\sigma _{\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].",
"Fix some $\\eta >0$ such that $-\\eta <\\min \\sigma (T)$ and define the spaces $\\mathcal {H}_s$ , $s\\in [0,2]$ , as the domains of the $\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\eta $ in $L^2(\\Omega )$ , $\\mathcal {H}_s:=\\operatorname{dom}\\bigl ((T+\\eta )^\\frac{s}{2}\\bigr ),\\qquad s\\in [0,2].$ Note that $\\mathcal {H}=\\mathcal {H}_0$ , $\\operatorname{dom}T=\\mathcal {H}_2$ and the form domain of $T$ is $\\mathcal {H}_1$ .",
"The spaces $\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\eta $ ; cf.",
"[40].",
"The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.",
"[9].",
"Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ such that $W\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds for some $s\\in (0,2]$ .",
"Then $A$ is non-negative over $ \\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r=\\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert w^{-1}\\Vert _\\infty \\quad \\text{and}\\quad \\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A+\\eta G_w^{-1} - it)^{-1} \\,dt\\,\\right\\Vert \\,<\\,\\infty .$ Moreover, $\\infty $ is not a singular critical point of $A$ .",
"In the next corollary the special case $\\Omega =\\mathbb {R}^n$ is treated; cf.",
"[9].",
"From now on we assume that $\\Omega =\\mathbb {R}^n$ and $\\Omega _\\pm =\\lbrace x\\in \\mathbb {R}^n:\\pm w(x)>0\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\in C^\\infty (\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.",
"Note that either $\\Omega _+$ or $\\Omega _-$ is bounded.",
"Corollary 4.4 Suppose that for some $s\\in (0,\\frac{1}{2})$ the Sobolev spaces $H^s(\\Omega _\\pm )$ are invariant under multiplication with $w\\!\\upharpoonright _{\\Omega _\\pm }$ .",
"Then the indefinite elliptic operator $A$ is non-negative over $\\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .",
"Moreover, $\\infty $ is not a singular critical point of $A$ .",
"A sufficient criterion for the invariance of $H^s(\\Omega _\\pm )$ in the above corollary can be deduced from [28].",
"Corollary 4.5 Suppose that the weight function $w\\!\\upharpoonright _{\\Omega _\\pm }$ belongs to some Hölder space $C^{0,\\alpha }(\\Omega _\\pm )$ , $\\alpha \\in (0,\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.",
"Then the claim in Corollary REF holds.",
"of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\operatorname{dom}A=\\operatorname{dom}T=H^2(\\mathbb {R}^n)\\qquad \\text{and}\\qquad \\mathcal {H}_s=H^s(\\mathbb {R}^n),\\qquad s\\in [0,2],$ by elliptic regularity and interpolation.",
"Since the Sobolev spaces $H^s(\\Omega _\\pm )$ are assumed to be invariant under multiplication with the functions $w\\!\\upharpoonright _{\\Omega _\\pm }$ for some $s\\in (0,\\frac{1}{2})$ it follows from [9] that also $H^s(\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds.",
"Furthermore, $G_w$ is uniformly positive in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\ge \\operatornamewithlimits{ess\\,inf}w^2 \\Vert f\\Vert ^2$ and $w^{-1}\\in L^\\infty (\\mathbb {R}^n)$ .",
"Now the assertion follows from Theorem REF .$\\Box $ Acknowledgements The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.",
"Contact information Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, [email protected] Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.",
"Juni 136, 10623 Berlin, Germany, [email protected] Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, [email protected]"
],
[
"Differential operators with indefinite weights",
"In this section we apply the results from the previous section to ordinary and partial differential operators with indefinite weights."
],
[
"Indefinite Sturm-Liouville operators",
"We consider Sturm-Liouville differential expressions of the form $\\mathcal {L}(f)(x) = \\operatorname{sgn}(x)\\big (-f^{\\prime \\prime }(x) + q(x)f(x)\\big ),\\quad x\\in \\mathbb {R}\\,,$ with a real-valued potential $q\\in L^\\infty (\\mathbb {R})$ and the indefinite weight function $\\operatorname{sgn}(\\cdot )$ .",
"The corresponding differential operator in $L^2(\\mathbb {R})$ is defined by $Af := \\mathcal {L}(f), \\qquad \\operatorname{dom}A := H^2(\\mathbb {R}).$ Here $H^2(\\mathbb {R})$ stands for the usual $L^2$ -based second order Sobolov space.",
"By $(\\cdot \\,,\\cdot )$ and $\\Vert \\cdot \\Vert $ we denote the usual scalar product and its corresponding norm in $L^2(\\mathbb {R})$ .",
"Let $J$ be the operator of multiplication with the function $\\operatorname{sgn}(x)$ .",
"This operator is obviously selfadjoint and unitary in $L^2(\\mathbb {R})$ .",
"Since the definite Sturm-Liouville operator $Tf := JAf = -f^{\\prime \\prime } + qf,\\quad \\operatorname{dom}T := H^2(\\mathbb {R}),$ is selfadjoint in $L^2(\\mathbb {R})$ , it follows that the indefinite Sturm Liouville operator $A$ is selfadjoint in the Krein space $(L^2(\\mathbb {R}),[\\cdot \\,,\\cdot ])$ , where $[f,g] := (Jf,g) = \\int _\\mathbb {R}\\,f(x)\\overline{g(x)}\\operatorname{sgn}(x)\\,dx,\\quad f,g\\in L^2(\\mathbb {R}).$ It is known that the operator $A$ is non-negative over some neighborhood of $\\infty $ , but no explicit bounds on the size of this neighborhood exist in the literature.",
"We first recall a theorem on the qualitative spectral properties of $A$ which can be found in a slightly different form in [8], [13], [39].",
"For this, denote by $m_\\pm $ the minimum of the essential spectrum of the selfadjoint operator $T_\\pm f_\\pm := - f_\\pm ^{\\prime \\prime } + q_\\pm f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ defined on $\\operatorname{dom}T_\\pm := \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ) : f_\\pm (0) = 0\\rbrace $ , where $q_\\pm $ is the restriction of the function $q$ to $\\mathbb {R}^\\pm $ .",
"The quantities $m_+$ and $m_-$ can be expressed in terms of the potential $q$ ; if, e.g., $q$ admits limits at $\\pm \\infty $ , then $m_\\pm =\\lim _{x\\rightarrow \\pm \\infty }\\,q(x).$ Theorem 4.1 The essential spectrum of the indefinite Sturm-Liouville operator $A$ is the union of the essential spectra of $T_+$ and $-T_-$ , the non-real spectrum of $A$ is bounded and consists of isolated eigenvalues with finite algebraic multiplicity.",
"Furthermore the following holds.",
"If $m_+ > -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ is finite; If $m_+\\le -m_-$ then $\\sigma (A)\\setminus \\mathbb {R}$ may only accumulate to points in $[m_+,-m_-]$ .",
"To the best of our knowledge explicit bounds on the non-real spectrum of indefinite Sturm-Liouville operators in terms of the potential $q$ do not exist in the literature.",
"One may expect that there is a relationship between the maximal magnitude of the non-real eigenvalues of $A$ and the lower bound of the selfadjoint operator $T = JA$ , see, e.g., the numerical examples in [12] and the conjecture in [56].",
"The following theorem confirms this conjecture and provides explicit bounds on the non-real eigenvalues of $A$ in terms of the potential $q$ .",
"Here only the nontrivial case $\\operatornamewithlimits{ess\\,inf}q<0$ is treated which corresponds to Theorem REF (ii).",
"Theorem 4.2 Assume that $\\operatornamewithlimits{ess\\,inf}q<0$ holds.",
"Then the indefinite Sturm-Liouville operator $A$ is non-negative over $\\overline{\\setminus K_r((-d,d)), wherer := 5\\Vert q\\Vert _\\infty ,\\quad \\text{and}\\quad d := -5\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x)>0.The non-real spectrum of A is contained inK_r((-d,d))\\cap \\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .",
"}\\begin{proof}The proof of Theorem~\\ref {t:SL2} is split into 4 parts.",
"The first step is preparatory and connects the present problemwith the abstract setting in Theorem~\\ref {t:main1}.",
"In the second part a Krein type resolvent formula is providedwhich is essential for the main estimates in the last two parts of the proof.\\end{proof}\\ \\\\{\\it 1.\\ Preparation: } The operator $ A0$,\\begin{equation*}A_0 f:=-\\operatorname{sgn}(\\cdot )f^{\\prime \\prime },\\qquad \\operatorname{dom}A_0=H^2(\\mathbb {R}),\\end{equation*}is selfadjoint and non-negative in the Krein space $ (L2($\\mathbb {R}$ ),[,])$.",
"Furthermore, we have $ (A0) = $\\mathbb {R}$$,and neither $ 0$ nor $$ is a singular critical point of $ A0$, see \\cite {cn}.",
"Hence $ A0$ satisfies theassumptions in Theorem~\\ref {t:main1}.",
"Define $ V$ as\\begin{equation*}Vf:=\\operatorname{sgn}(\\cdot ) qf,\\qquad \\operatorname{dom}V=L^2(\\mathbb {R}),\\end{equation*}so that $ V$ is a bounded selfadjoint operator in $ (L2($\\mathbb {R}$ ),[,])$ with$$\\min \\sigma (JV) = \\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}\\,q(x)\\quad \\text{and}\\quad \\Vert V\\Vert = \\Vert q\\Vert _\\infty .$$By Theorem \\ref {t:main1} the operator$$A = A_0 + V$$is non-negative over $Kr((-d,d))$, where$$r = \\frac{1+\\tau _0}{2}\\,\\Vert q\\Vert _\\infty ,\\qquad d = -\\frac{1+\\tau _0}{2}\\,\\operatornamewithlimits{ess\\,inf}_{x\\in \\mathbb {R}}q(x),$$and$$\\tau _0 = \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt\\,\\right\\Vert .$$Therefore, we have to show\\begin{equation}\\tau _0\\le 9\\end{equation}and\\begin{equation}\\sigma (A)\\setminus \\mathbb {R}\\,\\subset \\,\\lbrace \\lambda \\in |\\operatorname{Im}\\lambda |\\le 2\\Vert q\\Vert _\\infty \\rbrace .\\end{equation}$ 2.",
"Krein's resolvent formula: In the following the resolvent of $A_0$ will be expressed via a Krein type resolvent formula in terms of the resolvent of the diagonal operator matrix $B_0:=\\begin{pmatrix} B_+ & 0 \\\\ 0 & B_- \\end{pmatrix}\\quad \\text{in}\\quad L^2(\\mathbb {R})=L^2(\\mathbb {R}^+)\\times L^2(\\mathbb {R}^-),$ where $B_\\pm f_\\pm :=\\mp f_\\pm ^{\\prime \\prime }$ are the selfadjoint Dirichlet operators in $L^2(\\mathbb {R}^\\pm )$ which are defined on $\\operatorname{dom}B_\\pm = \\lbrace f_\\pm \\in H^2(\\mathbb {R}^\\pm ):f_\\pm (0)=0\\rbrace $ .",
"Here and in the following we will denote the restrictions of a function $f$ defined on $\\mathbb {R}$ to $\\mathbb {R}^\\pm $ by $f_\\pm $ .",
"For $\\lambda = re^{it}$ , $r > 0$ , $t\\in [0,2\\pi )$ , we set $\\sqrt{\\lambda } := \\sqrt{r}e^{it/2}$ .",
"For $\\lambda \\in \\mathbb {R}$ define the function $f_\\lambda (x) :={\\left\\lbrace \\begin{array}{ll}e^{i\\sqrt{\\lambda }x}, &x> 0,\\\\e^{-i\\sqrt{-\\lambda }x}, &x < 0.\\end{array}\\right.",
"}$ The function $f_\\lambda $ is a solution of the equations $\\mp f_\\pm ^{\\prime \\prime } = \\lambda f_\\pm $ in $L^2(\\mathbb {R}^\\pm )$ .",
"From $\\overline{\\sqrt{\\lambda }} = -\\sqrt{\\overline{\\lambda }}$ it follows that $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ holds for all $\\lambda \\in \\mathbb {R}$ .",
"Let us now prove that for all $f\\in L^2(\\mathbb {R})$ and $\\lambda \\in \\mathbb {R}$ we have $(A_0 - \\lambda )^{-1}f = (B_0 - \\lambda )^{-1}f - \\frac{[f,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda .$ Since $\\sigma (A_0)=\\mathbb {R}$ and $B_0$ is selfadjoint in the Hilbert space $L^2(\\mathbb {R})$ it follows that the resolvents of $A_0$ and $B_0$ in (REF ) are defined for all $\\lambda \\in \\mathbb {R}$ .",
"In particular, for $\\lambda \\in \\mathbb {R}$ there exists $g\\in \\operatorname{dom}B_0$ such that $f=(B_0-\\lambda )g$ holds.",
"The right hand side in (REF ) has the form $h:=g-\\frac{[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_\\lambda $ and we will show that the function $h$ belongs to $\\operatorname{dom}A_0=H^2(\\mathbb {R})$ .",
"As $g_\\pm $ and $f_{\\lambda ,\\pm }$ are elements of $H^2(\\mathbb {R}^\\pm )$ the same is true for $h_\\pm $ .",
"Moreover, $g$ and $f_\\lambda $ are continuous at 0 and so is $h$ .",
"Hence it remains to check that $h^\\prime $ is continuous at 0.",
"For this note first that $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=\\int _0^\\infty (-g_+^{\\prime \\prime }-\\lambda g_+)\\overline{f_{\\overline{\\lambda },+}}\\, dx- \\int _{-\\infty }^0 (g_-^{\\prime \\prime }-\\lambda g_-)\\overline{f_{\\overline{\\lambda },-}}\\, dx$ and since $f_{\\overline{\\lambda },\\pm }$ are solutions of $\\mp f_\\pm ^{\\prime \\prime }=\\overline{\\lambda }f_\\pm $ , integration by parts yields $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]=g_+^\\prime (0)\\overline{f_{\\overline{\\lambda },+}(0)}-g_+(0) \\overline{f^\\prime _{\\overline{\\lambda },+}(0)}-g_-^\\prime (0)\\overline{f_{\\overline{\\lambda },-}(0)}+g_-(0) \\overline{f^\\prime _{\\overline{\\lambda },-}(0)}.$ As $g\\in \\operatorname{dom}B_0$ we have $g_\\pm (0)=0$ and together with $f_{\\overline{\\lambda },\\pm }(0)=1$ we find $[(B_0-\\lambda )g,f_{\\overline{\\lambda }}]= g_+^\\prime (0)-g^\\prime _-(0).$ Therefore we obtain for the derivatives $h_\\pm ^\\prime $ on $\\mathbb {R}^\\pm $ of the function $h$ from (REF ): $h^\\prime _\\pm =g^\\prime _\\pm -\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}f_{\\lambda ,\\pm }^\\prime $ and as $f^\\prime _{\\lambda ,+}(0)=i\\sqrt{\\lambda }$ and $f^\\prime _{\\lambda ,-}(0)=-i\\sqrt{-\\lambda }$ we conclude $h^\\prime _+(0)-h^\\prime _-(0)= \\bigl (g_+^\\prime (0)-g^\\prime _-(0)\\bigr )-\\frac{g_+^\\prime (0)-g^\\prime _-(0)}{i(\\sqrt{\\lambda }+\\sqrt{-\\lambda })}\\bigl (f^\\prime _{\\lambda ,+}(0)-f^\\prime _{\\lambda ,-}(0)\\bigr )=0,$ that is, $h^\\prime $ is continuous at 0 and therefore $h\\in \\operatorname{dom}A_0$ .",
"Now a straightforward computation shows that $(A_0-\\lambda )h=(B_0-\\lambda )g=f$ holds and hence the resolvent of $A_0$ is given by (REF ).",
"3.",
"Proof of (): Let $f\\in L^2(\\mathbb {R})$ and $t > 0$ .",
"Then (REF ) and $\\sqrt{it} + \\sqrt{-it} = i\\sqrt{2t}$ yield $\\begin{split}[(A_0 + it)^{-1}f + (A_0 - it)^{-1}f,f] = \\,& [(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\\\& + \\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]).\\end{split}$ With $g(x) := |f(x)| + |f(-x)|$ , $x\\in \\mathbb {R}^+$ , we have $\\big |[f,f_{it}][f_{-it},f]\\big |\\,\\le \\,\\left(\\int _0^\\infty \\,e^{-x\\sqrt{t/2}}g(x)\\,dx\\right)^2$ and thus for $n\\in \\mathbb {N}$ $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt&\\le \\int _{1/n}^n\\,\\int _0^\\infty \\,\\int _0^\\infty \\,g(x)g(y)\\,\\frac{e^{-(x+y)\\sqrt{t/2}}}{\\sqrt{2t}}\\,dy\\,dx\\,dt\\\\&= 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\left(e^{-\\frac{x+y}{\\sqrt{2n}}} - e^{-\\frac{\\sqrt{n}(x+y)}{\\sqrt{2}}}\\right)\\,dy\\,dx\\\\&\\le 2\\,\\int _0^\\infty \\,\\int _0^\\infty \\,\\frac{g(x)g(y)}{x+y}\\,dy\\,dx.$ From Hilbert's inequality (see, e.g., [29]) it follows that $\\int _{1/n}^n\\,\\frac{|[f,f_{it}][f_{-it},f]|}{\\sqrt{2t}}\\,dt\\,\\le \\,2\\pi \\Vert g\\Vert _{L^2(\\mathbb {R}^+)}^2\\,\\le \\,4\\pi \\Vert f\\Vert ^2.$ and therefore we have $\\frac{1}{\\pi }\\int _{1/n}^n\\,\\frac{2}{\\sqrt{2t}}\\,\\operatorname{Re}([f,f_{it}][f_{-it},f]) \\,dt\\,\\le \\,8\\Vert f\\Vert ^2.$ Denote by $E_0$ ($E_\\pm $ ) the spectral function of the operator $A_0$ ($B_\\pm $ , respectively).",
"Moreover define $J_{n,\\pm } := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}\\right)\\,dt,$ and, in addition, $J_{n,0} := \\frac{1}{\\pi }\\,\\int _{1/n}^n\\,\\left((A_0 + it)^{-1} + (A_0 - it)^{-1}\\right)\\,dt.$ As $\\pm B_\\pm $ is non-negative, it follows that $J_{n,\\pm }$ converges strongly to $\\pm I_{L^2(\\mathbb {R}^\\pm )}$ when $n\\rightarrow \\infty $ , cf.",
"Lemma REF .",
"Moreover, by $(B_\\pm + it)^{-1} + (B_\\pm - it)^{-1}= 2B_\\pm (B_\\pm + it)^{-1} (B_\\pm - it)^{-1}$ the sequence $\\pm J_{n,\\pm }$ is an increasing sequence of non-negative selfadjoint operators in $L^2(\\mathbb {R}^\\pm )$ .",
"This implies $\\frac{1}{\\pi }\\,\\int _{1/n}^n\\,[(B_0 + it)^{-1}f + (B_0 - it)^{-1}f,f]\\,dt&= (J_{n,+}f_+,f_+) - (J_{n,-}f_-,f_-)\\\\&\\le \\Vert f_+\\Vert _{L^2(\\mathbb {R}^+)}^2 + \\Vert f_-\\Vert _{L^2(\\mathbb {R}^-)}^2 = \\Vert f\\Vert ^2.$ Now, from (REF ) and (REF ) it follows that $[J_{n,0}f,f]\\,\\le \\,\\Vert f\\Vert ^2 + 8\\Vert f\\Vert ^2 = 9\\Vert f\\Vert ^2,\\quad f\\in L^2(\\mathbb {R}).$ Hence, as $J = \\operatorname{sgn}(\\cdot )$ is unitary in $L^2(\\mathbb {R})$ , we finally obtain $\\Vert J_{n,0}\\Vert = \\Vert J J_{n,0}\\Vert \\le 9$ and thus $\\tau _0 = \\limsup _{n\\rightarrow \\infty }\\,\\Vert J_{n,0}\\Vert \\le 9$ .",
"That is, () holds.",
"4.",
"Proof of (): Let $\\lambda \\in +$ .",
"Using (REF ), for $f\\in L^2(\\mathbb {R})$ we obtain $\\Vert (A_0 - \\lambda )^{-1}f\\Vert \\,\\le \\,\\Vert (B_0 - \\lambda )^{-1}f\\Vert + \\frac{|[f,f_{\\overline{\\lambda }}]|\\,\\Vert f_\\lambda \\Vert }{|\\sqrt{\\lambda } + \\sqrt{-\\lambda }|}\\,\\le \\,\\frac{\\Vert f\\Vert }{|\\operatorname{Im}\\lambda |} + \\frac{\\Vert f_\\lambda \\Vert ^2\\Vert f\\Vert }{\\sqrt{2|\\lambda |}}\\,.$ Here we have used the identity $f_{\\overline{\\lambda }} = \\overline{f_\\lambda }$ .",
"Let $r > 0$ and $t\\in (0,\\pi )$ such that $\\lambda = re^{it}$ .",
"Then $\\Vert f_\\lambda \\Vert ^2&= \\int _0^\\infty \\,\\left|e^{i\\sqrt{r}e^{it/2}x}\\right|^2\\,dx + \\int _{-\\infty }^0\\,\\left|e^{-i\\sqrt{r}e^{i(t+\\pi )/2}x}\\right|^2\\,dx\\\\&= \\int _0^\\infty \\,e^{-2\\sqrt{r}\\sin (t/2)x}\\,dx + \\int _{-\\infty }^0\\,e^{2\\sqrt{r}\\cos (t/2)x}\\,dx\\\\&= \\frac{\\cos (t/2) + \\sin (t/2)}{\\sqrt{r}\\,\\sin (t)}\\,\\le \\,\\frac{\\sqrt{2}}{\\sqrt{r}\\,\\operatorname{Im}(\\lambda /r)} = \\frac{\\sqrt{2|\\lambda |}}{|\\operatorname{Im}\\lambda |}.$ Therefore, $\\Vert (A_0 - \\lambda )^{-1}\\Vert \\,\\le \\,\\frac{2}{|\\operatorname{Im}\\lambda |}\\,.$ The same estimate holds for $\\lambda \\in -$ .",
"Now, assume that $|\\operatorname{Im}\\lambda | > 2\\Vert q\\Vert _\\infty =2\\Vert V\\Vert $ .",
"Then $\\Vert V(A_0 - \\lambda )^{-1}\\Vert < 1$ , and hence $A - \\lambda = A_0 - \\lambda + V = \\left(I + V(A_0 - \\lambda )^{-1}\\right)(A_0 - \\lambda )$ is boundedly invertible, which proves ().$\\Box $"
],
[
"Second order elliptic operators",
"Let $\\Omega \\subset \\mathbb {R}^n$ be a domain and let $\\ell $ be the \"formally symmetric\" uniformly elliptic second order differential expression $\\ell (f)(x):=-\\sum _{j,k=1}^n \\left( \\frac{\\partial }{\\partial x_j} a_{jk}\\frac{\\partial f}{\\partial x_k}\\right)(x)+a(x)f(x),\\quad x\\in \\Omega ,$ with bounded coefficients $a_{jk}\\in C^\\infty (\\Omega )$ satisfying $a_{jk}(x)=\\overline{a_{kj}(x)}$ for all $x\\in \\Omega $ and $j,k=1,\\dots ,n$ , the function $a\\in L^\\infty (\\Omega )$ is real valued and $\\sum _{j,k=1}^n a_{jk}(x)\\xi _j\\xi _k\\ge C\\sum _{k=1}^n\\xi _k^2$ holds for some $C>0$ , all $\\xi =(\\xi _1,\\dots ,\\xi _n)^\\top \\in \\mathbb {R}^n$ and $x\\in \\Omega $ .",
"With the differential expression $\\ell $ we associate the elliptic differential operator $Tf:=\\ell (f),\\qquad \\operatorname{dom}T=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\ell (f) \\in L^2(\\Omega )\\bigr \\rbrace ,$ where $H^1_0(\\Omega )$ stands for the closure of $C_0^\\infty (\\Omega )$ in the Sobolev space $H^1(\\Omega )$ .",
"It is well known that $T$ is an unbounded selfadjoint operator in the Hilbert space $(L^2(\\Omega ),(\\cdot ,\\cdot ))$ with spectrum semibounded from below by $\\operatornamewithlimits{ess\\,inf}\\,a$ ; cf.",
"[24].",
"Let $w$ be a real valued function such that $w,w^{-1}\\in L^\\infty (\\Omega )$ and each of the sets $\\Omega _+:=\\bigl \\lbrace x\\in \\Omega : w(x)>0\\bigr \\rbrace \\quad \\text{and}\\quad \\Omega _-:=\\bigl \\lbrace x\\in \\Omega : w(x)<0\\bigr \\rbrace $ has positive Lebesgue measure.",
"We define a second order elliptic differential expression $\\mathcal {L}$ with the indefinite weight $w$ by $\\mathcal {L}(f)(x):=\\frac{1}{w(x)}\\,\\ell (f)(x),\\qquad x\\in \\Omega .$ The multiplication operator $G_w f=w f$ , $f\\in L^2(\\Omega )$ , is an isomorphism in $L^2(\\Omega )$ with inverse $G_w^{-1}f=w^{-1}f$ , $f\\in L^2(\\Omega )$ , and gives rise to the Krein space inner product $[f,g]:=(G_w f,g)=\\int _\\Omega f(x)\\overline{g(x)}\\,w(x)\\,dx,\\qquad f,g\\in L^2(\\Omega ).$ The differential operator associated with $\\mathcal {L}$ is defined as $Af=\\mathcal {L}(f),\\qquad \\operatorname{dom}A=\\bigl \\lbrace f\\in H^1_0(\\Omega ):\\mathcal {L}(f) \\in L^2(\\Omega )\\bigr \\rbrace .$ Since for $f\\in H^1_0(\\Omega )$ we have $\\ell (f)\\in L^2(\\Omega )$ if and only if $\\mathcal {L}(f)\\in L^2(\\Omega )$ it follows that $\\operatorname{dom}A=\\operatorname{dom}T$ and $A= G_w^{-1}T$ hold.",
"Hence $A$ is a selfadjoint operator in the Krein space $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ .",
"In order to illustrate Theorem REF for the indefinite elliptic operator $A$ we assume from now on that $\\min \\sigma _{\\rm ess}(T)\\le 0$ holds.",
"This also implies that the domain $\\Omega $ is unbounded as otherwise $\\sigma _{\\rm ess}(T)=\\varnothing $ .",
"A discussion of the cases $\\sigma _{\\rm ess}(T)=\\varnothing $ and $\\min \\sigma _{\\rm ess}(T)> 0$ is contained in [9], see also [26], [49].",
"Fix some $\\eta >0$ such that $-\\eta <\\min \\sigma (T)$ and define the spaces $\\mathcal {H}_s$ , $s\\in [0,2]$ , as the domains of the $\\frac{s}{2}$ -th powers of the uniformly positive operator $T+\\eta $ in $L^2(\\Omega )$ , $\\mathcal {H}_s:=\\operatorname{dom}\\bigl ((T+\\eta )^\\frac{s}{2}\\bigr ),\\qquad s\\in [0,2].$ Note that $\\mathcal {H}=\\mathcal {H}_0$ , $\\operatorname{dom}T=\\mathcal {H}_2$ and the form domain of $T$ is $\\mathcal {H}_1$ .",
"The spaces $\\mathcal {H}_s$ become Hilbert spaces when they are equipped with the usual inner products, the induced topologies do not depend on the particular choice of $\\eta $ ; cf.",
"[40].",
"The following theorem is a direct consequence of Theorem REF and the considerations in [21] and [19]; cf.",
"[9].",
"Theorem 4.3 Let $A$ be the indefinite elliptic operator in (REF ), $\\eta >0$ as above, and assume that there exists a bounded uniformly positive operator $W$ in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ such that $W\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds for some $s\\in (0,2]$ .",
"Then $A$ is non-negative over $ \\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r=\\eta \\,\\frac{1+\\tau _\\eta }{2}\\,\\Vert w^{-1}\\Vert _\\infty \\quad \\text{and}\\quad \\tau _\\eta := \\frac{1}{\\pi }\\,\\limsup _{n\\rightarrow \\infty }\\,\\left\\Vert \\int _{-n}^n\\,(A+\\eta G_w^{-1} - it)^{-1} \\,dt\\,\\right\\Vert \\,<\\,\\infty .$ Moreover, $\\infty $ is not a singular critical point of $A$ .",
"In the next corollary the special case $\\Omega =\\mathbb {R}^n$ is treated; cf.",
"[9].",
"From now on we assume that $\\Omega =\\mathbb {R}^n$ and $\\Omega _\\pm =\\lbrace x\\in \\mathbb {R}^n:\\pm w(x)>0\\rbrace $ consist of finitely many connected components with compact smooth boundaries, and that the coefficients $a_{jk}\\in C^\\infty (\\mathbb {R}^n)$ and their derivatives are uniformly continuous and bounded.",
"Note that either $\\Omega _+$ or $\\Omega _-$ is bounded.",
"Corollary 4.4 Suppose that for some $s\\in (0,\\frac{1}{2})$ the Sobolev spaces $H^s(\\Omega _\\pm )$ are invariant under multiplication with $w\\!\\upharpoonright _{\\Omega _\\pm }$ .",
"Then the indefinite elliptic operator $A$ is non-negative over $\\overline{\\mathbb {C}} \\setminus K_r((-r,r))$ , where $r$ is as in in Theorem REF .",
"Moreover, $\\infty $ is not a singular critical point of $A$ .",
"A sufficient criterion for the invariance of $H^s(\\Omega _\\pm )$ in the above corollary can be deduced from [28].",
"Corollary 4.5 Suppose that the weight function $w\\!\\upharpoonright _{\\Omega _\\pm }$ belongs to some Hölder space $C^{0,\\alpha }(\\Omega _\\pm )$ , $\\alpha \\in (0,\\frac{1}{2})$ , and that outside of some bounded set $w$ is equal to a constant.",
"Then the claim in Corollary REF holds.",
"of Corollary REF The assumptions on the coefficients $a_{jk}$ imply $\\operatorname{dom}A=\\operatorname{dom}T=H^2(\\mathbb {R}^n)\\qquad \\text{and}\\qquad \\mathcal {H}_s=H^s(\\mathbb {R}^n),\\qquad s\\in [0,2],$ by elliptic regularity and interpolation.",
"Since the Sobolev spaces $H^s(\\Omega _\\pm )$ are assumed to be invariant under multiplication with the functions $w\\!\\upharpoonright _{\\Omega _\\pm }$ for some $s\\in (0,\\frac{1}{2})$ it follows from [9] that also $H^s(\\mathbb {R}^n)$ is invariant under multiplication with $w$ , that is, $G_w\\mathcal {H}_s\\subset \\mathcal {H}_s$ holds.",
"Furthermore, $G_w$ is uniformly positive in $(L^2(\\Omega ),[\\cdot ,\\cdot ])$ since $[G_w f,f]=(G_w^2 f,f)\\ge \\operatornamewithlimits{ess\\,inf}w^2 \\Vert f\\Vert ^2$ and $w^{-1}\\in L^\\infty (\\mathbb {R}^n)$ .",
"Now the assertion follows from Theorem REF .$\\Box $"
],
[
"Acknowledgements",
"The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.",
"Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, [email protected] Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.",
"Juni 136, 10623 Berlin, Germany, [email protected] Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, [email protected] The support from the Deutsche Forschungsgemeinschaft (DFG) under the grants BE 3765/5-1 and TR 903/4-1 is gratefully acknowledged.",
"Jussi Behrndt: Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria, [email protected] Friedrich Philipp: Institut für Mathematik, MA 8-1, Technische Universität Berlin, Straße des 17.",
"Juni 136, 10623 Berlin, Germany, [email protected] Carsten Trunk: Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany, [email protected]"
]
] | 1204.1112 |
Subsets and Splits